Seventh IUTAM Symposium on Laminar-Turbulent Transition: Proceedings of the Seventh IUTAM Symposium on Laminar-Turbulent Transition, Stockholm, Sweden, 2009

Seventh IUTAM Symposium on Laminar-Turbulent Transition IUTAM BOOKSERIES Volume 18 Series Editors G.M.L. Gladwell, Un...

Author: Philipp Schlatter | Dan S. Henningson

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Seventh IUTAM Symposium on Laminar-Turbulent Transition

IUTAM BOOKSERIES Volume 18 Series Editors G.M.L. Gladwell, University of Waterloo, Waterloo, Ontario, Canada R. Moreau, INPG, Grenoble, France Editorial Board J. Engelbrecht, Institute of Cybernetics, Tallinn, Estonia L.B. Freund, Brown University, Providence, USA A. Kluwick, Technische Universität, Vienna, Austria H.K. Moffatt, University of Cambridge, Cambridge, UK N. Olhoff Aalborg University, Aalborg, Denmark K. Tsutomu, IIDS, Tokyo, Japan D. van Campen, Technical University Eindhoven, Eindhoven, The Netherlands Z. Zheng, Chinese Academy of Sciences, Beijing, China

Aims and Scope of the Series The IUTAM Bookseries publishes the proceedings of IUTAM symposia under the auspices of the IUTAM Board.

For other titles published in this series, go to www.springer.com/series/7695

P. Schlatter • D.S. Henningson Editors

Seventh IUTAM Symposium on Laminar-Turbulent Transition Proceedings of the Seventh IUTAM Symposium on Laminar-Turbulent Transition, Stockholm, Sweden, 2009

Editors P. Schlatter KTH Department of Mechanics SE-100 44 Stockholm Sweden

D.S. Henningson KTH Department of Mechanics SE-100 44 Stockholm Sweden

ISSN 1875-3507 e-ISSN 1875-3493 ISBN 978-90-481-3722-0 e-ISBN 978-90-481-3723-7 DOI 10.1007/978-90-481-3723-7 Springer Dordrecht Heideberg London New York Library of Congress Control Number: 2009940791 © Springer Science+Business Media B.V. 2010 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 987654321 springer.com

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Part I Invited Presentations A Gradient-based Optimization Method for Natural Laminar Flow Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Hanifi, O. Amoignon, J. O. Pralits, M. Chevalier

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A Thermodynamic Lower Bound on Transition-Triggering Disturbances . 11 Paolo Luchini Hypersonic boundary layer transition and control . . . . . . . . . . . . . . . . . . . . 19 A. A. Maslov, T. Poplavskaya, D. A. Bountin Instabilities of Miscible Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Eckart Meiburg Large-eddy simulations of relaminarization due to freestream acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Ugo Piomelli, Carlo Scalo Reduced-order models for flow control: balanced models and Koopman modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Clarence W. Rowley, Igor Mezić, Shervin Bagheri, Philipp Schlatter, Dan S. Henningson The description of fluid behavior by coherent structures . . . . . . . . . . . . . . . 51 Peter J. Schmid Instability of uniform turbulent plane Couette flow: spectra, probability distribution functions and K − Ω closure model . . . . . . . . . . . . . . . . . . . . . . 59 Laurette S. Tuckerman, Dwight Barkley, Olivier Dauchot

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Part II Oral Presentations Sensitivity to base-flow variation of a streamwise corner flow . . . . . . . . . . . 69 Frédéric Alizard, Jean-Christophe Robinet, Ulrich Rist Transition Control Testing in the Supersonic S2MA Wind Tunnel (SUPERTRAC project) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 J.-P. Archambaud, D. Arnal, J.-L. Godard, S. Hein, J. Krier, R. S. Donelli, A. Hanifi Breakdown of Low-Speed Streaks under High-Intensity Background Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Masahito Asai, Motosumi Yamanouchi, Ayumu Inasawa, Yasufumi Konishi Numerical Study on Transition of a Channel Flow with Longitudinal Wall-oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Takashi Atobe, Kiyoshi Yamamoto Direct Numerical Simulation of the Mixing Layer past Serrated Nozzle Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Andreas Babucke, Markus J. Kloker, Ulrich Rist Receptivity of a supersonic boundary layer to shock-wave oscillations . . . 99 Andreas Babucke, Ulrich Rist Roughness receptivity studies in a 3-D boundary layer – Flight tests and computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Andrew L. Carpenter, William S. Saric, Helen L. Reed DNS investigations of steady receptivity mechanisms on a swept cylinder . 111 G. Casalis, E. Piot Experimental Study of the Incipient Spot Breakdown Controlled by Riblets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 V. Chernoray, G. R. Grek, V. V. Kozlov, Y. A. Litvinenko Control of Stationary Cross-flow Modes Using Patterned Roughness at Mach 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Thomas Corke, Eric Matlis, Chan-Yong Schuele, Stephen Wilkinson, Lewis Owens, P. Balakumar Secondary optimal growth and subcritical transition in the plane Poiseuille flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Carlo Cossu, Mattias Chevalier, Dan S. Henningson Disturbance evolution in rotating-disk boundary layers: competition between absolute instability and global stability . . . . . . . . . . . . . . . . . . . . . . 135 Christopher Davies, Christian Thomas

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Instabilities due a vortex at a density interface: gravitational and centrifugal effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Harish N. Dixit, Rama Govindarajan Wave Packets of Controlled Velocity Perturbations at Laminar Flow Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Alexander Dovgal, Alexander Sorokin Linear Stability Analysis for Manipulated Boundary-Layer Flows using Plasma Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 A. Duchmann, A. Reeh, R. Quadros, J. Kriegseis, C. Tropea Stripy patterns in low-Re turbulent plane Couette flow . . . . . . . . . . . . . . . . 159 Yohann Duguet, Philipp Schlatter, Dan S. Henningson Characterization of the three-dimensional instability in a lid-driven cavity by an adjoint based analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Flavio Giannetti, Paolo Luchini, Luca Marino Bi-global crossplane stability analysis of high-speed boundary-layer flows with discrete roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Gordon Groskopf, Markus J. Kloker, Olaf Marxen Time-resolved PIV investigations on the laminar-turbulent transition over laminar separation bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Rainer Hain, Christian J. Kähler and Rolf Radespiel Control of transient growth induced boundary layer transition using plasma actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Ronald E. Hanson, Philippe Lavoie, Ahmed M. Naguib, Jonathan F. Morrison Laminar Flow Control by Suction at Mach 2 . . . . . . . . . . . . . . . . . . . . . . . . 189 S. Hein, E. Schülein, A. Hanifi, J. Sousa, D. Arnal Decay of turbulent bursting in enclosed flows . . . . . . . . . . . . . . . . . . . . . . . . 195 Kerstin Hochstrate, Jan Abshagen, Marc Avila, Christian Will, Gerd Pfister Local and Global Stability of Airfoil Flows at Low Reynolds Number . . . . 201 L. E. Jones, R. D. Sandberg, N. D. Sandham Numerical simulation of riblet controlled oblique transition . . . . . . . . . . . . 207 S. Klumpp, M. Meinke, W. Schröder Transition Movement in the Wake of Protruding and Recessed Three-Dimensional Surface Irregularities . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 V. S. Kosorygin, J. D. Crouch, L. L. Ng

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Plasma Assisted Aerodynamics for Transition Delay . . . . . . . . . . . . . . . . . . 219 Marios Kotsonis, Leo Veldhuis, Hester Bijl Experimental study on stability of the laminar and turbulent plane jets . . 225 V. V. Kozlov, G. R. Grek, G. Kozlov, Y. A. Litvinenko, A. Sorokin Evolution Of Traveling Crossflow Modes Over A Swept Flat Plate . . . . . . 231 Thomas Kurian, Jens H. M. Fransson, P. Henrik Alfredsson Computational Analysis for Roughness-Based Transition Control . . . . . . . 237 Fei Li, Meelan M. Choudhari, Chau-Lyan Chang, Jack R. Edwards Statistics of turbulent-to-laminar transition in plane Couette flow . . . . . . . 243 Paul Manneville Spectra of Swirling Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Xuerui Mao, Spencer J. Sherwin Localized edge states for the transition to turbulence in shear flows . . . . . . 253 D. Marinc, T. M. Schneider, B. Eckhardt Active steady control of vortex shedding: an adjoint-based sensitivity approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Olivier Marquet, Denis Sipp Feedback control of transient energy growth in subcritical plane Poiseuille flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Fulvio Martinelli, Maurizio Quadrio, John McKernan, James F. Whidborne Linear and non-linear disturbance evolution in a compressible boundary-layer with localized roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Olaf Marxen, Gianluca Iaccarino, Eric S. G. Shaqfeh Experimental Study of Boundary Layer Transition Subjected to Weak Free Stream Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Masaharu Matsubara, Kota Takaichi, Toshiaki Kenchi Open-loop control of compressible afterbody flows using adjoint methods 283 Philippe Meliga, Denis Sipp, Jean-Marc Chomaz Direct Numerical Simulation of a Swept-Wing Boundary Layer with an Array of Discrete Roughness Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Takafumi Nishino, Karim Shariff Wave packet pseudomodes upstream of a swept cylinder . . . . . . . . . . . . . . . 295 Dominik Obrist, Peter J. Schmid Bypass Transition prediction using a model based on transient growth theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Olivier Vermeersch, Daniel Arnal

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Flow in a Slowly Divergent Pipe Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Jorge Peixinho In-flight experiments on active TS-wave control on a 2D-laminar wing glove . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Inken Peltzer, Kai Wicke, Andreas Pätzold, Wolfgang Nitsche Global nonlinear dynamics of thin aerofoil wakes . . . . . . . . . . . . . . . . . . . . . 319 Benoˆıt Pier, Nigel Peake Riccati-less optimal control of bluff-body wakes . . . . . . . . . . . . . . . . . . . . . . 325 Jan Oscar Pralits, Paolo Luchini Asymptotic theory of the pre-transitional laminar streaks and comparison with experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Pierre Ricco Roughness-induced transition of compressible laminar boundary layers . 337 J. A. Redford, N. D. Sandham, G. T. Roberts On receptivity and modal linear instability of laminar separation bubbles at all speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 D. Rodr´ıguez, J. A. Ekaterinaris, E. Valero, V. Theofilis Hypersonic instability waves measured on a circular cone at M=12 using fast-response surface heat-flux and pressure gauges . . . . . . . . . . . . . . . . . . . 349 T. Roediger, H. Knauss, B. V. Smorodsky, D. A. Bountin, A. A. Maslov, E. Kraemer, S. Wagner Interaction of noise disturbances and streamwise streaks . . . . . . . . . . . . . . 355 Philipp Schlatter, Enrico Deusebio, Luca Brandt, Rick de Lange Experimental study on the use of the wake instability as a passive control in coaxial jet flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 ¨ u, P. Henrik Alfredsson, Alessandro Talamelli Antonio Segalini, Ramis Orl¨ Numerical and Experimental Investigations of Relaminarizing Plane Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Daisuke Seki, Takayuki Numano, Masaharu Matsubara Linear control of 3D disturbances on a flat-plate . . . . . . . . . . . . . . . . . . . . . 373 Onofrio Semeraro, Shervin Bagheri, Luca Brandt, Dan S. Henningson Experimental study of stability of supersonic boundary layer on swept wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 N. V. Semionov, A. D. Kosinov, Y. G. Yermolaev

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Comparison of Direct Numerical Simulation with the Theory of Receptivity in a Supersonic Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . 385 Vitaly G. Soudakov, Ivan V. Egorov, Alexander V. Fedorov Instability of high Mach number flows in the presence of hightemperature gas effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Christian Stemmer Spatially localised growth within global instabilities of flexible channel flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Peter S. Stewart, Sarah L. Waters, John Billingham, Oliver E. Jensen Global stability of a plane liquid jet surrounded by gas . . . . . . . . . . . . . . . . 403 Outi Tammisola, Fredrik Lundell, Daniel Söderberg, Atsushi Sasaki, Masaharu Matsubara Instabilities of flow in a corrugated pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Jianjun Tao The Late Nonlinear Stage of Oblique Breakdown to Turbulence in a Supersonic Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Dominic von Terzi, Christian Mayer, Hermann Fasel Turbulence stripe in transitional channel flow with/without system rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Takahiro Tsukahara, Yasuo Kawaguchi, Hiroshi Kawamura, Nils Tillmark, P. Henrik Alfredsson Direct Numerical Simulation and Theoretical Analysis of Perturbations in Hypersonic Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 Anatoli Tumin, Xiaowen Wang, Xiaolin Zhong Flow Transition in Free Liquid Film Induced by Thermocapillary Effect . 433 Ichiro Ueno, Toshiki Watanabe, Toshihiro Matsuya Boundary layer transition by interaction of streaks and Tollmien– Schlichting waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 Tamer A. Zaki, Yang Liu, Paul A. Durbin Numerical Investigation of Subharmonic Resonance Triads in a Mach 3 Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 Marcus Zengl, Dominic von Terzi, Hermann Fasel Part III Poster Presentations Transient Growth on the Homogenous Mixing Layer . . . . . . . . . . . . . . . . . . 453 Cristobal Arratia, Sarah Iams, Jean-Marc Chomaz, Colm-Cille Caulfield

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Closed-loop control of cavity flow using a reduced-order model based on balanced truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 A. Barbagallo, D. Sipp, P. J. Schmid On the asymptotic solution of the flow around a circular cylinder . . . . . . . 461 Iago C. Barbeiro, Ivan Korkischko, Karl P. Burr, Julio R. Meneghini, J. A. P. Aranha Investigations of Suction in a Transitional Flat-Plate Boundary Layer . . . 465 Stefan Becker, Jovan Jovanovic Global three-dimensional optimal perturbations in a Blasius boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 S. Cherubini, J.-C. Robinet, A. Bottaro, P. De Palma Quantifying sub-optimal transient growth using biorthogonal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Nicholas Denissen, Edward White, Robert Downs III Model reduction using Balanced Proper Orthogonal Decomposition with frequential snapshots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 G. Dergham, D. Sipp, J.-C. Robinet Control of a trapped vortex in a thick airfoil by steady/unsteady mass flow suction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 R. S. Donelli, F. De Gregorio, M. Buffoni, O. Tutty Receptivity of compressible boundary layer to kinetic fluctuations . . . . . . 485 Alexander V. Fedorov, Sergei N. Averkin Effect of transport modeling on hypersonic cooled wall boundary layer stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 Kenneth Franko, Sanjiva Lele Modeling Supersonic and Hypersonic Flow Transition over Three-Dimensional Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Song Fu, Liang Wang, Angelo Carnarius, Charles Mockett, Frank Thiele Amplitude threshold in the wake transition of an oscillating circular cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 Rafael S. Gioria, Julio R. Meneghini Certain Aspect of Instability of Flow in a Channel with Expansion/Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 Ayumu Inasawa, Masahito Asai, Jerzy M. Floryan Some properties of boundary layer under the joint effect of external flow turbulence and surface roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 Pavel Jonásˇ, Oton Mazur, Václav Uruba

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Wave forerunners of localized structures at the boundary layer . . . . . . . . . 509 M. Katasonov, V. Gorev, V. V. Kozlov Experiments on the wave train excitation and wave interaction in spanwise modulated supersonic boundary layer . . . . . . . . . . . . . . . . . . . . . . 513 A. D. Kosinov, N. V. Semionov, Y. G. Yermolaev Laminar-Turbulent Transition and Boundary Layer Separation on wavy surface wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 Viktor V. Kozlov, Ilya Zverkov, Boris Zanin Investigation of Thermal Nonequilibrium on Hypersonic BoundaryLayer Transition by DNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 Jens Linn, Markus J. Kloker Global sustained perturbations in a backward-facing step flow . . . . . . . . . 525 Olivier Marquet, Denis Sipp Large Reynolds number streak description using RNS . . . . . . . . . . . . . . . . 529 Juan A. Mart´ın, Carlos Martel Optimal disturbances with iterative methods . . . . . . . . . . . . . . . . . . . . . . . . 533 ˚ Antonios Monokrousos, Espen Akervik, Luca Brandt, Dan S. Henningson Connection between full-lifetime and breakdown of puffs in transitional pipe flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 ¨ ur Ertunç, Antonio Delgado Mina Nishi, Ozg¨ Effect of oblique waves on jet turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 ¨ u, Antonio Segalini, Alessandro Talamelli, P. Henrik Alfredsson Ramis Orl¨ The effect of a single three-dimensional roughness element on the boundary layer transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 Igor B. de Paula, Werner Würz, Marcello A. F. Medeiros Experimental study of resonant interactions of modulated waves in a non self-similar boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 I. B. de Paula, W. Würz, E. Krämer, V. I. Borodulin, Y. S. Kachanov High Reynolds Number Transition Experiments in ETW (TELFONA project) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 J. Perraud, J.-P. Archambaud, G. Schrauf, R. S. Donelli, A. Hanifi, J. Quest, S. Hein, T Streit, U. Fey, Y. Egami Entropy generation rate in turbulent spots in a boundary layer subject to freestream turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 Brendan Rehill, Ed J. Walsh, Kevin Nolan, Donald M. McEligot, Luca Brandt, Philipp Schlatter, Dan S. Henningson

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The Effect of a Particle travelling through a Laminar Boundary Layer on Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 Conny Schmidt, Trevor M. Young, Emmanuel P. Benard Flow past a plate with elliptic leading edge: layer response to free-stream vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 Lars-Uve Schrader, Luca Brandt, Catherine Mavriplis, Dan S. Henningson Fluctuation Measurements in the Turbulent Boundary Layer of a Supersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 Anne-Marie Schreyer, Uwe Gaisbauer, Ewald Krämer Experimental characterization of the transition region in a rotating-disk boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 Muhammad Ehtisham Siddiqui, Benoˆıt Pier, Julian Scott, Alexandre Azouyi, Roger Michelet Nonlinear Interaction Between Wavepackets in Plane Poiseuille Flow . . . . 577 Homero G. Silva, Ricardo A. C. Germanos, Marcello A. F. Medeiros Effects of Passive Porous Walls on Hypersonic Boundary Layers . . . . . . . . 581 Sharon O. Stephen, Vipin Michael Global Instabilities in Wall Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 Gayathri Swaminathan, A Sameen, Rama Govindarajan Spatial Optimal Disturbances in Three-Dimensional Boundary Layers . . . 589 David Tempelmann, Ardeshir Hanifi, Dan S. Henningson Influence of turbulence scale and shape of leading edge on laminar-turbulent transition induced by free-stream turbulence . . . . . . . . 593 M. V. Ustinov, S. V. Zhigulev Bifurcation characteristics of the channel flow on a rotating system undergoing transition under the influence of the Coriolis force . . . . . . . . . . 597 Venkatesa I. Vasanta Ram, Burkhard Müller Linear Stability Investigations of Flow Over Yawed Anisotropic Compliant Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 Marcus Zengl, Ulrich Rist Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Preface

The origins of turbulent flow and the transition from laminar to turbulent flow are the most important unsolved problems of fluid mechanics and aerodynamics. Besides being a fundamental question of fluid mechanics, there are numerous applications relying on information regarding transition location and the details of the subsequent turbulent flow. For example, the control of transition to turbulence is especially important in (1) skin-friction reduction of energy efficient aircraft, (2) the performance of heat exchangers and diffusers, (3) propulsion requirements for hypersonic aircraft, and (4) separation control. While considerable progress has been made in the science of laminar to turbulent transition over the last 30 years, the continuing increase in computer power as well as new theoretical developments are now revolutionizing the area. It is now starting to be possible to move from simple 1D eigenvalue problems in canonical flows to global modes in complex flows, all accompanied by accurate large-scale direct numerical simulations (DNS). Here, novel experimental techniques such as modern particle image velocimetry (PIV) also have an important role. Theoretically the influence of non-normality on the stability and transition is gaining importance, in particular for complex flows. At the same time the enigma of transition in the oldest flow investigated, Reynolds pipe flow transition experiment, is regaining attention. Ideas from dynamical systems together with DNS and experiments are here giving us new insights. The area of transition control is moving away from ad-hoc techniques to ones soundly based on modern design and control theories, e.g. the design of Riccatibased optimal control and estimation algorithms applied to disturbance control in boundary layers. The related adjoint techniques are also gaining in use, for example in the important receptivity problem, where it in the next few years is likely to provide a basis for rapid new developments. In short, there is a regained interest in Laminar-Turbulent Transition and many new developments are coming about. Thus, by 2009 it was high time to have another Symposium on laminar-turbulent transition under the auspices of the International Union of Theoretical and Applied Mechanics (IUTAM). The first IUTAM Symposium, Laminar-Turbulent Transition I, was organised in 1979 in Stuttgart by German scientists who were active in this research field. xv

xvi

Preface

The second IUTAM Symposium was held in Novosibirsk in 1984 where the leading Soviet centre in transition research is located. The third IUTAM Symposium, Laminar-Turbulent Transition III, was organised by French scientists at ONERA in Toulouse. The IUTAM Symposium Laminar-Turbulent Transition IV was held in Sendai in 1994. The fifth Symposium was held in Sedona, Arizona in 1989 and the sixth meeting in Bangalore, India in 2004. At that meeting a request was made to organise the seventh meeting in Stockholm, Sweden. These meetings have served as milestones in the development of the research area for 30 years now and as such they attract large attention in the community. The objectives of the current Symposium are to deepen the fundamental knowledge of stability and laminar-turbulent transition by providing a forum for discussions between the leading scientists from Europe, Asia, and the Americas. An IUTAM Symposium is the ideal mechanism for bringing together such a group. The scope of the Symposium covers the broad area of flow instabilities and transition to turbulence. Areas of emphasis include: • • • • • • • •

Novel approaches to receptivity analysis and transition modeling. Non-normal effects and global modes. Stability of complex flows, such as non-Newtonian and miscible-interface flows. Transition in simple shear flows and its relation to properties of non-linear dynamical systems. Modern feedback control and design techniques applied to transition. Transition in high-speed flows. Direct and Large-Eddy Simulation of transition. Applied Laminar Flow Control.

The symposium programme included for each of these areas an invited lecture from an international experts in that respective field, in addition to the normal conference programme with oral and poster presentations. The Symposium was held on the KTH Main Campus in the heart of Stockholm, the capital of Sweden. The beauty of Stockholm in the summer time just a few days after “Midsommar” and the sunny weather provided an excellent setting for this IUTAM Symposium. We would also like to thank all the graduate students from KTH Mechanics who made sure that the technical aspects of the Symposium went very smoothly. In addition, the scientific committee and the local organising committee is thanked for their contribution during the planning of the meeting; in particular the reviewing of the submitted abstracts. Qiang Li is specifically thanked for his valuable help in compiling this proceedings book. All the sponsors are gratefully acknowledged for making the Symposium possible, including Springer for their help in printing these proceedings. Finally, we thank all the authors, session chairmen and all the participants, whose active involvement and contributions defined the conference and made it a success. Stockholm, August 2009

Philipp Schlatter Dan S. Henningson

Preface

Scientific Committee Prof. Dan Henningson (Chairman) Linné Flow Centre, KTH Mechanics, Stockholm, Sweden Prof. Patrick Huerre (IUTAM Representative) LadHyX, Ecole Polytechnique, Paris, France Dr. Danial Arnal ONERA, Toulouse, France Prof. Bruno Eckhardt Philipps Universität Marburg, Marburg, Germany Prof. Rama Govindarajan Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore, India Prof. Alessandro Bottaro University of Genova, Genova, Italy Prof. Masaharu Matsubara Shinshu University, Nagano, Japan Prof. Tom Mullin Manchester Centre for Nonlinear Dynamics, Manchester, United Kingdom Prof. William Saric Texas A&M University, College Station, U.S.A.

Local Organising Committee Prof. Dan Henningson (Chairman) Dr. Philipp Schlatter (Scientific Secretary) Prof. Henrik Alfredsson Dr. Luca Brandt Dr. Jens Fransson Prof. Ardeshir Hanifi

xvii

xviii

Preface

Symposium Sponsors We wish to thank the following sponsors for their contribution to the success of the Symposium: • • • • • • •

International Union of Theoretical and Applied Mechanics (IUTAM) Royal Institue of Technology KTH Stockholm Vetenskapsr˚adet (Sweden) Linné FLOW Centre Dantec Dynamics Cambridge University Press Springer

Symposium Participants During the Symposium, a total of 8 invited talks (45 minutes), 68 oral presentations (17 minutes) and 39 poster presentations were delivered. The 142 participants came from 21 different countries:

Austria 1 Brazil 2 Canada 4 China 1 Czech Republic 1 Denmark 1 France 22 Germany 27 India 2 Ireland 3 Islamic Republic of Iran 1

Italy Japan The Netherlands Russian Federation Serbia and Montenegro Spain Sweden Switzerland United Kingdom United States

4 10 1 10 1 3 18 2 12 16

This volume of the contributions presented at the Symposium contains the material of the 8 invited talks (8 pages), 64 oral presentations (6 pages) and 38 poster presentations (4 pages).

IUTAM Symposium on Laminar-Turbulent Transition, June 23-26, 2009, Stockholm, Sweden

Preface xix

Alfredsson, P. Henrik Alizard, Frédéric Archambaud, Jean-Pierre Arnal, Daniel Arratia, Cristobal Asai, Masahito Atobe, Takashi Bagheri, Shervin Barbagallo, Alexandre Becker, Stefan Braga de Paula, Igor Brandt, Luca Casalis, Grégoire Chernoray, Valery Cherubini, Stefania Chomaz, Jean-Marc Corke, Thomas Cossu, Carlo Crouch, Jeffrey Davies, Christopher Dennissen, Nicholas Dergham, Grégory Donelli, Raffaele Dovgal, Alexander Duchmann, Alexander Duguet, Yohann Eckhardt, Bruno Fedorov, Alexander Franko, Kenneth

KTH Mechanics SINUMEF ENSAM Paris PARISTECH ONERA ONERA LadHyX, Ecole Polytechnique Tokyo Metropolitan University Japan Aerospace Exploration Agency KTH Mechanics LadHyX, Ecole Polytechnique University Erlangen Universität Stuttgart KTH Mechanics ONERA Chalmers University of Technology Arts et Metiers ParisTech and Politecnico di Bari Ecole Polytechnique - CNRS University of Notre Dame CNRS - Ecole Polytechnique The Boeing Company Cardiff University Texas A&M University ONERA CIRA Institute of Theoretical and Applied Mechanics TU Darmstadt KTH Mechanics Philipps-Universität Marburg Moscow Institute of Physics and Technology Stanford University

Sweden France France France France Japan Japan Sweden France Germany Germany Sweden France Sweden France France United States France United States United Kingdom United States France Italy Russian Federation Germany Sweden Germany Russian Federation United States

[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]

xx Preface

Friederich, Tillmann Garcia-Villalba, Manuel Garzon, Andres Gjelstrup, Palle Govindarajan, Rama Groskopf, Gordon Hain, Rainer Hanifi, Ardeshir Hanson, Ronald Healey, Jonathan Hein, Stefan Henningson, Dan Herbert, Mats Hochstrate, Kerstin Inasawa, Ayumu Jonas, Pavel Jones, Lloyd Kelterer, Maria Elisabeth King, Rudolph Kleiser, Leonhard Kloker, Markus J. Kosinov, Alexander Kosorygin, Vladimir Kotsonis, Marios Kozlov, Viktor Kurian, Thomas Lavoie, Philippe Li, Fei Lindgren, Rune (Texas)

Universität Stuttgart University of Karlsruhe Aerion Corporation Dantec Dynamics A/S Jawaharlal Nehru Centre Universität Stuttgart Universitaet der Bundeswehr Muenchen KTH Mechanics and FOI University of Toronto Keele University DLR KTH Mechanics Vidix University of Kiel Tokyo Metropolitan University Institute of Thermomechanics AS CR University of Southampton Graz University of Technology NASA Langley Research Center ETH Zurich Universität Stuttgart ITAM SB RAS Institute of Theoretical and Applied Mechanics SB Delft University of Technology Institute of Theoretical and Applied Mechanics KTH Mechanics University of Toronto NASA Langley Research Center KTH Mechanics

Germany Germany United States Denmark India Germany Germany Sweden Canada United Kingdom Germany Sweden Sweden Germany Japan Czech Republic United Kingdom Austria United States Switzerland Germany Russian Federation Russian Federation Netherlands Russian Federation Sweden Canada United States Sweden


Preface xxi

Lingwood, Rebecca J. Linn, Jens Luchini, Paolo Manneville, Paul Mao, Xuerui Marino, Luca Marquet, Olivier Martin, Juan Martinelli, Fulvio Marxen, Olaf Matsubara, Masaharu Mazlov, Anatoly McEligot, Donald M. Medeiros, Marcello Meiburg, Eckart Meinke, Matthias Meliga, Philippe Meneghello, Gianluca Meneghini, Julio Michael, Vipin Monchaux, Romain Monokrousos, Antonios Müller, Burkhard Najafi, Mehdi Nishi, Mina Nishino, Takafumi Obrist, Dominik ¨ u, Ramis Orl¨ Peixinho, Jorge

KTH Mechanics Universität Stuttgart University of Salerno Ecole Polytechnique Imperial College London University La Sapienza ONERA Universidad Politecnica de Madrid LadHyX, Ecole Polytechnique Stanford University Shinshu University ITAM U. Arizona/U. Idaho University of Sao Paulo University of California RWTH Aachen LadHyX, Ecole Polytechnique LadHyX, Ecole Polytechnique University of So Paulo University of Birmingham ENSTA KTH Mechanics Bochum University Sharif University of Technology University of Tokyo NASA Ames Research Center ETH Zurich KTH Mechanics City College of New York

Sweden Germany Italy France United Kingdom Italy France Spain France United States Japan Russian Federation United States Brazil United States Germany France France Brazil United Kingdom France Sweden Germany Islamic Republic of Iran Japan United States Switzerland Sweden United States


xxii Preface

Peltzer, Inken Pfister, Gerd Pier, Benoˆıt Piomelli, Ugo Pralits, Jan Oscar Rasuo, Bosko Redford, John Reeh, Andreas Rehill, Brendan Ricco, Pierre Rist, Ulrich Robinet, Jean-Christophe Rödiger, Tim Rodriguez, Daniel Rowley, Clarence Sakaguchi, Takeshi Sandham, Neil Saric, William Schlatter, Philipp Schmid, Peter Schmidt, Conny Schrader, Lars-Uve Schreyer, Anne-Marie Seki, Daisuke Semeraro, Onofrio Semionov, Nickolay Siddiqui, Muhammad Ehtisham Soudakov, Vitaly Stemmer, Christian

TU Berlin Germany [email protected] Christian Albrechts Universität zu Kiel Germany [email protected] LMFA France [email protected] Queen’s University Kingston Canada [email protected] Universit di Salerno Italy [email protected] University of Belgrade Serbia and Montenegro [email protected] University of Southampton United Kingdom [email protected] TU Darmstadt Germany [email protected] University of Limerick Ireland [email protected] King’s College London United Kingdom [email protected] Universität Stuttgart Germany [email protected] Arts et Métiers ParisTech France [email protected] Universität Stuttgart Germany [email protected] School of Aeronautics, UPM Spain [email protected] Princeton University United States [email protected] Shinshu University Japan [email protected] University of Southampton United Kingdom [email protected] Texas A&M University United States [email protected] KTH Mechanics Sweden [email protected] LadHyX, Ecole Polytechnique France [email protected] University of Limerick Ireland [email protected] KTH Mechanics Sweden [email protected] Universität Stuttgart Germany [email protected] Shinshu University Japan [email protected] KTH Mechanics Sweden [email protected] ITAM SB RAS Russian Federation [email protected] LadHyX, Ecole Polytechnique France [email protected] TsAGI Russian Federation vit [email protected] Technische Universität München Germany [email protected]

Preface xxiii

Stephen, Sharon University of Birmingham United Kingdom [email protected] Stewart, Peter The University of Nottingham United Kingdom [email protected] Streit, Thomas DLR Germany [email protected] Sunderland, Robert EADS Innovation Works United Kingdom [email protected] Swaminathan, Gayathri Jawaharlal Nehru Centre India [email protected] Swaters, Gordon University of Alberta Canada [email protected] Takaichi, Kota Shinshu University Japan [email protected] Tammisola, Outi KTH Mechanics Sweden [email protected] Tao, Jianjun Peking University China [email protected] Tempelmann, David KTH Mechanics Sweden [email protected] Theofilis, Vassilis Universidad Politécnica Madrid Spain [email protected] Tsukahara, Takahiro Tokyo University of Science Japan [email protected] Tuckerman, Laurette PMMH-ESPCI France [email protected] Tumin, Anatoli University of Arizona United States [email protected] Ueno, Ichiro Tokyo University of Science Japan [email protected] Ustinov, Maxim TsAGI Russian Federation [email protected] Vasanta Ram, Venkatesa Lyengar Ruhr University Bochum Germany [email protected] von Terzi, Dominic University of Karlsruhe Germany [email protected] Walsh, Edmond Univeristy of Limerick Ireland [email protected] Wang, Liang Technische Universität Berlin Germany [email protected] White, Edward Texas A&M University United States [email protected] Willis, Ashley Ecole Polytechnique France [email protected] Würz, Werner Universität Stuttgart Germany [email protected] Zaki, Tamer Imperial College London United Kingdom [email protected] Zengl, Marcus Universität Stuttgart Germany [email protected] Zverkov, Ilya Institute of Theoretical and Applied Mechanics Russian Federation [email protected]

xxiv Preface

Part I

Invited Presentations

A Gradient-based Optimization Method for Natural Laminar Flow Design A. Hanifi∗,† , O. Amoignon∗ , J. O. Pralits∗,‡ , and M. Chevalier∗

Abstract A gradient-based optimization method for minimization of the total drag of an airfoil is presented. The viscous drag is minimized by delaying the laminarturbulent transition. The gradients are obtained solving the adojoint of the Euler, boundary-layer and stability equations. The optimization is subjected to constraints such as restrictions on geometry, lift and pitch moment. The geometry is parametrised using radial basis functions.

1 Introduction Drag reduction for high-speed vehicles is a challenging task. In the past, optimization of airfoil mostly aimed at decreasing the pressure drag only neglecting the contribution from viscous drag. However, recent requirements on significant reduction of CO2 and NOx have resulted in increased interest in laminar airfoil design. Since laminar-turbulent transition in the boundary-layer flows is usually caused by breakdown of small unstable perturbations, the flow control methods for delay of transition aim at reducing the growth rate of these perturbations. The amplification of boundary-layer disturbances can be analyzed using linear stability theory. The growth rate of the disturbances can then be used to predict the transition location using the so-called eN method, see e.g. van Ingen [12]. Here, it is assumed that transition occurs when the disturbance amplification exceeds an empirically defined threshold. The most common approaches for transition control investigated for the aeronautic applications are wall-suction and shape optimization. The latter is usually denoted as Natural Laminar Flow (NLF) design.

∗ Swedish

Defence Research Agency, FOI, SE-164 90 Stockholm, Sweden. e-mail: [email protected] † Linn´ e Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden ‡ DIMEC, Universit` a di Salerno, Fisciano (SA), Italy P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_1, © Springer Science+Business Media B.V. 2010

3

4

A. Hanifi, O. Amoignon, J. O. Pralits, M. Chevalier

CFD-based design optimization has proved to be successful in reducing the pressure drag at transonic flow regime, see e.g. Jameson [7]. However, attempts to minimize the total drag by regular CFD-based design optimization have relied on a fixed laminar-turbulent transition point or on the assumption that the flow is fully turbulent as in Nemec & Zingg [9]. Linear stability analysis has been used in a number of investigations for NLF design. In Green & Whitesides [5] , a target pressure is found, based on a simplified relation between pressure and disturbance amplification (N-factor), which is used to state an inverse problem in order to find a geometry that may delay transition. Streit & Liersch [11] also used an inverse method to design a transonic wing with natural laminar flow. In Manning & Kroo [8] , a surface panel method was coupled with an approximative boundary layer calculation and linear stability analysis. Iuliano et al. [6] used a genetic algorithm for NLF design of a supersonic transport jet wing-body. However, none of these investigations calculated the sensitivities based on the linear stability analysis in order to formulate optimality conditions. The method for NLF design presented here is gradient-based and utilizes the adjoints of the Euler, boundary-layer and stability equations to calculate the required gradients. Here, CFD analyses are coupled to boundary-layer stability computations. With this approach, the geometry, here an airfoil, can be optimized with respect to the disturbance amplification in order to delay the laminar-turbulent transition.

2 Governing equations The most common transition prediction methods are based on the amplification rate of harmonic disturbances imposed to the boundary layer. Once the boundary-layer profiles are given the convective instability of these disturbances can be obtained with good accuracy using, for example, the nonlocal stability theory solving the Parabolized Stability Equations (PSE), see e.g. Bertolotti et al. [4]. The amplification rates of perturbations are known to be sensitive to accuracy of the computed profiles. Usually, the required accuracy is not achieved through Navier-Stokes (NS) computations due to the high resolution needed. Therefore, often the profiles are obtaind as solutions of the boundary-layer equations performed using pressure distribution given by Euler or Navier-Stokes computations as input. This approach has been used here. The governing equations in symbolic form can be written as Le (w, Γ ) = 0 , Lble (Q, w, Γble ) = 0 ,

(1) (2)

L pse (q, Q, Γpse ) = 0 .

(3)

Here, Le , Lble and L pse represent operators corresponding to Euler, boundarylayer and disturbance equations, respectively. Further, w denotes the inviscid flow field, Q the viscous boundary-layer profiles and q the perturbation quantities. The

A Gradient-based Optimization Method for Natural Laminar Flow Design

5

whole geometry is notated by Γ while , Γble and Γpse represent the part of geometry considered for boundary-layer and stability computations, respectively. The viscous-inviscid interaction is neglected here. In this way the Euler equations Le only depend on the shape Γ at given flow conditions (angle of attach, Mach and Reynolds numbers). The inviscid flow w on Γble provides the boundary conditions for the boundary-layer equations (Lble ). The flow in the laminar boundary layer Q and geometry Γpse define the coefficients of the PSE, for given flow conditions and for chosen disturbance parameters (frequency and wave number).

3 Optimization problem As mentioned above the laminar flow control problem and the aerodynamic shape optimization can be defined as an optimization problem which mathematically can be formulated as:  f j (q, w, Γ ) ≤ 0 , 1 ≤ j ≤ m ,     Le (w, Γ ) = 0 ,  Lble (Q, w, Γble ) = 0 , minn J (q, w, Γ ) subject to (4)  a∈R d  L (q, Q, Γ ) = 0 ,  pse pse   S (Γ , a) = 0 ,

where J is a cost function, f j are the constraints. The cost function J and the constraints f j are explicit functionals of one or several of the variables {q, w, Γ }. Further, S denotes the parameterization of the geometry which defines the geometry Γ for given control variables a. Since we aim to minimize the pressure drag as well as the friction drag the considered objective function should include measures of both of them. The friction drag is thought to be minimized by delaying the laminar-turbulent transition. This is addressed by reducing the amplification of boundary-layer perturbations as much as possible through variation of geometry (pressure gradient). Usually, different types of disturbances (e.g. cross-flow vortices and Tollmien-Schlichting waves) are present in the boundary layer simultaneously. These disturbances react differently to changes in pressure distribution. Further, perturbations of the same type are dominating at different streamwise positions. Therefore, the measure of disturbance amplitude should include contribution of different perturbations. The measure of disturbances used here is based on the kinetic energy of a number of perturbations integrated over a defined domain E=

1 K ∑ 2 k=1

Z

Ω pse

qHk Mqk d Ω ,

(5)

where M is a weighting matrix. This measure has successfully been used in optimization of wall-suction distribution for transition control, see Pralits & Hanifi[10] . Then, the objective function J is defined as

6


Fig. 1 Schematic of optimization procedure.

J = λU E + λDCD ,

(6)

where CD is the inviscid pressure drag. Here, λU and λD are weight parameters. The optimizations can also be constrained by requirements on values of lift and pitch moment. Constraints are further imposed on the geometry features as constant cross-sectional area, fixed trailing edge, and a fixed region of the airfoil around the leading edge. These constraints can be treated as equality or inequality constraints like minimum thickness and minimum lift. As mentioned before the optimization alghorithms used here are gradient-based. The gradients are efficiently obtained solving adjoints of the Euler, boundary-layer and disturbance equations. The gradient of the disturbance energy is computed using the chain rule

∂E ∂E ∂Q ∂w = × × . ∂a ∂Q ∂w ∂a

(7)

The gradients in the right hand side of the expression above represent solutions of the adjoint equations. This expression is equivalent to the following chain of computations APSE → Euler. | {z ABLE} → adjoint | {z } cont.

discr.

Adjoint of the stability equations (APSE) and boundary-layer equations (ABLE) are derived in contineous form and then discretised while adjoint of the Euler equations are derived directly from the discretised ones. Details of derivation and implementation of these equations can be found in references [3, 2]. The schematic of the optimization procedure is given in Fig. 1.

4 Results Here, results of the optimization of two-dimensional airfoils are presented. The aim is to demonsterate the behavior of our method. The geometries correspond to the tip and root sections of a low speed aircraft at a free-stream Mach number of M∞ = 0.374. The initial geometry is a NASA TP 1786 airfoil with 17% thickness. The optimization problem is formulated as:


7

20 Original geometry Opimised geometry

NE

15

10

5

0

0

0.1

0.2

0.3 x/C

0.4

0.5

0

0.5 x/C

1

Fig. 2 NLF airfoil optimization at M∞ = 0.374 and Re = 7 × 106 corresponding to the tip section. Right: Pressure coefficients −Cp and unscaled geometry. Left: NE -factor curves. The dashed lines denote initial design and the solid lines the final design. Symbols denotethe target disturbane.

min log (E) + 0.1 Γ

CD CD 0

subject to CL ≥ CL 0 , Cm ≥ Cm 0 , t ≥ t min

(8)

where the superscript 0 indicates values at the initial design (baseline), t the airfoil thickness and superscript min refers to the minimum avalue allowed. Further, as the value of the disturbance energy E varies a lot in the course of optimization the log (E) is used instead. Here, the Radial Basis Functions (RBF) have been used to parametrize the geometry. The details of its implementation can be found in reference [1]. In Fig. 2 results of optimization of the airfoil corresponding to the tip section of the wing are presented. Here, the minimum thickness is t min /C = 12% and the Reynolds number based on the chord length C is Re = 7 × 106 and CL0 = 0.3. The optimization aimed at reducing the disturbance growth on the upper side of the airfoil only. The targeted disturbance is a two-dimensional Tollmien-Schlichting (TS) wave at 5 kHz (marked with symbols). The NE -factor curves plotted in Fig. 2 are defined as ˆ E(x) , (9) NE (x) = 0.5 · ln ˆ 0) E(x

where Eˆ is the kinetik energy of disturbances and x0 refers to the location the disturbance energy first starts to grow. Here, the critical value of NE -factor is assumed to be around 10. As can be seen in Fig. 2, the disturbance amplification is reduced significantly and its value is well below the citical one every where. The boundary layer computations stop around x/C ≈ 0.55. The final geometry has a an accelerating flow in a large area of the upper side. This is known to have a damoing effect on the growth of TS-waves. Plots in Fig. 3 show the history of the computations corresponding to the results presented in Fig. 2. In Fig. 4 results of optimization of an airfoil corresponding to the root section of the wing are presented. Here, the minimum thickness is t min /C = 16% and the

8


Fig. 3 Optimization history corresponding to the results of Fig. 2.

Reynolds number based on the chord length Re = 12.1 × 106 and CL0 = 0.4. As for the tip section, the energy of a two-dimensional TS wave at 5 kHz on the upper surface of the wing is to be reduced. Besides the results for final geometry, results for an intermediate one are also plotted there. The pressure distribution on both sides are similar to that on the optimized tip section. However, here the pressure gradient on the upper side is stronger resulting in a more pronounced stabilisation of disturbances. As mentioned above, the optimization is based on the inviscid computations. In order to analyse the correct behaviour of the optimized airfoil, RANS computations with prescribed transition point were performed. The location of transition was predicted as the streamwise position where the envelope of the N-factor curves reached a value of 10. Results of these computations, in terms of CL and CD , are given in Fig. 5. Here, CD includes both the pressure and viscous drag. As can be observed there, for low and moderate lift coefficients the performance of the optimized airfoil is better than the initial geometry. However, the optimized airfoil loses lift at higher angle of attack. This is due to the fact that at these angle of attacks boundary-layer separation occurs earlier on the optimized airfoil than on the initial one. This clearly shows the need for inclusion of the viscous effects in the optimization procedure. In the previous examples only the upper side of the airfoil has been considered for laminar flow optimization. The resulted lower surface pressure distributions have stronger adverse gradients than that from the original geometries, resulting in earlier transition on the lower side of the optimized geometries than on the original ones. Therefore, we examined the possibility to extend the laminar flow on both side of the airfoil. Results of the computations are given in Fig. 6. Here, the objective function includes the contributions from disturbances on both side of the airfoil. As can be seen there, the disturbances are highly damped on both upper and lower surfaces of the optimized airfoil.


9

30 Initial geometry Intermediate solution Final geometry

25

NE

20

15

10

5

0

0

0.1

0.2

0.3 x/C

0.4

0.5

0

0.5 x/C

1

Fig. 4 NLF airfoil optimization at M∞ = 0.374 and Re = 12.1 × 106 corresponding to the root section. Upper side optimized for NLF. Right: Pressure coefficients −Cp and unscaled geometry. Left: NE -factor curves. The dashed lines denote initial design and the solid lines the intermediate and final design (thick line). Symbols denote the target disturbane . 2.0

CL

1.5

1.0 Initial geometry Intermediate geometry Final geometry

0.5 0

0.01

0.02 CD

0.03

0.04

Fig. 5 Lift and drag coeeficients for airfoil given in Fig. 6. RANS computations with transition point given by stability analyses.

5 Conclusions Utilizing the adjoint equations, the gradients of amplification of the boundarylayer disturbances with respect to geometry variations have been efficiently computed. The method couples the Euler, boundary-layer and stability equations as well as their adjoints to compute the gradients. The results of optimization of twodimensional airfoils have been presented indicating that the method is able to produce NLF airfoils. Further, RANS computations on the optimized geometries with prescribed transition location showed that inclusion of viscous effects in optimization procedure may be necessary in order to account for the effects of flow separation.

10

A. Hanifi, O. Amoignon, J. O. Pralits, M. Chevalier 20

30 Final geometry Initial geometry

Final geometry Initial geometry

25

15

NE

NE

20

10

15

10 5 5

0

0

0.1

0.2

0.3 x/C

0.4

0.5

0

0

0.1

0.2

0.3 x/C

0.4

0.5

0

0.5 x/C

1

Fig. 6 NLF airfoil optimization at M∞ = 0.374 and Re = 12.1 × 106 corresponding to the root section. Both upper and lower side are optimized for NLF. Right: Pressure coefficients −Cp and unscaled geometry. Middle: NE -factors curves on the upper side. Left: NE -factor curves on the lower side. The dashed lines denote initial design and the solid lines the final design. Symbols denote the target mode.

Acknowledgements Parts of this work have been funded by European Commission through SUPERTRAC project (contract no: AST4-CT-2005-516100) and CESAR project (contract no: AIP5CT-2006-030888).

References 1. O. Amoignon. AESOP - A Numerical Platform for Aerodynamic Shape Optimization. J. Optimization and Engineering, DOI 10.1007/s11081-008-9078-7, 2008. 2. O. Amoignon and M. Berggren. Adjoint of a median-dual finite-volume scheme: Application to transonic aerodynamic shape optimization. Technical Report 2006-013, Department of Information Technology, Uppsala University, Uppsala, Sweden, 2006. 3. O. Amoignon, J. O. Pralits, A. Hanifi, M. Berggren, and D. S. Henningson. Shape optimization for delay of laminar-turbulent transition. AIAA J., 44(5):1009–1024, 2006. 4. F. P. Bertolotti, Th. Herbert, and S.P. Spalart. Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech., 242:441–474, 1992. 5. B. E. Green and J. L. Whitesides. A method for the constrained design of natural laminar flow airfoils. AIAA Paper, (96-2502), 1996. 6. E. Iuliano, R. Donelli, D. Quagliarella, I. Salah El Din, and D. Arnal. Natural laminar flow design of a supersonic transport jet wing-body. AIAA paper 2009-1279, 2009. 7. A. Jameson. Optimum aerodynamic design using CFD and control theory. AIAA Paper, (951729), 1995. 8. V. M. Manning and I. M. Kroo. Multidisciplinary optimization of a natural laminar flow supersonic aircraft. AIAA Paper, (99-3102), 1999. 9. M. Nemec and D.W. Zingg. Towards efficient aerodynamic shape optimization based on the Navier–Stokes equations. AIAA Paper, (2001-2532), 2001. 10. J. O. Pralits and A. Hanifi. Optimization of steady suction for disturbance control on infinite swept wings. Phys. Fluids, 15(9):2756–2772, 2003. 11. T. Streit and C. Liersch. Design of a transonic wing with natural laminar flow for the EC project NACRE. KATnet II Drag Reduction Workshop. Oct. 14-16, 2008. Ascot, UK., 2008. 12. J. L. van Ingen. A suggested semiempirical method for the calculation of the boundary layer transition region. Technical Report VTH-74, Department of Aeronautical Engineering, University of Delft, 1956.

A Thermodynamic Lower Bound on Transition-Triggering Disturbances Paolo Luchini

1 Oscillators versus Amplifiers A clear distinction must be made, in the linear analysis of flow instabilities, between instabilities that behave as self-sustained oscillators, i.e. grow unceasingly in time from arbitrarily small initial disturbances even in an infinitely quiet environment, and those that behave as spatial amplifiers, i.e. amplify disturbances that come at one spatial location from the environment to a possibly large amplitude at another location, but cease if external disturbances are switched off. In slightly more mathematical terms, the former arise from initial conditions in time and the latter from boundary conditions (or other external sources) in space. This distinction, which applies without ambiguity to spatially bounded flows of any shape, transforms into the distinction between absolute and convective instabilities in the context of spatially infinite homogeneous flows. Oscillators in unbounded inhomogeneous flows develop global modes, eigenfunctions of the linear problem that behave like outgoing waves at infinity in all directions. By contrast amplifiers require an incoming wave (or a localized forcing) in at least some direction, and no unstable global modes arise despite spatial amplification.

1.1 Receptivity The concept of receptivity finds its original and still probably most relevant application in elongated flows (such as boundary layers) where the evolution of fluiddynamic instabilities can be described by a local approximation with slowly varying properties. When such flows behave as amplifiers (i.e. when their local approximation is convectively unstable) the region of linear amplification spatially connects Dept. of Mechanical Engineering, Università di Salerno, Italy e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_2, © Springer Science+Business Media B.V. 2010

11

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P. Luchini

a downstream region where nonlinearity is important and actual transition occurs with an upstream region where environmental disturbances provide the input to be amplified. When the local behaviour is characterized by a dominant mode, the sensitivity of this mode’s amplitude to the external disturbances is named receptivity. Notice, however, that the amplifier paradigm makes perfect sense also when there is no dominant mode and algebraic growth takes place, or in intermediate cases (such as Görtler and crossflow vortices) where amplification is algebraic over a part of the spatial range and modal over another. It turns out that, even in the case of the algebraic growth in a Blasius boundary layer where there is no a-priori reason for this to occur, the downstream disturbance has a well defined recurring shape and therefore an amplitude represented by a single number.

1.2 Can transition be indefinitely delayed? A striking consequence of some fluid instabilities’ behaving as amplifiers is that the spatial position of transition to turbulence depends on the product of receptivity of the instability and amplitude of the initial disturbances. It has always been known to experimentalists, even before the theory was fully developed, that boundary-layer transition occurs at different positions in low-noise and high-noise environments. So the question arises: can transition (at least in principle) be pushed indefinitely downstream through a carefully designed environment? The answer turns out to be a no. A physically unsurmountable lower threshold to external disturbances is provided by molecular fluctuations, the same that underlie Brownian motion or electronicappliance noise. The statistical mechanics of fluctuations [1] dictates that every dissipation term in the macroscopic equations is accompanied by a white random noise, representing the macroscopic effect of microscopic motion, with amplitude determined by the requirement that at global equilibrium (at rest) every macroscopic degree of freedom possess a fluctuation energy 12 kB T just like the microscopic ones. Under local equilibrium, when the fluid is in motion governed by the Navier-Stokes equations, these fluctuations will still be there with an amplitude that is the same function of the local temperature. Receptivity couples thermodynamic fluctuations to a downstream amplified disturbance, much in the same way as the Nyquist resistor noise is made audible by an electronic amplifier.

2 Fluctuation theory 2.1 One-dimensional: Langevin According to statistical physics, at thermodynamic equilibrium any (microscopic or macroscopic) object undergoes velocity fluctuations of Boltzmann probability

A Thermodynamic Lower Bound on Transition-Triggering Disturbances

distribution exp(−mv2 /2kB T ) (where kB = 1.3806503 × 10−23 J/K). Thus

m v2 = kB T.

13

(1)

At the same time in macroscopic dynamics an object is subject to friction: m

dv +σv = 0 dt

(2)

In contradiction with (1), eq.(2) prescribes that all fluctuations should eventually decay to zero. This contradiction was resolved by Langevin who postulated that the effect of microscopic degrees of freedom upon macroscopic ones can be represented by a random force with a characteristic time smaller than all macroscopic ones, i.e.a delta-correlated white noise. Eq.(2) is thus replaced by the Langevin equation m

dv +σv = f dt

(3)

with f a white gaussian noise characterized by its correlation function R f f = h f (t1 ) f (t2 )i = F δ (t1 − t2 ) and spectrum S f f (ω ) =

Z ∞

−∞

R f f (t)e−iω t dt = F.

The precise value of the spectral density F can be obtained most straightforwardly from the following 2.1.1 Fluctuation-dissipation relationship in the frequency domain Remembering that when a stochastic signal is passed through a linear system its spectrum gets multiplied by the square modulus of the corresponding transfer function, from the Fourier transform of (3): iω mv + σ v = f , one obtains Svv =

F |iω m + σ |

2 Z v =

∞

−∞

2

=

Svv

whence, on equating v2 from (5) and (1),

F ( ω 2 m2 + σ 2 )

dω F = 2π 2σ m

F = 2kB T σ

(4) (5)

14

P. Luchini

which is known as the fluctuation-dissipation theorem. according to this theorem, the spectral density of the white forcing and the friction coefficient are not independent but determine each other. 2.1.2 Fluctuation-dissipation relationship in the time domain For the purpose of later generalization to multiple dimensions, it is useful to rederive the fluctuation-dissipation theorem without recurring to a Fourier transform. The velocity correlation function Rvv (t2 − t1 ) = hv(t1 )v(t2 )i obeys, if its first argument t1 is treated as a constant, the same differential equation (3) as v itself: m

dRvv + σ Rvv = Rv f dt

(6)

On the other hand, if t2 is treated as a constant, the cross-correlation Rv f (t2 − t1 ) = hv(t1 ) f (t2 )i obeys the same equation again but with a sign reversal in the first term: −m

dRv f + σ Rv f = R f f = F δ (t) dt

(7)

Eqs. (6-7) can be combined in the single second-order equation −m2

d2 Rvv + σ 2 Rvv = F δ (t) dt 2

(whose Fourier transform (4) appeared in the former derivation). The solution bounded at infinity of this equation is Rvv (t) = F/(2σ m) exp(−σ |t| /m), whence mRvv (0) = F/(2σ ) = kB T just as before.

2.2 Multi-dimensional fluctuation theory: Onsager The generalization of the Boltzmann distribution to a closed system, in thermodynamic equilibrium under given values of a certain number of macroscopic state variables, rules that all such variables xi undergo statistical fluctuations of probability distribution exp(S(xi )/kB ), where S is the total entropy of the given state. Since entropy is maximum at equilibrium, a Taylor expansion around the equilibrium values (taken as the origin xi = 0) gives S − S0 = − 12 αi j xi x j , whence by known properties of gaussian distributions,

xi x j = kB βi j with β = α −1 . (8) Macroscopic friction can, for small perturbations, be described by the linear system dxi + Ai j x j = 0 (9) dt


15

which as before states (if stable) that all perturbations should decay to zero, in contradiction with (8). The resolution of this contradiction is the multi-dimensional Langevin equation dxi + Ai j x j = f i dt

with

fi (t1 ) f j (t2 ) = Fi j δ (t2 − t1 )

Introducing the correlation matrix xi (t1 )x j (t2 ) and repeating the argument of eqs. (6-7) gives the Sylvester equation F = A hx(0)x(0)i + hx(0)x(0)i AT .

(10)

The next ingredient is time reversibility (what in the kinetic theory of gases is also known as detailed balance). Because microscopic mechanics is reversible, all correlation functions including those of macroscopic degrees of freedom must be equal for positive and negative times:

xi (0)x j (t) = xi (0)x j (−t) = xi (t)x j (0) As a consequence, µ = A hx(0)x(0)i = Aβ is a symmetric matrix. This is the essence, with qualifications for different time symmetries, of Onsager relations. But, in addition, it follows that the two terms of (10) are equal, and therefore eq.(10) simplifies to F = 2A hx(0)x(0)i = 2kB µ . which is the multi-dimensional fluctuation-dissipation theorem. Matrix µ has a simple relationship with entropy production. From S − S0 = 1 − 2 αi j xi x j , β = α −1 : dS ∂ S dxi dxi = = −Xi dt ∂ xi dt dt

with Xi = αi j x j ;

xi = βi j X j

Since the macroscopic equations of motion (9) can be written as dx = f − Ax = f − Aβ X = f − µ X, dt the entropy production becomes dS = µi j Xi X j . dt

(11)

2.3 Infinite-dimensional fluctuation theory: Landau-Lifschitz For a fluid

dS = dt

Z

τi j ∂ v i qi ∂ T − 2 T ∂ x j T ∂ xi

dV.

(12)

16

P. Luchini

With the addition of Langevin-like forcing[1], the constitutive relationships become ∂ vi ∂ v j 2 ∂ vk ∂ vk τi j = µ + − δi j + λ δi j + si j , ∂ x j ∂ xi 3 ∂ xk ∂ xk qi = −k

∂T + gi . ∂ xi

Eq.(12) is a quadratic functional in the variables ∂∂ xvij and ∂∂ xTi . Identifying the latter as the thermodynamic potentials Xi of eq.(11) allows the corresponding µ coefficients to be directly read fron eq.(12). The fluctuation-dissipation theorem then tells us that the correlations between the Langevin terms si j and gi must be given by [1]

si j (t, x)shk (t ′ , x′ ) = 2 = 2kB T µ δih δ jk + δik δ jh + λ − µ δi j δhk δ (t − t ′ )δ (x − x′ ), 3

gi (t, x)g j (t ′ , x′ ) = 2kB T 2 kδi j δ (t − t ′ )δ (x − x′ ). 2.3.1 Fluctuations in macroscopic motion The functional dependence of the standard macroscopic constitutive coefficients (viscosity, thermal conductivity) is provided by the local equilibrium assumption: under macroscopic motion, when hvi 6= 0, the statistical distribution of microscopic degrees of freedom is the same that would prevail at rest under the local values of the state variables. It follows that under macroscopic motion, when hvi 6= 0, also the spectral intensity of the Langevin random forces is the same function of the local thermodynamic properties as it is in global equilibrium.

3 Boundary-layer receptivity to thermodynamic fluctuations Boundary layer receptivity is mostly characterized by quadratic effects that obtain the required phase matching from the beat of two different external sources. Molecular agitation enters the picture as an additional, linear receptivity mechanism that already contains all frequencies and wavenumbers. An interesting estimate was put forward in 1961 by Betchov [2]. In [3], pointed out to me by Yuri S. Kachanov, we read According to Smith and a recent review by Schlichting (...), transition occurs when the total amplification amounts to about e9 , which corresponds to an energy gain of about 80 dB. It has been pointed out by Betchov[2] that the velocity fluctuations associated with thermal agitation (Brownian motion etc.) are 80 to 100 dB below the level of turbulence fluctuations. This suggests that laminar instability could amplify thermal agitation to the


17

point of reaching a threshold beyond which a catastrophic nonlinear process would cause transition. This threshold corresponds roughly to velocity fluctuations of the order of 1% of the free-stream velocity.

A quantitative calculation of the receptivity to molecular agitation for the temporal stability problem was described in [4]. Since molecular agitation appears in the equations of motion through a stress forcing term, the main ingredient is the stress +(1) receptivity Ri j = −∂ q j /∂ xi , where q+ is the leading eigensolution of the adjoint Orr-Sommerfeld problem

∂ q+ i = 0, ∂ xi

ρ

iω q+j − iα Uq+j + δ2 j

∂ 2 q+j dU + ∂ c+ q1 − −µ = 0. dx2 ∂xj ∂ xi ∂ xi

The r.m.s.velocity fluctuation at time t turns out to be given by the dimensionless formula: " # (1) Z ZZ D E λ 1 − e−2 Im(ω )t dα dβ 2 −3 ∗ |v| = Re (Ri j + R ji ) (Ri j + R ji ) dy , (13) (1) h 2π 2π 2 Im(ω ) where all quantities inside the integral are nondimensionalized with the external velocity V and (transverse) reference length h used to define the Reynolds number, and the characteristic length λ = kB T /(ρν 2 ) is a material property; for standard air at 300 K and atmospheric pressure, λ = 1.508 × 10−11 m.

3.1 The spatial stability problem For spatial stability we use our own version of the method of multiple scales [5]. Equation (13) then turns out to be replaced by D

Z ZZ Z X h(X) 2

dω dβ |v| (Ri j + R ji ) (Ri j + R ji ) dy dξ 2π 2π 0 Z X where h(X) = exp α0 (ξ ′ ) + [first-order correction] dξ ′ . 2

E

λ = Re−3 h

h(ξ )

∗

X0

An porder-of-magnitude estimate runs as follows: for a boundary-layer thickness h = xν /V ≈ 1.5mm (corresponding to δ99 ≈ 7.5mm) the ratio λ /h ≈ 10−8 ; if at the neutral curve Re ≈ 500, Re−3 accounts for another factor of 10−8 ; the remaining integral is the square of the product of an order-unity receptivity factor times the

1/2 exponential amplification eN . For a final amplitude ur.m.s. = |v|2 = 10−2 we N 6 thus obtain e = 10 , N = 14. A precise computation for the Blasius boundary layer, on the other hand, yields the r.m.s.velocity fluctuation amplitude reported in Fig. 1.

18

P. Luchini

8

0.1

9

N 11

10

12

13

14

urms

0.01

0.001

1e-04

1e-05

3

3.5

4

4.5

5

Rex x 10

5.5

6

6.5

7

-6

Fig. 1 Longitudinal-velocity fluctuation urms vs. local Reynolds number and N-factor.

4 Conclusions • Molecular agitation is a non-negligible source of boundary-layer instabilities. • Molecular agitation sets a thermodynamic lower bound to the level of disturbances that a flow is subjected to. For a flat plate, even with zero external disturbances, transition occurs no later than Rex = 6 × 106 or N = 13 (113 dB). • Such values of Rex and N are not much higher than the usually encountered transition thresholds, and it cannot be excluded that they might actually be encountered in low-disturbance athmospheric flight (as opposed to in a wind tunnel).

References 1. L.D. Landau & E.M. Lifschitz Hydrodynamic fluctuations. Zh. Eksp. Teor. Fiz. 32:618, 1957. [Sov. Phys. JE TP 5:512, 1957]. 2. R. Betchov Thermal Agitation and Turbulence, in Rarefied Gas Dynamics, ed. L. Talbot (Academic Press 1961) 308-21 3. R. Betchov & A. Szewczyk Stability of a shear layer between parallel streams, Phys. Fluids 6, 1391 (1963) 4. P. Luchini The role of molecular agitation in boundary-layer transition, Open Transition Forum Study Group, 38th AIAA Fluid Dynamics Conference, Seattle, 23-26 June 2008 5. P. De Matteis, R.S. Donelli & P. Luchini Application of the ray-tracing theory to the stability analysis of three-dimensional incompressible boundary layers, in Atti del XIII Congresso Nazionale AIDAA, Roma, 11-15 Sept. 1995, Vol. 1, pp. 1–10

Hypersonic boundary layer transition and control A. A. Maslov, T. Poplavskaya, and D. A. Bountin

Abstract In the present paper an overview of the recent studies of a hypersonic laminar boundary layer and shock layer receptivity and stability are presented. Main attention is paid to investigations of nonlinear wave interaction. New active and passive methods of hypersonic stability control are described too.

1 Experiments on nonlinear instability of hypersonic boundary layers at moderate Mach numbers At the last stage of transition to turbulence nonlinear interactions play determinative role. The most essential experiments for understanding of hypersonic boundary layer nonlinear instability were carried out by Stetson et al [1], where spectra of natural disturbances on sharp cone in Mach 8 flow were measured. Understanding of physical processes of nonlinear interaction in hypersonic boundary layer was obtained by means of bispectral analysis measured harmonic interaction (Kimmel & Kendall [2], Chokani [3]). Present work is based on approaches [2,3]. It continues investigation of nonlinear mechanisms leading to turbulence.

Maslov A. Khristianovich Institute of Theoretical and Applied Mechanics SB RAS (ITAM), Russia, Novosibirsk, Institutskaya str.4/1, e-mail: [email protected] Poplavskaya T. ITAM and Novosibirsk State University, Russia, Novosibirsk, e-mail: [email protected] Bountin D. ITAM and Novosibirsk State University, Russia, Novosibirsk, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_3, © Springer Science+Business Media B.V. 2010

19

20

A. A. Maslov, T. Poplavskaya, D. A. Bountin

1.1 Experimental equipment The experiments are performed in the T-326 hypersonic wind tunnel of ITAM SB RAS at a free-stream Mach number M = 5.95, free-stream Reynolds number Re1 = 12 × 106 m−1 and the Reynolds number based on boundary-layer edge values Re1e = 15.8 × 106 m−1 . The fluctuations of the mass flow are measured by a custom built constant-current hot-wire anemometer with a frequency range up to 600kHz. The model is a 7 degree half angle steel cone 0.5m long, sharp nosed. One longitudinal half of the cone is solid, the other longitudinal half is porous. The porosity is laser perforated sheet with hole diameter of 50µ m. The x coordinate is counted from the model tip along the cone generatrix. The model is mounted at zero incidence. To excite artificial disturbances in the boundary layer, the model is equipped with a Synthetic Jet Source [4]. The frequency of artificial disturbances is 290kHz, which belongs to the second-mode frequency band in the present conditions.

1.2 Statistical and bispectral analysis To identify the effects of nonlinearity, the known fact that the Gaussian signal passing through a nonlinear system deviates from the normal distribution is used [5,6]. To determine this deviation, skewness S and kurtosis K are calculated. For the Gaussian distribution, S = 0 and K = 3. To reveal details of nonlinear interaction one has to calculate a bicoherence [2,3,7]. The bicoherence amplitude characterizes the degree of quadratic phase coupling of waves with frequencies f 1 , f 2 , and f3 = f1 + f2 . Bicoherence is bound by 0 (completely independent waves) and 1 (completely coupled waves). In the present work bicoherence is obtained from non-overlapping FFT blocks of length 512; the frequency resolution of the bicoherence is 9.8kHz.

1.3 Experimental results Fig.1 shows the evolution of the Fourier spectra of the signal obtained in the maximum rms disturbance location. The first spectrum (empty triangles) obtained in a linear wave evolution region, the second spectrum (filled triangles) obtained in the nonlinear wave evolution region. First spectrum clearly shows the first ( f ≈ 50 − 220kHz) and second ( f ≈ 230 − 380kHz) modes. The peak corresponding to the second mode is seen to be shifted toward low frequencies in downstream direction. The reason is the wavelength of the second-mode is tuned to the boundary-layer thickness. The figure also shows sharp peak at a frequency of 290kHz corresponding to the artificial disturbances. The disturbances √ of the first and second mode increase up to the stability Reynolds number R = Re1e x = 2170 (x = 297mm) [7], and then redistribution of the spectral energy is initiated by nonlinear processes: the spectrum becomes more uniform, and the second-mode amplitude decreases (Fig. 1).

Hypersonic boundary layer transition and control

21

12000

f

A, V

20000 276 mm (R = 2090)

A, V f

315 mm (R = 2230)

4000

10000

400

R

5000 4000

= 2090 = 276 mm

300

3000

0.020 0.042 0.064 0.086

2000

0.13 0.15 0.17 0.20

2

f , kHz

0.11

200

1000

0.22 0.24

100

0.26 0.28 0.31 0.33

500

0.35

0

100

200

300

f, kHz

Fig. 1 Fourier spectra evolution.

400

0

0

100

200

300

400

4000

12000 A, V f

f , kHz 1

Fig. 2 Bicoherence.

To understand the nonlinear processes the bicoherence analysis has been done (Fig. 2). The dot-and-dashed line in the Fig.2 is the axis of symmetry of the graph, the solid line is described by the equation f1 + f2 = fII , where fII is the frequency of the local maximum in the Fourier spectrum corresponding to the second mode. The Fourier spectrum is shown at the top and on the right of the plot of the bicoherence. Interactions occur in three frequency regions: 1. ( f1 , f2 ) = (300kHz, 300kHz) = ( fII , fII ). Hence, nonlinear mechanism generates harmonics of the second-mode waves: ( f 1 ≈ fII ) + ( f2 ≈ f II ) = ( f3 = f1 + f2 ≈ 2 fII ). This type of interaction was previously obtained by Kimmel & Kendall [2] and Chokani [3]. 2.Along the line f1 ≈ 295kHz. The interacting waves have frequencies f1 ≈ 280 − 310kHz and f2 ≈ 50 − 150kHz. The first interval of frequencies lies in the region of the second-mode maximum in the Fourier spectrum, and the second interval of frequencies almost completely covers the frequency range corresponding to the first mode. 3. In a wide frequency range along the line f1 + f2 = f II , that corresponds to the subharmonic resonance. As was shown in [7, 8] the subharmonic resonance is dominating in a maximum rms disturbance location and most likely this interaction leads to the spectral energy redistribution forming turbulent flow. Measurements across the boundary layer show that nonlinear interactions are presented below and higher the maximum rms disturbance location. Fig. 3 shows skewness S, kurtosis K, mean voltage V and rms voltage values < e > distribution in the last measurement location (x = 315mm). There are no regions in the boundary layer with K and S corresponding to the Gaussian distribution. In this x location nonlinear processes (it can be seen from the deviation of the values S and K from the Gaussian distribution) is extended outside the boundary layer and vanishes at the height y/δ = 1.55, where δ is a boundary layer thickness. Bicoherence also shows strong nonlinear interactions close to the model wall and to the boundary layer edge. Weaker nonlinear interactions are detected by the bicoherence out of the boundary layer (Fig. 4). The fact that the nonlinear processes and mass-

22


f

A , V

12000

2.2

2.4

2.6

2.8

2

3.0

0.0

0.5

1.0

1.5

2 4000

E, V

400

S

R = 2230, y/

skewness

<e>

= 1.23

300

kurtosis

E

0.020 0.042 0.064 0.086 0.11

1

0.13

200

0.15 0.17 0.20

y/

2

f , kHz

1

0.22 0.24

100

0.26 0.28 0.31 0.33 0.35

0

0

0 0

500

1000

1500 (a)

2

4

6

8 (b)

0

100

200

300

400

4000

12000 A, V f

f , kHz 1

<e>, mV

Fig. 3 R = 2230 (x = 315mm).

K

Fig. 4 Bicoherence out of the boundary layer: R = 2230 (x = 315mm).

flow fluctuations (Fig. 3a) go outside the boundary layer definitely suggests that a turbulent boundary layer starts to form [7]. The nonlinear aspects of the stabilization of the second-mode disturbance using a passive, ultrasonically absorptive coating (UAC) - Fedorov-Malmuth method, are studied using bispectral analysis. The bispectral measurements show that the subharmonic and harmonic resonances of the second mode are significantly modified [9]. The harmonic resonance, which is quite pronounced in the latter stages of the hypersonic boundary layer on solid surfaces, is completely absent on the porous surface. The degree of nonlinear phase locking that is associated with the subharmonic resonance and identified on the solid surface is substantially weakened on the porous surface. This nonlinear interaction persists farther downstream on the porous surface than on the solid surface; however, unlike on the solid surface, there are no strongly preferred interaction modes. The spectral measurements, made in previous work, show that the first mode is moderately destabilized on the porous surface [4]. The bispectral measurements identify a nonlinear interaction that is associated with the destabilized first mode; however, this is observed to be a very weak nonlinear interaction that has no deleterious effect on the performance of the UAC. Experimental investigation of UAC influence on laminar-turbulent transition shows that it is possible to delay the laminar run for sharp and blunted cones. Summary1: Bispectral and statistical analysis in hypersonic laminar boundary layer on solid and ultrasonically absorptive coating is conducted. Subharmonic and harmonic resonances were discovered. Second mode was established to play major role in all nonlinear processes in maximum rms pulsation layer on a solid wall. Importance of the nonlinear processes above and below critical layer was shown. Nonlinear processes in the regions above and below the maximum rms layer are fairly intense even when nonlinear interactions (of the quadratic type) have almost disappear in the layer of the maximum rms voltage fluctuation. At the late stages of the transition, the nonlinear processes reach beyond the boundary layer, forming a


23

turbulent boundary layer. According to the theoretical prediction and experimental data second mode disturbances are absorbed by porous coating [4]. Second mode on porous side almost disappears. Sufficient decrease of a second mode amplitude leads to the weakness of nonlinear interactions in maximum rms pulsation layer. Harmonic and subharmonic resonance becomes much weaker, so nonlinear interactions below and higher maximum rms pulsation layer start to play major role.

2 Active control of hypersonic shock layer instability: direct numerical simulation and experiments The present part describes comprehensive experimental and numerical investigations of evolution of disturbances generated in the hypersonic viscous shock layer on a flat plate by external acoustic waves and by perturbations introduced into the shock layer from the surface of model. The flow in a hypersonic shock layer has some specific features: the influence of a closely located shock wave on the evolution of disturbances, significant nonparallelism of the flow, and a high degree of rarefaction. Understanding the mechanisms of receptivity and instability of the viscous shock layer is a necessary condition for the development of effective methods for controlling the laminar-turbulent transition on flying vehicles in a hypersonic flow.

2.1 Experimental equipment and diagnostic methods All experiments are performed in the hypersonic nitrogen wind tunnel T-327A based at ITAM with a free-stream Mach number M = 21 and unit Reynolds number Re1 = 6 × 105 m−1 . A flat plate model 240mm long with a sharp leading edge 100mm wide was inserted into a hypersonic flow with stagnation temperature T0 = 1200K. The temperature factor of plate surface was Tw /T0 = 0.26. The distributions of mean density and the characteristics of density fluctuations were measured by the method of electron-beam fluorescence of nitrogen [10]. The external acoustic disturbances excited in the experiments were slow acoustic waves generated by the turbulent boundary layer formed on the nozzle walls [11] and fast acoustic waves generated by a spark discharge. Internal perturbations localized near the leading edge were generated by an obliquely cut cylindrical aerodynamic whistle located under the plate [12].

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2.2 Numerical simulation Direct numerical simulations of disturbances propagation are performed by solving two-dimensional Navier-Stokes equations with the use of high-order shockcapturing schemes [13]. The system is closed by the equation of state for a perfect gas. The computational domain is a rectangle. A part of the lower side coincides with the plate surface. The height of the computational domain is chosen such that the bow shock emanating from the leading edge does not interact with the upper boundary. An uniform computational grid consisted of 1050 cells in the streamwise direction and 240 in the crossflow direction. The computations involved operation of up to 20 processors in the Siberian Supercomputer Center. In numerical simulations of the problem of shock-layer interaction with external acoustic disturbances, the variables on the left boundary of the computational domain were set in the form of a superposition of the steady main flow and a plane monochromatic acoustic wave characterized by the amplitude A, frequency f , and angle of propagation θ [11]. The artificial perturbations introduced in the experiments by the oblique cylindrical whistle were perturbations similar to periodic blowing and suction organized locally, near the leading edge of the plate. In solving the problem numerically, they were simulated by setting the boundary condition for the transverse mass flow on a certain part of the plate surface. After introduction of disturbances, the Navier-Stokes equations were integrated until the unsteady solution reached a steady periodic regime.

2.3 Results The previous investigation [11,14 ] showed that the main specific feature inherent in the formation of the field of density fluctuations during shock-layer interaction with external acoustic disturbances and internal disturbances of the periodic blowing– suction type is the mechanism of generation of entropy-vortex disturbances and their propagation inside the shock layer. This is also confirmed by the linear theory of interaction of plane waves with the shock wave [15 ]. It was shown that computed and experimental data characterizing the mean flow field, intensity of density fluctuations, and their growth rates are in good agreement [11,14]. The spatial structures of disturbances under the action from inside and outside the shock layer are fairly similar (see Fig5a,b). Under these conditions, it is possible to use active methods of disturbance control, which work well in subsonic boundary layers. Oscillations generated by external perturbations can be suppressed by artificial perturbations, if an appropriate phase and amplitude are chosen for blowing and suction (Fig.5c). The possibility of such control was demonstrated numerically [16] and experimentally [17].The idea of the experiment for fast external acoustic waves is so. Also two types of disturbances were used in the experiment. First, controlled periodic disturbances of blowing-suction type were introduced into the shock layer by an


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Fig. 5 Instantaneous density fluctuations in the shock layer (a,b,c) and rms density fluctuations in the cross section x = 0.8 (e,f,g) for M = 21, ReL = 1.44 × 105 and f = 38.4kHz: a,e) blowingsuction of the gas from the plate surface A = 0.0694; b,f) fast acoustic wave A = 0.0017, θ = 0◦ ; c,g) anti-phase action.

oblique cylindrical gas-dynamic whistle located under the plate in the vicinity of the leading edge. Second, fast acoustic waves were generated by a spark discharge in settling chamber of the nozzle and introduced by the whistle signal with a certain time delay ∆ τ . This delay is responsible for the phase difference between these two types of disturbances. Fig.6 shows the measured amplitudes of density fluctuations in the cross section x = 0.63 at the boundary-layer edge as functions of the delay of the spark signal at the whistle operation frequency. It is seen that a certain delay ensures almost complete suppression of density fluctuations at the boundary-layer edge. For comparison computational data are shown, which are in good agreement with experimental data.

Fig. 6 Normalized amplitudes of root-mean-square density fluctuations on the boundary layer edge versus the delay between the whistle signal and discharge initiation for f = 37.5kHz. The symbols are the measured data; the solid curve is the dependence for interference of two sinusoidal waves; the crosses are the DNS data.

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Summary2: It is shown numerically and experimentally that the main wave processes in the hypersonic shock layer are connected with the generation entropyvortex mode. Interference method of control of hypersonic shock layer disturbances was demonstrated. Acknowledgements This work was supported by the Russian Foundation for Basic Research (projects 09-08-00557 and 09-08-00679) and by ADTP RNP 2.1.1/3963.

References 1. Stetson, K., Thompson, E., Donaldson, J., Siler, L. (1983) Laminar boundary layer stability experiments on a cone at Mach 8, part 1: sharp cone. AIAA Paper 83-1761. 2. Kimmel, R. L., Kendall, J. M. (1991) Nonlinear disturbances in a hypersonic boundary layer. AIAA Paper 91-0320. 3. Chokani, N. (1999) Nonlinear spectral dynamics of hypersonic laminar boundary layer flow. Phys. Fluids. 12, 3846-3851. 4. Fedorov, A., Shiplyuk, A., Maslov, A., Burov, E., Malmuth, N. (2003) Stabilization of a hypersonic boundary layer using an ultrasonically absorptive coating. J. Fluid Mech. 479, 99-124. 5. Papoulis A. (1965) Probability, random variables and stochastic processes. McGraw-Hill Book Company. 6. Nikias, C. L., Raghuveer, M. R. (1987) Bispectrum estimation: a digital signal processing framework. Proc. of the IEEE., 75, 7, 869-891. 7. Bountin, D., Shiplyuk, A., Maslov, A. (2008) Evolution of nonlinear processes in a hypersonic boundary layer on a sharp cone. J. Fluid Mech. 611, 427-441. 8. Shiplyuk A.N., Bountin D.A., Maslov A.A., Chokani N. (2003) Nonlinear mechanisms of the initial stage of the hypersonic boundary layer transition. J. Appl. Mech. Tech. Phys. 44(5),654659. 9. Chokani, N., Bountin, D., Shiplyuk, A., Maslov A. (2005) Nonlinear Aspects of Hypersonic Boundary-Layer Stability on a Porous Surface. AIAA J.43(1), 149-155. 10. Mironov, S. G., Maslov, A. A. (2000) An experimental study of density waves in hypersonic shock layer on a flat plate. Phys. Fluids A. 12(6), 1544-1553. 11. Kudryavtsev, A.N., Mironov, S.G., Poplavskaya, T.V., Tsyryulnikov, I.S. (2006) Experimental study and direct numerical simulation of the evolution of disturbances in a viscous shock layer on a flat plate. J. Appl. Mech. Tech. Phys. 47(5), 617-627. 12. Maslov, A. A., Mironov, S. G. (1996) Experimental investigation of the hypersonic lowdensity flow past a half-closed cylindrical cavity. Fluid Dynamics 31, 928-932. 13. Kudryavtsev, A.N., Poplavskaya, T.V., Khotyanovsky, D.V. (2007) Application of highorder accuracy schemes to numerical simylation of unsteady supersonic flows. Math. Modelling.19(7), 39-55. 14. Maslov, A. A., Kudryavtsev, A.N., Mironov, S. G., Poplavskaya, T.V., Tsyryulnikov, I.S. (2007) Numerical simulation of receptivity of a hypersonic boundary layer to acoustic disturbances. J. Appl. Mech. Techn. Phys. 48(3), 368-374. 15. Mckenzie, J. F., Westphal, K. O. (1968) Interaction of linear waves with oblique shock waves. Phys. Fluids 11, 2350-2362. 16. Fomin, V.M., Kudryavtsev, A.N., Maslov, A. A., Mironov, S. G., Poplavskaya, T.V., Tsyryulnikov, I.S. (2007) Active Control of Disturbances in a Hypersonic Shock Layer. Doklady Physics. 52(5), 274-276. 17. Maslov, A. A., Mironov, S. G., Kudryavtsev, A.N., Poplavskaya, T.V., Tsyryulnikov, I.S. (2008) Controlling the disturbances in a hypersonic flat-plate shock layer by unsteady action from the surface. Fluid Dynamics 43(3), 471-479.

Instabilities of Miscible Interfaces Eckart Meiburg

Abstract An overview is presented over instabilities in fluid flows with miscible interfaces. Specifically, the discussion focuses on recent results for a) interfacial instabilities in miscible core-annular flows, b) gravitational and viscously driven instabilities in Hele-Shaw cells, and c) interfacial instabilities in both homogeneous and heterogeneous porous media displacements.

1 Miscible core-annular flows Stability investigations of immiscible multi-layer flows date back to the early work of Yih [40], whose linear stability analysis considered the plane Couette-Poiseuille case for fluid layers of different viscosities. By means of an asymptotic long wave analysis, he was able to show that such flows can be linearly unstable to an interfacial mode for all non-zero Reynolds numbers. [15] extended Yih’s work to flows in cylindrical tubes under both axisymmetric and helical perturbations, and to fluids of different densities. Following these early works, Joseph and coauthors performed a number of stability investigations on immiscible core-annular flows. Motivated by the application of lubricated oil pipelines, their focus was on a more viscous core. Comprehensive summaries of much of their work are provided in the monograph by [19], and in the review by [18]. Miscible flow instabilities, which are the focus of this contribution, to date have received much less attention than their immiscible counterparts. [23], [3], [6], [20] [1] and [38] recently investigated miscible, variable viscosity displacements in capillary tubes. They recorded the front propagation velocity, and the thickness of the film left behind on the tube wall, as a function of the viscos-

Eckart Meiburg Department of Mechanical Engineering, University of California, Santa Barbara, CA, 93106, USA, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_4, © Springer Science+Business Media B.V. 2010

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ity ratio, the Péclet number, and the orientation of the tube, as an extension to the classical work by [36] and [8] for immiscible displacements. In [33], the linear stability of variable viscosity, miscible core-annular flows far behind the displacement front is investigated. Consistent with pipe flow of a single fluid, the authors find that the flow is stable at any Reynolds number when the magnitude of the viscosity ratio is less than a critical value. This is in contrast to the immiscible case without interfacial tension, which is unstable at any viscosity ratio. Beyond the critical value of the viscosity ratio, the flow can be unstable even when the more viscous fluid is in the core. This is in contrast to plane channel flows with finite interface thickness, which are always stabilized relative to single fluid flow when the less viscous fluid is in contact with the wall. If the more viscous fluid occupies the core, the axisymmetric mode usually dominates over the corkscrew mode. For a less viscous core, the corkscrew mode is inviscidly unstable, while the axisymmetric mode is unstable for small Reynolds numbers at high Schmidt numbers. The switchover in their relative dominance occurs at an intermediate Schmidt number. The occurrence of inviscid instability for the corkscrew mode is shown to be consistent with the Rayleigh criterion for pipe flows. In some parameter ranges, the miscible flow is seen to be more unstable than its immiscible counterpart, and the physical reasons for this behavior are discussed. In [34], this stability investigation is extended to spatially evolving perturbations, and the convective/absolute nature of the instability of miscible core-annular flow with variable viscosity is addressed via linear stability analysis and nonlinear simulations. From linear analysis, it is found that miscible core-annular flows with the more viscous fluid in the core are at most convectively unstable. On the other hand, flows with the less viscous fluid in the core exhibit absolute instability at high viscosity ratios, over a limited range of core radii. Nonlinear direct numerical simulations in a semi-infinite domain display self-excited intrinsic oscillations if and only if the underlying base flow exhibits absolute instability (figure 1). This oscillatortype flow behavior is demonstrated to be associated with the presence of a nonlinear global mode. Both the parameter range of global instability and the intrinsically selected frequency of nonlinear oscillations, as observed in the simulation, are accurately predicted from linear criteria. In convectively unstable situations, the flow is shown to respond to external forcing over an unstable range of frequencies, in quantitative agreement with linear theory. As discussed in an accompanying experimental investigation [9], self-excited synchronized oscillations were also observed in the laboratory.

2 Miscible Hele-Shaw displacements The instability that forms when one fluid displaces another one of larger viscosity in a Hele-Shaw cell or a porous medium was first analyzed half a century ago ([16], [31], [7]). In the decades since then, the viscous fingering instability has triggered a large body of experimental, theoretical and computational research (cf. the review

Instabilities of Miscible Interfaces

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Fig. 1 Snapshot of the nonlinear evolution of an absolutely unstable, miscible core-annular flow (from [34]).

by [17]) that sheds light on many of its fascinating facets. Many experimental studies of viscous fingering have employed Hele-Shaw configurations, rather than ‘true’ porous media. The main reason for this approach lies in the ease with which the flow can be visualized in such apparatuses. It is well-known that the analogy between a true porous medium and a HeleShaw cell is incomplete ([17]), due to the different nature of flow-induced dispersion effects in the two setups ([35]). Thus a comparison between Darcy results and Hele-Shaw experiments is not straightforward. To date, a closed-form description of dispersion in variable viscosity and density displacements within Hele-Shaw cells or porous media remains elusive. However, several attempts have been made to in-

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corporate its effects into Darcy-based, linear stability investigations, most of them based on the model of a passive tracer fluid [35]. A similar approach towards modeling dispersive effects was taken in the nonlinear simulations by [22]. [39], on the other hand, explicitly address Stokes flows in the Hele-Shaw geometry. They show by way of an asymptotic analysis that the approach of employing a convectiondispersion equation for the volume averaged concentration is of limited value at high Péclet numbers. [41] also hint at the breakdown of this formalism as the plausible reason for the seemingly unphysical instability they observe for sharp base concentration profiles. The above discussion reflects our currently incomplete understanding of the analogy between displacements in Hele-Shaw cells and porous media, respectively, with regard to a quantitative description of the miscible viscous fingering instability. In order to address this deficit, [12] conduct two-dimensional Stokes flow simulations for plane channels, and subsequently investigate the linear stability of the quasisteady base states thus obtained, with respect to spanwise perturbations. The resulting dispersion relations can then be compared with corresponding findings for Darcy flows, in order to identify both similarities and discrepancies between the two. Earlier Stokes flow simulations for miscible displacements in plane channels are reported by [24] for Péclet numbers up to O(500), based on the BGK lattice Boltzmann approach. The authors identify parameter regimes in terms of Péclet number and mobility ratio for which a sharp concentration front develops. Their results compare well with the asymptotic predictions of [39]. They also determine the shape of the displacement front in the limit of high mobility ratios, and they find good agreement with the results of [25] for the corresponding case of immiscible displacements at high capillary numbers. [12] extend the results of [24] to significantly higher Péclet numbers and mobility ratios, which brings to light some new and unexpected effects. Two-dimensional simulations of miscible displacements in a gap indicate the existence of a quasisteady state near the tip of the displacement front for sufficiently large Péclet numbers and viscosity ratios, in agreement with earlier work by other authors. The front thickness of this quasisteady state is seen to scale with Pe−1/2 , while it depends only weakly on the viscosity ratio. The nature of the viscosity-concentration relationship is found to have a significant influence on the quasisteady state. For the exponential relationship employed throughout most of their investigation, the authors find that the tip velocity increases with Pe for small viscosity ratios, while it decreases with Pe for large ratios. The simulation results suggest that in the limit of high Pe and large viscosity contrast, the width and tip velocity of the displacement front asymptote to the same values as their immiscible counterparts in the limit of large capillary numbers. In a subsequent step, the authors examine the stability of the quasisteady front to spanwise perturbations, based on the three-dimensional Stokes equations. A close inspection of the instability eigenfunction reveals the presence of two sets of counterrotating, roll-like structures, with axes aligned in the cross-gap and streamwise directions, respectively. The former lead to the periodic acceleration and deceleration of the front, while the latter result in the widening and narrowing of the front. These roll-like structures are aligned in such a way that the front widens where


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it speeds up, and narrows where it slows down. The findings are compared with their Darcy counterparts. [13] extend this line of investigation to variable density displacements in vertical Hele-Shaw cells. The corresponding gravitational instability is investigated theoretically by [14] and experimentally by [10]. Those authors consider the situation of a heavier fluid placed above a lighter one in a vertically arranged Hele-Shaw cell. The experiments and nonlinear simulations indicate the existence of a low Rayleigh number HeleShaw instability mode, along with a high Rayleigh number gap mode whose dominant wavelength is on the order of three times the gap width. These findings are in disagreement with linear stability results based on the gap-averaged Hele-Shaw approach, which predict much smaller wavelengths. Similar observations have been made for immiscible flows as well ([21]). In order to address this discrepancy, [14] perform a linear stability analysis based on the full three-dimensional Stokes equations. They formulate a generalized eigenvalue problem, whose numerical solution yields both the growth rate and the two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise wavenumber, an interface thickness parameter, and the Rayleigh number. For large Rayleigh numbers, the dispersion relations confirm that the optimally amplified wavelength is about five times the gap width, with the exact value depending on the interface thickness. The corresponding growth rate is in very good agreement with the experimental data as well. The eigenfunctions indicate that the predominant fluid motion occurs within the plane of the Hele-Shaw cell. However, for large Rayleigh numbers purely two-dimensional modes are also amplified, for which there is no motion in the spanwise direction. The investigation by [14] is extended to variable viscosity fluids by [11]. These authors find that, compared to the constant viscosity case, for fluids of variable viscosity the maxima of the eigenfunctions are shifted towards the less viscous fluid. Surprisingly, they observe that for high viscosity ratios, the largest growth rates occur for intermediate interface thicknesses. While the gap-averaged Hele-Shaw analysis also captures this optimal growth for intermediate interface thicknesses, the growth rates differ substantially from the full Stokes equations. Compared to the Hele-Shaw results, growth rates obtained from the modified Brinkmann equation are seen to yield better quatitative agreement. [32] extend these results further to nonmonotonic viscosity profiles.

3 Miscible porous media displacements The stability of interfaces separating fluids of different viscosities in porous media has been the subject of numerous investigations over the years. By means of experiments and, more recently, numerical simulations, the nonlinear interfacial dynamics has been studied as well, using a variety of physical models and geometries, cf. the review by [17]. In many physical applications, the basic instability due to the viscosity contrast is influenced by density and permeability variations as well.

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The overall displacement process is then governed by a combination of these three contributions. For unidirectional base flows, a number of numerical investigations have addressed the above issues in both two and three dimensions, among them [42], [29], [28], [37], [30] and [2]. On the other hand, problems involving line source injection, which occurs frequently during secondary oil recovery processes, have not received comparable attention. In doubly periodic arrangements of injection and production wells such as the well-known quarter five-spot configuration, spatially varying base flows exist that give rise to effects absent in unidirectional flows, cf. the two-dimensional investigations by [4] and [5]. The investigation by [26] extends this line of work to three dimensions, by focusing on variable viscosity and density, miscible displacements in homogeneous permeability, quarter five-spot domains. These authors carry out high accuracy, threedimensional numerical simulations of miscible displacements with gravity override, with special emphasis on describing the influence of viscous and gravitational effects on the overall displacement dynamics in terms of the vorticity variable. Even for neutrally buoyant displacements, three-dimensional effects are seen to change the character of the flow significantly, in contrast to earlier findings for rectilinear displacements. At least in part this can be attributed to the time dependence of the most dangerous vertical instability mode. Density differences influence the flow primarily by establishing a narrow gravity layer (figure 2).

Fig. 2 Snapshot of an unstable displacement in a homogeneous porous medium (from [26]).

[27] extend this work to miscible displacements in heterogeneous porous media. They perform detailed simulations in the regimes of viscous fingering, channeling,


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and resonant amplification, and they analyze the computational results for spatially periodic and random permeability distributions in detail with respect to the individual vorticity components. This enables them to identify the mechanisms dominating specific parameter regimes. Nominally axisymmetric displacements such as the present quarter five-spot configuration are particularly interesting in this respect, since some of the characteristic length scales grow in time as the front expands radially. This leads to displacement flows that can undergo resonant amplification during certain phases, while being dominated by fingering or channeling at other times.

References 1. Balasubramaniam, R., Rashidnia, N., Maxworthy, T., Kuang, J.: Instability of miscible interfaces in a cylindrical tube. Phys. Fluids 17, 052,103 (2005) 2. Camhi, E., Ruith, M., Meiburg, E.: Miscible rectilinear displacements with gravity override. Part 2. Heterogeneous porous media. J. Fluid. Mech. 420, 259 (2000) 3. Chen, C.Y., Meiburg, E.: Miscible displacement in capillary tubes. Part 2. Numerical simulations. J. Fluid Mech. 326, 57 (1996) 4. Chen, C.Y., Meiburg, E.: Miscible porous media displacements in the quarter five-spot configuration. Part 1. The homogeneous case. J. Fluid. Mech. 371, 233 (1998) 5. Chen, C.Y., Meiburg, E.: Miscible porous media displacements in the quarter five-spot configuration. Part 2. Effects of heterogeneities. J. Fluid. Mech. 371, 269 (1998) 6. Chen, C.Y., Meiburg, E.: Miscible displacements in capillary tubes: Influence of Korteweg stresses and divergence effects. Phys. Fluids 14, 2052 (2002) 7. Chouke, R., Meurs, P., Poel, C.d.: The instability of slow, immiscible, viscous liquid-liquid displacements in permeable media. Trans. AIME 216, 188 (1959) 8. Cox, B.G.: On driving a viscous fluid out of a tube. J. Fluid. Mech. 14, 81 (1962) 9. DOlce, M., Martin, J., Rakotomalala, N., Salin, D., Talon, L.: Convective/absolute instability in miscible core-annular flow. Part 1. Experiments. J. Fluid Mech. 618, 305 (2009) 10. Fernandez, J., Kurowski, P., Petitjeans, P., Meiburg, E.: Density-driven, unstable flows of miscible fluids in a Hele-Shaw cell. J. Fluid Mech. 451, 239 (2002) 11. Goyal, N., Meiburg, E.: Unstable density stratification of miscible fluids in a vertical HeleShaw cell: influence of variable viscosity on the linear stability. J. Fluid Mech. 516, 211–238 (2004) 12. Goyal, N., Meiburg, E.: Miscible displacements in Hele-Shaw cells: Two-dimensional base states and their linear stability. J. Fluid Mech. 558, 329 (2006) 13. Goyal, N., Pichler, H., Meiburg, E.: Variable density, miscible displacements in a vertical Hele-Shaw cell: linear stability. J. Fluid Mech. 584, 357 (2007) 14. Graf, F., Meiburg, E., Haertel, C.: Density-driven instabilities of miscible fluids in a HeleShaw cell: linear stability analysis of the three-dimensional Stokes equations. J. Fluid Mech. 451, 261 (2002) 15. Hickox, C.E.: Instability due to viscosity and density stratification in axisymmetric pipe flow. Phys. Fluids 14, 251 (1971) 16. Hill, S.: Channeling in packed columns. Chem. Eng. Sci. 1, 247 (1952) 17. Homsy, G.M.: Viscous fingering in porous media. Ann. Rev. Fluid Mech. 19, 271–311 (1987) 18. Joseph, D.D., Bai, R., Chen, K.P., Renardy, Y.Y.: Core-annular flows. Ann. Rev. Fluid Mech. 29, 65 (1997) 19. Joseph, D.D., Renardy, Y.Y.: Fundamentals of Two-Fluid Dynamics. Part II: Lubricated Transport, Drops and Miscible Liquids. Springer-Verlag, New-York (1992) 20. Kuang, J., Maxworthy, T., Petitjeans, P.: Miscible displacements between silicone oils in capillary tubes. Europ. J. Mech. 22, 271 (2003)

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21. Maxworthy, T.: Experimental study of interface instability in a Hele-Shaw cell. Phys. Rev. A 39, 5863 (1989) 22. Petitjeans, P., Chen, C.Y., Meiburg, E., Maxworthy, T.: Miscible quarter five-spot displacements in a Hele-Shaw cell and the role of flow-induced dispersion. Phys. Fluids 11, 1705 (1999) 23. Petitjeans, P., Maxworthy, T.: Miscible displacements in capillary tubes. Part 1. Experiments. J. Fluid Mech. 326, 37 (1996) 24. Rakotomalala, N., Salin, D., Watzky, P.: Miscible displacement between two parallel plates: BGK lattice gas simulations. J. Fluid Mech. 338, 277 (1997) 25. Reinelt, D., Saffman, P.: The penetration of a finger into a viscous fluid in a channel and tube. SIAM J. Sci. Stat. Comput. 6, 542 (1985) 26. Riaz, A., Meiburg, E.: Three-dimensional miscible displacement simulations in homogeneous porous media with gravity override. J. Fluid. Mech. 494, 95 (2003) 27. Riaz, A., Meiburg, E.: Vorticity interaction mechanisms in variable viscosity, heterogeneous miscible displacements with and without density contrast. J. Fluid. Mech. 517, 1 (2004) 28. Rogerson, A., Meiburg, E.: Numerical simulation of miscible displacement processes in porous media flows under gravity. Phys. Fluids A 5, 2644 (1993) 29. Rogerson, A., Meiburg, E.: Shear stabilization of miscible displacement processes in porous media. Phys. Fluids A 5, 1344 (1993) 30. Ruith, M., Meiburg, E.: Miscible rectilinear displacements with gravity override. Part 1. Homogeneous porous medium. J. Fluid. Mech. 420, 225 (2000) 31. Saffman, P., Taylor, G.: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 312 (1958) 32. Schafroth, D., Goyal, N., Meiburg, E.: Miscible displacements in Hele-Shaw cells: Nonmonotonic viscosity profiles. Eur. J. Mech. B/Fluids 26, 444 (2007) 33. Selvam, B., Merk, S., Govindarajan, R., Meiburg, E.: Stability of miscible core-annular flow with viscosity stratification. J. Fluid Mech. 592, 23 (2007) 34. Selvam, B., Talon, L., Lesshafft, L., Meiburg, E.: Convective/absolute instability in miscible core-annular flow. Part 2. Numerical simulations and nonlinear global modes. J. Fluid Mech. 618, 323 (2009) 35. Taylor, G.: Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186 (1953) 36. Taylor, G.I.: Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10, 161 (1960) 37. Tchelepi, H., Orr, F., Rakotomalala, N., Salin, D., Woumeni, R.: Dispersion, permeability heterogeneity, and viscous fingering: Acoustic experimental observations and particle-tracking simulations. Phys. Fluids A 5, 1558 (1993) 38. Vanaparthy, H., Meiburg, E.: Variable density and viscosity, miscible displacements in capillary tubes. Eur. J. Mech. B/Fluids 27, 268 (2008) 39. Yang, Z., Yortsos, Y.C.: Asymptotic solutions of miscible displacements in geometries of large aspect ratio. Phys. Fluids 9, 286–298 (1997) 40. Yih, C.S.: Instability due to viscous stratification. J. Fluid Mech. 27, 337 (1967) 41. Yortsos, Y.C., Zeybek, M.: Dispersion driven instability in miscible displacement in porous media. Phys. Fluids. 31, 3511–3518 (1988) 42. Zimmerman, W.B., Homsy, G.M.: Viscous fingering in miscible displacements: Unification of effects of viscosity contrast, anisotropic dispersion, and velocity dependence of dispersion on nonlinear propagation. Phys. Fluids A 4, 2348 (1992)

Large-eddy simulations of relaminarization due to freestream acceleration Ugo Piomelli and Carlo Scalo

Abstract The reversion of a turbulent flow to a laminar or quasi-laminar state may take different paths, depending on the cause of the relaminarization; modelling these flows is quite difficult. Large-eddy simulations (LES) may be quite effective, since the large, momentum carrying eddies are accurately resolved; the unresolved, subgrid-scale (SGS) stresses, however, may still play an important role. We report here the results of a study of the SGS stresses in a boundary layer subjected to freestream acceleration leading to relaminarization. The a priori test results are accompanied by actual LES, which highlight the effect of grid resolution and modelling errors on the dynamics of the relaminarization.

1 Introduction While the transition to turbulence of a laminar flow has received, historically, very much attention, the reverse case, i.e., a turbulent flow that reverts to a laminar state, has been studied less frequently. This problem, however, is also quite important in applications: relaminarization can be observed, for instance, in stratified, rotating or accelerating boundary layers. An important feature of relaminarizing flows is the fact that in most cases turbulent fluctuations do not vanish completely, whereas the correlation between streamwise and vertical fluctuations decreases, reducing turbulent momentum transport: turbulent motions persist, but become inactive. Understanding the state of the relaminarized flow in these cases is very important, since once the cause of the relaminarization is removed, the flow may transition back to turbulence, in a way that depends critically on the residual turbulence levels. Both transitioning and relaminarizing flows pose significant modelling challenges: turbulence models for the Reynolds-Averaged Navier-Stokes equations are Ugo Piomelli and Carlo Scalo Department of Mechanical and Materials Engineering, Queen’s University, Kingston, ON, K7L 3N6, Canada e-mail: [email protected], [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_5, © Springer Science+Business Media B.V. 2010

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U. Piomelli, C. Scalo

usually based on high-Reynolds-number turbulence physics, and are often unable to predict the non-equilibrium state that characterizes these flows without ad hoc adjustments. Subgrid-scale (SGS) models for large-eddy simulations (LES) are also based on high-Reynolds number physics, and do not include transitional effects; some models, however, have the property of giving vanishing eddy-viscosity when the flow is laminar. SGS models based on dynamic modelling ideas [2] or the spectral-dynamic model [4] have this property, and have been used successfully in simulations of this type. In LES of laminar-to-turbulent transition, however, grid resolution becomes an issue: the instability mechanisms must be resolved by the mesh itself, which may require very significant computational resources. The simulation must, in such cases, approach the resolution of a DNS, since coarse meshes tend to promote early transition [12]. In relaminarizing flows the grid requirements are less stringent than in transitional cases, since the turbulent eddies tend to become more elongated, and the smallest eddies are dissipated first. The SGS model, however, must still be able to account properly for the factors causing relaminarization; very dissipative models, for instance, are expected to result in excessively rapid reversion. In their review article, Narasimha and Sreenivasan [10] consider various mechanisms for reversion to a laminar state, and distinguish between cases in which the turbulent kinetic energy (TKE) decays during the re-laminarization process, from those in which it remains “frozen”, while the vertical and horizontal velocity components become decorrelated. Here, we will examine the reversion to laminar flow in a turbulent boundary layer subjected to strong acceleration (an example of the latter type of relaminarization). By performing well-resolved direct numerical simulations (DNS) we will be able to verify the accuracy of the LES model, and also understand the physics of the energy transfer from large to small scales. This goal shall be achieved by a priori tests, in which the DNS data is filtered and the “exact” SGS stresses, forces and dissipation are computed, and by a posteriori comparisons of the DNS data with the actual LES results.

2 Problem formulation The governing equations solved in this problem are either the incompressible flow equations of conservation of mass and momentum, or their filtered counterpart:

∂uj = 0; ∂xj

∂ ui ∂ u j ui ∂ 2 ui 1 ∂ p ∂ τi j + =ν − − ∂t ∂xj ∂ x j ∂ x j ρ ∂ xi ∂ x j

(1)

The subgrid-scale (SGS) stresses, τi j = ui u j − ui u j , are modelled using either the Plane-Averaged (DEV) [2, 6], or the Lagrangian-Averaged Dynamic EddyViscosity model (LDEV) [8]. The numerical model used is a well-validated finite-difference code [3], based on a staggered grid. Second-order central differences are used for both convective and

Large-eddy simulations of relaminarization due to freestream acceleration

37

diffusive terms, and a semi-implicit time-advancement scheme is used: the CrankNicolson scheme is used for the wall normal diffusive term, while a low-storage 3rd-order Runge-Kutta method is applied to the remaining terms. The solution of the Poisson equation is obtained by Fourier-transforms in the streamwise and spanwise directions, followed by a direct solution of the resulting tridiagonal matrix. The code is parallelized using the MPI protocol. Boundary conditions are periodic in the spanwise direction z. No-slip conditions are used at the walls. At the inflow, the recycling/rescaling method by Lund et al. [7] is used, while at the outflow a convective condition is applied [11]. At the freestream, a profile of the streamwise time-averaged velocity U∞ (x) is assigned; the mean freestream wall-normal velocity component, V∞ , is derived from mass conservation to satisfy V∞ (x) = U∞

dδ ∗ dU∞ + (δ ∗ − h) dx dx

(2)

(where δ ∗ is the displacement thickness and h the domain height) and homogeneous Neumann conditions are applied to the fluctuating velocity components. The Reynolds number Re∗ , based on Uo and δo∗ (the freestream velocity and displacement thickness at a reference location, x = 0, located downstream of the rescaling plane) is 800 (Reθ = 500 based on momentum thickness θ ). We report the results of three calculations: a DNS using 4096×257×384 grid points (in the streamwise, wall-normal and spanwise directions, respectively), and two LES with 1536×192×192 points, one with the DEV model, the other with the LDEV one. The grid was equispaced in x and z, and stretched in the wall-normal direction y. The useful part of the domain (downstream of the recycling region) extended from x/δo∗ = 0 to x/δo∗ ≃ 300. The computational domain extended to x/δo∗ ≃ 450, but the resolution near the outlet was marginal (in wall units), due to the increase in the friction velocity, making the results quantitatively unreliable.

3 Results and discussion Figure 1 shows the streamwise development of freestream velocity, acceleration parameter K = (ν /U∞2 )(dU∞ /dx), and skin-friction coefficient C f = τw /(ρ U∞2 /2). K exceeds significantly the threshold above which relaminarization is expected to occur (K > 3 × 10−6 ). One indication of the reversion to the laminar state is the decrease of the skin-friction coefficient observed between x/δo∗ ≃ 200 and 280350. This is due to a drastic re-organization of the near-wall eddies, that results in decorrelation between the streamwise and wall-normal fluctuations, and lower Reynolds shear stress (see the discussion in [9, 13, 1]). Only after the acceleration ends, at x/δo∗ ≃ 300, the flow begins to retransition to a turbulent state. The early increase of C f observed in the LES is due to the insufficient grid resolution in this region, as shown in Ref. [12].

38 Fig. 1 Streamwise development of (a) freestream velocity, (b) acceleration parameter and (c) skin-friction coefficient. ◦ DNS; LES, LES, DEV model; LDEV model.


(a)

(b)

(c)

The agreement between LES and DNS is fairly good, at least until the beginning of retransition. We defer a comparison of turbulent statistics until later, to examine first the a priori prediction of the behaviour of the subgrid, unresolved, scales, which can be gleaned from the DNS data. The DNS velocity fields were filtered using a trapezoidal filter with width ∆ f = 4∆ g (where ∆ g is the grid size). We examine several quantities: first, the subgrid-scale (SGS) stresses, τi j , and dissipation εsgs = −τi j Si j (Si j is the large-scale strain-rate tensor), which represents the net transfer of energy between resolved and unresolved scales. In addition, we computed the (resolved) structure parameter a1 = −hu′ v′ i/q2 (where q2 = hu′i u′i i is twice the turbulent kinetic energy), which represents the efficiency with which turbulence transports momentum for a given turbulent kinetic energy level. We also evaluated an SGS counterpart of this parameter, a1,sgs = −hτ12 i/q2 . The a priori prediction of the subgrid scale quantities is shown in Figure 2. The SGS shear stress is initially significant through the lower third of the boundary layer. As the flow relaminarizes, and the boundary layer becomes thinner, τ12 decreases and becomes confined to a thinner region near the wall. At the beginning of retransition the SGS stress increases again, and remains confined to a thinner region near the wall (reflecting the lower boundary layer thickness in the retransition region). The higher values of τ12 observed in the retransition region are due to the fact that the Reynolds number here is higher (since U∞ and uτ are nearly three times larger than in the inflow region), and a wider range of scales is formed; since the filter size is the same as in the ZPG region, however, the expanded small-scale range will increase the SGS contribution to momentum transport. The SGS dissipation follows similar trends; the effect of the shear, however, is to confine εsgs closer to the wall, compared to τ12 . The structure parameter a1 and its SGS counterpart, are also shown in Figure 2. Through most of the zero-pressure-gradient (ZPG) region a1 is approximately equal to 0.17, close to the value expected in equilibrium boundary layers. In the relaminarization region a1 increases slightly, and remains elevated at


39

Fig. 2 A priori prediction of the various measures of subgrid-scale activity. The thick line shows the boundary-layer edge.

the beginning of the transition region. Note that here the region of significant a1 extends well out of the boundary layer, and begins further away from the wall than in the initial equilibrium ZPG flow (two items worthy of further investigation). The behaviour of the SGS structure parameter, although generally similar to that of a1 also shows some differences. Its value, as expected, is significantly lower (the SGS stress with this filter is about 5% of the resolved one). It also decreases as the acceleration begins: by x/δo∗ ≃ 200 its maximum value is reduced by 50% (while the maximum value of a1 has increased by 20%). This reflects the fact that the small scales decay more rapidly than the large ones (which determine the resolved stress and q2 ). After retransition begins, a1,sgs increases rapidly, reflecting the increased SGS contribution to the momentum transport. Profiles of mean velocity (normalized by the freestream velocity U∞ ) as a function of y/θ are shown in Figure 3. The LES agree fairly well with the DNS. Note the existence of a logarithmic layer in the initial zero-pressure-gradient region, followed by a region in which a well-mixed layer is established with a higher value of the von Kármán constant, as observed in other studies (see, for instance, [1]). The region of re-laminarization is characterized by a low value of the friction velocity, and a laminar-like profile (x/δo∗ ≃ 280 − 350). The same profiles, when plotted in wall units, show some discrepancy between LES and DNS results, due to the over-prediction of the wall stress by the LES, which results in lower intercept of the logarithmic layer. This is a fairly common outcome of numerical simulations with marginal resolution and low dissipation. In the present case, the numerical scheme, which uses a staggered mesh, is fully conservative, and the SGS models used are not very dissipative. The prediction of the Reynolds stresses by the LES is also reasonably good. Figure 3 shows the streamwise turbulent intensities at several locations in the boundary

40


(a)

(b)

Fig. 3 Profiles of (a) mean velocity and (b) streamwise rms fluctuations. ◦ DNS; model; LES, LDEV model.

LES, DEV

layer. The decrease of the turbulent kinetic energy through the initial part of the acceleration region is predicted well. After x/δo∗ = 250 the peak TKE is overpredicted by both LES models. Figure 4 shows the development of the negative of the SGS dissipation, −εsgs . The DEV model is slightly more dissipative than the Lagrangian-averaged counterpart. The LDEV model has a more physical response to the acceleration: we observe a significant decrease of SGS activity, which mimics the decreased Reynolds stresses observed in this region, as turbulence becomes inactive (see the discussion of this issue in Narasimha and Sreenivasan [10], and also in De Prisco et al. [1]). Both models predict increased dissipation in the region where the flow retransitions, small scales are generated, and the grid becomes excessively coarse. In Figure 5 we show contours of the streamwise fluctuating velocity component. Both DNS and LES show the decreased turbulent activity in the acceleration region. Turbulent eddies are, however, still present, consistent with the view that “inactive” turbulence advected from upstream characterizes this type of reversion. The retran-

(a)

Fig. 4 Contours of the SGS dissipation. (a) DEV model; (b) LDEV model.

(b)


41

Fig. 5 Contours of u′ /U∞ in an xz−plane: (top) DNS; (bottom) LES

sition to turbulence takes place through the formation of turbulent spots, spatially related to instabilities of the near-wall streaks, which grow as they advect, until they merge to form a fully turbulent flow. The LES results show generally fewer turbulent spots and a more abrupt establishment of retransition. Notice that the grid resolution after x = 350 is marginal even in the DNS, and the numerical results should be considered qualitative only. Moreover, the computational domain is not sufficiently large to contain more than a single turbulent spot; this may also affect the dynamics of the retransition process.

4 Conclusions We have performed numerical simulations of a flow in which reversion from a turbulent to a laminar state occurs due to the freestream acceleration. First, a finely resolved DNS was performed, which gave a priori indications on the physical response of the unresolved scales, and also supplied data to evaluate the accuracy of SGS models, which were applied in actual LES. The SGS models used (the plane-averaged dynamic eddy-viscosity model, DEV, and the Lagrangian-averaged dynamic eddy-viscosity model, LDEV) do not predict the response of the SGS eddies precisely; however, the trends are correct (e.g., the eddy viscosity and SGS dissipation decrease in regions of reversion, for instance). The agreement between LES and experimental and DNS data, at least for the loworder statistics, is acceptable. The LES was also able to capture the behaviour of the coherent eddies during relaminarization and, to a lesser extent, during the retransition to turbulence, although the fine details of the streak instability leading to the formation of a turbulent spot are not resolved. At the level of resolution used, the SGS contribution to the momentum transport was less than 10%, and the accuracy of the models tested was comparable. In the accelerating boundary layer case the use of plane averaging is not justified, since the flow is highly inhomogeneous in the streamwise direction. However, because of the small contribution given by the SGS stresses to the turbulent momentum transport, the DEV model still gave results in good agreement with the DNS despite the modelling errors. Perhaps the most significant source of error, in the present

42


LES, was the grid resolution, which resulted in high levels of turbulent activity near the wall. One limitation of the present study is the Reynolds number of the calculations. The desire to perform companion DNS limited the achievable Reynolds number. As a follow up, we plan to perform simulations of the accelerating boundary layer at higher Reynolds numbers to further evaluate the SGS models a posteriori. Acknowledgements The authors thank the High Performance Computing Virtual Laboratory (HPCVL), Queen’s University site, for the computational support.

References 1. G. De Prisco, A. Keating, and U. Piomelli. Large-eddy simulation of accelerating boundary layers. AIAA Paper 2007-0725, 2007. 2. M. Germano, U. Piomelli, P. Moin, and W. Cabot. A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A, 3:1760–1765, 1991. 3. A. Keating, U. Piomelli, K. Bremhorst, and S. Neˇsić. Large-eddy simulation of heat transfer downstream of a backward-facing step. J. Turbul., 5(20):1–27, 2004. 4. E. Lamballais, O. Métais, and M. Lesieur. Spectral-Dynamic Model for Large-Eddy Simulations of Turbulent Rotating Channel Flow. Theor. Comput. Fluid Dyn., 12(3):149–177, 1998. 5. D. K. Lilly. The representation of small scale turbulence in numerical simulation experiments. In Proceedings of the IBM Scientific Computing Symposium on Environmental Sciences, pages 195–210, 1967. 6. D. K. Lilly. A proposed modification of the Germano subgrid-scale closure method. Phys. Fluids A, 4:633–635, 1992. 7. T. S. Lund, X. Wu, and K. D. Squires. Generation of inflow data for spatially-developing boundary layer simulations. J. Comput. Phys., 140:233–258, 1998. 8. C. Meneveau, T. S. Lund, and W. H. Cabot. A Lagrangian dynamic subgrid-scale model of turbulence. J. Fluid Mech., 319:353–385, 1996. 9. R. Narasimha and K. R. Sreenivasan. Relaminarization in highly accelerated turbulent boundary layers. J. Fluid Mech., 61:417–447, 1973. 10. R. Narasimha and K. R. Sreenivasan. Relaminarization of fluid flows. In Adv. Applied Mech., volume 19, pages 221–309, New York, 1979. Academic Press Professional, Inc. 11. I. Orlanski. A simple boundary condition for unbounded hyperbolic flows. J. Comput. Phys., 21:251–269, 1976. 12. V. Ovchinnikov, U. Piomelli, and M. M. Choudhari. Inflow conditions for numerical simulations of bypass transition. AIAA Paper 2004-0491, 2004. 13. U. Piomelli, E. Balaras, and A. Pascarelli. Turbulent structures in accelerating boundary layers. J. Turbul., 1(1):1–16, 2000. 14. F. Porté-Agel, C. Meneveau, and M. B. Parlange. A scale-dependent dynamic model for large-eddy simulation: application to a neutral atmospheric boundary layer. J. Fluid Mech., 415:261–284, 2000. 15. J. Smagorinsky. General circulation experiments with the primitive equations. I. The basic experiment. Mon. Weather Rev., 91:99–164, 1963.

Reduced-order models for flow control: balanced models and Koopman modes Clarence W. Rowley, Igor Mezić, Shervin Bagheri, Philipp Schlatter, and Dan S. Henningson

Abstract This paper addresses recent developments in model-reduction techniques applicable to fluid flows. The main goal is to obtain low-order models tractable enough to be used for analysis and design of feedback laws for flow control, while retaining the essential physics. We first give a brief overview of several model reduction techniques, including Proper Orthogonal Decomposition [3], balanced truncation [8, 9], and the related Eigensystem Realization Algorithm [5, 6], and discuss strengths and weaknesses of each approach. We then describe a new method for analyzing nonlinear flows based on spectral analysis of the Koopman operator, a linear operator defined for any nonlinear dynamical system. We show that, for an example of a jet in crossflow, the resulting Koopman modes decouple the dynamics at different timescales more effectively than POD modes, and capture the relevant frequencies more accurately than linear stability analysis.

1 Introduction The ability to effectively control a fluid would enable many exciting technological advances, including modifying the stability of laminar flows, and delaying transition from laminar to turbulent flow. Many of the tools available for analysis and design of control systems require knowledge of a model in terms of a system of differential equations, and the equations governing a fluid, though known, are too complex for these tools to apply. Model reduction addresses this problem: one obtains approxC.W. Rowley Princeton University, USA, e-mail: [email protected] I. Mezić University of California, Santa Barbara, USA e-mail: [email protected] S. Bagheri, P. Schlatter, and D.S. Henningson KTH, Stockholm, Sweden, e-mail: [email protected], [email protected], [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_6, © Springer Science+Business Media B.V. 2010

43

44

C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, D. S. Henningson

imate models that are computationally tractable and capture the essential physics, but neglect details that are not critical for the problem at hand. Here, we present a brief overview and comparison of various methods used for obtaining reduced-order models, aimed at applications in flow control. We then present a new method for analyzing fluid flows, based on spectral analysis of an object called the Koopman operator, and apply the method to a jet in crossflow.

2 Model reduction techniques Many of the methods used for model reduction involve projecting known, highdimensional dynamics onto a set of modes. For instance, for a state variable q(t) (which could be a flow field at a specified time t), we begin with known dynamics q(t) ˙ = f (q) (for instance, the Navier-Stokes equations). Then we expand q(t) in terms of a set of basis functions, or modes, ϕk : n

q(t) =

∑ ak (t)ϕk .

(1)

k=1

If the modes ϕk are orthonormal, we obtain projected dynamics as a˙k (t) = h f (q(t)), ϕk i. If the modes are not orthonormal (e.g., if they are eigenmodes of a non-normal operator A),

then we often have a complementary set of adjoint modes ψ j that satisfy ϕ j , ψk = δ jk (e.g., eigenmodes of adjoint operator A∗ ). In this case, we still have the expansion (1), but the projected dynamics are given by a˙k (t) = h f (q(t)), ψk i. In such projection methods, the main choices for obtaining reduced-order models are therefore how to choose the modes ϕ j and ψk .

2.1 Proper Orthogonal Decomposition and its limitations A common approach is to determine the modes ϕk by Proper Orthogonal Decomposition (POD) of a certain dataset [3]. While this approach is optimal for capturing the energy in a given dataset, this choice is often not appropriate for obtaining reducedorder models, since low-energy modes can be critically important for capturing the dynamics. A striking example of the importance of low-energy modes is shown in [4], in which POD modes are computed from snapshots of the transient growth of a disturbance in a linearized channel flow. The first five POD modes capture over 99% of the energy, and yet the corresponding reduced-order model completely misses the transient energy growth. If, however, low-energy modes 10 and 17 are used in place of modes 4 and 5, the resulting five-mode model performs very well, and captures the transient growth nearly perfectly. While POD models can work well, this ex-

Reduced-order models for flow control

45

ample illustrates that they often require careful tuning, and one must be aware that low-energy modes can be critically important to the dynamics.

2.2 Balanced models An alternative approach, popular in the control theory community, is balanced truncation [8]. This approach is applicable to linear systems, and has a priori error bounds that are close to the minimum possible error from any reduced-order model. In [9], an approximation of balanced truncation is introduced, called Balanced POD. In this method, the direct modes ϕ j and adjoint modes ψk are computed from snapshots taken from a simulation of the original (linear) system and an adjoint system. Once these modes are known, the reduced-order models are computed as described above. This method, which approaches exact balanced truncation as the number of snapshots is increased, also corresponds to POD of a particular dataset (an impulse response) with respect to a particular inner product (called the observability Gramian). This inner product may be regarded as a measure of dynamic importance, and thus, the notion of “energy” in POD is simply redefined to refer to dynamic importance, rather than the usual (physical) energy. In practice, this method often dramatically outperforms the standard POD method, particularly for highly non-normal systems. For instance, for the transient growth problem in [4] mentioned above, a 3-mode balanced model captures the transient growth nearly perfectly, and the models consistently improve as more modes are included, without any of the ad hoc tweaking necessary for the POD models. It is also worth noting that balanced models may also be obtained using the Eigensystem Realization Algorithm (ERA) [5]. In fact, it has recently been shown that for linear systems, ERA produces reduced-order models that are identical to those from Balanced POD [6]. This method does not involve adjoint simulations, and hence can be used with experimental data. However, unlike Balanced POD, the Eigensystem Realization Algorithm does not produce modes ϕ j , ψk , which can be useful for a variety of purposes, such as projection of nonlinear dynamics, or retaining parameters in the models. For more information about the advantages and disadvantages of ERA, see [6].

3 Spectral analysis of nonlinear flows In this section, we describe a new method for analyzing the dynamics of nonlinear systems, based on spectral analysis of an object called the Koopman operator. The Koopman operator is a linear operator defined for any nonlinear system, but it is not based on linearization: indeed, it captures all of the dynamics of the full nonlinear system. The Koopman operator describes the evolution of observables on the phase space. For instance, an observable may be a 2D slice of velocity vectors obtained

46


from an experiment using Particle Image Velocimetry (PIV). Below, we define this operator, and the Koopman modes associated with a particular observable. For a more detailed explanation of this operator and its use, see [7, 10].

3.1 Koopman operator and Koopman modes Consider a dynamical system evolving on a manifold M such that, for xk ∈ M, xk+1 = f(xk ),

(2)

where f is a map from M to itself. The Koopman operator is a linear operator U that acts on scalar-valued functions on M in the following manner: for any scalar-valued function g : M → R, U maps g into a new function Ug given by Ug(x) = g(f(x)).

(3)

Although the dynamical system is nonlinear and evolves on a finite-dimensional manifold M, the Koopman operator U is linear, but infinite-dimensional. The idea is to analyze the flow dynamics governed by (2) only from available data—collected either numerically or experimentally—using the eigenfunctions and eigenvalues of U. To this end, let ϕ j : M → R denote eigenfunctions and λ j ∈ C denote eigenvalues of the Koopman operator, U ϕ j (x) = λ j ϕ j (x),

j = 1, 2, . . .

(4)

and consider a vector-valued observable g : M → R p . For instance, if x ∈ M contains the full information about a flow field at a particular time, g(x) is a vector of any quantities of interest, such as a velocity measurements at various points in the flow. If each of the p components of g lies within the span of the eigenfunctions ϕ j , then as in [7], we may expand the vector-valued g in terms of these eigenfunctions, as g(x) =

∞

∑ ϕ j (x)v j .

(5)

j=1

We typically think of this expression as expanding the vector g(x) as a linear combination of the vectors v j , but we may alternatively think of this expression as expanding the function g(·) as a linear combination of the eigenfunctions ϕ j of U, where now v j are the (vector) coefficients in the expansion. In this paper, we will refer to the eigenfunctions ϕ j as Koopman eigenfunctions, and the corresponding vectors v j in (5) the Koopman modes of the map f, corresponding to the observable g. Note that iterates of x0 are then given by g(xk ) =

∞

∞

j=1

j=1

∑ U k ϕ j (x0)v j = ∑ λ jk ϕ j (x0)v j .

(6)


47

The Koopman eigenvalues, λ j ∈ C, therefore characterize the temporal behavior of the corresponding Koopman mode v j : the phase of λ j determines its frequency, and the magnitude determines the growth rate. Note that, as described in [7], for a system evolving on an attractor, the Koopman eigenvalues always lie on the unit circle.

3.2 Properties of Koopman modes and eigenvalues It is not immediately clear why the Koopman modes and eigenvalues might be of interest in studying fluid flows. However, these modes are in fact related to objects routinely used in fluid mechanics, such as global eigenmodes (for linear systems) and the discrete Fourier transform (for periodic solutions of (2)). For a more detailed presentation, see [10], but here we summarize the main results: • For a linear system (xk+1 = Axk ), if the observable is the full state g(x) = x, then the eigenvalues of A are also Koopman eigenvalues, and the corresponding eigenvectors of A are Koopman modes. • For a nonlinear system with a periodic orbit, if we restrict the phase space to the periodic orbit, then the Koopman modes are given by the discrete Fourier transform of the vectors that make up the periodic orbit. In particular, if we have a set S = {x0 , . . . , xm−1 } that forms a periodic solution of (2), such that xk+m = xk for all k, then the discrete Fourier transform defines a new set of vectors {ˆx0 , . . . , xˆ m−1 } that satisfy xk =

m−1

∑ e2π i jk/m xˆ j ,

j=0

k = 0, . . . , m − 1.

(7)

Then the vectors xˆ j are Koopman modes, with corresponding eigenvalues λ j = e2π i j/m . Thus, the phase 2π i j/m of the Koopman eigenvalue λ j is the frequency of the corresponding mode xˆ k . The Koopman modes therefore provide a framework that unifies linear stability theory (for transients of linear systems), and the discrete Fourier transform (for periodic solutions of nonlinear systems).

3.3 Computing Koopman modes from snapshots The Koopman eigenvalues and modes would not be useful for practical applications if they could not be computed. It turns out that approximations to these modes and eigenvalues may be computed directly from snapshots of an observable, using a Krylov subspace method, a variant of the commonly used Arnoldi iteration [11]. The algorithm is in fact identical to that referred to as Dynamic Mode Decomposition in [12]. For the details of the algorithm, and its relation to Koopman modes, see [10].

48


3.4 Example: jet in crossflow In this section we compute the Koopman modes for a jet in crossflow, and show that they directly allow an identification of various phenomena present in this flow. The parameters and the numerical code are the same as in the DNS performed in [2]; the jet inflow ratio is R ≡ Vjet /U∞ = 3, the Reynolds number is Reδ0∗ ≡ U∞ δ0∗ /ν = 165 and the ratio between the crossflow displacement thickness and the jet diameter is δ0∗ /D = 1/3. Approximate Koopman eigenvalues λ j and eigenvectors v j are computed from a sequence of flow-fields {u0 , u1 , . . . , um−1 } = {u(t = 200), u(t = 202), . . . , u(t = 700)} with m = 251, using the algorithm mentioned in Section 3.3. The transient time (t < 200) is not sampled, and only the asymptotic motion in phase space is considered. 400

kv j k

300

200

100

Fig. 1 The magnitudes of the Koopman modes v j , as a function of their nondimensional frequency St = ω D/(2πVjet ).

0 0

0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 St

The magnitudes of the modes, defined by the global energy norm kv j k, are shown in Figure 1, as a function of the corresponding frequency ω j = Im{log(λ j )}/∆ t (with ∆ t = 2 in our case). Only the ω j ≥ 0 are shown, since the eigenvalues come in complex-conjugate pairs. Ordering the modes with respect to their magnitude, the first (2–3) and second (4–5) pair of modes oscillate with St2 = 0.141 and St4 = 0.136 respectively, whereas the third pair of modes (6-7) oscillate with St6 = 0.017. All linear combinations of the frequencies excite higher modes: for instance, the nonlinear interaction of the first and third pair results in the fourth pair, i.e. St8 = 0.157 and so on. The frequencies of these dominant Koopman modes agree very well with frequencies obtained from point measurements of the DNS, taken from the shear layer and near-wall regions, respectively, as shown in Table 1. The streamwise velocity component u of Koopman modes 2 and 6 are shown in Figure 2. Each mode represents a flow structure that oscillates with one single frequency, and the superposition of several of these modes results in the quasiperiodic


49

global system. The high-frequency mode 2 (Figure 2(a)) can be associated with shear layer vortices; along the jet trajectory there is first a formation of ring-like vortices that eventually dissolve into smaller scales due to viscous dissipation. Also visible are upright vortices: on the leeward side of the jet, there is a significant structure extending towards the wall. This indicates that the shear-layer vortices and the upright vortices are coupled and oscillate with the same frequency. On the other hand, the low-frequency mode 6 shown in figure 2(b) features largescale positive and negative streamwise velocity near the wall, which can be associated with shedding of the wall vortices. However, this mode also has structures along the jet trajectory further away from the wall. This indicates that the shedding of wall vortices is coupled to the jet body, i.e. the low frequency can be detected nearly anywhere in the vicinity of the jet since the whole jet is oscillating with that frequency. (a)

(b)

Fig. 2 Positive (red) and negative (blue) contour levels of the streamwise velocity components of two Koopman modes. The wall is shown in gray. (a) Mode 2, with kv2 k = 400 and St2 = 0.141. (b) Mode 6, with kv6 k = 218 and St6 = 0.0175.

3.5 Comparison with linear global modes and POD modes The linear global eigenmodes of the Navier-Stokes equations linearized about an unstable steady state solution were computed by [2] for the same flow parameters as the current study. They computed 22 complex-conjugate unstable modes using the Arnoldi method combined with a time-stepper approach. The frequency of the most unstable (anti-symmetric) mode associated with the shear-layer instability was St = 0.169, not far from the value St = 0.14 observed for the DNS. However, the mode with the lowest frequency associated with the wall vortices was St = 0.043, far from the observed frequency of St = 0.017. These frequencies are summarized in Table 1. The global eigenmodes capture the dynamics only in a neighborhood of the unstable fixed point, while the Koopman modes correctly capture the behavior on the attractor. We also compared the Koopman modes with modes determined by Proper Orthogonal Decomposition (POD) of the same dataset. The POD modes themselves are shown in [1], and capture similar spatial structures to the Koopman modes shown in Figure 2. The most striking distinction is in the time coefficients: while a single

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C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, D. S. Henningson Mode DNS Global POD Koopman Shear layer 0.141 0.169 0.138, 0.158, 0.121 0.141 Wall 0.017 0.043 0.0188, 0.0094, 0.158, 0.121 0.017

Table 1 Comparison of the frequencies (St = f D/Vjet ) obtained from DNS probes; the global eigenmodes of the linearized Navier-Stokes; POD modes 1 and 6, corresponding to mainly shearlayer and wall oscillations, respectively; and Koopman modes.

Koopman mode contains, by construction, only a single frequency component, the POD modes capture the most energetic structures, resulting in modes that contain several frequencies. For situations such as the jet in crossflow where one is interested in studying the dynamics of low-frequency oscillations (such as wall modes) separate from high-frequency oscillations (such as shear-layer modes), the Koopman modes are thus more effective at decoupling and isolating these dynamics. Acknowledgments This work was supported by the National Science Foundation, award CMS-0347239, and the Air Force Office of Scientific Research, awards FA9550-05-1-0369 and FA9550-07-1-0127.

References 1. S. Bagheri, P. Schlatter, and D. S. Henningson. Self-sustained global oscillations in a jet in crossflow. Theor. Comput. Fluid Dyn., (submitted), 2009. 2. S. Bagheri, P. Schlatter, P. Schmid, and D. S. Henningson. Global stability of a jet in crossflow. J. Fluid Mech., 624:33–44, 2009. 3. P. Holmes, J. L. Lumley, and G. Berkooz. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge, UK, 1996. 4. M. Ilak and C. W. Rowley. Modeling of transitional channel flow using balanced proper orthogonal decomposition. Phys. Fluids, 20:034103, March 2008. 5. J.-N. Juang and R. S. Pappa. An eigensystem realization algorithm for modal parameter identification and model reduction. J. Guid. Contr. Dyn., 8(5):620–627, 1985. 6. Z. Ma, S. Ahuja, and C. W. Rowley. Reduced order models for control of fluids using the eigensystem realization algorithm. Theor. Comput. Fluid Dyn., (submitted), 2009. 7. I. Mezić. Spectral properties of dynamical systems, model reduction and decompositions. Nonlin. Dyn., 41:309–325, 2005. 8. B. C. Moore. Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Trans. Automat. Contr., 26(1):17–32, February 1981. 9. C. W. Rowley. Model reduction for fluids using balanced proper orthogonal decomposition. Int. J. Bifurcation Chaos, 15(3):997–1013, March 2005. 10. C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. S. Henningson. Spectral analysis of nonlinear flows. J. Fluid Mech., (accepted), 2009. 11. A. Ruhe. Rational Krylov sequence methods for eigenvalue computation. Linear Algebra Appl., 58:391–405, 1984. 12. P. Schmid and J. Sesterhenn. Dynamic mode decomposition of numerical and experimental data. 61st Annual Meeting of the APS Division of Fluid Dynamics, November 2008.

The description of fluid behavior by coherent structures Peter J. Schmid

1 Introduction and motivation The goal of many scientific investigations of fluid systems is the accurate description of the dominant fluid behavior. This has commonly been accomplished by computing the stability of the flow, expressed in coherent structures known as modes. The local support of the modes and their temporal or spatial evolution provide critical information about the prevalent dynamical features of the flow. The computation of modes for simple geometries with multiple homogeneous coordinate directions involves the discretization of the linearized equations and the direct solution of the resulting eigenvalue problem. For even moderately complex geometries this procedure becomes prohibitively expensive. Instead, iterative techniques to extract the dispersion relation from numerical simulations are used to compute global modes for complex flow configurations [2, 10]. For flows that are dominated by multiphysics processes (such as the presence of shear and acoustic phenomena) additional transformations may be necessary to access various parts of the spectrum by iterative techniques [4]. For open and semi-open flows, modal structures that describe the dynamics away from the wall or from free shear layers can be extracted in an efficient manner. For simple flows, whose freestream velocity field is governed by a constant-coefficient differential equation, this results in oscillatory eigenfunctions associated with an continuous spectrum. For cases where the freestream velocity field still exhibits a dependence on one or multiple spatial coordinates, modal structures prevail that can be described asymptotically as wavepackets within a WKB-like framework [11, 6]. This type of analysis accomplishes the extraction of global stability behavior at a computational cost comparable to a local analysis. The dispersion relation is given in algebraic form, parameterized by the characteristics of the wavepacket solution. Peter J. Schmid Laboratoire d’Hydrodynamique (LadHyX), CNRS-Ecole Polytechnique, 91128 Palaiseau, France, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_7, © Springer Science+Business Media B.V. 2010

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Most numerical techniques that compute stability information or any characteristics of flow behavior rely on a mathematical model that governs the fluid flow. This model is commonly based on the Navier-Stokes equations (or variants thereof) which are subsequently discretized and implemented on a computer. Iterative algorithms then call upon this model to determine the global stability behavior of the flow under investigation. In experiments, on the other hand, no such model is available; instead, only measured flow-fields, i.e., data, can be extracted. By reformulating the commonly applied algorithms from a model-based to a data-based setting, the description of experimental fluid structures in terms of dynamic modes is possible. This type of decomposition into coherent structures, which are also known under the name Koopman modes [5, 7], give a valuable representation of the global flow dynamics in terms of its most dominant structures [8, 9]. Common to all of the above examples is the reduced and compact description of flow behavior in terms of organized fluid elements. Different applications, conditions and circumstances support the use of different structures. In the examples that follow we will briefly touch upon a variety of these situations.

2 Global stability analysis of compressible flow about a swept parabolic body The compressible flow about a swept parabolic body — a common model for the flow over the swept wings of high-performance aircrafts — is marked by a variety of physical phenomena, including attachment-line instabilities near the stagnation-line, the appearance of cross-flow vortices further downstream from the leading edge, the presence of acoustic modes and the interaction of the bow shock with flow structures near the leading edge. The global spectrum reflects this complexity and poses great challenges to iterative algorithms designed to extract modal information. Conformal maps of the complex eigenvalue plane, such as the Cayley transformation, have to be used together with iterative solutions of the resulting linear system to access specific and user-defined parts of the spectrum and to isolate the associated physical processes. Information about the underlying linearized dynamics is furnished by a direct numerical simulation via a Jacobian-free framework which approximates the required matrix-vector operations by repeated calls to the simulation program. This interplay of direct numerical simulations (as data-provider) and iterative eigenvalue algorithms (as data-analyzer) is a powerful concept that promises to provide stability, receptivity and sensitivity information for any flow that can be simulated with a sufficient degree of fidelity and accuracy. A DNS-based global stability solver has been applied to compressible flow over a (spanwise infinite) swept wing with a sweep Mach number of Mas = 1.24, a sweep angle of Λ = 30o and a spanwise wavenumber of β = 0.244. Figure 1 displays the flow configuration, coordinate systems, governing parameters and computational grid. Flow fields have been computed using a high-order compact discretization of the compressible Navier-Stokes equations on a curvilinear moving mesh where the

The description of fluid behavior by coherent structures

Fig. 1 Geometric configuration for flow about a swept parabolic body, displaying the computational grid, the Cartesian (x − y − z) and body-fitted (s − n − z) coordinate systems and some of the governing parameters such as the nose radius R, the freestream velocity q∞ , the sweep angle Λ, and the resulting freestream sweep velocity w∞ .

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y

w∞

q∞ Λ

z

x n

s

R

bow shock represents the inflow boundary. A Cayley-transformed Arnoldi method together with an ILU-preconditioned BiCGStab algorithm [4] has been used to compute the global spectrum. Two representative global modes are shown in Figure 2. The first mode has been extracted from the branch representing boundary-layer instabilities. It clearly shows features of attachment-line modes and crossflow vortices: near the stagnation line, chordwise vortices are observed that match the leaststable modal structures of local flow models for the attachment-line boundary layer (such as swept Hiemenz flow), whereas further downstream typical crossflow vortices arise which are reminiscent of the well-studied dominant structures of threedimensional boundary layers. It is important to realize, however, that this global mode establishes a link between these two local flow phenomena and represents the characteristics of these two instabilities as localized components of one single global mode [3]. The second mode, displayed in Figure 2(b), is taken from the fast acoustic branch. It shows typical wave-like patterns of a characteristic size and direction which are accentuated near the nose of the parabolic body and penetrate into the boundary layer. For particular parameter combinations, the co-existence of a shear and acoustic instability has been observed.

3 Microlocal stability analysis of swept attachment-line boundary layer flow In many complex flow configurations, localized modal structures appear in the form of wavepackets. These structures often describe the flow behavior near the edge of a boundary layer or in the freestream of flows with non-vanishing ambient mean

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Fig. 2 Global modes for compressible flow about a swept parabolic body. (a) Perspective view of a global mode from the boundary layer branch showing local flow features typical of attachment-line and crossflow instabilities. (b) s-n-cut of a global mode from the acoustic branch, visualized by pressure contours. A spanwise wavenumber of β = 0.244, a sweep Reynolds and Mach number of Res = 500 and Mas = 1.24, and a sweep angle of 30o have been chosen.

(a)

(b) crossflow vortices

attachment line

shear. These modal structures can efficiently be described by an ansatz known as microlocal [11]. This approach is closely related to a WKB-analysis but uses only qualitative boundary behavior instead of exactly enforced boundary conditions. We will demonstrate this novel technique on a simplified model of swept attachmentline flow which describes uniform modes in form of the scalar differential eigenvalue problem [6] [∂y2 −V ∂y +V ′ − γ 2 − iγ ReW ]uˆ = −iReλ uˆ

(1)

with the mean velocity field (V (y),W (y)) given by the similarity solution for swept Hiemenz flow, γ as the spanwise wavenumber and Re denoting the Reynolds number. Assuming the eigenfunctions uˆ in the form of a general wavepacket located at y∗ with a characteristic wavelength of β∗ , i.e., uˆ ∼ eiβ∗ y/ε eC(y−y∗ )

2 /ε

ε = Re−1/2 ,

(2)

substitution into the eigenvalue problem yields an algebraic dispersion relation, known as the symbol λ = f (y∗ , β∗ ), parameterized by y∗ and β∗ . The condition of a true wavepacket, i.e., Real(C) < 0, translates into an additional inequality constraint on the dispersion relation f referred to as the twist condition. These two equations — the symbol curve and the twist condition — describe regions in the complex plane where wavepacket solutions to the governing equations exist. Figure 3(a) shows the regions in the (y∗ , β∗ )-parameter space where wavepacket solutions can be found. In the freestream we have both inward (β∗ < 0) and outward (β∗ > 0) propagating wavepackets, whereas in the boundary layer (y∗ . 3) only wavepackets propagating toward the wall exist. The mapping of the permissible region in (y∗ , β∗ )-space under the symbol λ = f (y∗ , β∗ ) is shown in Figure 3(b). It traces regions of the complex eigenvalue plane

The description of fluid behavior by coherent structures (a)

(b)

8

0

7

Real(C) < 0

55

λ = f (y∗ , β ∗ )

−0.05

6

−0.1

y∗

λr

5

4

−0.15 3

2

−0.2

Real(C) > 0

1

−0.25 0 −0.5

−0.4

−0.3

−0.2

−0.1

β∗ 0

0.1

0.2

0.3

0.4

0.5

0

0.5

λi

1

1.5

Fig. 3 Microlocal analysis of uniform modes for swept Hiemenz flow. (a) Regions in the parameter space (y∗ , β∗ ) where wavepacket solutions to the dispersion relation exist. (b) Mapping of the wavepacket region Real(C) < 0 under the symbol λ = f (y∗ , β∗ ).

where eigenfunctions to the dispersion relation of swept Hiemenz flow (for uniform modes) take the shape of wavepackets. The requirement of general wavepacket solutions, i.e., Real(C) < 0, is not restrictive enough to satisfy the boundary conditions on the eigenfunctions; rather, a measure needs to be introduced that specifies the compliance of the wavepacket solutions with the imposed boundary conditions at the wall and in the freestream. This final restriction, which leads to an area of continuous wavepacket modes in the λ -plane, results in a full representation of the dynamics in the freestream and boundary layer layer in terms of localized eigenmodes. It can be demonstrated that these type of modes are responsible for the interaction of boundary layer and freestream modes [6], and it is believed that they are crucial in describing receptivity mechanisms for swept Hiemenz flow (or any other flow with a non-constant freestream velocity field).

4 Dynamic mode analysis of the wake behind a flexible membrane Remarkable advances in data-acquisition and imaging techniques (such as timeresolved particle-image-velocimetry TR-PIV) have resulted in an accurate representation of fluid processes by physical experiments; analysis techniques similar to their computational equivalents are, however, lacking. The common statistical evaluation of the flow fields lags behind the capabilities of model-based techniques as far as the quantitative description of fluid processes and flow mechanisms is concerned. Even the widely applied proper orthogonal decomposition (POD) method [1], which computes an energy-ranked hierarchy of coherent structures from the spatial or temporal

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P. J. Schmid (a)

(b)

DMD1

0.01

αr

0

DMD2

(c)

(d)

−0.01

t

−0.02 −0.03 −0.04

−0.25

−0.2

−0.15

−0.1

−0.05

0

αi

0.05

0.1

0.15

0.2

0.25

y

Fig. 4 Dynamic mode decomposition of wake flow behind a flexible membrane. (a,b) Two dominant dynamic modes from a temporal analysis of the time-resolved PIV-data, visualized by the streamwise velocity component. (c) Spatial DMD-spectrum based on the same experimental data. (d) Dominant spatial dynamic mode, displayed in the (y,t)-plane, visualized by the streamwise velocity component.

covariance of the sampled flow fields, often does not provide the necessary insight into the dynamics of the flow. This is due to the loss of critical information during the averaging process for the flow-field covariance or due to the incompatible use of energy as a measure of coherence or importance; in particular, in acoustic applications the sound waves carry little energy content but are nevertheless pivotal in the overall dynamics of the flow. A novel decomposition technique [8, 9] that is based on data fields only and still retains the temporal dynamics of the flow has been developed by modifying an Arnoldi technique and by extracting a linear mapping between elements of a snapshot sequence. Mathematically, this mapping is equivalent to a finite-dimensional representation of the Koopman operator [5, 7]. The eigenvalues and eigenvectors of this mapping can then be interpreted as the dominant structures that best represent the dynamics of the flow by a linear process. Since the algorithm for this mapping only relies on data, it can be applied to numerical and experimental flow fields alike. Furthermore, the data do not need to show any spatial ordering, i.e., spatially unstructured data and data from subdomains of the entire flow can be processed, which provides great flexibility in assessing regions of the flow where localized flow phenomena and instabilities occur. Moreover, a spatial ordering of the snapshots allows the analysis of spatially evolving flows, a feature that is of particular interest to experimentalists. If a linear process is sampled, the resulting eigenvectors of the extracted mapping will correspond to the global modes of the flow. This dynamic mode decomposition (DMD) technique will be illustrated on timeresolved PIV-data from the wake flow behind a flexible membrane [8]. The latex membrane (38 mm × 66 mm) is attached to a thin U-shaped steel frame over which a uniform flow (with 14 m/s) passes. The flow in the wake of the membrane has been captured by snapshots at a rate of 2000 Hz which, after an adaptive cross-correlation

The description of fluid behavior by coherent structures

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PIV-treatment, have been fed to the dynamic mode decomposition algorithm. Two dominant frequencies have been detected, and the associated dynamic modes are displayed in Figures 4(a,b), visualized by the streamwise velocity component. A dominant streamwise wavelength can be observed in either figure, as can a spatial decay in amplitude in the streamwise direction. A great advantage of a decomposition that is purely based on flow data and that identifies a linear mapping between snapshots of a nonlinear process is the fact that the direction in which these snapshots are arranged determines the type of flow analysis. By traditionally arranging a temporal sequence of spatial flow fields, the extracted mapping carries information about the temporal evolution of coherent structures. The same set of flow fields, however, can also be reorganized into a spatial sequence of spatio-temporal flow fields [8]. In effect, we generate (t, y)-dependent flow fields at equispaced x-locations. In this case, the dynamic mode decomposition determines a linear mapping between snapshots at two adjacent spatial locations; and the eigenvalues of this mapping provide information about the spatial evolution of coherent structures. The spatial spectrum for the wake flow behind a flexible membrane is shown in Figure 4(c) which identifies a spatial wavenumber and spatial decay rate that match very well with the spatial characteristics of the corresponding temporal mode (Figure 4(a)). The associated spatial dynamic mode (in Figure 4(d)) shows a distinct temporal frequency, again in very good agreement with the corresponding temporal dynamic mode analysis.

5 Summary and conclusions More insight into the underlying physical mechanisms of fluid flow can be gained by the decomposition and reorganization of gathered data into coherent structures that contribute significantly to the governing temporal and/or spatial processes. By concentrating on these structures a clearer picture emerges that allows the description of complex flow behavior by a reduced set of characteristic structures. This article presented three examples of coherent structures: global stability modes extracted from direct numerical simulations of compressible flow about a swept parabolic cylinder, wavepacket pseudomodes determined from a microlocal analysis of flow near the attachment-line of a swept body, and dynamic modes identified from experimental data of flow in the wake of a flexible membrane within a temporal or spatial framework. Even though the techniques and approximations have varied greatly, the common theme has been the low-dimensional description of fluid behavior as a prerequisite for advancing our understanding of fluid behavior in complex configurations. Acknowledgements I would like to warmly thank Christoph Mack (LadHyX) and Dominik Obrist (ETH) for their many contributions and their fruitful and enjoyable discussions. Financial support from the chaires d’excellence program of the Agence Nationale de la Recherche (ANR) and the Alexander-von-Humboldt Foundation is gratefully acknowledged.

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References 1. Berkooz G, Holmes P, Lumley JL (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Ann. Rev. Fluid Mech. 25:539-575 2. Edwards WS, Tuckerman LS, Friesner RA, Sorensen DC (1994) Krylov methods for the incompressible Navier-Stokes equations. J. Comp. Phys. 110:82-102 3. Mack CJ, Schmid PJ, Sesterhenn JL (2008) Global stability of swept flow about a parabolic body: connecting attachment-line and crossflow modes. J. Fluid Mech. 611:205-214 4. Mack CJ, Schmid PJ (2009) A preconditioned Krylov technique for global hydrodynamic stability analysis of large-scale compressible flows. J. Comp. Phys. (submitted) 5. Mezic I (2005) Spectral properties of dynamical systems, model reduction and decompositions. Nonl. Dyn. 41:309-325 6. Obrist D, Schmid PJ (2009) Algebraically decaying modes and wavepacket pseudomodes in swept Hiemenz flow. J. Fluid Mech. (submitted) 7. Rowley CW, Mezic I, Bagheri S, Schlatter P, Henningson DS (2009) Spectral analysis of nonlinear flows. J. Fluid Mech. (accepted) 8. Schmid PJ (2009) Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. (submitted) 9. Schmid PJ, Li L, Juniper MP, Pust O (2009) Decomposition of experimental data into dynamic modes. Theor. Comp. Fluid Dyn. (submitted) 10. Theofilis V (2003) Advances in global linear instability analysis of nonparallel and threedimensional flows. Prog. Aero. Sci. 39:249-315 11. Trefethen LN (2005) Wave packet pseudomodes of variable coefficient differential operators. Proc. Roy. Soc. Lond. A 561:3099-3122

Instability of uniform turbulent plane Couette flow: spectra, probability distribution functions and K − Ω closure model Laurette S. Tuckerman, Dwight Barkley, and Olivier Dauchot

Abstract Near transition, plane Couette flow exhibits statistically steady bands of alternating turbulent and laminar flow. We simulate these patterns numerically and show that they can be quantified via the spatial Fourier component corresponding to the pattern wavevector and its probability distribution. The trigonometric nature of the turbulent-laminar pattern suggests that it emerges from a linear instability of the uniform turbulent state, and we attempt to verify this hypothesis via the K − Ω closure model. We calculate steady 1D solution profiles of the K − Ω model and their linear stability to 3D perturbations, but find no correspondence between this analysis and the onset of turbulent-laminar bands in experiment and simulation.

1 Turbulent-laminar bands Near transition, plane Couette flow exhibits a remarkable statistically steady state containing alternating oblique bands of turbulent and laminar flow which are regular and statistically steady. Discovered experimentally by Prigent and Dauchot (GITSaclay) [1, 2] and subsequently simulated numerically by Barkley and Tuckerman [3, 4, 5, 6], these patterns seem to be an intrinsic feature of the transition to turbulence in shear flows, since they are also seen experimentally in counter-rotating Taylor-Couette flow [1, 2] and the stator-rotor configuration [7], as well as in simulations of plane Poiseuille flow [8], and pipe flow [9].

Laurette S. Tuckerman PMMH-ESPCI (UMR 7636, CNRS, Univ. Paris 6 & 7), 10 rue Vauquelin, 75231 Paris, France, e-mail: [email protected] Dwight Barkley Mathematics Institute, Univ. Warwick, United Kingdom, e-mail: [email protected] Olivier Dauchot CEA-Saclay, SPEC-GIT, URA 2464, 91191 Gif-sur-Yvette, France, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_8, © Springer Science+Business Media B.V. 2010

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Fig. 1 Turbulent-laminar pattern at Reynolds number 350. Isosurfaces of instantaneous streamwise vorticity. This visualization is constructed by tiling a large domain with many repetitions of our computational domain.

Figure 1 shows a perspective plot of a turbulent-laminar patterned flow computed by our simulations at Re = 350. The upper and lower plates are located at ±h and move in the streamwise direction at ±U; the conventional Reynolds number is Uh/ν . In the flow of figure 1, the width of the bands is 40 in units of h, and they are oriented at an angle of θ = 24◦ from the streamwise direction, so that the pattern wavevector is oriented at 66◦ . This width and angle are within the typical range of these patterns [1, 2]. We focus on the transition from uniform turbulence to turbulent-laminar patterns with decreasing Reynolds number, which takes place near Re = 400. This low Reynolds number is quite accessible to direct numerical simulation of the NavierStokes equations. We first describe three numerically computed flows spanning the transition region, and characterize these flows by means of their Fourier transforms along the pattern wavevector. Our simulations were carried out with the spectral-element/Fourier code Prism [10]. We use between 8 and 20 modes per unit length, leading to a resolution of about 106 gridpoints or modes. More details concerning our numerical methods can be found in [3, 4, 5, 6].

Instability of uniform turbulent plane Couette flow

61

2 Analysis of Fourier spectra

Fig. 2 Top row: timeseries of the spanwise velocity at 32 points along a line midway between the plates and oriented along the pattern wavevector. Bottom row: time-average of the power spectrum along the pattern wavevector of the spanwise velocity. The m = 1 component corresponds to the pattern wavelength of 40. Left column: uniform turbulence at Re = 500. Middle column: intermittent state at Re = 410. Right column: statistically steady turbulent-laminar pattern at Re = 350.

We now describe the onset of these patterns as the Reynolds number is decreased. The upper portion of figure 2 shows timeseries of the spanwise velocity for 32 points along a line located midway between the plates and oriented in the direction of the pattern wavevector. (In this figure, as in some of our previous work [4, 5, 6], z denotes the direction of the pattern wavevector, but in section 3 we will use z to denote the spanwise direction instead.)

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Figure 2 shows that the turbulence is uniform for Re = 500 and banded for Re = 350. At the intermediate value Re = 410, the turbulence is intermittent, with a pattern appearing and disappearing sporadically. We then process this data by taking the square modulus of the Fourier transform along the pattern wavevector |spanm (t)|2 at each instant t, and then averaging over time for each Re. This yields the 1D power spectra avg |spanm (t)|2 ) of the spanwise velocity, shown in the lower t

portion of figure 2. These spectra display a very prominent feature: the m = 1 Fourier component corresponding to wavelength 40 emerges from the rest of the spectrum at Re = 350 and, to a lesser extent, at Re = 410. The emergence of the m = 1 Fourier component suggests using it as an order parameter for the transition from uniform to banded turbulence, as shown in figure 3.

Fig. 3 Bifurcation diagram for transition from uniform to banded turbulence using m = 1 component of averaged power spectra, as in figure 2.

To obtain a sharp threshold, however, it is necessary to return to the instantaneous component a(t) ≡ |span1 (t)|. Rather than averaging these over time, we treat the instantaneous values as statistical samples and, by binning these, construct their probability distribution function p(a). The resulting probability distribution functions for Re = 500, 410 and 350 are shown in figure 4. We can distinguish different regimes: For Re & 440, the most probable value of a is zero, and this is where p(a) has its maximum. At Re = 440, p(a) changes curvature and, for Re . 440, p(a) has a local minimum at a = 0 and a maximum at a finite value amax > 0. Although a Gaussian provides an extremely good fit to p(a) for Re = 500, the functional form which best fits p(a) for Re = 410 and 350 as well is: ln p(a) = c0 + c1 a + c2 a2

(1)

rather than the more usual quartic. The functional form (1) gives amax = −c1 /(2c2 ) as the most probable value. Both amax and c1 are shown on the right part of figure 4.


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Fig. 4 Left: probability distribution functions p(a) for pattern Fourier component. Re = 500 (squares), Re = 410 (triangles), and Re = 350 (circles). Solid curves are least-squares fits to ln p(a) = c0 + c1 a + c2 a2 , dashed curves to ln p(a) = c0 + c2 a2 + c4 a4 , Right: maximum amax of PDFs as a function of Reynolds number (solid dots) and coefficient c1 /10 (hollow dots) from least-squares fit.

3 Stability analysis of K − Ω Model The sequence seen in figures 2-4, in particular the emergence of a single trigonometric mode, suggests that these patterns result from a linear instability of the uniform turbulent state, whose temporal average depends only on a single spatial variable: the cross-channel coordinate y. We decompose the velocity field into its short-time mean and its fluctuating parts: ¯ + U′ U=U

¯ ≡ hUi where U

and h U′ i = 0

(2)

¯ is governed by the Reynolds-averaged equations: U ¯ = − ∇P − (U ¯ · ∇) U ¯ +F+ ∂t U

¯ =0 ∇·U

1 ¯ ∆U Re

(3a) (3b)

where the Reynolds stress force is F ≡ −h (U′ · ∇) U′ i. These are subject to the boundary conditions which define plane Couette flow: ¯ y = ±1, z) = ±ex U(x, ¯ + λx , y, z) = U(x, ¯ y, z + λz ) = U(x, ¯ y, z) U(x

(3c) (3d)

where x, y, z are the streamwise, cross-channel and spanwise directions, respectively.

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Since (3a) contains the Reynolds stress force, which in turn depends on the fluctuations U′ which we seek to bypass, we require a closure model relating F directly ¯ The K − Ω model, as described in [11, 12, 13] is that thought to the mean flow U. best adapted to walls and to low Reynolds numbers. Equations (3) are closed by approximating F as: F = ∇ · (νT 2S)

νT ≡

K Ω

2Si j ≡ ∂iU j + ∂ jUi

where K and Ω are scalar fields which evolve according to: νT 2 ∗ ¯ ∂t K = −U · ∇K + νT S − β (Ω K) + ∇ · ν+ ∇K σK νT 2 2 ¯ ∂t Ω = −U · ∇Ω + α S − β Ω + ∇ · ν+ ∇Ω σΩ

(4a)

(4b) (4c)

with typical parameters of α = 5/9, β = 3/40, β ∗ = 9/100, σK = σΩ = 2. K is to be interpreted as the kinetic energy density and has boundary conditions: K(x, y = ±1, z) = 0

(4d)

Ω is meant to account for the presence of the boundary and is subject to various phenomenological boundary conditions, such as: Ω (x, y = ±1, z) = 1/(∆ y)2

(4e)

where ∆ y is the numerical grid spacing at the boundary. We have computed steady 1D solutions of (3)–(4) for various Reynolds numbers. ¯ y, z) = U¯ (y)ex and ∂t = ∂x = ∂z = 0. We discretized the y To do so, we set U(x, direction over a grid of Ny +1 = 61 points with a Chebyshev spacing y j = cos jπ /Ny , ¯ which concentrates points at the boundaries, We then solved for U(y), K(y) and ¯ K, and Ω profiles Ω (y) via Newton’s method. The left portion of figure 5 shows U, ¯ for Re = 100, 300 and 500, while the right portion compares U(y)−y to the averaged turbulent profiles from our full 3D simulations at Re = 500. Like [14], we find that the K − Ω model exhibits a bifurcation from laminar flow (K = 0, U(y) = y) to turbulent flow (K > 0)) at Re ≈ 100, illustrated in the left portion of figure 6. (This is already inconsistent with actual plane Couette flow, in which the lowest Re at which turbulence has ever been observed is 220 in certain numerical simulations [4, 15] and more typically above 300 [3, 16].) For Re = 300 and 500, the U profiles have S-shapes and the K profiles show flattening in the bulk, both features found in actual turbulent channel flows. We then carried out linear stability analysis by substituting     ¯ ¯  U(y)e x u K(y) + k eσ t+2π i(x/λx +z/λz ) (5)     Ω (y) ω


65

Fig. 5 Profiles from K − Ω model. Right: Re = 100 (solid curves), Re = 300 (long-dashed curves) ¯ and Re = 500 (short-dashed curves). U(y) ranges between −1 and +1, K(y) has a bulge in the center. Ω (y) (not shown) is O(106 ) at boundaries and decreases sharply to O(1) in the bulk. Right: comparison of U(y) − y between K − Ω and full DNS. Solid and dashed curves show DNS and K − Ω profiles, respectively, for Re = 500.

into the full 3D equations (3)–(4), linearizing, and calculating the eigenvalues σ . In the ranges 90 ≤ Re ≤ 500 and 10 ≤ λx , λz ≤ 1000, we find that the eigenvalues σ (λx , λz , Re) are all negative. The table on the right of figure 6 shows the maximum σ for each Re, maximized over the wavelengths λx , λz . The largest (least stable) eigenvalue is found for Re = 120 at (λx , λz ) = (60, 20). This corresponds to a pattern angle of θ = atan(λz /λx ) = 18◦ to the streamwise direction and a pattern wavelength of λz cos(θ ) = 19. (see the Appendix of [5]). In contrast, turbulent-laminar patterns in plane Couette flow are observed experimentally at 300 ≤ Re ≤ 420 with λx = 110 and 50 ≤ λz ≤ 80, corresponding to an angle between 25◦ and 37◦ and a pattern wavelength of between 46 and 60. It is therefore unlikely that the instability of the K − Ω model bears any relationship to that undergone by uniform turbulent flow in plane Couette flow. A similar calculation using the simpler Prandtl mixing-length model also shows no instabilities. This analysis underscores the inadequacy of turbulence closure models, in particular for transitional wall-bounded flows at low Reynolds numbers. Quantitative prediction of turbulent-laminar banded patterns would provide an extremely stringent test of a future closure model.

References 1. Prigent, A., Grégoire, G., Chaté, H., Dauchot, O., van Saarloos, W.: Large-scale finitewavelength modulation within turbulent shear flows. Phys. Rev. Lett. 89, 014501 (2002)

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Re 90 100 110 120 150 200 300 400 500

λx 1000 90 60 60 100 500 500 500 400

λz 500 30 20 20 30 1000 1000 1000 1000

σ −0.0179 −0.0190 −0.0135 −0.0081 −0.0181 −0.0148 −0.0141 −0.0964 −0.0855

Fig. 6 1D and 3D instabilities of the K − Ω model. Left: Steady 1D solution to the K − Ω model as a function of log Re. Shown is K(0), the turbulent kinetic energy in channel center. A bifurcation from the laminar state K = 0 is seen at Re ≈ 100. Right: Leading eigenvalues from linear stability analysis of 1D. Shown is the maximum eigenvalue for each Re, varying 10 ≤ λx , λz ≤ 1000. 2. Prigent, A., Grégoire, G., Chaté, H., Dauchot, O.: Long-wavelength modulation of turbulent shear flows. Physica D 174, 100–113 (2003) 3. Barkley, D., Tuckerman, L.S.: Computational study of turbulent laminar patterns in Couette flow. Phys. Rev. Lett. 94, 014502 (2005) 4. Barkley, D., Tuckerman, L.S.: Turbulent-laminar patterns in plane Couette flow. In: Mullin, T., Kerswell, R. (eds.) IUTAM Symposium on Laminar-Turbulent Transition and Finite Amplitude Solutions, pp. 107–127. Springer (2005) 5. Barkley, D., Tuckerman, L.S.: Mean flow of turbulent-laminar patterns in plane Couette flow. J. Fluid Mech. 576, 109–137 (2007) 6. Tuckerman, L.S., Barkley, D., Dauchot, O.: Statistical analysis of the transition to turbulentlaminar banded patterns in plane Couette flow. J. Phys.: Conf. Ser. 137, 012029 (2008) 7. Cros, A., Le Gal,P.: Phys. Fluids 14, 3755–3765 (2002) 8. Tsukahara, T., Seki, Y., Kawamura, H., Tochio, D.: DNS of turbulent channel flow at very low Reynolds numbers. In Proc. 4th Int. Symp. on Turbulence and Shear Flow Phenomena, pp. 935–940 (2005) 9. Moxey, D., Barkley, D., to be published 10. Henderson, R.D., Karniadakis, G.E.: Unstructured spectral element methods for simulation of turbulent flows. J. Comput. Phys. 122, 191–217 (1995) 11. Menter, F.R.: Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal 32, 1598–1609 (1994) 12. Durbin, P.A., Pettersson Reif, B.A.: Statistical Theory and Modeling for Turbulent Flows. Wiley (2001) 13. Wilcox, D.C.: Re-assessment of the scale-determining equation for advanced turbulence models, AIAA Journal 26, 1414–1421 (1988) 14. Templeton, J.: Stability Analysis of the k-omega Turbulence Model for Channel Flow. Stanford University, 5 June 2000. 15. Schmiegel, A., Eckhardt, B.: Persistent turbulence in annealed plane Couette flow, Phys. Fluids 51, 395–400 (2000). 16. Bottin, S., Daviaud, F., Manneville, P., Dauchot, O.: Discontinuous transition to spatiotemporal intermittency in plane Couette flow, Europhys. Lett. 43, 171–176 (1998).

Part II

Oral Presentations

Sensitivity to base-flow variation of a streamwise corner flow Frédéric Alizard ∗ , Jean-Christophe Robinet ∗ , and Ulrich Rist +

Abstract The stability of flow formed by intersection of two perpendicular flatplates is revisited through a study of the sensitivity to the base flow variation. After a brief presentation of the asymptotic regime, sensitivity functions underlying corner mode (concentrated close to the intersection) and Tollmien-Schlichting modes, with different obliqueness angles, are computed. With this consideration, associated mechanisms as well as active regions are identified, which further confirm that the sensitivity area of the corner mode arises along the intersection of flat plates. Then, an optimization technique shows that a small deviation of the reference field in the area of uncertainty observed in experiments leads to decrease critical Reynolds number. A hypothesis based on the onset of an inflectional mechanism is thus proposed to explain the experimental results.

1 Introduction Viscous flow along a corner formed by intersection of two semi-infinite perpendicular flat plates has been under investigation for several decades. In particular, experimental results highlighted a transitional Reynolds number based on the distance from the leading edge of about 104 which is much lower than the critical Reynolds number of the classical Blasius flat plate boundary layer ≈ 105 [2]. Furthermore, local linear stability studies didn’t allow to explain the experimental results [1]. Nevertheless, although the theory provides that the instability of a zero pressure gradient corner layer is dominated by the classical Tollmien-Schlichting (noted TS hereafter) viscous modes, an inviscid mode strongly localized in the corner is also observed. Moreover, experiments exhibit a large variety of flows really close to the corner and some discrepancies between the theoretical base flow and the experi∗

SINUMEF laboratory 151 [email protected] and uni.stuttgart.de

+

Boulevard de l’Hopital 75013 Paris, e-mail: fredPfaffenwaldring 21, 70569 Stuttgart e-mail: rist@iag-

P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_9, © Springer Science+Business Media B.V. 2010

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F. Alizard, J.-C. Robinet, U. Rist

Fig. 1 Numerical solution of the self-similar corner equations. The shaded area indicates the region where a hypothetical laboratory uncertainty might appear.

mental data. Consequently, the purpose of this paper is to reconsider the problem through a stability analysis taking into account a certain degree of uncertainty associated with the base-flow in Figure 1. After a brief presentation of the theory, it will be shown that a small deviation of the base flow close to the corner allows to destabilize the inviscid mechanism derived from the corner mode.

2 Temporal asymptotic linear stability. Reference lengths are based on the maximum streamwise velocity and the distance from the leading edge. Disturbances q p =t (u p , v p , w p , p p ) are assumed of the following form: (1) q p = qˆ (y, z) e[i(α x−Ω t)] The space and time behaviour of a small perturbation is thus governed by the opˆ U) = 0 derived from the linearized Navier-Stokes equations. In erator L ({Ω , q}, a temporal framework, the wave number is fixed real, thus the mode is allowed to grow temporally with the growth rate and the circular frequency equal to Ωi and Ωr , respectively. The system of equations reduces to a large generalized eigenvalue problem which can be written : (A − iΩ B) qˆ = 0

(2)

with iΩ eigenvalues and qˆ eigenfunctions. The system (2) is discretized using a Chebyshev/Chebyshev spectral collocation method in the z and y directions. Classical no-slip boundary conditions are imposed at walls and Neumann conditions in the far-field for the velocity components. The use of symmetry conditions leads to two possible solutions: even-symmetric and odd-symmetric modes. Finally, an Arnoldi algorithm, based on ARPACK, combined with a shift-invert method is used to approximate the most relevant part of the spectrum. A spectral grid (65 × 65) is employed in our next computations.

Sensitivity to base-flow variation of a streamwise corner flow 0.01

0.32 12

0

4

5

0.28

7

6

0.26

8 TS modes

0.24

Corner Mode

αr

-0.01

Ωi

TS corner flow

0.3

3

even-modes odd-modes Full spectrum

0.22 0.2

-0.02

0.18 0.16 0.14

-0.03

TS Blasius

0.12 100000

0.06

71

0.07

0.08

Ωr

0.09

0.1

200000

300000 Rex

400000

500000

(b)

(a)

Fig. 2 2(a) Temporal spectrum at Rex = 2.5 × 105 and α = 0.2. 2(b) Neutral curve of the most unstable TS mode.

(a) u: ˆ corner mode.

(b) uˆ of TS 1: A-S.

(c) uˆ of TS 2: S.

(d) uˆ of TS 4: S.

Fig. 3 TS and corner mode. S:symmetric, A-S: anti-symmetric.

Typical eigenvalue spectra for Rex = 2.5 × 105 and α = 0.2 is depicted in Figure 2. A branch of eigenvalues may be observed which can be attributed to Tollmien-Schlichting (TS-) modes with different transverse wave lengths, i.e., different obliqueness angles with respect to the free-stream flow. The most unstable one corresponds to the classical (two-dimensional) TS-instability mode of the flat-plate boundary layer. Aside of this branch we get an isolated mode whose eigenfunction is dominantly concentrated near the corner line and rapidly decays along y and z. This is the so-called corner mode. The most unstable TS-mode is compared with the classical TS mode of a Blasius boundary layer through a neutral curve in the plane (Rex , α ) in Figure 2(b). The influence of the corner on this specific mode is weak and it is stabilizing the flow. The critical Reynolds number based on the streamwise position Rex equals to 1.17× 105 which is consistent with the value for Blasius flow ≈ 9.1 × 104 . Furthermore, the corner mode is observed to be always temporally stable in the parameters space which is analysed. These results are in good agreement with those of Parker & Balachandar [1] which validate our numerical methods. From the above discussion, it seems clear that local theory cannot explain why experimentalists observe a premature laminar-turbulent transition of corner flows compared to flat-plate boundary layers. Therefore, in order to take into account the extreme sensitivity in the corner region observed in experimental measurements,

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we reconsider this problem through a stability analysis where a certain degree of uncertainty associated with the base flow is theoretically investigated.

3 Sensitivity analysis 3.1 Sensitivity functions A sensitivity function Gu may be constructed by a projection of the perturbated operator L along the adjoint mode as follows: δ Ω =

Z Ly Z Lz 0

0

t

Gu δ U dzdy where

δ U is a small variation of U [3]. We take as a representative case the corner flow at Rex = 8 × 104 and α = 0.18. Gu is displayed in Figures 5 with respect to the modes depicted in the spectrum 4(a). One may observe that the corner mode is the

0.004

0

r=0 r2 =10-7 2 -7 r =2.10 2 -6 r =1.10 2 -6 r =5.10

0.002 Corner mode TS 1

TS 2

Corner mode

0

Ωi

Ωi

TS 3

-0.01

-0.002 TS1

TS2

-0.004

-0.02

0.07

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Ωr

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0.1

(a) Spectrum.

TS3

-0.006 0.07

0.075

0.08

Ωr

0.085

0.09

(b) Optimized spectrum.

Fig. 4 Rex = 8 × 104 and α = 0.18.

most sensitive one to any base-flow modification around the uncertain area, which demonstrates that this last one is the best candidate to provide an explanation to the low-Reynolds number observed in experiments through a small base-flow deviation. Therefore, on the basis of above results, it seems justified to further explore the influence of mean-flow modification having the most effect on the corner mode with respect to a deviation of a given magnitude.

3.2 Optimal deviation and physical mechanism The small deviation of the mean-flow is measured through an energy-like norm:

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(a) GU : corner mode.

(b) GV : corner mode.

(c) GU : TS 1.

(d) GV : TS 1.

(e) GU : TS 2.

(f) GV : TS 2.

(g) GU : TS 3.

(h) GV : TS 3.

Fig. 5 Sensitivity functions of the corner and TS modes. Rex = 8 × 104 and α = 0.18.

r2 =

Z Ly Z Lz 0

0

U −Ure f

2

+ V −Vre f

2

+ W −Wre f

2

dy dz

(3)

where re f refers to the theoretical base flow. We will focus on the deviation which maximizes the growth rate of this corner mode. A similar variational approach as by Bottaro et al. [3] is employed by introducing the functional H(U, λ ) = Ωi (U) − Z Ly Z Lz 2 2 2 λ r2 − U −Ure f + V −Vre f + W −Wre f dy dz 0

(4)

0

with λ a Lagrange multiplier. A constraint-optimization process classically employed in control theory is then introduced to maximize Ωi by successive iterations on the control (U,V,W ). An example of optimization is depicted in Figure 4(b). It appears that a small deviation of the base flow leads to destabilize the corner mode. Furthermore, from the velocity bissector profile displayed in Figure 6(a), it seems clear that the emerging instability mechanism derived from the optimal distorted base flow is strongly connected to the inflection point along the bissector. Finally, we track the optimal deviation of the corner mode in the parameter space (α , r, Rex ). Figure 6(b) shows the neutral curve which indicates the lowest Reynolds number for which a positive amplification rate occurs over a reasonably small deviation of the ideal mean flow. Typically, here the modification is between 0.1% and 1%. It may be observed that the critical Reynolds number varies in inverse proportion to the disturbance amplitude. Nevertheless, it appears that even for a lower Reynolds number than ≈ 104 , i.e. one order of magnitude lower than the critical Reynolds number associated with the TS waves in supercritical regime, the opti-

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mal distorted base flow is able to experience exponential growth through an inviscid mechanism. 1

0.2

0.8

-1

10

-2

-0.1

0.2

2

4 s

6

-0.2 8

(a) Velocity profile along the bissector.

Unstable area

r

0.4

4

U ref

0

10 ∆u

0.1

0.6

0 0

10

10

Rec TS

-3

Stable area

-4

10 10000

30000

50000 70000 Rex

90000

110000

(b) Neutral curve with respect to (r,Rex )

Fig. 6 Optimization results. In 6(a), the related optimal deviation for r2 = 2 × 10−7 with respect to Figure 4(b) is ploted.

4 Conclusion A study of sensitivity functions for TS and corner modes underlying a threedimensional corner flow reveals the high sensitivity of the mean-flow close to the intersection. An optimization technique applied to the corner mode shows the influence of a weak deviation of the reference base-flow in the area of uncertainty of the latter. It illustrates the possibility of destabilizing the corner flow at Reynolds numbers ranging from an order of magnitude lower than the critical Reynolds number associated with the classical Blasius boundary layer. A hypothesis associated with the onset of an exponential instability via a inviscid mechanism may be proposed to explain the low transitional Reynolds number observed in experiments.

References 1. Parker J., and Balachandar, S. (1999) Viscous and inviscid instabilities of flow along a streamwise corner. Theoret. Comput. Fluid Dynamics, 13:231–270. 2. Zamir M. (1980) Similarity and stability of the laminar boundary layer in a streamwise. Proc. R. Soc. Lond. A, 377:269–288. 3. Bottaro A., Corbett, P. and Luchini, P. (2003) The effect of base flow variation on flow stability. J. Fluid Mech., 476:293–302.

Transition Control Testing in the Supersonic S2MA Wind Tunnel (SUPERTRAC project) J.-P. Archambaud, D. Arnal, J.-L. Godard, S. Hein, J. Krier, R. S. Donelli, and A. Hanifi

Abstract This paper presents an attempt to control the laminar-turbulent transition on a swept wing in supersonic flow. The first part deals with the control of the transition due to crossflow (CF) instabilities present on a swept wing, by using MicronSized Roughness elements (MSR); the second one is focused on the prevention of the turbulent contamination along the Leading Edge (LE), using appropriate AntiContamination Devices (ACD). The work includes both numerical and experimental investigation. Key words: transition, supersonic, swept wing, CrossFlow, micron-sized roughness, leading edge contamination

1 Introduction Drag reduction by extending laminar boundary layer on the aircraft wings is not as well-known in supersonic flow as in transonic one. From a general point of view, the transition occurs by leading edge contamination, or by development of longitudinal Tollmien-Schlichting instabilities (TS) or crossflow instabilities (CF). In the present study, two control investigations have been carried out in supersonic environment: the control of the CF instabilities by means of distributed Micron-Sized Roughness elements (MSR) and the control of the leading edge contamination by appropriate Anti-Contamination Devices (ACD) [1]. The results presented in this paper are a part of the SUPERTRAC (SUPERsonic TRAnsition Control) project funded by the European Commission. J.-P. Archambaud ONERA, e-mail: [email protected] D. Arnal, J-L. Godard (ONERA), S. Hein (DLR), J. Krier (IBK), R. S. Donelli (CIRA), A. Hanifi (FOI) P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_10, © Springer Science+Business Media B.V. 2010

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2 Wing Definition For the first investigation (MSR), the pressure gradient along the chord has to be negative (CF instability) and the leading edge boundary layer laminar. On the contrary, for the second investigation (ACD), the leading edge boundary layer must be turbulent upstream of the device and the studied phenomenon is local. The two investigations are performed separately, but with the same wing. Figure 1(a) shows the Cp distribution which presents a significant negative pressure gradient all along the chord, favourable to the development of CF waves. The

(a) Cp distribution on the profile

(b) Leading edge contamination problem

Fig. 1 Characteristics of the profile (pressure gradient and LE contamination state)

figure 1(b)) shows the plot of the contamination Reynolds number R¯∗ [2] versus the sweep angle φ . R¯∗ increases with the LE radius, the sweep angle and the unit Reynolds number. Curves and symbols represent the estimation of the computed R¯∗ relative to our profile at Mo = 2.0, using respectively a simplified formula (underestimation) and the exact formula. We remark easily that MSR tests can be performed with a laminar leading edge at φ < 30o while ACD tests requiring a turbulent leading edge are possible at φ > 55o , for the considered range of total pressure.

3 Control by Micron-Sized Roughness elements (MSR) On a swept wing with a strong flow acceleration, the transition is triggered by natural stationary vortices due to CF instabilities. The principle for controlling this type of instability, due to W.S. Saric [3], requires the knowledge of the most amplified natural vortices (Target mode, computed by linear stability theory). The second actor is the Killer mode corresponding to the artificial vortices generated by Micron-Sized Roughness elements arranged along the leading edge, near the critical abscissa of the Target mode and spaced by the wavelength λ 2. Non linear stability computation

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demonstrates that interactions can occur between the two modes, as shown in the figure 2(a) (killer mode as dashed lines, target mode as full lines; λKiller = 2/3 λTarget ). The artificial augmentation of the initial amplitude of the killer (due to MSR) increases the level of this mode while the target mode level decreases, delaying potentially the transition. According to the flow conditions, the succes of such a scenario depends on the pressure gradient and on the choice of the killer wavelength λ 2. The previous nonlinear computations have provided several active scenarios. Five

(a) Example of active scenario

(b) Wing into the S2MA test section

Fig. 2 Control by MSR (principle and experimental set-up)

of them have been implemented on the wing leading edge, using rows of roughness elements of 10 µ m height and 0.2 or 0.15 mm in diameter (see figure 3(a)). The measurement of the transition position is achieved by four infrared cameras looking at resin patches on the two wing sides (figure 2(b)). The figure 3(b) presents an ex-

(a) Picture of MSR rows Fig. 3 View of MSR realization and example of IR image

(b) Example of IR image

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ample of infrared picture, showing the transition location along spanwise direction Y. Unfortunately, we observe the absence of gain, the transition moving upstream behind MSR rows. This trend common to all the test points is probably due to a rather bad adjustment of the roughness size to the required initial amplitude of the killer mode, as suggested by simplified receptivity computations performed after the experiments.

4 Control of the leading edge contamination (ACD) In a first step, a numerical approach has been carried out. Some partners performed RANS computations with different codes using different turbulence models. The ACD was a variety of simple bracelets surrounding the leading edge of a cylinder of 10 mm in diameter, and terminated by short legs on the two sides of the wing. Conditions of the numerical approach were : M0 = 1.7, φ = 65o , Runit = 20x106 m−1 , R¯∗ = 440. The main criteria of efficiency considered in this numerical preliminary study were that the ACD must provoke a significant stagnation point on its front face and that ACD should not induce large perturbations like separation zones. Finally, seven shapes of ACD have been retained for the tests. They are shown in the figure 4(a). ACD 1 is a basic rectangular shaped one; ACD 2 and 3 are characterized by a hollow shape, less perturbing a priori; ACD 4 and 5 are more sophisticated, the

(a) Picture of the seven ACD tested

(b) Wing in test configuration

Fig. 4 View of the ACD and the experimental set-up into S2MA wind tunnel.

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legs stretching in the incoming flow direction; ACD 6 and 7 are homothetic, with a trapezoidal shape. The figure 4(b) presents the wing mounted into the S2MA test section, with a sweep angle φ = 65o . ACD are successively placed at the middle point of the leading edge span. Three hot films are measuring the wall shear stress fluctuations in order to indicate the nature (laminar or turbulent) of the boundary layer at three points close to the attachment line : the first two hot films are located respectively just upstream and downstream of the ACD and the third one near the wing tip. Experimental results at M0 = 2.7 are plotted in the figure 5(a) in terms of RMS values of the hot film signals versus the contamination parameter R¯∗ (obtained by the variation of the total pressure). The hot film located just upstream of the ACD shows the behaviour of the natural boundary layer. The curves indicate a low level for R¯∗ < 250 that means Pt < 0.55 bar (except for ACD 6 and 7 which are massive and very perturbing), then the level increases from approximately R¯∗ ≃ 270 (onset of contamination) to R¯∗ = 305 before to decrease and to reach the turbulent level at the end of the transition at about R¯∗ ≃ 340 ie. Pt = 0.9 bar. Just downstream of the ACD (middle), the majority of them are rapidly going from the laminar level (transition onset at about R¯∗ ≃ 255) to the turbulent one except ACD 5 which keeps the laminar level up to R¯∗ ≃ 395 ie. Pt = 1.2 bar. Finally, the hot film located near the wing tip demonstrates that the boundary layer is also laminar at this position, till R¯∗ ≃ 390. In conclusion, ACD 5 is efficient, shifting substantially the critical value of R¯∗ . One may assume that the optimization of this ACD (adjustment of the height for example) could further improve the gain.

(a) Fig. 5 Contamination state along the leading edge depending on R¯∗

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5 Conclusion A wing has been defined, suitable for both control of CF instabilities and leading edge anti-contamination in supersonic flow. Concerning the control of the CF waves, a certain number of scenarios have been characterised with non linear PSE approach for the different test conditions. MicronSized Roughness rows have been manufactured on the leading edge of the wing. Valuable MSR test results have been obtained, but no demonstrated gain was highlighted, probably due to the inaccurate sizing of the roughness elements. For further test, the first recommendation is to concentrate on the good estimation and realization of the right size of the roughness elements. Concerning the anti-contamination investigation, a numerical approach (RANS computations) has helped to estimate good shapes and sizes of potentially efficient ACD. According to these results, seven devices have been designed and tested for two supersonic Mach numbers and a continuous variation of the total pressure. The majority of the ACD were not satisfactory, but one well-shaped device was particularly efficient, delaying the transition from the natural R¯∗ value of 250 to 380 at Mach 2.7 and 320 at Mach 1.7. As outlook, it will be interesting to test this ACD at transonic speed. Besides, the two ”hollow” ACD were not clearly efficient but it should be interesting to optimise the shape of this type of devices.

6 Acknowledgements The work presented was part of the European research project SUPERTRAC performed under contract No. AST4-CT-2005-516100. The authors are grateful also to the other SUPERTRAC partners for their contributions.

References 1. D. Arnal, SUPERTRAC: Final technical report. Report No D6.9, 2008. 2. D.I.A. Poll, Boundary layer transition on the windward face of space shuttle during reentry. AIAA Paper 85-0899, 1985. 3. W.S. Saric and H.L. Reed, Crossflow instabilities. Theory and technology. AIAA Paper 20030771, 2003.

Breakdown of Low-Speed Streaks under High-Intensity Background Turbulence Masahito Asai, Motosumi Yamanouchi, Ayumu Inasawa, and Yasufumi Konishi

Abstract Breakdown of low-speed streaks under high-intensity background turbulence is studied experimentally through artificially generating spanwise-periodic low-speed streaks in a boundary layer downstream of two-dimensional local suction. The suction is applied to a developed turbulent boundary layer such that nearwall turbulence structures are completely sucked out but most of turbulent vortices in the original outer layer survive the suction. Spanwise-periodic low-speed streaks are generated in such a quasi-laminar boundary layer by using a periodic array of small screens. The results show that the suction-survived turbulence soon excites sinuous motion of low-speed streaks leading to the streak breakdown. Under the present energetic background turbulence, the streak breakdown is found to be not caused by the linear instability but by the transient disturbance growth.

1 Introduction The near-wall streaky structure is one of the most dynamically-important coherent structures in transitional and turbulent boundary layers. When the near-wall Masahito Asai Tokyo Metropolitan University, 6-6 Asahigaoka, Hino, Tokyo 191-0065, Japan, e-mail: [email protected] Motosumi Yamanouchi Tokyo Metropolitan University, 6-6 Asahigaoka, Hino, Tokyo 191-0065, Japan e-mail: [email protected] Ayumu Inasawa Tokyo Metropolitan University, 6-6 Asahigaoka, Hino, Tokyo 191-0065, Japan e-mail: [email protected] Yasufumi Konishi Tokyo Metropolitan University, 6-6 Asahigaoka, Hino, Tokyo 191-0065, Japan e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_11, © Springer Science+Business Media B.V. 2010

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M. Asai, M. Yamanouchi, A. Inasawa, Y. Konishi

streaks are intensified, inflectional velocity distributions develop across each lowspeed streak in the normal-to-wall and spanwise directions and thus the near-wall streaks become unstable, generating quasi-streamwise or hairpin-shaped vortices. This instability is called the streak instability and is considered to be a key mechanism of generating and sustaining wall turbulence [3, 5]. We examined such a streak instability both for a single low-speed streak [1] and spanwise-periodic low-speed streaks [4] generated artificially in a laminar boundary layer and obtained the detailed instability characteristics for sinuous and varicose modes. In order to further understand the mechanism of wall turbulence generation, the present paper studies breakdown of low-speed streaks under high-intensity turbulence. In strong disturbance condition, the breakdown mechanism might be different from that observed in such well-controlled experiments. In this concern, Schoppa and Hussain [6] emphasized that under high-intensity background turbulence, transient growth is more responsible for the breakdown of low-speed streaks than the linear instability . In the present experimental study, spanwise-periodic low-speed streaks are artificially produced in a quasi-laminar boundary layer just downstream of local suction applied to the otherwise turbulent boundary layer [2] and the response of the low-speed streaks to suction-survived turbulence is examined.

2 Experimental setup and procedure The experiment is conducted in a wind tunnel with a rectangular test section of 300×200 mm2 . A schematic of the test section is illustrated in Fig. 1. We focus on a boundary layer developing on the lower wall of the test section. The boundary layer is disturbed by cylinder roughness at an upstream station to generate a developed turbulent boundary layer at the downstream observation region. A suction strip of 120×180 mm2 is mounted flush with the wall surface, 500 mm downstream of the roughness. Over the suction area of 100×160 mm2 , 0.3 mm-diameter holes are drilled at an interval of 1mm in the streamwise and spanwise directions in a staggered configuration. The detail is described in [2]. As for the coordinate system, x is the streamwise distance measured from the downstream end of the suction strip, y the normal-to-wall distance, and z the spanwise distance.

Fig. 1 Schematic of test section and geometry of wire-gauze screens (dimensions in mm).

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The free-stream velocity U∞ is fixed at 4 m/s in the whole experiment. In the absence of suction, the Reynolds number Rθ based on the momentum thickness of boundary layer ranges from 750 to 900 in the observation region. Boundarylayer suction is applied with the suction velocity Vs of 0.1U∞ . In this case, nearwall low-speed streaks disappear almost completely immediately downstream of the suction strip but most of turbulent eddies in the original outer layer survive the suction. Seven pieces of wire-gauze screen, whose geometry is shown in Fig. 1, are set normal to the wall immediately downstream (35mm downstream) of the suction strip at an interval of 7mm to generate spanwise-periodic low-speed streaks in such a quasi-laminar boundary layer. Here the interval of the screen array is chosen to be the average streak spacing in the developed wall turbulence, i.e., λ + =100. Velocity measurements are carried out by means of hot-wire anemometer and PIV.

3 Results and discussion In the absence of suction, the turbulent boundary layer caused by the cylinder roughness is fully developed in the observation region, x ≥ 0. The time-mean velocity distributions exhibit the log-law profile. The boundary-layer suction mentioned above is applied over the region x = -100–0mm. Figs. 2(a) and (b) illustrate the y-distributions of U and the r. m. s. value of u-fluctuations u′ , respectively at x = -20mm, 50mm and 150mm for Vs /U∞ = 10% by comparing with the corresponding distributions (at x = 50mm) in the absence of suction. Passing through the suction area, the r. m. s. value u′ near the wall decreases down to 6% of U∞ . This value of u′ is a half the maximum r. m. s. value u′m without suction. The Reynolds number Rθ is about 400 at x = 50mm. After the suction, near-wall fluctuations soon start to grow as seen in the y-distribution of u′ at x = 50mm though the fluctuations away from the wall is decaying slowly.

Fig. 2 The y distributions of mean velocity U in (a) and r.m.s. value of velocity fluctuation u′ in (b) at x = -20mm, 50mm and 150mm, with suction of Vs /U∞ = 10%. Solid circles denote the distrubutions at x = 50mm in the absent of suction.

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Figs. 3(a) and (b) visualize instantaneous near-wall velocity fields measured by PIV in terms of iso-contours of the streamwise velocity U + u in the (x, z) plane with and without a periodic array of screens, respectively. The figure covers the flow field over x =40–120mm. The velocity field in Fig. 3(a) is measured at y =1mm which corresponds to about 13 wall units for the turbulent boundary layer (without suction). Near-wall low-speed streaks which once disappear immediately downstream of the suction strip start to reappear for x > 60mm. The growth of the near-wall peak of u′ shown in Fig. 2(b) corresponds to the occurrence and growth of near-wall low-speed streaks. In Fig. 3(b), seven pieces of small screens are set at x = 35mm to generate spanwise-periodic low-speed streaks in the quasi-laminar flow downstream of the suction strip. We see distinct spanwise-periodic low-speed streaks developing behind the screens. These low-speed streaks soon begin to oscillate laterally around x = 45mm (10mm downstream of the screens). Beyond there, the low-speed streaks break down and some of them are merged. Fig. 4(a) illustrates such artificiallygenerated low-speed streaks in terms of the distribution of mean streamwise velocity U in the (y, z) cross-section at x =42.5mm, 7.5mm downstream of the screens. The velocity defect due to drag of each screen is about 70% of U∞ . Fig. 4(b) shows the corresponding distribution of r. m. s. value u′ at x = 42.5mm. The distribution of u′ takes maxima on the left and right vertical shear layers of each low-sped streak and u′ vanishes at the mid-span of each low-speed streak. This indicates the characteristic amplitude distribution of sinuous streak instability mode. Then, in order to examine the streamwise development of the disturbances in more detail, spectral components are extracted from power spectra of u-fluctuations and their r. m. s. values, e.g., u′0−20 , u′20−40 , are obtained. Here u′0−20 and u′20−40 are the r. m. s. values of u-fluctuations between 0.5–20Hz and 20.5–40Hz, respectively. Fig. 5 illustrates the (y, z) distributions of u′20−40 , u′40−60 , etc at x = 42.5mm. All the components are also of sinuous-type, and u′40−60 are the largest in magnitude among them though the frequency selectivity for the amplification is not so strong. Fig. 6(a) illustrates the streamwise development of u-fluctuations by plotting the maximum

Fig. 3 Instantaneous streamwise velocity near the wall downstream of screens. (a) Without screens (y =1 mm). (b) With screens (y =1.2 mm).

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of u′ in the (y, z) plane against x. The disturbance growth is only observed up to the location 10mm downstream of the screens. Importantly, the streamwise growth is not exponential but algebraic. To demonstrate the algebraic disturbance growth more 2 clearly, Fig. 6(b) plots the disturbance energy (u′40−60 , etc) against x, showing that the disturbance energy of each spectral component increases in linear proportion to x initially and soon attains the maximum value at and around x= 42mm, 7mm downstream of the screens. The streamwise wavelength of the most amplified (4060Hz) disturbance is 33-50mm assuming that its convection velocity is half the freestream velocity. So the disturbance growth occurs only within a short distance much smaller than its wavelength. These facts indicate that under energetic turbulence just like in turbulent boundary layer, the streak breakdown is governed by transient growth rather than by linear instability, as pointed out by Schoppa and Hussain [6]. It is also important to mention that the breakdown due to the transient growth is highly dependent on the spectral contents in the high disturbance environments.

Fig. 4 Distributions of mean velocity U/U∞ and r. m. s. amplitude of excited disturbance u′ /U∞ in the (y, z) plane at x = 7.5 mm. (a) U/U∞ (contour lines range from 0.1 to 1.0), (b) u′ /U∞ (contour lines range from 0.02 to 0.2).

Fig. 5 Distributions of r. m. s. amplitude of each spectral componentin the (y, z) plane at x = 42.5mm. (a) u′0−20 /U∞ , (b) u′20−40 /U∞ , (c) u′40−60 /U∞ , (d) u′60−80 /U∞ , (e) u′80−100 /U∞ , (f) u′100−120 /U∞ . Interval of contours is 0.0075.

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Fig. 6 Streamwise development of disturbances. (a) r. m. s. value of u-fluctuations, (b) square of r. m. s. value of each spectral component.

4 Conclusions In the present study, breakdown of low-speed streaks caused by high-intensity turbulent fluctuations is studied experimentally through artificially generating spanwiseperiodic low-speed streaks in a re-transitional boundary layer downstream of local suction. The results show that the suction-survived turbulence with magnitude of u′ /U∞ ≈0.05 soon causes sinuous motion of low-speed streaks leading to the streak breakdown within a very short distance by the transient growth mechanism. Frequencies of the disturbances causing the streak breakdown are lower than that of the most amplified mode calculated from the linear stabyility theory[6], and thus the streak breakdown is highly dependent on the spectral components of background energetic turbulence. Acknowledgements This work was in part supported by the Grant for the Scientific Research from Tokyo Metropolitan Government.

References 1. Asai, M., Minagawa, M., Nishioka, M.: The instability and breakdown of a near-wall lowspeed streak. J. Fluid Mech. 455, 289–314 (2002). 2. Asai, M., Konishi, Y., Oizumi, Y., Nishioka, M.: Growth and breakdown of low-speed streaks leading to wall turbulence. J. Fluid Mech. 586, 371–396 (2007). 3. Jiménez, J., Pinelli, A.: The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359 (1999). 4. Konishi, Y., Asai, M.: Experimental investigation of the instability of spanwise-periodic lowspeed streaks. Fluid Dyn. Res. 34, 299–315 (2004). 5. Schlatter, P., Brandt, L., Lange, H.C.de, Henningson, D.S.: On streak breakdown in bypass transition. Phys. Fluids. 20, 101505 (2008). 6. Schoppa, W., Hussain, F.: Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108 (2002).

Numerical Study on Transition of a Channel Flow with Longitudinal Wall-oscillation Takashi Atobe and Kiyoshi Yamamoto

Abstract Laminar-turbulent transition of a channel flow with longitudinal walloscillation is numerically investigated. In a model flow, two walls are oscillated in phase and Reynolds number based on a half width of the walls and the global maximum of the mean flow of non-oscillating channel flow is fixed to 10000. Direct numerical simulation shows that, due to the wall-oscillation, in some case the transition to turbulence is accelerated in comparison with the non-oscillating channel flow, and in other case is not. It is revealed from visualization of the vorticity that the streaks existing near the walls are divided in the accelerated case. It is found that the transition is suppressed in a certain area of the parameter space. In this case, it seems that the wall oscillation prevents the growth of T-S wave in the channel flow. Furthermore, the phenomena have correlation with the stability of Stokes layer flow.

1 Introduction It is well known that spanwise wall-oscillation on turbulent channel flow shows drastic drag reduction. This phenomenon was first pointed out by Jung et al.1 , and Quadrio and Ricco2 recently demonstrated by a numerical study that a maximum drag reduction is of 44.7%. Owing to the succeeding study3−6 , the fundamental mechanism has steadily been elucidated. Whereas there are relatively less study for the flow with longitudinal oscillation of the wall, nevertheless this system has an analogous to the pulsated flow. The flow in the blood vessel or in the exhaust pipe is basically pulsating and such flow is optimized due to the pulsation. In some case, Takashi Atobe Japan Aerospace Exploration Agency, 7-44-1 Jindaiji-Higashi, Chofu, Tokyo, Japan, e-mail: [email protected] Kiyoshi Yamamoto Japan Aerospace Exploration Agency, 7-44-1 Jindaiji-Higashi, Chofu, Tokyo, Japan, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_12, © Springer Science+Business Media B.V. 2010

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these pulsating flow can be seen a flow with wall-oscillation. Thus the purpose of the present paper is to study the effects of longitudinal wall-oscillation on a laminarturbulent transition of a channel flow by direct numerical simulation (DNS). Also, linear stability analysis is employed in order to describe the phenomena from the view point of flow stability. Model flow investigated here is constructed by linear combination of two-dimensional Poiseuille flow and Stokes layer because of the linearity of the governmental equation.

2 Numerical Analysis 2.1 Model Flow Overview of the numerical simulation and parameter region investigated are shown in Fig.1. Here, Ω and Uw are frequency and amplitude of the wall-oscillation. Parameters describing this system are Ω , Uw , and the Reynolds number defined as Re ≡ hU/ν , where U is the maximum value of mean flow, ν the kinematic viscosity and 2h the distance between the walls. In the present study, Re is fixed as 10000 for convenience. In DNS, x and y directions are expanded by the Fourier expansion, and for z direction the Chevichev-collocation method is employed as the follows. ˜ x , ky , z,t) exp[i(kx + ky )]. u(x, y, z,t) = Σ u(k

(1)

Based on this formulation, the energy component of each Fourier modes can be estimated as: Z 1 ˜ x , ky , z,t)|2 dz. E(kx , ky ,t) ≡ |u(k (2) 4 In the present study, transition to turbulence is evaluated using this energy of each Fourier mode.

2Uw, Ω U: Mean Flow

2Uw, Ω

Fig. 1 Overview of numerical simulation and parameter region.

Re = 10000（Fixed） Uw = 0.0 – 0.3 Ω = 0.0 – 0.3

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2.2 Results Some results of energy variation of each Fourier mode are shown in Figs.2-4. All of them are simulated with initial disturbances of order 109 . Since transition of the channel flow under considering Re number occurs dominated by the TollmienSchlichting wave, which corresponds to E(1,0) mode in the present formulation, order of 105 T-S mode is added to initial disturbances in order to save the time-cost. In Fig.2, non-oscillating case, the energy of the T-S mode increases with time. Other modes also increase and 10 the energy level of those merges with 10 E(0,0) of the T-S mode. Finally, E(0,0) mode 10 corresponding to the turbulent mean E(1,0) flow exceeds and then, the flow tran10 E sits to turbulent state. 10 On the other hand, typical cases 10 with wall-oscillation are shown in 10 Fig.3 and 4. In the case of Fig.3 with the parameters of (Ω ,Uw ) = 10 (0.2,0.3), although the energy of each 0 50 100 150 200 250 300 mode oscillates affected by the wallFig. 2 Time evolution of energy of non-oscillating oscillation, transition occurs almost t case. same as non-oscillating case of Fig.2. However, in the case of Fig.4 with (0.1,0.1), the flow rapidly transits to turbulent state. 0

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It is found from such parametric study that the situations can roughly be grouped in three patterns depending on the parameter. Result of this parametric study is

90


shown in Fig.5. The circles correspond to the acceleration cases, the diamonds to the deceleration, and square to the less affected cases. The accelerated cases exist in small Ω region. Figure 6 shows a contour of instantaneous vorticity for (a) at t = 150 of Fig.4, and (b) at t = 50 of Fig.4. The white color corresponds to the streamwise component, and the dark color corresponds to spanwise component. In the case without wall-oscillation, relatively large scale coherent structures so called gstreakh or glambda shaped vortexh exists near the wall. However, at least at this instant, the small scale spanwise vortices exist instead of the streak. It might be conjectured that the absence of the streamwise coherent structure leads to this acceleration of the transition.

0.35 0.3

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3 Linear Stability Analysis 3.1 Quaisi steady analysis In order to investigate the previous phenomena in detail, a model flow is analyzed by the linear stability theory. The flow near the walls might be thought as the stokes layer flow. Since the both of the channel flow and the Stokes layer are solution of linearized Navier-Stokes equation under the assumptions of two-dimensional and parallel flow, the basic flow dealt here can be described as a superposition of a Stokes layer onto a two-dimensional Poiseuille flow. Thus, the stremawise velocity U is written as the follows, U = 1 − z2 +Uw Re[ here κ = (1 + i)k, k =

p

cosh(κ z) exp(iΩt)], cosh(κ )

Ω /2ν . Then, the Orr-Sommerfeld equation,

(3)

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1 2 (D − α 2 )2 w¯ + i(ω − α U)(D2 − α 2 )w¯ + iα D2 w¯ = 0, R

(4)

is employed with velocity profile obtained from equation (3). Here α is the streamwise wavenumber and ω is the complex frequency of small disturbance, and D is a differential operator in z direction. ω can be solved from this eigenvalue problem with the eigenfunction w. ¯ The flow can be estimated as stable if imaginary part of ω is negative, and unstable if it is positive.

3.2 Results Velocity profiles of the basic flow for the case of (Ω ,Uw )=(0.1,0.5) is shown in Fig.7 at arbitrary instant. Upper half of the profiles are not shown because of its symmetricity. Using eq.(4) with these velocity profiles, stability of the flow can be estimated. Results of the stability analysis for the case of (Ω ,Uw )=(0.01,0.1) under some Re number are shown in Fig.8. It can be found that stability periodically varies with the period of the wall-oscillation. When Re number increases, amplitude of intensity of the stability increases but there is little change on the shape of over all. Thus, it seems that tendency of the results shown in Fig.5 does not change drastically for the certain range of the Re number. Oscillating Poiseuille Up=0.5, Omega=0.1 0

1/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8

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Because of the linearlity of the governmental equation, character of the present flow might correlate with the instability of the Stokes layer. Reynolds number of its flow is defined by, q Rδ ≡

2Uw2 /νΩ .

(5)

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Kerczek and Davis7 has investigated the critical Reynolds number Rδc of the Stokes layer, and reported the Rδc is 86 or 182. The former number was estimated by the use of the most unstable eignevalue during one period, and the latter by the integrated value of the eigenvalue during a period. Two curves in Fig.4 are written using the relation (5) and those Rδc . It seems that there is some correlation between the character of the present flow and the instability of the Stokes layer.

4 Conclusion The effects of longitudinal wall-oscillation on a laminar-turbulent transition of a channel flow is studied by direct numerical simulation (DNS) and linear stability analysis. Model flow investigated here is constructed by linear combination of twodimensional Poiseuille flow and Stokes layer because of the linearity of the governmental equation. From the present study, some new findings are obtained as the follows. • From the parametric study by DNS, parameter space built by frequency and amplitude of the wall-oscillation can be grouped in accelerated, decelerated and changeless regions. • Oscillating walls suppress the produce of the streak like structures near the wall. • Linear stability analysis suggests that these tendency might be kept in certain range of Reynolds number. • There is some correlation between stability of the wall-oscillating channel flow and the Stokes layer from the comparison with critical Reynolds number analysis of the Stokes layer.

References 1. Jung, W. J., Mangiavacchi, N. and Akhavan, R., gSuppression of turbulence in wall-bounded flows by high-frequency spanwise oscillationsh, Phys. Fluids A 4 (8), 1992, pp.1605-1607. 2. Quadrio, M. and Ricco, P., gCritical assessment of turbulent drag reduction through spanwise wall oscillationsh, J. Fluid Mech., 2004, pp.251-271. 3. Akhavan, R., Jung, W. and Mangiavacchi, N., gControl of wall turbulence by high frequency spanwise oscillationsh, AIAA Paper, 1993, 93-3282. 4. Choi, K.-S., DeBisschop, J. R. and Clayton, B. R., gTurbulent boundary-layer control by means of spanwise-wall oscillationsh, AIAA. J. 36 (7), 1998, pp.1157-1163. 5. Choi, K.-S. and Clayton, B. R., gThe mechanism of turbulent drag reduction with wall oscillationh, Int. J. Heat and Fluid Flow 22, 2001, pp.1-9. 6. Segawa, T., Li, F., Yoshida, H., Murakami, K. And Mizunuma, H., gSpanwise oscillating excitation for turbulence drag reduction using alternative suction and blowingh, AIAA Paper, 2005, 05-488. 7. Kerczek, C. Von and Davis, S. H., gLinear stability theory of oscillatory Stokes layersh, J. Fluid Mech. 62, 1974, pp.553-773.

Direct Numerical Simulation of the Mixing Layer past Serrated Nozzle Ends Andreas Babucke, Markus J. Kloker, and Ulrich Rist

Abstract The effect of serrations on the mixing layer past a thin splitter plate is investigated using spatial direct numerical simulation (DNS) with direct sound computation. Two different geometries are considered which yield a spanwise deformation of the Kelvin-Helmholtz rollers, streamwise vortices and subsequent breakdown of large-scale coherent structures, strongly affecting the noise emission. The results reveal that the spanwise extent of the serration is the driving parameter for sound reduction while its actual shape is less important.

1 Introduction and Numerical Method Noise reduction is an important issue in aviation. Especially jet noise is one of the major acoustic sources of an aircraft. Modification of the trailing edge shape is a recent approach to reduce jet noise [3], often explained by an increased mixing behind the nozzle end. However the underlying physical mechanisms are not yet well understood. The considered flow is made of two laminar boundary layers with the same freestream temperatures and MaI = 0.8 and MaII = 0.2 above and below the splitter plate, respectively. The Reynolds number Re = 1000 is based on the displacement thickness δ1,I of the upper boundary layer at the inflow. Two different types of serrations are considered: a non-symmetric serration (figure 2) and a rectangular notch with half the spanwise extent (figure 3). The depth of the serrations is 10 for both cases being roughly three times the boundary-layer thickness δ99 at the trailing edge. The extent in z-direction defines the fundamental spanwise wavenumber γ0 = 0.2. The flow is forced at the inflow of the upper boundary layer with a 2-d Tollmien-Schlichting wave (1, 0) of frequency ω0 = 0.0688 and |u| ˆ = 5 · 10−3 , and −4 a low-amplitude oblique wave (1, ±1) with |u| ˆ = 5 · 10 . In the following, disturAndreas, Babucke, Markus Kloker, Ulrich Rist IAG Universität Stuttgart, Germany, e-mail: [babucke]/[kloker]/[rist]@iag.uni-stuttgart.de P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_13, © Springer Science+Business Media B.V. 2010

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bances are denoted as (h, k), where h and k are the multiples of the fundamental frequency ω0 and the fundamental spanwise wavenumber γ0 , respectively. The simulations are carried out using the DNS-code NS3D which solves the 3-d unsteady compressible Navier-Stokes equations, see [1] for details. The serration is implemented by defining a region without wall at the respective grid points. There, the spatial derivatives in normal direction are recomputed, using values from the domain at the other side of the splitter plate as well. This is done by explicit finite differences of 8th order which are designed such that their numerical properties closely match those of the 6th -order compact scheme used in the rest of the domain. With spanwise filtering due to the spectral ansatz being applied only to the time derivatives of the flow variables, the no-slip condition is held completely at the wall.

2 Results For the reference case with the straight trailing edge, shown in figure 1, the flow is dominated by 2-d Kelvin-Helmholtz vortices. As shown in figure 2, the nonsymmetric serration causes a spanwise deformation of the Kelvin-Helmholtz vortices similar to the experimental results of Kit et al. [4]. The resulting shape of the vortex corresponds to the trailing-edge shape. At the location of the strongest spanwise gradients, longitudinal vortex tubes are generated which are directed towards the center of the notch, see figure 4. Further downstream these interact with the spanwise rollers leading to a breakdown to small scale structures. In case of the rectangular serration with half the spanwise extent (figure 3) the initial Kelvin-Helmholtz vortex at x ≈ 50 is modulated along the spanwise direction as well. Due to the double number of the serrations, the first roller is modulated with spanwise wavenumber 2 · γ0 , here. As shown in figure 5, longitudinal vortices exist as well but they are directed much more downward, compared to the wider nonsymmetric notch. Although more streamwise vortex tubes are generated, the 2-d structure of the spanwise rollers is preserved longer. Yet the dominance of smallscales occurs for both cases at x ≈ 120. The spectral decomposition in figure 6 a) shows that the non-symmetric serration generates steady spanwise modes (0, k) up to |v| ˆ max = 6 · 10−3 at the trailing edge. With increased amplitude, they interact with the 2-d TS-wave (1, 0) and its higher harmonics, generating oblique modes (h, 1), see figure 6 b). With doubled number of serrations, only steady disturbances with spanwise wavenumbers ≥ 2γ0 are introduced by the rectangular serrations (figure 7 a)). Mode (0, 1) stays on the lower level defined by the interaction of (1, 0) and (1, ±1) in the upper boundary layer. Accordingly, oblique waves with wavenumber 2γ0 are generated past the trailing edge and mode (1, ±1) is smaller by one order of magnitude as given in figure 7 b). Yet at the end of the integration domain, it reaches a similar level as for the non-symmetric case due to amplification inside the mixing layer. High-level 2-d waves (1, 0)-(3, 0) show a similar behavior for both serrations which is also the case for their subharmonics (not shown here).

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Fig. 1 Flow structures for the reference case with straight trailing edge, visualized by the isosurface Λ2 = −0.001 along two spanwise periods. Grey scales indicate the wall-normal distance.

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Fig. 3 Flow structures as in figure 2 but for the rectangular notch with half the spanwise extent.

The difference of the overall sound-pressure level ∆ L p = 20 · log10 (p′rms /pre f ) level with respect to the reference solution is shown in figures 8 a) and b) for both serrations. Directly behind the trailing edge, additional oblique waves yield an increase of pressure fluctuations by some 5 dB. On the other hand, missing large-scale

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Fig. 4 Top view (snapshot) on the mixing layer downstream of the non-symmetric trailing edge along two spanwise periods. Visualization by isosurface Λ2 = −0.005.

Fig. 5 Same as figure 4 but for the rectangular-notch trailing edge with half the spanwise extent.

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structures lead to lower fluctuations further downstream in the mixing layer. The reduction of the actual sound is determined by the difference in the farfield. The nonsymmetric serration provides a noise reduction in the range of 6 to 10 dB. In case of the narrow rectangular notch a difference of only some 5 dB is reached. Since the result for the non-symmetric serration is quite similar to the one of the corresponding rectangular serration [1], it seems that the actual shape of the notch is of

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Fig. 8 Reduction of the overall sound-pressure level in dB compared to the reference case: a) nonsymmetric serration and b) narrow rectangular serration, respectively. The location of the pressure spectra in figure 9 is marked by +.

minor importance. The dominant parameter of the serration is its spanwise extent / wavenumber γ as found also in the experiment by Bridges & Brown [2]. For a closer look the pressure spectra in the slow-speed stream (location indicated by + in figures 8 a) and b)) are compared with the straight trailing edge in figure 9. A notable reduction of the dominant low-frequency noise (ω ≤ ω0 ) is found for both serrations which is due to the decreased level of subharmonics compared to the straight trailing edge, see [1]. However for the narrow notch, sound with some 2ω0 is almost as high as for the straight trailing edge. Together with the partially increased level of | p| ˆ in the range of 4 to 20 ω0 the narrow serration yields a less efficient reduction of the emitted sound.

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Fig. 4 Snapshot of density disturbances in the presence of shock oscillations imposed at the freestream boundary.

A temporal Fourier analysis of the data, see Fig. 6 a) yields amplitude beatings in streamwise direction which are caused by the simultaneous presence of several modes. These are further identified in a spatial Fourier transform (using a Hanning window to render the data periodic) of the fundamental amplitudes of the frequency spectrum. An example is given in Fig. 6 b). It indicates two maxima at wave numbers

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Fig. 5 Snapshot of density disturbances 14 cycles after stopping of forcing.

α ≈ 9 and α ≈ 11.5, respectively. Using linear stability theory [1] two corresponding modes are found for these wavenumber-frequency combinations: one at α ≈ 9 and the other α = 11.3. The first one is damped while the second one is unstable according to linear stability theory.

Fig. 6 Fourier spectra of density fluctuations at y = 0.1. (a)Streamwise evolution of the fundamental disturbance amplitude ω and its higher harmonic 2ω . (b) Spatial spectrum for ω at three different intervals (14 – 15 cycles after end of forcing).

Comparison of the amplitude profiles extracted from the DNS with eigenfunctions of linear stability theory at the constant streamwise station Rx = 1575 which corresponds to the ω -amplitude maximum in Fig. 6 a) are shown in Fig. 7. They confirm that an instability wave has been generated by the incoming shock oscillations. Deviations occur due to the presence of the other mode.

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Fig. 7 Comparison of spectral fluctuation amplitudes (ω = 10, solid lines) at Rx = 1575 with eigenfunctions of the most unstable eigenmode according to linear stability theory (lines with symbols). η = y · Re/Rx .

4 Conclusions An oscillating shock wave can generate instability waves in the region of shockwave/boundary-layer interaction. Thus, a supersonic boundary layer is receptive to external disturbances even in the absence of surface roughness. Because all disturbances are convected out of the domain when the forcing at the free stream ends, the present separation bubble is clearly identified as an ‘amplifier’ in contrast to an ‘oscillator’ (cf. [2], for definition of these terms).

References 1. Mack, L.M. (1975). Linear stability theory and the problem of supersonic boundary-layer transition. AIAA Journal 13(3), 278–289. 2. Marquet, O., Sipp, D, Chomaz, J.-M. & Jacquin, L. (2008). Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework. J. Fluid Mech., 605, 429–443. 3. Pagella, A., Rist, U. & Wagner, S. (2002). Numerical investigations of small-amplitude disturbances in a boundary layer with impinging shock wave at Ma=4.8. Phys. Fluids, 14(7), 2088–2101. 4. Pagella, A., Babucke, A. & Rist, U. (2004). Numerical investigations of small-amplitude disturbances in a boundary layer at Ma=4.8: compression corner vs. impinging shock wave. Phys. Fluids, 16(7), 2272–2281.

Roughness receptivity studies in a 3-D boundary layer – Flight tests and computations Andrew L. Carpenter, William S. Saric, and Helen L. Reed

Abstract The receptivity of 3-D boundary layers to micron-sized, spanwise-periodic Discrete Roughness Elements (DREs) was studied. The DREs were applied to the leading edge of a 30-degree swept-wing at the wavelength of the most unstable disturbance. In this case, calibrated, multi-element hotfilm sensors were used to measure disturbance wall shear stress. The roughness height was varied from 0 to 50 microns. Thus, the disturbance-shear-stress amplitude variations were determined as a function of modulated DRE heights. The computational work was conducted parallel to the flight experiments. The complete viscous flowfield over the O-2 aircraft with the SWIFT model mounted on the port wing store pylon was successfully modeled and validated with the flight data. This highly accurate basic-state solution was incorporated into linear stability calculations and the wave growth associated with the crossflow instability was calculated.

1 Introduction This paper describes a series of flight tests conducted at the Texas A&M Flight Research Laboratory (FRL) to study the effects of micron-sized, spanwise-periodic, discrete roughness elements (DREs) on a 30-degree swept-wing. A swept-wing test article was mounted vertically to the port pylon of a Cessna O-2A aircraft and operated at a chord Reynolds number, Rec , of 7.5 million to match the flight conditions for the Sensorcraft aircraft. The focus of the current set of experiments is to understand the receptivity mechanism due to the introduction of artificial roughness near the leading edge of the model. The test article is described here as the Swept-Wing In-Flight Testing model or SWIFT model. The details of previous work and the successes of the DREs are found in [1, 2, 3, 4]. The SWIFT model has a leading-edge sweep of ΛLE = 30° and a chord of 1.37 m to provide the targeted Rec between 6.5 and 7.5 million within the performance capabilities of the O2A during a dive. The span of the SWIFT model was nominally 1.07 m at the leading edge. Figure 1 shows Andrew L. Carpenter, William S. Saric, Helen L. Reed Dept. of Aerospace Eng., Texas A&M University. e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_15, © Springer Science+Business Media B.V. 2010

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the test aircraft with the SWIFT model mounted to the port wing and a hotwire sting mounted on the starboard wing. An Aeroprobe, conical-tip, five-hole probe was used for all freestream measurements of total pressure, static pressure, aircraft angle of attack, and aircraft sideslip angle. The five-hole probe was mounted to the non-testside of the SWIFT model. Rhodes et al. [5] showed that there was a spanwise flow variation between the port wing and starboard wing, requiring localized measurements at the SWIFT model. Freestream turbulence levels were recorded prior to flying with the SWIFT model. During the test conditions, the turbulence levels were approximately 0.05% of the freestream velocity in the 1Hz to 10 kHz passband. Two panel meters were added to the instrumentation panel to display model AoA and Rec . The pilot maintained these two parameters constant while on test conditions during the dive.

Fig. 1 Cessna O-2A aircraft with SWIFT model and hotwire sting

2 C p Measurements The primary objective of the computational analysis [5] was to compare the Cp measurements recorded in flight with the calculated Navier-Stokes solution. Once these matched, within experimental error, then the computations proceeded to provide boundary-layer and stability calculations for the experiments. The primary objective for conducting experimental C p measurements on the SWIFT model was to determine which AoAs produced sufficient crossflow, resulting in a crossflow-dominated transition. An accelerated boundary layer back to the pressure minimum at 70% chord was the desired test point for studying the effect of DREs on transition. Figure 2(a) shows Cp values for the root pressure port row for three representative AoA cases. The SWIFT model contains two rows of pressure ports. However, the receptivity measurements described here are only conducted at the top pressure port row. The bottom half of the model was used for demonstrating the LFC strategy with the DREs, and is described in [6]. This paper focuses on two AoA conditions for the receptivity measurements. Figures 2(a) and 2(b) show a comparison between the experimental data, with error bars, and the calculated C p distribution from [5]. Figure 2(a) is at -2.61 deg AoA which demonstrated laminar flow back to 80% x/c. Figure 2(b) at -4.69 deg AoA shows early transition and strong crossflow growth. Both Fig. 2(a) and Fig. 2(b) show good agreement between the experimental and calculated Cp .

Roughness receptivity studies in a 3-D boundary layer – Flight tests and computations

(a) Comparison at -2.61 deg AoA

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(b) Comparison at -4.69 deg AoA

Fig. 2 Cp Comparison – Root Pressure Port Row

Infrared (IR) thermography was used to visualize boundary-layer transition on the SWIFT model. A large range of AoAs between -0.5 and -5.5 degrees has been tested here to get a broad sampling of the crossflow instability. A FLIR SC3000 camera was used for all IR flights. The model was first cold soaked at 10,500 feet until a uniform temperature was reached on the model. At the end of the dive it was possible to generate as much as a 3 °C differential between the laminar and turbulent regions. See [1, 6] for IR images. Transition data were first collected with a polished leading edge. The surface roughness was measured to be 0.33 µm rms with 2.20 µm average peak-to-peak.

3 Critical DREs LST calculations showed that one of the most unstable crossflow vortex spacing was λcrit = 4.5 mm, measured parallel to the leading edge. Roughness elements were placed 4.5 mm apart at the calculated neutral point of 2.21% x/c at the root pressure port row and 2.41% x/c at the tip pressure port row. The LST calculations showed that the spanwise variation in the Cp distribution resulted in a spanwise variation in the neutral point. As a result, the DREs were placed along a line that was not at a constant x/c, but rather shifted downstream at the tip to account for the spanwise variation. The initial roughness elements used in these experiments were dots and will be referred here as appliqué DREs. These appliqué DREs are approximately 6 µm high and the cross-sectional shape are idealized as right-circular cylinders. The heights of the roughness elements can be increased at 6 µm (as measured by a confocal lens system) increments by stacking individual elements. For the following test conditions k will be specified in all figures. However, it is also instructive to present roughness data with a Reynolds number based on roughness height, Rek which is calculated using the absolute velocity at the top of the roughness height, uk , the kinematic viscosity also at the roughness height, νk , and the characteristic length k. Furthermore, the roughness height can be represented as a fraction of the

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boundary layer height, k/δ . The boundary layer height, δ , is approximately 250 µm in the location of the DREs. The objective of the receptivity work was to determine coupling between surface roughness height and crossflow disturbance amplitude. This was necessary to provide the proper initial amplitudes to the Nonlinear Parabolized Stability Equations (NPSE) computations. Thus, receptivity was the missing link to the NPSE being used as an independent tool for calculating crossflow stability behavior. The variable roughness design used here was first demonstrated in [7] and proved to be effective at controlling transition in a low-turbulence wind tunnel. The idea was to create a pressurized chamber where a flexible membrane would seal off holes at the top of the surface. As pressure is applied to the chamber, the flexible membrane would stretch and generate a roughness element. The height of the element would be proportional to the applied pressure and could be controlled in flight.

4 Receptivity Measurements It was desired here to measure the disturbance shear stress generated by DREs placed at the neutral point upstream of a hotfilm array, in order to resolve the importance of roughness shape and height. Therefore, a calibration scheme was needed for calibrating the hotfilms in flight and making measurements of the disturbance amplitude generated by modulating DRE heights. A Preston tube is used for shear stress calibration. A custom hotfilm was designed for measuring the crossflow amplitude and spacing on the SWIFT model. With a maximum crossflow spacing of 4.5 mm, a multielement hotfilm array was needed for measuring the high and low shear stress regions in between areas of strong crossflow growth. The spacing of each hotfilm sensor needed to be small enough to resolve the shape of the crossflow wavelength, in addition to having a long enough span to capture at least one period. The 4.5 mm crossflow spacing is measured parallel to the leading edge. When each hotfilm sensor is placed perpendicular to the inviscid streamline, the crossflow spacing at the hotfilm is approximately 3.9 mm, resulting in an even finer spacing required for the hotfilm design. The multi-element hotfilm used in these experiments was constructed by Tao Systems. Sensors with a 50 µm width and 510 µm length were used with the smallest lead widths available. A total of 20 elements were used. Thus, there were 8 elements per wavelength. The smallest dimensions available for the design resulted in a staggered hotfilm where the line intersecting the center of each sensor was swept back 42 degrees. If this line where swept back at 30 degrees, then the sweep of each sensor would have matched the ΛLE , and each sensor would be located at the same x/c location on the model. However, the limitations in manufacturing tolerances forced the sensors to be along a 42-degree swept line resulting in a change in x/c location of 0.2% for 20 channels. The change in disturbance growth over 0.2% x/c is calculated to be negligible. The spacing between the surface mounted thermocouple, static pressure port, Preston tube, and first hotfilm channel were all 22 mm along a line of constant x/c.

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The flight experiments here are subject to large temperature excursions during the dive profile. As a result, a comparison of the raw voltages from each sensor would be meaningless. A Preston tube was placed next to the hotfilm to provide the known measure of shear stress for the calibration scheme. Bechert [10] provides the most accurate calibration method based on a comparison between experimental measurements of wall shear stress, τw , and computations. The Preston tube used here was a 28-gauge hypodermic stainless steel needle with a flat, round mouth. The outer diameter was 0.3mm which was 36% of the boundary layer height at the 34% x/c location. Each channel on the hotfilm array was adjusted with a gain of 10 and an overheat ratio of 1.2 to provide sufficient sensitivity during flight. With this gain, the zero flow voltage, E0 , of each channel would change by approximately -0.45 V/°C. The temperature drift agrees with the measurements made in [11] where a linear temperature drift was observed with CTAs and hotwires. The Preston tube data were recorded during three phases of each flight. During the cold soak, while the crew was waiting for the model to cool down uniformly, the pilot would increase the airspeed at five knot increments at a constant AoA. Once the model was at a uniform cold temperature, the dive was initiated, again at a constant AoA. During the return to base, the pilot would perform another calibration run, but this time the temperature was higher at the lower altitude. The calibration data showed obvious hysteresis loops which need to be corrected before a calibration can be attempted. Radeztsky et al. [11] suggest that the temperature drifts recorded for each hotfilm channel can be used to linearly correct the measured voltages to a compensated voltage based on a standard temperature. See [1] for details. The sensitivity slope was adjusted for each channel until the calibration curve for the top and bottom of the dive eventually collapsed on top of each other into one curve, indicating a temperature invariant calibration curve. The in situ adjustment of the slope for each flight was expected because the heat conduction lost to the model is different for each flight. This is in agreement with [12], where heat conduction to the substrate is the biggest problem in calibrating surfacemounted hotfilms, and must be accounted for on each model for every test. Once each temperature drift value is finalized, a calibration curve is generated that is now temperature invariant. Once the temperature compensation was accounted for, the hotfilms were calibrated using a third-order polynomial in E 2 for the shear stress.

5 Receptivity Results Two AoA conditions have been tested thus far. The test point at -2.61 degrees AoA proved to have very little crossflow growth at the 34% x/c location. Even with the variable height DREs pressurized to 50 µm, no crossflow structure is apparent in the hotfilm signal. On the other hand, the test point at -4.69 degrees AoA shows strong signs of crossflow, even without the DREs pressurized or evacuated. A definite structure spaced 4.5 mm is apparent in the signal. Furthermore, the signal gets stronger as the DREs are pressurized. A strong 4.5 mm structure was also observed.

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Figure 3 shows a comparison of the skin-friction coefficient, c f , versus spanwise position parallel to the leading edge between two flights with pressurized DREs.

Fig. 3 Comparison of different disturbance amplitudes with DREs spaced 4.5 mm apart

Figure 3 shows that the hotfilm covered two crossflow vortices. In Figure 3, the first curve at an AoA of -4.76 deg is in phase with the second curve with the DREs pressurized. All test points at the -4.69 deg AoA test condition show a strong 4.5 mm structure that is always in phase with the DREs on and off. Thus, a comparison can now be made between disturbances generated by a 2 µm bump versus a 50 µm bump. It appears as though there is a non-linear relationship between the roughness height and the disturbance amplitude measured downstream of the roughness elements. The signal generated by the 2 µm roughness element is close to the signal generated by the 50 µm bump, despite having a much smaller height. However, the signals measured at -4.69 deg AoA are measured just upstream of transition, so the amplitudes may have already saturated. Nevertheless, the technique outlined in this paper for calibrating hotfilms produces good results and more flights can be conducted to further understand the receptivity process due to DREs. Acknowledgements This work was supported by AFOSR Grant FA9550-05-0044.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Carpenter AL, Saric WS, Reed HL. AIAA Paper No. 2008-7335 (2008) Saric WS, Carrillo RB, Reibert MS. AIAA 98-0781 (1998) Saric WS, Reed HL, Banks DW. NATO-RTO-MP-AVT-111/RSM (2005) Saric WS, Reed HL, White EB. Ann. Rev. Fluid Mech. 35:413-40 (2003) Rhodes RG, Carpenter AL, Reed HL, Saric WS. AIAA 2008-7336 (2008) Saric WS, Carpenter AL, Reed HL. AIAA 2008-3834 (2008) White EB, Saric WS. AIAA 2000-0283 (2000) Radeztzky RH, Reibert MS, Saric WS, Takagi S. AIAA 93 -0076 (1993) Saric WS. 2007 Boundary-Layer Stability and Transition. Springer Handbook of Experimental Fluid Mechanics Springer-Verlag Berlin Heidelberg, Ed: Cameron Tropea, Alexander Yarin, John F. Foss. Chapter C.12, Section 12.3 pp. 886-96, 20 10. Bechert DW. AIAA J. 34(1):205-6 11. Radeztsky RH, Reibert MS, Takagi S. Proc. 3rd Intl Symp. on Thermal Anemometry (1993) 12. Sarma GR, Moes TR. Rev. Sci. Instruments 76 (2005)

DNS investigations of steady receptivity mechanisms on a swept cylinder G. Casalis and E. Piot

1 Geometry, main assumptions The main geometry is displayed in figure 1. Some roughness elements are placed periodically along a line parallel to the swept cylinder leading edge. The swept angle is ϕ = 60◦ , the incoming velocity norm Q∞ = 50 m/s and the cylinder radius Rc = 0.1 m. The unit Reynolds number is then closed to 3 106 m−1 . The attachment line Reynolds number is R¯ = 500, i.e. a value less than the critical one (582) corresponding to the Görtler-Hämmerlin instability threshold, so that a laminar flow at least in the vicinity of the leading edge is expected. The flow is assumed to be periodic with respect to the spanwise coordinate z. The numerical domain is then limited such as it only contains a single roughness element. The streamwise coordinate is denoted by s, the normal to the wall by n. In the following the streamwise locations will be given by the angle s/R in degree, the value s = 0◦ corresponding to the leading edge. The full Navier-Stokes equations are then solved with the ONERA in-house made code SABRINA, see [3], using a fourth order scheme in space and a third order one in time. The computations are done in two successive steps. First the cylinder is considered without roughness element. The flow is initialized by the analytical potential solution for the flow field around a swept cylinder. After a transient phase, a laminar steady flow is obtained, which is independent of the z coordinate as expected. Actually the obtained viscous flow is very close to the theoretical swept Hiemenz solution at least in the vicinity of the leading edge, as long as the flow is linearly accelerated with respect to s. This steady flow is stored in a file and will be referenced as the “basic” flow in the following. For the second step the roughness

Grégoire Casalis ONERA, 2, av. Ed. Belin, BP4025, Toulouse Cedex, FRANCE, e-mail: [email protected] Estelle Piot ONERA, 2, av. Ed. Belin, BP4025, Toulouse Cedex, FRANCE, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_16, © Springer Science+Business Media B.V. 2010

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element is placed on the cylinder. The simulation is initialized by the basic flow obtained at the end of the first step.

2 Unexpected unsteady fluctuation Two streamwise positions of the roughness element have been considered. The locations sr of the upstream and downstream streamwise limits of the roughness element, its height based Reynolds number Reh and the relative height h/δ1 with respect to the boundary layer displacement height are given in table 1. Taken into Table 1 Main characteristics of the two considered cases for the roughness element abscissa.

sr Reh h/δ1

Case A

Case B

[7.3◦ ;

[13.5◦ ; 14.1◦ ] 44.6 0.575

43.7 0.58

7.9◦ ]

account the spanwise Lz extent of the domain, it is possible to define a spanwise wave number β = 2π /Lz . For this wave number, for the zero frequency (expected steady crossflow mode) and for the used incoming flow field Reynolds number, the parallel linear stability theory provides a critical distance from the leading edge corresponding to the neutral line. This critical abscissa is located at 13.8◦ , so that the roughness element is placed upstream of the critical position in case A and exactly at the critical abscissa in case B.

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As detailed in [2], an unexpected unsteady perturbation appears just downstream of the roughness element. The origin can be related to the forcing induced by imposing abruptly a roughness element in the steady flow field at the beginning of the second step (see section 1). Indeed, a particular type of absolute instability (with imposed spanwise periodic boundary conditions) seems to occur whose characteristics (frequency, temporal growth rate) are in very good agreement with the behavior analyzed in the DNS, see [2] for further details on this point. Figure 2 shows the

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unsteady behavior of the transverse velocity in the two considered cases at the same numerical sensor located downstream of both roughness elements. As clearly indicated by this figure, if the unsteady perturbation seems neither to grow nor to damp in case A, it is damped in case B. For that reason, the results presented hereafter correspond to case B and are obtained after some transient phase, as soon as the unsteady perturbation is supposed to be small enough.

3 Steady perturbation The steady perturbation is obtained by the following procedure. First the remaining unsteadiness is eliminated by calculating a mean value of the flow field upon several time periods. Then the basic flow is algebraically suppressed from this mean flow field. As the flow is forced to be 2π /Lz -periodic in the z direction, the steady perturbation quantities (velocity components and pressure) are decomposed into a Fourier expansion in exp(2iπ z/Lz ). From this analysis, it can be deduced that at least up to s ≃ 20◦ , only the first fondamental term has a significant amplitude. Neither uniform disturbance nor harmonics seem to exist between 14◦ and 20◦ . Consequently only the term in exp(2iπ z/Lz ) which depends on s and n will be analyzed hereafter.

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3.1 Considered shapes of the perturbation Four different shapes of the roughness element (B1 , B2 , B3 and B4 ) have been investigated. Capital letter B refers to the streamwise location B as explained above. The characteristics of the four shapes are listed in table 2, knowing that the streamwise Table 2 Varying height and spanwise extent lz of the roughness element.

h/δ0 lz /δ0

Case B1

Case B2

Case B3

Case B4

0.1 1.625

0.075 1.625

0.1 4.875

0.1 9.75

extent is the same for all : lx /δ0 = 1.625 and that we have Lz /δ0 = 13, where δ0 is the boundary layer thickness at s = 0◦ . In cases B1 and B2 the roughness element normal projection on the cylinder is thus approximately a square. Case B1 is our reference, case B2 uses a smaller height with the same spanwise extent, conversely cases B3 and B4 use the same height (as for B1 ) but a greater spanwise extent.

3.2 Steady crossflow mode Let us begin with the analysis of the obtained perturbation in the reference case B1 . Following the procedure described above, the steady perturbation is identified as the algebraic difference between the time-average value of the instantaneous flow field calculated with the roughness element and the basic flow (obtained with the smooth cylinder, without roughness element). This leads to the three steady velocity components associated to the perturbation and to the steady fluctuation of the pressure. An example is given on figure 3. The roughness element section is indicated by the black rectangle, which is not a square owing to the different scales used for the streamwise and the spanwise axes. At a short streamwise distance of the roughness element and up to about 21◦ the perturbation seems fully Lz -periodic w.r.t. z and exhibits an oscillatory behavior in s associated to an amplification in the same direction. This particular evolution looks like a standard crossflow instability wave. In order to confirm this conclusion, the computed velocity components after a Fourier transform in z are compared to the linear stability theory (LST) predictions. The latter is performed using the basic flow computed at the end of the first step. The result for s = 17.1◦ is given in figure 4. The left hand part provides the comparison with the Orr-Sommerfeld equation results, neglecting non parallel and curvature terms, whereas these terms are taken into account in the two first leading orders of the Multiple Scale Analysis. In both cases, the amplitude of the LST velocity has been adjusted in order to match the DNS prediction. A quasi perfect agreement is obtained in the second case. It proves first that the computed perturbation corresponds

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exactly to a standard crossflow instability mode except maybe in a very short region close to the roughness element. This also proves that at least in that configuration including non parallel and curvature terms induces a visible effect.

3.3 Influence of the roughness element shape As indicated in table 2, four different sizes of the roughness element have been investigated. In all cases a crossflow instability wave is obtained as shown in the previous paragraph. The amplitude (based on the streamwise velocity component) is plotted in figure 5 as function of the streamwise coordinate for each of the four considered cases. It appears clearly that except in the case B4 the roughness induced perturbations are identical, only the amplitude differs. If the amplitude is normalized

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0.12 0.1 0.08 0.06

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by the one corresponding to case B1 , the amplitude for cases B2 and B3 obtained by DNS are : 0.7 and 1.8 resp. On the other hand, assuming a linear receptivity mechanism and thus an amplitude proportional to the Fourier transform of the roughness element shape, the expected amplitudes are : 0.75 and 2.4. It may thus finally be concluded that in the considered configuration, the influence of the roughness height is nearly linear (comparison between B1 and B2 ) whereas the influence of the spanwise extent is non linear (comparison between B1 and B3 ).

References 1. E. Piot. Simulation numérique directe et analyse de stabilité de couches limites laminaires en présence de micro-rugosités. PhD, SUPAERO, Université de Toulouse, September 2008. 2. E. Piot, G. Casalis. Absolute stability mechanism of a swept cylinder laminar boundary layer with imposed spanwise periodic boundary conditions. Physics of Fluids, vol. 21, 2009. 3. M. Terracol. Développement d’un solveur pour l’aéroacoustique numrique : SABRINA. Technical Report RT 5/07383 DSNA, Rapport technique ONERA, 2003.

Experimental Study of the Incipient Spot Breakdown Controlled by Riblets V. Chernoray, G. R. Grek, V. V. Kozlov, and Y. A. Litvinenko

Abstract The evolution of the Λ -vortex and its transformation into the turbulent spot in a flat plate boundary layer is investigated experimentally. Extensive measurements are performed to visualize the Λ -structure transformation into the turbulent spot on a smooth and ribbed surface of a flat plate. The flow behavior in the course of the spatial evolution of the breakdown is highlighted and discussed. Specific features of the breakdown development on the smooth and ribbed surfaces are demonstrated. In particular the following phenomena are highlighted: the suppression by the riblets of the Λ -vortex transformation into the turbulent spot, appearance of the coherent structures of the Λ -vortex type within of ensemble-averaged turbulent spot, and oblique wave generation both by the Λ -vortex and the turbulent spot.

1 Introduction The transition from laminar to turbulent flow in a Blasius boundary layer and Poiseuille channel flow usually occurs through the formation of so-called turbulent spots. Boundary layer transition at high free stream turbulence is certainly connected to occurrence of turbulent spots as well. Investigation of turbulent spots developed at high free stream turbulence can be found in work [1]. It is well-known that a turbulent spot can be generated artificially by a strong initial disturbance in the wall shear flow without stage of the instability wave amplification even in the subcritical region. This method to generate and investigate turbulent spots was used in study [2]. In other well-known scenario turbulent spots appear due to the downstream evoluV. Chernoray Applied Mechanics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden, e-mail: [email protected] G.R. Grek, V.V. Kozlov, Yu.A. Litvinenko Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Division of the Russian Academy of Sciences, 630090 Novosibirsk, Russia P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_17, © Springer Science+Business Media B.V. 2010

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tion of Λ -structures. In study [3] a method of triggering of the phase-synchronized breakdown is described and detailed study of the breakdown process is performed. The results of work [3] were extended further in current study by using the method of space-time hot-wire visualization. This paper presents detailed experimental investigations of the control by riblets of the Λ -vortex and its transformation into the turbulent spot. Experiments were performed by using modern methods of experimental data acquisition and processing. Spatial phase-synchronized measurements made it possible to obtain the spatiotemporal patterns of the Λ -structure development on smooth and ribbed surfaces and represent them as hot-wire visualizations of the flow structure evolution in space and time.

2 Experimental Setup and Measurement Procedure Experiments were performed in a low-turbulence wind tunnel on a flat plate model. The plate was mounted and oriented streamwise in the working part of the wind tunnel. A laminar boundary layer with the Blasius mean velocity profile was developed on the plate in the region of measurements. The Λ -vortex was generated by using a blowing by a dynamic loudspeaker through a 3-mm hole located at the center of the plate 435 mm from the leading edge. The loudspeaker was driven by an electric signal with the form of rectangular pulses supplied at a repetition rate of 4 Hz. The pulses provided the generation of spatially localized perturbations. Of particular importance for this study was to produce a fully controlled disturbance for using the phase averaging technique. Thus the primary pulses were additionally modulated by a secondary high-frequency perturbation of small amplitude and frequency of 240 Hz. The phase-synchronized flow pattern until the very final stages of the Λ -structure breakdown was successfully obtained due to the presence of such modulation. A riblet insert was mounted at a distance of 25 mm from the point of the disturbance injection and represented a rectangle with dimensions 200×100 mm2 . The insert was flash-mounted in such a way that the valleys of the riblets were leveled with the plate surface. The riblet height was 0.65 mm, spacing between the neighboring peaks was 1.3 mm, and the rib top width was 0.1 mm. Flow velocity in the wind tunnel was 8.9 m/s and the turbulence level did not exceed 0.04%. A single-wire hot-wire probe was used to measure the streamwise velocity component. The probe was equipped with a gold-plated tungsten wire of 1.2 mm active length and 5 µ m in diameter. All measurements were performed in an automated mode with use of a traversing system moving the probe in three dimensions by specially developed software. In different measurement sets about 15000 to 20000 spatial points were mapped. Measured voltage traces were converted into velocity traces and then phase averaged over 100 disturbance periods. By subtracting the base undisturbed flow the spatial-temporal distribution of the disturbance velocity was obtained.

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3 Results and Discussion The study of the flow structure in the boundary layer on the smooth surface of the flat plate showed that the flow is laminar and the velocity profile is close to the Blasius velocity profile. Patterns of generated perturbations at the point of the beginning of measurements (40 mm from the source and 15 mm from the riblet insert) on both smooth and ribbed surfaces were carefully analyzed and it was found that the amplitude and shape of the initial disturbance was same in both cases. This means that at the initial stage of the disturbance development the riblets did not produce any noticeable effect on the flow structure. Figure 1 demonstrates isosurfaces of the streamwise velocity fluctuations in a three-dimensional space for the ensemble-averaged Λ -structure and turbulent spot on a smooth flat plate surface. It is clearly seen that the Λ -structure generated initially (at x=100 mm) transforms downstream into a complex pattern consisting of a number of coherent structures at x=200 mm. These coherent structures are very similar to Λ -structures presenting determined large-scale formations of the turbulent spot. The similar conclusion that the turbulent spot contains numerous hairpin eddies arranged in an arrowhead formation has been made in work [5]. Comparison of the disturbance evolution on the smooth and ribbed flat plate surfaces at two downstream positions is shown in Fig. 2. This figure shows threedimensional visualizations of the isosurfaces of the streamwise velocity fluctuations. In the figure the results of measurements at various downstream positions are assembled together. One can clearly see that at initial station, x=40 mm, the structure and the amplitude of disturbances in both cases is essentially same. However, at development downstream, the Λ -structure on smooth surface transforms into a complex pattern characterized by a large number of regions of positive and negative velocity fluctuation and by a progressive increase of the streamwise and spanwise perturbation scales. Eventually, the Λ -structure gradually transforms into a solitary turbulent spot observed at x=175 mm. It is also necessary to note a longitudinal stretching of the Λ -structure (x=75 mm) which is typical when the Λ -vortex transforms into a socalled hairpin vortex in a shear flow. In contrast to this pattern, the structure of the Λ -vortex moving downstream on the riblet surface exhibits rather small variations. The spanwise and streamwise scales change insignificantly and finally at x=175 mm

Fig. 1 Hot-wire visualization of the ensemble-averaged Λ -structure and turbulent spot on smooth surface. The amplitude levels of isosurfaces is 2.5% of freestream velocity. The dark and the light isosurfaces correspondingly used for positive and negative values.

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Fig. 2 Hot-wire visualizations of the Λ -structure evolution on the smooth and ribbed surfaces at x = 40, 75 and 175 mm.

it is seen that the Λ -structure on the ribbed surface the disturbance amplitude is essentially reduced. Figure 3 shows the dependencies of the intensity of perturbations developed on smooth and ribbed surfaces against the longitudinal coordinate. Peak-to-peak amplitude is shown. One can observe that from same initial value of 19% at x=40 mm the amplitude of perturbations on the smooth surface sharply increases up to 47% at x=200 mm. The decay which is typical for turbulent breakdown is observed further downstream. In the presence of riblets the growth of the perturbation amplitude is stopped at about 31% of freestream velocity at x=140 mm and after this position the disturbance amplitude falls at about the same rate until it reaches the initial value at the end of the riblet insert (19% at x=210 mm). Thus, the riblets are responsible in decrease of the perturbation intensity more than two times, thereby stabilizing the flow. Overall the results of measurements showed that the riblets provide an effective opportunity of controlling the development of the Λ -structure breakdown to turbulence. On a higher level of investigations, with computer controlled acquisition, processing, and spatiotemporal representation of experimental data, obtained results confirmed the conclusion made in work [4] about the stabilizing effect of riblets on the development of non-stationary vortex structures. The quantitative information

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Fig. 3 Streamwise variation of disturbance amplitudes: (1) smooth surface,(2) riblet surface.

about the mechanism of suppression of the process of Λ -structure transformation into the turbulent spot on a riblet surface is obtained. Thus, the hypothesis concerning the stabilizing effect of riblets on the development of coherent structures in a viscous sublayer of the turbulent boundary layer is once more confirmed in experiments on the controlled development of coherent structures of the laminar-turbulent transition in the boundary layer.

4 Conclusions In current experiments we visualize the development of incipient spot into a turbulent spot under controlled experimental conditions and demonstrate that the riblets can be used to suppress the flow breakdown in this case. It is found that the riblets prevent the transformation of the Λ -structure into the turbulent spot and lead to the decay of this perturbation. It is found that the intensity of the incipient spot on riblet surface initially increases and then decays to a half of the level observed in the case of perturbation development on a smooth surface.

References 1. Alfredsson, P.H., Bakchinov, A.A., Kozlov, V.V., Matsubara, M.: Laminar-turbulent transition at high level of free stream turbulence, Proc. of the IUTAM Symposium on Nonlinear Instability and Transition in Three-Dimensional Boundary layer (Eds. P. W. Duck and P.H. Hall), Kluwer Academic Publishers, 423–436 (1996). 2. Gad-El-Hak, M., Blackwelder R.F., Riley J.J.: On the growth of turbulent regions in laminar boundary layers, J. Fluid Mech. 110, 73–95 (1981).

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3. Grek, G.R., Katasonov, M.M., Kozlov, V.V., Chernoray, V.: Experimental study of mechanism of high-frequency breakdown of Λ -structure, Thermophysics and Aeromechanics, 6 (4) 427– 442 (1999). 4. Grek, G.R., Kozlov, V.V., Titarenko, S.V.: An experimental study on the influence of riblets on transition, J. Fluid Mech., 315, 31–49 (1996). 5. Wygnanski, I., Haritonidis, J.H. and Zilberman H., On the spreading of a turbulent spot in the absence of a pressure gradient, J. Fluid Mech., 123, 69–90 (1982).

Control of Stationary Cross-flow Modes Using Patterned Roughness at Mach 3.5 Thomas Corke, Eric Matlis, Chan-Yong Schuele, Stephen Wilkinson, Lewis Owens, and P. Balakumar

Abstract Spanwise-periodic roughness designed to excite selected wave lengths of stationary cross-flow modes was investigated in a 3-D boundary layer at Mach 3.5. The test model was a sharp-tipped 14◦ right-circular cone. The model and integrated sensor traversing system were placed in the Mach 3.5 Supersonic Low Disturbance Tunnel (SLDT) equipped with a “quiet design” nozzle at NASA Langley RC. The model was oriented at a 4.2◦ angle of attack to produce a mean cross-flow velocity component in the boundary layer over the cone. Three removable cone tips have been investigated. One has a smooth surface that is used to document the baseline (“natural”) conditions. The other two have minute “dimples” that are equally spaced around the circumference, at a streamwise location that is just upstream of the linear stability neutral growth branch for cross flow modes. The azimuthal mode numbers of the dimpled tips were selected to either enhance the most amplified wave numbers or to suppress the growth of the most amplified wave numbers. The results indicate that the stationary cross-flow modes were highly receptive to the patterned roughness.

Thomas Corke University of Notre Dame, e-mail: [email protected] Eric Matlis University of Notre Dame e-mail: [email protected] Chan-Yong Schuele University of Notre Dame e-mail: [email protected] Stephen Wilkinson NASA Langley Research Center e-mail: [email protected] Lewis Owens NASA Langley Research Center e-mail: [email protected] P. Balakumar NASA Langley Research Center e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_18, © Springer Science+Business Media B.V. 2010

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1 Experimental Approach The objective of the research is to excite stationary cross-flow modes that are outside of the most amplified band of wave numbers, but could amplify sufficiently to modify the mean flow, and thereby suppress the more amplified cross-flow modes, and thereby extend the transition Reynolds number. This follows the approach found to be effective in subsonic swept-wing experiments. [1, 2, 3]. The model used in our experiments was a sharp-tipped circular cone with a halfangle of φc = 7◦ that was previously used in transition studies[4, 5]. The length of the cone is 35.56 cm. The model was placed inside the 0.5 m Mach 3.5 axisymmetric slow expansion nozzle at an angle of attack of 4.2◦ . It was positioned so that a majority (86%) of it was inside the “quiet” zone. The cone was supported by a sting which also held a motorized 3-D traversing mechanism that was used to make velocity measurements in the boundary layer. Schlieren flow visualization indicated that there were no reflected shock waves emanating from the nozzle side walls to impinge on the model and affect the boundary layer. A detailed linear stability analysis of the cross-flow instability for the conditions of the experiment was performed. This was used to determine the range of most amplified azimuthal mode numbers (m), and the location of the first growth of the modes (xI ). For the stationary cross flow modes, the most amplified band was near m = 50, and xI was 1.27 cm (0.5 in.) from the cone tip. Based on this, we chose patterned roughness at two azimuthal wave numbers of m = 45 and 68. The former is close to the band of most amplified wave numbers. The latter is 1.5 times higher and not a harmonic of the most amplified stationary mode, which are the conditions recommended to suppress transition by Saric and Reed[3]. The patterned roughness was located on removable tips of the cone. The roughness consisted of minute indentations (“dimples”) that were evenly spaced around the circumference at xI . We chose to use dimples rather than bumps because we thought that they could be produced with better uniformity. In a private communication, Saric[6] indicated that bumps and dimples worked about the same. He also indicated that the ratio of the diameter to wavelength (d/λ ) should be greater than 0.4 to be most effective. A smooth tip was also used to establish the baseline condition. Regular and magnified images of the m = 68 tip are shown in Figure 1. Surface profile measurements indicate that the width of the holes including the rim, were approximately 200µ m (0.0079in.) . The depth was approximately 86µ m (0.0034in). The ratio of the diameter to the wavelength (d/λ ) was then 0.43.

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Fig. 1 Photographs of removable cone tip with m = 68 “dimples: located 0.5 in. from the tip. Right photograph is a magnified view of the “dimples”.

2 Results 2.1 Flow Visualization Surface flow visualization was first used to obtain a global indication of the state of the stationary cross-flow modes for the different passive roughness cases. This used a mixture of 15 parts 1000cSt Silicon oil, 5 parts Oleic acid and 1 part TitaniumDioxide powder. An example of the surface flow visualization for the baseline smooth tip is shown on the left part of Figure 2. The presumption is that the corotating vortices that develop through the stationary cross-flow instability will cause the surface visualization material to reorganize into light and dark bands, with the light bands being an accumulation of the surface marker, and the dark bands being a depletion of the surface marker.Such bands are clearly visible. The flow visualization images were analyzed to extract azimuthal mode number information. A sample of the image spectral analysis is presented in the right part of Figure 2. This corresponds to the flow visualization image shown in the figure for the baseline smooth tip. The brighter colors indicate the highest amplitudes. The spectral analysis of the baseline image indicated that the dominant azimuthal mode number in the image was near the linear theory most amplified value, m ≃ 50.

Fig. 2 Photograph of surface flow visualization for baseline (smooth) tip (left) and corresponding spectral analysis of the image (right).

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2.2 Mean Velocity Measurements Velocity measurements were obtained using a miniature Pitot probe that was attached to the 3-D traversing mechanism. Our intention was to measure changes in the mean flow that would be indicative of the mean flow distortion produced by the stationary cross-flow modes. This would then further quantify the results from the surface flow visualization. An indication of the degree of control that the “dimples” provided is shown in Figure 3. This shows a comparison of the repeatability of the azimuthal surveys obtained with the Pitot probe for the baseline (smooth) tip (left) and the m = 68 dimpled tip (right). For these cases, the flow in the wind tunnel was stopped and re-started between the measurement surveys. For the baseline condition, both the amplitudes and locations of the mean flow distortion associated with the stationary cross-flow mode changed between runs. This was undoubtedly due to uncontrolled experimental conditions possibly produced by minute dust particles on the model. In contrast, with the m = 68 dimpled tip, the amplitudes and locations of the mean flow distortion were perfectly re-producable. This indicates a high degree of receptivity of the cross-flow instability to the controlled roughness condition.

Fig. 3 Multiple measurements of mean flow distortion from cross-flow instability for baseline natural roughness (left), and with m = 68 roughness (right).

Azimuthal surveys like those in Figure 3 were obtained at a fixed height above the cone surface at different streamwise locations. These surveys indicated that the periodic mean flow distortion first appeared at larger azimuthal positions, closer to the windward side of the cone. These progressed towards the top (lee side) of the cone with increasing downstream distance. Wavelet analysis of the azimuthal surveys was used to display the azimuthal variation in the amplitude of mode numbers for the different patterned roughness. Examples at the furthest downstream position that we could measure on the cone (x = 28.4 cm) for the baseline and the m = 68 roughness are shown in Figure 4. These reveal the dominant mode numbers, and the azimuthal angles at which they concentrate. For the baseline (smooth tip) in the left part of Figure 4, the dominant

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mode number is near m = 50, which was predicted from linear theory and also observed in the flow visualization. This appears closer to the leeward side of the cone at θ ≃ 30◦ .

Fig. 4 Wavelet analysis of mean flow distortion at x = 28.4 cm for baseline natural roughness (left), and with m = 68 roughness (right).

The wavelet analysis for the m = 68 tip is shown in the right part of Figure 4. This shows that the azimuthal location of the dominant mode had moved away from the leeward side of the cone to be centered at θ ≃ 70◦ . Linear theory N-factor calculations for the base flow have N-factor magnitudes that progressively increase in the azimuthal direction towards the lee side of the cone. Therefore we expect transition to turbulence to occur closer to the leeward side of the cone, at smaller azimuthal angles (theta). This is consistent with the wavelet analysis for the baseline case. The wavelet analysis for the m = 68 case indicates that the dominant mode is appearing at larger azimuthal angles on the cone where, if the linear analysis is still valid, the N-factors are less. This suggests that the transition to turbulence might be retarded. However the proof of this will require more detailed measurements that are planned for the next wind tunnel entry of the experiment.

3 Conclusions The first steps in this experimental program were performed to demonstrate that the cross-flow instability at supersonic Mach numbers could be controlled with patterned roughness. The circular cone at an angle of attack proved to be an excellent geometry for this experiment. The surface flow visualization was extremely effective in marking the presence of the stationary cross-flow modes. Image analysis provided a means of obtaining quantitative results of the cross-flow instability azimuthal mode numbers. The Pitot probe surveys provided another quantitative check of the azimuthal mode numbers of stationary cross-flow modes. In addition, they added information on their streamwise development. Based on these two measurement approaches, we have concluded that the stationary cross-flow mode was receptive to the passive patterned roughness. This indicates

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that control of transition to turbulence in stationary cross-flow dominated conditions should be possible. The next phase of this research will involve replacing the passive roughness with active “plasma bumps”. The location and spacing between the “plasma bumps” will follow those used with the passive roughness that was shown to be effective. As with the passive roughness, the effect of the “plasma bumps” will be documented initially through surface flow visualization and Pitot probe surveys. These will be processed in similar fashion to quantify that the desired azimuthal wave number stationary modes are produced. This will then be applied to determine their effect on controlling the transition Reynolds number. Acknowledgements This work was supported under the NASA NRA/Research Opportunities in Aeronautics Program Cooperative Agreement NNX08AB22A.

References 1. Reibert, M. Saric, W., Carrillo, R. and Chapman, K. 1996. Experiments in nonlinear saturation of stationary cross-flow vortices in a swept-wing boundary layer. AIAA Paper 96-0184. 2. Radeztsky, R., Reibert, M. and Saric, W. 1999. Effect of isolated micron size roughness on transition on swept-wing flows. AIAA J., 37, 11, pp. 1370-77. 3. Saric, W. and Reed, H. 2004. Toward practical laminar flow control - remaining challenges. AIAA Paper 2004-2311. 4. Corke, T. C., Cavalieri, D. A. & Matlis, E. 2002. Boundary layer instability on a sharp cone at Mach 3.5 with controlled input. AIAA J. 40, 5, 1015, 2002. 5. Matlis, E. 2003. “Controlled Experiments on Instabilities and Transition to Turbulence on a Sharp Cone at Mach 3.5” Ph.D. Thesis, University of Notre Dame. 6. Saric, W. 2008. Private communication.

Secondary optimal growth and subcritical transition in the plane Poiseuille flow Carlo Cossu, Mattias Chevalier, and Dan S. Henningson

Abstract Nonlinear optimal perturbations leading to subcritical transition with minimum threshold energy are searched in the plane Poiseuille flow at Re = 1500. To this end we proceed in two steps. First a family of optimally growing primary streaks U issued by the optimal vortices of the Poiseuille laminar solution is computed by direct numerical simulation for a set of finite amplitudes AI of the primary vortices. An adjoint technique is then used to compute the maximum growth and the finite time Lyapunov exponents of secondary perturbations growing on top of these primary base flows. The secondary optimals take into full account the non-normality and the local instabilities of the tangent operator all along the temporal evolution of the primary flows. The most amplified optimal perturbations are sinuous and realized in correspondence of streaks that are locally unstable. The combinations of primary and secondary perturbations optimal for transition are then explored using direct numerical simulations. It is shown that the minimum initial energy is realized by a large set of these combinations, revealing new paths to transition. Surprisingly we find that transition can be efficiently obtained even using secondary perturbations alone, in the absence of primary optimal vortices.

1 Introduction and background We are interested in finding the initial conditions having minimum energy to undergo subcritical transition in the plane Poiseuille flow at Re = 1500. In particular, Carlo Cossu ´ Laboratoire d’Hydrodynamique (LadHyX), CNRS-Ecole Polytechnique, Palaiseau FRANCE email: [email protected] Mattias Chevalier The Swedish Defence Research Agency (FOI), Stockholm, SWEDEN Dan S. Henninsgon Linné Flow Center, KTH Mechanics, Stockholm, SWEDEN P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_19, © Springer Science+Business Media B.V. 2010

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we look for ways to approach the nonlinear optimal initial conditions for transition [1] that can be applied to the Navier-Stokes equations. Reddy et al. [4] found that one of the most efficient ways to get outside the basin of attraction of the lami(opt) nar state is to use initial conditions of the type U0 + AI uI + AII uII where U0 is (opt) the Poiseuille flow, uI are the optimal vortices maximizing the transient energy growth on U0 , and uII is noise of amplitude AII = 0.1 AI . In this transition scenario (called SV) the initial vortices, of finite amplitude AI , evolve into large amplitude streamwise streaks by lift-up effect. These streaks are prone to sinuous secondary instability if their amplitude is large enough. Reddy et al.[4] find that transition is obtained if the streaks are sufficiently unstable. From the mathematical point of view, their analysis consists in the computation of a set of nonlinear unsteady solutions U of the full Navier-Stokes equations parametrized by the initial amplitude of the optimal vortices AI . The local (in time) stability of these solutions is then analyzed by computing the eigenvalues of the local tangent operator L (U, Re), i.e. of the Navier-Stokes equations linearized in correspondence to the actual velocity profiles U. In [4] it is then verified that transition is observed when the eigenvalues of the local tangent operator L (U, Re) are sufficiently unstable for a sufficiently long time. Two findings, however, suggest that even lower energies for transition might be attained. First, the subcritical nature of the instability of the streaks has been revealed by Waleffe [7, 8] who has been able to follow subcritical branches originating from unstable eigenvalues and modes of L (U, Re) in the forced problem. Secondly, Schoppa & Hussain [5] have argued that that non-normal transient growths should be considered not only for the laminar base flow but also on top of already developed streaks. Hoepffner & Brandt [3] have then computed the optimal transient growth sustained by frozen streaks in the Blasius boundary layer. From the mathematical point of view, in [3] the norm of the fictitious propagators P[τ ,0] = eτ L (t) built on the tangent operator frozen at t is computed for selected t along selected trajectories U. This kind of analysis takes into account the local (in time) non-normality of the system but this approach can be questioned because the times on which the secondary transient growths develop is of the same order of the time on which the primary streaky base flow itself evolves. Here we take a different approach based on the computation of the norm of the global (in time) linear propagator of the perturbations. Given the generally unsteady trajectory U, one can integrate in time the Navier-Stokes equations linearized near U. The perturbed field u′ at time t is related to the initial perturbation u′0 by the propagator P[t,0] (U, Re). The propagator is a linear operator which is function of the whole base flow history from 0 to t. The locally tangent operator to the propagator is always L . In particular, if U is unsteady, the propagator is not simply the exponential of L because L depends on time. The optimal growth is defined in the standard way as G(t) = supu′ ku′ k/ku′0 k. The (maximum) finite time Lya0 punov exponent is defined as λ (t) = [ln(G(t)]/t and the usual Lyapunov exponent is Λ = limt→∞ λ (t). All these measures take into account the global (in time) dynamic near the reference trajectory U. In particular if that trajectory remains near the laminar fixed point, then probably G(t), λ (t) and the associated optimal pertur-

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bations will mainly reflect the non-normality of L linearized near the laminar state. If, on the contrary, the reference U gets close to an unstable saddle point and stays there for a long time, then G(t) and λ (t) will reflect the unstable dynamics of L linearized near the saddle point. In general a mix of all these types of effects is taken into account in G(t) and λ (t).

2 Results and discussion In the same spirit of Reddy et al. [4] we consider (primary) base flows U(x,t, AI ) (opt) (opt) issued from the initial conditions U0 + AI uI , where uI are the optimal perturbations of the laminar Poiseuille flow with the optimal treamwise and spanwise wavenumbers α = 0 (streamwise uniform perturbations) and β h = 2 (where h is the channel half-width). The base flows U(x,t, AI ) are computed by integrating the fully nonlinear Navier-Stokes equations with typical resolutions of 65 × 32 points in the y − z cross-stream plane. The computation of G(t) and the associated optimal † † perturbations is based on power iterations applied to P[t,0] P[t,0] , where P[t,0] is the adjoint propagator. Each iteration requires to integrate the linearized equations forward in time and then integrate backward in time the adjoint equations (more details can be found in Ref. [2]). Typically few iterations (10 to 15) are needed for convergence. When AI is small, the maximum growths supt G(t) are obtained by streamwise uniform perturbations that are very similar to the optimal perturbations of the laminar flow U0 . The growths are also comparable to the ones of the laminar solution. For these small AI therefore essentially the same physical mechanism important to the laminar flow is at play (the lift-up effect) and therefore no new directions in phase space are given by the corresponding optimal perturbations. For larger values of AI , however, the maximum growth is reached by three-dimensional perturbations (α 6= 0). These perturbations realize the best compromise by initial nonnormal growth and successive local instability along the temporal evolution of the primary base flow. The secondary optimals having the largest growths correspond to the streamwise wavenumber α h = 0.4 and have the spanwise wavenumber β h = 2 enforced by the basic flow. We investigate the relevance of these secondary perturbations by looking at the combinations of primary and secondary optimal perturbations that is most efficient in inducing transition. Transition thresholds are sought using initial perturbations in (opt) (opt) the subspace U0 + AI uI + AII uII . A bisection method based on direct numerical simulations of the fully nonlinear Navier-Stokes equations is used to determine the threshold value AII for each considered AI . The DNS are performed in a domain with dimensions 5π h × 2h × π h (corresponding to β h = 2 and α h = 0.4) discretized with 32 × 65 × 32 collocation points in respectively the streamwise, wall-normal and spanwise directions.

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When uII is used instead of noise as initial secondary perturbation, lower transition thresholds are obtained, as shown in Figure 1. The optimal secondary perturbations therefore effectively optimize the ‘classical’ SV scenario. Above the SV threshold level of AI , very small values of AII are sufficient to induce transition, in accordance with the idea of the secondary instability of the streaks. However, transition can be still obtained for values of AI below the SV-threshold, and therefore for locally stable primary streaks, but now for larger threshold values of AII . This is in accordance with the idea that the streak instability is itself subcritical [7, 8]. A surprising result is that transition can be obtained even in the extreme limit of no primary perturbations (AI = 0) with a finite threshold level for AII . In other words transition can be obtained even in the absence of initial primary vortices with only secondary perturbations having the same energy required for transition in the SV scenario. This suggests that the subcritical nature of the streaks instability is at least as important as the mechanism of streaks transient growth (STG). In the STG scenario [5] a secondary transient growth riding on top of the primary growth of the streaks is invoked in order to explain transition for low amplitudes AI where the streaks are locally (in time) stable. However, if this explanation applies for moderate AI , the STG scenario is actually unable to explain our observation that transition can be found even in the complete absence of primary streaks growth (when AI = 0).

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Fig. 3 Initial critical conditions (top row) for respectively AI = 0.01 (left), AI = 0.0075 (center) and AI = 0.0025 (right) and the corresponding critical AII . The corresponding solutions at the time of maximum deviation from the Poiseuille flow are reported in the bottom row. Only the bottom half of the channel is considered. In green is reported the surface where the streamwise velocity is 40% of its maximum value in the whole channel. In respectively blue and red are reported the surfaces where the streamwise vorticity is ±55% of its maximum value.

The amplitude ku′ k(t) curves for transitioning and relaminarizing perturbations are reported in Figure 2 for respectively AI = 0.01, AI = 0.0075 and AI = 0.0025. These three cases respectively correspond to a standard SV-type scenario (full instability of the streaks), a subcritical SV-scenario, or STG scenario, where the streaks go to transition with a sufficiently large sinuous perturbation, and a wildly subcritical case, where the streaks of the base flow are well stable along all their evolution. For all cases, just before separation of the relaminarizing and transitioning perturbations, the trajectory of the system in the phase space remains on the hyper-surface limiting the basin of attraction of the laminar state referred to as the ‘edge of chaos’ [6]. A sketch of the critical initial conditions and of their evolution at the first peak of rms-energy is given in Figure 3 for the same three cases. From this figure it is apparent that different threshold combinations of AI and AII result in the exploration of different regions of the edge hyperplane. At the time of maximum growth (see Fig. 2) in particular both the amplitude of the streaks and their sinuous bending differ, depending on the initial condition that is used. A confirmation of this is obtained by redrawing the results (Figure 4) in the pseudo-phase-space given by the streamwise (u′ ) and spanwise (w′ ) rms components of u′ . These two components give a rough measure of respectively the amplitude of the streaks and the amplitude of their sinuous bending. In the SV-type case (AI = 0.01 left plot), the streamwise component ku′ k is the first to grow, followed by the growth of three-dimensional perturbations (kw′ k). In this case the relaminarizing and transitioning trajectories stay close up to a first point where they ‘loop’, then they jointly visit a second point where they then separatevery fast. These two points probably corresponds to saddles in the phase space. A different path is followed in the SV-STG case (AI = 0.0075, center plot), where kw′ k begins to grow before large values of ku′ k are attained, in accordance with the STG scenario and with the idea that the instability of the streaks is subcritical. Trajectories separate at different points in the phase space. A very different path is followed in the wildly subcritical case (AI = 0.0025, right plot), where kw′ k grows fast and early to then decrease before separating at a point very different from the previous ones. In this last case the landing on the edge hyperplane happens at a very different region.

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These results indicate that in addition to the SV and related scenarios where the most dangerous perturbations are streamwise vortices, other paths to transition exists which require at least as low initial energy as the SV scenario for transition. Future research dedicated to control of subcritical transition should therefore consider these additional paths and not only the standard streamwise optimal vortices.

References 1. Cossu, C.: An optimality condition on the minimum energy threshold in subcritical instabilities. C. R. Mécanique 333, 331–336 (2005) 2. Cossu, C., Chevalier, M., Henningson, D.: Optimal secondary energy growth in a plane channel flow. Phys. Fluids 19, 058,107 (2007) 3. Hœpffner, J., Brandt, L., Henningson, D.S.: Transient growth on boundary layer streaks. J. Fluid Mech. 537, 91–100 (2005) 4. Reddy, S.C., Schmid, P.J., Baggett, J.S., Henningson, D.S.: On the stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech. 365, 269–303 (1998) 5. Schoppa, W., Hussain, F.: Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108 (2002) 6. Skufca, J., Yorke, J., Eckhardt, B.: Edge of Chaos in a Parallel Shear Flow. Phys. Rev. Lett. 96(17), 174,101 (2006) 7. Waleffe, F.: Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140– 4143 (1998) 8. Waleffe, F.: Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102 (2001)

Disturbance evolution in rotating-disk boundary layers: competition between absolute instability and global stability Christopher Davies and Christian Thomas

Abstract Results obtained from the numerical simulation of linearized disturbance evolution in various rotating disk boundary-layers can be modelled using impulse solutions of the Ginzburg-Landau equation. An explanation may thus be given of why there appears to be no unstable linear global mode for the von Kármán boundary layer, even though it is subject to a strong form of absolute instability. The stability effects of applying suction at the disk surface, or imposing an axial magnetic field, have also been investigated. In both of these cases, numerical simulation results indicated that local stabilization, which had previously been predicted to lead to a postponement of absolute instability to higher Reynolds numbers, could in fact be associated with the introduction of a new form of global instability. The modelling approach, based on comparisons with solutions of the Ginzburg-Landau equation, provides some insight into how such behaviour can arise.

1 Introduction The von Kármán rotating disk boundary-layer can be shown to be absolutely unstable, using an analysis that deploys the usual ‘parallel-flow’ approximation, where the base flow is simplified by taking it to be homogeneous along the radial direction [1]. But for the genuine radially inhomogeneous base flow, numerical simulations [2] show that the absolute instability does not give rise to any unstable linear global mode, in full agreement with the behaviour found in recently conducted physical experiments [3]. This is despite the fact that the temporal growth rates for the absoChristopher Davies School of Mathematics, Cardiff University, Cardiff, United Kingdom, e-mail: [email protected] Christian Thomas School of Mathematics and Statistics, University of Western Australia, Perth, Australia, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_20, © Springer Science+Business Media B.V. 2010

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lute instability display a marked increase with the radial distance from the rotation axis [4].

2 Modelling of numerical simulation results The apparent disparity between the radially increasing strength of an absolute instability and the absence of any global instability can be understood by considering analogous behaviour in impulse solutions of the linearized complex GinzburgLandau equation [5]. Such an approach was first utilized by Davies, Thomas and Carpenter [4] to provide a theoretical explanation of the numerical simulation results obtained for the von Kármán rotating disk boundary layer. In the present study, we consider an extension to cases where there is mass transfer at the disk surface or an applied axial magnetic field. By studying the parametric dependence that arises when the mean flow within the rotating boundary layer is varied in a systematic fashion, we can expose a broader range of possible behaviour, thus affording more scope for making sense of the balances that can persist between local and global instability. The linearized Ginzburg-Landau equation may be written in the form:

∂ψ ∂ψ ∂ 2ψ +U = µψ + γ 2 , ∂t ∂r ∂r

(1)

where ψ (r,t) is some measure of the disturbance amplitude at the radial location r and time t. The terms multiplied by the quantities µ , U and γ (where Re(γ ) > 0) are used, respectively, to model the effects of temporal growth/oscillation, flow convection and diffusion/dispersion. Usually, all three of these quantities are taken to be constant, in order to provide a model for the development of disturbances in a spatially homogeneous flow. But by setting µ = µ (r) we can obtain a simple, if somewhat crude, model for disturbances that evolve in a flow that is spatially inhomogeneous. At any given location, the real part of the derivative d µ /dr can be interpreted as a local measure of the radial variation in the temporal stability properties of the flow. The imaginary part of d µ /dr quantifies the corresponding spatial dependency in the temporal frequencies at which disturbances can be excited. The simplest non-trivial radial variation in µ that may be considered is the case where d µ /dr is prescribed to be a constant µ1 , so that we have the linear dependency

µ = µ0 + µ1 r ,

(2)

with µ0 another constant. For this form of µ , the Greens function G(r,t) solution of the Ginzburg-Landau equation (1) can be shown to be [5]: 1 (r −Ut)2 1 1 G(r,t) ∼ √ exp µ0t − + µ1 rt + µ12 γ t 3 . (3) 4γ t 2 12 t

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This represents the response to a localized impulse ∼ δ (t) δ (r). (For convenience, we consider an impulse centred at r = 0. The impulse location may be translated to r = re when comparisons are made with numerical simulation results.) Identifying the instantaneous complex growth rate of a disturbance with the logarithmic derivative 1 ∂G ρ= , (4) G ∂t it may easily be seen that, for large times, the impulsively excited disturbance grows at a rate 1 2 Re(ρ ) ∼ (µ1r − µ1i2 )γr − 2 µ1r µ1i γi t 2 , (5) 4 where µ1r and µ1i denote the real and imaginary parts of µ1 , respectively, and similarly for γ . Using the fact that γr > 0, it may be deduced that when the measure of the frequency variation µ1i is large enough in magnitude, we can get a globally stable response even if µ1r is positive, which would correspond to a flow that becomes locally more and more unstable as the radius increases. Thus the behaviour of the impulse solution (3) indicates how detuning, arising from the spatial variation of the temporal frequency of an absolute instability, may be sufficient to globally stabilize disturbances. It suggests that it is possible for an absolutely unstable rotating-disk boundary layer flow to remain globally stable, depending on the balance that exists between radial increases in growth rates and corresponding shifts in frequencies. But there is another intriguing possibility. If the combination of parameters enclosed in the square brackets in equation (5) happens to be positive, then a novel form of global instability may arise, with a positive growth rate ∝ t 2 for the impulse response at large times. Before we move on to discuss the detection of such an instability in our numerical simulations, it should also be noted that the large time behaviour of the imaginary part of the logarithmic growth rate: Im(ρ ) ∼

1 2 (µ1r − µ1i2 )γi + 2µ1r µ1i γr t 2 , 4

(6)

implies that, in general when µ1r and µ1i are non-vanishing, there is no selection of any dominant temporal frequency in the impulse response.

3 Simulation results for cases with mass transfer Globally stability, associated with convectively unstable disturbance development, was found to be prevalent in numerical simulations that were conducted for cases where the von Kármán boundary layer was modified by the introduction of the mass injection at the rotating disk surface. Thus there was no qualitative alteration from the behaviour obtained in the no mass flow numerical simulations that were reported in [2] and then modelled in [4] using Ginzburg-Landau impulse solutions. More interestingly, it was discovered that globally unstable behaviour could be promoted when suction was applied. This contrasts with the stabilizing influence

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of suction that had previously been identified using the approximation of a radially homogenized base flow [6]. For the inhomogeneous flow, impulsively excited disturbances were found to display an increasingly rapid temporal growth around the radial position of the impulse, albeit without any selection of a dominant frequency, as would be more usually expected for an unstable global mode. The occurrence of this kind of novel behaviour is consistent with the range of possibilities that may be delimited by considering the idealized impulse solutions (3) of the Ginburg-Landau equation. Moreover, it turns out that these relatively simple solutions can be matched with the numerical simulation results to a surprising degree of accuracy, given the apparent crudeness of the modelling that led to them. (More details are given in [4] and [7]. In essence, the constants µ0 , U and γ were fixed using numerical simulation

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results obtained for the artificially radially homogenized flow. The qualitatively different form of behaviour that was found for the genuine inhomogeneous flow was then captured by selecting just one additional complex constant, µ1 .) An example of the close matching that can be obtained is given in Figures 1 and 2, which display the evolution with time of locally determined temporal frequencies and growth rates for a disturbance that is excited near to the radial position associated with the critical point of absolute instability. Numerical simulation results are shown together with corresponding results that were derived using impulse solutions of the GinzburgLandau equation. A similar form of global destabilization was traced in the results obtained from numerical simulations where a uniform axial magnetic field was applied to the radially inhomogeneous rotating-disk flow. Again, global effects overcame the locally stabilizing effects that had been found in a previous study for the homogenized version of the flow [8].

4 Summary We have briefly described how impulse solutions of the Ginzburg-Landau equation can be used to model the behaviour of absolutely unstable disturbances developing in rotating disk boundary layers. In particular, we have discussed how radial variations in the character of the absolute instability can lead to either a stabilization of the global response to an impulse (in the absence of suction or an axial magnetic field, and in the presence of mass injection) or to a new form of global instability (in the presence of suction or an axial magnetic field).

References 1. R. J. Lingwood. Absolute instability of the boundary layer on a rotating disk. J. Fluid Mech., 299:17–33, 1995. 2. C. Davies and P. W. Carpenter. Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer. J. Fluid Mech., 486:287–329, 2003. 3. H. Othman and T. C. Corke. Experimental investigation of absolute instability of a rotating-disk bounday layer. J. Fluid Mech., 565:63–94, 2006. 4. C. Davies, C. Thomas and P. W. Carpenter. Global stability of the rotating-disk boundary layer. J. Eng. Math., 57:219–236, 2007. 5. R. E. Hunt and D. G. Crighton. Instability of flows in spatially developing media. Proc. R. Soc. Lond. A, 435:109–128. 1991. 6. R. J. Lingwood. On the effects of suction and injection on the absolute instability of the rotatingdisk boundary layer. Phys. Fluids, 9:1317–1328, 1997. 7. C. Thomas. Numerical simulations of disturbance development in rotating boundary-layers. Ph.D. Thesis, Cardiff University, 2007. 8. H. A. Jasmine and J. S. B. Gajjar. Convective and absolute instability in the incompressible boundary layer on a rotating disk in the presence of a uniform magnetic field. J. Eng. Math., 52:337–353, 2005.

Instabilities due a vortex at a density interface: gravitational and centrifugal effects Harish N. Dixit and Rama Govindarajan

Abstract A vortex placed at an initially straight density interface winds it into an ever-tightening spiral. This flow then displays rich dynamics, due to inertial effects caused by density stratification (non-Boussinesq effects), and gravitational effects. In the absence of gravity we showed recently that the flow is subject to centrifugal Rayleigh-Taylor and spiral Kelvin-Helmholtz instabilities. The latter grows slightly faster than exponentially. In this paper we present computations including gravity with and without and with inertial effects. Gravity modifies the spiralling process and contributes to the breakdown of the vortex. When both effects are allowed to operate together, the resulting flow has a complex radial character, with small-scale structures near the vortex core attributed to non-Boussinesq effects, and large scale roll-up due to gravity followed by breakdown.

1 Introduction Vortical structures in stratified flows display a range of interesting instabilities and non-monotonic behaviour, see e.g. [1] and [2]. The present study is twodimensional, of a lone vortex with its axis perpendicular to the plane of density stratification, with and without gravity. An initially flat density interface is wound up into an increasingly tightened spiral by the vortex, similar to how it would advect a patch of passive scalar (see for example [5]). In the absence of gravity, centrifugal forces are predominant, and we showed recently [4] that two kinds of instabilities, of a centrifugal Rayleigh-Taylor (CRT) and spiral Kelvin-Helmholtz (SKH) types are Harish N. Dixit Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, INDIA, e-mail: [email protected] Rama Govindarajan Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, INDIA e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_21, © Springer Science+Business Media B.V. 2010

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triggered. The former arises from a mechanism similar to the Rayleigh-Taylor instability of a vortex with a heavy core, as studied by [9] and [6]. The latter arises purely from the fact that the density interface, being spiral, is not quite circular. Both instabilities would be missed upon making a passive-scalar approximation, i.e., upon neglecting the inertial effects of density stratification. Gravity is unnecessary in this process, but since the density stratification is stable with respect to gravity to begin with, it would be interesting to see the effect that gravity would have on these instabilities, and on the resulting breakdown into a possibly turbulence-like state. We show that gravity has a profound effect on the dynamics and hastens the final breakdown into a turbulence-like state. We present simulations (i) including centrifugal effects (non-Boussinesq) but without gravity, and (ii) gravity present, under the Boussinesq approximation, and (iii) of the two combined. The vorticity and density balance equations are given in the inviscid, infinite Peclet number limit by

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Dρ = 0, (2) Dt where D/Dt ≡ ∂ /∂ t + u · ∇, and u = (ur er , uθ eθ ), Ω , ρ and g denote the velocity vector (of radial and azimuthal components), the vorticity, the density and gravity respectively. The flow is taken to be incompressible, so ∇ · u = 0. The two cases we present in sections 2 and 3 correspond to the neglect of the second and the first term on the right hand side of (1) respectively, while the full equations are solved in the third case.

2 Centrifugal effects A brief description of the flow is given here, further details are available in [4]. Consider a point vortex of circulation 2πΓ located at an initially horizontal density interface, with a difference ∆ ρ in density across it. The point vortex causes a spiralling of the density interface, as in figure 1. For large time or small radius, the spacing between successive turns of the spiral scales as r3 λ∼ . (3) Γt When a small amount of diffusivity, κ is present in the flow, the prominent lengthscales in the flow can be expressed in terms of the Peclet number, Pe = Γ /κ , of the flow, as shown in fig.(1). If we assume that these jumps are circular, then the flow is analogous to the planar Rayleigh-Taylor instability, except that the gravitational potential arises from the centrifugal forces. Density jumps from ρh to ρl corresponding to heavy and light fluids. Just as a heavy fluid above a light one is unstable due

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to gravity, a heavy fluid inside a light one is unstable to centrifugal forces, here we refer to this as the CRT instability. It is shown analytically for inviscid flow with κ = 0 that although stabilising and destabilising density jumps occur alternately, the net effect is destabilising, even in the case of a light core, see fig.(2). We have so far approximated the interface to be circular, when in reality, the interface is actually in the form of a spiral. Baroclinic torque is created when a density interface is not strictly perpendicular to centrifugal acceleration, and this torque produces vorticity here since the interface is not perfectly circular. For a point vortex, any passive interface around it would take the form of a Lituus spiral, which can be represented parametrically as

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so the assumption of a circular interface is better at smaller radii or late times. Returning to the vorticity equation (1), neglecting gravity, and assuming that the effect of the circulation Γ of the point vortex is far greater on the basic flow than of that which is newly created, we may write 2 DΩ ∇ρ U ≃− × er . (6) Dt ρ r At high θs , i.e., at large time at a given radius, the integral of the above may be approximated as 2Γ t ∆ Uθ ≃ ∓A U log( 2 ). (7) r At every unstable density jump, negative vorticity is created and vice versa, resulting in heavier fluid travelling faster and lighter fluid slower. A spiral Kelvin-Helmholtz (SKH) instability thus ensues, and combines with the centrifugal Rayleigh-Taylor (CRT) instability. Given that ∆ Uθ increases logarithmically in time, we have a spiral KelvinHelmholtz instability with a slightly faster than exponential growth rate, i.e., ur ∼ at bt where a and b are constants. While a perfectly sharp density interface is increasingly unstable to increasing wavenumber, a small but finite thickness of the interface determines that the fastest growing wave has a wavelength comparable to the thickness, as expected. The replacement of a point vortex by a Rankine with smoothed-out edges does not change the answer qualitatively either, except within the vortex core. Numerical simulations are obtained as follows. The 2D Navier-Stokes equations including non-Boussinesq effects are solved using the Fourier pseudospectral method in space. Inviscid (with hyperviscosity) results are presented here but viscous results are qualitatively no different. Figure 3 shows the vorticity and density fields including non-Boussinesq effects in the absence of gravity. Vorticity of alternating sign is produced in the form of two interwoven spirals along the density interfaces, and instabilities ensue, consistent with our theoretical predictions.

3 Gravitational effects We now consider the effects of gravitational effects alone and employ the Boussinesq approximation. These are the first results from an ongoing study [3]. In the linear framework, such a flow has been treated in [7]. Figure 4(a) shows vorticity and density contours from simulations when the Froude number is 4.6. Centrifugal effects are not included, due to which the spiralling process of the interface is impeded. In general, gravity becomes important when g > Γ 2 /r3 . An estimate of this ratio for realistic trailing vortices was given by [8].

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The inclusion of gravity results in the formation of two satellite vortices, whereas the non-Boussinesq equations (without gravity) results in the formation of a small scale instability. When the two effects are combined, features of both the cases can be seen as shown in fig.(5). The small scale non-Boussinesq effects are seen near the vortex center, whereas the large scale overturning occurs further away from the center. More details on these effects can be seen in [3].

4 Conclusions and outlook We have shown that interesting dynamics emerges when the passive-scalar approximation is not made, as usually done for small Atwood number flows. When density variation is very sharp, gradients in density can lead to significant barloclinic torque, making the Boussinesq equations incomplete. Gravity impedes the original spiralling process, but gives rise to satellite spirals instead. Both speed up the de-

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Fig. 5 Density field in a full simulation combining effects of gravity and inertia related density effects. Small scale undulation near the center of the vortex is due non-Boussinesq terms, and large scale overturning seen at an angle of 600 is due to gravitational effects.

struction of the identity of the vortex, and result in a turbulence-like state. More work is in progress and will appear in a future study ([3]). Another area of interest would be influence of these instabilities in multiple vortex scenarios, as in trailing vortices and vortex merger problems.

References 1. P. Billant and J. M. Chomaz. (2000) Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 418, pp.167-188. 2. L. K. Brandt and K. K. Nomura. (2007) The physics of vortex merger and the effects of ambient stable stratification. J. Fluid Mech. 592, pp.413-446. 3. Harish N Dixit and Rama Govindarajan. Vortex in a density stratified flow: gravitational and centrifugal effects. Preprint 4. Harish N Dixit and Rama Govindarajan. (2009) Vortex-induced instabilities and accelerated collapse due to inertial effects of density stratification. Submitted to J. Fluid Mech., and movies at http://www.jncasr.ac.in/rama/rg research.htm 5. P. Flohr and J. C. Vassilicos. (1997) Accelerated scalar dissipation in a vortex. J. Fluid Mech. 348, pp.295-317. 6. L. Joly, J. Fontane and P. Chassaing. (2005) The RayleighTaylor instability of twodimensional high-density vortices. J. Fluid Mech. 537, pp.415-431. 7. J. C. S. Meng and J. W. Rottman. (1988) Linear internal waves generated by density and velocity perturbations in a linearly stratified fluid. J. Fluid Mech. 186, pp.419-444. 8. P. Orlandi, G. F. Carnevale, S. K. Lele and K. Shariff. (1998) DNS study of stability of trailing vortices. CTR, Proc. Summer Program. 9. D. Sipp, D. Fabre, S. Michelin and L. Jacquin. (2005) Stability of a vortex with a heavy core. J. Fluid Mech. 526, 67-76.

Wave Packets of Controlled Velocity Perturbations at Laminar Flow Separation Alexander Dovgal and Alexander Sorokin

Abstract Experimental data on controlled time-periodic disturbances of the laminar flow separating at a 2D backward-facing step on a flat plate are reported. Windtunnel results were obtained at low subsonic velocity through hot-wire measurements. It is found that vorticity perturbations generated locally behind the step contaminate an extended flow region downstream and upstream of their origin. One expects this could provide a feedback involved in self-sustained oscillations of the separation bubble.

1 Introduction A number of theoretical, numerical and experimental studies testify to local regions of laminar flow separation combining amplification of the external-flow perturbations and self-excitation. In experimental work, the intrinsic dynamics of separation bubbles is explained in terms of the ”shedding” type instability [12] or the ”wake mode” of velocity perturbations [7], emphasizing their essential difference from the convective instability of the separated shear layer. Theoretically, the oscillations synchronized over the entire separated-flow region are approached through stability analysis of local velocity profiles within the concept of their absolute / convective instability and global stability solutions [2, 6, 9, 10, 13, 14, 15]. In experiments [4], the self-excited unsteadiness competing with amplification of the shear-layer disturbances was observed at laminar flow separation behind a 2D backward-facing step on a flat plate. At the step height h comparable with the Alexander Dovgal Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, 630090 Novosibirsk Russian Federation e-mail: [email protected] Alexander Sorokin Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, 630090 Novosibirsk Russian Federation e-mail: am [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_22, © Springer Science+Business Media B.V. 2010

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boundary-layer thickness and the Reynolds number Reh of about 1700, the above instability phenomena appeared as two distinct spectral bands of velocity perturbations. In the present work we proceed with boundary-layer separation in the same flow configuration. The objective is to clarify a feedback mechanism of the global dynamics as it could be under the low-Re-number conditions of [4]. For this purpose, we deal with controlled perturbations of the separation bubble.

2 Experimental set-up Research data were obtained in T-324 low-turbulent subsonic wind tunnel of the Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk. The facility has a closed test section of 1 × 1 × 4 meters with the free-stream turbulence level less than 0.04%. The experimental model was composed of two acrylic flat plates, each of 10-mm thickness and 995-mm width, installed vertically in the wind-tunnel test section close to its centerplane, Fig. 1. The plates were attached to each other with a 3-mm step at 300 mm from the model leading edge. To minimize background velocity fluctuations in the near-wall layer, the rear plate was equipped with a trailing-edge flap. Harmonic disturbances of the separation bubble were generated by an external loudspeaker through 0.5-mm holes spaced in the central part of the model surface upstream and downstream of the step as is sketched in Fig. 1. Thus, the wave packets of oscillations excited in the pre-separated boundary layer and in the separation region could be compared. To examine the separated-flow velocity characteristics, a hot-wire anemometer with one-wire probes was used. The hot-wire signal digitized by an analog-to-digital converter was processed in MATLAB environment. In what follows, x is the streamwise distance from the step, y is normal coordinate measured from the surface of the rear plate, z is the spanwise coordinate with z = 0 at the position of the disturbances generator.

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3 Results and discussion The experimental runs were carried out at the external-flow velocity above the step U0 = 8.8 m/s corresponding to the Reynolds number on the step height Reh = 1670. The separated-flow pattern is illustrated by mean-velocity distributions and spectra of natural perturbation in Fig. 2. In the left-hand section of Fig. 2a, the boundarylayer profile measured close to the separation point at x/h = −1.3 is shown with the displacement and momentum thickness making δ ∗ /h = 0.31 and θ /h = 0.13 respectively. Behind the step, the separated boundary layer tends to reattachment and the bubble length is estimated as xr /h ≈ 20. Under quiet wind-tunnel conditions of the present experiment, the total intensity of background velocity perturbations u′ /U0 grows from about 0.3% at the step to 3–4% at reattachment so that the transition to turbulence takes place downstream of the separation region. In the disturbances spectra of Fig. 2b, two scales of velocity oscillations are well seen. The high-frequency band of perturbations is a packet of shear-layer instability while the low-frequency one corresponds to long-wave oscillations of the separation bubble. These flow features are very similar to those we observed earlier in [4] where they were examined in more details. For the following investigation of controlled disturbances, the separated flow was excited at two frequencies St = f h/U0 = 0.027 and 0.099 representing the dominant spectral components of the natural velocity perturbations. In each case, amplitude characteristics of the wave packets were determined by filtering the hot-wire signal at the excitation frequency through a narrow band of St = 0.00136. Hot-wire data obtained under different conditions of the disturbances generation are summarized in Figs. 3 and 4 where contours of their maximum rms amplitude are shown as percentage of the external-flow velocity U0 . The oscillations forced in the attached boundary layer are transported downstream of the excitation point in a narrow spanwise region and then amplify penetrating into the separation bubble from upstream (Fig. 3). Much different behavior of the disturbances is found when they are excited behind the step, so that the flow is perturbed all around their gener-

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ator (Fig. 4). An exception is the case of Fig. 4b where the forcing level proved to be too small to resolve the induced perturbations in the upstream sections, however, a certain flow response to the excitation is also found aside the generator in a wide range over the transverse coordinate.

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One expects different routes of the disturbances propagation to explain the flow contamination which is observed under the excitation applied behind the separation point. These could be upstream transportation of vorticity perturbations in the nearwall region and generation of the amplifying separated-layer disturbances at the step by the pressure waves, emanating from the source of oscillations. To clarify these options, we plot in Fig. 5 the scatter of signal-to-noise ratio measured in the range of −10 ≤ z/h ≤ 10, indicating rms maxima of the disturbances normalized by local amplitudes of the background fluctuations at the periodic forcing switched off. It is seen that the ratio is about unity in the upstream part of separation bubble so that no response to the excitation is found, whereas approaching the source of oscillations, the controlled perturbations grow much faster than the background noise. Apparently, such a behavior of the disturbances can hardly be attributed to the acoustic excitation of the convective shear-layer instability when the controlled and natural perturbations would have comparable spatial growth rates. Thus, under the present experimental conditions, the separated flow forced behind the point of separation is most likely contaminated through the vorticity field perturbed by the disturbances generator. Finally, we notice that the wind-tunnel results reported in this paper are obtained at the mean velocity in the backflow region near the wall as low as several percent of U0 only. To estimate the backflow component, which could not be accurately resolved with the hot-wire method of the present study, PIV data on flow separation behind backward-facing steps seem helpful [3, 7, 11]. The most close to the present conditions are those of experiments [3] where the maximum of backflow was found as 5% of U0 at Reh = 1060. Also we refer to theoretical findings on the intrinsic dynamics of separation bubbles which is associated with onset of the absolute instability of local velocity profiles at the backflow making 15–20% of the external-flow velocity and even higher, e.g. [1, 5, 6]. If so, localized separated-flow perturbations grow in time and contaminate the surrounding fluid, generating the self-sustained oscillations at boundary-layer separation. However, the present experimental approach to the problem is quite different from the theoretical one, the latter focusing

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on impulse response of a parallel flow. Moreover, along with instability effects of the time-averaged flow, one expects rather complicated mechanisms of the feedback in separation bubbles related to their instantaneous characteristics [8]. Acknowledgements The present study was supported by the Ministry of Education and Science of the Russian Federation (grant No. RNP.2.1.2.541).

References 1. Alam, M., Sandham, N.D.: Direct numerical simulation of ”short” laminar separation bubbles with turbulent reattachment. J. Fluid Mech. 410, 1–28 (2000) 2. Barkley, D., Gomes, M.G.M., Henderson, R.D.: Three-dimensional instability in flow over a backward facing step. J. Fluid Mech. 473, 167–190 (2002) 3. Boiko, A.V., Dovgal, A.V., Hein, S., Henning, A., Sorokin, A.: Particle image velocimetry of streaky structures in a laminar separation bubble. DLR–IB—224–2008 A 20, Goettingen (2008) 4. Dovgal, A.V., Sorokin, A.M.: Instability of a laminar separation bubble to vortex shedding. Thermophysics and Aeromechanics. 8, 179–186 (2001) 5. Gaster, M.: Stability of velocity profiles with reverse flow. In: Hussaini, M.Y., Kumar, A., Streett C.L. (eds.) Instability, Transition and Turbulence, pp. 212–215. Springer, Berlin (1992) 6. Hammond, D.A., Redekopp, L.G.: Local and global instability properties of separation bubbles. Eur. J. Mech. B/Fluids. 17, 145–164 (1998) 7. Hudy, L.M., Naguib, A., Humphreys, W.M.: Stochastic estimation of a separated-flow field using wall-pressure-array measurements. Phys. Fluids. (2007) doi: 10.1063/1.2472507 8. Jones, L.E., Sandberg, R.D., Sandham, N.D.: Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence. J. Fluid Mech. 602, 175–207 (2008) 9. Kaiktsis, L., Monkewitz, P.A.: Global destabilization of flow over a backward-facing step. Phys. Fluids. 15, 3647–3658 (2003) 10. Marquillie, M., Ehrenstein, U.: On the onset of nonlinear oscillations in a separating boundary-layer flow. J. Fluid Mech. 490, 169–188 (2003) 11. Scarano, F., Riethmuller, M.L.: Iterative multigrid approach in PIV image processing with discrete window offset. Exp. Fluids. 26, 513–523 (1999) 12. Sigurdson, L.W., Roshko, A.: The structure and control of a turbulent reattaching flow. In: Liepmann, H.W., Narasimha, R. (eds.) Turbulence Management and Relaminarization, pp. 497–514. Springer, Berlin (1988) 13. Theofilis, V.: Advances in global linear instability analysis of nonparallel and threedimensional flows. Progr. Aerospace Sci. 39, 249–315 (2003) 14. Theofilis, V., Hein, S., Dallmann, U.: On the origins of unsteadiness and three-dimensionality in a laminar separation bubble. Phil. Trans. Roy. Soc. Lond. A. 358, 3229–3246 (2000) 15. Wee, D., Yi, T., Annaswamy, A., Ghoniem A.F.: Self-sustained oscillations and vortex shedding in backward-facing step flows: Simulation and linear instability analysis. Phys. Fluids. 16, 3361–3373 (2004)

Linear Stability Analysis for Manipulated Boundary-Layer Flows using Plasma Actuators A. Duchmann, A. Reeh, R. Quadros, J. Kriegseis, C. Tropea Abstract This paper presents the implementation of a method for linear stability analysis (LSA) and its application to investigate transitional boundary-layer flows affected by dielectric-barrier discharge (DBD) actuators. These flow-control devices are used to influence the process of boundary-layer transition by electrohydrodynamic coupling of momentum to the surrounding fluid molecules. The boundarylayer profile and its stability characteristics are changed. Linear stability analysis is applied to numerical and experimental data and helps to understand the effective mechanisms of these flow-control actuators when applied for transition control. Amplification rates in the linear growth stage are diminished and the critical as well as the local Reynolds number are affected by DBD actuation, leading to considerable delay of transition.

1 Introduction Dielectric-barrier discharge actuators consist of at least two electrodes separated by a dielectric material. If a high AC voltage is applied between the electrodes, a periodic charge buildup on the dielectric surface occurs and causes ionization of surrounding fluid molecules. The charged molecules are accelerated in the electromagnetic field of the weakly ionized plasma and by collision with neutral molecules transfer momentum into the fluid. In quiescent air above a solid surface, this leads to a flow tangential to the wall. If applied in a boundary-layer flow, the actuator adds momentum in the proximity of the wall. As the actuator represents a zeronet mass-flux device, the tangential acceleration is accompanied by a smaller induced wall-normal velocity component. Since the physical mechanisms involved are not yet thoroughly understood, ongoing investigations concentrate on formulating a phenomenological model for the effect of DBD actuators on flow phenomena [3]. Although the plasma is excited by an alternating power supply at frequencies of several kHz, in steady operation mode a quasi-steady body force generates a constant wall jet with an effective origin close to the physical actuator. It was shown that this wall jet can be used to stabilize a laminar boundary layer and delay the transition to turbulence [1],[2]. The stabilizing effect of the actuator is similar to that of a favorable pressure gradient. By controlled modulation of the supply-voltage frequency, a pulsed operation mode of the DBD actuator is obtained. If the actuator is operated at the frequency of incoming disturbances and a correct phase relation Alexander Duchmann Center of Smart Interfaces, TU Darmstadt, Germany, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_23, © Springer Science+Business Media B.V. 2010

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is adjusted, TS-waves can be attenuated by destructive interference and transition can be significantly delayed [2],[4]. Additionally, like in the case of steady operation, the mean boundary-layer profiles are affected, resulting in altered stability characteristics. Currently, researchers at TU Darmstadt are engaged with experimental, numerical and theoretical investigations for a better understanding of the plasma actuator’s active principle and an extension of the flow-control applications. Both described methods of actuator operation for transition control are investigated. The major aim of the present work is to investigate the changes of the boundary-layer stability properties related to the different operation modes, enabling deeper insight into the flow-control mechanisms. Controlled boundary-layer stability experiments are conducted to investigate the actuators ability to delay laminar-turbulent transition. The setup chosen for both the experimental as well as the numerical investigations is a flat-plate centered in a wind tunnel. The shape of the wind-tunnel walls are chosen to obtain an adverse pressure gradient (APG) in order to accelerate the process of disturbance amplification. The APG has a destabilizing effect on the boundary-layer flow such that all consecutive stages of transition can be observed within the limited length of the wind-tunnel test section. In both experiment and numerical simulation, the freestream velocity at the inlet is constant and in this study, results based on investigations at U∞ = 8m/s are presented. 400mm behind the elliptical leading edge of the flat plate, a pulsed disturbance source induces reproducible two-dimensional waves in the base flow. In the experiment, a vibrating ribbon creates these waves which are similar in amplitude and frequency ( f = 110Hz) to natural TollmienSchlichting waves amplified in boundary-layers at such velocities. 100mm further downstream, a DBD actuator is situated on the surface of the flat-plate. It can be operated in pulsed or steady mode and the effect on the boundary-layer flow downstream is investigated experimentally by hotwire measurements. Details about the experimental and numerical setups can be reviewed in [1], [2] and [4].

2 Linear Stability Analysis In order to investigate the plasma actuator’s influence on flow stability and to validate existing experimental and numerical results, modal linear stability analysis is conducted. The local manipulation of the boundary-layer profiles by the plasma actuator is expected to have a stabilizing effect similar to a decreased pressure gradient. The stability calculations are intended to show the flow’s reduced affinity to become unstable. The linear stability analysis performed in the present work assumes low environmental disturbances and a parallel, two-dimensional, steady base flow U = (U(y), 0, 0T , which is superposed with small wavelike disturbances u′ and p′ . The description of such small three-dimensional disturbances can be reduced to a system of two linear disturbance equations (see Reeh [5]) for the wall-normal velocity component v′ and the wall-normal vorticity component Ω ′ . The initial growth of the small disturbances may be described by the wave modes iα x+β z−ω t and Ω ′ (x, y, z,t) = Ω ˆ (y)ei(α x+β z−ω t . Introduction of v′ (x, y, z,t) = v(y)e ˆ

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the Fourier-mode approach into the two disturbance equations gives rise to the wellknown dimensionless Orr-Sommerfeld and Squire equations d 2U 1 2 2 2 2 2 [(−iω + iα U) D − k − iα 2 − D −k vˆ = 0 dy Re 1 dU −iω + iα U − D 2 − k2 Ωˆ = −iβ v, ˆ Re dy

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This boundary-value problem obeys to the homogeneous boundary conditions vˆ = D vˆ = Ωˆ = 0 at the wall and in the free stream. The resulting relation between the disturbance behavior in time (expressed byp the angular frequency ω ) and space (characterized by the wave number k = |k| = α 2 + β 2 ) physically represents a dispersion relation of the type D(α , β , ω , Reδ1 ) = 0,

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which can only be solved numerically. In general the quantities α , β and ω are complex. In order to enable eigenvalue analysis two of the three quantities must be known. In the present work the resulting eigenvalue problem is solved for the spatial framework providing the complex streamwise wave number α = αr + iαi . The angular disturbance frequency ω is given by the frequency of the excitator f and the spanwise wave number β equals zero due to the two-dimensional excitation. The spatial approach is appropriate to describe the downstream evolution of instabilities since TS-waves grow spatially in proportion to ur ′ ∝ e−αi x . However, the eigenvalue α appears up to fourth order in equation (1). To reduce the nonlinear eigenvalue problem to a linear one, two consecutive transformations are necessary, an exponential variable transformation and a companion matrix method, which is described by Reeh [5]. Eventually a generalized eigenvalue problem of the form Lspatial eˆ = α Mspatial eˆ is obtained. In order to discretize this boundary-value problem for a numerical solution a spectral Chebyshev collocation method is chosen. The dependent variables vˆ and Ωˆ are represented by truncated sums of Chebyshev polynomials and a Gauss-Lobatto grid is used for the discrete presentation of the independent variable y. Since the Gauss-Lobatto points are only defined on the finite domain ξ ∈ [−1, 1] the algebraic function y(ξ ) = δ2 (1 + ξ ) maps the grid points into the physical boundary-layer domain y ∈ [0, δ ]. The resulting matrix equations are solved with a built-in MATLAB routine.

3 Results Active wave cancelation (AWC) by means of DBD actuators has been investigated numerically by Quadros [4]. He used a finite volume method to describe the flow domain. The CFD-code FASTEST (Flow Analysis Solving Transport Equations with Simulated Turbulence) enables large eddy simulations (LES) based on the Germano

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method. A semi-empirical model of the body force is used to describe the influence of an DBD actuator. Numerically, the actuator operation can be modeled as an additional body force term in the Navier-Stokes equations. The simulations proved the positive effects observed in Grundmann’s experiments [2]. Active wave cancelation could be effectively used to delay Tollmien-Schlichting wave dominated transition. The investigations performed in this work utilize the mean boundary-layer profiles derived from Quadros’ numerical simulations. These are investigated in the streamwise region 0.4m < x < 0.6m. The stability calculations are performed for two cases. In the first case only the disturbance source is active. The plasma actuator is switched off in order to obtain the flows stability behavior with uncontrolled Tollmien-Schlichting wave growth as reference. In the other case, the DBD actuator is operated to induce momentum in opposite phase to the arriving waves, thereby controlling transition. The total transition delay emerges from two effects, the active wave cancelation and the stabilization of the mean base-flow profiles. However, only the latter effect can be investigated by LSA. 6

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Modal analysis investigates the stability properties locally for single boundarylayer profiles. An example for such a modal solution is given in Figure 1 showing slight shape changes in the streamwise eigenfunctions caused by the actuator. However, these eigenfunctions do not allow quantitative statements since they are normalized with their maximum values. In general the shape of the mean-flow profile changes as the boundary layer develops in the flow domain. This affects the local displacement thickness δ1 and the edge velocity Uδ . Thus, the local Reynolds numU δ ber Reδ1 = δν 1 and the dimensionless angular disturbance frequency ω = 2πUf δ1 δ vary. Both serve as input parameters for the dispersion relation. Figure 2 demon-

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strates the streamwise evolution of the local Reynolds number. Differences between the mean-velocity profiles with and without DBD actuation are small and modifications become apparent in the wall-near part of the first and second derivatives. However the LSA is highly sensitive to those shape changes. For a global demonstration of the plasma actuator’s stabilizing effect, neutral curves of the two considered cases are compared in the ω -Re-parameter space at three different streamwise positions in Figure 3. The observed decline of the instability area downstream of the actuator is similar to the effect of a strong decrease in the streamwise pressure gradient ∂P ∂ x . In order to access the stability properties over a certain streamwise distance, a Exp Actuator OFF Exp Actuator ON Num Actuator OFF Num Actuator ON


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stability calculation for each profile is necessary. Figure 4 clearly demonstrates the positive effect of the plasma actuator over the investigated region with remarkably reduced growth rates for the primary instabilities. The instabilities amplify much less intensely during DBD actuation and thus transition is delayed. Figure 5 shows the N-factor development, integrating the dimensional instability growth rates obtained from the numerical data, starting from the point of neutral stability. A local change in phase velocity can be observed in Figure 6. Actuator OFF Actuator ON 0.5

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Controlled boundary-layer stability experiments were conducted by Duchmann [1] on a flat-plate in a open-circuit windtunnel. 500mm behind the leading edge a DBD actuator in steady operation was powered with 10kV AC at a frequency of 6kHz. Hotwire measurements of the boundary-layer profiles confirmed a delay of transition for approx. 100mm along the streamwise coordinate as observed from the H12 shape-factor development. The velocity profiles (Figure 7) are analyzed with the

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LSA algorithm and compared to the numerical results described above. Although experimental uncertainties cause certain deviation of the encountered profiles for the same streamwise position 50mm downstream of the actuator, the similarity of the laminar profiles is rather good. The calculated development of the amplification rates are contrasted to the numerical results in Figure 4. The coarse grid of the experimental data results from the limited accessability of the domain by hotwire measurements, yet the same trend can be observed as from the numerical data.

4 Discussion The linear stability method was successfully applied on numerical and experimental boundary-layer data yielding insight into the mechanism of transition delay through DBD actuation. The manipulations of the mean boundary-layer profiles caused by the actuation have been investigated in this linear stability analysis, not taking into account the wave canceling mechanism. It was clearly shown that the stabilizing effect of the plasma actuator on the mean boundary-layer profiles is an important factor in transition manipulation. Wave damping through destructive interference is an additional superimposed and effective mechanism. The wall-normal component induced by the actuation is relatively small as compared to the streamwise effect (V < 0.01Uδ ) such that the assumption of wall-parallel flow for LSA is not remarkably violated. The numerical and experimental data are in good agreement and for both, LSA delivers physically reasonable results. Some uncertainties like the freestream turbulence level, leading edge suction-peak or the surface roughness due to the geometrical obstacle of the actuator may explain the deviation between simulation and experiment. Additionally, the experimental investigation lacks some more refined data next to the actuator to describe its local influence. Since the direct vicinity of the plasma is not accessible using hotwire probes, future investigation of boundary-layer profiles by means of optical measurement techniques are foreseen.

References 1. Duchmann, A. (2007) Experimentelle Untersuchung der Transitionsbeeinflussung mit Hilfe von Plasma Aktuatoren, Bachelor’s thesis, Institute of Fluid Mechanics and Aerodynamics, TU Darmstadt. 2. Grundmann, S. (2008): Transition Control using Dielectric-Barrier Discharge Actuators. Ph.D. Thesis Institute of Fluid Mechanics and Aerodynamics, TU Darmstadt. 3. Lagmich, Y., Callegari, T., Pitchford, L., Boeuf, J.-P. (2008) Model description of surface dielectric barrier discharges for flow control. Journal of Physics D: Applied Physics, 41, Issue 9, pp. 095205 4. Quadros, R., Grundmann, S., Tropea, C. (2008): Numerical investigations of the boundarylayer stabilization using a phenomenological plasma actuator. Journal of Flow, Turbulence and Combustion (accepted). 5. Reeh, A. (2008): Development and Implementation of a Method for Linear Stability Analysis in Natural and Manipulated Boundary-Layer Flows, Bachelor’s thesis, Institute of Fluid Mechanics and Aerodynamics, TU Darmstadt.

Stripy patterns in low-Re turbulent plane Couette flow Yohann Duguet, Philipp Schlatter, and Dan S. Henningson

Abstract We present for the first time a complete bifurcation diagram of plane Couette flow based on direct numerical simulation of the full Navier-Stokes equations. The use of an unusually large computational domain (800h × 2h × 356h) is crucial for the determination of transition thresholds, because it allows to reproduce spatiotemporal intermittency structures such as transient spots, turbulent bands, and laminar holes. The threshold in Re (based on the half-gap) is found to be Rec = 324 ± 1, in very good agreement with available experimental data. This work points out that, at the onset of transition in Re, fragmented oblique patterns always emerge from the interaction of growing neighbouring spots. An analogy with thermodynamical phase transition seems relevant to describe the whole transition process.

1 Introduction Plane Couette flow (pCf), the flow between two parallel plates moving in opposite directions, is the simplest canonical example of the effect of shear on a viscous fluid. The only nondimensional parameter ruling the flow is the Reynolds number, here defined as Re = Uh ν , where ±U is the velocity of the two walls, h is the half-gap between them, and ν is the kinematic viscosity of the fluid. We are interested in the way sustained turbulence appears in this system around the onset of transition. Now pCf does not belong to the class of fluid systems undergoing transition through successive losses of stability of the base flow (as, for example, in Rayleigh-Bénard convection). The laminar base flow happens to be linearly stable for all values of Re [1], hence transition is subcritical and is necessarily due to a finite-amplitude instability of the base flow. Experimental investigation in a large set-up [2] has shown that the value of Rec has to be a statistical one, the suggested Yohann Duguet, Philipp Schlatter and Dan S. Henningson Linné Flow Centre, KTH Mechanics, Osquars backe, 18, SE-10044 Stockholm, Sweden e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_24, © Springer Science+Business Media B.V. 2010

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value being Rec = 323 ± 2. Importantly, the flow during transition shows clear signs of spatio-temporal intermittency. Experiments, focusing on the ability of localised disturbances to trigger the whole flow to become turbulent [3, 4, 2], yield other values of Rec between 325 and 370 corresponding to the development of a turbulent spot. Later experiments by the Saclay team in an even larger domain lead to a more accurate regime diagram parametrised by Re [5, 6]. By adiabatically reducing Re from a turbulent field they identified a stable turbulent regime with an alternance of turbulent and laminar regions, forming for Re = 325 a regular or sometimes fragmented pattern, oblique with respect to the direction of motion of the plates. This regime seems analogous to the ’spiral turbulence’ observed experimentally in Taylor-Couette experiments [7, 8]. In order to study the formation of these largescale patterns, we chose to perform direct numerical simulation of pCf in a periodic domain which is unusually large in the two in-plane directions. Let Lx (resp. Lz ) be the streamwise (resp. spanwise) extent of the numerical domain, and let L be their common order of magnitude. Quantitatively, L ∼ O(5h) allows to reproduce the local dynamics of a pair of streaks and the self-sustaining process responsible for the maintainance of a turbulent flow [9]. L ∼ O(20h) allows to simulate the collective dynamics of neighbouring streaks. For L ∼ O(40 − 100h), localised turbulence and laminar-turbulent interfaces appear, allowing for instance for reproducing the initial stage of the growth of turbulent spots [10] and the structure of a single turbulent band [11]. The collective behaviour of spots, as well as the pattern formation resulting from their interaction, requires L to be at least an order of magnitude larger. In the framework of spatio-temporal intermittency, the formation of turbulent patterns corresponds to the competition between a laminar and a turbulent phase [12]. Extending the size of the domain to study the phase transition thus corresponds to an extra-step towards the ’thermodynamic limit’.

2 Results 2.1 Numerical method We present here a numerical experiment in a periodic domain of size (Lx , Ly , Lz ) = (800h, 2h, 356h). For the sake of comparison, the set-up used by Prigent [5] and Bottin [2] respectively have size (770h, 2h, 340h) and (70h, 2h, 380h). We used a spectral code to integrate the full incompressible Navier-Stokes equations, where the velocity field is expanded on a basis of Fourier modes (in the x and z directions) and Chebyshev polynomials (in the wall normal direction y) [13]. The boundary conditions are periodicity in x and z, and no-slip at the walls (y = ±h). The numerical resolution is 2048 collocation points in x, 33 in y and 1024 in z, representing a drop of 6 decades in the energy spectrum. Each simulation was run on up to 256 parallel processors using MPI techniques, and corresponds to grossly one year of

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2.2 Formation of stripes We first describe the transient dynamics observed in the case 300 ≤ Re ≤ 323. Initially the noisy velocity is quickly dissipated and the energy of the perturbation relaxes, followed by the rapid temporal growth of streamwise streaks. Localised zones of spatial disorder, characterised by a higher perturbation energy, emerge on top of the regular streaky velocity field. These spots appear at random locations. This instantaneous nucleation of spots suggests a local competition between the viscous decay of streaks (on a time-scale of O(Re)) and their instability, induced by the residual noise and leading to their breakdown. As time evolves, all these localised structures take an ellipse-like shape, encircling an active zone of streaks of larger in-

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tensity. For Re ≤ 323, the lifetime of each of these structures is finite and the whole flow eventually returns to a global laminar state. In our numerical experiments, the mean lifetime of transient spots clearly increases with Re, consistently with statistical analysis [2]. However, statistical analysis of our data is numerically too costly to allow for statistical ensemble averages. Note that the number of spots nucleating from our initial condition increases with Re and that their individual lifetimes seem uncorrelated. The early stages of the numerical experiments at Re ≥ 325 are identical to the Re = 320 case, where the spots would all eventually die. However, while some of the spots at Re ≥ 325 are seen to transiently decay, the strongest survive and directly start to grow. The spots initially grow in a spanwise-symmetric way but later start to lose their symmetry and grow obliquely, forming large turbulent domains delimited by a sharp unsteady interface. When two neighbouring growing spots approach each other, the bands adjust themselves until one single band is formed. Both positive and negative angles are observed. For Re ≤ 330, the growth in size of these turbulent domains slows down and stops, see Fig. 2.2(top). The average angle with respect to the streamwise direction of the stripes decreases with increasing Re. From Re = 350 to Re = 370, a pattern, consisting of alternating laminar and turbulent bands, occupies the whole numerical domain. The average angle of the bands is close to the angle of the diagonal of computation domain. Depending on the value of Re, various spatial arrangements of the stripes are observed, despite a similar initial condition. A statistically steady pattern is reached after O(1000 h/U ), which consists of fragmented turbulent stripes. Visualisation of the cross-sectional flow (not shown), shows that the pattern is essentially two-dimensional, with a shear-induced distortion in the (x − y) plane. The fronts separating the two phases are sharp on a large scale, though on a small scale they correspond to a vague non-abrupt envelope formed by the tails of the streaks in the turbulent phase. The very regular pattern observed in pCf [5] as well in Taylor-Couette experiments, and modelled in Ref. [11], where all the stripes are parallel, is not observed here when starting from random noise. This highlights the strong sensitivity of the flow to the initial conditions. Nevertheless, if the pattern observed does not seem to be universal, its locally oblique nature seems unavoidable once the computational domain is large enough and Re lies in the narrow band [330 − 400]. This same sequence noise → streaks → spot nucleation → stripes was observed for all values of Re between Re = 325 and 400. However, from Re = 370 on, the steady regime attained after the initial phase is now dominated by the turbulent phase. Until Re = 420, many small laminar holes are visible in a (x − z) plane. The largest holes are present at Re = 370, their interface displays the orientation of the turbulent bands observed at lower values of Re. As Re is increased up to 420, only individual transient holes –of the width of one to two streaks– survive for a finite time in the flow. This sugests a unsteady contamination process of local laminar domains by the turbulent ones, as discussed in Ref. [14] and [15]. The typical width of a streak, of order O(3h), is the smallest spatial scale above which it is still possible to distinguish between turbulent and laminar dynamics, which makes the definition of an upper threshold for uniform turbulence rather ambiguous.

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Fig. 2 Patterns appearing for various Re starting from noise. From top to bottom : Re = 330, Re = 350, Re = 370.

Post-processing of the velocity field was done in order to measure as accurately as possible the turbulent fraction FT as a function of Re when a steady regime is reached. Assuming a weak dependence on the wall normal coordinate, we chose to compute the turbulent production in the mid-plane y = 0. The turbulent fraction FT was determined after repeated applications of a median filter, and is shown in Figure 1 (bottom). The turbulent fraction corresponding to the transient spots has been indicated (open circles) though the relevant value as time goes to infinity is FT = 0. Above Re = 325, there is a sharp jump of FT , due to the formation of sustained turbulent structures of finite spatial extent. The values of FT evolve smoothly toward unity as Re approaches 420, and no clear threshold can be determined for the transition to a ’uniform’ regime. Figure 1 (bottom) can be interpreted as a bifurcation

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diagram for pCf. It is qualitatively and quantitatively very analogous to the one by Bottin et Al. [14], however the range of existence of striped patterns is here clearly identified. Within the accuracy inherent to costly numerical simulations (including the lack of ensemble averages), the results clearly support a discontinuous (firstorder) phase transition near Rec ∼ 324 and a smooth (second-order) transition to uniform turbulence near Re > 400. The discontinuity of FT near Rec is linked to the nucleation process and corresponds to the minimal size of non-transient spots. The slow increase in FT with Re illustrates a weakening of the (anisotropic) regulatory mechanisms limiting the propagation of the laminar/turbulent interfaces. Acknowledgements: Computer time provided by SNIC (Swedish National Infrastructure for Computing) is gratefully acknowledged.

References 1. Romanov, V.A.: Stability of plane-parallel Couette flow. Funct. Anal. Appl. 7, 137–146 (1973) 2. Bottin, S. and Chaté, H.: Statistical analysis of the transition to turbulence in plane Couette flow. European Phys. J. B 1, 143–155 (1998) 3. Tillmark, N. and Alfredsson, P.H.: Experiments on transition in plane Couette flow. J. Fluid Mech. 235, 89–102 (1992) 4. Daviaud, F. and Hegseth, J.J. and Bergé, P.: Subcritical transition to turbulence in plane Couette flow. Phys. Rev. Letters. 69, 2511–2514 (1992) 5. Prigent, A.: PhD, La spirale turbulente: motif de grande longueur d’onde dans les coulements cisaills turbulents, Universit Paris Sud - Paris XI (2003) 6. Prigent, A. and Grégoire, G. and Chaté, H. and Dauchot, O. and Van Saarlos, W.: Large-scale finite-wavelength modulation within turbulent shear flows, Phys. Rev. Letters. 89, 014501 (2002) 7. Coles, D.: Transition in circular Couette flow. J. Fluid Mech. 21, pp–pp 385–425 (1965) 8. Van Atta, C.W.: Exploratory measurements in spiral turbulence. Journal. 25, 495–512 (1966) 9. Hamilton, J. and Kim, J. and Waleffe, F.: Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348 (1995) 10. Lundbladh, A. and Johansson, A.V.: Direct simulation of turbulent spots in plane Couette flow. J. Fluid Mech. 229, 499-516 (1991) 11. Barkley, D. and Tuckerman, L.S.: Computational Study of Turbulent Laminar Patterns in Couette Flow. Phys. Rev. Lett. 94, 014502 (2005) 12. Pomeau, Y.: Front motion, metastability and subcritical bifurcations in hydrodynamicstitle. Journal. Physica D 23, 3–11 (1986) 13. Chevalier, M. and Schlatter, P. and Lundbladh, A. and Henningson, D. S., SIMSON - A Pseudo-Spectral Solver for Incompressible Boundary Layer Flows, Technical report, TRITAMEK 2007:07, KTH Mechanics, Stockholm, Sweden (2007) 14. Bottin, S. and Daviaud, F. and Manneville, P. and Dauchot, O.: Discontinuous transition to spatiotemporal. Europhys. Lett. 43, 171–176 (1998) 15. Manneville, P.: Spatiotemporal perspective on the decay of turbulence in wall-bounded flows. Phys. Rev. E. 79, 025301 (2009)

Characterization of the three-dimensional instability in a lid-driven cavity by an adjoint based analysis Flavio Giannetti, Paolo Luchini, and Luca Marino Abstract In this paper we investigate the three-dimensional stability of the 2D flow generated in a cavity by the motion of two facing walls. An adjoint-based analysis of the most unstable global mode will be performed in order to localize the core of the instability.

1 Introduction Vortex instability is a complex phenomenon and plays a fundamental role in many physical process occurring in nature, ranging from boundary layers to geophysical flows. It is well known that 2D vortices are prone to several type of instabilties, such as centrifugal [14] or elliptical [8], which may lead to transition to complex 3D configurations. Many studies have been undertaken to better understand the mechanism of these instabilities. In this context open flows, are rather difficult to analyze since they develop both in time and space. Confined flows, on the other hand, like those inside a lid-driven cavity, are much easier to study and at the same time show many of the features which are also present in unbounded configurations. The main goal of this paper is the adjoint based analysis of the generalized lid-driven cavity. In the past the attention was paid on the classical square cavity, with a single moving lid. Among the others, the linear stability problem has been studied by [13, 6, 1, 15]. In these papers the three-dimensional stability of a two-dimensional base flow has been considered and the critical value of the Reynolds number is about 786 with a corresponding spanwise wave number approximatively 15.8 [1]. Recently [2, 3, 9, 10] the stability analysis has been exteded also to the case of rectangular cavities, with different aspect ratio, where the motion is driven by the sliding of one or two facing walls and a different instabilities rose up when the driving parameters (aspect ratio and Reynolds numbers) are changed. In this context the adjoint based analysis showed to be a very useful tool to complete the understanding of the mechanisms tha lead to the instability and to characterize the sensitivity of complex flows [5, 7, 11, 12]. Flavio Giannetti, Paolo Luchini DIMEC, Universitá di Salerno, Fisciano (SA) e-mail: [email protected] , e-mail: [email protected] Luca Marino Dip. Meccanica e Aeronautica, Universitá ”La Sapienza”, Via Eudossiana 18, I-00184 Roma email: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_25, © Springer Science+Business Media B.V. 2010

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2 Governing equations and solution method We consider an incompressible Newtonian flow in a rectangular cavity with walls of sizes Lx and Ly , in the x, y-plane. The cavity extends infinitely in the spanwise z direction. The fluid is driven by motion of one or two opposite walls with constant velocity V1 ,V2 and is governed by the classical Navier-Stokes equations. The problem description is completed by the three non-dimensional parameters Re1 = V1 Ly /ν , Re2 = −V2 Ly /ν and Γ = Lx /Ly , where ν is the kinematic viscosity. The stability analysis is carried out by means of a flow field decomposition in a base, stationary and two-dimensional flow (¯v, p) ¯ and a small amplitude, time dependent, three-dimensional perturbation (v′ , p′ ). The base flow (¯v, p) ¯ satisfies the steady, 2D Navier-Stokes equations with boundary conditions v¯ (y = ±Ly /2) = 0 and v¯ (x = ±Lx /2) = ±Re1,2 ey while the perturbation field (v′ , p′ ) is governed by the linearized Navier-Stokes equation (LNSE), with homogeneous boundary conditions. Solutions of the LNSE are searched as normal mode ′ v (x, y,t), p′ (x, y,t) = [ˆv(x, y), p(x, ˆ y)] exp[σ t + iβ z] + c.c. (1) where σ ∈ C is the eigenvalue, β ∈ R is the spanwise wave number and c.c. means the complex conjugate of the preceding expression. Following this assumptions by the following generalized eigenvalue problem is obtained (A + σ B)qˆ = 0,

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where qˆ = (ˆv, p). ˆ Both the base flow and the stability equations are discretized by mean of a second order centered finite differences scheme on a staggered grid. The grid follows a tanh law to have refinement close to the walls. The nonlinear set of the discretized base flow equations is solved adopting a Newton-Raphson iteration numerical scheme, and a sparse LU-solver, while the solution of the eigenvalue problem is achieved by means of an inverse iteration algorithm [7]. In particular the inverse iteration approach here adopted gives simultaneously also the left eigenvector, solution of the problem ˆ + σ B) = 0. (3) p(A Among the consequences of the non-normality of the Navier-Stokes equations we recall that the eigenvectors can experience a strong sensitivity to forcing terms and the eigenvalues can show a large receptivity to structural perturbation [5, 7, 11]. Here we do not report the details of the adjoint analysis but we recall that adjoint global mode quantifies the response to forcing terms and its spatial distribution shows the region of the flow where the global mode is sensitive to localized initial perturbation. In addition the product of the global and adjoint modes gives the sensitivity of the eigenvalues to spatially localized perturbation of the structure of the operator of the LNSE.

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3 Results In this section we report the results of a complete analysis carried out for the generalized li-driven cavity flow. Different aspect ratio and walls motion have been considered. For each case we show only the product of direct and adjoint and the local energy production [v′ · (∇¯v )] · v′ . In fact the transfer of the kinetic energy between the base flow and the perturbation has been adopted as a possible tool to a better understanding of the physical process wich can trigger the instability [2, 10]. The streamlines of the base flow are also plotted. The distribution of the leading global mode and the corresponding adjoint global mode are only described. The analysis has been carried out at Reynolds numbers just above the critical values, so all the cases correspond to unstable flows. Figure 1 shows the classical case of the square cavity with one lid moving. As known, at this value of the Reynolds number a big vortex is present in the center of the cavity and two small recirculating regions can be found in the corners between the fixed walls. The perturbation is localized near the fixed wall on the side where the base flow approaches the moving lid in a thin layer. The adjoint mode is instead localized near the opposite wall where the flow goes away the moving lid and so reduce its speed. The direct-adjoint product presents two spatially localized regions where the sensitivity of the eigenvalue is noticeable. A comparison with the local energy production reveals that this quantity is localized in the same place of the mode, in so failing to localize the sensitivity of the mode to forcing and initial conditions (adjoint) and the part of the sensitivity of the eigenvalue to structured perturbation. Figure 2 refers to a square cavity with two opposite walls moving in the same direction. The scenary is completly different, and apart from the obvious symmetry, the global mode is located almost all in the center of the two vortices. The adjoint distribution shows a sensitivity of the mode near the moving walls and, more pronounced, in the upper region of the cavity, where the streamlines leave the top wall. The eigenvalue sensitivity shows two precise areas of importance, in proximity of the vortex centers and in the upper part of the symmetry line. The local energy transfer reproduces quite well the sensitivity of the upper part of the cavity, but is more weak near the vortex centers, with respect to the directadjoint product. Figure 3 reports a situation corresponding to an instability called elliptical in literature [2, 10]. For this case is less difficult to give an explanation of the mechanism that causes the instability to rise up. Both the global mode and the corresponding adjoint present the maximum in the center of the vortex, with a deformation due to the vortex strain. As a consequence the eigenvalue sensitivity and the energy production are localized in the cavity center. Reducing the aspect ratio (Fig. 3) changes the instability characteristics, pushing almost all the mode towards the walls. This behaviour is recognized also for the adjoint even if a sort of lag in the maximum, respect to the mode, can be noticed. While the direct-adjoint product is not particularly localized the local energy transfer presents two clearly confined regions where this quantity is appreciable. Figure 4 corresponds to a high aspect ratio (2.5) case with antiparallel motion. Referring to [2] the instability is classified as centrifugal and the mode distribution is analogous to that of the classical one liddriven square cavity. In such a case both the adjoint and the direct-adjoint product

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have the same qualitatively spatial distribution as well as the local energy transfer plot. Figure 5 ends this series of studied cases and refers to an elliptical instability. In this case the global mode and the adjoint are located in different region of the cavity, the resulting product is almost confined in the centers of the two vortices. The energy transfer instead is strongly influenced by the direct global mode and is located at the center of the cavity, between the two vortex centers.

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References 1. S. Albensoeder, H.C. Kuhlmann, H.J. Rath. Three-dimensional centrifugal-flow instabilities in the lid-driven cavity problem, Phys. Fluids , 6, 121–135, 2001. 2. S. Albensoeder and H. Kuhlmann. Linear stability of rectangular cavity flows driven by antiparallel motion of two facing walls. J. Fluid Mech.,458:153–180, 2002. 3. S. Albensoeder and H. Kuhlmann. Strained vortices in driven cavities. Phys. Fluids A 15:2453–2456, 2003. 4. S. Albensoeder and H. Kuhlmann. Nonlinear three-dimensional stability flow in the lid-driven square cavity. J. Fluid Mech.,569:465–480, 2006. 5. Chomaz, J.-M.. Global Instabilities in Spatially Developing Flows: Non-Normality and Nonlinearity, Ann. Rev. Fluid Mech., 156, 209-240 , 2005 6. Y.Ding, M. Kawahara. Three dimensional linear stability analysis of incompressible viscous flows using the finite element method. Int. J. Num. Meth Fluids , 31, 451–479, 1999. 7. F. Giannetti and P. Luchini. Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech.,581:21–53, 2006. 8. R.R. Kerswell. Elliptical instability. Ann. Rev. Fluid Mech., (34):83–113, 2002. 9. H.C. Kuhlmann, M., Wanschura, H.J. Rath. Elliptic instability in two-sided lid-driven cavity flow. Eur. J. Mech. B/ Fluids , 17, 561–569, 1998. 10. H. Kuhlmann and S. Albensoeder. Strained vortices in driven cavities. ZAMM-Z.Angew Math. Mech.,85, No. 6:387–399, 2005. 11. L. Marino and P. Luchini. Adjoint analysis of the flow over a forward-facing step. Theor. Comput. Fluid Dyn., 23:37–54, 2009. 12. O. Marquet, M. Lombardi, J-M., Chomaz, D. Sipp and L. Jacquin. Direct and adjoint global modes of a recirculatiion bubble: lift-up and convective non-normalities. J. Fluid Mech., (622):1–21, 2009. 13. N. Ramanan, G.M. Homsy. Linear stability of lid-driven cavity flow. Phys. Fluids,6, 2690– 2701, 1994. 14. D. Sipp and L. Jacquin. Three-dimensional centrifugal-type instabilities of two-dimensional flows in rotating systems. Phys. Fluids, (12):1740–1748, 2000. 15. V. Theofilis, P.W. Duck and J. Owen. Viscous linear stability analysis of rectangular duct and cavity flows. J. Fluid Mech., (505):249–286, 2004.

Bi-global crossplane stability analysis of high-speed boundary-layer flows with discrete roughness Gordon Groskopf, Markus J. Kloker, and Olaf Marxen Abstract Bi-global secondary linear stability theory (B-SLST) is applied to a laminar Mach 4.8 flat-plate boundary-layer flow altered by discrete roughness elements, roughly 0.5 undisturbed boundary-layer thicknesses high. The steady primary state gained by DNS with the immersed-boundary technique shows a counter-rotating vortex pair (CVP) generating a low-speed streak in the roughness’ centerline wake. The resulting wall-normal and spanwise shear promotes 1st-mode instability, increasing the growth rate roughly by a factor of four. Two modes are obtained: a symmetric y-mode and an antisymmetric z-mode. A comparison with an unsteady DNS, where time-periodic 2-d perturbations are introduced upstream of the roughness, shows well matching growth rates and amplitude distribution of the y-mode.

1 Introduction If a flow is dominated by localized structures in its crosscut, e.g., by longitudinal vortices, strong wall-normal and spanwise gradients exist, and thus, the (secondary) instability is localized in the flow crosscut rather than monoharmonic, like it is assumed in classical SLST. An eigenvalue problem with 2-d eigenfunctions results, i.e., we have a bi-global approach (B-SLST). See [7], [4] and [1] for B-SLST applied to incompressible flows with (crossflow-)vortices. In high-speed flows laminar-turbulent transition of the boundary layer is often induced or promoted by discrete 3-d roughness. B-SLST can be used to identify instabilities in the wake of roughness elements that is dominated by a longitudinal CVP and a subsequently developing low-velocity streak. The same holds for setups of effusion cooling (see, e.g., [6]). The modifying mechanisms for disturbance receptivity and instability caused by 3-d roughness elements are still unclear, depending on their shapes, heights, distance, etc. Recently, [2] also started B-SLST work for the investigation of devices to trip hypersonic laminar boundary layers.

2 Bi-global secondary linear stability theory & numerics The applied B-SLST solver is based on the compressible 3-d Navier-Stokes equations in primitive variables: q = [ρ u v w T ]T , where density, velocity components in directions x (streamwise, u), y (wall-normal, v), z (spanwise, w), and temperature are Gordon Groskopf, Markus J. Kloker Universität Stuttgart, Germany, e-mail: (last name)@iag.uni-stuttgart.de Olaf Marxen Stanford University, United States, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_26, © Springer Science+Business Media B.V. 2010

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splitted into steady primary-state (index 1) and unsteady perturbation (superscript ′ ): Φ (x, y, z,t) = Φ1 (x, y, z) + Φ ′ (x, y, z,t). A calorically perfect-gas flow with constant Prandtl number is assumed at this stage. The usual assumptions and requirements for (primary) linear stability theory are applied. Specialties are: • B-SLST allows v1 6= 0 as long as it does not contribute to boundary-layer growth. Hence the mean value or, in spectral space, the zeroth mode of v1 is zero. ˆ z) · ei(α x−ω t) , q(y, ˆ z) is the • In the modal perturbation ansatz q′ (x, y, z,t) = q(y, complex 2-d amplitude distribution; α and ω describe the spatial wavenumber in x-direction and the frequency, respectively. Thus a linear eigenvalue problem results for the temporal approach (α = αr , ω = ωr +i· ωi ), solved using the implicitly restarted Arnoldi method (IRAM) implemented in ARPACK (see [5]). Spatial eigenvalues can be obtained by either Gaster’s transformation or an iteration of the general solution to ωi → 0. Eighth-order finite differences are used for the primary state as well as for the perturbation. Near boundaries in y-direction the order is reduced to four. The primary flow and perturbation are assumed to be periodic in z. At the wall the no-slip condition is prescribed. The temperature perturbation is set to zero. In the freestream all perturbations are assumed to vanish. More details can be found in [3]. Despite the assumption ∂∂x ≡ 0, all variables are taken here directly from the DNS of the primary flow.

3 Primary state A roughness element is placed on an adiabatic flat plate using the immersed boundary method (IBM) in the DNS ([3], [8], [9]). Note that the stability results rely on the primary state, demanding a careful temporal as well as spatial convergence study in computing the primary flowfield. The investigated roughness configuration is a square-box type element in a cold, supersonic flow with Ma∞ = 4.8, T∞ = 55.4 K (T0 = 311 K). Prandtl number and adiabatic exponent are fixed to 0.71 and 1.4, respectively. The Reynolds number based on the reference length L∗ is Re = 105 , thus ρ∞∗ L∗ = 4.87 · 10−4 mkg2 Ns m ∗ with µ∞∗ = 3.49 · 10−6 m 2 and u∞ = 716.15 s . The element is placed at x = 15.0, Rx = 1225, downstream of the leading edge of the plate with a height h = 0.55δu of the unperturbed flow at that position. Fig. 1 and Table 1 show the setups. IIa/b

z

x

z e

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ψ

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case e b I blunt 2.2δu 2.2 √δu IIa sharp 2.2 δ 2.2 2δu u √ IIb small sharp 1.1 2δu 2.2δu

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Fig. 1 Sketch of the investigated roughness settings. Case I: “blunt” box of width e and spanwise spacing sz , Cases IIa/b: “sharp” box. Table 1 Overview of box settings for the investigated cases. Symbols refer to Fig. 1.

x

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sz /b h ψ 3.0 0.55δu 0◦ 2.1 0.55δu 45◦ 3.0 0.55δu 45◦

Fig. 2 indicates the trailing vortices. Between the inner counter-rotating vortices low-momentum fluid is lifted up towards the boundary-layer edge, forming a low-

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speed streak behind the element. The lift-up of vortices in streamwise direction additionally increases the shear near the boundary-layer edge providing the base for increased instability. Cases IIa/b differ from case I in spanwise spacing of the vortices. The u-amplitude of the velocity streak is shown in Fig. 3. Compared to the decaying vorticity ωx of the CVP in streamwise direction it persists along x. The reversal of flow in the near-wake region possibly causing absolute or global instability plays no significant role. Downstream the roughness element the streamwise derivative of the u-velocity is more than one order of magnitude lower than the crosswise derivatives, justifying the B-SLST assumptions.

z

-0.4 -0.2 0 0.2 0.4

Fig. 3 Maximum amplitude of 3-d u-velocity deformation of the flow, streamwise vorticity ωx and u-velocity gradients in (y-z)-planes over streamwise direction x for case I.

max. amplitude

15 16 18 19 20 14 x 17 Fig. 2 Top view of vortex structures in the wake of the roughness element for case I (λ2 criterion, λ2 = −0.04). Flow is from left to right. The shading indicates the height y of the λ2 -isosurface above the wall (the lighter the higher). The meshed (black) isosurface shows the dimension of the recirculation zones (u < 0). Thick solid lines show the borders of the roughness elements.

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4 Results The B-SLST analysis is carried out in (y-z)-crosscut planes at various x-positions in the wake of the roughness element. The spatial growth rates are obtained applying Gaster’s transformation according to [4], see also [1]. Two relevant instability modes have been found. Called y-(or even) mode and z-(or odd) mode they are related to the wall-normal or the spanwise gradients in the flowfield, respectively. Streamwise analysis As a first guess αr,t is chosen corresponding to the most amplified 1st-mode instability. The wavelength is about 6δu ≤ λx ≤ 10δu , hence we set αr,t = 5.0, equivalently αr,t δu = 1.0. Spatial amplification rates αi,S and corresponding N factors for the most amplified y- and z-mode of the investigated cases are shown in Fig. 4. The frequency is a result of the theory and thus, varies along x (max. ∆ ωr,t = ±10%). The maximum amplification rate of both modes is of similar magnitude for all cases, and reaches roughly four times the maximum primary amplification rate of the flat-plate flow at x = 18 (Rx = 1341). However, the characteristics of y- and z-mode differ. While the y-mode exhibits high amplification in the near wake and decays rather quickly in streamwise direction, the z-mode reaches its peak amplification further downstream, and high values of the growth rate seem to persist downstream, which apparently is consistent with ∂∂ uz of the primary state (see

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Fig. 3). Remarkably, the z-mode of case IIb already shows high amplification in the near wake. Up to x = 21 the largest integral growth is observed for the y-mode of R cases I and IIa (Fig. 4, right). Recall that N = αi,S dx, and A/A0 = eN . Assuming that the growth rates of the z-modes persist, their integral growth will exceed the ymodes already slightly further downstream. We note that the analysis at streamwise stations upstream the element did not show notable amplifications. 0.1

1.5

1.5

0.06 0.04 0.02

N factor

−αi,S δu

0.08 1

0.5

1

0.5

0 0

0

15 16 17 x 18 19 20 15 16 17 x 18 19 20 21 Fig. 4 Spatial amplification rates over streamwise direction x (left) and corresponding N factors (right) for the most amplified y-mode (solid) and z-mode (dashed) at x-positions indicated by symbols. Wavenumber αr,S δu = αr,t δu = 1.0. Case I: squares, case IIa: diamonds, case IIb: circles.

Eigenfunctions The moduli Φ ′ = |Φˆ | of the complex eigenfunctions for case I (Fig. 5) are normalized by the highest value occurring in the u′ -distribution. The phase distribution for p shows that the y-mode is symmetric to z = 0 while the zmode is antisymmetric. All shown distributions share that the maxima are located above the sonic line in the supersonic regime at the edge of the deformed boundary layer. For more details about the eigenfunctions, especially regarding the differences between the three investigated roughness setups, see [3]. Wavenumber dependence A scan over αr,t has been carried out at x = 18. The covered range is 0.25 ≤ αr,t δu ≤ 3.0. Results are displayed in Fig. 6. The z-mode is most amplified for a wavenumber smaller than 1. The maximum amplification rate of the z-modes is virtually identical for the three investigated cases. Apart from that, the curves for identical modes show similar behavior. The value αr,t δu = 3.0 is typically associated with the second-/acoustic mode having a streamwise wavelength of about λx = 2δu in a supersonic flat-plate boundary-layer flow. But amplification for this wavenumber was not found in any case for any mode. We note that the 2-d second mode is not amplified in the roughnessless flow at this position either. Comparison with unsteady DNS Concurrent to the theoretical stability analysis, a disturbance calculation using the DNS has been set up for case I. Disturbances are forced using monofrequent 2-d blowing and suction within a spanwise disturbance strip at the wall, a short distance upstream the roughness element. The frequency is ωr,DNS = 4.1, obtained by stability analysis for αr,t δu = 1.0 at x = 18. Due to the symmetric forcing only excitation of symmetric modes, i.e. the y-mode, could be expected. Fig. 7 displays beatings caused by superposition of multiple eigenmodes of identical frequency but slightly different streamwise wavenumbers. A comparative superposition of the amplitude growth from DNS with the N-factor development from stability theory (Fig. 7) shows good agreement. The u′ -amplitude distribution extracted from DNS at x = 18 and normalized by the maximum value in the crosscut shows very good agreement with the B-SLST re-

Stability analysis of high-speed flows with roughness u′

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0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

0.3

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0

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0.2

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0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

y

0.2

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0.1 0

0.1

-0.2

-0.1

0

z

0.1

u′ ,

T′

0

-0.3 -0.2 -0.1 0 z 0.1 0.2 0.3 Fig. 5 Eigenfunctions of and Contours show the modulus of perturbation amplitude Φ ′ = |Φˆ |. y-mode (left) and z-mode (right) are displayed at position x = 18.0 for case I. Thin solid lines are isolines of u beginning with u = 0.1 near the wall and ending with u = 0.95. Thick solid lines mark the sonic line. Last row: Phase distribution tan Ψ ( p) ˆ = ppˆˆri for the p′ eigenfunctions. -0.3

0.2

0.3

p′ .

Fig. 6 Spatial amplification rates over given streamwise wavenumbers indicated by symbols for the most amplified y-mode (solid) and z-mode (dashed) at x = 18.0. Case I: squares, case IIa: diamonds, case IIb: circles.

0.1 0.08

−αi,S δu

0.06 0.04 0.02 0 0.02 0

0.5

1

1.5

2

αr,S δu

2.5

3 1.5

max. amplitude

0.003

1

0.002

0.5

0.001 0

15

16

17

x

18

19

20

21

N factor

Fig. 7 Streamwise amplitude development (max. over y and z) for y-mode (long dash) from unsteady DNS of case I (log scale), regression line (dash-dot) based on maxima (circles) and N-factor development from B-SLST (squares) for y-mode of case I (linear scale).

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sult, especially the position of amplitude maxima is excellently reproduced (Fig. 8). u′

0.3

0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95

0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95

0.2

y

y

0.2

u′

0.3

0.1

0.1

0

0

-0.3 -0.2 -0.1 0 z 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 z 0.1 0.2 0.3 Fig. 8 Comparison: DNS vs. B-SLST. y-mode for case I at position x = 18.0. Left: normalized u-velocity perturbation amplitude u′ from DNS. Right: Contour of modulus of normalized u-perturbation amplitude u′ = |u| ˆ from theory. Thin solid lines are isolines of u beginning with u = 0.1 near the wall and ending with u = 0.95. Thick solid lines mark the sonic line.

5 Conclusions A Mach 4.8 laminar boundary-layer flow altered by discrete 3-d roughness has been investigated regarding its instability behavior. The square-box type roughness element is found not to cause sudden bypass-transition. The horseshoe vortex plays virtually no role for the instability increase. Neither was the recirculation found to cause absolute instability. Rather, the inner pair of counter-rotating vortices past the element and the resulting low-speed streak in the downstream centerline cause a pronounced convective instability with a growth rate roughly four times the primaryinstability value without roughness. For the velocity streak, relevant growth downstream the roughness element was not found. So the roughness acts as an amplifier for 1st-mode instabilities. According to the B-SLST the sharp-type roughness arrangement is slightly more dangerous to laminar flow than the blunt setting. Generally, y- and z-mode exhibit similar amplification rates. But as the amplification of the z-mode persists further downstream, this mode might be more important for transition. It also has a somewhat smaller streamwise wavenumber, between 0.5 and 1.0, non-dimensionalized by the boundary-layer thickness, for the most amplified modes. The existence of the y-mode could be confirmed by an unsteady DNS, and comparison of the disturbance shapes shows excellent agreement. The growth rates could only be compared approximately, but also good agreement was found.

References 1. B ONFIGLI G. & K LOKER M. J. 2007 Secondary instability of crossflow vortices: validation of the stability theory by direct numerical simulation. J. Fluid Mech. 583, 229–272. 2. C HOUDHARI M. M., L I F. & E DWARDS J. 2008 Advanced stability analysis pertaining to roughness effects on laminar-turbulent transition in hypersonic boundary layers. NASA AAP Quarterly Highlights Hypersonics Project, Vol. 2, No. 2, April 2008. 3. G ROSKOPF G., K LOKER M. J. & M ARXEN O. 2008 Bi-global secondary stability theory for high-speed boundary-layer flows. In Proceedings of the 2008 Summer Program, CTR, Stanford, Calif., July 6–August 1. 4. KOCH W., B ERTOLOTTI F. P., S TOLTE A. & H EIN S. 2000 Nonlinear equilibrium solutions in a threedimensional boundary layer and their instability. J. Fluid Mech. 406, 131–174. 5. L EHOUCQ R. B., S ORENSEN D. C. & YANG C. 1998 ARPACK User’s Guide. SIAM, Philadelphia, Penn. 6. L INN J. & K LOKER M. J. 2008 Numerical investigations of film cooling and its influence on the hypersonic boundary-layer flow. NNFM 98, final reviewed papers of the HGF virtual institute RESPACE Key technologies for re-usable space systems (ed. Guelhan, A.), Springer-Verlag, Berlin, 151–169. 7. M ALIK M. R. & C HANG C.-L. 1994 Crossflow disturbances in three-dimensional boundary layers: nonlinear development, wave interaction and secondary instability. J. Fluid Mech. 268, 1–36. 8. M ARXEN O. & I ACCARINO G. 2008 Numerical simulation of the effect of a roughness element on high-speed boundary layer instability. Thirty-eighth AIAA Fluid Dynamics Conference & Exhibit, 23-26 June / Seattle, Wash. AIAA Paper 2008-4400. 9. NAGARAJAN S., L ELE S. K. & F ERZIGER J. H. 2003 A robust high-order method for large eddy simulation. Physics of Fluids A 4 (4), 710–726.

Time-resolved PIV investigations on the laminar-turbulent transition over laminar separation bubbles Rainer Hain, Christian J. Kähler, and Rolf Radespiel

Abstract At Reynolds-numbers Re = 2 · 104 − 1 · 105 a laminar separation bubble occurs typically on the suction side of a thin airfoil like the SD 7003. The transition in the shear layer of the separation bubble shows a dynamic behavior which was investigated by means of the time-resolved Particle Image Velocimetry in the past. The development of vortices which arise from Kelvin-Helmholtz instabilities is studied in detail in this paper. Different experimental setups and angles of attack were evaluated in order to get detailed information of the structures that occur in the region of the separation bubble.

1 Introduction Laminar separation bubbles (LSBs) have been subject of many experimental and numerical investigations in recent years [1, 2, 5, 6, 9, 10, 11]. There are different reasons for the strong interest in this flow phenomenon. One reason is the influence of such a bubble on the flow around an airfoil. On one hand, the pressure distribution is altered and on the other, the separation behavior at high angles of attack is strongly modified. This may lead to problems in the case of the bubble bursting [3, 7] which means an abrupt increasing of the size of the bubble. This leads to an abrupt stall

Rainer Hain Institute of Fluid Mechanics and Aerodynamics, Bundeswehr University Munich, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany, e-mail: [email protected] Christian J. Kähler Institute of Fluid Mechanics and Aerodynamics, Bundeswehr University Munich, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany, e-mail: [email protected] Rolf Radespiel Institute of Fluid Mechanics, TU Braunschweig, Bienroder Weg 3, 38106 Braunschweig, Germany, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_27, © Springer Science+Business Media B.V. 2010

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R. Hain, C. J. Kähler, R. Radespiel

and therefore to a loss of lift. In addition, the pressure drag of an airfoil is increased when a LSB is present. Another reason for the numerous investigations on laminar separation bubbles is the complicated transition which is observed when a bubble occurs. A vortex development in the shear layer in the region of the bubble is observed, which leads in a later stage to the development of a turbulent flow with small scale turbulent structures. A phenomenon which sometimes is observed in the presence of bubbles is the so called flapping. This constitutes to a low-frequency motion of the separated shear layer in the direction normal to the wall. Flapping is not fully understood so far. It is observed in many experimental investigations as well as in some numerical simulations. However, in the case of experiments as well as in the case of numerical simulations it cannot be excluded that this phenomenon is caused by boundary conditions. Therefore a detailed analysis was performed for the measurements presented here in order to determine the origin of flapping and the dynamics of the laminar separation bubble. The bubble was investigated at different angles of attack. Here, the results for α = 8◦ , obtained using a time-resolved PIV system, are discussed. The results for other angles of attack and further details are given in [5, 6, 13].

2 Experimental setup and data evaluation For the time-resolved PIV measurements presented here the light-sheet was aligned normal to the SD 7003 airfoil and parallel to the main flow direction. A continuouswave Argon-Ion laser was used for illumination of the seeding particles (hollow glass spheres with a mean diameter of d = 10 µ m). The recording was done with a pco.1200 hs highspeed camera from PCO. These measurements were performed in a water tunnel with a contraction of 4 and a 1250 mmL × 250 mmW × 330 mmH test section. The turbulence level of this facility is 0.28 % at a flow velocity of 0.31 ms−1 . This flow velocity leads, with the chord length of the airfoil of c = 200 mm, to a Reynolds number of Re = 66000. The evaluation of the PIV data was performed by means of the state-of-the-art software DaVis 7.2 from LaVision. In order to enhance the accuracy, the evaluation approach presented by [4] was applied in addition. This method allows an increase of the dynamic velocity range and thus a reduction of the relative measurement error. The acquisition frequency was 1798.5 Hz and 900 vector fields per second were computed. The total length of the image sequence is 2.24 s.

3 Results Time-resolved PIV allows capturing the dynamics of flow structures in a plane. Obviously such data can be used for the analysis of the dynamics in the time domain.

Time-resolved PIV investigations on laminar separation bubbles

179

In addition, this data can be transformed in the frequency domain for analyzing the most important frequencies and the increasing of amplitudes as a function of the location. Visualizations in the time domain are given in Figs. 1 and 2. Fig. 1 shows an instantaneous velocity field of the flow in the region of the bubble. The flowing-off vortex which was developed in the shear layer can clearly by identified by means of the streamlines. In addition, the recirculation area becomes visible. In order to visualize the time-dependence of the flow, the λ2 -criterion [8] was applied. Isosurfaces of λ2 are given in Fig. 2 as a function of the normalized airfoil coordinates and the normalized time tnorm = t · U∞ /hLSB . By means of the λ2 vortex detection criterion, the periodicity of the vortices which are developed in the separated shear layer can be observed. Only a section of the whole sequence is shown in this figure. However, the strong periodicity is clearly seen. As shown in [5, 6] for an angle of attack of α = 4◦ this strong periodicity is not observed. The most likely reason are variations of the disturbances in the oncoming flow. According to the linear stability theory, these variations are similar to variations of the initial amplitude. As shown

Fig. 1 Instantaneous velocity field with streamlines. Every 4th vector in x- and every 2nd vector in y-direction shown. Re = 66000 and α = 8◦ .

Fig. 2 Iso-surfaces of the λ2 -criterion. Re = 66000 and α = 8◦ .

180


Fig. 3 Dominant Strouhal number of u′ /U∞ (left hand side) and dominant Strouhal number of v′ /U∞ (right hand side). Re = 66000 and α = 8◦ .

Fig. 4 Amplitude of u′ /U∞ in dependence on the Strouhal number and y/c for x/c = 0.12 (left hand side) and x/c = 0.17 (right hand side). The smallest y/c-location is given by the airfoil surface. Re = 66000 and α = 8◦ .

by [9], even a small variation of the initial amplitude has a strong influence on the size of the separation bubble. Thus, a variation of the initial amplitude leads to a flapping of the bubble. Such variations of amplitudes in dependence on the time have also been observed in free-flight conditions [12]. Seitz et al [12] investigated Tollmien-Schlichting waves and found that they occur in packets. This may lead to a flapping in free-flight conditions as well. Performing Fourier transforms at every vector location separately for the velocity components in x- and y-direction leads to Fig. 3 which shows the dominant Strouhal number, Sr = f · hLSB /U∞ , with the corresponding magnitude. The dominant Strouhal number is this one with the largest magnitude in the Fourier space. For the given flow conditions of Re = 66000 and α = 8◦ , a strong increase of the magnitude of the dominant Strouhal number (Sr = 0.25) at x/c ≈ 0.15 is observed. This corresponds to the location where the strong vortex shedding starts, see Fig. 2.

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Upstream of this location the dominant Strouhal number near the airfoil is in the range of Sr ≈ 0.004. At this frequency, low amplitude variations in the turbulence spectrum of the water tunnel were observed, see [6]. At larger distances from the airfoil no dominant frequency is found, which leads to the noisy structure of the dominant Strouhal number in this area. Fourier spectra for locations x/c = 0.12 and x/c = 0.17 are given in Fig. 3. The magnitude is color-coded in this view. For location x/c = 0.12 a peak in the Fourier spectrum is observed at Sr = 0.004; y/c = 0.0555 (Fig. 4 left hand side). The peak at Sr = 0.25 is quite weak at this location. Further downstream at x/c = 0.17 the peak at Sr = 0.25 has increased strongly (Fig. 4 right hand side) while this one at Sr = 0.004 remains nearly constant. Additional peaks at other locations also occur at location x/c = 0.17. This is due to the turbulence which is developed. However, their magnitudes are much smaller than that one at Sr = 0.25. A decrease of the magnitude at Sr = 0.25; y/c = 0.059 can be seen in this figure. This results from the flow off of the vortices. Along their centers they do not induce a velocity fluctuation in x-direction. This was already observed in Fig. 3. Contrarily a velocity fluctuation is induced by the vortices also in y-direction along the paths of the vortex centers.

4 Conclusion The results presented clearly reveal that time-resolved PIV is well suited for measuring dynamic phenomena in a plane. An image sequence obtained by means of TR-PIV can be used for analyzing the phenomena in the time as well as in the frequency domain by performing a Fourier transform at each vector location. The visualization of the vortices in the time domain showed a periodic development of vortices in the shear layer of the separation bubble due to Kelvin-Helmholtz instabilities. By means of the frequency spectra, a single dominant vortex shedding frequency was found for the angle of attack of α = 8◦ . This is different from the observations for e.g. α = 4◦ where not only a single vortex shedding frequency occurs, see [5, 6]. Looking at the spatial distribution of the amplitude at Sr = 0.25 for the angle of attack of α = 8◦ a strong growth at x/c = 0.15 is observed. At this location the shedding of large vortices takes place. Acknowledgements The authors greatly acknowledge the support of the German Research Foundation (DFG) in the priority program SPP 1147 ”Bildgebende Messverfahren für die Strömungsanalyse”.

References 1. Alam M, Sandham ND (2000) Direct numerical simulation of ‘short’ laminar separation bubbles with turbulent reattachment. J Fluid Mech 410:1–28

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2. Burgmann S, Brücker C, Schröder W (2006) Scanning PIV measurements of a laminar separation bubble. Exp Fluids 41:319–326 3. Gaster M (1966) The structure and behaviour of laminar separation bubbles. AGARD CP4:813–854 4. Hain R, Kähler CJ (2007) Fundamentals of multiframe particle image velocimetry (PIV). Exp Fluids 42:575–587 5. Hain R (2008) Untersuchungen zur Dynamik laminarer Ablöseblasen mit der zeitauflösenden Particle Image Velocimetry. PhD thesis, TU Braunschweig, published by Shaker Verlag, Aachen, Germany 6. Hain R, Kähler CJ, Radespiel R (2009) Dynamics of laminar separation bubbles at lowReynolds-number aerofoils. J Fluid Mech 630:129–153 7. Horton H (1968) Laminar Separation Bubbles in Two and Three Dimensional Incompressible Flow. PhD thesis, Department of Aeronautical Engineering, Queen Mary College, University of London 8. Jeong J, Hussain F (1995) On the identification of a vortex. J Fluid Mech 285:69–94 9. Jones LE, Sandberg RD, Sandham ND (2008) Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence. J Fluid Mech 602:175-207 10. Lang M, Rist U, Wagner S (2004) Investigations on controlled transition development in a laminar separation bubble by means of LDA and PIV. Exp Fluids 36:43–52 11. Ol MV, Hanff E, McAuliffe B, Scholz U, Kähler C (2005) Comparison of laminar separation bubble measurements on a low Reynolds number airfoil in three facilities. 35th AIAA fluid dynamics conference and exhibit, Toronto, Canada, AIAA Paper 2005-5149, June 6-9 12. Seitz A, Horstmann KH (2006) In-Flight Investigations of Tollmien-Schlichting Waves. In: Meier GEA, Sreenivasan KR, Heinemann HJ (eds) IUTAM Symposium on One Hundred Years of Boundary Layer Research. Springer, Dordrecht, Netherlands 13. Zhang W, Hain R, Kähler CJ (2008) Scanning PIV investigation of the laminar separation bubble on a SD7003 airfoil. Exp Fluids 45:725–743

Control of transient growth induced boundary layer transition using plasma actuators Ronald E. Hanson, Philippe Lavoie, Ahmed M. Naguib, and Jonathan F. Morrison

Abstract This study investigates an actuation scheme that can be readily implemented and integrated as part of a feedback control system in the laboratory for the purpose of negating the effect of the transient growth instability in order to delay boundary layer transition. The actuators investigated here consist of a spanwise array of symmetric plasma actuators, which are capable of generating spanwise periodic counter-rotating vortices. Two different actuator geometries are investigated, resulting in a reduction of the total disturbance energy by 46% and 68%. It is demonstrated that the control authority of an actuator can be significantly improved by optimizing the geometry of the array.

1 Introduction The location at which a boundary layer transitions to turbulence depends on several factors including free-stream turbulence, surface roughness and pressure gradient. At higher disturbance levels, perturbations can manifest themselves as three dimensional instabilities undergoing transient growth that bypass the classical transition pathway and result in transition at sub-critical Reynolds numbers. More details on boundary layer transition can be found in Saric et al. [10]. Transient growth results from the coupling of the Orr-Sommerfeld and Squire equations, which describe the evolution of linear, three-dimensional perturbations. The optimal perturbation (causing the greatest growth over a selected streamwise length) takes the form of counter-rotating streamwise vortex pairs [6], which maniRonald E. Hanson, Philippe Lavoie Institute for Aerospace Studies, University of Toronto, Canada e-mail: [email protected]; [email protected] Ahmed M. Naguib Dept. of Mech. Engineering, Michigan State University, USA e-mail: [email protected] Jonathan F. Morrison Dept. of Aeronautics, Imperial College, UK e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_28, © Springer Science+Business Media B.V. 2010

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fests as a spanwise periodic modulation of the streamwise velocity in the boundary layer. The corresponding eigenmodes are non-orthogonal [1] resulting in algebraic growth of the energy followed by an exponential decay [8, 11]. This mechanism can be simulated in the laboratory using a spanwise array of roughness elements [3]. The steady disturbance introduced by a roughness array occurs at a predefined wavenumber and amplitude [12] defined by the roughness element geometry (height, diameter, etc) and spacing. Using this method, the authority of an actuator on the disturbance can be systematically studied before integration with a feedback control system. Actuator development is often considered to be the pacing item in many flow control studies [4]. The focus of the study is toward the development of actuators for future active boundary layer control studies in the laboratory. The present research is inspired by the work of Jacobson & Reynolds [4] who demonstrated some success in cancelling a stationary vortex pair imbedded in a boundary layer using a mechanically generated synthetic jet that produced opposing vorticity. For this study, single-dielectric-barrier-discharge (SDBD) plasma actuators, herein referred to as plasma actuators, are investigated due to their many advantages. These include no moving parts, they can be mounted flush to the wall and have demonstrated ability to generate streamwise vorticity [9].

2 Experimental Details Measurements were made in an open-loop wind tunnel at Michigan State University for which the working section is 0.35 m × 0.35 m and 2.8 m long. The test section walls diverge by an angle of 0.13 degrees with respect to the centerline to minimize the pressure gradient along the working section. The turbulence intensity of the test section is approximately 0.05%. A baseline laminar boundary layer was established on an acrylic plate, 0.635 m long, spanning the width of the test section. The plate was mounted between 1/3 and 1/4 of the test section height to minimize the potential effects from contraction-induced secondary flows. The leading edge was machined from aluminum to a knife edge and a 0.152 m long adjustable flap was used to ensure the stagnation point was located on the measurement side of the plate. The laminar boundary layer followed the zero pressure gradient Blasius solution. This was verified quantitatively using the shape factor, H12 = δ ∗ /θ , where δ ∗ is the displacement thickness and θ is the momentum thickness. The shape factor remained between 2.59 and 2.64 over the length of the plate for U∞ = 5 m/s that was constant for all experiments. To induce transient growth, five cylindrical roughness elements with a diameter of d = 5 mm and height of h = 1.29 mm were spaced 20 mm apart in the spanwise direction (∆ z) and located at 150 mm from the leading edge. The spanwise position of the roughness element could be adjusted to align the roughness and actuator arrays. A schematic of the apparatus is shown by Fig. 1(a). The dimensionless fundamental spanwise wavenumber (β = 2πδ /∆ z, where δ is the Blasius similarity length scale) at the roughness elements is ≈ 0.196 such that the disturbance energy grows over the full length of the plate in a manner representative of transient growth [5].

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In this work an array of plasma actuators in a symmetric electrode configuration was manufactured (see Fig. 1b). The actuator was oriented such that it generates pairs of impinging wall jets causing spanwise periodic streamwise vorticity [9]. The leading edge of the actuator array was located 100 mm downstream of the roughness array and is centered behind the middle roughness element such that the high voltage (HV) electrodes are located between the roughness elements. The plasma actuator had four surface mounted HV electrodes that were 40 mm long and were spaced 20 mm apart (∆ z). Two different actuator geometries, differing only in the width of the HV exposed electrodes, were tested. Actuator geometry A had a HV exposed electrode width (WHV ) of 5 mm, while WHV = 7 mm for actuator geometry B. Two layers of Kapton film tape are used as the dielectric, each with a total thickness of 90 µ m. Copper foil tape is used for the electrodes and was 74 µ m thick. An adjustable wall plug was used to ensure the actuator array was flush mounted with the boundary layer plate. Details concerning plasma actuator operation, physics and application can be found in the reviews by Corke et al. [2] and Moreau [7]. a)

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3 Results A Contour plot of the velocity disturbance generated at 300 mm downstream of the roughness array is plotted in Fig. 2(a), where u = U− , U is the local velocity and is the spanwise-averaged velocity at the corresponding wall normal distance. This plot is obtained from 32 evenly spaced wall normal profiles of the streamwise velocity using a single hot-wire probe. Spanwise averages were performed over two wavelengths of the disturbance. The presence of high- and lowspeed streaks downstream of the roughness array is clearly observed. Similar to the results of Fransson et al. [3], the high-speed streaks are located downstream of the elements. At the location of the maximum disturbance energy (η ≈ 2.1), the power spectrum of the streamwise velocity, φu , is calculated along the spanwise direction and is used to identify the modal content of the disturbance (see Fig. 2(b), where φu is normalized with U∞ ). As shown in the figure, the disturbance is composed primarily of three modes. The first-mode wavelength is defined by the spacing of the roughness elements and the second two are harmonics of the first. Narrow-band spatial filtering of u/U∞ at η ≈ 2.1 is used to reconstruct the first three spanwise modes and are plotted in Fig. 2(c).

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Fig. 2 Velocity contours of the streamwise disturbance (u/U∞ ) at x = 450 mm for (a) the roughness array, (d) produced by actuator A only and (g), with actuator A operating on the disturbance from the roughness array. (b, e, h) Power spectra calculated at η ≈ 2.1 and corresponding with each adjacent contour plot. (c, f, i) First three fundamental modes, — mode 1, – – 2, – · – 3.

Actuator A was driven by an AC ramp signal with amplitude Vpp = 4.4kV and frequency f a = 3 kHz to counteract the disturbance. The velocity disturbance produced by this actuator on the boundary layer in the absence of the roughness array upstream is shown in Fig. 2(d). From this figure it is evident that there is higher wavenumber content, which results in the double peaked low-speed region of the central low-speed streak. This is more clearly observed from the power spectrum (Fig. 2e) and the spatial signature of the individual modes (Fig. 2f) at η = 2.1, where the first three modes are plotted. As shown by this figure, the first mode of the actuator is nearly out of phase but at a higher amplitude than the disturbance. The actuation frequency was lowered to 2.75 in order to provide a nearly identical disturbance amplitude. A contour plot of u/U∞ , given in Fig. 2(g), shows attenuation of the disturbance by the actuator array when both occur simultaneously. It is evident from Fig. 2(h) that the first mode targeted by the actuation is attenuated. The maximum amplitude of the first mode is reduced by 77%. The observed phase shift of the residual of the first mode (compare Fig. 2i to 2c) is attributed to a combination of phase and amplitude misalignment between the actuator and roughness-element disturbances. The second mode in the controlled case is amplified for this actuator geometry since this mode is in phase for both the roughness and actuator arrays. However, the second mode corresponds to a larger value of β and is thus more likely to be stable than the first mode [5]. The third mode was out of phase causing destructive interference. However that component produced by the actuator was nearly twice that of the initial disturbance resulting in a residual of similar initial amplitude but of opposite phase. The specific modal content of the disturbance produced by the plasma actuator array varies depending on the actuator geometry. For example, the central low-speed streak generated by actuator A was comparatively wide to the central high-speed streak generated by the roughness array and was modified by increasing the width of the upper HV electrodes to 7 mm. The disturbance velocity plot, shown by Fig.

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Fig. 3 Velocity contours of the streamwise disturbance (u/U∞ ) at x = 450 mm for (a) actuator B only and (d), with actuator B operating on the disturbance from the roughness array. (b, e) Power spectra calculated at η ≈ 2.1 and corresponding with each adjacent contour plot. (c, f) First three fundamental modes, — mode 1, – – mode 2, – · mode 3 at η ≈ 2.1.

3(a), confirm a better qualitative match of the disturbance and counter disturbance since the wider HV electrodes caused a narrowing of the central low-speed streak. Actuator geometry B was driven by an AC ramp signal with amplitude Vpp = 4.4kV and frequency fa = 3 kHz. For this case, the resulting amplitude of the first mode (see Fig 3c) is within 5% of that caused by the spanwise array. The actuator did not generate a dominant second mode, however a strong third mode persists. The third mode is nearly out of phase resulting in destructive interference. With the roughness array in place, a strong level of cancellation is achieved, as shown by Fig. 3(d). The first mode maximum amplitude is reduced by 84% for this case. The above discussion focused on one wall-normal location. The overall control authority of each actuator can be quantified by comparing the average power spectrum of the normalized streamwise velocity disturbance present inside the boundary layer for the initial disturbance and the control cases using actuator A (Fig. 4a) and B (Fig. 4b). For practical reasons, the average power spectrum is calculated over the wall normal location where the disturbance level is greater than ±20% of the maximum value disturbance. The total decrease of the energy in the control case was 46% and 68% by actuator A and B, respectively. It is evident, from Fig. 4(a) and (b), that both of the actuators investigated caused a strong attenuation of the first mode defined by ∆ z. The second mode generated by actuator A was in phase with that caused by the disturbance and is therefore responsible for the increase in energy associated with that mode. Actuator B, however, did not produce significant energy at that the second mode, and therefore caused negligible effects. In both cases an increase in the strength of the third mode occurred.

4 Conclusions and Outlook Plasma actuators were used to attenuate the transient growth of the disturbance produced by an array of roughness elements. It was demonstrated that this type of actuator can effectively negate the disturbance produced by transient growth. Roughness induced transient growth is primarily composed of a mode related to the spanwise spacing of the roughness elements. It is also composed of additional higher wavenumber modes, which are weaker. The spanwise array of plasma actu-

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Fig. 4 Average power spectrum of u/U∞ calculated over ±20% of the maximum streamwise disturbance velocity range at X = 450 mm for (a) Actuator A, with WHV = 5 mm and (b) Actuator B, with WHV = 7 mm.

ators used here are found to generate a primary mode predetermined by the spacing between the upper exposed HV electrodes and additional weaker modes at integer multiples of the first, similar to that of the roughness array. Two actuators (A and B), with different geometry, were found to attenuate the energy of the first mode by as much as 94% in one instance and 96% in the other. The corresponding broadband (multi-modal) reduction was 46% and 68% for actuators A and B respectively. This improvement is attributed to the geometry modifications suggesting that these actuators can be optimized to minimize energy input at secondary wavelengths.

References 1. Butler KM, Farrell BF (1992) Three-dimensional optimal perturbations in viscous shear flow. Phys Fluids A 4(8):1637–1650 2. Corke TC, Post ML, Orlov DM (2007) SDBD plasma enhanced aerodynamics: concepts, optimization and applications. Prog Aero Sci 43:193–217 3. Fransson JHM, Brandt L, Talamelli A, Cossu C (2004) Experimental and theoretical investigation of the nonmodal growth of steady streaks in a flat plate boundary layer. Phys Fluids 16(10):3627–3638 4. Jacobson SA, Reynolds WC (1998) Active control of streamwise vortices and streaks in boundary layers. J Fluid Mech 360:179–211 5. Lavoie P, Naguib AM, Morrison JF (2008) Transient growth induced by surface roughness in a Blasius boundary layer. XXII ICTAM, Adelaide, Australia 6. Luchini P (2000) Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J Fluid Mech 404:289–309 7. Moreau E (2007) Airflow control by non-thermal plasma actuators. J Phys D 40:605–636 8. Reshotko E (2001) Transient growth: a factor in bypass transition. Phys. Fluids 13(5):1067– 1075 9. Roth JR, Sherman DM, Wilkinson SP (2000) Electrohydrodynamic flow control with a glowdischarge surface plasma. AIAA J 38:1166–1172 10. Saric WS, Reed HL, Kerschen EJ (2002) Boundary-layer receptivity to freestream disturbances. Annu Rev Fluid Mech 34:291–319 11. Schmid PJ, Henningson DS (2001) Stability And Transition In Shear Flows. Springer-Verlag, New York 12. White EB (2002) Transient growth of stationary disturbances in a flat plate boundary layer. Phys Fluids 14(12):4429–4439

Laminar Flow Control by Suction at Mach 2 S. Hein, E. Schülein, A. Hanifi, J. Sousa, and D. Arnal

Abstract Laminar flow control by suction of crossflow-dominated laminar-turbulent transition at Mach 2 was studied experimentally and numerically as part of the EU project SUPERTRAC. The measurements were performed in the Ludwieg Tube Facility (RWG) at DLR Göttingen. With a suction panel manufactured from porous sinter material a significant delay of laminar-turbulent transition could be achieved. Transition N-factors in the range of 5 to 7.5 were obtained for both the reference model and the model with suction system. Thus, a reduction of the transition Nfactor due to increased roughness and the non-uniform suction distribution on the suction panel was not noted. Moreover, the experimental and numerical results indicated that transition was dominated by travelling crossflow instabilities.

1 Introduction For subsonic and transonic flight Mach numbers the feasibility and potential benefits of laminar flow control (LFC) by suction have been demonstrated in several wind tunnel and flight experiments already. Comparable know-how on LFC by suction of supersonic boundary layers is not available yet. A comprehensive overview of the different experiments on LFC by suction can be found in [3]. Therefore, experimental and numerical studies on LFC by suction of crossflow-dominated transition on an infinite swept-wing configuration at Mach 2 were performed recently as part of the EU project SUPERTRAC (SUPERsonic TRAnsition Control) [1]. The present paper summarizes some major results from these studies. S. Hein, DLR, Bunsenstr.10, 37073 Göttingen, Germany, e-mail: [email protected] E. Schülein, DLR, Bunsenstr.10, 37073 Göttingen, Germany, e-mail: [email protected] A. Hanifi, FOI, SE-164 90, Stockholm, Sweden, e-mail: [email protected] J. Sousa, IST, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal, e-mail: [email protected] D. Arnal, ONERA, 2, Av. Ed. Belin, 31055 Toulouse, France, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_29, © Springer Science+Business Media B.V. 2010

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2 Experimental Set-Up The Ludwieg Tube Facility (RWG) at DLR Göttingen equipped with the Mach 2 nozzle and a test chamber cross-section of 0.34 m × 0.35 m was chosen for these experiments. Consequently, the relatively small dimensions of the wind tunnel model required rather large unit Reynolds numbers which led to rather thin boundary layers. Therefore, the wind tunnel model was designed based on the results of a detailed numerical trade-off study. In view of the very limited space for integration of the suction system in the wind tunnel model only single suction chamber designs were investigated. The finally built wind tunnel model is based on a symmetric arc-shaped aerofoil with relative thickness of t/c = 0.13, sharp leading edge and a nominal chord length of c = 300 mm, but on the side which was used for the measurements it follows the nominal contour up to 70 percent nominal chord only (Fig. 1). The suction panel located between x/c = 0.05 and x/c = 0.20 was manufactured from a 3 mm thick sinter plate SIKA-R10AX with a volume porosity of 43 %. According to the specifications provided by the manufacturer of the sinter material this corresponds to an equivalent hole diameter of 17 µ m. The mean roughness after polishing was approximately between 10 and 20 µ m on the suction panel and about 1 µ m elsewhere. Because the sinter material used has a comparable equivalent hole diameter, further polishing could not improve the surface quality without affecting its permeability. The residual bluntness of the nominally sharp leading edge was approximately 0.1 mm. A blunt leading edge geometry with nose radius of e.g. 1 mm would have required stronger suction even closer to the leading edge where there would have been not enough integration space for the suction system and where the boundary layer is even thinner. The model was equipped with pressure transducers for comparison with the numerically predicted surface pressure distribution and was mounted in the test section at zero angle of attack with a fixed sweep angle ϕ of either 20 or 30 degrees. Even at 30 degrees sweep angle there was still a large enough spanwise region for the measurements where it could be assumed that infinite swept-wing conditions hold. Details of the wind tunnel model set-up and the implementation of the suction system can be found in [4].

3 Experimental Results In a first wind tunnel test campaign the reference model without suction system was used. The unit Reynolds number was changed systematically for both sweep angles considered. Up to 20 percent chord, i.e. on the metallic part of the model surface, the transition location was detected by oil-film interferometry. Further downstream, where the model surface is manufactured from black PLEXIGLAS with low thermal conductivity, infrared thermography was used. The transition location xtr was defined as the position of the local minimum in the skin friction and the wall heat flux followed by a subsequent rapid rise. A smooth upstream movement of the mea-

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Fig. 1 Wind-tunnel model with suction panel in the open test section of the RWG facility (from [4])

sured transition location with increasing unit Reynolds number was noted even at 20 percent chord where the transition detection technique changes suggesting that both techniques provided consistent results. Then the suction system and the suction panel made from porous sinter material were integrated between 5 and 20 percent chord. Based on the experimental results for the reference model without suction panel three different freestream conditions for the subsequent studies on the effects of suction were selected. In this second test campaign only IR thermography could be used for transition detection. The transition locations as a function of the mean suction velocity are summarized in Fig. 2. Note that the results with zero suction velocity in Fig. 2 are for the reference model measured before the suction system was integrated in the workshop. Measurements on the effects of the increased surface roughness due to the suction panel but with zero suction velocity were not possible with the current set-up. Moreover, the suction mass flow rate could not be measured accurately enough during the experiment due to the rather small total mass flow and the short run time of the experiment. Thus, the given suction velocities are reference values which were only approximately determined from the measured pressure drop over the suction panel at 14 percent chord using previously obtained calibration data [4]. Fig. 2 shows that a significant delay of laminar-turbulent transition by suction could be achieved in this supersonic wind tunnel experiment. Above a certain level of suction a further increase of the mean suction velocity did not move the transition location further downstream, however. This observation might be related to test section side wall effects. The measured surface pressure distributions for both sweep angles showed small deviations from the calculated ones in the rear part of the wind tunnel model independent of the suction process [4]. Oil flow visualizations indicated that the bow shock of the model is reflected by the wind tunnel walls and impinges on the model surface at x/c ≈ 0.50. Therefore, downstream of x/c ≈ 0.50 the observed transition locations may be affected by those side wall effects.

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4 Numerical Analyses The pressure distribution on the model surface was calculated by shock-expansion methods or by Euler computations. A first-order boundary-layer code was used to compute the laminar boundary-layer profiles. Then, the instability characteristics of these profiles were studied both by classical linear local stability theory and by nonlocal theory based on linear parabolized stability equations (PSE). The constant frequency – constant spanwise wavenumber N-factor integration strategy was used and both frequency and spanwise wavenumber were varied systematically. As usual, surface curvature effects were taken into account in the nonlocal approach only. The computations have been performed for selected data points of Fig. 2. The nonlocal transition N-factors Ntr corresponding to the transition locations in the experiments with different mean suction velocities Vs are compiled in Fig. 3 and Fig. 4. For the two cases with 30 degrees sweep angle the nonlocal transition N-factors based on total kinetic energy are approximately in the range of 5 to 7.5 if both stationary and travelling crossflow modes are taken into account. With suction one usually would expect lower transition N-factors compared to the reference wing due to additional disturbances introduced by the increased surface roughness and the non-uniformity of the suction velocity on the perforated suction panel. No such tendency is visible in the data for sweep angles of 30 degrees, however. For the case with a sweep angle

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of 20 degrees the transition N-factor without suction panel is higher than those obtained with suction. Therefore, for the case with 20 degrees sweep there might be a reduction in transition N-factor by suction, but this interpretation strongly depends on the reliability on the single reference data point without suction. The transition N-factor values for the other 20 degrees cases, i.e. those with non-zero suction, are in the same range as those found for the 30 degrees cases. If only stationary crossflow modes were considered, the resulting transition N-factors would be unrealistically small. Qualitatively similar results were obtained from classical local stability theory and thus are not presented here. 10 All modes Stationary modes

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5 Conclusions An experiment on laminar flow control by suction of crossflow-dominated transition at supersonic Mach number was successfully performed in the Ludwieg Tube Facility (RWG) at DLR Göttingen. With the current set-up a substantial gain in laminarity could be achieved. A single suction chamber covered by porous sinter material was used. Suction panels manufactured from sinter material to our knowledge have never been used successfully before. The analysis of the experimental results provided transition N-factor values with relatively small variations independent of level of suction rate, sweep angle and unit Reynolds number. Similar values were also found for the reference configuration, at least for the cases with larger sweep angle. Hence, the additional disturbances introduced by the increased relative height of geometrical surface roughness of the suction panel and by the non-uniform suction distribution seemed to be negligible in the current experiments. The latter may also be attributed to the rather small equivalent hole velocities of the suction panel used. This suggests that, in the current experiments, the transition process is dominated by non-stationary crossflow disturbances which are not sensitive to surface inhomogeneities. Further evidence for this assumption is provided among others by the fact that the computed transition N-factors for stationary crossflow vortices are very small and by the experimental observation that transition Reynolds number increases with unit Reynolds number. An opposite tendency would be expected if the transition process were dominated by stationary crossflow vortices as those are promoted by surface roughness effects and the relative roughness height increases with increasing unit Reynolds number since the boundary layer becomes thinner. This observation is consistent with results of Cattafesta et al. [2] for a swept wing at Mach 3.5. They also concluded that the transition process in their wind tunnel experiment was probably dominated by travelling crossflow disturbances. Acknowledgements The work presented was part of the European research project SUPERTRAC performed under contract No. AST4-CT-2005-516100. The authors are grateful also to the other SUPERTRAC partners for their contributions.

References 1. Arnal, D., Unckel, C.G., Krier, J., Sousa, J.M., Hein, S.: Supersonic laminar flow control studies in the SUPERTRAC project. Proc. 25th Congress of the International Council of the Aeronautical Sciences ICAS 2006, 3-8 Sept. 2006, Hamburg, Germany. 2. Cattafesta III, L.N., Iyer, V., Masad, J.A., King, R.A., Dagenhart, J.R.: Three-dimensional boundary-layer transition on a swept wing at Mach 3.5. AIAA J., vol.33, no.11, pp.2032-2037, Nov. 1995. 3. Joslin, R.D.: Aircraft laminar flow control. Ann. Rev. Fluid Mech., vol. 30, pp. 1-29, 1998. 4. Schülein, E.: Experimental investigation of laminar flow control on a supersonic swept wing by suction. 4th AIAA Flow Control Conference, 23 – 26 June, 2008, Seattle, WA. AIAA 20084208.

Decay of turbulent bursting in enclosed flows Kerstin Hochstrate, Jan Abshagen, Marc Avila, Christian Will, and Gerd Pfister

Abstract In order to shed light on the transition to turbulence in shear flows, many investigations have analyzed the decay of localized turbulent structures. At low Reynolds numbers, exponential distributions of survival times have been observed in plane Couette, pipe and Taylor-Couette flows. We present a new flow state in the counter-rotating Taylor-Couette system that becomes unstable to transient turbulent bursting, which also shows an exponential distribution of lifetimes. In contrast to previous works, the turbulent state is here excited without disturbing the flow externally and appears in the centrifugally unstable regime, thus competing with several coherent states in phase space. In our combined experimental and numerical study we analyze the spatiotemporal properties and the lifetime behavior of this flow.

1 Introduction The origin and the transition to turbulence in wall-bounded shear flows is one of the outstanding problems of classical physics. In pipe and plane Couette flows the transition to turbulence occurs suddenly without any intermediate states, when the amplitude of a perturbation is large enough [5]. In former investigations of pipe [12, 14, 10], plane Couette [3] and recently also in Taylor-Couette flows [6], an external perturbation was applied to disturb the laminar flow and induce localized turbulent structures. Subsequently, the evolution of these structures was observed and their decay recorded to collect turbulent lifetimes. The distribution of these lifetimes is exponential, which has been related to the escape from a chaotic saddle in the phase space of the Navier–Stokes equations [7, 13]. Kerstin Hochstrate, Jan Abshagen, Christian Will and Gerd Pfister University of Kiel, 24118 Kiel, Germany, e-mail: [email protected] Marc Avila Max Planck Institute for Dynamics and Self Organization, 37073 Göttingen, Germany P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_30, © Springer Science+Business Media B.V. 2010

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In a closed system we have found a flow state that also shows transient turbulent behavior and an exponential distribution of lifetimes, but that can be observed without introducing an external perturbation. It appears in the Taylor-Couette system for counter-rotating cylinders as an instabilty of a rotating wave, so-called wavy vortex flow (WVF). Interestingly, the onset of the bursting is related to the precession speed of the WVF approaching zero. Here we analyze the appearing flow states in the Taylor-Couette system and take advantage of unlimited observation times in experiments and well defined boundary conditions for numerical calculations of the Navier-Stokes equations.

2 Methods The Taylor-Couette system consists of a fluid of kinematic viscosity ν confined between two concentric rotating cylinders. The inner cylinder is machined of stainless steel with a radius of ri = (12.50 ± 0.01) mm, the outer one is made of optically polished glass with a radius of ro = (25.00 ± 0.01) mm, defining a constant radius ratio of η = ro /ri = 0.5. The system is non-dimensionalized with the gap width d = ro − ri (length) and diffusion time τ = d 2 /ν . The Reynolds numbers of the cylinders are defined as Rei,o = ri,o d Ωi,o /ν (where Ωi,o are the angular speeds of inner and outer cylinders). The axial length of the system L is determined by the position of two stationary rigid end plates and it is fixed at Γ = L/(ro − ri ) = 8 in these investigations. The Navier–Stokes equations are solved using a second order time-splitting method. The spatial discretization is via a Galerkin-Fourier expansion in θ and Chebyshev collocation in r and z (see [2] for further details, including the treatment of the boundary conditions at the junctions where the stationary endwalls meet the rotating cylinders).

3 Results In contrast to plane Couette and pipe the flow in the Taylor-Couette system becomes linearly unstable at low Reynolds numbers and a sequence of bifurcations arises. In the considered parameter regime, Reo = [−200, −160], the laminar flow becomes linearly unstable to spiral vortex flow in the infinite system, plotted as a black dotted line in the stability diagram of Fig. 1. In the finite system, solid end walls confine the flow in the axial direction and induce localized spirals (thin gray lines) slightly below this threshold [9]. Well above the primary instability these flow states restabilize to the well-known Taylor vortex flow (TVF) at the dash dotted line (measurements marked by △). Subsequently increasing Rei , the TVF loses stability to wavy vortex flow (WVF) via a Hopf bifurcation (black solid line •). The turbulent bursting revealed in this paper appears from the WVF at higher Rei .

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In order to analyze the underlying bifurcation structure leading to turbulent behavior we quasi-statically increase Rei at a constant Reo . A characteristic measurement (LDV) of the axial velocity vz , following the path in parameter space marked by the vertical gray solid line in Fig. 1, is plotted in Fig. 2. Starting from TVF consisting of n = 12 cells Rei is increased in steps of ∆ Rei = 1, measuring at each step for 19τ (position: z = L/2, r = ri + 0.2d). At Rei = 190 the TVF becomes unstable to the primary wavy mode, for which we find excellent agreement with numerical simulations (deviation is within 1%) for onset and frequency (≈ 4τ −1 ). The wavy mode propagates in the same direction as the rotation of the inner cylinder and consists of n = 12 cells and an azimuthal wavenumber of m = 3, illustrated in the picture next to the measurement showing the isosurface of zero angular momentum. Further increasing Rei the frequency decreases, i.e. the azimuthal propagation of the wavy mode slows down until it almost stops in the laboratory frame. At this Reynolds

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number Rei,t , the wavy mode becomes unstable to turbulent bursting, distinguishable by the suddenly occurring high velocities with huge gradients. After a lifetime of τ , the turbulent bursts cease and the flow relaxes to a new wavy mode. This one propagates in the opposite direction as the rotation of the inner cylinder, which is indicated by negative frequencies. Its isosurface of zero angular momentum is plotted in the right panel of Fig. 2, showing an azimuthal wave number of m = 2 and n = 12 vortices. Compared to the primary wavy the wave number has changed and the surface of zero angular momentum penetrates here further in the system.

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Several repetitions of this measurement show the same critical Rei,t for the onset of the bursting, but the lifetimes of this flow state change statistically. Following the idea of investigations in plane Couette and pipe we measure the time for the decay of the turbulent bursting for different changes in the Reynolds number. Therefore we start from the wavy-mode (n = 12, m = 3) at Reo = −180 and Rei = 250, which is slightly below Rei,t , and accelerate the inner cylinder suddenly to either 250 or 300. This causes the onset of the bursting and we measure the time until the turbulent bursts disappear and the other wavy mode (n = 12, m = 2) appears. In the experimental time series in Fig. 3 (left picture on the top) the bursting decays after a lifetime of 9τ indicated by the vertical gray line and the onset of the oscillation of the wavy mode can be clearly seen. A comparison with the time series of numerical simulations below shows the same characteristics but with a lifetime of approximately 14τ (marked by the vertical gray line). Each jump width (i.e. from Rei = 250 to 300) is repeated between 50 and 100 times in the experiments until a clear scaling behavior is obtained. We observe an exponential probability distribution of lifetimes with different slopes depending on the jump width in Rei , shown in Fig. 3 (right panel). At a constant Reo = −180 lifetimes become longer for decreasing distances in Rei from the onset of the bursting. In phase space, an exponential

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probability distribution of lifetimes is typical for the decay from a chaotic saddle [13].

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In addition to the statistical analyses of the bursting for a sudden jump in Reynolds number, we can also observe the spatio-temporal development of the flow. In the experiments we monitor the flow with a HD-camera and convert the video in successive images. From this images we take a characteristic vertical line and plot it over the time to produce a space-time diagram as shown in Fig. 4. The harmonic oscillation of the primary wavy mode (n = 12, m = 3, see also Fig. 2) at Rei = 250 can be clearly seen. After accelerating the inner cylinder to Rei = 280, the wavymode slows down and becomes modulated, so that vortices start to oscillate in the axial direction, until turbulent bursts appear abruptly (τ ≈ 7). After a lifetime of approximately 5τ the bursting decays and the other wavy mode (n = 12, m = 2, see also Fig. 2) appears and remains stable in time. The right panel in Fig. 4 shows a 3dsnapshot of the radial velocity during the bursting from the numerics. The inward flow (colored in dark gray) shows small localized patches and big structures, similar to TVF. The radial outward jets are colored in a light gray and it can be seen that two jets collapse, joining together, which temporary changes the number of cells.

4 Conclusion and outlook We have found a flow state in the counter-rotating Taylor-Couette system which shows transient turbulent bursting at low Reynolds numbers. It appears above the centrifugal instability without an external perturbation and therefore it competes with coherent states of the system. We observe the same spatio-temporal character-

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istics of the turbulent bursting for quasi-statical increases of the Reynolds number, as well as for sudden jumps in Rei . An analysis of lifetimes for the second case shows an exponential distribution, which is often related to the existence of a chaotic saddle. Since Taylor-Couette flow is a closed system and considered as one of the best controllable experiments in hydrodynamics, it is ideally suited to investigate the asymptotic stability of a flow state. On the other hand, stable and also unstable solutions can be determined via calculations of the Navier-Stokes equations in quantitative agreement with experiments [1]. The purpose of current and future research is to shed light on the dynamical picture underlying the turbulent bursting revealed here. Although this is a considerable task, the Taylor system seems amenable to theoretical and experimental treatment, which may be then extended to more complicated shear flows where secondary states are always unstable. The experimental work was financially supported by the Deutsche Forschungsgemeinschaft (project AB 336/1). MA was funded by the Max Planck Society.

References 1. J. Abshagen, J. M. Lopez, F. Marques, and G. Pfister. Bursting dynamics due to a homoclinic cascade in Taylor-Couette flow. J. Fluid Mech., 613:357–384, 2008. 2. M. Avila, M .Grimes, J. M. Lopez, and F. Marques. Global endwall effects on centrifugally stable flows. Phys. Fluids, 20(10), 104104, 2008. 3. S. Bottin, and H. Chate. Statistical analysis of the transition to turbulence in plane Couette flow. Eur. Phys. J. B, 6(1):143–155, 1998. 4. R. M. Clever, and F. H. Busse. Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech., 344:137–153, 1997. 5. A. G.Darbyshire, and T. Mullin. Transition to turbulence in constant-mass-flux pipe flow. J. Fluid Mech., 289:83–114, 1995. 6. D. Borrero-Echeverry, and M. F. Schatz. Transient Turbulence in Taylor-Couette Flow. arXiv,0905.0147v1, 2009. 7. B. Eckhardt, B. Turbulence transition in pipe flow: some open questions. Nonlinearity, 21(1):T1–T11, 2008. 8. H. Faisst, and B. Eckhardt. Transition from the Couette-Taylor system to the plane Couette system. Phys. Rev. E, 61, 7227, 2000. 9. M. Heise, J. Abshagen, D. Küter, K. Hochstrate, G. Pfister, and C. Hoffmann. Localized spirals in Taylor-Couette flow. Phys. Rev. E, 77, 026202, 2008. 10. B. Hof, A. Lozar, D. J. Kuik, and J. Westerweel. Repeller or Attractor? Selecting the Dynamical Model for the Onset of Turbulence in Pipe Flow. Phys. Rev. Lett., 101, 214501, 2008. 11. M. Nagata. Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech., 217:519–527, 1990. 12. J. Peixinho, and T. Mullin. Decay of Turbulence in Pipe Flow. Phys. Rev. Lett., 96, 094501, 2006. 13. T. Tél, and Y.–C. Lai. Chaotic transients in spatially extended systems. Phys. Rep., 460:245– 275, 2008. 14. A. P. Willis, and R. R. Kerswell. Critical Behavior in the Relaminarization of Localized Turbulence in Pipe Flow. Phys. Rev. Lett., 98, 014501, 2007.

Local and Global Stability of Airfoil Flows at Low Reynolds Number L. E. Jones, R. D. Sandberg, and N. D. Sandham

Abstract Previous direct numerical simulations of the flow around a NACA-0012 airfoil have observed the presence of three instability mechanisms: convective instability, a three-dimensional absolute instability of the naturally occurring vortex shedding, and an acoustic feedback instability involving sound production at the airfoil trailing-edge . In this study the behaviour and relative importance of these mechanisms are documented for a range of compressible low Reynolds number airfoil flows exhibiting laminar separation bubbles.

1 Introduction It has long been conjectured as to whether regions of absolute instability exist within laminar separation bubbles, as have been proven to occur in shear layers (Hanneman & Oertel, 1989) and bluff-body wakes (Huerre, 1990); and if so what the resultant effect is upon the global flow dynamics. Linear stability analysis of analytically constructed velocity profiles (e.g. Rist & Maucher, 2002) has determined that absolute instability can occur if either the reverse flow is sufficiently large, or if the separated shear layer lies above a certain wall-normal distance. Nevertheless the presence of absolute instability in the classical sense has not been proven for either experimental or numerical laminar separation bubble flows. It should be noted however that Marquillie & Eherenstein (2003) did observe absolute instability for a laminar separation bubble that was subject to reacceleration via a surface bump. In recent years advances have been made with two-dimensional linear-stability analysis, termed ‘BiGlobal’ analysis (Theofilis, 2003). Theofilis applied BiGlobal analysis to the case of separation bubble flow and observed global instability modes that are not predictable by classical one-dimensional linear stability analysis. Growth rates of the global modes were small in comparison to convective instabilities, howL.E. Jones University of Southampton, Southampton, S017 1BJ, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_31, © Springer Science+Business Media B.V. 2010

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ever it was suggested that the global modes may be relevant to the vortex shedding behaviour that is observed to occur for many separation bubble flows. Computer performance has increased sufficiently such that now DNS of airfoil flows with separation bubbles are possible. Jones, Sandberg & Sandham (2008) performed DNS of the flow around a NACA-0012 airfoil exhibiting a laminar separation bubble, at Re = 5 × 104 , α = 5◦ , the results of which form the starting point for the current study. In two dimensions, due to an adverse pressure gradient, the upper surface boundary layer separates near the airfoil leading-edge. Further downstream the separated shear-layer becomes unstable, and rolls-up into a periodic vortexshedding behaviour. As the vortices convect over the airfoil trailing-edge they generate acoustic waves via a trailing-edge scattering process. In three-dimensions, the upper surface boundary layer again separates near the airfoil leading-edge, and the separated shear-layer undergoes transition to turbulence. The turbulent boundary layer reattaches and convects over the aft section of the airfoil, and the turbulent structures convecting over the airfoil trailing-edge generate broadband self-noise. The lower boundary layer is subject to a favorable pressure gradient and remains laminar and steady in both two and three-dimensional simulations. When the stability of the airfoil flows was investigated the laminar separated region was found to be convectively unstable, as expected. The separation bubble exhibited selfsustained transition to turbulence however, even though linear stability analysis of time-averaged velocity profiles found no regions of absolute instability (Jones et al. 2008). The self-sustained transition was instead attributed to a three-dimensional absolute instability of the periodic vortex shedding that occurs naturally for the two-dimensional flow (Maucher, Rist & Wagner 1997, Jones et al. 2008). The mechanism results in the temporal growth of three-dimensional perturbations at a point fixed in space, and produces spanwise periodic structures in the ‘braid’ regions between vortices (figure 1, left). Additional studies (Jones, Sandberg & Sandham 2009) determined that the time and span-averaged flowfields of the two and three-dimensional simulations were globally unstable via an acoustic-feedback instability, illustrated schematically in figure 1 (right). When the separation bubble is perturbed, hydrodynamic instabilities are generated which are convectively amplified (A). The instability waves convect over the trailing-edge and generate acoustic waves (B). The acoustic waves propagate upstream (C), and upon reaching the airfoil leading-edge, further hydrodynamic disturbances are generated within the upper surface boundary layer by a receptivity process (D). These perturbations are again convectively amplified within the separated shear layer, and the cycle repeats with increasing amplitude. It is clear that laminar separation bubbles exhibit complex stability behaviour and hence it is not surprising that accurate prediction of unsteady separation bubble behaviour remains difficult. In this study the stability characteristics of a range of airfoil flows exhibiting laminar separation will be investigated using DNS and stability analysis, with the intention of determining the relevance of the three instability mechanisms identified above. Simulations are primarily conducted for a NACA-0012 airfoil at Re = 5 ×104 , M = 0.4, however the effects of geometry, compressibility and Reynolds number are also investigated. The computational method

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employed for DNS is documented in Jones et al. (2008), and the methods used to determine the stability characteristics are documented in Jones et al. (2008, 2009).

2 Behaviour with incidence The behaviour of the three instability mechanisms will be considered first for twodimensional simulations of the flow around a NACA-0012 airfoil at Re = 5 × 104 , M = 0.4, at several angles of attack. As the airfoil incidence is increased, the onset of unsteadiness moves upstream and the bubble becomes shorter in length. As the airfoil incidence is decreased, the onset of unsteadiness moves downstream. At zero degrees incidence the flow over the airfoil is steady, whilst the wake exhibits roll-up into bluff-body vortex shedding. Temporal growth rate exponents of the instability mechanisms are plotted in figure 2 (left). For the case of convective instability growth, a pseudo-temporal growth rate is plotted as NCph /∆ x, where N is the maximum disturbance N-factor over the laminar region with length ∆ x, and Cph is the phase-speed of the most unstable mode. The acoustic feedback instability is weakly unstable for α ≥ 5◦ (noting that α = 8.5◦ was the largest angle of attack considered). At the smallest angle of attack considered, α = 3◦ , the feedback loop is weakly stable. It is apparent that the acoustic feedback instability possesses small growth rates compared to the other two instability mechanisms, and hence is considered not to be the mechanism behind the transition seen in the three-dimensional simulations, although it may be relevant to the two-dimensional vortex-shedding process. The convective instability mechanism exhibits the largest growth rates, however growth rates of the three-dimensional absolute instability are of comparable magnitude. The fact that positive growth is observed at zero-degrees angle of attack suggests that the three-dimensional absolute instability mechanism is directly related to instability mechanisms in bluff body wakes. Growth rates of the convective instability and the three-dimensional absolute instability increase with incidence. For the former this is attributed to the increased adverse pressure gradients present at high incidence, and associated changes in boundary layer structure. For the latter, the increased reverse flow magnitudes present at higher incidence is responsible. Figure 2 (right) illustrates the preferred frequency of the instability mechanisms and the dominant vortex shedding frequency. For angles of attack α ≤ 5◦ the vortex shedding is regular and tonal. For angles of attack α ≥ 7◦ the vortex shedding is more broadband in content, and the dominant frequency is obtained by inspecting pressure spectra on the upper airfoil surface. It is clear that the most convectively unstable modes possess significantly higher frequencies than the vortex shedding for all cases. The preferred frequency of the acoustic feedback loop is however comparable to that of the vortex shedding behaviour, and it can be seen that for cases where the feedback loop is unstable (5◦ ≥ α ≥ 8.5◦ ) the frequency of the instability more closely matches that of the vortex shedding. This suggests that the

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acoustic feedback loop may well play a role in frequency selection for the naturally occuring vortex shedding. Time and span-averaged flowfields of three-dimensional simulations are also available at incidence α = 5◦ and α = 7◦ . The flow at α = 5◦ is unstable to the feedback loop, with similar growth rates to that of the two-dimensional simulation and preferred frequency f ≈ 3.3, whereas the flow at α = 7◦ is weakly stable.

3 Influence of compressibility The acoustic feedback loop has been further analysed for the flow around a NACA0012 airfoil at Re = 5 × 104 , α = 5◦ and Mach numbers M = 0.2, 0.3 and 0.6, in order to determine the influence of compressibility. At low Mach numbers the amplitude of trailing-edge noise is expected to decrease and the wavelength of acoustic waves will increase, resulting in a greater separation of scales between the acoustic and hydrodynamic fields. These changes will make the feedback instability less likely to occur, hence it is unsurprising that the acoustic feedback loop is found to be stable for M = 0.2 and weakly stable for M = 0.3. Surprisingly the feedback loop is also found to be stable at M = 0.6. The flow at M = 0.6 exhibits very low frequency shedding ( f = 0.5) from a large separation bubble, and the time-averaged flowfield appears quite different to that at other Mach numbers. These changes make direct comparison more difficult. It appears that the feedback loop is sensitive to compressibility, occurring only at certain Mach numbers for a given flow configuration.

4 Influence of Reynolds Number and Airfoil Geometry In order to further determine the importance of the three-dimensional absolute instability the flow around a NACA-0012 airfoil at Re = 1 × 106 , M = 0.4 and α = 5◦ is analysed. In two dimensions the flow exhibits periodic vortex shedding, with frequency f = 3.8. The two-dimensional flow is again found to be unstable to the three-dimensional absolute instability, possessing growth rate σ ≈ 5. This is significantly less than for the equivalent case at Re = 5 × 104 , and is attributed to the reduced reverse flow present at Re = 1 × 105 (15% c.f. 28%), which is necessary to sustain the absolute nature of the instability. Although the NACA-0012 airfoil is widely used for research purposes, industrial applications commonly employ thinner airfoil geometries, hence analysis of the flow around a NACA-0006 airfoil at Re = 5 × 104 , M = 0.4, α = 7◦ has been conducted. In comparison to the equivalent NACA-0012 airfoil case, the flow around the NACA-0006 airfoil exhibits a shorter separation bubble. The two-dimensional simulation exhibits vortex shedding, but the frequency content of the vortex shedding behaviour is broadband and it is difficult to determine a dominant shedding

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frequency. Despite the irregularity of the vortex shedding, the three-dimensional absolute instability is again present and possesses a higher amplitude temporal growth rate than for the equivalent NACA-0012 case (σ = 17 c.f. σ = 13). DNS of the three-dimensional flow has been performed (figure 3) which again exhibits self-sustained transition to turbulence via this mechanism. The time-average of the two-dimensional simulation is unstable to the acoustic feedback instability, at a frequency of f ≈ 2.4, however the time and span-averaged flowfield from the threedimensional simulation is not.

5 Conclusion A range of compressible airfoil flows have been investigated in terms of their local and global stability characteristics. All of the airfoil flows exhibited separated regions which roll-up into vortex shedding. The separated regions are convectively unstable, and the convective instability growth rates increase with airfoil incidence. All of the airfoil flows, including a NACA-0006 geometry and higher Reynolds number flow, have been found to be absolutely unstable via a three-dimensional instability of the vortex shedding. Growth rates of the absolute instability mechanism are a similar order of magnitude to those of convective instability within the separated shear layer, and the absolute instability mechanism is responsible for the self-sustained transition to turbulence observed in three-dimensional simulations in the absence of boundary layer tripping or forcing. The time-averaged flowfields of several cases were also found to be globally unstable via an acoustic feedback instability. The feedback instability was found to occur only above a certain onset incidence and at M = 0.4. Growth rates of the feedback instability are low in comparison to convective instability, however it is likely that the mechanism plays a role in frequency selection of the vortex shedding, and potentially for the generation of particular tones in three-dimensional flows.

Fig. 1 Contours ωx at an arbitrary time within the three-dimensional absolute instability cycle for the airfoil flow at Re = 5 × 104 , M = 0.4, α = 5◦ , with background showing contours ωz = ±200 (left).Schematic of the acoustic feedback instability mechanism (right).

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Fig. 3 Contours of ωz = ±200 for the mid-span of the three-dimensional flow over a NACA-0006 airfoil at Re = 5 × 104 , M = 0.4, α = 5◦

References 1. Hannemann, K. & Oertel Jr, H. (1989) Numerical simulation of the absolutely and convectively unstable wake, Journal of Fluid Mechanics, 199, 55–88 2. Huerre, P. & Monkewitz, P.A. (1985) Absolute and convective instabilities in free shear layers, Journal of Fluid Mechanics, 159, 151–169 3. Jones, L.E. & Sandberg, R.S. Sandham, N. (2008) Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence, Journal of Fluid Mechanics, 602, 175–207 4. Jones, L.E. & Sandberg, R.S. Sandham, N. (2009) Stability and receptivity characteristics of a laminar separation bubble on an airfoil, Journal of Fluid Mechanics, Manuscript under review 5. Marquillie, M. & Ehrenstein, U.W.E. (2003) On the onset of nonlinear oscillations in a separating boundary-layer flow, Journal of Fluid Mechanics, 490, 169–188 6. Maucher, U. & Rist, U. & Wagner, S. (1997) Secondary instabilities in a laminar separation bubble, Notes on Numerical Fluid Mechanics, 60, 229–236 7. Rist, U. & Maucher, U. (2002) Investigations of time-growing instabilities in laminar separation bubbles, European Journal of Mechanics B: Fluids, 21 (5), 495–509 8. Theofilis, V. (2003) Advances in global linear instability analysis of nonparallel and threedimensional flows., Progress in Aerospace Sciences, 39 (4), 249–315

Numerical simulation of riblet controlled oblique transition S. Klumpp, M. Meinke, and W. Schröder

Abstract To analyze the fundamental physical mechanism which determines the damping effect of a riblet surface on three-dimensional oblique transition numerical simulations of a spatial evolving zero-pressure gradient boundary layer above a clean and a riblet wall are performed. The laminar flow is excited by two oblique waves to force the oblique transition scenario. The occurring three-dimensional structures, i.e, Λ - and streamwisely aligned vortices are found to be damped and their breakdown to turbulence is damped by the riblets compared to a clean surface. The investigation of the near-wall flow structures reveals secondary flows induced by the riblets.

1 Introduction Surface structures, so-called riblets, consisting of tiny grooves aligned with the main flow direction, are well known for reducing friction drag in turbulent flow. Various experimental and numerical investigations [2, 6] show drag to be reduced if the riblet spacing in non-dimensional wall units s+ = suτ /ν is below 30. At an optimum riblet geometry a drag reduction of about 10% has been achieved [2]. The influence of riblets on the transition from laminar to turbulent flow is, however, not clear, yet. Neumann and Dinkelacker [16] have found the transition of the flow on a body of revolution to be delayed, if the body was covered by riblets of a certain size compared to a smooth surface. Ladd et al. [11] have experimentally investigated the effect of riblets on transition of a zero-pressure gradient flow and have determined an accelerating influence. Grek et al. [8] have measured the influence of riblets on TS waves as well as on three-dimensional instabilities. They have found the TS waves to be amplified by riblets, whereas three-dimensional Λ - and S. Klumpp · M. Meinke · W. Schröder Institute of Aerodynamics, RWTH Aachen University, Germany e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_32, © Springer Science+Business Media B.V. 2010

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hairpin vortices have been damped. Further experimental analyses of Litvinenko et al. [13] and Chernorai et al. [5] indicate riblets not only to damp Λ - and hairpin vortex structures but streamwisely aligned vortices to occur in laminar boundary layers in general. Luchini and Trombetta [14] have predicted a destabilizing effect of riblets on TS waves, but a stabilizing on the Görtler instability based on a standard e9 -model. In [7] Ehrenstein shows by linear stability analysis the laminar channel flow over riblets to be more unstable than the parabolic Poiseuille profile at smooth walls. However, to the best of the authors’ knowledge in no paper a detailed near wall flow field at the transition stage is evidenced, since the flow structure has not been highly resolved in the near-wall region. A delayed transition by riblets would provide an additional drag reduction capability of riblets. To gain insight in the damping effects of riblets, large-eddy simulations (LES) of spatially evolving oblique transition in a flat plate zero-pressure gradient boundary layer above a clean and a riblet wall have been performed.

2 Numerical Method The Navier-Stokes equations for three-dimensional unsteady compressible flow are solved based on a large-eddy simulation (LES) formulation using the MILES (monotone integrated LES) approach [4]. The discretization of the inviscid terms consists of a mixed centered-upwind AUSM (advective upstream splitting method) scheme [12] at second-order accuracy, whereas the viscous terms are discretized second-order accurate using a centered approximation. The temporal integration is done by a second-order explicit 5-stage Runge-Kutta method. A detailed description of the flow solver and a thorough discussion of the quality of its solutions in fully turbulent flow are given, e.g., in [1, 15, 17, 18]. To validate the capability of the numerical method to capture transition, a simulation of a temporally evolving channel flow is performed. Following the setup of Schlatter [19], the Reynolds number based on the centerline velocity Ucl and the channel half-height h is Recl = 5000. To initialize the simulation a Poiseuille flow with a superposed two-dimensional TS wave and two three-dimensional oblique waves following the setup of Schlatter [19] is chosen. The streamwise wavelength of the two-dimensional TS wave is λx,2d = 5.61h and that of the three-dimensional wave is λx,3d = 5.61h in the streamwise and λz,3d = 2.99h in the spanwise direction. The amplitude is set to 0.03Ucl for the two-dimensional TS-wave and 0.001Ucl for the three-dimensional waves. When this initialization is used Λ -vortices will occur immediately and K-type transition will take place. Figure 1 shows the distribution of the friction velocity based Reynolds number Reτ as a function of the dimensionless time compared with the data from [19]. The point of transition occurs at the same dimensionless time as predicted in [19]. This result shows the method to be able to accurately compute transition.

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3 Flow configuration and computational mesh The computational domain, shown in Figure 2, extends Lx /δi = 440, Ly /δi = 14, and Lz /δi = 26 in the streamwise x, wall normal y, and spanwise direction z. Here, δi denotes the displacement thickness at the inlet boundary. The riblets rise at xs /δi = 47 and possess a tip spacing of s/δi = 0.9. The riblet geometry was chosen to be similar to that used by Litvinenko et al. [13], where a transition delay was observed, and to be feasible in a dimension appropriate for technical applications, as shown by Hirt and Thome [9]. Following the setup of Grek et al. Reδ ,i = U∞ δi /ν is set to 618. The inlet boundary condition is given by a laminar Blasius boundary layer profile with two superposed oblique waves with a wave number of α = 0.23/δi in the streamwise direction and β = ±0.242/δi in the spanwise direction and an amplitude of uˆ = 2 × 10−2U∞ . For the clean wall case the domain is resolved by 831 × 42 × 581 grid points, whereas for the riblet case 831 × 42 × 609 grid points are used. Based on the friction velocity uτ = 0.05U∞ , which is typical for the considered turbulent flow, the + grid spacings in wall units are ∆ x+ = 16.4, ∆ y+ wall = 0.83, and ∆ z = 1.39.

4 Results The flow structure of the clean wall and riblet cases from the inflow boundary to x/δi ≈ 80 is characterized by Λ -structures, which are generated by the two incoming oblique waves. Further downstream, the Λ -structures deform into streamwisely aligned vortices followed by turbulent breakdown. The entire instantaneous flow fields at the clean and riblet surface cases are visualized by the λ2 -criterion [10] in Figure 3. The distribution of the spanwise averaged instantaneous skin-friction coefficient c f over the streamwise position is given in Figure 4. Downstream of x/δi ≈ 80 the skin-friction coefficient deviates from the laminar state caused by the downwash of faster fluid towards the wall by the streaks, which is typical for oblique transition [3]. The breakdown to turbulence at the clean surface case at x/δi ≈ 300 is evidenced by a dramatic increase of c f which is emphasized by an overshoot compared to the empirical c f -distribution of a turbulent flat plate flow. The development of the skin-friction coefficient c f of the riblet surface case is also shown in Figure 4. The peak at x/δi = 47 is caused by the first appearance of the riblets. At x/δi ≈ 350 the skin-friction coefficient increases dramatically evidencing the breakdown to turbulence. Compared to the clean surface the development of the turbulent state is delayed by ∆ x/δi ≈ 200, which yields a maximum drag reduction of 77% at x/δi ≈ 350. The total drag of the flat plate is lowered by approximately 30%. The damping effect of the riblets is evidenced by comparing the time averaged maximum wall-normal |v|max and spanwise velocity |w|max in each wall-normal cross section on the clean and the riblet surface in the Figures 5 and 6. In the region 50 ≤ x/δi ≤ 250, which is characterized by streamwisely aligned vortex structures,

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the spanwise velocity on the riblet surface is on the average about 25% lower than that on the clean surface and the wall-normal velocity even 35%. Since these maximum velocities represent the strength of the near-wall vortex structures, damped velocities mean a damped downwash of faster fluid and as such a reduced drag in the interval 50 ≤ x/δi ≤ 250. The direct impact of the riblets on the disturbances leading to transition is illustrated in Figure 7, where the flow structure is visualized be streamlines above the riblet surface. The large vortex represents one streamwisely aligned vortex emerging at this state of oblique transition. Besides the expected wave-like overflow of the riblets tips some recirculation zones and vortices with a diameter equal to the riblet spacing are observed. Acknowledgements The authors gratefully acknowledge the financial support of the joint project “RibletSkin” by the Volkswagen Foundation, Hannover, Germany.

References 1. Alkishriwi, N., Meinke, M., Schröder, W.: A large-eddy simulation method for low mach number flows using preconditioning and multigrid. Comp. Fluids 35(10), 1126–1136 (2006) 2. Bechert, D.W., Bruse, M., Hage, W., van der Hoeven, J.G.T., Hoppe, G.: Experiments on drag-reducing surfaces and their optimization with an adjustable geometry. J. Fluid Mech. 338, 59–87 (1997) 3. Berlin, S., Lundbladh, A., Henningson, D.: Spatial simulation of oblique transition in a boundary layer. Phys. Fluids 6, 1949–1951 (1994) 4. Boris, J., Grinstein, F., Oran, E., Kolbe, R.: New insights into large eddy simulation. Fluid Dyn. Research 10, 199–228 (1992) 5. Chernorai, V., Kozlov, V., Loefdahl, L., Grek, G., Chun, H.: Effect of riblets on nonlinear disturbances in the boundary layer. Thermophys. and Aeromech. 13(1), 67–74 (2006) 6. Choi, H., Moin, P., Kim, J.: Direct numerical simulation of turbulent flow over riblets. J. Fluid Mech. 255, 503–539 (1993) 7. Ehrenstein, U.: On the linear stability of channel flow over riblets. Phys. Fluids 8, 3194–3196 (1996) 8. Grek, G.R., Kozlov, V., Titarenko, S.: An experimental study of the influence of riblets on transition. J. Fluid Mech. 315, 31–149 (1996) 9. Hirt, G., Thome, M.: Rolling of functional metallic surface structures. CIRP Annals - Manufacturing Technology 57(1), 317–320 (2008) 10. Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995) 11. Ladd, D.M., Rohr, J.J., Reidy, L.W., Hendricks, E.W.: The effect of riblets on laminar to turbulent transition. Exp. in Fluids 14, 1–2 (1993) 12. Liou, M.S., Steffen Jr., C.J.: A New Flux Splitting Scheme. J. Comput. Phys. 107, 23–39 (1993) 13. Litvinenko, Y.A., Chernoray, V.G., Kozlov, V.V., Loefdahl, L., Grek, G.R., Chun, H.H.: The influence of riblets on the development of a Λ structure and its transformation into a turbulent spot. Physics - Doklady 51, 144–147 (2006) 14. Luchini, P., Trombetta, G.: Effects of Riblets upon Flow Stability. Applied Scientific Research 54, 313–321 (1995) 15. Meinke, M., Schröder, W., Krause, E., Rister, T.: A comparision of second- and sixth-order methods for large-eddy simulations. Comp. Fluids 31, 695 – 718 (2002)

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16. Neuman, D., Dinkelacker, A.: Drag measurements on v-grooved surfaces on a body of revolution in axial flow. Applied Scientific Research 48(1), 105–114 (1991) 17. Renze, P., Schröder, W., Meinke, M.: Large-eddy simulation of film cooling at density gradients. Int. J. Heat Fluid Flow 29, 18–34 (2008) 18. Rütten, F., Meinke, M., Schröder, W.: Large-eddy simulations of frequency oscillation of the dean vorticies in turbulent pipe bend flows. Phys. Fluids 17, 035,107–1–035,107–11 (2005) 19. Schlatter, P., Stolz, S., Kleiser, L.: Computational simulation of transitional and turbulent shear flow. In: Progress in Turbulence, Proc. ITI Conference on Turbulence, (2005)

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Transition Movement in the Wake of Protruding and Recessed Three-Dimensional Surface Irregularities V. S. Kosorygin, J. D. Crouch, and L. L. Ng

Abstract A study is conducted to investigate the initial movement of transition downstream of protruding and recessed surface irregularities. Experimental data is provided on the transition movement, and on the steady and unsteady boundarylayer characteristics leading to the transition movement. The transition Reynolds numbers collapse reasonable well when plotted in terms of the roughness height non-dimensionalized by the boundary-layer displacement thickness. Local stability analysis is shown to capture many of the features of the pre-transitional flow. The upstream movement of the transition can be represented by a reduction in the critical N-factor, similar to earlier studies on surface roughness or two-dimensional steps.

1 Introduction Aerodynamic surfaces designed for laminar flow inevitably have geometric imperfections, which arrise as a consequence of the manufacturing process, design demands, and operational conditions. Any surface imperfection impacts the unsteady processes in the boundary layer and may accelerate the laminar-turbulent transition. From a practical point of view, it is of great importance to be able to predict the transition location in different situations and to provide an approximate estimate for the surface imperfection sizes that can be tolerated without affecting transition. Due to the important role of the surface imperfections in boundary-layer transition, many studies have been devoted to this topic (see, for example, the reviews V.S. Kosorygin Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, Ul. Institutskaya, 4/1, Novosibirsk, 630090 Russia, e-mail: [email protected] J.D. Crouch The Boeing Company, Seattle, WA U.S.A. e-mail: [email protected] L.L. Ng The Boeing Company, Seattle, WA U.S.A. e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_33, © Springer Science+Business Media B.V. 2010

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by Tani [1] and von Doenhoff & Braslow [2]). These papers summarize empirical knowledge gained at that time, and still in use. Tani [1] primarily describes results from experiments with two-dimensional roughness, but also includes data related to three-dimensional roughness elements. Doenhoff & Braslow [2] gathered data about the effects of isolated three-dimensional roughness of different shapes. They show a correlation of three-dimensional roughness transition (transition essentially at the roughness location) in terms of the local critical Reynolds number Rek = hU(h)/ν and the roughness shape parameter D/h for some restricted range of heights h and diameters D. There is a cloud of experimental data but there are no details on the transition movement or on the underlying physics. The data demonstrates some tendency to a lowering of the critical Reynolds number with the increase of the roughness shape parameter. However, it is unclear what happens with transition for lower Rek and what are the physical processes governing the transition movement. Kendall [3] observed a distortion in mean flow downstream of a single cylindrical obstacle in a Blasius boundary layer, and has noted the distortion can produce an alteration of stability within some region downstream. More recently, White & Ergin [4] and Fischer & Choudhari [5] have linked the distortion to transient growth effects in the wake of the obstacle. de Paula, Wurz & Medeiros [6] investigated the interaction of a two-dimensional Tollmien-Schlichting (TS) wave with a three-dimensional irregularity of small height. For larger-amplitude TS waves the interaction resulted in a highly three-dimensional wave pattern, while for smaller TS-wave amplitudes the flow remained more two dimensional. In all these studies transition was absent over the plate length and an estimation of the transition movement could not be done. The current paper considers an investigation of the initial movement of the transition location downstream of protruding and recessed surface irregularities. Local stability analysis is used to help interpret the results. The data is used to assess the potential of variable N-factor methods for predicting the transition movement – similar to that used to model the effects of surface roughness [7] or two-dimensional steps [8].

2 Transition Measurements Experiments are conducted in the T-324 wind tunnel at Khristianovich Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences (Novosibirsk). This facility is a low-turbulence closed-circuit wind tunnel with a 1m x 1m x 4m test section and is well suited for receptivity, stability, and transition experiments. The flat-plate model used in the experiments is 0.996m wide and 2.0m long. A surfacepressure variation is created by means of wall bump. The position of the stagnation line and the pressure distribution in the vicinity of the leading edge are controlled by a trailing-edge flap during the tests. The minimum of the pressure distribution is located at a longitudinal distance x = 200mm. An insert with the surface imperfection could be placed at either x = 150mm or x = 300mm. Free-stream velocity

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was varied from 18.3m/s to 22.6m/s. All measurements are conducted under natural background disturbances. The integral free-stream turbulence had the intensity ε ≤ 0.00039 (bandwidth 2-4000 Hz). The transition location is determined by means of a 1 − mm round Preston tube which is moved along the surface. The measured dynamic pressure can be interpreted in terms of the velocity profile slope close to wall (i.e. the skin friction). The transition location, xT , is estimated based on the minimum of the dynamic pressure distribution, with an uncertainty in xT of about 10mm. Velocity profile measurements are made using a single-wire probe built in-house with 5µ m x 1mm Pt-plate tungsten wire, and connected to a DANTEC constant-temperature anemometer. The pressure distribution and the mean boundary-layer profiles are shown in figure 1. The boundary-layer profiles show very good agreement between the experiments and the calculated boundary-layer solution used for the stability analysis. Figure 2 shows the movement in the transition front resulting from a D = 7.2mm surface irregularity with different heights at x = 150mm. The smooth-surface transition location is shown by the line labeled h = 0. The protrusion at h = 0.53mm results in a modest forward movement of the transition location extending over ∆ z ≈ 200mm ≈ 25D. An increase in the height to h = 0.62mm, results in transition moving almost up to the irregularity on the centerline behind the protrusion. The transition Reynolds number, ReT = UxT /ν measured on the centerline, collapses reasonable well with Rek for the protruding irregularities. The data shows a critical value of Rek = 300 − 700 for transition to jump forward to the irregularity location – in agreement with well know criteria for local bypass transition. For recessed irregularities, Rek is not meaningful so an alternative measure is needed. Figure 3 shows the transition Reynolds number as a function of the irregularity height, non-dimensionalized by the displacement thickness, h/δ ∗ for both protruding and recessed irregularities. The circular symbols correspond to rounded-head irregularities, the larger symbols correspond to D = 10.5mm − 13.7mm, the smaller symbols correspond to D ≈ 7.2mm, and the thin-line symbols correspond to irregularities at x = 150mm, and the thick-line symbols to irregularities at x = 300mm. For protruding irregularities, the data collapse is as good as, or better than, using Rek – showing a critical value of h/δ ∗ = 0.7 − 1 for a complete loss of laminar flow. For recessed irregularities the critical value is less definitive, but shows h/δ ∗ > 2. The recessed irregularities also exhibit a higher tolerance for smaller diameters.

3 Stability Calculations The forward movement of transition observed in figure 3 can result from enhanced receptivity or enhanced growth. To examine the potential change in growth characteristics, local stability analysis is applied to the steady velocity profiles measured on the centerline downstream of the surface perturbations. Figure 4a shows the measured distortion profiles at different streamwise locations for a protruding irregularity at x = 150mm. Near the perturbation, there is a significant velocity deficit. This

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quickly changes to a velocity surplus well within 14 diameters. This same characteristic has been seen in the study of Fischer and Choudhari [5]. The distortion profiles of figure 4a are superposed on the calculated boundarylayer profiles for a smooth surface, and used to calculate the local change in stability characteristics. Streamwise and spanwise variations in the profiles are neglected. In this framework, the instabilities are governed by the Orr-Sommerfeld equation, with fixed dimensional frequency ω and spanwise wavenumber β . Figure 4b shows the calculated growth rates for the unperturbed boundary layer (lines) and the perturbed boundary layer (symbols) for two frequencies. Near the surface irregularity, both frequencies are highly destabilized. Farther downstream, both are weakly stabilized. This sparse representation is not sufficient to calculate an integrated impact on the disturbance growth, but a simple piecewise estimate shows the potential for a large impact on the n-factor, leading to a forward movement in transition. Although the local stability analysis is a simplified approximation to the real flow, it does capture many of the observed changes resulting from the surface irregularities. The calculated mode shape with the distortion profile at x = 158mm is in good agreement with the experiment. In the absence of surface perturbations, the experiment shows a dominant frequency of f = 183Hz leading to transition, and the calculations show f ≈ 200Hz. With the flow distortion of figure 4a, the experiment shows a forward movement in transition and a shift in dominant frequency to f = 236Hz. This forward movement shifts the calculated dominant frequency to f ≈ 240Hz. The TS-wave amplification n-factors are calculated based on quasiparallel theory, in the absence of the surface irregularity. The amplification n-factor nT S (x) is defined as the maximum of the amplification curves for all physical frequencies and spanwise wavenumbers. The transition N-factors NT S are given by the value of nT S at the experimentally-observed transition location. To determine the delta N-factor needed for variable N-factor predictions, we take the difference of NT S with and without the surface irregularity. Figure 5 shows the ∆ NT S as a function of h/δ ∗ for the different surface irregularities considered in figure 3. For protruding irregularities, there is a negligible ∆ NT S for h/δ ∗ < 0.2. The ∆ NT S increases to 2.5 before the complete loss of laminar flow near h/δ ∗ ≈ 1. The results for protruding irregularities show no significant bias for varying diameter, head shape, or streamwise position. For recessed irregularities, the larger diameters have a greater impact on the critical N-factor. The results show ∆ NT S as large as 4 before the complete loss of laminar flow around h/δ ∗ ≈ 2.

4 Conclusions The initial movement of transition in the wake of localized surface irregularities appears to result from altered TS waves. Velocity profiles in the near wake of the irregularity show a significant destabilization, and a shift toward higher frequencies. The initial transition movement can be characterized by a reduction in the critical N-factor for transition – similar to earlier works.

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References 1. I. Tani. Effect of two-dimensional and isolated roughness on laminar flow. In.: Boundary Layer and Flow Control, vol.2, Ed.: G.V.Lachmann, Pergamon Press, Oxford, 1961, pp.637656. 2. A.E. von Doenhoff, and A.F. Braslow. The effect of distributed surface roughness on laminar flow. In.: Boundary Layer and Flow Control, vol.2, Ed.: G.V.Lachmann, Pergamon Press, Oxford, 1961, pp.657-681. 3. J.M. Kendall. Laminar boundary layer velocity distortion by surface roughness: Effect upon stability. AIAA Paper 81-0195, 1981. 4. E.B. White, and F.G. Ergin. Receptivity and transient growth of roughness-induced disturbances. AIAA Paper 2003-4243, 2003. 5. Fischer, P. and Choudhari, M. Numerical simulation of roughness-induced transient growth in a laminar boundary layer. AIAA Paper 2004-2539, 2004. 6. de Paula, I.B., Wurz, W. and Medeiros, M.A.F. Experimental study of a Tollmien-Schlichting wave interacting with a shallow 3D roughness element. J. Turbulence, 9, pp.1-23 (2008). 7. Crouch, J.D. and Ng, L.L. Variable N-factor method for transition prediction in threedimensional boundary layers. AIAAJ., 38, pp.211-216 (2000). 8. Crouch, J.D., Kosorygin, V.S. and Ng, L.L. Modeling the effects of steps on boundary-layer transition. In.: Sixth IUTAM Symposium on Laminar-Turbulent Transition, Ed.: R. Govindarajan, Springer, 2006, pp.37-44.

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Plasma Assisted Aerodynamics for Transition Delay Marios Kotsonis, Leo Veldhuis, and Hester Bijl

Abstract This work involves the numerical investigation of Dielectric Barrier Discharge (DBD) actuators used as wave cancellation devises for transition delay of subsonic flows. The operation of the DBD/plasma actuator is described based on a first-principles electrodynamic model. Several cases of different geometrical and electrical configurations are studied. In all cases the electrodynamic model indicates the formation of a weak near-wall body-force component. The research concept involves the utilization of this body-force through a wave superposition method for dampening unsteady Tollmien-Schlichting waves in a laminar boundary layer. To study this effect, the output body-force distribution of the plasma model is planned to be coupled with a Direct Numerical Simulation study of the transition process.

1 Introduction The concept of transition delay via cancellation of the TS waves through superposition of ’counter-waves’ has been proved both experimentally [6],[7] and numerically [2]. A new and promising technology that has recently emerged in flow control research is the use of DBD/plasma actuators [3]. The relatively simple construction employs two thin electrodes, separated by a dielectric barrier. High alternating voltage applied to the electrodes creates a strong electric field, which ionizes the air creating a local region of plasma. Due to collisional processes, momentum transfer from the charged particles to the air induces a body force on the flow. This force can Marios Kotsonis Delft University of Technology, Delft, The Netherlands e-mail: [email protected] Leo Veldhuis Delft University of Technology, Delft, The Netherlands e-mail: [email protected] Hester Bijl Delft University of Technology, Delft, The Netherlands e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_34, © Springer Science+Business Media B.V. 2010

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be used in a variety of ways for flow control, one of which is TS wave cancellation [4]. This work is aimed at the modeling of the plasma operation and the set-up for coupling the output to a DNS study which is currently being developed for simulating laminar-turbulent transition. This is expected to provide insight into the interaction of the body-force with the unsteady TS modes.

2 Theoretical Background For the modeling of the plasma operation a first-principles model [5] is preferred over empirical equivalents due to better approximation of the actual plasma formation mechanisms. A shortcoming of this approach is the high computational cost, although not a major issue for the current research work. For transition analysis and the correct representation of the ’counter-wave’ introduction and interaction with the TS modes, a method that correctly captures the non-parallel, non-homogeneous character of the process has to be chosen. A Direct Numerical Simulation study is proposed with the inclusion of the body-force distribution as an unsteady disturbance quantity [1].

2.1 Plasma model The model incorporated in this study is based on fluid equations describing the movement of charged species under the influence of an electric field. Additionally, the mechanisms responsible for the creation and destruction of the species such as ionization and ion-electron recombination are accounted for. The plasma is considered as a fluid comprised of two types of primary species, namely, ions (i) and electrons (e). The working fluid is air. • Continuity equations The continuity equations consider the development over time of the number densities of the species (n) taking into account ionization (S) and recombination (r) processes. ∂ ne + ∇(ne ve ) = ne Sie − rni ne , (1) ∂t ∂ ni + ∇(ni vi ) = ne Sie − rni ne . (2) ∂t • Momentum equations The plasma actuator is intended for use in atmospheric pressure, a regime of relatively high pressure compared to typical plasma formation conditions. Based on this the momentum equation can be reduced to the drift-diffusion form, neglecting all the inertial and unsteady terms. The momentum equation gives the species

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fluxes (Γ ) used in (1) and (2) based on the electric field (E), diffusion coefficients (S) and species mobilities (µ ) −ne µe E − ∇(ne De ) = ne ve = Γ e ,

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• Electric field equation For coupling the electric field with the species number densities the Poisson equation is solved using the fundamental electron charge (e), dielectric material and free space relative permitivities (εd and ε0 respectively) ∇(εd E) =

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Equations (1) - (5) can be solved to provide the number densities of electrons and ions at a given time instant. In conjunction with the local electric field the exerted body force (F p ) in the plasma volume can be expressed using the definition of the electric charge (q) and Coulomb’s law1 F p = E ∑ qk nk .

(6)

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3 Results Several cases have been tested for the operation of the plasma actuator. The 2D domain has been discretized using a mesh of 51 × 51 points and a high-order finite difference scheme is used for the calculation of the spatial derivatives. Due to the extreme variation of characteristic time-scales of the various processes arising during the discharge, the time advancement is performed using a split operator scheme. More specifically, a global time step is used for the slower processes such as ion drift and an additional sub-cycling using a smaller time step is used for the faster processes such as ionization and recombination.

Fig.1 presents the distribution of the potential in the 2D domain for a typical test case. It is evident that the general behavior of the potential field is regulated by the asymmetric configuration of the electrodes. This has a direct impact on the distribution of the electric field seen in Fig.2. The potential and the electric field are coupled with the simple relation E = −∇Φ . This explains the existence of a strong electric field near the inner edge of the electrodes which in turn is the region were the highest intensity of plasma is observed [3].

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Fig. 1 Potential distribution at Tt = 0.25. Vapp = 2kVpp , F = 4kHz, upper electrode length = 10mm, lower electrode length = 15mm. Thick lines denote electrode positions while thin line denotes dielectric barrier position

Fig. 2 Electric field distribution at Tt = 0.25. Vapp = 2kVpp , F = 4kHz, upper electrode length = 10mm, lower electrode length = 15mm.

Fig.3 and Fig.4 show the time-averaged spatial distribution of the body-force F p as calculated from relation 6. It is apparent that the body-force is highly directional2 towards the lower electrode and towards the wall. Additionally it attains larger values in a thin near wall region just downstream of the inner edge of the upper electrode. These features render the plasma actuator an ideal control devise for influencing TS waves since the involved spatial scales are directly comparable to the respective scales of low-velocity boundary layers. Apart from gaining insight into the actual mechanisms and morphology of the body-force components, the numerical study can be extended into a practical parametric study as an extra tool for understanding the capabilities and limitations of the plasma actuator. Several test cases have been executed investigating different geometrical and electrical variations and their influence on the body-force. Fig.5 presents the horizontal component of the body-force distribution for six cases of different electrode lengths. It is evident that the length of the lower electrode has a large influence on the spatial extend. Specifically, two limits can be distinguished. An upper limit evident in the cases of longer electrode lengths (cases a,d,f) were the body-force fails to extend over the entire length of the lower electrode. In this case the electric field becomes to weak to initiate any ionization processes. On the contrast, in cases with short lower electrodes (cases b,c) the extent of the body-force seams to be limited by the electrode length itself. While the electric field is suffi2

The positive x direction is from the upper electrode towards the lower electrode while the positive y direction is from the lower electrode towards the upper electrode

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ciently strong in this area, the ions cannot be ’pulled’ in distances further the lower electrode due to the lack of a low potential ’attractor’.

Fig. 3 Time-averaged spatial distribution of the horizontal component of the body-force Fpx . Vapp = 2kVpp , F = 4kHz, upper electrode length = 10mm, lower electrode length = 10mm

Fig. 4 Time-averaged spatial distribution of the vertical component of the body-force Fpy . Vapp = 2kVpp , F = 4kHz, upper electrode length = 10mm, lower electrode length = 10mm

4 Conclusions and Future work A numerical study has been conducted for modeling the operation of a DBD/plasma actuator. Several test cases indicate the formation of a local, near-wall directional body-force which could be used as a cancellation mechanism for unsteady TS waves growing in a laminar boundary layer. The attained body force distributions are planed to be inserted into a DNS analysis in the form of known harmonic disturbance quantities. Results are expected to provide insight into the interaction between the body-force component and the unstable TS waves. Furthermore, the entire numerical investigation will act as a theoretical companion to ongoing experiments on TS wave cancellation using plasma at the TU Delft.

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Fig. 5 Time-averaged spatial distribution of the horizontal component of the body-force Fpx for several test cases of different upper (lu ) and lower (ll ) electrode lengths, respectively. (a): [lu = 5, ll = 10], (b): [10, 5], (c): [5, 5], (d): [10, 10], (e): [15, 10], (f): [10, 15] (all lengths in mm).

References 1. Bertolotti, F.P., Herbert, T., Spalart, P.R.: Linear and nonlinear stability of the Blasius boundary layer. Journal of Fluid Mechanics 242, 441–474 (1992) 2. Bower, W.W., Kegelman, J.T., Pal, A., Meyer, G.H.: A numerical study of two-dimensional instability-wave control based on the Orr–Sommerfeld equation . Physics of Fluids 30, 998– 1004 (1987) 3. Corke, T.C., Post, M.L., Orlov, D.M.: SDBD plasma enhanced aerodynamics: concepts, optimization and applications. Progress in Aerospace Sciences 43, 193–217 (2007) 4. Grundmann, S., Tropea, C.: Active cancellation of artificially introduced TollmienSchlichting waves using plasma actuators. Experiments in Fluids 44, 795–806 (2008) 5. Jayaraman, B., Cho, Y., Shyy, W.: Modeling of dielectric barrier discharge plasma actuator. 38th AIAA Plasmadynamics and Lasers Conference (2007) 6. Sturzebecher, D., Nitsche, W.: Active cancellation of Tollmien-Schlichting instabilities on a wing using multi-channel sensor actuator systems. International Journal of Heat and Fluid Flow 24, 572–583 (2003) 7. Thomas, A.: Control of boundary layer transition using a wave-superposition principle. Journal of Fluid Mechanics 137, 233–250 (1983)

Experimental study on stability of the laminar and turbulent plane jets V. V. Kozlov, G. R. Grek, G. Kozlov, Yu. A. Litvinenko, and A. Sorokin

1 Laminar plane jet with the abrupt mean velocity profile at the nozzle exit Experimental investigation of plane-jet transition to turbulence is presented in [5]. Sinusoidal (asymmetric) and varicose (symmetric) modes of the jet instability were considered, as both of them can occur quite upstream. Origin of each mode strongly depends on initial mean velocity profiles [1]. Varicose mode is related to the presence of so-called abrupt mean velocity profiles at the nozzle exit, while more developed parabolic mean velocity profiles result in predominance of the sinusoidal oscillations. z

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Fig. 1 Scheme of the classical plane jet (short nozzle) downstream evolution

As is known (see the scheme in Fig. 1, where 2h is a slit width), the abrupt mean velocity distribution gradually transforms in the streamwise direction to the parabolic one that is accompanied by disappearance of the jet core. Patterns of a smoke visualization of the plane jet with the abrupt mean velocity profile on the nozzle exit [2] are shown in Fig. 2. The general view (left) demonstrates alternation V. V. Kozlov Khristianovich Institute of Theoretical and Applied Mechanics SB RAS, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_35, © Springer Science+Business Media B.V. 2010

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V. V. Kozlov, G. R. Grek, G. Kozlov, Y. A. Litvinenko, A. Sorokin plane jet cross-section in xz-plane

general view of the plane jet

z x

симметричная мода неустойчивости (f=30Hz) Гц) symmetric mode instability (f=30

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Fig. 2 Smoke visualization patterns of the plane jet with an abrupt mean velocity profile at the nozzle exit

of dark and light stripes on some distance from the nozzle exit in xy-plane and their absence in region near of the nozzle exit. Cross-sections in xz-plane are shown on the right. It is seen that alternation of the stripes observed in Fig. 2 (left) reflects in oscillation of the jet as a single whole [2] in the region where, apparently, the mean velocity profile becomes parabolic. Note, that the disturbances induced by acoustics at f = 30 Hz result in oscillations of the vortex street developing as the varicose (see Fig. 2, on the right above), and sinusoidal (see Fig. 2, on the right below) instabilities. However, only instability of sinusoidal type dominates downstream. Different flow pattern occurs in the region where the mean velocity distribution preserves the abrupt shape (see Fig. 3, left). Cross-section in xz-plane of the natural plane jet (without acoustic) is shown in Fig. 3 (right). The region where the jet is subject to oscillatory process as a single whole is situated downstream of the region with abrupt mean velocity distribution. The last represents two very thin shear layer and extensive area of a jet core. A major role for the downstream evolution of the jet plays streaky structures [3, 4] observed in the unforced case directly at an jet exit from the nozzle, or generated by the roughness elements pasted on the internal wall of the nozzle exit. U/U0 1,0

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Fig. 3 Mean velocity profile in a cross-section near the nozzle (left) and smoke visualization pattern of the plane jet general view (right)

Pattern of the streaky structures smoke visualization generated by the roughness elements (Fig. 4) and a result of their interaction with the shear layer (Fig. 4b) are shown in Fig. 4. The streaky structures can be observed in Fig. 4a. The Λ or Ω vortices generated by interaction of the streaky structures and the shear layer at

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Fig. 4 Smoke visualization patterns of the general view of the plane jet with an abrupt mean velocity profile at the nozzle exit (a) and the jet cross-section in yz-plane (b)

that side of the nozzle where the roughness elements are placed can be observed in Fig. 4b. In the other shear layer, where the artificially generated streaky structures are absent, the Λ or Ω -vortices are also absent, indicating that the disturbances developing in both shear layers are quite independent of each other. It is shown, that the region of the plane jet with the abrupt mean velocity distribution at the nozzle exit is more sensitive to the acoustic disturbances of high rather than low frequency. High-frequency acoustics result in intensification of the jet mixing with ambient gas through the Λ or Ω -vortex eddies and acceleration of the jet transition into the turbulent state. However, observation of the far field of the plane jet, where the velocity profile gradually approaches to the parabolic shape, shows that the jet became more sensitive to low-frequency disturbances. The last results in predominance of the sinusoidal oscillations of the jet as a single whole and in selection of a vortex street (see Fig. 2). Thus, in the plane jet with the abrupt mean velocity distribution at the nozzle exit it is possible to select three instability regions: two instability regions related to the presence of two shear layers developing separately from each other in a near-field and an instability region of the jet as a single whole in the far field. Note also, that the streaky structures are developed quite far downstream and can be observed even in the region of sinusoidal vortex street.

2 A laminar and turbulent plane jet with a parabolic mean velocity profile at the nozzle exit Laminar parabolic mean velocity profile and turbulent mean velocity profile at the nozzle exit obtained in [2] are shown in Fig. 5. In case of the abrupt mean velocity distribution, in accordance with the jet downstream evolution, there is the continuous competition of these two modes of instability. In the initial stage of the jet development, the symmetric instability mode prevails. As the jet core becomes downstream more narrow, the mean velocity profile gradually approaches to the parabolic shape, the asymmetric mode becomes prevailing and, finally, the symmetric instability mode ceases to dominate in the breakdown mechanism of the jet.

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V. V. Kozlov, G. R. Grek, G. Kozlov, Y. A. Litvinenko, A. Sorokin 2,0 U/Umean •

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Fig. 5 Laminar and turbulent mean velocity profiles at the nozzle exit

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In the present experiment the parabolic mean velocity profile is formed directly at the nozzle exit. Hence, one should expect a predominant development of asymmetric sinusoidal instability mode. Smoke visualization in xz-plane of the laminar plane jet with the parabolic mean velocity profile are shown in Fig. 6. In absence of acoustic forcing ( f = 0 Hz), it is subject to sinusoidal oscillations in the xz-plane. Existence of self-sustaining strong sinusoidal oscillations in the plane jet for the first time has been noted in [6] where it was shown, that they resemble absolutely unstable modes. When the plane jet had parabolic mean velocity distribution directly at the nozzle exit, it was possible to observe sinusoidal oscillation of the jet as a single whole. This process can be caused by the absolute instability of the plane jet. At the acoustic forcing of the jet with the same sound intensity but at different frequencies (from 30 to 150 Hz, see Fig. 6) it is seen that low frequencies of excitation (30–70 Hz) promotes clear observation of the asymmetric vortical structures. Downstream spreading angle of the jet α = 18◦ (without the acoustic forcing) and α = 30◦ (with the forcing), i.e. acoustics promotes growth of the transverse expansion of the plane jet and accelerates its turbulization. As the excitation frequency grows from 80 to 150 Hz , a breakdown of the sinusoidal periodicity of the vortical structures, suppression of the jet oscillations as a single whole and, at finally, a breakdown of the

f = 0 Hz

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f = 150 Hz

Fig. 6 Smoke visualization patterns of the laminar plane jet cross-section with parabolic mean velocity profile at the nozzle exit in xz-plane for different frequencies of the acoustic effect

Experimental study on stability of the laminar and turbulent plane jets

229

plane jet to two ‘sleeves’ is seen, Fig. 6. This result confirms conclusions of [5] that the plane jet with a parabolic mean velocity profile is subject to sinusoidal instability and is more sensitive to low-frequency disturbances.

f = 0 Hz

f = 30 Hz

f = 40 Hz

f = 50 Hz

f = 60 Hz

f = 100 Hz

Fig. 7 Smoke visualization patterns of the turbulent plane jet cross-section in xz-plane for different frequencies of the acoustic effect.

Sinusoidal oscillations of the plane turbulent jet as a single whole caused by the acoustic effect are shown in Fig. 7. It is seen that low-frequency acoustic forcing (30–60 Hz) of the turbulent jet results in its sinusoidal oscillation in xz-plane, similarly to the laminar plane jet. However, at f = 0 Hz and at acoustic forcing of the jet at higher frequency ( f = 100 Hz) this effect is not revealed. It is interesting to consider influence the streaky structures on the downstream evolution of the laminar plane jet. In contrast to the plane jet with abrupt mean velocity distribution at the nozzle exit where two relatively independent shear layers are formed, in the present case the parabolic mean velocity profile at the nozzle exit represents the whole shear layer. Hence, the streaky structures interacts with cylindrical vortices of the sinusoidal vortex street.

3 Plane jet approaching to the microjet When lateral dimension of the nozzle exit was decreased up to 2.5 mm, the laminar plane jet is subject to longitudinal sinusoidal oscillations as a single whole similarly to the case of the nozzle with lateral size of 14.5 mm. Further decreasing of the lateral size of the nozzle exit to 1 mm and below result in origin of the so-called microjet important in different engineering applications. However understanding of physics of process of the formation and downstream evolution of microjet, especially in experimental aspects, remains quite restricted and scattered. Results of a qualitative experimental studies of the plane microjet formed at the nozzle exit with the lateral size of 700 mkm are illustrated by smoke visualizations of the jet evolution in a xz-plane in Fig. 8. It is seen that process of the jet oscillation as a single whole with formation of a sinusoidal vortex street qualitatively remains the same as in the previous experiments where the plane jet was generated from the nozzle with the lateral size more than 1 mm. Acoustic forcing of the jet, as earlier, results in change of periodicity of the sinusoidal vortex street.

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V. V. Kozlov, G. R. Grek, G. Kozlov, Y. A. Litvinenko, A. Sorokin x

x

f = 30 Hz

nozzle is 700 m

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Fig. 8 Smoke visualization patterns of the plane jet in xz-plane outgoing from the nozzle with lateral size of 2h = 0.7 mm (U0 = 0.3 m/s (Reh = 15).

4 Conclusions The laminar and turbulent plane jets generated at the nozzle exit of Hagen–Poiseuille channel were modeled in physical experiment at the same Reh = 3577 (U0 = 3.7 m/s). It was observed that both jets experience as a whole longitudinal sinusoidal oscillations. It was shown that acoustics promotes a selection of cylindrical vortices of the asymmetric vortex street of the laminar jet. As the acoustic frequency grows, the oscillation period of the laminar and turbulent jets decreases.The streakystructure interaction with the cylindrical vortices of the laminar jet results in generation of azimuthal Λ or Ω -vortices, which intensify mixing of the jet with ambient gas and accelerate its turbulization. It was found that in contrast to a classical plane jet, the streaky structures generated on one side of the nozzle for the laminar jet interact both with right and left cylindrical vortices of the vortex street. It was shown that both the structure and the evolution mechanism of a plane microjet and the conventional jet are quite similar. Acknowledgements This work was supported by the Russian Foundation for Basic Research (grant No. 08-01-00027), the Ministry of Education and Science of the Russian Federation (project RNP 2.1.2.3370) and by the grants of President of the Russian Federation (NSh-454.2008.1 and MK-420.2008.1).

References 1. A. V. Boiko, A. V. Dovgal, G. R. Grek, and V. V. Kozlov. Origin of turbulence in near-wall flows. Springer–Verlag, Berlin, 2002. 2. G. V. Kozlov, G. R. Grek, . M. Sorokin, and Yu. . Litvinenko. Effect of initial conditions at the nozzle exit on the flow structure and stability of the plane jet. Bull. Novosibirsk State Univercity. Ser.: Physics, 3(3):25–37, 2008. In Russian. 3. V. V. Kozlov, G. R. Grek, L. L. Loefdahl, V. G. Chernoray, and M. V. Litvinenko. Role of localized streamwise structures in the process of transition to turbulence in boundary layers and jets (review). J. Appl. Mech. Tehn. Phys., 43(2):349–363, 2002. 4. M. V. Litvinenko, V. V. Kozlov, G. V. Kozlov, and G. R. Grek. Effect of streamwise streaky structures on turbulisation of a circular jet. J. Appl. Mech. Tehn. Phys., 45(3):349–357, 2004. 5. H. Sato. The stability and transition of two-dimensional jet. J. Fluid Mech., 7:53–80, 1960. 6. M-H. Yu and P. A. Monkewitz. Oscillations in the near field of a heated two-dimensional jet. J. Fluid Mech., 225:323–347, 1993.

Evolution Of Traveling Crossflow Modes Over A Swept Flat Plate Thomas Kurian, Jens H. M. Fransson, and P. Henrik Alfredsson Abstract An experimental investigation has been carried out to examine the growth of traveling crossflow instabilities over a swept flat plate mimicking the FalknerSkan-Cooke boundary layer. Different turbulence generating grids were placed upstream of the leading edge to vary incoming parameters. Hot-wire measurements were taken for one component of velocity and compared with linear PSE analysis. These showed a decrease in the growth rate for increasing turbulence intensity, which was most likely cause by nonlinear effects. Streamwise correlation measurements were also taken. All the cases except one triggered the same spanwise integral length scale inside the boundary layer. Receptivity coefficients are needed to do the PSE calculations and to see the need for nonlinear PSE.

1 Introduction In industrial applications knowledge about transition criteria is highly sought after, but still not fully understood. Currently, the eN -method is widely used even in cases where surface roughness is present and the free stream is affected by background noise. Today, this is known to trigger a different transition scenario with algebraic disturbance growth at sub-critical Reynolds numbers considering classical stability theory [1] [4]. More experiments are needed to enhance the fundamental knowledge of the receptivity mechanisms, which can lead to the establishment of receptivity coefficients.

2 Experimental Setup An experimental investigation has been undertaken to study the growth of traveling crossflow instabilities over a swept flat plate under different turbulence intensities and characteristic length scales. The experiments were carried out in the MTL wind tunnel at KTH Mechanics with a free stream velocity at the leading edge, U∞,LE , of 18 m/s. The leading edge of the flat plate was a modified super-ellipse at a sweep angle of 25◦ . The base flow was designed to mimic a Falkner-SkanThomas Kurian, Jens H.M. Fransson, P. Henrik Alfredsson Linne Flow Center, Osquars Backe 18, SE-10044 Stockholm SWEDEN e-mail: [email protected], [email protected], [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_36, © Springer Science+Business Media B.V. 2010

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T. Kurian, J. H. M. Fransson, P. H. Alfredsson C 600

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Fig. 1 (a) C p contours over the swept flat plare. The dashed lines are aligned with the leading edge. (b) Shows the mean free stream velocity over the plate for U∞,LE = 13.2 m/s. Index t and n correspond to coordinate systems aligned with the incoming flow and the leading edge, respectively. In both figures, the first dashed line represents the location of the leading edge

Cooke (FSC) boundary layer which is obtained by imposing U∞ (x) = Cxm with a crossflow velocity component in the governing equations. Taking into account the expected growth of the crossflow mode as well as the thickness of the boundary layer a displacement body, designed to give a strong acceleration with a Hartree parameter, βH ≡ 2m/(m + 1) = 0.17, was built. Measurements in the free stream were done with an X-probe to measure the streamwise and spanwise component. The probe could also be turned accurately to measure the wall-normal component. Fig. 1a shows the quality of the base flow with contour lines of constant C p parallel with the sweep angle up to approximately one metre from the leading edge. The acceleration along the centreline matches with βH = 0.17 (Fig. 1b). Measurements inside the boundary layer were done with a single hot-wire probe aligned with xt .

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Table 1 Grid parameters at the leading edge. Grid #

Mesh Width (mm) Tu%

Λx (mm)

Symbol

LT1 LT2 LT3 LT4 LT5

0.96 1.65 1.80 2.56 4.20

4.18 5.54 5.28 5.52 6.50

0.23% 0.28% 0.31% 0.36% 0.58%

◦ • ⋄

2.1 Turbulence generation Low free stream turbulence levels were varied by placing different turbulence generating grids upstream of the leading edge. By changing the grid (thus the mesh and bar widths), it is possible to vary the incoming free stream turbulence intensity, Tu, as well as the integral length scale, Λx [3]. The parameters for the grids are shown in Table 1.

3 Results 3.1 Velocity and Disturbance Profiles Velocity profiles were taken from a region 450 mm to 1000 mm from the leading edge in xt . This is the region where the boundary layer was thick enough to be able to resolve the disturbance peak. The free stream velocity was fitted to a power law to be able to determine the Hartree parameter. βH was then used to solve the FSC profile and this was then fitted to the measured profiles to determine the wall position as well as the boundary layer thickness. The boundary layers for case LT5 are shown 8

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5 Increasing X

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in Fig. 2a with the free stream velocity inset. This shows that the boundary layer accelerates and gets thicker for increasing xt , as expected. Also in Fig. 2b is the fit to the Ut component of the FSC profile which shows excellent agreement. The wall normal disturbance profiles of urms,t for two grids LT1 and LT5 are shown in Fig. 3. These profiles are representative of all cases tested. There is noticeable disturbance growth for both cases with xt . For very small levels of disturbance, filtering was needed around 13 Hz, which was caused by the vibration of the traversing mechanism.

3.2 Spanwise Correlations Two-point spanwise correlations were taken at one xt -location 1 m from the leading edge for all 5 grids and are plotted in Fig. 4. The minimum value is well known to correspond to the averaged half-spanwise wavelength, λz /2. For grids LT2 − LT5 , the resulting wavelength is nearly identical despite an incoming streamwise length scale varying from 5.28 mm to 6.50 mm. It is clear that grid LT1 with significantly smaller integral length scale also generates a smaller spanwise wavelength inside the boundary layer, Table 2. Table 2 Resulting spanwise wavelengths for all grids Grid #

Λx (mm)

λz (mm)

β (m−1 )

LT1 LT2 LT3 LT4 LT5

4.18 5.54 5.28 5.52 6.50

6.76 9.22 9.93 9.32 9.75

929 681 632 674 644

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3.3 Disturbance Growth The resulting disturbance growth curves, Fig. 5a shows disturbance levels between 1-10%. In Fig. 5b the natural logarithm of the disturbance growth, N, is plotted normalized with the value at xt = 450 mm. Interestingly a lower turbulence intensity induces a faster growth rate. Since the spanwise correlations are the same for four of the cases, we cannot conclude that this is the result of different modes growing at different rates. This result is believed to be attributed to nonlinear growth resulting from the redistribution of energy between certain modes. Comparison with linear parabolized stability equations (PSE) is shown in Fig. 6. This shows poor agreement with measured cases and may be caused by a few factors. Firstly, it is clear from Fig. 5 and Fig. 6 that we have not resolved the location of branch I. For this reason, even assuming linear growth in the early region, receptivity coefficients cannot be calculated until the location of branch I is resolved. Another cause for deviation could be that the disturbance levels have saturated. It 0.1

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Fig. 5 (a)Disturbance energy growth for all grids. (b) The natural logarithm of the disturbance normalized by the value at xt = 450 mm. See Tab. 1 for symbols.

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linear PSE curve

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2 1 0 0

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400 600 x (mm)

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has been shown that linear PSE calculations start to deviate from measured values when saturation begins [2].

4 Conclusions and Outlook Hot-wire measurements were done on a swept flat plate with contoured side walls to mimic an infinitely swept wing for approximately 1 m. Several different turbulence generating grids were used to vary the parameters of integral length scale and free stream turbulence intensity at the leading edge. The same spanwise wavelength was triggered in the boundary layer for a range of streamwise wavelengths with only the smallest initial integral length scale triggering a different wavelength. Growth curves of the disturbance peak show that the growth rate decreased with increasing free stream turbulence intensity. These were also not very well predicted by linear PSE. For the future more calculations will need to be done to obtain the receptivity coefficients. Acknowledgements This investigation has been performed within the project TELFONA (TEsting for Laminar Flow On New Aircraft) under contract number AST4-CT-2005-516109. The PSE calculations were performed by David Tempelmann at KTH.

References 1. Bippes H. (1991) Basic experiments on transition in three-dimensional boundary layers dominated by crossflow instabiltity. Progress in Aerospace Sciences, 35:363-412 2. Haynes T.S., Reed H.L. (2000) Simulation of swept-wing vorticies using nonlinear parabolized stability equations. J. Fluid Mech, 435:1-23 3. Kurian T., Fransson J.H.M. (2009) Grid-generated turbulence revisited. Fluid Dyn. Res, 41:021403 4. Saric W.S., Yeates L.G. (1985) Experiments on the Stability of Crossflow Vortices in SweptWing Flows. AIAA Paper 85-0493

Computational Analysis for Roughness-Based Transition Control Fei Li, Meelan M. Choudhari, Chau-Lyan Chang, and Jack R. Edwards

Abstract Suitably placed discrete roughness elements are known to delay or hasten the onset of transition, depending on requirements. In this paper, 2D eigenvalue analysis is used to study the effects of surface roughness in the context of transition delay over subsonic and supersonic swept wing configurations, as well as boundarylayer tripping on the forebody of a hypersonic air breathing vehicle.

1 Introduction Surface roughness is known to have a substantial impact on the aerodynamic or aerothermodynamic predictions for a flight vehicle, regardless of the state of the boundary layer. When the incoming boundary-layer flow is laminar, the presence of 3D surface roughness tends to accelerate the laminar-turbulent transition process, although carefully placed spanwise periodic discrete roughness elements can delay the onset of transition in crossflow dominated boundary-layer flows. Passive control via roughness elements provides an attractive avenue for drag reduction on subsonic and supersonic swept wing configurations. On the other hand, in scramjet applications, artificial roughness is often employed for tripping the boundary layer flow on the forebody of the vehicle to prevent engine unstart and to minimize the flow non-

Fei Li NASA Langley Reseacrh Center, Hampton, VA 23681 e-mail: [email protected] Meelan M. Choudhari NASA Langley Reseacrh Center, Hampton, VA 23681 e-mail: [email protected] Chau-Lyan Chang NASA Langley Research Center, Hampton, VA 23681 e-mail: [email protected] Jack R. Edwards North Carolina State University, Raleigh, NC 27693 e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_37, © Springer Science+Business Media B.V. 2010

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F. Li, M. M. Choudhari, C.-L. Chang, J. R. Edwards

uniformities at the entrance to the combustor inlet. This paper uses spatial, 2D eigenvalue analysis to explore the physics of roughness effects on boundary layer transition across subsonic through hypersonic regimes via a study of selected flow configurations.

2 Methods The equations governing the unstable small-amplitude perturbations are obtained by linearizing the Navier-Stokes equations about a specified basic state, such as a finite amplitude crossflow vortex developing in a swept-airfoil boundary layer or longitudinal streaks behind a spanwise-periodic array of boundary layer trips. For a basic state that is slowly varying along one of the spatial coordinates (e.g. the crossflow vortex axis), using a wave ansatz in that direction reduces the spatial dimension of the problem governing the disturbance quantities by one, resulting in a set of two-dimensional, linear partial differential equations at the leading order. The 2-D eigenvalue problem in a spatial framework is solved to characterize the amplification characteristics of high-frequency, secondary instabilities of crossflow vortices as well as the instabilities of the stationary streaks trailing a periodic array of roughness elements. The spatial stability equations are formulated in a non-orthogonal coordinate system in order to properly account for the direction of spatial growth and the spanwise periodicity of the unstable perturbations (see Li and Choudhari 2008).

3 Results Computations are carried out to study the effects of surface roughness in the context of three specific configurations, viz., (1) transition delay over a subsonic swept wing at chord Reynolds number of Rec = 7 × 106 (Carpenter et al. 2008); (2) transition control over a swept wing at Mach 2.4 and Rec = 16×106 (Saric and Reed 2002) and (3) boundary layer tripping over a scaled model of the Hyper-X forebody at Mach 6 (Berry et al. 2001). For cases (1) and (2), the amplitude that is often referred to below is that of the velocity component in the direction of the normal chord as a fraction of the freestream velocity, and for case (3) it refers to the velocity component in the direction of the freestream. In case (1), the effect of periodically spaced roughness elements is simulated by introducing stationary crossflow vortices with a spanwise wavelength of 2.25 mm (the control mode) to suppress the growth of the linearly more unstable stationary crossflow mode with a spanwise wavelength of 4.5 mm (the target mode). Computations based on nonlinear Parabolized Stability Equations (PSE) have shown that

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the amplitudes of the target mode can be significantly reduced via a control mode with sufficiently large initial amplitudes. With the initial amplitude of the 4.5 mm primary stationary crossflow vortex mode fixed at 10−4 , the effect of increasing the amplitude of the 2.25 mm control mode is shown in Figure 1 (a). With zero control amplitude, it is seen that both of the dominant Y- and Z-mode secondary instabilities undergo strong modal amplification. As the control amplitude increases, modal amplification of both the dominant Y- and Z-modes is weakened. Furthermore, as shown in Figure 1 (a), the presence of the control mode apparently affects the Zmode of secondary instability much more strongly than the Y-mode. This indicates that, if indeed transition is caused by the growth of the Z-mode as some experiments seem to suggest (Kawakami, M. et al. 1999, White and Saric 2005), the DRE is a very effective means to achieve transition delay in the present case. There is, however, a possibility that the 2.25 mm control mode could become susceptible to secondary instability and may itself break down to cause an earlier onset of transition (i.e. overcontrol). Secondary instability analysis for the 2.25 mm control mode is carried out to assess the likelihood of this scenario. The peak N-factors for Z-mode of secondary instability are less than 1 for all three control inputs. Of the three initial amplitudes of 0.002, 0.005 and 0.01, only the last gives rise to a maximum Y-mode secondary instability N-factor of approximately 11 at a chordwise location close to x/c = 0.25. For the other two amplitudes of control input, the Y-mode N-factors remain less than or equal to approximately 8. Therefore, even if the Y-mode can also lead to transition, the possibility of premature transition due to overcontrol can be avoided by keeping the initial control mode amplitudes below a

Fig. 1 (a) Subsonic swept wing configuration.Envelops of secondary N-factor curves for Y- and Zfamiles. Nonlinear control of most unstable mode (4.5 mm) via control input (2.25 mm) with different initial amplitudes. (b) Mach 2.4 supersonic swept wing configuration. Crossflow vortex amplitude curves. Nonlinear control of most unstable mode (3 mm) via control input (1.5 mm) with different amplitudes

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certain threshold, which in this case appears to be around 0.005 or slightly larger. In case (2), similar analyses as those in case (1) are carried out for a swept wing at a Mach number of 2.4. The wavelength of the control mode is 1.5 mm, and that of the target mode is 3.0 mm. The rapid amplitude rise of the 3 mm target mode is progressively delayed with increasing initial amplitude of the 1.5 mm control mode (Fig. 1 (b)). Computations of secondary instability confirm the accompanying delay in the amplification of the secondary instability modes on the 3 mm crossflow vortex. More interestingly, however, the secondary instability analysis on the 1.5 mm control mode reveals the increased likelihood of premature transiton via overcontrol, since the N-factors of both the Y- and the Z-modes of secondary instability reach relatively large values (with the Y-mode continuing to be the most dominant one). For an initial control amplitude of 0.001, maximum N-factors for the two modes are approxiamtely 8 and 6, respectively. When the amplitude is increased to 0.002, the peak N-factors approach 13 and 9, respectively. In case (3), the roughness array is employed to trigger an earlier transition in a hypersonic boundary layer instead of delaying it. The computations described herein model the Hyper-X Mach 6 flow configuration with three flat ramps to provide the necessary compression ahead of the scramjet engine (see Berry et al. 2001). The trip elements produce strong trailing streaks (Fig. 2), which are susceptible to instabilities similar to those riding on finite amplitude Görtler vortices over a concave surface. The roughness array is placed on ramp 1 at 7.4 inches from the leading edge, and the two corners joining the three ramps are at 12.4 and 17.7 inches, respectively. The results presented herein correspond to a trip spacing of δ = 0.081” and a peak height of h = 0.060”. The mean flow is computed with a Navier-Stokes solver, using the immersed boundary technique (see Ghosh et al. 2008). Figure 2 shows that the streamwise streaks with strong spanwise boundary-layer displacement in the wake of the trip array would persisit for long distances over the Fig. 2 Steamwise streaks produced behind the trip array on Hyper-X model (visualized via u-velocity contours at x = 6”, 8”,10”, 12” and 14”, respectively, flow is from bottom left to top right and, for visual clarity, the wall normal and spanwise coordinates Y and Z have been modified relative to X coordinate). The underlying light surface corresponds to the isosurface of streamwise velocity for u = 1m/s.

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forebody surface if the boundary-layer flow were to remain laminar. Since the spanwise and wall-normal length scales of these streaks are comparable with each other, the modified boundary-layer flow has a strongly inhomogeneous character in both Y and Z directions. Therefore, its stability characteristics are more appropriately studied by solving a 2D eigenvalue problem. It is found that multiple modes of instability co-exist, which is typical of boundary-layer flows modified by finite amplitude streaks. The growth rates of the two dominant modes of instability at x = 7.9” have been plotted against the disturbance frequency in Fig. 3(a), which also indicates the representative mode shapes for the magnitude of the u-velocity perturbation associated with each mode. The spanwise period of these modes is equal to the array spacing δ . Despite the presence of strongly inflexional boundarylayer profiles in the wall-normal direction, the more unstable mode (i.e., mode 1) from abovementioned modes is found to be driven by the spanwise (Z) shear of the basic state (i.e., corresponds to an odd mode, which induces sinuous motions of the underlying stationary streaks). Since the spanwise shear occurs solely because of the trip array, this dominant, odd (or Z) mode of streak instability would not have existed without the roughness elements. The subdominant mode 2 is found to be an even (Y) mode. N-factors and growth rates of fixed frequency disturbances belonging to mode 1 family have been plotted in Figs. 3(b) and 3(c), respectively. As seen from Fig. 3(c), the growth of streak instability ceases well upstream of the end of the model ramp 1 (x=12.4”) before resuming again over the second compression ramp. The absence of growth in the immediate vicinity of the compression corner is attributed to the rapid decrease in streak amplitudes just ahead of the corner. As seen from Fig. 3(b), the odd mode (mode 1) disturbances near f = 90 kHz reach an N-factor of approximately 7 across an amplification region of just 2.5 inches. In a previous set of experiments in the same facility (Horvath et al. 2002), transition onset on a smooth, flared cone model had been found to correlate with N ≈ 4. Thus, if a similar value of N is

Fig. 3 For steamwise streaks produced behind the trip array on Hyper-X model. (a) Growth rates and mode shapes of modes 1 and 2 at plane x = 7.9”. (b) Mode 1 N-factors. (c) Mode 1 growth rate.

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assumed to correlate with the onset of roughness-induced transition on the Hyper-X model, then the predicted onset of transition will be within a distance of ∆ x/L = 0.05 behind the trip (where L=48” denotes the reference length of the vehicle at the model scale). The phosphor thermography measurement in the experiment (Berry et al. 2001) is also suggestive of transition onset at a short distance behind the trip array.

4 Conclusions By applying artificial roughness elements, transition can be either delayed or hastened, depending on the specific requirements. In the former case, carefully designed roughness elements produce long streamwise structures that suppress the growth of the more dangerous crossflow vortices and hence weaken the high frequency secondary instabilities that would otherwise cause transition to occur earlier. The 2D eigenvalue analysis plays an important role in this analysis by delineating an optimal range of control input magnitudes. In the case of tripping over compression surfaces, roughness elements generate stationary streaks that amplify across the compression corner and, furthermore, enhance the growth of nonstationary streak instabilities that are expected to trigger an earlier onset of transition as observed in the experiments.

References 1. Berry, S.A., Auslender, A.H., Dilley, A.D. and Calleja, J.F., “Hypersonic Boundary-Layer Trip Development for Hyper-X,” J. Spacecraft and Rockets, vol. 38, No. 6, pp. 853-864, Nov.-Dec. 2001. 2. Carpenter, A.L. Saric, W.S., and Reed, H.L., “Laminar Flow Control on A Swept Wing With Distributed Roughnes,” AIAA Paper 2008-7335, 2008. 3. Ghosh, S., Choi, J.-I., and Edwards, J.R. “RANS and hybrid LES/RANS Simulation of the Effects of Micro Vortex Generators using Immersed Boundary Methods,” AIAA Paper 20083728, June, 2008 4. Kawakami, M., Kohama, Y. and Okutsu, M. “Stability Characteristics of Stationary Crossflow vortices,” AIAA Paper 99-0811, 1999. 5. White E. B., and Saric W. S., “Secondary Instabilty of Crossflow Vortices,” J. Fluid Mech. Vol. 525, pp. 275-308, 2005. 6. Li, F., and Choudhari, M., “Spatially Developing Secondary Instabilities and Attachment Line Instability in Supersonic Boundary Layers,” AIAA Paper 2008-590, 2008. 7. Saric, W.S. and Reed, H.L., “Supersonic Laminar Flow Control on Swept wings Using Distributed Roughness.” AIAA paper 2002-147, 2002.

Statistics of turbulent-to-laminar transition in plane Couette flow Paul Manneville

Abstract A study of turbulence decay in plane Couette flow at large aspect ratio suggests an alternative spatiotemporal interpretation to the current chaotic transient paradigm. It is supported by numerical simulations of a reduced semi-realistic two-dimensional partial-differential model. Evidence comes from the distribution of the sizes of laminar domains nucleating in the turbulent state. Statistical analysis shows that, above some well-defined Reynolds number Rlow , its variance stays finite while it diverges below. Accordingly, at large aspect ratio, turbulence has vanishingly small probability to decay when R ≥ Rlow , while decay is unavoidable for R < Rlow .

1 Introduction Wall-Bounded laminar flows such as the Blasius boundary layer flow, plane Poiseuille flow, plane Couette flow, or Poiseuille pipe flow, are conditionally stable below their linear stability threshold, if any. Laminar flow may indeed coexist with bifurcated, nontrivial, nonlinear, locally chaotic/turbulent flow at moderate values of the Reynolds number R. This coexistence can be understood in the framework of low dimensional dynamical systems as phase-space attractor coexistence, but finds a better expression in physical space as turbulent spots, patches, bands, immersed in laminar flow. Here we focus on the turbulent-to-laminar transition. With some success, the decay of turbulence has been interpreted in terms of chaotic transients associated to stochastic repellers. The main limitation is that the approach is best adapted to confined systems, i.e. a few Minimal Flow Units, while relevant experiments rather take place in large aspect ratio systems. Consult contributions to [1] for a more detailed account of these issues. The very existence of sustained Paul Manneville ´ Laboratoire d’Hydrodynamique (LadHyX), CNRS UMR 7646, Ecole Polytechnique, Palaiseau, France, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_38, © Springer Science+Business Media B.V. 2010

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turbulence for arbitrary long times has even been questioned, a conclusion mostly based on experimental results obtained for Poiseuille pipe flow [2]. Our findings strongly suggest that, at least for plane Couette flow, sustained turbulence does exist beyond some threshold Rlow . They have been obtained by deliberately turning to a spatiotemporal approach in line with Pomeau’s early proposal of connecting subcritical bifurcations in hydrodynamics with first-order thermodynamic phase transitions and spatiotemporal intermittency (STI), a process by which transient local chaos is converted into sustained global spatiotemporal chaos beyond some percolation threshold [3].

2 Model and Results The model used has been derived from the Navier–Stokes equations by standard Galerkin methods, here unfreezing the space-time in-plane (x, z) dependence of the velocity field while keeping the wall-normal (y) dependence limited to a few modes [4]. At lowest truncation order, it already contains the main ingredients of turbulence-sustaining processes [1] and mimics plane Couette flow in extended geometry (in-plane dimensions large compared to the gap). Preliminary studies at moderate aspect ratio have pointed out a subcritical bifurcation at Rlow ≈ 170 and comparisons of small and large aspect ratio systems have shown that the large systems can support laminar domains several times larger than the size of the small systems without decaying. Whereas a small system cannot recover from the occurrence of a sizable laminar fraction, the large system does, just because regions still turbulent can contaminate the laminar domain back to turbulence, which is precisely the basic process at work in STI. To be statistically conclusive, numerical experiments need very large aspect ratios and very long simulations, which in turn requires a lowering of the space-time resolution just above the limit where the physics would be qualitatively changed. A domain of size 1536h × 2h × 1536h has been considered (to be campared to the size of the minimal flow unit ∼ 6h × 2h × 4h, 2h is the gap). Simulations have confirmed the STI character of the bifurcation. It has further been shown that the large-size tail of distributions of laminar domains follow power-laws, i.e. ∝ S−α , where S is the surface of a laminar patch, Fig. 1. Values of α of the order of 3 have been found. The first and second moments of the distributions converge for α > 2 and 3, respectively. A change of behaviour of the variance is thus expected for α = 3. Turbulence is sustained provided that no wide laminar patch is present in the system, which turns out to be the case with probability essentially equal to 1 for R = 171.5 (α ≈ 3.4). On the contrary, for R = 171 a wide laminar domain nucleates with probability 1 since the variance of the distribution diverges (α ≈ 2.8). This wide domain then invades the system so that turbulence decays, which helps one to define the threshold Rlow , without ambiguity [5]. A limitation of the model is that, due to its low wall-normal resolution implying underestimated energy transfer toward (and dissipation in) small scales, the transi-

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4

size histogram

10

2

10

2.80±0.05 0

10

3.40±0.14 R=171 R=171.5

−2

10

1

10

2

10

laminar cluster sizes

Fig. 1 Histograms of sizes of laminar domains for R = 171.5 (squares, red) and R = 171.0 (circles, blue); straight lines in log-log scale indicate power law behaviour; slope smaller than 3 in absolute value signals a divergent second moment of the size distribution.

tional range is lowered by a factor of 2 down to R ≈ 170 instead of being spread on the interval R ∼ 320–380 as in the laboratory experiments [6]. Though it was observed under exceptional circumstances [4], the characteristic formation of oblique turbulent bands in the upper part of this interval was generally absent, whereas it was reproduced in fully resolved direct numerical simulations in a long but narrow domain oriented at an appropriate angle with the streamwise flow direction [7]. These oblique bands have also been observed recently in unconstrained geometry at large aspect ratio [8]. To our surprise, we could easily obtain them [9] in simulations with periodic boundary conditions at distances of the order of the streamwise and spanwise wavelengths measured in the experiments (∼ 110h and ∼ 70h, respectively) using a public domain software [10]. We now intend to combine results from the model and from the full equations to get a deeper understanding of the mechanism that govern the breakdown of turbulence. To conclude, the statistics of decaying turbulence has been studied for plane Couette flow in a large aspect-ratio limit still hardly accessible to fully resolved direct numerical simulations but of interest for a comparison with laboratory experiments. Our study supports the STI scenario, as initially suggested by Pomeau and our main conclusion is that whether or not lifetimes of chaotic local structures diverges for Rlow < ∞ becomes irrelevant in regard of the fact that these lifetimes are increasing functions of R and can be tuned to be sufficiently large for spatial coupling to be effective in propagating local temporal chaos and converting it into global spatiotemporal chaos. Owing to its generic character, this conclusion should not have to be basically revised, even though ongoing work on the upper transitional range might force one to reformulate it slightly. It should also hold more generally when laminar/turbulent domains coexist at intermediate Reynolds numbers, e.g. for Poiseuille pipe flow, provided that the long time, large aspect-ratio limit is considered.

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Acknowledgements Members of the Instability & Turbulence Group at Saclay are collectively thanked for their contribution to the author’s understanding of the problem. At LadHyX, M. Lagha played an important role in the development of the reduced model. Discussions with Y. Duguet have been greatly appreciated. H. Chaté deserves special thanks for his insightful remarks about the statistics of laminar domain distributions. Simulations were performed thanks to CPU time allocations of IDRIS (Orsay) under projects #61462 and #72138.

References 1. T. Mullin, R. Kerswell (eds.). Laminar-Turbulent Transition and Finite Amplitude Solutions , Springer (2005). 2. B. Hof et al.: Repeller or attractor? selecting a dynamical model for the onset of turbulence. Phys. Rev. Lett. 101 214501 (2008) 3. Y. Pomeau: Front motion, metastability and subcritical bifurcations in hydrodynamics. Physica D 23 3–11 (1986) 4. M. Lagha, P. Manneville: Modeling transitional plane Couette flow. Eur. Phys. J. B 58, 433– 447 (2007) 5. P. Manneville: A spatiotemporal perspective on the decay of turbulence in wall-bounded flows. Phys. Rev. E 79 025301(R) & 039904(E) (2009) 6. A. Prigent et al.: Long-wavelength modulation of turbulent shear flows. Physica D 174 100– 113 (2003) 7. D. Barkley, L.S. Tuckerman: Computational study of turbulent laminar patterns in Couette flow. Phys. Rev. Lett. 94, 014502 (2005); L.S. Tuckerman, this conference 8. Ph. Schlatter, Y. Duguet: Stripy patterns in low Reynolds number Couette turbulence; Y. Duguet, this conference and private communication 9. J. Rolland, P. Manneville: Oblique turbulent bands in plane Couette flow, from visual to quantitative data. 16th International Couette-Taylor Workshop, Princeton, Sept. 2009 10. J. Gibson: ChannelFlow: a spectral Navier-Stokes simulator in C++ texttthttp://www.cns.gatech.edu/channelflow/, 1999–200x.

Spectra of Swirling Flow Xuerui Mao and Spencer J. Sherwin

Abstract The Batchelor vortex is adopted as the mathematical model of swirling flow. The modes of the Batchelor vortex fall into three broad categories: core modes, algebraic modes and continuous modes. The core modes have been extensively documented but the last two modes have received little attention. The energy growth of those modes are studied by employing asymptotical instability analysis and transient growth analysis. The spectra map of the Batchelor vortex is obtain and the variation of radial distribution of the continuous modes with wave numbers is investigated.

1 Introduction There are several mathematical models for swirling flow, such as the Batchelor vortex, the Burgers vortex and the Lamb-Oseen vortex. The Bachelor vortex is adopted here because it models not only swirling flow but also trailing or jet-like vortices. In cylindrical coordinates (x, r, θ ), the axial, radial and azimuthal velocity components of the Batchelor vortex are defined in non-dimensional form as: 2

U(r) = a + e−r ,

V (r) = 0,

2

W (r) = q/r(1 − e−r ),

(1)

where q represents the swirl strength and a is the free stream velocity (Batchelor, 1964). The Reynolds number of the Batchelor vortex is defined as Re = △UR/ν ,

Xuerui Mao Department of Aeronautics, Imperial College London, e-mail: [email protected] Spencer J Sherwin Department of Aeronautics, Imperial College London, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_39, © Springer Science+Business Media B.V. 2010

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where ∆ U is the dimensional velocity excess in the core of the vortex, R is a measure of the core radius, typically defined as the point of minimum tangential velocity, corresponding to a value of R ≈ 1.121, and ν designates the kinematic viscosity. In this paper, Re is set to be 1000 if not specially stated. The modes of swirling flow fall into three broad categories: 1) Core modes, which decays exponentially from the core of the vortex to the free stream. This mode has been extensively studied, including helical instability modes (Lessen et. al., 1974) and viscous center modes (Heaton, 2007). 2) Algebraic modes, whose structures exist out of the vortex core and decay algebraically with r. This mode has received little attention. 3) Continuous modes, which oscillate in the free stream. The continuous modes of the one-variable Orr-Sommerfeld equation in boundary layer flow has been investigated by Zaki (2009) and Gustavsson (1979) , but it has not been introduced into the swirling flow, which has three velocity components and the governing equations cannot be reduced to one variable. The energy distribution of the three types of modes along the radial direction is illustrated in figure 1.

2 Linear asymptotical stability analysis The incompressible NS equations can be written as:

∂t u = −uu · ∇ u − ∇ p + ν ∇2 u ,

with ∇ · u = 0,

(2)

where u = (u, v, w)(z, r, θ ,t) is the velocity field and p is the modified pressure. The variables u, v, w are the velocity components in the axial, radial and azimuthal directions respectively. Equation (2) can be linearized by decomposing the velocity components and pressure into the summation of a base flow and a perturbation: U · ∇ )uu′ − (uu′ · ∇ )U U − ∇ p ′ + Re−1 ∇2 u ′ , ∂t p′ = −(U

(3)

Fig. 1 Energy distribution in the radial direction. From left to right: core mode, algebraic mode and continuous mode. The dashed lines represent the core radius of the Batchelor vortex.

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where u ′ and p′ are the perturbation velocity vector and pressure, and U is the base flow velocity vector given by equations (1). The action of equation (3) on an initial perturbation u ′ (0) over time interval τ may be stated as u ′ (τ ) = M(τ )uu′ (0); the asymptotic/large-time behaviour of linear perturbations is exponential and governed by the leading eigenmodes of M. If the maximum eigenvalue of M is positive, the base flow U is unstable and vice versus. The Batchelor vortex has two main types of unstable modes, inviscid helical instability modes and viscous center instability modes. Both of them are core modes but the growth rates of viscous centre modes are one-order smaller than those of helical modes. The structures of helical modes and center modes are illustrated in figure 2. The energy of those modes concentrates in the vortex core, especially the center mode.

Fig. 2 Typical core modes. Left: viscous centre mode at Re = 14000, m = −1, k = 0.268. Right: inviscid helical mode at m = −2, k = 1.2.

3 Optimal transient growth analysis In the axisymmetric condition, where the bubble vortex breakdown occurs, the Batchelor vortex is asymptotically stable, so transient growth analysis is employed to examine if there is strong transient growth to provide by-pass routes to vortex breakdown. Transient growth is especially suited to flows which are asymptotically stable or only weakly unstable. So it is a natural tool to the asymptotically stable axisymmetric modes of the Batchelor vortex. We take a direct approach to computing initial conditions that lead to optimal transient growth. As is typical, we define transient growth with respect to the energy norm of the perturbation flow, derived from the L2 inner product 2E(uu′ ) = (uu′ , u ′ ) ≡

Z

Ω

u ′ · u ′ dV,

(4)

where E is the kinetic energy per unit mass of a perturbation, integrated over the full domain. If the initial perturbation u ′ (0) is taken to have unit norm, then the transient energy growth over interval τ is E(τ )/E(0) = u′ (τ ), u′ (τ ) = M(τ )uu′ (0), M(τ )uu′ (0) = u′ (0), M ∗ (τ )M(τ )uu′ (0) ,

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where we introduce M ∗ (τ ), the adjoint of the forward evolution operator M(τ ). The action of M ∗ (τ ) is obtained by integrating the adjoint linearized NS equations U · ∇)uu∗ + (∇ ∇U )T · u∗ − ∇ p∗ + Re−1 ∇2 u∗ , −∂t u∗ = −(U

with

∇ · u∗ = 0

(5)

backwards over interval τ . The action of the symmetric operator M ∗ (τ )M(τ ) on u ′ is obtained by sequential time integration of M(τ ) and M ∗ (τ ). Typically G(τ ) = max(λ j ) is used to denote the maximum energy growth obtainable at time τ , while the global maximum is denoted by Gmax = maxτ G(τ ) (Barkley et. al., 2007; Balckburn et. al., 2008). The optimal perturbation and its outcome at τ = 60 are illustrated in figure 3. The energy of the optimal perturbation concentrates in the region between the vortex core and the free stream and the radii of structures increases for increasing τ . The energy distribution of the optimal perturbations indicate that they are algebraic modes. The algebraic modes are asymptotically stable but experience strong transient growth before eventually decaying. We see from figure 3 that the slope of the envelope dGdmax becomes less than 0.005 and a long time interval is required to reach the τ maximum growth, which can be up to 106 ( Heaton and Peak, 2007).

4 Map of spectra ˆ p)exp(ikx Considering perturbations in the form (u′ , p′ ) = (u, ˆ + imθ + σ t), the governing equations (3) can be written as: Lvv Lvw v Rvv Rvw v v v σ = or σ =D , (6) Lwv Lww w Rwv Rww w w w The growth rate σ is the eigenvalue of matrix D. In free stream where both the core modes and algebraic modes decay and only the continuous modes survive, Lvw = Lwv = Rvw = Rwv = 0, so v and w are decoupled. In the region between the

Fig. 3 Transient growth of algebraic modes at axisymmetric condtion. Left top: optimal perturbation at τ = 60; left bottom: outcome of the perturbation at t = 60. The dashed line represents the core radius of the Batchelor vortex. Middle: 3d view of the outcome. Right: optimum envelope together with transients evolved from optimal perturbations at τ = 5, 10 and 20.

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core and the free stream, where the algebraic modes dominate, the exponential terms of the base flow have decayed and Lvw = Lwv = Rwv = 0, so v and w are partially decoupled, or in other words, the azimuthal velocity evolves independently, but it feeds into the evolution of the radial velocity.

Fig. 4 Left: spectra of D. Right: typical eigenvectors of D, from left to right: algebraic mode, core mode and continuous mode.

The full spectra of matrix D and typical eigenvectors are illustrated in figures 4. We can see that the three types of modes correspond to different branches on the map and all the eigenvalues of D are located on the left half plane, which is consistent with the result in section 2 that the Batchelor vortex is asymptotically stable at m = 0. The continuous spectra in figure 4 can be written as σ = −ν (n2 + k2 ), where n can be interpreted as radial wave number. As Reynolds number tends to infinity or axial wave number tends to zero, the continuous spectra reach the imaginary line and make the base flow neutrally unstable. The continuous modes are sheltered by the vortex core at large axial wave numbers, but they penetrate into the vortex core as the decrease of axial wave numbers and increase of radial wave numbers, as can be seen in figure 5.

Fig. 5 Continuous spectra penetrate into the vortex core. Left: n = 2π , k = 10, 1, 0.5 from left to right. Right: k = 0.5 and n = π /2, 3π /2, 2π from left to right.

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5 Conclusion The Batchelor vortex has been taken as the mathematical model of swirling flow. The modes of the Batchelor vortex can be divided into three groups according to the energy distribution in the radial direction: core modes, algebraic modes and continuous modes. Core modes concentrate in the vortex core, algebraic modes decays in the core as well as the free stream and continuous modes oscillate in the free stream. Each type of modes correspond to separated branches on the spectra map of the governing equations. Core modes can be unstable at km < 0. The unstable core modes include helical instability modes and viscous center instability modes. The algebraic modes are asymptotically stable but show strong transient growth in the axisymmetric condition. Continuous modes are the only modes in the free stream and they are neutrally unstable in invsicid condition or when the axial wave considered is infinitely long. Continuous modes are shielded by the vortex core. As the increase of radial wave number and decrease of axial wave number, continuous modes penetrate into the vortex core.

References 1. Barkley, D., Blackburn, H. M. and Sherwin, S. J. 2007 Direct optimal growth analysis for timesteppers. Int. J. Numer. Fluids. 231, 1-20. 2. Batchelor, G. K. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20, 645-658. 3. Blackburn, H. M., Barkley, D. and Sherwin, S. J. 2008 Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271-304. 4. Heaton, C. J. & Peake, N. 2006 Algebraic and expoential instability of inviscid swirling flow. J. Fluid Mech. 565, 279-318. 5. Heaton, C. J. 2007 Centre modes in inviscid swirling flows and their application to the stability of the Batchelor vortex. J. Fluid Mech. 576, 325-348. 6. Khorrami. M. R. 1991 On the viscous modes of instability of a trailing line vortex. J. Fluid Mech. 225, 197-212. 7. Lessen, M., Singh P. J. and Paillet, F. 1974 The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 63, 753-763. 8. Zaki, T. and Saha, S. 2009 On shear sheltering and the structure of vortical modes in singleand two-fluid boundary layers. J. Fluid Mech. 626, 111-147. 9. Gustavsson, L. 1979 Initial-value problem for boundary layer flows Phys. Fluids 22(9), 16021605. Acknowledgements We would like to thank Dr. Tamer Zaki of Imperial College London for discussing about the continuous spectrum. Xuerui Mao would like to acknowledge the Student Opportunity Fund at Imperial College for financial support while Spencer J Sherwin wishes also to acknowledge financial support from an EPSRC Advanced Research Fellowship.

Localized edge states for the transition to turbulence in shear flows Daniel Marinc, Tobias M Schneider and Bruno Eckhardt

Abstract In parallel shear flows like pipe flow or plane Couette flow, laminar and turbulent dynamics coexist. The boundary between the two types of dynamics shows up clearly in studies that monitor the life time of a perturbation, i.e. the time it takes to relaminarize. Initial conditions that neither become fully turbulent nor return to the laminar state live in the boundary between laminar and turbulent flow. They typically approach a relative attractor called the edge state. The edge state together with its stable manifold then defines the boundary between laminar and turbulent motion. Edge states determined in small domains are infinitely extended when extrapolated to larger domains and are not compatible with the observation that a local perturbation suffices to produce turbulence. Studies of pipe flow, however, show that in long domains also localized edge states can exist. We here present results from direct numerical simulations of plane Couette flow which show that in narrow but long domains the edge states are localized and similar to the ones found in pipe flow.

1 Introduction The linear instabilities of the laminar state in fluids heated from below or fluids between rotating cylinders support Landau’s vision of a transition to turbulence that Daniel Marinc Aerodynamisches Institut, RWTH [email protected]

Aachen,

D-52062

Aachen,

Germany

e-mail:

Tobias M Schneider School of Engineering and Applied Sciences, Harvard University, Cambrige, Ma 02138, USA e-mail: [email protected] Bruno Eckhardt Fachbereich Physik, Philipps-Universität Marburg, 35032 Marburg, Germany, and Laboratory for Aero- and Hydrodynamics, TU Delft, 2928 CA Delft, The Netherlands e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_40, © Springer Science+Business Media B.V. 2010

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is connected with sequences of bifurcations that add more and more spatial and temporal complexity to the dynamics until the fully turbulent state is reached [12, 11]. The transition to turbulence in parallel shear flows does not fit such a scenario: the laminar profile in pipe flow and plane Couette flow is stable for all Reynolds numbers, and in plane Poiseuille flow it becomes unstable at a Reynolds number well above the observed transition [18, 10, 5]. In such a situation the transition to turbulence requires the crossing of two thresholds, one in Reynolds number and one in perturbation strength. The threshold can be determined by mapping out the amplitude-Reynolds number plane with different initial conditions and monitoring their evolution. The boundary between the ones that relax to the laminar profile and the ones that become turbulent then defines the above thresholds. A good indicator function for determining the boundary is the lifetime of a perturbation, defined as the time it takes to return to the laminar profile [19, 8, 21]. Typically, this lifetime varies smoothly with parameters when the flow relaminarizes, and it shows rapid variations when the flow becomes turbulent. In typical life time plots one notes a rather drastic increase which suggested the name ’edge of chaos’ [25]. This behaviour is seen in studies on plane Couette flow [19], TaylorCouette flow [7] and pipe flow [3, 21] and seems to be a generic feature of the transition in linearly stable shear flows. Approaching the transition from the laminar side, there is a first point where the lifetime diverges: it belongs to a trajectory that neither relaxes to the laminar profile nor becomes turbulent. Observations show that this trajectory is attracted to an invariant object, called the edge state [25, 22]. The significance of the edge state lies in the fact that it together with its stable manifold defines the boundary between laminar and turbulent motions in the state space of the system [5, 27]. The dominant feature in the edge states velocity field are downstream vortices, but they have to be supplemented by three-dimensional flow structures, since translationally invariant perturbations to parallel shear flows cannot be sustained and decay. The full, 3-dimensional state can be stationary or be dynamically active, as a traveling wave, a heteroclinic loop or even a chaotic object [5, 27]. The edge states determined in sort segments of a pipe or in small domains in plane Couette flow can be periodically continued to edge states that extend over the full domain, but they are fully extended and would suggest that a perturbation also has to be spatially extended. This contrasts with the possibility to trigger turbulence with localized perturbations [6, 13, 26, 2] and the observation of localized and spatially structured turbulence in pipe flow and plane Couette flow [28, 1], which suggest that in spatially extended domains also a localized perturbation should suffice to trigger turbulence. Localized edge states have been calculated in a model of pipe flow [4], in fully resolved pipe flow [16] and in plane Couette flow [24]. We here present results of a direct numerical simulation in a narrow and long plane Couette cell, where we expect localized structures in the downstream direction. The calculations are based on the Diplomarbeit of Daniel Marinc, [15].

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2 Numerical methods and results We study edge states in plane Couette flow, with the coordinates x, y, z pointing in the downstream, normal and spanwise directions, respectively, and a Reynolds number based on half the velocity difference and half the gap withs d. All lengths and widths will be given in units of d. For the numerical representation of the NavierStokes equation we use the code Channelflow [9], in the form developed and tested by John F. Gibson. The resolution in the normal diection is 33 Chebychev modes. In the spanwise direction, we keep 16/π modes per length and in the downstream direction 4/π or 8/π modes per length. The present results are for a Reynolds number Re = 400. The tracking of the edge state was achieved using the protocol described, for instance, in [23] for the small domains. The calculations in wider domains show that it suffices to monitor the energy and that it is not necessary to track additional measures related to the size of the turbulent patch. The vortices in the edge states in plane Couette flow are about 2π wide. Increasing the domain widths one expects and finds that more vortices fit into it. Increasing the width further, three localized states appear: a stationary state with up-down reflection symmetry consisting of four vortices, and a pair of asymmetric traveling waves with three vortices. The estimated widths of these states is about 7π . Therefore, the box width of 2π used in the present study is too narrow to give a localization in the spanwise direction. Increasing the lengths of the domain to 16π , the nature of the edge state changes: while it was a stationary state before, it now develops some time-dependence. This is clearly detected in the edge state tracking protocol, which converges to a constant energy for either a stationary state or a travelling wave, only. In the small domains, Fig. 1 (left) this is the case, but in the larger domains Fig. 1 (right) there is a time-dependence. This shows that by increasing the length of the domain the system exploits the additional spatial degrees of freedom to begin developing temporal degrees of freedom, indicative of a step towards a chaotic edge state. 0.3

0.32

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Fig. 1 Comparison of edge state tracking in the short domain (left) and in the long domain (right). Shown are the trajectory segments that become turbulent. The edge state corresponds to the lower envelope of the time traces. The convergence of the edge state tracking algorithm to a constant energy in the left shows the convergence to a stationary state or a traveling wave, the residual timedependence for the long domain on the right shows that the state remains dynamically active and is attracted towards a relative chaotic attractor.

Applying a Newton method to these states one finds in the short domain convergence to the expected stationary patterns with only a single unstable direction. In

256


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the long domain, the Newton algorithm converges to different stationary states, all of which have more than one unstable direction: this shows that while they cannot act as edge states, they can be part of the chaotic relative attractor on the edge. Interestingly, there are states that are rather uniform in the downstream direction (Fig. 2 (left)) but there are others modulated on half the box length (Fig. 2 (right)).

0.0585

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Fig. 2 Energy density of stationary states for Re = 400 in a domain of length 16π and width 2π , obtained by applying a Newton method to states from the edge. Both states have several unstable directions and cannot be edge states. The weak modulation on top of the mean energy density of 0.6 indicates a symmetry breaking in the downstream direction and is a first weak indicator for the emerging localization.

For longer domains the modulations increase and a localized pattern emerges: for a domain length of 32π the edge state is clearly localized with a length of about 10π , and the same state can be realized in a domain twice as long. The independence of the localization length on the downstream extension of the domain confirms that the localization is a dynamical process, determined by the intrinsic dynamics of the Navier-Stokes equation and not influenced by boundary conditions. The snapshot of the state in Fig. 3 shows the localization, the dominance of downstream streaks in the interior, active region of the state, and the small scale structures that seem to drive the persistent dynamics. Similar structures are found in the chaotic edge state in pipe flow, see [20].

Fig. 3 Snapshot of the edge state for Re = 400, Lx = 64π and Lz = 2π . Shown is the normal velocity in the mid plane. The state is dominated by two pairs of vortices in the downstream direction. The innermost parts of the edge state show a non-periodic transverse modulation that reflects the 3-d flow elements as well as the perpetual dynamics.

An integrated measure of the edge state is the energy content in a cross section perpendicular to the flow, hu2 iz,y . It clearly shows the localization of the structure within a length of about 40 (Fig. 4 (left)). A semilogarithmic representation of this

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state shows that the localization is exponential (Fig. 4 (right)). The slight differences between the left and right hand side cannot be dynamically significant because of the symmetries of the flow and are an instantaneous property of the state. 0.1

u2 iy;z (x) h~

u2 iy;z (x) h~

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Fig. 4 Energy densities of the localized edge states Re = 400 in a domain of length 64π and width 2π . The semilogarithmic plot on the right shows that the localization is exponential in the downstream direction. The flucutations and asymmetries are not significant as the data is for an instantaneous snap shot of the state. Upon averaging in time symmetric and smooth variations can be expected.

3 Conclusions The edge states that are localized in the downstream direction in narrow but long Couette cells demonstrate a quality of edge states that is specific to spatially extended systems: they capture – within the edge state concept – the experimental observation that a localized perturbation suffices to trigger turbulence. The states discussed here are very similar to axially localized edge states determined for pipe flow [4, 16]. Both states consist of downstream vortices, their internal dynamics is chaotic, and their localization is exponential in the downstream direction. Remarkably, their lengths are also similar when measured in units of the length over which the gradient is present: then 20 radii in pipe flow correspond to 20 times the gap or 40 times half the gap in Couette flow. The localization length discussed here introduces a length scale that is larger than the typically rather short wave lengths of the coherent states, but shorter than the extend of turbulent patches. It is indicative of how large a region has to be perturbed before the flow can become turbulent. In order to determine the exact sizes it would be necessary to follow the stable manifolds as well. Localized edge states are a step towards exploring transitions in sub-critical spatially extended systems where aspects of spatio-temporal chaos, non-equilibrium nucleation phenomena and mechanisms of localization come together [17, 14]. These dynamical systems connections hold the promise to lead to a much improved understanding of the formation and dynamics of localized turbulent patches in internal and external flows. We thank the Deutsche Forschungsgemeinschaft for support.

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References 1. S. Bottin, F. Daviaud, P. Manneville, and O. Dauchot. Discontinuous transition to spatiotemporal intermittency in plane Couette flow. Europhys. Lett., 43(2):171 – 176, 1998. 2. Sabine Bottin, Olivier Dauchot, and Francois Daviaud. Intermittency in a locally forced plane Couette flow. Phys. Rev. Lett., 79:4377–4380, 1997. 3. A.G. Darbyshire and T. Mullin. Transition to turbulence in constant-mass-flux pipe flow. J. Fluid Mech., 289:83–114, 1995. 4. Y. Duguet, A. P. Willis, and R. R. Kerswell. Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech., 613:255–274, 2008. 5. Bruno Eckhardt, Tobias M Schneider, Bjorn Hof, and Jerry Westerweel. Turbulence transition in pipe flow. Annual Review of Fluid Mechanics, 39:447–468, 2007. 6. H W Emmons. The laminar-turbulent transition in a boundary layer. Journal Aeronautical Science, 18:490–498, 1951. 7. Holger Faisst and Bruno Eckhardt. Transition from the Couette-Taylor system to the plane Couette system. Phys. Rev. E, 61:7227–7230, 2000. 8. Holger Faisst and Bruno Eckhardt. Sensitive dependence on initial conditions in transition to turbulence in pipe flow. J. Fluid Mech., 504:343–352, 2004. 9. John F Gibson. www.channelflow.org. Technical report, 2004. 10. S Grossmann. The onset of shear flow turbulence. Rev. Mod. Phys., 72(2):603–618, 2000. 11. E. L. Koschmieder. Bénard Cells and Taylor Vortices. Cambridge University Press, 1993. 12. L.D. Landau. On the problem of turbulence. C.R. Acad. Sci. USSR, 44:311–314, 1944. 13. Anders Lundbladh and Arne v. Johansson. Direct simulation of turbulent spots in plane Couette flow. J. Fluid Mech., 229:499–516 (18 pages), 1991. 14. Paul Manneville. Spatiotemporal perspective on the decay of turbulence in wall-bounded flows. Phys. Rev. E, 79:025301(R), 2009. 15. Daniel Marinc. Localised edge-states in plane Couette flow. Diploma thesis, PhilippsUniversität Marburg, 2008. 16. Fernando Mellibovsky, Alvaro Meseguer, Tobias M Schneider, and Bruno Eckhardt. Transition in localized pipe flow turbulence. Phys. Rev. Lett., 103:054502, 2009. 17. Y. Pomeau. Front motion, metastability and subcritical bifurcations in hydrodynamics. Physica D, 23:3–11 (9 pages), 1986. 18. P. J. Schmid and D. S. Henningson. Stability and Transition of Shear Flows. Springer, New York, 1999. 19. Armin Schmiegel and Bruno Eckhardt. Fractal stability border in plane Couette flow. Phys. Rev. Lett., 79:5250–5253, 1997. 20. Tobias M Schneider and Bruno Eckhardt. Edge of chaos in pipe flow. Chaos, 16:041103 (1 page), 2006. 21. Tobias M Schneider and Bruno Eckhardt. Lifetime statistics in transitional pipe flow. Phys. Rev. E, 78:046310 (10 pages), 2008. 22. Tobias M Schneider, Bruno Eckhardt, and James A Yorke. Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett., 99:034502, 2007. 23. Tobias M Schneider, John F Gibson, Maher Lagha, Filippo deLillo, and Bruno Eckhardt. Laminar-turbulent boundary in plane Couette flow. Phys. Rev. E, 78:037301 (4 pages), 2008. 24. Tobias M Schneider, Daniel Marinc, and Bruno Eckhardt. Localized edge states that nucleate turbulence in plane Couette flow. submitted, 2009. 25. Joseph D Skufca, James A Yorke, and Bruno Eckhardt. Edge of chaos in a parallel shear flow. Phys. Rev. Lett., 96:174101 (4 pages), 2006. 26. Nils Tillmark and P Henrik Alfredsson. Experiments on transition in plane Couette flow. J. Fluid Mech., 235:89–102, 1992. 27. Jürgen Vollmer, Tobias M Schneider, and Bruno Eckhardt. Basin boundary, edge of chaos, and edge state in a two-dimensional model. New Journal of Physics, 11:013040 (23pp), 2009. 28. I J Wygnanski and F H Champagne. On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug. J. Fluid Mech., 59:281–335, 1973.

Active steady control of vortex shedding: an adjoint-based sensitivity approach Olivier Marquet and Denis Sipp

Abstract The effect of steady wall forcing on the stability of the circular cylinder wake is investigated for Reynolds number close to Rec = 47. An adjoint-based sensitivity approach is developed in the framework of global stability theory to help in determining positions and type of actuation (blowing or suction) that efficiently alter the flow stability.

1 Introduction Despite many years of research control of flow transition is still a challenging problem in fluid mechanics. In recent years closed-loop control strategies have focused the attention of scientists in particular with the introduction of a new framework, the input-output analysis [1]. Yet the implementation of such closed-loop control to realistic flow cases remains quite challenging. On the other hand open-loop control strategies remain attractive owing to the simplicity of their implementation. They aim to design an actuator without a priori taking into account its effect on the flow. The challenging problem is then to find the optimal design of the actuator that enables us to reach a given objective, for instance the suppression of vortex shedding behind a circular cylinder. In general this is achieved empirically by measuring a posteriori the effect of a chosen actuator onto the flow. Such an empirical approach is extremely time-consuming, especially when the optimal design is not obvious. A systematic approach for the passive control of global instabilities have first been proposed by [3] and recently rediscovered by [4]. It has been successfully applied to re-stabilize the cylinder wake. The objective of the present study is to extend this Olivier Marquet ONERA-DAFE, 92190 Meudon, France, e-mail: [email protected] Denis Sipp ONERA-DAFE, 92190 Meudon, France, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_41, © Springer Science+Business Media B.V. 2010

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approach to the active control of global instabilities by means of steady wall forcing. It will be demonstrated on the cylinder wake at low Reynolds number.

2 Global stability of the cylinder flow The two-dimensional stability of the circular cylinder flow is investigated using a global stability analysis. The flow variables, non-dimensionalized with the cylinder diameter D and the upstream uniform velocity U∞ , are decomposed into a twodimensional base flow (U, P ) and an infinitesimal unsteady perturbation (u′ , p′ ). The base flow is sought as a solution of the non-linear steady Navier-Stokes equations (∇U) · U + ∇P − Re−1 ∇2 U = 0 , ∇ · U = 0 (1)

where Re = U∞ D/ν is the Reynolds number. No-slip conditions U = 0 are applied on the cylinder wall Γw . The perturbation is decomposed onto a set of twodimensional temporal modes (u, ˆ p) ˆ exp[σ t] which are solutions of the generalized eigenvalue problem ˆ · U + ∇ pˆ − Re−1 ∇2 uˆ = 0 , ∇ · uˆ = 0 σ uˆ + (∇U) · uˆ + (∇u)

(2)

A complex global mode (u, ˆ p) ˆ is associated to a complex eigenvalue σ = λ + iω . The real (λ ) and imaginary (ω ) parts of a complex eigenvalue are respectively the growth rate and pulsation of the corresponding global mode. The global modes of largest growth rate are obtained by solving the generalized eigenvalue problem via a shift and invert strategy. Figure 1(a) depicts the spectrum computed for Re = 60. The leading eigenvalue, i.e. the eigenvalue of largest growth rate, is unstable (λ > 0). The long-time linear flow dynamics is governed by this leading global mode (not shown here). Figure 1(b) displays the evolution of the growth rate and pulsation of the leading mode as a function of the Reynolds number. It shows that the flow becomes globally unstable for Re > Rec ∼ 47. (a)

(b) 0.2

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Fig. 1 (a) Global spectrum at Re = 60. (b) Growth rate λ and pulsation ω of the leading global mode as a function of Re.

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The objective of the present study is to alter the flow dynamics via a steady forcing Uw on the cylinder surface Γw . For supercritical flows (Re > Rec ) the control objective can be either to stabilize the flow (by shifting the leading eigenvalue in the stable-half plane) or to modify the frequency of the instability.

3 Adjoint-based sensitivity approach The present open-loop control strategy is based on a sensitivity approach. Let us consider any eigenvalue σ as a function of the steady wall forcing Uw and perform the Taylor expansion

σ (Uw + δ Uw ) = σ (Uw ) + [ ∇Uw σ , δ Uw ]w + O(k δ Uw k22 )

(3)

R

where [a , b]w = Γw aH ·b is the inner product of two complex vectors defined on the wall Γw . H denotes here the conjugate and transpose operation. In the above expression, the amplitude of a is the L2 norm k a k2 = [a , a]1/2 w . An alternative definition of the amplitude is given by the L∞ norm k a k∞ = max(x,y)∈Γw a(x, y). For small amplitude of the steady wall forcing, the eigenvalue variation induced by any steady wall forcing can be straightforwardly computed once the complex field ∇Uw σ is determined. This field is the gradient of the eigenvalue with respect to the steady wall forcing and is called the sensitivity to steady wall forcing. A Lagrange-multiplier technique has been used to determine the expression of this sensitivity. More details concerning the derivation can be found in [4] where a similar technique is used to determine the sensitivity to steady bulk forcing. The determination of the sensitivity to steady wall forcing requires to solve two adjoint problems. The first problem is the adjoint generalized eigenvalue problem

σ + uˆ + + (∇U)T · uˆ + − (∇uˆ + ) · U + ∇ pˆ+ − Re−1 ∇2 uˆ + = 0 , ∇ · uˆ + = 0

(4)

where {uˆ + , pˆ+ } is an adjoint global mode associated to an eigenvalue σ + = σ ∗ . The knowledge of a direct global mode and its associated adjoint global mode enables to compute the sensitivity of the corresponding eigenvalue to base flow modifications ˆ uˆ + ) = −(∇u) ˆ H · uˆ + + (∇uˆ + ) · uˆ ∗ S(u,

(5)

The second adjoint problem is defined by the non-autonomous linear equations ˆ uˆ + ) , ∇ · U+ = 0 −(∇U+ ) · U + (∇U)T · U+ − ∇P+ − Re−1 ∇2 U+ = S(u,

(6)

where (U+ , P+ ) is a complex field called the adjoint base flow. The sensitivity to steady wall forcing is defined on the wall Γw by ∇Uw σ = P+ n + Re−1 (∇U+ ) · n

(7)

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where n is the unit vector normal to the wall Γw and oriented from the body to the fluid. The variation of the eigenvalue resulting from a small amplitude steady wall forcing is then given by

δ σ = [∇Uw σ , δ Uw ]w =

Z

Γw

P+ n + Re−1 (∇U+ ) · n

H

· δ Uw d Γ

(8)

This eigenvalue variation is the sum of two distinct contributions. The first contribution given by the product of the adjoint base flow pressure and the wall base flow velocity is an inviscid effect. Note that only steady wall forcing oriented normal to the wall lead to a variation of the eigenvalue through this inviscid contribution. The second contribution weighted by the inverse of the Reynolds number takes into account the viscous effects. A velocity tangential to the wall can induce a variation of the eigenvalue only through this term. Finally the variations of the growth rate and pulsation are given by

δλ = δω =

Z

Z

Γw

∇Uw λ T · δ Uw d Γ , ∇Uw λ = Pr+ n + Re−1 (∇U+ r )·n

(9)

∇Uw ω T · δ Uw d Γ , ∇Uw ω = −Pi+ n − Re−1 (∇U+ i )·n

(10)

Γw

where the subscripts r and i denote respectively the real and imaginary parts. The growth rate ∇Uw λ and pulsation ∇Uw ω sensitivities are thus obtained by considering the real and imaginary parts of the complex adjoint base flow fields. Note that a sign minus appears in the expression of the pulsation sensitivity because of the conjugate transpose operation in (8). How to compute the sensitivity to steady wall forcing? ˆ p}) 1. select a global mode (σ ; {u, ˆ in the spectrum, 2. solve the adjoint eigenvalue problem (4) and select the corresponding adjoint global mode (σ ∗ , {uˆ + , pˆ+ }), 3. compute the sensitivity to base flow modifications (5) 4. solve the adjoint base flow equations (6) 5. compute the sensitivity to steady wall forcing (7)

4 Results The sensitivity to steady wall forcing has been computed for all the eigenvalues shown in figure 1(a). The L∞ -norm of the growth rate and pulsation sensitivity is respectively plotted in figure 2(a) and 2(b) as a function of the eigenvalues sorted by increasing order of |λ |, i.e. the distance to the real axis in the spectrum. The

Active steady control of vortex shedding: an adjoint-based sensitivity approach (a)

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Fig. 2 L∞ norm of the (a) growth rate and (b) frequency sensitivity of the eigenvalues shown in the spectrum. The eigenvalues are sorted by increasing value of |λ |. Re = 60.

open circles correspond to the stable eigenvalues and the black circle stands for the unstable eigenvalue. It is interesting to note that there is no correspondence between the sensitivity of an eigenvalue and its growth rate. So the sensitivity of an eigenvalue can not be inferred from its position in the spectrum. Nevertheless we note that the unstable eigenvalue is one of the most sensitive eigenvalues. For the computed spectrum only one eigenvalue is more sensitive to a wall steady forcing and this eigenvalue is strongly damped. Therefore we may expect that an appropriate steady wall forcing will shift the unstable eigenvalue in the stable halfplane without shifting stable eigenvalues in the unstable half-plane. Comparison of figures 2(a) and 2(b) shows that the magnitude of the pulsation sensitivity is larger than the magnitude of the growth rate sensitivity. This suggests that modifying the instability frequency requires less energy than suppressing the flow instability. In the case of flow induced vibration problems a control method could be to modify the frequency of the flow instability rather than to suppress it. The growth rate and frequency sensitivities of the unstable eigenvalue are plotted in figure 3. The vectors depict these sensitivity fields whereas the lines are the streamlines of the base flow. The numbers correspond to the L∞ -norm of each sensitivity field. We note that the two sensitivity fields are symmetric with respect to the x axis. Therefore any anti-symmetric steady wall forcing of small amplitude has no effect on the flow stability. Let us consider first the growth rate sensitivity (left figure). Its orientation is normal to the cylinder surface for almost all locations on this surface. This shows that a normal actuation (blowing or suction) will alter the flow stability using less energy as compared to a tangential actuation. The only location where a tangential actuation should be preferred is the separation point of the base flow. The magnitude of the growth rate sensitivity is large at the base of the cylinder and at lateral positions, upstream the separation points. For a base actuation, blowing has a stabilizing effect (δ λ < 0) because the actuation and the sensitivity field are in opposite directions. On the contrary suction has a destabilizing effect (δ λ > 0). This is in qualitative agreement with previous results [2] obtained by direct numerical simulations of a circular cylinder flow controlled with a fixed wall velocity profile centered around the cylinder base. For a lateral actuation, blowing has a destabilizing effect (δ λ > 0) whereas suction has a stabilizing effect (δ λ > 0). This shows how the

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Fig. 3 Growth rate (left) and frequency (right) sensitivity of the unstable eigenvalue. The number indicates the L∞ -norm of each sensitivity field. Re = 60.

sensitivity can be used as a guide to locate an actuator and to determine the type of actuation (blowing or suction) which is appropriate to suppress the flow instability. Now let us consider the pulsation sensitivity (right figure). Its magnitude is almost nil at the base of the cylinder. Consequently a steady wall forcing of small amplitude at the base of the cylinder does not modify the frequency instability. A maximum of the frequency sensitivity is obtained at the lateral positions and the orientation is almost normal to the wall. It shows that a strong modification of the frequency instability can be obtained only via inviscid effects. The present results are only valid for small amplitude of steady wall forcing. Further investigations will be dedicated to extend this approach to finite-amplitude steady wall forcing. An optimal control algorithm based on the knowledge of the sensitivity function [5] will be designed to address this question.

References 1. Bagheri S, Henningson D S, Hoepffner J and Schmid P J (2008) Input-Output analysis and control design applied to a linear model of spatially developping flows. Appl. Mech. Rev. (in press). 2. Delaunay Y and Kaiktsis L (2001) Control of circular cylinder wakes using base mass transpiration. Phys. Fluids 13:3285 3. Hill D C (1992) A theoretical approach for analysing the restabilization of wakes. AIAA 92-0067 4. Marquet O, Sipp D and Jacquin L (2008) Sensitivity analysis and passive control of cylinder flow. Journal of Fluid Mechanics 615:221–252 5. Pralits J O, Hanifi A and Henningson D S (2002) Adjoint-based optimisation of steady suction for disturbance control in incompressible flows. Journal of Fluid Mechanics 467:129–161

Feedback control of transient energy growth in subcritical plane Poiseuille flow Fulvio Martinelli, Maurizio Quadrio, John McKernan, and James F. Whidborne

Abstract Subcritical flows may experience large transient perturbation energy amplifications, that could trigger nonlinear mechanisms and eventually lead to transition to turbulence. In plane Poiseuille flow, controlled via wall blowing/suction with zero net mass flux, optimal and robust control theory has been recently applied to a state-space representation of the Orr-Sommerfeld-Squire equations, leading to reduced transient growth as well as increased transition thresholds. However, to date no feedback control law has been found that is capable of ensuring the closed-loop Poiseuille flow to be monotonically stable. The present paper addresses first the possibility of complete feedback suppression of the transient growth mechanism in subcritical plane Poiseuille flow when wall actuation is available, and demonstrates that closed-loop monotonic stability cannot be achieved in such a case. Secondly, a Linear Matrix Inequality (LMI) technique is employed to design controllers that directly target the energy growth mechanism. The performance of such control laws is quantified by using Direct Numerical Simulations of transitional plane Poiseuille flow, and the increase in transition thresholds due to the control action is assessed.

Fulvio Martinelli ´ LadHyX, Ecole Polytechnique, Palaiseau, France, e-mail: [email protected] Maurizio Quadrio Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Milano, Italy John McKernan ECLAT, King’s College, London, UK James F. Whidborne Department of Aerospace Sciences, Cranfield University, Cranfield, UK P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_42, © Springer Science+Business Media B.V. 2010

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1 Introduction Transient energy growth has recently been recognized as a possible mechanism explaining subcritical transition in wall-bounded flows. In fact, subcritical flows may experience large transient amplifications of the energy of perturbations, that could trigger nonlinear mechanisms and lead to transition to turbulence [1]. In plane Poiseuille flow, optimal and robust control theory was applied to a statespace model derived from the Orr-Sommerfeld-Squire equations, by Bewley & Liu [2] for a single wavenumber pair and by Högberg et al. [3] for a large array of wavenumber pairs. This led to a reduction of the maximum transient growth as well as to an increase in transition thresholds. However, to date no feedback control law has been ever found that is capable of ensuring closed-loop monotonic stability. In the present paper, it is shown first that it is impossible to design a linear statefeedback controller ensuring the plane Poiseuille flow, controlled via distributed zero-net-mass-flux transpiration with any velocity component at the walls, to be monotonically stable. Furthermore, a design technique – based on a Linear Matrix Inequality (LMI) approach – is described; this technique enables the synthesis of feedback laws that directly target the transient growth mechanism. Feedback controllers designed with the technique are tested in nonlinear simulations of transitional plane Poiseuille flow, evaluating the control performance with different initial conditions in terms of increase in transition threshold.

2 Discretization We consider the linearized dynamics of three-dimensional perturbations to the laminar Poiseuille flow in a plane channel. The governing equations, written in v − η form, are discretized spectrally by Fourier expansion in streamwise and spanwise directions, and by Chebyshev expansion in the wall-normal direction. For each wavenumber pair (α , β ), a lifting procedure [3] is employed to account for nonhomogeneous boundary conditions at the two channel walls; in the most general case, the three components of the perturbation velocity vector can be assigned at each wall, so that there are six degrees of freedom to actuate on the system. The linear perturbation dynamics can be written in standard state-space form as: x˙ = Ax + Bu

(1)

where the input vector u = (u˙u , u˙l , v˙u , v˙l , w˙ u , w˙ l )T accounts for “vectorized” transpiration at both the upper and the lower wall, and the state vector x has been rescaled so that the perturbation energy is written as the Euclidean norm ||x||2 = xH x. The time-invariant matrices A and B characterize the system dynamics; they are functions of the wavenumber pair (α , β ) and the Reynolds number Re.

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3 System properties It has been shown by Whidborne & McKernan [4] that a static state-feedback control law u = Kx exists such that the closed-loop system is monotonically stable (i.e. energy decays monotonically from all initial conditions x0 ), if and only if B⊥ A + AH B⊥H < 0 or BBH > 0, (2)

where B⊥ is the left null space of B. It is immediate to verify that the second criterion in eq. 2 is never satisfied for the controlled Poiseuille flow described by eq. 1, as the system is underactuated. The first criterion is verified numerically, by computing the maximum (real) eigenvalue λmax of the hermitian matrix B⊥ A + AH B⊥H as a function of (α , β , Re). Upon comparison of the region where λmax (α , β , Re) < 0 with the region where the uncontrolled flow admits transient energy growth, portions of the (α , β , Re) parameter space can be identified where a feedback controller would suppress the transient growth phenomenon. Figure 1 (a) shows the present result along with the wellknown result on the transient growth dependence on (α , Re) in plane Poiseuille flow [1] (i.e. the open-loop case), when β = 0 and wall actuation is performed with the v-component at the two walls. The white area corresponds to the domain where the open-loop system is monotonically stable, while the shaded area is the region where the open-loop system admits transient energy growth. The level curve of λmax = 0 lies on the very boundary between shaded and white areas, implying that the form B⊥ A + AH B⊥H is indefinite when the open-loop system is not monotonically stable. This means that a linear state-feedback controller cannot be designed to ensure the closed-loop Poiseuille flow to be monotonically stable, when the corresponding open-loop flow is not. A similar result is shown in fig. 1 (b) for Re = 120, where the three-dimensional case – with actuation on the three velocity components at both walls – is considered. These results lead to the conclusion that, even if complete knowledge of the instantaneous flow state were available, transient growth suppression by feedback is not achievable through wall actuation.

4 Control design An LMI-based technique [5, 6] can be employed to design state-feedback control laws minimizing an upper bound on the maximum transient energy growth. Additionally, a control constraint in the form maxt≥0 ||u(t)||2 < µ 2 is considered to tune the maximum control effort during the operation of the controller. The resulting LMI problem is stated as follows:

268


(a)

(b)

Fig. 1 Numerical verification of the first criterion in eq. 2. Lines are contours at constant λmax (B⊥ A + AH B⊥H ). λmax > 0: solid line; λmax = 0: thick solid line; λmax < 0: dashed line. (a): Wall actuation with the v-component at both walls, β = 0. (b): Wall actuation with all components at both walls, Re = 120.

min γ : AQ + QAH + BY +Y H BH < 0, Q = QH > 0 I < Q < γI Q YH >0 Y µ 2I

(3)

and the optimal compensator gains are obtained by K = Y Q−1 . This is a linear optimization problem over a convex set, can be solved numerically by standard algorithms [8].

5 Results The linear evolution of the perturbation energy is compared, for the controlled and uncontrolled flow, in fig. 2; parameters for these simulations are Re = 2000 and µ = 100. In particular, optimal initial conditions for the open-loop Poiseuille flow are assigned to both the controlled and uncontrolled flow, and actuation is performed with different velocity components. Fig. 2 shows that the controller is capable of reducing the energy growth amplitude, with different degrees of success depending on the actuation component used. Specifically, when the oblique wave (α = 1, β = 1) case is considered as initial condition, the most effective wall actuation component turns out to be v, whereas u and w behave similarly (as it should be, since their ef-

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fect on the oblique wave is symmetric). Furthermore, when the streamwise vortex (α = 0, β = 2) initial condition is considered, w actuation performs slightly better than v actuation, and both outperform u actuation, that in this situation acts in a weakly controllable direction.

800

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After these numerical experiments in the linear setting, performance of LMIbased controllers in terms of transition delay has been verified using full Direct Numerical Simulations of transitional channel flow at Re = 2000, using the code described in [7]. Controllers are tested against initial conditions in the form of a pair of oblique waves (α0 = 1, β0 = ±1) and streamwise vortices (α0 = 0, β0 = 2). Random noise, in the form of Stokes modes and having 1% of the total perturbation energy, is added on the wavenumber array (0, ±1, ±2)α0 and (0, ±1, ±2)β0 . Design of the controllers is performed on the same array, and the control effort tuning parameter is µ = 100. The performance of control laws is quantified by in(thres) (thres) troducing an improvement factor E0,control /E0, f ree , defined as the ratio between the transition threshold energy computed in the controlled case over that corresponding to the uncontrolled flow. A summary of the results is reported in table 1, where it is shown that, for initial conditions in the form of both streamwise vortices and oblique waves, wall-actuation with the v-component is more effective than actuation with other components. Furthermore, the improvement factor measured for the oblique wave is an order of magnitude larger than that obtained with streamwise vortices. This result is coherent with previous findings using LQR control laws [3], and it is associated to the fact that targeting oblique waves mitigates the formation of streamwise vortices, therefore reducing the entity of subsequent streak instabilities. Finally, it is worth noting that acting with u on streamwise vortices or acting with w on oblique waves does not increase the threshold energy.

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Table 1 Improvement factor in transition thresholds for different initial conditions and actuation.

Oblique waves Streamwise vortices

u

v

w

≈ 6.0 1

≈ 20.7 ≈ 2.0

1 ≈ 1.6

6 Conclusions In this paper, an algebraic criterion for the prediction of feedback suppression of the transient growth mechanism – once a state-space representation of the system dynamics is available – has been presented. This criterion has been exploited to demonstrate that, even if complete knowledge of the instantaneous flow state were available, a feedback controller actuating with all velocity components at the two channel walls would not be able to ensure closed-loop monotonic stability in plane Poiseuille flow. Furthermore, a design technique for the synthesis of feedback controllers directly targeting the transient growth mechanism, by minimization of an upper bound to the maximum growth, has been presented. Controllers designed with the technique have been tested in a nonlinear case, and the resulting increase of transition threshold has been quantified. When using wall-normal velocity forcing, results are qualitatively similar to those LQR-based reported in literature [3]; further, results indicate that wall actuation with u and w is less effective than forcing with v at the walls. Acknowledgements Prof. P. Luchini is gratefully acknowledged for providing the unlimited availability of his computing facility at the University of Salerno.

References 1. Reddy, S.C., Henningson, D.S.: Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238 (1993) 2. Bewley, T.R., Liu, S.: Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365, 305–349 (1998) 3. Högberg, M., Bewley, T.R., Henningson, D.S.: Linear feedback control and estimation of transition in plane channel flow. J. Fluid Mech. 481, 149–175 (2003) 4. Whidborne, J.F., McKernan, J.: On the Minimization of Maximum Energy Growth. IEEE Trans. Autom. Cont. 52(9) 1762–1767 (2007) 5. McKernan, J.: Control of plane Poiseuille flow: a theoretical and computational investigation. PhD thesis, Dept. Aerospace Sciences, School of Engineering, Cranfield University (2006) 6. Whidborne, J.F., McKernan, J., Papadakis, G.: Minimising transient energy growth in plane Poiseuille flow. Proc. IMechE J. Syst. Contr. Eng. 222, 323–331 (2008) 7. Luchini, P., Quadrio, M.: A low-cost parallel implementation of direct numerical simulation of wall turbulence. J. Comp. Phys. 211(2) 551–571 (2006) 8. Nesterov, Y., Nemirovskii, A.: Interior point polynomial algorithms in convex programming. Stud. Appl. Math. SIAM 13 (1994)

Linear and non-linear disturbance evolution in a compressible boundary-layer with localized roughness Olaf Marxen, Gianluca Iaccarino, and Eric S. G. Shaqfeh

Abstract A numerical investigation of disturbance amplification in a laminar compressible flat-plate boundary layer with a localized two-dimensional roughness is carried out. Both linear and weakly non-linear disturbance evolution are considered. The non-linear case exhibits a secondary subharmonic resonance. In addition to deterministic simulations, a stochastic approach is applied. The random parameter is chosen to be the height of the roughness in the linear case, while in the non-linear case the amplitude of the primary disturbance is considered a random parameter.

1 Introduction Laminar-turbulent transition in compressible, high-speed boundary layers is currently not well understood. This is particularly true for transition on surfaces with a localized two-dimensional unevenness, such as a roughness element [1, 7]. Cases with geometrical changes are relevant for a number of applications, including the heat-shield of vehicles (re-)entering a planetary atmosphere and the inlet to scram jet combustors for hypersonic cruise vehicles. While transition is undesirable on the heat shield due to an associated increase in temperature, it is desirable for scram jets to ensure proper mixing within the combustor. In all theses applications an accurate prediction of the transition location is advantageous for the vehicle design. Presently, many transition prediction methods rely on a deterministic description, in which a fixed transition location is computed. Examples are correlation based methods and the eN -method [5]. Typical N-factors at transition lie in the range of 5 − 10 (see examples in Ref. [6]). Recently, Ref. [2] proposed to account for growth modifiers, such as twodimensional steps, by adding a ∆ N to the N-factor. We will follow this idea and conCenter for Turbulence Research, Building 500, Stanford University, Stanford, CA 94305-3035, United States, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_43, © Springer Science+Business Media B.V. 2010

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sider the linear evolution of an explicitly forced perturbation in a flat-plate boundary layer, with a 2-D roughness acting as a growth modifier. In addition to a deterministic description, we will take a probabilistic approach to account for uncertainties in the height of the roughness. Numerical simulations with explicitly forced perturbations, the so-called controlled transition, have been criticized as not being representative of situations were such forcing is absent, the so-called natural transition. While the former approach has often been adopted in fundamental investigation of transition mechanisms, the latter situation is regarded as more relevant in engineering applications. A commonly accepted way to treat the natural transition has not yet emerged. Let us assume that the essential difference between the two situations lies in the fact that the disturbance spectrum remains unspecified for the natural transition. Under this assumption, stochastic simulations, in which a number of deterministic simulations with explicitly forced perturbations are carried out, offer an attractive method to handle the uncertainty with respect to the disturbance spectrum. As a first step, we will consider the secondary instability [4], provided that we only have a statistical description of the amplitude of the primary perturbation.

2 Mathematical model and numerical method Numerical simulations are based on an algorithm described in Ref. [8]. Solutions to the compressible Navier-Stokes equations are obtained, applying sixth-order compact finite-differences together with third order explicit Runge-Kutta time stepping. The numerical discretization is constructed on a structured, curvilinear grid using staggered variables. No explicit shock-capturing scheme is present as occurring shocks are sufficiently weak so that they can be treated by applying a high-order compact numerical filter. The fluid is assumed to be a calorically perfect gas, and the viscosity is computed from Sutherland’s law. The height of the roughness is on the order of, but typically smaller than, the boundary-layer thickness of the smooth flow. Disturbances of fixed frequency and fixed spanwise wave number, are triggered via blowing and suction at the wall [7] close to the inflow boundary (in 2-D: (ρ v)w = A2−D × f (x) × g(t) with f , g ∈ [0, 1]). v Two different cases are studied. First, a linear case is considered in which the evolution of the single forced 2-D disturbance is independent of its amplitude. Second, a weakly non-linear case of secondary instability is investigated. In addition to a finite-amplitude primary 2-D wave, a pair of small-amplitude oblique waves, with half the frequency of the primary wave and a non-zero spanwise wave number, is forced (with a fixed phase difference between the two). Due to subharmonic resonance, the oblique waves can experience stronger amplification than in the absence of the 2-D wave, if the amplitude of this 2-D wave is sufficiently large.

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3 Uncertainty quantification Both cases shall depend parametrically on a (case-dependent) random variable s, respectively. This parameter s is assumed to lie within a certain interval: s ∈ [smin , smax ]. It is distributed either uniformly or according to a beta distribution. Based on a random number q ∈ [0, 1], s can be obtained from a linear transformation: (1)

s(q) = q × (smax − smin ) + smin .

For the beta distribution, the probability density function (pdf) for q is defined as follows, here with a choice (α , β ) = (4, 4): q ∼ beta(α , β ), with pdf(q; α , β ) = R 1 0

qα −1 (1 − q)β −1

uα −1 (1 − u)β −1 du

.

(2)

We apply non-intrusive stochastic collocation. In this technique, a value for s, i.e. sm , is chosen and a simulation is performed. Then, the quantity of interest p is computed based on the simulation result (a definition for p will be given in sec. 6). We perform a sequence of M independent simulations, giving M deterministic values pm . A nested rule is applied to chose the collocation points for sm =s(qm ) on the abscissae. We employ the Clenshaw-Curtis rule [9]: qm = (1 − cos (π (m − 1)/(M − 1)))/2 with m = 1 . . . M .

(3)

In a final step, the obtained values pm are used to compute a pdf for the quantity p. This is achieved by means of Monte Carlo sampling. For this sampling standard random generators are used, delivering a random number q ∈ [0, 1]. The corresponding s is computed from eq. (1). The response p(s) is approximately built as an interpolant of the collocation points using a Lagrange polynomial: M M k m k pm Lm (s) with Lm (s) = Πk=1,k6 p(s) ≈ pCM = Σm=1 =m (s − s )/(s − s ) .

(4)

All the results p(s), here we have computed 100, 000 based on as many samples s, are sorted into 10 bins in the range p1 to pM to form a pdf histogram.

4 Mean flow Only a brief overview of the mean flow field (Re∞ = 105 , Pr∞ = 0.71, Ma∞ = 4.8, γ∞ = 1.4, Sutherland’s law with T˜S /T˜∞ = 1.993, adiabatic wall) shall be given, more details can be found in Ref. [7]. A representation of the mean flow field is given in figure 1 by means of a numerical Schlieren image. Boundary-layer separation occurs both upstream and downstream of the roughness. For hR = 0.1, the ratio between the boundary-layer thickness δ99 , based on u/ ˜ u˜∞ =0.99 for the flat-plate at the center location of the roughness (x = 15), and the roughness height is hR /δ99 ≈ 0.55.

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Fig. 1 Contours of the absolute wall-normal density gradient |∂ ρ /∂ y| for roughness hR = 0.1.

5 Deterministic disturbance evolution The typical evolution of disturbances in the flow field with a 2-D roughness shall be visualized based on selected deterministic simulations (isothermal wall with Tw = Tw,adiabatic ). Results for the streamwise velocity are Fourier transformed in time t and span z with a fundamental circular frequency F = 2π f˜ µ˜ /(ρ˜ u˜2 ) ∞ = 10−4 and a fundamental spanwise wave number γ = 2π /λz = 10.4. Then, the maximum over y is computed to yield amplitudes uˆmax (h,k) . The notation (h, k) is used to refer to a disturbance with a frequency h × F and a spanwise wave number k × γ . In the linear case, the modification in disturbance evolution caused by the roughness shall be quantified based on the change in N-factor, defined as:

∆ N = ln ((A(hR ))/(A(flat plate))) with A = uˆmax (1,0) .

(5)

Figure 2 (left) compares the 2-D disturbance evolution, mode (1, 0), between the √ case with height hR = 0.125 and a flat plate (Rx = Re∞ × x). The roughness acts as a local growth modifier, causing the disturbance amplitude downstream to be significantly larger as compared to the flat plate. Far downstream, the same amplification as for the flat-plate case is recovered, and ∆ N approaches a constant value. Further details regarding the linear disturbance evolution can be found in Ref. [7]. In the non-linear case, the modification in disturbance evolution caused by the finite-amplitude 2-D wave shall be quantified. To reflect the importance of absolute disturbance amplitude for transition, we define a transition parameter χ as follows:

χ = log (uˆmax (1/2,±1) ) .

(6)

If χ reaches values larger than a certain threshold χc , transition may be imminent (the exact threshold for transition remains to be determined). This may make xT = x|χ =χc a very simple criterion for the transition location xT . In figure 2 (right), a difference between the non-linear case with 2-D roughness and the linear case for a flat plate can not only be seen in the vicinity of the roughness. Instead, the growth of

Disturbance evolution in a compressible boundary-layer with roughness

max. amplitude

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(1/2,±1))

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-4

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2−D = 4.17 × 10−4 ) for h = 0.125 (solid line) and a flat plate (dashed line), Fig. 2 Left: uˆmax R (1,0) (Av

2−D = 0.035 together with the respective ∆ N (dash-dotted line). Right: uˆmax (h,k) for hR = 0.125 with Av

(solid line) and for a flat plate with A2−D = 0.0 (dashed line). The vertical lines mark the center v location of the roughness.

the oblique waves continues to be larger due to the presence of the large-amplitude 2-D wave due to subharmonic resonance. For a similar setup, Ref. [3] did not observe a significant subharmonic resonance due to their smaller 2-D amplitudes.

6 Stochastic disturbance evolution For the linear case the height of the roughness is taken as the random parameter s = hR , and extrema in eq. (1) are chosen as smin = 0.075 and smax = 0.125. The quantity of interest is the change in N-factor caused by the roughness, eq. (5), sufficiently far downstream of the roughness p = ∆ N|Rx =1750 , and we have chosen M=17. The resulting pdf’s are presented in figure 3 (left) for a uniform distribution of roughness heights and a β -distribution centered around hR = 0.1. In both cases, the pdf’s possess a local maximum at a value ∆ N close to the deterministic result for hR = 0.1. For the uniform distribution, this is a direct result of an inflection point of the function ∆ N(hR ) at approximately hR = 0.105 . . . 0.11, ∆ N = 1.2. For the non-linear case, the forcing amplitude of the 2-D wave is the random parameter s = A2−D , with smin = 0.005 and smax = 0.035. The quantity of interest is v the parameter χ , eq. (6), again downstream of the roughness p = χ |Rx =1550 (M=33). The pdf’s for the non-linear case (figure 3, right) show two interesting features. First, non-linear effects can lead to a damping. This causes a significant probability that the amplitude of the oblique wave is smaller than in the linear case. Second, the right-hand flank of the pdf is even qualitatively different for the two considered amplitude distributions. This suggests that great care has to be taken in choosing a distribution if it should lead to an accurate prediction of transition for a given case. Note that since the solution is linear in the disturbance amplitude of the oblique wave, the position of the pdf relative to the origin is, in fact, arbitrary.

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0.4 uniform distribution beta distribution

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Fig. 3 Left: pdf’s for ∆ N at Rx = 1750 for the linear case. The line marks the deterministic value for hR = 0.1. Right: pdf’s for χ at Rx = 1550 for the non-linear case. The lines mark the value for linear disturbance evolution, A2−D = 0.0, for a flat plate (dashed) and for hR = 0.125 (solid). v

7 Conclusion The evolution of disturbances in a Ma=4.8 flat-plate boundary layer with a 2-D roughness element has been investigated. Deterministic simulations show that the 2-D roughness acts as an amplifier for convective disturbances, and the resulting increased disturbance amplitude can enhance a secondary instability. As one possible first step towards a probabilistic description of transition, a stochastic approach has been applied to quantify the probability for an increased or decreased amplification. Acknowledgements Financial support from NASA, contract #NNX07AC29A, is gratefully acknowledged. We thank Sanjiva Lele for useful discussions and for providing the simulation code.

References 1. P. Balakumar. Transition in a supersonic boundary-layer due to roughness and acoustic disturbances. AIAA 2003–3589, 2003. 2. J. D. Crouch. Modeling transition physics for laminar flow control. AIAA 2008–3832, 2008. 3. W. Eissler and H. Bestek. Spatial numerical simulations of linear and weakly nonlinear wave instabilities in supersonic boundary layers. Theor. Comp. Fluid Dyn., 8(3):219–235, 1996. 4. T. Herbert. Secondary instability of boundary layers. Ann. Rev. Fluid Mech., 20:487–526, 1988. 5. M. R. Malik. Prediction and control of transition in supersonic and hypersonic boundary layers. AIAA J., 27(11):1487–1493, 1989. 6. M. R. Malik. Hypersonic flight transition data analysis using parabolized stability equations with chemistry effects. J. Spacecraft Rockets, 40(3):332–344, 2003. 7. O. Marxen and G. Iaccarino. Numerical simulation of the effect of a roughness element on high-speed boundary-layer instability. AIAA 2008–4400, 2008. 8. S. Nagarajan, S. K. Lele, and J. H. Ferziger. A robust high-order method for large eddy simulation. J. Comput. Phys., 191:392–419, 2003. 9. L. N. Trefethen. Is Gauss quadrature better than Clenshaw–Curtis? SIAM Review, 50(1):67–87, 2008.

Experimental Study of Boundary Layer Transition Subjected to Weak Free Stream Turbulence Masaharu Matsubara, Kota Takaichi, and Toshiaki Kenchi

Abstract An experimental investigation of laminar turbulent transition in a flat plate boundary layer subjected to weak free stream turbulence has been done with focusing on transition scenarios. Flow visualization clearly indicated that a packet of Tollmien-Schlichting(T-S) wave emerges in the boundary layer and immediately breaks down to turbulence with deformation to Λ -shape structures. In a much weaker turbulence intensity case, the transition assumes more complication that a short streak breakdown coexists with the T-S wave breakdown in addition to interaction between the wave packets and streaky structures considerably elongated in the streamwise direction. There exist, at least, three scenarios of boundary layer transition due to free stream turbulent: wavy motion of the elongated streaky structure, the Λ -shape deformation of the T-S wave packet and the short streak breakdown to turbulence. In order to accomplish a proper prediction method for the boundary layer transition, further classification of boundary layer transition scenarios is needed.

1 Introducution It is known that in a flat plate boundary layer subjected to free stream turbulence of a few % turbulence intensity, longitudinally elongated streaks appear in the boundary layer and then break down to turbulence with their wavy motions [1, 2]. The Masaharu Matsubara Department of Mechanical Systems Engineering, Shinshu University, 4-17-1 Wakasato, Nagano, 380-8553, Japan, e-mail: [email protected] Kota Takaichi Department of Mechanical Systems Engineering, Shinshu University, 4-17-1 Wakasato, Nagano, 380-8553, Japan, e-mail: [email protected] Toshiaki Kenchi Department of Mechanical Engineering, Gifu National College of Technology, 2236-2 Kamimakuwa, Motosu, 501-0495, Japan, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_44, © Springer Science+Business Media B.V. 2010

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experimental results showed that the profile of the streamwise velocity fluctuation has a peak at the middle of boundary layer and that the disturbance energy increases in proportion to the streamwise distance from the leading edge. These results are sufficiently consistent with the non-modal theory [3, 4]. The streaks were also observed in a numerical simulation [5], though trigger of breakdown to transition was due to a backward jet from the outside of the boundary layer. On the other hand, experimental studies on boundary layer transition due to weak free stream turbulence are limited. An early experiment was performed by Schubauer and Skramstad [6], who first confirmed T-S waves with comparison of velocity oscillations to the linear theory. It was confirmed by Kendall [7] with flow visualizations and observation of hot-wire signals that T-S waves locally appear in the case of free stream turbulence much lower than 1 %. Though growth envelopes of the T-S wave do not agree with the linear theory, the streamwise emergent position and the frequency of the local T-S wave are consistent with the linear theory prediction. Recently Kenchi et al. [8] ascertained that in a boundary layer subjected to the free stream turbulence of 0.77 % wall-normal velocity fluctuation, a locally emerging T-S wave packet breaks down after its deformation to Λ -shape structure. The present study were performed by smoke flow visualization and a hot-wire measurement in a certain range of weak free stream turbulence with focusing on transition scenarios and dominant disturbances to trigger breakdown to turbulence.

2 Experimental Set-up The present experiments were made in a closed wind tunnel. A diagram of the experimental set-up is shown in Fig. 1. Air from a settling chamber flows through a three-dimensional 9 : 1 contraction into a test section of a 400 mm width, a 600 mm height and a 4 m length. An aluminum test plate of a 2.1 m length, a 580 mm width and a 10 mm thickness is mounted vertically in the test section with a 100 mm separation from one sidewall of the test section. This plate has a 10 : 1 elliptic leading edge of a 20 mm minor axis. The leading edge is located at 1250 mm from the exit of the contraction. The sidewall faced to the test surface and a trailing-edge flap were carefully adjusted to prevent separation around the leading edge and to obtain uniform streamwise distribution of pressure in the free stream. The coordinate system is denoted by streamwise x, wall-normal y and spanwise z with the origin at the center of the leading edge. The free stream turbulence level after high pass filter of 13.8 Hz is 0.03 % at free stream velocity U∞ = 10 m/s. For artificial generation of free stream turbulence, two turbulence grids are used. Table 1 shows properties of the grids, where xg is the streamwise grid position measured from the leading edge. Grid A is inserted in exit of the contraction and grid B is mounted upstream of the contraction. As shown in Fig. 2, pipes of grid B have 2 mm diameter holes oriented upstream. Air pressure inside the pipes is adjusted at the pressure difference ∆ P = 410 Pa so that the upstream jets from the holes increase the turbulence intensity.

Experimental Study of Boundary Layer Transition Subjected to Weak FST

Flap

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Fig. 2 Detail of the turbulence grid B.

Fig. 1 Experimental set-up. Table 1 Properties of turbulence grids. Grid

Streamwise position xg

Mesh size

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Bar width

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A B

−1500 mm −3800 mm

30 mm 120 mm

square circular

6 mm 13 mm

36.0 % 20.5 %

Flow visualization is made with alcohol mist carefully induced through a 2 mm wide spanwise slot at x = 210 mm. For the video recordings, a 0.31-megapixel digital video camera of 120 frames per second and 500 W floodlights are used. A sensor of the hot-wire anemometer is made of 2.5 µ m diameter platinum wire of a 1.5 mm length. An X-probe is used for measurement of the free stream turbulence, and single-probe is used for the boundary layer measurement. A three axis robot arm enables precise movements of the hot-wire probe in streamwise, spanwise and wallnormal directions. Three conditions of the free stream turbulence are adopted in the present experiment. One is with grid A at U∞ = 4 m/s. In the other two cases grid B are used with or without the jet blowing at the free stream velocity U∞ = 12 m/s or 14 m/s, respectively. Fig. 3 shows spectra of the free stream turbulence at the leading edge. The horizontal axis is the normalized frequency F = 2π f ν × 106 /U∞2 , where f is the dimensional frequency and ν is the kinetic viscosity. The vertical axis is premultiplied by f so that the area corresponds with the energy. In the case of grid A, the streamwise component urms,∞ in low frequency range is slightly higher than the lateral component vrms,∞ , indicating weak anisotropy. In both grid B cases, the vertical component is much higher than the streamwise component except in high frequency range. The peak frequency of the streamwise component shifts with the jet blowing, while that of the lateral component is constant. In the case of grid A, urms,∞ /U∞ and vrms,∞ /U∞ at the leading edge are 2.55 % and 2.11 %, respectively. In the case of grid B with the jet blowing, urms,∞ /U∞ is 0.30 % and vrms,∞ /U∞ is 0.67 %, and without the jet blowing urms,∞ /U∞ and vrms,∞ /U∞ are 0.20 % and 0.30 %, respectively. The condition of the case of grid B with the jet blowing is similar to that of Kenchi et al. [8] investigation.

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30 grid A urms,∞

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0 −1 10

0

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Fig. 3 Spectra of the velocity components in the free stream at x = 0.

3 Hot-wire Measurement Result and Flow Visualization Profiles of the mean streamwise velocity in the case of grid B without the jet blowing are shown in Fig. 4. The wall-nomal position y is normalized by a boundp ary layer displacement thickness δ ∗ = 1.72 x ν /U∞ . The upstream mean velocity profile is in good agreement with the Blasius profile shown in a solid line. At Rex (= U∞ x/ν ) = 8.4 × 105 , the profile begins to deviate from the Blasius profile and the downstream profile presents a typical turbulent one. The fluctuation distributions shown in Fig. 5 have a peak at the middle of boundary layer in upstream region, suggesting existence of a non-modal growth disturbance. Downstream of Rex = 6.5 × 105 , however, the velocity fluctuation suddenly increases near the wall. This peak shift was also observed by Kenchi et al. [8]. The downstream distributions of the velocity fluctuation also have features of a typical turbulent one. Fig. 6 shows flow visualization in a transitional boundary layer in the case of grid B with the jet blowing. The streamwise range of the picture is 3.48 × 105 ≤ Rex ≤ 7.24 × 105 . A turbulent spot and Λ -shape structures are seen in the flow visualization. Streamwise tone stripes of the mist can be also observed though their typical spanwise spacing of about 50 mm is more than 5 times as large as the optimal disturbance predicted by the non-modal growth theory [4]. In the case of grid B without the jet blowing as shown in Fig. 7, there also coexist Λ -shape structures and streaks. Figs. 8 and 9 are the sequences extracted from the video recordings with 8.3 ms intervals. The streamwise range of the picture is 4.06 × 105 ≤ Rex ≤ 8.45 × 105 . In Fig. 8, a locally emerging packet of ripples forms the Λ -shape structure and then

Experimental Study of Boundary Layer Transition Subjected to Weak FST 0.2

1 0.9

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Fig. 4 Mean velocity profiles for grid B without the jet blowing.

Fig. 5 Velocity fluctuation distributions. Symbols are same of Fig. 4.

Fig. 6 Flow visualization in grid B case with the jet blowing.

Fig. 7 Flow visualization in grid B case without the jet blowing.

Fig. 8 Wave packet breakdown with the Λ shape structure forming.

Fig. 9 Breakdown of the short streak.

immediately break down to turbulence. A wave length of the ripples and their phase velocity are in agreement with the linear theory. Therefore this packet is considered

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the T-S wave packet. In long term video observation the wave packet is sometimes distorted by streaky structure. The wave packet breakdown was also seen in the case of grid B with the jet blowing. As shown in Fig. 9, there exists a different transition scenario that a short streak suddenly appears and breaks down to turbulence. This short streak is very different from an ordinary elongated streak due to high intensity free stream turbulence, in terms of its sudden appearance and very short streamwise scale. This type of breakdown was very rare to observed in the case of grid B with the jet blowing.

4 Conclution In the case of the weak free stream turbulence of vrms,∞ /U∞ = 0.67 %, a packet of T-S wave triggers transition to turbulence with its deformation to the Λ -shape structure and subsequent generation of a turbulent spot. In a much weaker turbulence intensity case, a short streak breakdown coexists with the T-S wave breakdown and the wave packets interact with streaky structures. There exist, at least, three scenarios of boundary layer transition due to free stream turbulent: wavy motion of the elongated streaky structure, Λ -shape deformation of the T-S wave packet and a short streak breakdown to turbulence. For accomplishment of proper prediction method for the boundary layer transition, further detail classification of boundary layer transition scenarios is needed.

References 1. Matsubara, M., Alfredsson, P.H.: Disturbance growth in boundary layers subjected to freestream turbulence, J. Fluid Mech, 430, 149–168 (2001) 2. Fransson, J.H.M., Matsubara, M., Alfredsson, P.H.: Transition induced by free-stream turbulence, J. Fluid Mech, 527, 1–25 (2005) 3. Andersson, P., Berggren, M, Henningson, D.S.: Optimal disturbances and bypass transition in boundary layers, Phys. Fluids, 11(1), 134–150 (1999) 4. Luchini, P.: Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations, J. Fluid Mech, 404, 289–309 (2000) 5. Jacobs, R.G., Durbin, P.A.: Simulations of bypass transition, J. Fluid Mech, 428, 185–212 (2001) 6. Schubauer, G.B., Skramstad, H.K.: Laminar boundary-layer oscillations and transition on a flat plate, NACA Technical Report, 909 (1948) 7. Kendall, J.M.: Experimental study of disturbances produced in a pre-transitional laminar boundary layer by week freestream turbulence, AIAA paper, 85–1695 (1985) 8. Kenchi, T., Matsubara, M., Ikeda, T.: Laminar turbulent transition in a boundary layer subjected to free stream turbulence, J. Fluid Sci and Tech, 3(1), 56–67 (2008)

Open-loop control of compressible afterbody flows using adjoint methods Philippe Meliga, Denis Sipp, and Jean-Marc Chomaz

Abstract We present the theoretical study of a compressible afterbody flow in the subsonic regime. We use sensitivity analyses developed in the framework of the linear global stability theory to predict beforehand the effect of a steady bulk and wall forcing on the growth rate of linear global modes. The sensitivity functions are derived analytically using adjoint methods, and presented for the global mode responsible for the onset of a periodic regime. Our results show that the global mode is sensitive to momentum forcing along the separation line, or to a localized heating in the core of the recirculating bubble.

1 Introduction Open-loop control relies on the simple idea that a fixed modification in the flow conditions is susceptible to affect the whole flow dynamics. In the context of the cylinder wake flow, Strykowski & Sreenivasan [8] have experimentally investigated how a small control cylinder could alter the vortex-shedding phenomenon if suitably placed in the lee of the main cylinder. These authors successfully identified flow regions where the addition of the control cylinder leads to a complete suppression of the phenomenon. Such an approach is however empirical as it relies on a ‘trial and error’ process. Consequently, it can be extremely time-consuming if the number of degrees of freedom is large. A more systematic approach for the open-loop control of vortex-shedding, based on sensitivity analyses, has been introduced by Hill [4] and revisited by Marquet et al. [5]. The main idea underlying these studies is that Philippe Meliga LadHyX, 91128 Palaiseau, France, e-mail: [email protected] Denis Sipp ONERA/DAFE, 92190 Meudon, France, e-mail: [email protected] Jean-Marc Chomaz LadHyX-CNRS, 91128 Palaiseau, France, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_45, © Springer Science+Business Media B.V. 2010

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the effect of a steady control is to modify the base flow on which the perturbation develops. The effect of the control on the flow stability can then be simply estimated from the computation of a gradient (or sensitivity function). These concepts are extended here to the case of nonparallel compressible flows and applied to the control of unsteadiness in a subsonic afterbody flow.

2 Theoretical framework We consider an axisymmetric body of revolution of diameter D and total length l = 9.8D, with a blunt trailing edge and an ellipsoidal nose of aspect ratio 3 : 1, placed into a uniform flow at zero angle of attack. Standard cylindrical coordinates (r, θ , z) with origin taken at the center of the base are used. The fluid is taken as a non-homogeneous compressible perfect gas with constant specific heat c p , thermal conductivity κ , and dynamic viscosity µ , related by a unit Prandtl number. The fluid motion is described by the state vector q = (ρ , u , T, p)T , where ρ is the density, T the temperature, p the pressure and u = (u, v, w)T the three-dimensional velocity field with u, v and w its radial, azimuthal and streamwise components. q obeys the unsteady compressible Navier-Stokes equations, thus leading to a set of six nonlinear equations (continuity, momentum, internal energy and perfect gas) formulated in non-conservative variables. These equations are made non-dimensional using the body diameter and the upstream flow quantities, and are written formally as B(qq)∂t q + M (qq, G ) = ( j, f , h, 0)T ,

(1)

where B and M are differential operators and G represents the set of control parameters (Reynolds and Mach numbers, angle of attack...) which remains constant here, so that the dependence in G is omitted so as to ease the notation. The right-hand side in (1) defines the bulk forcing, f (resp. j et h) being the volumetric momentum flux (resp. volumetric heat and mass fluxes) associated to the control. The effect of wall forcing is also taken into account by adding a subsonic inlet condition at the base, whose surface is denoted Γc : u = uW , T = TW .

(2)

In the following, the Mach number is set to M = 0.5. The state vector is split into an axisymmetric steady base flow q 0 and a three-dimensional perturbation q 1 of small amplitude. We consider here the case of an axisymmetric steady control only, so that the base flow equations read M0 (qq0 ) = (m, f , h, 0)T ,

u 0 = uW , T 0 = TW on Γc ,

(3)

where M0 is the axisymmetric form of the evolution operator M . Perturbations are chosen under the form of normal modes

Open-loop control of compressible afterbody flows using adjoint methods

285

q 1 = qˆ 1 (r, z)e(σ +iω )t+imθ + c.c. ,

(4)

where qˆ 1 is the so-called global mode, characterized by an integer azimuthal wavenumber m, a growth rate σ and a pulsation ω . qˆ 1 and the complex pulsation λ = σ + iω are solutions of a generalized eigenvalue problem reading

λ B(qq0 )qˆ 1 + Am (qq0 )qˆ 1 = 0 ,

uˆ 1 = 0 , Tˆ 1 = 0 on Γc ,

(5)

where Am is the differential operator obtained by linearization of operator M about q 0 and substitution of the ∂θ terms by the product by im. We use a finite elements method to discretize equations (3) and (5) on a computational domain Ω corresponding to the azimuthal plane θ = 0, whose boundary is denoted Γ = Γw ∪ Γ∞ , where Γw corresponds to the afterbody walls. All pressure terms are eliminated and replaced by their expressions issuing from the perfect gas equations for p0 et pˆ1 . The base flow equations (3) are solved using an iterative Newton method (Barkley [1]), and the disturbance equations (5) are solved using a Shift and Invert Arnoldi method. In the present compressible regime, the choice of appropriate far-field radiation conditions may be particularly involved. Consequently, we use sponge zones where all fluctuations are progressively damped to negligible levels through artificial dissipation before they reach the boundary of the computational domain. The boundary conditions satisfied by the base flow and the disturbances are then deduced from that satisfied by the state vector q = (ρ , u , T )T : u = (0, 0, 1)T , ρ , T = 1 on Γ∞ ,

u = 0 , ∂n T = 0 on Γw \Γc .

(6)

We consider now the unforced flow, for which j = 0,

f = 0,

h = 0,

uW = 0 ,

TW = 1 +

γ −1 2 M . 2

(7)

The effect of a small-amplitude forcing on the flow stability is assessed by investigating the variation of a given eigenvalue δ λ = δ σ + iδ ω . In the present linear framework, the eigenvalue variation resulting from the introduction of the forcing can be written as the scalar product between the forcing term and a sensitivity function or gradient. We thus introduce the complex vectors ∇ j λ , ∇ f λ and ∇ h λ defining the sensitivity of the eigenvalue to a source of mass, momentum and internal energy, respectively. Similarly, ∇ uW λ and ∇ TW λ define the sensitivity of the eigenvalue to a wall velocity and temperature. The analytical expression of these gradients is derived using a Lagrangian method relying on the definition of adjoint variables, an approach similar to that widely used in the context of optimization problems (Gunzburger [3]). We obtain ∇ j λ , ∇ f λ , ∇ h λ )T = (ρ 0† , u 0† , T 0† )T , (∇ (8a) 1 2 T ∇ · u 0† ) I + ∇ uˆ 0† + ∇ uˆ 0† ∇ uW λ = ρ 0 ρ 0† n + − (∇ · n (8b) , Re 3 γ ∇ TW λ = ∇ T 0† · n , (8c) PrRe

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where n is the normal to the afterbody wall oriented from the body towards the fluid. q 0† = (ρ 0† , u 0† , T 0† )T is termed the adjoint base flow, and is solution of the forced linear problem A0† (qq0 )qq0† = −λ ∗ R † (qq0 , qˆ 1 )qˆ 1† − Sm† (qq0 , qˆ 1 )qˆ 1† ,

(9)

where the superscript ∗ designate the complex conjugate. In (9), A0† is the adjoint of the axisymmetric operator A0 , obtained by integrating by parts the axisymmetric, steady form of the disturbance equations (5). Similarly, operators R † and Sm† are the adjoints of operators R and Sm , defined as R(qq0 , qˆ 1 ) = ∂ (B(qq0 )qˆ 1 )/∂ q 0 ,

Sm (qq0 , qˆ 1 ) = ∂ (Am (qq0 )qˆ 1 )/∂ q 0 . (10)

Finally, qˆ 1† is the adjoint global mode, solution of the adjoint stability problem

λ ∗ B(qq0 )qˆ 1† + Am† (qq0 )qˆ 1† = 0 .

(11)

3 Results The global stability analysis shows that the unforced axisymmetric base flow sustains two subsequent instabilities: a first bifurcation occurs at the critical Reynolds number ReA = 483.5, and involves a stationary global mode A of azimuthal wavenumber m = 1 and frequency ω = 0 (not shown here). A Hopf bifurcation then occurs at ReB = 983, involving a m = 1 global mode B oscillating at the frequency ωB = 0.399 (St = f D/U∞ = 0.06). Mode B exhibits positive and negative velocity perturbations alternating downstream of the body, in a regular, periodic way, as illustrated by the streamwise velocity distribution shown in figure 1. These results are in agreement with the global stability analysis carried out by Natarajan & Acrivos [7] in the context of the incompressible flow past disks and spheres, for which mode B dominates the dynamics of the fully 3D flow at large Reynolds numbers, and triggers the occurrence of a fully 3D periodic state (Fabre et al. [2]; Meliga et al. [6]). Consequently, in the following, we focus on the effect of open-loop control on the growth rate σB of the oscillating mode B.

Fig. 1 Streamwise velocity component of the oscillating global mode B at the threshold of instability - ReB = 983.0, M = 0.5 (the background grey hue stands for vanishing perturbation amplitudes).

Open-loop control of compressible afterbody flows using adjoint methods

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We study first the effect of a small control device, chosen here as a ring of width e and radius rc , mounted at the rear of the main body, at a distance zc of the base. As in the studies of Hill [4] and Marquet et al. [5], we consider that the flow exerts a localized drag force on the control ring, and that the ring exerts in return an opposite force on the flow, modeled as 1 δ f (r, z) = − Ceρ 0 (r, z)kuu0 (r, z)kuu0 (r, z)δ (r − rc , z − zc ) , 2

(12)

where C is a drag coefficient depending on the value of the Reynolds number Ree built from the ring width and the local flow velocity. Typical values of Ree in the recirculating bubble are of order Ree ≃ 30, so that we choose here C = 1, an empirical value estimated from the drag coefficient of a circular cylinder in this range of Reynolds numbers. To each position of the ring (rc , zc ) corresponds a variation of the growth rate δ σB that can be expressed simply as the scalar product between the induced force δ f and the sensitivity function ∇ f σB . Figure 2(a) presents the spatial distribution of the growth rate variation δ σB (rc , zc ). Since the global mode is marginally stable, negative variations δ σB < 0 (resp. positive variations δ σB > 0) correspond to a stabilization (resp. a destabilization) of the global mode. We find that the ring induces a strong stabilization if placed along the separation line. However, it should be noted that the effect of such momentum forcing is complex, since several regions contributing either to a weak stabilization or destabilization of the global mode are visible around the main stabilizing region. We consider now the effect of a localized heat source modeled by

δ h(r, z) =

1 δ hˆ δ (r − rc , z − zc ) . 2π r c

(13)

Physically, δ hˆ is the flux of internal energy imposed by the control, so that a positive (resp. negative) value of δ hˆ corresponds to a heating (resp. a cooling) of the flow. Again, to each position of the source corresponds a growth rate variation δ σB given by the scalar product between the forcing term δ h and the sensitivity function ∇ h σB . We show on figure 2(b) the results obtained for δ hˆ = 10−2 , i.e. the flow is heated and the cost of the control represents 1% of the internal energy flux of the incoming flow. We find that heating the flow within the recirculating area has a stabilizing effect, whatever the position of the source. The maximum stabilizing effect obtained by this method corresponds to a variation δ σB = −0.06, and is however less important than that achieved using the control ring.

4 Perspectives Currently, we investigate the effect of fluid blown in the wake through the base, a strategy also called base bleed. We also aim at using this formalism to interpret the

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Fig. 2 (a) Spatial distribution of the growth rate variation δ σB (rc ,zc ) obtained when the presence of a small control ring is modeled by the force (12). (b) Same as (a) when using a heat source modeled by (13) with δ hˆ = 10−2 - Re = 983, M = 0.5. The background grey hue stands for vanishing variations.

stabilizing effects documented here. This is done by carrying out similar sensitivity analyses, where the growth rate variation is no more investigated as a function of the forcing, but as a function of the base flow modification induced by the forcing. The results obtained so far suggest that all control strategies investigated act in the same way, namely they modify the base flow momentum distribution, which results in an increase of the disturbances advection.

References 1. Barkley, D., Gomes, M.G.M. & Henderson, R.D. (2002). Three-dimensional instability in flow over a backward-facing step. J Fluid Mech 473, 167–190. 2. Fabre, D., Auguste, F. & Magnaudet, J. (2008). Bifurcations and symmetry breaking in the wake of axisymmetric bodies. Phys Fluids 20, 051702 1–4. 3. Gunzburger, M.D. (1999). Sensitivities, adjoints and flow optimization. Int J Numer Meth Fluids 31, 53–78. 4. Hill, D.C. (1992). A theoretical approach for analyzing the restabilization of wakes. NASA Tech Report 103858. 5. Marquet, O., Sipp, D. & Jacquin, L. (2008). Sensitivity analysis and passive control of the cylinder flow. J Fluid Mech 615, 221–252. 6. Meliga, P., Chomaz, J.-M. & Sipp, D. (2009). Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion. J Fluid Mech 633, 159–189. 7. Natarajan, R. & Acrivos, A. (1993). The instability of the steady flow past spheres and disks. J Fluid Mech 254, 323–344. 8. Strykowski, P. J. & Sreenivasan, K.R. (1990). On the formation and suppression of vortex shedding at ’low’ Reynolds numbers. J Fluid Mech 218, 71–107.

Direct Numerical Simulation of a Swept-Wing Boundary Layer with an Array of Discrete Roughness Elements Takafumi Nishino and Karim Shariff

Abstract Direct numerical simulations of crossflow-induced boundary layer transition are performed for a 45-degree swept wing with an array of discrete roughness elements placed near the leading edge, at a chord Reynolds number of 2.4 million. A large part of the wing upper surface in the chordwise direction (but with only less than two-percent chord in the spanwise direction) is resolved to simulate the growth and breakdown of crossflow vortices on the wing. The procedures, initial results and some difficulties encountered during the study are described.

1 Introduction Control or delay of laminar-to-turbulent transition in swept-wing boundary layers is a promising area of research for reducing skin-friction drag and thus improving the fuel efficiency of aircraft. In general, transition on a swept wing may be induced by several different types of instabilities, such as attachment-line instabilities, streamwise or Tollmien-Schlichting (T-S) instabilities, centrifugal or Taylor-Görtler (T-G) instabilities, and crossflow instabilities [3]. The nonlinear interactions of these instabilities are rather complicated and have not been explored in detail, although it is widely recognized that the dominant mechanism of the swept-wing boundary layer transition in most practical flight conditions is the breakdown to turbulence due to the secondary instability of stationary crossflow vortices [8]. Given the dominance of stationary crossflow vortices or modes in the process of swept-wing boundary layer transition, the main challenge here is how to suppress or delay the growth of such “critical” crossflow modes that lead to the breakdown Takafumi Nishino NASA Ames Research Center, Moffett Field, CA, USA, e-mail: [email protected] Karim Shariff NASA Ames Research Center, Moffett Field, CA, USA, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_46, © Springer Science+Business Media B.V. 2010

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to turbulence (NB: another possible strategy to delay the transition would be to directly suppress the secondary instability by manipulating the structure of saturated crossflow vortices [1]; however the present study does not go into this). A series of experiments by Saric et al. [7] has shown that the growth of critical crossflow modes can be temporarily suppressed and thus the breakdown to turbulence can be delayed by introducing artificial “subcritical” crossflow vortices, which were created in their experiments using a spanwise periodic array of discrete roughness elements (DRE) attached near the wing leading edge with a spanwise spacing shorter than the wavelengths of the critical crossflow modes. The validity of these experimental results has been confirmed theoretically by Malik et al. [4] and Haynes and Reed [2], who applied nonlinear parabolized stability equations (NPSE) to calculate the nonlinear growth of stationary crossflow disturbances. The effectiveness and robustness of DRE-based transition control in practical applications, however, is still unclear since the physics underlying it has not been clarified in detail. One of the key issues is the receptivity of the boundary layer to freestream disturbances as well as to wall roughnesses [8], which determines the initial state (or more specifically the initial amplitude ratio, and thus the subsequent growth) of stationary and traveling crossflow disturbances. It should be noted that although the final breakdown to turbulence at low freestream turbulence is known to be triggered by the secondary instability of the stationary modes, the influence of the traveling modes on the growth of (both critical and subcritical) stationary modes is still unclear. Only recently have some of these issues been investigated using direct numerical simulations (DNS) of the Navier-Stokes equations [9][11][12]; however the application of these DNS studies in recent years is still limited to a swept flat plate (neglecting wall curvature effects) and not a swept wing. The first goal of the present (ongoing) study is to examine the feasibility of DNS to solve the whole transition process on a practical swept wing (rather than a swept flat plate). A possible approach to perform such DNS is described in this article, along with some initial results and difficulties encountered during the study. The second and more important goal to be achieved in the future is to further investigate the physics of DRE-based transition control for swept wings.

2 Flow Configuration and Numerical Method The flow configuration investigated is a 45◦ swept NLF(2)-0415 airfoil at −4◦ angle of attack, which was studied experimentally by Saric et al. [7][6]. The Reynolds number based on the chord of the√swept wing, C = 1.83m, is 2.4 million (note that the unswept airfoil chord c = C/ 2). This configuration provides a long favorablepressure-gradient region up to about 70% chord on the airfoil upper surface, which is ideal for experimentally observing the growth and saturation of crossflow disturbances. When considering the cost of DNS for reproducing the whole transition process, however, this is a rather demanding case since the final breakdown to turbulence occurs far downstream of the airfoil leading edge, i.e., a large computational

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Fig. 1 Two-dimensional computation of flow past a 45-degree-swept NLF(2)-0415 airfoil preliminarily performed in order to obtain the boundary and initial conditions for the DNS: (a) contours of the eddy viscosity ratio, µt / µ , and (b) pressure distributions on the airfoil upper surface.

Fig. 2 Schematic of the computational domain for the DNS, configurations of discrete roughness elements and an unsteady disturbance strip near the leading edge, and the coordinate system.

domain in the chordwise direction is required and hence the domain size in the wallnormal and spanwise directions is rather restricted. Figure 1 shows a two-dimensional (three-velocity-component) computation that was preliminarily performed to obtain the boundary (and initial) conditions for the DNS in this study. Since the DNS can be performed only near the top surface of the wing, the influence of wind-tunnel blockage was calculated in this preliminary computation and was imposed on the DNS as outer boundary conditions. Note that the Spalart-Allmaras eddy-viscosity model [10] was turned on in this two-dimensional computation (in order to avoid laminar separation bubbles) but only around and behind the trailing edge, which is outside the DNS domain. This can be seen in Fig. 1(a), showing contours of the calculated eddy viscosity ratio, µt /µ , as well as the computational domain for the DNS. The obtained pressure distribution on the wing top surface is in good agreement with the experiments, as shown in Fig. 1(b). Figure 2 describes the computational domain for the DNS and its coordinate system. The present DNS is performed using a modified version of a compressible Navier-Stokes code developed by Nagarajan et al. [5], which uses a sixth-order compact finite difference scheme on a staggered grid for the spatial discretization

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and a second-order fully-implicit Beam-Warming scheme for the time integration. The difficulty here is to give a proper grid resolution; since the crossflow vortices to be solved are not aligned with the chordwise (ξ ) direction due to the large swept angle, a relatively fine grid is required not only in the spanwise (ζ or z) direction but also in ξ direction all over the domain. Meanwhile, the spanwise (periodic) domain size needs to be large enough to at least “harmonize” with the wavelengths of physically important crossflow modes (12mm for the critical or naturally dominant mode and 8mm for the subcritical mode in this study) to simulate the interactions among them. In this study we use ∆ ξ = 0.5mm and ∆ ζ = 0.25mm for most of the wing top surface and a spanwise domain size of 24mm; however the resolution and domain size effects are still unclear and need to be clarified in future studies. Figure 2 also shows the configurations of artificial roughness elements applied near the leading edge (at x/c = 0.023). Note that the shape of roughnesses considered in this study is a very shallow square box of 3mm×3mm×6µ m, and also of 4mm×4mm×6 µ m for comparison, whereas that used in the experiments is a circular disk of 3.7mm diameter and 6µ m height. Since the height of these roughnesses is extremely small (the roughness Reynolds number is about 0.1), they are not meshed but modeled in this study by applying linearized boundary conditions, i.e., no-slip conditions on the roughness surface are extrapolated to the original (smooth) wing surface via a first order Taylor series expansion. In addition to these roughness elements that induce stationary crossflow disturbances, unsteady wave-like disturbances are also introduced via a zero-net-mass-flux blowing/suction strip placed just upstream of the roughness elements; however the results for these unsteady disturbances are still preliminary and are not described in detail here.

3 Initial results and future work The first computation was conducted with 3×3mm roughness elements without unsteady forcing. The roughness spacing was set at 12mm in the spanwise direction,

Fig. 3 Initial development and convection of the “front edge” of crossflow vortices, visualized using isosurfaces of the Q-criterion. Roughness spacing is 12mm (without unsteady forcing). The plotted domain size is doubled in z direction for visual purposes.

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Fig. 4 Breakdown of traveling crossflow vortices, visualized using isosurfaces of the Q-criterion. Roughness spacing is 12mm, with unsteady forcing of multiple wavelengths (24, 12, 8mm) and frequencies (100, 200, 400, 800, 1600, 3200Hz). The plotted domain size is doubled in z direction for visual purposes.

which was investigated experimentally by Saric et al. [7] as a baseline “critical forcing” case. The computation was initiated with the aforementioned two-dimensional laminar flow and the roughness model was turned on at time zero. This “emergence” of the roughness elements at time zero created some initial pulse-like disturbances, which then evolved into large-amplitude “front edges” of crossflow vortices, as shown in Fig. 3. These initial disturbances, however, just convected downstream and then the flow field slowly converged to a steady laminar state with small-amplitude stationary crossflow vortices. The amplitude of these steady crossflow disturbances was found to be more than an order of magnitude smaller than that measured in the experiments, indicating that the roughness receptivity simulated in the present DNS was different from the experiments. An additional computation employing larger roughness elements of 4×4mm was also performed for comparison; however the effects of the roughness size were found to be very small. Hence possible causes of the difference between the experiments and the present DNS are: (i) insufficient grid resolution to resolve the roughness receptivity, (ii) influence of the roughness shape (circular disk vs. shallow square box), and (iii) influence of freestream disturbances. Further investigations are needed to clarify these issues. Since one of the differences between the experiments and the above DNS is the presence of freestream disturbances that may induce unsteadiness of the upstream boundary layer approaching the roughness elements, another series of computations are currently being performed with unsteady wave-like disturbances introduced via a zero-net-mass-flux blowing/suction strip just upstream of the roughness elements (cf. Fig. 2). The first test case with some large-amplitude unsteady forcing of multiple wavelengths and frequencies, however, turned out to be a different transition scenario from the experiments, i.e., the unsteady forcing itself induced strong traveling crossflow disturbances and the transition to turbulence occurred due to the breakdown of those traveling crossflow vortices, as shown in Fig. 4.

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In summary, the present study examined the feasibility of DNS to investigate crossflow-induced boundary layer transition on a swept wing at a high Reynolds number. A spatial grid resolution just high enough to resolve the structure of crossflow vortices was given for most part of the wing upper surface; however the computational costs were still high, which severely restricted the spanwise size of the computational domain. Also of possible importance are the grid resolution effects on the roughness receptivity results, which were not examined in this study. Since the stationary disturbance amplitude behind the roughness elements calculated in the present DNS was much smaller than that measured in the experiments, it seems probable that an even finer grid resolution is required to simulate the roughness receptivity process correctly. Further investigations on the receptivity to freestream disturbances are also necessary to fully understand the effectiveness and robustness of artificial roughness elements on transition control. Acknowledgements This work has been sponsored by NASA’s Fundamental Aeronautics Program at Ames Research Center, and T. Nishino is supported by the NASA Postdoctoral Program (NPP) administrated by Oak Ridge Associated Universities (ORAU). The authors would like to thank Prof. W. S. Saric of Texas A&M University for providing the experimental details, and Dr. M. Rogers of NASA Ames for useful discussions.

References 1. Friederich, T. A. and Kloker, M. J. 2008. Localized blowing and suction for direct control of the crossflow secondary instability. AIAA Paper 2008-4394. 2. Haynes, T. S. and Reed, H. L. 2000. Simulation of swept-wing vortices using nonlinear parabolized stability equations. J. Fluid Mech. 405, 325–349. 3. Joslin, R. D. 1998. Aircraft laminar flow control. Annu. Rev. Fluid Mech. 30, 1–29. 4. Malik, M. R., Li, F., Choudhari, M. M. and Chang, C.-L. 1999. Secondary instability of crossflow vortices and swept-wing boundary-layer transition. J. Fluid Mech. 399, 85–115. 5. Nagarajan, S., Lele, S. K. and Ferziger, J. H. 2007. Leading-edge effects in bypass transition. J. Fluid Mech. 572, 471–504. 6. Reibert, M., Saric, W. S., Carrillo, R. B. Jr. and Chapman, K. L. 1996. Experiments in nonlinear saturation of stationary crossflow vortices in a swept-wing boundary layer. AIAA Paper 96-0184. 7. Saric, W. S., Carrillo, R. B. Jr. and Reibert, M. S. 1998. Nonlinear stability and transition in 3-D boundary layers. Meccanica 33, 469–487. 8. Saric, W. S., Reed, H. L. and White, E. B. 2003. Stability and transition of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 35, 413–440. 9. Schrader, L.-U., Brandt, L. and Henningson, D. S. 2009. Receptivity mechanisms in threedimensional boundary-layer flows. J. Fluid Mech. 618, 209–241. 10. Spalart, P. R. and Allmaras, S. R. 1994. A one-equation turbulence model for aerodynamic flows. La Recherche Aerospatiale 1, 5–21. 11. Wassermann, P. and Kloker, M. 2002. Mechanisms and passive control of crossflow-vortexinduced transition in a three-dimensional boundary layer. J. Fluid Mech. 456, 49–84. 12. Wassermann, P. and Kloker, M. 2003. Transition mechanisms induced by travelling crossflow vortices in a three-dimensional boundary layer. J. Fluid Mech. 483, 67–89.

Wave packet pseudomodes upstream of a swept cylinder Dominik Obrist and Peter J. Schmid

Abstract The theory of wave packet pseudomodes provides tools for studying the (pseudo-)modes of complex flow configurations. We use these tools to investigate the eigenmodes supported by the flow field upstream a swept cylinder which is an extension to the thoroughly studied leading-edge boundary layer.

1 Introduction The flow upstream of a swept cylinder can be split into a potential flow field and a boundary layer (Fig. 1). A fluid particle, which is moving toward the leading edge of the swept cylinder, passes from a parallel flow (outer layer) through a swept stagnation point flow (matching layer) into a three-dimensional boundary layer known as swept Hiemenz flow (inner layer). We study the stability of this flow configuration, or more specifically: the stability of the flow field in the vicinity of the symmetry plane spanned by the wall-normal y-axis and the spanwise z-axis. The global stability problem for the swept Hiemenz flow only (which does not account for the parallel flow far upstream of the cylinder) has been studied extensively by various authors (e.g. [1, 2, 3, 4, 5, 6]). It has been shown [7] that this stability problem supports modes of the form (u, v, w) = (pN (x)u(y), ˆ pN−1 (x)v(y), ˆ pN−1 (x)w(y)) ˆ exp[i(γ z − λ t)] where pN (x) is a polynomial of order N ≥ 0. There exists an unstable mode (Görtler–Hämmerlin mode) within the boundary layer which is of order N = 1 Dominik Obrist Institute of Fluid Dynamics, ETH Zurich, SWITZERLAND e-mail: [email protected] Peter J. Schmid Laboratoire d’Hydrodynamique (LadHyX), CNRS-Ecole Polytechnique, Palaiseau, FRANCE email: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_47, © Springer Science+Business Media B.V. 2010

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Fig. 1 Flow around a swept cylinder with radius R swept by the angle ϕ .. The coordinate x is oriented in the chordwise direction, the wallnormal coordinate y points in the upstream direction, and the spanwise coordinate z is parallel to the sweep flow W .

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and decays super-exponentially fast with increasing y. The linear stability problem supports also modes which decay algebraically in the wall-normal direction. These modes are mostly located in the free-stream. It is a matter of debate, however, whether these modes are physically relevant, how they connect to the outer flow field and what their eigenvalue spectrum looks like. It is the aim of the present work to shed some light on these questions. For simplicity, we limit our investigation to the class of uniform modes (N = 0) which are constant in the chordwise direction and have no wall-normal and spanwise velocity component, i.e. (u, v, w) = (u(y), ˆ 0, 0) exp[i(γ z − λ t)]. For these modes the global stability problem reduces to an ordinary differential equation, 2 ∂y −V ∂y +V ′ − γ 2 − iγ ReW uˆ = −iReλ uˆ (1) with the boundary conditions u(0) ˆ = u(y ˆ → ∞) = 0. Figure 3 shows the numerically evaluated eigenvalues of the discretized stability problem (1). Whereas the super-exponentially decaying modes form a discrete spectrum and a continuous line spectrum along Re{λ } = γ , the algebraic eigenvalues are highly sensitive to the applied discretization. It appears that these modes belong to a continuous spectrum which covers a large area of the lower complex half-plane. It can be shown that algebraic modes of the uniform stability problem decay like u(y → ∞) ∼ y−ν −1

(2)

where ν = −[N +2+ γ 2 +iRe(γ − λ )]. Obrist & Schmid[4] argued that the exponent −ν −1 must be negative which limits the spectrum of the algebraic eigenvalues from above by 1 + γ2 Im{λ } < − < 0. (3) Re We will show later that this limit does not hold anymore if we include the outer layer in our baseflow. To this end, we replace the Hiemenz base flow V (y) by the wall-normal flow profile shown in Fig. 2. This profile is the result of an asymp-

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totic matching of the Hiemenz boundary layer solution with the potential flow field around a swept cylinder. But even with this modification, the numerically computed eigenvalues spread out over a large area of the lower complex half-plane without any apparent structure (similar to Fig. 3). We conclude that classical eigenvalue solvers are not a useful tool for studying the algebraic spectrum.

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To make progress, we apply the theory of wave packet pseudomodes [8, 9] to the stability problem. A wave packet pseudomode is an asympotitically good solution to an eigenvalue problem. These modes have the shape of a wave packet which is located at y = y∗ with the dominant wave number β = β∗ /ε (where ε is a small parameter). The eigenvalue λ associated to a wave packet pseudomode is approximated by the symbol f (y∗ , β∗ ) of the operator in (1). The symbol f corresponds formally to the eikonal equation [10] for the stability problem evaluated at y = y∗ (where we approximate the the leading order term S0 (y → y∗ ) by iβ∗ y/ε +C(y − y∗ )2 + h.o.t.). In addition, we require that the real part of C in the leading order term is negative. This translates directly into the twist condition

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The symbol and the twist condition define a map which connects wave packet pseudomodes to their (approximate) spectrum λ = f (y∗ , β∗ ).

2 Wave packet pseudomodes for the uniform stability problem The symbol of the uniform eigenvalue problem (1) is f (y∗ , β∗ ) = −iβ∗ 2 + ε V (y∗ )β∗ + iε 2 [V ′ (y∗ ) − γ 2 ] + γ W (y∗ ) (5) √ with ε = 1/ Re. Figure 5(a) shows the symbol curves λ = f (y∗ , −∞ . . . ∞) which illustrate the mapping from (y∗ , β∗ ) to λ . According to the twist condition (4), wave packet pseudomodes exist almost everywhere and for every wave number. Only in the boundary layer, wave packet pseudomodes exist only for β∗ < 0. For y∗ → ∞ we obtain a limiting symbol curve f (y∗ → ∞, β∗ ) = −i(β∗ 2 + ε 2 γ 2 ) + εβ∗ Re/ tan ϕ + γ . All other symbol curves (for y∗ in the free-stream) lie below this curve. 0

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Fig. 4 Symbol curves for Re = 1000, γ = 0.3, and ϕ = 45◦ : narrow parabolas for y∗ inside the boundary layer and wide parabolas in the free-stream (broken lines indicate that the twist condition is not satisfied). The labels ① ,② and ③ indicate the symbol values for the pseudomodes shown in Fig. 5 and 6.

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We can test the quality of the map λ = f (y∗ , β∗ ) by computing numerically the optimal pseudomode1 for the estimated eigenvalue λ . Figure 5 and 6 show optimal modes for three different (y∗ , β∗ ). They confirm that these pseudomodes are indeed wave packets and that their position and dominant wave number are well approximated by y∗ and β∗ . Some pseudomodes (see, for instance, pseudomode ① in Fig.6) feature two wave packets (one in the boundary layer and one in the freestream) which is directly related to the intersection between symbol curves from the boundary layer and from the free-stream (Fig. 4). 1

Optimal pseudomodes are found by computing the singular value decomposition of the resolvent of the discretized eigenvalue problem.

Wave packet pseudomodes upstream of a swept cylinder Fig. 5 Optimal modes for ① y∗ = 10, β∗ = 0.1, ② y∗ = 100, β∗ = 0.01, and ③ y∗ = 1000, β∗ = 0.0001 (the corresponding λ = f (y∗ , β∗ ) are indicated in Fig. 4). The peak locations and the dominant wave numbers of the optimal modes are ① y = 10.19, β = 0.1008/ε , ② y = 100.3, β = 0.009825/ε , and ③ y = 1001.3, β = 0.0001079/ε .

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The optimal modes follow roughly the predicted algebraic decay within the matching layer (Fig. 6). This trend persists until we enter the domain of the outer layer where the amplitudes of the modes decay exponentially fast. Pseudomode ③ shows that we may even have modal solutions which grow algebraically fast within the matching layer. This indicates that the spectrum of the algebraic modes upstream of a swept cylinder is not limited by (3) but rather by the limiting symbol curve for y∗ → ∞ (Fig. 7).

Fig. 7 Shape of the continuous spectrum for algebraic modes upstream of a swept cylinder (Re = 1000, γ = 0.3, ϕ = 45◦ ) as predicted by the limiting symbol curve. (The broken line shows the upper limit according to (3).)

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For y∗ in the free-stream, the resolvent norm k[L − λ (y∗ , β∗ )I]−1 k of the discretized eigenvalue problem tends toward infinity (given that the twist condition is satisfied). This indicates that these pseudomodes are as good as actual eigenmodes and that the corresponding symbol curves are as good as actual eigenvalues.

3 Conclusions Whereas classical eigenvalue solvers fail to explain the structure of the algebraic eigenvalue spectrum, the theory of wave packet pseudomodes lays out a clear structure of the linear stability properties of the flow field upstream of a swept cylinder. The symbol f (y∗ , β∗ ) and the twist condition (4) define a map between the (estimated) eigenvalue λ and the position y∗ and wave number β∗ of a wave packet pseudomode. The extension from the classical swept Hiemenz boundary layer to a baseflow which includes the parallel flow upstream of a swept cylinder gave us new insight into eigenvalue spectrum of the algebraic modes and revealed how these modes connect with the outer flow. We found that the modes decay algebraically fast only within the matching layer (stagnation point flow). As soon as the base flow goes over into a parallel flow the modes decay exponentially fast. Therefore, the boundary conditions at y → ∞ can also be satisfied for modes which grow algebraically within the matching layer (e.g. mode ③ ). This connection of algebraically growing modes with the outer flow supports a result by Danak & Stuart [11] who studied the stability of stagnation point flow in connection with external disturbances. As a direct consequence of this result, there are now strong indications that the algebraic modes form a continous spectrum (Fig.7) which is limited from above by a parabola defined by the limiting symbol curve.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

P. Hall, M. Malik, D. Poll, Proc. R. Soc. Lond. A 395, 229 (1984) P. Hall, M.R. Malik, J. Fluid Mech. 163, 257 (1986) R.S. Lin, M.R. Malik, J. Fluid Mech. 311, 239 (1996) D. Obrist, P.J. Schmid, J. Fluid Mech. 493, 1 (2003) D. Obrist, P.J. Schmid, J. Fluid Mech. 493, 31 (2003) A. Guégan, P. Schmid, P. Huerre, J. Fluid Mech. 603, 179 (2008). DOI 10.1017/S0022112008001067 V. Theofilis, A. Fedorov, D. Obrist, U.C. Dallmann, J. Fluid Mech. 487, 271 (2003) L.N. Trefethen, M. Embree, Spectra and pseudospectra: The behavior of nonnormal matrices and operators (Princeton Univ. Press, 2005) L.N. Trefethen, Proc. Roy. Soc. A 461(2062), 3099 (2005) C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw–Hill, 1978) M. Dhanak, J. Stuart, Phil. Trans. R. Soc. Lond. A 352, 443 (1995)

Bypass Transition prediction using a model based on transient growth theory Olivier Vermeersch and Daniel Arnal

Abstract A simplified model describing the dynamics of the Klebanoff modes inside a laminar boundary layer is presented. This model is coupled with a criterion in order to estimate the bypass transition location induced by the amplification of the Klebanoff modes. Applications to different experimental data are described. Key words: Transition, Bypass, Klebanoff modes, Streaks, Transient growth

1 Introduction In many cases, like for instance boundary layers subjected to significant free stream turbulence level, transition occurs at lower Reynolds numbers than those predicted by the classical linear stability theory. This suggests that another amplification process may exist. This process, called ’transient growth’, results from the nonnormality of the eigenfunctions, which allows interactions and can lead to a significant rise of perturbations energy. This phenomenon was called ”Lift-up effect” by Landahl. A longitudinal vortex superimposed to the boundary layer shear pushes up low speed particles from the wall to the top of the shear layer, and pulls down high speed particles toward the wall, leading to a spanwise alternation of backward and streamwise jet streaky structures called Klebanoff modes. An early laminar turbulent transition can be triggered if the energy of the Klebanoff modes significantly grows : this is the so-called Bypass transition, meaning that the classical process, driven by the TS waves, has been short-circuited. The main objective of this paper is to present a new simplified model describing the

Olivier Vermeersch and Daniel Arnal Department Modelling Aerodynamics and Energetics, Transition and instability Group. ONERA, 2, avenue Edouard Belin, B.P. 4025 31055 TOULOUSE Cedex 4, FRANCE P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_48, © Springer Science+Business Media B.V. 2010

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streaks dynamic. The model will be applied to predict transition for boundary layers subjected to free high stream turbulence.

2 Governing equations If the free stream turbulence (FST) level is high, the influence of the streaks on the mean flow cannot be neglected even in the laminar zone. Instantaneous quantities are split into a mean part and a fluctuating one, and in order to make the equation dimensionless, typical length and velocity scales are introduced. Klebanoff modes consist in streaky streamwise elongated structures : therefore, a typical scale of the geometry L (for instance the length of the flat plate) is used to normalize the streamwise coordinate. In the wall normal direction, a typical boundary layer thickness √ δ = L/ ReL is applied to characterize the diffusion process. These assumptions for two-dimensional incompressible boundary layers lead to the following system of equations :  ∂ U˜ ∂ V˜   + =0    ∂x ∂y   2 ˜ ′ ′ ′ ′ ′ ′ ˜ ˜    U˜ ∂ U + V˜ ∂ V = Ue dUe + ν ∂ U + γ ∂ hu v it + (1 − γ ) ∂ hu u ist + ∂ hu v ist ∂x ∂y dx ∂ yy ∂y ∂x ∂y | {z } | {z }    Turbulence Transient growth   2 ′  ′ ∂ Uu ˜ ′  ˜ ∂u ∂ u′ ∂ u ∂ 2 u′  ′ ∂U  ˜  + +V +v =ν + ∂t ∂x ∂y ∂y ∂ yy ∂ zz (1) It is important to distinguish the two mechanisms implicitly included in these equations. In the laminar zone, where the intermittency factor γ is equal to zero, the terms multiplied by (1 − γ ) describes the transient growth phenomenon, i.e. the Klebanoff modes amplification. The other terms, multiplied by γ , traditionally appear considering turbulent equations. They numerically interfere when γ stops to be zero, i.e. when the turbulent spots appear, and within the fully turbulent region when γ is equal to one ; the apparent turbulent stress < u′ v′ >t is computed with a classical mixing length model. In this study, the relationship for the intermittency function proposed by Arnal and al [1] has been used. The computation of the transition location is based on a criterion established by Biau [2] : −ρ · hu′ v′ i st max (2) =C ∀y µ ∂∂Uy From a physical point of view, this relationship, derived from the Van Driest and Blumer’s criterion [9], expresses the fact that transition occurs only when the ratio between the driving term of streak formation hu′ v′ ist and the dissipative one µ (∂ U/∂ y) reaches a given threshold.

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In order to close the system (1) in the laminar region, the wall normal velocity fluctuation has to be modelled. This aims at expressing the ’Lift-up’ effect in accordance with the fact that a wall normal velocity perturbation in a shear flow brings about emergence and amplification of streamwise velocity fluctuations.

3 Wall normal velocity fluctuation modelling Recent experimental data [3, 5] and DNS [6] exhibit a monotonic profile for v′ from the wall to the boundary layer outer edge. From these considerations Biau [2] proposed the function g to represent the normalized wall normal velocity fluctuation :

 

y2 e−α ·y if y ≤ δ99 g(y) = maxy |y2 e−α ·y |  cst if y > δ99

 g(y = 0) = 0 (No slip condition)          ∂g = 0 (Continuity equation) ∂ y y=0       2 ∂g   ⇒ = 0 (Derivability at the border of the BL) α = δ99 ∂ y y=δ99

Besides, Westin has shown that for boundary layers subjected to an isolated disturbance [10], the streak amplitude is decreasing in contrast to the growing u′ perturbations in the FST case [11]. He concluded that the downstream development of the streaks depends on a large extent on the continuous forcing provided by the free stream turbulence all along the boundary layer edge and is not restricted to the vicinity of the leading edge. This phenomenon is taken into account by assuming that at the edge of the boundary layer, the v′ amplitude is directly proportional to the local free stream turbulence level Tue : v′ = A × g(y) × (Ue × Tue )

(3)

4 Results The parameters A in (3) and C in (2) have to be determined. The calibration has been performed from the T 3A case of Roach’s experiments [8] : mean flow quantities and fluctuations have been measured by a hot-wire technique inside a boundary layer developing on a flat plate subjected to grid turbulence, such as at the leading edge Tu0 = 3%, with an external constant velocity Ue = 5.3 [m/s]. The constant A has been fixed in such a way that the numerical streak amplitude matches the measurements (Fig 1(a)). The corresponding streaks profiles are plotted in figure 1(b). The numerical curve of the momentum Reynolds number (figure 1(c)) shows that taking into account the Klebanoff modes effect on the mean flow improves the results. The

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transition threshold has also been calibrated on the T 3A case so that the numerical transition position corresponds to the experimental one. The model has first been

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U e [m/s]

5.5

0.004

0.05 0.002

4

0.5

X [m]

(a) Ue

1

0.5

X [m]

1

0

(b) Streak Amplitude

0.5

X [m]

1

(c) C f

Fig. 2 Influence of the pressure gradient on the bypass transition location. Comparison between the T 3A and T 3C3 cases.

applied to the T 3C3 case which is characterized by an initial accelerated flow followed by a deceleration (Figure 2(a)). The free stream turbulence level at the leading edge is the same that in the T 3A case : Tu0 = 3%. The figure 2(b) clearly shows that a favorable pressure gradient has a deep stabilizing effect. Thus, at x = 0.4 [m], which corresponds to the T 3A transition location, the amplitude of the streaks is only 7% of the external velocity whereas it reaches 13% for the zero-pressure gradient T 3A case. Besides, we can see that, as soon as the external flow starts to be decelerated, the streak amplitude increases rapidly and triggers transition. This is illustrated by the evolution of the skin friction coefficient, see figure 2(c). This figure also shows that the intermittency function γ gives a rather good description of the transition region. The propagation of the streaks inside the boundary layer is also influenced by the properties of the external turbulence. When the external free stream turbulence level increases, the location of the bypass transition moves toward the

Bypass Transition prediction using a model based on transient growth theory

305

leading edge (comparison between T 3B case and T 3A case of the table 1). Moreover, Jonàsˇ’experimental data [7] , have highlighted the fact that Klebanoff modes propagation depended not only on the level but also on the scales of the free stream turbulence. The dissipative length scale is defined by [4] :

Le =

3/2 − u′2

Ue ·

e d u′2

(4)

e

dx

A free stream turbulent flow with a large length scale has a smaller rate of dissipation; and thus, decays less rapidly in the flow direction (see figure 3(a)). The present model using the local turbulence level in the formulation (3), in agreement with the measurements, takes into account the influence of the dissipation parameter : the onset of bypass transition moves upstream for decreasing dissipative length scales. The present simulation has been applied to several cases of Roach experiments in

L=2.2 [mm]

0.006

L=3.8 [mm]

0.024

L=5.9 [mm]

0.022 0.02

0.004

Cf

Tu

0.018 0.016 0.002

0.014 0.012 0.01

0

0

0.5

1

X [m]

(a) Tue

1.5

0

0.5

1

X [m]

1.5

(b) C f

Fig. 3 Influence of the dissipation length scale Le . ( Symbols : Jonàsˇ ’measurements)

order to compare the numerical transition location with the measured one. The first two cases T 3A and T 3B concern zero pressure gradient flows with respective inlet turbulence levels Tu0 = 3% and 6%. The five other cases are characterized by a favorable negative pressure gradient in the first part of the plate. These cases aim at roughly simulate flow over turbomachinery blades. The T 3C1 case is highly perturbed with Tu0 = 6.6%. For the other cases Tu0 = 3% and the inlet velocity is gradually reduced to delay the abscissa of transition. Results are summarized in the table 1.

5 Conclusion A simplified model based on the resolution of a transport equation for the Klebanoff modes, and aiming at predicting the bypass transition location has been presented.

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O. Vermeersch, D. Arnal

Table 1 Comparison between experimental and numerical results. Case T 3A T 3B T 3C1 T 3C2 T 3C3 T 3C4 T 3C5

U∞ [m/s] 5.4 9.4 5.9 5 3.7 1.2 8.4

dP/dx Tu [%] 0 3 0 6 6= 0 6.6 6= 0 3 6= 0 3 6= 0 3 6= 0 3

xT Exp. Num. 0.38 0.42 0.10 0.07 0.18 0.10 0.80 0.7 1.10 1.1 1.25 1.4 (Separation) 0.32 0.22

This model is not focused on describing all the physical mechanisms which trigger transition; in particular, the linearity of the two transport equations does not allow capturing the secondary inflectional instability which leads to the breakdown into turbulent spots. Nonetheless, the dynamics of Klebanoff modes within the laminar zone has been numerically determined and used to estimate the transition location. The present model gives coherent results for boundary layers subjected to high external turbulence levels, in agreement with experimental data. The model has been theoretically developed for compressible boundary layers but at the time being, it has just been tested in the limit of incompressible ones for which several experimental data exist. The logical following task is to extend this study to higher Mach number flows for which transient growth still exists. In the same way, the same modelling technique will be applied to study roughness-induced transition.

References D. Arnal, La Recherche Arospatiale 4 (1988) D. Biau, D. Arnal, O. Vermeersch, Aerospace Science and Technology 11 (2007) J.H.M. Fransson, K.J.A. Westin, Exp. Fluids 32 (2002) P.E. Hancock and P. Bradshaw, J. Fluid Mech 205 (1989) A. Inasawa, F. Lundell, M. Matsubara, Y. Kohama, P.H. Alfredsson, Exp. Fluids 34 (2003) R.G. Jacobs, P.A. Durbin, J. Fluid Mech 428 (2001) P. Jonàsˇ, O. Mazur, V. Uruba, Eur. J. Mech B-Fluids 19 (2000) P.E. Roach, D.H. Brierley Numerical Simulation of Unsteady Flows and Transition to Turbulence, (Cambridge University Press), pp. 319-347 (1990) 9. E.R. Van Driest, C.B. Blumer, AIAA journal 1 (1963) 10. K.J.A. Westin, A.A. Bakchinov, V.V. Kozlov, P.H. Alfredsson , Eur. J. Mech. B/Fluids 17 (1998) 11. K.J.A. Westin, A.V. Boiko, B.G.B. Klingmann , J. Fluid Mech 281 (1994) 1. 2. 3. 4. 5. 6. 7. 8.

Flow in a Slowly Divergent Pipe Section Jorge Peixinho

Abstract The results of an experimental investigation of a constant mass flux flow in a slowly diverging cylindrical pipe expansion section or diffuser flow are presented. At low flow rate, no recirculation region is detected. For medium flow rate, a stable recirculation region is observed. For large flow rate, the recirculation region or axisymmetric state becomes time dependent and with a further increase of the flow rate, a subcritical transition for localised transient turbulence arises. Additional experiments on the quenching of turbulence by reducing the Reynolds number and observing the decay of disordered motion allowed to build a stability diagram.

1 Introduction Flow transition in straight pipes is well documented and quantitative comparisons between experiments and numerical studies are now possible. The flow in diffusers or slowly diverging pipes (i.e. cylindrical pipes of slowly increasing diameter along the pipe axis), as sketched in figure 1, are not well documented despite some fundamental and practical features. This geometry arises in the context of microfluidic when transferring low viscosity liquid using slowly diverging pipettes and in the context of physiological flows. The general two-dimmensional problem of flow stability between two plane walls meeting at a source point with an angle is known as the Jeffery-Hamel problem. There are several theoretical and numerical developments where bifurcations are found by Sobey and Drazin [1], Tutty [2], Kerswell et al. [3] and others. All these works indicate a rich and diverse set of solutions even for small diverging angles. Thus, there is the important question of solution selection. Recently, Putkaradze and

Jorge Peixinho Levich Institute, City College of City University of New York, New York, NY 10031, USA e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_49, © Springer Science+Business Media B.V. 2010

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J. Peixinho

d

D L

Fig. 1 Schematic of a diverging tube (drawn to scale). L is the length of the diverging section, α is the total diverging angle, d is the inlet diameter and D is the outlet diameter

Vorobieff [4] observed, using particle image velocimetry measurements, the multiple vortex flow régime [3]. In the limit case of two-dimensional channel sudden 1:3 expansion flow, it was shown by Fearn et al. [5] that the asymmetry arises at a critical Reynolds number, Re, based on the inlet height, through a pitchfork bifurcation at Re ≃ 80. They were able to measure the degree of asymmetry due to small imperfections of the experimental apparatus and compared it with numerical results. The present work considers the case of axisymmetric circular pipe expansion. The limit case is an abrupt circular pipe 1:2 expansion was studied in a modern investigation using high resolution magnetic resonance imaging by Mullin et al. [6] and a bifurcation point at Re = 1, 139 ± 10 where the recirculating flow becomes axisymmetric and moves towards a sidewall was uncovered. Numerical predictions of the pressure drop of axisymmetric diffusers laminar flow are available (see e.g. Rosa and Pinho [7] for α from 0◦ to 90◦ and D/d of 1.5 and 2). The stability of the axisymmetric Jeffery-Hamel flow (α = 3◦ , L/d = 120 and D/d ≈ 7.3) has been investigated in a numerical work by Sahu and Govindarajan [8]. They found the flow is linearly unstable from Re = 150. This is a surprising result since flow in a straight pipe is believed to be linearly stable. In another axisymmetric numerical work, Sherwin and Blackburn [9] considered a long (≈ 70D) straight pipe with a smooth axisymmetric constriction, so that the velocity profile at the inlet of the divergent section (α ≈ 45◦ , L = d and D/d = 2) is almost flat, the steady flow exhibits a laminar recirculation region and the subcritical transition to turbulence occurs at Re = 722. The experimental setup is presented in §2 prior results and a summary stability diagram in §3.

2 Experimental Setup A schematic of the experimental setup is given in figure 2(a). It consists of a vertical pipe. The flow is controlled using a syringe pump (TSE Systems Model 540230) together with 100 ml glass syringes. The device pulls the fluid at a constant mass flux along the pipe. Hence, even if the motion becomes turbulent, the mass flux through

Flow in a Slowly Divergent Pipe Section

309 Reservoir

(b)

(c)

(d )

(a) d

L

D

Fig. 2 (a) Schematic of the experimental setup (drawn to scale). The piston moves at a controlled speed and the fluid is pulled through the pipe at constant mass flux. Flow is from top to bottom. (b − d) Flow visualisation photographs: (b) Laminar flow at Re = 150 (c) Relaminarisation of a turbulent localised puff at Re = 750 and (d) Turbulent localized puff at Re = 1, 100.

Piston

the pipe is unaffected so that Re remains constant. Also, the pressure gradient is mainly the hydrostatic pressure above the pipe. Two slowly diverging acrylic pipes, with α = 4 ◦ (or π /30 rad) and 6 ◦ (or π /45 rad) were used. The 4 ◦ divergent has an inlet diameter is d = 2.14 ± 0.1 mm and the outlet diameter is D = 21.4 ± 0.1 mm over a length of L = 275.8 ± 0.1 mm. The 6◦ divergent has a constriction prior the diverging section. The inlet diameter is d = 3 ± 0.1 mm and the outlet diameter is D = 21.4 ± 0.2 mm over L = 178.1 ± 0.1 mm. Downstream the divergent, the pipe extends over 270 mm with an acrylic straight pipe. Flow visualisation was performed using Mearlmaid Pearlessence into the deionised water and a vertical light sheet through the axes to illuminate the flow. Observation were made in the direction orthogonal to the flow using a camera (MotionPro). Examples of flow visualisation are shown in figure 2(b − d). The Reynolds number is build on d: Re = Ud/ν , where U is the mean flow, d is the inlet diameter and ν is the kinematic viscosity of the fluid. The other parameters of the pipe are the expansion ration: E = D/d between the outlet and intlet diameters and the non-dimensional length of the diverging section: β = L/d. For the 4◦ and 6◦ pipes, (E, β ) = (10, 129) and (7, 59) respectively.

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J. Peixinho

3 Results and Discussions

Fig. 3 Centreline Uc /Ucd velocity versus x/d

First, figure 3 presents the centreline velocity versus the axial position along the 4◦ pipe assuming the velocity profile is parabolic all along the pipe. x = 0 is the inlet of the divergent. Within the expansion section the velocity scales as 1/x2 . A centreline fluid particle experiences a rapid deceleration. In experiments, for Re < 400, no recirculation are detected. The inset in figure 3 is a schematic of the streamlines and velocity profiles. For Re > 400, a the recirculation region appears at the outlet corner of the divergent and grows upstream and downstream. Note the recirculation region may be of a toroidal shape. There are many works on laminar separation recirculation region on flat plate boundary layer [10], however only Dovgal et al. [11] studied an axisymmetric recirculation region around an axisymmetric object and described its stability depending on the initial perturbation location and frequency. The boundary conditions of the problem assume a fully developed Poiseuille velocity profile at the inlet and at the outlet. A question which may be asked about the effect of the recirculation region is: does it behave as a amplifier or resonator with regard to perturbation? These ideas were discussed by Marquet et al. [12] for a recirculation flow in a curved channel. In the diverging pipe, our observation indicates the recirculation region grows with Re. For instance, for Re = 550, the recirculation region becomes asymmetric and the reconnection position can be 200d from 0. This shows how far downstream the recirculation region may extend. Moreover, the intensity of the recirculation is observed to be one order of magnitude smaller than the centerline flow.

Flow in a Slowly Divergent Pipe Section

No Separation 400

311

Axisymmetric Bubble 500

600

Fig. 4 Schematic of the flow patterns for the

Unstable Bubble 700

4◦

800

Localized Turbulence 900

1000

Re

diverging pipe expansion as a function of Re

With a further increase of the flow rate: Re > 720, the recirculation region could give raise to localised turbulent patches. These localised turbulent patches have similarities with the localised puffs observed in cylindrical straight pipe flow transition. Indeed, they have a definite lengths for a given Re and a decaying wave at the front since the downstream Re is too small to sustain turbulence. These localised turbulent patches are subcritical and depend on the accuracy of the flow rate. In order the estimate this stability domain, relaminarisation experiments [13] were performed. In practice, a localised turbulent patch is first created and after some time the flow rate would be reduce to observe the relaminarisation. In summary, the subcritical domain for appearance of turbulent patches is from Re = 720 to 860 in the 4◦ diverging pipe as shown in figure 4. A quantitatively similar diagram is obtain for the 6◦ pipe.

4 Conclusions An experimental investigation on the flow in slowly diverging pipe section of 4 and 6 ◦ is presented. At low flow rate, no recirculation region is observed. For larger flow rate, stable laminar recirculation region is observed and expands downstream the divergent. With further increase of the flow rate, a domain of unstable turbulent patches are uncovered. Using slowing-down of the flow rate experiments, a bifurcation diagrams is obtained. It is believed this results can be included in a large parameter space depending on α , E and β . Solution space may include secondary separation as observed by Durst et al. [14]. Note that E and α can be combined in only one additional parameter E ∗ = 2β tan (α /2) − 1.

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J. Peixinho

Finally, it is suggested this geometry can be the place for the detail study of a fixed localized puff. Indeed, the study of puffs properties is usually made complicate because its advective nature. Here the puff is fixed and measuring velocity statistics and lifetimes for different Re could be easily done.

Acknowledgements The experiments were conducted in the Fluid Engineering Laboratory at the Department of Mechanical Engineering of the University of Tokyo, Japan. The financial support of the Japanese Society for the Promotion of Science and the hospitality of the Fluid Engineering Laboratory are gratefully acknowledged as well as encouragement from Rich Kerswell, Jeff Morris and Tom Mullin.

References 1. Sobey I J and Drazin P G (1986) Bifurcations of two-dimensional channel flows, J Fluid Mech 171 263-287 2. Tutty O R (1996) Nonlinear development of flow in channels with non-parallel walls, J Fluid Mech 326 265-284 3. Kerswell R R, Tutty O R and Drazin P G (2004) Steady nonlinear waves in diverging channel flow, J Fluid Mech 501 231-250 4. Putkaradze V and Vorobieff P (2006) Instabilities, bifurcations, and multiple solutions in expanding channel flows, Phys Rev Lett 97 144502 5. Fearn R M, Mullin T and Cliffe K A (1990) Nonlinear flow phenomena in a symmetric sudden expansion, J Fluid Mech 221 595-608 6. Mullin T, Seddon J R T, Mantle M D and Sederman A J (2009) Bifurcation phenomena in the flow through a sudden expantion in a circular pipe, Phys Fluids 21 014110 7. Rosa S and Pinho F T (2006) Pressure drop coefficient of laminar Newtonian flow in axisymmetric diffusers, Int J Heat Fluid Flow 27 319-328 8. Sahu K C and Govindarajan R (2005) Stability of flow through a slowly diverging pipe, J Fluid Mech 531 325-334 9. Sherwin S J and Blackburn H M (2005) Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows, J Fluid Mech 533 297-327 10. Gaster M (2004) Laminar separation recirculation regions. In Proc. Sixth IUTAM Symp. Laminar–Turbulent Transition, R Govindarajan (ed) Springer 1-14 11. Dovgal A V, Kozlov V V and Michalke A (1996) On disturbances excited by a point source in an axisymmetric laminar separation recirculation region, Eur J Mech, B/Fluids 15 (5) 651-664 12. Marquet O, Sipp D, Chomaz J M and Jacquin L (2008) Amplifier and resonator dynamics of a low-Reynolds-number recirculation recirculation region in a global framework, J Fluid Mech 605 429-443 13. Peixinho J and Mullin T (2006) Decay of turbulence in pipe flow, Phys Rev Lett 96 094501 14. Durst F, Maxworthy T and Pereira J C F (1989) Piston-driven, unsteady separation at a sudden expansion in a tube: Flow visualization and LDA measurements, Phys Fluids 1(7)1249-1260

In-flight experiments on active TS-wave control on a 2D-laminar wing glove Inken Peltzer, Kai Wicke, Andreas Pätzold, and Wolfgang Nitsche

Abstract In-flight measurements to delay laminar-turbulent transition by means of active Tollmien-Schlichting (TS) wave cancellation were carried out on a 2Dlaminar wing glove for a sailplane. The sensor-actuator system attached to the wing glove consisted of an array of surface hot-wire reference sensors to detect oncoming TS-waves upstream of a membrane actuator and surface hot-wire error sensors downstream of the actuator. The method applied was based on the dampening of naturally occurring instabilities through superimposition of a counter wave, which was calculated by a fast digital signal processor (DSP), using a closed loop feed-forward control algorithm. The flight experiments validated this system under varying atmospheric conditions successfully. Further attention was directed to the dampening of instabilities in the span-wise direction.

1 Introduction The paper describes in-flight experiments to investigate the two-dimensional laminar boundary layer, particularly one active method to keep the boundary layer laminar for a longer distance. The basic idea of this active method is the cancellation of TS-instabilities by superimposing proper anti-waves. The first successful investigations were done in the early 1980s by Milling [6] and Liepmann [5]. A large number of experimental and numerical investigations dedicated to this topic followed [4, 8, 3]. The experiments were carried out primarily using artificial disturbances. Experiments to actively control natural TS-instabilities were less frequent but extensively investigated in wind tunnel tests [10, 1, 2]. Since the growth of TS-instabilities depends strongly on the ambient flow conditions, this experimental research aims to investigate the transition under flight conditions in free atmosphere. In contrast to wind tunnel experiments the TS-waves in flight tests can also propagate in the span-wise direction. Therefore a 3D-sensor-actuator system is mandatory [7].

Inken Peltzer et al. Berlin Institute of Technology, Institute of Aeronautics and Astronautics Secr. F2, Marchstr.12-14, 10587 Berlin, Germany, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_50, © Springer Science+Business Media B.V. 2010

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I. Peltzer, K. Wicke, A. Pätzold, W. Nitsche

2 Experimental setup A laminar wing glove, which was developed for the Grob G103 TWIN II two-seater sailplane at the Institute for Aeronautics and Astronautics, Berlin Institute of Technology, is used for these in-flight measurements. The glove has a 2D central part with a 1.0m span and a chord length of 1.22m with an exchangeable wing segment. That segment has a width of 0.6m and can be equipped with various sensors and actuators depending on the specific measuring task. The measuring equipment, such as the constant temperature anemometer, voltage amplifier, digital signal processor (DSP), the data acquisition system and pressure transducers are located externally in an equipment support box beneath the glove. A Prandtl probe, which is used to obtain the free stream velocity, is mounted below the glove. A thermocouple and a manometer are used to measure the air temperature and absolute pressure, respectively. These parameters are continuously recorded to obtain the free stream boundary conditions. The experiments were carried out at flight velocities from 21m/s to 27m/s. For damping the TS-instabilities, sensors are needed to measure the instabilities in the very early amplification stage. That means very sensitive sensors with a high cut-off frequency have to be used. Furthermore, span-wise information of growth and phase of the instabilities are necessary. Therefore a surface hot-wire array was designed that included the reference as well as the error sensors. The layout was produced by standard photo-etching techniques on a flexible circuit board with a pocket to integrate the actuator. The exact positions of the sensors and actuators are shown in Fig. 1. Furthermore, an actuator capable of generating a proper counter wave in the frequency range of the TS-instabilities with a minimal time delay is needed. The response of the actuator should be linear. Sturzebecher et al. [9] carried out experiments on active wave control with two different actuator concepts (slot and membrane actuators). The best damping rates were achieved with membrane actuators. In-flight experiments showed the unstable TS-frequencies on the glove between 500Hz and 1000Hz. A further challenge was the restricted installation height of 15mm within the glove. These were the parameters to develop a qualified actuator.

Fig. 1 Positions of sensors and actuator

In-flight experiments on active TS-wave control

315

The actuator was equipped with a pre-stressed and flush-mounted membrane over a span-wise slot of 3mm, which was 25% of the average TS-wave length. A plunger was fixed between the membrane and the surfaces of the speakers. The modified speakers were used to move the plunger and the membrane respectively (Fig. 2).

Fig. 2 View of actuator and sectional view with plunger and speakers

3 Control algorithm Natural TS-wave packets occur randomly. Therefore, high attenuation can only be achieved by a closed loop feed-forward control algorithm. The flow chart of the used control algorithm is shown in Fig. 3 left. The reference sensor upstream of the actuator measures the oncoming TS-instabilities (x). The actuator signal (y) is calculated directly from this signal by a convolution with the primary FIR-filter (finite impulse response, FIR 1). The primary transfer function is continuously adapted in order to minimize the error signal (e) measured downstream of the actuator. The path between the actuator and error sensor has to be adapted beforehand in a pre-adaptation calculation. This secondary path (FIR 2) emulates the actuator behaviour and wave superimposition between the actuator and the error sensor. The primary FIR-filter (FIR1) is continuously changed following the steepest descent of the quadratic error signal (e2 ). The gradient method used for the filter update is based on the least mean square (LMS) algorithm. The modified filtered-x-LMS algorithm procedure considers the secondary path to derive a filtered reference signal (y2). A correlated error signal (r) is derived from the filtered reference and error signal that is applied to perform the LMS-algorithm. The internal transfer function (FIR filter) that is used is a digital filter which is linear and non-recursive. The output signal is the continuously calculated dot product of the continuously shifting input signal and the filter coefficients. The FIR filter is able to precisely model the frequency selective amplification and convection of TS-instabilities. The FIR filter is described as follows: M−1

y(n) =

∑ hi · x(n − i)

i=0

A digital signal processor (DSP, dspace DS 1103) was employed to achieve realtime performance of the control algorithm. This processor has direct access to a 16-channel A/D converter with a sampling rate of 1MHz and an 8-channel D/A converter. The analogue input and output channels have a resolution of 16-bits. This control algorithm was operated autonomously for several pairs of reference sensor

316


(RS) - actuator - error sensor (ES). In the in-flight experiments, four autonomous control loops were performed. The span-wise distance of the entire system was about 100mm. A principle scheme of the model is shown in Fig 3 right.

Fig. 3 Flow chart of the FX-LMS algorithm (left) and scheme of the autonomous control of spanwise arranged sensor (RS)-actuator-sensor (ES) pairs (right)

4 Results For comparison of cases without and with active wave control (AWC) the flight velocity as well as the spectra of the reference sensor had to agree to guarantee the same boundary conditions. Therefore, Fig. 4 shows power spectra of reference and error sensors with and without AWC as one result at a flight velocity of 26.5m/s. It can be seen that the reduction of the amplitudes is quite different at the different span-wise control systems. Since the autonomous model was used, each actuator generates an independent signal at each span-wise position. Nevertheless, the amplitudes of the higher harmonic frequencies (1000Hz - 1600Hz) are damped almost completely and the best attenuation of the fundamental TS-instabilities of about 12 dB is reached. This corresponds to a local amplitude reduction of the fundamental frequencies up to 75%. The higher harmonic frequencies are damped almost completely. With that control scheme a local TS-amplitude reduction on average of 68% (10dB) at one span-wise position was reached for all mentioned velocities. It was reproducible for all investigated cases.

Fig. 4 Power spectra of the reference (left) and error signal (right) without and with active wave autonomous control algorithm at a flight velocity of 26.5m/s

In-flight experiments on active TS-wave control

317

Fig. 5 Time traces of the reference, actuator and error signal without (left) and with active wave autonomous control algorithm (right) at a flight velocity of 26.5m/s

Fig. 5 shows time traces of measured reference signals, the actuator signal and the error signal without and with the control activated. The time traces of the reference sensors show the typical TS-wave packets with relative small amplitudes with respect to the amplitudes of the error sensor. The actuator signal is only a flat line in the left diagram, meaning the control is off. When turning on the control an actuator signal is generated; this is to be seen in the right diagram. Furthermore, the amplitudes of the error sensor are decreased significantly in correlation to the signal amplitude of the error sensor without control (left graph). The amplitude of the actuator is not comparable to the sensor amplitudes. It only documents the generated signal, which is given to the actuator and could be used to analyse the time and phase shift. By looking at the error sensor signals of a span-wise row, the span-wise development of the amplitudes and phase shift of the TS-wave packet can be investigated. Fig. 6, left side, shows a contour plot of such a span-wise row without control. A single, locally limited TS-wave packet can be seen clearly. It is obvious that the amplification varies in the span-wise direction. In spite of this variation, the phase shift of the adjacent sensors is very small. Nevertheless that explains that the damping rate differs in the span-wise direction. The measured amplitudes on the reference

Fig. 6 Contour plots of the spanwise row of error sensors without control (left) and with active wave autonomous control (right) at a flight velocity of 26.5m/s

318


sensors are so small that actually no damping is reasonable at that stage of transition. Therefore the generated actuator signal is very low and the effect in the spectra is low (see again Fig. 4). The right side of the figure shows the contour plot of the error sensors with AWC. The reduction of the amplitudes can be seen clearly. Furthermore the span-wise variation of the amplitudes is less. That could be an advantage for a second sensor-actuator system (cascaded system) behind the first system as it has already been implemented in wind tunnel experiments by Sturzebecher et al. [9].

5 Conclusions In-flight experiments to reduce the amplitudes of TS-instabilities were successfully carried out on a 2D-laminar wing glove for a sailplane. The basic idea was the damping of the natural occurring TS-instabilities by superimposing them on a counter wave. Therefore, a sensor-actuator system was developed to measure the oncoming TS-waves and couple the counter wave into the boundary layer. Because of the random occurrence of locally limited TS-waves four sensor-actuator-sensor pairs were used and actuated autonomous with the FX-LMS algorithm. With this system a damping of the local amplitudes of the instabilities of up to 75% was reached, reproducibly. The span-wise variation of the TS-waves on the glove has an important effect on the possible damping rates. Therefore, the 3D sensor-actuator system with span-wise autonomous control algorithms should be implemented in the glove in a cascaded manner for further experiments.

References 1. M. Baumann, D. Sturzebecher, and W. Nitsche. Active control of TS-instabilities on an unswept wing. In H.F. Fasel and W.S. Saric, eds, Laminar-Turbulent Transition: IUTAMSymposium on Laminar-Turbulent Transition 1999 in Sedona USA, pp. 155–160. Springer Verlag, 2000. 2. M. Engert, A. Ptzold, R. Becker, and W. Nitsche. Active Cancellation of Tollmien-Schlichting Instabilities in Compressible Flows Using Closed-Loop Control. In D.M.; Lavoie P. Morrison, J.F.; Birch, eds, IUTAM Symposium on Flow Control and MEMS, vol. 7 IUTAM Bookseries, pp 319–331. Springer Verlag, 2008. 3. S. Grundmann and C. Tropea. Active canellation of artificially introduced TollmienSchlichting waves using plasma actuators. Exp. Fluids, 44(5):795–806, Mai 2008. 4. Y. Li and M. Gaster. Active control of boundary layer instabilities. J. Fluid Mech., 550:185– 205, 2006. 5. H.W. Liepmann, D.L. Brown, and D.M. Nosenchuck. Control of laminar instability waves using a new technique. J. Fluid Mech., 118:187–200, 1982. 6. R.W. Milling. Tollmien–Schlichting wave cancellation. Phys. Fluids, 24(5):979–981, 1981. 7. I. Peltzer. Comparative in-flight and wind tunnel investigation of the development of natural and controlled disturbances in the laminar boundary layer of an airfoil. Exp. Fluids, 44(6):961–972, 2008. 8. P. Pupator and W. Saric. Control of random disturbances in a boundary layer. AIAA Paper 89-1007, AIAA, 1989. 9. D. Sturzebecher and W. Nitsche. Active cancellation of Tollmien-Schlichting instabilities on a wing using multi-channel sensor actuator systems. Int. J. Heat Fluid Flow, 24:572–583, 2003. 10. A.S.W. Thomas. The control of boundary-layer transition using a wave superposition principle. J. Fluid Mech., 137:233–250, 1983.

Global nonlinear dynamics of thin aerofoil wakes Benoˆıt Pier and Nigel Peake

Abstract In the present investigation of thin aerofoil wakes we compare the global nonlinear dynamics, obtained by direct numerical simulations, to the associated local instability features, derived from linear stability analyses. A given configuration depends on two control parameters: the Reynolds number Re and the adverse pressure gradient m (with m < 0) prevailing at the aerofoil trailing edge. Global instability is found to occur for large enough Re and |m|; the naturally selected frequency is determined by the local absolute frequency prevailing at the trailing edge.

1 Introduction The global dynamics of spatially developing shear flows is known to closely depend on its local stability features. Previous linear stability analyses [11, 10] have established that thin aerofoil wakes display local absolute instability near the trailing edge. In the present investigation we revisit this flow and examine its fully nonlinear régime within the framework of nonlinear global mode theory [9, 4]. With x and y denoting streamwise and cross-stream coordinates respectively, we consider the semi-infinite flow domain (x > 0, −∞ < y < +∞) corresponding to the region downstream of the aerofoil trailing edge, where a symmetric velocity profile is imposed, represented by the double Falkner–Skan boundary layer solutions with negative pressure gradient m. The parameter m is varied from m = 0 (double-Blasius wake) to m = −0.09 (near flow separation); Reynolds numbers from Re = 50 up Benoˆıt PIER Laboratoire de mécanique des fluides et d’acoustique (CNRS — Université de Lyon) ´ ´ Ecole centrale de Lyon, 36 avenue Guy-de-Collongue, 69134 Ecully cedex, France. Nigel PEAKE Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OWA, UK. P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_51, © Springer Science+Business Media B.V. 2010

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to Re = 5000 are considered. The entire study is carried out for two-dimensional incompressible flows governed by the Navier–Stokes equations.

2 Global nonlinear dynamics obtained by direct numerical simulations Global flow dynamics are obtained by direct numerical simulation of the governing equations, implementing a method similar to that used in [8]. Time-integration is performed via a fractional step method of second-order accuracy in time. Spatial discretization combines fifth-order finite differences in the x-direction and Chebyshev collocation points in the y-direction. The streamwise grid is uniformly stretched in the downstream direction and the collocation points are mapped onto the entire cross-stream domain via an algebraic transformation. Non-reflecting boundary conditions [5] are used at the downstream boundary of the computational domain. At low values of Re the wake is globally stable and the system converges towards a steady y-symmetric solution of the Navier–Stokes equations. Above a critical Reynolds number, which depends on the value of the parameter m, global instability occurs, and time-dependent fluctuations develop in the wake. After a transient régime, a fully developed self-sustained downstream-propagating vortex-street is observed. Figure 1 shows snapshots of streamwise and cross-stream velocity fields of such a saturated régime obtained with Re = 5000 and m = −0.085. y (a) 20 10 0 x −10 −20 0 50 100 150 200 y (b) 20 10 0 x −10 −20 0 50 100 150 200 Fig. 1 Snapshot of fully developed nonlinear régime at Re = 5000 and m = −0.085. Equispaced isocontours of (a) streamwise and (b) cross-stream velocity fields.

The naturally selected vortex-shedding frequencies are obtained by recording time-series at different locations and computing the associated frequency spectra. These spectra display a wide range of harmonics but obtaining them at different spatial locations demonstrates that the entire flow is tuned to a single fundamental frequency. The dependence of the selected global frequency on both control parameters is given in figure 2. ¿From these curves it is concluded that, for given m, the global frequency is nearly independent of Re, except near onset. However, the

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global frequency significantly depends on m: the vortex shedding slows down when m is decreased and separation of the inlet velocity profile is approached. m = −0.082 ω 0.190

m = −0.084

0.185

m = −0.086

0.180

m = −0.088

0.175 0.170 0.165

m = −0.090 0

1000

2000

3000

4000

5000 Re

Fig. 2 Reynolds number dependence of the naturally selected global frequencies for different values of the parameter m.

3 Basic flow and local absolute frequencies This section investigates the unperturbed basic flow and its local stability features, covering the same parameter ranges as the previous section. The basic flow is, by definition, a steady solution of the Navier–Stokes equations. Basic flow fields may be computed by imposing a symmetry condition along the x-axis and considering only the domain y ≥ 0. At very large Reynolds numbers, however, this symmetry condition is not enough to stabilize the basic flow, and the selective frequency damping method [1] has therefore also been implemented. Thus, the basic flows are readily obtained by numerical integration of the governing equations, using the same code as in the previous section with only minor modifications. For given values of Re and m, local stability characteristics of the associated basic flow are then derived at a given streamwise station by freezing the x-coordinate and studying the equivalent parallel shear flow of corresponding velocity profile. Such an approach can be justified by rigorous asymptotic analyses based on the assumption of slow streamwise development of the basic flow. In the case of thin aerofoil wakes, this is certainly a reasonable assumption, except very close to the trailing edge; and even in the very near wake region, the assumption of weak spatial inhomogeneity only breaks down for small values of both Re and |m|. Local linear instability waves developing on these velocity profiles are then governed by the Orr–Sommerfeld equation, which yields the local linear dispersion relation ω = Ω (k; x; Re, m) (1)

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between the complex frequency ω and complex wavenumber k at streamwise station x for the basic flow characterized by Re and m. In the context of open shear flows, a crucial feature is the complex absolute frequency defined for parallel flows as the frequency observed at a fixed spatial location in the long-time linear response to an initial impulse. The local absolute frequency ω0 (x; Re, m) is derived in classical fashion by applying a pinch-point criterion [3, 2], equivalent to a vanishing group velocity condition

ω0 (x; Re, m) = Ω (k0 ; x; Re, m) with

∂Ω (k0 ; x; Re, m) = 0. ∂k

(2)

Figure 3 illustrates the streamwise evolution of the local absolute growth rate ω0,i and real absolute frequency ω0,r . It is seen that the largest absolute growth rate ω0,i occurs near the trailing edge at x = 0. The growth rate increases with (adverse) pressure gradient |m|, and the size of the absolutely unstable domain (where ω0,i > 0) increases with Reynolds number. All basic wake flows of this investigation are at least marginally absolutely unstable at x = 0. However, for moderate values of m, the extent of the absolutely unstable domain is rather limited. It is only for stronger adverse pressure gradients, close to detachment, that relatively strong absolute instability is obtained.

50 R e=

100

ω0,r 0.23 0.22 0.21 0.20 0.19 0.18 0.17 0

(a)

= Re

ω0,i 0.02 0.01 0.00 −0.01 −0.02 −0.03 −0.04

100

150

200 x

Re = 5 000 50 0

Re =

10 00

(b) 000 Re = 5

Re =

50

Re =

10 0

100

150

200

x 200

Fig. 3 Streamwise variation of (a) imaginary and (b) real parts of local absolute frequencies obtained with m = −0.085 and Re = 100, 200, 300, 400, 500, 700, 1000, 1500, . . . , 5000.

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4 Discussion The spatial structure of the global mode is the following. The upstream region, close to the trailing edge, is covered by small-amplitude perturbations that propagate downstream. While propagating downstream, their amplitude grows exponentially until nonlinear saturation occurs. Further downstream, a finite-amplitude wavetrain prevails, the amplitude and wavenumber of which slowly vary with downstream distance. In the far wake, the wave amplitude slowly decays while small-scale features may develop for high-Reynolds-number configurations. A comparison of the vortex shedding frequency measured by DNS and the real part of the absolute frequency ω0,r (x = 0), prevailing at the trailing edge, is shown in figure 4. This plot shows that the naturally selected frequency is in excellent agreement with the absolute frequency prevailing at the trailing-edge, thereby demonstrating that the global dynamics of thin aerofoil wakes is closely linked to the local absolute stability characteristics of the associated base flow. ω 0.20 0.19 0.18

ωg ω0,r (x = 0)

0.17 0.16 −0.090

−0.088

−0.086

−0.084

−0.082

−0.080

m

Fig. 4 Comparison of measured vortex shedding frequency ωg and trailing-edge absolute frequency ω0,r (x = 0).

A nonlinear saturation station xsat may be defined as the streamwise position where the envelope of the global mode reaches 90% of its maximum. Figure 5 shows how the thus computed xsat depends on Re for selected values of m. The saturation length appears to converge towards a constant value for Re → ∞ while it would diverge at the critical Reynolds number for onset of global instability. A somewhat intriguing result is also derived from this plot: the nonlinear saturation may occur downstream of the absolutely unstable domain. It has been verified that this observation does not depend on the length of the computational domain. Nevertheless, we are currently implementing a completely different numerical method to verify the robustness of this counterintuitive result. We are also studying the controllability of this flow by applying small-amplitude harmonic forcing near the trailing edge. Preliminary results indicate that the openloop control technique developed in a different context [6, 7] is capable of modifying the present flow.

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xsat m = −0.086

100

m = −0.084

m = −0.088

75

m = −0.090

50

0

1000

2000

3000

4000

5000 Re

Fig. 5 Variation of nonlinear saturation station with control parameters Re and m.

Computing resources from Fédération lyonnaise de calcul hautes performances (FLCHP) and Institut national de physique nucléaire et de physique des particules (IN2P3) are gratefully acknowledged.

References ˚ 1. Akervik, E., Brandt, L., Henningson, D.S., Hœpffner, J., Maxen, O., Schlatter, P.: Steady solutions of the Navier–Stokes equations by selective frequency damping. Phys. Fluids 18, 068102 (2006) 2. Bers, A.: Space-time evolution of plasma instabilities — absolute and convective. In: M. Rosenbluth, R. Sagdeev (eds.) Handbook of plasma physics, pp. 451–517. North–Holland, Amsterdam (1983) 3. Briggs, R.J.: Electron-stream interaction with plasmas. M.I.T. Press, Cambridge, Mass. (1964) 4. Chomaz, J.M.: Fully nonlinear dynamics of parallel wakes. J. Fluid Mech. 495, 57–75 (2003) 5. Jin, G., Braza, M.: A nonreflecting outlet boundary condition for incompressible unsteady Navier–Stokes calculations. J. Comp. Phys. 107, 239–253 (1993) 6. Pier, B.: Open-loop control of absolutely unstable domains. Proc. R. Soc. Lond. A 459, 1105– 1115 (2003) 7. Pier, B.: Primary crossflow vortices, secondary absolute instabilities and their control in the rotating-disk boundary layer. J. Eng. Math. 57, 237–251 (2007) 8. Pier, B.: Local and global instabilities in the wake of a sphere. J. Fluid Mech. 603, 39–61 (2008) 9. Pier, B., Huerre, P., Chomaz, J.M.: Bifurcation to fully nonlinear synchronized structures in slowly varying media. Physica D 148, 49–96 (2001) 10. Taylor, M.J., Peake, N.: A note on the absolute instability of wakes. Eur. J. Mech. B/Fluids 18, 573–579 (1999) 11. Woodley, B.M., Peake, N.: Global linear stability analysis of thin aerofoil wakes. J. Fluid Mech. 339, 239–260 (1997)

Riccati-less optimal control of bluff-body wakes Jan Oscar Pralits and Paolo Luchini

Abstract In this paper we propose a new method to solve the optimal control problem in which the feedback matrix K is computed in an efficient way for complex flows, with large number of degrees of freedom, using an approach similar to adjoint-based control optimization. The idea is to consider the direct-adjoint system as an input-output problem where the input is given by the current state and the output is the control. Since the control has much smaller dimension than the state, the feedback matrix K can be efficiently obtained from the solution of the adjoint of the direct-adjoint system. It can further be shown using the symplectic product that the direct-adjoint system is self adjoint. As a consequence the new adjoint system is equivalent to the direct-adjoint system with suitable initial and terminal conditions. With this method the optimal control problem can be solved efficiently for any value of the control penalty l 2 . Results are presented of this novel technique as applied to suppressing the vortex shedding behind a circular cylinder, and compared to the minimal-energy feedback control presented in [4].

1 Background Modern optimal control algorithms, based on the matrix Riccati equation, are usually difficult to apply to complex flows such as the wake behind a cylinder because of the large number of degrees of freedom originating from the discretized NavierStokes equations. An approximate method to overcome this problem, which has received attention in the literature, is to use reduced-order modeling. However, here we will present an exact method which does not rely on such modeling. Jan O. Pralits DIMEC, Università di Salerno, Fisciano (SA) Italy, e-mail: [email protected] Paolo Luchini DIMEC, Università di Salerno, Fisciano (SA) Italy, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_52, © Springer Science+Business Media B.V. 2010

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The optimal control problem is to find the control u which minimizes the cost function Z i 1 Th H J= x Qx + l 2 uH Ru dt, (1) 2 0 where superscript H denotes conjugate transpose, and the state x and the control u are related via the state equation

∂x = Ax + Bu on 0 < t < T ∂t

with

x = x0

at t = 0.

(2)

The result depends on the initial state x0 , the final time T , the choice of the matrices Q and R, and the real valued parameter l. To increase the value of l means to ascribe a higher cost to the control, and vice versa. This problem can be solved using a gradient based method, and the gradient can be efficiently evaluated using the adjoint of (2). The adjoint equations are here derived using Lagrange multipliers. If we introduce the adjoint variable p then the cost function can be written Z Z T i ∂x 1 Th H J= x Qx + l 2 uH Ru dt + pH − Ax − Bu dt. (3) 2 0 ∂t 0 Integration by parts yields 1 J= 2

Z Th 0

Z i x Qx+l u Ru dt + H

2

T

H

0

∂p H H H H x − − A p + u B p dt +[pH x]T0 (4) ∂t

Nullifying the functional derivative of J with respect to x gives

∂p = −AH p + Qx on 0 < t < T ∂t

with

p=0

at t = T.

(5)

Nullifying the functional derivative of J with respect to u gives l 2 Ru − BH p = 0.

(6)

At this point we can distinguish between two different approaches to solve the optimal control problem: in the first, the optimal control u corresponding to the state existing at each time step is computed in real time. This approach is generally combined with a finite horizon (value of T ) to make it tractable. In the second, considering a feedback rule u = Kx and a system which is time invariant, the feedback matrix K is computed once and for all off-line. In this case we can rewrite equation (6) as 1 Kx = 2 R−1 BH p. (7) l This problem is usually solved using a relation between the state vector x = x(t) and adjoint vector p = p(t) via a matrix X = X(t) such that p = Xx in order to write the two-point boundary value problem, given by the direct (2) and adjoint (5) equations, as one differential equation for X. The resulting equation is usually

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denoted differential Riccati equation. However, such a frame work is not suitable for numerical calculations when considering flow problems involving a large number of degrees of freedom, given the size of the matrix X. There are, as shown by [1] and [4], at least some cases where a mathematically rigorous optimal control can become a reality. They considered a minimal-energy stabilizing feedback rule u = Kx (using the problem definition above) in the limit as l 2 → ∞. In this limit the eigenvalues of the closed-loop system A + BK are given by the union of the stable eigenvalues of A and the reflection of the unstable eigenvalues of A into the left-half plane. They showed, by considering the system in modal form, that the feedback gain matrix K is a function solely of the unstable eigenvalues and the corresponding left eigenvectors. It was further demonstrated that the feedback matrix K, which is computed once and for all, works well even when applied to the complete nonlinear system. So far no approach has been set forth in order to compute K for complex flows when the parameter l is allowed to take any value. A new approach to solve this problem is given in the next section.

2 Riccati-less optimal control In this section the aim is to compute the feedback matrix K such that it is independent of the initial condition x0 and time invariant. This can in theory be done by investigating an number of initial conditions corresponding to the dimension of the state x. However, this is often computationally expensive and it is therefore of interest to find an alternative method. For any linear system where the dimension N0 of the output is much smaller than the dimension Ni of the input the sensitivity can be computed efficiently from its adjoint. This can be understood by considering that N0 computations of the adjoint completely replace Ni computations of the original system. In the optimzation problem that leads from x0 to u0 the linear system is given by the direct and adjoint equations (2) and (5). Since the dimension of u0 is much smaller than x0 it is favorable to compute the sensitivity with respect to the initial condition using the adjoint of the direct-adjoint system. The new adjoint is solved using an initial condition of small dimension, u+ 0 , and its output, which is the sensitivity with respect to the initial condition, is of large dimension and corresponds to a row of the feedback matrix K. The adjoint of the direct-adjoint system is derived by introducing the adjoint variables x+ and p+ which are multiplied with the equations (2) and (5), respectively, and integrated in time from t = 0 to t = T . This can be written Z T Z T +H ∂ x −1 H +H ∂ p H x − Ax − BR B p dt + p + A p − Qx dt = 0. (8) ∂t ∂t 0 0

328


Using integration by parts the differentiation operators are shifted from the direct to the adjoint equations. Considering that both R and Q are symmetric, we obtain Z T + Z T + ∂p ∂x H + −1 H + H H + + − p − Ap + BR B x dt − x + A x + Qp dt ∂t ∂t 0 0 T T + pH p+ 0 + xH x+ 0 = 0. (9) If we now define the new adjoint equations as

∂ p+ = Ap+ − BR−1 BH x+ , ∂t ∂ x+ = −AH x+ − Qp+ , ∂t

(10) (11)

with x+ (t = T ) = 0 in equation (11), and use the terminal condition p(t = T ) = 0, the remaining terms in expression (9) are written p+H (0)p(0) + x+H (0)x(0) = 0.

(12)

The optimality condition (7) can now be imposed one at a time by comparing each of its rows with the general identity (12). In particular, setting p+H (t = 0) equal to one row of R−1 BH we shall obtain that −x+H (t = 0) shall equal the corresponding row of K. In order to compute K we now need to solve the coupled system of linear equations (10)-(11) with the initial and terminal conditions x+ (T ) = 0 and p+H (t = 0) equals one row of R−1 BH , respectively. However, if let x+ → −p and p+ → x then these equations become the same as the direct-adjoint system (2) and (5). In other words, with respect to the symplectic product, the Hamiltonian direct-adjoint system is self-adjoint. This means that the adjoint of the direct-adjoint system can be obtained by solving the coupled system of linear equations (2) and (5) with an initial condition given by one row of R−1 BH . With the “optimal control” u so obtained directly gives one row of K. In order to obtain the minimum of (3) for t → ∞ we iteratively search for a sufficiently large T . The possible storage problems posed by the need for storing x(t) on [0, T ] during the forward march in order to reuse it during the adjoint can be avoided using a checkpointing algorithm, see [2] and [3], which saves x(t) occasionally on the forward march and then recomputes x(t) as necessary from these checkpoints during the backward march of the adjoint.

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It is important to note that the derivation shown in this section is made for a control of dimension one. This means that the initial condition for xH (t = 0) is given by the a row of = R−1 BH . Further, one row of the feedback matrix is obtained as pH (t = 0). In a general case, however, the above solution procedure must be performed for each control variable.

3 Application Results obtained from the new “Riccati-less” approach to compute the feedback matrix K are shown here in comparison with K computed assuming l 2 → ∞, as in [4], when the flow past a cylinder at Re = 55 is considered, and angular oscillation of the cylinder is used as the control variable. The Reynolds number is based on the free-stream velocity and cylinder diameter. Both the base flow and the linearized equations are discretized using second-order finite differences over a staggered, stretched, Cartesian mesh. An immersed-boundary technique is used to enforce the boundary conditions on the cylinder. Both the nonlinear and linearized Navier-Stokes equations are solved using the Adams-Bashforth/Crank-Nicholson scheme. Further, the adjoint equations are derived from the discretized form of (2) and are exact to machine precision. In figures 1 and 2 the u and v components of K are shown for the cases in which l 2 = 1 and l 2 → ∞, respectively. In both cases the u and v components are, respectively, anti symmetric and symmetric with respect to the horizontal axis, and the maximum values of both sensitivity components is situated close to the cylinder. Note that the maximum value for the case in which l 2 = 1 is larger compared to the l 2 → ∞ case. The effect of the control on the lift and drag forces on the cylinder is shown in figure 3 in comparison with the forces in a stationary flow and in the fully developed unstationary flow in the absence of control. It can be seen that the drag force approaches the values for stationary flow as the control is applied. This is of course obtained more quickly in the case in which l 2 = 1.

Fig. 1 K for l = 1, Re = 55; (left) Ku , (right) Kv . Solid contours inidicate positive values and dashed negative values.

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1.45

0.15

1.44

0.1

1.43

u=0 l2 → ∞ l2 = 1 stationary

1.42

Lift

Drag

Fig. 2 K for l → ∞ (Minimum energy control), Re = 55; (left) Ku , (right) Kv . Solid contours inidicate positive values and dashed negative values.

0.05 0

1.41 1.4

u=0 l2 → ∞ l2 = 1

-0.05

0

50

100

t

150

200

250

-0.1

5

10

15

20

25

t

30

35

40

45

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Fig. 3 Time trace of forces; (left) horizontal, (right) vertical

References 1. T. R. Bewley, J.O. Pralits, and P. Luchini. Minimal-energy control feedback for stabilization of bluff-body wakes. Proceeding FEDSM2002 31048 In BBVIV5, Fifth Conference on Bluff Body Wakes and Vortex-Induced Vibrations, Bahia, Brazil, December 2007. 2. J. Kim, and T. R. Bewley. A linear systems approach to flow control. Annu. Rev. Fluid Mech., 1:1–33, 2007 3. P. Luchini, and M. Quadrio. Adjoint DNS of turbulent channel flow. In 2002 ASME Fluids Engineering Division Forum, Montreal, Quebec, Canada, July 2002. 4. J. O. Pralits, T. R. Bewley, and P. Luchini. Feedback stabilization of the wake behind a steady cylinder. In 7th ERCOFTAC SIG 33 - FLUBIO WORKSHOP on Open Issues in Transition and Flow Control, Genova, Italy, October 2008.

Asymptotic theory of the pre-transitional laminar streaks and comparison with experiments Pierre Ricco

Abstract The response of the Blasius boundary layer to free-stream vortical disturbances of the convected gust type is studied. The vorticity signature of the boundary layer is computed through the boundary-region equations, which are the rigorous asymptotic limit of the Navier-Stokes equations for low-frequency disturbances. The method of matched asymptotic expansion is employed to obtain the initial and outer boundary conditions. The gust viscous dissipation and upward displacement due to the mean boundary layer produce significant changes on the fluctuations within the viscous region. The boundary-layer response induced by a threedimensional gust with spanwise wavelength comparable with the boundary-layer thickness is computed with first-order accuracy in the outer part of the boundary layer and with second-order accuracy in the core of it. The boundary-layer fluctuations of the streamwise velocity match the corresponding free-stream velocity component so that a realistic streak profile is obtained. A good agreement with the experimental data by [4] is found.

1 Introduction The present research work focusses on a mixed theoretical-numerical approach targeted at the modeling of the laminar streaks (or Klebanoff modes), namely the streamwise-elongated, low-frequency disturbances appearing in pre-transitional laminar boundary layers as a consequence of a medium-to-high level of free-stream turbulence. The streaks may be responsible for bypass transition, i.e. the breakdown to turbulence via some mechanism which is alternative to the classical instability route involving Tollmien-Schlichting waves. Experimental evidence indeed shows

Pierre Ricco Department of Mechanical Engineering, King’s College London WC2R 2LS London, United Kingdom, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_53, © Springer Science+Business Media B.V. 2010

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that, when the free-stream is sufficiently disturbed by vortical fluctuations, the laminar streaks quickly evolve into turbulence spots. The objective of the work is to obtain a mathematically rigorous description of the laminar streaks by means of perturbation methods. The work by [1] has shown that the spanwise velocity component of the free-stream vortical disturbances is responsible for the streak generation in the core of the boundary layer, while here we explain the mechanism of the penetration and confinement of the free-stream disturbances in the outer portion of the boundary layer. This analysis is important because (i) a more realistic streak profile may be obtained, (ii) the disturbances in the outer layer have been linked to the streak instability in recent numerical simulations [2], [3]. A further aim is to compare the results with the available experimental data by [4]. Further details and results of the present work can be found in [5].

2 Mathematical formulation The dynamics of the laminar streaks is governed by the unsteady linearized boundary region equations (LUBR), which are the rigorous asymptotic limit of the NavierStokes equation for low-frequency disturbances (see [1] and [5]). The gist of the problem resides in the specification of the initial and outer boundary conditions, which properly account for the continuous mutual interaction between the boundary-layer and the free-stream disturbances. A flow of uniform velocity U∞ past an infinitely-thin flat plate is considered. Superimposed on U∞ are homogeneous, statistically-stationary vortical fluctuations. These perturbations are of the gust type, i.e. they are convected by the mean flow. The flow is assumed to be incompressible and is described in terms of a Cartesian coordinate system, x, y, z. Mathematically, the vorticity fluctuations can be represented as a superposition of sinusoidal disturbances: u − î = ε u∞ (x − t, y, z) = ε uˆ ∞ ei(k·x−k1 t) + c.c., where ε ≪ 1 indicates the amplitude of the gust. The problem is formulated for a single Fourier component of the free-stream turbulence. We focus on low-frequency (i.e. long-wavelength) disturbances with k1 ≪ 1 as these are the ones that can penetrate the most into the boundary layer to form the laminar streaks. Following LWG, the flow is studied at downstream locations where the boundary-layer thickness δ ∗ = O(λz∗ ) (where λz∗ is the spanwise wavelength of the gust), which means that the diffusion in the spanwise direction is of the same order as that in the wall-normal direction. We assume that the amplitude of disturbances is much smaller than the amplitude of the mean flow, so that the equations can be linearized. A schematic of the flow domain is shown in Figure 2. The solution in the boundary layer is expressed as: {u, v, w, p} = {U,V, 0, −1/2}

Asymptotic theory of the laminar streaks and comparison with experiments

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ε u∞ (x, y,z,t)

λz∗ U∞

δ∗ λx∗ Fig. 1 Schematic of flow domain.

(

2xk1 +ε u0 (x, η ), Rλ

1/2

)

v0 (x, η ), w0 (x, η ), p0 (x, η ) ei(k3 z−k1 t) + c.c. + . . . ,

where U and V and the streamwise and wall-normal component of the Blasius boundary layer, x = 2π x∗ /λx∗ , z∗ is scaled by λz∗ and the time by λ ∗ /U∞ , where λ ∗ is the gust streamwise wavelength. The velocity and pressure disturbances are expressed as n o  {u0 , v0 } = C(0) u(0) , v(0) + (ik3 /k1 )C{u, v},     (0) (0) (1) w0 = −(ik1 /k3 )C w +Cw,     p0 = (k1 /Rλ )C(0) p(0) + iκ (k1 /Rλ )1/2 Cp,

p where {κ , κ2 } ≡ k3 , k2 /(k1 Rλ )1/2 = 2πνλx∗ /U∞ /{λz∗ , λy∗ } = O(1), C(0) ,C are order-one constants related to the free-stream gust. The terms proportional to the components {u, v, w, p} have been studied by LWG and represent the dominant part of the vorticity and pressure fluctuations in the core of the boundary layer. The terms proportional to the components {u(0) , v(0) , w(0) , p(0) } indicate the second-order part in the middle of the boundary layer and the leading-order part of the Klebanoff modes at the outer edge of the boundary layer. They are solved here for the first time. The outer boundary conditions for the boundary region equations are: u(0) =

(0) 2 2 (0) eix κ2 eiκ2 y −(κ +κ2 )x − i|κ |e−|κ |y , κ2 − i|κ | 1/2

+

∂ v(0) iβ κ 2 eix−|κ |(2x) η + |κ |(2x)1/2 v(0) → ∂η (κ2 − i|κ |) (2x)1/2 ! iκ2 β κ22 − κ 2 1/2 2 2 − i + κ 2 + κ22 eix+iκ2 (2x) η −(κ +κ2 )x , (2x)1/2 κ 2 + κ22

(2)

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∂ w(0) iβ |κ |3 eix−|κ |(2x) + |κ |(2x)1/2 w(0) → ∂η κ2 − i|κ |

1/2 η

− 1/2 η

∂ p(0) iβ |κ |eix−|κ |(2x) + |κ |(2x)1/2 p(0) → − ∂η 2x (κ2 − i|κ |) and

1/2 η −

2β κ 2 κ22 eix+iκ2 (2x) κ 2 + κ22 1/2

β κ 2 eix+iκ2 (2x) η −(κ + 2 x κ 2 + κ22

(κ 2 +κ22 )x

2 +κ 2 2

(3) )x , (4)

u = 0,

(5)

1/2 2 2 ∂v + |κ |(2x)1/2 v → −eix+iκ2 (2x) η −(κ +κ2 )x , ∂η

(6)

1/2 2 2 ∂w + |κ |(2x)1/2 w → iκ2 (2x)1/2 eix+iκ2 (2x) η −(κ +κ2 )x , ∂η

(7)

∂p + |κ |(2x)1/2 p → 0, ∂η

(8)

as η → ∞. Here η = η − β , where β is the Blasius displacement constant. The LUBR equations are parabolic in the streamwise direction and elliptic in the spanwise direction, so that they can be solved by marching downstream by applying asymptotically rigorous initial and outer boundary conditions, and the wall no-slip condition. A second-order, implicit finite-difference scheme is employed with the pressure terms computed on a grid staggered along the wall-normal direction with respect to the grid for the velocity components with the purpose of avoiding the pressure decoupling phenomenon. The linear system is solved by a standard blockelimination algorithm.

3 Results The generation of the low-frequency laminar streaks by free-stream vortical disturbances may be explained through two different, independent physical mechanisms. In the first one, discovered by [1], the spanwise velocity component w of the freestream gust is balanced by the wall-normal component v through continuity and is responsible for the amplification of the streamwise velocity perturbation in the core of the boundary layer. The gust streamwise velocity u and the pressure p play no role. The outer boundary conditions (5)-(8) exemplify this interaction. The second mechanism operates as follows. The gust streamwise velocity u(0) is balanced by a low-frequency pressure disturbance p(0) , which, in turn, drives a spanwise velocity fluctuation w(0) . The latter is finally balanced by a wall-normal velocity fluctuation v(0) through continuity. All these velocity components and the pressure fluctuation drive the laminar streaks in the outer portion of the Blasius boundary layer. This outer-layer mechanism is synthesized in the outer boundary conditions (2)-(4) and schematically represented in figure 3.

Asymptotic theory of the laminar streaks and comparison with experiments

335

u(0) - free stream u(0) - Blasius displacement

p(0) v(0)

continuity

w(0)

u

v

Klebanoff modes

w

z momentum

p

Fig. 2 Schematic of outer-layer mechanism for formation of first-order Klebanoff modes in outer part of boundary layer.

Figure 3 (left) shows the downstream evolution of the streamwise velocity profile of the laminar streaks. The peak inside the boundary layer is due to the first mechanism, while the velocity perturbation near the free-stream is caused by the second one. The result is a realist streak profile, similar to the ones often encountered in the experimental studies [6]. Figure 3 (right) presents the comparison between the experimental profiles of the fluctuating energy by [4] and our calculations, rescaled by the maxima of each experimental profile. The agreement is good and the lowfrequency fluctuations penetrate more deeply into the boundary layer and are amplified more than the high-frequency fluctuations, which are confined in the outer portion of the boundary layer. Realistic assumptions based on the experimental campaign have been made, such as small streak amplitude and homogeneous free-stream turbulence along planes perpendicular to the mean flow direction.

4 Conclusions and Outlook The present research effort has been successful to gain further insight into the formation of the laminar streaks by free-stream vortical fluctuations. Further work should be directed at investigating if and how the boundary layer disturbances in

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P. Ricco

0.3

1.0 1.5 2.0 2.5 3.0 4.0 5.0

0.25

0.2

10-2

10-3

7.5 7.5 35 35 75 75 150 150 220 220

|u|2

|u|

10-4

0.15

10-5

0.1

0.05

0

0

1

2

3

4

5

6

7

8

10

-6

10

-7

0.5 1 1.5 2 2.5 3 3.5 4 Y = y/δ Y = y/δ Fig. 3 Amplitude of streamwise velocity profiles for {κ , κ2 } = {1, 1}, k1 = 2πλz∗ /λx∗ = 0.1, uˆ∞ 3 = −0.2 at x ≥ 1 (left), and comparison with experimental profiles of fluctuating energy for 6 ω ∗ ν /U 2 (right). Lines denote the numerical calculations and symbols the exˆ frequencies F=10 ∞ perimental data at x∗ = 500 mm for U∞ = 8 m/s shown in figure 12a at page 210 of [4] (ν is the ∗ kinematic viscosity, uˆ∞ 3 is the spanwise velocity of the gust, ω is the frequency).

the outer layer trigger streak instability and lead the streaks to the breakdown into turbulent spots. It is our hope that the present work will motivate further experimental investigation. For example, the accurate study by [7] may be extended to study the boundary layer response to a single three-dimensional Klebanoff mode, which would allow a detail comparison with our results and the ones by [1]. As opposed to a straight ribbon as in [7]’s work, a wavy, vibrating (or rotating) thin ribbon could be used, thereby allowing the precise specification of the gust amplitude, frequency, and its wall-normal and spanwise wavelengths.

References 1. S. J. Leib, D. W. Wundrow, and M. E. Goldstein, Effect of free-stream turbulence and other vortical disturbances on a laminar boundary layer. J. Fluid Mech., 380:169–203, 1999. 2. R. G. Jacobs, and P. A. Durbin, Simulations of bypass transition. J. Fluid Mech., 428:185– 212, 2001. 3. P. Schlatter, and L. Brandt, and H. C. de Lange, and D. S. Henningson, On streak breakdown in bypass transition. Phys. Fluids, 20 101505, 2008. 4. K. J. A., Westin, and A. V. Boiko, and B. G. Klingmann, and V. V. Kozlov, and P. H. Alfredsson, Experiments in a boundary layer subjected to free stream turbulence. Part 1. Boundary layer structure and receptivity. J. Fluid Mech., 281:193–218, 1994. 5. P. Ricco, The pre-transitional Klebanoff modes and other boundary layer disturbances induced by small-wavelength free-stream vorticity. Accepted for publication in J. Fluid Mech.. 6. M. Matsubara, and P. H. Alfredsson, Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech., 430:149-168, 2001. 7. A. J. Dietz, Local boundary-layer receptivity to a convected free-stream disturbance. J. Fluid Mech., 378:291–317, 1999.

Roughness-induced transition of compressible laminar boundary layers J. A. Redford, N. D. Sandham, and G. T. Roberts

Abstract Direct numerical simulation is used to study flow over a roughness element in a flat plate boundary layer at high speed. The roughness element is modeled as a continuous bump with a height that is approximately half the boundary-layer thickness. Simulations with a variety of Mach numbers and wall temperatures have been run over a range of Reynolds numbers, and the results are used to establish a criterion for turbulent breakdown.

1 Introduction Transition from laminar to turbulent flow in a boundary layer is important when trying to quantify surface heat transfer and drag for supersonic or hypersonic flight vehicles. The influence of surface roughness is at present only partially understood [4, 5]. The flow perturbation made by the roughness element either causes fluctuations to grow and breakdown to turbulence or else there is a relaxation back to the spanwise uniform laminar flow [1]. Roughness elements distort the flow creating streamwise vorticity and an unstable detached shear layer. The shear layer contains an inflection point which corresponds to the location of amplified disturbances, indicating that there is a Kelvin-Helmholtz type instability [1]. The growth of fluctuations within the shear layer is quick but for transition to be complete the strong perturbations or maybe even turbulence must also penetrate into the near wall region and cause the formation of boundary layer turbulence. Eventually the turbulence causes a breakup of the highly non-uniform roughness wake. Results from experiments [1, 2] show that the shear layer perturbations must reach considerable amplitude before they trigger boundary layer turbulence. The Reynolds number is the most significant parameter in transition and it is well accepted in the incompressible flow literature [6] that the appropriate Reynolds Dr. John Redford University of Southampton, Southampton, UK, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_54, © Springer Science+Business Media B.V. 2010

337

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J. A. Redford, N. D. Sandham, G. T. Roberts

number for an isolated roughness element is Reh = ρhUh h/µh where h is the height of the roughness element, and ρh , Uh and µh are the fluid density, streamwise velocity and dynamic viscosity at height h in the laminar boundary layer away from the roughness element. At low Reh the flow is stable and the perturbation does not cause breakdown, but beyond a critical Reh (for a hemispherical roughness element 300 to 400 in incompressible flow [2]) the roughness causes transition to turbulence. Tani [6] suggests that the critical Reynolds number is reduced by the introduction of freestream turbulence. At high Mach numbers acoustic disturbances will be prevalent, particularly in enclosed wind-tunnel experiments. Schneider [5] shows results suggesting that, below Mach 4, the critical roughness Reynolds number is unaffected by compressibility, but at greater Mach numbers there is a large increase in critical Reynolds number. The present numerical study aims to provide more data to help understand roughness-induced transition.

2 Method The numerical method used here is the same low-dissipation method that was used by Krishnan and Sandham [3] to study turbulent spots. To save computational effort in simulating the leading edge flow the inlet condition is fixed using a velocity and temperature profile from a compressible similarity solution at distance x0 from the leading edge. All parameters are normalized by the inlet displacement thickness δ0∗ , and the free-stream density ρ∞ , velocity U∞ , viscosity µ∞ and temperature T∞ . The lateral boundary condition is periodic, but the spacing is such that there is virtually no interaction between the wakes of the roughness elements. In the free-stream there are expansions and shock waves originating from the roughness element, and characteristic boundary conditions are used at the top and the outlet boundaries to reduce the reflections from these and other disturbances. The computational domain is lx × ly × lz = 325 × 16 × 64 in size with a bump centered at (xh , zh ) = (125, 32) on the plate and defined by the function y0 = A 2 2 2 2 (tanh(rh /B − 1) + tanh(−rh /B − 1)), where rh = (x − xh ) + (z − zh ) , A = 1.71 and B = 0.04, making a smooth three-dimensional shape with a bump height of h = 1.3 and a half-width of 5.9. A body-fitted grid is generated using a series of analytical functions that maintaining a grid that is close to orthogonal, while ensuring there are sufficient points in each part of the domain. Grid stretching is used to place more points within the boundary layer. A set of simulations have been run with a range of Reynolds numbers Re = ρ∞U∞ δ0∗ / µ∞ , free-stream Mach numbers M and wall temperatures Tw (Table 1). The stability of the flow is strongly affected by the Reynolds number, so under each condition a range of Reh are simulated, aiming for one stable and at least two unstable cases for each M and Tw combination.√Along with the roughness Reynolds number, a roughness Mach number Mh = Uh / γ RTh can be defined and is also indicated in Table 1. Higher Reynolds number cases require more nodes Nx , Ny , Nz to ensure sufficient resolution. A detailed study has been conducted to establish that there is little dependence of the results on resolution. The

Roughness-induced transition of compressible laminar boundary layers

339

‘hot wall’ cases shown in Table 1 correspond to a fixed wall temperature equal to the adiabatic wall temperature of the incoming laminar boundary layer. Table 1 Flow parameters and occurrence of transition in cases run. Case

Tw

M

Re

Rex0 × 105

Reh

Mh

δh∗

δh

Transition

Nx , Ny , Nz

Hot3A Hot3B Hot3C Cold3A Cold3B Cold3C Hot6A Hot6B Hot6C Hot6D Cold6A Cold6B

2.517 2.517 2.517 1 1 1 7.027 7.027 7.027 7.027 1 1

3 3 3 3 3 3 6 6 6 6 6 6

1500 2250 3000 1100 1500 1750 4000 8000 9500 11000 3000 3500

2.8 4.8 7.3 3.0 5.0 6.4 6.0 14 17 21 6.9 8.7

324 595 920 561 820 985 117 440 620 810 774 975

1.2 1.4 1.5 1.7 1.8 1.8 1.1 1.7 1.8 2.0 3.1 3.2

1.8 1.6 1.4 1.3 1.3 1.2 2.6 1.9 1.8 1.7 1.5 1.5

2.9 2.5 2.3 2.9 2.8 2.7 3.1 2.3 2.2 2.1 2.3 2.2

N Y Y N Y Y N Y Y Y N N

780, 150, 150 1248, 210, 240 1248, 210, 240 780, 150, 150 1092, 180, 210 1248, 210, 240 780, 150, 150 1092, 180, 210 1248, 210, 240 1248, 210, 240 780, 150, 150 1092,180, 210

The present simulations require a perturbation that will act as a trigger for transition. If no disturbance is present in the numerical simulations the flow can remain laminar even at high Re. Here a disturbance is introduced within the freestream in the form of an acoustic line source centered at (xc , yc ) = (100, 8) for M = 3 and (75, 8) for M = 6, within a radius r = R defined as R = 1, where r2 = (x − xc )2 + (y − yc )2 . The forcing function is F(r,t) = A(r)B(t), where A(r) = Nf a 2 (1 − cos (π (1 − r/R))), (a = 1/80), and B(t) = ∑n=1 sin (2π f n nt + φ (n)), (N f = 8 frequencies are applied in increments of fn = 0.02), and the phase angles φ (n) are random numbers between 0 and 2π . F(r,t) is added to the continuity equation.

3 Flow structure Placing a roughness element within the flow causes a strong downstream perturbation. Fig. 1 shows an example of a flow where the perturbations become unstable and breakdown to turbulence. Along the centreline there is a negative velocity perturbation and either side of this there is flow at greater velocity than in the flow without a roughness element. The flow close behind the roughness element is laminar with streaks developing in the velocity field. Within the streaky flow behind the roughness, fluctuations develop that eventually cause transition. At x = 160 the perturbed flow starts to breakdown and at the same time extra streaks develop. The resulting turbulence breaks up the streaky structure formed by the roughness. When the flow becomes turbulent the lateral spreading of the roughness wake visibly increases. The boundary layer already contains spanwise vorticity, but roughness element will cause the introduction of streamwise and vertical vorticity. The vorticity from

340


the unperturbed laminar boundary layer typically wraps around the obstacle to form a ‘necklace vortex’, but this effect is weak in the smooth bump case simulated here. More important are the two streamwise vortices that trail the roughness element, causing slow-moving fluid to be lifted up into the quicker moving region of the boundary layer (Fig. 2 (a) and (b)) and faster fluid to move toward the wall. It is evident from the flow visualization that instability arises first in the shear layer away from the wall (Fig. 2 (c)). Further flow visualization of the unstable region shows that hairpin vortices are formed when the streamwise vorticity and shear layer instability interact. The hairpin vortices then travel downstream and eventually breakdown into turbulence.

4 Critical threshold A number of simulations with widely varying conditions are presented here and using the results it is possible to look for correlations involving the main flow parameters. Instability and transition are strongly dependent upon the local Reynolds number Reh . However, more parameters are needed because some of the simulations do not pass through transition whilst having a high Reh . This is the case at high Mach number, so a dependence on the local Mach number Mh is evident. One more parameter is needed because the Mach 6 cold wall cases do not undergo transition while some of the Mach 6 hot wall simulations do, despite both having a similar Reh . Therefore, the wall temperature Tw /T∞ is also important. In Fig. 3 (a) the information from all simulations is collected together by plotting Mh T∞ /Tw against Reh since with this choice there is a clear dividing line Reh = 300 + 700 3 Mh T∞ /Tw between the turbulent and non-turbulent cases, which provides a dependence of critical Reynolds number on compressibility. As has already been mentioned the upper shear layer can become unstable but it can be some distance downstream before the perturbations will penetrate the near wall region and form boundary layer turbulence. The transition start location xt is

Fig. 1 Case Hot3C isosurfaces of velocity perturbation u′ ± 0.1U∞ from laminar base flow at z = 0. The light regions are accelerated flow and the dark regions are deficits. The isosurfaces are plotted only for x > 150, when grid skew is negligible.

Roughness-induced transition of compressible laminar boundary layers

341

the point at which the skin-friction begins to rise significantly, corresponding to the location in Fig. 2 (c) where flow structures impact on the wall. Fig. 3 (b) shows that transition start, plotted here as δh /(xt − xh ) (i.e. an inverse transition length, where δh is the boundary layer thickness at xh and z = 0), is brought forward by an increase in roughness Reynolds number. Using the critical Reh threshold found with Fig. 3 (a) arranges the points into laminar (no transition) and transitional results. Increasing M (rather than Mh ) delays the transition.

5 Conclusion Flow over a three-dimensional roughness element has been simulated for a range of high-speed flows. A correlation is proposed, based on roughness height Reynolds number, Mach number and wall temperature, that delineates transitional from laminar cases.

(a) 4

y

(b) 12

2 0 100

150

200

250

x

(c) Fig. 2 Hot3B particle paths (a) side view and (b) front view calculated from a single velocity field. All particles start at x = 105 where there are filled symbols at (y, z) = (0.5, 32.5) —, (y, z) = (1, 32.5) – · –, (y, z) = (1, 34) – –. The end of a particle path is marked by an open symbol and the bump is the shaded area. The y axis is stretched by a factor of 16. (c) Shear stress τ = µ∂ u/∂y along the domain centreline z = 32, contours from 1 to 4 increasing in increments of 0.5. The y axis is stretched by a factor of 4.

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Acknowledgements This work is funded by the EU through the ATLLAS project. Time on the UK HPCx supercomputer was provided by the UK Applied Aerodynamics Consortium (EPSRC Grant EP/F005954/1).

References

4

0.08

3

0.06 h

δ /(x −x )

t

2

0.04

h

MhT∞/Tw

1. F.G. Ergin and E.B. White. Unsteady and transitional flows behind roughness elements. AIAA J., 44(11):2504–2514, 2006. 2. P.S. Klebanoff, W.G. Cleveland, and K.D. Tidstrom. On the evolution of a turbulent boundary layer induced by a three-dimensional roughness element. J. Fluid Mech., 237:101–187, 1992. 3. L. Krishnan and N.D. Sandham. Effect of Mach number on the structure of turbulent spots. J. Fluid Mech., 566:225–234, 2006. 4. E. Reshotko. Transition issues for atmospheric entry. J. Spacecraft Rockets, 45(2):161–164, 2008. 5. S.P. Schneider. Effects of roughness on hypersonic boundary-layer transition. J. Spacecraft Rockets, 45(2):193–209, 2008. 6. I. Tani. Boundary-layer transition. Annu. Rev. Fluid Mech., 1:169–196, 1969.

1 0

0.02

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200

400

600 Reh

(a)

800

1000

0

0

200 400 600 Reh − 700/3 MhT∞/Tw

800

(b)

Fig. 3 Transition correlations (a) transition threshold is given by Mh T∞ /Tw = 3(Reh − 300)/700. (b) Normalized transition start location. Hot3 △, Cold3 ◦, Hot6 , Cold6 ⋄, and filled symbols are cases that are transitional.

On receptivity and modal linear instability of laminar separation bubbles at all speeds D. Rodr´ıguez, J. A. Ekaterinaris, E. Valero, and V. Theofilis

Abstract Global linear modal instability analysis of laminar separation bubble (LSB) flows has been performed. The pertinent direct and adjoint eigenvalue problems have been solved in order to understand receptivity, sensitivity and instability of the global mode associated with adverse-pressure-gradient-generated LSB, as well as that formed by shock/boundary-layer-interaction (SBLI) and over a finiteangle wedge in supersonic flow. In incompressible flow the global mode of LSB has been found to be at the origin of the experimentally-observed phenomena of U-separation and stall-cell formation. Qualitatively analogous results have been obtained in compressible subsonic flow, but convergence of the corresponding eigenspectra in supersonic flow continues posing computational challenges, despite use of state-of-the-art algorithms and the largest European supercomputing facility.

1 Introduction and Motivation Global linear instability analysis tools were introduced about a decade ago to study modal perturbations in LSB flows and have led to the first unequivocal demonstration of the existence of self-sustained oscillations of laminar flow in the archetypal boundary-layer [14] and backward-facing step [3] configurations. Ever since, both the degree of geometrical irregularity of the configuration addressed and the theoretical sophistication of the global analysis and prediction tools applied to the problem of global LSB flow instability have kept increasing continuously, [7, 2, 1], in tune with the advancing computing hardware capabilities for the accurate soluDaniel Rodr´ıguez, Eusebio Valero and Vassilis Theofilis School of Aeronautics, Universidad Politécnica de Madrid, Pza. Cardenal Cisneros 3, E-28040 Madrid, Spain, e-mail: [email protected], [email protected], [email protected] John Ekaterinaris School of Mechanical and Aerospace Engineering, University of Patras, 26500 Patra, Greece, email: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_55, © Springer Science+Business Media B.V. 2010

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D. Rodr´ıguez, J. A. Ekaterinaris, E. Valero, V. Theofilis

Fig. 1 Isolines of stream function in incompressible LSB flow [11] (left). Isolines of density for compressible LSB flows generated by SBLI (middle) and a 45o −degree wedge [6] (right).

tion of the partial-derivative-based eigenvalue problems (EVP). From a numerical point of view, after one decade of global instability analysis work in LSB flows in several groups one may assert that it is relatively straightforward to describe the global modes of LSB, which are localized in the vicinity of the respective basic states. The situation is different when compressibility is considered; in supersonic flow LSB flows may be generated as the result of interaction of a single shock wave with the boundary layer on a flat surface or as part of the complex interaction pattern formed in the neighborhood of geometric discontinuities, such as (finite-angle) wedge flows. The recent renewed interest in hypersonic flight has resulted in the need to understand (and, if possible, control) basic instability mechanisms inaccessible to classic local modal linear analysis. First efforts in the viscous linear global instability analysis of compressible flow exist in the open cavity [4], transonic [5] and supersonic regimes [10]. The present contribution intends to shed new light on the characteristics of global modes of laminar separation bubbles at all flow speeds. In the incompressible regime the basic state is obtained by an inverse boundary layer methodology; a comparison of the basic flow analyzed with an established result is shown in the left part of figure 1. A novel residual distribution algorithm [16] is used in order to study compressibility effects on this basic flow [15]. In the supersonic regime, the basic states corresponding to LSB flows generated by shock/boundarylayer-interaction (SBLI) and in the vicinity of a 45o −degree wedge are calculated by a high-order finite-difference method [6]; representative steady flows obtained at Re = 5000 and M = 2 and 2.5, respectively, are shown in the middle- and right parts of figure 1, respectively. Notewhorthy findings associated with instability, receptivity and sensitivity analysis of the global mode pertaining to such LSB flows, as well as with three-dimensional topological flowfield reconstructions, are briefly outlined next.

2 Theory and Numerical Methods Instability and receptivity analysis commences with the decomposition of the state vector q = (u, v, w, p)T into linear perturbations, superposed to the basic flow according to the Ansatz q = q¯ + ε q′ . The direct and adjoint eigenvalue problems are respectively defined by

On receptivity and modal linear instability of laminar separation bubbles at all speeds

ˆ A · qˆ = ω B · q, The condition

A † · qˆ = ω B · qˆ

(ωi − ω j ) · hB qˆ i , q˜ j i = 0

345

(1) (2)

ensures (bi-)orthogonality of the direct eigenvectors qˆ i against all but one of the adjoint eigenvectors, q˜ j . Here A and B result from spatial discretization of the linearized (incompressible or compressible) equations of motion and the inner product is defined by Z hB qˆ i , q˜ j i =

Ω

(B qˆ i )† · q˜ j d Ω = Ci, j ,

(3)

where † indicates complex conjugation. Note that Ci, j = 0 if i 6= j. Flow reconstruction is accomplished in the present BiGlobal context by: ¯ y) + ∑ Ai qˆ i (x, y) · exp [i (β z − ω t)] , q(x, y, z,t) = q(x,

(4)

i

where the initial amplitudes are obtained from the initial perturbation field f projected on the adjoint eigenmodes: Ai =

hB · f, q˜ i i Ci,i

(5)

q˜ i shows the spatial region of higher receptivity of the mode i. The spatial location of the receptivity-instability feed-back, or ”maximum structural sensitivity region” is obtained from the kernel of the inner product:

λi = (B qˆ i )† · q˜ i .

(6)

3 Results The direct and adjoint eigenvalue problems have been solved by the massively parallel solver discussed by Rodr´ıguez and Theofilis [13] in the incompressible regime. The compressible work has been performed using a novel immersed-boundary method based on 4th −order finite-differences in both the interior of the domain and the boundary closure.

3.1 Incompressible flow 3.1.1 Receptivity and sensitivity of LSB flow Figure 2 shows contours of the dominant component of the adjoint eigenvector and those of the kernel of the inner product defined in (6), both superimposed upon

8

8

7

7

6

6

5

5 y / δ*

Fig. 2 Streamlines of primary separation and greyscalecoded contours of the u−component ˜ of the adjoint eigenvector (left) and the kernel of the direct-adjoint coupling λ (right).


y / δ*

346

4 3

4 3

2

2

1

1

0 180

200

220

240

260 x / δ*

280

300

320

0 180

340

200

220

240

260 x / δ*

280

300

320

340

the primary incompressible LSB flow over a flat plate. The results demonstrate that the maximally-receptive region coincides with the line of primary separation, while the maximal feedback region between direct and adjoint eigenmodes [8] practically coincide with the primary recirculation center. 3.1.2 U-separation and Stall Cells The self-excited nature of the global mode leads to spanwise-periodic topological changes of the two-dimensional basic state in line with the Ansatz (4). O(1) modifications of the basic state then give rise to the well-known pattern of U-separation, identified in a multitude of flow configurations; details are discussed elsewhere [11]. The Kutta condition imposes a qualitative difference on the spatial structure of the 0

Fig. 3 Left: Eigenspectrum of LSB flow on a flat plate, on which the global mode is highlighted. Right: 3d flowfield reconstruction showing U-separation as a result of the global mode[11].

−0.02

Im (ω)

−0.04 −0.06 −0.08 −0.1 −0.12 −0.2

−0.1

0 Re (ω)

0.1

0.2

global mode pertinent to semi-infinite and finite domains. Performing a topological analysis and three-dimensional flowfield reconstruction on a stalled NACA0015 airfoil analogous to that on the flat-plate LSB flow, namely superposing the leading global mode highlighted in the left part of figure 4 upon the steady laminar base flow, surface streamline patterns are recovered, which are strongly reminiscent of the experimentally-observed stall-cells [12].

0.5

0 Im (ω)

Fig. 4 Left: Eigenspectrum of LSB flow on a stalled NACA0015 airfoil with the global mode highlighted. Right: 3d flowfield reconstruction showing stall cells as a result of the amplified global mode[12].

−0.5

−1

−8

−6

−4

−2

0 Re (ω)

2

4

6

8

On receptivity and modal linear instability of laminar separation bubbles at all speeds

347

3.2 Compressible flow 3.2.1 Subsonic flow: the effect of compressibility The effect on the basic wall-shear of Mach number variations in the range 0.1 < M < 0.5 is shown in figure 5 and discussed at some length elsewhere[15]. Qualitatively the LSB flows obtained by the application of the residual distribution scheme [16] subject to boundary conditions consistent with those of the incompressible work on a flat plate are analogous. At a quantitative level the extent and strength of recirculation are found to diminish with increasing Mach number. This leads to the conjecture that more stable flow is to be expected, at least as far as its leading global eigenmode is concerned, since this instability has been found to scale with the strength of primary flow recirculation. However, no qualitative differences of the instability results associated with the global mode are expected in the subsonic regime; instability analyses are underway in order to verify this point.

Fig. 5 Effect of compressibility on wall-shear of the LSB basic state subject to Howarth free-stream deceleration [15].

3.2.2 Supersonic regime: SBLI and supersonic finite-angle wedge flows The situation has been found to be different in supersonic flow, in both classes of SBLI- and geometry-generated LSB flows shown in figure 1. Although rather high resolutions have been utilized (nominally a factor five larger number of points than comparable work in the case of SBLI [10]) neither eigenspectrum result could be converged in a satisfactory manner in the Reynolds number range 5 × 103 < Re < 2 × 104 examined. Figure 6 shows the leading eigenmodes corresponding to an analysis in which the entire basic flow calculation domain, including the shock systems shown in figure 1, has been considered. Superposed are the streamlines of the basic

Fig. 6 Spanwise disturbance velocity component of the leading eigenmode of the SBLI (left) and wedge (right) .

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state, showing the coincidence of the peaks of the spanwise disturbance velocity component with the peaks of the vortical structures. These results demonstrate that solution of the problem of global instability in arbitrary realistic supersonic LSB flows may still be some way into the future. Acknowledgements The material is based on research partially sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under agreement number FA865506-1-3066 to nu modelling s.l., entitled Global instabilities in laminar separation bubbles. The Grant is monitored by Dr. Douglas Smith of AFOSR and Dr. S. Surampudi of the EOARD. The views and conclusions contained herein are those of the author and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the U.S. Government. Computations have been performed on the Magerit (www.cesvima.upm.es) and Blue-Gene/P (Forschungzentrum Jüelich - www.fzjuelich.de) supercomputing facilities.

References 1. N. Abdessemed, S. J. Sherwin, V. Theofilis. Linear instability analysis of low-pressure turbine flows. J. Fluid Mech., 628:57-83, 2009. ˚ 2. E. Akervik, J. Hœpffner, U. Ehrenstein, D. S. Henningson. Optimal growth, model reduction and control in a separating boundary-layer flow using global eigenmodes. J. Fluid Mech., 579:305-314, 2007. 3. D. Barkley, M. G. M. Gomes, R. D. Henderson. Three-dimensional instability in a flow over a backward-facing step. J. Fluid Mech., 473:167-190, 2002. 4. G. A. Bres, T. Colonius. Three-dimensional instabilities in compressible flow over open cavities. J. Fluid Mech. 599:309-339, 2008. 5. J. D. Crouch, A. Garbaruk, D. Magidov. Predicting the onset of flow unsteadiness based on global instability. J. Comput. Physics 224(2):924-940, 2007. 6. J. A. Ekaterinaris. Performance of high-order-accurate, low-diffusion numerical schemes for compressible flow. AIAA J., 42(3):493-500, 2004. 7. F. Gallaire, M. Marquillie, U. Ehrenstein. Three-dimensional transverse instabilities in detached boundary layers. J. Fluid Mech., 571:221-233, 2007. 8. F. Giannetti, P. Luchini. Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581:167–197, 2007. 9. V. Kitsios, D. Rodr´ıguez, V. Theofilis, A. Ooi, J. Soria. BiGlobal stability analysis in curvilinear coordinates of massively separated lifting bodies. J. Comput. Phys., to appear 2009. 10. J. Ch. Robinet. Bifurcations in shock-wave/laminar-boundary-layer interaction: global instability approach. J. Fluid Mech. 579: 85-112, 2007. 11. D. Rodriguez, V. Theofilis. Structural changes induced by global linear instability of laminar separation bubbles. J. Fluid Mech. submitted, 2009. 12. D. Rodriguez, V. Theofilis. On the birth of stall cells on airfoils. Theor. Comp. Fluid Dyn. submitted, 2009. 13. D. Rodriguez, V. Theofilis. Massively parallel numerical solution of the BiGlobal linear instability eigenvalue problem using dense linear algebra. AIAA J. to appear, 2009. 14. V. Theofilis. Global linear instabilities in laminar separated boundary layer flow. IUTAM Laminar-turbulent Transition Symposium IV, Springer, Sedona, AZ, pp. 663-668, 2000. 15. E. Valero, V. Theofilis. Compressibility effects in Howarth’s separation bubble on a flat plate. AIAA-Paper 2008-0593, 46th Aerospace Sciences Meeting and Exhibit, 2008. 16. E. Valero, J. de Vicente, G. Alonso (2009) The application of compact residual distribution schemes to two-phase flow problems. Comput. Fluids, doi: 10.1016/j.compfluid.2009.06.002

Hypersonic instability waves measured on a circular cone at M=12 using fast-response surface heat-flux and pressure gauges T. Roediger, H. Knauss, B. V. Smorodsky, D. A. Bountin, A. A. Maslov, E. Kraemer, and S. Wagner

1 Introduction Conical boundary layers are prevalent on many hypersonic vehicles and stability measurements on sharp cones have been conducted since the 1970’s. Schneider [1] gives a review of experimental and numerical studies on the subject and provides an overview of instability mechanisms on circular cones including a comprehensive list of references. Up to now, experimental stability investigations of a conical boundary layer (BL) at free stream Mach numbers larger than M=8 are very sparse. The following experiments present spectral wave amplitude distributions at M=12 measured by a staggered array of surface-mounted, fast-response pressure and ALTP heat-flux gauges. Both gauges cover a frequency range up to 1 MHz and allow highly time-resolved experimental studies where conventional measurement techniques like hot-wires cannot be used due to their limited durability, limited overheat ratio and temporal resolution. The data quality allows the calculation of amplification rates, which are compared with predictions from LST computations (2D) and amplification rates obtained with the same model in two different Mach-6 facilities.

2 Experimental Setup The experiments are carried out in the AT-303 facility of ITAM. A special asset of this tunnel is the high purity of the working gas and the sufficient running time of 40 - 500 ms for obtaining a steady flow in the test chamber. A more detailed description of the AT-303 can be found in Kharitonov et al. [2]. Stagnation conditions between 89 bar ≤ p0 ≤ 258 bar and 894 K ≤ T0 ≤ 1068 K are realized during the present measurement campaign. Hence, a unit Reynolds number range of 4.3 × 106 /m≤ Reunit ≤ 23.8 × 106 /m is covered. T. Roediger · H. Knauss · E. Kraemer · S. Wagner Institute of Aerodynamics and Gas Dynamics, Universität Stuttgart, Pfaffenwaldring 21, 70569 Stuttgart, Germany. e-mail: [email protected] B.V. Smorodsky · D.A. Bountin · A.A Maslov Khristianovich Institute of Theoretical and Applied Mechanics (ITAM), Russian Academy of Sciences, Novosibirsk, Russian Federation. P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_56, © Springer Science+Business Media B.V. 2010

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A 7-degree, half-angle cone with a sharp nose tip was installed in the test section (see Fig. 1). The 500 mm long cone is made from aluminum with the exception of the stainless steel nose. The diameter of the tip is estimated to be about 6 micrometers by means of a microscope. The cone-model alignment into the flow is carried out in successive runs by comparison of the maximum frequency of the second-mode instability (SM) at rotation angles of 0, 90 and 180 degrees. The model is adjusted to a frequency shift of fSM,max < 10 kHz between the three angular positions. The cone is instrumented with a staggered array of ALTP and PCB pressure gauges (see Fig. 1 and Table 1). The output signal of the ALTP is directly proportional to heat flux density and covers a frequency range up to the 1 MHz range [3], [4], [5]. The active area of the ALTP gauges used in the present experiments is 2 × 0.4 mm2 limiting the spatial resolution in the streamwise direction to 0.4 mm. The amplitudefrequency response (AFR) characteristics of the measurement system is taken into account by means of a dynamic laser calibration procedure [3] in order to determine absolute values of wave amplitudes. Commercial pressure sensors of type M131A32 manufactured by PCB Piezotronics are used in the experiments [6]. The sensors are flush mounted in the model surface and the diameter of their sensing area is 3.18 mm. Two synchronized 4 channel, 12-bit data-acquisition cards are used for capturing of the voltage signals of the ALTPs and pressure gauges. The TiePie 4 card has a maximum sampling rate of 128 KS/s and the L-card is operated with the maximum rate of 500 KS/s. A time period of up to 300 ms is evaluated for the ALTP spectra. A period of 65 ms is processed for the pressure gauges limited by the maximum Position arc length (Number) x [mm] 5 200 6 225 7 270 8 295 9 320 10 345 11 370

Sensor Sensitivity s Sensitivity s (Serial number) [µ V/(W/cm2 )] [µ V/Pa] ALTP 941 80.0 PCB 1 (4437) 20.3 ALTP 942 110.1 PCB 2 (4342) 22.0 ALTP 943 101.1 PCB 3 (4343) 24.5 ALTP 1004 117.0

Table 1 Specification of installed ALTPs and PCB pressure gauges.

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sampling length of the data acquisition card. For the calculation of the amplification rates between the single-point sensors, constant exponential amplitude growth is assumed: −αi =

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Fig. 2 displays typical amplitude spectra simultaneously obtained from (a) ALTP and (b) pressure gauges in an early stage of BL transition (Reunit =6.67×106 /m). A characteristic shift of the second-mode frequencies towards lower values with increasing x-location and hence rising BL thickness is clearly visible in both diagrams. The quality of the amplitude spectra of the ALTP is better due to the longer time period of 300 ms used for averaging in comparison with the pressure spectra that only use a time signal of 65 ms in length. Fig. 3 displays the reduced amplification rates calculated from (a) ALTP heat flux spectra between x=270 mm and x=320 mm and (b) pressure spectra between x=295 mm and x=345 mm. The amplification rates are normalized by relevant BL edge condition (subscript e) in order to allow better quantitative comparison of growth rates. The growth rates of the

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second mode (SM) collapse for approximately the same reduced frequency in both diagrams. The quantitative comparison of the maximum growth rates in Fig. 3 (a), (b) show that the amplification rates measured by the pressure sensors are systematically slightly lower than the one detected by ALTPs for the same Ree range (see also Fig. 4 for direct comparison at specific Reunit = 6.67 × 106 /m).√The peaks in Fig. 3 (a) detected by the ALTP in a range between 0.025 ≤ f x/(ue Ree ) ≤ 0.04 resembles very likely the growth rates of the first higher harmonic of the SM. Its maximum amplification rates are significantly lower than the one of its fundamental. Whereas, the harmonic detected by the pressure gauges in a similar frequency range appears to be slightly stronger amplified (Fig. 3(b)). For comparison, 2-D LST computations are carried out for the specific conditions of the experiments. The axisymmetric conical BL of a sharp cone at zero angle of attack is computed by means of self-similar flat-plate compressible BL equations. Fig. 4 compares the amplification rates detected by the ALTP and pressure gauges for a fixed unit Reynolds number of 6.67 × 106 /m. The maximum growth rates of the SM predicted by LST are pretty well matched by the experimental data; the maximum rates detected by the ALTPs are slightly larger and the ones measured by the pressure gauges are slighly smaller than the LST predictions. The amplified frequency range in the experiments is broader and extends into the low fre-

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−αi from ALTP heat flux spectra between x=270 and x=320; Ree=3.61E6 LST; Ree=3.61E6 −αi from PCB pressure spectra between x=295 and x=345; Ree=3.92E6 LST; Ree=3.92 E6

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quency range, where also certain dominant features are visible that might indicate first-mode growth or amplified disturbances originating from tunnel noise. In the higher frequency range, LST predicts a noticeable amplification of a third instabil√ ity mode at M=12 between 0.04 ≤ f x/(ue Ree ) ≤ 0.05. The experimental data in this extended frequency range are suspect to higher uncertainties due to limited signal to noise ratio. Yet the growth rates determined by the ALTPs are in fairly good agreement with LST predictions. The growth rates obtained from the pressure spectra are attributed to even larger scatter in this frequency range, but show also a trend towards amplification. It should be noted that the pressure spectra are not AFR corrected which could have an significant effect on the quantitative value of amplification rates measured in such a high frequency range. In addition, Fig. 5 displays growth rates detected by ALTPs at a higher unit Reynolds of 9.23 × 106 /m that also indicate the amplification of a third mode. Thus, the results present evidence for the first experimental detection of a third-mode instability at M=12 to the authors’ best

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/(U e Re )

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knowledge. Fig. 6 compares the growth rates in the early stages of BL transition for the present M=12 results and partly previously published transition studies at M=6 with the same cone model at the Mach-6-Quiet-Tube (M6QT) of Purdue University [5] and the Hypersonic Ludwieg Tube Braunschweig (HLB) [7]. In these two Mach-6 facilities (both operated under noisy flow), good agreement of the growth rates of the SM and its harmonic is found in the early transition stages. For better comparison with the current M=12 results, the reduced frequency range in the diagram is adjusted in such a way that the the maximum rates of the SM collapse on the abscissa. The absolute values are not relevant since the stability diagrams of M=6 and M=12 differ significantly. However, the ratio of the amplification rate of the harmonic and its fundamental show a very different behavior. At M=6, the growth is found to be approximately equal for both pressure and heat flux fluctuations. At M=12, however, the harmonic shows significantly lower amplification rates.

References 1. Schneider SP (2004) Hypersonic laminar- turbulent transition on circular cones and scramjet forbodies. Progress in Aerospace Sciences 40:1–50 2. Kharitonov A, Zvegintsev V, Chirkashenko V, Vasenev, L (2005) Commissioning and Acceptance Testing of the New Hypersonic Wind Tunnel at ITAM RAS. AIAA-2005-3328, 13th International Space Planes and Hypersonics Systems and Technologies Conf. 3. Roediger T, Jenkins S, Knauss H, v. Wolfersdorf J, Gaisbauer U, Kraemer E (2008) TimeResolved Heat Transfer Measurements on the Tip Wall of a Ribbed Channel Using a Novel Heat Flux Sensor - Part I: Sensor and Benchmarks. J. Turbomachinery 130(1):011018 4. Knauss H, Roediger T, Bountin DA, Smorodsky BV, Maslov AA, Scrulijes J (2009) A Novel Sensor for Fast Heat Flux Measurements. J. Spacecraft&Rockets 46(2):255–265 5. Roediger T, Knauss H, Estorf M, Schneider SP, Smorodsky BV (2009) Hypersonic instability waves measured using fast-response heat-flux gauges. J. Spacecraft&Rockets 46(2):266–273 6. PCB Piezotronics, Pressure and Force Sensors Division(2007) Pressure Catalog. Avail. at. http://www.pcb.com/Linked Documents/Pressure/PFScat.pdf. Cited 20 Aug 2007 7. Heitmann D, Roediger T, Kaehler C, Knauss H, Radespiel R, Kraemer E (2008) DisturbanceLevel and Roughness-Induced Transition Measurements in a Conical Boundary Layer at Mach 6. AIAA-2008-3951, 26th Aerodyn. Meas. Techn. & Ground Test. Conf.

Interaction of noise disturbances and streamwise streaks Philipp Schlatter, Enrico Deusebio, Luca Brandt, and Rick de Lange

Abstract The evolution of disturbances in boundary layers modified through spanwise periodic, steady streamwise streaks is studied via numerical simulations. The disturbances are introduced via random two- and three-dimensional noise of various amplitudes close to the inlet (Rex ≈ 60000). The aim of the present work is to determine the impact of the interaction of streaks and noise on the arising flow structures and, eventually, on the location and details of the breakdown to turbulence. It is shown that large-scale 2D noise can be controlled via streaks, whereas the more general 3D noise configuration is prone to premature transition due to increased instability of the introduced streaks. It is interesting to note that the latter transition scenario closely resembles the flow structures found in bypass transition. Transition in true bypass transtion forced by ambient free-stream turbulence is also promoted by the addition of streamwise streaks in the laminar part of the boundary layer.

1 Introduction and Numerical Method The reduction of viscous drag acting on streamlined bodies in motion is an active field of research with great potential for technical applications. The corresponding energy saving can either be accomplished by a direct modification of the (usually turbulent) mean flow profiles, or by delaying the changeover from laminar to turPhilipp Schlatter Linné Flow Centre, KTH Mechanics, Stockholm, Sweden, e-mail: [email protected] Enrico Deusebio Linné Flow Centre, KTH Mechanics, Stockholm, Sweden, e-mail: [email protected] Luca Brandt Linné Flow Centre, KTH Mechanics, Stockholm, Sweden, e-mail: [email protected] Rick de Lange TUe Mechanical Engineering, Eindhoven, The Netherlands, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_57, © Springer Science+Business Media B.V. 2010

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Fig. 1 Frequency spectrum of large-scale and small-scale noise. The maximum spanwise scale based on the local boundarylayer thickness δ99 at the position of forcing (Rex = 60000) for the 3D noise cases is chosen as λz,max = 2.25δ99 and λz,max = 0.75δ99 for the large and small-scale noise, respectively.

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bulent flow. The latter method is persued in the present study. A recent theoretical and numerical study by Cossu and Brandt [3] has shown that a substantial stabilisation of a boundary layer subject to essentially two-dimensional disturbances (i.e. Tollmien-Schlichting (TS) waves) can be achieved by a spanwise modulation of the mean flow, i.e. via superimposed streamwise streaks on the laminar Blasius flow. In particular, it has been shown both experimentally via finite-amplitude roughness [4] and later via large-eddy simulation (LES, [7]) that transition to turbulence can effectively by moved to a more downstream position via this essentially passive control mechanism. However, the disturbances considered in the mentioned studies have all had their maximum energy in two-dimensional (spanwise invariant) modes. It is therefore interesting to examine the interaction of streamwise streaks with disturbences of a more general nature, i.e. 2D and 3D random noise at various frequencies and (spanwise) wavenumbers. In addition, the general case of ambient free-stream turbulence acting on the boundary layer is also considered.

2 Numerical Method The present study uses a spectral numerical method [2] together with the large-eddy simulation (LES) approach based on high-order filtering (ADM-RT model [8]), in a similar setup as presented in Refs. [7, 6]. In particular, the computational domain starts at Rex = 32000 extending up to Rex = 590000; the domain is discretised with 512 × 121× 128 grid points in a sufficiently wide and high domain. Periodic streamwise boundary conditions are achieved via a fringe region connecting outlet and inlet. The laminar streaks, characterised by a three-dimensional disturbance field, are introduced at the inlet as optimal disturbances computed from PSE (parabolised stability equations), subsequently evolving nonlinearly inside the domain. Conversely, the noise is forced within the computational domain at Rex = 60000 by a volume force acting in the wall-normal direction close to the wall. Two frequency spectra of the noise are shown in Fig. 1, specifying the frequencies and spanwise scales denoted “large-scale” and “small-scale” in the following. Note that for all simula-

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tions additional low-amplitude three-dimensional noise is included to enable random transition to turbulence. The amplitude of the noise is determined for each case in such a way that transition to turbulence, i.e. the appearance of a turbulent patch, could be observed inside the computational box (Rex < 590000) for the uncontrolled setup (no streaks). The same noise parameters were then used to investigate the interaction of the noise in the presence of streamwise streaks. Depending on the nature of the noise (twoor three-dimensional, scales), different amplitudes had to be chosen: 2D, fine-scale noise proved to be more efficient for reaching transition, requiring urms,noise ≈ 2.6%. On the other hand, 2D large-scale noise lead to transition only with the significantly larger amplitude of 18.3%. The amplitudes necessary for the 3D noise cases lie in between these values.

3 Results Directly connected to the different amplitudes required for the noise disturbances in the uncontrolled cases is the predominant transition scenario observed in the flow. Sample visualisations are shown in Figs. 2a) and 3a). It becomes clear that twodimensional small-scale noise in fact leads to the appearance of spanwise uniform waves, similar to TS-waves, see Fig. 2a). The growth associated to the TS-waves is sufficient to lead to non-linear breakdown within the computational box even for the mentioned low amplitudes. On the other hand, large-scale 2D noise does not excite

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Fig. 4 Blow-up of the region in the middle of Fig. 3 identifying the flow structures just prior to turbulent breakdown. Compare to Figs. 8c) and 12d) in Ref. [5].

any growing instability in the boundary layer at the considered Reynolds numbers due to the low frequency F < 80 and the two-dimensionality of the disturbances (i.e. no lift-up mechanism). Therefore, intermittent turbulent spots are directly triggered at the forcing position, which then grow downstream (not shown). For the cases with 3D noise, the dominant instability mechanism is non-modal growth based on the lift-up mechanism, generating streamwise streaks. Again, a dependence of the arising flow structures on the scales of the disturbances is observed: Fine scales tend to decay quickly, whereas larger scales lead to a flow with significant streamwise streaks, which then might get unstable developing a growing wave packet [5] and finally break down into triangle-shaped turbulent spots as seen in Fig. 3a) and as a blow-up in Fig. 4. The flow structures appearing in these cases are similar to observations in bypass transition induced by ambient free-stream turbulence and to an impulse response on a steady streak. Corresponding figures to be compared with Fig. 4 are given in, e.g., Figs. 8c) and 12d) in Ref. [5]. Note that in the presence of strong streaks – irrespective of whether they are actively introduced in the flow or naturally arise due to e.g. free-stream turbulence, the turbulent spots do not feature a clear triangular shape. The efficiency of the imposed streamwise streaks to damp the amplification of disturbances in the boundary layer is naturally strongly dependent on the nature of these disturbances and the respective transition scenario. The damping abilities of streaks has been demonstrated for cases with dominant two-dimensional waves, see the references mentioned above [3, 4, 7]. Consequently, for the case which leads to TS-wave dominated transition (2D small-scale noise), transition delay can indeed be observed when streamwise streaks are added, see Fig. 2b). However, as opposed to cases with clean TS-waves (for example F = 120 as in Ref. [7]) no complete stabilisation of the boundary layer can be achieved; intermittent turbulent spots are appearing further downstream as an instability of the strong streaks. This instability appears to be similar to the one discussed above and in Fig. 4, i.e. a wavepacket travelling on a (low-speed) streak. Quantitative data is given in Fig. 5 showing the disturbance growth (maximum streamwise disturbance urms,max ) inside the boundary layer. The top row shows the cases with 2D noise; transition delay is clearly seen for the case with small-scale noise (top right). The cases in which large-scale 2D noise is forced are not stabilised by the streaks. This is mainly related to large amplitudes needed for the noise (see above), which directly destabilises the boundary layer without any significant region of disturbance growth.

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On the other hand, three-dimensional noise produces a change of the disturbance growth mechanism in the boundary layer from modal (TS-waves) to non-modal (liftup, streaks). The spanwise periodic base flow created by forcing large-amplitude streaks does not lead to any transition delay, independent of the scales of the noise. To the contrary, turbulent spots can be observed more frequently and they appear further upstream, see Fig. 3b). As for the uncontrolled case, the transition scenario is a secondary instability of the streak, characterised by a growing wave packet riding on the streak [5]. However, due to the larger amplitude of the streaks in the case with superimposed steady streaks at the inlet, the streak breakdown occurs further upstream. Quantitativelz, in the bottom row of Fig. 5 these cases with 3D noise are shown; the breakdown location is clearly moving upstream as soon as streaks are introduced. It is interesting to note that an increase of the streak amplitude directly leads to an upstream movement of the transition location, which clearly supports the finding that a streak instability is the dominant cause of breakdown. In a last part, the interaction of streamwise streaks and ambient free-stream turbulence was studied. LES of bypass transition with a similar setup as in [6, 1] were performed. Sample results are shown in Fig. 6. Irrespective of the imposed streak amplitude, the transition location is always moved upstream compared to the uncontrolled case.

Fig. 6 Intermittency γ for bypass transition, with instensity of ambient free-stream turbuno lence Tu ≈ 4.7%. streaks (“uncontrolled”), streaks (amplitude 19%). Note that γ is defined based on laminar and turbulent correlations of the skin friction coefficient c f which explains values above unity during transition.

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4 Conclusions It can be concluded that the passive control mechanism based on a spanwise modulated base flow (streamwise streaks) and the subsequent damping of the growth of TS-waves [3] is very efficient for transition scenarios based on the modal growth of essentially two-dimensional disturbances. On the other hand, if transition is induced by (non-modal) growth related to bypass transition, then the addition of a strong base-flow modulation might lead to premature transition. These results have been obtained for oblique transition (no shown), a boundary layer subject to threedimensional noise disturbances and free-stream turbulence induced transition (bypass transition). Acknowledgements Computer time provided by SNIC (Swedish National Infrastructure for Computing) is gratefully acknowledged.

References 1. L. Brandt, P. Schlatter, and D. S. Henningson. Transition in boundary layers subject to freestream turbulence. J. Fluid Mech., 517:167–198, 2004. 2. M. Chevalier, P. Schlatter, A. Lundbladh, and D. S. Henningson. SIMSON - A Pseudo-Spectral Solver for Incompressible Boundary Layer Flows. Technical Report TRITA-MEK 2007:07, KTH Mechanics, Stockholm, Sweden, 2007. 3. C. Cossu and L. Brandt. Stabilization of Tollmien-Schlichting waves by finite amplitude optimal streaks in the Blasius boundary layer. Phys. Fluids, 14(8):L57–L60, 2002. 4. J. H. M. Fransson, A. Talamelli, L. Brandt, and C. Cossu. Delaying transition to turbulence by a passive mechanism. Phys. Rev. Lett., 96(064501):1–4, 2006. 5. P. Schlatter, L. Brandt, H. C. de Lange, and D. S. Hennginson. On streak breakdown in bypass transition. Phys. Fluids, 20(101505):1–15, 2008. 6. P. Schlatter, H. C. de Lange, and L. Brandt. The effect of free-stream turbulence on the growth and breakdown of Tollmien-Schlichting waves. In J. M. L. M. Palma and A. Silva Lopes, editors, Advances in Turbulence XI, pages 179–181. Springer, Berlin, Germany, 2007. 7. P. Schlatter, H. C. de Lange, and L. Brandt. Numerical study of the stabilisation of TollmienSchlichting waves by finite amplitude streaks. In R. Friedrich, N. A. Adams, J. K. Eaton, J. A. C. Humphrey, N. Kasagi, and M. A. Leschziner, editors, Turbulence and Shear Flow Phenomena 5, pages 849–854, 2007. 8. P. Schlatter, S. Stolz, and L. Kleiser. LES of transitional flows using the approximate deconvolution model. Int. J. Heat Fluid Flow, 25(3):549–558, 2004.

Experimental study on the use of the wake instability as a passive control in coaxial jet flows ¨ u, P. Henrik Alfredsson, and Alessandro Talamelli Antonio Segalini, Ramis Orl¨

Abstract The present paper verifies experimentally the theoretical result by Talamelli and Gavarini (Flow, Turbul. & Combust., 2006), who proposed that the wake behind the separation wall between the two streams of a coaxial jet creates the condition for an absolute instability. This absolute instability, by means of the induced vortex shedding, provides a continuous forcing mechanism for the control of the flow field. The potential of the found control mechanism as well as its robustness has been verified and its ability to enhance the turbulence intensity within the inner and outer shear layer has been demonstrated.

1 Introduction Over the past decades a variety of passive and active flow control devices has been tested and applied in a number of canonical as well as applied flow cases. An example for the latter is the coaxial jet flow, which has mainly been investigated regarding its receptivity to active flow control strategies (see e.g. [5] and [1]). Passive control strategies, albeit attractive in single jet flows, have mainly been disregarded, probably due to the complexity of the flow field in the near-field region of coaxial jet flows [3]. Physical and numerical experiments (see e.g. [4] and [8]) have established that the vortical motion in coaxial jet flows is dominated by the vortices emerging from the outer shear layer. The frequency of these vortices is related to the KelvinHelmholtz instability as predicted by linear stability analysis for single jets. The Antonio Segalini · Alessandro Talamelli Dept. of Mechanical & Aerospace Eng. (DIEM), Univeristy of Bologna, Forl´ı, Italy, e-mail: [email protected], [email protected] ¨ u · P. Henrik Alfredsson Ramis Orl¨ Linné FLOW Centre, KTH Mechanics, Stockholm, Sweden, e-mail: [email protected], [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_58, © Springer Science+Business Media B.V. 2010

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vortices in the inner shear layer, on the other hand, are trapped in the spaces left free between two consecutive outer shear layer vortices, and are therefore sharing the frequencies of the most amplified modes of the outer shear layer and do not relate to the values one would expect from linear stability analysis. This fact has become known as the “locking phenomenon”, which describes the mutual interaction of both shear layers. Nevertheless it is believed that only the outer shear layer is able to significantly control the evolution of the inner shear layer [10], which may explain the focus of control strategies on the outer shear layer. Besides the velocity ratio of the two streams, ru , and the Reynolds number, the thickness of the wall separating the inner and outer jet plays an important role in the evolution of transitional coaxial jet flows. Two trains of alternating vortices are found to be shed from both sides of the inner wall with a well-defined frequency, which scales with the wall thickness and the average velocity of the two streams [3]. In a recent study Talamelli and Gavarini [9] showed, by means of linear stability analysis, that the alternate vortex shedding behind the inner wall can be related to the presence of an absolute instability, which exists for a specific range of velocity ratios and for a finite thickness of the wall separating the two streams. The authors proposed that this absolute instability may provide a continuous forcing mechanism for the destabilisation of the whole flow field. The proposed idea of Talamelli and Gavarini [9], namely to test whether the vortex shedding behind a blunt inner wall of a coaxial jet nozzle can be utilised as a continuous forcing mechanism and hence as a passive flow control device for the near-field of coaxial jet flows, has motivated us to experimentally verify the proposed idea. The present paper briefly reviews the authors recent findings [7] and supplies further experimental results on the character of the instability as well as its parameter range, and thereby verifies the proposed idea and underlines its potential as a robust tool to passively control coaxial jet flows.

2 Experimental arrangement The experiments were carried out in the Coaxial Air Tunnel (CAT) facility in the laboratory of the Second Faculty of Engineering at the University of Bologna in Forl´ı by means of hot-wire anemometry and flow visualisations. The experimental details regarding the facility and measurement technique can be found in [7], to which the interested reader is referred to. In the present experiment, two different types of separation walls have been used. The first one has a thickness (t) of 5 mm and ends in a rectangular geometry, whereas the second one ends with a sharp trailing edge making the wall thickness negligible (t ≈ 0 mm) with respect to the sum of the side boundary layers thicknesses. These two separating walls will in the following be denoted as thick and sharp, respectively. The sharp and thick wall cases represent the flow cases in the absence and presence of the vortex shedding phenomenon, respectively, as can also be evinced from flow visualisation snapshots shown in figure 1.

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Fig. 1 Smoke flow visualisation at Uo = 4 m/s and ru = 1 for the sharp (left) and thick (right) wall.

r/Di = 0.6

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3 Results As apparent from the aforementioned snapshots, the presence of the vortex shedding, changes the flow topology not only in the vicinity of the nozzle exit, but over the entire near-field region. Spectral measurements, shown in figure 2 behind both the inner and outer duct wall confirm these observations. Furthermore, the “locking phenomenon”, which is commonly said to be present only for ru = Uo /Ui > 1 (with Uo and Ui denoting the mean exit velocity of the outer and inner jet streams, respectively) and used to denote the dominance of the outer shear layer on the inner one, is here shown to be reversible, i.e. the vortex shedding within the inner shear layer dictates its dominant instability frequencies on the outer shear layer. The state of the art understanding in coaxial jet flows, as for instance summarised in [2] or [6], presumes that the dominant frequencies of the inner and outer shear layer relate to the Kelvin-Helmholtz instability of the outer shear layer (particularly for ru > 1). However, as shown by linear stability analysis [9] and confirmed by recent hot-wire measurements by the present authors [7], a different scenario is found in the presence of an absolute instability, i.e. the presence of vortex shed-


364 2500

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Fig. 3 Fundamental frequency as function of Uo behind the inner and outer wall at ru = 1 and x/Di = 0.26 for the sharp (a) and thick (b) wall. The dashed line, denoted with StV S = 0.24, depicts the relation corresponding to the vortex shedding frequency.

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v′ /Uo [-] Fig. 4 Rms of radial velocity fluctuations along r/Di = 1 at Uo = 4, 8 and 12 m/s for the sharp (a) and thick (b) wall. In the present paper ru is kept constant at unity, i.e. Uo = Ui = Um .

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ding within a certain velocity ratio behind the inner separating wall. In this case, as clearly demonstrated in figure 3(b), the dominant frequencies of both the inner and outer shear layers vortices coincide and scale with the thickness of the separating wall, t, and the average velocity of both streams, Um , i.e. StV S = fmax t/Um = 0.24, whereas no clear relation is present for the sharp wall case. Spectral analysis, as for instance shown in figure 2, indicates that the energy content of the flow, in the presence of the vortex shedding, is enhanced not only by means of the emergence of stronger organised structures, but also by a drastic increase in the incoherent background turbulence [7]. This trend can also be seen in terms of integral quantities like the rms of the radial velocity fluctuations along r/Di = 1 for Uo = 4, 8 and 12 m/s as shown in figure 4. The results presented so far are restricted to a velocity ratio of unity, where the outer shear layer—according to the prevalent understanding (see e.g. [6])—is not necessarily dominating the evolution of the inner shear layer. Nevertheless, it is shown that the presence of the vortex shedding not only alters the structural composition within the inner and outer shear layer, but clearly controls the evolution of the vortices within the near-field of coaxial jet flows. What remains to be checked in order to verify the theoretical work by Talamelli and Gavarini is to identify the

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Fig. 5 Power spectral density distribution of the radial velocity fluctuations at x/Di = 0.25 and r/Di = 0.5 for different axisymmetric excitation frequencies indicated through the dashed line at different velocity ratios, viz. (a) ru = 0.25, (b) ru = 1.0, (c) ru = 1.6, and (d) ru = 4. Curves for different excitation frequencies are shifted upwards for clarity.

nature of the instability present in the coaxial jet configuration employing the thick separating wall. While blunt body wakes are absolutely unstable, mixing layers and detached free shear layers are convectively unstable, i.e. they act as noise amplifiers. This characteristic provides an ideal indicator in order to identify the character of the instability present behind a blunt separating wall in a coaxial jet configuration. By acoustically forcing the inner shear layer, spectral analysis can expose the nature of the instability within the present flow case. For this reason a loudspeaker has been placed at the rear end of the inner settling chamber, and the inner jet has been acoustically forced in an axisymmetric mode for a variety of velocity ratios at various frequencies. In the presence of an absolute instability the evolution of the dominant frequencies should—despite the applied excitation—follow the vortex shedding frequency, i.e. the clear linear relation given in figure 3(b). In the case of a convective instability, however, the vortical structures constituting a large fraction of the total energy in the flow, should organise or at least be affected by the frequency of the acoustic excitation. Figure 5 depicts the spectral distribution of the radial velocity fluctuations for four different excitation frequencies illustrated through dashed lines. While clear imprints of the excitation are present downstream the inner duct wall for ru = 0.25 and 4 (figures 5(a) and (d)), the vortex shedding frequency dominates and controls the evolution of the vortices for ru = 1.0 and 1.6 (figures 5(b) and (c)). The presence of an absolute instability by means of the vortex shedding mechanism behind a blunt inner separating wall explains the found insensitivity (figures 5(b) and (c)) of the present flow to the external forcing.

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4 Conclusions In summary we have demonstrated that the vortex shedding behind a blunt inner separating wall can—within a certain velocity ratio (0.4 ≤ ru ≤ 1.6 for the present coaxial jet configuration)—be related to an absolute instability, which in turn can be exploited to control the evolution of the vortices in both shear layers and hence the whole near-field region. The present study thereby verifies the theoretical work by Talamelli and Gavarini [9]. Furthermore it was shown, that in the presence of an absolute instability • the “locking phenomenon” is reversible i.e. the inner shear layer can trigger and dominate the evolution of the vortices in the outer shear layer, • the dominant frequencies in coaxial jet flows, do not have to relate to the KelvinHelmholtz instability of the outer shear layer, • the vortex shedding behind a blunt inner separating wall can be exploited as an robust and effective passive control device, which • increases the turbulence intensity within the inner and outer shear layers and thereby the mixing between the two coaxial jet streams as well as the annular jet with the ambient fluid. Acknowledgements The cooperation between KTH and the University of Bologna is supported by The Swedish Foundation for International Cooperation in Research and Higher Education (STINT), which is greatly acknowledged. Furthermore, Prof. G. Buresti is acknowledged for placing the coaxial jet facility to the disposal of the Second Faculty of Engineering of the University of Bologna.

References 1. Angele K P, Kurimoto N, Suzuki Y and Kasagi N (2006) Evolution of the streamwise vortices in a coaxial jet controlled with micro flap-actuators. J. Turbulence, 6:1–19 2. Balarac G, Métais O and Lesieur M (2007) Mixing enhancement in coaxial jets through inflow forcing: A numerical study. Phys. Fluids, 19:075102 3. Buresti G, Talamelli A and Petagna P (1994) Experimental characterization of the velocity field of a coaxial jet configuration. Exp. Thermal Fluid Sci., 9:135–146 4. Dahm W J A, Frieler C E and Tryggvason G (1992) Vortex structure and dynamics in the near field of a coaxial jet. J. Fluid Mech., 241:371–402 5. Kiwata T, Ishii T, Kimura S, Okajima A and Miyazaki K (2006) Flow visualization and characteristics of a coaxial jet with a tabbed annular nozzle. JSME Int. J. Ser. B, 49:906—913 6. Lesieur M (2008) Turbulence in fluids. Springer. ¨ u R, Segalini A, Alfredsson P H and Talamelli A (2008) On the passive control of the 7. Orl¨ near-field of coaxial jets by means of vortex shedding. Proc. of the Int. Conf. on Jets, Wakes and Separated Flows (ICJWSF-2), 1:1–7 8. da Silva C B, Balarac G and Métais O (2003), Transition in high velocity ratio coaxial jets analysed from direct numerical simulations. J. Turbulence, 4:1–18 9. Talamelli A and Gavarini I (2006) Linear instability characteristics of incompressible coaxial jets. Flow, Turbul Combust, 76:221–240 10. Wicker R B and Eaton J K (1994) Near field of a coaxial jet with and without axial excitation. AIAA J, 32:542–546

Numerical and Experimental Investigations of Relaminarizing Plane Channel Flow Daisuke Seki, Takayuki Numano, and Masaharu Matsubara

Abstract Dominant disturbances in a relaminarizing plane channel flow were investigated numerically and experimentally. Flow visualization demonstrates that with decreasing the Reynolds number the fully developed turbulent flow becomes more intermittent and forms patches of small-scale turbulent structures. In place of the turbulence, longitudinally elongated streaks appear just above the minimal Reynolds number. A hot wire measurement indicates that the energy of the streamwise velocity fluctuation is maintained for the long streamwise distance for the condition in which the longitudinal streaks are dominate. The numerical simulation result reveals that the longitudinal streaks is characterized as large-scale vortical structure placed in the center region of the channel. It is concluded that the streak disturbance is one of sustainable nonlinear modes in a plane channel flow.

1 Introduction In a plane channel flow, the critical Reynolds number is predicted at Re(= Um D/ν ) = 7696 by the linear stability theory, where Um is bulk mean velocity, D is a channel width and ν is the kinematic viscosity. When initial disturbances are adequately strong, the flow undergoes a transition to turbulence and turbulnet state is sustained even below the critical Reynolds number. With further decreasing Re, Daisuke Seki Department of Mechanical Systems Engineering, Shinshu University, 4-17-1, Wakasato, Naganoshi, Nagano, 380-8553, Japan. e-mail: [email protected] Takayuki Numano Hitachi Engineering & Services Co., Ltd., 3-2-2, Saiwai-cho, Hitachi-shi, Ibararki, 317-0073, Japan. Masaharu Matsubara Department of Mechanical Systems Engineering, Shinshu University, 4-17-1, Wakasato, Naganoshi, Nagano, 380-8553, Japan. e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_59, © Springer Science+Business Media B.V. 2010

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the disturbances in this turbulent state decay and the flow inversely transits to laminar flow if Re is less than a certain value. This value, in this paper, is defined as the minimal Reynolds number. It seems that phenomena around the minimal Reynolds number is exceedingly significant for not only theoretical exploration of the most persistent nonlinear mode against viscosity, but also understanding a sustain mechanism of disturbances in a quasi-equivalent turbulent, or disturbed, flow. It is also expected that in such a marginal disturbed flow there exist veiled key phenomena that inspire us to describe fully developed turbulent share flows. Carlson et al. [1] investigated development of an artificially triggered turbulent spot in a plane channel flow and confirmed that the minimum Re for the spot generation is 1300. Alavynoon et al. [2] extended the Re range of Carlson et al. [1] to higher Re, then they found that for Re < 1500 the spot cannot be generated even if the initial disturbance is very large. Nishioka and Asai [3] also demonstrated the minimal transition Re is of 1300. Iida and Nagano [4] numerically simulated a relaminarizing plane channel flow and evaluated the turbulent statistics. In the present investigation, flow visualization, a hot-wire measurement and a direct numerical simulation (DNS) were performed to observe the characteristic disturbance structure in the relaminarizing plane channel flow qualitatively.

2 Numerical Method and Experimental Setup 2.1 Numerical Method The lattice kinetic scheme (LKS) [5], which is an improved scheme of the lattice Boltzmann method (LBM) [6], is adopted for the DNS. The LBM is based on the microscopic models and the mesoscopic kinetic equations. The advantages of the LBM are simplicity of the algorithm and suitability for the parallel computing in addition to its robustness and flexibility. The LKS reduces the variables to be calculated so that the computational memory can be saved. Details of the LKS algorithm are in Ref.[5]. For the present simulation, no-slip boundary conditions are imposed at the wall, and periodic boundary conditions are for the streamwise and spanwise boundaries. The coordinate system is defined as streamwise x, wall-normal y and spanwise z, and it is in common with those in the experiment with the origin at the inlet of the test section. The computational domain sizes are 12 D for x and 3 D for z. A 1440 × 120 × 360 grid system in x, y and z is used, respectively.

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Table 1 Specification of the water and air channel facilities.

Channel width D Test-section length Test-section span width Expantion rate

Water channel

Air channel

D = 10 mm 350 D 60 D 20 %

D = 5 mm 560 D 52 D 37 %

2.2 Experimental Facility and Measurement Technique In this study, two channel flow facilities were used. One is a circulating water channel and the other is an open air channel. Both channels consist of nozzle, entrance, expansion and test sections. Working fluids of water and air are circulated by frequency-controlled pump and blower, respectively. Two tripping wires of 0.5 mm diameter are placed 100 mm downstream from the exit of the nozzle in order to obtain the fully developed flow in the entrance section. The Reynolds number decreases by 2 degree divergence for each end wall in the expansion section. Dimensions of the water and air channels are shown in Table 1. For the flow visualization 8 µ m pearl flakes were added to the water. A 500 W halogen lamp for lighting and a 0.31-megapixel video camera of a 60 frames per second were used. A hot-wire measurement was made in the air channel. A sensor part of the hot-wire probe is made of 2.5 µ m diameter platinum wire of a 1.5 mm length. A traverser with a slanting mechanism enables both streamwise and wallnomal movements of the probe in the test section.

3 Results Fig.1 shows flow visualization results at Re = 1400, 1450, 2000 and 3000. The flow direction is from left to right. At Re = 3000, the small-scale disturbances, which are considered as turbulent structure, are filled with the whole flow. With decreasing Re to 2000, there coexist streaks of a relatively large scale and the small-scale turbulent structure. With decreasing Re, the ratio of the large-scale streak area as well as the spanwise scale of the streaks increases. At Re = 1450, the streaks are dominant and patches filled with the small-scale turbulent structure are formed. Quasi-laminar areas appear and the small-scale turbulent structure completely disappears at Re = 1400. The fully laminar state of the flow at Re = 1250 is confirmed. One can have a question whether the large-scale streak is sustainable even at low Re where the turbulent structure can not survive. To answer this, the streamwise variations of the streamwise velocity component energy are measured from both hot wire result and DNS. The energy estimated with both DNS and hot wire measurements is normalized by the value E0 at x/D = 120 and tUm /D = 120, respectively. The DNS results show the sustainable flow for Re > 1450 as well as the

370


Fig. 1 Flow visualization of the relaminalizing plane channel flow. (a) Re = 1400; (b) Re = 1450; (c) Re = 2000; (d) Re = 3000. Images in the area of 24D × 18D are at x/D = 330.

measurements. The energy variations for Re 6 1400 in the DNS are underestimated in comparison with those of the experiment. It seems that this disagreement is due to the limited sizes of the computational domain and the periodic boundary conditions. In experimental results the energy variation at Re = 1500 is almost constant. At Re = 1400 the energy still keeps constant for more than 200D after temporal decay until x/D = 250. For Re 6 1300, the energy decays exponentially. From the constant energy in the streamwise direction, one can straightforwardly conclude that the large-scale streaky structure is sustainable at Re = 1400. For comparison between the turbulent and streaky structures, the disturbance velocity fields in the DNS are surveyed with typical velocity vector maps in the y-z plane and instantaneous streamwise velocity distributions in the x-y plane at Re = 2320 and 1430 as shown in Figs. 3 and 4. The streamwise velocity distribution at Re = 2320 considerably resembles to the flake pattern in the flow visualization at Re = 3000, in terms of fullness in the flow of the small-size streaks of the spanwise scale less than half channel width. As seen in the y-z velocity vector map, the streaks has vortical structures and are positioned close to one of the walls. In contrast to the turbulent case, the disturbance at Re = 1430 is characterized as a longitudinally elongated vortex as shown in the streamwise velocity distribution and the vector map of Fig. 4. The vortex core is located in the center region of the channel.

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Fig. 2 The streamwise variations of the streamwise velocity component energy.

Fig. 3 Distribution of the instantaneous streamwise velocity in the x-y plane (upper) at y = 0.5 D and velocity vector map of the wall-nomal and spanwise velocity components in the y-z plane (bottom) at Re = 2320.

It is concluded that the sustainable large-scale streaky structure has different properties and is distinguishable from the turbulent structures in the plane channel flow.

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Fig. 4 Distribution of the instantaneous streamwise velocity in the x-y plane (upper) at y = 0.5 D and velocity vector map of the wall-nomal and spanwise velocity components in the y-z plane (bottom) at Re = 1430.

4 Conclusion In the present study, flow visualization, a hot wire experiment and a DNS were carried out for investigation in a relaminarizing plane channel flow. A longitudinally elongated vortex disturbance just above the minimal Reynolds number are observed and the energy of the streamwise velocity fluctuation is maintained for the long streamwise distance more than 200 D. The disturbance is characterized as a largescale vortical structure located in the center region of the channel. It is concluded that this disturbance is one of sustainable nonlinear disturbances in a plane channel flow.

References 1. Carlson, D.R., Windall, S.E., Peeters, M.F.: A flow-visualization study of transition in plane Poiseuille flow, J. Fluid. Mech., 121, 487–505 (1982) 2. Alavyoon, F., Henningson, D.S., Alfredsson, P.H.: Turbulent spots in plane Poiseuille flowflow visualization, Phys. Fluids, 29, 1328–1331 (1986) 3. Nishioka, M., Asai, M.: Some observations of the subcritical transition in plane Poiseuille flow, J. Fluid Mech., 150, 441–450 (1985) 4. Iida, O., Nagano, Y.: The relaminarization mechanisms of turbulent channel flow at low Reynolds numbers, Flow Turbul. Combust. 60, 193–213 (1998) 5. Inamuro,T.: A lattice kinetic scheme for incompressible viscous flows with heat transfer, Phil. Trans. R. Soc. Lond. A, 360, 477–484 (2002) 6. Chen, S., Doolen, G.D. : Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 30, 329–364 (1998)

Linear control of 3D disturbances on a flat-plate Onofrio Semeraro, Shervin Bagheri, Luca Brandt, and Dan S. Henningson

Abstract Using a number of localized sensors and actuators, a feedback controller is designed in order to reduce the growth of three-dimensional disturbances in the flatplate boundary layer. A reduced-order model of the input-output system (composed of the linearized Navier–Stokes equations including inputs and outputs) is computed by projection onto a number of balanced truncation modes. It is shown that a model with 50 degrees of freedom captures the input-output behavior of the highdimensional (n ∼ 107 ) system. The controller is based on a classical LQG scheme with a row of three sensors in the spanwise direction connected to a row of three actuators further downstream. The controller minimizes the perturbation energy in a spatial region defined by a number of (objective) functions.

1 Input-output configuration The three-dimensional input-output configuration considered is the extension of the two-dimensional case studied in [2, 3]. We focus on the dynamics and control of small amplitude perturbations about a steady base flow. The main numerical tool is a pseudo-spectral code that provides solutions of the linearized Navier-Stokes equations and its associated adjoint equations. For further details of the code, boundary conditions etc, we refer to [1, 2] . The computational domain has the dimensions (Lx , Ly , Lz ) = (500, 20, 160) and a resolution of 384 × 81 × 80 grid points; the fringe region starts at x = 400. The Reynolds number is Re = U∞ δ0∗ /ν = 1000, where δ0∗ is the inflow displacement thickness. The plant (shown schematically in Fig. 1), written in an input-output form reads,

Linné Flow Centre, KTH Mechanics, Stockholm, Sweden. e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_60, © Springer Science+Business Media B.V. 2010

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u˙ = Au + B1 w + B2 u z = C1 u + lu v = C2 u + α g

(1) (2) (3)

where u is the velocity field, A ∈ Rn×n is the discretized and linearized NavierStokes equations, whereas the vector B1 ∈ Rn×1 and the matrix B2 ∈ Rn×3 provide the spatial distributions of the incoming disturbance upstream and the (three) actuators downstream. The output signals are extracted via the matrices C1 ∈ Rk×n and C2 ∈ R3×n that define the spatial distributions of the sensors. The scalar quantities α and l are penalties of measurements noise g(t) and control signal u(t) (see [2]). The system (1) is stable since all the eigenvalues of A are strictly to the left of the imaginary axis on the complex plane. However, the system is characterized by sensitive dynamics as it acts as an amplifier of disturbances. The upstream disturbance consist of the optimal localized initial condition computed by Monokrousos et al. [4], that provides the largest energy growth over a given time. For long time, the optimal initial condition is a three-dimensional wave-packet of Tollmien-Schilichting (TS) waves triggered by upstream tilted structures exploiting the Orr-mechanism. As the three actuators, we use localized volume forcing in the form of TS-like wavepackets. The three sensors in the spanwise direction are also modelled as localized wavepacket structures. The k sensors C1 , located further downstream are used to define the objective functional

Fig. 1 Sketch of the input-output configuration considered. The disturbance (B1 ) is modelled as a localized TS-wavepacket located upstream at (xB1 , yB1 , zB1 ) = (20, 1, 0). A spanwise row (C2 ) of three sensors equally spaced along z (∆z = 40) are used for estimation; the center sensor is located at (xC2 , yC2 , zC2 ) = (150, 1, 0). The actuator row (B2 ) has a similar configuration with the center actuator located at (xB2 , yB2 , zB2 ) = (200, 1,0). A centralized controller is designed, i.e. all actuators are connected to sensors as shown by the inset figure. Further downstream a region, that is spanned by a number of basis functions (C1 ), is used to evaluate the disturbance dynamics and thus acts as an “objective function”.

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Fig. 2 Streamwise velocity component (positive is shown in black and negative in gray) of the basis functions C1,k with k = 1, . . . , 4 given in equation (5) projected onto divergence-free subspace.

kzk2 =

Z T 0

|C1 u|2 + l 2 |u|2 dt.

(4)

The aim of the controller is to determine the input signal u(t), based on noisy sensor measurement v(t) such that the above objective is minimized. Note that in this inputoutput framework, the controller minimizes the disturbance energy in a subspace of the domain, spanned by the basis {C1,1 , . . . , C1,k }. One choice of basis (the so-called output projection [5]) are the POD modes obtained from the impulse response of all the inputs. This basis is empirical, i.e. it accurately represents the data used to generate it. Using output projection with few leading POD modes, the controlled system shows significantly smaller output signals z(t) compared to the open-loop system. However, for three-dimensional disturbances, this does not correspond to an actual reduction of the total kinetic energy of the perturbation. Instead of including a very large number of POD modes, alternatively, a set of spanwise Fourier modes (see Fig. 2) localized in the streamwise and wall-normal directions can be used as to define a basis, of the form Z 2π (k − 1)z 2 2 2 2 u dxdydz. (5) C1,k u = (0, exp −(x − x0 ) /σx − y /σy , 0 cos Nz Ω Four modes – from k = 1 to k = 4 – localized around x = 300 are used in the present configuration.

2 Model reduction In order to design a feedback controller, it is sufficient to capture the input-output (I/O) behavior of the system, rather than the entire perturbation dynamics. For a small number of inputs and outputs, the I/O behavior of a linear system can be

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Fig. 3 Streamwise velocity (positive is shown in black and negative in gray) of the first (top row) and 10th (bottom row) balanced mode (left) and their associated adjoint mode (right).

described by a reduced-order model obtained via the balanced truncation method [6]. The reduced model retains the states affected easily by the inputs (controllable states) and the states that contribute the most to the outputs (observable states). Essentially, the method amounts to an oblique projection of the system (1), onto a number of so-called balanced modes which can be computed for high-dimensional plants using the snapshots method proposed in [5]: snapshots are collected from the impulse response of each input via a forward simulation, and of each output via a simulation of the adjoint system followed by one singular value decomposition (of the size of number of adjoint snapshots times forward snapshots). Two balanced modes (first and 10th) and their associated adjoint modes are shown in Fig. 3. The first balanced mode is nearly two-dimensional and takes the shape of a TS wave-packet with a large amplitude downstream. This spatial structure is trigged with the least energy by the input B1 . Its corresponding adjoint mode is essentially two-dimensional with its largest amplitude upstream. This structure, on the other hand, generates the largest response in the sensors C1 . The higher balanced modes (bottom row in Fig. 3) look similar to the first mode, but are mainly characterized by different spatial wavelengths. A reduced-order model of order 50 is found to capture the behavior between all the inputs and all the outputs of the Naiver-Stokes system of order 107 . An example of the performance of the reduced-order model is shown in Fig. 4. With an impulse in B1 , a (optimal) disturbance is introduced in the boundary-layer upstream that grows as it is convected in the downstream direction. The sensor outputs z1 (t) and z2 (t) extracted by the sensors C1,1 and C1,2 respectively, is shown with black solid lines. After an time-delay the sensors register a wave-packet; the signal eventually decays to zero as the disturbance leaves the computational box. In the same figure, the output signals computed using the reduced-order model is shown (circles) , where an impulse in the input of the reduced-order model (Bˆ 1 ) results in the same response (extracted via the reduced sensors Cˆ1,1 and Cˆ1,2 ) as the full NavierStokes system, albeit the significant order reduction. The approximate Hankel sin-

Linear control of 3D disturbances on a flat-plate

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gular values (not shown here) decay rapidely and the leading singular values come in pair similar to observations in previous studies [2]. A thorough analysis of the performance of the reduced-order and the model-reduction error will be presented elsewhere.

3 Controller design The reduced-order model can be used to design a controller of low-order that will run “online”, next to the numerical experiments. Here, a classical Linear-QuadraticGuassian (LQG) (see e.g. [7] for introduction in control theoretical tools from a fluid mechanics viewpoint) is designed, where all three sensors used for estimation (C2 ) are connected to all three actuators B2 . Such a centralized controller minimizes the energy of the output signals (4) and more importantly the resulting closed-loop is guaranteed to be stable. A de-centralized controller – when the control signal of each actuator is based only on the output from the sensor located upstream and at the same spanwise location – was found both by RGA analysis [8] and by numerical experiments to result in an unstable closed-loop. This is partly due to the fact that localized disturbance introduced in the boundary-layer spreads (or widens) in the spanwise direction as it is convected downstream, resulting in a strong coupling in the spanwise direction. The performance of the controller is shown in Fig. 5. The r.m.s. (streamwise velocity component integrated in spanwise and wall-normal directions and time) when forced upstream with temporal white noise is compared for three linearized DNS; in solid black the exponential growth of optimal TS wave-packet in the streamwise direction is observed; the dash-dotted and dashed lines show the disturbance development when the controller is active, i.e. when the measurements from the three sensors (C2 ) upstream are fed into a controller that provide the three actuators further downstream (B2 ) a control signal. The dashed line represents a “cheap” controller with l = 10, whereas the dashed-dotted line is “expensive” controller with l = 100 (for both controllers α = 0.1). Near the location (x = 200) of the actuators the growth of perturbations is transformed into a decay; further downstream the perturbations again begin to growth, but their overall amplitude is reduced.

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Fig. 5 R.M.S of the streamwise velocity component of the uncontrolled system (black line), the cheap controller (dashed) and expensive controller (dotted-dashed line).

In summary, a reduced-order model of order 50 is able to capture the input-output behavior between three-dimenstional disturbances, actuators, sensors and “objective functions”. Using this model, efficient control strategies can be designed in order to damp the growth of small-amplitude perturbations inside the boundary-layer. A number of improvements are currently under investigation. The spatial structure of the sensors and actuators will be chosen in order to reflect what actually can be achieved in a practical experimental implementation (for instance with plasma actuators). Also, the choice of basis defining the objective functions in this study, was rather arbitrary and can be improved.

References 1. Chevalier, M., Schlatter, P., Lundbladh, A. and Henningson, D. S. A pseudo spectral solver for incompressible boundary layer flows. Technical Report, Trita-Mek 7, 2007. 2. Bagheri, S., Brandt, L., Henningson, D.S., Input-output analysis, model reduction and control of the flat-plate boundary layer. J. Fluid Mech, Vol 620,263-298, 2009. ˚ 3. Bagheri, S., Akervik, E., Brandt, L., Henningson, D.S., Matrix-free methods for the stability and control of boundary layers. AIAA J., Vol 47,1057-1068, 2009. ˚ 4. Monokrousos, A., Akervik, E., Brandt, L., Henningson, D.S., Global optimal disturbances in the Blasius boundary-layer flow using time-steppers. Submitted to the J. of Fluid Mech, 2009. 5. Rowley, C.W. Model reduction for fluids using balanced proper orthogonal decomposition. Int. J. of Bif. Chaos, 15(3):997-1013, 2005. 6. Moore, B. Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Trans. Aut. Control, 26:1732, 1981. 7. Bagheri, S., Hœpffner, J., Schmid, P.J., Henningson, D.S., Input-Output Analysis and Control Design Applied to a Linear Model of Spatially Developing Flows. Appl. Mech. Rev., Vol 62 (2), 2009. 8. T. Glad and L Ljung, Control Theory, multivariable and Nonlinear Methods. Taylor and Francis.

Experimental study of stability of supersonic boundary layer on swept wing N. V. Semionov, A. D. Kosinov, and Yu. G. Yermolaev

Abstract The paper is devoted to an experimental study of laminar-turbulent transition and instability disturbances evolution in a three-dimensional supersonic boundary layer on swept wing with sharp leading edge. The detailed data of natural disturbances development are obtained for the first time. Characteristic zones of disturbances evolution are determined. A position of instability region of secondary flow is experimentally defined. Some features of disturbances evolution, characteristic only for a supersonic boundary layer are revealed. It was shown experimentally that secondary cross-flow instability plays the main role in laminar-turbulent transition in 3-D supersonic boundary layers.

1 Introduction The attention of researchers in various countries is focused on the problem of transition to turbulence in spatial boundary layers. This interest arises from the practical applications of this phenomenon, in particular, because similar boundary layers are observed in the flow around a swept wing of an airplane. Most theoretical and experimental results on stability and transition control of a three-dimensional boundary layer are obtained for subsonic flow. However very few theoretical investigations [1, 2, 3] of supersonic 3 D boundary layer stability have been fulfilled up N.V. Semionov Institute of Theoretical and Applied Mechanics, Novosibirsk, Institutskaya 4/1, Russia, e-mail: [email protected] A.D. Kosinov Institute of Theoretical and Applied Mechanics, Novosibirsk, Institutskaya 4/1, Russia e-mail: [email protected] Yu.G. Yermolaev Khristianovitch Institute of Theoretical and Applied Mechanics, Novosibirsk, Institutskaya 4/1, Russia, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_61, © Springer Science+Business Media B.V. 2010

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to date.Linear stage of cross-flow instability in relation to stationary and unsteady disturbances was investigated theoretically by Gaponov and Smorodsky [4]. Direct quantitative comparison of theory with our experiments [5] was presented. A good agreement of the theory with measurements performed in wing tunnel T–325 has been obtained only for spanwise scales of cross-flow vortices. However computed growth rates differ significantly from measurements. Stability of supersonic boundary layer on swept wing was studied experimentally only in ITAM [5, 6, 7].Evolution of natural fluctuations in the boundary layer on a swept wing was studied by Ermolaev et. al. [6].It was shown that the character of distribution of the mean and fluctuating characteristics of the boundary layer is similar to the case of subsonic velocities. It was obtained at M = 2, that the disturbances growth in three-dimensional boundary layer occurs much faster, than in the flat plate case. The results of an experimental study of evolution of controlled disturbances on a swept-wing model for Mach number M = 2 are presented by Semionov et. al. [5, 7]. The wave characteristics of traveling waves are obtained. The evolution of disturbances at frequencies of 10, 20, and 30 kHz is similar to the development of traveling waves for subsonic velocities. The angle of inclination of the wave vector for energy-carrying disturbances is directed across the flow, and the group-velocity vector is aligned with the steady cross-flow disturbance. In this paper some results of experimental study of stability of supersonic boundary layer on swept wing are presented.

2 Experimental equipment The experiments were conducted at the Institute of Theoretical and Applied Mechanics of the Siberian Division of the Russian Academy of Sciences in the wing tunnel T–325 supersonic wind tunnel with test-section dimensions 0.2 × 0.2 × 0.6 m at Mach numbers M=2.0 and unit Reynolds number Re1 = 5 × 106 m−1 . The model length was 0.4 m, its width was 0.2 m, and the maximum thickness was 12 mm. The model was mounted at zero incidences in the central section of the test section of the wind tunnel. The oscillations were measured by a constant-temperature hot-wire anemometer. Single-wire tungsten probes of diameter 10 µ m and length 1.2 mm was used. The overheat ratio of the wire was 0.8, and the measured disturbances corresponded to mass-flow fluctuations. The fluctuating and mean characteristics of the flow were measured by an automated data acquisition system. The fluctuation signal from the hot-wire anemometer was measured by a 12–bit A/D converter with a digitization on time 1.33 µ s, and a mean voltage was fixed by a voltmeter. The length of each realization was 65536 points. With the help of the discrete Fourier transform (DFT) on time t the amplitude-frequency spectra were determined: e′f (x′ , z′ , y) =

1 T

∑ e′f (x′ , z′ , y, tk ) exp[iω tk ]∆ tk k

(1)

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381

where T — length of digital time trace, ∆ tk = tk−1 − tk and e′ (x′ , z′ , y, tk ) — digital oscillogram of a pulsation signal from a hot–wire anemometer. Values of the mass flow fluctuations ρ U were determined by the method described in [8].

3 Stability Up to now stability of supersonic boundary layer on swept wing was studied experimentally only in ITAM. Some interesting data on evolution of natural and controlled disturbances were presented in [5, 6, 7]. But all these data were obtained for wing with 7.8 % profile, where transitional Reynolds number is equal Retr = 106 . Complex structure of disturbances and very fast changing of structure of traveling and stationary disturbances was observed in [5, 6, 7]. All this strongly complicates research of stability of a three-dimensional boundary layer. Therefore it has been decided to spend new experiments on a thin wing and at low unit Reynolds number. In this case the distance from a leading edge up to a point of transition becomes more in some times in comparison with the previous experiments [5, 6, 7]. It has allowed investigating in detail disturbances evolution in supersonic boundary layer on swept wing, especially at an initial linear stage.

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Evolution of natural disturbances in supersonic boundary layer of swept wing was investigated in detail for the first time. Oscillogram, amplitude-frequency spectra, mean velocity profiles, pulsation profiles and statistical diagrams of natural fluctuations were obtained. Streamwise disturbances evolution is shown in fig.1. Characteristic regions of disturbances development were determined. Growth of disturbances was observed approximately from x=100 mm, that corresponds to

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Rex ≥ 0.5 × 106 . it is shown from fig.1 that laminar-turbulent transition took place approximately at x/c ≈ 0.7 ( Retr ≈ 1.3 × 106 ), where c - chord of the wing, x longitudinal coordinate from the leading edge. Measurements were spent in the field of stable fluctuations of supersonic boundary layer on swept wing (50 mm < x < 100 mm, Rex = 0.25 × 106 ÷ 0.5 × 106 ) at this experimental set-up for the first time. Amplitude-frequency spectra are plotted in fig.2 at Rex = 0.35 × 106 (x=70 mm) and fig.3 at Rex = 0.4 × 106 (x=80 mm) for several meaning of normal coordinate. As a result of measurements it was revealed, that on an initial stage (Rex = 0.25 × 106 ÷ 0.35 × 106 ) spectra of disturbances remind a case of a flat plate. But some excitation of pulsations in a range of frequencies from 10 up to 30 kHz was detected at x=80 mm at values of normal coordinate close to the critical layer (y=0.53 mm). These disturbances correspond to an instability mode of cross flow and are observed at Rex ≈ 0.35 × 106 for the first time. So it is possible to say that experimentally defined a position of instability region of cross flow. At Rex ≥ 0.5 × 106 growth of traveling disturbances was observed near to the surface of the wing, and at Rex ≥ 0.7 × 106 across all boundary layer. Amplitude-frequency spectra of disturbances in dependence of longitudinal coordinate x are presented in fig.4. The measurements were conducted in the layer of maximum fluctuations. At increasing of Reynolds number there was an intensive excitation and growth of pulsations in a range of frequencies from 10 up to 30 kHz on an initial stage of disturbances development till 10 ÷ 80 kHz near to transition location. The excitation of high-frequency pulsations was detected. Nonlinear effects were observed at Rex ≥ 1.0 × 106 , and above the critical layer too. Nonlinear effects is planned to study with the help of bispectral analysis. Measurements across boundary layer were made for all mentioned above zones of disturbances evolution. Obtained distributions of mass flux pulsation < m′ > in

Experimental study of stability of supersonic boundary layer on swept wing

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dependence of normal coordinate y are presented in fig. 5. Dependencies of m versus y have two maxima, first corresponds to critical layer, second - subsonic layer. Two maxima in dependencies were observed in the case of flat plate too. But in supersonic boundary layer on flat plate second maximum was lesser than maximum in critical layer. Fast growth of disturbances corresponding to the second maximum was observed in the case of swept wing in nonlinear region of disturbances evolution. And in region close to transition location second amplitude of disturbances approximately the same for both maxima. On the other hand distribution of mean voltage E from hot-wire output in dependence of normal coordinate y have some difference near surface of the swept wing from the ones for the case of flat plate. The reason of this difference is cross flow. So we can say that that cross-flow instability plays the main role in laminar-turbulent transition in supersonic boundary layers on swept wings.

4 Conclusions The detailed data of disturbances development up to transition location are obtained for the first time. Characteristic zones of disturbances evolution are determined. A position of instability region of secondary flow is experimentally defined. Some features of disturbances evolution, characteristic only for a supersonic boundary layer are revealed. It is confirmed, that the basic mechanism of occurrence of turbulence in a supersonic boundary layer on a swept wing - instability of cross flow.

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Acknowledgements This work has been supported by the RFBF grant 08–01–00124.

References 1. M.R. Malik, F. Li, C.L. Chang, in: Nonlinear instability and transition in three-dimensional boundary, edited by P.W.Duck and P.Hall (Kluwer Academic Publishers, Dodrecht etc., 1996), pp. 257–266. 2. Ch. Mielke, L. Kleiser, in: Laminar-turbulent Transition, edited by W.Saric and H.Fasel ( Springer-Verlag, Berlin, 2000), pp. 397–402. 3. L.N. Cattafesta, V. Iyer, J.A. Masad, R.A. King, J.R. Dagenhart, Three-dimensional boundary-layer transition on a swept wing at Mach 3.5, textitAIAA J, textbf33(11), 2032– 2037 (1995). 4. S.A. Gaponov, B.N. Smorodsky, ”On linear instability of three-dimentional compressible swept-wing boundary layer”, in: Proc. of the Int. Conf. on Methods of Aerophys. Research, Pt 4, Novosibirsk, 2007, pp. 28–34. 5. N.V. Semionov, Yu.G. Ermolaev, A.D. Kosinov, V.Ya. Levchenko, Experimental investigation of development of disturbances in a supersonic boundary layer on a swept wing, Thermophysics and Aeromechanics, Vol. 10, No 3, 2003, pp. 347–358. 6. A.D. Kosinov, N.V. Semionov, Yu.G. Ermolaev, V.Ya. Levchenko, Experimental study of evolution of disturbances in a supersonic boundary layer on a swept wing model under controlled conditions, J.App. Mech. and Tech. Phys., Vol. 41, No 1, 2000, pp. 44–49. 7. N.V. Semionov, A.D. Kosinov, Yu.G. Ermolaev, Evolution of disturbances in a laminarized supersonic boundary layer on a swept wing, J. App. Mech. and Tech. Phys. Vol. 49, No 2, 2008, pp. 40–47. 8. A.D. Kosinov, N.V. Semionov, Yu.G. Yermolaev, Disturbances in test section of T-325 supersonic wind tunnel. Preprint Institute of Theoretical and Applied Mechanics, No 6–99, Novosibirsk, 1999, 24 p.

Comparison of Direct Numerical Simulation with the Theory of Receptivity in a Supersonic Boundary Layer Vitaly G. Soudakov, Ivan V. Egorov, and Alexander V. Fedorov

Abstract Two-dimensional direct numerical simulation (DNS) of receptivity to acoustic disturbances radiating onto a flat plate with a sharp leading edge in the Mach 6 free stream is carried out. Different angles of incidence of fast and slow acoustic waves are considered. DNS results are compared with theoretical modeling of leading-edge receptivity and downstream propagation of boundary-layer disturbances.

1 Introduction In quiet free streams, transition on aerodynamically smooth surfaces includes receptivity, linear phase and nonlinear breakdown to turbulence. Receptivity refers to the mechanism by which free-stream disturbances enter to the laminar boundary layer and generate unstable waves of certain initial amplitudes [9]. Because the boundary layer behaves as a disturbance amplifier, knowing the initial amplitudes is an equally critical component of transition prediction as the amplification rates. Theoretical studies [8] and stability experiments [6] showed that the second mode becomes the dominant instability mode at sufficiently high Mach numbers M (for the boundary layer on an insulated wall at zero pressure gradient, this occurs for M>4). In contrast to the first mode, the growth rate of second mode is maximal for two-dimensional waves. A theoretical model of receptivity to acoustic disturbances radiating a sharp leading edge of a flat plate in supersonic flow was developed in [4, 5]. This model proVitaly G. Soudakov TsAGI, Zhukovsky, RUSSIA, e-mail: vit [email protected] Ivan V. Egorov TsAGI, Zhukovsky, RUSSIA e-mail: ivan [email protected] Alexander V. Fedorov TsAGI, Zhukovsky, RUSSIA, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_62, © Springer Science+Business Media B.V. 2010

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vides insight into the physics of receptivity associated with diffraction and scattering of acoustic waves in the leading edge vicinity. The boundary-layer mode excited near the leading edge by a fast acoustic wave can be referred as mode F, and by a slow acoustic wave as mode S. A series of numerical studies related to receptivity and stability of high-speed flows over a flat plate conducted in [7]. Two-dimensional DNS of receptivity to acoustic waves was carried out in [7] using a high-order shock-fitting scheme. However this scheme is not applicable to a small region near the leading edge, which may play important role in receptivity, especially when acoustic waves radiate the leading edge from below. DNS [10] of receptivity in a boundary layer over a sharp wedge was carried out with two-dimensional perturbations at free stream Mach number 8. The perturbation flow field was decomposed into normal modes with the help of the biorthogonal eigenfunction system. Filtered-out amplitudes of two discrete normal modes and of the fast acoustic modes are compared with the linear receptivity problem solution. In this paper, we discuss DNS of receptivity to slow and fast acoustic waves radiating a flat plate in Mach 6 free stream. DNS results are compared with theoretical modeling [4, 5] of leading-edge receptivity and downstream propagation of boundary-layer disturbances.

2 Problem formulation Viscous two-dimensional unsteady compressible flows are governed by the NavierStokes equations. The fluid is a perfect gas with the specific heat ratio γ = 1.4 and Prandtl number Pr = 0.72. The viscosity-temperature dependence is approximated by the power law µ ∗ /µ∞∗ = (T ∗ /T∞∗ )0.7 . Calculations are carried out for supersonic flow over a flat plate with sharp leading edge at the free-stream Mach number M∞ = 6 and the Reynolds number Re∞ = ρ∞∗ U∞∗ L∗ /µ∞∗ . Hereafter ρ∞∗ is free-stream density, U∞∗ is free-stream velocity, L∗ is plate length, asterisks denote dimensional variables. Flow variables are made nondimensional using steady-state free-stream parameters and plate length, pressure is made nondimensional using doubled dynamic pressure. The computational domain is a rectangle with its bottom side corresponding to the plate surface. The no-slip boundary conditions are imposed on the plate surface. The wall temperature corresponds to the adiabatic condition for the steadystate solution. At first the steady-state solution of the supersonic flow over flat plate is calculated with required accuracy. Numerical solutions are obtained using the implicit second-order finite-volume method described in [2, 1]. The Navier-Stokes equations are approximated by the conservative TVD shock-capturing scheme. The computational grid has 2001 × 301 nodes. In the boundary-layer region, the grid nodes are clustered in the direction normal to the body surface. Then, an acoustic wave is induced in the free stream and the unsteady problem for Navier-Stokes equations is solved.

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q 0, z ∈ C belongs to the ε -pseudospectrum if ||(zI − A)−1 || ≥ ε −1 where I is an identity matrix or operator of appropriate dimension [7]. The quantity ||(zI − A)−1 || is known as the resolvent norm. More formal definitions can be found in [7]. The eigenvalues of the matrix or operator correspond to the points z ∈ C for which ε = 0. For a normal matrix or operator the ε -pseudospectrum would take the form of closed circles of radius ε surrounding the eigenvalues. However, when the operator is non-normal the ε -pseudospectrum may be much larger. Our choice of numerical method generates a number of non-physical eigenvalues. We identify these by comparing them to the eigenvalues of the corresponding adjoint problem under a given inner product: those which are not identical are deemed spurious. We denote the space spanned by these physical eigenfunctions as W and scale each eigenfunction to have unit 2-norm. The matrix of normalised eigenvectors spanning this space is denoted VW . We project the operator S onto this space, where it is denoted as SW , as follows. Following [8], we apply the Gram–Schmidt procedure to the normalised eigenvectors to construct an orthonormal basis for W and construct the square matrix UW which relates the expansion coefficients in the regular and orthonormal bases. By construction, the diagonal matrix of eigenvalues of the reduced operator, denoted DW , is the matrix representation of SW in the eigenfunction basis. Therefore, the matrix representation of SW in the orthonormal basis is −1 SW = UW DW UW . (10) In reducing the operator to eliminate the unphysical modes we also incorporate the spatially neutral mode (wavenumber k = 0, denoted B in Fig. 1a) not generated by our numerical method. This mode is equivalent to planar Womersley flow [9], which is fundamental to pressure-driven flows in channels of finite length [3]. The corresponding eigenfunction is given by φ = A y − (m sinh(m))−1 (cosh(my) − cosh(m(1 − y))) + c , (11)

ˆ 1/2 and we choose c to where A is a normalisation constant, m = exp(−iπ /4)(ωˆ R) ensure φ (0) = 0. Transient growth of non-normal temporal operators is discussed at length in Chap. 14 of [7]. The maximal spatially localised growth of the long-wavelength Orr–Sommerfeld operator (9) for each point along the channel length is given by G(x) ˆ = ||eiSW xˆ ||2 .

(12)

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4 Results We illustrate the eigenvalues of both the full (6) and long-wavelength (9) Orr– Sommerfeld systems in Fig. 1(a), using R = 400, ω = 4.2654, the parameters corresponding to operating point u1 in [4]. The pseudospectrum of the long-wavelength operator (incorporating the spatially neutral mode, (12)) is shown in Fig. 1(b), which indicates strong non-normality; this is an approximate indication of the pattern of non-normality of the full Orr–Sommerfeld operator. However, for these parameter values the long-wavelength approximation fails to capture the Tollmien–Schlichting mode accurately (Fig. 1a). The corresponding maximal spatially localised growth is shown as the dashed line in Fig. 2(a), which becomes constant toward the downstream end of the channel due to the spatially neutral Womersley mode. Spatially localised growth from a representative initial condition, perturbing only even modes, is illustrated by the streamlines in Fig. 2(b), whilst the corresponding spatially localised growth is shown in Fig. 2(a) (solid line). The subsequent decay is dominated by the most unstable even mode (k = 3.327 + 0.574i). For these parameter values in the full spectrum this would be the Tollmien–Schlichting mode.

5 Discussion By constructing the pseudospectra and spatially localised growth of the longwavelength Orr–Sommerfeld operator, we have highlighted a possible mechanism for the growth of vorticity waves in a rigid channel downstream of an oscillating membrane. However, for the parameter values under consideration the long-

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y

5

0

0

2

4

x

6

8

10

1

0.5

0

0

2

4

x

6

8

10

Fig. 2 Spatially localised growth of disturbances governed by the Orr–Sommerfeld operator measured in the 2-norm for ω = 4.2654 and R = 400 (a) growth from an initial condition which disturbs only even modes (solid line) and the maximal growth at every point along the channel (dashed line); (b) the corresponding streamlines for the growth of the even perturbation illustrated in (a).

wavelength approximation to the eigenvalues is comparatively poor (Fig. 1a); we must await computations of the spatial ε -pseudospectrum for finite wavenumbers to validate our predictions. Nevertheless, if we consider the snapshot of the oscillation taken from [4] shown in Fig. 1(c), we observe a vorticity wave of wavelength λ ≈ 1.429 (approximately seven wavelengths per ten spatial units), which corresponds with that of the Tollmien–Schlichting mode (λ = 1.400). It remains to be seen whether the vorticity wave contributes in an essential manner to global instabilities of flow through finite length flexible-walled channels. Acknowledgements PSS acknowledges support from BBSRC. The authors are very grateful to Prof. K. A. Cliffe for helpful discussions.

References 1. Luo, X.Y., Pedley T.J., A numerical simulation of unsteady flow in a two-dimensional collapsible channel, J. Fluid Mech. 314, 191-225 (1996) 2. Stewart, P.S., Waters, S.L., Jensen, O.E., Local and global instabilities of flow in a flexiblewalled channel, Eur. J. Mech. B/Fluids 28, 541-557 (2009) 3. Jensen, O.E., Heil, M., High-frequency self-excited oscillations in collapsible-channel flow, J. Fluid Mech. 481, 235-268, (2003) 4. Luo, X.Y., Cai, Z.X., Li, W.G., Pedley, T.J., The cascade structure of linear instability in collapsible channel flows, J. Fluid Mech. 600, 45-76 (2008) 5. Stephanoff K.D., Pedley, T.J., Lawrence, C.J., Secomb, T.W., Fluid flow along a channel with an asymmetric oscillating constriction, Nature 305, 692-695 (1983) 6. Biau, D., Bottaro, A., Transient growth and minimal defects: Two possible paths of transition to turbulence in plane shear flows, Phys. Fluids 16, 3515-3529 (2004) 7. Trefethen, L.N., Embree, M.:Spectra and pseudospectra, Princeton University Press (2005) 8. Reddy, S.C., Schmid, P.J., Henningson, D.S., Pseudospectra of the Orr–Sommerfeld operator, SIAM J. Appl. Math. 53 15-47 (1993) 9. Womersley, J.R., Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, J. Physiol. 127, 553-563 (1955)

Global stability of a plane liquid jet surrounded by gas Outi Tammisola, Fredrik Lundell, Daniel Söderberg, Atsushi Sasaki, and Masaharu Matsubara

Abstract The global stability of a liquid sheet in gas is studied. The global 3D stability problem for a 2D base flow is formulated, including surface tension of the interface, and the viscosity and density of both phases. The implementational requirements are clarified, and met by using a parallel code for eigenvalue computations based on the mathematical software libraries PARPACK and ScaLAPACK. Preliminary eigenvalue spectra and eigenmodes are presented for the case of a water jet surrounded by air.

1 Introduction A plane liquid sheet is a liquid jet with a narrow width and a large spanwise extent, so that the mean flow can be considered as two-dimensional. Here, global stability of a liquid jet surrounded by gas will be studied numerically. The flow case is sketched in figure 1 (a). If the sheet is surrounded by a viscous gas and the interface has surface tension, the flow case is governed by four non-dimensional parameters: the Reynolds number Re, the Weber number We, the density ratio ρ ∗ and the viscosity ratio µ ∗ : Re =

ρ l Um a µl

We =

ρ l Um2 a γ

ρ∗ =

ρg ρl

µ∗ =

µl . µg

(1)

Outi Tammisola, Fredrik Lundell and L. Daniel Söderberg Linné FLOW Centre, KTH Mechanics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden, e-mail: [email protected] Atsushi Sasaki and Masaharu Matsubara Shinshu University, Wakasoato 4-17-1, Nagano 380-8553, Japan e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_65, © Springer Science+Business Media B.V. 2010

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O. Tammisola, F. Lundell, D. Söderberg, A. Sasaki, M. Matsubara

log10We air

a)

log10We

log10Re

c) ω0 i

water air

x/Re

ω0 i log10Re

x/Re

Fig. 1 a) Illustration of the flow geometry. b) Convective wave growth in a water sheet in air. c) The absolutely unstable region as a function of x and Re (or equivalently, We), in the study of [4].The solid line represents the absolute instability boundary for sinuous, and dashed for varicose disturbances.

where a is the nozzle thickness, Um the mean velocity of the liquid at the nozzle exit, ρ the density, µ the dynamical viscosity and γ the surface tension at the interface between the liquid and gas. Indices l and g refer to liquid and gas, respectively. The coordinate system is x, y and z for the streamwise, sheet normal and spanwise directions, respectively, and the origin is positioned at the nozzle exit on the jet centreline. All lengths are scaled with a and velocities with Um . Figure 1 (b) shows a flow visualization of disturbance waves on a liquid sheet interface. In this particular case, the sheet is forced at a given frequency and the disturbance is of a convective nature, growing in amplitude in the downstream direction. The instability mechanism in this case is of Kelvin-Helmholtz type, caused by the velocity difference between the liquid and the surrounding gas. This is called aerodynamic instability, as opposed to the Rayleigh-Taylor instability of axisymmetric liquid jets caused by surface tension [6]. Surface tension is always stabilising for this type of instabilitities. However, a viscous liquid sheet with a uniform velocity profile in inviscid gas exhibits a curious type of instability, called pseudoabsolute instability, which appears only in flows where surface tension dominates over inertia: We < 1 [2]. This instability is hard to interpret physically — apart from being neutrally absolutely stable, it has an infinite wavelength, infinite extent and zero frequency. Nevertheless, [2] argue that it might explain the explosive rupture of a liquid sheet seen in the experiments of [3]. Furthermore, a genuine absolute instability for a relaxational1 liquid sheet with a parabolic inlet profile (figure 1 (a)) has been reported [4] (see figure 1 (c), where the flow is locally absolutely unstable if ω0,i > 0). A region of absolute instability is a necessary, but not a sufficient, condition to obtain a linear global mode[1]. From the absolute instability results shown in figure 1, it is not evident in which cases a global mode might be obtained, since 1

The sheet relaxes from a parabolic velocity profile at the nozzle to near a plug profile further downstream.

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Fig. 2 The streamwise velocity of the base flow indicated with grayscale for Re = 316 (above), and Re = 1000 (middle) - dark colour represents high velocity. The Re = 316 base flow develops faster. Velocity profiles in three streamwise positions are shown on top of the pictures. In the lowermost row of pictures, velocity profiles for the upper half of the jet are shown for both cases in x − positions scaled with the position where the jet is thinnest in each case. It is seen that near the inlet there is a difference in the profiles, but downstream they become more self-similar.

there might be an interplay between the length of the absolutely unstable region, mode wavelengths, and their growth rates. Also, the most unstable eigenvalue was only given at the stability boundary, and not in the region from the nozzle to this boundary. In the present work, we begin the search of linear global modes in plane liquid sheets. We formulate the global stability problem, and compute preliminary 2D linear global modes for the relaxational liquid sheet for which the local absolute instability is shown in figure 1 (c). The mean flows at Re = 316, 1000 are shown in figure 2.

2 Flow modelling, numerical implementation and verification Here, a brief description of the formulation and numerical implementation will be given. The full procedure can be found in Tammisola (2009)[5]. We solve the 2D eigenvalue problem obtained by linearising the disturbance equations around the steady base flow shown in figure 2. Since the fluids are immiscible due to surface

406


tension, the disturbed position of the interface has to be solved for together with the disturbance velocities and pressure. The solution (eigenvalue spectrum and corresponding eigenmodes) is obtained by splitting the problem domain along the interface between liquid and gas. The two subdomains are coupled so that the velocities and stresses (normal and tangential) are continuous at the interface. After transformation to rectangular boxes, each subdomain is discretised by Chebyshev polynomials in the streamwise and sheetnormal directions. The discretised eigenvalue problem is solved by a parallel variant of the Arnoldi Package, PARPACK, together with Mathematical Kernel Library (MKL), the latter package used to solve the equation systems in the algorithm. The implementation of the two domains, the mapping from curved to rectangular subdomains and the stress conditions together with the oscillation of the surface between the liquid and gas are verified as follows. A single phase flow case (a wake flow is used for the purpose) is used as a reference. The global stability of this flow can be studied both with one single domain and by splitting the domain along a streamline, along which the interface conditions must be valid. Setting the surface tension along this interface to zero, the results should be similar. The result of such a verification is shown in figure 3 and the eigenvalues are seen to agree to the third digit. The eigenmode distribution is also the same, except for the random phase.

Fig. 3 The most unstable mode in the wake problem using: a) one domain with Chebyshevdiscretisation in both directions, and b) two domains with a streamline of the base flow as a boundary, and coupling by the same type of interface conditions as in the liquid jet problem. The interface position (in this case passively following the flow) shown with a white line.

Global stability of a plane liquid jet surrounded by gas coarser grid finer grid

407

0.04

0.05 0.02 0

0

−0.05 ωi

ω −0.02 i

−0.1

−0.04

−0.15

−0.06

−0.2

−0.08

−0.25

−1.9

−1.8

−1.7 ωr

−1.6

−1.5

−0.1 0

0.5

1

ωr

1.5

2

Fig. 4 (a): The demonstration of grid convergence for the candidates for unstable global modes (above dashed line) at Re = 1000. (b) Full spectrum.

Fig. 5 One mode along the most unstable branch at Re = 1000 together with the corresponding free surface position (white), and a close-up of the oscillation in the middle region (below).

3 Results and discussion The numerical tool described above has been applied to study the global stability of a relaxational plane liquid jet shown in figure 2. The results are shown in figures 4 and 5. First it is verified that the spectra are converged in terms of resolution in 4 (a). Here, spectra with 500 and 600 points in the streamwise direction are compared and the eigenvalues are seen to agree well. In figure 4 (b), a full spectra at

408


Re = 1000, We = 25 is shown. The eigenvalues appear along continuous lines. An example mode (indicated with an arrow in the spectrum) is shown in figure 5. The results in figures 4 and 5 have been obtained with a streamwise length of the computational domain of 500. Even though the mode in figure 5 has a positive growth rate, convergence with respect to the domain length is not assured yet and the tendency is that longer domains give more stable eigenvalues. Thus, the interpretation of the spectra in terms of disturbance dynamics is not clear, and a transient growth analysis might shed more light to this. However, so far no unstable global mode corresponding to the low Reynolds number absolute instability described in figure 1 has been found.

4 Conclusions and outlook The global stability of a plane liquid jet (liquid sheet) surrounded by gas has been studied. A numerical code has been developed and the model includes the full linearised Navier-Stokes equations for the disturbance in the liquid as well as in the gas. In particular, the global stability of a relaxational liquid jet has been studied. It has been hypothesised that the rapid breakdown of the jet is due to an absolute instability[2, 4]. Eigenvalue spectra together with global modes have been calculated. However, no solitary unstable mode have been found and the dynamics of the liquid sheet continues to be an elusive problem. Future studies will increase the parameter range studied and include transient dynamics.

References 1. Huerre, P., Monkewitz, P.A.: Local and global instabilities in spatially developing flows. Ann.Rev. of Fluid Mech. 22, 473–537 (1990) 2. Lin, S.P., Lian, Z.W., Creighton, B.J.: Absolute and convective instability of a liquid sheet. Journal of Fluid Mech. 220, 673–689 (1990) 3. Lin, S.P., Roberts, G.: Waves in a viscous liquid curtain. Journal of Fluid Mech. 112, 443–458 (1981) 4. Söderberg, L.D.: Absolute instability of a relaxational plane liquid jet. Journal of Fluid Mech. 493, 689–737 (2003) 5. Tammisola, O.: Linear stability of plane wakes and liquid jets: global an local approach. Licentiate thesis, Department of Mechanics, Royal Institute of technology, Stockholm (TRITAMEK 2009:04) 6. Rayleigh, L.: On the instability of jets. Proc. of London Society of Mathematics. s1-10(4), 4–13 (1878)

Instabilities of flow in a corrugated pipe Jianjun Tao

Abstract Flow in a corrugated pipe is considered. Different from previous studies, both the corrugation amplitude and wavelength are much smaller than the pipe diameter. Results of the multi-scale analysis show that the mean flow modulated by the surface corrugation becomes unstable to three-dimensional travelling waves at moderate Reynolds numbers, and the wave with one azimuthal period is found to be the most unstable mode.

1 Introduction The laminar-turbulent transition in pipe flows has been an open problem since the original experiments of Reynolds [1]. The experimental data shown in Moody’s chart [2] and Nikuradse’s chart [3] illustrate clearly that the pipe roughness has dominant effect on the laminar-turbulence transition and the turbulent states. A corrugated surface may be the simplest model of a rough wall. The two-dimensional (no variation in the azimuthal direction) instability in a corrugated circular pipe has been investigated recently [4], and the flow was found to be unstable at moderate Reynolds numbers. It should be noted that in all previous works the axial characteristic lengths of the wall variation are of the same order as the pipe diameter, however, and much larger than the scale of a typical roughness. Therefore, the problem: how small-scale wall variation induces instability in pipe flows, is thus still open. For shear flows, one mechanism to sustain unstable disturbances [5] is: when the mean flow profile has mildly deviation with respect to the stable basic flow profile, the flow may turn to be unstable. This mechanism is referred as mean flow instability hereafter. By studying the parabolic profiles of pipe flows distorted by axisymmetric and non-axisymmetric azimuthally periodic deviations, it was found that the flows may become unstable [6, 7, 8]. Nevertheless, in previous studies the LTCS and CAPT, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_66, © Springer Science+Business Media B.V. 2010

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J. Tao

mean flow instability can lead to streamwise vortices and streak-type structures, but those profile distortions are not solutions of the Navier-Stokes equations and have no real physical origins.

2 A model of corrugated pipe and stability analysis

r* Ro

Fig. 1 Sketch of the setup. The pipe surface has fine corrugations as R0 [1 + ε sin(n∗ x∗ )], where the relative roughness ε ≪ n2∗ πRo ≪ 1 .

o

x*

R0

εR

0

2 π /n*

The rough pipe is simulated by a circular pipe with fine surface corrugations and is described by cylindrical coordinates x∗ , r∗ and φ ∗ . Its corrugation amplitude and its wavelength are both much smaller than the average pipe radius R0 (see Fig. 1). The Reynolds number is defined as Re = 2Rν0U0 , and the flow rate Q = π R20U0 . Using R0 as length scale and the velocity U0 as velocity scale, we obtain the dimensionless governing equations for the velocity field u(x, r, φ , τ ) with components u, v and w in the x, r and φ directions, respectively. The boundary conditions are no-slip on the pipe wall where R(x) = 1 + ε sin(nx) and boundedness at the centerline. ε is 2π the relative roughness. For fine roughness, the corrugation wavelength δ = n∗R = 0 2π n ≪ 1 is required. When ε ≪ δ or the shape factor S = nε ≪ 2π the flow is nearly parallel both in the center region and near the corrugated wall. Therefore, parallel flow approximation is used to get the steady and axisymmetric basic flow solutions U (x, r) and V (x, r). The smaller ε and S are, the smoother the pipe is. First, we study the basic flow or the corrugation flow. By scaling we know that 2 2 ∂ 2U the nonlinear inertia term U ∂∂Ux = O( Uδ ) and the viscous term Re = O( ReUε 2 ), ∂ r2 so the inertia terms cannot be ignored if δ is not much larger than O(Reε 2 ). Hence the governing equations for the basic flow are simplified as:  ∂U ∂U ∂P 2 ∂ 2U 1 ∂ U   U +V =− + ( 2 + )    ∂x ∂r ∂ x Re ∂ r r ∂r   ∂P (1) ≈0  ∂r       ∂ U + 1 ∂ (V r) = 0 ∂x r ∂r

Instabilities of flow in a corrugated pipe

411

with boundary conditions U(x, R) = V (x, R) = 0. In previous study [4] the basic flow was calculated using a finite-element method. However, and are not applicable for present pipe model Since we are interested in the asymptotic behavior of fine roughness (e.g. the corrugation wavelength δ is third or fourth orders smaller than the pipe diameter), it becomes a tremendous hard task to use numerical [4] or theoretical methods [9, 10] to obtain such basic flow solution. In addition, numerical simulations are also not convenient for general analysis in a wide parameter space. Instead, we use an approximate solution of (1), which was solved with an integral method for a converging-diverging tube [11]: U(x, r) =

2 r2 Re dR 8 8 r2 (1 − ) + [ − R2 R2 R3 dx 225 225 R2 4 r3 4 r4 − + ]. 15 R3 15 R4

(2)

V can be calculated easily based on the continuity equation. When the pipe wall is smooth, R(x) = 1, the basic flow solution (2) reduces to the classical HagenPoiseuille flow solution. Since the corrugation wavelength δ ≪ 1, the axial length scale of basic flow structure described by (2) is much smaller than the average radius R0 . The latter is the scale of typical structures (e.g. waves and axial vortices) observed in laminar-turbulent transitions. Therefore, it is interesting to find how these small-scale structures affect the mean flow field Uˆ . Flows in practical circular pipes will appear (in a large axial-scale sense) to be steady, axisymmetric and parallel at moderate Reynolds numbers, though we know that surface roughness always exists and in fact the velocities are mean values. By ˆ ˆ2 scaling we know that the nonlinear inertia term Uˆ ∂∂Ux = O( Ul ) and the viscous term 2 ∂ 2Uˆ Re ∂ r2

ˆ

U = O( Re ), so the inertia terms can be ignored because l could be infinitely large for mean flow. Therefore, the governing equations for mean flow are:  ∂ Pˆ 2 ∂ 2Uˆ 1 ∂ Uˆ   + ( 2 + )=0 − ∂ x Re ∂ r r ∂r . (3)  ∂ Uˆ   U(1) ˆ = 0, |r=0 = 0 ∂r

∂ Pˆ ∂x

is a constant for laminar smooth-pipe flows, but will be a weak function of r for rough pipe. In both the large-scale flow (mean flow) and the small-scale flow (basic flow) the pressure term must remain to balance the viscous terms and to maintain the flow rate. Therefore, the pressure term is the bridge to connect different axialscale flows in a rough pipe. In this paper, the mean pressure gradient is estimated with the basic flow solution by ignoring ε 2 and higher order terms without Re:

∂ Pˆ ∂P 2 ∂ 2U 1 ∂ U ∂U ∂U 16 Re +C = = ( 2 + ) − (U +V ) = − − (nε )2 f (r) ∂x ∂x Re ∂ r r ∂r ∂x ∂r Re 2

412

J. Tao 2

0.011

( a)

( b)

0.01 1.5

Re

0.009

-2

0.008

S

^

U

1

0.007

0.5

0

0.006

0

0.2

0.4

r

0.6

0.8

1

0.005 1850

2050

2250

Re

2450

2650

2850

Fig. 2 (a) Velocity profiles of the mean flows. The red curve is the solution of Hagen-Poiseuille flow. (b) Critical shape factor S required to cause instability as a function of Reynolds number for m = 1 mode. The mean flow profiles at critical states for S = 0.006 and 0.008 are shown in (a) as the dotted and dashed curves.

The constant C is chosen to guarantee a fixed flow rate, and the average value is R defined as ψ = δ1 0δ ψ dx. It is noted that the basic flow solution is axially periodic with wavelength δ . The mean flow solution is resolved as Re Uˆ = 2(1 − r2 ) − ( S)2 g(r), 2

(4) R

g(r) = h(1)(1 − 2r2 ) + h(r) + 4(r2 − 1) 01 h(r)rdr and h(r) = f (r1 )r1 dr1 dr. The first term on the RHS of (4) is just the Hagen-Poiseuille flow solution for smooth pipes, and the second term is the contribution of the pipe-wall corrugation. Some velocity profiles of the mean flows are shown in Fig. 2(a). To our knowledge, it is the first time to illustrate theoretically that how small-scale wall variation or fine roughness affects the mean velocity profile. It is also interesting to note that the mean flow is not affected by the relative roughness only but by the Reynolds number and the shape factor S = nε , which divided by 2π is the ratio of the corrugation amplitude to the wavelength. The mean flow stability analysis has been carried out based on the mean flow solution (4). The form of disturbance to be considered is where R R

r1 r 0 r 0

u, ˜ v, ˜ w, ˜ p˜ = R[F(r), iG(r), H(r), J(r)]ei(mφ +α x−ωτ ) .

(5)

The imaginary part of the complex frequency ω determines the stability of the pipe flow to particular disturbance. For details of the solving method we refer to previous papers [12, 13]. According to stability analysis, the mean flow becomes unstable to a threedimensional non-axisymmetric travelling wave mode (m = 1) at moderate Reynolds numbers when the fine roughness exists. It is shown in Fig. 2(b) that the critical Reynolds number decreases with the increase of S. This is because larger shape factor S represents stronger non-parallel feature of the basic flow and larger mean flow modulation, which make the flow more unstable. The mode with m = 0 will be unstable but its critical Reynolds number is too high to be meaningful in understand-

Instabilities of flow in a corrugated pipe

413

ing the transition of rough pipe flows. In addition, different Reynolds numbers and shape factors are examined and no unstable m = 2 and 3 modes are found, though the possibility of existence of unstable m ≥ 2 mode is not excluded. The mean velocity profiles at the critical states of m = 1 mode for S = 0.006 and 0.008 are shown in Fig. 2 as the dotted and dashed curves, respectively. Except slight deviations from the parabolic profile there are no inflection points on these curves, so the mean flow instability revealed in this paper belongs to an viscous mechanism. By using small impulsive jets and push-pull disturbances from holes in a pipe wall, it was found experimentally [14] that the disturbance amplitude threshold required to cause transition scaled as Re−1 (jets) and Re−1.3 or Re−1.5 (push-pull disturbance). Note that they are just local disturbances and are not periodically distributed in the axial direction. According to present fine-roughness model, the threshold of shape factor S = nε scales as Re−2 as shown in figure 2(b), which should be examined by future experiments. (a )

( b)

Fig. 3 The disturbing flow fields of critical states at S = 0.006. The frames (a)(b) are separated with △x = 2πα , and the contours indicate the values of disturbing axial velocity u˜ (dashed lines are for negative values).

The disturbing flow patterns for m = 1 mode are shown in Fig. 3. Since the critical wavelength is 24.74 for S = 0.006, the three-dimensional wave is a long flow structure. The main feature is the remarkable axial high/low-speed streaks and a pair of axial narrow vortices, which are quite close to the wall. It is noted that the streaks here don’t like the ordinary ones because they are spirals along the axis. Because of symmetry the modes with ±m will share the same critical parameters, and superposition of the ±m modes will produce ordinary streaks. Another feature is that the disturbing velocity on the axis is not zero but a finite value with a rotating direction. Especially, stability analysis concludes that it is the shape factor S not the relative roughness ε determines the critical properties. In fact, this conclusion has been implied by previous experimental data. In Nikuradse’s experiments the roughness was simulated by uniform-size sand grains (the same shape), and the fiction factor data almost coincide with each other at the start stage of transition though the relative roughness ε (the ratio of grain size to pipe radius) varies from 1/15 to 1/507.

414

J. Tao

3 Conclusions Different from previous studies, this paper constitutes the first bridge for pipe flow between the modulated velocity profile required by mean flow instability and its physical origin, which is the fine roughness in present model. The high/low speed streaks, which are the precursors of bypass transition in boundary-layer flows, and the axial vortices shown in Fig. 3 are expected to obviously influence the laminarturbulent transition process. Acknowledgements I acknowledge valuable discussions with Friedrich H. Busse, Fazle Hussain, Mingde Zhou and Zhensu She. This work has been supported by the NSFC (10672003).

References 1. Reynolds O (1883) An experimental investigation of the circumstances which determine whether motion of water shall be direct or sinuous and of the law of resistance in parallel channels. Proc R Soc Lond A 35: 84-99 2. Moody L F (1944) Friction factors for pipe flow. Trans ASME 66: 671 3. Nikuradse J (1933) VDI Forschungsheft 361 [in English, in Technical Memorandum 1292, National Advisory Committee for Aeronautics (1950)] 4. Cotrell DL, McFadden GB, Alder BJ (2008) Instability in pipe flow. PNAS 105: 428 5. Gill AE (1965) A mechanism for instability of plane Couette flow and of Poiseuille flow in a pipe. J Fluid Mech 21: 503-511 6. Gavarini MI, Bottaro A and Nieuwstadt FTM (2004) The initial stage of transition in pipe flow: role of optimal base-flow distrotions. J Fluid Mech 517: 131 7. Ben-Dov G, Cohen J (2007) Critical Reynolds Number for a natural transition to turbulence in pipe flows. Phys Rev Lett 98: 064503 8. Ben-Dov G, Cohen J (2007) Instability of optimal non-axisymmetric base-flow deviations in pipe Poiseuille flow. J Fluid Mech 588: 189 9. Floryan JM (2003) Vortex instability in a diverging-converging channel. J Fluid Mech 482:1750 10. Floryan JM (2007) Three-dimensional instabilities of laminar flow in a rough channel and the concept of hydraulically smooth wall. Euro J Mech B/Fluids 26: 305-329 11. Forrester JH, Young DF (1970) Flow through a converging-diverging tube and its implications in occlusive vascular disease-I Theoretical development. J Biomechanics 3: 297 12. Batchelor GK and Gill AE (1962) Analysis of the stability of axisymmetric jets. J Fluid Mech 14: 529-551 13. Tao J, Le Quéré P, Xin S (2004) Spatio-temporal instability of the natural-convection boundary layer in thermally stratified medium. J Fluid Mech 518: 363 14. Peixinho J, Mullin T (2007) Finite-amplitude thresholds for transition in pipe flow. J Fluid Mech 582: 169-178

The Late Nonlinear Stage of Oblique Breakdown to Turbulence in a Supersonic Boundary Layer Dominic von Terzi, Christian Mayer, and Hermann Fasel

Abstract Oblique breakdown to turbulence was initiated by low amplitude forcing in a laminar flat-plate boundary layer at Mach three. The growth of the instability waves was investigated using spatial Direct Numerical Simulations (DNS). Excellent agreement with theory was obtained in the linear stage corroborating that the entire transition process from the linear regime to the final breakdown was captured. A fully turbulent flow was reached demonstrating that this transition scenario is a viable path to turbulence. Key events in the late nonlinear stage of breakdown are studied in detail.

1 Introduction In boundary layers at supersonic speeds, complete transition from the disturbance introduction at low amplitudes in the laminar flow to fully developed turbulence is difficult to study. Previous investigations therefore limited their efforts on selected regimes within this process, mostly on either the linear and early nonlinear stages or the fully-developed turbulent flow. For the latter, it is then unclear what mechanism actually caused the breakdown to turbulence. For the first, it is not certain if a fully-developed turbulent flow will be reached or whether the flow relaminarizes or another breakdown mechanism eventually takes over. In the following, results from a set of six Direct Numerical Simulations (DNS) using up to 266 million grid points are presented for a flat-plate boundary layer at Mach three. The data cover the complete transition process from the linear regime to the fully turbulent boundary layer. Disturbances were introduced at low amplitudes in a controlled fashion, hence enabling comparisons with theory and experimental findings. After a brief summary of Dominic von Terzi∗ · Christian Mayer · Hermann Fasel Department of Aerospace and Mechanical Engineering, The University of Arizona, U.S.A. e-mail: [email protected], [email protected], [email protected] ∗ Present affiliation: Institut f¨ ur Thermische Strömungsmaschinen, Universität Karlsruhe, Germany P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_67, © Springer Science+Business Media B.V. 2010

415

xL(∼ 14.5λx ) (CASE 3 & 4)

yH (∼4.7δ) (CASE 1)

xL(∼ 13.5λx )(CASE 2) xL(∼ 13.1λx ) (CASE 5 & 6) xL(∼ 11.3λx ) (CASE 1) x2 (∼ 3λx ) x1(∼ 2λ x )

0.03

e)

b)

0.02

*

d)

y,m

a)

0.01

3000 points

Fig. 1 Computational setup for all cases: (a) computational domain based on streamwise wavelength of the forced waves and the (laminar) boundary layer thickness at the outflow, (b) grid used for CASE 3 & 4, for clarity only every 14th point in x and every 6th point in y are plotted, (c) grid clustering in the streamwise direction, (d) location and amplitude distribution of forcing slot, (e) grid stretching in the wall-normal direction

yH (∼10δ) (CASE 5)

D. von Terzi, C. Mayer, H. Fasel yH (∼5δ) (CASE 2,3,4 & 6)

416

0

c)

CASE CASE CASE CASE

2000

0.00 200 400 points

1 2 3, 4 & 5 6

1000 0

0.3

0.4

0.5

0.6

0.7 *

0.8

0.9

1.0

1.1

x,m

Table 1 Main simulation parameters that differ between cases Parameter

Unit

CASE 1

CASE 2

CASE 3

CASE 4

CASE 5

Forcing amplitudes of mode [h, k] (h in time and k in the lateral direction): A[1,0] (−) 0.0 0.0 0.0 1.0E−4 0.0 A[1,1] (−) 3.0E−3 3.0E−3 3.0E−3 3.0E−3 3.0E−3 Domain size (see Fig. 1): xL (m) 0.948 1.087 1.145 1.145 1.050 yH (m) 0.026 0.029 0.030 0.030 0.059 Grid size (see Fig. 1): nx × ny × nz (−) 9.7E6 80.4E6 211.6E6 211.6E6 200.9E6 Streamwise resolution at outflow (based on wavelength of forced wave): [1,1] points per λx (−) ∼ 170 ∼ 350 ∼ 440 ∼ 440 ∼ 440

CASE 6 0.0 3.0E−3 1.065 0.032 266E6 ∼ 550

the computational setup, it is confirmed that the so-called ‘oblique breakdown’ scenario of [1] was initiated in our setup. The subsequent analysis focuses on the late nonlinear stage of the transition process. Finally, it is demonstrated that a turbulent flow state is attained.

2 Computational Setup Matching an existing experimental facility, the unit Reynolds number based on the free-stream velocity and free-stream viscosity at the inflow was Re = 2.181 × 106 m−1 and the free-stream temperature was T∞∗ = 103.6 K. The setup is illustrated in Fig. 1 and the main parameters of the simulations are compiled in Table 1. The simulations were performed with the flow solver NSCC developed in our CFD Laboratory at the University of Arizona that solves the compressible Navier–Stokes equations. The code employs fourth-order split finite differences in the wall-normal

The Late Nonlinear Stage of Oblique Breakdown to Turbulence a)

8

1.0e-04

b)

1.5

417

0.04

1+1/M

*

-1

*

-1

PSE: max u’, β =196.2m

DNS: max u’, β =196.2m

0.0e+00

*

DNS: max u’, β =211.52m

1-1/M 0.5 0.0

0.03

-1

-1.0e-04 0.06

0.2 ω = FRx

0.4

*

LST: β =211.51m

-1

-1

-1

0

0.00 0.0

4

*

0.02 0

*

1.0

-αi , m

αi Loo

cph,x

*

*

DNS: wall p’, β =211.52m

0.2 ω = FRx

0.4

Fig. 2 Stability behavior of fast (−·−) and slow (—) discrete eigenmodes with β = 0.0: (a) phase velocity and (b) streamwise amplification rate over frequency ω = F Rx ; the vertical line indicates the end of the longest domain (CASE 3)

0.4

0.8 * x,m

1.2

Fig. 3 Comparison of amplification rate of forced waves from DNS with theory using different criteria

and streamwise directions and a pseudo-spectral approach for the lateral direction with periodic boundary conditions. Integration in time is carried out using a fourthorder Runge–Kutta method. Additional information on the setup, governing equations, boundary conditions, disturbance generation and numerical method are provided in [2, 3].

3 Initiating the Oblique Breakdown Mechanism A pair of symmetric oblique waves at low amplitudes with frequency f ∗ = 6.36 kHz (F = ω /Rx = 3 × 10−5 ) is introduced in the supersonic flat-plate boundary layer through a blowing and suction slot at the wall (see Fig. 1 and Table 1). The domain width was chosen to match the spanwise wavelength of this fundamental wave ∗ = λ ∗ = 2π /β ∗ ≃ 0.03 m). Periodicity is assumed in the lateral direction. Anal(zW z ysis using Linear Stability Theory (LST) showed that a possible synchronization of the fast and the slow mode can be disregarded for our setup (see Fig. 2). Other transition mechanisms, like subharmonic resonances, were either excluded by choice of the domain or strongly reduced in importance by only forcing the oblique waves with frequency f ∗ = 6.36 kHz. In general, however, a subharmonic resonance can occur for a Mach three boundary layer as explained in [5] and might also be initiated by the round-off error in the present simulation. In addition, DNS with a considerably reduced forcing amplitude was performed such that the oblique waves remained in the linear regime within the entire computational domain. The data from this simulation serve as reference to establish linear flow behavior including non-parallel effects. The results were compared to theoretical results from LST and calculations employing the linear Parabolic Stability Equations (PSE). In Fig. 3, excellent agreement for a critical quantity, the streamwise growth rate, is demonstrated. This confirms that the resolution, domain height and accuracy of boundary conditions of the simulations are sufficient to capture the linear stage. Similar agreement was obtained for amplitude and phase distributions of various quantities [2]. Increasing the forcing amplitude to 0.3% of the free-stream velocity caused an ini-

418

D. von Terzi, C. Mayer, H. Fasel

1.0e-01

(a)

0.004

1.0

turbulent CASE 2 CASE 3 CASE 4 CASE 5 laminar

u

0.5

1.0e-03

1.0e-04

0.4

0.6

*

0.8

1.0

x,m

Fig. 4 Streamwise development of maximum u-velocity disturbance for selected spanwise modes with the forced (fundamental) frequency

0.0 0

(b)

4

t/Tforcing

8

12 cf

[1,1], CASE 1 [1,3], CASE 1 [1,5], CASE 1 [1,1], CASE 2 [1,3], CASE 2 [1,5], CASE 2 [1,1], linear

max

0.002

1.0 0.5

u

1.0e-02 |u|[h,k]

0.0 0

4

t/Tforcing

8

12

Fig. 5 Time signals of the uvelocity at two locations, i.e. the two vertical lines in Fig. 6 (CASE 3, y∗ > 2.15 mm, y+ ≈ 50). a x∗ = 0.942 m. b x∗ = 1.104 m

0.000

0.4

0.6

0.8 * x,m

1.0

Fig. 6 Streamwise development of the skin-friction coefficient for various simulations; reference data from [4]

Fig. 7 Top and sideview of iso-surfaces of the vortex identification criterion Q (CASE 2); left: Q = 10, 000, 0.798 m < x∗ < 0.924 m; right: Q = 40, 000, 0.924 m < x∗ < 1.051 m

tial linear growth of the instability waves with a departure from linear behavior well downstream of the inflow boundary of the computational domain (see [2] for details). This is shown in Fig. 4 for two simulations with different domain sizes and resolutions. Corresponding flow structures from the early nonlinear stage to the breakdown are visible in the left plot of Fig. 7.

4 The Late Nonlinear Stage of Breakdown After departure from the linear behavior, the oblique waves still grow, but wave numbers with higher spanwise modes appear (see Fig. 4), as well as higher harmonics and steady flow structures of streamwise orientation that are typical for the oblique breakdown scenario (not shown here). Some of these other modes reach amplitude levels comparable to those of the fundamental waves. This mixture finally results in a flow that “looks” turbulent (see right plot in Fig. 7 and Fig. 8), however, the flow field is still periodic in time which was corroborated by Fourier analyses and demonstrated in Figs. 5a and 8. In physical space, the mean skin friction rises (Fig. 6) and localized instantaneous flow structures with high shear are formed. These structures are apparently linked to the rise of additional waves in the

The Late Nonlinear Stage of Oblique Breakdown to Turbulence

419

(a)

2.0

y*, m *10-2

(b)

0.0

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

x*, m

1.00

1.02

1.04

1.06

1.08

1.10

Fig. 8 Contours of spanwise averaged wall-normal density gradient at two instants in time one forcing period apart; Fourier analysis shows periodicity breaking for x∗ > 1.05 m (CASE 3)

spectrum that are not directly generated by the oblique scenario [2]. The small-scale vortical structures “riding on top” of the high-shear structures on the right-hand-side in the left plot in Fig. 7 appear to be a manifestation of these additional waves. For the clean setup studied here, a single pair of harmonic oblique waves is continuously forced at a single fixed location. As a consequence, signature events in the transition process were occurring always at the same streamwise location and at the same phase with respect to the forcing. One example is the local formation of the small-scale structures discussed above. This location (x∗ ≈ 0.9 m) coincides with the second plateau in the rising skin-friction coefficient plotted in Fig. 6 and marks the beginning of the final stage of oblique breakdown to turbulence. Finally, once the peak in skin friction is surpassed, periodicity in time breaks down (see Fig. 5b) and the process of transition seems to have come to an end.

5 Reaching Fully Developed Turbulence For our setup, the turbulent Mach number is sufficiently small (< 0.32). Therefore, Morkovin’s hypothesis is likely to hold and comparison with incompressible reference data is justified after proper transformation. In Fig. 9, the transformed skinfriction coefficient downstream of its maximum compares well with correlations for turbulent boundary layers from the literature and, for the transformed mean streamwise velocity in near-wall coordinates, a logarithmic layer can clearly be discerned (Fig. 10). Momentum thickness and shape factor (not shown) also compare reasonably well with data in the literature. Power spectra of the streamwise velocity taken at locations inside of the log-layer exhibit the onset of the −5/3 and clearly show the −7 slopes predicted by theory (Fig. 11). More details of the fully turbulent regime, including the transformations employed, and additional comparisons with data from the literature are presented in [3].

420

D. von Terzi, C. Mayer, H. Fasel 25

0.004

20

0.002

CASE 3 laminar boundary layer turbulent boundary layer

0.001 0 6e+05

uc

cf,i

0.003

+

0.005

15

+

y + 2.44ln(y )+5.2 CASE 4 CASE 3

1.0e-03

1.0e-06

Eαα

10

1.0e-09

5

6.4e+05 Rex,i

6.8e+05

Fig. 9 Van Driest II transformed skin-friction coefficient for comparison with incompressible correlations in [4] (CASE 3, x∗ > 1.047 m)

0 0.1

1

10 + y

100 1000

Fig. 10 Van Driest transformed mean u-velocity in near-wall coordinates at x∗ = 1.104 m; vertical line indicates position of spectra in Fig. 11

1.0e-12 0.1

u v w -5/3 ~β -7 ~β 1.0

*

β , mm

10.0

-1

100.0

Fig. 11 Time-averaged power spectra in the lateral direction for all velocity components (CASE 3, x∗ = 1.087 m, y+ ≈ 49)

6 Conclusions The downstream development of two oblique instability waves and the concomitant process of laminar to turbulent transition was investigated numerically using DNS. It was shown that the oblique breakdown mechanism was initiated and it was demonstrated that this scenario is a viable path to turbulence. The focus of our analysis was on the late nonlinear stage of the breakdown. Key events of this stage are the formation of localized instantaneous flow structures of high shear that break up into small-scale vortical structures, a rapid filling of the spectrum, the flow surpassing its maximum of skin friction, and the breaking of periodicity with respect to the forcing frequency. Acknowledgements This work was funded by the Air Force Office for Scientific Research under grant FA9550-08-1-0211 with Dr. John Schmisseur serving as program manager. The authors want to thank Drs. Hein, Rist and Kloker for enabling the PSE stability analysis. Computer time provided by NASA Ames is gratefully acknowledged.

References 1. Fasel, H., Thumm, A., Bestek, H.: Direct numerical simulation of transition in supersonic boundary layer: Oblique breakdown. In: L.D. Kral, T.A. Zang (eds.) Transitional and Turbulent Compressible Flows, no. 151 in FED, pp. 77–92. ASME (1993) 2. Mayer, C.S.J., von Terzi, D.A., Fasel, H.F.: DNS of complete transition to turbulence via oblique breakdown at Mach 3. AIAA paper 2008-4398 (2008) 3. Mayer, C.S.J., von Terzi, D.A., Fasel, H.F.: DNS of complete transition to turbulence via oblique breakdown at Mach 3: Part II. AIAA paper 2009-3558 (2009) 4. White, F.M.: Viscous Fluid Flow. McGraw-Hill (1991) 5. Zengl, M., von Terzi, D., Fasel, H.: Numerical investigation of subharmonic-resonance triads in a Mach three boundary layer. In: Proceedings 7th IUTAM Symposium on Laminar-Turbulent Transition. Springer (2009)

Turbulence stripe in transitional channel flow with/without system rotation Takahiro Tsukahara, Yasuo Kawaguchi, Hiroshi Kawamura, Nils Tillmark, and P. Henrik Alfredsson

Abstract We report direct numerical simulations and experiments conducted in two types of plane channel flows—plane Poiseuille flow and plane Couette flows without/with system rotation—considering the subcritical-transition regime with using large aspect-ratio computational domains and channels. Both flows give rise to coexisting laminar and turbulent equilibrium regions in the form of oblique stripes, which are tilted by a certain angle with respect to the mean flow. When subjected to a stabilizing rotation, the Couette turbulence is locally quenched and exhibits the stripe pattern as similar to non-rotating cases. In this context, the turbulence stripe can be seen as an intrinsic phenomenon in the reverse transition in a channel flow.

1 Introduction Reverse transition to laminar in wall-bounded turbulent-flow systems has been the subject of ongoing debate and study by a number of researchers. There exist two critical Reynolds numbers as the upper and lower boundaries of the transition range. The upper one is the threshold above which the steady flow is unstable for some infinitesimal perturbations. For instance, a plane Poiseuille flow (PPF) is linearly stable up to Rec = 5772 (defined as Rec = uc δ /ν : uc , center-line velocity; δ , channel half width; ν , kinematic viscosity), while a pipe flow and a plane Couette flow (PCF) are stable even for infinite Reynolds number. The other is the lower critical Reynolds number below which there is no self-sustained turbulence. The former Reynolds number can be derived by the linear stability theory. The prediction of the latter one, on the other hand, is difficult and unsolved, because it is a nonlinear Takahiro Tsukahara, Yasuo Kawaguchi, Hiroshi Kawamura Tokyo Univ. of Science, 2641 Yamazaki, 278-8510 Japan, e-mail: [email protected] Nils Tillmark, P. Henrik Alfredsson Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_68, © Springer Science+Business Media B.V. 2010

421

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T. Tsukahara, Y. Kawaguchi, H. Kawamura, N. Tillmark, P. H. Alfredsson

Table 1 Computational condition: Li , domain size; Ni , number of grids.

PPF PPF PPF PPF PCF PCF PCF

Reτ

Rec

Rep

Rew

Lx × Ly × Lz

Nx × Ny × Nz

56.0 64.0 80.0 95.0 25.1 27.3 32.0

1070 1170 1440 1680 — — —

268 293 360 420 — — —

— — — — 350 375 425

327.68δ × 2δ × 128δ 327.68δ × 2δ × 128δ 327.68δ × 2δ × 128δ 204.8δ × 2δ × 64δ 204.8δ × 2δ × 102.4δ 204.8δ × 2δ × 102.4δ 204.8δ × 2δ × 102.4δ

4096 × 96 × 2048 4096 × 96 × 2048 4096 × 96 × 2048 4096 × 96 × 2048 2048 × 96 × 1024 2048 × 96 × 1024 2048 × 96 × 1024

phenomenon and often gives rise to spatio-temporal intermittent behaviors in the form of large-scale unique patterns. Therefore the lower critical Reynolds number is not yet proven to be unique, although has been believed to be about Rec = 1000 for PPF by experiments [1]. It has been well known that, in the transition Reynolds-number range where transition between laminar and turbulence occurs in pipe flow, the flow is partially turbulent with co-existing laminar regions [2]. This turbulence exist within an equilibrium turbulent puff, which preserves itself in the laminar background and occupies the entire cross-section of the pipe. For PPF, Tsukahara et al. [3] performed direct numerical simulations (DNS) at transitional Reynolds numbers, with a moderate aspect-ratio domain of Lx × Ly × Lz = 51.2δ × 2δ × 22.5δ (x, y and z indicate the streamwise, wall-normal and spanwise direction, respectively), and demonstrated that in this case there existed a large-scale stripe pattern of laminar and turbulent regions. Similar patterns in PCF have been found experimentally by Prigent et al. [4, 5] and numerically by Barkley & Tuckerman [6, 7]. However, the selection mechanism of the size (wave-length) of the turbulence stripe as well as the preferred inclination are not yet understood. In this work, we have carried out DNS and flow-visualization experiments on PPF and PCF with/without system rotation. In Sect. 3 we first report DNS performed for PPF and PCF with applying large enough domains compared to a prospective size of the turbulence stripe. We then report experiments conducted in each flow system to verify the result obtained by DNS. Our experimental studies on PCF rotating around its spanwise axis as well as non-rotating PPF and PCF, confirmed the existence of the stripe pattern in the transitional flow.

2 Numerical and experimental descriptions The configurations for DNS are the fully-developed flow fields of PPF and PCF, which are driven by a uniform pressure gradient and by a relative movement of both walls, respectively. Some details of the numerical method have been given in our previous papers [3, 8]. The computational conditions, such as the domain size and the grids number, are given in Table 1. Here the Reynolds number as a

Turbulence stripe in transitional channel flow with/without system rotation

423

control parameter for PPF is Reτ = uτ δ /ν , where uτ is the friction velocity; and the parameter for PCF is Rew = Uw δ /ν , where Uw is the half of the relative velocity of the walls. In addition, we introduce a unified Reynolds number of Re (= Rec /4) for PPF, which is based on the half scale of the relevant shear layer, i.e. uc /2 and δ /2, and corresponds to Rew for PCF by considering the analogy. Flow-visualization experiments on PPF and PCF, both with a large aspect-ratio channel (for PPF, 800δ × 2δ × 80δ , δ = 5 mm; for PCF, 150h × h × 40h, h = 10 mm), have been made by adding reflective-flake particles to the water. The test sections were illuminated by a fluorescent light and photographed by a still camera. The reflected light from the particles clearly reveales areas of turbulent and perturbed laminar flow, while the reflective light in a laminar flow is steady. This method of flow visualization has previously been used to visualize vortex structures and turbulent spots [1, 4, 5, 9]. In the case of PPF, a turbulence grid was installed at the entrance of the channel test section. The flow visualization domain of 70δ × 70δ in an (x, z) plane was 760δ downstream from the turbulence grid. Also, the study on PCF with/without system rotation was based on observations made through flow visualization. The Couette apparatus with counter-moving walls was basically the same as in the experiment by Tillmark & Alfredsson [9]. A turntable, on which the Couette apparatus was placed, was driven by a DC motor and rotates around the vertical zaxis. See Alfredsson & Tillmark [10] for a description of the rotating PCF (RPCF) apparatus. The control parameters of the RPCF are Rew and a non-dimensional rotation number Ω = 2Ωz δ 2 /ν (Ωz is the rotation rate of the turntable).

3 Results 3.1 DNS on PPF and PCF In the DNS, we initialized a featureless turbulent flow at a moderate Reynolds number and subsequently decreased Reτ stepwise down to an aimed value. At Reτ = 95, the flow revealed an unpatterned turbulent state (not shown here). For Reτ ≤ 80 for PPF, we have observed the stripe pattern of oblique bands, alternating between turbulence and laminar flow, as shown in Fig. 1(a). It can be found that small-scale streamwise streaks as well as vortices (not visualized in the figure) appear in the large-scale turbulent bands. In the case of Reτ = 80, the streamwise and spanwise wave-lengths of the bands are λx = 66δ and λz = 26δ , respectively; and the inclination angle with respect to streamwise direction is α = tan−1 (λz /λx ) = 21◦ . Also for PCF, it can be clearly seen that the flow organized into a distinct periodic pattern with three pairs of laminar and turbulent bands, as given in Fig. 1(b), with λx = 68h, λz = 34h, and θ = 27◦ . Note that there was no trace of a large-scale pattern at Rew = 425, which is in agreement with previous findings [4, 6]. In Table 2, we summarize the DNS results about properties of the turbulence stripe, for both PPF and PCF. The wave-lengths for each flow at same level of Rep or

424


(a) Plane Poiseuille flow at Reτ = 80

(b) Plane Couette flow at Rew = 375

Fig. 1 Turbulence stripe obtained by DNS. Gray and dark-gray isosurfaces indicate positive (u′ /uτ > 2.5) and negative (u′ /uτ < −2.5) streamwise velocity fluctuations, respectively. Mean flow direction is from bottom-left to top-right.

Rew are nearly equal when normalized by outer units, i.e. δ for PPF and h for PCF: for instance, see the cases of PPF at Reτ = 80 (Rep = 360) and PCF for Rew = 350– 375. Both λx and λz in the outer units are elongated at lower Reynolds numbers, but they are roughly scaled in wall units. Although the inclination angles of the stripes clearly differ between PPF and PCF, it is to be noted that wave-lengths and angles are more or less affected by the finite computational domain and its aspect ratio. In the present flow only a few turbulent stripes could be observed simultaneously due to the rather low aspect-ratio of the present domain, even though unusually large computational domains were employed in our present DNS. It is worthy of note, however, that another DNS using a spectral code with about six times of the domain has performed by Duguet & Schlatter [11], exhibiting multiple turbulent stripes of similar wave-lengths and angle.

3.2 Flow visualization on PPF and RPCF Figures 2–4 display typical snapshots of each flow which reveals an inhomogeneous brightness distribution, since laminar regions and turbulent regions are unevenly distributed: the lighter areas correspond to a laminar flow, while the turbulent regions appear dark because of the completely disordered motion of the flakes in these area. Here Rem for PPF is based on the bulk mean velocity and h. A negaTable 2 Wave-length and inclination angle of turbulence stripe: λi+ is normalized as λi uτ /ν .

PPF PPF PPF PCF PCF

Reτ

Rep or Rew

λx

λx+

λx

λz+

α

56.0 64.0 80.0 25.1 27.3

268 293 360 350 375

110δ 82δ 66δ 68h 51h

6100 5200 5200 3400 2800

43δ 32δ 26δ 34δ 26δ

2400 2100 2100 1700 1400

21 21 21 27 27

Turbulence stripe in transitional channel flow with/without system rotation - x Fig. 2 Typical snapshot of turbulence stripes, which is ? z obtained by flow visualization on PPF taken near the outlet for Rem = 2000. Mean flow direction is from left to right. The reflected light intensity is high in a quasi-laminar region, whereas it is relatively low in a turbulent region. Thus two laminar bands are 0 10δ observed.

425

40δ

- x ?

Fig. 3 Turbulence stripe obtained by flow visualization on RPCF at Rew = 460 and Ω = −4.9 (stabilizing rotation).

z

0

10h

30h

60h

0

10h

30h

60h

- x ?

z

Fig. 4 Same as Fig. 3, but for Rew = 750 and Ω = −16.

tive value of Ω for RPCF is equivalent to the stabilizing rotation. For PPF, the laminar/turbulent band was inclined about α = 20◦ –30◦ against the streamwise direction with λx = 60δ –70δ and λx = 30δ –40δ (see Fig. 2). For RPCF, the inclination angle is same or slightly large, e.g. α ≈ 37◦ in Fig. 3, compared to that in PPF, and the wave-lengths are λx = 40h–70h and λx = 30h–40h). The Reynolds number in Fig 4 is 750, at which turbulent PCF without rotating should be uniform turbulence [6, 8]. In PCF with system rotation the Coriolis force will either be stabilizing or destabilizing across the full channel and for negative rotation the Coriolis force can suppress turbulence, resulting in stripe-shaped spatial intermittency similar to the non-rotating transition case. Moreover, it can be conjectured that the turbulent stripes in the RPCF (as well as PCF) and the spiral turbulence in a Taylor-Couette flow are similar: cf. Prigent et al. [4]. The TaylorCouette flow has also a strong effect of the centrifugal force, so this force can affect the flow stability. It is of interest that the same phenomenon was observed also in the PCF, which is not affected by any body force, and also in RPCF, where turbulent motions are quenched locally by the stabilizing effect due to the Coriolis force.

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4 Conclusions The spatio-temporal intermittencies observed in a plane Poiseuille flow and a rotating plane Couette flow (also in non-rotating case) have been investigated using direct numerical simulations and also through flow-visualization experiments. It was found that the turbulence stripe occurred spontaneously at Reynolds numbers of Rep or Rew ≤ 375 in the case without system rotation, and also stipes were observed in the plane Couette flow at higher Reynolds numbers when subjected to strong stabilizing rotations. Its wave-lengths, which are normalized by each relevant outer units (i.e., the channel half width for the Poiseuille flow, and the whole width for the Couette flow), and its inclination angle were roughly same for these flow systems. The turbulence stripe flow seems to be an intrinsic regime in the reverse transition (from uniform featureless turbulence to laminar) irrespectively of flow systems with/without system rotation. Acknowledgements This work has been partially supported by KAKENHI (#20860070), and part of this study was conducted in “Research Center for the Holistic Computational Science (Holcs)” supported by MEXT. The present simulations were performed with the use of SX-9 at Cyberscience Center, Tohoku University.

References 1. Carlson, D.R., Widnall, S.E., Peeters, M.F.: A flow-visualization study of transition in plane Poiseuille flow. J. Fluid Mech. 121, 487–505 (1982) 2. Wygnanski, J., Sokolov, M. and Friedman, D.: On transition in a pipe. Part 2. The equilibrium puff. J. Fluid Mech. 69, 283–304 (1975) 3. Tsukahara, T., Seki, Y., Kawamura, H., Tochio, D.: DNS of turbulent channel flow at very low Reynolds numbers. In: Humphrey, J.A.C. et al. (eds.) Proceedings of Fourth International Symposium on Turbulence and Shear Flow Phenomena, Williamsburg, USA, Jun. 27-29, pp. 935–940 (2005) 4. Prigent, A., Grégoire, G., Chaté, H., Dauchot, O., van Saarloos, W.: Large-scale finitewavelength modulation within turbulent shear flows. Phys. Rev. Lett. 89, 014501 (2002) 5. Prigent, A., Grégoire, G., Chaté, H., Dauchot, O.: Long-wavelength modulation of turbulent shear flows. Physica D 174, 100–113 (2003) 6. Barkley, D., Tuckerman, L.S.: Computational study of turbulent laminar patterns in Couette flow. Phys. Rev. Lett. 94, 014502 (2005) 7. Barkley, D., Tuckerman, L.S.: Mean flow of turbulent-laminar patterns in plane Couette flow. J. Fluid Mech. 576, 109–137 (2007) 8. Tsukahara, T., Kawamura, H., Shingai, K.: DNS of turbulent Couette flow with emphasis on the large-scale structure in the core region. J. Turbulence 7, 019 (2006) 9. Tillmark, N., Alfredsson, P.H.: Experiments on transition in plane Couette flow. J. Fluid Mech. 235, 89–102 (1992) 10. Alfredsson, P.H., Tillmark, N.: Instability, transition and turbulence in plane Couette flow with system rotation. In: Mullin, T., Kerswell, R.R. (eds.) Laminar Turbulent Transition and Finite Amplitude Solutions, pp. 173–193, Springer, Netherlands (2005) 11. Duguet, Y., Schlatter, P.: Stripy patterns in low Reynods number Couette turbulence. In: Abstracts of Seventh IUTAM Symposium on Laminar-Turbulent Transition, Stockholm, Sweden, Jun. 23-26, pp. 72–73 (2009)

Direct Numerical Simulation and Theoretical Analysis of Perturbations in Hypersonic Boundary Layers Anatoli Tumin, Xiaowen Wang, and Xiaolin Zhong

Abstract Direct numerical simulations of receptivity in a boundary layer over a flat plate and a sharp wedge were carried out with two-dimensional perturbations introduced into the flow by periodic-in-time blowing-suction through a slot. The free stream Mach numbers are equal to 5.92 and 8 in the cases of adiabatic flat plate and sharp wedge, respectively. The perturbation flow field was decomposed into normal modes with the help of the multimode decomposition technique based on the spatial biorthogonal eigenfunction system. The decomposition allows filtering out the stable and unstable modes hidden behind perturbations having another physical nature.

1 Introduction The progress being made in computational fluid dynamics provides an opportunity for reliable simulation of such complex phenomena as laminar-turbulent transition. The dynamics of flow transition depends on the instability of small perturbations excited by external sources. Computational results provide complete information about the flow field that would be impossible to measure in real experiments. Recently, a method of normal mode decomposition was developed for two- and three-dimensional perturbations in compressible and incompressible boundary layers [6, 2, 7]. The method was applied to analysis of DNS data for perturbations introduced through the wall in the vicinity of the actuator [9]. The analysis demonstrated very good agreement between amplitudes of the modes filtered out from the A. Tumin The University of Arizona, Tucson, AZ 85721, USA, e-mail: [email protected] X. Wang University of California, Los Angeles, CA 90095, USA, e-mail: [email protected] X. Zhong University of California, Los Angeles, CA 90095, USA, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_69, © Springer Science+Business Media B.V. 2010

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A. Tumin, X. Wang, X. Zhong

DNS data and linear theory of the flow receptivity to blowing-suction through the wall. In the present work, we apply the multimode decomposition to DNS results in a boundary layer past a flat plate and a sharp wedge downstream from the actuator in order to compare amplitudes of the modes found from the computations with the predictions of the linear stability theory.

2 Outline of the method The method of multimode decomposition of perturbations having a prescribed frequency is based on the biorthogonal eigenfunction system for linearized NavierStokes equations [7]. For the clarity of further discussion, we reproduce the main definitions necessary for discussing the present work. We consider a compressible two-dimensional boundary layer in the Cartesian coordinates, where x and z are the downstream and spanwise coordinates, respectively, and coordinate y corresponds to the distance from the wall. We write the governing equations (the linearized Navier-Stokes equations) for a periodic-in-time perturbation (the frequency is equal to zero in the case of a roughness-induced perturbation), ∼ exp (−iω t), in matrix form as ∂ ∂A ∂A ∂A ∂A L0 + L1 = H1 A + H2 + H3 + H4 A (1) ∂y ∂y ∂y ∂x ∂z where vector A is comprised of velocity components, pressure, temperature, and some of their derivatives; L0 , L1 , H1 , H2 , H3 , and H4 are 16 × 16 matrices (their definitions are given in Ref. [8]). Matrix H4 originates from the nonparallel character of the flow. It includes terms with the y-component of the mean flow velocity and derivatives of the mean flow profiles with respect to the coordinate x. In the quasi-parallel flow approximation, the solution of the linearized NavierStokes equations can be expanded into normal modes of the discrete and continuous spectra {Aαβ , Bαβ } [7], where Aαβ and Bαβ are eigenfunctions of the direct and adjoint problems. Subscripts α and β indicate the eigenfunctions corresponding to the streamwise, α , and spanwise, β , wavenumbers, respectively. The eigenfunction system {Aαβ , Bαβ } has an orthogonality relation given as ∞

Z H2 Aαβ , Bα ′ β ≡ H2 Aαβ , Bα ′ β dy = Γ ∆αα ′

(2)

0

where Γ is a normalization constant, ∆αα ′ is a Kronecker delta if either α or α ′ belongs to the discrete spectrum, and ∆ αα ′ is a Dirac delta function if both α and α ′ belong to the continuous spectrum. In a weakly nonparallel flow, one can employ the method of multiple scales by introducing fast (x) and slow (X = ε x, ε ≪ 1) scales. The mean flow profiles depend

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on y and X only, whereas the perturbation will depend on both length scales. In the case of a discrete mode, solution of the linearized Navier-Stokes equation is presented in the form h i R R (1) Aβ (x, X, y) = Dν (X) Aαν β (X , y) ei αν (X)dx + ε Aαν β (X, y) ei αν (X)dx + . . . (3)

where the function Dν (X ) has to be determined. After substitution of Eq. (3) into (1) Eq. (1), we arrive in order O(ε ) at an inhomogeneous equation for Aαν β . The solvability condition of this equation allows finding Dν (X) (one can find details and relevant references in [8]).

3 Numerical Approach In the present work, we used the DNS results for flows past a flat plate and a sharp wedge with periodic-in-time perturbations introduced through the wall [10, 11]. The flow is assumed to be thermally and calorically perfect. The governing equations are the Navier-Stokes equations for a compressible gas in the conservative form. The fifth-order shock-fitting finite difference method of Zhong [12] is used to solve the governing equations in a domain bounded by the bow shock and the flat plate (or wedge). In other words, the bow shock is treated as a boundary of the computational domain. The Rankine-Hugoniot relations across the shock and a characteristic compatibility relation coming from downstream flow field are combined to solve the flow variables behind the shock. The shock-fitting method makes it possible for the Navier-Stokes equations to be spatially discretized by high-order finite difference methods. Specifically, a fifth-order upwind scheme is applied to discretize the inviscid flux derivatives. By using the shock-fitting method, the interaction between the bow shock and the wall forcing induced perturbations is solved as a part of solutions with the position and velocity of the shock front being solved as dependent flow variables. A second-order TVD scheme [13] is applied to simulate the steady base flow in a small region including the leading edge to supply inlet conditions for the shock-fitting simulation. The same numerical method was used in Refs. [3, 4, 5]. Both cases correspond to the adiabatic wall boundary condition.

4 Results 4.1 Flat plate Free-stream flow conditions: Mach number M∞ = 5.92, temperature T∞ = 48.69 K, pressure p∞ = 742.76 Pa. The Prandtl number and the specific heats ratio are 0.72 and 1.4, respectively. The periodic-in-time blowing-suction has been applied

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through a slot having coordinates of the leading and trailing edges at 33 mm and 37 mm from the leading edge, respectively. The frequency of the perturbation was 100 kHz. Analyses of the mean flow velocity, temperature profiles and their derivatives have shown that they agree well with the self-similar solution for a boundary layer over a flat plate. Therefore, self-similar profiles have been used in the stability equations. In order to deal with the two-dimensional perturbations within the solver of Refs. [7, 8], the spanwise wave number β scaled with the Blasius length scale, L = (µ∞ x/ρ∞U∞ )1/2 , was chosen equal to 10−5 .

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In order to illustrate further analysis of DNS results, features of the spectrum should be introduced. Figure 1 shows the branches of the continuous spectrum and two discrete modes at x = 0.08 m. One of the discrete modes is labeled as mode F (fast), the other is labeled as mode S (slow). The modes’ names stem from their phase velocity features in the vicinity of the leading edge. One can see in Fig. 2 that mode S is synchronized with the slow acoustic wave (cr = 1 − 1/M∞ ), whereas mode F is synchronized with the fast acoustic wave (cr = 1 + 1/M∞ ). At the chosen flow parameters, mode F is always stable, and mode S is the unstable mode. One can see that mode F is synchronized with vorticity/entropy modes having dimensionless phase velocity cr = 1 at x ≈ 0.25 m. The significance of the decaying mode F stems from its synchronization with mode S, where the decaying mode can give rise to the unstable mode (switching of the modes), which may lead to the transition [1]. Figure 3 shows pressure perturbation on the wall (scaled with the free stream pressure) obtained in the DNS (indicated as “measured”) and projections on the discrete modes F and S. Amplification and decay of the discrete modes has been evaluated including the nonparallel flow effects as it is outlined in section 2. We do not show the amplitudes of the modes calculated within the quasi-parallel flow approximation. In the case considered, the nonparallel flow effect is significant. The “measured” data for the wall pressure perturbation have wiggles near the actuator

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region due to input from the various modes presented in the signal. The filtered out amplitude of the unstable mode S is smooth, and it is in good agreement with the theoretical prediction on the whole interval. It is interesting to look at the filtered out decaying mode F. It is in good agreement with the theoretical prediction (thin solid line) up to x ≈ 0.25 m. After that, it has a jump and the amplitude becomes comparable with the amplitude of the mode S. The result can be attributed to the (1) next term in the expansion (3). The second term, Aαν β (X, y), can be expanded into the eigenfunction system. It is a standard problem of finding eigenfunctions of a perturbed operator using the unperturbed basis. For the non-resonance case when eigenvalues of modes F and S are distinct (αS 6= αF ), it is straightforward to find (1) a projection of Aα β (X, y) on AαF β (X, y) (indices S and F indicate slow and fast S discrete modes, respectively): D E (0) ∂ AS (0) ¯ 4 A(0) , B(0) H2 ∂ X , BF + H F S DS (X) D E CF (X) = (4) (0) (0) i ( αF − αS ) H2 AF , BF ¯ 4 = ε −1 H4 . For the purpose of brevity, we use only indices F and S indicatwhere H ing the fast and slow modes, respectively. The input of mode F into the second term of Eq. (3) has a wave number (and phase speed) corresponding to mode S. We call this contribution of the mode F as “S2F centaur” in order to emphasize the twofold character of the term. The wall pressure perturbation associated with “S2F centaur” is shown in Fig. 3 as well. Although the theoretical result for mode F downstream from the point of synchronism demonstrates qualitatively the same behavior as the amplitude of the DNS projection onto the mode F, there is a quantitative discrepancy that has yet to be understood.

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4.2 Wedge Free-stream flow conditions: Mach number M∞ = 8, temperature T∞ = 54.8 K, pressure p∞ = 389 Pa. The Prandtl number and the specific heats ratio are 0.72 and 1.4, respectively. The periodic-in-time blowing-suction has been applied through a slot having coordinates of the leading and trailing edges at 51.84 mm and 63.84 mm from the leading edge, respectively. Frequency of the perturbation is 104.44 kHz.The wedge half-angle is equal to 5.3o degrees. In this example, we use the velocity and temperature profiles obtained in the computation. The result of the DNS data projection onto the mode S and comparison with theoretical results is shown in Fig. 4. The results of sections 4.1 and 4.2 illustrate how the multimode decomposition technique may serve as a tool for gaining insight into the flow dynamics in the presence of perturbations belonging to different modes. Acknowledgements This work was sponsored by the AFOSR/NASA National Center for Hypersonic Research in Laminar-Turbulent Transition and by the Air Force Office of Scientific Research, USAF, under Grants No. FA9550-08-1-0322 (A.T.), FA9550-07-1-0414 (X.Z. and X. W.) monitored by Dr. J. D. Schmisseur.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

A. Fedorov, A. Khokhlov, Theor. Comp. Fluid Dyn. 14 P. Gaydos, A. Tumin, AIAA J. 42 Y. Ma, X. Zhong, J. Fluid Mech. 488, pp. 31-78 Y. Ma, X. Zhong, J. Fluid Mech. 532, pp. 63-109 Y. Ma, X. Zhong, J. Fluid Mech. 488 A. Tumin, Phys. Fluids 15 A. Tumin, J. Fluid Mech. 586 A. Tumin, J. Spacecraft and Rockets 45 A. Tumin, X. Wang, X. Zhong, Phys. Fluids 19 X. Wang, X. Zhong, AIAA Paper 2005-5025 X. Wang, X. Zhong, Phys. Fluids 21 X. Zhong, J. Comput. Phys. 144 X. Zhong, T. Lee, AIAA Paper 1996-1856

Flow Transition in Free Liquid Film Induced by Thermocapillary Effect Ichiro Ueno, Toshiki Watanabe, and Toshihiro Matsuya

Abstract The authors introduce unique flow patterns realized in a thin free liquid film of O(0.1-1.0 m) in thickness exposed to a temperature gradient parallel to the free surfaces. A thermocapillary effect drives the fluid over the free surfaces, and leads two types of flow transitions; from two-dimensional ’basic’ flow to threedimensional ’oscillatory’ flow, and then to three-dimensional chaotic flow. The flow indicates a different bifurcation scenario strongly depending upon the film thickness. The authors describe their flow structures as well as occurring conditions.

1 Introduction The authors have been inspired by a series of scientific performances by Dr. Donald Pettit, a NASA astronaut, on The International Space Station during his stay in 2003. Among his invaluable performances, one can see a unique behavior of free liquid film of O(10 cm) in diameter sustained in a metal ring [1]. A microgravity condition would enable a liquid film to be sustained stably with its surface tension and wettability of the sustainer against the liquid concerned. This unique system is a potential way to realize a new kind of crystallization process of materials. A free liquid film has a great feature that a contact area with a solid sustainer can be minimized. This is a vigorous advantage to avoid the liquid contaminated by imIchiro UENO Tokyo University of Science (TUS), 2641 Yamazaki, Noda, Chiba 278-8510, Japan, e-mail: [email protected] Toshiki WATANABE Graduate School at TUS, 2641 Yamazaki, Noda, Chiba 278-8510, Japan, e-mail: [email protected] Toshihiro MATSUYA Graduate School at TUS, 2641 Yamazaki, Noda, Chiba 278-8510, Japan, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_70, © Springer Science+Business Media B.V. 2010

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purities supplied from the solid. It is of great importance to accumulate knowledge on a thermal-fluid dynamics of a free film of a liquid with a non-zero temperature coefficient of the surface tension exposed to a non-uniform temperature field. In the case of a thin liquid film form in a rectangular cavity (i.e., a liquid film with a single free surface) with a temperature difference between both-end walls, Smith & Davis [2] predicted that a new type of thermal-fluid instability would arise in the film beyond a critical temperature difference. Under a temperature difference smaller than the critical value, a two-dimensional steady flow parallel to the imposed temperature gradient is induced in the film. After the onset of the transition, roll structure emerges in the film and propagate in a direction inclined to the imposed temperature gradient. This unique instability was named as ’hydrothermal wave instability,’ and was revealed by an experimental approach by Riley & Neitzel [3]. After their beautiful experimental work, a number of experimental and numerical works have been carried out with a liquid film with a single free surface for a decade [4]. In the case of free liquid films, on the other hand, little knowledge has been accumulated on a flow dynamics in such a thin film with two free surfaces; one can easily imagine that a long-lasted stable free liquid film is hardly formed in a terrestrial experiment. In the present study, the authors carry out a three-dimensional numerical simulation and a series of experiments with their special attention to a flow field inside the free surface under non-uniform temperature distribution; that is, a thermocapillary-driven flow induced in a free thin liquid film under a temperature gradient parallel to the free surfaces.

2 Target Geometry Target geometry is shown in Fig. 1. A thin liquid film with two free surfaces parallel to x-z plain is concerned. Length in the direction of the temperature gradient is defined as Lx , and that in the span-wise direction Lz . The thickness of the film is presented as d. In the present geometry, two kinds of aspect ratios are defined; that is, Γx = Lz /Lx and Γy = Lx /d. One end wall is maintained at a temperature of Th , and the other at Tc (Th > Tc ). Thus the film is exposed to a temperature difference ∆ T = Th − Tc . Intensity of the thermocapillary effect imposed to the film is described by non-dimensional Marangoni number; Ma =

σT ∆ T d 2 , ρνκ Lx

(1)

where σT is the absolute value of the temperature coefficient of the surface tension, ρ the density, ν the kinematic viscosity, κ the thermal diffusivity. The Marangoni number can be described as the product of the thermocapillary Reynolds number Reσ and Prandtl number Pr.

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Fig. 1 Target geometry: a free liquid film exposed to a temperature gradient.

3 Numerical Simulation In the numerical simulation, the authors solve following non-dimensional equations of continuity, momentum and energy by a finite differential method. ∇·u = 0 ∂u Pr 2 + (u · ∇)u = −∇P + ∇ u ∂t Ma ∂T Ma( + (u · ∇)T ) = ∇2 T ∂t

(2)

Free surfaces are assumed to be flat without any dynamical deformation. Stress balance on the free surface leads so-called Marangoni boundary conditions. Nonslip condition is applied to the both of hot and cold walls. The Prandtl number is fixed at 10.0 through the simulation. The present numerical code is validated under the condition of the single-free-surface thin film by comparing the incident angle, the wave number and the travelling velocity of the HW with the theoretical prediction by Smith & Davis [2].

4 Experiment In order to realize a stable free liquid film in a series of experiments under the normal gravity, the authors prepared a rectangular hole in aluminum plates of 0.6, 1.0 and 1.2 mm in thickness, and the one of iron of 0.2 mm in thickness as the liquid film holder. A liquid film is formed and sustained inside the rectangular hole. Figure 2 shows one example of the prepared holders; one side of the plate is heated, and the other cooled to expose the hole to a designated temperature difference. Silicone oils of 2 and 5 cSt were employed as the examined fluids. Temperature near the end walls sustaining the free liquid film was measured with thermocouples to evaluate the temperature difference ∆ T . Induced flow was visualized by suspending gold-coated acrylic particles of 15 µ m in diameter. The visualized flow was detected by a CCD camera whose frame

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Fig. 2 Example of liquid film holder (right). Liquid film is formed in a square region indicated as ’test area.’ Thickness of the holder in the figure is of 0.2 mm.

rate was 30 frame/s. Shutter speed was 1/60 s. Temperature variation over the top free surface was measured by use of an infrared camera of 22 frame/s in frame rate.

5 Results & Discussion Three dimensional numerical simulation indicates a two-dimensional flow inside the film is realized in which the fluid returns in the middle region of the film under a small temperature gradient. By increasing the temperature difference between the end walls, instability takes place to realize a three-dimensional flow. Figure 3 indicates (1) top view of the deviation of the temperature over the top free surface and (2) cross-sectional view of temperature and velocity in x-y plane of ’oscillatory’ flows. Case (a) indicates a single free surface (top surface is free) and case (b) double free surfaces under the same intensity of the thermocapillary effect. In the figure right wall is hot, and left wall cold. Lines in the film in the frames (2) indicate the isotherms. In the case of (b) free liquid film, one can clearly observe a typical inclined thermal wave propagating with a certain angle. There exists a slight difference in the propagating angle, the speed of the hydrothermal wave between the cases (a) and (b). Clear difference is emerged in the cross-sectional view; the free film has a double-layered structure of rolls with a constant phase shift. That is, an effective thickness of the liquid film to realize the hydrothermal wave instability must be different for the free liquid film case. Typical examples of the induced flow field observed from above are shown in Fig. 4. These images are obtained by integrating 30 frames, or for 1 s. Under small-enough temperature difference for d = 0.6 mm, a similar basic flow is realized inside the film (Fig. 4 -(a)-(1)). That is, thermocapillary effect drives the free surfaces towards the cold wall, and the fluid returned towards the hot wall in the middle region of the film between the free surfaces. By increasing the temperature difference between the walls, a three-dimensional oscillatory flow emerged in the film (Fig. 4 -(a)-(2)). The oscillatory flow, however, never exhibits an ordered hydrothermal wave instability with a certain wavelength and propagation speed as predicted by the simulation aforementioned. The film is divided into two cells in a span-wise direction. That is, the flow indicates a three dimensional pattern. The particles in one cell seldom penetrate into another cell. This span-wise cellar pattern has not been indicated by the present numerical simulation. It should be noted that such a pattern has not been reported in the case of the thin liquid film in the cavity. In the case of thin liquid film of 0.2 mm in thickness, on

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Fig. 3 Typical example of ’oscillatory’ flow; (1) top view of the deviation of the temperature over the top free surface and (2) cross-sectional view in x-y plane in the cases of (a) a thin liquid film in the cavity (a single free surface (top surface is free)) and (b) double free surfaces under the same intensity of the thermocapillary effect. Right-hand side is the hot wall, and the opposite is the cold wall for each frame. Marangoni number Ma is kept constant at 1 × 103 .

the other hand, two-dimensional basic flow is hardly realized; even under a small thermocapillary effect, or at small Marangoni number, a single-layered span-wise cellular pattern emerges in the film shown in Fig. 4 -(b)-(1). Further increasing ∆ T , the flow exhibits a chaotic behavior as shown in Fig. 4 -(b)-(2). This could be the first capture of the route to the chaotic flow in a free liquid film by the present authors group, to the best of their knowledge. Nonetheless, a double-layered hydrothermal wave instability as predicted by the numerical simulation (Fig. 3-(b)) hardly emerges in the film in the experimental runs so far. The authors continue further investigation on occurring conditions of induced flow patterns and mechanism of these instabilities by both of numerical and experimental approaches.

Fig. 4 Typical example of flows observed from above at (a): (∆ T [K], Ma [-]) = (1) (4.7, 1.6 × 102 ) & (2) (37.7, 2.1 × 103 ) for d = 0.6 mm, and (b): (1) (2.9, 1.1 × 101 ) & (2) (17.9, 6.8 × 101 ) for d = 0.2 mm. Right end wall is hotter for each frame. Each frame is obtained by integrating for 1 s.

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6 Concluding Remarks Flow patterns and their transitions in a free thin liquid film induced by thermocapillary effect are investigated with numerical and experimental approaches. The authors indicate, through both approaches, that a double-layered two-dimensional return flow emerges as a ’basic’ flow, then a double-layered three-dimensional ’oscillatory’ flow, and then a drastic transition into a single-layered span-wise cellular flow emerges in increasing the thermocapillary effect. In the case of thinner liquid film, by the experiment, a double-layered basic state never appears in the film; a single-layered span-wise cellular flow emerges even with a very tiny temperature difference.

References 1. http://science.nasa.gov/headlines/y2003/25feb nosoap.htm Cited 4 July 2009. 2. Smith, M. K. & Davis, S. H. (1983), Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities, J. Fluid Mech. 132: 119-144. 3. Riley, R. J. & Neitzel, G. P. (1998), Instability of thermocapillarybuoyancy convection in shallow layers. Part 1. Characterization of steady and oscillatory instabilities, J. Fluid Mech. 359: 143-164. 4. e.g., Kawamura, H., Tagaya, E. & Hoshino, Y. (2007), A consideration on the relation between the oscillatory thermocapillary flow in a liquid bridge and the hydrothermal wave in a thin liquid layerInt. J. Heat Mass Trans. 50: 1263-1268.

Boundary layer transition by interaction of streaks and Tollmien–Schlichting waves Tamer A. Zaki, Yang Liu, and Paul A. Durbin

Abstract Transition to turbulence in zero-pressure-gradient boundary layers is studied using direct numerical simulations (DNS). Both discrete and discrete-continuous mode transition are considered by prescribing particular eigenmodes of the Orr– Sommerfeld (OS) equation at the inlet plane of our computational domain. The downstream evolution of the modes, secondary instability, and non-linear breakdown are computed using DNS. The natural, or orderly, route is simulated starting from the discrete, Tollmien– Schlichting (TS) waves. Introducing streaks, which are forced by low-frequency continuous modes, promotes breakdown of the boundary layer. Transition location is controlled by two competing effects: In the presence of a streak distortion of the base flow, the growth rate of the primary TS wave is reduced. However, their secondary instability is enhanced. The first effect dominates for narrow streak distortions, and transition can be delayed. For wide streaks, direct resonance between the three-dimensional streaks and the secondary instability mode promotes transition. Simulations with a fully turbulent free-stream and TS waves are also carried out. For low turbulence intensity, transition via secondary instability of the TS waves is promoted, and proceeds in the same manner as the canonical simulations of continuous-discrete mode interactions.

Tamer A. Zaki Mechanical Engineering, Imperial College, Exhibition Road, London SW7 2AZ, UK e-mail: [email protected] Yang Liu Knight Capital, USA Paul A. Durbin Aerospace Engineering, Iowa State University, Ames, IA 50011, USA e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_71, © Springer Science+Business Media B.V. 2010

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1 Introduction Boundary layer breakdown to turbulence is generally classified as orderly or bypass. The orderly route is characterized, in the early stages, by the amplification of Tollmien–Schlichting (TS) instability waves. At the critical Reynolds number, the first linear instability waves are two-dimensional. With increasing downstream distance, TS waves amplify, develop secondary instability [4], and finally breakdown to turbulence [7]. The proceedings of bypass transition are less precise. Indeed the term bypass refers to all transition scenarios that deviate from the orderly description [10], but is most commonly used in reference to transition beneath vortical disturbances [2]. The initial stages of bypass transition also involve the amplification of initial disturbances, which are elongated in the streamwise direction and reach high-amplitude relative to the free-stream turbulence intensity [6]. These disturbances, referred to as Klebanoff distortions, are not due to an exponential instability. Instead, they have been explained in terms of displacement of mean momentum [11]. Downward displacement of high-momentum fluid towards the wall generates high-speed perturbation streaks; upward displacement of low-momentum fluid away from the wall generates low-speed streaks. The low-speed streaks become unstable when exposed to free-stream turbulent forcing, and locally breakdown into turbulent patches [5]. The inception of turbulent spots marks the start of non-linear breakdown. The above description of bypass transition presupposes that TS waves are absent, or do not play a role in breakdown. This is consistent with observation of boundary layer breakdown at high free-stream turbulence levels. At lower turbulence intensities and in adverse pressure gradient, however, discrete instability waves emerge at lower Reynolds numbers and can coexist with the streaks. Whether Klebanoff distortions promote breakdown of TS waves is a matter of active research. Some results indicate a stabilizing influence [1], and others a destabilizing effect [12]. Here, the notion of stability must be qualified. First, for a base state which is a Blasius profile distorted by steady streaks, the growth rate of the TS-type waves is reduced relative to the conventional discrete modes in the absence of streaks. However, the secondary instability of these waves is increased [9]. In this paper, the interaction of streaks and TS waves is studied using direct numerical simulations. First, orderly transition is simulated starting from the discrete Tollmien–Schlichting waves. When acting alone, a discrete instability wave amplifies with increasing Reynolds number, but subsequently decays. No secondary instabilities are excited due to the absence of any background noise in our simulations. A low-level of free-stream turbulence intensity, Tu ∼ 0.1% is included in order to seed the “natural” secondary instability of the TS waves. The development of Λ -structures is consistent with the analysis of Herbert (1988) [4], and is followed by breakdown to turbulence. In order to evaluate the interaction of Klebanoff streaks and TS waves, two approaches are pursued. First, the streaks are modeled by the boundary layer response to a single continuous Orr–Sommerfeld mode. This choice allows direct control of

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the streak amplitude and spanwise scale. These idealized simulations are followed by DNS of fully-turbulent free-streams interactions with TS waves.

2 Interaction of streaks and Tollmien–Schlichting waves Direct numerical simulation of the amplification of a single, two-dimensional Tollmien–Schlichting wave was carried out. The non-dimensional frequencypof the TS wave is F ≡ 106 ω /Re = 33 and the inflow plane was located at R ≡ Ux/ν = 400. The amplitude of the mode was AT S = 1%. For these parameters, the inflow disturbance is unstable, and amplifies downstream on the inlet plate. However, in the absence of any background disturbances, the TS wave remains two-dimesional and decays when it crosses the second branch of the stability curve. When a very low level of background disturbances is introduced, Tu ∼ 0.1%, the two-dimensional Tollmien–Schlichting waves develop a sub-harmonic secondary instability, illustrated in the top view of the perturbation field in figure 1. The staggered Λ pattern corresponds to b ≡ 103 β /R = 0.35, which is the spanwise wavenumber of maximum secondary instability growth rate [4].

Fig. 1 Plane view of a boundary layer undergoing natural transition. The plane is colored by contours of v−perturbations, and the Λ -structures are visualized using an iso-surface Q = −0.01 where Q ≡ ∂i u j ∂ j ui . The surface of the Λ ’s is colored by the mean streamwise velocity.

Continuous-discrete mode interactions In order to study the influence of streaks on the evolution of Tollmien–Schlichting waves and their breakdown, a single low-frequency Orr–Sommerfeld continuous mode is introduced with the unstable TS disturbance at the inlet. Various amplitudes and spanwise wavenumbers of the continuous mode were considered, which resulted in streaks of different intensities and spanwise scales. While the streaks and the Tollmien–Schlichting waves acting alone are innocuous, when they are both included at the inlet to the computational domain, their downstream interaction caused transition. This is despite a reduction in the Tollmien–Schlichting growth rate due to the streak distortion [1, 8]. Increasing the streak amplitude resulted in seemingly conflicting results: For one class of streaks transi-

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tion was promoted, despite a further reduction in the TS growth rate; for the second class of streaks, transition moved further downstream. The skin friction curves of figure 2 capture this dependence of transition location on streak amplitude.

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Fig. 2 The influence of increasing the streak amplitude on transition location. At left, the effect of wide streaks and, at right, the influence of narrow streaks.

These trends were explained using a Floquet analysis [9], and are due to a competition between (a) the decrease in the primary TS growth rate and (b) an enhanced secondary instability in the presence of streaks. For wide streaks, the enhanced secondary instability promotes transition, while for narrow streaks, the stabilizing effect on the primary TS wave is more significant1 . In the DNS of the interaction between wide streaks and TS waves, transition proceeded by the formation of Λ -structures. This is suggestive of a secondary instability of the TS wave. However, the width of these structures differed from those in conventional secondary instability of Tollmien–Schlichting waves. They were modulated by the streaks: the Λ ’s were of the same spanwise size as the Klebanoff mode (see figure 3). Floquet analysis of the base flow, which is a superposition of the Blasius profile, a streak distortion and a primary TS wave demonstrates that fundamental resonance in the spanwise wavenumber yields an unstable secondary instability mode. Therefore, this mode is amplified in the DNS, which explains the spanwise scale of the Λ -structures in figure 3. The secondary instability is further enhanced by higher amplitude streaks, and transition moves upstream. The interaction of TS waves and narrow streaks was less straightforward. Indeed Λ -structures emerged upstream of transition, and were responsible for breakdown. However, the width of these Λ ’s did not match the natural secondary instability of TS waves, nor did it coincide with the spanwise size of the streaks (see figure 4). Floquet analysis demonstrates that the fundamental mode in the spanwise direction, based on the streak width, is stable. Instead, the most unstable secondary instability is a detuned mode. This provides an explanation of the size of the Λ ’s in the case of narrow streaks. 1

In Liu et al. [8], the streaks were named according to the number of wavelength in the spanwise extent of the domain. The wide streaks, mode 2, correspond to β δ ⋆ = 11.6; the narrow streaks, mode 5, correspond to β δ ⋆ = 4.6.

Streak–TS wave interactions

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Fig. 3 Top: Threedimensional view showing the Λ ’s generated by the interaction of TS waves and wide Klebanoff streaks [3]. Right: Zoomed-in plane view. The Λ ’s are visualized using an iso-surface Q = −0.01.

Fig. 4 Time-sequence showing the evolution of Λ ’s due to the interaction of TS waves and narrow streaks. The width of the Λ ’s does not match the Klebanoff distortion [3].

The influence of free-stream turbulence The above simulations concerned the interaction of idealized streaks and TS waves. Here, we consider streaks which arise due to free-stream turbulent forcing. Three conditions were simulated, the first at Tu = 1.5% and two at 1.0%. The lower intensity cases differed in integral scale. The smaller scale caused faster decay of the free-stream turbulence, and generated weaker streaks. At Tu = 1.5%, bypass transition can take place at lower Reynolds number than in the current simulations. However, the choice of the turbulence level is aimed at studying the interaction of TS waves and high-intensity streaks, which is possible in adverse pressure conditions. An instance of the interaction of streaks and TS waves is presented in figure 5. The plane view shows contours of u-perturbations, and the iso-surface Q = −0.01. At the highest Tu = 1.5, narrow Λ ’s emerge. Their spanwise wavenum-

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ber, b ∼ 0.80 − 0.85, is much higher than the most-unstable secondary instability of TS waves [4]. They are, however, the same width as the streaks. They are therefore fundamental, resonant, secondary instabilities [8]. As the turbulence intensity is reduced, two sizes of Λ ’s become apparent. The wider Λ ’s correspond to b ∼ 0.35−0.5, much nearer to the “natural” secondary instability of TS waves. They can arise naturally due to the background disturbance environment, or be triggered via a spanwise harmonic instability in the presence of streaks. As a result, they are observed earlier upstream than in figure 1 or “natural” transition.

Fig. 5 Plane view of the u-perturbation velocity, and iso-surface of Q = −0.01. Top: Tu = 1.5%; middle:Tu = 1.0%; bottom: Tu = 1.0% and smaller integral scale of the free-stream turbulence.

References 1. Cossu C, Brandt L (2004) On Tollmien–Schlichting-like waves in streaky boundary layers. Eur. J. Mech. B/Fluids 23:815–833. 2. Durbin P, Wu X (2006) Transition beneath vortical disturbances. Annu. Rev. Fluid Mech. 30:107–128. 3. Durbin P A, Zaki T A, Liu Y (2009) Interaction of discrete and continuous boundary layer modes to cause transition. Intl. J. Heat Fluid Flow 30:403–410. 4. Herbert T (1988) Secondary instability of boundary layers. Annu. Rev. Fluid Mech. 20:487– 526. 5. Jacobs R G, Durbin P A (2001) Simulations of bypass transition. J. Fluid Mech. 428:185–212. 6. Klebanoff P S (1971) Effect of freestream turbulence on the laminar boundary layer. Bull. Am. Phys. Soc. 10:1323–1334. 7. Kleiser L, Zang T A (1991) Numerical simulation of transition in wall-bounded shear flows. Annu. Rev. Fluid Mech. 23:495–537. 8. Liu Y, Zaki T A, Durbin P A (2008) Boundary-layer transition by interaction of discrete and continuous modes. J. Fluid Mech. 604:199–233. 9. Liu Y, Zaki T A, Durbin P A (2008) Floquet analysis of secondary instability of boundary layers distorted by Klebanoff streaks and Tollmien–Schlichting waves. Phys. Fluids 20:124102. 10. Morkovin M V (1969) On the many faces of transition. In: Wells C S (ed) Viscous drag reduction. Plenum, New York. 11. Phillips O M (1969) Shear-flow turbulence. Annu. Rev. Fluid Mech. 1:245–264. 12. Wu X, Choudhari M (2003) Linear and nonlinear instabilities of a Blasius boundary layer perturbed by streamwise vortices. Part 2. Intermittent instability induced by long-wavelength Klebanoff modes. J. Fluid Mech. 483:249–286.

Numerical Investigation of Subharmonic Resonance Triads in a Mach 3 Boundary Layer Marcus Zengl, Dominic von Terzi, and Hermann Fasel

Abstract The possibility of a subharmonic-resonance triad in the transition process for a flat-plate boundary layer at Mach three was demonstrated using Direct Numerical Simulations (DNS). The parameters controlling such a resonance triad were investigated in detail. Finally, the interaction of this triad with the so-called ‘oblique breakdown’ scenario was studied. To this end, the mechanisms were invoked separately and in combination. By itself the oblique breakdown leads to a faster growth of instability waves than the subharmonic resonance. However, in a combination with a subharmonic-resonance triad, the dominant modes of the oblique breakdown experience a slower growth, whereas those of the subharmonic resonance are more amplified. Overall, the interaction promotes transition and the rise in skin friction and the drop in shape factor are moved upstream when compared to the oblique breakdown.

1 Introduction Wind-tunnel experiments, theoretical studies and Direct Numerical Simulations (DNS) of a flat-plate boundary layer at Mach two indicate that subharmonicresonance triads can play a role in supersonic laminar-turbulent transition [4, 3]. For Mach three, numerical investigations demonstrated that the so-called ‘oblique breakdown’ scenario of [2] is a transition mechanism that by itself can lead to a fully developed turbulent boundary layer [5, 8]. So far it was not clear whether subharmonic resonances can occur in the boundary layer at Mach three and, how their

Marcus Zengl1 · Dominic von Terzi2 · Hermann Fasel Department of Aerospace and Mechanical Engineering, The University of Arizona, U.S.A. e-mail: [email protected], [email protected], [email protected] 1 Present affiliation: Institut f¨ ur Aerodynamik und Gasdynamik, Universität Stuttgart, Germany 2 Present affiliation: Institut f¨ ur Thermische Strömungsmaschinen, Universität Karlsruhe, Germany P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_72, © Springer Science+Business Media B.V. 2010

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presence would affect the laminar-turbulent transition process. This was investigated using DNS, and results are presented in this paper.

2 Computational Setup The computational setup is analogous to [5, 8], where the oblique breakdown was studied from its linear inception to a fully turbulent flow. The present study is complementary and focuses on the possibility of subharmonic resonances and their interaction with oblique breakdown for the same flow conditions. The main differences in the setup are in the forcing of instability waves and in the domain size and required resolution. For the present study, it was sufficient to compute only up to the early nonlinear regime. Therefore, results have shown grid-independence using a coarser resolution. Three flow solvers were employed to integrate the compressible Navier–Stokes equations. All DNS codes use a fourth-order Runge–Kutta method for time integration and incorporate finite differences in the streamwise and wall-normal directions and a pseudo-spectral approach for the periodic spanwise direction. A linearized DNS (LDNS) code described in [6] facilitated to establish linear behavior including non-parallel effects and, moreover, to find instability waves that satisfy the resonance condition (Sect. 3). Simulations investigating the characteristics of the resonance for the chosen waves (Sect. 4) were performed using the flow solver NSCC described in [7]. The interaction between the oblique breakdown and subharmonic scenarios (Sect. 5) were investigated with the computer program NS3D [1]. The main difference between the codes is that NSCC employs fourth-order split finite differences and NS3D and the LDNS code use sixth-order compact finite differences. In addition, NS3D computes the second derivatives of the diffusion terms directly, whereas the other codes employ twice a first derivative approximation, i.e. they solve the governing equations in conservative form. Selected simulations were repeated, with both NSCC and NS3D, and identical results on the same grids were obtained. Domain height and resolution studies ensured that the results are independent of the computational grid and timestep. For the studies in Sects. 3 and 4, very small forcing amplitudes were chosen such that the amplitudes of the dominant waves always remained in the linear regime. A streamwise extent of the domain of 1000 ≤ Rx ≤ 1450 was sufficient to study the linear and weakly nonlinear flow behavior due to a subharmonic resonance, where Rx denotes the square root of the Reynolds number based on distance to the leading edge of the plate. An equidistant stepsize of approximately 45 grid points per fundamental wavelength was used in the streamwise direction (x). In the wall-normal direction (y) a non-equidistant grid with 110 points was employed where 40 grid points resided in the boundary layer. The spanwise direction (z) was resolved with 16 Fourier modes. In order to study the interaction of the subharmonic resonance with the oblique breakdown (Sect. 5), in the downstream direction, the domain was extended to Rx = 1675 and the grid was refined towards the outflow. In addition,

Subharmonic Resonance Triads in a Mach 3 Boundary Layer βS1 / βF 1

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Fig. 1 Subharmonic resonance triad for F = 1.5 × 10−5 at Rx = 1400, M = 3. Axes are arranged such that the resonance condition is satisfied at the intersection of the two curves. Vertical lines denote presented computational setups (tuned and detuned)

1000

1100

1200

Rx

1300

1400

Fig. 2 Development of the streamwise wavenumber in the x-direction for the wave triplet fulfilling the resonance condition at Rx = 1400, i.e. the fundamental wave (F) and the two subharmonic waves S1 & S2 marked by the vertical dashed line in Fig. 1

the lateral resolution was increased. Altogether this resulted in 180 points per fundamental wavelength in the streamwise direction and 32 spanwise Fourier modes.

3 Subharmonic Resonance Triad We define a subharmonic-resonance triad as a triplet of waves satisfying the resonance conditions for streamwise and spanwise wavenumbers and frequency, i.e.

αS1 + αS2 = αF , βS1 + βS2 = βF and ωS1 = ωS2 = ωF /2 ,

(1)

respectively, with F representing the fundamental wave and S1 and S2 two appropriate subharmonic waves. Employing Linear Stability Theory (LST) and LDNS a suitable fundamental wave was chosen with a normalized frequency of F = ων∞ /U∞2 = 3 × 10−5 and spanwise wavelength λz = 6.48 × 104 ν∞ /U∞ . In order to find possible subharmonic-resonance triads, calculations using LDNS were carried out to obtain the dispersion relation for waves with a frequency of F/2. In Fig. 1, the results are shown for a given streamwise location. The streamwise wavenumber is plotted over the spanwise wavenumber, both scaled by the corresponding value of the fundamental wave. The two curves show possible α for a given β of a subharmonic wave. The lower and left axis belong to the curve linking diamond symbols and the upper and right axis belong to the curve linking square symbols. The axes are arranged such that the resonance condition of the three waves is met at locations where the curves intersect. A possible triad is therefore found for spanwise wavenumbers of subharmonic waves with a ratio of approximately 6/7 and 13/7 of βF which are denoted as waves S1 and S2, respectively. In Fig. 2, the development in the x-direction of the streamwise wavenumbers αr of the fundamental wave (denoted as F) and the subharmonics S1 and S2 are shown.

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800

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Fig. 3 Amplitude distribution in the x-direction of the streamwise velocity fluctuation for the fundamental and two subharmonic waves; LDNS results are included to indicate linear behavior

900

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Fig. 4 Development in the x-direction of the streamwise wavenumber for the fundamental and two subharmonic waves; LDNS results are included to indicate linear behavior

In addition, the difference in αr between the fundamental and the subharmonic S1 is plotted. As can be seen, this difference matches the curve of the subharmonic S2 extremely well indicating that the resonance condition of Eq. (1) for the selected waves is met within the entire range of the computational domain.

4 DNS of Oblique-Subharmonic Resonance Triad Detailed investigations on the behavior of the resonance were carried out using spatial DNS and introducing instability waves at low amplitude levels through a blowing and suction slot at the wall, i.e. forcing the instability waves. These investigations included studies on the effects of forcing amplitude and phase of the waves participating in the resonance triad, dependence on the forcing location, feedback between two waves, and detuning of the resonance condition. Within the computational domain, one of the two subharmonic waves (S1) was amplified according to LST, while the other wave (S2) was at first only slightly amplified and farther downstream damped. It was shown that in the simulations a resonance occurred since the streamwise development of the two subharmonic waves deviated strongly from linear behavior (see Figs. 3 and 4). Which of the three waves was amplified depended on the parameters studied above. In simulation 4.1 all three waves were forced. In simulation 4.2 only S2 was forced, and in 4.3 only F & S1 were forced. The amplitude of the wave S2 in simulation 4.1 can be viewed as a linear superposition of S2 from simulations 4.2 and 4.3. Depending on the phase of the forcing, different outcomes can be achieved, where forcing S2 in phase or in counter-phase to the nonlinearly generated S2 yields the envelope for the amplitude distribution (dotted lines in Fig. 3). Therefore, low-amplitude forcing of the wave S2 in addition to F and S1 is superfluous, since it does not influence its own amplitude farther downstream once the resonance has taken over. Furthermore, a resonance was observed even if

Subharmonic Resonance Triads in a Mach 3 Boundary Layer

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Fig. 5 Interaction of breakdown mechanisms: Amplitude ratios of simulation where both mechanisms are invoked simultaneously (u′1 ) to simulations with pure oblique or pure obl.-subh. resonance (u′2 )

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Fig. 6 Skin friction coefficient and shape factor for various simulations: 5.1: ±S1 forced, 5.2: ±F forced, 5.3: ±F & ±S1 forced, and 5.4: +F & -S1 forced

the triad was slightly detuned. In this case a higher amplification of the amplified subharmonic but a lower amplification of the other subharmonic were attained.

5 DNS of the Interaction of Oblique-Subharmonic Resonance with Oblique Breakdown The interaction of oblique breakdown with the subharmonic-resonance triad was studied for the detuned case indicated by the dot-dashed vertical line in Fig. 1. This simulation is more cost effective than computing the exact triad (by a factor of seven), but parameter studies indicated that it is likely to show the same effects for the present purpose. Several simulations were carried out with results shown in Figs. 5 and 6: In simulation 5.1 and 5.2, a pure oblique breakdown setup is realized for either the subharmonic S1 or the fundamental F, respectively. Hereby, the oblique breakdown setup for the fundamental leads to the faster growth and the route to transition follows the oblique breakdown mechanism as studied in detail in [5, 8]. In simulation 5.3, both F and S1 are forced symmetrically, i.e. both the oblique breakdown and the detuned subharmonic resonance mechanisms are at work at the same time and can interact. Finally, in simulation 5.4, only the subharmonic resonance is forced. Invoking only the resonance, leads to substantially lower growth of instability waves than for the oblique breakdown scenario for the fundamental wave. Whereas, for the combined forcing (case 5.3), as shown in Fig. 5, it turned out that the presence of the subharmonic resonance leads to a lower amplification of the modes generated by the oblique breakdown mechanism (closed symbols) alone. On the other hand, the growth of the subharmonic modes (open symbols) is accelerated. Figure 6 shows that the overall result is an acceleration of transition, i.e. the rise in skin-friction coefficient and the decline in shape factor move upstream. A possible explanation is that the steady modes generated by the oblique breakdown

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strongly interact with the subharmonic waves. However, to support this conjecture, additional investigations are necessary.

6 Conclusions It was demonstrated that a subharmonic-resonance can occur in a flat-plate boundary layer at Mach three. The parameters controlling the resonance were investigated. It was shown that the presence of a subharmonic resonance has a significant effect on the oblique breakdown mechanism. The growth of dominant modes of the oblique breakdown scenario is diminished, whereas the subharmonic resonance modes are enhanced. Overall, transition is accelerated, by the presence of an oblique-subharmonic resonance, i.e. the rise in skin friction and the drop in shape factor are shifted upstream. As a consequence, for an accurate transition prediction in supersonic flows, interactions of different transition mechanisms need to be accounted for. Acknowledgements Most of the research was carried out while the first author (MZ) was at the University of Arizona with an exchange scholarship of the DAAD. This work was funded by the Air Force Office for Scientific Research under grant FA9550-08-1-0211 with Dr. John Schmisseur serving as program manager. The authors are indebted to Drs. Andreas Babucke and Richard Sandberg for providing their DNS/LDNS codes and to Dr. Ulrich Rist for providing additional computational resources.

References 1. Babucke, A.: Direct Numerical Simulation of Noise–Generation Mechanisms in the Mixing Layer of a Jet. Dissertation, Universität Stuttgart (2009) 2. Fasel, H., Thumm, A., Bestek, H.: Direct numerical simulation of transition in supersonic boundary layer: Oblique breakdown. In: L.D. Kral, T.A. Zang (eds.) Transitional and Turbulent Compressible Flows, no. 151 in FED, pp. 77-92. ASME (1993) 3. Fezer, A., Kloker, M.: Spatial Direct Numerical Simulation of Transition Phenomena in Supersonic Flat-Plate Boundary Layers. In: H.F. Fasel, W.S. Saric (eds.) Proceedings 5th IUTAM Symposium on Laminar-Turbulent Transition, pp. 415-420. Springer (2000) 4. Kosinov, A.D., Tumin, A.: Resonance Interaction of Wave Trains in Supersonic Boundary Layer. In: P.W. Duck, P. Hall (eds.) Proceedings IUTAM Symposium on Nonlinear Instability and Transition in Three-Dimensional Boundary Layers, pp. 379-388. Kluwer (1996) 5. Mayer, Ch.S.J., von Terzi, D.A., Fasel, H.F.: DNS of Complete Transition to Turbulence via Oblique Breakdown at Mach 3: Part II. AIAA paper 2009-3558 (2009) 6. Sandberg, R.D.: Numerical Investigation of Transitional and Turbulent Supersonic Axisymmetric Wakes. Dissertation, The University of Arizona (2004) 7. von Terzi, D.A.: Numerical Investigation of Transitional and Turbulent Backward-Facing Step Flows. Dissertation, The University of Arizona (2004) 8. von Terzi, D., Mayer, Ch., Fasel, H.: The Late Nonlinear Stage of Oblique Breakdown to Turbulence in a Supersonic Boundary Layer. In: Proceedings 7th IUTAM Symposium on LaminarTurbulent Transition. Springer (2009)

Part III

Poster Presentations

Transient Growth on the Homogenous Mixing Layer Cristobal Arratia, Sarah Iams, Jean-Marc Chomaz, and Colm-Cille Caulfield

Abstract We compute the three-dimensional (3D) optimal perturbations of an homogeneous mixing layer. We consider as a base state both the hyperbolic tangent (tanh) velocity profile and the developing two-dimensional (2D) Kelvin-Helmholtz (KH) billow. For short enough times, the most amplified perturbations on the tanh profile are 3D and result from a combination between the lift-up and Orr mechanisms[1]. For developing KH billows, there are different mechanisms that prevail depending on the initial amplitude of the billow, the spanwise wavenumber and the time of the response observed. We determine when the largest transient growth at a particular time is associated with an optimal response reminiscent of the elliptic or hyperbolic instability.

1 Introduction Homogeneous (constant density) mixing layers are ubiquitous sheared flows which are known to exhibit the KH instability. The KH instability may lead to the rollup of spanwise vortices, the KH billows, which are the outcome of the non-linear saturation of the growing unstable KH mode. These essentially 2D KH billows can be subject to 3D instabilities and may exhibit transition to turbulence [2]. Much research has addressed the linear instability mechanisms for transition on the KH

Cristobal Arratia, Jean-Marc Chomaz Laboratoire d’Hydrodynamique, [email protected]

Ecole

Polytechnique-CNRS,

e-mail:

arra-

Sarah Iams, Colm-Cille Caulfield BP Institute, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK S. Iams now at the Center for Applied Mathematics, Cornell University C. Caulfield also at Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_73, © Springer Science+Business Media B.V. 2010

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billows [3] and several unstable eigenmodes have been found to be mainly localized in two different flow regions: • the vortex core, where the elliptic instability occurs on the strained closed streamlines. • the braid, where the hyperbolic instability occurs on the region separating consecutive billows. The hyperbolic flow stretches vorticity generating streamwise vortices. However, because of the non stationarity of the KH billows, classic modal analysis relies on the separation of time scales implied by freezing the basic state. Moreover, because of the non-normality of the linearized Navier-Stokes equations, linear growth of 3D perturbations should be described by the non-modal stability theory. We characterize the appearance of 3-dimensionality on the mixing layer by computing the optimal transient growth of perturbations u(x,t) during a finite time interval [0, T ] over a basic state U. The optimal perturbations that attain the optimal transient growth are solutions of the maximization problem G(T ) = maxu0 ku(T )k2 /ku0 k2 . To find the optimal perturbations we use a direct adjoint iterative technique [4] in which we solve the direct and adjoint Navier-Stokes equations with a pseudo-spectral code. The basic states considered and the results are discussed in the following.

2 Hyperbolic Tangent Velocity Profile We consider U(y) = tanh(y)ex as the base state. We can write the perturbation velocity as u = u′ (y,t)ei(kx x+kz z) and we characterize the gain of the optimal perturbations by their mean growthrate σm (kx , kz , T ) = log(G)/2T . As shown in figure 2(a) for T = 7, for short optimization times (T < 14) the most amplified optimal perturbations are 3D. They correspond to a combination of the Orr and the lift-up mechanisms as is common for the transient growth in shear flows

σm

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[1]. As the optimization time increases, the most amplified optimal perturbations become 2D as the unstable KH modes start to dominate (see circles in fig. 2(b)). As T increases further, the optimal responses approaches the KH mode and the perturbations approach the corresponding adjoint mode. The dots in figure 2(b) show how the mean growthrate of the most amplified optimal perturbation approaches from above the largest growthrate of the KH instability σKH . k =0.69813 z

t=0 18

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Fig. 3 G(kz , T )) for Tsat = 35 (a) and Tsat = 15 (b). Symbols for each T are indicated on plots.

3 Developing KH billow We consider the fully nonlinear development of the 2D KH instability as our time dependent basic state. As initial condition for the basic state we use the most unstable linear KH mode, the initial amplitude being characterized by the saturation time Tsat , time at which the amplitude of the 2D field superposed on the tanh profile reaches its maximum energy. The basic flow fixes the streamwise wavenumber at kx = 0.44 (the most unstable KH mode) and supresses the subharmonic instability. The perturbative field now is u = u′ (x, y,t)ei(kz z) and the optimal gain becomes a function of 3 variables G(kz , Tsat , T ). We calculated optimal perturbations for Tsat = 35 and Tsat = 15. The optimal responses for Tsat = 35 and different kz and T reveal both the elliptic and hyperbolic mechanisms. The largest gain for T > 50 occurs at kz ≈ 0.7 and is associated to an elliptic type of response as seen on fig. 2(a). Hyperbolic response dominates for larger kz . For Tsat = 15, the optimal response is of hyperbolic type for all times and wavenumbers explored, except at large time (T = 40) for the 2 smallest kz computed. The gain of the calculated optimal perturbations is shown in figure 3. With these computations we have determined the mechanisms that have the greater potential for linear growth on mixing layers.

References 1. Farrell, B.F., Ioannou, P.J.: Perturbation growth in shear flow exhibits universality. Phys. Fluids A, 5, 2298–2300 (1993) 2. Caulfield, C.P., Peltier, W.R.: The anatomy of the mixing transition in homogeneous and stratified free shear layers. J. Fluid. Mech. 413, 1–47 (2000) 3. Fontane, J., Joly, L.: The stability of the variable-density Kelvin-Helmholtz billow. J. Fluid. Mech. 612, 237–260 (2008) 4. Corbett, P., Bottaro, A.: Optimal linear growth in swept boundary layers. J. Fluid. Mech. 435, 1–23 (2001)

Closed-loop control of cavity flow using a reduced-order model based on balanced truncation A. Barbagallo, D. Sipp, and P. J. Schmid

Abstract The application of control tools to fluid problems often requires model reduction to correctly capture the input-output behavior of the associated initial-value problem. In this study a model combining global modes (for the unstable subspace) and balanced modes (for the stable subspace) is considered. We show that this model succeeds in removing the global instability of the flow over an incompressible cavity. Comparison with other reduced models clearly demonstrates the superiority of this approach for control problems.

1 Introduction Over the last decade substantial effort has been placed on suppressing instabilities in unstable flows. To achieve this goal, the tools developed in control theory, such as optimal control, seem quite promising. However, since the typical size of discretized problems in fluid mechanics is generally too high to apply these techniques directly, a reduced-order model (ROM) of the flow is needed. A reduced model is usually obtained by a Petrov-Galerkin projection of the discretized equations onto either global modes [1], proper orthogonal decomposition (POD) modes [2] or balanced modes [3]. Each of these bases captures a different feature of the flow and thus accomplish a different goal. For control problems, the Alexandre Barbagallo Laboratoire d’Hydrodynamique (LadHyX), CNRS-Ecole Polytechnique, Palaiseau, France. e-mail: [email protected] Denis Sipp ONERA-DAFE, 8 rue des Vertugadins, Meudon, France. e-mail: [email protected] Peter J. Schmid Laboratoire d’Hydrodynamique (LadHyX), CNRS-Ecole Polytechnique, Palaiseau, France. e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_74, © Springer Science+Business Media B.V. 2010

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relevant quantity of interest is the input-output relation which can be optimally captured by balanced truncation [4]. This method, which determines modes that show the same amount of controllability and observability, became recently computationally feasible with the so-called balanced proper orthogonal decomposition technique [5].

2 Configuration, modeling and model reduction We consider the incompressible flow over an open square cavity. The Reynolds number based on the inlet velocity and the depth of the cavity is 7500. For this Reynolds number, the flow is globally unstable [6], and no boundary layer instabilities are present. We control the flow using an actuator located at the upstream edge of the cavity and measure the shear stress at the downstream edge. The stability analysis and simulations are performed using the same discretization (finite elements) which results in equations with approximately 900,000 degrees of freedom. The equations are linearized about a base flow computed via a Newton method. (a)

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Fig. 1 (a) First balanced mode. (b) Corresponding adjoint balanced mode.

Since the balanced POD technique is only valid for stable flows, we model the stable and unstable subspace independently (see [7]). The unstable subspace is represented by the eight unstable global modes (see the computation described in [8]); the stable subspace is captured by BPOD modes. The BPOD modes are computed according to the technique introduced by [5], combining 1001 equispaced snapshots from the direct and adjoint simulations (see figure 1). We point out that only one direct (resp. adjoint) simulation is required with the control function (resp. the measurement function) as an initial condition.

3 LQG control Using partial-state information, the flow is stabilized provided a sufficient number of BPOD modes is taken into account. In figure 2(a) the energy is represented versus time for reduced-order models (ROMs) of varying numbers of BPOD modes. The energy still increases for reduced models comprising up to six BPOD modes; for seven BPOD modes the energy remains bounded and starts to decay for a higher

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numbers of modes. This performance is related to an improved capture of the inputoutput behavior as the number of BPOD modes increases. (a)

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For a reduced model comprising 8 unstable global modes and 13 BPOD modes, the unstable subspace is exactly modeled and the input-output behavior of the stable subspace is very well captured: the measurement signal of the reduced model matches accurately the measurement signal of the DNS (see figure 2(b)). This allows us to study the stability of the compensated problem by substituting the DNS (plant) by this reduced model, thereafter called the reduced plant. Considering the eigenvalues of the coupled system: reduced plant + compensator based on an arbitrary reduced model, a positive real part of the least stable eigenvalue σmax indicates an unstable compensated system (see details in [8]). This quantity is displayed on figure 3 (black dots) versus the number p of modes modeling the stable subspace. We notice that the compensated problem is stable when at least 7 BPOD modes are considered for the stable subspace (the vertical full line indicates that σmax < 0). This is in agreement with the energy curves on figure 2(a).

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4 Comparison with reduced model based on POD and global modes In this section, we use the same technique to investigate the stability of compensated systems with reduced models based on POD or global modes (see figure 3). The POD modes are computed using snapshots from an impulse response of the control function (see [5]). They correspond to the most controllable modes and are ranked by decreasing Hankel values. The global modes are ranked by decreasing damping rate. While only 7 BPOD modes are required to stabilize the system, reduced models based on POD modes need at least 28 modes to achieve this goal (see the dashed vertical line). Whatever the number of global modes used, they cannot stabilize the system.

5 Conclusions and Outlook In this paper the stabilization of a globally unstable flow has been achieved via a reduced-order model based on balanced modes. The success of this control strategy can be attributed to and is crucially dependent on the accurate representation of the input-output behavior by balanced modes (as opposed to global modes or traditional POD modes). This fact makes reduced-order models based on BPOD the method of choice for a wide range of control problems.

References ˚ 1. E. Akervik, J. Hoepffner, U. Ehrenstein and D. S. Henningson. Optimal growth, model reduction and control in a separated boundary-layer flow using global modes. J. Fluid Mech., 579, 305-314, 2007. 2. J. Delville, L. Cordier and J. P. Bonnet. Large-scale structure identification and control in turbulent shear flows. Flow Control: Fundamentals and Practice, 199-273, Springer Verlag, 1998. 3. M. Ilak and C. W. Rowley. Modeling of transitional channel flow using balanced proper orthogonal decomposition. Phys. Fluids, 20, 034103, 2008. 4. S. Bagheri, L. Brandt and D. S. Henningson. Input-output analysis, model reduction and control of the flat-plate boundary layer. J. Fluid Mech., 620, 263-298, 2009. 5. C.W. Rowley. Model reduction for fluids using balanced proper orthogonal decomposition. Int. J. Bifurc. Chaos, 15(3), 997–1013, 2005. 6. D. Sipp and A. Lebedev. Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech., 593, 333-358, 2007. 7. K. Zhou, G. Salomon and E. Wu. Robust and Optimal Control. New Jersey: Prentice Hall, 2002. 8. A. Barbagallo, D. Sipp and P. J. Schmid. Closed loop control of an open cavity flow using reduced-order models. submitted to J. Fluid Mech.

On the asymptotic solution of the flow around a circular cylinder Iago C. Barbeiro, Ivan Korkischko, Karl P. Burr, Julio R. Meneghini, and J. A. P. Aranha

Abstract This study presents new advances concerning the asymptotic solution derived for the viscous flow around a circular cylinder. It provides arguments, based on numerical and experimental (DPIV) results, to suport the validity of the solution for Reynolds numbers far beyond the critical value of the first Hopf bifurcation (Recr ≈ 46). Results are then related to recent studies regarding the construction of reduced models employing POD techniques.

1 Introduction The flow around a circular cylinder presents a complex sequence of régimes, from attached stationary to full turbulent, see [1]. They are mostly defined by their Reynolds1 number range but may also depend on boundary conditions like the blockage ratio, cylinder span dimension, free surface effects among others. The stationary range goes up to a critical Reynolds number (Recr ≈ 46), when there is a pair of conjugated eigenvalues that crosses the imaginary axis in a Hopf bifurcation, see [2] and [3], giving rise to the periodic two-dimensional régime. The bifurcation theory allows the construction of an asymptotic solution for the weakly oscilating flow that should be valid at least in the vicinity of Recr . The aim of this study is to provide arguments, based on numerical and experimental results, to suport the validity of such solution for Re >> Recr . This sort of asymptotic solution proposes a compact mechanism to represent the nonlinear effects that takes the flow from its unstable stationary solution to its stable limit-cycle, the Landau model. It is also considered that a Laplacian term can be added to the solution to set the spanwise coupling and give place to weak Iago C. Barbeiro, Ivan Korkischko, Karl P. Burr, Julio R. Meneghini and J. A. P. Aranha NDF, Escola Politécnica, University of São Paulo, e-mail: [email protected] Re = UD ν , where U is the incident flow velocity, D is the diameter of the cylinder and ν is the kinematic viscosity of the fluid. 1


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three-dimensionalities, the Ginzburg term. This is the Ginzburg-Landau equation, one of the most celebrated reduced models for this flow, see [5] and [6]. The solution presented by [4] can be grouped in terms of same frequency and each group sets a fourier harmonic since their frequencies are all natural multiples of a main one, the Strouhal frequency fs ( f 1 = fs , f2 = 2 fs , f3 = 3 f s , ...). It is important to note the amplitude hierarchy followed by the groups (harmonics): the solution predicts an aproximatelly constant reduction ratio from the first to the second harmonic, and then to the third and so on. The harmonic structure and the amplitude hierarchy are strong characteristics of the asymptotic solutions and it is part of this study to quantify them after numerically calculated and experimentally measured time series of the velocity field. The numerical time series are generated using a two-dimensional Finite Element code implemented for this purpose. The experiments are carried out in a recirculating water channel and a DPIV (Digital Particle Image Velocimetry) apparatus is employed to take measure of the velocity field at the desired cross section. POD (Proper Orthogonal Decomposition2 ) is a set of heuristic techniques that can be used to extract the hierarchical most energetic coherent structures of the flow allowing the construction of efficient reduced models, see [8] and [9]. The following results comes also to elucidate the relation between the POD modes and the fourier harmonics of this flow.

2 Numerical Simulations The discrete Navier-Stokes equations are obtained by a finite element method using linear shape functions for both variables, velocity and pressure. A penalty method, analyzed by [7], is employed to extract the pressure from the discrete system and ensure the incompressibility condition. The time integration scheme is implicit and of second order, allowing good precision and numerical stability with a time-step of 10−2 . The fourier harmonics are calculated on the fly during the time evolution and a very fine temporal precision is attained with more than one thousand points per vortex shedding cycle, even the third harmonic is well precise. After the simulations reach the periodic régime, ten shedding cycles are considered to obtain the harmonics.

3 Experiments and DPIV Mesurements The flow was reproduced in the recirculating water channel of the NDF’s experimental facility, at the University of São Paulo. The channel has a 7.5m long test

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section being 0.7m in breadth and up to 0.8m in deepness. It is an open channel and its debit goes from 0.01m3 /s to 0.40m3 /s. The DPIV equipement consists of a Nd:YAG pulsed laser (Brilliant Twins - Quantel), a pair of 2 megapixel digital cameras and a workstation for data processing (LaVision package). The maximum acquisition rate is 15Hz and the cameras can store up to 345 pairs of pictures per run (enough to calculate 345 velocity fields). For Re = 100, a 6.35mm diameter cylinder was used and the incident flow velocity was set to its minimum of 15.90mm/s, giving a vortex shedding period of 2.43s. This is the best found configuration, giving the best teporal and spatial precision: the slowest shedding cycle with the larger possible diameter.

4 Results: harmonic fields and amplitudes Numerical and experimental results are robust and their fine agreement for Re = 100 (see figures 1 and 2) can confirm the two-dimensionality of the experiment and the convergence of the simulations. The numerical two-dimensional simulations are extended up to Re = 600 to account the nonlinear effects in this range but it is worth to remember the reader that the real flow becomes weakly three-dimensional after Re ≈ 190.

(a) u1,r,x [−0.41; 0.41] (b) u1,r,y [−0.56; 0.64] (c) u2,r,x [−0.12; 0.12] (d) u2,r,y [−0.13; 0.13]

(e) u1,r,x [−0.34; 0.33] (f) u1,r,y [−0.56; 0.51] (g) u2,r,x [−0.08; 0.07] (h) u2,r,y [−0.10; 0.10] Fig. 1 1st and 2nd harmonics of the velocity fields (only the real part is displayed) for Re = 100. Above: numerical simulation; Bellow: DPIV measurements.

The harmonics are global structures that oscilate in a synchronized movement. One could take them as a well organized reduced base to the velocity space and the curves in figure 2 suggest that the three first harmonics (also including the meanflow) would give a good approximation of the real flow. The result would be close to what was obtained by [8], where the POD analysis has found three pairs of leading modes oscilating in three different frequencies ( fs , 2 fs and 3 fS )3 . Results are even 3

the frequency ratio of the diferent POD modes can be extracted from the Lissajous figures provided in [8]

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closer saying that the harmonics are all orthogonal to each other considering the same metric of the POD scheme, as it was numerically checked. Back to the initial proposition, the two features predicted by the asymptotic solution of [4] were observed in both ways, numerically and experimentally: the harmonic structure and the amplitude hierarchy stands for Re >> Recr . The POD techniques have been widely employed to extract powerfull reduced models for the flow around the circular cylinder and this study has shown how it can be associated to the former asymptotic solution giving interesting insights into the nonlinear mechanisms of the flow. ●

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5 Acknowledgements The authors have grants from FINEP-CTPetro, FAPESP and Petrobras.

References 1. A. Roshko (1954) On the development of turbulent wakes from vortex streets. NACA 1191 2. C. P. Jackson (1987) A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182:23–45 3. J. I. H. Lopez , J. R. Meneghini , F. Saltara (2008) Discrete approximation to the global spectrum of the tangent operator for flow past a circular cylinder. App. Num. Math. 58(8):1159–67 4. J. A. P. Aranha (2004) Weak Three Dimensionality of a Flow Aroud a Slender Cylinder: The Guinzburg-Landau Equation. J. Braz. Soc. Mech. Sci. Eng. XXVI-n4 : 355–367 5. M. Provansal, C. Mathis, L. Boyer (1987) Benard-von Karman instability: Transient and forced regimes Equation. J. Fluid Mech. 182:1–22 6. P. Albarede, P.A. Monkewitz (1992) A model for the formation of oblique shedding and “chevron” patterns in cylinder wakes. Phys. Fluids A 4, 744–756 7. Max D. Gunzburger (1989) Finite element methods for viscous incompressible flows: a guide to theory, practice and algorithms. Academic Press 8. X. Ma, G. E. Karniadakis(2002) A low-dimensional model for simulating three-dimensional cylinder flow. J. Fluid Mech. 458:181–190 9. B. R. Noack, K. Afanasiev, M. E. Morzynski, G. Tadmor, F. Thiele(2003) A hierarchy of lowdimensional models for transient and post-transient cylinder wake. J. Fluid Mech. 497:335–63

Investigations of Suction in a Transitional Flat-Plate Boundary Layer Stefan Becker and Jovan Jovanovic

Abstract For the maintenance of a laminar boundary layer flow on transonic wings, it is necessary to integrate a boundary layer suction unit in the nose region. This concept of the Hybrid Laminar Flow control is realized through a suction area adapted to the outer pressure distribution by an array of suction holes. With this, a stabilization of the boundary layer is reached, although cross-flow instabilities are present in the flow. The goal of the experimental examinations was to widen the understanding of the results of flow topology at a single suction hole on hole arrays. Of particular interest was the influence of the hole size, the hole spacing and the suction flow rate on the receptivity of the boundary layer flow. The work was conducted in a refractive index channel, in which laser Doppler velocity measurements in the boundary layer close to the wall are possible. The investigations were carried out under natural transition conditions to clarify the physical mechanisms of the transition acceleration as well as deceleration in dependence of the suction rate. The results show that the transition process follows completely a strong stochastic behavior.

1 Introduction Engineering attempts to maintain the laminar flow conditions at higher Reynolds numbers or larger downstream distances than those at which the flow is normally turbulent or transitional is called Laminar Flow Control (LFC). In most cases, LFC entails sucking a portion of the boundary layer flow through the wall. Linear stability theory shows that continuous suction can stabilize the boundary layer and Stefan Becker University of Erlangen-Nuremberg, Institute of Fluid Mechanics, Cauerstr. 4, e-mail: [email protected] Jovan Jovanovic University of Erlangen-Nuremberg, Institute of Fluid Mechanics, Cauerstr. 4, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_76, © Springer Science+Business Media B.V. 2010

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substantially increase the critical Reynolds number at which transition begins. This stabilization is accomplished by decreasing the boundary layer thickness, thereby decreasing the effective boundary layer Reynolds number. Unfortunately, continuous wall suction is inpractical from a fabrication viewpoint, and suction is usually implemented via discrete holes or sometimes slots in the wall surface. Discrete holes can lead to hole-to-hole interactions, trailing vortices, and additional suction parameters / variables. The aim of the present work is to understand the critical parameters affecting boundary layer stabilization via suction through discrete holes. In the literature differences exist in the experimental results (Goldsmith [1], Magnus and Eaton [2]) and in results from Direct Numerical Simulation (DNS) (Meitz [3], Messing and Kloker [4]). The simulation from the literature formed the basis of the chosen experimental parameters of the suction flow in the boundary layer. This work provides a comprehensive data set to validate the numerical prediction and explain the physical mechanisms of the transition process behind the suction holes.

2 Experimental setup The quantification of boundary layer transitional flows requires measurements very close to the wall for an accurate determination of the wall shear stress. Laser Doppler measurements (LDA) usually suffer from optical interference or blockage of the laser beams, especially when systems for two and three component measurements are employed. The Institute of Fluid Mechanics (LSTM) has utilized several methods to improve this situation. LDA (non-intrusive) was employed in a matched index-of-refraction (MIR) facility (no optical distortion) of very large size (higher spatial measurement resolution) to study suction LFC. However, so far no refractive index matched flow facility existed that would have permitted reasonably-sized flat plate boundary layers to be set up. Hence, this provides the basic test facility to study LFC of the laminar to turbulent transition control in detail. A comprehensive study of the test rig with experimental results is given in Stoots et al. [5] and Becker et al. [6]. The transition process in the flat boundary layer is consequently described by the natural defined spectrum of the test facility. Therefore, the low turbulence level of the test section is perfectly suited for this purpose. The design and location of the suction holes, the suction flow rates and the velocity around the holes were calculated on the basis of the Reynolds number and the ratio of the velocities around the suction to the free stream velocity as in the DNS in [4]. The row of suction holes consists of 43 holes with a diameter of 3.5 mm and a hole spacing of 6.74 mm. Consequently, the row of suction holes includes 48% of the total width of the test plate. The Reynolds number of the experiments was about RexL = 5.67 x 105 based on the distance of the suction holes from the leading edge of the plate. The Reynolds number Reθ , due to the displacement thickness of the boundary layer, equaled 1224. For the experiments, a fibre cable based LDA system was used. The complete system was equipped with a 2D-LDA probe, so just two components of the velocity

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field were measured simultaneously. The data of the third component were taken after turning the probe. The processing of the measuring signals was done with a Burst Spectrum Analyser (BSA) and the corresponding software provided by DANTEC. In figure 1 gives a schematic overview of the measuring setup.

3 Results The measurements took place in different planes upstream and downstream of the suction holes. Figure 2, the velocity profiles and the turbulence intensity distributions are represented for a subcritical and supercritical suction flow rate in comparison to the profiles without suction.

Fig. 1 Experimental setup

Fig. 2 Velocity and turbulence intensity profiles for subcritical (top) and supercritical (bottom) suction flow rate in comparison to measurements without suction

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This figure reveals that a large disturbance peak develop behind the suction holes in the boundary layer that gradually moves away from the wall in the downstream direction. For the subcritical test case, this peak is damped out, amplification occurs with supercritical suction flow rate and with an accompanying acceleration of the transition process. This characteristic behavior of the increase disturbances is also to be found in the v-component. The power density spectrum shows no dominant frequencies for either both test case. On the basis of the data rate, the highest resolvable frequencies lay in the area of 50 Hz however and consequently in the low area of Tollmien-Schlichting-instabilty. Nevertheless the results suggest that the transition process is not introduced by a Tollmien-Schlichting instability but completely follows a stochastic scenario.

4 Conclusion The results show that the transition process downstream of the suction holes is different from what it is known from the classical linear stability theory. Downstream of the suction holes the turbulence intensity profiles show a large disturbance peak, which is damped out in the case of the subcritical flow rate further downstream. For the supercritical suction rate the disturbance peak increase in the flow direction and cause the transition process of the boundary layer flow. Additional FFT analysis of the velocity records leads to the conclusion that the transition process is not dominated by a specific frequency. The transition from laminar to turbulent flow tends to be a stochastic process and seems to be described in a better way by using the tools of the invariant theory (see Jovanovic [7]).

References 1. Goldsmith, J.: Critical Laminar Suction Parameters for Suction into an Isolated Hole or a Single Row of Holes. NAI-57-529 (BLC-95). Northrop Aircraft Inc. (1957) 2. MacManus, D.G., Eaton, J.A.: Flow physics of discrete boundary layer suction - measurements and predictions. J. Fluid Mech. 417, 47–75, (2000) 3. Meitz, H.L.: Numerical investigations of suction in a transitional flat-plate boundary layer, PhD-thesis, Department of Aerospace and Mechanical Engineering, University of Arizona (1996) 4. Messing , R. and Kloker, M.: DNS-Untersuchung der diskreten Absaugung durch EinzellochArrays, 2. Zwischenbericht zum Forschungsvorhaben 20A 9505G, University Stuttgart (1998) 5. Stoots, C., Becker, S., Condie, K., Durst, F. and McEligot, D.: A large-scale matched index refraction flow facility for LDA studies around complex geometries, Experiments in Fluids 30, Springer Verlag (2001) 6. Becker, S., Stoots, C.M., Condie, K., Durst, F. and McEligot, D., 2002: LDA Measurements of Transitional Flows Induced by a Square Rib. J. of Fluid Engineering 124, 108–117 (2002) 7. Jovanovic, J.: The Statistical Dynamics of Turbulence. Springer, Berlin, Heidelberg, New York (2002)

Global three-dimensional optimal perturbations in a Blasius boundary layer S. Cherubini, J.-C. Robinet, A. Bottaro, and P. De Palma

Abstract The three-dimensional global optimal and near-optimal perturbations in a flat-plate boundary layer are studied by means of an adjoint-based optimization, and their non-linear evolution is investigated by means of DNS.

1 Introduction Since the early observations [1], many studies have been dedicated to the interior structure of turbulent spots, their shapes, spreading rates, and mechanisms of growth [2]. However, minor efforts have been dedicated so far to identify the initial, localised states which most effectively bring the flow on the verge of turbulent transition via the formation of a spot. In this work a new attempt is made to identify optimal initial disturbances capable to induce breakdown to turbulence in a boundary layer. The optimization is not restricted to an initial state (at x = 0 or t = 0) characterised by a single wavenumber, but it considers a wave packet, localised in the streamwise (and eventually spanwise) direction, formed by the superposition of monochromatic waves. In order to assess whether such an optimal localised flow state is effective in provoking breakdown, direct numerical simulations (DNS) are then performed, highlighting the importance of non-linear effects which lie at the heart of the initiation of a turbulent spot. S. Cherubini DIMEG, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy e-mail: [email protected] SINUMEF Laboratory, Arts et Metiers ParisTech, 151, Bd. de l’Hopital, 75013 Paris, France J.-C. Robinet SINUMEF Laboratory, Arts et Metiers ParisTech, 151, Bd. de l’Hopital, 75013 Paris, France A. Bottaro DICAT, University of Genova, Via Montallegro 1, 16145 Genova, Italy P. De Palma DIMEG, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_77, © Springer Science+Business Media B.V. 2010

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2 Problem formulation The linear behaviour of a perturbation, q = (u, v, w, p)T , evolving in a laminar incompressible flow past a flat-plate is studied by employing the 3D Navier-Stokes (NS) equations linearized around the 2D steady state Q = (U,V, 0, P)T . Dimensionless variables are defined with respect to the inflow boundary layer displacement thickness δ ∗ and to the freestream velocity U∞ , so that the Reynolds number is Re = U∞ δ ∗ /ν . The linearized NS equations are integrated by a fractional step method using a second-order-accurate centered space discretization on a staggered grid [3]. In order to investigate the global optimal perturbations, a Lagrange multiplier technique is employed, where the perturbation kinetic energy integrated over the whole domain is chosen as the objective function. By imposing the linearized NS equations as a constraint and following a procedure similar to the one described in [4], the adjoint equations are recovered and integrated in time together with the direct equations until convergence to the optimal solution is achieved. Finally, the non-linear evolution of the optimal perturbation is investigated by means of DNS.

3 Results and discussion Computations have been performed at Re = 610, for a domain of dimensions Lx = 400, Ly = 20, Lz = 10.5, where the value of Lz has been chosen in order to obtain the largest amplification. The energy gain, G(t) = E(t)/E(0), has been found to reach approximately G(t) ≈ 736 at time tmax ≈ 247. The optimal spatially localised initial disturbance is characterized by a counter-rotating vortex pair in the y − z plane, reminiscent of the one predicted by the local optimization for a perturbation with the streamwise wavenumber α = 0. Indeed, in the present case, a modulation is found in the x direction, the perturbation being composed by upstream-elongated structures with x-alternated-sign velocity components, as shown in Figure 1 (a). The time evolution of such an optimal solution shows that the perturbation is tilted in the mean flow direction by means of the Orr mechanism [5], while being amplified by the lift-up mechanism, resulting at the optimal time in streaky structures alternated in the x direction (see Figure 1 (b)). Similar structures have been found for lower Reynolds numbers and different domain lengths. In particular, for large spanwise domain lengths, the optimal perturbation results to be a single-spanwise-wavenumber perturbation similar to the one in Figure 1 extended in the whole spanwise direction. However, in nature, disturbances are always characterized by a wide spectrum of frequencies, and often are localized in wave packets, so that a localized perturbation would be more suitable to represent the disturbance which is more likely to lead the flow to a chaotic behaviour. Therefore, an artificial wave packet has been constructed for Lz = 180 by multiplying the optimal single-wavenumber perturbation by an envelope which varies like exp(−z2 /Lz ), chosen as the initial configuration for the optimization process. A partially optimized perturbation, shown in Figure 3 (a), has been extracted at the level

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Fig. 1 Iso-surfaces of the streamwise optimal perturbation for Re = 610 at t = 0 and t = tmax .

of convergence e1 = (E(t)(n) − E(t)(n−1) )/E(t)(n) = 10−3 . Analyzing such an intermediate solution, it is possible to notice that the perturbation is still localized in the spanwise direction, although its shape is changed. In particular, the streak-like structures at the edge of the wave packet is inclined with respect to the z axis, resulting in oblique-like waves bordering the wave packet. Furthermore, such a near-optimal solution, although different in the spanwise direction with respect to the optimal one, can be amplified up to very high values of the energy gain, reaching a value which differs of less than 1% from the optimal one (G(t)e1 = 728). In order to investigate the capability of the three-dimensional optimal perturbation to induce transition in a boundary-layer flow, non-linear simulations are performed and compared to those arising from the evolution of the local optimal (with α = 0) and suboptimal (with α = 2π /Lz ) disturbances. Figure 2 shows the mean skin friction coefficient measured in the simulations initialized with three initial energies, E0(a) = 0.5 , E0(b) = 2 and E0(c) = 10, as well as the theoretical trend of the laminar and turbulent skin friction coefficient in a boundary layer. As shown by the curves, the global optimal disturbance is the most effective in inducing transition, followed by the local suboptimal one. Such results confirm those in [6], in which the suboptimal perturbation is found more effective in inducing transition than the local optimal one, but also assesses that a localized x-modulated array of elongated structures is able to lead a subcritical boundary layer to chaos for an even lower value of the initial energy.

Fig. 2 Mean skin friction coefficient: theoretical values for a laminar (lowest dotted line) and turbulent (highest dotted line) boundary layer and computed values for the considered flow perturbed with the global three-dimensional optimal (solid line), the local optimal (dashed line), and the suboptimal (dashed-pointed line), at E0a = 0.5 (a), E0b = 2 (b) and E0c = 10 (c).

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Fig. 3 Contours of the evolution of the streamwise near-optimal velocity for y = 1 at T = 0 (a), T = 220 (b), T = 330 (c) and T = 420 (d).

In order to generalize such a result to larger and more realistic domain lengths, the non-linear evolution of the near-optimal wave packet has been investigated. The perturbation has been extracted at three instants of time from a DNS computation with E0 = 0.5. Figure 3 (b) shows that at t = 220 the wave packet has been convected downstream, amplifying itself and saturating its energy, so that the streaky structures seem to partially merge, and two kinks appear at the leading edge of the most amplified streaks. It has been observed that, at this time both the scenarios of quasi-sinuous and quasi-varicose breakdown (see [7]) could be identified in the present case due to the staggered arrangement of the streaks. As a result, at least four streaks experience breakdown at the same time, explaining the efficiency of the global optimal and near-optimal perturbation in inducing transition. Due to streak breakdown, at t = 330 the most amplified elongated structures in the middle of the wave packet have already experienced transition, (see Figure 3 (c)), and at a sufficiently large time the linear wave packet has totally disappeared and the disturbance has taken the form of a localized turbulent spot (see Figure 3 (d)). The optimal and near-optimal wave packets computed by means of the threedimensional direct-adjoint optimization, which have been found more effective than the local optimal perturbations in inducing transition, could thus represent a linear precursor of a turbulent spot, and the transition mechanism here investigated could represent an optimal way to transition.

References 1. 2. 3. 4. 5. 6. 7.

H. W. Emmons Journal of Aeronautical Science, 18, 490–498, 1951. I. Wygnanski, M. Sokolov and D. Friedman J. Fluid Mech., 78, 785–819, 1976. R. Verzicco and P. Orlandi, J. Comp. Phys., 123 402–414, 1996. S. Zuccher, A. Bottaro and P. Luchini, Eur. J. Mech. B/Fluids, 25, 1-17, 2006. P. J. Schmid and D. S. Henningson, Springer, Berlin, 1990. D. Biau, H. Soueid and A. Bottaro J. Fluid Mech., 596, 133–142, 2008. L. Brandt, P. Schlatter and D. S. Henningson J. of Fluid Mech., 517, 167–198, 2004.

Quantifying sub-optimal transient growth using biorthogonal decomposition Nicholas Denissen, Edward White, and Robert Downs III

Abstract Receptivity calculations are shown for physically-realizable, roughnessinduced disturbances. These calculations give the continuous-spectrum amplitude distribution for sub-optimal disturbances for both DNS and experimental data. A technique for computing the amplitude distribution with only streamwise data is shown, and gives the vortex behavior of transiently growing disturbances.

1 Motivation The role of surface roughness in boundary layer transition-to-turbulence is of interest in many contexts. For many years attempts were made to understand roughness effects through the modal growth mechanism of the Orr–Sommerfeld/Squire (OSS) equations with little success. Transient growth may provide a non-modal description of so-called “bypass transition” consistent with linear stability theory[1]. The solutions that arise from the continuous spectrum of the OSS operator generate algebraic, non-modal, energy growth from a combination of purely decaying eigenfunctions in flow regimes with no traditional instability. Roughness-induced transient growth has been observed experimentally[2, 3], and via DNS[4]. However, these results are at odds with “optimal” calculations[5] that predict growth over much longer distances than what is observed. Given that roughnes-induced disturbances are sub-optimal, the focus becomes what combination of decaying modes is generated. For modal instabilities, receptivity can be quantified by a single (complex) number and growth rates do not depend on receptivity. For transient growth, the receptivity process determines the initial amplitude of boundary-layer perturbations, as well as their subsequent non-modal growth. In this case, the amplitude is not a single number assigned to a particular mode. Rather, it is a continuous function of the complex streamwise wavenumber, α . Receptivity analysis must partition the physically realized perturbation into its Nicholas Denissen, Edward White, Robert Downs III Department of Aerospace Engineering, Texas A&M University. e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_78, © Springer Science+Business Media B.V. 2010

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constituent modal components[6]. The present work shows results of a decomposition for a physically realized setup, a flat plate with periodic, cylindrical roughness elements (Re ≈ 200 based on roughness height), with DNS (from Rizzetta and Visbal[4]), as well as a new technique for extracting the complex amplitude function when only experimental data is available (from Ergin and White[3]).

2 Method The governing equations for the roughness induced perturbations of present interest are the three-dimensional incompressible Navier–Stokes equations. The equations are linearized for small perturbations about a parallel-flow, zero-pressure-gradient basic state. Taking Fourier transforms in x (α ), z (β ) and t (ω ) the linearized equations become the Orr–Sommerfeld/Squire system: Dφˆ = LOS φˆ where D = ∂ /∂ y and φˆ = [u, ˆ ∂ u/ ˆ ∂ y, v, ˆ p, ˆ w, ˆ ∂ w/ ˆ ∂ y]T (the hats denote variables in Fourier space). LOS is the Orr–Sommerfeld/Squire operator, a function of y, and parameterized by α , β , ω , and R. In the present case, the focus is on spatial growth of steady disturbances, ω = 0. The values of α are dictated by the continuous spectrum solutions, which are bounded but do not decay as y → ∞. For steady disturbances they are purely imaginary, with αi > β 2 /Re. When a direct numerical simulation (DNS) can be performed of a specific flow geometry, it is possible to obtain complete information about all the flow variables at any point of interest. From this, a decomposition of the flow into its continuous spectrum modes can be directly computed[6]. Using the modal solutions to the adjoint OSS equation, ψˆ , and a vector of data, φˆ0 obtained at an upstream location x0 , a relation for the amplitude function, Cα can be found as a function of streamwise wavenumber: Z −i ∞ ∂ LOS ˆ Cα = φ0 ψˆ α dy (1) Qα 0 ∂ α In this equation Qα is a normalization constant that depends on α but not y. Once decomposed at x0 , the amplitude coefficients completely determine the flow behavior at all downstream locations. Experiments do not provide the complete information required to evaluate Eq. 1. While DNS can be performed for certain simple geometries, it remains resource intensive, and computing realistic roughness is not currently possible. Thus experiments will remain the primary source of new data for distributed roughness. Hotwire anemometry can be used to obtain detailed measurements of the streamwise perturbation but other components are very difficult to obtain. The result is that in the initial condition vector, the streamwise velocity components can be used, but the unmeasured components must be left as integrals over the appropriate α of the eigenmodes and unknown amplitude function. T Z Z Z Z ˆ φ0 = uˆ0 , Duˆ0 , Cα vˆα , Cα pˆα , Cα wˆ α , Cα Dwˆ α α

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Substituting Eq. 2 into Eq. 1 and discretizing the α spectrum yields an algebraic equation in the amplitudes: M j α Cα = F j (3) where M jα contains integrals of the direct and adjoint solutions, and F j contains integrals of the known, streamwise components. Due to the missing information about v, ˆ w, ˆ and p, ˆ Eq. 3 is ill-posed, and a straightforward solution is unsuccessful. Overcoming this problem is both challenging and essential as vˆ and wˆ are the components of the decaying streamwise vortex that initiate transient growth. To solve, M and F are augmented to include multiple streamwise locations, resulting in an overdetermined system. The usual approach to such a system would find a solution that satisfies Eq. 3 in the least-squares sense. However, this does not overcome the ill-posedness and a regularization approach based on Tikhonov[7] is used. The regularization approach requires finding the minima of a functional, M , with other constraints imposed in addition to least-squares. Terms are added to control the norm and smoothness of the solution, and the regularization parameters γ1 and γ2 are chosen to minimize the error from the experimental energy evolution. M = ||M jα Cα − Fj ||2 + γ1 ||Cα ||2 + γ2 ||Γ Cα ||2

(4)

In Eq. 4 the first term is the standard least squares solution, the second term controls the overall norm, and in the third term the Γ operator controls the smoothing (i.e., penalizes the derivative ∂ Cα /∂ α ). This procedure not only extracts the amplitude function Cα , but also reveals the behavior of vˆ and wˆ from measurements of uˆ only.

3 Results Figure 1 shows a comparison of streamwise and wall-normal velocity profiles 25 mm (39 δ ) downstream of the array of cylinders for the dominant spanwise mode (3rd harmonic). The right figure shows how well the decomposition is able to capture the velocity profiles as the symbols from the DNS and the lines of the decomposition line up nearly atop one another. The figure at left shows the reconstruction from the experimental profiles and the partial data technique. It is important to emphasize that these vˆ and wˆ profiles were generated using only uˆ as an input. The regularization procedure is able to find the correct spanwise and wall-normal velocities profiles, giving not only the receptivity information, but also the vorticity that drives the transient growth. The amplitude function obtained by decomposing the 3rd harmonic of the DNS is found in figure 2. This function represents the complete receptivity solution, describing the relative amplitudes of all continuous spectrum modes. It accurately captures the sub-optimal behavior at all points downstream. R Figure 3 shows results experimental energy evolution E(x) = u(x)2 dy from Ergin and White[3] for the 3 dominant spanwise modes (the 2nd harmonic contains almost no energy). λk is the spanwise spacing of the roughness. The lines represent the energy evolution found via regularization. The set of amplitudes found do an excellent job reproducing the disturbance energy over the entire domain.

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Acknowledgements The authors would like to acknowledge support by the NSF and the AFOSR/NASA National Center for Hypersonic Research in Laminar-Turbulent Transition.

References 1. 2. 3. 4. 5. 6. 7.

Reshotko, Eli. Phys. Fluids 13:1067–1075 (2001) White, Edward B. Phys. Fluids, 14:4429–4439 (2002) Ergin, F. G. and White, E. B. AIAA J. 44:2504–2514 (2006) Rizzetta, D. P. and Visbal, M. R. AIAA J. 45:1967–1976 (2007) Andersson P. et al. Phys. Fluids. 11:134–150 (1999) Tumin, A. Phys. Fluids. 15:2525–2540 (2003) Tikhonov, A.N. et al. Numerical Methods for the Solution of Ill-Posed Problems. (1995)

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Model reduction using Balanced Proper Orthogonal Decomposition with frequential snapshots G. Dergham, D. Sipp, and J.-C. Robinet

Abstract Many of the tools of flow control theory require model reduction to correctly capture the input-output behavior at stake. In this paper, we consider a model reduction based on balanced truncation using the method of snapshots. The particularity of this work is that these snapshots are computed in the frequency domain, allowing a reduction of their required number. This model is applied on the twodimensional incompressible flow over a rounded backward facing step. We show that this model is able to catch the input-output behavior of the full system, a comparison with standard POD clearly demonstrates the superiority of this approach.

1 Introduction Model reduction involves finding low-dimensional models that approximate the full high-dimensional dynamics. In terms of flow control, the systems are too large to apply optimal control tools so that a reduced-order model of the flow is necessary. Here we are interested in reduced order models based on Petrov-Galerkin projections. One can use global modes [1], proper orthogonal decomposition (POD) modes [2] or balanced modes [3] depending on the desired purpose. For control problems, the relevant quantity of interest is the input-output relation which can be optimally Grégory Dergham SINUMEF Laboratory, 151 boulevard de l’Hôpital, 75013 Paris, France. ONERA-DAFE, 8 rue des Vertugadins, 92190 Meudon, France. e-mail: [email protected] Denis Sipp ONERA-DAFE, 8 rue des Vertugadins, 92190 Meudon, France. e-mail: [email protected] Jean-Christophe Robinet SINUMEF Laboratory, 151 boulevard de l’Hôpital, 75013 Paris, France. e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_79, © Springer Science+Business Media B.V. 2010

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captured by balanced truncation [4]. This latter remains not yet computationally tractable for very large systems so that an approximated technique called balanced POD [5] has been introduced. It is based on a discretization of the controllability and observability gramians using snapshots. The present method relies on a frequential definition of the gramians and consequently frequential snapshots. BPOD and POD modes are computed in the case of a backward facing step flow to illustrate their ability to model an input-output relation.

2 Flow configuration We consider the incompressible flow over a rounded backward facing step. The step consists of a circular part and is considered infinite in the spanwise direction. A uniform and unitary velocity field is prescribed at the inlet boundary and a laminar boundary layer starts developing on the lower boundary at (x = −2, y = 1). The upstream velocity and the step height are used to make all quantities non-dimensional. The Reynolds number is fixed to 600 so that the flow is globally stable to twodimensional perturbations and no boundary layer instabilities are present. The stability analysis and simulations are performed using the same discretization (finite elements) which results in equations with approximately 400 000 degrees of freedom. The equations are linearized about a base flow (see figure 1) computed via a Newton method [7]. The input of the system is a vertical momentum impulse localized on a gaussian centered just in front of separation. The output is measured by the shear stress just after reattachment. Input

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3 Model reduction The dynamics of perturbations is given by the linear input-output system x˙ = Ax + Bu y = Cx

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where x(t) is the full state vector, A is the linearized Navier-Stokes operator, B is the control operator, C is the measure operator, u(t) is the scalar input and y(t) is the scalar output. The associated controllability and observability Gramians are then defined by

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The discretization of the integrals is performed in the frequency domain namely on the r.h.s. of (2). Thus, the i th direct snapshot is given by ( jωi I − A)−1 B and its adjoint by (− j ωi I − A∗ )−1C∗ . We have represented these snapshots on figure 2(a) and 2(b) for the pulsation ωi = 2. The BPOD modes are computed according to the technique introduced by [5] combining 279 equispaced snapshots from the direct and adjoint computations. The first BPOD modes are represented in figure 2(c) and 2(d). POD modes are computed in a similar manner (see [5]) using only the direct frequential snapshots.

(a) Direct snapshot with ωi = 2.

(b) Adjoint snapshot with ωi = 2.

(c) First balanced mode.

(d) First adjoint balanced mode.

Fig. 2 Snapshots and BPOD modes for the backward facing step flow.

4 Input-output response The reduced models are computed from a Petrov-Galerkin projection of the system discretized equations. We have computed the inpulse response of the system as done by Barbagallo [8]. The temporal response is shown in figure 3(a) and its fourier transform in 3(b). The performance of the reduced order models to capture the inputoutput behavior is improved as the number of BPOD (or POD) modes increases. In both cases we succeeded in capturing the impulse response: the measurement signal of the reduced model matches accurately the measurement signal of the DNS. While only 10 BPOD modes are required, reduced models based on POD modes need at least 60 modes to achieve this goal.

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Fig. 3 Impulse responses for the different models versus two-dimensional direct numerical simulations. G(t) = m(t) when u(t) = δ (t).

5 Conclusions and Outlook In this paper, balanced POD has been performed on the flow over a backward facing step using snapshots in the frequency domain. With this technique, the controllability and observability gramians are accurately computed using fewer snapshots, reducing the computational time of the process. We showed that 10 BPOD modes (or 60 POD modes) are sufficient to catch the full input-output behavior of the system. These results show encouraging signs in terms of flow control (see [8]) and forecast models.

References ˚ 1. E. Akervik, J. Hoepffner, U. Ehrenstein and D. S. Henningson. Optimal growth, model reduction and control in a separated boundary layer flow using global eigenmodes. J. Fluid Mech., 579, 305-314, 2007. 2. J. Delville, L. Cordier and J. P. Bonnet. Large-scale structure identification and control in turbulent shear flows. Flow Control: Fundamentals and Practice, 199-273, Springer Verlag, 1998. 3. M. Ilak and C. W. Rowley. Modeling of transitional channel flow using balanced proper orthogonal decomposition. Phys. Fluids, 20, 034103, 2008. 4. S. Bagheri, L. Brandt and D. S. Henningson. Input-output analysis, model reduction and control of the flat-plate boundary layer. J. Fluid Mech., In Press, 2008. 5. C.W. Rowley. Model reduction for fluids using balanced proper orthogonal decomposition. Int. J. Bifurc. Chaos, 15(3), 997–1013, 2005. 6. T. Duriez. Application des générateurs de vortex au contrôle d’écoulement décollés. Phd thesis, Diderot University, january 23 2009. 7. D. Sipp and A. Lebedev. Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech., 593, 333-358, 2007. 8. A. Barbagallo, D. Sipp and P. J. Schmid. Closed loop control of an open cavity flow using reduced-order models. submitted to J. Fluid Mech.

Control of a trapped vortex in a thick airfoil by steady/unsteady mass flow suction R. S. Donelli, F. De Gregorio, M. Buffoni, and O. Tutty Abstract Experimental results of trapping vortices on a thick airfoil by using a cavity and an inside unsteady mass flow suction have been investigated emphasizing drawback and benefits of the use of forcing suction compared to a simple steady suction system.

1 Introduction Modern airplanes lifting surfaces are claimed to show high lift-to-drag ratios to minimize the emissions and the fuel burn. The tendency to design commercial aircrafts of larger dimensions, or innovative configurations such as Blended-Wing-Body airplanes, requires innovative solutions in the field of wing structures. In order to carry a larger payload, to have thick wings would be beneficial. The drawback of such type of airfoils is the low efficiency due to the high value of their drag coefficient. Thick airfoils are characterised by an early flow separation phenomenon even for small incidence angles. The potentially separated flow can be forced to remain attached by using an intense vortex anchored in a cavity. At the end of 2005, VortexCell2050, an European funded research project, was launched with the aim of investigating the possibility to control the flow separation using trapped vortex cavities [1, 2, 3, 4] equipped with a suction system aimed to stabilize the vortex. In this work, the activity regarding the stabilization of the vortex inside the cavity by using unsteady mass flow suction is presented and results have been compared to the results achieved by using steady mass flow suction.

2 Experimental Setup and Measurements Instrumentation The experimental measurements have been performed in the CIRA CT-1 open wind tunnel, showed in fig. 1. The CT-1 test section sizes are 305x305x600 mm, while the maximum wind tunnel speed is of 55 m/s. An airfoil, named test bed, has been installed on the bottom wall of the test section as shown in fig. 1. The test bed has been equipped with a cavity mounted close to the flow separation location. The vortex cavity has been realized by transparent material in order to allow the illumination of the flow region inside the cavity and perform PIV measurements. The flow R. S. Donelli, · F. De Gregorio CIRA, Italian Center for Aerospace Research, Capua, Italy, 81043 e-mail: [email protected] M. Buffoni, · O. Tutty University of Southampton, Southampton, United Kingdom, SO17 1BJ P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_80, © Springer Science+Business Media B.V. 2010

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field measurements have been carried out by means of Particle Image Velocimetry technique. The recording region is illuminated by two Nd-Yag resonator heads providing a laser beam energy of about 300 mJ each at a wave length of 532 nm. In order to measure simultaneously the upper and lower region of the model a double light sheet configuration has been adopted. One CCD camera has been used for recording the flow field above the separation region of the model. The camera has been mounted horizontally on a 2D linear traversing system that allows the camera to move accordingly with the planned zone to record. The cavity is equipped with suction system obtained by making, in the internal zone of the cavity, 906 passing holes with a diameter of 1 mm. These holes are connected to three different collectors in order to apply different suction mass flow values. Each collector is connected by dedicated circuit to a vacuum pump. Each circuit is provided of a partial valve in order to vary the mass flow rate and of a flow meter.

3 Experimental Test Campaign The objectives of the experimental investigation at the CIRA CT-1 facility mostly consisted in the evaluation of: • • • • •

the vortex trapping feasibility the design and optimization tools (cavity design) the optimal cavity location with respect to the laminar/turbulent separation the internal flow control to stabilize the trapped vortex the unsteady forcing signals to minimize mass flow suction.

3.1 No-Suction and Steady Suction Passive control has been investigated testing the behavior of the TVC without any mass flow suction. The results showed that the TVC did not induce flow reattachment due to the vortex shedding. The vortex is characterized by an elliptical shape

Fig. 1 CIRA CT1 open wind tunnel and test bench installed in CT1 test section

Control of a trapped vortex in a thick airfoil by steady/unsteady mass flow suction 15 m/s, 740rpm − Max./Zero constant suction: old vs. new

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Fig. 2 Cp comparisons for different signals and phases

located in the shear layer region inducing flow separation. Successively, massive steady suction has been applied inducing flow reattachment due to the stabilization in the cavity of an intense vortex. It has been observed that increasing the mass flow rate, the recovery-pressure increases and when a mass flow suction rate of 25.8 m3 /h is applied, the flow is attached up to a 95% of the chord. The vortex is steadily located in the centre of the cavity and the flow results fully attached.

3.2 Unsteady Suction Numerical tests showed that using unsteady suction the effective mass flow rate could be reduced. A third experimental test campaign was performed to verify the numerical results. PIV measurements have been performed focusing on the region of the cavity. Several forcing signals, with different amplitudes and frequencies were tested. Changes in phase of the forcing signals between the different porous regions were also investigated. Three proportional electric control valves governed the flow in the three sets of suction holes, giving variable suction rates. Surface pressure sensors were placed to collect data for future closed-loop control studies. Data were collected through a National Instruments data acquisition PCI card (M-Series PCI6259) plugged into a PCI bus slot. Control of the valve’s signals and data collection were performed using NI LabView 8.6 Professional Development + Real-Time Module. Tests were performed for wind tunnel speed of 15 m/s and with the model placed at 7.66◦ . The suction mass flow rate was varied in time applying to the control valves sinusoidal, triangular, sawtooth and square voltage signals. Frequencies of 1, 5, 10, 20 and 50 Hz were used during the test in order to investigate their influence in trapping the vortex for a particular suction rate.

4 Results Figure 2 shows comparisons of the pressure distributions for different signals and frequencies applied to the control valves. No big differences in Cp are observed for changes in frequency and signal, as well as for changes in frequency and phase for a given signal. Moreover, fig. 2 shows that for a mean suction mass flow rate of about 20 m3 , the recovery-pressure remains almost the same compared to the case of maximum con-

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Fig. 3 Mean velocity field on the test bed for the two investigated zones

stant suction. The reasons are due to the limited pressure drop achievable from the vacuum pump and to the additional pressure losses introduced by the proportional electric control valves. The valves operate like a low pass filter damping the mass flow amplitude. The PIV flow field measurements have been phase locked to the electric valve forcing signals. Figure 3 shows the mean flow velocity and streamlines in the cavity respectively in correspondence of the maximum, minimum and middle opening of the valve. In all the images, the vortex is located in the centre of the cavity and the flow results fully attached, when a mean mass flow suction rate of 21.2 m3 is applied. No main differences are evident between the three flow fields due to the limited amplitude oscillations achievable.

5 Conclusions An extended test campaign has been carried out in order to investigate the trapped vortex behaviour on a thick airfoil. The TVC has been tested as a passive flow control device and a predetermined active flow control device. This last has been tested in steady and unsteady conditions. The results show that the TVC without mass flow suction blowing is not able to induce flow reattachment. Steady flow suction generates a stable vortex in the cavity inducing a clear flow reattachment. For unsteady suction test, no-conclusions can be drawn since the limited mass flow amplitude and frequency achievable with the available experimental setup. However, the learned lesson gained by the development of the open-loop oscillatory system shall be propaedeutic for the future test campaign at POLITO large wind tunnel. The Vcell2050 project foresees to perform a test campaign on a two dimensional NACA0024 airfoil equipped with a trapped vortex.

References 1. Vortex cell2050 european project - description of work (dow) (2005) 2. Donelli, R.S., Iannelli, P., Chernyshenko, S., Iollo, A., Zanetti, L.: Flow models for a vortex cell. AIAA Paper accepted 3. Kruppa, E.W.: A wind tunnel investigation of the kasper vortex concept. AIAA (1977) 4. Riddle, T.W., Wadcock, A.J., Tso, J., Cummings, R.M.: An experimental analysis of vortex trapping techniques. Journal of Fluids Engineering 121 (1999)

Receptivity of compressible boundary layer to kinetic fluctuations Alexander V. Fedorov and Sergei N. Averkin

Abstract This paper addresses the fundamental issue: What is the upper limit of transition Reynolds number relevant to “absolutely quiet” free stream, in which the only source of unsteady disturbances is unavoidable thermal noise? To simulate thermal sources generating unstable boundary-layer modes, Langevin terms are included into the stress tensor and the heat transfer vector of the Navier-Stokes equations. Excitation of unstable wave-packets and their downstream propagation is analyzed using combined asymptotic and numerical approach for low-speed (15 m/s) and high-speed (Mach=6.8) boundary-layer flows. The transition onset Reynolds numbers are estimated for both cases.

1 Introduction The laminar-turbulent transition locus essentially depends on receptivity of the boundary-layer flow to freestream disturbances. The latter are decomposed to pressure fluctuations (acoustic noise), vortical fluctuations (turbulence) and entropy fluctuations (temperature spottiness). Numerous receptivity studies have been performed to understand mechanisms by which the aforementioned disturbances excite unstable boundary-layer modes. However, surprisingly small attention has been paid to the fundamental issue: What is the upper limit of transition Reynolds number relevant to “absolutely quiet” free stream? In this case the only source of unsteady disturbances is a thermal noise associated with kinetic fluctuations. Betchov [1] was the first who assumed that the thermal noise could trigger boundary-layer disturbances in sufficiently quiet environment. Zavolsky & Reutov [2] developed a theoretical model describing excitation of Tollmin-Schlichting (TS) waves by kinetic Alexander V. Fedorov Moscow Institute of Physics and Technology, Zhukovsky, RUSSIA, e-mail: [email protected] Sergei N. Averkin Moscow Institute of Physics and Technology, Zhukovsky, RUSSIA P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_81, © Springer Science+Business Media B.V. 2010

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fluctuations in the incompressible boundary layer on a flat plate. Solving the spatial receptivity and stability problems they evaluated the root-mean-square (rms) amplitude of thermally induced TS wave-packets for the experimental conditions [6]. The predicted rms amplitude of streamwise velocity fluctuations was approximately 20 times smaller than that reported in [2] for most quiet freestream conditions. Luchini [3] considered the temporal problem for the parallel incompressible boundary layer and concluded that the transition onset point owing to the thermal excitation of TS waves corresponds to the amplification factor N ≈ 9 that is “strikingly on top of the empirical value of 9÷10 used in aerodynamics.” These results encouraged us to extend the model [2] to compressible boundary-layer flows, evaluate the transition onset Reynolds numbers and corresponding N-factors for unstable modes induced by kinetic fluctuations, and compare results with predictions of [2, 3].

2 Analysis and results Consider unsteady disturbances in 2D compressible laminar boundary-layer flow at the Reynolds number R = U∞∗ ∆ ∗ /ν∞∗ , where ∆ ∗ ∼ the boundary-layer thickness. Velocities (u, v, w), time t, pressure p, temperature T and kinematic viscosity ν are referenced to the free-stream velocity U∞∗ , time ∆ ∗ /U∞∗ , doubled dynamic pressure ρ∞∗ U∞∗2 , temperature T∞∗ and ν∞∗ , respectively. To account for kinetic fluctuations, the Langevin (noise) terms are added to the stress tensor and the heat transfer vector. The space-time correlations of these terms are given in [4]. Small 3D disturbances are expressed in the vector form εLΨ (x, y, z,t) = 1/2 2 (u, ∂y u, v, p, T, ∂y T, w, ∂y w)T , where εL = kB T∞∗U∞∗ /ν∞∗3 ρ∞∗ /R is small parameter characterizing the level of shear stresses induced by thermal fluctuations, kB – Boltzmann constant, asterisks denote dimensional quantities. Luchini [3] noticed that λ ∗ ≡ kB T∞∗ /ν∞∗2 ρ∞∗ can be treated pas a characteristic length scale associated with thermal fluctuations, then εL = λ ∗ /(R3 ∆ ∗ ). Performing the Fourier transform F(x, y; β , ω ) =

+∞ R +∞ R

−∞ −∞

Ψ (x, y, z,t) exp(iω t − iβ z)dzdt, we obtain the linearized

Navier–Stokes equations for the disturbance amplitude function F, with the righthand side containing Fourier transforms of the normalized Langevin sources. Imposing the standard boundary conditions: (F1 , F3 , F5 , F7 ) = 0, at at y = 0 and |F| < ∞ for y → ∞, we solved the inhomogeneous problem using the theoretical model [5] developed for deterministic forcing. With this solution we evaluated the spectral correlation of unstable mode “n” generated by thermal fluctuations. Finally, the rms r amplitude of the corresponding wave-packet is arms (x) = f2 =

max f 2 , where y

q +∞ +∞ R 2 εL2 R π 2 2N(β ,ω ;x) d β , d ω a (x , β , ω ) 0 2 F σx (x0 ) q (x0 )e 2π −∞ 0

N=

Rx

x0

σ dx,

σ (x, β , ω ) = −Im(αn ) is the spatial growth rate, x0 (β , ω ) is the neutral point, q2 (x0 ) is the correlation spectral density characterizing receptivity in the neutral point

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487

vicinity, aF is amplitude of a physical quantity calculated using the eigenfuction components Fj . If 2D waves (β = 0) are most unstable, then asymptotic integration over ω and β provides the maximum amplitude of the wave-packet with a central frequency ωm , which is observed at the station x r ε 2 a2 (x0 , 0, ωm )q2 (x0 ) π max f 2 ≈ q L F e2N(0,ωm ;x) . y σ (x ) x 0 2π Nβ β (0, ωm ; x)Nωω (0, ωm ; x)

First series of calculations were performed for TS waves excited in the boundary layer on a flat plate at U∞∗ = 15 m/s, T∞∗ = 300 K, pressure P∞∗ = 1.013 × 105 Pa, ν∞∗ = 1.58 × 10−5 m2 /s. These parameters correspond to the experimental conditions [6] and the numerical example considered in [2]. The quantity arms represents a maximal amplitude of u-velocity disturbance, aF = max |F1 |. The distriy

butions of arms (R) and the amplification factor N(R) are shown in Fig. 1, herep after ∆ ∗ = (ν∞∗ x∗ /U∞∗ )1/2 and R = U∞∗ x∗ /ν∞∗ . The result [2] (black square) agrees well with our prediction (lines). The criterion acr = 1% gives the transition onset Reynolds number Retr ≈ 6.8 × 106 that corresponds to N ≈ 14.4. The experimentally observed Reynolds number Retr ≈ 5 × 106 correlates with N = 12. Zavolsky & Reutov [2] assumed that the nonparallel effect can substantially increase the TS amplification. Our calculations did not confirm this assumption. In Fig. 1, the distribution of arms (R) resulted from the weakly nonparallel analysis (dashed line) is close to that predicted by the local-parallel theory (solid line). 0.1

16 N=14.4

arms

arms including nonparallel effect Zavlsky & Reutov N factor

14 Retr=5E+6

12

N=12

0.01

Retr=6.8E+6

10 8

arms

N

6

1E-3

4 2 1E-4 1000

1500

2000

2500

0 3000

R

Fig. 1 Distributions of arms (R) and N(0, ωm , R) for TS Fig. 2 Transition onset Reynolds waves. number vs. freestream turbulence intensities.

Luchini [3] reported much larger amplitudes of TS waves corresponding to N ≈ 9. Presumably, Luchini did not account for the multiplier Nβ β (0, ωm ; x1 )Nωω (0, ωm ; x1 ) −1/4 ∼ 10−2 that is quite small due to selective amplification of TS waves. Figure 2 shows the transition onset Reynolds number as

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a function of the freestream turbulence Tu ≡ u∗rms /U∞∗2 . Symbols represent experimental data, which were obtained in low-turbulence wind tunnels and systemized by Spangler & Wells [6]. Apparently, transition due thermal fluctuations corresponds to the perfectly “clean” case and should provide the upper limit of Retr . Our prediction (Retr = 6.8 × 106 shown by the upper arrow) agrees with this statement, whereas Retr predicted by Luchini [3] (the lower arrow) is essentially smaller than some of the data (squares and triangle). Similar calculations were carried out for the Mack second mode on a flat plate at the conditions relevant to shock tunnel: M∞ = 6.8, T∞∗ = 227.5 K, the walltemperature ratio Tw∗ /T∞∗ = 2.82 and the unit Reynolds number Re1∞ = 2 × 107 m−1 . With the assumption that the maximal amplitude of streamwise mass-flux disturbance acr ≈ 1% at the transition onset criterion, the predicted Retr ≈ 42.8 × 106 corresponds to N ≈ 14.8 that is again essentially larger than the empirical N = 10.

3 Conclusions The foregoing analysis showed that thermal fluctuations effectively excite unstable wave-packets in the vicinity of the lower neutral branch. However, as the wave packet propagates downstream, it is strongly narrowed down in ω − β space due to selective amplification of its components. As a result, integration over ω and β dramatically reduces rms amplitudes of instability, and the corresponding amplification ratio is significantly larger than the empirical value N = 9 ÷ 10. Nevertheless, the thermal mechanism may be important for highly unstable shear layers or boundary layers at unfavorable pressure gradients. Acknowledgements This work is supported by Russian Foundation for Basic Research under Grant 06-08-01214 and partially supported by the AFOSR/NASA National Center for Hypersonic Research in Laminar-Turbulent Transition.

References 1. Betchov, R.: Thermal agitation and turbulence. Proc. Second Intern. Symp. Rarefied Gas Dynamics. Academic Press, New York, 307–321 (1961). 2. Zavolsky, N.A., and Reutov, V.P.: Thermal agitation of waves in boundary layer. Izv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza. 5, 21–29 (1983) (in Russian). 3. Luchini, P.: The role of microscopic fluctuations in transition prediction. arXiv:0804.2067v1 [physics.flu-dyn], 13 April 2008. 4. Landau, L.D., and Lifshits, E.M.: Course of Theoretical Physics V: Statistical Physics. Pergamon Press, Oxford, 1978 5. Fedorov, A.V., and Khokhlov, A.P.: Receptivity of hypersonic boundary layer to wall disturbances. Theoret. Comput. Fluid Dynamics. 15, 231–254 (2002) 6. Spangler, J.G., and Wells, C.S.: Effects of freestream turbulence on boundary-layer transition. AIAA J. 6(3), 543–545 (1968)

Effect of transport modeling on hypersonic cooled wall boundary layer stability Kenneth Franko and Sanjiva Lele

Abstract Boundary layer transition to turbulence on blunt entry/reentry vehicles leads to increased heating and additional thermal protection system weight. The boundary layers on the blunt face contain large temperature gradients which require the modeling of complex multi-species flows including detailed transport models and non-equilibrium chemistry. In this work, the sensitivity of stability characteristics to different transport and thermodynamic models is investigated. The flat plate boundary layer is used as a model problem using representative temperature profiles and Mach numbers for temperature ranges without dissociation. It is shown that the stability characteristics, both linear and transient growth, are sensitive to the models used for calculating the transport and thermodynamic properties.

1 Introduction Accurate prediction of boundary layer transition on hypersonic vehicles would allow for less conservative vehicle designs. In order to computationally predict transition, thermodynamic and transport models are needed to solve the mean flow and disturbance equations. Any differences or uncertainties in the models could lead to inaccurate predictions of transition and vehicle heating. Traditional stability calculations consider the exponential growth of instability waves in the boundary layer. However, this mechanism is not the dominant one for the transition to turbulence in all cases. One alternative theory is transient growth where disturbances grow algebraically. Tumin and Reshotko[3] have proposed this as potential mechanism for blunt body transition to turbulence. Since both processes may be important, the senKenneth Franko Stanford University, Stanford, CA, USA. e-mail: [email protected] Sanjiva Lele Stanford University, Stanford, CA, USA. e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_82, © Springer Science+Business Media B.V. 2010

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sitivity of these two types of linear stability prediction to different chemistry models is considered. A previous study by Lyttle and Reed [5] was undertaken for missile configurations with high edge Mach numbers and temperatures. They compared different models for transport coefficients, thermal properties and chemical kinetic parameters on 2nd mode instabilities. The largest variation was due to the difference in transport models for viscosity and thermal conductivity. The current work focuses on lower speed flows with large temperature gradients for temperature ranges where dissociation and thermal non-equilibrium are negligible. These temperature ranges were chosen in order to make an initial estimate of the effects of different transport and thermodynamic models without having to consider the effects of chemical reactions.

2 Methodology In order to calculate the stability of a boundary layer, the mean flow solution must first be determined. In this work, it is calculated using a similarity solution obtained from the boundary layer equations using the Levy-Lees transformation. The set of non-linear ODEs is solved using a spectral collocation method with a Newton iteration. Two different linear processes are considered, exponential growth (classic linear stability theory) and transient growth. The compressible linear stability equations are obtained using the standard assumptions of disturbance linearity, parallel flow approximation, and wavelike solutions. A Chebyshev spectral collocation method is used and the resulting eigenvalue problem is solved numerically using the QZ algorithm. Details of the equations and method can be found in [4]. Transient growth is algebraic growth due to the nonorthogonality of eigenmodes of the Orr-Sommerfeld operator and the equivalent compressible stability operator. Due to this nonorthogonality, the superposition of eigenmodes can lead to short term algebraic growth even if all of the modes are exponentially stable. Details of the energy norm, superposition of modes, and calculation of the maximum transient growth rate can be found in [3] for waves which grow in the spatial coordinate. The exponential stability code was validated using a shooting code and the transient growth calculations were validated using [3] and [2]. The transport coefficients and the specific heat (∂ h/∂ T = Cp ) are determined using three different models of increasing complexity. The first model assumes constant Cp and Prandtl number. The viscosity is calculated using Sutherland’s law. The thermal conductivity is determined using the Prandtl number relation. The second model determines C p using the thermodynamic properties of each species and weighs their contribution by their mass fraction. The Cp includes the contributions from translational, vibrational, and rotational energy. The species viscosity is determined using a Blottner curve fit and the mixture viscosity is determined using Wilke’s mixture rule. The species thermal conductivity is determined using Eu-

Effect of transport modeling on hypersonic cooled wall boundary layer stability Mach Number

Model

Adiabatic

Te = 600K

Te = 1000K

Te = 1500K

0.5 0.5 0.5

1 2 3

0.004879 0.004894 0.004876

0.01842 0.01880 0.01816

0.03497 0.03527 0.03314

0.04734 0.04678 0.04321

1.0 1.0 1.0

1 2 3

0.004938 0.004999 0.004929

0.008022 0.008160 0.008000

0.01195 0.01214 0.01163

0.01486 0.01491 0.01409

1.5 1.5 1.5

1 2 3

0.005021 0.005149 0.005005

0.005940 0.006030 0.005951

0.007543 0.007693 0.007485

0.008716 0.008833 0.008528

491

Table 1 Maximum spatial transient growth over ω (frequency) and β (spanwise wavenumber) scaled by Re2 for different Mach numbers, temperature boundary conditions and chemistry models.

cken’s relation and mixture thermal conductivity is determined using Wilke’s mixture rule. The third model uses the mutation library [1]. It calculates the Cp in the same manner as the second model. The mixture viscosity and thermal conductivity are determined using the Chapman-Enskog expansion method with a LaguerreSonine polynomial expansion used to solve the resulting integral equations. The equations depend on the reduced collision integrals.

3 Results In order to initially evaluate the effect of different transport and thermodynamic models, the temperature range is limited so that the composition is constant. The boundary layer edge Mach numbers are chosen to be representative of the edge Mach numbers over a blunt reentry body. The wall boundary condition is either adiabatic or a constant wall temperature of 300K. The boundary layer edge temperature (Te ) is varied between 300K and 1500K in order to simulate increased wall cooling without dissociation. The differences between stability predictions for spatial exponential growth using the three models can be found on figures 1 and 2. The Reynolds transition number is determined using the eN method with a value of N=9. The largest difference in prediction is for the highest edge temperature of 1500K where the difference between models 3 and 2 is 35%. For each Mach number, temperature boundary condition, and chemistry model, the maximum spatial transient growths optimized over α and β are tabulated on table 3. As wall cooling is increased the maximum transient growth increases. This effect is less pronounced as Mach number increases. It has been shown previously that for higher Mach numbers [3], cooling can actually be stabilizing for transient growth. The different chemistry models lead to a maximum difference of 10% for transient growth prediction for the M=0.5, Te = 1500K case.

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0.0040 Model 1 Model 2

3.0e+06

Model 3

Retr

0.0035

2.0e+06

0.0030

0.0025

1.0e+06

0.0020 900000

1100000

1300000

Re

1500000

1700000

Fig. 1 Stability curves over parameters ω (frequency) and Re at β = 0 (spanwise wavenumber) using the three different models for Te = 1500K. Contours are spatial growth rates at increments of 0.0002.

300

700

Te

1100

1500

Fig. 2 Transition Reynolds number as a function of edge temperature. Obtained using eN method by integrating growth over Reynolds number for different ω (frequency) and β (spanwise wavenumber) and choosing lowest transition Reynolds number.

4 Conclusion The present work has shown that the different transport and thermodynamic models lead to significantly different predictions of exponential and transient growth for Mach numbers and temperature profiles representative of hypersonic blunt body reentry. This suggests that the model used is also important in the case where chemical and thermal non-equilibrium effects are present. Acknowledgements We thank Dr. Thierry Magin for providing the mutation code and guidance. Support for first author is provided by Fannie and John Hertz Foundation Fellowship and Stanford Graduate Fellowship.

References 1. Magin, T., Degrez, G. (2004) Transport algorithms for partially ionized and unmagnetized plasmas. Journal of Computational Physics, Vol 198, pp 424-449. 2. Hanifi, A., Schmid, P.J., and Henningson, D.S. (1996) Transient growth in compressible boundary layer flows. Physics of Fluids, Vol 8, pp 826-837. 3. Tumin, A., Reshotko, E. (2001) Spatial theory of optimal disturbances in boundary layer. Physics of Fluids, Vol 13, No 7, pp 2097-2104. 4. Malik, M.R. (1990) Numerical methods for hypersonic boundary layer stability. Journal of Computational Physics, Vol 86, pp 376-413. 5. Lyttle, I.J., Reed, H.L. (2005) Sensitivity of second-mode linear stability to constitutive models within hypersonic flow. AIAA-2005-889.

Modeling Supersonic and Hypersonic Flow Transition over Three-Dimensional Bodies Song Fu, Liang Wang, Angelo Carnarius, Charles Mockett, and Frank Thiele

Abstract In this study we propose a local-variable-based laminar-turbulence transition model that considers the effects of different instability modes existing in highspeed aerodynamic flows. This model is validated with a number of available experiments on transition including supersonic and hypersonic flows past straight/flared cones at small incidences and elliptic cones at zero incidences. Computed results are in good agreement with experimental data.

1 Introduction Understanding, predicting and controlling laminar-turbulent transition in supersonic and hypersonic boundary layers play an key role in design of future space vehicles operating at sustained supersonic and hypersonic speeds. It is generally believed that two instability mechanisms, the second-mode and the crossflow instabilities, are both important for such transition processes, but whether they interact with each other or not and how they exhibit influence on transition process are still topics of active research. The goal of this investigation is to identify the dominant instability mechanism in the high-speed flow transition over three-dimensional (3-D) bodies, with the aid of a recently proposed RANS model [1]. To the author’s knowledge, the corresponding experimental data for such transitional flows have never been explained, though various attempts have been made to correlate subsets of the data.

Song Fu School of Aerospace, Tsinghua University, Beijing, CHINA, e-mail: [email protected] Liang Wang Institute of Fluid Mechanics and Engineering Acoustics, TU-Berlin, Berlin, GERMANY, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_83, © Springer Science+Business Media B.V. 2010

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2 Model formulation Due to limited space, only the summary of such model is presented here. It consists of a transport equation for the intermittency factor, γ , which is coupled with the well-known SST k − ω eddy-viscosity model in the solution procedure. The fluctuating kinetic energy k is considered to include the non-turbulent as well as turbulent components. γ then acts as a weighting factor between these components in Pk and Pω , i.e. the production terms of k and ω equations. This approach is thereby centred upon the determination of an effective viscosity of the non-turbulent fluctuations, µnt , where the effects of first-mode, second-mode and crossflow instabilities are all considered. The value of µnt controls the evolution of the kinetic energy of nonturbulent fluctuations, kL , in the pre-transitional region. Moreover, a function in the source term of the novel γ equation proposed, as determined by the development of kL and the mean flow, is used to trigger the transition onset. It is demonstrated that the present model can be successfully applied to the 2-D aerodynamic flow transition with a reasonably wide range of Mach numbers [1].

3 Results and discussion The present model is firstly applied to simulate transition in super-/hypersonic flows past straight/flared cones at small angle of attack (AOA), and all the experimental data are obtained from quiet nozzles [2, 3]. For the supersonic case, at 0 ◦ AOA the calculated transition onset location xt = 0.192m compares well with the measured xt = 0.21m. Fig. 1(a) compares the measured and computed lines of xt . It is well predicted on the windward ray, but over-predicted on the leeward ray because this model over-emphasizes the effect of the 2nd-mode instability [1]. Moreover, if the crossflow mode is excluded in such model, the corresponding calculations (denoted NC) show a strong change in the slope of the transition-onset line, as shown in Fig. 1(b). This indicates that the crossflow instability plays a key role here, which is also verified in experiments [2]. For the hypersonic case, the calculated xt at different AOA are given in table 1, which agree well with the experimental data (0 ◦ and 2 ◦ AOA cases), as the maximum relative error is 4.2%. However, it is seen that on the leeward ray, xt at 2 ◦ is smaller than at 4 ◦ , which is counter-intuitive. Moreover, the comparison of these two cases shows that with the same increase of AOA, the decrease of xt on the leeward ray in the supersonic case is greater than in the hypersonic case; the increase of xt on the windward ray in the supersonic case is smaller than in the hypersonic case. Table 1 Variation of transition onset location with angle of attack (AOA). AOA 0◦ 1◦ 1◦ (NC) 2◦ 2◦ (NC) 4◦ 4◦ (NC) Leeward ray 0.396 0.348 0.363 0.321 0.337 0.342 0.358 Windward ray 0.396 0.406 0.406 0.428 0.422 0.463 0.456

Modeling Supersonic and Hypersonic Flow Transition over Three-Dimensional Bodies

0.1

x (m)

0.2

-0.02

angle of attack: 2 o

-0.04

0.3

0.1

x (m)

adiabatic wall 0.8 0.6 8 6

0.66

0

0.2

Level γ 9 7 5 3 1

0.88 0. 88

excluding the 5 o half-angle cone crossflow mode 0.23

-0.04

0.02

1

-0.02

0.04

0.45 0.45

0

Re∞ = 3.74E7 m -1 Ma∞ = 3.5 T ∞ = 92.3 K FSTI = 0.1% Level γ 9 0.88 Leeward 0.4 7 0.66 0. 5 88 0 5 0.45 0. 0.0 .23 6 6 1 3 0.23 Windward 1 0.01

0.0

y (m)

0.02

y (m)

0.04

495 0.88 0.66 0.45 0.23 0.01

0.3

Fig. 1 Intermittency factor isolines near the cone surface computed by the present model (left) and the model excluding crossflow effect (right). The solid line shows the measured transition onset.

0.4

0.01 0.23 5 0 .4 0.6

x (m)

0.8

θ = 0o

9 7 5 3 1

0.88 0.66 0.45 0.23 0.01

0.2

elliptic cone

0.1 0

1

angle of attack: 0 o

e=2

excluding the crossflow mode

0.4

FSTI = 1.2%

45

0

0.66 0.88

0.3

Level γ

01 0.

0.1

θ = 90o

0.6

θ = 90o

8 0.8

-1 Ma∞ = 7.95 Re∞ = 2.03E6 m T w = 303 K T∞ = 52.3 K 7 o half angle ( minor axis )

y (m)

0.2

0.

y (m)

0.3

0.23

0.8

x (m)

θ = 0o

Level γ 9 7 5 3 1

0.88 0.66 0.45 0.23 0.01

1

Fig. 2 Intermittency factor isolines near the elliptic cone surface computed by the present model (left) and the model excluding crossflow effect (right).

To explain all these phenomena, this paper considers the test case of hypersonic flow over an elliptic cone of aspect ratio 2 at 0 ◦ AOA, where the transition process is only affected by the second-mode and the crossflow instabilities [4]. The crossflow is established by the pressure difference between the major and minor axes, which causes flow from the higher-pressure major axis (azimuth angle θ = 90◦ ) to the minor axis (top and bottom centerlines, θ = 0◦ ). Fig. 2(a) gives the calculated isolines of γ near the elliptic cone surface. It indicates that the transition takes place first on the minor-axis ray, in agreement with experimental observations. Moreover, excluding the crossflow mode in this model leads to strong distorted results, as shown in Fig. 2(b). As mentioned, the faster k develops in the pre-transitional region, the earlier transition happens in the present approach. Since the wavelength of the secondmode fluctuation is approximately twice the boundary layer thickness, δ , and its phase velocity is equal to the local mean velocity at the generalized inflection point, the corresponding timescale, τnt2 , scales as Ue /δ , where Ue stands for the edge velocity. Therefore, using the magnitude analysis based on the boundary layer assumption, the second-mode component of production of k scales as (Pk /k)nt2 =

"

′′ u′′ ρ¯ ug i j

k

!

S˜i j

#

nt2

¯ nt2 ∼ ρτ

∂U ∂y

2

∼ ρ¯

δ Ue

Ue δ

2

= ρ¯

Ue δ

(1)

where ρ¯ is the density of the mean flow. Since δ is about one order of magnitude greater on the minor axis compared to the major axis, which far exceeds the corresponding variations of ρ¯ and Ue , k increases faster on the major-axis ray. This is why the opposite is predicted, as shown in Fig. 2(b), if only the second mode is considered. And by analogously analyzing the crossflow component, it is found that it increases with the increase of δ , which leads the occurrence of transition first on the minor-axis ray, as shown in Fig. 2(a). It demonstrates the importance of the crossflow model and that crossflow instability dominates such transition process. Similar work has furthermore been performed in the 3-D super-/hypersonic cone cases, which delivers complete explanations to the phenomena mentioned above.

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For the hypersonic case, since the half-angle of cone is only 5 ◦ , the mean flow is affected significantly p by AOA. In the windward region, the crossflow effect is very weak, thus δ ∼ 3υe x/Ue . With the increase of AOA, the shock loss increases, then Ue deduces, resulting in the decrease of Ue /δ , which is much larger than the increase of ρ¯ . Therefore, (Pk /k)nt2 decreases based on Eq. (1), i.e. the increase of AOA results in the increase of xt . For the supersonic case, however, since the 2ndmode effect is very weak at Ma∞ = 3.5, there is much smaller increase of xt . This is the explanation why the increase of xt on the windward ray in the supersonic case is smaller than in the hypersonic case. In the leeward region, due to the strong crossflow effect, the mean flow becomes much more complex. For the hypersonic case, with the increase of AOA, thep shock loss deceases. Then at 1 ◦ and 2 ◦ AOA, δ is approximately proportional to 3υe x/Ue , thus (Pk /k)nt2 slightly increases according to Eq. (1), resulting in the decrease of xt ; but at 4 ◦ AOA, the increase of δ is much larger than the variation of ρ¯ and Ue , even leading (Pk /k)nt2 to reduce. That is why the decrease of xt at 2 ◦ is larger than at 4 ◦ . Furthermore, it is found that at 2 ◦ AOA, the δ value on the leeward ray is as 3 times as that on the windward ray, and the maximum crossflow intensity is 5% the edge velocity, which are both much smaller than the corresponding values for the elliptic cone case as 10 (times) and 11%. It suggests that the effect of the crossflow instability at 2 ◦ AOA case may not dominate transition process, as it is much weaker than that in the elliptic cone case. On the contrary, for the supersonic case, the effect of second-mode instability is so weak that the crossflow instability dominates transition process. Therefore, with recalling that in the hypersonic case (Pk /k)nt2 slightly increases, i.e. the 2nd-mode instability leads to the decrease of xt , it is explained why the decrease of xt on the leeward ray in the supersonic case is greater than in the hypersonic case.

4 Conclusion A new k − ω − γ transition/turbulence model based on instability modes is applied to simulate the 3-D supersonic and hypersonic flow transition over circular and elliptic cones. Computed results are in good agreement with experimental data, which have been further explained for the first time.

References 1. Fu, S., and Wang, L. 2008 Modeling the flow transition in supersonic boundary layer with a new k − ω − γ transition/turbulence model. In: 7th International Symposium on Engineering Turbulence Modelling and Measurements, Limassol, Cyprus. 2. Doggett, G. P., Chokani, N., and Wilkinson, S. P. 1997 Effects of angle of attack on hypersonic boundary layer stability in a quiet wind tunnel. AIAA J. 35, 464-470. 3. King, R. A. 1991 Mach 3.5 Boundary-layer transition on a cone at angle of attack. AIAA Paper 91-1804. 4. Kimmel, R. L., Poggie, J., and Schwoerke, S. N. 1999 Laminar-turbulent transition in a Mach 8 elliptic cone flow. AIAA J. 37, 1080-1087.

Amplitude threshold in the wake transition of an oscillating circular cylinder Rafael S. Gioria and Julio R. Meneghini

Abstract In this work we investigate the effect of the amplitude of oscillation on the secondary transition of the wake of an oscillating circular cylinder. In order to carry out this investigation, Floquet stability analysis is employed for Reynolds numbers 200 and 300, and the amplitude of oscillation is varied from 0.0d to 0.3d, where d is the cylinder diameter. A threshold amplitude is observed: if the oscillation amplitude is smaller than the threshold the flow behaves in a manner similar to the flow around a fixed circular cylinder. Above this threshold, the oscillation affects the three-dimensionalities of the wake depending on the amplitude of oscillation.

1 Introduction The phenomena of the secondary wake transition in the flow around a circular cylinder is still subject of several studies. Questions regarding how the transition occurs and its physical mechanism are of great interest. This investigation is concerned with the oscillation effect on the wake transition and with the minimum amplitude for which the oscillation influences the three-dimensionalities in the wake. Therefore, we chose the amplitude of oscillation as a parameter and investigate cases with Reynolds number (Re), based on the cylinder diameter, Re = 200 and 300. A few publications have dealt with the transition of the wake of an oscillating circular cylinder. Gioria & Meneghini [4] have presented direct numerical simulations of the flow around an oscillating circular cylinder. They show that, for Re = 200 and an amplitude A = 0.4d, the wake remains two-dimensional. Leontini et al. [7] carRafael S. Gioria NDF, Dept. Mech. Eng., POLI, University of São Paulo, Brazil e-mail: [email protected] Julio R. Meneghini NDF, Dept. Mech. Eng., POLI, University of São Paulo, Brazil e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_84, © Springer Science+Business Media B.V. 2010

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ried out Floquet stability analysis of the flow in the same situation. Amongst their results, they showed that the flow is stable to three-dimensional perturbations for Reynolds number around 200 and amplitudes as low as 0.1d. Gioria et al. [3] also investigated this problem and, in agreement, showed that there is a range of amplitudes for which the wake is absolutely stable to three-dimensional perturbations. None of these cited investigations were concerned with an estimate of a threshold amplitude as this present work. In order to investigate the existence of a threshold amplitude, in the present paper we carried out calculations with very low amplitudes. In this paper, by threshold amplitude we mean the minimum amplitude of oscillation which suppresses the three-dimensionalities of the wake. We chose the frequency of oscillation to be exactly the Strouhal frequency of the fixed circular cylinder for Reynolds numbers Re = 200 and Re = 300 so the wake is synchronized at very low amplitude of oscillation. The spectral/hp method [6] is employed to numerically solve the incompressible Navier-Stokes equations. 9th -degree polynomials are employed as basis functions in the two-dimensional simulations. In the stability analysis to three-dimensional perturbations, the same polynomials are used in the cross-sectional planes while Fourier expansion is used in the spanwise direction. A second order stiffly-stable time-stepping scheme as described is employed to advance the solution in time. Floquet stability analysis is employed to evaluate linear stability analysis of the wake. Barkley & Henderson [1] first applied the Floquet stability analysis to the wake of a fixed circular cylinder. Other applications to the wake of circular cylinders can be found in Leontini et al. [7], Carmo et al. [2] and Gioria et al. [5].

2 Results To investigate the threshold amplitude of oscillation, we obtain the two-dimensional flow field through direct numerical simulation. In order to maintain the wake synchronized to the body movement at very low amplitudes of oscillation, we impose the the body oscillation at exactly the Strouhal frequency. The two Reynolds numbers chosen, 200 and 300, are due to the unstable modes observed in a fixed circular cylinder wake. For Re = 200, only mode A is observed whilst for Re = 300, both modes A and B are observed [1]. Therefore, for each Reynolds number, we varied the amplitude of oscillation (chosen as our investigated parameter) from 0.0d to 0.3d, where d is the circular cylinder diameter. We expect that the three-dimensionalities of the wake of a circular cylinder are suppressed as the amplitude of oscillation is higher than a certain threshold. The suppression of three-dimensionalities was observed in Gioria & Meneghini [4] through DNS and Leontini et al. [7] for moderate amplitudes of oscillation, but there were no attempts to estimate a threshold amplitude for the suppression of the threedimensionalities. To observe the stabilizing effect of the body oscillation, we start the stability analysis from limit case A → 0 which is the fixed cylinder case.

Amplitude threshold in the wake transition of an oscillating circular cylinder

499

5 4 3 2

Wavelength in diameters

MODE A

1

4 2

3

MODE A

1

Wavelength in diameters

5

For a fixed circular cylinder and Re = 200, the unstable three-dimensional mode A has a typical wavelength in the spanwise direction of approximately 4.0d [1]. Here we present a stability map in Fig. 1(a). The map presents the parameter space of the analysis: the amplitude of oscillation × wavelength of instability. The neutral stability curve is highlighted and the cross-hatched region is the unstable region of the parameter space. Fig. 1(a) shows that the threshold amplitude of oscillation for Re = 200 is around 0.03d. This means the three-dimensionalities are suppressed when the amplitude of oscillation is higher than 0.03d. This effect on the wake of an oscillating cylinder is observed in a range of amplitudes 0.03d < A < 0.65d. This upper limit of the range, not shown in this paper, can be found in Leontini et al. [7] and Gioria et al. [5]. When the amplitude of oscillation is lower than the threshold amplitude 0.03d, the observed unstable mode has mode A symmetry characteristics but its wavelength is less than 3.98d (see [1]).

0

0

MODE B

0

0.02

(a)

0.04

0.06

Amplitude in diameters

0.08

0.1

0

(b)

0.05

0.1

0.15

0.2

0.25

0.3

Amplitude in diameters

Fig. 1 Amplitude of oscillation × wavelength of instability stability map. The cross-hatched area is the unstable region for the parameters. (a) Reynolds number 200, (b) Reynolds number 300.

In the case with Re = 300 for a fixed circular cylinder, not only mode A is present but also mode B, with wavelength approximately 0.8d, is unstable [1]. The results with Re = 300 are shown in Fig. 1(b), which is the same kind of stability map presented for Re = 200: the parameter space is the amplitude of oscillation × wavelength of instability. Three-dimensional instabilities are observed for amplitudes of oscillation lower than approximately 0.28d. Therefore the threshold amplitude of oscillation for Re = 300 is 0.28d. In other words, for amplitudes of oscillation in the small range 0.28d < A < 0.35d, the oscillation inhibits the growth of three-dimensional perturbations. When the amplitude of oscillation is lower than the threshold amplitude, mode A is observed with a shorter wavelength than the fixed cylinder case while mode B has a wavelength of the same order 0.8d.

3 Conclusions The amplitude of oscillation has strong influence on the transition of the wake of an oscillating circular cylinder. At low Reynolds number, it is possible to suppress the

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three-dimensionalities of the flow by means of oscillation of the circular cylinder. There is a range of amplitude of oscillation that the three-dimensionalities are suppressed: 0.03d < A < 0.65d with Re = 200 and 0.28d < A < 0.35d with Re = 300. We showed that, for amplitude of oscillation less than approximately 0.03d with Reynolds number 200, the flow is similar to the fixed cylinder case at the same Re. The instability has mode A characteristics with a slightly shorter wavelength. When the amplitude is higher than threshold The oscillation inhibits the growth of three-dimensional perturbations if in this range 0.03d < A < 0.65d. The threshold amplitude of oscillation with Re = 300 is estimated as 0.28d. This means that, in the small range of amplitude 0.28d < A < 0.35d, the oscillation inhibits the growth of three-dimensional perturbations with Reynolds 300. When the amplitude os oscillation is smaller than the threshold 0.28d, the flow is observed with a similar behavior to the fixed cylinder case with the same Re = 300, presenting both unstable three-dimensional modes A and B. As far as the authors know, there are no other publications on the subject in which thresholds of amplitude of oscillation for the suppression of three-dimensionalities in the wake of a circular cylinder are estimated.

4 Acknowledgements R.S.G. and J.R.M. would like to acknowledge the financial support by Fapesp, FINEP/CTPetro, CNPq, and Petrobras during the development of this work. We would also like to thank Prof. Peter W. Bearman and Prof. Spencer J. Sherwin for their useful comments regarding this investigation.

References 1. Barkley, D., Henderson, R.D.: Three-dimensional Floquet stability analysis of the wake of a circular cylinder. Journal of Fluid Mechanics 322, 215–241 (1996) 2. Carmo, B.S., Sherwin, S.J., Bearman, P.W., Willden, R.H.J.: Wake transition in the flow around two circular cylinders in staggered arrangements. Journal of Fluid Mechanics 597, 1–29 (2008) 3. Gioria, R.S., Carmo, B.S., Meneghini, J.R.: Floquet stability analysis of the flow around an oscillating cylinder. BBVIV-5, Brazil (2007) 4. Gioria, R.S.,Meneghini, J.R.: Three-dimensionalities of the flow around an oscillating circular cylinder. IUTAM Symposium on Unsteady Separated Flows and their Control, Corfu, Greece, (2007) 5. Gioria, R.S., Jabardo, P.J.S., Carmo, B.S., Meneghini, J.R.: Floquet stability analysis of the flow around an oscillating cylinder. Journal of Fluids And Structures 25, 676–686 (2009) 6. Karniadakis, G.E., Sherwin, S.J.: Spectral/hp element methods for CFD. Oxford University Press, New York (2005) 7. Leontini, J.S., Thompson, M.C., Hourigan, K.: Three-dimensional transition in the wake of a transversely oscillating cylinder. Journal of Fluid Mechanics 577, 79–104 (2007)

Certain Aspect of Instability of Flow in a Channel with Expansion/Contraction Ayumu Inasawa, Masahito Asai, and Jerzy M. Floryan

Abstract The effect of the beginning of the roughness patch on the evolution of two-dimensional Tollmien-Schlichting waves is considered using theory and direct numerical simulations. The roughness effects may be divided into two elements: (i) change in the average position of the wall and (ii) the roughness shape. The former effect is studied by considering channel with a simple expansion/contraction. It is shown that evolution of base flow towards the plane Poiseuille flow downstream of the step is found to be represented by using slowly-decaying stationary eigenmodes, similar to that for the channel entrance flow. It is also found that the influence of sudden expansion/contraction to the stability of flow persists far downstream.

1 Introduction Flows over rough walls have been studied since the early works of Hagen [9] and Darcy [4], which were focused on the turbulent regimes. Reynolds [11] was the first to pose the problem in the context of laminar-turbulent transition. While the issue of characterization of the effects of distributed surface roughness and the question when a rough wall can be considered as being hydraulically smooth are of fundamental importance, their rational resolution is still lacking. In recent years, the effects of wall roughness on the stability of flow have been studied both theoretAyumu Inasawa Tokyo Metropolitan University, 6-6 Asahigaoka, Hino, Tokyo 191-0065, Japan, e-mail: [email protected] Masahito Asai Tokyo Metropolitan University, 6-6 Asahigaoka, Hino, Tokyo 191-0065, Japan, e-mail: [email protected] Jerzy M. Floryan The University of Western Ontario, London, Ontario, N6A 5B9 Canada, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_85, © Springer Science+Business Media B.V. 2010

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Fig. 1 Schematics of flow domain.

ically and experimentally by its geometry being represented in terms of Fourier expansions [6]. Floryan [7] considered two-dimensional TS waves and determined critical stability conditions. The experimental verification is made in [2]. Threedimensional analysis [8] showed that the critical stability mode could have the form of either a two-dimensional TS wave or a streamwise vortex depending on the roughness amplitude. The roughness models used above studies were periodic in the streamwise direction and thus were unable to assess the role played by the origin of the roughness patch. A typical wall, on the other hand, may consist of a smooth segment followed by a rough segment (roughness patch), and the form of transition from the smooth wall to the rough wall may play a role in the evolution of disturbances [3]. In the present study, to better understand a spatial response of the two-dimensional TS wave to change in the wall geometry, the stability of flow in a channel with sudden expansion/contraction is investigated by theory and direct numerical simulations.

2 Numerical procedure We consider two-dimensional incompressible flow in a channel with a sudden expansion/contraction, as sketched in Fig. 1. The coordinate systems based on the flow fields upstream and downstream of step are denoted by (x, y) and (X,Y ), respectively. The Navier-Stokes and continuity equations are solved using finite difference method in combination with the Simplified Marker and Cell (SMAC) algorithm on a staggered grid system. Chebyshev SOR is used to solve the Poisson equation for the pressure. Fourth-order, fully conservative scheme that preserves local energy for the discretized quantities [10] is adopted for the spatial discretization. Third-order Adams-Bashforth method is employed for the time advancement. As for the boundary conditions, a parabolic velocity profile of the plane Poiseuille flow is given at the channel inlet and thus the mass flow rate is preserved in the whole flow field. Under this constraint, the Reynolds numbers are the same far upstream and far downstream of the step, i.e. Re = huUc,u /ν = hd Uc,d /ν where ν is kinematic viscosity. In order to introduce disturbance wave at the upstream boundary, the Orr-Sommerfeld mode is superposed on the base flow with its amplitude being 0.001Uc,u . Non-reflecting boundary condition is applied to the outgoing TS wave at the outlet boundary [5]. Non-uniform Cartesian mesh of 2306 × 210 grid points is used in the direct simulations. In this study, the step height s for expansion/contraction is chosen to be

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503

s = ±0.006hu , ±0.01hu and ±0.02hu . Here positive and negative values correspond to channel expansion and contraction. The Reynolds number is fixed at Re = 5000 and dimensionless frequency of TS mode is ωu = 0.27.

3 Results and Discussion First, we demonstrate the development of base flow downstream of the step by considering stationary eigenmodes for the plane Poiseuille flow [12] as ∞

U(X ,Y ) = 1 −Y 2 + ∑ U (n) (Y )exp(β (n) X/Re).

(1)

n=1

Figure 2(b) compares Y -distribution of the perturbation velocity (deviation from the Poiseuille flow) ∆ U at X = 96.9 with that predicted by the theory. Here, first-three symmetric eigenmodes whose decay rates are β (2) = −28.221, β (4) = −86.284 and β (6) = −175.94 are used to represent the perturbation velocity (Fig. 2a). Streamwise variation of the perturbation velocity at Y =0 is also compared in Fig. 2(c). The agreement between DNS and theory is fairly good indicating that evolution of base flow downstream of the step is well represented by these slowly-decaying stationary eigenmodes, similar to that for the channel entrance flow [1]. Linear stability analyses for the base flow described by Eq. (1) are implemented at each X station and compared with DNS. Parallel flow assumption is applied in the analysis since the stationary modes decay slowly in X (see Fig. 2c). Figures 3 (a) and (b) show streamwise variations of the growth rate and wave number of TS wave, respectively. The linear stability analyses are able to reproduce wave behaviors produced by DNS at streamwise locations beyond X ≈ 50. In addition, the growth rate and wave number have not yet attained asymptotic values inside the computational domain. The X locations where growth rate reaches 95% of that for the plane Poiseuille flow, denoted by Xc , are plotted against magnitude of step height |s| in Fig. 3(c), showing that Xc depends on |s| both for expansion and contraction. The result also indicates that the influence of small change in position of the wall on the flow stability persists surprisingly far downstream. Acknowledgements This work was in part supported by the Grant-in-Aid for Scientific Research (No. 21560820) from Japan Society for the Promotion of Science (JSPS). The first author would like to acknowledge support of FY 2008 Researcher Exchange Program between JSPS and the NSERC of Canada. The SHARCNET of Canada provided the computing resources.

References 1. Asai, M. and Floryan, J.M.: Certain aspects of channel entrance flow, Phys. Fluids, 16-4, 1160–1163 (2004).

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Fig. 2 First-three symmetric stationary modes in (a). The Y -distribution of perturbation velocity at X = 96.9 and its streamwise decay are compared in (b) and (c) (s = 0.02hu ).

Fig. 3 Streamwise variation of growth rate in (a) and wave number in (b). Symbols and lines are DNS and theory, respectively. Distance from the step to the location where growth rate reaches 95% of that for the plane Poiseuille flow is plotted in (c).

2. Asai, M. and Floryan J.M.: Experiments on the Linear Instability of Flow in a Wavy Channel, Eur. J. Mech./B Fluids, 25, 971–986 (2006). 3. Corke T.C., Sever A.B. and Morkovin, M.V.: Experiments on transition enhancements by distributed roughnessh. Phys. Fluids, 29, 3199–3213 (1986). 4. Darcy H.: Recherches expérimentales relatives au mouvement de l’eau dans les tuyaux, Paris, Mallet-Bachelier, (1857). 5. Fasel, H.: Investigation of the Stability of Boundary Layers by a Finite-Difference Model of the Navier-Stokes Equations, J. Fluid Mech., 78, 355–383 (1976). 6. Floryan, J.M.: Stability of wall-bounded shear layers in the presence of simulated distributed roughness, J. Fluid Mech., 335, 29–55, (1997). 7. Floryan J.M.: Two-dimensional Instability of Flow in a Rough Channel, Phys. Fluids, 17, 044101/8 (2005). 8. Floryan, J.M.: Three-Dimensional Instabilities of Laminar Flow in a Rough Channel and the Concept of Hydraulically Smooth Wall, Eur. J. Mech. B/Fluids, 26, 305–329 (2007). 9. Hagen, G.: Uber den Einfluss der Temperatur auf die Bewegung des Wasser in Röhren, Math. Abh. Akad. Wiss., Berlin, 17–98 (1854). 10. Morinishi, Y., Lund, T.S., Vasilyev, O.V. and Moin, P.: Fully Conservative Higher Order Finite Difference Schemes for Incompressible Flow, J. Comp. Phys., 143, 90–124 (1998). 11. Reynolds, O., An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the wall of resistance in parallel channels, Philos. Trans. R. Soc. London, 174, 935–82, (1883). 12. Sadri, R. and Floryan, J.M.: Entry Flow in a Channel, Comp. & Fluids, 31, 133–157 (2002).

Some properties of boundary layer under the joint effect of external flow turbulence and surface roughness Pavel Jonásˇ, Oton Mazur, and Václav Uruba

Abstract Experimental investigation of the boundary layer development on the rough plate (80-grits sandpaper) under external turbulent flows with a grid turbulence of various turbulence scales proved existence of the laminar/transitional region, regardless the surface roughness. Both the outset and the length of the transitional region are affected by the external flow turbulence.

1 Introduction The effect of either of the two, the surface roughness and the free stream turbulence, on laminar turbulent boundary layer transition is known for a long time. In general, they both accelerate the transition process under otherwise equal conditions. Experimental investigations of the effects in question are beneficial even if they are acting separately however their joint action occurs in many engineering applications and still has not been explored enough. The effect of surface roughness on the boundary layer is determined by the ratio r s ν τw ∂U )w s+ = ; δν = ; uτ = ; τw = µ ( (1) δν uτ ρ ∂y where s is the representative length (height) of the roughness elements and δν indicates the viscous length scale, µ ,ν denote molecular viscosity, kinematic viscosity

Pavel Jonásˇ Institute of Thermomechanics v.v.i., Academy of Sciences of the Czech Republic, Dolejˇskova 5, Prague 8, e-mail: [email protected] Oton Mazur, Václav Uruba both: Institute of Thermomechanics v.v.i., Academy of Sciences of the Czech Republic, Dolejˇskova 5, Prague 8, e-mail: [email protected], e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_86, © Springer Science+Business Media B.V. 2010

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and density of fluid, τw (x) and uτ are the local wall shear stress and the friction velocity. The effect of roughness is important in decreasing the viscous length scale δν and in a vertical shift of the mean velocity profile from the level of the roughness spikes, y′ = 0, inside the layer of roughness grains, on the level y′ = −y′0 ≤ 0. A deeper analysis is given in e.g. [1] ÷[5]. The effect of the free stream turbulence (FST) level Tu on the location of transition onset is known as very important and it has been investigated permanently since the 40th. Later, the effect of the turbulence length scale was also proved [6]. The intensity Iue of the longitudinal velocity fluctuations and the dissipation length parameter Le in the leading edge plane (x = 0) characterize the FST in this paper Iue =

q

(u2 )e /U e ;

3/2

Le = −(u2 )e /U e

∂ (u2 ) . ∂x

(2)

2 Experiment and results The flat plate boundary layer (grad Pe = 0) was investigated experimentally on the plate (2.75 m long and 0.9 m wide) covered with the 80-grit sandpaper (K-type roughness) in the close circuit wind tunnel (0.5 x 0.9) m2 . The rough plate leading edge has an elliptic shape (a x b = 60 x 20 mm2 ). The maximum size of grains was employed as the representative length (height) of the roughness elements s = 0.343 mm ± 0.009 mm. The features of FST were controlled by means of square mesh plane grids screens chosen from the family of grid generators, developed in the IT AS CR [7]. Six different FST were generated. The values of Iue and Le are shown in the Table 1 together with the relevant symbols used in diagrams. More details on experimental set up are in [5], [6]. The mean velocity profiles were measured by means of the flattened Pitot probe (0.18 x 2.95 mm2 ) and round nosed static pressure probe (diameter = 0.18 mm). The MacMillan’s correction, e.g. [8] of the total pressure readings was applied. Presented results relates to the mean flow velocity 5.2 m/s (± 3%). Table 1 Characteristics of FST in the leading edge plane and the relevant signs used in diagrams Grid No

GT 0

GT 8A

GT 8B

GT 1

GT 4

GT 5

GT 8C

Iue [1] ILe [mm]

0.003 ∼ 10

0.01 5.7

0.03 3.0

0.03 7.0

0.03 16.2

0.03 30.5

0.05 2.2

symbol

triangle

circle

diamond square

Solid triangle Solid circle

Solid square

Analyzing the mean velocity profiles, they were evaluated as functions of the displacement thickness Reynolds number Re1 : the following distributions of the

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shape factor H12 (Figure 1), the skin-friction coefficient C f (Figure 2), the surface roughness parameter s+ (Figure 3) and the origin of the coordinate y, where the mean velocity became zero and the roughness function ∆ u+ describing the shift of the universal velocity distribution in the inertial sub-layer below the course in case of turbulent boundary layer on a smooth surface. The applied nomenclature is in general usage so it is not necessary give the definitions.

Fig. 1 Distributions of skin friction coefficient. Comparison with Blasius solution (dashed line) and with Ludwieg & Tillmann formulae (dotted line) (meaning of the symbols see Table 1)

Fig. 2 Distributions of shape factor (meaning of the symbols see Table 1)

The evaluated distributions of the surface roughness parameter s+ are shown in Figure 3. The values of s+ stay near the lower bound of the transitional roughness region and they are continuously decreasing with the increasing distance x from the leading edge after completing the transition.

3 Conclusions The surface roughness affects the boundary layer development more than the external flow turbulence but the effect of turbulence is not negligible. The increase of

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Fig. 3 Distribution of surface roughness parameter (meaning of the symbols see Table 1)

external turbulence intensity creates shortening of both the quasi-laminar region and the transitional region. Further, it has been ascertained: the investigated case belongs to transitional roughness; the velocity zero level (U = 0) is of (0.09 ± 0.03) mm below the top plane of the highest roughness elements (0.34 0.01) mm; the roughness function ∆ u+ , derived from the log-law interpolation moderately decreases with increasing Reynolds number from about 2.3 at Re1 = 1000 up to about 1.3 at Re1 = 2000 and it mildly increases with the external flow intensity. Investigations of transitional intermittency distributions and flow over plate covered with 40-grits and 120-grits sandpaper respectively are in progress. Acknowledgements This work is carried out under the project no. A200760614 supported by the Grant Agency of the Academy of Sciences of the Czech Republic. The support is greatly acknowledged.

References 1. Rotta, J. C. (1962) Turbulent boundary layer in incompressible flow, in: Progress in Aeronautical Sciences (A. Ferri, D. Kchemann & L. H. G. Sterne eds.) Pergamon Press, Oxford, pp. 1-220. 2. Rotta, J.C. (1972) Turbulente Strmungen. B.G. Teubner, Stuttgart. 3. Pope, S. B. (2000) Turbulent flows. Cambridge University Press, Cambridge. 4. Schlichting, H. & Gersten, K. (2000) Boundary-Layer Theory. Springer, Berlin. 5. Jonásˇ, P., Mazur, O. and Uruba, V. (2008) Preliminary study on the effect of the wall roughness and free stream turbulence on the boundary layer development. Proc. Conf. Engineering Mechanics 2008, Institute of Thermomechanics, Academy of Sciences of the Czech Republic, v.v.i., Prague, ISBN 978-80-87012-11-6, CD ROM, 1-13. 6. Jonásˇ, P., Mazur, O. and Uruba, V. (2000) On the receptivity of the by-pass transition to the length scale of the outer stream turbulence. Eur. J. Mech. B - Fluids 19, 707-722. 7. Jon, P. (1989) Control of free stream turbulence by means of passive devices. In: Proc. of International Seminar ”Problems of simulation in wind tunnels”. Pt.2 ITPM SO AN SSSR, Novosibirsk, 160-174. 8. Tropea, C., Yarin, A. L. & Foss, J. F. (Eds.) (2007) Springer Handbook of Experimental Fluid mechanics. Springer-Verlag Berlin.

Wave forerunners of localized structures at the boundary layer M. Katasonov, V. Gorev, and V. V. Kozlov

1 Introduction Under the conditions of high free stream turbulence level, the laminar-turbulent transition is due to the action of disturbances proceeding from the external flow on the boundary layer; as a result, in the boundary layer, streamwise-oriented structures consisting of regions with an excess and deficit of the longitudinal velocity are formed. These structures provide preconditions for the development of highfrequency wave disturbances such as secondary instability and forerunners, which can further transform into turbulent spots. As a result, the boundary layer flow goes over from the laminar to the turbulent state [1, 2]. At present day, investigations performed under ”natural” conditions do not provide an exhaustive answer to the questions posed, because, in boundary layers, disturbances arise at random and it is hardly possible to trace the behavior of a particular disturbance. In the studies aimed at the comprehensive investigation of localized structures, the structures are artificially modeled. This work is devoted to the study of wave packets - forerunners formed in boundary layers in the regions that precede a drastic change in the flow velocity inside the boundary layer (fronts of localized disturbance) [3].

2 Experimental set-up The investigations were carried out in the subsonic wind tunnel MT-324, ITAM SB RAS with the test section of 200 × 200 × 700 mm size. The velocity of the oncoming flow was U∞ = 5 m/s. The high level of free stream turbulence was generated from grid located at the beginning of the wind tunnel test section. The tested models M. Katasonov Khristianovich Institute [email protected]

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RAS,


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were straight and swept wing airfoils with chords C1 = 290 mm and C2 = 410 mm respectively. The models were set at zero angle of attack at the middle of the test section. The blowing-suction method was used to introduce disturbances into the boundary layer through the 40 mm width slot on the model surface. The duration of the blowing (suction) was controlled by a high-speed electromagnetic valve synchronized with the system of signal recording. Measurements were carried out using a constant-temperature hot-wire anemometer with a single-wire probe. Both the average U and fluctuation u components of the longitudinal velocity were measured.

3 Results As shown in [3], the forerunners observed in the boundary layer of the straight wing are Tollmien - Schlichting (T-S) wave packets. Actually, the forerunner is excited by the impact of a rectangular pulse (time-dependent blowing or suction) on the boundary layer. As a result of dispersion, the rectangular pulse front spreads generating a broad spectrum, the most unstable ones oscillations being enhanced by the boundary layer. The appearance of forerunner is highly related with local velocity gradient at the front of localized structure. If the velocity gradient △u/△t is ”high”, one can see the appearance the forerunner, which is grow downstream, Fig. 1 (a). In opposite, with the ”low” △u/△t near the front of the localized structure the forerunner is not appear, Fig. 1 (b). In other words, wave forerunner is accompanying the process of drastic local change of velocity in the boundary layer. ∆u/∆t1

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The experiments shown, that the localized structures have different effect upon the boundary layer depending on the way of their generation. Once they are excited by blowing, the filling of velocity profile is reduced that makes the flow receptive to external perturbations. Thus, more intensive should be development of the wave packet at the back front of the structure. In contrast, at boundary-layer suction the flow receptivity becomes lower so that the wave packet (forerunner) at the leading

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front dominates. It was found that, developing downstream in the boundary layer, the forerunners transform into - structures and lead further downstream to turbulence through the formation of turbulent spots [3]. It was observed that the forerunners are strongly amplified in the adverse pressure gradient flow being much influenced by local velocity gradients. The curves of increase of the disturbances magnitude in a boundary layer along a chord of straight wing under the low and high free stream turbulence level had been investigated. In Fig. 2 (a) an distributions of speed fluctuations for a case of the ”natural” disturbances, appearing in a boundary layer at a low and high free stream turbulence level are presented (in this case any artificial generated structures were absent). Distributions were represented at a maximum level of disturbances (or wave packets) at Y axis inside the boundary layer. From these curves it is possible to see, that under influence of high free stream turbulence level, the beginning of laminar-turbulent transition is delayed, i.e. shifted to downstream direction. It is explaining on the basis of, as has been shown earlier [3], at the high level of free stream turbulence the wave packets can exist and intensively grow, resulting to the laminar-turbulent transition. The high level of free stream turbulence influences on a boundary layer, creating in it the streaky structures which suppress growth of wave packets at the initial stage of their development, and result is delay of laminar-turbulent transition. To check this point, in the current experiments an artificial Tollmien - Schlichting wave was introduced in to the boundary layer and its propagation downstream was analyzed in cases of low and high level of free stream turbulence. As were expected, Fig. 2 (b), at the linear stage of development (0.3 < X /C1 < 0.5) occurs a stabilization of T-S wave, that also were observed in work [4]. In Fig. 2 (c) and (d) the distributions of velocity fluctuations along the wing chord for the forerunners are presented. The analysis of distribution of velocity fluctuations in a boundary layer shows, that in case of a low-amplitude forerunner (∼ 0.4U∞ near the source), Fig. 2 (c), at the initial stage of development in the area of a favorable pressure gradient (X/C1 < 0.5) the amplitude of fluctuations is being slightly decreased, but, in the field of an adverse pressure gradient - intensively

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grows. The forerunners magnitude at the high free stream turbulence level is grow faster than at low free stream turbulence. With the larger initial magnitude of the wave forerunner the beginning of the laminar-turbulent transition becomes earlier. The behavior of the magnitude of swept wing forerunners downstream the boundary layer flow under the high free-stream turbulence level (Tu = 2.31U∞ ) is similar with the straight wing case. The maximum of distribution of the forerunner magnitude downstream the flow, shifts toward the wing profile leading edge under the influence of high free-stream turbulence level. The amplitude of the forerunner near the trailing front of localized disturbance decaying downstream under the low free stream turbulence level, but grow under the high free stream turbulence level in area of adverse pressure gradient.

4 Conclusions The results obtained in the present study has been shown, that wave packets - forerunners of fronts of the localized disturbances (streaks) can exist and result of turbulence formation in gradient flow under the high free stream turbulence level. It is found, that for a straight wing the forerunner amplitude growth under the high free stream turbulence level begins earlier, than at low free stream turbulence level. With the larger initial magnitude of the wave forerunner the beginning of the laminarturbulent transition becomes earlier. At the swept-wing boundary layer, the forerunners and the localized structures become asymmetric, but the behavior of the magnitude of swept wing forerunners downstream the boundary layer flow under the high free-stream turbulence level is similar with the straight wing case. Acknowledgements The work was supported by Russian Foundation for Basic Research (grant No. 08-01-00027), the Council of the President of the Russian Federation for the Support of Young Russian Scientists and Leading Scientific Schools (NSh-454.2008.1, MK-101.2007.1), and the Ministry of Education and Science of the Russian Federation (project RNP .2.1.2.541)

References 1. A. V. Boiko, A. V. Dovgal, G. R. Grek, and V. V. Kozlov. Origin of turbulence in near-wall flows. Springer–Verlag, Berlin, 2002. 2. V. V. Kozlov. in Proceedings of the EUROMECH Colloquium 353: Dynamics of Localized Disturbances in Engineering Flows, Karlsruhe, 15–16, 1996. 3. V. N. Gorev, M. M. Katasonov. Origination and development of precursors on the fronts of streaky structures in the boundary layer on a nonswept wing. Thermophys. Aeromech., 11(3):391–403, 2004. 4. A. V. Boiko, K.J. A. Westin, B. G. B. Klingmann, V. V. Kozlov, P. H. Alfredsson. Experiments in boundary layer subjected to free stream turbulence Part 2: The role of TS-waves in the transition process. J. Fluid Mech., 281:219–245, 1994.

Experiments on the wave train excitation and wave interaction in spanwise modulated supersonic boundary layer A. D. Kosinov, N. V. Semionov, and Yu. G. Yermolaev

Abstract The experimental investigation of wave train development in the spanwise modulated flow of the supersonic flat plate boundary layer are conducted in comparison with smooth plate. It was found that the spanwise flow modulation can increase the receptivity features of supersonic boundary layer to the surface excited controlled pulsations.

1 Introduction Usage of the controlled disturbances technique for the research of the nonlinear interaction mechanisms of unstable waves in a supersonic boundary layer still is a preferable method in contrast to the natural disturbance investigation. However, controlled disturbances can interact with random natural oscillations in nonlinear region of the supersonic boundary layer and to trigger the earlier transition. So, the experiments become more complicated. Nevertheless, we were able to detect by this method the following nonlinear mechanisms: (1) the parametric subharmonic amplification of strongly inclined waves [1]; (2) the abnormal growth of quasi two dimensional disturbances of large amplitude [2]. There is also third wave interaction mechanism named as oblique breakdown predicted by DNS [3, 4] which was also detected in our experiments [3, 4]. It is appear in the interaction of oblique stationary A.D. Kosinov Khristianovitch Institute of Theoretical and Applied Mechanics, Novosibirsk, Institutskaya 4/1, Russia, e-mail: [email protected] N.V. Semionov Khristianovitch Institute of Theoretical and Applied Mechanics, Novosibirsk, Institutskaya 4/1, Russia, e-mail: [email protected] Yu.G. Yermolaev Khristianovitch Institute of Theoretical and Applied Mechanics, Novosibirsk, Institutskaya 4/1, Russia, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_88, © Springer Science+Business Media B.V. 2010

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wave and inclined fundamental waves. Perhaps this mechanism will play an important role in laminar – turbulent transition at the presence of a periodic spanwise modulation of flow.

2 Experiments set-up The experiments were conducted in T–325 low noise supersonic wind tunnel of ITAM SB RAS on a flat plate with the sharp leading edge at Mach number 2 and unit Reynolds number Re1 = (6.73 ± 0.04) × 106 m−1 . The rectangular stickers from a scotch tape about 60 microns in thickness, approximately 1 mm in width and 5 mm in length were applied to induce the spanwise modulation of mean flow in the boundary layer. Twenty stickers were spanwise located on a distance of 25 mm from the leading edge and before an aperture of the source (38 mm) with step about 4 mm. It is shown in Figure 1 (a). The experiments were executed for the single model installation (and fixed source intensity) in three phases. First phase was the initial amplitude choice for a smooth surface (see Figure 1 (b)). Next step was the measurements in the artificially modulated boundary layer (Figure 1 (a)) and then we repeated the measurements for a smooth plate. The controlled disturbances development was measured by CTA. The tungsten hot-wire of 10 micron in diameter and of 1.5 mm in length was used. AC and DC signals from CTA were written to the PC by 12–bit ADC with sampling rate 750 kHz and by DC voltmeter. The measurements were synchronized with glow discharge which was ignited with fundamental frequency of 20 kHz. 256 time traces of 1024 points in length were measured and averaged in each space position. The spanwise measurements were made at the fixed normal distance from the model surface and at y/δ ≈ const for each X position. Fig. 1 Flat plate with periodical roughness (a) and without it (b)

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the flow occurred for the data at X=120 mm. As Re1 was approximately constant and the stagnation temperature decreased, the mass flow rate also reduced. The reason of stronger distortion of the mean flow in center of the distribution probably appeared because of the influence of the source hole in the model surface.

3.1 Initial Amplitudes of Wave Trains Figure 3 presents the initial mass flow pulsation distributions at X=60 mm for different frequencies. We introduce mainly the fundamental disturbances for smooth flat plate and wide spectrum of pulsations is generated for artificially modulated boundary layer. Note that the electrical power of the glow discharge was the same in both cases. So this change for the excited pulsations may be explained also by changes in the receptivity as the measurements were conducted in near field from the source. The result was unexpected and only subsequent measurements without roughness have confirmed that the disturbance source had earlier selected initial intensity

3.2 Nonlinear Wave Train Development We have observed the subharmonic resonance at X=120 mm for the flat plate boundary layer on the smooth surface that was obtained before [1]. The spanwise flow modulation changes the nonlinear development of wave trains. 3D subharmonic resonance still takes place, but 2D waves grow very rapidly. The development of pulsations looks similar to the anomalous nonlinear propagation of high amplitude wave trains. If the oblique breakdown mechanism takes place, we would observe a

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fast growth of high β part of the amplitude spectra for fundamental or subharmonic frequency. We found out such a behavior for the results presented in figure 4. The effect is not so big in comparison to subharmonic resonance or 2D wave growth. Fig. 4 Relative growth factors for fundamental and subharmonic disturbances in modulated boundary layer

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4 Conclusions The results give a direction for the further investigations. We should change the experiment setup in order to test other roughness locations as well as the size and shape. It was found that the spanwise flow modulation can lead to increasing the receptivity features of supersonic boundary layer to the controlled pulsations exciting from the surface. At least three nonlinear mechanisms of breakdown appeared. Acknowledgements This activity is executed at support of RFBR grant 09-01-00767.

References 1. Kosinov A.D., Semionov N.V., Shevel’kov S.G., Zinin O.I. Experiments on the nonlinear instability of supersonic boundary layers. In: S.P.Lin et.al.(eds) Nonlinear Instability of Nonparallel Flows. (Springer–Verlag, 1994) p 196–205 2. Kosinov A.D., Ermolaev Yu.G., Semionov N.V. Experimental study of anomalous wave processes in supersonic boundary layer. (Proceedings of 9 International Conference on the Methods of Aerophysical Research, Novosibirsk, 1998), part 2, p.106–111 3. Mayer C. S. J, Wernz S., Fasel H. F. Investigation of oblique breakdown in a supersonic boundary layer at Mach 2 using DNS. AIAA Paper 2007–0949, (2007) 4. Mayer C. S. J, Fasel H. F. Investigation of asymmetric subharmonic resonance in a supersonic boundary layer at Mach 2 using DNS. AIAA Paper 2008–0591, (2008)

Laminar-Turbulent Transition and Boundary Layer Separation on wavy surface wing Viktor Kozlov, Ilya Zverkov, and Boris Zanin

Abstract Boundary layer parameters on wavy-surface wings (below wavy wing) at the angle of attack α = 0◦ and the chord Reynolds number Re = 1.7 •105 are studied experimentally. Substantial differences are found in the transition position and in the transition scenarios realized on the wavy-surface wing along its streamwise grooves and humps.

1 Introduction Nowadays, micro air vehicles (MAV) find new military and civil applications. Miniaturization of the electronic equipment promotes the creation of progressively smaller unmanned aircraft. At the moment, many researchers pay much attention to MAVs with an overall mass below 0.5 kg and a chord-based Reynolds number between 104 and 105 . In spite of certain success, the wide application of MAVs is limited by their rather modest aerodynamic performances. Important MAV parameters, such as the critical angle of attack and lift-to-drag ratio, are mainly deteriorated by changes in the boundary-layer flow character at low Reynolds numbers. Wing surface modification is one of the major resources of separation control and aerodynamic performance improvement. The one of most aspects is to consider the effect of spanwise-periodic modifications to the wing surface on the laminar boundaryViktor Kozlov Khristianovich ITAM SB RAS, 630090, Institutskaya 4/1, Novosibirsk, RUSSIA, e-mail: [email protected] Ilya Zverkov Khristianovich ITAM SB RAS, 630090, Institutskaya 4/1, Novosibirsk, RUSSIA, e-mail: [email protected] Boris Zanin Khristianovich ITAM SB RAS, 630090, Institutskaya 4/1, Novosibirsk, RUSSIA, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_89, © Springer Science+Business Media B.V. 2010

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layer separation. As in our previous work was demonstrated [1], the stall angle of attack wavy wing (below wavy wing) is 1.5 times that of classical surface wing and turbulent boundary layer separation was not observed. The present paper describes a further detailed investigation of the boundary layer on the wavy wings.

2 Experimental Apparatus All experiments were performed in a T-324 subsonic wind tunnel based at the Khristianovich Institute of Theoretical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences. The contraction ratio of the convergent channel is 16:1. The turbulence level measured by a single I-type hot-wire anemometer sensor is lower than 0.04 percent in the frequency range of 1 - 5,000 Hz. The free-stream velocity is U∞ = 12 m/s, and the chord-based Reynolds number is Re = 1.7 • 105 . The fiberglass wing models with a 195-mm chord and a 200-mm span were used in the experiments. The airfoil TsAGI R-3a-12 was the same both along the grooves and the humps with the hump height f = 2 mm (0.01 chord), see Fig. 1. The hotwire data were obtained with an AN-1003 constant-temperature anemometer (AA Labs) and an I-type single-wire boundary-layer probe.

Fig. 1 Middle section of the wavy wing.

3 Results The hot-wire measurements along groove and along hump has been performed. The data in Fig. 2 illustrate the rms disturbance growth along the line passing through the disturbance profile maxima. Two important features are worth noting. The first one is the high level of disturbances near the leading edge along the wavy-wing groove, which is later reduced. The second one is insignificant growth of disturbances on the hump, whereas the transition in the groove is very intense and the disturbances start decaying at 0.5 of the chord. Apparently, the disturbances on the hump did not yet reach their maximum at 0.74 of the chord. Together with the distributions

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of the integral parameters, this fact indicates that the transition on the hump is not completed to 0.74 of the chord. The results of the spectral analysis of disturbances along the hump show the development of exponentially growing instability wave packets with the central frequency Fc = 800 Hz.

Fig. 2 Streamwise disturbance growth.

4 Conclusion An essentially different structure of the boundary layer was observed along the groove and the hump of the wavy wing. On the hump, the laminar-turbulent transition begins at 20 percent of the chord downstream, as compared with the groove. The transition scenarios along the groove and hump are also different. On the hump, the transition occurs without flow separation and generation of a pronounced instability wave packet.

References 1. Zverkov I. D. , Zanin B. Yu. (2001). Wing Form Effect on Flow Separation. Thermal physics and Aeromechanics, 10(2), 197–204.

Investigation of Thermal Nonequilibrium on Hypersonic Boundary-Layer Transition by DNS Jens Linn and Markus J. Kloker

Abstract High-temperature gas effects are known as the physical processes which result in a deviation of the behavior of the calorically perfect gas in the hypersonicflow regime. These effects may have a significant impact on laminar-turbulent transition of the boundary layer, and thus affect the heat loads of hypersonic vehicles. In the present paper we deal with thermal effects, i.e. rotational and vibrational energy relaxation. A fundamental-breakdown scenario of a Mach-6.8 flat-plate boundary layer at flight conditions is simulated by high-order DNS using a calorically perfect gas, or thermal equilibrium, or a nonequilibrium model. A similar behavior is found for the calorically perfect gas and the thermal equilibrium case. In contrast, a stabilizing effect is observed in the thermal nonequilibrium case, leading to a fall off of fundamental breakdown.

1 Gas Models The Navier-Stokes Equations are altered in two ways to deal with thermal nonequilibrium (TNEQ), i.e. rotational and vibrational energy relaxation. First, a bulk viscosity or volumetric viscosity µv⋆ is added [3] to approximately account for rotational relaxation. Secondly, a vibrational-energy equation for each species has been added. In this case we deal with two temperatures. The translational temperature is T ⋆ = T ⋆ tra and the vibrational temperature Ti⋆ vib for each species, where the superscript ⋆ denotes dimensional quantities. The definition of the internal energy now reads: e⋆ = e⋆ trans + e⋆ rot + e⋆ vib Ri⋆ · θi⋆ vib 3 = R ⋆ T ⋆ + R ⋆ T ⋆ + ∑ ci · e⋆i vib , with e⋆i vib = , (1) 2 exp(θi⋆ vib /Ti⋆ vib ) − 1 i Institut für Aero- und Gasdynamik, Universität Stuttgart, Pfaffenwaldring 21, D-70550 Stuttgart, GERMANY · e-mail: [email protected]; [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_90, © Springer Science+Business Media B.V. 2010

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where R ⋆ is the specific gas constant, θi⋆ vib the characteristic vibrational temperature of the species, and ci is the mass fraction of species i. The vibrational energy for each species i is modelled by the Landau-Teller equation [3]: h i → − ∂ e⋆i vib 1 ⋆ vib,eq = div ϑi⋆ vib · ∇Ti⋆ vib , (2) + ∇ · e⋆i vib u⋆ + ⋆ vib e⋆i vib − ei ⋆ ∂t τi where e⋆i vib,eq = e⋆i vib,eq (T ) is the vibrational energy in equilibrium and τi⋆ vib is the relaxation time. A detailed description of the rotational and vibrational energy relaxation models can be found in [3].

Calorically perfect gas (CPG) In the case of CPG the specific heat c⋆v is constant with temperature, and vibrational excitation is neglected. Equation (1) simplifies to: 3 5 e⋆ = R ⋆ T ⋆ + R ⋆ T ⋆ = R ⋆ T ⋆ = c⋆v T ⋆ , (3) 2 2 ⋆ vib ⋆ with e = 0; µv can be turned on or off. Thermal equilibrium (TEQ) In the thermal equilibrium case, the relaxation time τi⋆ vib tends to zero, hence the term 1/τi⋆ vib · (e⋆i vib − e⋆i vib,eq ) is dominating eq. (2) and leading the numerical time-step limit of the whole set of equations. Alternatively, the vibrational temperature Ti⋆ vib is set to the translational temperature T ⋆ , with eq. (2) discarded. Test DNS have shown virtually identical results for τi⋆ vib small enough. Thermal nonequilibrium (TNEQ) Equation (2) is solved employing an algebraic equation for the relaxation time τi⋆ vib as given by Anderson [1]. Thus the temperatures can be different, and the bulk viscosity model is applied: ⋆ ⋆ 2 µ ⋆ (T ⋆ ) µv T − 293.3 µv⋆ = λ ⋆ + µ ⋆ , with v⋆ ⋆ = · exp , (4) 3 µ (T ) µ ⋆ T =293.3K 1940 where λ ⋆ is the coefficient of bulk viscosity (Lamé’s constant); µv⋆ scales the di− latation effect, i.e. the term µv⋆ div(→ u ⋆ ) is added to the normal stresses underlying Stokes’ hypothesis.

2 Results The DNS is based on 6th -order compact finite differences in streamwise and wallnormal directions, with the spanwise derivatives computed by a Fourier ansatz [2, 4]. For the investigation of the rotational and vibrational energy relaxation for a fundamental breakdown scenario, we simulate a Mach-6.8 flat-plate boundary layer at atmospheric flight conditions as investigated by Fezer and Kloker [4]. The freestream ⋆ ⋆ temperature is T ⋆ = 220K, and a cold isothermal wall with Twall = 975K ≈ 12 Trec is ⋆ ⋆ vib prescribed. At the wall, in TEQ holds Tw = Ti , and the temperature disturbances are zero, T ′⋆ = Ti′⋆ vib = 0. The global Reynolds number is ReL = 105 and the unit Reynolds number Re⋆unit = 5.71 · 106 1/m, thus L⋆ = 0.0175 m. In figure 1a, the temperature profiles (left) of the baseflow are shown for all three models. At the inflow boundary the case TNEQ is set to case CPG. For the cases TEQ and TNEQ the maximum temperature in the boundary layer is smaller than

Investigation of Thermal Nonequilibrium on Hypersonic Boundary-Layer Transition 0.3 a)

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in case CPG. The heat capacity c⋆v is a function of the temperature for TEQ, increasing with temperature. The deviation from equilibrium in case TNEQ is shown in figure 1b. The strongest nonequilibrium region is at the boundary-layer edge. However, the nonequilibrium energy is only a small fraction of the total energy e (Fig. 1c). The energy in the vibrational motion has its maximum at the wall, but only is approximately 13% of the total energy. Note that the effect of µv⋆ 6= 0 is negligible for the baseflows. The downstream development of the u-velocity amplitude for cases CPG, TEQ, and TNEQ is presented in figures 2 and 3. In a disturbance strip at Rex ≈ 14.1 · 105 (Rx = 1187), an amplified 2-d 2nd -mode disturbance (1,0) is introduced by blowing/suction. (h, k) indicates a wave with frequency h · F and the spanwise wavenumber k · γ . The frequency parameter is F = 2π f ⋆ L⋆ /(u⋆∞ ReL ) = 10−4 ( f ⋆ = 184kHz). The influence of the different gas models and the bulk viscosity on the disturbance evolution is shown in Fig. 2. µv 6= 0 always damps, and case TEQ shows the earliest amplitude decay after strongest growth, whereas TNEQ has the lowest initial growth. For the case shown in figure 3, the amplitude of (1,0) is increased, and a damped oblique 1st -mode disturbance (1,1) is added. The spanwise wavenumber is γ = 11 (γ ⋆ = γ /L⋆ = 628.6/m). The largest amplitude of (1,0) is reached in case CPG, and the originally damped 3-d disturbance (1,1) undergoes resonant amplification at Rex ≈ 21 · 105 . Somewhat further downstream, higher harmonics reach

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high amplitudes as well and the fundamental breakdown begins. In case TNEQ, (1,1) gets in resonance as well, but the amplitude of (1,0) is smaller and hence the growing of the amplitude (1,1) is not as strong as for the other case. As a consequence, the fundamental-breakdown scenario eventually falls off. The scenario in the TEQ case is slightly weaker but similar to the CPG case. 10

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Fig. 3 Downstream development of u′(h,k) (wall-normal max.) in a Mach-6.8 boundary layer for the three gas models (CPG, TEQ: µv⋆ = 0; TNEQ: µv⋆ 6= 0).

3 Conclusions DNS were performed to investigate the influence of different gas models on a fundamental-breakdown scenario at Mach 6.8 at flight conditions. Three different gas models were used namely calorically perfect gas (CPG), thermal equilibrium (TEQ), and thermal nonequilibrium (TNEQ). The baseflow shows only slight differences for the translational temperature profiles. The CPG and TEQ cases show approximately the same behavior for the fundamental-breakdown case, with lower amplitudes for case TEQ. For case TNEQ the growth rates of the disturbances are lower and breakdown does not occur. Albeit the influence of the gas models on the baseflow is small, their impact on the disturbance evolution can be significant.

References 1. Anderson, J.D. (1989) Hypersonic and High Temperature Gas Dynamics, MacGraw-Hill 2. Babucke, A., Linn, J., Kloker, M. and Rist, U. (2006). Direct numerical simulation of shear flow phenomena on parallel vector computers In: M. Resch, et al. (Eds.), High Performance Computing on Vector Systems 2005, pp. 229–247. Springer. 3. Bertolotti, F.B. (1998) The influence of rotational and vibrational energy relaxation on boundary-layer stability. J. Fluid Mech., vol. 372, pp. 93-118. 4. Fezer, A., Kloker, M.J. (1999) Transition process in Mach 6.8 boundary layers at varying temperature conditions investigated by spatial direct numerical simulation. In: New Results in Numerical and Experimental Fluid Mechanics II., W. Nitsche et al. (eds.) NNFM, vol. 72, Vieweg, pp. 138-145.

Global sustained perturbations in a backward-facing step flow Olivier Marquet and Denis Sipp

Abstract The two-dimensional stability of a backward-facing step flow is investigated at the low Reynolds number Re = 500. A method is proposed to determine the optimal energy growth of linear perturbations sustained by harmonic forcing. The maximum sustained energy growth is of order 105 at frequency fS ∼ 0.08 and is related to convective instabilities growing in the shear layer of the separated flow.

1 Introduction The instability mechanisms at the origin of the transition to turbulence in separated flows are only partially understood even for low Reynolds number. Whether the transition onset is due to intrinsic or extrinsic dynamics remains an open question. Global stability theory, for which two or three inhomogeneous directions are considered, is a promising approach to investigate the stability of non-parallel flows. The classical global stability analysis, based on a temporal modal decomposition of perturbations, is convenient to describe the intrinsic dynamics. However, the extrinsic dynamics, defined as the ability of the spatially evolving flow to strongly amplify external perturbations, cannot be captured by a single global mode. For instance the energetic transient amplification of initial perturbations results from the cooperation of a finite-number of global modes [2] which are non-orthogonal because of the nonnormality of the global linearized Navier-Stokes operator. A different approach, not relying on the computation of global modes, but based on the temporal response of the global linearized Navier-Stokes equations to initial perturbations has recently been proposed and performed on a rounded backward-facing step flow [3] and on the present backward-facing step flow [1]. In the latter study large optimal transient Olivier Marquet ONERA-DAFE, 92190 Meudon, France, e-mail: [email protected] Denis Sipp ONERA-DAFE, 92190 Meudon, France, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_91, © Springer Science+Business Media B.V. 2010

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Fig. 1 Flow configuration and base flow at Re = 500.

energy growths have been computed and linked to the existence of a strong local convective instability. In the present study, the frequency response of this flow is investigated. Linear perturbations are here sustained by external harmonic forcing. A method is proposed to determine the optimal sustained energy growth by solving an eigenvalue problem.

2 Global sustained perturbations The linear stability of the backward-facing step flow

∂t u′ + U · ∇u′ + u′ · ∇U + ∇p′ − Re−1 ∇2 u′ = f′ , ∇ · u′ = 0.

(1)

where f′ represents a forcing term sustaining the flow perturbations (u′ , p′ ) which develop on the base flow. This forcing term is looked for as an harmonic forcing f′ (x, y,t) = 1/2 { f(x, y)eiω t + c.c. } characterized by a real forcing pulsation ω and a complex spatial structure f = ( f , g)T . A similar decomposition is used for the flow perturbations, (u′ , p′ )(x, y,t) = 1/2 { (u, p)(x, y)eiω t + c.c. } where (u, p) is the spatial structure of the perturbation sustained by the forcing f. These decompositions are introduced into (1). The relation between the forcing and the velocity perturbation is governed by the equation u = R f , R = P T iω PP T − L

−1

P

(2)

where R is the resolvant operator, P the prolongation operator and L the linearized Navier-Stokes operator. These two last operators are defined by I2,2 −U · ∇ − () · ∇U + Re−1 ∇2 −∇ P= ,L = (3) 01,2 ∇ · () 0 with ∇ = ( ∂x ∂y )T . By discretizing the continuous problem (2) on a finite-element basis, one obtains u = R f , R = PT S−1 P Q , S = iω PQPT − L

(4)

where the velocity (resp. forcing) field u (resp. f) is now a vector containing all the degrees of freedom of the two components of velocity (resp. forcing). R, P and L are the discrete version of the continuous operator defined above and Q designate the mass-matrix.

Global sustained perturbations in a backward-facing step flow

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Fig. 2 (a) Optimal energetic gain λ12 and (b) sub-optimal as a function of the frequency fS

Optimal sustained growth and eigenvalue problem - The kinetic energy of the perturbation uH Qu is chosen to measure the growth of the perturbation. The symbol H denotes here the discrete conjugate transpose operation. The energetic gain of the perturbation u sustained by the forcing f is given by G(ω , f) =

uH Qu (Rf)H QRf = fH Qf fH Qf

(5)

where the last part is obtained by considering the relations (4). Introducing RA , the adjoint operator of the resolvant (with respect to the discrete energy inner product) H A which satisfies uH Qu2 for any vectors u1 and u2 , the energetic 1 QRu2 = R u1 gain can be expressed as a Rayleigh quotient H −1 RA R f Qf G(ω , f) = , RA = PT SH PQ fH Qf

(6)

ˆ RA R ˆfi = λi2 ˆfi , ˆfH i Q fi = 1

(7)

We consider the eigenvalue problem

The operator RA R being symmetric, the eigenvalues are positive (and denoted λi2 ) and the eigenvectors ˆfi are orthogonal with respect to the energy inner product. For a fixed pulsation ω , the maximal growth is given by the largest eigenvalue λ12 = maxf G(ω ) and the optimal forcing corresponds to the associated eigenvector ˆf1 . The optimal sustained perturbation is defined as uˆ 1 = R ˆf1 /λ1 . Numerical method - The largest eigenvalues of the problem (7) are determined by using the routine znaupd of the ARPACK’s library in regular mode. It requires to compute y = RA Rx for some given vector x. This is done in two steps: first solve v = Rx and then solve y = RA v. Each of these steps requires to solve a linear problem. This is achieved by first forming explicitly the appropriate sparse matrices, and then inverting them through a direct LU solver (MUMPS package). In terms of storage, only the matrix S has to be LU decomposed since this matrix is involved in both inverse. Hence, the cost of this algorithm is approximately given by the cost of the LU decomposition of a large sparse complex matrix. In terms of time, it is very efficient, since after the LU decomposition is achieved the successive inverses are cheap.

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(a) (b)

Fig. 3 Spatial structure of (a) the forcing and (b) the perturbation associated to the optimal energetic gain λ12 at frequency fS = 0.075 (real part of the vertical component).

3 Results Figure 2 depicts the optimal energetic gain λ12 as a function of the frequency. It shows a sharp peak centered around the frequency fS = 0.075 where the largest optimal gain is equal to λ12 = 5.6 107 . This frequency is the optimal frequency. At the optimal frequency, the sub-optimal gains (not shown here) are four orders of magnitude smaller than the optimal gains. Therefore, the flow perturbation will be dominated by the spatial structure of the optimal perturbation. The spatial structures of the optimal forcing and perturbation at the optimal frequency are respectively shown in figure 3(a) and 3(b). The forcing is strongly localized at the step edge and its magnitude decays quickly further downstream. In the shear layer of the lower recirculation bubble, the forcing exhibits structures of opposite sign which scale with the size of the shear layer. We can conclude that the forcing activate a KelvinHelmholtz mechanism in the shear layer of the recirculation bubble. Moreover, we note that these structures are inclined in the opposite direction to the shear of the base flow. This also suggests that an Orr mechanism is active and contributes to the energetic amplification. Compare to the forcing, the perturbation is located further downstream and is much less localized. Its maximal magnitude is reached at the station x ∼ 25. The spatial structures are of opposite sign and scale with the downstream channel height 2. The imaginary part of the vertical velocity (not shown here) is in quadrature of phase with the real part. This indicates that the perturbation is a travelling wave with phase speed approximatively to 2ω ∼ 0.24. Further investigations will be dedicated to compare results of the present analysis with results from a direct numerical simulation continuously forced by a white-noise temporal signal located upstream from the backward-facing step.

References 1. Blackburn H M, Barkley D and Sherwin S J (2008) Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603:271–304 2. Ehrenstein U and Gallaire F (2005) On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. J. Fluid Mech. 536:209–218 3. Marquet O, Sipp D, Chomaz J M and Jacquin L (2008) Amplifier and resonator dynamics of a low-Reynolds number recirculation bubble in a global framework. J. Fluid Mech. 605:429– 443

Large Reynolds number streak description using RNS Juan A. Mart´ın and Carlos Martel

Abstract We use the Reduced Navier-Stokes (RNS) equations for the simulation of the nonlinear evolution of streaks in a flat plate boundary layer. The RNS are asymptotically derived from the Navier Stokes equations for Re ≫ 1, and they are appropriated for flow configurations with one slow scale and two short scales. We derive the RNS with the appropriate boundary conditions, comment the details of the numerical method used, and compare our 3D streak simulations with the results present in the literature (computed using DNS and PSE).

1 RNS streak formulation We consider a flat plate boundary layer at zero angle of incidence with a spanwise periodic array of counter-rotating steady streaks developing in the streamwise direction, see Figure1. The velocities are made nondimensional with the reference velocity the free stream flow, U∞ , the spatial scales with a characteristic length L, and the resulting Reynolds number is defined in the usual form, Re = U∞ L/ν , where ν is the kinematic viscosity. The Reduced Navier Stokes equations (RNS) are obtained from the full 3D steady incompressible Navier Stokes equations in the limit of large Reynolds number. The asymptotic structure of the streaks for Re ≫ 1 exhibits, as sketched in Figure 1, a slow spatial dependence in the streamwise direction, and two short spatial scales, in the normal and spanwise direction. This scaling is similar to that of the standard 3D Boundary Layer equations (BL) [1] but with two short scales instead of just one. Juan A. Mart´ın ETSI Aeronáuticos, Universidad Politécnica de Madrid, e-mail: [email protected] Carlos Martel ETSI Aeronáuticos, Universidad Politécnica de Madrid, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_92, © Springer Science+Business Media B.V. 2010

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Fig. 1 Streak scale structure for Re ≫ 1.

The appropriate expansions for the flow variables are of the form √ √ xˆ = x, (y, ˆ zˆ) = (y, z) / Re, uˆ = u + . . . , (v, ˆ w) ˆ = ((v, w) + . . .)/ Re, pˆ = p0 + p1 /Re + . . . , which, once inserted into the Navier Stokes equations yields, at first order, p0 y = 0

and

p0 z = 0.

The pressure is then p0 = p0 (x), given by the prescribed inviscid pressure at the upper edge of the boundary layer (as in the standard BL formulation). And, at next order, the full set of RNS is obtained: ux + uy + uz = 0, uux + vuy + wuz = −p0 x + uyy + uzz , uvx + vvy + wvz = −p1 y + vyy + vzz ,

(1)

uwx + vwy + wwz = −p1 z + wyy + wzz . The RNS are a truly parabolic system in x that can be solved by marching techniques [2, 3, 4]. In contrast with the standard 3D BL formulation, now the second order y and z momentum equations are required to complete the system, and the pressure correction term, p1 , has to be computed in order to solve problem. The RNS equations have been previously used to compute high Re number micro channel and micro tube flow (see, e.g. [3]), where we have to impose no slip boundary conditions on the surface of the tube, and the gradient of p0 is obtained as an eigenvalue at each transverse section. The compressible version of the RNS have also been applied to the computation of high Mach number external boundary layer flow, but mostly in 2D or axisymmetric configurations (see [2, 4] and references therein). In the context of 3D incompressible boundary layers, the linearized RNS have been recently applied to analyze the linear growth of small 3D streaks from the 2D Blasius boundary layer [5, 6, 7]. We integrate the full 3D RNS system (1) to compute the nonlinear spatial evolution of streaks on a flat plate boundary layer. The appropriate boundary conditions for this case are periodicity in z (i.e., we consider a spanwise periodic array of streaks, see Figure 1), together with no slip at the bottom wall, (u, v, w) = (0, 0, 0) at y = 0, and, at the upper edge of the boundary layer,

Large Reynolds number streak description using RNS

(u, v, w) → (U∞ , < v >z , 0)

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as y → ∞,

where < · >z denotes the mean value in the spanwise direction. This last boundary condition states that, in order to match with the outer inviscid flow, all components of v must vanish at y → ∞ except for its mean value < v >z , which is computed as part of the solution. Notice that this is quite different from the standard 3D BL case, where no boundary condition at all can be imposed on v as y → ∞. Finally, in order to completely determine the correction of the pressure p1 , we add the following extra condition: < p1 >z = 0 at y = 0. The applied numerical method advances in the streamwise direction and solves the RNS in the plane y, z for each station in x. A first order one step Euler scheme is used to march in x (the nonlinear terms are treated explicitly, and the pressure and viscous terms are integrated implicitly to avoid instabilities), and, in the transverse plane, we use second order finite difference approximations. The RNS formulation allows us to perform 3D streaky BL computations with much less CPU cost than previous 3D DNS computations[8], which require high Re number. More recently streaks have also been computed using PSE [9]. The RNS are again better suited for this task because they do not have the numerical problems of the PSE, i.e., divergence of the results for small ∆ x [10], and blow-up of the solutions when the deviation from Blasius is not small [9].

2 Results and Conclusions We compare first the RNS results with those from the linearized problem around the Blasius flow. To this end we run the RNS starting from x0 = 0.4 with a initial condition that consists of the Blasius flow plus a small perturbation taken from [7]. As it is clearly appreciated from Figure 2a, the RNS results agree very well with the linear theory predictions. In Figure 2b we present the RNS results for the simulation of fully nonlinear streaks. We plot the streamwise evolution of the amplitude of the streak, As (x) = (max(u) − min(u))/2 for the same four streaks that were computed using DNS in y,z

y,z

[8], and PSE in [9]. The integration is again started at the station x0 = 0.4, and the initial profile data is taken from [9]. The results show that the agreement with the PSE data is quite good: the difference between both results grows with the amplitude of the streak, but never exceeds 3%. Note that PSE results are systematically below those of the RNS. This could be probably due to the fact that the PSE formulation includes extra dissipative terms in order to stabilize the computations. We have also marked with thick blue lines in Figure 2b the DNS computed maximum values of the streak amplitude from [8]. It is interesting to mention that the RNS results are always closer to the DNS than the PSE, and that the PSE was simply not able to complete the computation of the largest streak (labeled E).

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Fig. 2 (a): Maximum values of the deviation from Blasius at each station in x. Linear theory predictions from [7] (red line) and RNS results (black dots). (b): Nonlinear streamwise streak amplitude evolution. Black line: RNS, dashed red line: PSE from [9], and thick blue line: maximum amplitude computed via DNS from [8].

In conclusion, the presented RNS scheme for computing nonlinear streaks is faster than DNS, and more robust than PSE. Our idea for the next future is to include the effect small depth streamwise grooves carved in the plate, to use them to induce the streaks, and to optimize their shape to find the ones that give the most stable streaks.

3 Acknowledgments The authors would like to thank Mar´ıa Higuera and Shervin Bagheri for kindly having allowed us access to their results.

References 1. H. Schlichting, Boundary Layer Theory, 7th edn. (McGraw-Hill, New York, 1979) 2. D. Anderson, J.C. Tannehill, R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer, (Hemisphere, New York, 1984) 3. C. A. J. Fletcher, Computational Techniques for Fluid Dynamics, vol II, 2nd edn. (SpringerVerlag Berlin, 1991) 4. S.G. Rubin, J.C. Tannehill, Annu. Rev. Fluid Mech. 24, (1992) 5. P. Luchini, J. Fluid Mech. 327 (1996) 6. P. Luchini, J. Fluid Mech. 404 (2000) 7. M. Higuera, J.M. Vega, J. Fluid Mech. 626 (2009) 8. C. Cossu, L. Brandt, European J. of Mechanics B/Fluids, 23 6 (2004) 9. S. Bagheri, A. Hanifi, Phys. of Fluids, 19 7 (2007) 10. P. Andersson, D.S. Henningson, A. Hanifi, J. of Eng. Math, 33 (1998)

Optimal disturbances with iterative methods ˚ Antonios Monokrousos, Espen Akervik, Luca Brandt, and Dan S. Henningson

1 Optimal disturbances with iterative methods The flat-plate boundary layer is a classic example of convectively unstable flows; these behave as broadband amplifiers of incoming disturbances. As a consequence, a global stability analysis based on the asymptotic behaviour of single eigenmodes of the system do not capture the relevant dynamics. From this global perspective all the eigenmodes are damped, and one has to resort to an input/output formulation in order to obtain the initial conditions yielding the largest possible disturbance growth at any given time and the optimal harmonic forcing. To this end, an optimisation procedure is adopted. We consider both the optimal initial condition leading to the largest growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response. The Lagrange-multiplier optimisation involves the solution of the linearised Navier–Stokes equations,

∂t u = −(U · ∇)u − (u · ∇)U + Re−1 ∆ u + ∇π ,

(1)

where u is the velocity vector and π the pressure. In compact form the system can be written as ∂t u = A u, where A represents the system matrix (see [2] and [3]). In both cases the evolution equations for the Lagrange multiplier p are the adjoint Navier–Stokes equations −∂t p = (U · ∇)p − (∇U)T p + Re−1 ∆ p + ∇σ ,

(2)

where p is the adjoint velocity vector (co-state) and σ the adjoint pressure. Similarly as for the forward system we write −∂t p = A † p. The approach proposed here is particularly suited to examine convectively unstable flows, where single global eigenmodes of the system do not capture the downstream growth of the disturbances. We solve the linearised Navier–Stokes equations using a spectral DNS code described in [1]. Two relevant eigenvalue problems are solved. First, the problem yielding the Linné Flow Centre, KTH Mechanics, Stockholm, Sweden P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_93, © Springer Science+Business Media B.V. 2010

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optimal initial condition exp(A † T ) exp(A T )u(0) = γ u(0)

(3)

where γ indicates the largest possible energy amplification. Second, we compute the flow optimal response to harmonic forcing (iω I − A † )−1 (iω I − A )−1 f = γ f .

(4)

In the case of problems involving complex flows the system matrix becomes prohibitively large and the matrix-free approach is preferable if not indispensable. This amounts to solving the eigenvalue problem of the composite forward and adjoint operator only using Direct Numerical Simulations and the largest eigenvalue is retrieved using power-iterations. This approach is commonly referred to as a timestepper technique [4].

2 Results We investigate the potential for growth of initial conditions with different spanwise wave-numbers β by solving the eigenvalue problem (3) for a range of instances of time T as well as the the response to harmonic forcing, eigenvalue problem (4), for different spanwise wave-numbers β and frequencies ω . The growth obtained for each case can be seen in figure (1). Initial Condition

(a)

Optimal forcing

(b) 1

0.8 0.7

0.8

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0.6

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Fig. 1 Contour maps of (a) the optimal transient growth versus spanwise wave-number β and optimisation time T ; and (b) optimal response to forcing with frequency ω and spanwise wavenumber β . The contour levels span different orders of magnitude from 10 to 104 . In both plots there is a maximum relevant to the combined Orr/TS mechanism, which also corresponds to the global maximum and one relevant to the lift-up mechanism.

Optimal disturbances with iterative methods

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(a)

(b)

(c)

Fig. 2 Evolution of streamwise velocity when initialising the system with the optimal initial condition at kz = 0.60. a) The wall-normal velocity of the optimal initial condition with surface levels at ten percent of its maximum value, b) spanwise velocity of the optimal initial condition and c) streamwise velocity of the response at time t = 600.

The optimal initial condition for spanwise wavelengths of the order of the boundary layer thickness appear for short optimisation times and are streamwise vortices exploiting the lift-up mechanism to create streaks. The relevant structures are shown in figure (2). For long spanwise wavelengths the optimal disturbance appears for long optimisation times. The relevant structures are shown in figure (3) where it can be seen that the initial disturbance is leaning against the shear of the base flow. The resulting instability exploits the Orr-mechanism to efficiently initialise a wave packet that later on propagates downstream and gains energy by exploiting the convectively unstable boundary layer. It is found that the latter mechanism is dominant for the relatively high Reynolds number and the long computational domain considered here. The spatial structure of the optimal forcing is similar to that of the optimal initial condition, and the response to forcing is also dominated by the Orr/oblique wave mechanism, however less so than in the former case. The lift-up mechanism is, as in the local approach using the Orr–Sommerfeld/Squire equations, most efficient at zero frequency and degrades slowly for increasing frequencies. The formulation is extended to determine an optimal localised initial condition. In this case, the initial perturbation is forced to exist only inside a certain region in space. The optimal shape contained within this region is determined by the optimisation procedure. For short optimisation times, the optimal disturbances consist of streaky structures propagating and elongating in the downstream without any significant spreading in the lateral direction. For long optimisation times, conversely, the optimal disturbances are characterised by wave-packets of TS-waves. These travel at lower streamwise speed and have much faster spreading rate in the spanwise direction. The latter can achieve the largest possible energy amplification.

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(a)

(b)

Fig. 3 Isosurfaces of streamwise component of disturbances at the spanwise wavenumber β = 0.05. Red/blue colour signifies isosurfaces corresponding to positive/negative velocities at 10 percent of the maximum. a) Streamwise component of optimal initial condition leading to the global optimal growth at time T = 1820. b) flow response at time t = 1600

3 Conclusions Three different destabilising mechanisms are considered in this study, all at work in the boundary layer flow. Although these could be explained using the OSS equations, they are analysed without any simplifying assumptions with a global approach. The present work is of a more general character. By choosing an objective function and using the full linearized Navier–Stokes equations as constraints we are not limiting ourselves to simple geometries. We show that the stability and receptivity to external excitations of any flow configuration can be investigated by means of a Navier-Stokes solver. To this end, it is necessary to implement a linearised version of the equations and the corresponding adjoint equations along with a wrapper built around these two ensuring the correct optimisation scheme. The method used here is therefore easily applicable to any geometrical configuration.

References 1. M. Chevalier, P. Schlatter, A. Lundbladh and D. S. Henningson, A pseudo spectral solver for incompressible boundary layer flows, KTH Technical Report, Trita-Mek, 2007-7. 2. G. Kreiss, A. Lundbladh, and D. S. Henningson, Bounds for threshold amplitudes in subcritical shear flows, J. Fluid Mech. 270, 175–198, 1994. 3. L. N. Trefethen, A. E. Trefethen , S. C. Reddy and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science, 261, 578–584, 1993. 4. L. S. Tuckerman and D. Barkley, Bifurcation Analysis For Timesteppers, Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, 453–566, 2000, Springer, New York.

Connection between full-lifetime and breakdown of puffs in transitional pipe flows ¨ ur Ertunç, and Antonio Delgado Mina Nishi, Ozg¨

Abstract Puff splitting was studied by direct measurement of the full-lifetime (LT f ull ) of transitional structures (puffs) in low Reynolds number pipe flows. During the investigations, a fully developed laminar pipe flow was triggered by an iris diaphragm with pre-defined amplitude and lapse time and the evolution of puffs was monitored by the transients of pressure drop along the pipe and the hot wire anemometry at the pipe exit. Those complementary measurements showed that a single puff favors to breakdown into two or more (splitting) puffs and later into slug-like puffs at Re ≈ 2300, where they are expected to have infinite LT f ull .

1 Introduction Wygnanski et al. [1] showed that at Re ≈ 2300 puff splittings start to occur. Nishi et al.[2] investigated the evolution of puffs to slugs through puff splittings which is termed breakdown of puffs in the present investigation. Recently, numerous studies focused on the dissipation phenomena of puffs in low Re pipe flow (e.g. [3], [4], [5] and [6]). One important parameter in those investigations is lifetime of puffs, which is an adopted notion from the definition of half-life in radioactive decay phenomenon. Lifetime was observed to diverge to infinity at Re = 1760, 1870 and 2250 by [4], [6] and [3] respectively and not to diverge in finite Re by [5]. Present study was carried out by a simple pressure drop measurement, developed by [7] to obtain a characteristic time of puffs (LT f ull ) and monitored the probability of puff splitting occurrence to establish the connection between the time required for the dissipation of puffs and the breakdown of puffs in low Re pipe flows. Mina Nishi Institute of Fluid Mechanics, Friedrich Alexander University of Erlangen-Nuremberg, Cauerstr.4 D-91058 Erlangen Germany, e-mail: [email protected] ¨ ur Ertunç Ozg¨ same as above, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_94, © Springer Science+Business Media B.V. 2010

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2 Experimental test rig The present experimental test rig, developed first by [8], modified later by [2] is schematically shown in figure 2. The test rig contains mainly a mass flow controller, a critical nozzle, a flow conditioner, an iris diaphragm triggering device, a pipe section, pressure measurement tabs and hot wire anemometry (HWA) measurement section. The mass flow rate was maintained constant and the natural transition occurs at Re = 11500 for the present test rig. To trigger a transition, an iris diaphragm system was applied, which creates an wall fence disturbance with pre-determined amplitude and lapse time and is operated by the computer. The pipe section was carefully constructed by connecting number of short pipes, which reveals the total length of L = 8.5 m (633 pipe diameter). The inner diameter of the pipe section was 15 mm and at L = 0.05 and 8.25 m, pressure tabs were facilitated so that the transient of the pressure difference of the two points was monitored. At the end of the pipe, an HWA was equipped to detect if the puff reaches to the end of the pipe.

Fig. 1 Experimental test rig for a full-lifetime measurement

3 Measurement of a full-lifetime of puffs and a rate of splitting puffs’ occurrence Figure 2 (a) shows an over 200 realizations ensemble transients of pressure difference (∆ p) between inlet and outlet of the pipe. Each realization, the HWA was activated to check if a puff reached to the pipe end. ∆ p first decreases when a puff

Connection between full-lifetime and breakdown of puffs in transitional pipe flows

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Fig. 2 (a) Pressure difference transients and definition of a full-lifetime of a puff, (b) Change of different lifetimes of puffs with increase in Reynolds number

Fig. 3 (a) Inverse of fulllifetime change with increase in Reynolds number, (b) Puffs’ splitting occurrence with increase in Reynolds number

was generated just after the operation of the iris diaphragm, then it increases rapidly while a puff develops (LTrec and LTdev ). After reaching the peak, ∆ p decreases to the value of which no transition exist in the pipe, i.e. the puffs dissipated completely (LTdiss ). Thus from the time between the iris diaphragm operation and ∆ p returning back to the original value is termed as full-lifetime (LT f ull ) in the present study. Figure 2 (b) shows the change of various lifetime with increase in Re. LTrec and LTdev remain more or less constant for all Re i.e. the creation of transitions remains the same. LTdiss and LT f ull increase with increase in Re. The result was replotted in the relation between Re and the inverse of LT f ull for Re < 1890 in figure 3 (a) which shows that LT f−1 ull decreases with increase in Re linearly and has the intercept of Re ≈ 2300 when LT f−1 ull = 0 i.e. LT f ull becomes very long or infinity. On the other hand, a separate sets of experiments were carried out by employing the HWA equipped at the pipe exit to study about the rate of an occurrence of splitting puffs, which was first reported as the necessary step of the transition from a puff to a slug by [2]. The occurrence rate of splitting puffs increased systematically with increase in Re as shown in figure 3 (b) which reveals that at Re = 2300 some puffs start to split and the puff splitting occurrence increases systematically with increase in Re then at around Re = 2700, all puffs split. For 2300 ≤ Re ≤ 2700, it was observed that some puffs remain as single puffs and some appeared as splitting puffs as shown in figure 3.

4 Conclusions and final remarks The full-lifetime measurements were successfully conducted by analyzing the transients of the pressure difference which exhibited possible routes for the development and/or the decay of puffs with increase in Re. The extrapolation of the inverse of the average LT f ull to zero reveals that LT f ull diverges to infinity at Re ≈ 2300. The oc-

¨ Ertunç, A. Delgado M. Nishi, O.

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currence probability of splitting puffs showed that they start to occur at Re ≈ 2300 within the pipe having a length of 633 pipe diameter, where they are expected to have very long or infinite LT f ull . Acknowledgements The authors would like to thank Prof. Bruno Eckhardt and Prof. Franz Durst who kindly gave valuable suggestions while proceeding the present investigations.

References 1. Wygnanski, I. J., Sokolov, M., Friedman, D.: On transition in a pipe. Part 2. The equilibrium puff. J. Fluid Mech. 69, 283–304 (1975) ¨ 2. Nishi, M., Unsal, B., Durst, F., Biswas, G.: Laminar-to-turbulent transition of pipe flows through puffs and slugs. J. Fluid Mech. 614, 425–446 (2008) 3. Faisst, H. and Eckhardt, B.: Sensitive dependence on initial conditions in transition to turbulence in pipe flow.: J. Fluid Mech. 504, 343352 (2004) 4. Peixinho, J. and Mullin, T.: Decay of turbulence in pipe flow. Meas. Sci. Technol. 96, 1-4 (2006) 5. Hof, B., Westerweel, J., Schneider, T. M. and Eckhardt, B.: Finite lifetime of turbulence in shear flows. Nature 443, 59-62 (2006) 6. Willis, A. P. and Kerswell, R. R.: Critical behavior in the relaminarization of localized turbulence in pipe flow. Phys. Rev. Lett. 98, 1-4 (2007) 7. Nishi, M.: Laminar to turbulent transition in pipe flow through puffs and slugs. PhD thesis (2009) ¨ 8. Durst, F., Unsal, B.: Forced laminar to turbulent transition of pipe flows. J. Fluid Mech. 560, 449–464 (2006) 9. Willis, A. P., Peixinho, J. and Kerswell, R. R., Mullin, T.: Experimental and theoretical progress in pipe flow transition. Phil. Trans. Roy. Soc. A. 366, 2671–2684 (2008)

Effect of oblique waves on jet turbulence ¨ u, Antonio Segalini, Alessandro Talamelli, and P. Henrik Alfredsson Ramis Orl¨

Abstract The paper describes experiments on acoustically excited axisymmetric turbulent jet flows. The investigation is based on the hypothesis that so called oblique transition may play a role in the breakdown to turbulence for an axisymmetric jet. For wall bounded flows oblique transition gives rise to steady streamwise streaks that break down to turbulence, as for instance documented by Elofsson & Alfredsson (J. Fluid Mech. 358). The scenario of oblique transition has so far not been considered for jet flows and the aim was to study the effect of two oblique modes on the transition scenario as well as on the flow dynamics. Even though it was not possible to detect the presence of streamwise streaks, for a certain range of the excitation frequencies, the turbulence intensity, at a fixed streamwise position, is found to be significantly reduced.

1 Introduction For many years investigations have been conducted in order to understand the flow instabilities that lead to transition in jet flows. Among the earliest, the inviscid linear stability analysis of Batchelor & Gill [1] showed that immediately at the jet exit, where the velocity profile has a ’top-hat’ behaviour, all instability modes are able to be exponentially amplified, while in the far field region only the helical mode seems to be unstable. The transition between these two different instability regions is still unclear and the analysis is complicated by the presence of several unstable modes embedded in the turbulence background. Therefore, a large number of linear ¨ u · P. Henrik Alfredsson Ramis Orl¨ Linné Flow Centre, KTH Mechanics, Stockholm, Sweden, e-mail: [email protected], [email protected] Antonio Segalini · Alessandro Talamelli Dept. of Mechanical & Aerospace Eng. (DIEM), Univeristy of Bologna, Forl´ı, Italy, e-mail: [email protected], [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_95, © Springer Science+Business Media B.V. 2010

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¨ u, A. Segalini, A. Talamelli, P. H. Alfredsson R. Orl¨

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stability analyses, simulations and experiments have been performed in order to understand the role played by a single or few modes in the evolution of the flow. Investigations in naturally and artificially excited jets have determined the importance of two instability lengthscales: one associated with the initial shear-layer (momentum-loss) thickness, θ0 , at the exit of the nozzle [7], and the other associated with the jet diameter, D, which governs the shape of the mean velocity profile at the end of the potential core [3]. The instability modes in the first region develop through continuous and gradual frequency and phase adjustments to produce a smooth merging with the second region. Axisymmetric excitation by means of acoustic forcing has been able to highlight several important aspects of the complex dynamics involved. So, for instance, have the roles played by the shear layer [7] and jet column mode [3] acting in the nearfield region of the jet at the nozzle exit and at the end of the potential core, respectively, been of particular interest. However, fewer works have been devoted to the investigation of higher azimuthal modes, particularly, due to the higher complexity of the excitation facility [2].

2 Motivation for experiments The motivation of the present work is to investigate the possibility that higher order modes in the jet can actually be the result of nonlinear combination of oblique waves. Such an hypothesis implies that oblique transition may play a role in the evolution of the coherent structures until the final breakdown to turbulence also for an axisymmetric jet. For wall bounded flows it was already shown that oblique transition gives rise to steady streamwise streaks that break down to turbulence [4]. The scenario of oblique transition has so far not been considered for jet flows and the aim was to study the effect of two oblique modes on the transition scenario as well as on the flow dynamics. The present experiments have been carried out in the Fluid Physics Laboratory of the Linné Flow Centre at KTH Mechanics. The facility used consists of a round jet rig with an exit diameter of 0.025 m equipped with 24 perpendicular aligned metal tubes for the acoustic excitation as shown in figure 1(a). The excitation has been imparted on the initially laminar flow by feeding the amplitude modulation

Fig. 1 (a) Close up of the excitation rig, together with the hole number, that corresponds to a certain (b) amplitude modulation for the generation of two opposing helical modes, i.e. m = ± 1.

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depicted in figure 1(b) to the nozzle exit in order to generate two opposite helical modes (m = ±1) in such a way that a standing wave pattern is formed. Details of the experimental setup and the employed measurement technique are given in [6].

3 Results and Discussion Figure 2 depicts flow visualisations from both an axisymmetric and oblique excitation. While the axisymmetry prevails for the m = 0 excitation, as apparent from figure 2(a) and (b), the m = ±1 excitation produces a clear helical structure, as would be expected. The presence of a helical mode can also be evinced by means of phase averaged hot-wire measurements [6]. An extensive parametric study in terms of frequencies and amplitudes for a selected range of Reynolds numbers was performed in order to focus on the most influential parameter set. Several combinations of Reynolds number and excitation frequency have been tested in order to deduce the effect on the streamwise rms level. In figure 3 the relative rms (compared to the unexcited case) as function of the dimensionless frequency, Stθ0 = f θ0 /U0 , is given, where U0 denotes the centreline exit velocity. It must be stated that similar Stθ0 have been obtained with different combinations of velocity and excitation frequency, which can be the reason for the presence of a non negligible scatter. Even though the scatter is significant, it is interesting to note that a clear reduction in the turbulence intensity can be observed around Stθ0 ≃ 0.012, which is close to the value found for an axisymmetrically excited jet and corresponds to the natural shear layer instability frequency [5]. This is

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(b)

(c)

(d)

Fig. 2 Smoke flow visualisations at ReD ≈ 8500, based on nozzle diameter and exit velocity: unexcited jet, across r/D = 0 (a) and 0.5 (b), excited jet, StD = 0.5 (Stθ0 ≈ 0.005) and m = ±1, across r/D = 0 (c) and 0.5 (d).

¨ u, A. Segalini, A. Talamelli, P. H. Alfredsson R. Orl¨

544

Fig. 3 Relative change in streamwise velocity rms at x/D = 3 between the excited (m = ± 1) and unexcited case for different exit velocities. The moving average serves to visualise the trend.

rel. rms [%]

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also in accordance with the flow visualisations shown in figure 2, where for the excited case a reduction in the incoherent (small-scale) turbulence can be anticipated. Whether or not the reduction of turbulence is connected to the scenario of oblique transition as described in [4], where the nonlinear interaction of two oblique waves in laminar boundary layers was found to generate streamwise streaks, remains unclear. In this first investigation, it was not possible to detect the presence of such streaks, and this aspect must be further investigated in the future in order to understand the connection between the turbulence reduction and the azimuthal forcing. Acknowledgements The cooperation between KTH and the University of Bologna is supported by The Swedish Foundation for International Cooperation in Research and Higher Education (STINT), which is greatly acknowledged.

References 1. Batchelor G, Gill A (1962) Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14:529–551 2. Corke T, Kusek S (1993) Resonance in axisymmetric jets with controlled helical-mode input. J. Fluid Mech. 249:307–336 3. Crow S, Champagne F (1971) Orderly structure in jet turbulence. J. Fluid Mech. 48:547–591 4. Elofsson P, Alfredsson P H (1998) An experimental study of oblique transition in plane Poiseuille flow. J. Fluid Mech. 358:177–202 5. Husain H, Hussain F (1995) Experiments on subharmonic resonance in a shear layer. J. Fluid Mech. 304:343–372 ¨ u R (2009) Experimental studies in jet flows and zero pressure-gradient turbulent boundary 6. Orl¨ layers. PhD thesis Royal Institute of Technology, Stockholm, Sweden 7. Zaman K, Hussain A K M F (1981) Turbulence suppression in free shear flows by controlled excitation. J. Fluid Mech. 103:133–159

The effect of a single three-dimensional roughness element on the boundary layer transition Igor B. de Paula1,2 , Werner Würz1 , and Marcello A. F. Medeiros2

Abstract Most studies of TS wave evolution consider a perfectly smooth surface, whereas studies of roughness induced transition consider a perfectly quiet flow. It appears that a combination of these scenarios could better represent a practical situation. This work investigates the effect of a small and medium sized (h/δ ∗ < 0.2) roughness on the evolution of a pre-existing T-S wave. A cylindrical element was used as a localized roughness. Several roughness heights were tested experimentally under different sets of flow parameters. The results indicated that under various circumstances the fundamental resonance was the dominant mechanism in the wake of the roughness. A physical model for the prediction of the effect the roughness on a non quiet environement was then proposed and verified.

1 Introduction The current work consists of a study on the boundary layer transition induced by a single three-dimensional roughness element. Such elements are often found on aircrafts surfaces and are well known to influence drag. In practical applications it is often assumed that protuberances have no influence on the boundary layer transition if a criterion is satisfied [1]. This criterion is normally a critical Reynolds number (Reh = Uh/ν ) based on the roughness height (h). In [1], it is suggested that protuberances with h < δ ∗ have a critical Reynolds number around 100. However, some works suggest that an interaction between a plane TS wave and a roughness element can enhance the secondary instability of the boundary layer [2, 3]. This effect could be potencially important even for very small roughnesses. The current work confirms this possibility and proposes physical model for the phenomenon. 1

Universität Stuttgart, Institute für Aerodynamik und Gasdynamik, Pfaffenwaldring 21, Stuttgart, Germany. 2 Universidade de S˜ ao Paulo - EESC, Depto de Eng. de Materiais, Aeronáutica e Automobil´ıstica, Av. Trabalhador Sancarlense 400, São Carlos, SP, Brazil. P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_96, © Springer Science+Business Media B.V. 2010

545

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I. B. de Paula, W. Würz, M. A. F. Medeiros

2 Experimental Set-up and Results The experiments were carried out on a 600mm chordlength airfoil placed in the Laminar Wind Tunnel of the University of Stuttgart [3]. The controlled disturbances that produced the T-S waves were introduced into the flow by a slit source mounted flush to the airfoil surface [3]. The cylindrical roughness element with 10mm in diameter was placed at the spanwise center of the airfoil and at a streamwise position equal to 40% of the chordlength. During the experiments, the roughness height was slowly oscillated with a frequency approximately 1500 times lower than the T-S wave frequency. Therefore, it could be considered as a quasi-steady roughness. In the secondary instability, the bandwidth of unstable 3D waves is dependent on the initial amplitude of the primary 2D T-S wave [4]. In the current experiments, the initial amplitudes of the incoming 2D T-S waves were chosen based on simulations of the secondary instability performed for a flat plate without roughness. A solver of the Parabolized Stability Equations ([5]) was used. From the simulations, different conditions were selected for the experiments. They included two T-S wave amplitudes [A0 = (0, 45; 0, 75) %U0 ] and two frequencies [700;900Hz]. They covered from stable to very unstable regimes of fundamental secondary instability. Also, the bandwidth of the unstable 3D modes changed significantly among the different regimes. The experimental outcome agreed with the expectations from the secondary instability calculations. For stable conditions, the 3-D structure caused by the roughness decayed downstream (figure 1), while for the unstable cases the opposite occurred (figure 2 and 3). The influence of the roughness was then modeled by assuming that it excited a flat spectrum of oblique waves with the frequency of the incoming waves. The evolution of the oblique waves was calculated with the PSE code and compared with the experiments. For both unstable and stable conditions good agreement was found (figure 2). Of particular importance was that the bandwidth of the unstable waves varied with the TS wave amplitude (figure 2 and 3). This strongly supported the assumption that, within the investigated conditions, the diameter of the roughness did not play a hole, particularly for humps below 0.2δ ∗ . Further analysis not shown here, indicated that the streaks formed behind the roughness are a key ingredient in the process.

3 Final remarks The results demonstrated that roughness elements with heights far below the critical limit could provide seeds of oblique modes for the secondary instability and, as a consequence, affect boundary layer transition. The evolution downstream of the hump depended on both the roughness height and the T-S wave amplitude and frequency. For roughness heights up to approximately 20% of δ ∗ it was possible

The effect of a single 3-D roughness on the boundary layer transition

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in normalised fluctuations of the velocity, density, temperature and pressure, assumptions about the characteristics of the flow field have to be made. In the free flow it can be assumed that the disturbance field is a pure acoustic field [5][11]. In turbulent constant pressure boundary layers, it can be assumed that pressure fluctuations are negligible compared with temperature, velocity and density fluctuations [2]. In a first step, the

Fluctuation Measurements in the Turbulent Boundary Layer of a Supersonic Flow

571

same assumption is applied here for the measurements downstream of the shock wave, although the applicability has yet to be proved.

4 Results and Discussion In Figure 1 the fluctuation profiles for measurements upstream and downstream of the deflection corner and hence the oblique shock wave are shown. The local Mach number is M > 1 for all shown data points. Temperature and density fluctuation profiles are about equal in magnitude due to the assumption of negligible pressure fluctuations. Their magnitude is about twice as high as the velocity fluctuations. In the centre part of the BL (y > 0.6δ ), the temperature and density fluctuation profiles measured upstream and downstream of the shock look qualitatively similar. Downstream of the shock wave, the disturbance level is generally increased by about 45%. Near the wall, the < T > and < ρ > fluctuation level downstream of the shock decreases considerable stronger than upstream of the shock. The velocity fluctuation profile changes considerably through the shock wave. Upstream of the shock, increases with decreasing distance y from the wall. Downstream, shows a distinct maximum, laying closer to the wall than the < T > and < ρ > maxima. A possible explanation is the separation bubble and increasing thickness of the BL at the deflection corner. The flow encounters a curved wall instead of the defined corner. Hence, multiple shock waves merge, causing a curved shock in the near wall region [1]. According to Crocco’s theorem the flow field behind a curved shock is rotational, which possibly causes the changed fluctuation profile. As expectable, the fluctuation levels in the free stream are much lower than in the BL: = 0.03%, < T >= 0.06% and < ρ >= 0.06%.

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572

A.-M. Schreyer, U. Gaisbauer, E. Krämer

Much higher fluctuation levels were found by Sagaut et al. [9] for M = 2.3. On the other hand, the shape and order of magnitude of the velocity fluctuation profiles in the undisturbed BL agree well with measurements by Kistler [2] for M = 1.72 and M = 3.56 and Reθ = 30000 (no figure shown). According to Lenz [6], the standard deviation in fluctuations with this measurement technique is 0.1 at low overheat ratios τ , getting even higher with increasing τ . Hence, at this point, the measurements should rather be seen as qualitative.

5 Conclusions and Outlook Velocity, temperature and density fluctuations within a turbulent supersonic boundary layer and in the free stream are discussed. Measurements in the boundary layer were performed for two positions upstream and downstream of a compression ramp. The measurements upstream of the deflection corner agreed well with measurements in an undisturbed boundary layer (not shown). Velocity, temperature and density fluctuations were calculated based on the assumption of negligible pressure fluctuations in a first step. To clarify the applicability of modal analysis in flows with pressure gradient, the static pressure fluctuations will be measured and compared with fluctuation levels measured via HWA.

References 1. Gaisbauer, U.: Untersuchung zur Stoss-Grenzschicht-Wechselwirkung an Doppelrampen unter verschiedenen Randbedingungen. Dissertation Universität Stuttgart (2004) 2. Kistler, A.L.: Fluctuation Measurements in a Supersonic Turbulent Boundary Layer. Physics of Fluids 2(3), 290-296 (1959) 3. Kovásznay, L.S.G.: The Hot-wire anemometer in supersonic flow. J. Aer. Sci. 17, 565-573 (1950) 4. Kovásznay, L.S.G.: Turbulence in supersonic flow. J. Aer. Sci. 20(10), 657-674 (1953) 5. Laufer, J.: Aerodynamic Noise in Supersonic Wind Tunnels. J. Aer. Sci. 28(9), 685-692 (1961) 6. Lenz, B.: Experimental Determination of Fluctuation Levels in Supersonic Boundary Layers via Hot-Wire Anemometry. Dissertation Universität Stuttgart, to be published. 7. Lenz, B., Gaisbauer, U., Krämer., E.: Fluctuation measurements in the boundary layer of a supersonic flow. In: Proceedings in Applied Mathematics and Mechanics (2007) 8. Morkovin, M.: Fluctuations and Hot-wire Anemometry in Compressible Flow. Agardograph 24 (1956) 9. Sagaut, P., Garnier, E., Tromeur, E., Larcheveque, L., Labourasse, E.: Turbulent Inflow Conditions for LES of Compressible Wall-Bounded Flows. AIAA Journal 42(3), 469-477 (2004) 10. Smits, A.J., Hayakawa, K., Muck, K.C.: Constant Temperature Hot-Wire Anemometer Practice in Supersonic Flows. Experiments in Fluids 1, 83-92 (1983) 11. Weiss, J.: Experimental Determination of the Free Stream Disturbance Field in the Short Duration Supersonic Wind Tunnel of Stuttgart University. Diss. Universität Stuttgart (2002)

Experimental characterization of the transition region in a rotating-disk boundary layer Muhammad Ehtisham Siddiqui, Benoˆıt Pier, Julian Scott, Alexandre Azouyi, and Roger Michelet

Abstract A series of experiments were performed to study the transition from laminar to turbulent flow for the boundary layer over a rotating disk and to compare with theoretical results. The mean flow profile measured in the laminar region is found to be in excellent agreement with analytical results, and the spatial growth of natural disturbances matches linear theory predictions. A hot-wire sensor was positioned at different spatial locations to determine the evolution of natural disturbances. Spectral analysis at high resolution and ensemble-averages of velocity time series have been carried out to distinguish different flow regimes.

1 Introduction The flow due to a rotating disk has been used as the canonical configuration for the study of 3D boundary layers [7]. It exhibits a self-similar exact solution [1]. The p basic-flow solution displays a constant boundary layer thickness δ = ν /Ω , where ν is the kinematic viscosity and Ω the disk rotation rate. Such boundary layers give rise to strong instabilities leading to the formation of cross-flow vortices, as already noticed experimentally by Gregory, Stuart and Walker [8]. Throughout this study, the axial coordinate Z and radial coordinate R are nondimensionalized by δ . The rotating-disk flow is known to display a sharp transition from laminar to turbulent regimes at a nondimensional critical radius R ≃ 510 [7, 3]. Lingwood [2] found this transition location to precisely coincide with onset of local absolute instability at Rca ≃ 507. More recently, a fully nonlinear analysis [4] has revealed additional aspects of primary and secondary instabilities and has further contributed to the understanding of the complex flow dynamics prevailing near the transition region. Muhammad Ehtisham SIDDIQUI ([email protected]) Laboratoire de mécanique des fluides et d’acoustique (CNRS – Université de Lyon). ´ ´ Ecole centrale de Lyon, 36 avenue Guy-de-Collongue; 69134 Ecully cedex, FRANCE. P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_103, © Springer Science+Business Media B.V. 2010

573

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M. E. Siddiqui, B. Pier, J. Scott, A. Azouyi, R. Michelet

2 Experimental setup The present rotating-disk facility consists of a glass disk, 500 mm in diameter, rotated at constant angular velocity Ω , up to 1500 rpm (see fig. 1). The maximum out-of-flatness was found to be less than 50 microns. The disk has constant angular velocity Ω to within 0.05%. A high-precision two-axes traversing mechanism is used for hot-wire sensor positioning with radial and axial precisions of 20 µ m and 2 µ m respectively. The traversing mechanism and velocity measurements are controlled and recorded by a dedicated computer. A constant-temperature hot-wire anemometer is used for velocity measurements.

Fig. 1 Left: experimental setup consisting of a glass disk and a traversing mechanism, right: a close-up of the traversing mechanism for positioning the hot-wire probe.

3 Mean-velocity profiles Circumferential mean-velocity profiles are measured at different radial locations. Fig. 2 shows measurements (symbols) obtained at Ω = 950 rpm, plotted together with the laminar similarity profile (solid line). Velocities are nondimensionalized by Rδ Ω . At the lower radii, R ≤ 470, the measured velocity profiles are found to precisely follow the laminar profile (fig. 2a). For 490 ≤ R ≤ 530 (fig. 2b), the velocity profiles are significantly different from the theoretical profile for 1 < Z < 4, and ≤ 10% flow correction is observed, but the flow is not yet fully turbulent. For R ≥ 550 (fig. 2c), a strong mean flow correction extends up to Z = 16, this boundarylayer thickening is characteristic of a fully turbulent regime.

4 Spectral analysis Fourier spectra were calculated from the azimuthal velocity time series for different radii at Z = 1, 1.5, 2, 2.5, 3, 4 and Ω = 950 rpm (fig. 3). At low radii R ≤ 450, flat spectra are observed with uniform background noise level. For R ≥ 470 and Z ≤ 2.5,

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there is a band of amplified frequencies, with a maximum amplitude near ω /Ω ≈ 30. The harmonics of the fundamental frequency start to appear for R ≥ 500, which is an indication that nonlinearities are developing. For R ≥ 530, the harmonic peaks give way to fully developed spectra, characteristic of the turbulent regime. Further away from the disk surface, the weakly nonlinear regime is bypassed and a sudden transition from laminar to turbulent flows occurs (fig. 3c). 1e+12

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Fig. 4 High resolution power spectra with Ω = 950 rpm at Z = 2: (a) for R = 500, (b) close-up, showing discrete spectral components. (c) for R = 450, 470, 500, 510, 520, 530, 550, 600.

frequency peaks are particularly visible at radial locations 500 ≤ R ≤ 550, where the low-resolution spectra display a weakly nonlinear harmonic regime. This discrete spectral component is not observed in the laminar boundary layer for R < 470 and appears to be suppressed by the turbulent regime for R ≥ 550.

5 Conclusions and outlook In the present experimental study we have distinguished different flow regimes. Mean azimuthal velocity measurements have been performed and were found to follow precisely the theoretical profile in the laminar flow regime. High-resolution spectra of azimuthal velocity time series, recorded over a long duration, are found to consist of continuous and discrete parts in which the discrete part of the spectra is made up of narrow peaks with the same periodicity as the disk. Work in progress consists in implementing a new control method [5, 6] to modify the naturally selected flow dynamics. This research is financially supported by the Agence nationale de la recherche (ANR, project ”Microsillon”).

References 1. 2. 3. 4. 5. 6. 7. 8.

T. von Kármán, Z. Angew. Math. Mech. 1, 232–252, 1921. R. J. Lingwood, J. Fluid Mech. 299, 17–33, 1995. R. J. Lingwood, J. Fluid Mech. 314, 373–405, 1996. B. Pier, J. Fluid Mech. 487, 315–343, 2003. B. Pier, J. Eng. Math. 57, 237–251, 2007. B. Pier, Proc. R. Soc. Lond. A, 459, 1105–1115, 2003. H. L. Reed and W. S. Saric, Annu. Rev. Fluid Mech. 21, 235–284, 1989. N. Gregory, J. T. Stuart, W. S. Walker, Phil. Trans. R. Soc. Lond. A 248, 155–199, 1955.

Nonlinear Interaction Between Wavepackets in Plane Poiseuille Flow Homero G. Silva, Ricardo A. C. Germanos, and Marcello A. F. de Medeiros

Abstract This abstract presents numerical results and analysis related to natural transition in a plane Poiseuille flow. The natural transition scenario was modeled by three-dimentional wavepackets. Natural transition is often studied in boundary layers, but the analysis in a parallel flow allows the use of the Reynolds number as a control parameter. The results indicated that the interactions between wavepackets lead to oblique transition which is already often linked to natural transition.

1 Introduction Several studies on natural transition in boundary layer indicate the presence of wavepackets prior to the occurrence of turbulent spot [1]. Other studies suggest the oblique transition is the nonlinear regime observed in natural transition [2]. However, the nonlinear regime of isolated wavepackets in boundary layer is largely dominated by secondary instability of the subharmonic type [3, 4]. There seems to be a missing link between these two observations. The present work indicates a possible explanation for these apparently conflicting results. The link stems from the interactions of wavepackets. Indeed, the current results indicate that the interaction of wavepackets evolving side by side lead to oblique transition, while isolated wavepacket of similar amplitude still display linear behavior. The isolated packet may well be a too simplified model. The current work involved numerical simulation of wavepackets in incompressible channel flow, but it is expected that, in general, the results would apply to boundary layer. Homero FACIP/UFU - Faculdades de Ciências Integradas do Pontal-Universidade Federal de Uberlândia, Brazil, e-mail: [email protected] Ricardo and Marcello EESC/USP - Departamento de Engenharia de Materiais, Aeronáutica e Automobil´ıstica-Escola de Engenharia de São Carlos-Universidade de São Paulo, Brazil e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_104, © Springer Science+Business Media B.V. 2010

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2 Equations, Numerical Methods and Initial Condition The three-dimensional Navier-Stokes equations written in a vorticity-velocity formulation was used to describe an incompressible three-dimensional flow between two plates. Periodic boundary conditions were adopted in both streamwise and spanwise directions of the flow. At the wall, non-slip and impermeability conditions were imposed. Numerical simulations of the above equations were performed using Fourier methods in the periodic directions and compact finite differences in the wall-normal direction. As the flow was parallel, a temporal analysis was considered. Details of the chosen numerical scheme and procedures can be found in [5]. A disturbance for v−velocity was introduced into the computational domain somewhat imitating the technique of blowing or suction used in experiments. The disturbances was given by v(x, y0 , z,t) = A(m,n) (1 − cos(2π

t − t1 n m (n,m) )) Re(ei(α x+β z+φ ) ), t2 − t1

(1)

where A(m,n) , α n , β m and φ (n,m) indicate, respectively, amplitude, streamwise and spanwise wavenumbers and phase of the disturbance mode (n, m). The symbol Re indicates that only the real part was considered. The parameter y0 assumes values 0 and 2H. Initially the disturbance produced a flat spectrum in both streamwise and spanwise wavenumbers. It covered the range 0 ≤ α ≤ 1.72, and −1.72 ≤ β ≤ 1.72. The Reynolds number based on the channel width was 8000. At this stage the maximum amplitude in physical space was 1 × 10−3 . The simulations of the isolated wavepacket used 256 Fourier modes in each periodic direction. This includes both positive and negative wavenumbers and the part of the spectrum that was filtered based on the 2/3 rule to avoid aliasing. The spectral discretization was 0.02 in the spanwise direction and 0.04 in the streamwise direction. In the wall normal direction 201 equally spaced points were used. For the non isolated case, the only modification was to change the spanwise resolution to 0.04. Since the simulation was periodic in the spanwise direction, this reduced the distance between adjacent packets and enabled the interaction.

3 Results Figure 1 displays, at (y = 0.11H) in both physical and Fourier domains, the temporal evolution of interacting wavepackets. During the linear regime, the wavepackets spreaded over a large portion of the flow field. Around t = 814, at the edges of the packets there were interactions between the very oblique waves of adjacent wavepackets. After, t = 1647, the results show mean flow distortions that amplified and dominated the flow. In the Fourier domain, initially the linear filter gradually selected the waves most amplified linearly. At time t = 1294, the first nonlinear signature was a band of

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Fourier modes at low streamwise wavenumbers and spanwise wavenumbers between 0.5 and 0.8. Later, these modes increased and dominated the nonlinear regime of the flow. The spectra is consistent with the generation of streaks observed in physical space. The generation of streaks from the interaction of oblique waves is characteristic of oblique transition. For reference, the evolution of an isolated packet was also simulated, Fig. 2. The simulation parameters were identical to that of the Fig. 1, except the spanwise resolution which was modified to provide more distance between packets. The results are shown for times according to Fig. 1. At the later stages, both in physical and Fourier space, the isolated packet did not show the level of nonlinear activity observed for the interacting packets. It seems that the nonlinear regime of isolated wavepackets requires much larger amplitudes than for the interacting ones.

4 Final Remarks The results demonstrated that the nonlinear regime arises earlier for interacting wavepackets than for isolated ones. The nonlinear regime observed was consistent with oblique type transition. Wavepackets are often observed in natural transition, but the expected subharmonic content is less often reported. This may well be related to the fact that in natural transition the packets are unlikely to occur isolated. The

Fig. 1 Temporal evolution of the u−velocity. Left, physical space. Right, Fourier space.

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relevance is further significant in view that oblique transition is often reported in natural transition experiments. Considering that compressible flows are more unstable to oblique waves, the scenario of oblique transition would be even more prevalent there. The authors acknowledge the financial support from the AFOSR, CNPq and FAPESP.

References 1. Gostelow J.P.: Effect of strong adverse pressure gradients and incident wakes on transition and calming, In R. Govindarajan and R. Narasimba, editors, IUTAM Symposium on laminarturbulent transition, Bangalore-India, December 2004. Springer-Verlag. 2. Westin, J.A., Boiko, A. V., Klingmann, B.G.B., Kozlov V. V. and Alfredson P.H., Experiments in a boundary layer subject to free stream turbulence. Part I, Boundary layer structure and receptivity. J. Fluid Mech., 281,193-218 (1994). 3. Breuer K.S. and Haritonids J.H.: The evolution of a localised disturbance in a laminar boundary layer-Part 1-weak disturbances, J. Fluid Mech., 220, 569-594 (1990). 4. Medeiros M. A. F., Gaster, M.: The production of sub-harmonic waves in the nonlinear evolution of wavepackets in boundary layers. J. Fluid Mech., 399, 301-318 (1999). 5. Silva, H. G., Medeiros M. A. F.: Nonlinear regime of a spanwise modulated wavetrain in a plane Poiseuille flow. Escola de Transiça˜ o e Turbulência-EPTT, São Carlos, Brazil (2008). Published in proceedings of the EPTT.

Fig. 2 The u−velocity in physical and Fourier domains of Wavepackets evolving separately.

Effects of Passive Porous Walls on Hypersonic Boundary Layers Sharon O. Stephen and Vipin Michael

Abstract A theoretical linear stability analysis is used to consider the effect of a porous wall on the first mode of a hypersonic boundary layer on a sharp slender cone. The effect of curvature and of the attached shock are included. The flow in the hypersonic boundary layer is coupled to the flow in the porous layer. The theoretical model of a porous wall developed by Fedorov and his co-workers is used for regular microstructures. The resulting transcendental equations are solved numerically. Neutral solutions are presented, indicating a destabilizing effect of the porous wall. The spatial growth rates determined demonstrate that the porous wall leads to a significant increase in growth rates.

1 Introduction Transition to turbulence in hypersonic flows is associated with amplification of the first and/or second modes. The first mode is the high speed counterpart of TollmienSchlichting waves The second mode is an inviscid instability and believed to be responsible for transition to turbulence on hypersonic slender bodies. Recent experiments by Fedorov et al. [3–5] have shown that a porous coating greatly stabilizes the second mode of the hypersonic boundary layer on sharp slender cones. The effect of the porous coating is to reduce the growth rates of the second mode to a level where they are comparable with those of the first mode (occurring at lower frequencies). In addition, the first mode is observed to be slightly destabilized by the presence of the porous coating. Thus, the first mode may now be more significant Sharon O. Stephen School of Mathematics, University of Birmingham, Birmingham B15 2TT, U.K., e-mail: [email protected] Vipin Michael School of Mathematics, University of Birmingham, Birmingham B15 2TT, U.K., e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_105, © Springer Science+Business Media B.V. 2010

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in the transition process, especially if a disturbance is triggered by wall roughness, since this mode is located close to the wall and governed by viscous effects. The numerical studies described in [3–5], using linear stability theory, also predict stabilization of the second mode for hypersonic boundary layer flow over porous walls. These numerical studies do not include the effects of cone surface curvature or conical divergence of streamlines. We consider the flow of a compressible, viscous fluid over a sharp cone with a porous boundary at hypersonic speeds, with magnitude U0 parallel to its axis. We consider an attached shock. The basic flow is as described in [6]. A boundary-layer solution is introduced in the region close to the surface of the cone. The boundary conditions are no-slip at the surface of the cone (coupled to the porous layer) and appropriate conditions at the shock location. The non-dimensional temperature and viscosity at the surface of the cone are taken to be Tw and µw , respectively. Following [3] and [5] we consider the porous layer on the cone surface to be a sheet perforated with cylindrical blind holes of equal spacing.

2 Linear Stability Problem The analysis follows the linear study of [6] for hypersonic flow over a rigid sharp slender cone. We consider small disturbances of wavenumber α , frequency Ω and azimuthal wavenumber n, for axisymmetric and non-axisymmetric modes. The governing compressible flow equations for large Reynolds number, Re, and large Mach number, M, (just behind the shock) are solved analytically for flow over a porous wall using asymptotic methods. Attention is focused at a location on the surface of the cone with non-dimensional scaled radius r = a and shock location r = rs , which may be described by a triple-deck structure. The asymptotic analysis differs for axisymmetric modes (n = 0) and for non-axisymmetric modes (n 6= 0). The resulting equations for axisymmetric disturbances are scaled following [1] and [2]. Solutions of the equations yields an eigenrelation relating the streamwise wavenumber α and frequency Ω , namely I0 (iα rs )K0 (iα a) − I0 (iα a)K0 (iα rs ) Ai′ (ξ0 ) = −(iα )1/3 AY + iα . Ai( ξ )d ξ I0 (iα rs )K1 (iα a) + I1 (iα a)K0 (iα rs ) ξ0

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Here ξ0 = −i1/3 Ω α −2/3 , Ai(ξ ) is the Airy function, Kn (z) and In (z) are the usual 1/8 3/8 modified Bessel functions, and Ay = Re−1/8 µw λ 1/4 Tw (M 2 − 1)3/8 AY . The admittance, Ay , is a function of the disturbance frequency and depends on the physical properties of the flow and the porous layer. In addition, λ is the boundary-layer skin friction. The angular frequency of disturbance propagation through the pore is −1/4 −3/4 ω = (R/Re)Re1/4 µw λ 3/2 Tw (M 2 − 1)1/4 Ω , where R is the Reynolds number based on displacement thickness.

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3 Results The eigenrelations (1) and (2) are solved with porous wall parameters chosen to correspond to the previous theoretical and numerical studies. Fig. 1 shows the neutral values of Ω . For axisymmetric modes these are slightly reduced from those corresponding to a solid wall. However, for larger values of the radius for nonaxisymmetric modes they are significantly lower. The porous wall does not significantly alter the neutral values of α . Spatial instability occurs when Im(α ) ≡ αi < 0. Fig. 2 shows the spatial growth rates for n = 0 and n = 1. The growth rates are significantly larger for a porous wall compared to a solid wall. Thus, the porous wall has a destabilising effect on the axisymmetric and non-axisymmetric modes.

4 Future Work Studies are underway to consider the effects of the porous wall parameters on the linear stability problem. The effects of rarefaction may be included in a straightforward manner. Comparisons of the resulting spatial growth rates will be made with

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the available experimental results. These will aid in determining whether the first mode may be important in the transition process. Acknowledgements This effort is partially sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655-08-1-3044. V.M. acknowledges additional financial support from the School of Mathematics, University of Birmingham for his PhD studies.

References 1. Cowley, S.J., Hall, P.: On the instability of the hypersonic flow past a wedge. J. Fluid Mech. 214, 17–42 (1990) 2. Duck, P.W., Hall, P.: On the interaction of Tollmien-Schlichting waves in axisymmetric supersonic flows. Q. J. Mech. Appl. Maths 42, 115–130 (1989) 3. Fedorov, A.V., Malmuth, N.D., Rasheed A., Hornung, H.G.: Stabilization of hypersonic boundary layers by porous coatings. AIAA J. 39, 605–610 (2001) 4. Fedorov, A., Shiplyuk, A., Maslov, A., Burov, E., Malmuth, N.: Stabilization of a hypersonic boundary layer using an ultrasonically absorptive coating. J. Fluid Mech. 479, 99–124 (2003) 5. Fedorov, A., Kozlov, V., Shiplyuk, A., Maslov, A., Malmuth, N.: Stability of hypersonic boundary layer on porous wall with regular microstructure, AIAA J. 44, 1866–1871 (2006) 6. Seddougui, S.O., Bassom, A.P.: Instability of hypersonic flow over a cone. J. Fluid Mech. 345, 383-411 (1997)

Global Instabilities in Wall Jets Gayathri Swaminathan, A Sameen, and Rama Govindarajan

1 Introduction A wall jet, produced when fluid is blown tangentially across a flat surface, occurs in various applications. The linear instability of this flow has been studied under the assumption of locally parallel flow [2,4] and weakly non-parallel flow, i.e., using the parabolized stability equations (PSE) [7] but to our knowledge, its global stability has not. Close to its origin, a wall jet may be approximated by the combination of a shear layer and a Blasius boundary layer, see e.g. [7]. Far downstream, the velocity assumes a similarity profile [1], with the maximum streamwise velocity Umax ∼ x−1/2 , and the jet thickness δ ∼ x3/4 , where x is the streamwise coordinate. The wall normal location y where U = 0.5Umax , and decreasing, is defined as δ . The former profile displays a dominant outer instability mode due to the inflexional region, and a viscous inner or near-wall mode [7]. The two exist, though sometimes less distinctly, in the Glauert profile too [2]. Experiments by [3] on fully developed wall jets show a dominance of large scale disturbances which peak in the outer region.

Gayathri Swaminathan Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Banglaore, India. e-mail: [email protected], A Sameen Dept of Aerospace Engineering, IIT Madras, Chennai, India. e-mail: [email protected], Rama Govindarajan Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Banglaore, India. e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_106, © Springer Science+Business Media B.V. 2010

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2 Formulation and Numerical Method A two-dimensional disturbance streamfunction of the form ψ (x, y,t) = φ (x, y)e(−ιω t) has been assumed where ω is the complex frequency whose imaginary part gives the disturbance growth rate. A linear stability equation is then derived from the Navier-Stokes equations in the usual manner. No-slip boundary condition at the wall and zero disturbances in the free-stream are applied. The streamwise boundary conditions are discussed below. The equation is discretized with chebyshev spectral method in both directions with suitable stretching, with n points in x and m points in y. This results in a generalized eigenvalue problem of size nm × nm for temporal stability, which is solved using the package LAPACK, which uses QZ algorithm. A heavy sponging is applied at the exit region of the domain to avoid spurious reflections from the exit boundary.

3 Results First, we solve the global equation but still retain the wave-like disturbance assumption, by enforcing the Robin boundary conditions (φx = ιαφ ) at the inlet and exit. Next, the wave-like assumption is released and a linearly extrapolated boundary condition at the inlet and exit of a long domain is imposed. This will help us understand whether a wave-like assumption is applicable in the streamwise direction. First, the disturbance is assumed to be wave-like. Domains as long as the wavelength 2π /α of the specified mode are considered with Robin boundary conditions. As compared to the parallel analysis the neutral boundary is shifted to higher Reynolds numbers (Re), and a clearer separation of the inflexional and near-wall modes is seen in figure 1. The near-wall modes go unstable first, unlike in 1D predictions, and there is no stable region at higher Re. While a weakly non-parallel analysis can predict only a viscous mode or an inflectional mode independently, a global analysis displays modes which have both the viscous and inflectional mechanisms operating simultaneously, as in figure 2. The dominant wavelength in the inflexional outer region is approximately three times that of the near-wall region. The results are in qualitative agreement with experiment [3]. Thus global stability analysis is handy for flows where two instability mechanisms operate simultaneously. Next,releasing the assumption of wave-like disturbances, we consider a long domain of length L = 120δ , with inlet Re = 200; n = 251; m = 41. The downstream 50% of L is sponged, bringing down Re at the exit to 10% of its inlet value. The scalings of the self-similar wall jet indicate that α should increase downstream as α ∼ x(3/4) . For a global mode, a local wavelength can be extracted from its wavelet transform. In contrast to the parallel results, we find global modes belonging to the same spectrum whose ‘wavenumbers’ are (i) constant (ii) functions of x, as well as (iii) functions of x and y. One typical example is shown in figure 3 where a single global mode has small scale structures close to the wall and large scale structures

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Fig. 4 Wavelet transform of the lower mode in fig 3 at two different y locations, indicating α is a function of y.

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away from the wall. The wavelet transform of the mode shown in the lower half of figure 3 at two different y locations is shown in figure 4. This indicates α is indeed a function of y and is not explicitly reported before, to our knowledge [8]. In the inflectional region we may write, to the lowest order i(ω − αφ ′ )D2 ∼ iαφ ′′′ . In the viscous region, three terms could become equally large, i(ω − αφ ′ )D2 ∼ R−1 (D4 ) [−ıαφ ′′′ ]. We have seen that one may have viscous and inflexional mechanisms contributing in equal measure to the same mode, which can be seen as the existence of two critical layers for a wall jet. This aspect is being studied.

4 Conclusions Global stability analysis with Robin boundary conditions cleary demarcates the inflectional and viscous modes in the neutral boundary. Co-existence of viscous and inflectional mechanisms in a single mode is captured and two ’critical layers’ are indicated. Wave-like boundary condition is not good for this flow, as the local wavenumber is a function of both x and y. Acknowledgements We gratefully acknowledge the Defence Research Development Organization, India for financial support.

References 1. M. B. Glauert. (1956) The wall jet. Journal of Fluid Mechanics. 1, (06) 2. D. H. Chun, and W. H. Schwarz. (1967) Stability of the Plane Incompressible Viscous Wall Jet Subjected to Small Disturbances. Phys. Fluids. 10(5) 3. R. A. Bajura and M. R. Catalano. (1975) Transition in a two-dimensional plane wall jet. Journal of Fluid Mechanics. 70, (04), pp.773-799 4. P. Mele and M. Morganti. (1986) Behavior of Wall Jet in Laminar-to-Turbulent Transition. AIAA 0001-1452,24(6): 938–939. 5. C. Cossu and J. M. Chomaz. (1997) Global Measures of Local Convective Instabilities. Phys. Review Letters. 78, 23, 4387-4390 6. U. Ehrenstein and F. Gallaire. (2005) On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. Journal of Fluid Mechanics. 536, pp 209-218 7. O. Levin, V. G. Chernoray, L. Lofdahl and D. S. Henningson. (2005) A study of the Blasius wall jet. Journal of Fluid Mechanics. 539: 313-347 8. G. Swaminathan, A. Sameen and R. Govindarajan. Two dimensional instabilities in fullydeveloped wall jets. Preprint.

Spatial Optimal Disturbances in Three-Dimensional Boundary Layers David Tempelmann†,1 , Ardeshir Hanifi∗,† , and Dan S. Henningson†

Abstract A parabolised set of equations is used to compute spatial optimal disturbances in Falkner-Skan-Cooke boundary layers. These disturbances associated with maximum energy growth initially take the form of vortices which are tilted against the direction of the mean crossflow shear. They evolve into bended streaks while traveling downstream and finally into crossflow disturbances when entering the supercritical domain of the boundary layer. Two physical mechanisms, namely the lift-up and the Orr-mechanism, can be identified as being responsible for nonmodal growth in three-dimensional boundary layers. A parametric study is presented where, amongst others, the influences of pressure gradient and sweep angle on optimal growth are investigated. It turns out that substantial disturbance growth is already found in regions of the flow where modal disturbances are damped.

1 Introduction In the case of a Blasius boundary layer the non-modal growth mechanism yielding streaks and the modal-growth mechanism yielding Tollmien-Schlichting waves were found to dominate at points well separated in the frequency-wavenumber plane (see e.g. [1], [2] and [3]). In the three-dimensional case however, modal and nonmodal growth complement each other as is stated by [4] who performed a study on temporal optimal disturbances in swept boundary layers. They argue that non-modal growth may provide proper initial conditions for modal disturbances and thus constitutes a preferential receptivity path for the selection of crossflow modes. A spatial framework is however needed to describe the initiation of modal instabilities due to non-modal growth. In the following we present a method to compute spatial optimal disturbances in three-dimensional boundary layers and show some results. †

Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden Swedish Defence Research Agency, FOI, SE-164 90 Stockholm, Sweden 1 e-mail: [email protected] ∗


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D. Tempelmann, A. Hanifi, D. S. Henningson

2 Methodology Our objective is to describe both modal and non-modal spatial disturbance growth in a 3D boundary layer using a parabolised set of equations. The well-known parabolised stability equations (PSE) (see e.g. [5]) can not be readily employed as they only describe the evolution of one modal disturbance. However, they can be modified to also account for non-modal growth which is known to result from a superposition of discrete modes. A standard procedure when deriving the PSE is to introduce disturbances of the form Zx q′ (x, y, z,t) = q(x, z) exp i α (x′ )dx′ + iβ y − iω t (1) x0

into the linear disturbance equations, where β represents the spanwise wavenumber, α the chordwise wavenumber and ω the frequency. (x, y, z) denote the chordwise, spanwise and wall-normal directions and (u,v,w) the respective disturbance velocities. p is the pressure and q = (u, v, w, p)T . Both the shape function q and the chordwise wavenumber α are functions of x. In order to resolve this ambiguity an auxiliary function is introduced in the classical PSE which ensures that the oscillations as well as the growth of the disturbance q′ are absorbed by the exponential part of (1). This auxiliary condition is usually enforced in an iterative manner using a local eigenmode and the corresponding complex α as an initial guess at the first station. However, this procedure does not allow for describing non-modal growth and a different approach is chosen here. We compute a real-valued wave vector iteratively as follows. I) We initially assume that the wave vector is perpendicular to the external streamline. II) The governing equations which we will arrive at in the following, are solved using the corresponding α . III) From the solution in II) a new wave vector is determined based on the dominating flow quantity. The latter is chosen to be the velocity component perpendicular to the wave vector (i.e. tangential to the corresponding line of constant phase) of the previous step as streaks and crossflow modes will dominate. Having obtained a new α we go back to step II) and repeat this procedure until the disturbance growth is converged. This way the oscillations of q′ are absorbed by the exponential part of (1) whereas the growth of the disturbance is absorbed by the shape function q. The next step towards obtaining a parabolised set of equations is to identify higher order terms. As we are interested in both non-modal and modal growth we apply a composite scaling. It consists of the boundary layer scaling which is usually associated with non-modal growth (see [1], [2]) and a PSE scaling related to the one used by [3] which is associated with modal growth. Both scalings assume a slow variation of q in x. Identifying and neglecting terms of higher order then leads to a parabolised set of equations of the form Aq + B(∂ q/∂ z) + C(∂ 2 q/∂ z2 ) + D(∂ q/∂ x) = 0, where A, B, C and D denote linear operators. The optimal disturbance q(x0 ) that yields maximum energy growth at a specific position x1 , for a given wavenumber β , frequency ω , Reynoldsnumber Re and sweep angle Λ are obtained employing adjoint based optimisation.

Spatial Optimal Disturbances in Three-Dimensional Boundary Layers

591

3 Results In the following we present results obtained for different Falkner-Skan-Cooke boundary layers which are defined through a velocity at the boundary layer edge given by Ue /Q = (x/L)βH /(2−βH ) where Q and L define the Reynolds number Re = QL/ν which was chosen as Re = 106 for all considered boundary layers. βH is the Hartree parameter referring to the pressure gradient. Figure 1 depicts the physical mechanisms responsible for non-modal growth in 3D boundary layers. These are the lift-up mechanism where momentum is transfered to the streamwise component due to vortical motion and the Orr-mechanism where the initially tilted vortices gain energy while being erected due to the mean crossflow shear. Figure 2 depicts N-factors of optimal growth of stationary disturbances. It shows that substantial growth is already yielded upstream of the supercritical domain of the boundary layer and that non-modal growth has an effect on the predicted dominating wavenumber (compare dashed and solid line). Figure 3 compares optimal growth to modal growth of corresponding crossflow modes. The difference between both can be related to the optimal initial amplification of modal disturbances which is shown in figure 3(b). A large potential for non-modal growth becomes apparent from figure 3(a). As the initial amplification is largest for the smallest sweep angle we may conclude that non-modal growth becomes more important for smaller sweep angles and thus more stable boundary layers. Employing Newton iterations also those parameters β , ω and x0 were computed which yield the maximum possible growth at a specified position x1 . These computations showed that maximum growth is obtained for non-stationary disturbances. As both freestream turbulence and wall-roughness are likely to cause disturbances similar in shape to the optimal disturbances non-modal growth may be related to a receptivity mechanism for modal instabilities in 3D boundary layers. Future work will therefore determine how optimal growth can be associated with receptivity coefficients. (b) 3

1.5 0 3

0

1.5

z

1.5

4

0 3

2

1.5

0 0 1 |us|,|vs|,|w|

z

z

z

3 z

(a)

−6

−3

0 ys

3

6

0 −0.02 V

s

0

−6

−3

0 y

3

6

s

Fig. 1 Optimal disturbances at different positions x. ω = 0, βH = 0.1 and Λ = 45◦ . Vectors denote vs and w. Contours represent us . The subscript s denotes a rotated coordinate system where xs is tangential to the external streamline. (a) Lift-up effect: x0 = 0.01 (upper), x1 = 0.13 (lower). Left: Absolute values of us (—), vs (---), w (· · ·). Right: Projection of q onto (ys , z)-plane. (b) Orr-mechanism: x0 = 0.01 (upper), x = 0.033 (center), x1 = 0.13 (lower). Left: Mean crossflow component. Right: Projection of q′ onto (ys , z)-plane.

592

D. Tempelmann, A. Hanifi, D. S. Henningson

(a)

(b)

1

1

3.05

2.5

0.8

3.35

2

3

1.5

2.15

β

β

3.65 4

0.6

2.6

0.8

3.5

0.6

3.95

4.5 4.25 5

0.4

0.4

4.55 3.35 2.75

0.2

0.2

0.4

x

0.6

0.8

0.2

1

0.2

0.4

1

0.6

x

0.8

1

1

Fig. 2 Contours of N-factors of optimal growth for steady disturbances where x0 = 0.01. Optimal spanwise wavenumber β at each position x1 considering optimal growth (---). Optimal β considering solely modal growth (—). The shaded region represents the supercritical domain. (a) βH = 0.1, Λ = 45◦ . (b) βH = −0.05, Λ = 45◦ .

(a)

6

35

non-modal + modal

°

35

25° 0,CF

25°

45

4000

E

opt

N

,N

PSE

2

°

6000

°

4 3

(b)

45°

5

1

2000

0

solely modal

−1 0.2

0.4

0.6 β

0.8

1

0

0.2

0.4

β

0.6

0.8

Fig. 3 (a) N-factors of optimal growth (upper curves) and solely modal growth of crossflow modes (lower curves) at x1 = 1 for x0 = 0.16, ω = 0, βH = 0.1 and three different sweep angles. (b) Optimal initial amplification of crossflow modes at x0 = 0.16.

Acknowledgements This work is supported by the European Commission through the FP6 project “Telfona” (Contract No AST4-CT-2005-516109).

References 1. P. Andersson, M. Berggren and D. S. Henningson: Optimal disturbances and bypass transition in boundary layers. Phys. Fluids. 11(1), 134–150 (1999) 2. P. Luchini: Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289–309 (2000) 3. Ori Levin and D. S. Henningson: Exponential vs Algebraic Growth and Transition Prediction in Boundary Layer Flow. Flow, Turb. and Comb. 70, 183–210 (2003). 4. P. Corbett and A. Bottaro: Optimal linear growth in swept boundary layers. J. Fluid Mech. 435, 1–23 (2001). 5. T. Herbert: Parabolized Stability Equations. Annu. Rev. Fluid Mech. 29, 245–283 (1997)

Influence of turbulence scale and shape of leading edge on laminar-turbulent transition induced by free-stream turbulence M. V. Ustinov and S. V. Zhigulev

Abstract Linear receptivity theory describing generation of boundary layer disturbances by FST accounting for final radius of leading edge was developed. Basic findings of this theory - enhanced receptivity of blunt-nosed-plate boundary layer and final amplification coefficient of FST induced perturbations in sharp-nosedplate boundary layer - were verified experimentally.

1 Introduction Study of laminar-turbulent trannsition induced by free-stream turbulence (FST) in a boundary layer is an urgent problem in applied engineering, for instance for prediction of transition on turbine blades or airplane wing. General consensus is that boundary layer disturbances in this conditions grow proportionally to Reynolds number based on the boundary layer thickness. It means that transition Reynolds number should be determined by the turbulence intensity only. However the discrepancy in published observations of transition [1-4] is substantial. From this it follows that transition location is not entirely determined by turbulence level, but it is influenced by several factors which are not taken into account. The most obvious is the influence of length scale of FST. Despite of several studies focused on this factor [2] there is no general agreement about influence of turbulence scale on transition. Theoretical work of M. Goldstein [4] showed that another important factor should be the shape of leading edge. Present work is devoted to investigation of influence of FST scale and shape of leading edge on the transition in flat-plate boundary layer.

M.V. Ustinov, S.V. Zhigulev TsAGI, 1 Zhukovsky str., Zhukovsky, Russia, e-mail: [email protected] P. Schlatter and D.S. Henningson (eds.), Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries 18, DOI 10.1007/978-90-481-3723-7_108, © Springer Science+Business Media B.V. 2010

593

594

M. V. Ustinov, S. V. Zhigulev

2 Linear receptivity theory to FST Consider the velocity perturbations produced by free-stream turbulence in a boundary layer at the flat plate with blunt leading edge. The oncoming flow has mean velocity u∞ and r.m.s. pulsations u′ = Tu u∞ , where turbulence level Tu is assumed to be small enough to ensure linear development of perturbations. We introduce non-dimensional variables using free-stream velocity and viscose length l = ν /u∞ as scales. The oncoming turbulence is assumed to be isotropic with Karman’s energy spectrum E(k) = Tu2 LF(k1 ) where k1 = kL is normalized wave-number. Turbulence scale L and radius of nose r are assumed to be large. At first the receptivity of boundary layer to spanwise- and time- periodic free-stream disturbances called as vertical modes was found. When the radius of nose is small with respect to L, the boundary layer is most receptive to streamwise vertical modes. In the opposite case of blunt nose when r >> L the most dangerous perturbations are vertical modes. Than, based on solution for single vortical mode the response of boundary layer to free-stream turbulence was found. For sharp-nosed plate solution is sumilar to found in [5] and takes form. √ u = Tu LG|| ′

√

x z ,√ L x

; G|| =

Z

∞

0

1/2 F(k1 ) z 2 x H|| k1 2 , √ dk1 k1 L x

(1)

where kernel function H|| is defined by integration of solution for perturbations from single streamwise mode over frequency and angle of wave-vector. Solution for velocity pulsations in the blunt-nosed plate (for r >> L) was found along the similar manner r u = Tu √ G⊥ L ′

√

x z ,√ L x

; G⊥ =

Z

∞ 0

1/2 F(k1 ) z 2 x H⊥ k1 2 , √ dk1 L x k12

(2)

Universal functions G|| and G⊥ describing the amplification of pulsations in the boundary layer are plotted in Figure 1. From this Figure it is seen, that disturbances produced by turbulence in boundary layer on the infinetely thin plate do not grow √ as x , but they reach maximum for x˜L2 and than decay. Maximal amplification √ 1 5 of pulsations is scaled as ˜0.06 L. Special analysis reveals that u′ ˜TuL− 3 x 3 √for x

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