IUTAM Symposium on Unsteady Separated Flows and their Control: Proceedings of the IUTAM Symposium (IUTAM Bookseries)

IUTAM Symposium on Unsteady Separated Flows and their Control IUTAM BOOKSERIES Volume 14 Series Editors G.M.L. Gladwe...

Author: Marianna Braza | K. Hourigan

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IUTAM Symposium on Unsteady Separated Flows and their Control

IUTAM BOOKSERIES Volume 14 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 http:www.springer.com/series/7695

Marianna Braza Editors

•

Kerry Hourigan

IUTAM Symposium on Unsteady Separated Flows and their Control Proceedings of the IUTAM Symposium “Unsteady Separated Flows and their Control”, Corfu, Greece, 18–22 June 2007

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Editors Marianna Braza Institut de Mécanique des Fluides de Toulouse Unité Mixte de Recherche CNRS 5502 Allée du Prof. Camille Soula 31400 Toulouse France [email protected]

Kerry Hourigan Monash University Division of Biological Engineering Clayton VIC 3800 Australia [email protected]

ISSN 1875-3507 e-ISSN 1875-3493 ISBN 978-1-4020-9897-0 e-ISBN 978-1-4020-9898-7 DOI 10.1007/978-1-4020-9898-7 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009928840 c Springer Science+Business Media B.V. 2009 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 Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This Volume is the Proceedings of the IUTAM Symposium on Unsteady Separated Flows and Their Control held in Corfu, Greece, 18–22 June 2007. This was the second IUTAM Symposium on this subject, following the symposium in Toulouse, in April 2002. The Symposium consisted of single plenary sessions with invited lectures, selected oral presentations, discussions on special topics and posters. The complete set of papers was provided to all participants at the meeting. The thematic sessions of this Symposium are presented in the following:

Experimental techniques for the unsteady flow separation Theoretical aspects and analytical approaches of flow separation Instability and transition Compressibility effects related to unsteady separation Statistical and hybrid turbulence modelling for unsteady separated flows Direct and Large-Eddy Simulation of unsteady separated flows Theoretical/industrial aspects of unsteady separated flow control

This IUTAM Symposium concerned an important domain of Theoretical and Applied Mechanics nowadays. It focused on the problem of flow separation and of its control. It achieved a unified approach regrouping the knowledge provided from theoretical, experimental, numerical simulation and modelling aspects for unsteady separated flows (incompressible and compressible regimes) and included efficient control devices to achieve attenuation or suppression of separation. The subject areas covered important themes in the domain of fundamental research as well as in the domain of applications. These themes received a great deal of impetus from international research groups, including research Institutes and industrial companies especially in Aeronautics in various countries and associating leading Government Programs. This symposium brought together scientific communities working on the problems related to the understanding of the prediction and control of unsteady, separated flows. The meeting addressed physical aspects of the dynamics of unsteady separation, as well as the state-of-the-art of modelling approaches, especially at high Reynolds numbers. The understanding of the physical mechanisms of the unsteady separation dynamics is a cornerstone for the purpose of efficient turbulence modelling of unsteady separated flows, a challenging issue in a number of engineering

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applications including aeronautics, aeroelasticity, space and land vehicles. A round table discussion took place on Monday 18th June 2007 between the IUTAM symposium participants and those of the symposium “Hybrid RANS-LES methods”, 17–18 June addressing the main problems between theoretical aspects and applications domain with respect to the Symposium’s theme. Attended this Symposium were scientists working in the experimental investigation of unsteady separated flows and those working in the numerical simulation and turbulence modelling of these flows, stimulating discussion and advancing the knowledge of the related physical mechanisms. In this way, the symposium contributed to a better insight of this important category of flows from a fundamental and applied research point of view, by means of a synergy between the three main approaches: theoretical, experimental and prediction methods. The main objectives of this symposium were fulfilled. Among the invited 17 Key-Note speakers, two were devoted to provide the major outcomes from the European research programs in aeronautics, dealing with unsteady separated flows, the DESIDER (Detached Eddy Simulation for Industrial aerodynamics) and the UFAST (Unsteady effects in Shock wave induced separation). Seventy-six oral presentations and six poster sessions were included in this Symposium. A best-poster award was attributed to S.C. Luo, T.T.L. Duong and Y.T. Chew of the National University of Singapore, Singapore for their poster, “Flow separation of a rotating cylinder” by the Scientific Committee during the gala dinner held in the Achilleion Palace and Museum of Corfu. The significant outcomes of this symposium are the following : Advances in the physical knowledge of the unsteady separated flows are achieved by means of the Particle Image Velocimetry (especially the three-component PIV and time-resolved PIV as well as simultaneous combination of both), that provides a detailed physical analysis of the unsteady separation and of the detached flow around fixed and moving body configurations. A considerable number of theoretical, numerical and experimental studies were devoted to the instability, transition and control of unsteady separation, including also the physics of compressibility. Among these aspects, the instability and transition related to VIV (Vortex Induced Vibrations), wall rotation effects and dynamic stall were an important issue in the topic, as well as analytical approaches that are able to capture and evaluate the separation. Configurations of cylinders in tandem, of elliptical cross-section cylinders, of oblate freely rising bodies, of impulsively starting motion around bluff bodies and of flows around curved or tappered cylinders and spheres were considered. An original issue of the Symposium related to this topic was the numerical study of flexible bodies leading to very complex wake structure in the context of fluid-structure interaction. A detailed insight of the sensitivity to external disturbances has been presented for separated boundary layers, leading edge separation on slender airfoils at incidence, separation associated with wall jets and in channels enforced by suction. Asymptotic analysis in the limit of large Reynolds number proved that the marginalseparation equations govern these vastly different situations. A novel separating shear layer scaling has been deduced from the unsteady motion and first analytical and numerical results were produced, representing the rotational/irrotational

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interaction of the separating shear layer with the external potential flow. Furthermore, amplitudes causing bubble bursting have been evaluated and found to increase with increasing forcing frequencies, as observed experimentally. The transverse instabilities in separating boundary layers have been studied, based on the perturbed Navier–Stokes system using Floquet theory. The stability and sensitivity of separated flows past freely moving disks, orbiting cylinders and in lid-driven cavities have been investigated, as well as the axisymmetric absolute instability of swirling jets. New analytical approaches in 2D and 3D and Navier–Stokes simulations have been presented for flows in vascular vessel bifurcations involving wall flexibility, and the role of separation associated with the longer or shorter length scales has been studied. The influence of 3D parturbations in the stability of 3D separated flows have been analytically studied for subsonic and supersonic flows, as well as the receptivity of separated boundary layers to external sound waves. Compressibility effects on unsteady separation were investigated by experimental, theoretical and numerical approaches, for transonic and supersonic flows. In particular, the unsteady shock-boundary layer interaction (SBWLI) and consequent separation have been analysed. The Proper Orthogonal Decomposition, POD, approach had a significant impact in analysing the organised modes yielded from SBWLI. The buffeting phenomenon was the object of a number of successful presentations. Efficient Reduced order modelling (ROM) for compressible and incompressible flows have been also presented. The organised mode development due to the compressibility effects and generation of Bénard-von Kármán instabilities interacting with buffet modes and their control have been studied in detail. The analysis of these effects is an important outcome of the Symposium, contributing to the fundamental research of compressibility phenomena arising in aeroelasticity. Significant advances in the flow physics of unsteady separation have been also achieved by using Direct Numerical Simulation, (DNS), and Large Eddy Simulation, (LES), in the low and moderate Reynolds number ranges. A number of DNS studies have analysed the unsteady separation and instability modes around bluff bodies and wings, as well as separation in biomechanical flows (aneurysms). Concerning turbulence modelling, in the context of LES, the regularisation subgrid modelling approach and the Variational Multi-Scale approach (VMS) have been investigated, as well as applications of LES and DNS for SBWLI analysis. Considerable achievements in the prediction of high-Reynolds number unsteady separated flows have been made by using the Detached Eddy Simulation, (DES) and adapted/modified statistical turbulence modelling approaches in the context of Unsteady Reynolds Averaged Navier–Stokes, (URANS), for both academic unsteady flows around bodies as well as flows around real aircraft configurations. Advanced statistical turbulence modelling was presented, sensitive enough to capture the coherent structures (OES, Organised Eddy Simulation) and involving the anisotropic eddy-viscosity concept. In the context of hybrid approaches, the embedded hybrid RANS-LES, the hybrid RANS-VMS (Variational Multi-Scale) and the Detached Eddy Simulation (DES) using among other models algebraic-stress modelling for the RANS part were investigated. Among the configurations considered, the Boeing 777 Nose-Gear Cavity, wing-body junction flows, dynamic stall wings as well as

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cavity flows including those of the stratospheric observatory (infrared astronomy – American-German research program SOFIA) were presented. The control of flow separation has been achieved by using electromagnetic forcing, plasma actuators, piezoelectric actuators and surface plasma. Vortex shedding control has been achieved by synthetic jets and Vortex Generators (VGS) and spanwise sinusoidal perturbation (SPPM). Efficient passively controlled studies by means of LES have been presented, concerning transonic cavity flows, as well as passive drag control of turbulent wakes. Biomimetic flows around fish and natural flyers have been studied and optimised by flow control. Flow control of unsteady separation in bluff bodies undergoing VIV has been achieved by small vibrating rods, active/passive ventilation methods, and rotational control. Small rotating rods have been also used for control of vortex breakdown in a closed cylinder. The flow mechanisms employed in actively controlled, fish-like bodies are investigated to optimise their locomotion and maneuvering performance. Concerning biomimetics, the hydrodynamics of beating cilia has been investigated by DNS involving immersed boundary method and the PALM coupler. This approach is useful for the control (passive or active) of the boundary layer flow for the suction of airfoils using a ciliated wall. Flow forcing techniques, as well feedback stabilisation techniques, were also efficiently used for separated bluff bodies. Stochastic feedback flow control has been applied to flexible flapping wing emulating natural flyers, to create thrust and to reduce drag. Optimal control of separated flows has been achieved by adjoint equations, as well as robust closed-loop control methods applied to highlift devices and sub-optimal control approaches applied in separated channel flows. Concerning feedback control strategies, the approach of linear proportional control has been applied in the flow over a sphere. A new strategy, “Multiscale retrograde estimation and forecasting of chaotic non-linear systems”, concerning multiscale complexity and model uncertainty, has been presented. The ensemble of these aspects constitutes major outcomes of the symposium concerning the state-of-the-art for control devices to attenuate/suppress instabilities and unsteady separation. Most challenging research outcomes examined in this symposium are the prediction of turbulent unsteady separated flows around bodies at high Reynolds number, involving also the complex of fluid-structure interaction phenomena. It has been seen that the experimental, DNS and LES approaches of complex unsteady separated flows have proven quite promising, despite the fact that the two latter ones are limited in the relatively low Reynolds number range. This difficulty is overcome by having associated URANS and LES in the context of hybrid turbulence modelling. These last approaches provided a very useful panel of complex flow physics to model unsteady separated 3D flows in the high Reynolds number regime, which is of priority interest for industrial applications. Therefore, a major outcome of this symposium concerning future issues is the fact that there is a need for increasing the knowledge of unsteady flow physics by the synergy of well focused physical experiments in the higher Re-range (e.g. tomographic PIV) and the knowledge coming from the DNS and LES studies. These efforts contribute to improve the above-mentioned modelling methodologies in the higher Reynolds number range, to increase the predictive capabilities of strongly detached flows and especially of the

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fluid-structure interaction phenomena around moving and deformable bodies. These topics are of a significant interest to applications in the domain of flow-induced vibrations (FIV), in unsteady aerodynamics and in aeroelasticity. CNRS-IMFT, France Monash University, Australia

Marianna Braza Kerry Hourigan

International Scientific Committee A. Bottaro A. Kluwick, IUTAM representative B. J. Geurts G. E. Karniadakis C. Norberg F. Smith F. Thiele G. Tzabiras

Università di Genova, Italy University of Vienna, Austria University of Twente, Netherlands Brown University, USA Lund Institute of Technology, Sweden University College, London Technical University, Berlin National Technical University of Athens, Greece

Local Organizing Committee M. Braza (Chairman) P. Chassaing G. Harran A. Sevrain S. Saintlos-Brillac G. Barbut R. Bourguet R. El Akoury G. Martinat M. Sabater D. Bourrel G. Martin Y. Exposito W. Haase Y. Hoarau G. Tzabiras K. Georgiou A. Papageorgiou, M. Koufioti, N. Rapti

CNRS-IMFT, France CNRS-IMFT, France CNRS-IMFT, France CNRS-IMFT, France CNRS-IMFT, France CNRS-IMFT, France CNRS-IMFT, France CNRS-IMFT, France CNRS-IMFT, France CNRS-IMFT, France CNRS-IMFT, France CNRS-IMFT, France CNRS-IMFT, France EADS, Germany University of Strasbourg NTUA, Greece NTUA, Greece Heliotopos Conferences, Greece

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Host Institutions The symposium was organised by the: Institut de Mécanique des Fluides de Toulouse (IMFT), affiliated with the: Centre National de la Recherche Scientifique (CNRS) Institut National Polytechnique de Toulouse (INPT) Ecole Nationale Supérieure d’Electrotechnique, Electronique, Informatique, Hydraulique et Télécommunications (ENSEEIHT) Université Paul Sabatier (UPS) National Technical University of Athens (NTUA).

Preface

IUTAM “Unsteady Separated Flows and their Control” Symposium, Corfu-Greece, 18–22 June 2007. Best poster award attribution during the gala dinner by Marianna Braza and Kerry Hourigan to the co-authors S.C. Luo, T.T.L. Duong and Y.T. Chew of the National Univ. of Singapore, for their poster “Flow separation of a rotating cylinder”, at the Achilleion Palace, on 20th June 2007.

Participants of the Symposium in the garden of the hotel “Chandris” – Corfu, next to the Conference room, Friday 22nd June 2007.

Detailed Programme

Sunday 17 June 2007: 18:00–20:00

Early registration

Monday 18 June 2007: 09:00–11:00 11:00–11:30

Registration Welcome address

11:30–12:45

Session I.1 Experimental techniques for the unsteady flow separation

11:30–12:15

Effects of unsteady separated flow phenomena in vortex-induced vibrations C.H.K. Williamson, Cornell Univ., USA; Opening lecture* PIV measurements of the flow around oscillating cylinders at low KC numbers D. Sumner, H.B. Hemingson, D.M. Deutscher & J.E. Barth (Univ. of Saskatchewan, Canada) Measurement of instantaneous velocity and surface topography of a cylinder at low Reynolds number A. Fouras, D. Lo Jacono, G. J. Sheard & K. Hourigan* (Monash Univ., Australia)

12:15–12:30

12:30–12:45

12:45–14:00

Lunch

14:00–16:00


14:00–14:45

Unsteady flow around impulsively stopped bluff bodies T. Leweke, M.C. Thompson, G.J. Sheard, L. Schouveiler & K. Hourigan, IRPHE, France - FLAIR, Australia; Invited keynote lecture*

For the presentations marked by () there is no article available

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14:45–15:00

15:00–15:15

15:15–15:30

15:30–15:45

Detailed Programme

Coherent and turbulent process analysis in the flow past a circular cylinder at high Reynolds number R. Perrin, M. Braza, E. Cid, S. Cazin, P. Chassaing, C. Mockett, T. Reimann & F. Thiele (IMFT, France; TU-Berlin, Germany) Wake behind a sphere: experimental and numerical investigations K. Gumowski, J. Miedzik, S. Goujon-Durand, G. Bouchet, P. Jenffer & J.E. Wesfreid* (ESPCI–PMMH, IMFS, France) Investigation of aerodynamic capabilities of flying fish in gliding flight H. Park & H. Choi (Seoul National Univ., Korea) Effect of unsteady separation on an automotive bluff-body in cross-wind M. Gohlke, J.F. Beaudoin, M. Amielh & F. Anselmet (PSA, IRPHE, France)

16:00–16:45

Coffee break

16:45–18:15

Session II.1 Statistical and hybrid turbulence modeling of unsteady separated flows

16:45–17:30

18:30–19:30

Prediction methodologies for non-stationary turbulent flows T. B. Gatski, Laboratoire d’Etudes Aérodynamiques, Poitiers, France - Center of Coastal Physical Oceanography and Ocean, Earth and Atmospheric Sciences, Old Dominion Univ. Norfolk; Invited keynote lecture* Flow prediction around an oscillating NACA0012 at ReD1000 000 O. Frederich, U. Bunge, C. Mockett & F. Thiele (TU-Berlin, IVM Automotive Wolfsburg GmbH, Germany) Two-velocities hybrid RANS-LES of a trailing edge flow J.C. Uribe, N. Jarrin, R. Prosser & D. Laurence (Univ. of Manchester, UK - EDF, France) Assessment of flow control devices for transonic cavity flows using DES and LES G. Barakos, S. Lawson, R. Steijl & Punit Nayyar (Univ. of Liverpool, UK) Round table discussion

20:00

Cocktail

17:30–17:45

17:45–18:00

18:00–18:15

Detailed Programme

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Tuesday, 19 June 2007: 08:30–10:30

Session III.1 Theoretical aspects & analytical approaches of flow separation

08:30–09:15

Near critical phenomena in laminar boundary layers A. Kluwick, S. Braun & E.A. Cox, Technical Univ. of Vienna, Austria - Univ. College Dublin, Irland; Invited keynote lecture State curves and flipping for an orbiting cylinder at low Reynolds numbers L. Baranyi (Univ. of Miskolc, Hungary) Study of a lid-driven cavity in an axisymmetric geometry R. Dasgupta & R. Govindarajan* (JNCASR, India) Global low-frequency oscillations in a separating boundary-layer flow U. Ehrenstein & F. Gallaire (IRPHE, Univ. de Nice-Sophia Antipolis, France) Axisymmetric absolute instability of swirling jets J.J. Healey (Keele Univ., UK) Global instability and control of laminar separation bubbles V. Theofilis (Univ. Politécnica de Madrid, Spain)

09:15–09:30

09:30–09:45

09:45–10:00

10:00–10:15

10:15–10:30

10:30–11:00

Coffee break

11:00–12:45

Session III.2 Theoretical aspects & analytical approaches of flow separation

11:00–11:45

New applications with unsteady flow separation analysis F.T. Smith & N.C. Ovenden, Univ. College London, UK; Invited keynote lecture* Instability of supersonic compression ramp flow R.P. Logue, J.S.B. Gajjar & A.I. Ruban (Univ. of Manchester, UK) Asymptotic theory of turbulent bluff-body separation: A novel shear layer scaling deduced from an investigation of the unsteady motion B. Scheichl & A. Kluwick (Vienna University of Technology, Austria) Cylinders with elliptical cross-section: Wake stability with variation in angle of incidence G.J. Sheard (FLAIR, Australia) Structural stability of the finite-amplitude vortex shedding behind a circular cylinder P. Luchini, F. Giannetti & J. Pralits (Univ. di Salerno, Italy)

11:45–12:00

12:00–12:15

12:15–12:30

12:30–12:45

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Detailed Programme

Posters of session III Influence of 3D perturbations on separated flows F. Alizard & J.C. Robinet* (ENSAM, France) Orbiting cylinder at low Reynolds numbers L. Baranyi (Univ. of Miskolc, Hungary) Global instability computations of separated flow B.V. Bharati Laxmi & J.S.B. Gajjar* (Univ. of Manchester, UK) A two-dimensional disturbed flows over a flat plate: theoretical and numerical approach K. Debbagh & S. Saintlos Brillac (IMFT, France) 12:45–14:00

Lunch

14:00–15:45

Session IV.1 Instability and transition

14:00–14:45

Forced wakes J.E. Wesfreid, ESPCI, France; Invited keynote lecture* Wake transition in the flow around two circular cylinders in staggered arrangements B. Carmo, S. Sherwin, P. Bearman & R. Willden (Imperial College London, UK) Wake dynamics of external flow past a curved circular cylinder with the free-steam aligned with the plane of curvature A. de Vecchi, S.J. Sherwin & J.M.R. Graham (Imperial College London, UK) Successive steps of 2D and 3D transition in the flow past a rotating circular cylinder at moderate Reynolds numbers R. El Akoury, G. Martinat, M. Braza, R. Perrin, Y. Hoarau, G. Harran & D. Ruiz (IMFT, IMFS, ENSEEIHT, France) Direct numerical simulation of vortex shedding behind a linearly tapered circular cylinder V.D. Narasimhamurthy, H.I. Andersson & B. Pettersen (NTNU, Norway)

14:45–15:00

15:00–15:15

15:15–15:30

15:30–15:45

15:45–16:15

Coffee break

16:15–18:30

Session IV.2 Instability and transition

16:15–17:00

Wake transition of oscillating bluff bodies M.C. Thompson, K. Hourigan & J. Leontini, FLAIR, Monash Univ., Clayton, Australia; Invited keynote lecture Dynamics of oblate freely-rising bodies P. Fernandes, P. Ern, F. Risso & J. Magnaudet (IMFT, France)

17:00–17:15

Detailed Programme

17:15–17:30

17:30–17:45

17:45–18:00

18:00–18:15

18:15–18:30

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Parametric study of the two degree-of-freedom vortex-induced vibrations of a cylinder in a two-dimensional flow D. Lucor & M.S. Triantafyllou (Massachusetts Institute of Technology, USA) Vortex dynamics associated with the impact of a cylinder with a wall L. Schouveiler, M.C. Thompson, T. Leweke and K. Hourigan (IRPHE, France - FLAIR, Monash Univ., Australia) Modification of the flow structures in a swirling jet K. Atvars, M.C. Thompson & K. Hourigan (FLAIR, Monash Univ., Australia) Thickness effect of NACA symmetric hydrofoils on hydrodynamic behavior and boundary layer states H. Djeridi, C. Sarraf & J.Y. Billard (Ecole Navale, France) Vortex formation in black-step flow A. Mihaiescu, H. Hangan, A. Straatman & J.E. Wesfreid (Univ. of Western Ontario, London - ON, N6G 2B9, Canada, ESPCI, France)

Posters of session IV Transition of boundary layer on a circular cylinder in uniform flow S. Behara & S. Mittal* (Indian Institute of Technology, Kanpur, India) Numerical simulation of the flow-induced vibration in the flow around two circular cylinders in tandem arrangements B. Carmo, S. Sherwin, P. Bearman & R. Willden* (Imperial College London, UK) Three-dimensionalities of the flow around an oscillating circular cylinder R.S. Gioria & J.R. Meneghini* (Univ. of Sao Paulo, Brazil) Quasi-steady self-excited angular oscillation of equilateral triangular cylinder in 2-D separated flow S. Srigrarom (Nanyang Technological Univ., Singapore) Wednesday, 20 June 2007: 08:30–10:00

Session V.1 Compressibility effects related to unsteady separation

08:30–09:15

Why do shock waves move in separated flows? J.P. Dussauge, IUSTI, France; Invited keynote lecture* Compressibility effects on turbulent separated flow in a streamwise-periodic hill channel - Part1 J. Ziefle & L. Kleiser (Institute of Fluid Dynamics, ETH Zurich, Switzerland)

09:15–09:30

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09:30–09:45

Detailed Programme

09:45–10:00

Global structure of buffeting flow on transonic airfoils J.D. Crouch, A. Garbaruk, D. Magidov & L. Jacquin (St-Petersburg Univ., Russia - ONERA, France) Low-order modeling for unsteady separated compressible flows by POD-Galerkin Approach R. Bourguet, M. Braza & G. Harran (IMFT, France)

10:00–10:30

Coffee break

10:30–12:15

Session V.2 Compressibility effects related to unsteady separation

10:30–11:15

European research on unsteady effects of shock wave induced separation - UFAST Projet P. Doerffer, IMP PAN, Gdansk, Poland; Invited keynote lecture* On the three-dimensionality of shock-wave / laminar boundary layer interaction J.C. Robinet (ENSAM, France) Unsteady flow organization of a shock wave/turbulent boundary layer interaction R.A. Humble, F. Scarano & B.W. Van Oudheusden (Delft Univ. of Technology, The Netherlands) Dependance between shock and separation in a shock wave/boundary layer interaction J.F. Debiève & P. Dupont (IUSTI, France) Effect of a serrated skirt on the buffeting phenomenon in transonic afterbody flows P. Meliga, P. Reijasse & J.M. Chomaz* (ONERA - LadHyX, Ecole Polytechnique, France)

11:15–11:30

11:30–11:45

11:45–12:00

12:00–12:15

Posters of session V Large Eddy Simulation of a supersonic turbulent boundary layer at MD2.25 A. Hadjadj & S. Dubos (INSA de Rouen, France) Film cooling mass flow rate influence on a separation shock in an axisymmetric nozzle P. Reijasse & L. Boccaletto (ONERA - CNES, France) 12:15–14:00

Lunch

14:00–15:30


Detailed Programme

14:00–14:45

14:45–15:00

15:00–15:15

15:15–15:30

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DESider - Detached Eddy Simulation for Industrial Aerodynamics W. Haase, EADS, Münich, Germany; Invited keynote lecture Detached Eddy Simulation of a nose landing-gear cavity R.B. Langtry & P.R. Spalart (The Boeing Company, USA) Experimental and numerical study of unsteady wakes behind an oscillating car model E. Guilmineau & F. Chometon (Ecole Centrale de Nantes - CNAM, France) Physical analysis of an anisotropic eddy viscosity concept for strongly detached unsteady flows R. Bourguet, M. Braza, R. Perrin & G. Harran (IMFT, France)

15:30–16:00

Coffee break

16:00–16:45


16:00–16:15

Dynamic stall of a pitching and horizontally oscillating airfoil G. Martinat, M. Braza, G. Harran, A. Sevrain, Y. Hoarau & D. Favier (IMFT - IMFS - LABM - France) Unsteady flow around a NACA0021 airfoil beyond stall at 60ı angle of attack R. El Akoury, M. Braza, Y. Hoarau, J. Vos, G. Harran & A. Sevrain (IMFT, IMFS, France - CFS engineering, Switzerland) Analysis of Detached-Eddy Simulation for the flow around a circular cylinder with reference to PIV data C. Mockett, R. Perrin, T. Reimann, M. Braza & F. Thiele (TU-Berlin, Germany)

16:15–16:30

16:30–16:45

16:45–19:30 19:30

Free Venue at Achilleion Palace/Museum - visit and gala dinner

Thursday, 21 June 2007: 08:30–10:00


08:30–09:15

On the use of LES for flow control: the compressible cavity flow case P. Sagaut, Univ. Paris VI, France; Invited keynote lecture On the coherent dynamics of turbulent junction flows J. Paik, C. Escauriaza & F. Sotiropoulos (Univ. of Minnesota, USA)

09:15–09:30

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09:30–09:45

09:45–10:00

Detailed Programme

Simulation of the unsteady cavity flow of the stratospheric observatory for infrared astronomy S. Schmid, T. Lutz & E. Krämer (IAG Univ. of Stuttgart, Germany) Simulation of bluff-body flows through a hybrid RANS/VMS-LES model M.V. Salvetti, B. Koobus, S. Camarri & A. Dervieux (Univ. di Pisa, Italy)

Poster of session II Numerical simulation of non-steady supersonic double ramp flow by URANS Approach I.A. Fedorchenko, N.N. Fedorova & U. Gaisbauer* (ITAM, Russia - IAG Univ., Germany) 10:00–10:30

Coffee break

10:30–13:00

Session VI DNS and LES of unsteady separated flows

10:30–11:15

Regularization modeling for large-eddy simulation of turbulent separated boundary layer flow B.J. Geurts, Univ. of Twente, The Netherlands; Invited keynote lecture* Direct numerical simulation of the bursting of a laminar separation bubble and evaluation of flow-control strategies O. Marxen & D.S. Henningson* (KTH, Sweden) LES of the flow around two cylinders in tandem G. Palau-Salvador, T. Stoesser, J. Fröhlich & W. Rodi* (Karlsruhe Univ., Germany) Large Eddy Simulation of impinging shock wave / turbulent boundary layer interaction at MD2.3 E. de Martel, E. Garnier & P. Sagaut (ONERA - LMM, Univ. Pierre et Marie Curie, France) Design and validation of a Large Eddy Simulation methodology for compressible shock-free flows on unstructured meshes L. Georges, J.F. Thomas, G. Winckelmans & P. Geuzaine* (CENAERO, Belgium) On coherent structures and low frequency motions of shock wave-turbulent boundary layer interactions via DNS M. Pino Martin & M. Wu (Princeton Univ. USA) Simulation and modelling of a laminar separation bubble on airfoils F. Richez, I. Mary, V. Gleize & C. Basdevant (ONERA, France)

11:15–11:30

11:30–11:45

11:45–12:00

12:00–12:15

12:15–12:30

12:30–12:45

Detailed Programme

12:45–13:00

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Undesirable haemodynamics in aneurysms G.J. Sheard, R.G. Evans, K.M. Denton & K. Hourigan* (Monash Univ., Australia)

Posters of session VI Unsteady separated flow around the Ahmed body J. Hoessler, J.F. Beaudoin & F. Perot* (PSA Peugeot Citroën, France) Unsteady RANS Calculation of Flow over Ahmed Car Model J. Yao, O. Mouzoun, Y.F. Yao & P. Mason (Kingston Univ., UK) 13:00–14:15

Lunch

14:15–15:45


14:15–15:00

Vortex shedding dynamics in the laminar wakes of various bluff bodies (cylinders, spheres and cones) M. Provansal & P. Monkewitz, IRPHE, France - LMF EPFL, Switzerland; Invited keynote lecture* Effect of velocity ratio on the streamwise vortex structures in the wake of a stack M.S. Adaramola, D. Sumner & D.J. Bergstrom (Univ. of Saskatchewan, Canada) Unsteady force measurements of an airfoil undergoing dynamic stall at low Reynolds number K.K.Y. Tsang, R.C.K. Leung & R.M.C. So* (The Hong Kong Polytechnic Univ., Hong Kong) Coherent structure eduction from PIV data of an electromagnetically forced separated flow T. Weier, C. Ciepka & G. Gerbeth (Forschungszentrum Rossendorf, Germany)

15:00–15:15

15:15–15:30

15:30–15:45

Posters of session I Detailed wake structure behind unsteady airfoils and characteristics of dynamic thrust M. Fuchiwaki & K. Tanaka* (Kyushu Institute of Technology, Japan) Flow separation of a rotating cylinder Best poster award S.C. Luo, T.T.L. Duong & Y.T. Chew (National Univ. of Singapore, Singapore) Unsteady flow behind a blunt based POD model S.D. Sharma & A.A. Kumar* (Indian Institute of Technology Bombay, India) 15:45–16:15

Coffee break

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Detailed Programme

16:15–18:30

Session VII.1 Theoretical/Industrial aspects of unsteady separated flow control

16:15–17:00

Control systems with large parameter spaces C.M. Ho, S. Ho, P.K. Wong & H. Nassef, Univ. of California, Los Angeles, Office of the Secretary of Defense, Washington, Univ. of Arizona, Fluidiqm Inc, San Francisco, USA; Invited keynote lecture* Control of vortex breakdown in a closed cylinder with a small rotating rod D. Lo Jacono, J.N. Sorensen, M.C. Thompson & K. Hourigan* (FLAIR, Monash Univ., Australia - Technical Univ. of Denmark) The wake dynamics of a cylinder moving along a plane wall with rotation and translation B. Stewart, K. Hourigan, M. Thompson & T. Leweke (Monash Univ. Australia - IRPHE, France) Sub-optimal control of unsteady separation in a channel K.W. Cassel, C. Sardesai, S. Braun & A.I. Ruban* (MMAE, Illinois I. of Tech., USA - I. of FMTH, Vienna Univ. of Tech., Austria - Univ. of Manchester, UK) Accurate POD reduced-order models of separated flows J. Favier, A. Kourta & L. Cordier* (IMFT - LEMTA, France) Simulation study of a robust closed-loop control of a 2D high-lift configuration B. Günther, R. Becker, A. Carnarius, F. Thiele & R. King (TU-Berlin, Germany) Linear proportional control of flow over a sphere S. Jeon & H. Choi* (Seoul National Univ., Korea)

17:00–17:15

17:15–17:30

17:30–17:45

17:45–18:00

18:00–18:15

18:15–18:30

Friday, 22 June 2007: 08:30–10:15


08:30–09:15

Multiscale retrograde estimation and forecasting of chaotic nonlinear systems T. Bewley, Univ. of California San Diego, USA; Invited keynote lecture* Spreading and vectoring of a subsonic axisymmetric air jet by plasma actuator : a preliminary study N. Benard, J. Jolibois, M. Forte, G. Touchard & E. Moreau (LEA, Univ. de Poitiers, France)*

09:15–09:30

Detailed Programme

09:30–09:45

09:45–10:00

10:00–10:15

xxiii

Simulation of the reduction of the unsteadiness in a passively-controlled transonic cavity flow P. Comte, F. Daude & I. Mary (ENSMA - ONERA, France) Passive drag control of a turbulent wake by local disturbances O. Cadot, B. Thiria & J.F. Beaudoin (ENSTA, France) The effect of zero-net-mass-flux jet geometry on active separation control of a NACA0015 airfoil T. Stephens, C. Atkinson & J. Soria* (Monash Univ., Australia)

10:15–10:45

Coffee break

10:45–12:45


10:45–11:30

Unsteady separated flows and their control M. Triantafyllou, Massachusetts Institute of Technology, USA; Invited keynote lecture* Active control of a cylinder wake using surface plasma T.N. Jukes & K.S. Choi (Univ. of Nottingham, UK) Active control of flow separation over an airfoil using synthetic jets D. You & P. Moin (Stanford Univ. USA) Flow control for rorocraft applications at flight Mach numbers H. Nagib, J. Keidaisch, D. Greenblatt, I. Wygnanski & A. Hassan* (Illinois I. of Tech., USA - TU-Berlin, Germany - Univ. of Arizona, USA - The Boeing Company, USA) Electromagnetic control of separation at hydrofoils* G. Mutschke, T. Weier, T. Albrecht, G. Gerbeth & R. Grundmann (Forschungszentrum Rossendorf - Dresden Univ. of Technology, Germany) A numerical study of ZNMF jet lift enhancement of a NACA 0015 airfoil V. Kitsios, A. Ooi, J. Soria & D.You* (Univ. Of Melbourne - Monash Univ., Australia)

11:30–11:45

11:45–12:00

12:00–12:15

12:15–12-30

12:30–12:45

12:45–14:00

Lunch

14:00–15:30


14:00–14:45

Hydrodynamics of beating cilia A. Dauptain, J. Favier & A. Bottaro, Univ. di Genova, Italy; Invited keynote lecture*

xxiv

14:45–15:00

15:00–15:15

15:15–15:30

16:00–16:30

Detailed Programme

Control of the separated flow behind a circular cylinder by low forcing – Experiments and computations E. Konstantinidis, C. Liang, G. Papadakis & S. Balabani* (Univ. of Western Macedonia, Greece - Iowa State Univ., USA King’s College London, UK) Vortex models for feedback stabilization of bluff body wake flows B. Protas* (McMaster Univ., Hamilton, Canada) Computational investigation of flow control over wings P.S. Vavilis & J.A. Ekaterinaris* (Univ. of Patras, Greece) Closing address

Posters of session VII Towards the control of cavitating flows R.E.A. Arndt & M. Wosnik* (Univ. of Minnesota, Minneapolis) Active control of flows with trapped vortices R.M. Kerimbekov & O.R. Tutty (Univ. of Southampton, UK) A three-dimensional numerical study into non-axisymmetric perturbations of the hole-tone feedback cycle M.A. Langthjem & M. Nakano (Yamagata Univ., Japan) Flow control in high-speed train applications A. Orellano & M. Schober (Bombardier, Germany) Flow control of annular jet expansion using cross-flow injection M. Vanierschot & E. Van den Bulck* (Katholieke Univ. Leuven, Belgium)

Acknowledgement

We express appreciation for the support for this symposium from the following sponsors: International Union of Theoretical and Applied Mechanics (IUTAM) Institut de Mécanique des Fluides de Toulouse, UMR 5502 CNRS-INP/ ENSEEIHT-UPS (CNRS–IMFT) Centre National de la Recherche Scientifique (CNRS) European Research Community on Flow Turbulence and Combustion (ERCOFTAC) European Aeronautic Defence and Space Company (EADS) The symposium chairmen are grateful to the IUTAM Committee for their support and for having attributed the IUTAM label to the symposium’s topic, as well as to the President of INPT (Institut National Polytechnique de Toulouse), Professor G. Casamatta, to the President of UPS (Université Paul Sabatier), Professor J.F. Sautereau, to the Director of ENSEEIHT (Ecole Nationale Supérieure d’Electrotechnique, Electronique, Informatique, Hydraulique et Télecommunications), Professor A. Ayache, to the Director of IMFT (Institut de Mécanique des Fluides de Toulouse), Dr. J. Magnaudet, for their contribution to the success of this Symposium. The Hellenic Touristic Development Company as well as M.M. Kritikos and Drougas are thanked for having allowed the organisation of the gala dinner in the Achilleion palace and Museum of Corfu, built by the Empress Elisabeth of Austria (1837–1898). This IUTAM symposium was held in the Chandris Hotel of Dassia–Corfu, in the main conference room “Odysseas” of the hotel, located on a very nice coast of the Ionian sea at Corfu. A friendly atmosphere was created among the participants, helped by the organisation of two social events, the cocktail and buffet-dinner on Monday 18th June, at the beach, jointly organised by the IUTAM symposium and

xxv

xxvi

Acknowledgement

the statistical-hybrid one and the gala dinner on 20th June in the superb “Achilleion Palace and Museum”, that was especially open for visiting by all the participants. A general outcome from this symposium was that the scientific communities working on experimental, theoretical and numerical approaches related to the unsteady separation and its control, have learned a lot from one another and the meeting has brought new research ideas to everyone. Toulouse, France Melbourne, Australia

Marianna Braza Kerry Hourigan

Contents

Part I

Experimental Techniques for the Unsteady Flow Separation

PIV Measurements of the Flow Around Oscillating Cylinders at Low KC Numbers : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : D. Sumner, H.B. Hemingson, D.M. Deutscher, and J.E. Barth

3

Coherent and Turbulent Process Analysis in the Flow Past a Circular Cylinder at High Reynolds Number : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 R. Perrin, M. Braza, E. Cid, S. Cazin, P. Chassaing, C. Mockett, T. Reimann, and F. Thiele Investigation of Aerodynamic Capabilities of Flying Fish in Gliding Flight : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 H. Park and H. Choi Effect of Unsteady Separation on an Automotive Bluff-Body in Cross-Wind : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 35 M. Gohlke, J.F. Beaudoin, M. Amielh, and F. Anselmet Part II Statistical and Hybrid Turbulence Modeling of Unsteady Separated Flows Flow Prediction Around an Oscillating NACA0012 Airfoil at Re = 1,000,000 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 49 O. Frederich, U. Bunge, C. Mockett, and F. Thiele Two-Velocities Hybrid RANS-LES of a Trailing Edge Flow : : : : : : : : : : : : : : : : : 63 J.C. Uribe, N. Jarrin, R. Prosser, and D. Laurence Assessment of Flow Control Devices for Transonic Cavity Flows Using DES and LES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 77 G.N. Barakos, S.J. Lawson, R. Steijl, and P. Nayyar

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Part III

Contents

Theoretical Aspects & Analytical Approaches of Flow Separation

Near Critical Phenomena in Laminar Boundary Layers : : : : : : : : : : : : : : : : : : : : 91 A. Kluwick, S. Braun, and E.A. Cox State Curves and Flipping for an Orbiting Cylinder at Low Reynolds Numbers : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 111 L. Baranyi Global Low-Frequency Oscillations in a Separating Boundary-Layer Flow : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 123 U. Ehrenstein and F. Gallaire Asymptotic Theory of Turbulent Bluff-Body Separation: A Novel Shear Layer Scaling Deduced from an Investigation of the Unsteady Motion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 135 B. Scheichl and A. Kluwick Structural Sensitivity of the Finite-Amplitude Vortex Shedding Behind a Circular Cylinder : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 151 P. Luchini, F. Giannetti, and J. Pralits Orbiting Cylinder at Low Reynolds Numbers : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 161 L. Baranyi A Two-Dimensional Disturbed Flows Over a Flat Plate: Theoretical and Numerical Approach :: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 167 K. Debbagh and S. Saintlos Brillac Part IV

Instability and Transition

Wake Dynamics of External Flow Past a Curved Circular Cylinder with the Free-Stream Aligned to the Plane of Curvature : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 175 A. de Vecchi, S.J. Sherwin, and J.M.R. Graham Successive Steps of 2D and 3D Transition in the Flow Past a Rotating Cylinder at Moderate Reynolds Numbers : : : : : : : : : : : : : : : : : : : : : : : : : : 187 R. El Akoury, G. Martinat, M. Braza, R. Perrin, Y. Hoarau, G. Harran, and D. Ruiz Direct Numerical Simulation of Vortex Shedding Behind a Linearly Tapered Circular Cylinder : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 201 V.D. Narasimhamurthy, H.I. Andersson, and B. Pettersen Dynamics of Oblate Freely Rising Bodies : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 213 P.C. Fernandes, P. Ern, F. Risso, and J. Magnaudet

Contents

xxix

Parametric Study of Two Degree-of-Freedom Vortex-Induced Vibrations of a Cylinder in a Two-Dimensional Flow : : : : : : : : : : : : : : : : : : : : : : : : 223 D. Lucor and M.S. Triantafyllou Vortex Dynamics Associated with the Impact of a Cylinder with a Wall : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 235 L. Schouveiler, M.C. Thompson, T. Leweke, and K. Hourigan Modification of the Flow Structures in a Swirling Jet :: : : : : : : : : : : : : : : : : : : : : : : 243 K. Atvars, M. Thompson, and K. Hourigan Thickness Effect of NACA Symmetric Hydrofoils on Hydrodynamic Behaviour and Boundary Layer States : : : : : : : : : : : : : : : : : : : 255 H. Djeridi, C. Sarraf, and J.Y. Billard Quasi-Steady Self-Excited Angular Oscillation of Equilateral Triangular Cylinder in 2-D Separated Flow : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 269 S. Srigrarom Part V

Compressibility Effects Related to Unsteady Separation

Compressibility Effects on Turbulent Separated Flow in a Streamwise-Periodic Hill Channel – Part 1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 285 J. Ziefle and L. Kleiser Global Structure of Buffeting Flow on Transonic Airfoils : : : : : : : : : : : : : : : : : : : 297 J.D. Crouch, A. Garbaruk, D. Magidov, and L. Jacquin Low-Order Modeling for Unsteady Separated Compressible Flows by POD-Galerkin Approach :: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 307 R. Bourguet, M. Braza, G. Harran, and A. Dervieux Unsteady Flow Organization of a Shock Wave/Turbulent Boundary Layer Interaction :: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 319 R.A. Humble, F. Scarano, and B.W. van Oudheusden Dependence Between Shock and Separation Bubble in a Shock Wave Boundary Layer Interaction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 331 J.F. Debiève and P. Dupont Large Eddy Simulation of a Supersonic Turbulent Boundary Layer at M D 2:25 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 343 A. Hadjadj and S. Dubos Film Cooling Mass Flow Rate Influence on a Separation Shock in an Axisymmetric Nozzle : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 349 P. Reijasse and L. Boccaletto

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Contents

Detached Eddy Simulation of a Nose Landing-Gear Cavity : : : : : : : : : : : : : : : : : 357 R. Langtry and P. Spalart Experimental and Numerical Study of Unsteady Wakes Behind an Oscillating Car Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 367 E. Guilmineau and F. Chometon Physical Analysis of an Anisotropic Eddy-Viscosity Concept for Strongly Detached Turbulent Unsteady Flows : : : : : : : : : : : : : : : : : : : : : : : : : : : : 381 R. Bourguet, M. Braza, R. Perrin, and G. Harran Dynamic Stall of a Pitching and Horizontally Oscillating Airfoil : : : : : : : : : : : : 395 G. Martinat, M. Braza, G. Harran, A. Sevrain, G. Tzabiras, Y. Hoarau, and D. Favier Unsteady Flow Around a NACA0021 Airfoil Beyond Stall at 60ı Angle of Attack :: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 405 R. El Akoury, M. Braza, Y. Hoarau, J. Vos, G. Harran, and A. Sevrain Analysis of Detached-Eddy Simulation for the Flow Around a Circular Cylinder with Reference to PIV Data : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 417 C. Mockett, R. Perrin, T. Reimann, M. Braza, and F. Thiele Simulation of Bluff-Body Flows Through a Hybrid RANS/VMS-LES Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 429 Maria Vittoria Salvetti, Bruno Koobus, Simone Camarri, and Alain Dervieux Part VI

DNS and LES of Unsteady Separated Flows

Large Eddy Simulation of Impinging Shock Wave/Turbulent Boundary Layer Interaction at M D 2 :3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 443 E. de Martel, E. Garnier, and P. Sagaut Simulation and Modelling of a Laminar Separation Bubble on Airfoils : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 457 F. Richez, I. Mary, V. Gleize, and C. Basdevant Unsteady RANS Calculation of Flow Over Ahmed Car Model : : : : : : : : : : : : : : 471 J. Yao, O. Mouzoun, Y. Yao, and P. Mason Effect of Velocity Ratio on the Streamwise Vortex Structures in the Wake of a Stack : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 477 M.S. Adaramola, D. Sumner, and D.J. Bergstrom Flow Separation of a Rotating Cylinder : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 487 S.C. Luo, T.T.L. Duong, and Y.T. Chew

Contents

Part VII

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Theoretical/Industrial Aspects of Unsteady Separated Flow Control

The Wake Dynamics of a Cylinder Moving Along a Plane Wall with Rotation and Translation :: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 495 B. Stewart, K. Hourigan, M. Thompson, and T. Leweke Simulation Study of the Robust Closed-Loop Control of a 2D High-Lift Configuration:: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 505 B. Günther, A. Carnarius, F. Thiele, R. Becker, and R. King Large-Eddy Simulation of Passively-Controlled Transonic Cavity Flow : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 517 P. Comte, F. Daude, and I. Mary Passive Drag Control of a Turbulent Wake by Local Disturbances : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 529 O. Cadot, B. Thiria, and J.-F. Beaudoin Active Control of a Cylinder Wake Using Surface Plasma : : : : : : : : : : : : : : : : : : : 539 T. Jukes and K.-S. Choi Active Control of Flow Separation Over an Airfoil Using Synthetic Jets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 551 D. You and P. Moin Electromagnetic Control of Separation at Hydrofoils :: : : : : : : : : : : : : : : : : : : : : : : 563 G. Mutschke, T. Weier, T. Albrecht, G. Gerbeth, and R. Grundmann Active Control of Flows with Trapped Vortices: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 575 R.M. Kerimbekov and O.R. Tutty A Three-Dimensional Numerical Study into Non-Axisymmetric Perturbations of the Hole-Tone Feedback Cycle: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 581 M.A. Langthjem and M. Nakano

Part I

Experimental Techniques for the Unsteady Flow Separation

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PIV Measurements of the Flow Around Oscillating Cylinders at Low KC Numbers D. Sumner, H.B. Hemingson, D.M. Deutscher, and J.E. Barth

Abstract The flow around cylinders of circular, square, and diamond cross-section oscillating in quiescent fluid was studied experimentally using particle image velocimetry (PIV). Phase-averaged measurements of the velocity field were obtained at the maximum-amplitude, zero-amplitude, and intermediate positions of the oscillation cycle. The experiments were performed at low Keulegan-Carpenter numbers, from KC D 1 to 3.5, and for moderate Stokes numbers, from ˇ D 250 to 376. Within this range of KC, the flow patterns remained symmetric about either side of the cylinders. For KC D 1, the flow remained close to the surface of the cylinders throughout the cycle. For KC D 1:5 to 3.5, an attached vortex pair formed behind the circular and square cylinders at the maximum-amplitude position. The distinct geometry of the diamond cylinder, with two fixed separation points, led to a unique but still symmetric vortex pattern. Keywords Unsteady flow Bluff body Cylinder Oscillation Particle image velocimetry

1 Introduction The flow around a two-dimensional circular cylinder of diameter, D, oscillating sinusoidally from side to side with amplitude, A, and frequency, f (or period, T D 1=f ), in quiescent fluid of kinematic viscosity, , can be considered a fundamental problem in unsteady fluid mechanics (Fig. 1). The flow is a useful starting point for the study of flow-induced vibrations and has applications in the behaviour of offshore structures in wave motion. The time-dependent cylinder position, x.t/, and velocity, u.t/, are given by

D. Sumner (), H.B. Hemingson, D.M. Deutscher, and J.E. Barth Department of Mechanical Engineering, University of Saskatchewan, 57 Campus Drive, Saskatoon, Saskatchewan, Canada, S7N 5A9 e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

3

4

D. Sumner et al.

Fig. 1 Circular cylinder of diameter, D, oscillating with amplitude, A, in a quiescent fluid

x.t/ D A sin .!t / and u.t/ D Umax cos .!t/ ;

(1) (2)

respectively, where t is time, !.D 2πf / is the angular frequency, and Umax is the maximum cylinder velocity, Umax D A! D 2Af D 2A=T:

(3)

Experimental [1,7–10,13,14] and numerical [3,12,15] studies of the oscillating circular cylinder have identified several flow regimes, which are primarily a function of the dimensionless amplitude parameter known as the Keulegan-Carpenter number, KC, where KC D Umax T =D D 2A=D: (4) A second influencing parameter is the Reynolds number, Re, which is defined using the maximum cylinder velocity, Re D Umax D= D 2ADf =:

(5)

The dimensionless Stokes number (or reduced frequency), ˇ, is often used instead of the Reynolds number and behaves in a similar manner, where ˇ D Re=KC D fD 2 =:

(6)

The number of flow regimes, and the limiting values of KC defining these regimes, are functions of Re or ˇ. For sufficiently high values of ˇ, the flow regime is almost solely determined by KC. The flow patterns are distinguished by the “wake re-encounter” phenomenon, where separated flow and vortices formed and shed by the cylinder during one half cycle return to interact with the cylinder during the next half cycle [5]. For KC 1, the flow does not appreciably separate from the cylinder and the extent of the ambient fluid affected by the moving cylinder is small. For 1 < KC < 4, the flow separates from the cylinder and a symmetric pair of attached vortices forms behind the cylinder during each half cycle. For 4 KC < 8, the attached vortex pair becomes asymmetric. For KC 8, vortex shedding occurs during each half cycle [5, 14]. Similar flow regimes and KC boundaries can be identified

PIV Measurements of the Flow Around Oscillating Cylinders at Low KC Numbers

5

for square and diamond-shaped cylinders, as shown in several experimental [1, 8, 9] and numerical [11, 12] studies. For the circular cylinder, the separation points are free to move over large distances during an oscillation cycle [10]. However, for the oscillating square cylinder, the separation points are fixed at the four corners; for the oscillating diamond cylinder, there are only two fixed separation points. There are relatively few experimental studies of oscillating cylinders which report velocity field measurements [2, 4, 6]. In the present study, particle image velocimetry (PIV) was used to measure the velocity field of oscillating circular, square and diamond cylinders at low KC numbers. The PIV measurements were phase locked with the cylinder position, at the zero-amplitude (corresponding to dimensionless time t D t=T D 0, 0.5), maximum-amplitude (t D 0:25, 0.75), and intermediate positions (t D 0:125, 0.375, 0.625, 0.875) in the oscillation cycle.

2 Experimental Approach The oscillating cylinder experiments were conducted in water in an X–Y towing tank (Fig. 2) with internal dimensions of 3.96 m long, 1.03 m wide, and 0.75 m deep. The glass side walls, end walls, and floor of the towing tank give optical access for the PIV system. The primary towing direction (X-direction, 3.5 m of travel) is along the length of the tank, where the main carriage straddles the tank width and moves on two parallel rails. A linear motion stage containing the secondary carriage is mounted on the main carriage for transverse movement across the tank (Y-direction, 260 mm of travel).

Fig. 2 Three-view drawing of the X–Y towing tank showing the circular cylinder suspended in the water beneath the Y-motion stage

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The motion control system for the X–Y towing tank includes a personal computer, a National Instruments (NI) PCI-7344 motion controller card, two Intelligent Motion Systems IM1007 micro-step drivers, and a stepping motor on each axis. Encoders provide closed-loop position feedback. The user interface was developed in the LabVIEW programming language. In the present study, only the secondary carriage (Y-direction) was used. A circular or square cylinder of D D 25:4 mm (for the square cylinder, D corresponds to the side length) was suspended in the water vertically beneath the Y-motion stage (Fig. 2). The cylinder aspect ratio ranged from AR D 25:5 to 26.7. The oscillation amplitude ranged from A D 4:04 to 14.15 mm (giving KC D 1 to 3.5). The oscillation frequency ranged from f D 0:38 to 0.60 Hz (giving ˇ D 250 to 376 and Re D 374 to 875); see Table 1. Velocity field measurements (u and v components, in the x and y directions, respectively) were made with a TSI PIV system. Laser light was supplied by a 120-mJ/pulse dual Nd:YAG laser. The light sheet was located 29.2 cm above the lower end of the cylinder. Images were acquired with a TSI PIVCAM 10–30 (1 Megapixel) camera in a fixed location beneath the tank. The timing was controlled by a TSI LaserPulse synchronizer and Insight 5 software and was phase-locked to a reference signal from the motion control system. The water was seeded with 8–12-μm hollow glass spheres. Image pairs were processed with a single-pass Nyquist grid algorithm, a FFT correlation algorithm, and a Gaussian peak detection algorithm. The interrogation window was 32 32 pixels. The field of view was approximately 80 80 mm (60 60 vectors with 50% overlap) giving a spatial resolution greater than [4] but lower than [6]. For each phase (cylinder position), an ensemble average of 250 instantaneous velocity vector fields was obtained. The cylinder completed at least 50 oscillation cycles before PIV measurements were made, to give sufficient time for the flow patterns to be established. The numerical simulations of [12] showed that at least 15 cycles were needed to reach equilibrium flow behaviour.

Table 1 Summary of oscillating cylinder experiments Cylinder Circular Circular Circular Circular Circular Circular Square Square Square

KC 1.0 1.5 2.0 2.5 3.0 3.5 1.0 1.5 2.0

ˇ 376 325 250 250 250 250 375 325 250

Re 376 488 499 625 751 875 375 488 500

A/D 0.159 0.239 0.318 0.398 0.477 0.557 0.159 0.239 0.318

Cylinder Square Square Square Diamond Diamond Diamond Diamond Diamond Diamond

KC 2.5 3.0 3.5 1.0 1.5 2.0 2.5 3.0 3.5

ˇ 250 250 250 374 325 251 250 250 250

Re 625 750 875 374 488 502 625 751 875

A/D 0.398 0.477 0.557 0.159 0.239 0.318 0.398 0.477 0.557


7

3 Results and Discussion For the range of KC numbers investigated, 1 KC 3:5, the flow fields were symmetric about the axis of motion and on opposite ends of the cylinder for subsequent half-cycles, for all three cylinder geometries. Selected results (only vorticity results are presented here) are shown in Figs. 3–5.

Fig. 3 Phase-averaged vorticity fields for an oscillating circular cylinder at the zero- .t D 0/ and maximum-amplitude .t D 0:25/ positions, cylinder moving to the right: (a) KC D 1:5; ˇ D 325; (b) KC D 2; ˇ D 250; (c) KC D 2:5; ˇ D 250; (d) KC D 3:5; ˇ D 250. Solid isovorticity contour lines represent positive (CCW) vorticity; dashed lines represent negative (CW) vorticity. Minimum vorticity contour of ! D 0:5, contour increment of ! D 0:5

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Fig. 4 Phase-averaged vorticity fields for an oscillating square cylinder at intermediate- .t D 0:125/ and maximum-amplitude .t D 0:25/ positions, cylinder moving to the right: (a) KC D 1; ˇ D 375; (b) KC D 2; ˇ D 250; (c) KC D 3; ˇ D 250. Contour lines as in Fig. 3


9

Fig. 5 Phase-averaged vorticity fields for an oscillating diamond cylinder at the zero- .t D 0/ and maximum-amplitude .t D 0:25/ positions, cylinder moving to the right: (a) KC D 1; ˇ D 374; (b) KC D 2; ˇ D 251; (c) KC D 3; ˇ D 250. Contour lines as in Fig. 3

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3.1 Circular Cylinder For the oscillating circular cylinder (Fig. 3), frame-by-frame analysis of the instantaneous velocity fields did not show any asymmetry on either side of the axis of motion up to KC D 3:5, consistent with [14, 15]. The flow at these KC numbers is characterized, for the most part, by the formation and movement of concentrations of vorticity and/or symmetric vortex pairs behind and about the cylinder. At KC D 1, the fluid is swept symmetrically around the circular cylinder from the front to the rear, the process repeating itself with minimal disturbance to the surrounding fluid as the cylinder reverses direction. The region of disturbed, separated flow is small and remains close to the cylinder surface throughout the range of motion, consistent with [5, 15]. The largest concentrations of vorticity are seen on either side of the cylinder at the zero-amplitude .t D 0/ position. As the cylinder moves toward the maximum-amplitude position .t D 0:25/, these concentrations are swept behind the circular cylinder and weaken. At KC D 1:5 (Fig. 2a), the flow field is generally similar to KC D 1, with the separated flow remaining close to the cylinder throughout the cycle. However, a small, weak attached vortex pair is observed behind the circular cylinder at the maximum-amplitude position .t D 0:25/, which is the primary feature of the flow regime for 1 < KC < 4. For KC D 2 (Fig. 2b), the flow behind the cylinder at maximum amplitude .t D 0:25/ contains a prominent attached symmetric vortex pair, similar to what is observed in the near-wake region of an impulsively started circular cylinder, where each vortex is of similar size and strength but of opposite sign. Behind this vortex pair is weaker detached vortex pair, with vortices of opposite sign to those that are attached. As the cylinder reverses direction .t D 0:375/, the attached vortex pair is now ahead of the moving cylinder, and it quickly diminishes in size and vanishes; this is consistent with the simulations of [3]. The flow is similar for KC D 2:5 (Fig. 2c), where a strong attached vortex pair forms at maximum amplitude .t D 0:25/. This is accompanied by the detached vortex pair of opposite sign extending further from the cylinder. As the cylinder reverses direction, the attached vortex pair disappears .t D 0:375/ while the detached vortex pair is split by the approaching cylinder .t D 0:5/. These two concentrations of vorticity are then entrained into the new attached vortex pair that forms behind the cylinder on the opposite side .t D 0:625/. At KC D 3 (Fig. 2d) and 3.5, the attached vortex pair is stronger and more prominent than at KC D 2 and 2.5. The behaviour is characterized by the movement of vortex pairs around the circular cylinder from the front to rear [12, 14]. As the cylinder reverses direction (t D 0:375, 0.5) the attached vortices no longer disappear. Instead, they detach from the cylinder and sweep around into the new wake that is developing behind the cylinder (t D 0:5, 0.625). As this happens, they pair up with the developing vortices that will form the attached vortex pair at the next maximum-amplitude position. The swept vortices eventually meet and pair up downstream of the attached vortex pair.


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3.2 Square Cylinder For the oscillating square cylinder (Fig. 4), the four sharp corners lead to welldefined separation points. Vortex formation from the leading corners leads to a wider region of disturbed flow compared to the circular cylinder [9]. For KC D 1 (Fig. 4a), the vortex structures that form during the cycle remain close to the cylinder surface. Movement of vorticity along the upper and lower surfaces of the cylinder results in the formation of a vortex pair at each of the rear corners at the maximum-amplitude position .t D 0:25/. For KC D 1:5 and 2 (Fig. 4b), an attached vortex pair forms behind the square cylinder, similar to what is observed for the circular cylinder, and consistent with [9]. The vortex pair does not meet at the flow centerline; rather, at the maximum-amplitude position .t D 0:25/, there is an arrangement of four vortices of alternating sign on the rear surface of the cylinder, and another pair of vortices of opposite sign at the rear corners (Fig. 4b). The flow pattern has distinct differences at KC D 3 (Fig. 4c) and 3.5 (the flow pattern for KC D 2:5 can be considered transitional). The pairs of vortices at the rear corners are less prominent. Flow separation from the front corners of the square cylinder leads to the formation of small corner vortices. The corner vortices from the previous half cycle enter the wake of the cylinder and form a second, detached vortex pair of opposite sign situated behind the main attached vortex pair (shown for t D 0:125 in Fig. 4c). The detached vortex pair weakens at the maximumamplitude position .t D 0:25/ and is absent when KC D 3:5.

3.3 Diamond Cylinder The flow around the oscillating diamond cylinder (Fig. 5) is distinct from the circular and square cylinders, being characterized by flow separation and vorticity production at the upper and lower sharp corners. In the direction of motion, the angled flat surfaces meet at the front sharp corner that forms the leading edge. Each flat surface pushes the fluid to the side and some of the fluid flows around the upper and lower corners into the wake. In the base region, the angled flat surfaces meet at the rear sharp corner that forms the trailing edge. For KC D 1 (Fig. 5a), appreciable fluid movement is confined mainly to the regions about the upper and lower corners, where large concentrations of vorticity are produced. The sign of the upper and lower corner vorticity concentrations alternates from one half cycle to the next. During the half cycle, these concentrations of vorticity move behind the diamond cylinder to positions on the angled flat surfaces on the trailing edge. At the maximum-amplitude position .t D 0:25/, an attached pair of opposite-sign vortices has formed on both the upper and lower angled flat surfaces on the rear of the diamond cylinder. Narrow rows of vorticity, resembling free shear layers, extend away from the diamond cylinder parallel to the axis of motion on the upper and lower sides of the flow field.

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For KC D 1:5, 2 (Fig. 5b), and 2.5, large concentrations of vorticity are produced at the upper and lower sharp corners. These vorticity concentrations are convected into the wake of the diamond cylinder. On the angled flat surfaces at the rear of the cylinder, each pairs up with a vorticity concentration of opposite sign. At the maximum-amplitude position .t D 0:25/, an attached vortex pair has formed on each of the rear angled flat surfaces. For KC D 3 (Fig. 5c) and 3.5, only a single, large attached vortex forms on each of the flat surfaces at the rear of the diamond cylinder. Weaker concentrations of vorticity of opposite sign form further downstream, but are not attached to the flat surfaces (as in the case of KC D 1:5 to 2.5).

4 Conclusions In the present study, PIV was used to study the velocity and vorticity fields for circular, square and diamond cylinders oscillating in quiescent fluid at low KeuleganCarpenter numbers (KC D 1 to 3.5) and moderate Stokes numbers (ˇ D 250 to 376). Phase-averaged velocity and vorticity fields were obtained at the maximumamplitude, zero-amplitude, and intermediate positions of the oscillation cycle. The flow patterns remained symmetric about either side of the oscillation axis. For KC D 1, the flow remained close to the cylinder surfaces throughout the oscillation cycle. For KC D 1:5 to 3.5, attached vortex pairs formed behind the cylinders at the maximum-amplitude position. The flow around the diamond cylinder was generally distinct from the circular and square cylinders, which can be attributed, in part, to its different base geometry. Acknowledgements The authors acknowledge the support of the Natural Sciences and Engineering Research Council (NSERC), the Canada Foundation for Innovation (CFI), and the Innovation and Science Fund of Saskatchewan. The assistance of M.G. Crane, O.J.P. Dansereau, J.L. Heseltine, and Engineering Shops is appreciated.

References 1. Bearman, P.W., Graham, J.M.R., Obasaju, E.D., and Drossopoulos, G.M., The influence of corner radius on the forces experienced by cylindrical bluff bodies in oscillatory flow. Appl. Ocean Res. 6 (1984) 83–89. 2. Dütsch, H., Durst, F., Becker, S., and Lienhart, H., Low-Reynolds-number flow around an oscillating circular cylinder at low Keulegan-Carpenter numbers. J. Fluid Mech. 360 (1998) 249–271. 3. Iliadis, G. and Anagnostopoulos, P., Viscous oscillatory flow around a circular cylinder at low Keulegan-Carpenter numbers and frequency parameters. Int. J. Num. Meth. Fluids 26 (1998) 403–442. 4. Lam, K.M. and Dai, G.Q., Formation of vortex street and vortex pair from a circular cylinder oscillating in water. Exp. Therm. Fluid Sci. 26 (2002) 901–915.


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5. Lin, X.W., Bearman, P.W., and Graham, J.M.R., A numerical study of oscillatory flow about a circular cylinder for low values of beta parameter. J. Fluid Struct. 10 (1996) 501–526. 6. Lin, J.C. and Rockwell, D., Quantitative interpretation of vortices from a cylinder oscillating in quiescent fluid. Exp. Fluids 23 (1997) 99–104. 7. Obasaju, E.D., Bearman, P.W., and Graham, J.M.R., A study of forces, circulation and vortex patterns around a circular cylinder in oscillating flow. J. Fluid Mech. 196 (1988) 467–494. 8. Okajima, A., Matsumoto, T., and Kimura, S., Force measurements and flow visualization of bluff bodies in oscillatory flow. J. Wind Eng. Ind. Aerod. 69–71 (1997) 213–228. 9. Okajima, A., Matsumoto, T., and Kimura, S., Force measurements and flow visualization of circular and square cylinders in oscillatory flow. JSME Int. J. B-Fluid T. 41 (1998) 796–805. 10. Sarpkaya, T. and Butterworth, W., Separation points on a cylinder in oscillating flow. J. Offshore Mech. Arct. 114 (1992) 28–35. 11. Scolan, Y.-M. and Faltinsen, O.M., Numerical studies of separated flow from bodies with sharp corners by the vortex in cell method. J. Fluid Struct. 8 (1994) 201–230. 12. Smith, P.A. and Stansby, P.K., Viscous oscillatory flow around cylindrical bodies at low Keulegan-Carpenter numbers using the vortex method. J. Fluid Struct. 5 (1991) 339–361. 13. Tatsuno, M. and Bearman, P.W., A visual study of the flow around an oscillating circular cylinder at low Keulegan-Carpenter numbers and low Stokes numbers. J. Fluid Mech. 211 (1990) 157–182. 14. Williamson, C.H.K., Sinusoidal flow relative to circular cylinders. J. Fluid Mech. 155 (1985) 141–174. 15. Zhang, H.-L. and Zhang, X., Flow structure analysis around an oscillating circular cylinder at low KC number: a numerical study. Comp. Fluids 26 (1997) 83–106.


Coherent and Turbulent Process Analysis in the Flow Past a Circular Cylinder at High Reynolds Number R. Perrin, M. Braza, E. Cid, S. Cazin, P. Chassaing, C. Mockett, T. Reimann, and F. Thiele

Abstract With the aim of providing a database useful for validation and improvement of turbulence models for strongly detached flows, the flow past a circular cylinder at high Reynolds number has been experimentally studied using PIV, stereoscopic PIV and Time Resolved PIV in the very near wake. As the presence of coherent structures and their non linear interactions with the turbulent motion have to be taken into account in a model, a particular attention was paid to the decomposition of the flow into a coherent and a turbulent part. This was achieved using phase averaging and also using Proper Orthogonal Decomposition. In a precedent study, it was found that the POD coefficients could be used to define a phase angle representative of the vortex shedding, and that defining the phase angle from the POD coefficients may alleviate the overestimation of the turbulent stresses due to phase jitter between the trigger signal and the velocity, compared to a definition of the phase angle from a wall pressure time trace. In this paper two new complementary data sets, which are resolved in time and space, are analysed with the objectives of, first, providing an evaluation of the performed conditional averaging and, second, to achieve a more complete description of the flow. The main results presented here are issued from Time Resolved PIV measurements which were carried out in the near wake. Some results of a Detached Eddy Simulation which have been validated against experiment, are also used. Keywords Cylinder wake Phase averaging POD Time Resolved PIV DES

R. Perrin (), M. Braza, E. Cid, S. Cazin, and P. Chassaing Institut de Mécanique des Fluides de Toulouse, Unité Mixte C.N.R.S.-I.N.P.T. 5502, Av. du Prof. Camille Soula, 31400 Toulouse, France Present address: Laboratoire d’Etude Aérodynamique (LEA), Université de Poitiers, ENSMA,CNRS, France R. Perrin, C. Mockett, T. Reimann, and F. Thiele Institut für Strömungsmechanik un Technische Akustik, TU-Berlin, Müller Breslau Str. 8, 10623 Berlin, Germany M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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1 Introduction Modelling and simulating unsteady turbulent flows past bluff bodies remains a very challenging task, given the strong separations, and the presence of coherent structures in non linear interactions with the turbulent motion, which have to be taken into account in the model used. A experimental data base concerning this class of flow, which can be used for validation or improvement of modelling, is therefore of significant relevance. In this study, the generic case of a circular cylinder was chosen, first because of the strong separations and the presence of coherent structures, and also, because of the symmetries of the flow, which allow adapted processing, and then a better visualisation of the flow. With the aim of allowing direct comparison possible with simulations, and therefore evaluations of the models themselves, the cylinder was placed in a confined environment (a square channel) with a high blockage coefficient and a low aspect ratio, thereby allowing simulations to be made on a domain of moderate size, corresponding to the experimental geometry, with the use of well defined boundary conditions. In precedent studies, the data base was achieved using pressure measurements, PIV, stereoscopic PIV and Time Resolved PIV, carried out near the separation and in the very near wake of the cylinder, [10]. The main limitation concerning the TRPIV measurements was the size of the domain which was smaller than that of the low data rate acquisitions, due to the low energy of the laser used. A particular attention was paid to the decomposition of the flow into a coherent part and a turbulent part. To this purpose, phase averaging, using pressure on the cylinder as a reference signal, and POD have been first applied and compared. Both decompositions have been analysed with the help of TRPIV measurements in the small domain, by comparing the contributions of coherent and turbulent fluctuations to the mean Reynolds stress tensor. It has been found, in agreement with many other studies (e.g. [1]), that phase averaging with pressure leads to an overestimation of the turbulent motion and a smoothing of the Kármán vortices, resulting from phase lags occurring at certain instants between the pressure signal and the velocity to be averaged in the wake. On the other hand, POD was found useful to analyse the different parts of the flow, but the main difficulty lies in the choice of the modes to reconstruct the coherent motion. An enhancement of the phase averaging was then achieved, using a definition of the phase angle based on the first two POD coefficients, and thereby alleviating the phase lags effects as the phase is determined directly from the velocity fields to be averaged. This phase averaging was obtained from low data rate PIV. Two new complementary data sets are analysed in this study. First, Time Resolved PIV measurements have been carried out in a domain of similar size as that of the low data rate PIV, using a cylinder of smaller physical size and a laser delivering more energy. Also, a Detached Eddy Simulation has been performed on a domain which corresponds precisely to the experiment. Then, the objectives of this study is both, to use these data resolved in space and time to evaluate the performed conditional averaging, and to achieve a better analysis of the flow.

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Section 2 briefly presents the configuration of the flow and the new measurements which were carried out, together which some comparisons with previous measurements. In Section 3, the instantaneous motion is analysed. Section 4 is devoted to the analysis of POD and phase averaging. Finally an analysis of coherent and incoherent quantities is conducted in Section 5.

2 Configuration and Experimental Set-Up The experiments were conducted in the wind tunnels S1 and S4 of IMFT. As mentioned in the preceding section, a high blockage coefficient and a low aspect ratio were employed to allow comparison with simulations carried out on a domain corresponding precisely to the experimental geometry without the use of ’infinite boundary conditions’. For the previous measurements, the cylinder had a diameter of 14 cm and was placed in a square channel of cross section 6767 cm, leading to a blockage coefficient of D=H D 0:21 and an aspect ratio of L=D D 4:8. Due to the low energy of the laser used, and to the access of the wind tunnel, the previous TRPIV measurements were limited to a small domain. To carry out TRPIV on a larger domain in the wake, the new measurements have been done in the wind tunnel S4 of IMFT, which has a cross section of 61 71 cm. To keep the same blockage coefficient as in the previous measurement, the diameter of the cylinder was chosen 12.5 cm, which results in a blockage coefficient of 0.21 and an aspect ratio of 5.7. Although this aspect ratio is different from the previous one and therefore the flow was not expected to be rigourously the same, it will be shown that a good agreement with the old measurements is achieved for the mean flow and the velocity spectra. A detailed description of the former measurements by low data rate PIV can be found in [11], as well as the procedure used for the reconstruction of the three components in stereoscopic measurements. The new measurements were done using a laser Darwin 220 mJ from Excel Technology, a camera CMOS APX (PHOTRON) with a resolution of 1024 1024 pixels, and DEHS as seeding particles (typical size 1 μm). The system allowed acquisition of image pairs at a rate of 1 kHz. The image pairs were analysed using an in-house code ‘PIVIS’ developed by the ‘Services Signaux Images’ of IMFT, which uses an algorithm based on a 2D FFT cross correlation function implemented in an iterative scheme with a sub-pixel image deformation ([5]). The flow was analysed by cross-correlating 50% overlapping windows of 32 32pixels, yielding fields of 61 57 vectors with a spatial resolution of 3 mm (0.0238D). Approximately 2% of the calculated vectors were detected as outliers using a sort based on the norm, the signal-to-noise ratio, and a median test filter, and these vectors were replaced using a second order least square interpolation scheme. Six temporal series of 3,072 images pairs have been acquired and analysed, each series containing approximately 85 vortex shedding periods. Before presenting results in the following sections, some comparisons between these new TRPIV measurements and the former measurements are necessary, given the small differences in the experimental set-up. First, some comparisons

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y/D = 0.5

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(a) iso contours of 21 (top: low data rate PIV, bot- (b) spectra of v at x=D D 1, and differtom: TRPIV) ent y=D Fig. 1 Comparison of new measurements with the old measurements

between the TRPIV measurements and the former measurements are made. Figure 1a shows a comparison of the mean 21 component of the rotation rate tensor. Although not shown here, the recirculation length is found 1.25 and agrees well with the value of 1.28 found with low data rate PIV. Figure 1b shows a comparison of velocity spectra at the same points issue from the new TRPIV measurements, and TRPIV performed in the S1 wind tunnel. A very good agreement is also achieved. Therefore, the influence of the aspect ratio which had to be modified comparing to the previous studies is found to not have a important effect in the middle span plane. Some results of a Detached Eddy Simulation, which has been performed on a domain that corresponds precisely to the experimental geometry, are also presented in this paper. This simulation, which have been validated against experiment using time averaging, phase averaging and POD in [12], will be presented in details in a companion paper [8] and the numerical details are not presented here. The data set analysed here consists in 14,000 instantaneous snapshots, corresponding to approximately 90 vortex shedding periods, the dimensionless time step being 0.0321.

3 Instantaneous Motion Figure 2 shows a sequence of instantaneous vorticity and velocity, corresponding approximately to half a period (one picture every three is represented and one velocity vector every two is represented). The vortex shedding is clearly shown, together with smaller vortices in the separated shear layer which are wrapped around the Kármán vortices. This behavior is in good agreement with the measurements of [7] at a lower Reynolds number, although the flow is more irregular, as could be expected at this high Reynolds number.

Coherent and Turbulent Process Analysis 0.6 W21 10 8 6 4 2 1 -1 -2 -4 -6 -8 -10

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(e) t=10.752

Fig. 2 Sequence of instantaneous fields (TRPIV)

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Fig. 3 Velocity spectra in the near wake (TRPIV)

Regarding the velocity spectra, shown in Fig. 3 at different locations, they classically exhibit a peak and harmonics which are linked with the vortex shedding, and a continuous part which corresponds to the turbulent motion. As expected, due to the absolute character of the von Kármán instability, the peak which corresponds to the vortex shedding is found at the same dimensionless frequency at every point (Strouhal number St D 0.21). On the rear axis, the first harmonic is predominant in the u spectrum due to the symmetry of the flow. It is also seen that the level of the second harmonic increases as x=D increases, especially for v.

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R. Perrin et al. 1

V

1

0.5 0 0

v,p

v,p

−0.5 −1

−1

−1.5 −2

−2

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P

−3 0

100

200

300

400

150

t*

200

250

300

t*

Fig. 4 Time signal of V velocity at x=D D 1, y=D D 0:5 and of the pressure signal at D 70ı from the simulation (left) and the experiment (right)

The same temporal behavior has been found in the DES simulation. Looking at the time signals of velocity and pressure obtained from the experiments and the simulation, it also appears that some irregularities are present at certain instants (Fig. 4 at t ' 350 and t ' 400 for the DES and t ' 240 for the experiment). During these instants, which do not appear with a regular frequency, the periodic component of the pressure and of the velocity in the very near wake seems to disappear and the ‘mean value’ of v tends to zero. By looking at the field on a larger domain, it appears that the formation of the vortices occurs further downstream in the wake, the velocity signal at a position further downstream still exhibiting a periodic component (Fig. 5).

4 POD and Phase Averaging The quasi periodicity of the vortex shedding allows the use of phase averaging, according to [13], using the triple decomposition: Ui D Ui C UQi C u0i

(1)

where Ui is the time-independent mean flow, UQi is the quasi periodic fluctuating component, u0i is the random fluctuating component, and hUi i D Ui C UQi is the phase averaged velocity. In [11], this was done using the pressure signal on the cylinder at D 70ı as a trigger signal. In [10], it was shown that due to phase lags occurring at certain instants between the pressure signal and the velocity in the wake, a residual periodic component remained in the ‘random fluctuation’, the spectra of which exhibited a small peak in addition to the continuous part. As a result, the contribution of the coherent motion to the time independent Reynolds stresses was underestimated


21

1

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+

+

+

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+

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350 355 360 365 370 375 380 385 390 395 400 405 410 415

t* 1

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350

360

370

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350 355 360 365 370 375 380 385 390 395 400 405 410 415

t*

t*

Fig. 5 Near wake behavior at instants where the shedding is regular (left) and irregular (right). top: DES, bottom: TRPIV

and that of the random motion was overestimated. To alleviate this problem, the phase angle was then defined using the coefficients associated with the two first POD modes, following [2, 14]. It was shown that the vortices was less smoothed by the averaging, and that the level of the contribution of the ‘random motion’ to the time independent Reynolds stresses was diminished. The data used here, which are resolved in space and time, allow to look more precisely at the time evolution of the POD coefficients, as well as the achieved decomposition from a spectral point of view. Figure 6a–c show the first POD modes, obtained from the TRPIV measurements. In agreement with many other studies in wakes ([3, 9]), the two first mode are linked with the von Kármán vortices. Spectra of the coefficients associated with these modes are represented in Fig. 6d. As expected, the spectra of the first two coefficients mainly exhibit a peak at the Strouhal frequency. The spectrum of the third coefficient is more important in the low frequency range. The temporal evolution of

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R. Perrin et al.

0.4

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(d) spectra of POD coefficients (e) time evolution of coeffi- (f) spectrum of v, VQ and v0 at cients a1 , a2 and 'POD x=D D 1:5, y=D D 0:5 Fig. 6 POD and phase averaging (TRPIV)

the first two coefficients, represented in figure 6e, confirms the possibility to define a phase angle representative of the vortex shedding using: p 22 a1 'POD D arctan. p / 21 a2

(2)

where 1 and 2 are the corresponding eigenvalues. The time evolution of this phase angle is represented on Fig. 6e. As this phase angle is defined directly from the velocity to be averaged, the effects of the phase lags occurring between the reference signal and the velocity are expected to be alleviated. For the phase averaging, the instants where the shedding is irregular discussed in the preceding section have been detected and removed. As a low amplitude of the first two POD coefficients q is observed during these instants, this was achieved by

applying a threshold to a12 C a22 . Approximately 20% of the signal was rejected. Although not shown here, this phase averaging have also been applied to the numerical data, and as simulation give access to the whole domain, the influence of the region on which is performed POD have been studied. It appears that the peak in the spectra are well represented by the two first POD modes when using a region which is extended up to approximately 5D. Taking a larger region leads to obtain more modes linked with the von Kármán vortices, and then it is difficult to define a phase angle. However, the definition of a phase angle when POD is performed in the near wake allows the phase averaging on all the domain, and the results appears similar to those achieved for the experiment.


23

The resulting phase averaged motion and fluctuation away from this phase average can be view from a spectral point of view. Figure 6f shows spectra of vQ and v0 at x=D D 1:5 and y=D D 0:5. A significant reduction of the residual peak in the v0 spectrum is achieved using POD coefficients, confirming that the effects of phase jitter are alleviated. The same conclusions have been obtained for the simulation.

5 Analysis of Phase Averaged and Turbulent Motion The phase averaged fields and phase averaged turbulent stresses obtained are presented in this section. Figure 7 shows the phase averaged velocity, the h21 i component of the rotation rate tensor, the turbulent stresses obtained from 2C-PIV and 3C-PIV, as well as the turbulent kinetic energy and the production term at the phase angle ' D 45ı . h21 i clearly exhibits the vortex shedding. The maximum value at the centre of the vortices is of order 3 at x=D ' 1 during their formation and decreases to 2 at

-3

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0.7

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(h) hki

Fig. 7 Phase averaged quantities at ' D 45ı

P 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

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(i) hP i

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24

R. Perrin et al.

x=D ' 2 at the beginning of their convection. hu2 i and huvi presents their highest values in the shear regions near the separation. When the vortices are formed, the two lobes of high values of hu2 i are transported towards the rear axis, and the centre of the vortices. At the beginning of the convection, the highest values of huvi are located in the shear regions near the saddles points between the vortices, where the deformation rate is important. Concerning the normal stresses in the near wake, they all exhibit their highest values near the centre of the vortices when they begin to be convected. High values of hv2 i and hw2 i are also present between the vortices, and can be supposed to be linked with the presence of longitudinal vortices connecting the primary one. Finally, regions of low values of stresses are identified in front of the vortices, corresponding to external fluid entering in the wake. It is also noticeable that a strong anisotropy is observed. The general topology of the stresses is found in good agreement with precedent studies in wakes ([1, 4, 6]). With the use of stereoscopic PIV, the turbulent kinetic energy can be evaluated without assumption on hw2 i, and its topology is compared here to the production term that appears in its equation. It appears, in agreement with the aforementioned studies, that while the production is mainly located near the saddles points in the shear regions, where the deformation rate and huvi are important, the turbulent kinetic energy is mainly located near the centre of the vortices, suggesting a transport of the turbulent energy.

6 Conclusion and Outlooks The very near wake of a circular cylinder at high Reynolds number was experimentally studied using PIV, stereoscopic PIV and TRPIV. Due to the organised and random character of the flow, a particular attention was paid to achieve a decomposition of the motion into a coherent and a turbulent part. While it was shown in precedent studies that phase averaging using the pressure signal on the cylinder could lead to an overestimation of the random motion, due to phase lags occurring between the reference signal and the velocity in the wake, it has been shown that the first POD coefficients could be used to define a phase angle directly from the velocity fields, then alleviating the overestimation of the turbulent motion. Two complementary data sets which are revolved in space and time, obtained by TRPIV and by a DES simulation, have been used to analysed the instantaneous motion, as well as to analyse the POD and phase averaging that were performed. Then, a cartography of the phase averaged mean motion and turbulent stresses in the near wake have been provided. A future study will be devoted to the analysis of the tridimensionality, and particularly the longitudinal vortices which connect the primary ones, using both the presented numerical results and also Time Resolved Stereoscopic measurements in planes along the spanwise direction. Acknowledgements The company ”Excel Technology France” is greatly acknowledged for loaning Darwin PIV laser system.


25

The authors also acknowledge the partial funding of the work presented here by the European Community during the DESider project (in the 6th Framework Program, under Contract No. AST3-CT-2003-502842) and by the German Research Foundation (DFG) within the scope of the Collaborative Research Center SFB 557.

References 1. B. Cantwell and D. Coles. An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder. J. Fluid Mech., 136:321–374, 1983. 2. M. Ben Chiekh, M. Michard, N. Grosjean, and J. C. Bera. Reconstruction temporelle d’un champ aéodynamique instationnaire partir de mesures PIV non résolues dans le temps. In 9e Congrès Francophone de Vélocimétrie Laser, 2004. 3. A. E. Deane, I. G. Kevrekidis, G. E. Karniadakis, and S. A. Orzag. Low-dimensional models for complex geometry flows: Application to grooved channels and circular cylinders. Phys. Fluids A, 3:2337–2354, 1991. 4. A. K. M. F. Hussain and M. Hakayawa. Eduction of large-scale organized structures in a turbulent plane wake. J. Fluid Mech., 180:193–229, 1987. 5. B. Lecordier and M. Trinite. Advanced PIV algorithms with image distortion - Validation and comparison from synthetic images of turbulent flows. In PIV03 Symposium, Busan, Korea, 2003. 6. A. Leder. Dynamics of fluid mixing in separated flows. Phys. Fluids A, 3(7):1741–1748, 1991. 7. A. Leder and M. Brede. Comparisons between 3D-LDA measurements and TR-PIV data in the separated flow of a circular cylinder. In 12th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 2004. 8. C. Mockett, R. Perrin, T. Reimann, M. Braza, and F. Thiele. Analysis of detached-eddy simulation for the flow around a circular cylinder with reference to piv data. In IUTAM Symposium on Unsteady Separated Flows and their Control, June 18–22, Corfu, Greece, 2007. 9. B. Noack, K. Afanasiev, M. Morzynski, G. Tadmor, and F. Thiele. A hierarchy of lowdimensional models for the transient and post-transient cylinder wake. J. Fluid Mech., 497:335–363, 2003. 10. R. Perrin, M. Braza, E. Cid, S. Cazin, A. Barthet, A. Sevrain, C. Mockett, and F. Thiele. Phase averaged turbulence properties in the near wake of a circular cylinder at high Reynolds number using POD. In 13th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 2006. 11. R. Perrin, E. Cid, S. Cazin, A. Sevrain, M. Braza, F. Moradei, and G. Harran. Phase averaged measurements of the turbulence properties in the near wake of a circular cylinder at high Reynolds number by 2C-PIV and 3-C PIV. Exp. Fluids, 42(1), 2007. 12. R. Perrin, C. Mockett, M. Braza, E. Cid, S. Cazin, A. Sevrain, P. Chassaing, and F. Thiele. Joint numerical and experimental investigation of the flow around a circular cylinder at high reynolds number. Topics in Applied Physics, PIVNET 2. To be published, 2007. 13. W. C. Reynolds and A. K. M. F. Hussain. The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech., 54:263–288, 1972. 14. B. W. van Oudheusden, F. Scarano, N. P. van Hinsberg, and D. W. Watt. Phase-resolved characterization of vortex shedding in the near wake of a square-section cylinder at incidence. Exp. Fluids, 39:86–98, 2005.


Investigation of Aerodynamic Capabilities of Flying Fish in Gliding Flight H. Park and H. Choi

Abstract In the present study, we experimentally investigate the aerodynamic capabilities of flying fish. We consider four different flying fish models, which are darkedged-wing flying fishes stuffed in actual gliding posture. Some morphological parameters of flying fish such as lateral dihedral angle of pectoral fins, incidence angles of pectoral and pelvic fins are considered to examine their effect on the aerodynamic performance. We directly measure the aerodynamic properties (lift, drag, and pitching moment) for different morphological parameters of flying fish models. For the present flying fish models, the maximum lift coefficient and lift-to-drag ratio are similar to those of medium-sized birds such as the vulture, nighthawk and petrel. The pectoral fins are found to enhance the lift-to-drag ratio and the longitudinal static stability of gliding flight. On the other hand, the lift coefficient and lift-to-drag ratio decrease with increasing lateral dihedral angle of pectoral fins. Keywords Flying fish Aerodynamic performance Gliding Static stability Wing morphology

1 Introduction Other than numerous birds and flying insects, several vertebrates have been observed to possess an ability to fly in air [1, 2]. Among them, marine flying fish has been noted for its excellent flight performance and a few early studies investigated the basic aerodynamics of flying-fish flight based on the field observations [3–5]. Morphologically the flying fish has hypertrophied pectoral and pelvic fins which it uses as wings for gliding flight. The flight of flying fish is quite remarkable, e.g., it can glide the distance over 400 m in successive flights which are enabled by the unique method for take-off, named ‘taxiing’. Recently several biologists like

H. Park () and H. Choi School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

27

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H. Park and H. Choi

Davenport [6–8] and Fish [9] conducted allometric studies in relation to the aerodynamic performance of flying fish. They measured the variations of morphometric parameters for various flying-fish species in live or preserved states and analyzed their influences on the flight characteristics. Although several aspects of flying-fish flight has been understood or conjectured in previous studies, any quantitative analysis about the flying-fish flight has not been conducted so far. Furthermore, it might be interesting to compare the aerodynamic properties of flying-fish flight with those of other fliers in nature. Therefore, in the present study, we directly measure the aerodynamic forces and moment on flyingfish models in a wind tunnel. To meet the real flight condition of flying fish, flyingfish models are made of the real darkedged-wing flying fishes (Cypselurus hiraii) that are stuffed in gliding posture. The force measurements are conducted at the freestream velocity of 12 m/s. The maximum flight speed of real flying fish is known to be 10–20 m/s [7, 9]. We also investigate the aerodynamic function of flying fish’s wing morphology.

2 Experimental Setup 2.1 Flying-Fish Models For our study, we collected about 40 darkedged-wing flying fish (Cypselurus hiraii) and stuffed four of them in appropriate gliding postures (Fig. 1). As shown in Fig. 1, Cypselurus hiraii has both enlarged pectoral and pelvic fins and we consider the wing configuration of each flying-fish model as follows: (a) both pectoral and pelvic fins enlarged (models 1–3), and (b) only pectoral fins enlarged with pelvic fins folded against the body (model 4). For four models, we measured several parameters representing the aerial morphology of flying fish as tabulated in Table 1. Definitions of these parameters are illustrated in Fig. 2. For models 1–3, the lateral dihedral angles .ˇ1 / of pectoral

Fig. 1 Pictures of the flying fish (Cypselurus hiraii) models considered in the present study

Investigation of Aerodynamic Capabilities of Flying Fish in Gliding Flight

29

Table 1 Morphometric parameters of the flying fish models Models Standard length (SL, mm) Aspect ratio (AR) Pectoral fin area .A1 ; mm2 / Pelvic fin area .A2 ; mm2 / Wing span (S, mm) Averaged chord length of pectoral fins (c, mm) Lateral dihedral angle of pectoral fins .ˇ1 ; ı / Incidence angle of pectoral fins .ˇ2 ; ı / Incidence angle of pelvic fins .ˇ3 ; ı /

1 205 8.5 7,468 1,858 252 29.6 22 12 2

2 209 9.1 7,392 2,020 260 28.4 12 15 2

3 203 9.8 6,946 2,131 261 26.6 7 12 5

4 199 9.6 5,639 – 233 24.2 5 8 –

Fig. 2 Definitions and orientations of morphometric parameters

fins are artificially changed such that models 1 and 3 have largest and smallest ˇ1 , respectively. On the other hand, other parameters are the original values of the specimen. The aspect ratio (AR) of the present flying fish is around 8.5–9.8 which is comparable to those of birds in general [2, 9]. Each flying-fish model is connected to the force/torque sensor to measure aerodynamic forces and moment.

2.2 Force Measurements Force measurements are performed in an open-circuit blowing-type low-speed wind tunnel as shown in Fig. 3. Here, x, y, z denote the streamwise, vertical and spanwise directions, respectively. The test section has the size of 3 0:3 0:6 m in the streamwise, vertical and spanwise directions, respectively and the maximum speed is 25 m=s. At the free-stream velocity of 10 m=s, the background turbulence intensity and the uniformity of the mean velocity is within 0.5%.

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Fig. 3 Schematic diagram of the force measurement in a wind tunnel

To measure the forces and moment, we use a six-axis force/torque sensor (NANO17, ATI) that measures three-components of forces and moments simultaneously. The resolutions of the sensor are 1=1,280 N and 1=256 N mm in measuring force and moment, respectively. Varying the attack angle (˛, see Fig. 3) in the range of 15ı < ˛ < 45ı , we measure the lift and drag forces, and pitching moment for models 1–4. To minimize the interference by the strut, we use very slender strut whose cross section is an ellipse with maximum thickness is 2.68 mm. The repeatability error of the measurement is within ˙1:5%. The raw signals from the sensor are digitized by the A/D converter (PXI-6259, NI) for 300 s at the sampling rate of 10 kHz. As the maximum flight speed of real flying fish has been reported to be 10–20 m=s [7, 9], force measurement is conducted at the free-stream velocity .u1 / of 12 m=s and the corresponding Reynolds numbers are Re D u1 c= D 20;000– 24;000, where u1 is the free-stream velocity, c is the averaged chord length of the pectoral fins and is the kinematic viscosity.

3 Results and Discussion From the measured lift .L/, drag .D/ and pitching moment .M /, we obtain lift .CL /, drag .CD / and pitching moment .CM / coefficients as a function of angle of attack as follows: CL D L=.0:5u1 2 A/:

(1)

CD D D=.0:5u1 A/:

(2)

CM D M=.0:5u1 2 A1 c/:

(3)

2


31

Fig. 4 Variations of the lift .CL / and drag .CD / coefficients with the attack angle: (a) CL ; (b) CD

where is the density of air, and A is the total wing area, A1 C A2 (see Fig. 2 and Table 1). Figure 4 shows the variations of the lift and drag coefficients with respect to the attack angle, ˛ C ˇ2 , for models 1–4. Here, ˇ2 is the incidence angle of the pectoral fins which is different for each model (Table 1). As shown in Fig. 4, the maximum lift coefficients are about 1.0–1.1 and the minimum drag coefficients are about 0.07–0.12, which are similar to those of the vulture and nighthawk [10]. The lift coefficients for models 1–3 (with both enlarged pectoral and pelvic fins) are smaller than that of model 4 (with only pectoral fins enlarged). Since the area of the pelvic fins is about 20% of the total wing area (Table 1), pelvic fins do not enhance the wing loading. On the other hand, the lift coefficient decreases with increasing lateral dihedral angle of pectoral fins .ˇ1 /. The variations of lift-to-drag ratio with the attack angle, ˛ C ˇ2 are given in Fig. 5. The maximum lift-to-drag ratio is about 4.4 (model 3) which is greater than that of hawk (3.8) and petrel (4.0) [10]. The lift-to-drag ratios for models 1–3 are larger than that of model 4, indicating that the pelvic fins enhance the gliding performance of flying fish by having smaller drag coefficient. Like the lift coefficient, the lift-to-drag ratio also decreases with increasing lateral dihedral angle of pectoral fins .ˇ1 /. For a gliding animal, flight stability is also one of the important issues. It is known that the nose-down pitching moment should increase with increasing attack angle (i.e., the slope of the pitching moment coefficient curve should be negative) for a glider to have a longitudinal static stability [11]. Also, the more negative the slope of pitching moment curve, the more stable the glider is. The variations of the pitching moment coefficient at the center of gravity are shown in Fig. 6. It is found that the gliding flight of flying fish is statically stable in longitudinal direction, and with pelvic fins (models 1–3) the longitudinal static stability is enhanced, which is similar to the function of tail plane in modern aircraft. On the other hand, the longitudinal static stability is more enhanced with decreasing lateral dihedral angle of pectoral fins .ˇ1 /.

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Fig. 5 Variations of the lift-to-drag ratio (L=D) with the attack angle

Fig. 6 Variations of the pitching moment coefficient .CM / with the attack angle


33

4 Conclusions In the present study, we investigated the aerodynamic properties of flying-fish flight by directly measuring the lift and drag forces, and pitching moment on real flyingfish models. The flying fish has a wing performance comparable to those of birds like the vulture, nighthawk and petrel in terms of the lift coefficient and lift-to-drag ratio. We also examined the effects of wing morphologies on the gliding performance of flying fish. Enlarged pelvic fins enhanced the lift-to-drag ratio and longitudinal static stability of the flying fish, but, the lift coefficient decreased due to enlarged pelvic fins. On the other hand, the lift coefficient and lift-to-drag ratio decreased with increasing lateral dihedral angle of pectoral fins. Acknowledgements This work has been supported by the National Research Laboratory Program of the Korean Ministry of Science and Technology.

References 1. Alexander, D.E., Nature’s Flyers, The Johns Hopkins University Press, Baltimore, MD (2002). 2. Vogel, S., Lift in Moving Fluids, Princeton University Press, Princeton, NJ (1994). 3. Breder, C.M., Jr., On the structural specialization of flying fishes from the standpoint of aerodynamics. Copeia 4 (1930) 114–121. 4. Mills, C.A., Source of propulsive power used by flying fish. Science 83 (1936) 80. 5. Hertel, H., Take-off and flight of the flying fish. In: Structure – Form – Movement, eds. M.S. Katz, Reinhold, New York (1966) 218–224. 6. Davenport, J., Wing-loading, stability and morphometric relationships in flying fish (Exocoetidae) from the north-eastern atlantic. J. Mar. Biol. Ass. U.K. 72 (1992) 25–39. 7. Davenport, J., How and why do flying fish fly? Rev. Fish Biol. Fish. 40 (1994) 184–214. 8. Davenport, J., Allomeric constraints on stability and maximum size in flying fishes: implications for their evolution. J. Fish Biol. 62 (2003) 455–463. 9. Fish, F.E., Wing design and scaling of flying fish with regard to flight performance. J. Zool. Lond. 221 (1990) 391–403. 10. Withers, P.C., An aerodynamic analysis of bird wings as fixed aerofoils. J. Exp. Biol. 90 (1981) 143–162. 11. Thomas, A.L.R. and Taylor, G.K., Animal flight dynamics. I. stability in gliding flight. J. Theor. Biol. 212 (2001) 399–424.


Effect of Unsteady Separation on an Automotive Bluff-Body in Cross-Wind M. Gohlke, J.F. Beaudoin, M. Amielh, and F. Anselmet

Abstract This paper is treating the dynamic effects of cross-wind on a 3D bluffbody. It is shown that contrary to the mean flow, it is not the upper vortex that predominates the forces, but the flow generated and piloted by the front lee-strut wake. It is made responsible for the appearance of a frequency peak in the bluff-body force spectra. Furthermore it is shown that the intensity as well as the frequency of this peak depend on the yawing angle. Experimental and numerical results are used to analyse the flow dynamics. Keywords Cross-wind Aerodynamics Bluff-body Vortices Wake

1 Introduction During many years automobile aerodynamic research was concentrated on drag-reduction and some major advances have been achieved. However, a new topic has emerged recently. As consumers demand bigger interior, cars become higher and larger. This leads to more side wind sensibility. Hence, it is necessary to understand the link between side-force, yawing-moment and rolling-moment, so that the driving-stability and -comfort can be enhanced. Howell [4] and Gillhaus and Hoffmann [3] have shown that in automobile applications it is essentially the side force and its repartition, causing the yawing moment, that are responsible for driving stability. The aim of the present study is to get some insight into the flow-structures that have an influence on these forces, which is a fundamental question that concerns many fields beyond the automotive aerodynamics community. To reduce the

M. Gohlke () and J.F. Beaudoin PSA Peugeot-Citroën, Department of Research and Innovation, Vélizy-Villacoublay, France e-mail: [email protected] M. Gohlke, M. Amielh, and F. Anselmet IRPHE, UMR CNRS 6594, Technopôle de Château-Gombert, Marseille, France M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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Fig. 1 Coordinate system and forces as applied to the bluff-body used for investigation, over all length used as reference length: Lref

complexity of the flow around a vehicle in cross-wind a simplified bluff-body in ground-proximity, introduced by Chometon et al. [1], has been chosen for a more generic analysis. Figure 1 shows the model, “Willy”, used for the present study, with the forces and moments in their reference system. In a cross wind situation the flow around this bluff body is basically characterised by its asymmetry, which is at the origin of additional forces and moments, namely side force and yawing moment. This paper mainly aims at giving some insight to the dynamics of the forces and their link to the flow structures. First of all a summary of the principal results, obtained from the steady flow analysis, are presented. In the main part, devoted to the unsteady character of the flow, first a brief description of the numerical simulation and the experimental setup is given. Some results obtained from these investigations are then analysed and finally the influence of the flow structures is discussed.

2 Recapitulation of Mean Flow Features In this section the main features of the steady flow field are presented. A more exhaustive overview of these results can be found in Gohlke et al. [2]. As mentioned in the introduction, the main interest is turned toward the side force distribution and the resulting yawing moment. To simulate cross wind, a simple but often applied method is used. The wind tunnel model is therefore turned in respect to the free stream by an angle ˇ, called the yaw angle. As this angle rises, the component of the oncoming flow becomes more important, simulating a stronger cross wind. In this study, the maximum yaw angle is 30ı and the upstream velocity is 40 m=s. Figure 2 shows the strong dependency of forces and moments on the yawing angle. While the global side force (Fig. 2a) has a rather linear rise, the slope of the yawing moment CN (Fig. 2b) shows an inclination at around 16ı , which, as discussed in the introduction, is favourable for cross-wind stability. To clarify the origin of this decrease, the lateral force is divided into front and rear side forces (Fig. 2c), which leads to the following two observations. The front contribution CYF

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is three times higher but it is the rear side force CYR that shows two slope changes, at 4ı and 16ı . The second one, occurring at yaw angles above 16ı , leads to a steeper slope which is thought to partially explain the decrease in yawing moment. In Gohlke et al. [2] these two slope changes are linked to the presence of two vortices (a) and (b) as shown by the sketch in Fig. 3. The vortex (b) is close to the body at small yaw angles and induces low pressure at the rear end of the model. As ˇ grows, this flow structure gains in size but is also orientated away from the body, hence it has less influence on the rear side force. The front strut wake plays herein an important role by imposing a direction to this vortex. At yaw angles around 15ı a second vortex (a) is created, due to detachment of the flow over the top. This vortex grows in size, as visible on the LDV-measurements shown in Figs. 4a and 4b, and induces a low pressure zone on the rear lee side, giving rise in CYR , and finally leading to a counter-rotating rear yawing moment. This local effect is shown in Fig. 4c, which is obtained from local static pressure measurements and has permitted the calculation of force contributions per region. It f can be observed that the strongest gain in local yawing moment C N is measured on the rear upper leeward side (). Figure 5a and b give an overview of the mean flow field of the lee side at 15ı and 30ı as obtained from numerical simulations. They illustrate the presence of the longitudinal vortices and their location in respect to the model.

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3 Unsteady Flow Analysis 3.1 Setup In this section the setup used to analyse the unsteady character of the flow is presented. First the numerical method employed and its setup is discussed and then the experimental method is outlined. The upstream flow velocity in both cases is 40 m=s. To simulate a cross flow, the model is turned around its z-axis at the strut centre in respect to the oncoming flow up to a yaw angle of ˙30ı . Numerical Investigation The numerical code used in this investigation is based on a lattice Boltzmann method. Two yaw angles have been studied, one at 15ı and one at 30ı . Different zones of spatial resolution are used, the finest grid being close to the model and around the cylindrical struts of the order of magnitude of 0:875 106 Lref . In the 30ı case a special refinement of 1:75 106 Lref is done in the zone


39

of the lateral vortices on the lee side. This zone is defined by the iso-surface of the vorticity norm, obtained from an earlier coarser numerical investigation. The total number of voxels used to define the flow field is of the order of 107 . 1:4 105 timesteps are simulated for both yaw angles, which correspond to a physical time of 0.56 s. To find a converged behaviour drag-, side-force- and lift coefficients are monitored and it is found that 2 104 timesteps are necessary to consider the simulation as converged. This leaves 0.48 s of physical time for dynamic flow analysis. The turbulence level and the boundary layer are adapted to fit the conditions in the wind tunnel experiments. The numerical simulation is confronted with experimental results and globally show a satisfactory correspondence. Exemplarily some of these confrontations are given here. The force measurements in Fig. 2 show that the steady forces are well reproduced (the numerical results are displayed by the stars). At the high yaw angle, the side force is slightly over estimated by the simulation with a difference of CY D 8:8%, but the front/rear distribution is well respected, which is confirmed by a good CN estimation differing by CN D 1:4%. A further confirmation of good coherence comes from the LDV measurements in a plane close to the blunt end. The vorticity level as well as the roll-up of the streamlines is quite similar for the 30ı case, shown in Fig. 6a and b. It is therefore stated, that the numerical simulation gives accurate results and can be amended with the experimental investigation, which is completer in angular resolution.

(a) LDV measurements 30ı

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Fig. 6 Vorticity in a plane obtained from LDV measurements and from numerical simulation at 30ı yaw

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Fig. 7 Model equipped with pressure transducers and their position during the different setups

Experimental Investigation One half of the hollow experimental model is equipped with 249 pressure holes with a diameter of 0.8 mm. They can be equipped with microphones to analyse local pressure fluctuation. The microphones are fixed on the inside of the model with the help of tight joints, that are glued to the shell. The measuring equipment used is capable to analyse 60 microphones at a time. Three different setups are hence used: one covering the upper part, one covering the lower part and a combination of both of them (see Fig. 7). This investigation has been undertaken for a yaw angle range between 30ı and C30ı with a step of 5ı at a measurement frequency of 8 kHz.

3.2 Results and Discussion It has been shown that the steady flow is mainly characterised by two longitudinal vortices on the lee side of the model and that they play an important role in the local side force and local yawing moment distributions (see Section 2). In this section, further results are presented, that aim at understanding the dynamics of the side force CY and the yawing moment CN . These results are based on two different approaches, on the one hand, a numerical simulation gives access to the unsteady forces and a very good spatial resolution. But, on the other hand, good yaw angle resolution can only be achieved with an experimental approach, where pressure fluctuations are measured on the model surface. These two setups are described in Section 3.1 Numerical Investigation To analyse the dynamics of the forces, numerical simulations were undertaken for two different yaw angles, 15ı and 30ı . As shown in Section 3.1 the comparison between the wind tunnel measurements and the numerical simulation is satisfactory. It can be seen that the steady forces and moments of the numerical simulation correspond fairly well. This is demonstrated by the stars in the forces versus yaw angle graphs in Fig. 2. On the bases of this good correlation

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between the numerical simulation and the wind tunnel measurements for the steady flow, it is assumed that the unsteady analysis using the numerical data will also produce reliable results. Force autospectra obtained from the numerical simulation of the 15ı case and 30ı case, as shown in Fig. 8 for the side force and the yawing moment, display overall simplicity. A first observation is the resemblance of these spectra between the two yaw angles. The spectra globally have the same energy level. At high frequencies they follow a power law, typical for generic turbulent flows, but the slope value of 4, could not be explained so far. Furthermore, all spectra display a frequency peak emergence, which is easiest to distinguish for the small yaw angle. These peaks have an amplitude that is about 10 dB higher for the 15ı case than for the 30ı case. Furthermore, these frequencies vary with yaw, most visible on the front side force, where f15CıYF D 317 Hz and f30CıYF D 363 Hz. To analyse the origin of this coherent part of the spectra, indications can be obtained from local force analysis undertaken on the cylindrical struts. They are shown in Fig. 9. Again, at high frequencies, the spectra show a decrease following a power law. Generally, peaks with identical frequencies, as observed for the whole body, emerge clearly at both yaw angles. The amplitude of the frequency peak is higher and the peaks are slimmer for the 15ı case. The global and the front side force again show a variation of the frequency which depends on the yaw angle: f15ı D 317 Hz and f30ı D 363 Hz. Only the frequency on the rear struts is independent of the yaw angle change: f15ı D f30ı D 317 Hz. Due to the fact that there is a strong coherent phenomenon in the spectra, with the same frequency as observed on the body force spectra, it is believed that the struts are at the origin of these frequencies. This can be explained by vortices shed in their wake, similar to the “von Karman vortex street”. This shedding is thought to induce its proper frequency on the body, by an alternation of local low and high pressure zones close to the body surface.

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The spectrum of CYF is presented in Fig. 9d, it has been obtained only from the force on the body (without the contribution from the struts). Again the frequency peak can be observed, which underlines the influence of the strut wake on the body forces. A first attempt is to verify whether the order of magnitude corresponds to the vortex shedding frequency of a cylinder. This is done by a simple comparison through the Strouhal number. The corresponding shedding frequency is calculated with the dimensions of the cylindrical struts and a Strouhal number of St D 0:21 (for a Re D U1 D= D 103 to 3 106 ). The local Reynolds number being Re.U1 ; d / D 8:2 104 this corresponds to f D 274:5 Hz. A detailed look at the oncoming flow velocity is shown in Fig. 10. It can be seen that the flow velocity upstream of the lee strut is at first slowed down, due to a blockage effect of the model, before it is strongly accelerated due to the curvature on the model nose, generating a convergent effect. So the velocity to be taken into account to estimate the vortex shedding frequency is about 10 m=s higher than the actual free stream velocity, so that the obtained frequency is about 343 Hz, corresponding to the 30ı yaw angle. This is to be compared with f30CıYF D 363 Hz and shows a good agreement.


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This comforts the idea that the vortex shedding from the leeward strut is responsible for the peaks found on the force spectra. Figure 10 also shows that the oncoming flow velocity changes with the yaw angle, leading to a change in the shedding frequency. The authors believe that this explains in part the variation of the frequency of the coherent phenomena observed in the spectra; but other factors such as the model geometry behind the strut might favour an acceleration of the strut wake, which then also influences the resulting frequency. Experimental Investigation The numerical investigation suggests that there is a yaw angle dependency of the coherent part of force spectra, showing frequency changes and amplitude variations. With the help of surface pressure measurements, as described in Section 3.1, an analysis can be undertaken over the whole yaw angle range from 30ı to C30ı , with a step of 5ı . The spectral analysis of the results from these local pressure measurements is shown in Fig. 11. Exemplarily, four different transducers 30, 31, 32 and 37 are analysed for the yaw angles between 0ı to 30ı (lee side). First of all, it has to be pointed out that these transducers are not only sensitive to turbulent fluctuations, but also to acoustic noise. The wind tunnel, being equipped with four turbines downstream of the model, generates a noise with a frequency of 86 Hz for a wind velocity of 40 m=s. Black lines in the illustrated graphs indicate this frequency and its harmonics, so that they can be clearly distinguished. It can be stated that their amplitude stays below 100 dB. Flow analysis will nevertheless be restricted to frequencies that do not correspond to these acoustic frequencies. A further observation on these measurements is a broad band bump in the spectra at frequencies around 2 kHz. This is explained by the way the transducers are installed and corresponds to resonance frequencies of the cavity between the outer shell of the model and the transducer. This frequency can vary slightly with the location and the fixation of the transducer. Figure 11a–c reproduces the results from three different pressure transducer at different heights on the lee ward side, No. 30 being the lowest, placed on the horizontal underbody, as indicated in Fig. 12a. This transducer No. 30 shows a yaw angle independent energy level. The further the transducer is located upward the lateral side of the model, the more this level depends on the yaw angle. Figure 11b shows a dependency up to a yaw angle of 10ı , beyond the energy levels remain constant. A strong dependency of this level on the yaw angle is demonstrated by transducer No. 32. It can also be pointed out that at 30ı yaw the global energy level is the same for all the positions. This means that the lateral side is more and more subject to the flow fluctuation and the lateral turbulence level rises constantly with the yaw angle, as a large lateral wake is building up. Furthermore, these spectra show a slope at high frequency of 7/3, this spectral decrease has often been observed on pressure measurements in homogeneous and isotropic turbulence. Most of the spectra show clear peaks at frequencies between 270 and 300 Hz with an amplitude mostly above 100 dB. Although this is not exactly the same value as observed in the numerical simulation, these frequencies are rather close. As already

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observed on the force spectra, the amplitudes and the corresponding frequencies, obtained from the pressure transducers, vary with the yaw angle. Figure 12b and c show this behaviour exemplarily for the transducers No. 31 and 37, the later one being just behind the front lee side strut (see Fig. 12a) and therefore measuring in its wake.


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It can be observed that at low ˇ the frequency and the amplitude rise with the yaw angle. The amplitude reaches its maximum for 10ı to 15ı , where the frequency attains a value of around 290 Hz. While the amplitude decreases with a further rise in yaw angle, the frequency seems established around 295 Hz. In addition to this, there is a further frequency bump arising around a frequency of 160 Hz at yaw angles of 25ı and 30ı , mainly visible on the autospectra of the transducer No. 31, which still has to be analysed in more detail. The amplitude change observed on the pressure transducer No. 37 is most likely due to its position in respect to the front lee cylinder wake. As the yaw angle grows the wake is more and more turned away from the measurement point, so that it is less sensible to the vortex shedding. The variation in amplitude of tap No. 31 on the other hand is believed to be linked to the observations made for the mean flow. At 0ı yaw the cylinder wake stays under the model and hence has little or no influence on the lateral side. As the angle rises, the cylinder wake interacts with the lateral flow and the peak appears. With a further rise, a lateral vortex (b) is generated due to an interaction of the underbody flow with the free flow and is piloted by the cylinder wake. As the angle becomes larger than approximately 15ı , this longitudinal structure detaches from the body and hence the coherent part of the fluctuations is orientated further away from the body, so that they have less effect on the body forces. This results in a decrease in the induced pressure fluctuations and hence the organised force fluctuation. This shows that it is mainly the front lee strut that pilots the flow on the lee side and induces coherence into the flow and hence onto the body forces.

4 Conclusion Summarising, the mean flow is characterised by two lee ward vortical structures (a) and (b), shown in Fig. 3. The upper vortex (a) plays an important role on the mean forces and is at the origin of a counter rotating yawing moment, created on the upper rear lee side. This moment has a positive effect on the global yawing moment, which it reduces. The work on the analysis of the force dynamics, as developed in this paper, has shown that a coherent frequency is induced to the model dynamic forces. It is shown that this is linked to the vortex shedding of the cylindrical struts, especially to the wake of the front lee strut. Furthermore, the frequency and the amplitude of this shedding phenomenon, are shown to depend on the yawing angle. The amplitude reaches its maximum around 15ı at a frequency around 290 Hz for the pressure measurements and 317 Hz for the numerical simulation. The wind tunnel measurements indicate that the frequency rises with the yaw angle. Beyond 20ı the frequency stays fairly constant around 295 Hz, but the amplitude decreases further. This phenomena is explained by the analysis from the steady flow, more precisely it is linked to the flow structure (b). This longitudinal vortex detaches from the model at yaw angles larger than 15ı , hence causing less coherence in the dynamic forces.

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So in the dynamic analysis it is not the upper vortex, that plays a predominant role at high yaw angles, as it is the case for the mean force, but it is especially the lower vortex that is strongly interacting with the front lee strut and has the most effect in the mid yaw angle range.

References 1. CHOMETON , F. ET AL.: Experimental Analysis of Unsteady Wakes on a New Simplified Car Model. Bluff Body Aerodynamics and Applications Congress, Ottawa (2004) 2. GOHLKE, M. ET AL.: Experimental Analysis of Flow Structures and Forces on a 3D-Bluff-Body in Constant Cross-Wind. Experiments in Fluids, 43(4), 579–594, Springer (2007) 3. GILHAUS, A.M. & RENN , V.E.: Drag and Driving-Stability-Related Aerodynamic Forces and Their Interdependence Results of Measurements on 3/8-Scale Basic Car Shapes. In: SAE, Society of Automotive Engineers, Warrendale, PA (1986) 4. HOWELL Shape Features which Influence Crosswind sensitivity. In: Vehicle Ride and Handling Conf., I. Mech. E. Paper C466/036/93

Part II

Statistical and Hybrid Turbulence Modeling of Unsteady Separated Flows


Flow Prediction Around an Oscillating NACA0012 Airfoil at Re D 1,000,000 O. Frederich, U. Bunge, C. Mockett, and F. Thiele

Abstract The maximum obtainable lift of a rotationally-oscillating airfoil is significantly higher than in the static or quasi-static case. The correct prediction of dynamic stall as the basis of the dynamically increased lift is essential to quantify the time-dependent load on the airfoil structure. This study applies unsteady RANS (URANS) and detached-eddy simulation (DES) with various turbulence models and parameter variations in order to capture the physics around an oscillating NACA0012 airfoil at a relatively high Reynolds number and to identify possible advantages and potential drawbacks of the given methods. The quality of the flow prediction is assessed primarily on the basis of integral force coefficients compared to experimental results, revealing the influence of resolution on maximum lift and the corresponding angle of incidence. Keywords NACA0012 Oscillating airfoil Dynamic stall URANS DES

1 Introduction Flows around rotationally-oscillating airfoils are characterised by two important physical phenomena. On the one hand a hysteresis develops in the curves of lift and drag versus angle of attack and on the other hand the maximum lift is much higher than in the static or quasi-static case. Both effects are associated with unsteady flow separation, which is also referred to as “dynamic stall”. A dynamically affected process occurs when the oscillatory frequency is significantly higher than the frequency of potential vortex shedding in the presence of high angles of attack.

O. Frederich (), C. Mockett, and F. Thiele Berlin University of Technology, Institute of Fluid Mechanics and Engineering Acoustics, Müller-Breslau-Str. 8, 10623 Berlin e-mail: [email protected] U. Bunge IVM Automotive Wolfsburg GmbH, Wolfsburger Landstr. 22, 38442 Wolfsburg M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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Such patterns appear in diverse engineering applications e.g. helicopter rotor blades, turbo-machinery and wind turbines. The configuration investigated here is a NACA0012 airfoil that oscillates sinusoidally around the quarter chord line and mean incidence of 15ı with an amplitude of 10ı at Reynolds number Rec D106 . With the reduced frequency of the oscillatory movement kosc D fosc c=u1 D 0:1 and the estimated Strouhal number for the vortex shedding (vsh) of a NACA0021 St D fvsh c=u1 D 0:2 [8], the ratio fosc =fvsh 18 confirms the existence of a dynamically affected stall. This also reveals that the large-scale dynamics in the stalled flow are dominated by the prescribed motion of the airfoil in contrast to the vortex shedding behind a steady airfoil at high angle of attack or behind a bluff body. The present case comprises an oscillation around the maximum lift, and thus the challenge to rise to is the massively-separated flow after stall and the accompanying highly unsteady flow physics. The time-accurate prediction of this is essential to enable a correct prediction of any fluid-structure interaction. This has already been outlined in [3], where turbulence-resolving and more advanced simulation methods such as detached-eddy simulation (DES) are recommended. With integral values in the focal point of interest though, unsteady RANS (URANS) is in some cases seen to be sufficient in this study due to the aforementioned motion-induced unsteadiness of the flow. This will be demonstrated by comparison with experimental results averaged over 200 cycles that have been published by Mcalister et al. [6].

2 Numerical Method and Parameters The flow field around the NACA0012 airfoil is discretised using block-structured grids with varying topology and spatial resolution. Concerning the spatial discretisation, a c-type grid arranged for URANS boundary layer predictions with 35;000 cells in each 2D slice is tested against an o-type grid specially prepared for DES of massive separation with 46;000 cells in 2D (Fig. 1). The spanwise domain size of one chord length is resolved for the c-type grid by 3, 11 and 21 nodes (variants C3, C11, C21), whereas the number of nodes for the o-type grids is 60. Two different o-type grids are considered (variants 01 and 02), which differ only in the wall-normal refinement of the boundary layer region. The 02 variant exhibits a fine resolution of y C < 1 throughout the oscillation cycle. A no-slip condition is applied to the physical wall with a universal hybrid treatment implemented to the turbulence equations allowing arbitrary values of y C . A periodic condition is applied to the lateral boundaries and a convective outflow condition to the downstream boundary. The whole grid is rotated sinusoidally using rigid body motion, thus a constant inflow profile with the bulk velocity u1 is sufficient. With focus on the integral force coefficients of relevance for engineers, URANS simulations were performed on the c-type grid and DES on all grid types. Turbulence models with varying degrees of complexity are used, such as the SALSA one-equation [9], the two-equation LLR k–! [10] and an explicit algebraic stress

Flow Prediction Around an Oscillating NACA0012 Airfoil at Re D 1,000,000

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Fig. 1 2D planes of the c-type and o-type grids.

model, the CEASM [5]. For details the reader is referred to [1], where a detailed description of the implementation of these models and their DES variants is given. The investigations were performed using the solver ELAN developed at the Berlin University of Technology. This numerical procedure uses a conservative finite-volume discretisation based on general curvilinear coordinates of the Navier–Stokes equations for incompressible and compressible flows. The spatial discretisation of the diffusive and convective terms is realised using central differencing, and a backward difference quotient of first or second order accuracy is used for the temporal derivative [12]. The code has been enhanced towards DES as well as an arbitrary Lagrange–Eulerian formulation (ALE) to capture moving and deforming grids [2], and is therefore able to handle all common simulation approaches such as URANS, DES, LES as well as direct numerical simulation for a wide range of engineering applications. The correct treatment of moving grids is realised with the space conservation law which is important for the presented case. In order to prevent unwanted encroachment of the LES mode of the DES inside attached turbulent boundary layers, the delayed DES (DDES) shield function of Spalart et al. [7] has been implemented to the CEASM-based DES.

3 Flow Physics For the configuration investigated, the airfoil exhibits a sinusoidal oscillation between 5ı and 25ı . Such harmonics around the maximum static lift are characterised by a forced rotatory movement of the wing or an equivalent unsteady inflow condition with viscous effects dominating the flow. The flow phenomena during the oscillation period and the pressure distribution on the central airfoil section predicted by a numerical simulation are visualised in Figs. 2 and 3.

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By rapidly increasing the incidence of the airfoil beyond the static separation angle the boundary layer on the upper side of the profile becomes fully turbulent. When further increasing the incidence a vortex starts to develop at the separation of


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the turbulent boundary layer near the profile’s leading edge. The vortex grows very fast and is convected downstream, while remaining close to the surface, increasing drastically the suction on the upper side. The maximum dynamic lift is created as

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the vortex is transported just beyond the streamwise midpoint of the profile, but before the maximum incidence is reached. The vortex suction is reduced further downstream until passing the trailing edge, thereby initiating full dynamic stall. The brief, wake-induced reattachment of the stalled region above the profile leads to an intermediate increase of the suction and the lift. Thereafter vortex shedding at the stalled airfoil occurs until the incidence has decreased such that the flow can reattach from the front to the rear.

4 Results In the following, the results of the numerical simulations are summarised and evaluated with respect to the variation of individual parameters. The force coefficients obtained using the o-grids are phase-averaged with respect to the sinusoidal oscillation of the airfoil (Section 4.5). The analysis of the differences between two solutions is mainly focused on the numerical and modelling reasons for the deviation.

4.1 Spatial Resolution The variation of the spatial resolution comprises two main parts. On the one hand there are two general grid topologies, and on the other hand the number of spanwise nodes is varied among the c-type grids as well as between c-type and o-type grids. The curves in Figs. 4 and 5 depict the lift coefficient predicted by URANS and DES with varying grids for different background models. Figure 4 clearly shows that for URANS the lift curve changes very slightly with increasing spanwise resolution (Fig. 4), whereas using DES the changes are significant, especially in the region of total stall. The flow predicted by URANS apparently tends to remain twodimensional without considering the resolution of the span. The DES modification of the same equations results in a much reduced eddy viscosity through a length scale based on the maximum grid dimension, i.e. the spanwise dimension in these McAlister 1982 SALSA C03 SALSA C11 SALSA C21

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highly under-resolved cases. Thus, the turbulent structures can more easily become three-dimensional, leading to a reduction of the secondary vortex shedding with increased spanwise resolution. This can be recognised in the lift curves of DES (Fig. 5) by the smoothing of the lift curve and the decreasing discrepancy compared to the experimental results. The difference of the curves between c-type and o-type grids in Fig. 5 results from the fact that originally the c-type grid was adapted for boundary layer flows to be predicted by URANS, whereas the the o-type grid is as uniform as possible to account for the grid resolution necessary in the LES region. This fact, together with the improved agreement achieved by the c-type grid suggests that for this case capturing the dominant wall effects is essential for the integral force coefficients. The convergence of the results with respect to the spanwise resolution is evident for URANS. In the case of DES clear convergence of the solution is not seen, but might be possible with further refined resolution of the span on the same grid topology.

4.2 Temporal Resolution The temporal discretisation is based on a varying number of 300, 600 and 1,000 timesteps t to resolve a single period of oscillation Tosc D c=.kosc u1 /. The

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influence of the timestep size is investigated using the c-type grid with the coarsest spanwise resolution C03. This has been used to keep the required simulation expense within affordable limits. The Figs. 6 and 7 show the lift curves for URANS and DES for fixed grid and background model. The results of the URANS simulations with the SALSA model clearly demonstrate convergence of the lift curve with refined temporal resolution. The same level of convergence can be recognised also for the SALSA DES as the timestep size is reduced. By contrast, the LLR k–! is far from demonstrating convergence with increasing temporal resolution, neither for URANS nor DES. It is


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evident that the simulation resolves more unsteadiness with reduced timestep size, but in case of the LLR k–! model the numerical results increasingly deviate from the experimental results in the detached flow region predicting an excessive secondary vortex shedding that appears to be unphysical. The temporal dependency for the CEASM cannot yet be concluded from the results available. The hypothesis that remains to be proven is that the LLR model constants have been derived based on the presence of excessive numerical dissipation (either from coarse grids or overly-dissipative numerical schemes), giving rise to a model that itself is insufficiently dissipative. A RANS model should show independence of the time step beyond a certain extent, as the Reynolds-averaged paradigm upon which it is derived dictates that fine turbulent structures remain modelled using eddy viscosity irrespective of spatial and temporal resolution. It is supposed however that the LLR model does not produce sufficient eddy viscosity to damp small scale structures when the time filtering effect of a large numerical timestep is reduced, resulting in the unphysical “resolution” of fine-grained structures corresponding neither to a RANS or an LES. Independent of the modelling approach used, the lift curves in Figs. 6 and 7 clearly show that the maximum lift and the corresponding angle of attack are reduced with increasing temporal resolution, which corresponds to better corroboration with the experimental results. The curves in Figs. 6 and 7 reveal that the dynamics of the flow is already captured correctly for the extensively used timestep size of t D T=300. Although the Courant–Friedrichs–Lewy criterion CFL D u t= x 1 is not met overall for this temporal resolution, the influence on the flow prediction is relatively small and has been neglected in order to achieve a wider range of parameter variations with available computational resources.

4.3 URANS Versus DES In the Figs. 8 and 9 the lift and drag curves are shown such that the model influence can be judged on a fixed grid. In the case of URANS, Fig. 8, the results demonstrate McAlister 1982 SALSA C03 LLR C03 SALSA C11 LLR C11 SALSA C21 LLR C21

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that the solution does not depend on the spatial resolution but on the turbulence model used. In addition, comparison to the experimental results reveals that the LLR k–! model agrees slightly better with these. In contrast to URANS, the results obtained using DES show on the one hand a significant dependency upon the spanwise resolution, and on the other a reduced dependency on the turbulence model used. This behaviour arises from the hybrid synthesis of RANS and LES, where the LES activity and RANS=LES blending location causes the differences in the results for varying spatial resolution and the RANS region is mainly responsible for the model influence, visible in Fig. 9.

4.4 Model Influence For the present configuration the maximum lift is experienced before the maximum incidence is reached. This behaviour is predicted by all models and for all grids (see Figs. 10, 11). However, the angle of attack where the dynamic stall occurs is predicted best by the one-equation SALSA model compared to the two-equation models, even though the angle on the c-type grid and the maximum lift on the o-type grid are predicted too highly. The latter aspects can also be observed for the LLR k–! model but with slightly degraded performance concerning the prediction of the maximum lift and the associated incidence. Nevertheless the agreement of the results for this model with the experimental results is somewhat better close to the maximum incidence, where secondary vortex shedding occurs. The k"-based CEASM has only been applied in the DDES variant and URANS has not yet been performed with this model. The force coefficients obtained for this model on the C21-grid, shown in Fig. 11, reveal the changing quality of the pre and post-stall prediction. On the one hand the predicted lift matches the experimental results during the upstroke until the dynamic stall vortex develops, whereas on the other hand the discrepancy with the experiments is much higher than for the other models, especially noticeable in the drag curve. The results obtained on the O2-grid show the opposite behaviour for the lift and drag; the drag agrees well with the

Flow Prediction Around an Oscillating NACA0012 Airfoil at Re D 1,000,000 McAlister 1982 SALSA DES C21 LLR DES C21 CEASM DDES C21

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experimental results in the pre-stall period, whereas the lift curve and the post-stall part of the drag are poorly predicted. Taking into account that the c-type grid resolution is focused close to the profile and for the o-type grid a uniform resolution in the wake was preferred, the conclusion is that capturing the physics near the profile is essential for the size of the maximum lift and the sufficient wake resolution is the basis for the correct dynamic stall angle. This thesis is supported by the results of the SALSA and LLR models, and partly those of the CEASM model.

4.5 Numerical Aspects The DES approach is a hybrid of RANS and LES. To obtain stable but also physical solutions for a given problem, it has been ensured that for both modes the favourable convection scheme will be used – in case of RANS upwind-based and for LES central schemes. Thus, the DES modification of the turbulence equations is connected with the usage of a hybrid convection scheme [11]. The blending of upwind-based and central schemes depends mainly on the local flow solution and is therefore not limited to DES.

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The importance of the hybrid convection scheme is demonstrated by the lift curves in Fig. 12. Depicted are URANS solutions obtained with an upwind-based TVD scheme, with the hybrid scheme and the respective DES solution. In case of the SALSA model, it is evident that the main improvement of DES relative to URANS is based on the convection scheme (at least for this relatively coarse grid). For the LLR k! a small improvement can also be recognised in the downstroke region. This comparison reveals that URANS flow predictions can be improved by adapted numerical schemes, especially in the presence of flow separation. Starting a testcase from scratch, the temporal behaviour of the flow solution passes several periodic cycles (2–4) until initial perturbations are decayed. As can be recognised in Fig. 13, the solution reaches a stable periodic state represented by

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the nearly perfect reproduction of the lift curve in successive cycles for the coarse C03-grid. With increasing spatial resolution of the span, the convergence of the hysteresis is reduced to the upstroke phase of the airfoil. Due to the variations in this phase for fine spanwise resolution, the integral quantities obtained for the o-grids are phase-averaged with respect to the oscillation of the airfoil. However, the consistent reproduction of the lift curve with minor variations for an arbitrary number of cycles is caused by the entire reattachment of the flow without vortex shedding at the minimum incidence, and assisted by the timestep size chosen as an exact partition ratio of the periodic time. Thus, the hysteresis converges also for such – relative to the averaged experimental data – seemingly unphysical results predicted by the LLR k–! at fine temporal resolution.

5 Conclusion and Outlook Numerical simulations using URANS and DES were performed for the flow around an oscillating NACA0012 airfoil in order to identify important parameters influencing the prediction of the dynamic load on the structure. Both approaches can be used to capture the dynamics of the dynamic stall phenomenon correctly with acceptable discrepancies to few experimental results, but with some restrictions according mainly to the spatio-temporal resolution. Moreover, the necessity of sufficient spatial resolution in the spanwise direction as well as a very good circumferential resolution and in the wake has been clearly demonstrated. In addition, the prediction of the stall angle and the maximum lift is improved with increased temporal resolution. The results reveal that the one-equation SALSA model seems to be more robust and captures most of the physics for this case than the two-equation models applied. The technology feedback from DES to URANS, namely the application of a hybrid convection scheme for URANS, can improve the flow prediction without additional resolution. The ongoing work and further research will concentrate especially on the refinement in time and space. Therefore, the circumferential node distribution in the O-type grid will be improved and c-type grids with increased spanwise resolution, e.g. C41 and C61, will be used. In this context, the impact of an increased spanwidth, refined temporal resolution (such that CFL 1 is achieved) as well as the hybrid convection scheme applied to URANS on a fine grid will be investigated. To side-step the problems experienced with the two-equation models, more established standard models such as the Wilcox k–!, will be considered. The hypothesis concerning the possible recalibration of the LLR k–! model should be further investigated, which would certainly require more comprehensive experimental results and additional testcases. Acknowledgements The support of the EU within the DESider project (Detached Eddy Simulation for Industrial Aerodynamics – http://cfd.me.umist.ac.uk/desider), contract-no. AST3-CT-200-502842 and the IBM pSeries 690 of the North German cooperation for High-Performance Computing (HLRN) facility is gratefully acknowledged.

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References 1. B UNGE , U., M OCKETT, C., T HIELE , F. (2007) Guidelines for implementing detached-eddy simulation using different models. Aerospace Science and Technology 11(5), 376–385. 2. B UNGE , U. (2004) Numerische Simulation turbulenter Strömungen im Kontext der Wechselwirkung zwischen Fluid und Struktur. Ph.D. thesis, ISTA, TU Berlin. 3. B UNGE , U., G URR , A., T HIELE , F. (2003) Numerical aspects of simulating the flow-induced oscillations of a rectangular bluff body. Journal of Fluids and Structures 18(3–4), 405–424. 4. F REDERICH , O. (2003) Numerical simulation of oscillating airfoils in turbulent flow with URANS and DES in comparison. Diploma thesis, ISTA, TU Berlin. ¨ 5. L UBCKE , H. (2001) Entwicklung expliziter Darstellungen zweiter statistischer Momente zur numerischen Simulation turbulenter Strömungen. Ph.D. thesis, ISTA, TU Berlin. 6. M C A LISTER , K.W., P UCCI , S.L., C ARR , L.W. AND M C C ROSKEY, W.J. (1982) An experimental study of dynamic stall on advanced airfoil sections. NASA TM-84245. 7. S PALART, P., D ECK , S., S HUR , M., S QUIRES , K., S TRELETS , M., AND T RAVIN , A. (2006) A new version of detached-eddy simulation, resistant to ambiguous grid densities. Theoretical and Computational Fluid Dynamics, 20:181–195. 8. S WALWELL , K.E., S HERIDAN , J. AND M ELBOURNE , W.H. (2003) Frequency analysis of surface pressure on an airfoil after stall. 21. AIAA Applied Aerodynamics Conference, Orlando, Florida, June 23–26. 9. RUNG , T., B UNGE , U., S CHATZ , M. AND T HIELE , F. (2003) Restatement of the SpalartAllmaras Eddy-Viscosity Model in Strain-Adaptive Formulation. AIAA Journal 41(7), 1396– 1399. 10. RUNG , T. AND T HIELE , F. (1996) Computational Modelling of Complex Boundary-Layer Flows. Proc. 9th Int. Symp. on Transport Phenomena in Thermal-Fluid Eng., Singapore, 321–326. 11. T RAVIN , A., S HUR , M., S TRELETS , M. AND S PALART, P. (2000) Physical and Numerical Upgrades in the Detached-Eddy Simulation of Complex Turbulent Flows. Proc. of the 412th Euromech Colloquium on LES and Complex Transitional and Turbulent Flows, Munich. 12. X UE , L. (1998) Development of an efficient parallel solution algorithm for the threedimensional simulation of complex turbulent flows. Ph.D. thesis, ISTA, TU Berlin.

Two-Velocities Hybrid RANS-LES of a Trailing Edge Flow J.C. Uribe, N. Jarrin, R. Prosser, and D. Laurence

Abstract The flow around a trailing edge is computed with a new hybrid method designed to split the influences of the averaged and instantaneous velocity fields. The model is first tested on channel flows at different Reynolds numbers and coarse meshes giving good predictions of mean velocities and stresses. On the trailing edge flow the predictions of the hybrid model are compared with those using DES-SST on the same coarse mesh. The results of the hybrid model are close to the reference fine LES in terms of mean velocity and turbulent content. Keywords RANS LES Trailing edge Hybrid methods

1 Introduction Large Eddy Simulation has been successfully applied to different kinds of flows, but its use in industry has remained scarce mainly due to the large constraint present in the mesh requirements of wall bounded flows, specially at high Reynolds numbers. For such flows, the size of the energy containing structures scales with Re and hence the number of grid points required to resolve accurately the near wall eddies scales approximately with Re 1:76 [6]. To circumvent this severe near wall requirement, LES can be restricted to the simulation of the outer flow eddies whereas a RANS like eddy viscosity model is used to model the dynamics of the near wall eddies. In recent years, such hybrid methods combining RANS and LES have received increased attention from groups around the world. J.C. Uribe (), N. Jarrin, and R. Prosser School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Manchester M60 1QD, UK e-mail: [email protected] D. Laurence School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Manchester M60 1QD, UK EDF R&D, 6 quai Watier, 78420 Chatou, France M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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In an attempt to ease the computational requirements in wall bounded flows, many approaches have been suggested. One method is to use so-called “wall functions” to bridge the viscous sublayer and provide a suitable boundary condition for the wall cells [20]. This can range from a log-law approximation [22] to a solution of a system of simplified equations in the near wall region [4]. Another approach is the use of RANS equations near the wall to provide the outer layer with correct information. The main problem of this type of approach is how to connect a statistically averaged flow (RANS) with the instantaneous filtered field (LES). A way to couple the two types of flows is the Detached Eddy Simulation (DES) ([24,30]) in which the turbulent lengthscale in the RANS equation is switched to a lengthscale based on the mesh filter width in order to reduce the viscosity in the separated region. The main idea of the DES approach is to solve the attached boundary layers with a RANS models and the separation region with a LES-like technique. One of the main issues of the DES approach is the fact that in regions near the wall where the grid is refined, the turbulent length scale dictated by the RANS model can become larger than the grid length scale therefore making the model reduce the turbulent viscosity and leading to a “grid induced separation”. A way to reduce the sensitivity to the grid induced separation was formulated by Menter et al. [15] using the blending function of the SST model to “shield” the boundary layer. As its name implies, DES should only be applied for separated flows and when the resolution of near wall fluctuations is required, another approach should be used. Other approaches are ‘zonal’, in which a part of the domain is set to be computed using RANS equations and the rest is computed with LES. Davidson and Peng [8] used the k! model in a region near the wall (y C 60) and a one equation model for the sub-grid stresses in the outer region. The location of the interface was fixed and Neumann boundary conditions were applied for the RANS variables. Another zonal approach has been developed by Temmerman et al. [28] where continuity of the turbulent viscosity is imposed at the interface. Different RANS models and different locations for the interface were tested. Using this constraint, the coefficient C is extracted from the interface and then adapted via exponential functions to increase the RANS contribution as the wall is approached. Hamba [12] tried coupling a near-wall RANS region with an outer LES by using a lengthscale computed from DNS. The approach resulted in an acceleration of the velocity profile at the interface, which was compared to another approaches where there is a similar effect ([8, 19]). In the zonal approach, the treatment of the interface has always been of importance for the success of the method since the RANS information does not provide correct turbulent fluctuations. Some ways to deal with this issue are the introduction of backscatter [21], damping the modelled stresses [27], the addition of fluctuations at the LES side of the interface [7] or the use numerical smoothing [31].

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2 The Hybrid Approach In many hybrid approaches the main problem is how to couple the two different velocity fields used in RANS (statistically averaged) and LES (filtered). This is often done by applying a matching criteria, i.e. same turbulent viscosity at the interface, same kinetic energy or dissipation etc. This poses a problem since these values represent totally different properties of different velocity fields. Instead of using one velocity field to couple RANS and LES, the model presented here attempts to adjust the resolved velocity field with information taken from the averaged velocity field. Many sub-grid models assume that the flow contains an inertial subrange and hence the sub-grid motions can be assumed to be isotropic. This is true only if the grid is small enough so that the anisotropy introduced by the mean shear can be neglected. At high Reynolds numbers, the refinement of the grid becomes too costly, therefore restricting the LES method to low Reynolds numbers flows. As the solid boundary is approached, the mean shear becomes high enough to introduce anisotropy across a diminishing range of scales. It is then necessary for the model to represent at the same time subgrid-scale contributions to the mean shear stress and isotropic dissipation effects. The approach presented herein differs form the previous ones by an attempt to separate these two issues in a way that allows the LES fluctuations to develop deeply in RANS layer without any perturbation on or from the mean flow characteristics. This allows to use coarser near wall meshes in all directions (i.e. very high aspect ratio cells as in classical near-wall RANS grids) while maintaining the turbulent characteristics of a resolved field on the whole domain.

2.1 Modelling The instantaneous velocity can be decomposed as U D hU i C u0

(1)

where hU i is the averaged velocity and u0 is the fluctuating one. Schumann [22] proposed to split the residual stress tensor into two, one “locally isotropic” part and one “inhomogeneous” part. The isotropic part is proportional to the fluctuating strain and does not affect the mean flow equations but determines the rate of energy dissipation. The inhomogeneous part is proportional to the mean strain and controls the shear stress and mean velocity profile: 2

ijr kk ıij D 2r .S ij hS ij i/ 3 ƒ‚ … „ locally isotropic

2a hS ij i „ ƒ‚ …

(2)

inhomogeneous

where h:i denotes ensemble averaging of the filtered equations. The viscosities r and a are based on fluctuating and mean strains. The isotropic part of the residual

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stress tensor has a zero time mean value. By refining the grid the residual stresses must tend to zero, therefore the inhomogeneous part must have a grid dependence parameter in the turbulent viscosity a . Schumann [22] used a mixing length model for a with the length scale computed as L D min. y; C10 /, where C10 is a constant that is difficult to prescribe for all types of flows. Schumann [22] and Grotzbach and Schumann [10] tried to derive a theoretical value for the constant but they were forced to introduced corrective constants to agree with a range of experiments. Moin and Kim [17] used the same principle of splitting the residual stress but in the mixing length model, they use the spanwise size of the cell as the length scale. They argue that for the near wall region in a channel flow, the important structures are streaks that are finely spaced on the spanwise direction. Therefore a coarse resolution in the spanwise direction would lead to larger eddies and a thicker viscous sublayer. Sullivan et al. [26] developed a similar approach for planetary boundary layer flows but chose a to match the Monin-Obukhov similarity theory [5]. Baggett [3] used a similar approach to compare two hybrid models, one “Schumann-like” and one “DES-like” but found excessive streamwise fluctuations leading to streaks that were much too large. In the context of hybrid LES–RANS, a blending function, fb , can be used to introduce a smooth transition between the resolved and the ensemble averaged turbulence parts. In the present study the total residual stress is written as: 2

ijr kk ıij D 2r fb .S ij hS ij i/ 2.1 fb /a hS ij i 3

(3)

In this way the averaged stress would be: 2 r

ij kk ıij D 2.1 fb /a hS ij i 3

(4)

which is just the RANS stress. This way the total shear stress would be 2.1 fb / a hS ij i C hu0 v0 i. It is therefore necessary that the blending function fb tends to one in the region where hu0 v0 i is resolved correctly and to zero in the region near the wall where the shear stress is under resolved due to the coarse grid. The total rate of transfer ˝of energy ˛from the ˝ filtered ˛ motions to the residual scales is given by (assuming that r S ij S ij r S ij S ij [18]) ˛ ˝ ij S ij D

˝ ˛ ˛ ˝ ˛ ˝ ˝ ˛ 2 r fb .S ij S ij /S ij C 2.1 fb / a S ij S ij

˛ ˝ ˛˝ ˛ ˝ ˛˝ ˛ ˝ D 2fb r . S ij S ij S ij S ij / C 2.1 fb /a S ij S ij

(5) (6)

which shows how the RANS viscosity contributes to dissipation in association with the mean velocity only, i.e. the resolved turbulent stresses are free to develop independently from the RANS viscosity.


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For the isotropic viscosity r , Schumann [22] used a model based on the sub-grid energy. Moin and Kim [17] used the standard Smagorinsky [23] model based on the fluctuating strain. Here the later approach is used: q r D .Cs /2 2sij0 sij0

(7)

sij0 D S ij hS ij i

(8)

In the frame of unstructured codes, the filter width is taken as twice the cell volume ( D 2Vol). In this study the elliptic relaxation model ' f of Laurence et al. [13] is used to calculate the RANS viscosity. This model solves for the ratio ' D v2 =k to compute the turbulent viscosity as: a D C 'kT (9) k p where T D max " ; CT " . Although for the channel flow calculations presented here, there is not much difference in the choice of the RANS models, the aim is to have a formulation that can be used in complex 3D flows, therefore the elliptic relaxation method seems a suitable choice to account for the wall effects. The blending function has been parameterised by the ratio of the turbulent length scale to the filter width: Lt n fb D tanh Cl (10) Here Cl D 1 and n D 1:5 are empirical constants. These values were chosen to match the shear stress profile based on channel flow results at Re D 395 with DNS data. When using the ' f model (Equation 9), the wall distance is not desirable and the blending function can be formulated using Lt D 'k 3=2 =". The blending function has been devised to connect the two length scales smoothly so its value is close to zero near the wall and one far from it. Similar functions have been used in other hybrid approaches (see Abe [1], Hamba [11] or Speziale [25]). Although the function in Equation (10) is totally empirical, it has been tested for a range of Reynolds numbers and grids and gave satisfactory results (not presented here). The function allows a higher contribution from the LES part as the grid is refined. Different coefficients have been used in the optimisation of the blending function, but the results are not greatly affected and anyway are always better than standard LES on the same mesh. In Equation (3) the averaged velocity has been calculated as a running average with an averaging window of about ten times the eddy turnover time. Although it is possible also to use plane averaged in the case of the channel flow, this was not done in order to keep the formulation applicable for 3D flows where no plane averaging is possible. In all computations shown here, a finite volume code Code Saturne [2] has been used with a second order time advancing scheme.

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3 Channel Flows Channel flow computations have been carried out at different Reynolds numbers on the same domain, a box of dimensions [0,6.4], [0,2], [0,3.2]. Here only few of the simulations are shown. For the channel flow at Re D 395 the coarsest mesh used has 40 40 30 cells (denoted C1). In Figs. 1 and 2 the resulting velocity and shear stress profiles for a standard Smagorinsky model and the hybrid model can be seen. The mesh is too coarse for a standard LES to capture all the small scales and therefore the shear stress is underpredicted, which in turn, produces an overestimation of the velocity magnitude. The hybrid model successfully blends the average shear stress and with the resolved one to produce the correct magnitude and therefore a better prediction of the velocity profile. The normal stresses can be seen in Fig. 3. The usual behaviour of overestimating the streamwise normal stress on coarse

30 DNS LES Hybrid

25 20 15 10 5

Fig. 1 Velocity profile at Re D 395

0

1

10

100

1000

Resolved Modelled Total DNS

0.8

0.6

0.4

0.2

Fig. 2 Shear stress at Re D 395

0

0

100

200 y+

300

400

Two-Velocities Hybrid RANS-LES of a Trailing Edge Flow Fig. 3 Normal stresses at Re D 395

69 DNS Hybrid LES

, , <w’w’>

10 8 6 4 2

0

100

Fig. 4 Velocity profiles at different Reynolds numbers

log-law Reτ = 395 Reτ = 590 Reτ = 1100 Reτ = 2000 Reτ = 4000

Cells 40x30x30 40x40x30 50x50x40 50x50x40 64x80x64

200 y+

+

Δx 59 88 140 256 400

300

400

1000

10000

+

Δz Case 39 C1 59 C4 88 C6 160 C8 200 C10

40

U+

30 20 10 0

1

10

100 y+

meshes by LES is corrected by the hybrid model. For higher Reynolds numbers the results have a similar trend. In Fig. 4 the velocity profiles are shown for different Reynolds number with coarse meshes. All the velocity profiles follow the log-law.

4 Trailing Edge Computations In the present study, we performed various hybrid simulations of the flow past an axisymmetric bevelled trailing edge. The geometry is described in detail in [33]. The Reynolds number based on the free stream velocity U1 and the hydrofoil chord, is 2:15 106 . The trailing edge tip angle is 25ı . Simulations were performed over the rear-most 38% of the hydrofoil chord with inlet conditions discussed below.

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The inlet Reynolds numbers based on the local momentum thickness and boundary layer edge velocity are 2;760 on the lower side and 3;380 on the upper side. These values were obtained from an auxiliary RANS calculation although some questions remain about their fidelity as mentioned in [32] and [33]. The computational domain is 0:5 h 41 h 16:5 h where h denotes the hydrofoil thickness. The grid was coarsened from 1;536 96 48 cells (claimed to be sufficient for a well resolved LES by Wang and Moin [33]) to 512 64 24 cells. The mesh has 192 cells in the streamwise direction on the wing both on the lower and on the upper side and 64 cells in the wake. The maximum grid spacing in wall unit is zC D 110 in the spanwise direction and x C D 230 in the streamwise direction. The mesh spacing in the streamwise direction is kept constant along the trailing edge when it is refined in the fine LES of Wang and Moing [33] ( zC D 55 and x C D 60). The minimum grid spacing in the wall normal direction is about y C D 2. The boundary conditions at the inlet are taken from Wang and Moin [33] using the following procedure. First, an auxiliary RANS calculation is conducted in a domain enclosing the entire strut. The resulting mean velocities, accounting for the flow acceleration and circulation associated with a lifting surface, are used as the inflow profiles outside the boundary layers on both side of the strut. Originally two RANS simulations were performed using the v2 f turbulence model of Durbin [9] and the Menter [16] SST model. In the present study the SST profiles have been used. The two turbulence model produce a noticeable difference in the velocity overshoot (undershoot) outside the upper (lower) boundary layer. SST results are associated with a smaller mean circulation, which is thought to promote trailing edge separation. Within the turbulent boundary layers the time series of inflow velocities are generated from two separate LES calculations of flat-plate boundary layers with zero pressure gradient, using the method described by Lund et al. [14]. The inflow generation LES matches the local boundary layer properties, including the momentum thickness and Reynolds number, with those from the RANS simulation. The boundary conditions for the RANS simulations are obtained from time averaging of the available samples. A no-slip condition is applied on the surface of the strut. The top and bottom boundaries are placed far away from the strut to minimise the impact of the imposed symmetry boundary condition. At the downstream boundary a standard exit boundary condition is applied. This case has been treated with dynamic wall models by Wang and Moin [34] and with similar methods in Tessicini et al. [29]. Two simulations using the same mesh are presented here. One with DES using the SST model as the RANS background model. The blending function F1 is used to avoid grid induced separation (see [15] for details). The second simulation is done with the hybrid method presented above. Velocity profiles and rms streamwise velocity fluctuations profiles are available from the fine LES [33] at locations of x=h D 0:625, 1:125, 1:625, 2:125, 3:125 on the aerofoil and at x D 0, 0:5, 1, 2 and 4 in the wakep(x=h D 0 is located at the trailing edge). Figure 5 shows the absolute velocity ( U 2 C V 2 ) profiles at the five stations on the aerofoil. The hybrid model presents a better agreement with the reference LES. It can be seen how the DES separates earlier and by the station at x=h D 1:125 there is a strong

Two-Velocities Hybrid RANS-LES of a Trailing Edge Flow Fig. 5 Velocity profiles over the aerofoil

71

x/h=−3.125 x/h=−2.125 x/h=−1.625 x/h=−1.125 x/h=−0.625

(y – yw) / h

DES + F1 shield function LES Wang & Moin (2000) Hybrid

0.4

0.2

0

Fig. 6 urms profiles over the aerofoil

0

1

2

3 |U| / Ue

4

5

x/h=−3.125 x/h=−2.125 x/h=−1.625 x/h=−1.125 x/h=−0.625

0.6

(y – yw) / h

0.5


0.4 0.3 0.2 0.1 0 0.2

0.4 urms / Ue

0.6

0.8

recirculation which is not present in the reference LES. The hybrid model predicts a slightly earlier separation but closer to the reference value. The turbulent content is also better predicted by the hybrid model as it can be seen from Fig. 6. The DES simulation does not sustain the fluctuations from the inlet and by the time the flow reaches the station at x D 3:125, the resolved fluctuations are very small. This is the normal behaviour of DES in an attached boundary layer, since it is designed to be used in massively separated flows. In general the structures predicted by DES are larger than what the hybrid model predicts. The structures are better resolved than with the DES approach because in the hybrid model the resolved stresses can develop independently from the RANS viscosity, i.e. the model associates the RANS viscosity with the mean flow only and links the resolved scale dissipation to the LES viscosity only (as shown in

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Equation (6)). This can be seen in Figs. 7 and 8 where the isosurfaces of fluctuating streamwise velocity are shown (C0.05 in red, 0.05 in blue). This is mainly due to the way in which each approach treats the boundary layer. DES uses RANS in a zone much larger near the wall, solving much of the boundary layer. On the other hand, the hybrid only uses full RANS in a zone very close to the wall and then it has a transition zone where both approaches contribute via different velocity fields. The blending function contours can be seen in Fig. 9. In Fig. 10 the zones where the two different length scales are used on the DES simulation can be seen.

Fig. 7 Iso-contours of u0 , hybrid model

Fig. 8 Iso-contours of u0 , DES model

Fig. 9 Blending function contours for the hybrid model


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Fig. 10 LES and RANS zones on DES simulation

Fig. 11 Velocity profiles at the wake

x/h=0.0 x/h=0.5 x/h=1.0 x/h=2.0 x/h=4.0

1

DES + F1 shield LES Wang & Moin (2000) Hybrid

y/h

0.5

0

−0.5

−1

0

1

2

3 U/U∞

4

5

The red region represents where the DES acts in LES mode, i.e. where the turbulent scale (k 3=2 =") is larger that the filter length scale (Cs ). In DES, the zone close to the wall is solved in RANS mode, therefore damping the fluctuations, whereas the hybrid model resolves the structures dictated by the size of the mesh, and introduces the effect of the wall via the averaged velocity. Although DES is acting in LES mode in the separated region, as it is designed to do, it is possible that the coarse resolution of the grid does not allow for a good resolved LES. Therefore a finer grid would probably lead to better results but this issue needs to be investigated further. In Fig. 11 the streamwise velocity profiles at the wake can be seen. Due to the better representation of the separated region by the hybrid model, the velocity profiles are in a better agreement than the DES. The rsm fluctuations can be seen in Fig. 12 where it can be seen that the levels are similar, although the spectral content is most certainly very different (as seen from Figs. 7 and 8) and this is important in aero-acoustic applications.

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Fig. 12 urms profiles at the wake

x/h=0.0 2

x/h=0.5

x/h=1.0

x/h=2.0

x/h=4.0


y/h

1

0

−1

0

0.1

0.2

0.3 0.4 urms / U∞

0.5

0.6

0.7

5 Conclusions A new hybrid method has been presented based on splitting the contributions from the averaged and fluctuating velocity fields. The method performs well in channel flows where the boundary layer is attached and on the separated flow over a trailing edge. The method has been compared with the DES method which is not designed to reproduce the behaviour of attached boundary layers. In the case of the separated flow around a trailing edge, the DES method behaves in LES mode in the separated region but is found to be not turbulent enough predicting a early separation from the curved surface. On the other hand, the hybrid method is capable of sustaining fluctuating behaviour only limited by the size of the cells. Although the mesh is too coarse to be able to reproduce the small structures, the model successfully includes the near wall effect on mean strain via the mean velocity field, allowing a separation of dissipative effects. This makes the model predict better separation and mean quantities compared to the DES simulation. There are many issues to investigate such as the sensitivity of the model to different meshes and different blending functions but the results obtained here are encouraging. Acknowledgements This work was financed by the DESider project (Detached Eddy Simulation for Industrial Aerodynamics) which is a collaboration between Alenia, ANSYS-AEA, Chalmers University, cnrs-Lille, Dassault, DLR, EADS Military Aircraft, EUROCOPTER Germany, EDF, FOI-FFA, IMFT, Imperial College London, NLR, NTS, NUMECA, ONERA, TU Berlin, and UMIST. The project is funded by the European Community represented by the CEC, Research Directorate-General, in the 6th Framework Programme, under Contract No. AST3-CT-2003502842.


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References 1. K Abe. A hybrid LES RANS approach using an anisotropy-resolving algebraic turbulence model. International Journal of Heat and Fluid Flow, 26:204–222, 2005. 2. F. Archambeau, N. Mechitoua, and M. Sakiz. A finite volume method for the computation of turbulent incompressible flows – industrial applications. International Journal on Finite Volumes, 1(1):1–62, 2004. 3. J.S. Baggett. On the feasibility of merging LES with RANS for the near-wall regions of attached turbulent flows. In Annual Research Briefs, pp. 267–276. Center for turbulence research, Stanford, CA, 1998. 4. E. Balaras, C. Benocci, and U. Piomelli. Two-layer approximate boundary conditions for large eddy simulations. AIAA Journal, 34:1111–1119, 1996. 5. J.A. Businger, J.C. Wyngaard, Y. Izumi, and E.E. Bradley. Flux-profile relationships in the atmospheric surface layer. Journal of Atmospheric Science, 28:181–189, 1971. 6. D. Chapman. Computational aerodynamics development and outlook. AIAA Journal, 17:1293– 1313, 1979. 7. L. Davidson and S. Dahlström. Hybrid LES-RANS: An approach to make LES applicable at high Reynolds number. International Journal of Computational Fluid Dynamics, 19:415–427, 2005. 8. L. Davidson and S.-H. Peng. Hybrid RANS-LES: A one equation sgs model combined with a k! model for predicting recirculating flows. International Journal of Numerical Methods in Fluids, 43:1003–1018, 2003. 9. P.A. Durbin. A reynolds stress model for near-wall turbulence. Journal of Fluid Mechanics, 249:465–498, 1993. 10. G. Grotzbach and U. Schumann. Direct numerical simulation of turbulence velocity-pressure, and temperature fields in channel flows. In Symposium on Turbulent Shear Flow, pp. 18–20, 1977. 11. F. Hamba. An attempt to combine large eddy simulation with the k" model in a channel flow calculation. Theoretical and Computational Fluid Dynamics, 14:323–336, 2001. 12. F. Hamba. A hybrid RANS LES simulation of turbulent flows. Theoretical and Computational Fluid Dynamics, 16:387–403, 2003. 13. D. Laurence, J.C. Uribe, and S. Utyuzhnikov. A robust formulation of the v2-f model. Flow, Turbulence and Combustion, 73:169–185, 2004. 14. T. Lund, X. Wu, and D. Squires. Generation of turbulent inflow data for spatially-developing boundary layer simulations. Journal of Computational Physics, 140:233–258, 1998. 15. F. Menter, M. Kuntz, and R. Langtry. Ten years of industrial experience with the SST model. In K. Hanjalić, Y. Nagano, and M. Tummers, eds., Turbulence, Heat and Mass Transfer 4, 4:625–632, 2003. 16. F.R. Menter. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, 32(8):1598–1605, 1994. 17. P. Moin and J. Kim. Numerical investigation of turbulent channel flow. Journal of Fluid Mechanics, 118:341–377, 1982. 18. F. Nicoud, J.S. Baggett, P. Moin, and W. Cabot. Large eddy simulation wall-modeling based on suboptimal control theory and linear stochastic estimation. Physics of Fluids, 13:2968–2984, 2001. 19. N.V. Nikkitin, F. Nicoud, B. Wasistho, K.D. Squires, and P. Spalart. An approach to wall modelling in large eddy simulations. Physics of Fluids, 10:1629–1632, 2000. 20. U. Piomelli and E. Balaras. Wall-layer models of large eddy simulation. Annual Review of Fluid Mechanics, 34:349–374, 2002. 21. U. Piomelli, E. Balaras, E. Pasinato, K.D. Squire, and P. Spalart. The inner-outer later interface in large eddy simulation with wall layer models. International Journal of Heat and Fluid Flow, 23:538–550, 2003. 22. U. Schumann. Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. Journal of Computational Physics, 18:376–404, 1975.

76

J.C Uribe et al.

23. J. Smagorinsky. General circulation experiments with the primitive equations: I the basic equations. Monthly Weather Review, 91:99–164, 1963. 24. P. Spalart, W. Jou, M. Strelets, and S. Allmaras. Comments of feasibility of les for wings, and on a hybrid RANS/LES approach. In International Conference on DNS/LES, Aug. 4–8, 1997, Ruston, LA, 1997. 25. C.G. Speziale. Turbulence modeling for time-dependant RANS and VLES: A review. AIAA Journal, 26:179–184, 1998. 26. P. Sullivan, J. McWilliams, and C. Moeng. A subgrid-scale model for large eddy simulation of planetary boundary layer. Boundary Layer Methodology, 71:247–276, 1994. 27. L. Temmerman and M. Leschziner. A priori studies of near wall RANS model whithin a hybrid LES/RANS scheme. In Rodi W. and Fueyo N., editors, Engineering Turbulence Modelling and Experiments, 5:317–327, 2002. 28. L. Temmerman, M. Hadˆziabdić, M. Leschziner, and K. Hanjalić. A hybrid two-layer URANSLES approach for large eddy simulation at high Reynolds numbers. International Journal of Heat and Fluid Flow, 26:173–190, 2005. 29. F. Tessicini, L. Temmerman, and M. A. Leschziner. Approximate near-wall treatments based on zonal and hybrid RANS-LES methods for LES at high Reynolds numbers. International Journal of Heat and Fluid Flow, 27:789–799, 2006. 30. A. Travin, M. Shur, M. Strelets, and P. Spalart. Detached-eddy simulation past a circular cylinder. Flow Turbulence and Combustion, 63:393–313, 1999. 31. P. Tucker and L. Davidson. Zonal kl based large eddy simulations. Computational Fluid Dynamics, 33:267–287, 2004. 32. M. Wang. Progress in les of trailing-edge turbulence and aeroacoustics. In Annual Research Briefs. Center for turbulence research, Stanford, CA, 1997. 33. M. Wang and P. Moin. Computation of trailing-edge flow and noise using large-eddy simulation. AIAA Journal, 38:2201–2209, 2000. 34. M. Wang and P. Moin. Dynamic wall modeling for large-eddy simulation of complex turbulent flows. Physics of Fluids, 14:2043–2051, 2002.

Assessment of Flow Control Devices for Transonic Cavity Flows Using DES and LES G.N. Barakos, S.J. Lawson, R. Steijl, and P. Nayyar

Abstract Since the implementation of internal carriage of stores on military aircraft, transonic flows in cavities were put forward as a model problem for validation of CFD methods before design studies of weapon bays can be undertaken. Depending on the free-stream Mach number and the cavity dimensions, the flow inside the cavity can become very unsteady. Below a critical length-to-depth ratio (L=D), the flow has enough energy to span across the cavity opening and a shear layer develops. When the shear layer impacts the downstream cavity corner, acoustical disturbances are generated and propagated upstream, which in turn causes further instabilities at the cavity front and a feedback loop is maintained. The acoustic environment in the cavity is so harsh in these circumstances that the noise level at the cavity rear has been found to approach 170 dB and frequencies near 1 kHz are created. The effect of this unsteady environment on the structural integrity of the contents of the cavity (e.g. stores, avionics, etc.) can be serious. Above the critical L=D ratio, the shear layer no longer has enough energy to span across the cavity and dips into it. Although this does not produce as high noise levels and frequencies as shorter cavities, the differential pressure along the cavity produces large pitching moments making store release difficult. Computational fluid dynamics analysis of cavity flows, based on the Reynolds-Averaged Navier–Stokes equations was only able to capture some of the flow physics present. On the other hand, results obtained with Large-Eddy Simulation or Detached-Eddy Simulation methods fared much better and for the cases computed, quantitative and qualitative agreement with experimental data has been obtained. Keywords Transonic cavity flow LES DES Flow control

G.N. Barakos, S.J. Lawson (), and R. Steijl CFD Laboratory, Department of Engineering, University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK P. Nayyar Aircraft Research Association, Manton Lane, Bedford MK41 7PF, UK M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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1 Introduction Numerous experimental investigations have been performed on cavity flows in an attempt to better understand the problem. Most cavity experiments were originally based on unsteady pressure measurements in wind tunnels. Ross et al. from QinetiQ [1,2] and Tracy et al. from NASA Langley [3] are two examples of researchers who have conducted a significant number of wind tunnel experiments on cavities of several configurations over a broad range of Mach and Reynolds numbers. Ross and Peto [4] also provided experimental data on cavity acoustical suppression methods including leading-edge spoilers and vortex-shedding rods amongst others. Recent experimental endeavours have exploited non-intrusive, optical techniques such as Schlieren Photography [5] and Optical Reflectometry [6] to study cavity flows. More advanced non-intrusive methods such as Particle Image Velocimetry (PIV) [7] and Laser Doppler Velocimetry (LDV) [8] are also beginning to appear in cavity flow studies as a means of obtaining high-fidelity, high-resolution data for the instantaneous flow-field and the velocity variations inside the cavity. Such methods are, however, expensive and are either restricted to low Reynolds number flows. Recent studies of cavity flows have therefore attempted to use Computational Fluid Dynamics (CFD) as an analysis tool, with most emphasis on the use of Reynolds-Averaged Navier–Stokes (RANS) equations in conjunction with various turbulence closures. Most applications of Unsteady RANS (URANS) to cavity flows have employed algebraic turbulence models, especially different versions of the Baldwin-Lomax models, due to their simplicity [9–11]. However, such simple eddy viscosity models were realised to be generally incapable of accurately predicting the turbulent cavity flow-field [12], and investigations with more advanced turbulence models such as the two-equation k" and k! models were made. Most references to cavity flow modelling with two-equation models were related to supersonic flow conditions [13, 14] while applications of two-equation models to transonic cavity flows are rare. A detailed survey of published works on cavity flow can be found in [15]. One of the major drawbacks of URANS is the difficulty in predicting the full spectrum of turbulent scales. In high Reynolds flows, for instance, a broad range of turbulent length and time scales persist. For the cavity, this intense turbulent environment is further coupled with strong acoustic radiation, the source of which is located at the downstream corner of the cavity. The acoustical signature in the cavity is composed of broadband noise (lower-frequency, lower-energy noise contributed by the free-stream and=or the shear layer) with the narrow-band noise (a combination of higher-frequency and lower-frequency noise of different magnitude contributed by vortex-vortex, vortex-wall, vortex-shear layer, shock-shear layer and shear layer-wall interactions) superimposed on it. The narrow-band spectrum comprises of discrete acoustic tones called Rossiter modes after J.E. Rossiter who developed a semi-empirical formula to calculate them [16]. Statistical turbulence models tend to predict well the larger scales associated with the lower-frequency discrete acoustic tones but fail to provide the same accuracy in capturing the smaller, higher-frequency and more intermittent time scales. The broadband noise is not

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captured by these models either. The presence of these multiple acoustic tones and of a large number of turbulent scales may mean that achieving a good level of accuracy and consistency with turbulence models is difficult for cavity flows. Previous computations with a cavity L=D ratio of 5, for example, showed that the results were sensitive to the grid used. In an attempt to reduce the grid sensitivity and obtain a grid-converged solution, grids finer than those employed in previous studies [15] were used. In addition, cavities of different dimensions were investigated to determine the range of applicability of the URANS method. This was accomplished by performing parametric studies in time and space on 2D cavities of different sizes, e.g. L=D D 2, L=D D 10 and L=D D 16. Ever since the problems associated with cavity flows were realised (such as high acoustics and buffetting) many experiments and computations were conducted with the aim of improving the cavity environment. Some control methods involved manipulating the cavity geometry by either modifying the angle at which the cavity walls are slanted, for instance, or by adding an external device to deliberately alter the flow inside the cavity (see Fig. 1). Such control techniques are referred to here as open-loop control because no feedback loop is implemented in the control method. Consequently these open-loop control methods are most effective at one particular stage in the aircraft’s flight profile. Rossiter [16] and more recently Ross [2], for example, have performed extensive wind tunnel experiments on the effectiveness of spoilers as open-loop flow control devices. The non-versatility of such openloop control devices over a larger proportion of the flight regime diverted attention

3D Cavity with Sloping Walls 3D Cavity Model

Rod spoiler

Flat spoiler

Fig. 1 Flow control devices used in combination with the clean cavity

Saw-tooth spoiler

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to closed-loop control methods, which continually adapt to the flight conditions making them more suitable for time-varying and off-design situations. Although open-loop control studies have dominated most control efforts in cavity flows, examples of closed-loop cavity control studies are also beginning to appear in literature. For more information on the cavity flow control, the reader is directed toward an excellent review by Cattafesta et al. in Ref. [17], which provides an elaborate account of different open-loop and closed-loop control strategies adopted by different researchers.

2 Outline of the Experiments by Ross et al. Experimental pressure measurements were obtained using the Aircraft Research Association Ltd (ARA) wind tunnel at Bedford, UK (see Ref. [2]). The ARA wind tunnel is a 9 by 8 foot continuous flow, transonic wind tunnel (TWT) with ventilated roof, floor and side walls. 3D clean cavities were first studied with or without doors (see Fig. 1). The doors prevented any leakage at the cavity edges in the span-wise direction forcing the flow to channel into the cavity. In this configuration, the flow behaves as if it were two-dimensional and is assumed to be well represented by modelling the cavity as 2D. Several flow control devices were subsequently added to the cavity as shown in Fig. 1. The L=D D 5 cavity model (with width-to-depth ratio [W/D] of 1) measured 20 in. in length and 4 in. in width and depth. The generic cavity rig model was positioned at zero incidence and sideslip and the wind tunnel was operated at a Mach number of 0.85 and atmospheric pressure and temperature. Unsteady pressure measurements were registered inside and outside the cavity via Kulite pressure transducers: ten pressure transducers were located inside the cavity aligned along the cavity rig center, two on the flat plate ahead of the cavity (see Fig. 1), one on the flat plate aft of the cavity, two on the front and rear walls and four on the port side walls [2, 18]. The data was sampled at 6 kHz using a high-speed digital data acquisition system. The measured data was presented in terms of Sound Pressure Level (SPL) and Power Spectral Density (PSD) plots. The SPL is an indication of the intensity of noise generated inside the cavity and can be obtained from the measurements using the following equation: SPL .dB/ D 20 log10 prms =2 105

(1)

where prms is the RMS pressure normalised by the International Standard for the minimum audible sound of 2 105 Pa with the RMS pressure denoted by: prms D

r X

p pmean

2 =N

(2)


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Fig. 2 SPL computed using the experimental data by Ross [4] for the cavity configurations shown in Fig. 1

Spectral analysis was performed using Fast Fourier Transform (FFT) to obtain the power spectral density, which presents the RMS pressure versus frequency and provides a measure of the frequency content inside the cavity. Indicative results of the SPL along the floor of the L=D D 5 cavity are shown in Fig. 2 where the relative effectiveness of the flow control devices is compared.

3 Results and Discussion Several sets of results have been obtained for this work. Starting from URANS calculations, CFD grids of approximately 4 million cells were used. Figure 3 presents an overview of such a grid. Care has been taken to maintain uniform cells near the cavity region and grid dependency has been checked [19]. The LES results required finer grids with further requirements for orthogonality and aspect ratio, so grids with more than 6 million cells were employed. For the cases where flow control devices were present, the region near the front of the cavity had to be refined so that spoilers can be embedded in the grid. The details of the employed CFD solver as well as the employed turbulence models are given in [20]. For all cases computed the Mach number at the free-stream ahead of the cavity was kept at 0.85. The Reynolds number based on the length of the cavity was 1 million, and the LES solutions were forced with noise at the free-stream which corresponded to 1% of turbulence intensity. The k! and Spalart–Almaras turbulence models were employed along with their Detached-Eddy Simulation (DES) versions. For LES the

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Fig. 3 3D view of the employed CFD grid for LES calculations. Almost uniform grid was applied within the cavity and around the walls, however, cell skewness had to be tolerated further away from the cavity

Table 1 Summary of the obtained URANS results and comparison with the experimental findings Location

Control device

Numerical data

X=L D 0:05

No device Spoiler Rear sloping wall

˙6;000 ˙500 ˙2;000

Experimental data (doors on) ˙4;000 ˙2;000 ˙1;000

Reasonable prediction? Yes No Yes

X=L D 0:55


˙10;000 ˙500 ˙4;000

˙6;000 ˙2;000 ˙1;500

Yes No No

X=L D 0:95


˙15;000 ˙500 ˙5;000

˙10;000 ˙4;000 ˙2;000

No No No

classic Smagorinsky sub-grid scale model was employed. Table 1 summarises the obtained URANS results and gives an overall assessment of these in comparison to the experimental data. As can be seen, the employed URANS method (based on the Spalart–Almaras and the k! models) did not capture the changes induced by the flow control devices in the overall flow topology and the reduction in noise. Inspection of the experimental data suggests that although the overall level of pressure fluctuations is lower for the cases where flow control devices are deployed, the distribution of acoustic energy in the various frequency-bands around each Rossiter mode is not significantly affected. For this reason further LES and DES results were obtained for cavities with the control devices deployed. Indicative results of these URANS computations are shown in Fig. 4 where the obtained RMS pressure at a station close to the downstream corner of the cavity is compared with the experimental data. It is evident that URANS based on the k! model was not able to capture even the magnitude of the pressure fluctuations measured in the wind tunnel.


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Fig. 4 URANS results using the k! turbulence model for the case where slanted walls have been deployed at the front and back of the cavity

3.1 LES for Cavities Without Flow Control Devices The first set of results were obtained for clean cavities with and without doors and these are compared against experimental data and Rossiter’s theory in Figs. 5 and 6. Given the complexity of the flow and the demanding conditions, fair agreement has been obtained with experiments, at least for the magnitude of the mean pressure and the pressure fluctuations for the lowest Rossiter frequencies. The results shown here suggest that the current LES is perhaps a good compromise for the analysis of these complex flows since it maintains good predictions for a range of cases (doors=no-doors) and compares well (at least in magnitude) with experimental data. For the cases shown in this paper, DES results were also obtained based on the Spalart-Almaras model but these are not shown here due to space. In contrast to the URANS results, the LES=DES results show substantial unsteadiness in the flow around the cavity and the instantaneous results have enough frequency content. This encouraging result suggests that for the flow controlled cases LES and DES could also be used. Further computations on finer and coarser grids revealed much less dependency of the obtained results on grid size which was not the case with the employed URANS models.

3.2 Cavities with Flow Control Devices Out of all controlled cases computed, results are shown here for the cavity with a slanted downstream wall. For this test case, Table 1 indicates that stations close to

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Fig. 5 LES results for the clean cavity case (no doors): RMS pressure at four different stations (g, h, i and j) along the cavity floor

Fig. 6 LES results for the clean cavity case (with doors): RMS pressure at four different stations (g, h, i and j) along the cavity floor


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the leading edge of the cavity were well-predicted by URANS. This, however, was not the case for downstream stations. Figure 4 compares the URANS predictions with experiments and as can be seen, substantial discrepancies are encountered. The overall characteristics of the URANS solution indicate that the shear layer formed just after the leading edge of the cavity remains coherent until it reached the downstream wall. This flow topology is in contrast to experimental observations and resulted in a less noisy cavity in comparison to measurements. The striking difference between LES (or DES) results and URANS for this case was the instantaneous flow-field predicted by LES and DES was not coherent downstream the middle of the cavity, and this resulted in a much more energetic flow-field with higher levels of noise. Figure 7 shows exactly this effect by comparing the RMS pressure and the pressure spectra with experiments. Although discrepancies still exist, the level of RMS pressure is much better predicted and most of the tones measured during experiments were also present in the CFD solutions (at least for the lower Rossiter frequencies). An instantaneous flow-field from our LES computations for the clean cavity case is shown in Fig. 8a while Fig. 8b presents results for the case where a downstream slanted wall was employed. Similar results have been obtained for the control devices shown in Fig. 1, however, these are not presented here.

RMS Pressure

Pressure Spectra

Fig. 7 RMS pressure at two different stations (e and f) along the cavity floor for the case where a slanted downstream wall was used as a flow control device

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(a) Clean cavity (no doors)

(b) Slanted downstream wall (no doors)

Fig. 8 Comparison between instantaneous Mach number fields for (a) the clean cavity and (b) the controlled cavity (slanted downstream wall)

4 Conclusions In this paper, results have been presented from URANS, LES and DES computations for transonic cavity flows. For clean (without control) cavity flow cases, LES and, to some extend, DES computations were in good agreement with experiments providing a reliable (though expensive in terms of computing time) method for the analysis of such complex turbulent flows. Having established confidence on the obtained results for clean cavities, several cases have been investigated where changes in the cavity geometry were introduced in an attempt to control the flow and reduce the level of noise radiated from the cavity. As was the case for clean cavities, URANS results (obtained using the Spalart–Almaras and the k! models) were in poor agreement with experiments and failed to predict essential pars of the cavity flow physics. LES based on wall-functions and the Smagorinsky sub-grid scale model, as well as DES results (using the Spalart–Almaras model) captured much better the energetic and unsteady flow field of the cavity and compared better with experimental data. The current set of computations generated significant amounts of data, which after further analysis could be used to better identify possible modifications to the employed URANS and DES approaches that could lead to better computing efficiencies. Further work is currently underway to obtain results using finer grids and different sub-grid models for LES so that confidence can be established in the adopted method. Acknowledgements The financial support of the Engineering and Physical Sciences Research Council through grant EP=C533380=1 is gratefully acknowledged. The authors would like to extend their gratitude to John Ross and Graham Foster of QinetiQ (Bedford) and Trevor Birch of DSTL for providing the experimental data.


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References 1. J.A. Ross and J.W. Peto. Internal Stores Carriage Research at RAE. Technical Report 2233, Royal Aircraft Establishment, January 1992. 2. J.A. Ross. Cavity Acoustic Measurements at High Speeds. Technical Report DERA/MSS/ MSFC2/TR000173, QinetiQ, March 2000. 3. M.B. Tracy, E.B. Plentovich and J. Chu. Measurements of Fluctuating Pressure in a Rectangular Cavity in Transonic Flow at High Reynolds Numbers. Technical Report 4363, NASA, June 1992. 4. J.A. Ross and J.W. Peto. The Effect of Cavity Shaping, Front Spoilers and Ceiling Bleed on Loads Acting on Stores, and on the Unsteady Environment Within Weapon Bays. Technical report, QinetiQ, March 1997. 5. H. Heller and J. Delfs. Cavity Pressure Oscillations: The Generating Mechanism Visualized. AIAA, 33(8):1404–1411, August 1995. 6. M.A. Kegerise, E.F. Spina and L.N. Cattafesta. An Experimental Investigation of Flow-Induced Cavity Oscillations. In 30th AIAA Fluid Dynamics Conference. AIAA, June/July 1999. AIAA Paper 99-3705. 7. S.A. Ritchie, N.J. Lawson and K. Knowles. An Experimental and Numerical Investigation of an Open Transonic Cavity. In 21st Applied Aerodynamics Conference, Orlando, Florida, June 23–26 2003. AIAA Paper 2003-4221. 8. M.J. Esteve, P. Reulet and P. Millan. Flow Field Characterisation within a Rectangular Cavity. In 10th International Symposium Applications of Laser Techniques to Fluid Mechanics, July 2000. 9. O. Baysal and S. Srinivasan. Navier–Stokes Calculations of Transonic Flows Past Cavities. Technical report, NASA, 1989. Contractor Report 4210. 10. P.D. Orkwis and P.J. Disimile. Transient Shear Layer Dynamics of Two- and ThreeDimensional Open Cavities. Technical Report ADA 298030, Air Force Office of Scientific Research, 1995. 11. C.-J. Tam, P.D. Orkwis and P.J. Disimile. Algebraic Turbulence Model Simulations of Supersonic Open-Cavity Flow Physics. AIAA, 34(11):2255–2260, November 1996. 12. D.P. Rizzetta. Numerical Simulation of Supersonic Flow Over a Three-Dimensional Cavity. AIAA, 26(7):799–807, July 1988. 13. S.H. Shih, A. Hamed and J.J. Yeuan. Unsteady Supersonic Cavity Flow Simulations Using Coupled k" and Navier–Stokes Equations. AIAA, 32(10):2015–2021, October 1994. 14. X. Zhang. Compressible Cavity Flow Oscillation due to Shear Layer Instabilities and Pressure Feedback. AIAA, 33(8):1404–1411, August 1995. 15. D. Lawrie. Investigation of Cavity Flows at Low and High Reynolds Numbers Using Computational Fluid Dynamics. Ph.D. thesis, University of Glasgow, 2004. 16. J.E. Rossiter. Wind Tunnel Experiments on the Flow over Rectangular Cavities at Subsonic and Transonic Speeds. Technical Report 64037, Royal Aircraft Establishment, October 1964. 17. L.N. Cattafesta, D. Williams, C. Rowley and F. Alvi. Review of Active Control of FlowInduced Cavity Resonance. In 33rd AIAA Fluid Dynamics Conference. AIAA, June 2003. AIAA Paper 2003-3567. 18. M.J. de C. Henshaw. M219 Cavity Case. Technical Report RTO-TR-26, Research and Technology Organization, October 2000. 19. P. Nayyar and G. Barakos. A Summary of Turbulence Modelling Approaches in CFD. Technical report, University of Glasgow, 2002. Aerospace Engineering Report 0206. 20. P. Nayyar, G. Barakos and K. Badcock. Numerical Simulation of Transonic Cavity Flows using LES and DES. The Aeronautical Journal, 111(1117):153–164, March 2007.


Part III

Theoretical Aspects & Analytical Approaches of Flow Separation


Near Critical Phenomena in Laminar Boundary Layers A. Kluwick, S. Braun, and E.A. Cox

Abstract Recent developments in the construction of airfoils and rotorblades are characterized by an increasing interest in the application of so-called smart structures for active flow control. These are characterized by an interplay of sensors, actuators, real-time controlling data processing systems and the use of new materials e.g. shape alloys with the aim to increase manoeuvrability, reduce drag and radiated sound. The optimal use of such devices obviously requires a detailed insight into the flow phenomena to be controlled and in particular their sensitivity to external disturbances. In this connection locally separated boundary layer flows are of special interest. Asymptotic analysis of boundary layer separation in the limit of large Reynolds number Re ! 1 has shown that in a number of cases which are of importance from a practical point of view solutions of the resulting interaction equations describing two-dimensional steady flows exist up to a limiting value c of the relevant controlling parameter only while two branches of solutions exist in a regime < c . The present study aims at a better understanding of near critical flows j c j ! 0 and in particular the changes of the flow behaviour associated with the passage of through c . Keywords Boundary layer theory Separation bubble Laminar-turbulent transition Fisher’s equation

1 Introduction Asymptotic analysis of high Reynolds number flows Re ! 1 has shown that there exist at least two different routes leading to the formation of a separated flow region inside an otherwise attached laminar boundary layer. Firstly, the presence A. Kluwick () and S. Braun Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, Resselgasse 3/E322, A-1040 Vienna, Austria e-mail: falfred.kluwick, [email protected] E.A. Cox School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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of an imposed adverse pressure gradient acting over a distance of order one on the typical boundary-layer length scale may cause the wall shear to decrease and finally become negative over a bounded distance before it recovers again. Examples of this so-called marginal separation are provided by the leading edge separation on slender airfoils at incidence, flow separation associated with the deflection of wall jets and flow separation in channels enforced by suction. Secondly, a firmly attached laminar boundary layer may be forced to separate due to the presence of a large adverse pressure gradient acting over a short distance caused, for example, by a surface mounted obstacle or the Kutta condition near the trailing edge of a slender airfoil. Although these scenarios ultimately resulting in the formation of separated flow differ vastly in detail they, nevertheless, share a number of common features. Most important, it is found that a uniformly valid description of the flowbehaviour close to separation requires the investigation of three layers or decks having substantially different properties. Viscosity plays a significant role inside a thin sublayer of the oncoming boundary layer (the lower deck) only while the dynamics of the flow further away from the wall is predominantly inviscid. The main portion of the boundary layer (main deck) primarily acts to transfer the displacement effects of the low speed flow inside the lower deck to the region outside the boundary layer (upper deck) and to transfer the resulting pressure response unchanged to the near wall region. While the leading order upper and main deck problems can be solved analytically, the study of the flow behaviour inside the viscous wall layer requires a numerical treatment in general. Specifically, for the 1st route it is found that the essential features of the lower deck region associated with marginally separated flows are captured by the integro-differential equation 3 2 Zx Z 2 @P .; z; t/ 1 @ P .; z; t/ 4 A2 x 2 C D C p d 5 d @ @z2 x 1 1 (1) Zx Zx 1 @.A h/ vw d d .x /1=4 @t .x /1=4 1

1

where A.x; z; t/ and P .x; z; t/ denote the (negative) perturbation displacement thickness and the pressure while the parameter represents a measure of the angle of incidence, the turning angle and the suction rate, respectively. Furthermore, x, z and t denote Cartesian coordinates in the streamwise and spanwise directions and the time while , , are positive constants. All quantities are suitably nondimensionalized and scaled. Finally, h.x; z; t/ and vw .x; z; t/ account for the effects of controlling devices such as surface mounted obstacles and suction stripes. In contrast, if boundary layer separation is approached along route 2 then the boundary layer equations in incompressible form @u @v @w C C D 0; @x @y @z

Near Critical Phenomena in Laminar Boundary Layers

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@u @u @u @P @2 u @u Cu Cv Cw D C 2; @t @x @y @z @x @y @w @w @w @P @2 w @w Cu Cv Cw D C 2 @t @x @y @z @z @y

(2)

together with the matching and boundary conditions u!y

for x ! 1 ; B.x; z; t/ ; w! y

u ! y C A h;

u D w D 0 ; v D vw

Zx B D 1

at y D 0

@P d @z

for y ! 1

(3)

have to be solved. Here u, v, w are the velocity components in x, y, z directions where y measures the distance from the solid wall. To close the problems (1) and (2), (3) a relationship P D F .A/ between A and the induced pressure P is required which is problem specific. Here we focus on incompressible flows where 1 P D 2

Z1 Z1 1 1

@.A h/.; ; t/=@ p dd : .x /2 C .z /2

(4)

Despite the fact that the form of the interaction law P D F .A/ depends on the specific problem under consideration, marginally separated flows exhibit a number of properties which appear to be universal. Most important, in all known cases of marginal separation it is found that two-dimensional steady state solutions exist up to a critical value c of only and that inside a range of values < c the problem is non-unique and admits two branches of solutions. As a specific example, Fig. 1 displays A.0/ versus for uncontrolled incompressible flow h D vw D 0

1.5

upper branch

1

0.5

A(0) 0 −0.5

¡c

lower −1 −1

−0.5

0

0.5

1

1.5

2

2.5

3

¡ Fig. 1 Fundamental curve of marginal separation; dashed line: local solution of classical boundary layer theory (asymptote for ! 1), dotted line: parabola approximation near the bifurcation point, see Section 2.1

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past the leading edge of a slender airfoil at incidence first studied by Ruban [17] and independently by Stewartson et al. [20] where c 2:66. Interestingly, similar phenomena are known to occur also in situations where a fully attached boundary layer separates due to rapid changes of the boundary conditions, e.g. subsonic trailing edge flow, Korolev [14], supersonic flow past flared cylinders, Gittler and Kluwick [10]. The nonexistence of steady two-dimensional solutions to equations (1) or (2), (3) supplemented with the interaction relationship P D F .A/ if the relevant controlling parameter exceeds a critical value raises a number of questions concerning the changes of the flow behaviour associated with its transition from sub-critical to super-critical values, e.g. Braun and Kluwick [4]. Their answer requires the investigation of unsteady, three-dimensional effects which poses an extremely difficult numerical task. To the authors knowledge it has been attacked so far for marginally separating flows only where Smith [19], Ryzhov and Smith [18], Elliott and Smith [7] noted that the evolution of unsteady two-dimensional disturbances above c inevitably leads to the formation of finite time singularities. Probably the most detailed calculations based on the Navier–Stokes equations have been carried out by Alam and Sandham [1] for the specific case of a channel flow designed such that boundary layer separation on the lower wall is enforced by the pressure increase resulting from suction at the upper wall. The results indicate that the flow inside the separation bubble becomes increasingly sensitive to disturbances as the suction rate increases ultimately leading to bubble bursting and, if the suction rate is sufficiently high, to repeated bubble bursts in the form of self-sustained oscillations. Also, it is found that a transition from laminar to turbulent flow then occurs near and downstream of reattachment which is characterized, among others, by the formation of -type vortices, Fig. 2, which is supported also by experimental evidence. Obviously, one then is confronted with the question if and how the singularities predicted by the asymptotic theories for Re ! 1 are related to flow structures for large but finite Reynolds number and how much of the dynamics emerging from Navier–Stokes calculations can be captured by considering truly unsteady, threedimensional effects described by (1) or (2), (3).

2 Bifurcation Analysis of Near Critical Flows 2.1 Route 1 Towards Separation As noted by Braun and Kluwick [2–4] the treatment of marginally separated flows simplifies considerably if differs only slightly from c or, more precisely, by focussing on the limit " D j c j1=4 ! 0. Appropriate expansions of A; h; and vw then are


95

x

y

m rea ean tt. z

m sep ean . Fig. 2 Instantaneous contours of the span-wise vorticity component !z D @v=@x @u=@y, [1]. -vortex structures within the time mean separated region are associated with the generation of moving singularities (white lines) immediately after blow-up events

A D A1c .x/ C "2 a1 .x; zN; tN/ C "4 a2 .x; zN; tN/ C : : : ; h D h1 .x/ C "4 h1 .x; zN; tN/ C : : : ;

(5)

vw D vw1 C " vw1 .x; zN; tN/ C : : : 4

Here zN D "z, tN D "2 t and the subscript ‘1c’ refers to steady two-dimensional critical flow conditions. Introducing the abbreviations Z1 I D x

@2 p d ; x @ 2 1

Z1 J D x

d p ; x

Zx K D 1

d ; .x /1=4

(6) substitution of expansions (5) into (1) and (4) yields .2A1c I /a1 D 0. Consequently, a1 D b.x/ c.Nz; tN/ where b.x/ denotes the right eigenfunction of the singular operator .2A1c I /: .2A1c I /b D 0 : (7) Solutions of the equation for a2 .2A1c I /a2 D sgn. c / b 2 c 2 C

J b @2 c @c Kb I h1 Kvw1 (8) 2 @Nz2 @tN

exist only if Fredholm’s alternative is met, i.e. if the yet unknown ‘shape’ function c.Nz; tN/ satisfies the evolution equation

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@2 c @c 2 C c 2 sgn. c /ı D gN : @tN @Nz

(9)

The constants , , ı and the function gN which accounts for the effects of controlling devices are uniquely defined in terms of b.x/ and the left eigenfunction n.x/: .2A1c I / n D 0. Here R 1 .2A1c I / denotes the adjoint of .2A1c I /. Using the notation h n; q i D 1 nq dx one obtains: D

h n; J b i ; 2h n; Kb i

gN D

D

h n; b 2 i ; h n; Kb i

ıD

h n; 1 i ; h n; Kb i

h n; I h1 i C h n; Kvw1 i : h n; Kb i

(10)

Numerical results for A1c .x/, b.x/ and n.x/ with h1 D vw1 D 0 are displayed in Fig. 3 and yield: p 3:0, 2:07, ı 1:60. If < c stationary points of (9) satisfy c D ˙cs , cs D ı= and correspond to upper and lower branch solutions for below critical flow conditions, p Fig. 1. Finally, by applying the transformation c.Nz; tN/ C cs D 2cs u.z; t/, zN D =.2cs / z, tN D t=.2cs /, g D g=.4ı/ N equation (9) assumes the parameter free form ut uzz D u u2 ‚. c /=2 C g.z; t/

(11)

known as the forced Fisher-Kolmogoroff, Petrovsky, Piscounoff (FKPP)-equation, e.g. Fisher [8]. Here ‚.s/ denotes the Heavyside function ‚ D 0 for s < 0 and ‚ D 1 for s > 0. 4

3

A∞c

b

2

1

n

0 −1 −4

−2

0

4

2

x

Fig. 3 Near critical marginally separated flows (h1 D vw1 D 0): (negative) perturbation displacement thickness A1c .x/, right and left eigenfunctions b.x/ and n.x/


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2.2 Route 2 Towards Separation The main ideas associated with the bifurcation analysis of marginally separated flows carry over unchanged although the details are considerably more complicated. To be specific, we consider the case of incompressible flow pastp an expansion ramp with ramp angle ˛ and slightly rounded corner h1 .x/ D ˛.x C x 2 C r 2 /, r 1. Generalizing equations (5) we now expand as Œu; v D Œuc ; vc .x; y/ C "2 Œu1 ; v1 .x; y; zN; tN/ C "4 Œu2 ; v2 .x; y; zN; tN/ C : : : ŒP; A D ŒPc ; Ac .x/ C "2 ŒP1 ; A1 .x; zN; tN/ C "4 ŒP2 ; A2 .x; zN; tN/ C : : : w D "3 w1 .x; y; zN; tN/ C : : :

(12)

B D " B1 .x; zN; tN/ C : : : 3

Œh; vw D Œh1 ; vw1 .x/ C "4 Œh2 ; vw2 .x; zN; tN/ C : : : where " D j˛ ˛c j1=4 and as before zN D "z, tN D "2 t. Steady two-dimensional flow fields have been computed first by Korolev [15] for r D 0 who found that numerical solutions cannot be obtained for ˛ > ˛c while two branches of solutions exist for 0 ˛ ˛c , Fig. 4. Similar to marginally separated flows deviations from the critical two-dimensional steady state but now characterized by the perturbations of the two velocity components u, v and the perturbation displacement !T

function A and expressed in terms of the vector r 1 D .u1 ; v1 ; A1 / can be writ! ! ! ten as r 1 .x; y; zN; tN/ D c.Nz; tN/ r .x; y/. Here r T .x; y/ D .ur ; vr ; Ar / represents the right eigenvector of a singular operator matrix M which depends on the unperturbed flow quantities only: 20

lb 15

10

5

0

®c

−6

−5.8

−5.6

−5.4

−5.2

−5

−4.8

® Fig. 4 Non-uniqueness of the planar ramp flow: bubble length lb versus ramp angle ˛ for r D 0:1 and vw1 D 0: ˛c 5:926, source: Zametaev (2003, private communication)

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0

1 Z1 2 0 1 1 d =d 2 B uc @x C ucx C vc @y @yy ucy C d C ur B CB C ! x ! C @ vr A D 0 : Mr DB 1 B C @x @y 0 @ A Ar 0 0 @y (13) As before the ‘shape’ function c.Nz; tN/ remains arbitrary at this level of approximation and is determined by the requirement that solutions for the higher order approxima!T

tions u2 , v2 , A2 exist. Introducing the left eigenvector l the resulting solvability condition assumes the form @c @tN

Z

Z mur dD C c 2

D

m.ur urx C vr ury / dD C D

Z sgn.˛ ˛c / D

@2 c @Nz2

.x; y/ D .m; n; q/ of M

Z nwr dD D

Z1 Z1 h2 m C d dD D n0 vw2 dx : x 1

(14)

1

which is again recognized as an equation of Fisher type. The quantity wr accounts for cross flow effects via the relationship w1 D wr .x; y/@c=@Nz and is obtained as the solution of uc

@2 wr @wr @wr C vc D Pr C ; @x @y @y 2

Z1 0 1 Ar C d : Pr d with Pr D y D 0 W wr D 0 ; x 1 1 (15) Numerical work in progress (Szeywerth, 2007) indicates that the problems for the right and left eigenvectors have a unique solution and that the integrals entering (14) exist which, therefore, can transformed into its canonical form (11) which will be taken as the basis for the following discussion. 1 y ! 1 W wr y

Zx

3 FKPP Equation Equations of Fisher’s type (heat equations with nonlinear source terms) are known from nonlinear wave propagation phenomena in gene populations, reactiondiffusion and heat conduction processes. Its appearance in the context of near critical flow phenomena forms one of the key observations of the present study. In contrast to previous applications where u.z; t/ is limited to positive values within the range Œ0; 1 or Œ0; 1/ no restrictions on the magnitude and sign of u exist in cases which are of interest here. As a consequence, the associated dynamics becomes considerably more complicated and only first steps towards a full understanding have been taken.


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3.1 2-D Unsteady Flows 2-D unsteady flows where further analytical progress is possible provide a natural starting point for a discussion of flow phenomena described by (11). In the case of unforced flow gN D 0 it reduces to Bernoulli’s equation which can be solved in closed form for both sub- and super-critical flows

< c W

u.t/ D

u0 C u0 tanhŒ.t t0 /=2 ; 1 C .2u0 1/ tanhŒ.t t0 /=2

(16)

> c W

u.t/ D

u0 C .u0 1/ tanŒ.t t0 /=2 : 1 C .2u0 1/ tanŒ.t t0 /=2

(17)

Here u0 D u.t0 / denotes the value of u imposed at time t D t0 . According to (16) the steady upper branch solution us D 1 is approached for initial conditions u0 > 0, Fig. 5. In contrast, for u0 < 0 i.e. for u0 below the steady lower branch solution us D 0, finite time blow-up occurs at the blow-up time t D t0 C 2 artanh Œ1=.1 2u0 / :

(18)

Still, however, the steady upper branch solution us D 1 is approached in the limit t ! 1. No such steady state exists for super-critical flow where Equation (17) predicts periodic blow-up, i.e. self-sustained oscillations of the separation bubble. The above interpretation of solutions to Equation (11) is watertight if u remains bounded for all times t t0 but hinges on tacit assumptions if finite time blow-up occurs, namely (i) that u.t/ can be extended beyond t and (ii) that the singular behaviour of u for t t ! 0 causes a singular response of u for t t ! 0C . Although no rigorous proof of (i) and (ii) exists at present, their validity appears to be supported by available numerical data and physical considerations. For example, as mentioned before, DNS calculations for marginally separated channel flows

4

u(t) 2

0 −2

t∗

−4 0

1

2

3

4

5

6

7

8

t Fig. 5 Solutions of (11) for unforced planar flow; sub-critical conditions < c , (16); dashed lines: steady states corresponding to upper and lower branch solutions; blow-up time t

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carried out by Alam and Sandham [1] predict that self-sustained bubble oscillations occur if the suction rate ˛ D VPs =VP1 where VP1 and VPs , respectively, denote the volume fluxes at the channel entry and the suction strip is sufficiently large. Specifically, such oscillations were observed in the range ˛ D 0:2 0:25 which is larger than but of the same order of magnitude as the critical suction rate ˛c 0:09 predicted by the asymptotic approach which is encouraging. Also, if a singular behaviour of u.t/ for t t ! 0 is accepted, conservation of mass immediately implies a related singular behaviour for t t ! 0C which in turn ‘selects’ unique solutions (16), (17) for arbitrary values u0 . Therefore, assumptions (i), (ii) will be adopted in the following considerations dealing with more general situations as for example unsteady 2-D forced flow where g.t/ is taken to be purely harmonic g.t/ D a‚.t/ sin !t. Introduction of the transformation u ! R:

1 R0 .tN/ ! u.t/ D 1C! ; tN D t (19) 2 R.tN/ 2 4 then leads to the canonical form of Mathieu’s equation R00 C Œp 2q cos.2tN/ R D 0

(20)

with p D 1=! 2 and q D 2a=! 2 . Taking into account (19), blow-up solutions of (11) are associated with zeros of solutions to (20) and multiple blow-up will be associated with periodic solutions of (20). For p D a0 .q0 / this equation has an even 2 periodic solution with no zeros R D c e0 .tNI q0 / where c is a normalization constant. Using the transformation R D c e0 .tNI q0 / .tN/ application of the theorem of Leighton [16] to the resulting equation for shows that repetitive blow-up occurs for < c if the forcing amplitude a > ac D q0 ! 2 =2 and is inevitable if > c . Evaluation of the relationship ac D ac .!/ as displayed in Fig. 6 together with its p limiting form ac != 2 as ! ! 1 shows that the danger of bubble bursting in sub-critical flows decreases with increasing values of ! in agreement with experimental observations (Ruban, 2005). Numerical solutions of (11) for < c , ! D 2 and two different values of a are depicted in Fig. 7 and seen to be in complete agreement with the prediction following from the analytical result q0 0:7268, ac 1:45216.

3.2 Further Analytical Solutions of the FKPP Equation Closed form solutions of (11) without forcing can be obtained also in the case of steady three-dimensional flow where it reduces to the integrated form of the Korteweg-de Vries equation if < c . Consequently, bounded solutions varying periodically with z are given in terms of the Jacobian elliptic functions cn .sjm/ and the integration constant ' 2 Œ0; =3:


101

1.6 1.4 1.2 1

ac(!)

0.8 0.6 0.4 0.2

0

0.5

1

1.5

2

! Fig. 6 Critical forcing amplitude for planar flow and p g.t / D a‚.t / sin.2t /: a > ac leads to repetitive blow-up; ac .0/ D 1=4, ac .! ! 1/ != 2 C (dashed line)

2

a = 1:44

1

u(t)

0 −1

g(t)=a 1:5

−2 0

5

10

15

t∗ 20

25

30

t Fig. 7 Planar forced flow: numerical solutions of (11) with g.t / D a‚.t / sin.2t /. Repeated blowup occurs for a > ac 1:45216

s !# " ˇ p ˇ 2 tan ' sin ' z 1 ˇp C 3 sin ' cn2 u.z/ D sin2 cos ' C p 2 2 3 2 ˇ 3 C tan ' (21) known from the theory of shallow water waves, Fig. 8(a). The limit ' D =3 yields the homoclinic orbit (solitary wave) '

u.z/ D 1

z 3 cosh2 ; 2 2

(22)

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a 1.2

¼=3

1

u(z)

1:02

0.8

0:9

0.6

¼=4

0.4

¼=6

0.2 0 −0.2

' = 0:2

−0.4 −0.6

0

2

4

6

8

10

12

14

z

b

2 1.5

u(z)

−1=216

0 1 0.5 0 −0.5

−0:01

−1 −1.5

−0:005

g3 = −0:05 0

2

4

6

8

10

12

14

z Fig. 8 Steady sub-critical solutions of (11) without forcing; dashed lines: us D 0; 1; (a) bounded, Equations (21), (22), (b) singular, Equation (24)

while in the limit ' ! 0, which corresponds to a linearization about the unperturbed planar steady state u D 0, one obtains (Stokes waves) p 3 ' cos z C O.' 2 / : u.z/ 2

(23)

An additional family of solutions which vary periodically with z is obtained if, as in the case of two-dimensional unsteady flow, the presence of singularities is accepted. It exists for sub-critical and, interestingly, also for super-critical flow conditions and can be expressed in terms of the Weierstrass elliptic function }.zI g2 ; g3 / with g2 D sgn. c /=12 while g3 remains arbitrary: u.z/ D 6}.zI sgn. c /=12; g3 / C 1=2 :

(24)


103

5

u(z) 4 3 2 1

1 0

2!2 -1

0

5

g3 = −1 10

15

20

z Fig. 9 Singular steady super-critical solutions; solid line: limiting case with maximum period 2!2max 10:2909 (g3 0:00423616)

For < c , Fig. 8b, the distance between consecutive singularities (streak spacing) varies between 0 and 1 and a non-periodic solution is found in the limit g3 D 1=216. In contrast solutions for > c , Fig. 9, are always periodic as they cannot decay to a two-dimensional steady state and there exists an upper bound z 10:2909 for the spacing of streaks. The soliton solution (22) exhibits the exponential decay for z ! 1 observed in numerical solutions for steady two-dimensional upper branch flows disturbed by an isolated three-dimensional surface mounted obstacle while the existence of periodic solutions (21) support the finding that the perturbation displacement thickness A exhibits an oscillatory behaviour for z ! 1 if one considers disturbances of steady planar states corresponding to the lower branch, Braun and Kluwick [2]. The physical meaning of singular solutions (24) and in particular the possible existence of super-critical steady states associated with a maximum streak spacing remains unclear at present. As a last class of exact solutions to the unforced version of (11) we consider travelling wave solutions u.z; t/ D v./ ;

D z U t 0 ;

(25)

where U and 0 denote the wave speed and an arbitrary constant. For < c one then obtains v00 C U v0 C v v2 D 0 : (26) Because of the invariance property u.z; tI U / D u.z; tI U / it is sufficient to consider right running waves U > 0 only. Except for U D 0 bounded solutions connect the steady states us D 1; 0, Fig. 10a. In contrast, singular travelling wave solutions are seen to deviate and return to the stable upper branch level u D us D 1, Fig. 10b.

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a

1

v(») U=3

2

0

0:5 0 0

10

20

30

40

»

b

10

v(») 5

0 0

0:2 2 −5 U=3 −10 −10

−5

0

5

10

» Fig. 10 (a) Bounded and (b) singular travelling (right running) wave solutions of Fisher’s equation (26) depending on the wave speed U

3.3 Blow-up in Unsteady Three-Dimensional Flow For two-dimensional unsteady flow the singular behaviour of u near blow-up is readily obtained from the exact solution (16), (17): u 1=.t t / as jt t j ! 0. In the case of three-dimensional flow, however, the analysis of the flow structure is considerably more complex and a fully self-consistent description has not been obtained yet. Work carried out by Hocking et al. [11] in a different context suggests the ansatz, Braun and Kluwick [4] u.z; t/

1 f .; / C ; t

z D p ; jtj

D ln jtj

(27)


105

as t ! 0 (where the blow-up point is assumed at t D 0 and z D 0 without loss of generality) and results in f C

sgn.t/ . c / f f 2 D f f f sgn.t/ e f C e2

g : 2 2

2

(28) Expansion of f .; / for ! 1 requires the introduction of logarithmic terms ! f1 ./ ln

ln2

: (29) C CO f .; / f0 ./ C g1 ./

2 Near blow-up exponentially small terms in (28) can be neglected to the order considered here which in turn allows for an analytical treatment and, furthermore, reveals the important symmetry property f .; / ! f .i; / if t ! t indicating that, similar to two-dimensional flows, also z-dependent solutions of (11) can be extended beyond blow-up and that the singular behaviour for t ! 0 forces a singular behaviour for t ! 0C : 16 c1 2 82 ln j8 ˙ 2 j : .8 ˙ 2 /2 (30) Here c1 is an arbitrary constant depending on initial conditions and the upper/lower sign corresponds to t ! 0 . According to (30) the focussing of u as the blow-up time is approached leads to the generation of a pair of vortices after blow-up moving along the paths p zs .t/ s . / D p D ˙ 8 C ; (31) t

f0 D

8 ; 8 ˙ 2

g1 D

10 2 ; .8 ˙ 2 /2

f1 D

Fig. 11, which is thought to provide a mechanism for the appearance of coherent structures (-vortices, see Fig. 2) in transitional separation bubbles, Braun and Kluwick [4]. Unfortunately, expansion (29) ceases to be valid in a vicinity of the moving singularities where all terms become of equal magnitude. This p deficiency can be partly corrected through the introduction of inner regions .˙ 8/ D O.1/ where the leading order terms represent singular travelling waves discussed in Section 3.2 in the limit of infinite wave speed. Higher order terms, however, cannot be matched with the outer solution (29) and, as a result, the flow description remains incomplete. Nevertheless, this analysis generalized by the p method of strained coordinates indicates that s . / may deviate from the values ˙ 8 by corrections of O.ln = / which appears to be supported by numerical evidence, Braun and Kluwick [4].

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u(z; t)

t

blow-up

z

Fig. 11 Local blow-up behaviour of solutions to Fisher’s equation (11) (schematic); for t > 0 the solution is shown in the right half plane only

3.4 Numerical Prediction of Blow-up: Influence of Forcing The forced FKPP equation (11) is one of many partial differential equations which have solutions that blow-up in a finite time. The blow-up typically involves the solution becoming infinite at an isolated point at a finite time. Near the blow-up point the solution develops a singular spike with both decreasing width and increasing height. To compute the solutions of blow-up accurately it is essential to employ an adaptive method which can then move points into the blow-up region. The adaptive meshing algorithm implemented here is based on the moving mesh methods of Huang, Ren and Russell [12, 13]. Defining a monitor function M.z; t; u.z; t// > 0 and a computational coordinate an equation for the evolving mesh is given by @ @z @2 @z M D 2 ; @ @ @ @t

(32)

where the parameter 1 is a relaxation time determining the time scale over which the mesh converges to a steady state. This is an approximation to the equation M z D 1 modified to include temporal smoothing. The approach is to discretise (32) on a uniform mesh and discretise the Fisher equation using Hermite cubic collocation on the nonuniform mesh generated from (32). The choice of function M is critical and motivation comes from requiring that the underlying self-similar scaling transformation exhibited by solutions of Fisher’s equation near blow-up should be a property also of the numerical algorithm. In the neighbourhood of the blow-up point the linear and forcing terms in (11) are insignificant in describing the leading blow-up structure. With these terms neglected the resulting equation is invariant under the rescaling t ! t C ˇ.t t /, u ! u=ˇ, z ! z C ˇ 1=2 .z z / where the


107

blow-up point is at .z ; t /. The choice of M D u then ensures that (32) is also invariant under these scalings near blow-up. While solutions of (11) near blow-up are not self-similar under these scalings they are approximately self similar, see Equations (27), (29), and (30). It can be shown then, Budd, Chen, Huang and Russell [6], that for the coupled problems (11), (32) the computed mesh points near blow-up have also the desirable property of lying on trajectories for which given by (27) is constant. This reveals the usefulness of the moving mesh method in inheriting the natural spatial structure of the original differential equation. An example of blow-up in finite time is shown in Fig. 12 where we see the 2 evolution to blow-up for the applied forcing g.z; t/ D 30 sin.2t/e100z with initial condition u.z; 0/ D 1. The scaling analysis above suggests that the evolution presented in Fig. 12 provides a typical structure that is independent of the precise forcing imposed in the neighbourhood of blow-up. The question is then: how are variations in the applied forcing reflected in blow-up formation? The formal asymptotics of Galaktionov, Herrero and Velázquez [9] indicate the possibility of an alternative ‘flatter’ asymptotic structure near blow-up described by the Herp mite polynomials Hm .y/, y D z= t t with m 4 and m even. We are able to generate results suggestive of this flatter blow-up structure through coalescing spike structures generated by a two-peak forcing of the form g.z; t/ D 2 2 30 sin.2t/.e100.zR/ C e100.zCR/ /. For R large there is blow-up at two points and R small at one point. However for R D R 1:92969 the blow-up at z D 0 is associated with the coalescing of two maxima. This blow-up pattern is seen in Fig. 13. More complicated and flatter blow-up patterns can also be generated by inclusion of additional peaks in the forcing and appropriate tuning of the peak separation distance.

1 0.8

(t−t∗)u

f0−

0.6

0.4 t = 3:75

0.2 0

1

2

3

4

5

´ Fig. 12 Evolution to blow-up for the applied forcing g.z; t / D 30 sin.2t /e100z with initial condition u.z; 0/ D 1. Consecutive time steps: t D 3:75; 3:752; 3:75349998; 3:7541998; estimated blow-up time t 3:75438944, blow-up profile f0 , (30) 2

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u(z; t) −2e+06

−3e+06 0

0.05

0.1

0.15

0.2

z Fig. 13 Flat blow-up structure for forcing g.z; t / D 30 sin.2t /.e100.zR/ C e100.zCR/ /, R D 1:92969 with initial conditions u.z; 0/ D 1. Consecutive time steps: t D 4:180145 C i t , i D 0; 2; 4; 6; 8; 10; 15; t D 107 2

2

4 Conclusions In the present study it has been shown that near critical flow phenomena are governed by the same evolution equation of Fisher’s type for both routes leading to the separation of high Reynolds number laminar flows analyzed in the past. Although this equation has been investigated over 70 years, the existing literature contains relatively little material which is of relevance in the present context. This is due to the fact that in most investigations carried out to date u.z; t/ is taken to be in the interval Œ0; 1 or Œ0; 1/, which is sufficient if one stays within its classical field of applications, e.g. population dynamics, but is too restrictive if it is used to study near critical separated flows. Here u may vary in the whole range .1; 1/ which significantly increases the richness of solutions. First steps towards the understanding of the associated new phenomena have been taken by Braun and Kluwick [4, 5]. Here a number of new solutions have been presented which are of interest both in the context of structure formation and flow control. For example, it has been shown that unsteady two-dimensional perturbations of a two-dimensional critical steady state caused by harmonic oscillations of a surface mounted hump starting at t D 0 can be expressed in terms of Mathieu functions leading in turn to a complete picture of the flow behaviour. Among others it is found that, in incompressible sub-critical flows, the critical value ac of the amplitude a causing bubble bursting increases with increasing forcing frequency as observed experimentally. In addition to analytical considerations a new numerical scheme allowing the study of general unsteady, z-dependent flows and specially designed to capture the phenomenon of bubble bursting in detail has been presented. Results obtained so far support existing analytical evidence that the flow properties for jt t j ! 0 where t denotes the bursting time are universal, i.e. independent of the specific form of


109

the adopted forcing term but work in progress also shows that the actual value of t is very sensitive to small changes of the forcing and thus can be controlled very effectively.

Acknowledgement Part of this work was supported by the Austrian Science Fund FWF (project number WK W008) which is gratefully acknowledged.

References 1. Alam, M. & Sandham, N.D. 2000 Direct numerical simulation of ‘short’ laminar separation bubbles with turbulent reattachment. J. Fluid Mech. 410, 1–28. 2. Braun, S. & Kluwick, A. 2002 The effect of three-dimensional obstacles on marginally separated laminar boundary layer flows. J. Fluid Mech. 460, 57–82. 3. Braun, S. & Kluwick, A. 2003 Analysis of a bifurcation problem in marginally separated laminar wall jets and boundary layers. Acta Mech. 161, 195–211. 4. Braun, S. & Kluwick, A. 2004 Unsteady three-dimensional marginal separation caused by surface-mounted obstacles and/or local suction. J. Fluid Mech. 514, 121–152. 5. Braun, S. & Kluwick, A. 2005 Blow-up and control of marginally separated boundary layer flows. In New Developments and Applications in Rapid Fluid Flows (Eds. J.S.B. Gajjar & F.T. Smith). Phil. Trans. R. Soc. Lond. A 363, 1057–1067. 6. Budd, C.J., Chen, J., Huang, W. & Russell, R.D. 1996 Moving mesh methods with applications to blow-up problems for PDEs. In Numerical Analysis 1995: Proceedings of 1995 Biennial Conference on Numerical Analysis (Eds. D.F. Griffiths & G.A. Watson). Pitman Research Notes in Mathematics, Longman Scientific and Technical, pp. 1–17. 7. Elliott, J.W. & Smith, F.T. 1987 Dynamic stall due to unsteady marginal separation. J. Fluid Mech. 179, 489–512. 8. Fisher, R.A. 1937 The wave of advance of advantageous genes. Ann. Eugenics 7, 355–369. 9. Galaktionov, V.A., Herrero M.A. & Velázquez, J.J.L. 1991 The space structure near a blow-up point for semilinear heat equations: a formal approach. USSR Comput. Math. Math. Physics, 31 (3), 399–411. 10. Gittler, Ph. & Kluwick, A. 1987 Triple-deck solutions for supersonic flows past flared cylinders. J. Fluid Mech. 179, 469–487. 11. Hocking, L.M., Stewartson, K., Stuart, J.T. & Brown, S.N. 1972 A nonlinear instability burst in plane parallel flow. J. Fluid Mech. 51, 705–735. 12. Huang, W., Ren, Y. & Russell, R.D. 1994 Moving mesh methods based on moving mesh partial differential equations. J. Comput. Phys. 113, 279–290. 13. Huang, W., Ren, Y. & Russell, R.D. 1994 Moving mesh partial differential equations (MMPDEs) based upon the equidistribution principle. SIAM J. Numer. Anal. 31, 709–730. 14. Korolev, G.L. 1990 Contribution to the theory of thin-profile trailing edge separation. Izv. Akad. Nauk SSSR: Mekh. Zhidk. Gaza 4, 55–59 (Engl. transl. Fluid Dyn. 24 (4), 534–537). 15. Korolev, G.L. 1992 Non-uniqueness of separated flow past nearly flat corners. Izv. Akad. Nauk SSSR: Mekh. Zhidk. Gaza 3, 178–180 (Engl. transl. Fluid Dyn. 27 (3), 442–444). 16. Leighton, W. 1949 Bounds for the solutions of a second-order linear differential equation. Proc. Natl. Acad. Sci. USA 35, 190–191. 17. Ruban, A.I. 1981 Asymptotic theory of short separation regions on the leading edge of a slender airfoil. Izv. Akad. Nauk SSSR: Mekh. Zhidk. Gaza 1, 42–51 (Engl. transl. Fluid Dyn. 17 (1), 33–41).

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18. Ryzhov, O.S. & Smith, F.T. 1984 Short-length instabilities, breakdown and initial value problems in dynamic stall. Mathematika 31, 163–177. 19. Smith, F.T. 1982 Concerning dynamic stall. Aeron. Quart. 33, 331–352. 20. Stewartson, K., Smith, F.T. & Kaups, K. 1982 Marginal separation. Stud. in Appl. Math. 67, 45–61.

State Curves and Flipping for an Orbiting Cylinder at Low Reynolds Numbers L. Baranyi

Abstract Sudden changes found in the time-mean and rms values of force coefficients of a circular cylinder in forced orbital motion placed in a uniform stream when plotted against ellipticity of the orbital path suggest that two solutions (states) exist. This 2D numerical simulation was performed in order to gain further evidence of this hypothesis through flipping of the solution. Time histories and limit cycle curves of force coefficients for stationary, in-line, and orbital paths around the time of the flip were investigated, as well as time-mean and rms values of lift, drag, and base pressure coefficients versus ellipticity for the flipped solution. Results provide evidence of the existence of two solutions. Keywords Orbiting cylinder Numerical simulation Flipping 2D flow Low Reynolds number

1 Introduction Although there are countless studies for flow around a circular cylinder, either stationary or oscillating in one direction, investigations concentrating upon orbital motions are still rather uncommon (see e.g. [10, 11]). Among these, studies of a cylinder in forced orbital motion in a uniform stream are relatively rare. Didier and Borges [7] were able to identify lock-in for in-line and transverse cylinder oscillation and for a cylinder orbiting in a circular path. Lu and Dalton [8], working with forced transverse cylinder oscillation and investigating the effect of oscillation frequency, found switches in flow patterns and sudden 180ı phase angle change between lift and cylinder displacement. Blackburn and Henderson [6] confirmed these findings, as well as identifying sudden changes in energy transfer between cylinder and fluid. L. Baranyi Department of Fluid and Heat Engineering, University of Miskolc, H-3515 Miskolc-Egyetemváros, Hungary e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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In earlier studies, the author identified sudden jumps in the time-mean and rms values of force coefficients when plotted against ellipticity of orbit. Since these sudden jumps occur between two envelope curves [2], the author’s hypothesis is that there are two solutions (states) that characterise the wake flow, and these represent changes in the structure of flow patterns. These jumps were investigated for several cases at different Reynolds numbers, and at different amplitudes of the in-line component of orbit [3]. A pre- and post-jump study also incorporated energy transfer, limit cycle curves, phase angle differences, and flow patterns [4]. All showed sudden switches, in agreement with the forced transverse results of [6] and [8]. This study attempts to gather further evidence of wake behaviour through flipping the solutions, as this has been recommended as one method of confirming the presence of two solutions [5].

2 Governing Equations and Numerical Method The dimensionless governing equations for an incompressible constant property Newtonian fluid flow around an orbiting circular cylinder are the two components of the Navier—Stokes equations, the continuity equation and pressure Poisson equation written in a non-inertial system fixed to the cylinder: @u @u @u @p 1 2 Cu Cv D C r u a0x ; @t @x @y @x Re

(1)

@v @v @p 1 2 @v Cu Cv D C r v a0y ; @t @x @y @y Re

(2)

@u @v C D 0; @x @y

@u @v @D @u @v r 2p D 2 @x @y @y @x @t DD

(3) (4)

In these equations r 2 is the 2D Laplacian operator, x; y are Cartesian co-ordinates, u; v are the x; y components of velocity in the system fixed to the cylinder, a0x ; a0y are the components of cylinder acceleration, p is the pressure, D is dilation. Here Re is the Reynolds number, Re D Ud= where d is the cylinder diameter, U is the free stream velocity and is the kinematic viscosity. Although in Equation (4) the dilation D D 0 by continuity (3), I retain its partial derivative with respect to time to reduce numerical errors. Equations (1), (2) and (4) will be solved while the continuity equation (3) is satisfied at every time step. No-slip boundary condition is used on the cylinder surface for the velocity and a Neumann-type condition is used for pressure p. A potential flow distribution is assumed far from the cylinder.

State Curves and Flipping for an Orbiting Cylinder at Low Reynolds Numbers

113

Boundary fitted coordinates are used to impose the boundary conditions accurately. Using unique, single-valued functions, the physical domain bounded by two concentric circles can be mapped into a rectangular computational domain where the spacing is equidistant in both directions. In the physical domain logarithmically spaced radial cells are used, providing a fine grid scale near the cylinder wall and a coarse grid in the far field. Using the mapping functions, not specified here, the governing equations and boundary conditions are transformed into the computational plane. The transformed equations are solved by using the finite difference method. For further details see [1]. The 2D code developed by the author has been extensively tested against experimental and computational results for a stationary cylinder and good agreement has been found [1]. The code was extended first for an oscillating and then for an orbiting cylinder. For this study the dimensionless time step was 0.0005 and the number of grid points 301 177. For all Re investigated in this study .Re D 120–180/ the solution was grid independent. The ratio of the radius of the outer computational domain and cylinder radius was 40. Figure 1 shows the flow arrangement. The motion of the centre of the cylinder with unit diameter is specified as follows: x0 .t/ D Ax cos .2f t/ ; y0 .t/ D Ay sin .2f t /

(5)

where f is the dimensionless oscillation frequency, Ax ; Ay are the dimensionless amplitudes of oscillations in x and y directions, respectively. In Fig. 1 U is the free stream velocity. Here the frequencies in the two directions are identical, which for nonzero Ax ; Ay amplitudes gives an ellipse, shown in the dotted line in the figure. If one of the amplitudes is zero, in-line or transverse oscillation is obtained. Ax alone yields pure in-line oscillation, and then as Ay is increased, the ellipticity e D Ay =Ax increases to yield a full circle at e D 1. The negative sign in y0 in Equation (5) makes the cylinder orbit clockwise (clw); by changing this sign of y0 an anticlockwise (aclw) orbit is obtained.

y

U

O

P1 1

d=

Ay

Fig. 1 Cylinder in orbital motion

Ax

x

P2

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3 Computational Results During each set of computations Re and Ax are fixed and f is kept constant at some percentage of Strouhal number St0 (the frequency of vortex shedding from a stationary cylinder at that Re). In this study this percentage was between 70–105 to ensure lock-in at moderate oscillation amplitudes. An interesting phenomenon was observed when looking at the time-mean value (TMV) and root-mean-square (rms) values of lift, drag and base pressure coefficients for an orbiting cylinder in a uniform flow. Abrupt jumps were found when these values were plotted against ellipticity e with Re and Ax kept constant, [3]. A typical example for the TMV of lift coefficient for both clockwise and anticlockwise direction of orbit is shown in Fig. 2a for Re D 160; Ax D 0:4; f D 0:85St0 . The filled triangles show results for a cylinder orbiting anticlockwise (aclw in the figure). Note that there are two envelope curves, which are roughly parallel with each other and of negative slope. On the other hand, the empty squares in Fig. 2a show results for a clockwise (clw) orbit, with the other parameters unchanged. The two envelope curves can be seen, again roughly parallel, but the slope is positive, and they are a mirror image of the envelope curves of the cylinder orbiting anticlockwise. Although it cannot be seen well at small e values in the figure, there are eight jumps or switches in state. For both directions of orbit the jumps occur at the same ellipticity values and computational points for the two cases lie on different state curves except for the values near e D 0. In all calculations made so far, CLmean has shown this pattern. Time histories of CLmean before and after the jumps are substantially different, [2]. The TMV and rms of drag and base pressure, further the rms of lift, behaved differently from CLmean , characterised by two state curves which are not parallel but intersect each other at e D 0. A typical example is shown in Fig. 2b. The main parameters (Re, Ax and f ) are the same as in Fig. 2a. From the sets of computations, it is clear that the two pairs of envelope or state curves are independent of the direction of orbit. Here the computational points belonging to the same e values coincide with each other and thus naturally lie on the same envelope curve. This is reassuring in two ways: (1) The code produces the same time-mean and rms results for two a

b

Re=160; Ax=0.4; 0.85St0

Re=160; Ax=0.4; 0.85St0

0.7 1.2

0.5 1

0.1 –0.1 0

0.2

0.4

0.6

0.8

1

1.2

–0.3

CLrms

CLmean

0.3

0.8 0.6 0.4

–0.5

0.2 0

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Fig. 2 Time-mean and rms values of lift versus ellipticity for clockwise (clw) and anticlockwise (aclw) direction of orbit .Re D 160I Ax D 0:4; f D 0:85St0 D 0:15997/


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different situations represented by the two directions of orbit, and this confirms that the code is consistent, and (2) the existence of envelope curves is proved by results obtained for two different cases. This finding also supports the idea that there are two states or solutions and the solution jumps from one state to the other and back. To sum up the findings, there appear to be two states between which the solution switches which indicates a strong possibility of bifurcation. The two solutions can be obtained (a) by using different initial conditions or (b) by flipping the solution. It was shown in [4] that changing the initial condition for the orbiting cylinder yields two different solutions. With flipping, if both states are obtained, this is additional evidence for a two-state solution. For this purpose the solution for a circular cylinder (stationary, moving in-line, or in orbital motion) placed in an otherwise uniform flow is flipped. This is done by replacing every quantity (velocity and pressure fields) by its mirror image values, at one instant . ! 2 / without changing the cylinder motion, where is the polar angle. Time-histories and limit cycles were plotted in order to compare pre-flip and post-flip solutions. Two types of limit cycles were produced: one for two components of flow velocity at a point, and the other for drag and lift coefficients. To check that the code for flipping was effective, the least complicated cases were attempted first.

3.1 Flipping for a Stationary Cylinder Computations were carried out for a stationary cylinder at Re D 180 for the dimensionless time interval of [0, 500] and the solution was flipped at t1 D 250 when the flow was already periodic (limit cycle). Time histories of lift and drag coefficients and those of u and v velocity components were stored at points P1 (2,1) and P2 .2; 1/, as these points have been shown to be reliable for experimental measurement of velocity signals (see [9]). Points P1 and P2 , shown in Fig. 1, are mirror images of each other and are located in the wake of the cylinder on the physical plane, where the origin of the coordinate system is fixed to the centre of the cylinder and coordinates are made dimensionless by the cylinder diameter d . Time history and limit cycle curves for velocity components at these points, i.e. .u1 ; v1 /; .u2 ; v2 / and limit cycle curves for force coefficients .CD ; CL / were plotted before and after the flip. It was found that: – All three limit cycle curves mentioned remain unchanged after flipping. – There was an approximately 180ı phase shift in CL .t/ at flipping and practically no phase shift in CD .t/. – For a stationary cylinder the shape of lift and drag coefficient signals are regular and this feature of the solution is preserved after the flipping as well. – Due to symmetry in the position of points P1 and P2 , limit cycle curves for the velocity components are mirror images of each other, i.e. .u1 ; v1 / D .u2 ; v2 / and .u2 ; v2 / D .u1 ; v1 /. All these expected results serve to show that the code works well. Due to lack of space no figures are included for the stationary cylinder.

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3.2 Flipping for In-Line Oscillation Computations were carried out for a cylinder oscillating in-line at Re D 180 with Ax D 0:3 amplitude and f D 0:9 St0 D 0:1737 frequency. Flipping was carried out when the oscillating cylinder reached its furthest downstream position .x0 D Ax /. In this case, velocity signals were measured at points P1 and P2 , which are fixed to the oscillating cylinder. In this way, the velocity components at the two points are relative velocities measured in the coordinate system fixed to the cylinder. – Figure 3 shows the CL .t/ signal around the flip .t1 D 247:5535/. The shape of the signal, after a short transitional period, is reversed, i.e. the more rounded peaks switch from bottom to top. This is evidence for the existence of two solutions. – Here, all limit cycle curves change with the flip. Still, some symmetries can be found between quantities before .t < t1 / and after .t > t1 / the flip, e.g. .u1 ; v1 j t < t1 / D .u2 ; v2 j t > t1 / (see Fig. 4a, b) .u2 ; v2 j t < t1 / D .u1 ; v1 j t > t1 / (see Fig. 4c, d) .CD ; CL j t < t1 / D .CD ; CL j t > t1 / (see Fig. 5) Figure 5 shows the relationship between CL and CD through the flip. The thin and thick closed curves show the .CD ; CL / limit cycle curves before and after the flip, respectively. Arrows show the orientation of the curves. The thin straight line represents the flip, when the solution jumps between the two states.

1.5

1

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−0.5 −1 −1.5 230

235

240

245

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255

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Fig. 3 Time history of lift coefficient in the vicinity of the flip

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Fig. 4 Limit cycles for in-line cylinder oscillation .Re D 180; Ax D 0:3I f D 0:1737I t1 D 247:5535/

Fig. 5 Relationship between CD and CL near flip (parameters as in Fig. 4)

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3.3 Flipping for Orbiting Cylinder To break the symmetry of in-line motion an orbiting cylinder was investigated, although due to lack of space we cannot go into detail here. A disturbance to in-line motion, the amplitude of transverse oscillation Ay , is chosen to be much smaller than Ax . In this way we expect that the symmetry features obtained for the cylinder oscillating in-line are not much distorted. The main parameters of the investigated case are: Re D 160I Ax D 0:3; Ay D 0:012 and f D 0:9 St0 D 0:16938; t1 D 247:5535. – The CL .t/ signal around the flip is reversed, similarly to the one shown in Fig. 5 after a short transitional period; i.e. the more rounded peaks switch from bottom to top. – Limit cycle curves show similar features, with the difference that relations mentioned for limit cycle curves in Section 3.2 are just approximately true. – Limit cycle .CL ; y0 / alters to a near-mirror image with the flip, while .CD ; x0 / hardly alters at all.

3.4 Effect of Flipping on Time-Mean and rms of Force Coefficients

CLmean

For all investigated cases the cylinder was orbiting in clockwise direction. Sets of computations were performed to investigate the effect of flipping on the TMV and rms values of different force coefficients. Out of the four sets investigated, two patterns have been identified. Representatives are shown in Figs. 6 and 7, where the CLmean curves are plotted against ellipticity e. The main feature of the first pattern (see Fig. 6) is that solutions flipped when the cylinder position is characterised by x0 x0max (‘3 o’clock’ position) roughly correspond to the solutions belonging to the anticlockwise direction of orbit (see also Fig. 2a), while keeping the other

0.2 0.15 0.1 0.05 0 −0.05 0 −0.1 −0.15 −0.2 −0.25 −0.3

0.1

0.2

0.3

e

0.4

0.5

aclw

0.6

t1=247.5535 x0≈x0max (3 o’clock)

flip

Fig. 6 Flipped and unflipped TMV of lift versus e .Re D 160; Ax D 0:3; f D 0:9 St0 D 0:16938/

CLmean

State Curves and Flipping for an Orbiting Cylinder at Low Reynolds Numbers 0.65 0.55 0.45 0.35 0.25 0.15 0.05 −0.05 0 −0.15 −0.25 −0.35

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e

clw

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t1=248.484 y0=y0max (12 o’clock)

flip

Fig. 7 Flipped and unflipped TMV of lift versus e .Re D 160; Ax D 0:4; f D 0:85 St0 D 0:15997/

0.65

CLmean

0.45 0.25

t1=248.484 y0=y0max (12 o’clock)

0.05 –0.15

0

0.2

0.4

0.6

1

0.8

1.2

–0.35

e

clw

flip back

Fig. 8 Original and doubly flipped TMV of lift versus e (as in Fig. 7)

parameters (Re; Ax and f ) unchanged. This means that CLmean has negative slope, and that the flipped results are basically complementary to the values belonging to the case before flipping. The discrepancy tends to become larger with increasing e, but at times returns to near zero. If the solution is flipped once more (flipped back), however, we obtained an almost perfect reproduction of the original curve after two flippings! Although only CLmean versus e is shown here, similar results were obtained for the other TMV and rms values. Figure 7 shows the other characteristic pattern found belonging to flipping time when the cylinder position is characterised by y0 D y0max (‘12 o’clock’ position) Interestingly in this case the flipped solutions approximate the results belonging to the clockwise direction of orbit. The location of the jumps is unchanged and the flipped results are complementary to the unflipped solutions (i.e. can be found on the other state curve). In this respect the effect of flipping is very similar to that of changing the initial conditions for the cylinder motion [4]. The other surprising thing is that the flipped solution reproduces the state curves very accurately over the whole investigated e domain. The flipped solution was flipped back; Figure 8 shows an almost perfect reproduction of the original curve after two flippings. At this stage it is unclear what kind of mechanism leads to either pattern 1 or pattern

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2. Hence further investigations are needed. It seems, though, that the position of the cylinder at the time when the flipping takes place has a crucial effect. Results for both patterns, however, give some extra evidence for the existence of a double solution, which seems as if it might be a case of bifurcation.

4 Conclusions The effect of flipping on flow features for a stationary, oscillating or orbiting cylinder in a uniform stream was studied. Limit cycle and time history curves were investigated, and results for a cylinder either stationary or in in-line motion gave evidence for the existence of two solutions (states). Simulations of an orbiting cylinder with small cross-wise amplitude further supported this conclusion. The time-mean and rms values of force coefficients were investigated versus ellipticity e for a cylinder orbiting clockwise, and two patterns were identified: (1) the flipped solution approximates the solution belonging to the anticlockwise orbit, discrepancy increasing with e, or (2) the flipped solution gave a very accurate complementary solution to the clockwise orbit, even when double-flipped. Further research is needed to explain why two patterns appear, and to further clarify the phenomenon causing sudden changes in time-mean and rms values of force coefficients. POD analysis is planned to identify the type of bifurcation. Acknowledgements The support provided by the Hungarian Research Foundation (OTKA, Project No. T 042961) is gratefully acknowledged. The author also thanks Prof. D. Barkley of Warwick University for his valuable advice, and Mr. S. Ujvárosi for his help in figure preparation.

References 1. Baranyi, L., Computation of unsteady momentum and heat transfer from a fixed circular cylinder in laminar flow. Journal of Computational and Applied Mechanics 4(1) (2003) 13–25. 2. Baranyi, L., Numerical simulation of flow past a cylinder in orbital motion. Journal of Computational and Applied Mechanics 5(2) (2004) 209–222. 3. Baranyi, L., Sudden jumps in time-mean values of lift coefficient for a circular cylinder in orbital motion in a uniform flow. In: 8th International Conference on Flow-Induced Vibration, Eds. E. de Langre and F. Axisa, Paris, II (2004) 93–98. 4. Baranyi, L., Energy transfer between an orbiting cylinder and moving fluid. In: Proceedings of the ASME Pressure Vessels and Piping Conference 9 (2007) 829–838. 5. Barkley, D., Private Communication, University of Warwick, December, 2005. 6. Blackburn, H.M., Henderson. R.D., A study of two-dimensional flow past an oscillating cylinder. Journal of Fluid Mechanics 385 (1999) 255–286. 7. Didier, E. Borges, A.R.J., Numerical predictions of low Reynolds number flow over an oscillating circular cylinder. In: Conference on Modelling Fluid Flow, Eds. T. Lajos and J. Vad, Budapest (2006) 165–172. 8. Lu, X.Y., Dalton, C., Calculation of the timing of vortex formation from an oscillating cylinder. Journal of Fluids and Structures 10 (1996) 527–541.


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9. Koide, M., Kubo, Y., Takahashi, T., Baranyi, L., Shirakashi, M., The vibration response of a cantilevered rectangular cylinder in cross-flow oscillation. Journal of Fluids Engineering 126, Trans. ASME (2004) 884–887. 10. Stansby, P.K., Rainey, R.C.T., On the orbital response of a rotating cylinder in a current. Journal of Fluid Mechanics 439 (2001) 87–108. 11. Williamson, C.H.K., Hess, P., Peter, M., Govardhan, R., Fluid loading and vortex dynamics for a body in elliptic orbits. In: Conference on Bluff Body Wakes and Vortex-Induced Vibration, Eds. P.W. Bearman and C.H.K. Williamson, Washington, DC (1998) 1–8.


Global Low-Frequency Oscillations in a Separating Boundary-Layer Flow U. Ehrenstein and F. Gallaire

Abstract A separated boundary layer flow at the rear of a bump is considered and two-dimensional flow states at increasing Reynolds numbers are computed using a nonlinear continuation procedure for the stationary Navier–Stokes system. The global instability analysis of the steady states is performed by computing two-dimensional temporal modes. The analysis reveals non-normal modes which are able to describe localized initial perturbations associated with large transient energy growth. At larger time a global low-frequency oscillation is found accompanied with periodic regeneration of the flow perturbation inside the bubble, as the consequence of non-normal cancellation of modes. The initial condition provided by the optimal perturbation analysis is applied to Navier–Stokes time integration and is shown to trigger nonlinear ‘flapping’ typical for separation bubbles. Keywords Boundary-layer separation Global instability Non-normality

1 Introduction There is general evidence that laminar detached boundary layers are likely to undergo two-dimensional low-frequency global oscillations [6] which have been observed both in experiments, whether the separation bubble is triggered by leadingedge geometries [4] or by adverse pressure gradients [8, 11]. The physical mechanisms at the origin of this type of instability which occurs above a critical Reynolds number are only partially understood. For instance, the frequencies associated with possible transition from convective to absolute instability of local velocity profiles U. Ehrenstein () ´ Aix-Marseille Université, 49, Rue Joliot-Curie, F-13384 Marseille Cedex 13, France IRPHE, e-mail: [email protected] F. Gallaire Lab. J. A. Dieudonné, Université de Nice-Sophia Antipolis, Parc Valrose, F-06108 Nice Cedex 02, France e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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with a certain amount of reverse flow appear to be higher than the typical frequencies of the oscillations known as ‘flapping’ cf. [9, 13]. By providing some evidence for the appearance of secondary recirculating flow regions at the rear of the main bubble, it has been conjectured [5, 16], that topological flow changes might be responsible for the overall instability behaviour. This possibility has been explored in [10] for a separated flow induced by a bump mounted on a flat plate but no clearcut confirmation of topological flow changes could be given, in the absence of a basic state above criticality. In this work we readdress the low-frequency oscillations for an elongated separation bubble induced by the bump geometry which has already been considered in [10]. We focus on the two-dimensional global instability behaviour. The threedimensional transverse instability characterized by a global steady and weakly growing eigenmode has recently been analyzed in [7]. The resulting longitudinal instability first takes place when increasing the Reynolds number, which has also been observed in backward-facing step flows [2, 3]. However, the low-frequency fluctuations which lead to an overall motion of the separation bubble is the dominant instability mechanism at higher Reynolds numbers. The analysis combines a quasi-Newton approach to determine steady states for Reynolds numbers above criticality and a numerical method suitable for computing two-dimensional temporal modes. Section 2 is devoted to the description of the numerical tools. The global stability results are discussed in Section 3 and an optimal perturbation analysis is performed in Section 4. Optimal initial conditions are then used in the direct numerical simulation procedure and are shown to produce low-frequency ‘flapping’ described in Section 5. Some conclusions are provided in Section 6.

2 Numerical Tools The flow domain is 0 x L; .x/ y H , with .x/ the lower boundary containing the bump which has already been considered in [10] (cf. Fig. 1 for the geometry). The height H has been chosen large enough in order to recover uniform flow, the Navier–Stokes system being made dimensionless using the displacement thickness ı at inflow, where the Blasius profile is described, and the uniform free stream flow velocity U1 . The dimensionless bump height is h D 2 and the mapping xN D

2 x 1; L

yN D

1 .1 /y C H

with D

H C H

(1)

transforms the domain into Œ1; 1 Œ1; 1. A Chebyshev–Chebyshev collocation discretization is used for the transformed system, together with a stretching in the wall-normal direction which redistributes the collocation points in order to take into account the boundary-layer structure.

Global Low-Frequency Oscillations in a Separating Boundary-Layer Flow

125

a

b

c

0

x

25

150

Fig. 1 Streamlines of flow states at (a) Re D 510, (b) Re D 620, (c) Re D 670

Here, we focus on nonlinear equilibrium states of the stationary Navier–Stokes system for increasing Reynolds number Re D ı U1 = and hence the flow velocity u D .u; v/ and pressure p are solution of

f .u; p; Re/ D .u r/u rp C .1=Re/r 2 uI r u D 0:

(2)

Homogeneous Neumann boundary condition for the flow field u are imposed at outflow x D L, whereas at y D H uniform flow u D .1; 0/ is prescribed and the no-slip condition applies on the wall boundary. At inflow the Blasius profile .U.y/; 0/ is imposed. The Chebyshev–Chebyshev discretization is efficient in terms of precision versus grid-size. The problem of spurious pressure modes associated to a Chebyshev discretization in the square is overcome using extra-conditions, by imposing the continuity of the normal derivative of the pressure at the corners and by eliminating the four modes for which the gradient vanishes due to the Chebyshev discretization (cf. [12]). A quasi-Newton method, here the Broyden rank-one update procedure (cf. [15]), is considered to solve the system (2) for the steady state by adding the Reynolds number as a parameter in an arc-length continuation procedure. The linear systems to be solved during the iterations in the quasi-Newton approach involve the Jacobian matrix A.u; Re/ D D.u;p/ f .u; Re/

(3)

which is evaluated at an initial guess of the flow field and a QR decomposition is performed. The rank-one update of the decomposition at successive iterations may easily be performed with little extra computational cost (cf. [15]). Once the solution converged to a steady state .us ; ps /, its stability is computed by considering twodimensional temporal modes u D u.x; O y/e i !t ;

p D p.x; O y/e i !t :

(4)

126


Taking into account in the Jacobian evaluated at a steady state us that the flow perturbation is zero at inflow, the modes are solution of the generalized eigenvalue problem i !Bq D A.us ; Re/q (5) with q D .u; O p/ O and Bq D .u; O 0/. The resulting large eigenvalue problem is solved using large-scale Krylov subspace projections together with the Arnoldi algorithm similar to the approach used in [1]. In all the computations the length of the computational domain is L D 300 and the height H D 30. The length of the box proved to be large enough to minimize effects of the box-size on the stability results. The modes are discretized using 250 collocation points in x and 40 collocation points in y and a Krylov subspace with dimension m D 1;600 has been considered.

3 Basic States and Global Instability Analysis The basic states at different Reynolds numbers are depicted in Fig. 1, the recirculation length increasing with the Reynolds number. In [10] no steady state could be obtained using time integration of the Navier–Stokes system above a Reynolds number of 610 and it has been speculated whether the observed global oscillations were somehow connected with topological flow changes [16]. With the present nonlinear continuation procedure flow states are computed for higher Reynolds numbers in the unstable range. The results to be discussed hereafter show that the flow at Re D 620 is indeed unstable. But even at Re D 670 (cf. Fig. 1c) there is no evidence of a change in topology. Once a flow state obtained, its stability is computed with the method outlined in Section 1. Figure 2a shows the spectrum at Re D 590: There are several weakly unstable modes and this Reynolds number hence appears to be slightly above the margin of instability. In Fig. 2b only the unstable parts of the spectrum are depicted for comparison at Re D 590; 620, the amplification rates increasing with the Reynolds number. The modes labelled from 1 to 6 are shown in Fig. 3, Fig. 3a depicting the streamlines of the steady state at Re D 590. One observes that the modes originate approximately at the center of the recirculation bubble. While the mode labelled .1/ in Fig. 2a at a lower frequency reaches the outflow boundary (cf. Fig. 3b), the modes .2/ to .6/ with equally spaced frequencies have the same type of structure, the spatial spreading of the modes decreasing with increasing frequency. The similarity of the eigenmode structure is a typical feature of non-normal operators and the next section will describe the implication of this.

4 Optimal Growth When addressing the possibility of growth in a flow system, the notions of optimal initial condition and non-normality of the underlying operators are essential [14]. We are looking for initial disturbances u0 that maximize the energy at time t


a

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G.t/ D max u0 ¤0

jju.t/jj2E jju0 jj2E

(6)

and a convenient form of this expression can be obtained by expanding the solution P in terms of the generalized eigenmodes u.t/ D N O l with uO l solution of lD1 l .t/u (5). Hence the flow dynamics is described by d D ƒ ; dt

.0/ D 0 ;

(7)

where is the vector of expansion coefficients and ƒ is a diagonal matrix whose elements are given by ƒl D i !l . The flow perturbation energy in this basis is jjujj2E D jjF exp.ƒt/ 0 jj22 , where F is the R Cholesky factor of the Hermitian energy measure matrix M with entries Mij D uO H O j dxdy. Hence, the maximum growth i u expressed in the basis of eigenmodes reads G.t/ D jjF exp .ƒt/F1 jj22 :

(8)

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a 0

x

300

b

c

d

e

f

g

Fig. 3 (a) Streamlines of basic state at Re D 590. (b–g) Streamwise velocity components of eigenfunctions corresponding to eigenvalues labelled (1)–(6) in Fig. 2

and the largest growth at time t is given by the largest singular value of F exp .ƒt/F1 and the optimal initial condition 0 is the corresponding right singular vector providing the optimal initial flow condition u0 through the eigenmode expansion. The optimal energy gains G.t/ are depicted in Fig. 4, for Re D 590 and Re D 620. The large Krylov subspace procedure (with m D 1;600) gives rise to a set of eigenvalues and its convergence in terms of optimal growth dynamics has been assessed. At Re D 590 for instance the results with N D 337 modes and the gain obtained for a much lower truncation at N D 150 (starting with the most unstable modes) are very close. One observes a fast initial energy growth of a magnitude of almost 109 followed by a global cycle with a period close to 200. This global undulation is accompanied by a weak energy growth due to the amplification rates of the individual modes. Inspecting Fig. 2, one observes that the real parts of the unstable eigenvalues in the right half of the spectrum (the modes labelled from (2) to (6)) are distant of about ı 0:03 and the structures of the corresponding eigenmodes are similar. The superposition of these modes as provided by the optimal perturbation analysis gives rise to cancelling leading to the global period of 2=ı 200. Similar nonnormal effects have also been described in a recent work on a separated flow in a cavity-like geometry [1].


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1e+18 1e+16 1e+14

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t=0 0

x t = 100

250

t = 250 t = 320 t = 370 t = 470 t = 520

Fig. 5 Streamwise velocity component of the perturbation for increasing time from top to the bottom, starting with the optimal initial condition (Re D 590). The vertical line shows the location of the reattachment point

To illustrate the dynamical behaviour, the time evolution of the perturbation in the eigenmode system is depicted in Fig. 5, starting with the optimal initial condition which is mainly located in the vicinity of the rear part of the bump. The perturbation evolves along the plate as a localized wavepacket accompanied with a tremendous increase of energy and at approximately t D 250 it is leaving the recirculation bubble. At the same time it reappears in the the rear part of the bubble, which is visible at t D 320. The perturbation then evolves downstream (cf. structure at t D 470) before it reappears again in the bubble (cf. t D 520) and the cycle restarts. The modes responsible for the oscillating flow pattern are individually weakly unstable which leads to the overall growth shown in Fig. 4 for t > 200.

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5 Direct Numerical Simulation Dynamics Focusing on the relation between optimal energy growth and the low-frequency oscillations observed in previous investigations of the separation bubble, the optimal initial condition has been considered in the direct numerical simulation (DNS) procedure of the Navier–Stokes system used in [10]. The flow at Re D 590 is considered, which according to the present analysis is slightly supercritical. As shown in [10], at this Reynolds number it was however possible to converge by time marching to a steady state using the DNS procedure. Weak stabilization effects might be attributed to the influence of the outflow boundary of the shorter domain considered in the DNS (with L D 200), where a classical convection condition is applied. The streamwise direction is discretized using fourth-order finite differences with 1,024 equidistant grid points. In y the region up to H D 80 is covered (with 97 Chebyshev-collocation points) for the solution to be uniform in the upstream region, a simple linear coordinate transform being employed. The gain in energy being expected to be close to 109 , the optimal initial condition has been affected with a small maximum amplitude of 5 105 (however above the residual noise of order of 106 in the time-marching towards the steady basic state ub ). The energy of the perturbation u0 D uub has been integrated in the whole domain and the gain is depicted in Fig. 6. Up to t D 200 the energy growth follows closely the curve provided by the eigenmode system before nonlinear saturation occurs. The integration has been pursued up to t D 3;000 and the flow is seen to remain in a nonlinear state. When entering the nonlinear saturation, the oscillations visible in the global energy curve are somehow reminiscent of the global oscillations in the envelope curve provided by the modes. The amplitude of the nonlinear flow perturbation, by simply subtracting the base flow, along the plate at increasing time has been computed and the results are shown in Fig. 7. At t D 100

1e+16 1e+14 1e+12

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Fig. 8 Power spectrum in time 500 t 3;000 of streamwise velocity component at x D 40; y D 1

0.0003 0.00025

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f

the perturbation is similar to that depicted in Fig. 5 for the dynamical system formed with the global modes. At t D 300 a regeneration behaviour of the perturbation is visible upstream the previous wavepacket at t D 250. The perturbation is highly nonlinear and is hence far from being a superposition of eigenmodes. In [10] low-frequency oscillations of the bubble have been reported at the supercritical Reynolds number Re D 650. In the present analysis similar fluctuations are observed as a consequence of the initial optimal perturbation at Re D 590. Figure 8 depicts the amplitude of the Fourier transform for the streamwise velocity profile at x D 60; y D 1, that is at a position which corresponds approximately to the center of the steady bubble. The low frequency f D ı=.2/ 0:005

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0


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Fig. 9 Instantaneous vorticity at t D 1;500

corresponds to the non-normal ‘beating’ frequency in the eigenmodes system and indeed the spectrum exhibits peaks in this frequency range, the nonlinear flow being highly aperiodic. Similar to the results reported in [10] a second dominant range of higher frequency is visible. Figure 9 shows an instantaneous vorticity field illustrating the vortex-shedding behaviour resulting from the two-dimensional global oscillations.

6 Conclusion In the aim of shedding new light on the phenomenon of global ‘flapping’ in separated wall-bounded flows, the two-dimensional flow over a bump has been considered, by computing nonlinear states of the stationary Navier–Stokes system. When increasing the Reynolds number, no topological changes in the flow structure are detected. The separation bubble becomes unstable with respect to two-dimensional temporal modes starting at the center of the bubble and extending more or less downstream the reattachment point. The corresponding eigenvalues cross the axis of marginal instability almost simultaneously and they exhibit equally spaced frequencies. By performing an optimal initial disturbance analysis, a periodic regeneration mechanism of the perturbation at the rear of the bubble, resulting from non-normal cancelling of eigenmode structures, has been detected. The time-integration of the Navier–Stokes system starting with the optimal initial condition leads to nonlinear aperiodic flow with however the reminiscence of a global low-frequency oscillation of the bubble. This gives some strength to the conjecture, that non-normal interaction of global modes is at the origin of the ‘flapping’ behaviour. Acknowledgements Parts of the computations have been performed on the NEC-SX8 of the IDRIS, France. The authors would like to thank Matthieu Marquillie for providing the DNS code.

References ˚ 1. E. Akervik, J. Hœpffner, U. Ehrenstein, and D. Henningson. Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech., 579:305–314, 2007. 2. D. Barkley, M. Gomes, and D. Henderson. Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech., 473:167–189, 2002.


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3. J.-F. Beaudoin, O. Cadot, J.-L. Aider, and J. Wesfreid. Three-dimensional stationary flow over a backward-facing step. Eur. J. Mech. B/Fluids, 23:147–155, 2004. 4. N.J. Cherry, R. Hiller, and M.P. Latour. Unsteady measurements in a separating and reattaching flow. J. Fluid Mech., 144:13–46, 1984. 5. U. Dallmann, Th. Herberg, H. Gebing, W.-H. Su, and H.-Q. Zhang. Flow field diagnostics: Topological flow changes and spatio-temporal flow structure. AIAA Paper 95-0791, 1995. 6. A.V. Dovgal, V.V. Kozlov, and A. Michalke. Laminar boundary layer separation: Instability and associated phenomena. Prog. Aerospace Sci., 30:61–94, 1994. 7. F. Gallaire, M. Marquillie, and U. Ehrenstein. Three-dimensional transverse instabilities in detached boundary-layers. J. Fluid Mech., 571:221–233, 2007. 8. C.P. Häggmark, A.A. Bakchinov, and P.H. Alfredsson. Experiments on a two-dimensional laminar separation bubble. Phil. Trans. R. Soc. Lond. A., 358:3193–3205, 2000. 9. D.A. Hammond and L.G. Redekopp. Local and global instability properties of separation bubbles. Eur. J. Mech. B/Fluids, 17:145–164, 1998. 10. M. Marquillie and U. Ehrenstein. On the onset of nonlinear oscillations in a separating boundary-layer flow. J. Fluid Mech., 490:169–188, 2003. 11. L.L. Pauley, P. Moin, and W.C. Reynolds. The structure of two-dimensional separation. J. Fluid Mech., 220:397–411, 1990. 12. R. Peyret. Spectral Methods for Incompressible Flows. Applied Mathematical Sciences 148, Springer, New York, 2002. 13. U. Rist and U. Maucher. Investigations of time-growing instabilities in laminar separation bubbles. Eur. J. Mech. B/Fluids, 21:495–509, 2002. 14. P.J. Schmid and D.S. Henningson. Stability and Transition in Shear Flows. Applied Mathematical Sciences 142, Springer, New York, 2001. 15. J. Stoer and R. Bulirsch. Introduction to Numerical Analysis, second edition. Texts in Applied Mathematics 12, Springer, New York, 1993. 16. V. Theofilis, S. Hein, and U. Dallmann. On the origins of unsteadiness and three-dimensionality in a laminar separation bubble. Phil. Trans. R. Soc. Lond. A, 358:3229–3246, 2000.


Asymptotic Theory of Turbulent Bluff-Body Separation: A Novel Shear Layer Scaling Deduced from an Investigation of the Unsteady Motion B. Scheichl and A. Kluwick

Abstract A rational treatment of time-mean separation of a nominally steady turbulent boundary layer from a smooth surface in the limit Re ! 1, where Re denotes the globally defined Reynolds number, is presented. As a starting point, it is outlined why the ‘classical’ concept of a small streamwise velocity deficit in the main portion of the oncoming boundary layer does not provide an appropriate basis for constructing an asymptotic theory of separation. Amongst others, the suggestion that the separation points on a two-dimensional blunt body is shifted to the rear stagnation point of the impressed potential bulk flow as Re ! 1 – expressed in a previous related study – is found to be incompatible with a selfconsistent flow description. In order to achieve such a description, a novel scaling of the flow is introduced, which satisfies the necessary requirements for formulating a self-consistent theory of the separation process that distinctly contrasts former investigations of this problem. As a rather fundamental finding, it is demonstrated how the underlying asymptotic splitting of the time-mean flow can be traced back to a minimum of physical assumptions and, to a remarkably large extent, be derived rigorously from the unsteady equations of motion. Furthermore, first analytical and numerical results displaying some essential properties of the local rotational/irrotational interaction process of the separating shear layer with the external inviscid bulk flow are presented. Keywords Coherent motion Gross separation Perturbation methods Turbulent boundary layers

1 Introduction The rational description of break-away separation of a statistically steady and two-dimensional incompressible turbulent boundary layer flow past an impermeable rigid and smooth surface in the high-Reynolds-number limit represents a B. Scheichl () and A. Kluwick Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, Resselgasse 3/E322, A-1040 Vienna, Austria M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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B. Scheichl and A. Kluwick

long-standing unsolved hydrodynamical problem. Needless to say that an accurate prediction of the position of separation, in combination with the local behaviour of the skin friction, has great relevance for many engineering applications, where e.g. internal flows, like those through diffuser ducts, or flows past airfoils play a crucial role.

1.1 Problem Formulation and Governing Equations The picture of such flows near separation is sketched in Fig. 1. As a basic assumption, the suitably formed global Reynolds number Re is taken to be asymptotically large, Q Q ! 1; D Re1 ! 0: Re D UQ L= (1) Q and UQ denote, respectively, the (constant) kinematic viscosity of the Herein , Q L, fluid, a reference length, typical for the geometry of the portion of the surface under consideration, and a characteristic value of the surface slip velocity impressed by the limiting inviscid stationary and two-dimensional irrotational bulk flow, hereafter formally indicated by D 0. All flow quantities are suitably non-dimensionalised Q UQ , and the (uniform) fluid density. Let t, p, x D .s; n; z/, and u D .u; v; w/ with L, be the time, the fluid pressure, and the position and the velocity vector. Here u, v, and w are the components of u in directions of the natural coordinates s, n, and z, respectively, along, normal to, and projected onto the separating streamline S, given by n D 0, of the flow in the limit D 0. Furthermore, ue .s/ denotes the surface slip velocity in that limit. The origin s D n D 0 is chosen as the location S where S departs from the surface. Thus, S coincides with the surface contour for s 0. Also, note that S has, in general, a curvature of O.1/ for jsj D O.1/. In coordinate-free form the Navier–Stokes equations then are written as r u D 0;

(2a)

Dt u D rp C u;

Dt D @t C u r;

a

D r r;

(2b)

b

n

u

S

u

u

s

n n

s

u

S

S ue

S −s

ue

Fig. 1 Time-mean flow near (a) smooth separation (the dotted streamline indicates possible backflow), (b) separation due to stagnation of the bulk flow, cf. [5]. The inviscid limit of u is shown dashed, and the turbulent shear flow is indicated by a shading

Asymptotic Theory of Turbulent Bluff-Body Separation

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where r is the gradient with respect to x. They are subject to the common noslip condition u D 0 holding at the surface. As a well-known characteristic, the stationary Reynolds-averaged turbulent flow can be expressed in terms of the time-averaged motion. In the following we employ the conventional Reynolds decomposition of any (in general, tensorial) flow quantity q into its time-mean component q, here regarded as independent of z, and the (in time and space) stochastically fluctuating contribution q 0 , 1 !1

q.x; t; : : :/ D q.x; y; : : :/ C q 0 .x; t; : : :/; q D lim

Z

=2

q.x; t C ; : : :/ d: =2

(3) Herein the dots indicate any further dependences of q apart from x and t. Reynoldsaveraging of (2b) then yields the well-established Reynolds equations (in the case @z 0 of planar time-mean flow), r u D 0; Dt u D rp r u0 u0 C u;

(4a) Dt D u r:

(4b)

It is further presumed in the subsequent analysis, that all components of the Reynolds stress tensor u0 u0 are, in general, of asymptotically comparable magnitude (assumption of locally isotropic turbulence). Most important, we disregard any effects due to free-stream turbulence. That is, the turbulent motion originates from the relatively thin fully turbulent boundary layer adjacent to the surface, which near S passes into an accordingly slender separated free shear layer along S for s > 0.

1.2 Motivation From an asymptotic point of view, three outstanding contributions to the solution of the problem under consideration have to be mentioned. Sychev [11,12] was the first who elucidated the question of the asymptotic structure of the oncoming boundary layer by proposing a three-layer splitting of the latter, sufficiently far ahead of S . This scaling, however, is at variance with the classical finding of a two-tiered boundary layer that is found to hold for firmly attached flow only (see, for instance, the pioneering work by Mellor [3]). We start the outline of both formulations by noting that each of them adopts the familiar description of the viscous wall layer close to the surface; the same holds for the flow descriptions discussed subsequently. On top of that region the Reynolds shear stress u0 v0 asymptotically equals the (local) wall shear stress, given by the square of the skin friction velocity u , and the streamwise velocity component u satisfies the celebrated logarithmic law of the wall. By using the conventional notation, it reads u=u 1 ln nC C C C ;

nC D n u Re ! 1;

(5)

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where the well-known constants and C C are quantities of O.1/. The match of the wall region with the adjacent layer then shows that the expansion Œu; u0 v0 =u2 Œu0 ; T0 .s; / ŒU1 ; T1 .s; / C O. 2 /;

D n=ı;

(6)

holds in the latter. Here, ı is a measure for the thickness of that layer, and, by introducing the so-called slip velocity us , the gauge function is seen to satisfy the skin-friction law D u =us = ln Re;

d=ds D O. 2 /;

us .s/ D u0 .s; 0/:

(7)

In the classical two-tiered description of the boundary layer, cf. [3], it is assumed that in the fully turbulent main region the (positive) streamwise velocity ‘defect’ with respect to the external potential flow, ue u, is asymptotically small. In turn, u0 .s; / us .s/ ue .s/, and in the boundary layer limit the momentum balance (4b) reduces to a balance between the linearised convective terms and @n .u0 v0 / in leading order, showing that the boundary layer thickness ı is of O. /. In contrast, according to [11, 12] the expansion (6) holds in the additionally introduced middle layer which meets the requirement that the velocity defect ue u and, consequently, ue us are quantities of O.1/. Thus, in the boundary layer approximation to (4b) the convective terms balance both @n .u0 v0 / and the imposed (adverse) pressure gradient ue due =ds, such that the thickness ı of the middle layer is of O. 2 /. This wake-type flow structure then allows for a significant decrease in the wall shear stress according to (7) when us tends to zero as s ! 0 and, moreover, for the occurrence of flow reversal further downstream by adopting a local turbulent/irrotational interaction strategy (without the need of a specific turbulence closure). One readily finds that the gradients @n u in the viscous wall layer and the adjacent layer, described by (6), match on the basis of (5) provided that @ u0 0. Unfortunately, this again gives u0 .s; / us .s/ ue .s/ and thus contradicts the original assumption of a large velocity defect in the middle layer. That inherent mismatch of the wall layer and the wake region was first noted by Melnik [4], who used mixing length arguments, in the second work to be highlighted. Therefore, Sychev’s approach can hardly be accepted as a self-consistent theory. However, Melnik also proposed a non-classical initially three-tiered boundary layer where the outermost part plays the role of the aforementioned middle layer. But most important, and in striking difference to any previous asymptotic treatment of turbulent shear flows, in [4] the slenderness of the latter is measured by some small non-dimensional parameter, denoted by ˛, which is regarded to be essentially independent of Re. Melnik’s motivation for the resultant two-parameter matched asymptotic expansions of the flow quantities merely relies upon the observation that any commonly employed shear stress closure includes a small number (a most familiar example is the so-called Clauser ‘constant’ ˛ 0:0168 in the algebraic Cebeci–Smith model) which is seen to measure the boundary layer thickness if the velocity defect in the


139

fully turbulent flow regime is taken to be of O.1/. This idea has been followed up and substantiated by order-of-magnitude reasoning by Scheichl and Kluwick [6, 7], where it is shown to provide a sound basis for developing a self-consistent theory of turbulent marginal separation. On the other hand, it is found that Melnik’s theory cannot be extended in order to describe the global separation process due to two serious shortcomings: (i) the proposed flow structure is strongly associated with the adopted coupling ˛ 1=2 ln Re D O.1/, which is apparently inconsistent with the original assumption on ˛ and, hence, does not allow for a correct formulation of the gradual transition from attachment to separation of the flow inside the wall layer; (ii) the impressed potential flow does not exhibit a free streamline departing smoothly from the surface, in order to avoid a Goldstein-type singularity encountered by the boundary layer solution that is evidently unsurmountable by assuming a firmly attached external bulk flow, cf. [7]. A different viewpoint was taken up in the third contribution to be noticed, by Neish and Smith [5]. They considered the streamwise development of a classical small-defect boundary layer where the irrotational external flow is indeed presumed to be strictly attached; that is, it exhibits a rear stagnation point, see Fig. 1(b). Interestingly, this concept is fully consistent with the following important finding elucidated in the subsequent analysis: in the case of smooth inviscid flow detachment, as depicted in Fig. 1(a), the associated singular behaviour of the surface pressure immediately upstream of the (a priori unknown) position of S does not trigger a significant change in the order of magnitude of the (initially small) velocity defect, which would be necessary to render smooth boundary layer separation possible. Consequently, within the framework of classical turbulent boundary layer theory separation is suggested to occur asymptotically close to the rear stagnation point as Re ! 1. Unfortunately, however, it is not addressed satisfactorily in [5] whether and how the small velocity defect may rather abruptly become of O.1/ due to the retardation of the potential flow as the stagnation point S is approached in order to ensure an uniformly valid flow description. As pointed out in the first part of the present study, the inviscid vortex flow induced in the immediate vicinity of the stagnation point S indeed appears to hamper severely the construction of a self-consistent asymptotic theory. This finding represents the starting point for the subsequent analysis, where it is shown how the closure-independent asymptotic formulation of a turbulent boundary layer having a finite thickness of O.˛/, ˛ 1, as Re ! 1 and which may undergo marginal separation, see [7], can be adapted to that of massive separation. Unlike the theories presented in [4, 5], here the formal limit ˛ D Re1 D 0 corresponds to the required class of inviscid flows with free streamlines. Furthermore, we demonstrate how the asymptotic scaling of the (oncoming) flow, which in [7] was based on rather heuristic arguments from a time-averaged point of view, can be deduced by means of a multiple-scales analysis of the equations of motion (2). We commence the investigation by considering the evolution of the boundary layer immediately upstream of the surface position S , indicating inviscid separation.

140


2 Limitations of the Small-Defect Approach The case where the streamwise velocity defect in the fully turbulent main region of the boundary layer is small, say, ue u D O./, 1, is considered first. To be more precise, we assume that D , according to (6) and (7) (although the more general assumption = D O.1/, including o.1/, would not alter the following analysis substantially). Therefore, the boundary layer thickness ı is of O. / and expanded as ı= D 0 .x/ C 1 .x/ C : (8) By setting U1 =ue D F00 .s; /, D O.1/, the leading-order streamwise momentum equation, supplemented with appropriate boundary and matching conditions, then reads ue Œd.ue 0 /=dsF000 0 @s .u2e F00 / D u2e T00 ; F0 .s; 0/ D T0 .s; 0/ 1 D 0; F00 .s; 1/

D

F00 1 ln C O.1/; F000 .s; 1/

(9a) ! 0;

D T0 .s; 1/ D 0:

(9b) (9c)

We note, that in this connection primes denote derivatives with respect to . Also, it will prove convenient to integrate (9a) with respect to by using (9b), which gives u2e Œd.ue 0 /=dsF00 @s .u3e 0 F0 / D u3e .T0 1/:

(10)

Finally, evaluation of (10) at the boundary layer edge and subsequent integration from some value s0 < 0 to s < 0 yields dŒu3e 0 F00 .s; 1/=ds D u3e ;

Z s ˇ Ds ˇ u3e ./0 ./F00 .; 1/ˇ D u3e ./ d: Ds0

(11)

s0

In order to assess the assumption of a small velocity defect holding in the oncoming flow with respect separation, we analyse (9) in the limit s ! 0 for the two different cases indicated by Fig. 1a and b, respectively. Without adopting a specific turbulence closure, we begin the analysis by considering the first case.

2.1 Flow Slightly Upstream of Smooth Separation It is well known that, under rather general conditions concerning the flow in the stagnant or backflow region where s > 0 and n < 0,

ue .s/=ue .0/ 1 C 2k.s/1=2 C .10k 2 =3/ .s/ C O .s/3=2 ;

s ! 0 ; (12)


141

in the inviscid limit D 0, cf. [1], for instance. Here the non-negative parameter k parametrises the class of smoothly separating flows as it depends on the position of S on the body contour. It gives rise to a locally adverse and unbounded pressure gradient ue due =ds k.s/1=2 . Therefore, the question arises if the latter provokes a significant increase of the velocity defect in the oncoming boundary layer, which is required for a correct description of flow reversal further downstream. In order to keep the analysis as general as possible, we only assume that ue .s/=ue .0/ 1 C .s/ C ;

jd=dsj ! 1;

s ! 0 :

(13)

This singular behaviour is expected to provoke a considerable growth of the turbulent velocity scale u (and, in turn, of the fluctuations), expressed through a gauge function '.s/, F0 '.s/G./ C ;

T0 ' 2 .s/R./ C ;

' ! 1;

s ! 0 : (14)

From (11), (14), and the fact that ue in (13) admits a finite limit, there follows a (intuitively rather unexpected) decrease of the boundary layer thickness of the form 0 D=', where D is a (positive) constant. Also note, that the term @s .u3e 0 F0 / in (10) is bounded for s ! 0 . Since ue is bounded too, the first term in (10) asymptotically equals Du3e .0/ G 0 ./ d.ln '/=ds. As the velocity defect and, in turn, G 0 are non-negative, that expression tends to 1 for s ! 0 . Then ' is seen to be proportional to .s/1=2 , as (10) reduces to a balance between that negative term and u3e ' 2 .s/R./. The latter term, however, is non-negative, as is the Reynolds stress T0 in the oncoming flow. From this contradiction one then infers that F0 , T0 , and 0 are finite for s ! 0 . Consequently, inspection of (10) and (13), subject to condition (9c), shows that (14) is to be replaced by a sub-expansion of expansion (6), ŒF0 ; T0 ; 0 ŒF00 ./; T00 ./; 00 C .s/ŒF01 ./; T01 ./; 01 C :

(15)

Therefore, the velocity defect does not change its order of magnitude. One then concludes that, by specifying .s/ in (13) in accordance with (12), the small-defect formulation represents an inadequate description of a turbulent boundary layer approaching smooth separation. Note, that the same conclusion can be drawn for turbulent separation at a trailing edge under angle of attack, where the external velocity admits a square-root behaviour akin to (12). More generally spoken, (15) holds if ue admits a finite limit according to (13). We add that it has been demonstrated numerically in [8] that even in case of a rather sharp step-like decrease of ue .s/ the velocity defect characterised by F0 , T0 , and 0 , remains bounded. Summarising, it is possible to give a rather comprehensive answer to an interesting question raised by Degani [2], namely, how the small-defect structure responds to different limiting forms of ue .s/ as s ! 0 : apparently, the only scenario that is compatible with a change of magnitude of the velocity defect, as it is required for an asymptotic description of separation, is that of a boundary layer approaching a stagnation point of the (otherwise attached) flow in the inviscid limit D 0. This is exactly the picture of separation originally proposed by Neish and Smith [5].

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2.2 Flow in the Vicinity of a Rear Stagnation Point Close to a rear stagnation point, see Fig. 1 (b), the potential flow is linearly retarded as u c s, v c n, where s; n ! 0 and c is a positive constant. Then ue cs, in contrast to (13). Substitution of this relationship into (10) and (11) then predicts a growth of both the boundary layer thickness and the velocity defect, as expressed by (14). Specifically, ı 0 DŒ ln.s/1=2 .s/;

'DD

ı˚ 2Œ ln.s/1=2 .s/2 ;

s ! 0 ; (16)

where D again is a positive constant, cf. [2,5]. It then follows from (16) that (10) reduces to the equation G 0 ./ D R./ for D O.1/. Since (16) is incompatible with the inhomogeneous boundary conditions (9b) required by the match with the viscous wall layer, on top of the latter a sublayer where D O.' 2 / has to be introduced. However, as that flow region appears to behave passively with respect to the further analysis, it is disregarded here. As a consequence of the growth of , see (16), the boundary layer approximation ceases to be valid close to the stagnation point S when the distance s and ı are of comparable magnitude. From (8) it then follows that this region is characterised by suitably rescaled coordinates .X; Y / D .s; n/= , where

D .D /1=2 Œ.ln /=21=4 . The resulting asymptotic splitting of the flow is depicted in Fig. 2 (a). In the new ‘square’ domain II of extent the flow quantities are expanded in the form

u v p pS ; ; c c .c /2

1 X2 C Y 2 1 C Œ@Y ; @X ; P C O

X; Y; ; 2 ln ln2 (17)

where p S is the (time-mean) pressure in S . Here the magnitude of the velocity defect is still asymptotically small and varies only logarithmically with . As an important

Y; s; n

a

k=0

± II III −X; s

I

b

¿

k>0

S

II I

S ¿

S

Fig. 2 (a) Asymptotic flow splitting near rear stagnation point S: oncoming boundary layer I with emerging sublayer I0 , resulting ‘square’ region II with sublayer II0 (not considered in text), viscous wall layer III (increase of thickness proportional to 1=s, s ! 0, not discussed here), separating streamline S of the stagnant potential flow; the dotted lines indicate the connection to the regions not considered in the analysis. (b) Smooth inviscid separation from a (here symmetrical) cylindrical body for different values of k in (12); note the flow showing a cusp-shaped closed cavity which neighbours the attached flow characterised by a rear stagnation point S


143

implication, the presence of the logarithmic terms in (16) is seen to prevent the Reynolds stresses, which are of O. 2 = ln2 /, to affect even the perturbed flow in leading order. Indeed, substitution of (17) into the momentum Equation (4b) shows that the perturbation stream function .X; Y / and the pressure disturbance P .X; Y / satisfy the Euler equations, linearised about the stagnant potential flow, @X .X @Y / Y @Y Y D @X P;

X @XX C @Y .Y @X / D @Y P:

(18)

By introducing the vorticity ˝ D .@XX C @Y Y / , elimination of P in (18) yields the vorticity transport equation, .X @X Y @Y /˝ D 0. Finally, integration gives .@XX C @Y Y / D ˝.X Y /;

(19a)

expressing the well-known property of two-dimensional steady inviscid flows that the vorticity is constant along a streamline. The match with the oncoming boundary layer flow according to (14), (16), and (17) then fixes both the vorticity ˝ and the boundary conditions supplementing (19a), ˝ D G 00 ./;

D X Y;

(19b)

.X; 0/ D 0; G./=X ; 2

D O.1/;

(19c) X ! 1:

(19d)

Also, the reuse of the boundary layer coordinate introduced before in (19b) shows that the edge n D ı of the turbulent flow region II here is given by ı =.X /, see Fig. 2(a). Stated equivalently, the curve X Y D 1 disjoins the turbulent from the (approximately) irrotational external region as ˝ D 0 for 1. We seek a solution of the Poisson problem (19) for for X < 0, Y 0. That is, in the present investigation we do not take into account the ‘collision’ of the oncoming flow with that approaching S for s ! 0C , cf. Figs. 1(b) and 2(a). We set D p C h , where p .X; Y / is a particular solution of (19a)– (19c) and the homogeneous contribution h .X; Y / satisfies Laplace’s equation, .@XX C @Y Y /h D 0, subject to (19c). By defining G./ WD G./, 0, and using standard methods, one obtains after integration by parts and some manipulations 1 p D 2π

Z

1

0

Z

1

G ./ 1

1

jjY d d .jX j 0/: (20) Œ 2 .X /2 C .jjY /2

The function p .X; Y / is found to vary with R2 for R2 D X 2 C Y 2 ! 1 and fixed values of # WD arctan.Y =X /. On the other hand, p H./=X 2 for X ! 1 where and, in turn, the function H./ (which is not stated explicitly here) are kept fixed. Since H 6 G, however, h .X; Y / must behave as h ŒG./ H./=X 2 ;

X ! 1;

D O.1/;

(21)

144


such that satisfies (19d). An asymptotic investigation of Laplace’s equation then shows that h R2 ŒA cos.2#/ C B sin.2#/, where A and B are constants, is the only possible behaviour for R ! 1. Unfortunately, this relationship does not meet the required match with (21) as # ! π , X ! 1. Thus, the problem (19) has no solution. Therefore, the asymptotic picture of separation taking place close to a rear stagnation point, as proposed in [5], must be regarded as at least questionable. The formal inconsistency outlined before has not been addressed by Neish and Smith [5]. Apparently, this is due to the neglect of the logarithmic terms in (16) in their discussion of the match with the ‘square’ region II. In turn, they propose a vortex flow there which exhibits a velocity defect relative to the stagnating external flow of O.1/, in striking contrast to the expansion (17). Consequently, in [2, 5] both the magnitude of the velocities and the extent of the emerging region II are of 1=2 . Thus, the flow there is governed by the full Reynolds equations (1), rather than their linearised form (18). It is that fully nonlinear stage which prompted the authors of [2, 5] to conclude that separation would occur a distance of O. 1=2 / upstream of S . Also, it is not explained in these papers how the flow region II is transformed into a turbulent shear layer along the separated streamline S, which then coincides with the Y -axis, see Fig. 2(a). A further uncertainty is raised by another issue put forward in [5]: it is argued that the position of smooth flow detachment approaches the rear stagnation point if one considers the limit k ! 1 in (12). The flow situation for different values of k is sketched in Fig. 2(b), cf. [1]: from a topological point of view, the only candidate for a flow exhibiting free streamlines around a cylindrical body that neighbours the completely attached potential flow with a rear stagnation point S is the one which embeds a vanishingly small interior (cusp-shaped) cavity/eddy in the vicinity of S . However, it has not been demonstrated convincingly so far that such a solution is associated with correspondingly large values of k. We note that the class of inviscid flows having free streamlines is currently under investigation.

3 The Large-Defect Boundary Layer and Smooth Separation The picture of separation considered in [2, 5] is apparently not in accordance with experimental findings. In fact, separation from a cylindrical body takes place a relatively short distance downstream of the location of its maximum cross-section, even for very large values of Re. This finding, together with the serious difficulties discussed in the previous section, then strongly suggests to abandon the assumption of a small-defect boundary layer in favour of a flow description where a streamwise velocity deficit of O.1/ is stipulated. As outlined in the introduction, such an asymptotic concept that (i) surmounts the difficulties in the matching procedure due to the logarithmic velocity distribution (5) encountered in Sychev’s theory [11, 12], and (ii) is corroborated by any commonly used turbulence closure, has already been proven successful in the description of turbulent marginal separation, see [6, 7].


145

In this novel flow description the boundary layer thickness ı is measured by a small parameter ˛ which is independent of Re as Re ! 1. This most remarkable characteristic anticipates the existence of a turbulent shear layer of finite width with a wake-type flow, even in the formal limit ˛ ! 0, D 0, included in (1). In that limit the unsteady flow in the wake region is presumably not affected significantly by the periodically occurring well-known wall layer bursts. As will turn out, this characteristic allows for an investigation of some properties associated with the unsteady motion on the basis of (2).

3.1 The Slender-Wake Limit In the wake region the Reynolds stresses are quantities of O.˛/. Then the nonlinearities in the momentum Equation (2b) suggest the expansions Œu; v; w; p Œu0 ; 0; 0; p0 .s; N / C ˛ 1=2 Œu01 ; v01 ; w01 ; p10 .t; s; N; : : :/ ˚ C ˛ Œu2 ; v2 ; 0; p 2 .s; N / C Œu02 ; v02 ; w02 ; p20 .t; s; N; : : :/ C O.˛ 3=2 /;

(22a)

ı=˛ D ı0 .s/ C O.˛ 2 /:

(22b)

In (22a) a suitable shear layer coordinate N D n=˛ is introduced, and the dots indicate dependences on inner spatial and time scales, which are specified later. Inserting (22) into (4) then gives rise to the shear layer approximation p 0 .s; N / D p0 .s/; @s u0 C @N v2 D 0;

dp0 =ds D ue due =ds;

u0 @s u0 C v2 @N u0 D dp0 =ds @N .u01 v01 /:

(23a) (23b)

The Equations (23a) and (23b) govern the turbulent motion along the separating streamline S to leading order sufficiently far from S , i.e. for jsj D O.1/, see Fig. 1(a). They are subject to the wake-type boundary conditions v2 .s; 0/ D u01 v01 .s; 0/ D 0; u0 s; ı0 .s/ ue .s/ D u01 v01 s; ı0 .s/ D 0: (23c) By excluding the apparent trivial solution u0 ue .s/, v2 u01 v01 0, which implies a velocity defect of o.1/, we seek non-trivial solutions u0 ; v2 ; ı0 of (23) with respect to separation. To this end, it is useful to consider (23b) and (23a) evaluated for N D 0,

d.u2s u2e /=ds D 2 @N .u01 v01 / .s; 0/: (24) Herein us .s/ D u0 .s; 0/ again denotes the slip velocity. Note, that separation is associated with flow reversal further downstream, which, in turn, requires us .0/ D 0. To gain first insight how the boundary layer behaves as s ! 0 , the problem (23) has been solved numerically, by adopting the same algebraic shear stress closure that was employed successfully for the boundary layer calculations in [7].

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We again discard the possibility that the impressed potential flow exhibits a rear stagnation point S , since inspection of (24), confirmed by the numerical study, shows that then us not necessarily approaches zero in the vicinity of S . Therefore, the picture of a ‘collision’ of two boundary layers is apparently not appropriate for describing turbulent separation. Consequently, separation is seen to be associated with a smoothly separating inviscid flow, according to the situation sketched in Fig. 2(b). As outlined in [1], only flows having k 0 are topologically possible. A suitable model for the surface velocity ue .s/ that exhibits the then required local behaviour (12) is given by ue .s/ D .3=2 C s/m Œ1 C k.2s/1=2 =.1 C k/, 1=2 s < 0, such that ue .1=2/ D 1. Here the exponent m represents an eigenvalue of the self-preserving solution for a given value of us .1=2/, which serves as the initial condition for the downstream integration of (23), cf. [7]. Specifically, : the value us .0/ D 0:95 has been imposed, yielding m D 0:3292. The distributions for the impressed adverse difference pressure p0 .s/ p0 .0/ and the resulting slip velocity us .s/ are plotted in Fig. 3 for different values of the control parameter k. It is found that for sufficiently small values of k the integration terminates in a singular manner at s D 0 where us assumes a finite limit, i.e. us .0/ > 0. For increasing values of k this threshold decreases, such that it finally vanishes for a critical value of k, say, k D kc . We note that near k D kc the numerical calculations are very sensitive to slight variations in the value of k; for the specific choice of ue .s/ adopted : here one finds that kc D 2:7. For k > kc , however, the solution admits a Goldsteintype singularity at a position upstream of s D 0 which is discussed in more detail in [7]. Here we add, that a thorough analytical study of the numerically observed singular behaviour of the boundary layer solutions, also expressed through (24), is a task of the current research. As a first, rather remarkable, result, the location of turbulent break-away separation in the double limit ˛ D D 0 is seen to be associated with a positive, presumably single-valued, value kc of k, which has to be found by means of iterative boundary layer calculations. This strikingly contrasts its laminar counterpart, where the so-called Brillouin-Villat condition fixes the position of inviscid flow detachment by the requirement k D 0, see [9, 10]. Furthermore, the downstream shift of that point for increasing values of k, sketched in Fig. 2(b), explains why, in general,

0 Δp0 −0.5 −1

2.5 us 2

k < kc : k = kc

k > kc

1.5

−1.5 −2

k > kc

−2.5

1 0.5

k < kc

: k = kc

0 −0.1 −0.08 −0.06 −0.04 −0.02 s 0 : Fig. 3 Local distributions of p0 D p0 .s/ p0 .0/ and us .s/ for k D 1:5, k D 2:7 D kc , and k D 3:4; the circles indicate the occurrence of singular points −3 −0.1 −0.08 −0.06 −0.04 −0.02 s 0


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turbulent separation from a cylindrical body takes place further downstream as it is the case when the flow is still laminar. Moreover, first investigations indicate that in the turbulent case the more precise determination of the location of separation for small but finite values of both ˛ and is determined by a locally strong rotational/irrotational interaction mechanism, analogous to that proposed in [11, 12].

3.2 Internal Structure ‘Derived’ from First Principles As the starting point, we consider the well-known transport equation for the time-mean value of the specific turbulent kinetic energy κ D u0 u0 =2 .u02 C v02 C w02 /=2, which results from Reynolds-averaging the inner product of u0 with (2b) by substituting (2a), Dt κ C r .κ C p 0 /u0 κ C "p D u0 u0 W ru ;

"p WD ru0 W ru0 :

(25)

Herein "p is commonly referred to as turbulent (pseudo-)dissipation. By taking into account (22), the least-degenerate shear layer approximation of (25) in the double limit ˛ ! 0, ! 0 is found to be @N .p10 v01 / C "p u01 v01 @N u0 :

(26)

We integrate (26) across the shear layer thickness, i.e. from N D 0 to N D ı0 . Then the net contribution of the diffusive term on the left-hand side of (26) is seen to vanish, whereas the resulting net turbulent ‘production’ on the right-hand side is positive and of O.1/ since both the Reynolds shear stress u01 v01 and the shear rate u0;N are apparently non-negative. Remarkably, then "p is a quantity of O.1/ in the formal limit D 0. The quantity "p is obtained by Reynolds-, or equivalently, time-averaging according to (3), the stochastically varying quadratic form ru0 W ru0 . By adopting the very weak assumption that the averaging process leaves its order of magnitude unchanged, we find, with some reservation, that ru0 D O. 1=2 /

(27)

holds for the predominant fraction of intervals of the time t. As the most simple description of the fluctuating motion, we next assume that the turbulent fluctuations are governed by a single spatial ‘micro scale’, denoted by (together with a correspondingly small time scale) apart from the ‘macro scales’, represented by a streamwise length of O.1/ and the shear layer thickness of O.˛/. It then follows from the estimate (27) in combination with (22) that appropriate ‘micro variables’ are given by .t 0 ; x 0 / D .t; x/= where x 0 D .s 0 ; n0 ; z0 / and D .˛/1=2 . That is, the smallest scales are measured by . Interestingly, they are asymptotically larger than the (non-dimensional) celebrated Kolmogorov length scale, which is commonly

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associated with the dissipative small-scale structure of turbulence and given by . 3 ="p /1=4 . Hence, the equations of motion (2b) are expanded in the sequence of ‘inviscid’ linear equations r 0 u0i D 0; D0t u0i C Ni01 D r 0 pi0 ;

N00 D 0;

D0t D @t 0 C u0 .s; N / @s 0 :

(28a) (28b)

Here and in the following i D 1; 2; : : :, u0i D .u0i ; v0i ; w0i /, and r 0 denotes the gradient with respect to x 0 . The inhomogeneous terms Ni0 in (28b) are defined by expanding the nonlinear convective operator in (2b) according to (22), .u r 0 u0 @s 0 /u0 ˛ 1=2 N10 C ˛N20 C :

(28c)

Then the vector Ni0 depends on the velocity fluctuations u0j where j D 1; 2; : : : ; i . By eliminating the pressure fluctuations pi0 in (28b), the vorticity fluctuations !0i are seen to satisfy the equations D0t !0i D r 0 Ni01 ;

!0i D r 0 u0i :

(28d)

Thus, D0t !01 D 0, so that !01 depends on the ‘micro variables’ 0 D s 0 u0 t 0 , n0 , and z0 , but not explicitly on t 0 . In principle, u01 then can be calculated from its Helmholtz decomposition, given by the distribution of !01 together with the vanishing divergence as expressed by (28a). Therefore, u01 and, in turn, N10 also show no explicit dependence on t 0 , giving !02 D C 0 .r 0 N10 /t 0 , where C 0 is a ‘constant’ of integration. The requirement that the expansion (22a) must be uniformly valid with respect to the ‘micro time’ t 0 then gives rise to the solvability condition r 0 N10 D 0. As a result, one recursively finds that D0t !0i D 0 in general, such that the velocity and pressure fluctuations u0i and, pi0 , respectively, depend on 0 , s 0 , n0 , and z0 , but, most important, not explicitly on t 0 , and are determined by r 0 Ni0 D 0;

r 0 pi0 D Ni01 :

(29)

Equation (29) describe a stationary motion with respect to 0 , i.e. in a frame of reference which moves with the time-mean streamwise velocity u0 .s; N / along the separating streamline S of the flow in the formal limit D 0, see Fig. 1(a). Note, that they comprise the full nonlinear steady Euler equations, satisfied by u01 and p20 . This transport of the stochastic fluctuations along with the time-averaged flow found from the ‘micro-scales’ analysis is commonly termed as ‘coherent motion’. As a further consequence of these considerations, the process of time-averaging according to (3) is seen to provide a filtering of the fluctuating motion with respect to 0 and, in turn, rather not only with respect to the ‘micro time’ t 0 but also to the streamwise ‘micro variable’ s 0 . The view that the statistically stationary turbulent flow depends on the spatial ‘macro variables’ s and N only is, therefore, supported by an asymptotic investigation of the Navier–Stokes equations (2b) and time-averaging.


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The relationships (28) are valid for i < I where the index I signifies contributions to (22) of O. 1=2 /. For i D I it follows from (2b) that the dynamics of these contributions are affected by the viscous term on the right-hand side of (2b). Also, the normal gradient @N u0 then enters the momentum balance as a consequence of the ‘macro scale’ ˛ which describes the time-mean shear layer approximation. This in turn suggests the introduction of a further set .t ˛ ; x ˛ / D .t; x/=˛ of ‘micro variables’. Let r ˛ denote the gradient with respect to x ˛ and es , en , and ez the unit vectors in the respective directions indicated by the subscripts. We then find r 0 u0I D r ˛ u01 ;

(30a)

D0t u0I C NI0 1 C D˛t u01 C es v01 @N u0 D r 0 pI0 r ˛ p10 C 0 u01 ; D˛t D @t ˛ C u0 .s; N / @s ˛ ;

0 D r 0 r 0 :

(30b)

From (29) it follows that p10 is independent of x 0 since N00 vanishes according to (28b). By taking the curl with respect to x 0 one then obtains from (30b) D0t !0I D r 0 NI0 1 D˛t !01 .@N u0 /.en @z0 ez @n0 /v01 C 0 !01 :

(30c)

The right-hand sides of both (30c) and (30a) do not explicitly depend on t 0 . With the same arguments leading to (29), then the Helmholtz decomposition of !0I suggests that u0I and, as a consequence of (30b), pI0 exhibit no explicit t 0 -dependence too. In turn, the right-hand side of (30c) must vanish. Therefore, (30c) not only determines the quantity u0I 1 , but can also be interpreted as a linear transport equation for the leading-order contribution !01 to the vorticity with respect to the newly introduced time t ˛ and x 0 . However, the motion which is affected by the viscous term in (2b) is presumably also governed by convective terms which are nonlinear in the leading-order contribution u01 to the velocity fluctuations. But, in view of (30b), this is only possible by introducing a set of ‘intermediate micro variables’ .tO; x/ O D .t; x/=˛ 3=2 . Thus, the associated new length scale of O.˛ 3=2 / is much larger than the viscosity-affected one, , but still smaller than the shear layer thickness of O.˛/. We close the analysis by noting that this new length scale serves as a measure for the size of the large eddies in the wake region, and, in turn, of the mixing length. This fully agrees with the scaling of the latter found from the time-mean analysis, cf. [7].

4 Conclusions and Further Outlook We have demonstrated that turbulent bluff-body separation requires a streamwise velocity defect of O.1/ as Re ! 1 in the fully turbulent main region of the oncoming boundary layer, as the classical assumption of a small velocity deficit is intrinsically tied to the idea of a firmly attached external potential flow, and, in turn, leads to a serious inconsistency in the asymptotic hierarchy of the flow, which

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originates from an asymptotically small vicinity of the rear stagnation point. On the other hand, for the large-defect boundary layer the limiting inviscid solution must be sought in the class of flows exhibiting a free streamline which departs smoothly from the surface. As one remarkable result strikingly contrasting a well-known finding in the theory of laminar separation, here the Brillouin-Villat condition is not met at the separation point. The formulation of the locally strong rotational/irrotational interaction of the separating flow with the external bulk flow is a topic of the current research. Future research activities include, amongst others, the asymptotic investigation of the unsteady motion. Most important, as a first step in this direction, it has been shown here how the underlying boundary layer concept is strongly supported by such an analysis. As one physically appealing result, an inner length of O.˛ 3=2 / reflecting the size of the large-scale eddies in the wake flow regime is found, which interestingly equals that of the mixing length given in [7]. Acknowledgements This research was granted by the Austrian Science Fund (project no. P16555-N12), which is gratefully acknowledged.

References 1. G. Birkhoff and E. H. Zarantonello. Jets, Wakes, and Cavities, volume 2 of Applied Mathematics and Mechanics. Academic Press, New York, 1957. 2. A. T. Degani. Recent Advances in Wall-Bounded Shear Flows. Survey of Contributions from the USA (Invited Lecure). In K. Gersten, editor, Asymptotic Methods for Turbulent Shear Flows at High Reynolds Numbers. Proceedings of the IUTAM Symp., Bochum, Germany, June 28–30, 1995, volume 37 of Fluid Mechanics and its Applications, pages 119–132, Dordrecht/Boston/London, 1996. Kluwer MA. 3. G. L. Mellor. The Large Reynolds Number, Asymptotic Theory of Turbulent Boundary Layers. Int. J. Engng. Sci., 10(10):851–873, 1972. 4. R. E. Melnik. An Asymptotic Theory of Turbulent Separation. Comput. Fluid, 17(1):165–184, 1989. 5. A. Neish and F. T. Smith. On turbulent separation in the flow past a bluff body. J. Fluid Mech., 241:443–467, 1992. 6. B. Scheichl and A. Kluwick. On turbulent marginal boundary layer separation: how the halfpower law supersedes the logarithmic law of the wall. Int. J. Comput. Sci. Math. (IJCSM), 1(2/3/4):343–359, 2007. Special Issue on Problems Exhibiting Boundary and Interior Layers. 7. B. Scheichl and A. Kluwick. Turbulent Marginal Separation and the Turbulent Goldstein Problem. AIAA J., 45(1):20–36, 2007. see also AIAA paper 2005-4936. 8. B. F. Scheichl. Asymptotic Theory of Marginal Turbulent Separation. Ph.D. thesis, Vienna University of Technology, Vienna, Austria, June 2001. 9. F. T. Smith. The Laminar Separation of an Incompressible Fluid Streaming Past a Smooth Surface. Proc. R. Soc. Lond. A, 356(1687):443–463, 1977. 10. V. V. Sychev. Laminar Separation. Fluid Dyn., 7(3):407–417, 1972. Original Russian article in Izv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza (3), 1972, 47–59. 11. Vik. V. Sychev. Asymptotic Theory of Turbulent Separation. Fluid Dyn., 18(4):532–538, 1983. original Russian article in Izv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza (4), 1983, 47–54. 12. Vik. V. Sychev. Theory of Self-Induced Separation of a Turbulent Boundary Layer. Fluid Dyn., 22(3):371–379, 1987. original Russian article in Izv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza (3), 1987, 51–60.

Structural Sensitivity of the Finite-Amplitude Vortex Shedding Behind a Circular Cylinder P. Luchini, F. Giannetti, and J. Pralits

Abstract In this paper we study the structural sensitivity of the nonlinear periodic oscillation arising in the wake of a circular cylinder for Re47. The sensibility of the periodic state to a spatially localised feedback from velocity to force is analysed by performing a structural stability analysis of the problem. The sensitivity of the vortex shedding frequency is analysed by evaluating the adjoint eigenvectors of the Floquet transition operator. The product of the resulting neutral mode with the nonlinear periodic state is then used to localise the instability core. The results obtained with this new approach are then compared with those derived by Giannetti & Luchini [8]. An excellent agreement is found comparing the present results with the experimental data of Strykowski & Sreenivasan [7]. Keywords Fluid mechanics Nonlinear global modes Structural sensitivity Adjoint

1 Introduction Spatially developing flows such as mixing layers, wakes and jets, may sustain in specific parameter ranges synchronised periodic oscillations over extended regions of the flow field, displaying there an intrinsic dynamics characterised by a sharp frequency selection. Under these conditions the whole flow field behaves like a global oscillator and the structure underlying the spatial distribution of the fluctuations is usually termed “global mode”. The spatio-temporal evolution of such flows has been clarified considerably only in recent years: progress was made through model equations, experiments, stability analysis and direct numerical simulations. A theoretical approach to this class of problems was formulated by Chomaz et al. [1], Monkewitz et al. [6] and Le Dizés et al. [4] in the context of flows with properties slowly varying in space. Relying only on a local analysis, they were able to show that such flows may exhibit internal resonance when a region of absolute instability P. Luchini (), F. Giannetti, and J. Pralits DIMEC, Università di Salerno, 84084 Fisciano (SA), Italy e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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of sufficient size develops. The resonance is self-excited and is characterised by a well defined frequency. The important link between the global and local instability properties, both in the linear and fully nonlinear regime, is obtained via a WKBJ approach: the theory identifies a specific spatial position in the absolutely unstable region which acts as a wavemaker, providing a precise frequency selection criterion and revealing some important insights pertaining to the forcing of these modes. In particular, in a linear setting, the complex global frequency !g is obtained by the saddle-point condition !g D !0 .Xs /

with

@!0 .Xs / D 0 @X

(1)

based on the analytic continuation of the local absolute frequency curve !0 .X / in the complex X-plane, with X denoting here the slow streamwise variable. Although this asymptotic theory yields accurate predictions for slowly evolving flows, in many real configurations the assumptions underlying the WKBJ approach are not met very closely. This is the case of bluff-body wakes, where strong non-parallel effects prevent us from using asymptotic theory. In such cases a numerical modal analysis must be used to determine the characteristics of the instability and to find its critical Reynolds number. One of the most common examples is given by the flow around an infinitely long circular cylinder. In order to study the global properties of such flow, Giannetti & Luchini [8] performed a 2D stability and receptivity analysis of the steady base flow using the properties of the adjoint eigenfunctions. The asymptotic theory developed by Chomaz et al. [1], Monkewitz et al. [6] and Le Dizés et al. [4] in the context of slowly evolving quasi-parallel flows endows the region around the saddle point with the fundamental role of wavemaker in the excitation of the global mode. In the context of a two-dimensional modal analysis a concept similar to that of wavemaker can be introduced by investigating where in space a modification in the structure of the problem is able to produce the greatest drift of the eigenvalue. Using this approach Giannetti and Luchini (2007) determined the regions where the feedback from velocity to force is maximum and consequently identified the regions were the instability acts. Qualitative agreement was obtained with the numerical and experimental data of Strykowski and Sreenivasan [7]. From a theoretical point of view a similar approach, being based on the properties of the steady base flow, is only valid in a neighbourhood of the neutral point. When Re > Rec 47 the flow becomes unsteady and a Karman vortex street develops. In this paper we extend the approach developed by Giannetti and Luchini (2007) to study finite-amplitude vortex shedding, in order to assess how unsteadiness and saturation can modify the previous results.

2 Problem Formulation We investigate the stability characteristics of the two-dimensional flow arising around an infinitely long circular cylinder invested by a uniform stream. A Cartesian coordinate system has its origin in the cylinder’s centre, with the x axis pointing in

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the flow direction. For Re < Rec;2 180 the fluid motion can be described by the two-dimensional unsteady incompressible Navier–Stokes equations 1 @u C u ru D rp C u @t Re ruD0

(2a) (2b)

where u is the velocity vector and p is the reduced pressure. Equations (2a,b) are made dimensionless using the cylinder diameter D as the characteristic length scale, the velocity of the incoming uniform stream u1 as the reference velocity and u2 1 as the reference pressure. Thus Re D

u1 D

(3)

is the Reynolds number based on the cylinder diameter. Equations (2a,b) must be supplemented by appropriate boundary conditions. In particular, on the surface of the cylinder c the no-slip and no-penetration conditions require both velocity components to vanish, while in the far field the flow approaches asymptotically the incoming uniform stream. For Re > Rec the steady symmetric flows becomes unstable and a Karman vortex street develops. In such conditions, after an initial transient, the flow becomes periodic: u.t C T / D u.t/ ; p.t C T / D p.t/

(4)

with period T , Strouhal number S t D 1=T and angular pulsation ! D 2=T . In order to locate the wavemaker of the instability, Giannetti and Luchini (2007) determined the space distribution of the sensitivity of the eigenvalue to a structural perturbation of the problem. The analogous quantity for the nonlinear periodic oscillation is the space distribution of the sensitivity of its frequency to a structural perturbation of the problem. This is the objective of the present paper. Suppose now we give a structural perturbation to this problem, in the form of a body force depending on the local velocity h.u/. If the perturbation is small the new solution will remain periodic but with a different period (in contrast with the corresponding linear problem whose frequency will in general become complex and bring about either amplification or damping). In order to be able to treat the problem perturbatively and avoid secular effects, it is convenient to scale the time variable on the period of the solution itself. Thus introducing the scaled time

D

t T

(5)

the equations can be rewritten as 1 1 @u C u ru D rp C u T @

Re ruD0

(6a) (6b)

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where T is an additional unknown and the period in the variable is constant and equal to 1. Writing the perturbed solution as fu0 . / C u. /; p0 . / C p. /g; where fu0 ; p0 g denote the unperturbed periodic flow and fu; pg have become the small perturbations induced by the added forcing, and inserting it into the equations together with the small external forcing h D Cu0 we obtain 1 1 @.u0 C u/ C .u0 C u/ r.u0 C u/ C r.p0 C p/ D .u0 C u/ C h T C ıT @

Re (7a) r .u0 C u/ D 0

(7b)

If the effect of the structural perturbation is small, we can linearize these equations and obtain an equation for the perturbation ıT @u0 1 1 @u C u0 ru C u ru0 C rp u D 2 Ch T @

Re T @

ruD0

(8a) (8b)

This linear problem can be studied through Floquet analysis, and as is well known the resulting perturbation will in general not be periodic, but modified by the Floquet exponent. The condition, implicit in the definition of , that a constant period equal to 1 be maintained, constitutes a compatibility condition determining ıT , which is exactly the variation of period induced by the structural perturbation h D Cu0 .

2.1 Adjoint Equations Just as in the corresponding linear stability problem, if we just wanted to determine the variation of period for a specific form of structural perturbation we could solve the problem as stated above; but we can obtain a much more powerful result, i.e. the sensitivity of the period to an arbitrary structural perturbation with the aid of adjoint equations. Key to this approach is the observation that the unperturbed equations (6a,b) have a non-unique solution, insofar as if u0 . / is a periodic solution, u0 . C ı / is as well. Linearizing with respect to ı we find that @u0 =@ is a solution of the linearized equations (8) in homogeneous form (e.g., with h D 0). Since Equations (8a,b) with periodic boundary conditions have a nontrivial solution with zero forcing and zero ıT , the original inhomogeneous linear problem only has a solution if a compatibility condition is satisfied. This compatibility condition can be derived through adjoint equations.


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The adjoint of the linearized Navier–Stokes operator (8) is defined, as usual, by multiplying both sides of the equation by suitably differentiable fields ff C ; mC g, integrating over all space and a period in time, and then using integration by parts to shift differentiation operators from the direct to the adjoint fields. We thus obtain: Z

ıT @u0 C h d3 x dt f T 2 @

Z

1 1 @u C C D f C u0 ru C u ru0 C rp u C m r u d3 x dt T @

Re Z 1 1 @f C C C C C r .u0 f / C ru0 f rm f C D u T @

Re pr f C d3 x dt (9) C

where periodicity eliminates finite terms in time, and spatial boundary conditions for the adjoint are assumed to eliminate finite terms in space that are not already eliminated by boundary conditions for the direct problem. Equation (9) constitutes a generalized Green’s identity (Morse and Feshbach [9]) for the LNSE. It is self-evident that if ff C ; mC g are chosen to nullify the r.h.s. of Equation (9) independently of u and p, i.e. to satisfy the adjoint equations, the l.h.s. must be zero as well. Recalling that h D Cu0 , we thus obtain the compatibility condition N

ıT D T

Z

f C Cu0 d3 x dt

Z where N D

fC

1 @u0 3 d x dt T @

(10)

Since ı!=! D ıT =T , we have obtained the structural sensitivity S of the oscillation frequency ! to the structural perturbation C.x/: ! ı! D SD ıC N

Z

u0 f C dt

(11)

Notice that C is a tensor quantity, relating a force to a velocity, and so is S. The notation u0 f C must be read as a dyadic product.

3 Numerical Approach The time dependent flow around the cylinder is solved by discretizing the equations with finite differences on a staggered Cartesian grid. The advancement in time is obtained by the classical Runge-Kutta Crank-Nicholson scheme of Rai and Moin. The presence of the cylinder is represented by an immersed-boundary technique similar to that used by Fadlun et al. [2]. Thus, the entire domain is covered by computational cells and there is no need for body-fitted coordinates. The boundary

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conditions on the surface of the cylinder c are imposed through a linear interpolation. Several interpolation procedures have been proposed in the past: in Fadlun et al. [2] the velocity at the first grid point external to the body is obtained by linearly interpolating the velocity at the second grid point (which is instead obtained by directly solving the Navier–Stokes equations) and the velocity at the body surface: in their numerical algorithm this condition is approximately enforced by applying momentum forcing inside the flow field. The interpolation direction is either the streamwise or the transverse direction, but the choice between them is not specified. Mohd-Yusof [5] used a more complex interpolation scheme which involved forcing the Navier–Stokes equations both inside and on the surface of the body. In particular the no-slip conditions were imposed at the point of the boundary touched by the wall-normal line passing through the closest internal point, using bilinear interpolations for this purpose. Finally, Kim et al. [3] introduced a mass injection forcing to satisfy the continuity equation for the cells containing the immersed boundary. A slightly different and easier approach has been used by Giannetti and Luchini (2007) to study the structural sensibility of the first instability of the cylinder wake. In this paper we follow this last approach: the interpolation is performed using the point closest to the body surface (which can be either an internal or an external point) and the following point on the exterior of the cylinder. The interpolation is performed either in the streamwise or transverse direction according to which one is closest to the local normal. The linear system of algebraic equations deriving form the discretization of the nonlinear equations, along with their boundary conditions is solved at each substep through a sparse LU decomposition. Both the nonlinear equations and the adjoint equations are marched in time until a time-periodic state is reached.

4 Numerical Results Figure 1 shows our typical result: a space distribution of the structural sensitivity S defined by Equation (11), in this case at a Reynolds number slightly above threshold. Since S is a tensor, various representative quantities may be chosen to be plotted. In Fig. 1 the choice is the spectral radius, which represents the sensitivity to a force of the worst possible direction. Other choices can be the Frobenius norm (sum of the squares of all four components) or the absolute value of the trace (sensitivity to a force locally aligned with velocity, i.e. a pure resistance). Similar data for Re = 80 and 100 are shown in Figs. 2 and 3. All three figures agree remarkably well with the experimental data of Strykowski and Sreenivasan [7], who introduced a small perturbing cylinder in the wake of a larger one and reported the variation in critical Reynolds number as a function of position of the perturbing cylinder (Fig. 4).


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y

Re=50 18

1.2

17

1

16

0.8

15

0.6

14

0.4

13

0.2

12 12

0 14

16

18

20

22

x

Fig. 1 Structural sensitivity of the periodic wake at Re D 50

Re=80 1.4

18

1.2

17

1

16 y

0.8 15 0.6 14

0.4

13 12 12

0.2 0 14

16

18

20

22

x


4.1 Comparison with the Linear Results In fact, it is a surprise that the structural sensitivity of the saturated periodic oscillation, even at the relatively low Reynolds number of 50, does not agree as satisfactorily with the structural sensitivity of the linear eigenmode as calculated by Giannetti and Luchini (2007) (Fig. 5). Actually, if attention is paid to the colour scale, it will be seen that the two are quite different in amplitude and not just in shape.

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P. Luchini et al. Re=100 1.6

18

1.4

17

1.2

y

16

1 0.8

15

0.6

14

0.4 13 12 12

0.2 14

16

x

18

20

22

0


Fig. 4 Experiment of Strykowski and Sreenivasan [7]

This was a puzzle until we realized that the frequency of the nonlinear oscillation can be influenced in two different ways: by a structural perturbation force determined by the fluctuating velocity alone (as implicitly assumed in our linear results), or by a force that responds both to the mean and to the fluctuating velocity. Neither is wrong: they serve different purposes. The structural perturbation depending on the fluctuation only was the appropriate tool to study the position of the wavemaker,


159

Re=50 0.25

18 17

0.2

16 y

0.15 15 0.1 14 0.05

13 12 12

0 14

16

18

20

22

x

Fig. 5 Structural sensitivity of the linear instability mode at Re D 50

Re=50 1.6

18

1.4

17

1.2

y

16

1

15

0.8 0.6

14

0.4 13 12 12

0.2 0 14

16

18

20

22

x

Fig. 6 Structural sensitivity of the linear instability mode at Re D 50 with the base-flow modification included

but the perturbation depending on the full velocity field is the one that was implicitly assumed in the present nonlinear results, and of course is the one that occurs in the experiments. Once this difference is identified, it is not difficult to extend the linear eigenmode calculation to account for the frequency variation induced by a perturbation influencing the mean flow. On so doing Fig. 5 becomes Fig. 6 and, all of a sudden, a more satisfactory agreement is recovered with both experiments and nonlinear sensitivity results.

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5 Conclusion The structural sensitivity map of the frequency of a periodically oscillating wake to a perturbing force locally proportional to velocity has been determined for twodimensional flow past a cylinder at various Reynolds numbers. The results, meant as an extension of the eigenmode structural sensitivity of Giannetti and Luchini (2007) have actually uncovered a dominant effect of the frequency variation induced by a modification of the base flow over the frequency variation induced by the direct structural perturbation of the eigenmode, thus clarifying that the former effect was also dominant in the experiments of Strykowski and Sreenivasan [7]. When the nonlinear results are epurated of the contribution of the mean flow, or vice versa this effect is included in the eigenmode calculation, agreement for near-threshold Reynolds number is actually recovered.

References 1. J.-M. Chomaz, P. Huerre, and L. Redekopp. A frequency selection criterion in spatially developing flows. Stud. Appl. Math., 84:119–144, 1991. 2. E.A. Fadlun, R. Verzicco, P. Orlandi, and J. Mohd-Yusof. Combined immersed-boundary finitedifference methods for three-dimensional complex flow simulations. J. Comp. Phys., 161:35–60, 2000. 3. J. Kim, D. Kim, and H. Choi. An immersed-boundary finite-volume method for simulations of flow in complex geometries. J. Comp. Phys., 171:132–150, 2001. 4. S. Le Dizs, P. Huerre, J.-M. Chomaz, and P.A. Monkewitz. Linear global modes in spatially developing media. Phil. Trans. R. Soc. Lond., 354:169–212, 1996. 5. J. Mohd-Yusof. Combined immersed-boundary/b-spline methods for simulations of flows in complex geometries. Annual research briefs, Center for Turbulence Research, NASA Ames and Standford University, 1997. p. 317. 6. P.A. Monkewitz, P. Huerre, and J.-M. Chomaz. Global linear stability analysis of weakly nonparallel shear flows. J. Fluid Mech., 251:1–20, 1993. 7. P.J. Strykowski and K.R. Sreenivasan. On the formation and suppression of vortex “shedding” at low Reynolds number. J. Fluid Mech., 218:71–107, 1990. 8. F. Giannetti and P. Luchini. Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech., 581:167–197, 2007. 9. P.M. Morse and H. Feshbach. Methods of Theoretical Physics. McGraw-Hill, 1953.

Orbiting Cylinder at Low Reynolds Numbers L. Baranyi

Abstract Sudden changes found in the time-mean and rms values of force coefficients of a circular cylinder in forced orbital motion placed in a uniform stream when plotted against ellipticity of the orbital path suggest that two solutions (states) exist. For a better understanding of the changes in state, some new factors are considered: the torque coefficient; computations for Reynolds numbers up to 300; the investigation of lock-in domain for Re D 160. Keywords Orbiting cylinder Lift Drag Torque Lock-in 2D flow Low Reynolds number flow

1 Introduction While investigating the flow around an orbiting cylinder placed in an otherwise uniform flow at low Reynolds numbers, a rather peculiar phenomenon appeared. When plotting the time-mean or root-mean-square (rms) values of lift, drag, and base pressure coefficients against ellipticity of the orbital path, sudden changes were found in the values in all coefficients investigated at certain ellipticity values [2]. Later energy transfer between cylinder and fluid was investigated by investigating limit cycles, time histories, phase angles and flow patterns confirming the phenomenon, [4, 8]. Here the dimensionless torque coefficient is added which also shows sudden jumps, and computations are repeated for different forced frequency ratios and for Reynolds numbers up to 300. The incompressible flow around an orbiting cylinder was simulated by a 2D code developed by the author [1], and is based on Finite Difference Method. Boundaryfitted coordinates are used and both the computational domain and the governing equations are transformed into a computational plane [1]. L. Baranyi Department of Fluid and Heat Engineering, University of Miskolc, H-3515 Miskolc-Egyetemváros, Hungary e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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2 Results for an Orbiting Cylinder The flow arrangement can be seen in Fig. 1. The orbital motion of the cylinder is created by a superposition of two forced harmonic oscillations with identical frequencies f [2]. In Fig. 1 U is the free stream velocity Ax and Ay are dimensionless oscillation amplitudes in x and y directions, respectively. Nonzero Ax and Ay values give an ellipse, shown in the dotted line in Fig. 1. Ax alone yields pure in-line oscillation, and then as Ay is increased, the ellipticity e D Ay =Ax increases to yield a full circle at e D 1. During each set of computations the Reynolds number Re (Re D Ud= where is the kinematic viscosity) and Ax are fixed and f is kept constant at some percentage of Strouhal number St0 (dimensionless vortex shedding frequency from a stationary cylinder at that Re), and amplitude of transverse oscillation Ay is varied to produce varying ellipticity e D Ay =Ax . Only cases were considered where lockin prevailed, even at zero ellipticity (in-line motion). In this study this was between 70–105% to ensure lock-in at moderate oscillation amplitudes. Here results will be shown only for a cylinder in clockwise direction of orbit. An interesting phenomenon was observed when looking at the time-mean value (TMV) and rms values of lift, drag and base pressure coefficients for an orbiting cylinder in a uniform flow. Abrupt jumps were found when these values were plotted against ellipticity e with Re and Ax kept constant [2, 3]. A typical example for the TMV of lift coefficient CLmean is shown in Fig. 2a for Re D 140, Ax D 0:4, f D 0:9St0 D 0:16389. Note that there are two envelope or state curves, which are roughly parallel with each other and of positive slope, and values jump between these two curves. The TMV and rms of drag and base pressure, as well as the rms of lift, behaved differently from CLmean , characterised by two state curves which are not parallel but intersect each other at e D 0. A typical example is shown in Fig. 2b. The main parameters (Re, Ax and f ) are the same as in Fig. 2a. The number and location of jumps are identical in these two figures. In this study investigations are extended to

y

U

O

d=1 x

Ay

Fig. 1 Layout of a cylinder in orbital motion

Ax

Orbiting Cylinder at Low Reynolds Numbers

a

163

0.95

0.3

0.9

CDrms

CLmean

orbital; clockwise; Re=140; Ax=0.4

b

orbital; clockwise; Re=140; Ax=0.4

0.4 0.2 0.1

0.85 0.8

0 −0.1 0

0.5

1

0.75

−0.2

0

0.5

e

1 e

0.9 St0 clw

0.9 St0 clw

Fig. 2 Time-mean value of lift (a) and rms value of drag (b) versus ellipticity

a

b

0.003

−0.001 0

0.2

0.4

0.6

0.8

1

1.2

−0.003 −0.005

tq rms

tq mean

0.001

0.0063 0.0058 0.0053 0.0048 0.0043 0.0038 0.0033 0.0028 0

−0.007

0.2

0.4

0.75St0

0.8St0

0.6

0.8

1

1.2

e

e 0.75St0

0.85St0

0.8St0

0.85St0

Fig. 3 Time-mean and rms values of torque coefficient versus e .Re D 160I Ax D 0:4/

include torque coefficient tq (measured clockwise), which is determined from the moment of the shear stress on the cylinder. Omitting details: 1 tq D 4

Z2 0

1

0 .'/d' D 4Re

Z2 !0 .'/d' 0

where ' is the polar angle, 0 and !0 are dimensionless wall shear stress and vorticity, respectively. A similar torque coefficient is defined in [7]. Figure 3a shows the TMV of tq for three frequency ratios of f =St0 D 0:75; 0.8; 0.85 versus ellipticity e for Re D 160 and Ax D 0:4. The TMV of torque coefficient tqmean shows a similar pattern to CLmean (see Fig. 2a), though the lines have negative slopes. It can be seen in Fig 3a that the distance between lower and upper envelope curves increase with f =St0 . Figure 3b shows the rms values of torque coefficient for the three cases shown in Fig. 3a. As can be seen in the figure, the number and location of jumps at the corresponding curves are identical with each other and the pattern of envelope curves in Fig. 3b is similar to that in Fig. 2b. By increasing the frequency of oscillation f the envelope curves shift upward.

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In earlier papers [2, 3] only results below Re D 190 have been shown. At this Reynolds number three-dimensional effects start to appear for stationary cylinders [5]. Experimental evidence from [6] for oscillating cylinders shows that lock-in increases the span-wise correlation of signals and the two-dimensionality of the flow compared to flow around stationary cylinders. Poncet [9] shows how the 3D wake behind a circular cylinder can be made 2D by using lock-in triggered by rotary oscillation of the cylinder. For this reason a 2D code is suitable even at higher Reynolds numbers than 190. In [8] a 2D code was used for simulation of flow around an oscillating cylinder at Re D 500. The effect of Re on TMS and rms curves for lower Re values was investigated in [3]. Time-mean and rms results are shown in Fig. 4a and b for orbiting cylinders at Re D 200; 250; 300. As seen in Fig. 4a, the distance between the two state curves decreases with Re. Figure 4b shows the rms of drag coefficient versus e for the same three Re numbers. The location and number of jumps in the corresponding curves in Fig. 4a and b are identical. Figure 4b shows that state curves shift upward with Re. Finally, the domains of [0.7,1.05] and [0.1,0.5] in f =St0 and Ax (see Fig. 5) was investigated using the CFD code mentioned for Re D 160 in order to determine

a

b

Ax=0.2; f=0.9St0; clockwise orbit 0.3

CD rms

CL mean

0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

1.2

−0.1

Ax=0.2; f=0.9St0; clockwise orbit

0.56 0.54 0.52 0.5 0.48 0.46 0.44 0.42 0.4 0.38 0

−0.2

0.2

0.4

0.6

Re=200

Re=250

0.8

1

1.2

e

e Re=200

Re=300

Re=250

Re=300

Fig. 4 Time-mean and rms of lift coefficient versus ellipticity for Re D 200; 250; 300

0.6 0.5

Ax

0.4 0.3 0.2 0.1 0 0.65 0.7 0.75 0.8 0.85 0.9 0.95

1

1.05 1.1

f/St0

Fig. 5 Range of lock-in for Re D 160

lock-in

not lock-in

partial lock-in

Orbiting Cylinder at Low Reynolds Numbers

165

where lock-in prevails over the whole e D 0–1:2 interval investigated. In Fig. 5 filled squares show full lock-in, empty diamonds represent cases where not a single point in the e interval is locked-in. Empty squares show cases where lock-in cease to exist within the e interval.

3 Conclusions To better understand the changes in state, some new factors are considered: –

Torque coefficient, which has similar features to those of the lift coefficient, (identical number and location of jumps). The distance between upper and lower state curves for the TMV of the torque coefficient increases and the rms state curves shift upward with frequency of cylinder oscillation f . – Investigating Reynolds numbers up to 300. The distance between upper and lower state curves for CLrms decreases with Re. – Systematic investigation of lock-in domain for Re D 160 over the ellipticity interval of e D 0–1:2. All computational results show similar patterns. The results of these investigations confirm the previous findings in this field and strengthen the case for the existence of a phenomenon behind these jumps. Acknowledgements The support provided by the Hungarian Research Foundation (OTKA, Project No. T 042961) is gratefully acknowledged.

References 1. Baranyi, L., Computation of unsteady momentum and heat transfer from a fixed circular cylinder in laminar flow. Journal of Computational and Applied Mechanics 4(1) (2003) 13–25. 2. Baranyi, L., Sudden jumps in time-mean values of lift coefficient for a circular cylinder in orbital motion in a uniform flow. In: 8th International Conference on Flow-Induced Vibration, Eds. E. de Langre and F. Axisa, Paris, II (2004) 93–98. 3. Baranyi, L., Abrupt changes in the root-mean-square values of force coefficients for an orbiting cylinder in uniform stream. In: 4th Symposium on Bluff Body Wakes and Vortex-Induced Vibrations, Eds. T. Leweke and C.H.K. Williamson, Santorini, Greece (2005) 55–58. 4. Baranyi, L., Energy transfer between an orbiting cylinder and moving fluid. In: Proceedings of the ASME Pressure Vessels and Piping Conference 9 (2007) 829–838. 5. Barkley, D., Henderson, R.D., Three-dimensional Floquet stability analysis of the wake of a circular cylinder. Journal of Fluid Mechanics 322 (1996) 215–241. 6. Bearman, P.W. and Obasaju, E.D., An experimental study of pressure fluctuations on fixed and oscillating square-section cylinders. Journal of Fluid Mechanics 119 (1982) 297–321. 7. Chen, M.M., Dalton, C. and Zhuang, L.X., Force on a circular cylinder in an elliptical orbital flow at low Keulegan-Carpenter numbers. Journal of Fluids and Structures 9 (1995) 617–638. 8. Lu, X.Y., Dalton, C., Calculation of the timing of vortex formation from an oscillating cylinder. Journal of Fluids and Structures 10 (1996) 527–541. 9. Poncet, P., Vanishing of B mode in the wake behind a rotationally oscillating cylinder. Physics of Fluids 14(6) (2002) 2021–2023.


A Two-Dimensional Disturbed Flows Over a Flat Plate: Theoretical and Numerical Approach K. Debbagh and S. Saintlos Brillac

Abstract The incompressible 2D flow over a flat plate with and without incidence is studied in respect of the propagation of spatial mode disturbances, by solving the unsteady Navier–Stokes equations. The solutions are compared with a theoretical analysis providing an analytical uniformly valid approach. The effect of the flow disturbances is studied by analytical and the numerical approaches. Keywords 2D disturbances Flat plate Incompressible flow

1 Introduction Following the ideas of Libby and Fox [7], a theoretical analysis of algebraic disturbances evolving spatially in the Blasius flow is lead. These disturbances, searched as self similar two-dimensional disturbances in the boundary layer, are built in the whole flow domain with the asymptotic method of the matched expansions for high Reynolds numbers. They are sought as a product of functions of power of x and functions of the similarity variable. For particular case without incidence of the Blasius flow, the first two-dimensional mode, named Stewartson mode, is retrieved. This asymptotic method, previously used in order to obtain a uniformly valid approximation for the mean Falkner-Skan flow [9], allows taking into account to the non-parallel effects of the disturbed flow and ensures the irrotationnality of the flow outside the boundary layer. The good agreement between numerical and theoretical solutions has been proved earlier [10]. An extension of this method to the disturbed flow is presented in this paper; it allows putting in evidence generic disturbances of this flow.

K. Debbagh and S. Saintlos Brillac () Institut de Mécanique des fluides, UMR CNRS 5502 INP/UPS, Allée du Pr. Camille Soula, 31400 Toulouse, France e-mail: email: [email protected], [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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2 2D Disturbances Over a Flat Plate 2.1 The Uniformly Valid Disturbed Solution Over a Flat Plate In this section, we consider a viscous bidimensional incompressible flow perturbations over a flat plate with an incidence angle '. We are interested particularly in the uniformly valid perturbed solution provided in [11]. By introducing the parameter " D p1re , one can write the semi-analytical solution for the wall-normal and streamwise velocity perturbation components (upt h ,vpt h ) and the pressure perturbation (ppt h ) in the form: upt h vpt h

ppt h

nC1 n1 nC1 2 sin D C "A 2 2

nC1 1n 0 n1 hn ./ C .hn ./ A / D "x 2 2 2 nC1 n1 nC1 2 cos C"A 2 2 1n n1 3n C 1 2 sin D "A 2 2

x h0n ./

p n1 ' , D arctan.y=x/, D x 2 C y 2 , D 1" x 2 y and hn ./ is a where n D ' similarity function which satisfies to the eigenvalue problem: .n h000 n ./C

C 1/ n1 00 0 0 Fn ./hn ./.nC/Fn ./hn ./C Fn00 ./hn ./ D 0 2 2

with boundary conditions: hn .0/ D 0I h0n .0/ D 0I lim h0n ./ D 0I lim hn ./ D A I h00n .0/ D 1 !1

!1

is the eigenvalue associated to the eigenfunction hn ./, A is a constant who depends on the numerical solution of the problem and Fn ./ is the Falkner-Skan function solution of the problem: Fn000 ./ C

.n C 1/ Fn ./Fn00 ./ C n.1 Fn2 .// D 0 2

with boundary conditions: Fn .0/ D 0I Fn0 .0/ D 0I lim Fn0 ./ D 1 !1

A Two-Dimensional Disturbed Flows Over a Flat Plate

169

3

3 ' = 15 o

' = 15 o

' = 5o ' = 0o ' = −5 o ' = −1 5 o

' = 5o ' = 0o ' = −5 o ' = −1 5 o

2

Y

Y

2

1

1

0 −2e−05 8e−05 2e−04 3e−04 4e−04 5e−04 6e−04

uth

0

0

5e−06

1e−05

1.5e−05

2e−05

uth

Fig. 1 Disturbance streamwise and wall-normal theoretical velocity profiles for the first mode with various incidence angles ' at x D 6. Re D 1;000 and grid mesh of 301 501

In Fig. 1, we show the theoretical streamwise and wall-normal velocity perturbation profiles for the first mode versus the incidence angle ' at x D 6 with a Reynolds number equal to 1,000. These more realistic solutions are used when analysing the destabilisation of the flow, especially under the adverse pressure gradient effects of high incidence. A numerical method to solve the Navier–Stokes equations of the disturbed flow in the vicinity of the basic flow has been developed. The vorticity-stream function formulation is used with a fourth-order compact scheme. The spatial discretisation uses the finite differences scheme with an equal mesh size in each direction and the temporal discretisation uses the Crank-Nicolson scheme which ensures an unconditional stability of the method. The resolution of the discretized equations is done by the fractional temporal step formulation adopted in [1–3] with an alternating implicit direction scheme. The method is second-order accurate in time and fourth-order accurate in space. Dirichlet and Neumann boundary conditions have been successfully tested, as well as a non reflecting outlet boundary condition [5]. Numerical tests have been performed on different problems such that the analytical flow solution of a Green-Taylor vortex, the lid-driven cavity flow, the Falkner-Skan flow and compared to the analytical and benchmark solutions found in the literature [1, 4, 6, 9]. This method is also adapted for the linearized Navier– Stokes equations around the two-dimensional basic solution. The uniformly valid solution provided in [9] is used for the basic flow. The equations are solved in the Cartesian coordinates (x,y) in a semi-infinite domain Œ1; 16 Œ0; 10 over the plate. The numerical simulation allows capturing with a very good agreement the transient stages as well as the steady-state reached solutions of the linearized (Fig. 2) and non linear Navier-Stokes equations. Other comparisons will be presented at the conference.

170


1.5

1.5

1

1

Y

2

Y

2

0.5

0

0.5

5

10

15

0

5

10

15

V

U

Fig. 2 Disturbance streamwise and wall-normal velocity contours of the theoretical solution (solid) and the numerical solution () that reached steady-state for an incidence angle ' D 5. Re D 1;000 and grid mesh of 301 501

2.2 2D Optimal Disturbances A well-known, the transition from laminar to turbulent flow is a critical process in many engineering applications. We study the growth of bidimensional optimal disturbances over a flat plate. The aim is to optimize the initial disturbance (uopt ,vopt ) at xi n D 1, the beginning of the interval, in order to achieve maximum possible amplification of the disturbance energy at xout D 16, the end of the interval. We define the growth G over the interval xi n x xout as the ratio between the disturbance energy E at the end and beginning of the interval: G.xi n ; xout ; Re/ D 1 Eu .x/ D 2

Z

1

E.xout / ; E.x/ D Eu .x/ C Ev .x/ E.xi n /

1 u .x; y/dy ; Ev .x/ D 2

Z

1

2

0

v2 .x; y/dy 0

The calculations of the optimal disturbances are carried out with an adjoint-based optimization procedure as in [2–8]. Figure 3 shows the wall-normal velocity profile and the kinetic energy growth of the optimal disturbance associated with initial zero streamwise velocity component. The maximum growth obtained Gmax D 1:74:106 Re at x D 1:42. The optimal disturbance is searched in a general way with both streamwise and wall-normal velocity components, as shown in Fig. 4. Preliminary results give the maximum growth Gmax D 1:22103 Re at x D 1:37, which is in the same order of the maximum growth found in a 3D optimal disturbances study (Gmax D 2:2:103 Re, [2]) and the streamwise velocity profil of 2D optimal disturbance (Fig. 4a) is similar to the spanwise velocity profil of the 3D optimal disturbance found in [8].

A Two-Dimensional Disturbed Flows Over a Flat Plate a

171

b

0.1

Eu /Re Ev /Re

1.5E-06

0.08

G/Re

vopt

0.06

1.0E-06

0.04

5.0E-07 0.02

0 0

20

40

60

80

0.0E+00

100

1

1.5

2

Y*

2.5

3

X

Fig. 3 Wall-normal velocity profile (a) and kinetic energy growth (b) of the 2D optimal disturbance in the case of initial zero streamwise velocity component. Re D 1;000 and grid mesh of 301 501 a

b

1

1.2E-03

0.8

Eu/Re

1.0E-03

uopt 0.6

Ev/Re

8.0E-04

vopt

G/Re 6.0E-04

0.4

4.0E-04 2.0E-04

0.2

0.0E+00

0

0

5

10

15

Y*

20

25

30

1

2

3

4

5

X

Fig. 4 Streamwise and wall-normal velocity profiles (a) and kinetic energy growth (b) of the 2D optimal disturbance. Re D 1;000 and grid mesh of 301 501

3 Conclusions A theoretical and numerical method is provided, allowing treating the problem of a flow over a semi-infinite flat plate with and without incidence by using the Navier– Stokes equations. The Navier–Stokes solver provided the time-marching solutions and is able to compute unsteady separated flows. The main objective of this study is to quantify the amplification of instabilities in this flow and their control. This numerical method is adapted to solve the adjoint Navier–Stokes equations in order to study the 2D optimal disturbances for this type of flows. Some preliminary results are compared to existing benchmark solutions found in 3D optimal disturbances studies.

172


References 1. Braza, M. 1981. Simulation numérique du décollement instationnaire externe par une formulation vitesse-pression. Application a` l’écoulement autour d’un cylindre. Thèse de DocteurIngenieur. Institut National Polytechnique de Toulouse. 2. Cathalifaud, P., Luchini, P. 2000. Algebraic growth in boundary layers: optimal control by blowing and suction at the wall. Eur. J. Mech. B – Fluids 19, 469–490 3. Douglas, J. 1962. Alternating direction methods for three-space variables. Numerische Mathematik 4, 41–63 4. Erturk, E., Gokcol, C. 2006. Fourth order compact formulation of Navier–Stokes Equations and driven cavity flow at high reynolds numbers. Int. J. Num. Mech. Fluids 50, 421–436 5. Jin, G., Braza, M. 1993. A non-reflecting outlet boundary condition for incompressible unsteady Navier–Stokes calculations. J. Comput. Phys. 107(2), 239–253 6. Kalita, J.C., Dalal, D.C., Dass, A.K. 2002. A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients. Int. J. Num. Methods Fluids 38, 1111–1131 7. Libby, P.A., Fox, H. 1963. Some perturbations solutions in laminar boundary-layer theory. Part1. J. Fluid. Mech. 17(3), 433–449 8. Luchini P. 2000. Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid. Mech. 404, 289–309 9. Saintlos, S., Bretteville, J. 2002. Approximation uniformément valable pour l’écoulement de Falkner-Skan. C.R. Mecanique 330, 673–682 10. Saintlos, S., Bretteville, J., Braza, M. 2004. Uniformly valid asymptotic solution for FalknerSkan flow and Navier–Stokes simulation. International Conference on Boundary and Interior Layers (BAIL 2004), Toulouse, France, July 2004. CD ROM Proceedings 11. Saintlos Brillac, S., Debbagh, K. 2007. Approximation asymptotique uniformément valable d’un e´ coulement perturbé sur une plaque plane. Accepté au 8eJme congrès de Mécanique. (El Jadida, Maroc 2007)

Part IV

Instability and Transition


Wake Dynamics of External Flow Past a Curved Circular Cylinder with the Free-Stream Aligned to the Plane of Curvature A. de Vecchi, S.J. Sherwin, and J.M.R. Graham

Abstract The fundamental mechanism of vortex shedding past a curved cylinder has been investigated at a Reynolds number of 100 using three-dimensional spectral=hp computations. Two different configurations are presented herein:in both cases the main component of the geometry is a circular cylinder whose centreline is a quarter of a ring and the inflow direction is parallel to the plane of curvature. In the first set of simulations the cylinder is forced to transversely oscillate at a fixed amplitude, while the oscillation frequency has been varied around the Strouhal value. Both geometries exhibit in-phase vortex shedding, with the vortex cores bent according to the body’s curvature, although the wake topology is markedly different. In particular, the configuration that was found to suppress the vortex shedding in absence of forced motion exhibits now a primary instability in the near wake. A second set of simulations has been performed imposing an oscillatory roll to the curved cylinder, which is forced to rotate transversely around the axis of its bottom section. This case shows entirely different wake features from the previous one: the vortex shedding appears to be out-of-phase along the body’s span, with straight cores that tend to twist after being shed and manifest a secondary spanwise instability. Further, the damping effect stemming from the transverse planar motion of the part of the cylinder parallel to the flow is no longer present, leading to a positive energy transfer from the fluid to the structure. Keywords Vortex shedding Bluff bodies Vortex-induced vibration

1 Introduction The intricate dynamics of vortex shedding past curved cylinders is particularly significant for many engineering applications, especially in the offshore industry: the increasing need to exploit deep-water reservoirs has highlighted the lack of a complete insight into the Vortex-Induced Vibrations (VIV) dynamics on long structures A. de Vecchi (), S.J. Sherwin, and J.M.R. Graham Department of Aeronautics, Imperial College London, London, SW7 2AZ, UK e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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such as the marine riser pipes used to convey fluids from the seabed to the sea surface. Steel catenary riser pipes and flexibles are being increasingly used offshore but in spite of their practical importance, curved configurations have received much less attention in the past than straight cylinders. For a straight circular cylinder the initiation of vortex shedding occurs at a Reynolds number of 47–49. The flow throughout the so called periodic laminar regime, which persists until Re 180–200, can remain two-dimensional if care is taken to manipulate the body’s end conditions such that vortices are shed parallel to the cylinder’s axis [1,2]. Among the first researchers who investigated the effect of curvature on the vortex shedding dynamics at low Reynolds numbers, Takamoto and Izumi [3] reported on the stable arrangement of vortex rings developed in experiments behind an axisymmetric ring. Leweke and Provansal [4] studied the flow past a ring at sufficiently large aspect ratios (ring perimeter=cross section diameter >30) that allow approximation of the flow patterns of a straight cylinder by periodic boundary conditions, hence discarding the influence of end effects; with smooth initial conditions parallel vortex shedding was found to be dominant. Their results compare favourably with the numerical simulations of flow past a quarter of ring performed by Miliou et al. [5], who observed in-phase parallel shedding when the flow was directed normal to the plane of curvature. However only few studies have been carried out with the flow aligned to the plane of curvature. Miliou et al. [6] investigated the same geometries of their previous publication exposed to a constant inflow parallel to the plane of curvature of the quarter ring and observed a fully three-dimensional vortex shedding at Re D 100 and Re D 500. When the inflow was applied onto the outside surface of the ring the vortex shedding in the upper part of the body was found to drive the wake dynamics in the lower end at one dominant shedding frequency for the whole cylinder’s span. In the case of an inflow directed towards the inside of the ring no vortex shedding was detected and the near wake reached a steady state at both Reynolds numbers: in this configuration a drag reduction of 12% was achieved. Miliou et al. related the stabilisation of the wake to the strong velocity component aligned to the cylinder’s axis: the shape of the stagnation face in this flow configuration gave rise to an axial flow directed towards the top of the cylinder, where the vortex shedding is expected to occur. This axial flow was associated with the production of streamwise and vertical vorticity components in the top region, which appeared to make the shear layer less susceptible to rolling up in a Von Karman street. These results can be compared with the outcome of the analysis carried out by Darekar and Sherwin [7] on bodies with a wavy leading edge. The authors observed a variation in the wake width along the span of the wavy cylinder: the flow deflected towards the most downstream section, named “geometrical minimum”, generated a wider separated shear layer as compared to the most upstream cross-sections. Different flow regimes could be identified depending on the type of waviness: the so called regime III A was found to suppress vortex shedding with a drag reduction of 16%, leading to a symmetric wake with respect to the cylinder centreline in analogy with the behaviour of the concave configuration investigated by Miliou et al. [6].

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The experimental work of Bearman and Owen [8] and Bearman and Tombazis [9] confirms that a sufficiently high wave steepness stabilises the near wake in a timeindependent state and results in drag reduction. In the previous studies on curved and wavy cylinders the body was kept fixed in a constant inflow. In the present work, numerical simulations of forced oscillation cases at Re D 100 have been performed on the same geometries tested in Miliou et al. [5] and Miliou et al. [6], with the aim to understand the wake dynamics of a highly three-dimensional configuration vibrating inside the the lock-in region: these conditions are meant to capture the fluid dynamic features of a freely vibrating pipe under the simplified assumptions provided by forced oscillations. The potential of the forced oscillation simulations to provide insight into the more complicated dynamics of freely vibrating structures has been the object of many studies on straight cylinders. Varying the amplitude and the frequency of the imposed oscillation Williamson and Roshko [10] compiled a map of the wake patterns induced by body motion and found that not all the parameter combinations ensured that the net energy transfer over a cycle was positive, as is required for a body to freely vibrate. More recently, Leontini et al. [11] carried out a numerical study of the parameter space for a two-dimensional cylinder, focussing on the wake modes and energy transfer mechanism; Kaiktsis et al. [12] performed a similar investigation showing that qualitative differences in the hydrodynamic forces and wake state occur according to whether the body is forced to vibrate below or above the resonant frequency. Carberry et al. [13, 14] identified two distinct wake states for a cylinder in forced pure sinusoidal vibration and they could replicate most of the features of a freely oscillating cylinder such as variations of the phase angle and switch in the timing of shedding (see also [15, 16]). However Al Jamal and Dalton [17] highlighted contrasting results concerning the phase angle variation in forced and free oscillation experiments. Section 2 presents the geometrical configurations and a brief overview of the computational techniques used in the present work. The results for the convex geometry follow in Section 3, focussing on the effect of two different types of motion on the vortex shedding dynamics, while the flow past the concave configuration is presented in Section 4. Finally the conclusive remarks are outlined in Section 5.

2 Problem Description and Numerical Method 2.1 Geometrical Configurations The computations herein presented involve two types of bodies, sketched in Fig. 1. According to whether the free-stream is applied on the outside or the inside of the quarter ring two different flow configurations can be identified, named respectively “convex” and “concave”. The convex cylinder in Fig. 1a includes a horizontal extension, 10 diameters long, that displaces the outflow plane downstream from the quarter ring in order to allow a full development of the wake; this condition is not

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10D θ

s /D=0

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1 R=

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Z s/D=19.6 X

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Fig. 1 Convex configuration (left) and concave configuration (right); the arrows indicate the freestream direction

necessary in the concave configuration in Fig. 1b, which only involves the ring part. A non-dimensional arc length, s=D, has been defined to identify the different spanwise locations: in both cases the end of the ring part is at s=D D 19:6 and the top section corresponds to s=D D 0.

2.2 Navier–Stokes Solver The three-dimensional computations have been performed using a spectral=hp element Navier–Stokes solver developed by Sherwin and Karniadakis [18]; Karniadakis and Sherwin [19]. The temporal discretisation is achieved by a stiffly stable splitting scheme which allows the primitive variables to be treated independently over a time step t. The solution at time tnC1 is obtained from the solution at tn over three sub-steps. First, the non-linear term is treated explicitly, then a Poisson equation for the pressure is obtained by taking the divergence of the pressure term and by enforcing the incompressibility constraint. Finally the diffusive term is treated implicitly in the last sub-step. Overall one Poisson equation for the pressure and one Helmholtz equation for each of the velocity components are solved for every time step. Further details on the splitting scheme can be found in Karniadakis and Sherwin [19] and Karniadakis et al. [20].

2.3 Numerical Boundary Conditions In both configurations the top boundary plane of the computational domain was du modelled using a symmetric boundary condition (i.e. w D 0, du dn dz D 0 and dv

dv D 0), while a fully developed zero stress condition was specified at the dn dz outflow. The free-stream velocity was imposed along all other boundaries, with the exception of the inflow in the concave configuration where the cylinder intersects the boundary: here an exponential term was added to the inflow velocity profile to achieve exponential decay within the boundary layer region around the cylinder.


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3 Convex Configuration 3.1 Forced Translation in the Transverse Direction In all the simulations performed the amplitude of motion has been kept fixed and equal to 0:5D, while the input frequencies for the forced oscillation have been varied from 0:9fs to 1:2fs , where fs is the Strouhal frequency for a fixed straight cylinder [2]. The values of input frequency and amplitude of oscillation correspond to a point in the parameter plane found by Williamson and Roshko [10] which lies close to the critical curve where the switching from a 2S to a 2P mode of shedding occurs. The wake topology at fi D 1:1fs is displayed in Fig. 2 (left) using 2 isosurfaces: the shedding at the top exhibits the features of a 2S mode, with two opposite-signed single vortices shed per oscillation cycle. Due to the three-dimensionality of the model, the vortex cores in the near-wake are bent according to the curvature of the body and start to distort further downstream. In proximity of the horizontal extension they become weaker and deform in a wavy fashion, developing lateral arms of vorticity. Figure 2 (right) illustrates the evolution in time of the lift coefficient isocontours in every section perpendicular to the cylinder’s axis. The vortex shedding appears to be in phase along the span: considering a fixed time instant in this plot no variation in sign of CFy occurs as s=D increases. This feature is strictly related to the type of motion and is absent when the body is kept fixed in an uniform flow. In this case Miliou et al. [6] found different shedding dynamics, shown in Fig. 3a: the vortex cores are straight and their distance from the body’s surface changes along the span, as the cylinder is curved. Therefore a gradual phase shift occurs as a consequence of the delay in the vortex shedding along the span of the cylinder. As expected from theoretical considerations [21, 22] if the body oscillates the correlation length of the vortices increases and the coupling of the flow in the spanwise direction becomes stronger, resulting in a more correlated form of shedding

Fig. 2 Left: Wake topology for the flow past the convex configuration at fi D 1:1fs visualised through isosurfaces at 2 D 0:1. Right: Time evolution of the sectional lift coefficient along the cylinder’s span

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Fig. 3 Isocontours of the v velocity component in the y-direction (positive out of paper) at Re D 1000: (a) Stationary cylinder case. (b) Oscillating cylinder case at fi D 1:1fs Fig. 4 Spanwise distribution of the time-averaged lift coefficient components in phase with the velocity (CLv ) and with the acceleration (CLa )

(Fig. 3b). Further, the distance of the first vortex from the body decreases when the cylinder is forced to oscillate: this is consistent with this case exhibiting higher forces on the body’s surface. The fact that the sectional forces in Fig. 2 (right) do not decrease with increasing s=D may appear in contrast to the weakening of the shedding in the lower part of the cylinder shown in Fig. 2 (left): however, in the horizontal extension (s=D > 19:6) the body undergoes only a drag type force in the y-direction due to the cross flow, since the inflow is parallel to the cylinder’s axis and does not generate vortex shedding. Therefore the horizontal part behaves like a slender body and provides a strong hydrodynamic damping to the whole structure. In Fig. 4 the time-averaged components of the lift coefficient in phase with the velocity, CLv , and with the acceleration, CLa , are plotted against the non-dimensional arc-length. CLv , which predicts when free vibration should take place, reaches the most negative value for s=D > 20 and increases towards the top sections: therefore the net energy per cycle is negative, as opposed to flow-induced motions which require positive energy transfer from the fluid to the cylinder. The wake topology at the other frequencies tested is substantially similar to the one presented above and a 2S type of shedding is observed in all cases. As expected


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from the theory, the formation length is found to decrease with increasing shedding frequency, resulting in higher forces on the cylinder’s surface. However these flow states lie outside the region of positive energy transfer and therefore predict that free vibrations would in fact not occur, in contrast to the results obtained for a straight cylinder with the same input parameters.

3.2 Forced Rotation About the Horizontal Extension Axis As shown in the previous simulations, the lower part of the body acts like a strong hydrodynamic damper in forced translation, preventing the whole structure from being excited by the flow. To avoid this mechanism an oscillatory roll motion about a horizontal extension axis through the lower part of the cylinder has been imposed as an alternative: the maximum amplitude, equal to 0:5D, is thus reached at the top section and linearly decreases with decreasing distance to the roll axis. At an input frequency equal to fi D 0:9fs , the 2 isosurfaces in Fig. 5 (left) show that the shed cores twist around their axes and exhibit spanwise waviness. Furthermore they are only slightly bent according to the cylinder’s curvature and detach from the main vortex at different spanwise locations for every time instant: the imposed motion leads to out of phase shedding and thus to the non uniform spanwise distribution of the sectional lift coefficient illustrated in Fig. 5 (right). In contrast to the translational motion previously considered, the main vortex is weaker and does not envelop the horizontal extension, which is now fixed. In Fig. 6 the timeaveraged components of the lift coefficient in phase with the velocity reach positive values for 2 < s=D < 10: in this interval the energy is conveyed from the fluid to the structure, which is subsequently excited (phase angle 2 Œ0; 180ı ]). Therefore the damping effect stemming from the lower part of the body is inhibited and the resulting energy transfer for the entire structure is positive.

Fig. 5 Left: Wake topology for the flow past the convex configuration at fi D 0:9fs visualised through isosurfaces at 2 D 0:1. Right: Time evolution of the sectional lift coefficient along the cylinder’s span

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Fig. 6 Spanwise distribution of the time-averaged lift coefficient components in phase with the velocity, CLv , and with the acceleration, CLa (left) and of the phase angle (right)

4 Concave Configuration Three-dimensional DNS of forced oscillation has been performed on the second configuration of Fig. 1 using the same input parameters as the convex case simulations. The vortices in the wake of the concave cylinder are visualised in Fig. 7 (left) through 2 isosurfaces. As the free-stream velocity is now directed towards the inside of the curved cylinder, the vortex cores are not interacting with the structure when they are convected downstream; however they appear to be bent according to the curvature of the leading edge in this case as well. Figure 7 illustrates for comparison the wake past a fixed cylinder in an uniform flow at the same Reynolds number [5, 6]. This case was found to suppress vortex shedding, giving rise to a steady wake without interaction between the shear layers. Moreover, in the absence of motion a variation of the wake width along the span was observed: the top section exhibited the widest wake, while the bottom one the narrowest. This phenomenon was related to the strong component of axial flow stemming from the stagnation face curvature and to the associated production of vorticity in the x- and y- direction in the developing shear layers at the top of the cylinder. As shown in Figs. 7 (left) and 8 (left), the transverse motion results in a disruption of this stabilising mechanism: the axial flow direction is not constant along the span and the formation of vorticity in the part of the cylinder most susceptible to periodic vortex shedding is weakened. The near-wake topology appears to be completely different from the one obtained for the convex configuration at the same input frequency. Figure 8 (left) shows that the shear layers are more contracted, generating higher forces on the body. Further, both the near and far wakes are much wider than in the convex configuration shown in Fig. 2 (left). This can be related to the induced velocity stemming from the circulation in the curved cores that is now directed towards the outside of the wake and


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Fig. 7 Comparison of the wake topology for the flow past the concave configuration. Left: forced oscillation at fi D 1:1fs , isosurfaces at 2 D 0:1. Right: steady wake in the stationary case visualised at 2 D 0:01 [6]

Fig. 8 Left: Spanwise vorticity isocontours overlaid on 2 D 0:1 isosurfaces in the case of forced translation (concave configuration). Right: Time evolution of the sectional lift coefficient along the cylinder’s span

not towards the centre-line as in the convex case. The secondary core between the shear layers is generated from the vorticity in the base region, which separates under the effect of motion; during one half cycle the strength of this vortex decreases until it fades out before reforming from the opposite-signed base vorticity in the remaining half of the motion. Finally we note that the absence of the horizontal extension allows the surface forces to decrease towards the lower part of the body, where the shedding is less strong. Figure 8 (right) shows that the shedding is in-phase along the whole length of the body and there is no switching in time.

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5 Concluding Remarks The vortex shedding past two different curved configurations has been investigated by imposing a sinusoidal motion on the body. In particular we have addressed the question of the role played in the near-wake dynamics by the orientation of the stagnation face and of the effect of two different types of forced oscillation on the forces distribution and energy transfer. In the convex configuration case the forced translation has led to in-phase shedding, with bent vortex cores that detach from the shear layers at the same time for every spanwise location. The strong spanwise correlation induced by the motion prevails on the effect of curvature that was responsible for the phase shift when the body was stationary. However the influence of curvature in forced vibration is significant in the lower region, where the cylinder is aligned with the inflow direction: the horizontal extension behaves in fact like a transversely oscillating slender body, causing a high hydrodynamic damping which ultimately results in a negative energy transfer. To replicate the features of a freely oscillating configuration, a transverse rotation about the horizontal extension has been imposed on this geometry: the amplitude of oscillation linearly decreases along the span, while the frequency is fixed. This kind of motion allows the body to be excited by the fluid, without the damping effect stemming from the horizontal extension motion. The wake topology results markedly different from the case of forced translation and exhibits straight vortex cores and out-of-phase shedding. Finally the effect of forced transverse vibration has been investigated in the concave configuration. In the absence of motion this geometry was found to suppress vortex shedding; the controlled oscillation disrupts the stabilising mechanism triggered by curvature and gives rise to a wide wake with staggered arrays of vortices shed alternately from the sides of the cylinder. The present work highlights the lack of a full correspondence between the flow states observed in forced oscillations for curved and straight cylinders; this suggests that a redefinition of the lock-in boundaries for more complex geometries should be undertaken in order to understand the combined influence of motion and curvature on the vortex shedding. Acknowledgement A. de Vecchi would like to acknowledge the Engineering and Physical Sciences Research Council (UK) who supports her position. This work has been carried out in the research group on Vortex flows of the Department of Aeronautics at Imperial College on the basis of CPU allocations of the computer facilities in the ICT cluster of Imperial College.

References 1. Eisenlohr, H., Ecklemann, H., 1989. Vortex splitting and its consequences in the vortex street wake of cylinders at low reynolds number. Physics of Fluids A1, 189–192. 2. Williamson, C., 1989. Vortex shedding in the wake of a circular cylinder at low reynolds numbers. Journal of Fluid Mechanics 206, 579–627.


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3. Takamoto, M., Izumi, K., 1981. Experimental observation of stable arrangement of vortex rings. Physics of Fluids 24, 1582–1583. 4. Leweke, T., Provansal, M., 1995. The flow behind rings – bluff-body wakes without end effects. Journal of Fluid Mechanics 288, 265–310. 5. Miliou, A., Sherwin, S., Graham, J., 2003. Fluid dynamic loading on curved riser pipes. ASME Journal of Offshore Mechanics and Arctic Engineering 125, 176–182. 6. Miliou, A., de Vecchi, A., Sherwin, S. J., Graham, J. M. R., 2007. Wake dynamics of external flow past a curved cylinder with the free-stream aligned to the plane of curvature. Journal of Fluid Mechanics 592, 89–115. 7. Darekar, R. M., Sherwin, S. J., 2001. Flow past a square-section cylinder with a wavy stagnation face. Journal of Fluid Mechanics 426, 263–295. 8. Bearman, P., Owen, J., 1998. Reduction of bluff-body drag and suppression of vortex shedding by the introduction of wavy separation lines. Journal of Fluid and Structures 12, 123–130. 9. Bearman, P., Tombazis, N., 1997. A study of the three-dimesional aspects of vortex shedding from a bluff body with a mild geometric disturbance. Journal of Fluid Mechanics 330, 85–112. 10. Williamson, C., Roshko, A., 1988. Vortex formation in the near wake of an oscillating cylinder. Journal of Fluids and Structures 2, 355–381. 11. Leontini, J., Stewart, B., Thompson, M., Hourigan, K., 2006. Wake state and energy transitions of an oscillating cylinder at low reynolds number. Physics of Fluids 18, 067101, 1–9. 12. Kaiktsis, L., Triantafyllou, G., Ozbas, M., 2007. Excitation, inertia, and drag forces on a cylinder vibrating transversely to a steady flow. Journal of Fluids and Structures 23, 1–21. 13. Carberry, J., Govardhan, R., Sheridan, J., Rockwell, D., Williamson, C., 2004. Wake states and response branches of forced and freely oscillating cylinders. European Journal of Mechanics, B23, 89–97. 14. Carberry, J., Sheridan, J., Rockwell, D., 2005. Controlled oscillations of a cylinder: forces and wake modes. Journal of Fluid Mechanics 538, 31–69. 15. Meneghini, J., Bearman, P., 1995. Numerical simulations of high amplitude oscillatory flow about a circular cylinder. Journal of Fluids and Structures 9, 435–455. 16. Blackburn, H., Henderson, R., 1999. A study of two-dimensional flow past an oscillating cylinder. Journal of Fluid Mechanics 385, 255–286. 17. Al Jamal, H., Dalton, C., 2005. The contrast in phase angles between forced and self-excited oscillations of a circular cylinde. Journal of Fluid and Structures 20, 467–482. 18. Sherwin, S., Karniadakis, G., 1996. Tetrahedral hp finite elements: algorithms and flow simulations. Journal of Computational Physics 124, 14–45. 19. Karniadakis, G. E., Sherwin, S. J., 2005. Spectral=hp Element Methods for CFD. Oxford University Press, Oxford. 20. Karniadakis, G. E., Israeli, M., Orszag, S. A., 1991. High-order splitting methods for the incompressible Navier–Stokes equations. Journal of Computational Physics 97, 414–443. 21. Bearman, P., Obasaju, E., 1982. An experimental study of pressure fluctuations on fixed and oscillating square-section cylinders. Journal of Fluid Mechanics 119, 297–321. 22. Bearman, P., 1984. Vortex shedding from oscillating bluff bodies. Annual Review of Fluid Mechanics 16, 195–222.


Successive Steps of 2D and 3D Transition in the Flow Past a Rotating Cylinder at Moderate Reynolds Numbers R. El Akoury, G. Martinat, M. Braza, R. Perrin, Y. Hoarau, G. Harran, and D. Ruiz

Abstract The flow past a rotating circular cylinder, placed in a uniform stream, is investigated by means of 2D and 3D direct numerical simulations, using the finite-volume version of the code ICARE/IMFT. The flow transition is studied for Reynolds numbers from 40 to 500, and for rotation rates ˛ (ratio of the angular and the free-stream velocities) up to 6. For a fixed Reynolds number, different flow patterns are observed as ˛ increases: Von-Kármán vortex shedding for low rotation rates, suppression of the vortex shedding at higher ˛, appearing of a second mode of instability for a high interval of ˛ where only counter clockwise vortices are shedd, and steady state flow for very high rotation speeds where the rotation effects keep the vortex structure near the wall and inhibit detachment. Three dimensional computations are carried out showing that the secondary instability is attenuated under the rotation effect. The linear and non-linear growth of the 3D flow transition are quantified using the Ginzburg-Landau global oscillator model. The analysis of the coherent structures under the rotation effect is performed by the proper orthogonal decomposition, as well the pattern reconstruction using the first POD modes. Keywords Rotation Transition DNS POD

1 Introduction The vortex dynamics of the flow around a fixed circular cylinder have been the objective of a considerable number of investigations. Comparatively, less work has been performed in the case of a rotating cylinder. This paper aims to study the R. El Akoury (), G. Martinat, M. Braza, R. Perrin, and G. Harran Institut de Mécanique des Fluides de Toulouse, CNRS/INPT UMR Nı 5502 Av. du Prof. Camille Soula, 31400 Toulouse, France Y. Hoarau Institut de Mécanique des Fluides et des Solides de Strasbourg, CNRS/ULP UMR Nı 7507 2 rue Boussingault, 67000 Strasbourg, France D. Ruiz ENSEEIHT, CNRS/IRIT UMR Nı 5505, Toulouse, France M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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Fig. 1 Schematic of the physical problem

transition in the wake past a circular cylinder under the rotation effect (Fig. 1). The flow depends mainly on two parameters: The Reynolds number Re D U1 D and D! , where U1 is the free-stream velocity, D the cylinder the rotation rate ˛ D 2U 1 diameter, the kinematic viscosity and ! the angular velocity of the cylinder. Earliest experiments on the flow past a circular rotating cylinder where performed by Reid [18], Prandtl [17] and Thom [21, 22]. More recently, the early phase of the establishment of the flow around a cylinder started impulsively into rotation and translation was investigated experimentally by Coutanceau and Ménard [9], and numerically by Badr and Dennis [3], for moderate Reynolds numbers (Re 1000). The same study was performed both theoretically and experimentally by Badr et al. [2], in higher Reynolds number range 103 Re 104 . The same initial stage of the vortex shedding was studied numerically by Chang and Chern [7], for 103 Re 106 end 0 ˛ 2, and later by Nair et al. [14] who provided detailed results for Re D 3800 and ˛ D 2. Concerning the established state, a number of investigation were performed at low and moderate Reynolds numbers, showing the suppression of the Von-Kármán vortex shedding when the rotation rate increases. Stojković et al. [19] were the first to notice the existence of a second shedding mode for 4:8 ˛ 5:15 at Re D 100. The two-dimensional numerical study of Stojković et al. [20] confirmed the existence of this second mode in the Reynolds number range 60 Re 200. Different flow regimes as rotation speed increases were also investigated numerically by Mittal and Kumar [13], at Re D 200, 0 ˛ 5. Later, Cliffe and Tavener [8] studied the effect of the rotation of a cylinder on the critical Reynolds and Strouhal numbers at the hopf bifurcation point. They noticed the restabilization of steady flows at large blockage ratios as the Reynolds number is increased even for non-rotating cylinders. The different flow regimes are studied in this paper for the flow around a rotating circular cylinder by means of 2D and 3D computations. The outlines of the finitevolume formulation of the ICARE code of the IMFT, Braza et al. [6], Persillon and Braza [16] are presented in Section 2. Detailed two-dimensional analysis for the flow transition is performed in Section 3 for Reynolds numbers from 40 to 500, and for rotation rates up to 6. Section 4 analyses the onset of the three-dimensional transition under the rotation effect. The amplification of the secondary instability is

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studied by means of the global oscillator model. The study of the coherent structures motion is next carried out by means of the proper orthogonal decomposition as well the pattern reconstruction in Section 5.

2 Principles of Numerical Method The 2D and 3D simulations were carried out using the code ICARE of the IMFT, in its finite-volume version. The governing equations are the continuity and the Navier-Stokes equations for an incompressible fluid, written in general curvilinear coordinates in the .x; y/ plane, while the z-component (in the spanwise direction) is in cartesian coordinates. The numerical method is based on a pressure-velocity formulation using a predictor-corrector pressure scheme of the same kind as the one reported by Amsden and Harlow [1], extended in the case of an implicit formulation by Braza et al. [6]. The temporal discretisation is done adopting the Peaceman and Rachford [15] scheme in an Alternating Direction Implicit formulation. The method is second-order accurate in time and space. The staggered grids by Harlow and Welch [10] are employed for the velocity and pressure variables. H-type grids are used because this kind of grid offers the possibility to introduce more physical boundary conditions on the external boundaries and it avoids branch-cut lines. A zoom of the grid around the obstacle is shown in Fig. 3. The characteristics of the different grids used are shown on Table 1. The grid used for the 3D simulation is (250 100 80) where the same grid is repeated in all d z sections. The spanwise length of the computational domain is 12D where D represents the cylinder diameter. A careful study of the numerical parameters and of the dimensions of the computational domain had been conducted for the final choice, in respect of the grid and the spanwise distance independence of the results, as well as for the 2D study. The boundary conditions are those specified in Persillon and Braza [16]. Concerning the spanwise free edges of the computational domain, periodic boundary conditions are applied.

Fig. 2 Computational domain

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Fig. 3 Magnification of the grid around the cylinder

Table 1 Characteristics of the different computational domains; N x and Ny: number of points in the x- and y-direction respectively; 2Nw : number of points on the cylinder surface; Xu and Xd : upstream and downstream length; 2Y u: vertical width of the domain; see also Fig. 2

Re Re Re Re

200 D 200; 4:35 < ˛ < 4:85 D 300 D 500

Nx 250 352 303 360

Ny 100 112 120 146

Nw 43 43 47 61

Xu 8.95 9.40 11.50 11.54

Xd 20.51 42.37 23.70 23.41

Yu 7.42 8.50 10.02 10.21

3 Successive Stages in the Two-Dimensional Transition The flow transition is analysed for Reynolds numbers 40 to 500 for different rotation rate numbers, ˛ varying from 0 to 6. The changes in the flow pattern are studied by means of the averaged and instantaneous streamlines, of the global parameters and of the vorticity fields.

3.1 Steady State Flow at Re D 40 For Re < 48, the flow remains steady for all the rotation rates investigated. In the fixed cylinder case, the flow shows two symmetric vortices attached to the cylinder. The rotation causes a loss of symmetry as shown in Fig. 4; stagnation points are attained in the upper part of the cylinder. As a consequence of the Magnus effect, the lift coefficient increases and the drag coefficient decreases.


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Fig. 4 Streamlines for different rotation rate values, Re D 40

Detailed flow computations were carried out for higher Reynolds numbers up to 500; similar flow regimes appear as ˛ increases. In the following section, the Re D 300 case is detailed.

3.2 Successive Stages of 2D Transition in the Flow Around a Rotating Cylinder at Re D 300 3.2.1 Different Flow Patterns For low rotation rates, ˛ < 2:5, the flow is unsteady, qualitatively similar to the fixed cylinder case, where the Von Kármán vortex shedding is observed, asymmetric towards the upper side of the cylinder due to the rotation sense. For higher rotation rates, the vortex shedding is suppressed; the flow remains steady until a rotation rate of 3.9 where a second mode of instability (mode II) appears for 3:9 < ˛ < 4:8. In this interval, only counter-clockwise vortices are shedd from the upper side of the cylinder in a periodic motion. This second mode of instability is due to the increasing of the rotation rate and of the velocity gradient between the two sides of the cylinder, in association with the strong viscous effect near the wall. Thus the streamlines that are closed around the cylinder start to have an oval-like form. The fluid flow forces this structure to be more elongated until detachment. Mode II disappears for higher rotation rates where the rotation effects keep the vortex structure near the wall and inhibit detachment. Figure 5 shows the different flow configurations for Re D 300 as ˛ increases.

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Fig. 5 Iso-vorticity contours for different states at Re D 300 0.2

R e =300

St

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0 0

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Fig. 6 Strouhal number St versus rotation rate ˛ at Re D 300

3.2.2 Global Parameters Figure 6 shows the evolution of the Strouhal number S t D UfD as a function of 1 ˛. For low rotation rates in mode I, S t is practically constant, and it shows a reduction as a function of ˛ before the first bifurcation. S t in mode II also decreases in the range of 3:9 ˛ 4:8. Therefore, the effects of increasing rotation have the tendency to diminish the instability mode and even to make it vanish. Concerning the lift and the drag coefficients, Fig. 7 represents the phase diagram .CD ; CL /. For ˛ D 0, the phase plot is like a figure of 8, because in each cycle, two vortices of equal strength are released from the upper and the lower side of the cylinder, and the frequency of the drag variation is twice that of the variation


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Fig. 7 Phase diagram .CD ; CL / for different values of ˛ at Re D 300

of lift. The rotation introduces asymmetry in the strength and the location of the positive and negative vortices. The phase plots form closed lobes in both mode I and mode II, corresponding to a periodic flow. The size of each lobe shows the amplitude of fluctuation of the global coefficients. It is shown that the amplitude of the drag coefficient increases as ˛ increases. In the mode II, large amplitudes for CD and CL are shown; in this case, the periodic flow is more complex than the mode I case.

3.3 Influence of the Reynolds Number Similar flow regimes appear as a function of the Reynolds number. The Von-Kármán vortex shedding is observed for low rotation rates ˛ < ˛L1 , ˛L1 increasing as Re increases. The flow is steady for ˛L1 < ˛ < ˛L2 . Mode II of instability appears in the range ˛L2 < ˛ < ˛L3 . ˛L2 and ˛L3 decrease as the Reynolds number increases where viscous effects are reduced, and the flow approaches the potential theory flow for lower rotation rates. The critical ˛ values are presented in Fig. 8

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for different Reynolds numbers in comparison with previous results of Mittal and Kumar [13] and Stojković et al. ([20], [19]). This figure shows also that the critical Reynolds number of appearance of the first flow unsteadiness increases with respect to ˛. Figure 9 shows the evolution of the Strouhal number for Re D 100, 200, 300 and 500.


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4 Three-Dimensional Transition This section analyses the onset of the 3D transition phenomena under the rotation effect concerning the coherent structures in the wake. Without rotation, at Reynolds number 200 the three-dimensionality starts from an amplification of the w component versus time in the near wake. This displays a linear amplification rate following a non-linear state that leads finally to a saturation state as reported by Persillon and Braza [16]. The w amplification announces the development of a secondary instability concerning the 3D modification of the von Kármán mode that starts to display a regular spanwise undulation (mode A). The spanwise undulation is shown in Fig. 10. The same kind of flow where wall rotation is applied (˛ D 1:5) displays however a total damping of the amplification mode (Fig. 11). Therefore the rotation attenuates the secondary instability and increases the critical Reynolds number of appearance of this instability. Indeed, at higher Reynolds number, Re D 300, the 3D undulation is clearly shown for ˛ = 0.5 (Fig. 14). The amplification of the above instability can be studied by means of the Landau global oscillator model, Mathis et al. [12]. @A D @t

r A „ƒ‚…

lr jA3 j C „ƒ‚…

li near growt h

non li near

@2 A r 2 @z „ ƒ‚ … 3Dund ulat i on

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Fig. 11 vanishing of the 3D undulation with the rotation, Re D 200, ˛ D 1:5

(1)

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The real part of the coefficients r and lr can be evaluated by the present DNS study, that provides the amplitude variation as a function of period. r can be evaluated by the log.A/ variation as a function of time. For Re D 200, ˛ D 0, r D 0:013 and for Re D 300, ˛ D 0:5, r D 0:034. The r evaluation allows furthermore the assessment of the non-linear growth coefficient lr : lr D r

A jA3 j

(2)

The sign of lr coefficient indicates the subcritical or supercritical nature of the present instability. lr can be evaluated near the saturation threshold. The values 3:055 and 2:314 are found for Re D 200 and 300 respectively, therefore the nature of the instability is subcritical (Handerson and Barkley [11]) It has been found that the rotation changes the instability nature comparing to the non-rotated case, where at Re D 300 the mode is supercritical, Bouhadji and Braza [5]. The evaluation of r and lr coefficients allows assessment of the real part of the Ginzburg Landau coefficient, r in respect to the 3D growth. In the saturation stage, the term @A van@t ishes. Therefore, r D e.r lr A2 /=2. This yields an assessment of dimensionless mur values: 4:43 103 and 7:26 103 for Re D 200 and 300 respectively. The above discussion provides the amplification characteristics of the global instability by means of the DNS approach and by simpler, global oscillator model. In the following section this paper aims at analysing the energy of the organised modes in space and time and to provide the pattern reconstruction of the coherent structures.

5 Analysis of the Organised Modes by the Proper Orthogonal Decomposition The analysis of the 2D and 3D organised modes under the rotation effect has been performed by the proper orthogonal decomposition, using the snapshot method, Berkooz et al. [4]. Figure 12 compares the energy of the first 20 P.O.D. modes for the different values of ˛ for Re D 200 in the 2D study A rapid energy decay is attained for both modes of instability. However, higher number of modes were needed to reproduce the flow pattern in the mode II, but a number of 20 modes are sufficient at this Reynolds number. The same study was performed for the 3D flow; Fig. 13 compares the energy of the first 20 modes of the 2D and the 3D cases. It can be seen that the 3D energy decay is less abrupt than the 2D case. This displays a more chaotic character that is captured by the 3D DNS. While 7 modes seems to be sufficient de reconstruct the 2D flow field in the mode I, higher number of modes, of order of 20, is needed in the 3D case as shown in Fig. 14.


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Fig. 12 Energy of the first 20 POD modes for different rotation rates, Re D 200

Fig. 13 Slope comparison of the energy contribution of the first POD modes in 2D and 3D simulations, Re D 200, ˛ D 0

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a instantaneous field

b reconstruction with the first 3 modes

c 19 modes

Fig. 14 Reconstructions of the instantaneous field with the first eigenmodes, Re D 300, ˛ D 0:5

6 Conclusion The present study analyses the successive transition steps in the flow around rotating circular cylinder for Reynolds numbers from 40 to 500, as the rotation rate ˛ increases from 0 to 6, by means of direct numerical simulations. For low Reynolds numbers (less than 48), the flow remains steady Increasing ˛ causes a loss of symmetry, an increasing of the lift coefficient, and a decreasing of the drag coefficient due to the Magnus effect. Detailed flow computations have been carried out for higher Reynolds numbers (up to 500). Similar flow regimes appear as a function of Re as ˛ increases. The Von-Kármán vortex shedding, asymmetric due to the rotation, disappears at a critical value of rotation rate, but a second mode of instability appears for a higher range of ˛ where only counter clockwise vortices are detached. For higher rotation rates, the flow is steady again because the rotation effects keep the vortex structure near the wall and inhibit detachment. Three-dimensional computations have been carried out in order to analyse the onset of the 3D transition phenomena under the rotation effect, concerning the coherent structures in the wake.


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It is shown that the rotation attenuates the secondary instability and increases the critical Reynolds number of appearance of this instability. The amplification characteristics of the global instability are analysed by means of the DNS approach and by simpler, global oscillator model. The analysis of the energy of the organised modes is carried out by proper orthogonal decomposition. For the 2D case, mode I reconstructions are satisfactory with an order of 9 modes, concerning the mode II, more modes were needed; an order of the 20 modes is sufficient. In the 3D case, about 15 modes are needed to capture the secondary instability. This work has a significant implications for the flow control strategies using rotating cylinders. Steady flows may occur for some intervals of rotation rates. Acknowledgements This work has been carried out in the research group EMT2 (Ecoulements Mono-phasiques, Transitionnels et Turbulents) of the Institut de Mécanique des Fluides de Toulouse. We are grateful to D. Faghani and A. Barthet concerning their collaboration in the P.O.D. approach. Part of this work is carried out on the basis of CPU allocations of the national computer centres of France CINES, CALMIP and IDRIS.

References 1. M. A. Amsden and F. H. Harlow. The SMAC method : a numerical technique for calculating incompressible fluid flows. Los Alamos Scientific Laboratory Report. L.A. 4370, 1970. 2. H. M. Badr, M. Coutanceau, S. C. R. Dennis, and C. Ménard. Unsteady flow past a rotating circular cylinder at reynolds numbers 103 and 104 . J. Fluid Mech., 220:459–484, 1990. 3. H. M. Badr and S. C. R. Dennis. Time dependent viscous flow past an impulsively started rotating and translating circular cylinder. J. Fluid Mech., 158:447–488, 1985. 4. G. Berkooz, P. Holmes, and J. Lumley. The proper orthogonal decomposition in the analysis of turbulent flows. Ann. Rev. Fluid Mech., 25:539–575, 1993. 5. A. Bouhadji and M. Braza. Compressibility effect on the 2d and 3d vortex structures in a transonic flow around a wing. ERCOFTAC Bull., 34:4–9, 1997. 6. M. Braza, P. Chassaing, and H. Ha-Minh. Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder. J. Fluid Mech., 165:79–130, 1986. 7. C. C. Chang and R. L. Chern. Vortex shedding from an impulsively started rotating and translating circular cylinder. J. Fluid Mech., 233:265–298, 1991. 8. K. A. Cliffe and S. J. Tavener. The effect of cylinder rotation and blockage ratio on the onset of the periodic flows. J. Fluid Mech., 501:125–133, 2004. 9. M. Coutanceau and C. Ménard. Influence of the rotation on the near wake development behind an impulsively started circular cylinder. J. Fluid Mech., 158:399–446, 1985. 10. F.H. Harlow and J.E. Welch. Numerical calculation of the time-dependent viscous incompressible flow of fluids with free surface. Phys. Fluids, 8:2182–2189, 1965. 11. R.D. Henderson and D. Barkley. Secondary instability in the wake of a circular cylinder. Phys. Fluids, 8:1683–1685, 1996. 12. C. Mathis, M. Provansal, and L. Boyer. Bénard-von kàrmàn instability: transient and forced regimes. J. Fluid Mech., 182:1–22, 1987. 13. S. Mittal and B. Kumar. Flow past a rotating cylindre. J. Fluid Mech., 476:303–334, 2003. 14. M. T. Nair, T. K. Sengupta, and U. S. Chauchan. Flow past rotating cylinders at high reynolds numbers using higher order upwiwnd scheme. Comput. Fluids, 27:47–70, 1998. 15. D.W. Peaceman and J.R. Rachford. The numerical solution of parabolic and elliptic differential equations. J. Soc. Indust. Appli. Math., 3:28, 1955.

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16. H. Persillon and M. Braza. Physical analysis of the transition to turbulence in the wake of a circular cylinder by three-dimensional navier-stokes simulation. J. Fluid Mech., 365:23–88, 1998. 17. L. Prandtl. Application of the “magnus effect” to the wind propulsion of ships. Die Naturwissenschaft, 13:93–108, 1925. Trans. NACA-TM-367, june 1926. 18. E. G. Reid. Tests of rotating cylinders. Technical Report NACA-TN-209, 1924. 19. D. Stojković, M. Breuer, and F. Durst. Effect of high rotation rates on the laminar flow around a circular cylinder. Phys. Fluids, 14:3160–3178, 2002. 20. D. Stojković, P.Schön, M. Breuer, and F. Durst. On the new vortex shedding mode past a rotating circular cylinder. Phys. Fluids, 8:1683–1685, 2003. 21. A. Thom. The pressure round a cylinder rotating in an air current. (ARC R. & M. 1082), 1926. 22. A. Thom. Experiments on the flow past a rotating cylinder. (ARC R. & M. 1410), 1931.

Direct Numerical Simulation of Vortex Shedding Behind a Linearly Tapered Circular Cylinder V.D. Narasimhamurthy, H.I. Andersson, and B. Pettersen

Abstract The three-dimensional transition to turbulence in the wake of a tapered circular cylinder with the taper ratio 75:1 has been analyzed by performing direct numerical simulation. The Reynolds number based on the uniform inflow velocity and the diameters at the wide and narrow ends were 300 and 102, respectively. The same Reynolds number range was previously studied by Parnaudeau et al. (J. Turbulence, 2007) but with a different taper ratio 40:1. The effect of taper ratio on the transition to turbulence was investigated in the present study. It was found that the Strouhal number versus Reynolds number curves nearly collapse, thereby indicating that a change in the taper ratio by a factor of two has only a modest effect on the Strouhal number. However, there still exists a significant contrast in the cellular shedding pattern. Flow-visualization of instantaneous 2 -structures and the enstrophy j ¨ j revealed that the mode A instability appeared around Re 200 and mode B around Re 250. Keywords Instability Transition Turbulence DNS Tapered cylinder

1 Introduction Three-dimensional flow over circular cylinders is a common phenomenon in many engineering applications which occurs behind oil-platform legs, chimneys, cooling towers and even tapered aircraft wings. Such three-dimensionalization of the separated flow is often induced by a spanwise variation of the cylinder diameter (e.g. tapered cylinders). A distinct feature of tapered cylinder wakes is that depending on the local Reynolds number (Relocal ) along the span, a range of flow-regimes V.D. Narasimhamurthy and H.I. Andersson () Norwegian University of Science and Technology (NTNU), Department of Energy and Process Engineering, 7491 Trondheim, Norway e-mail: [email protected] B. Pettersen NTNU, Department of Marine Technology, 7491 Trondheim, Norway M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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(e.g., laminar unsteady wake L3, transition in the wake TrW or shear layer TrSL, etc.) may exist side by side in the same geometry. Three-dimensional instabilities in the wake of tapered circular cylinders (L3 regime) was previously studied by Papangelou [1], Piccirillo and Van Atta [2], Vallés et al. [3] and more recently by Narasimhamurthy et al. [4]. In comparison, remarkably few investigations of the TrW regime for tapered cylinders has appeared. Recently Parnaudeau et al. [5] performed Direct Numerical Simulation (DNS) in the TrW regime with a taper ratio, RT D l=.d2 d1 / D 40:1 (where l is the length of the circular cylinder and d2 and d1 denote the diameter of its wide and narrow ends, respectively). In the present study RT D 75:1, which implies a more modest tapering than considered by Parnaudeau et al. [5] and the Reynolds numbers are same in both the cases. Thereby, the effect of tapering on the transition to turbulent process is investigated in the present study. In this DNS study, an in-depth exploration of the frequency spectra and the instantaneous vortical structures was carried out to understand both the evolution of large-scale structures (vortex dislocation or vortex splits) and the small-scale structures (mode A and mode B). Qualitative comparisons of the present results with the earlier numerical study in L3 regime [4], and in-house PIV (Particle Image Velocimetry) measurements by Visscher et al. [6] (where they studied TrSL regime) are also made.

2 Flow Configuration and Numerical Method The computational domain was as shown in Fig. 1. All dimensions were normalized by d2 . The mean diameter dm D 0:67. The aspect ratio (a D l=dm ), RT , and the Reynolds numbers Re2 , Re1 , Rem , based on the uniform inflow velocity (U D 1) and the diameters d2 , d1 , dm , respectively were as shown in Table 1. 22 49.5

SIDE WALL TOP WALL SIDE WALL d 1 = 0.34

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Case Present simulation Parnaudeau et al. [5]

a 74 40

RT 75:1 40:1

203 Re2 300 300

Re1 102 100

Rem 201 200

The Navier–Stokes (N–S) equations in incompressible form were solved in 3-D space and time using a parallel Finite Volume code [4, 10]. The code uses staggered Cartesian grid arrangement. Time marching was carried out using a third order explicit Runge-Kutta scheme for the momentum equations and an iterative SIP (Strongly Implicit Procedure) solver for the Poisson equation. Spatial discretization was carried out using a second order central-differencing scheme. The total number of grid points used was equal to 15 106 . The time step t D 0:003d2=U and the number of Poisson iterations per time step was equal to 50. A uniform inflow velocity profile U D 1 was fixed at the inlet without any freestream perturbations. A free-slip boundary condition was applied on both the side walls, top wall and the bottom wall (see Fig. 1). At the outlet, Neumann boundary condition was used for velocities and pressure was set to zero. The no-slip boundary condition on the cylinder body was implemented by using a direct forcing Immersed Boundary Method (IBM) [4, 9]. The computations were performed on a SGI Origin 3800 parallel computer. The total consumption of CPU-time was approximately equal to 12,000 h.

3 Results and Discussion 3.1 Frequency Analysis The time evolution of the instantaneous velocity components U, V, W and the instantaneous pressure, P, were sampled along two lines parallel to the axis of the cylinder and located 2dm and 12dm downstream the axis in X-direction, respectively. Both lines were offset by 1dm in Z-direction. The time traces of U, V, W and P were plotted in Figs. 2–5, respectively. The figures clearly indicate the oblique and cellular shedding pattern. Quantitative investigations of the frequency spectra were carried out by spectral analysis of the cross-stream velocity component (W) time trace. In Fig. 6a the Strouhal number (S t D f dm =U ) versus Relocal (D Udlocal =; dlocal is the local diameter) curve from Parnaudeau et al. [5] was compared against the present result. It is surprising to see that the curves nearly collapse, thereby indicating that a change in the RT by a factor of two does not affect the Strouhal number much. However, there still exists a significant contrast in the distribution of constant-frequency cells along the span. The vortex dislocations in the TrW state of flow behind a uniform circular cylinder typically occur at the location of mode A instability [8]. In contrast, these large-scale structures occur spontaneously along the whole span for tapered circular cylinders (see Fig. 6b). The two discontinuities in the local Strouhal number

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Fig. 2 Time evolution of the U velocity along the entire span. Y = 0 corresponds to Re2 and Y = 49.5 corresponds to Re1

Fig. 3 Time evolution of the V velocity along the entire span (see Fig. 2 for details)

Direct Numerical Simulation of Vortex Shedding Behind a Tapered Cylinder

Fig. 4 Time evolution of the W velocity along the entire span (see Fig. 2 for details)

Fig. 5 Time evolution of the pressure P along the entire span (see Fig. 2 for details)

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(S tlocal D f dlocal =U ) versus Relocal curve in Fig. 6b for the uniform circular cylinder correspond to change over of eddy-shedding mode from laminar-mode A and mode A-mode B, respectively [8]. However, for the tapered case it seems that the vortex dislocations depend primarily on RT . In Fig. 7 the S tlocal curve from the present simulation (TrW regime) was plotted together with the results from Narasimhamurthy et al. [4] (L3 regime) and the


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PIV measurements by Visscher et al. [6] (TrSL regime). In all the three studies RT D 75:1. The Strouhal number initially increases with Reynolds number and then decreases with the increase in Rem , similar to the uniform circular cylinder [11]. Piccirillo and Van Atta [2] in their experimental study (L3 regime) observed that the shedding cell size increases with the local diameter. A similar observation was also reported by Parnaudeau et al. [5]. In Fig. 7 it can be seen that shedding cell size clearly increases towards the large diameter for low Reynolds numbers (L3 and TrW). It seems that this observation is only valid for low Reynolds numbers as the curves for TrSL does not follow this trend.

3.2 Instantaneous Vortical Structures In order to identify the topology and the geometry of the vortex cores correctly the 2 -definition due to Jeong and Hussain [12] was used. 2 corresponds to the second largest eigenvalue of the symmetric tensor Sij Sij C ij ij , where Sij and ij are respectively the symmetric and antisymmetric parts of the velocity gradient tensor. Figure 8 shows the iso-surfaces of negative 2 at different instances in time, t. A three-dimensionality in the form of waviness in the spanwise vortex cores

Fig. 8 Three-dimensional 2 contours (negative 2 ) showing the topology and geometry of the vortex cores at different instances in time, t. Y-axis corresponds to the axis of the cylinder. The flow direction is from bottom to top

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(primary Karman vortices) is evident even in the L3 regime. The snapshots clearly illustrate the time-evolution of the vortex dislocations (around Y D 25–40) and the small-scale streamwise structures (mode A and mode B) along the span. However, vortex dislocations are more clearly visible in Fig. 9. Negative 2 , vorticity magnitude or enstrophy j ! j and the vorticity components evaluated at the same instant in time were plotted together. The vortex dislocations formed between spanwise cells of different frequency when the primary vortices move out of phase with each other are visible at Y 12, 23, and 40. The development of helical twisting of vortex tubes is visible in the vicinity of the vortex dislocations. Williamson [7] concluded that these helical twistings are the fundamental cause for the rapid spanwise spreading of dislocations, and indeed for the large-scale distortion and break-up to turbulence in a natural transition wake.


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Fig. 9 Three-dimensional vortical structures at the same instant in time, t D 47d2 =U as vortex dislocation occurs along the span. The flow direction is from bottom to top. (a) Negative 2 ; (b) Enstrophy j!j; (c) streamwise vorticity !x ; (d) spanwise vorticity !y ; (e) cross-stream vorticity !z . The surfaces colored yellow and red mark a particular value of positive and negative vorticity, respectively

In uniform circular cylinder wakes the Reynolds number will be constant along the whole span and therefore the individual modes of three-dimensionality (either mode A or mode B) exist along the entire span of the cylinder. Barkley and Henderson [13] from their Floquet stability analysis predicted the critical Reynolds number for the uniform circular cylinder to be 188:5 ˙ 1:0 and the wavelength of mode A equal to 3.96 diameters. However, in the present tapered cylinder study the Relocal varies along the span and both mode A and mode B co-exist in the same geometry. Thereby only a small span of the cylinder is available for each of the modes

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Fig. 9 (continued)

to develop (especially for mode A) and it is hard to pin-point the exact Reynolds number at which these modes start to appear. Parnaudeau et al. [5] reported that the mode A behind their tapered cylinder occurred in the L3 regime. The present flowvisualizations revealed that mode A appeared around Relocal 200 and mode B around Relocal 250.

4 Conclusions The effect of taper ratio on the transition to turbulence was investigated in the present study. It was found that the Strouhal number versus local Reynolds number curves nearly collapse, thereby indicating that a change in the taper ratio by a factor of two does not affect the Strouhal number much. However, there still exists a significant contrast in the cellular shedding pattern. Spot-like vortex dislocations in the TrW regime of uniform circular cylinders correspond to change over of eddy-shedding mode from laminar-mode A and mode A-mode B [8], but for tapered circular cylinders dislocation primarily depends on RT . In the present investigation, it was observed that the shedding cell size increases with the local diameter, which is in agreement with the previous studies at low Reynolds numbers [2, 4, 5]. Both mode A and mode B were found to co-exist in the same geometry but only in a small span of the cylinder. It was hard to pin-point


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the exact Reynolds number at which these modes develop. However, from the flowvisualization it can be concluded that the mode A appeared around Relocal 200 and mode B around Relocal 250. Acknowledgments This work has received support from The Research Council of Norway (Programme for Supercomputing) through a grant of computing time. The first author was the recipient of a research fellowship offered by The Research Council of Norway.

References 1. Papangelou, A., Vortex shedding from slender cones at low Reynolds numbers. J. Fluid Mech. 242 (1992), 299–321. 2. Piccirillo, P. S. and Van Atta, C. W., An experimental study of vortex shedding behind linearly tapered cylinders at low Reynolds number. J. Fluid Mech. 246 (1993), 163–195. 3. Vallès, B., Andersson, H. I. and Jenssen, C. B., Oblique vortex shedding behind tapered cylinders. J. Fluids Struct. 16 (2002), 453–463. 4. Narasimhamurthy, V. D., Schwertfirm, F., Andersson, H. I. and Pettersen, B., Simulation of unsteady flow past tapered circular cylinders using an immersed boundary method. In: Proc. ECCOMAS Computational Fluid Dynamics 06, Eds. P. Wesseling, E. Oñate, J. Périaux, TU Delft, The Netherlands, Egmond aan Zee (2006). 5. Parnaudeau, P., Heitz, D., Lamballais, E. and Silvestrini, J. H., Direct numerical simulations of vortex shedding behind cylinders with spanwise linear nonuniformity. J. Turbulence 8 (2007), No. 13. 6. Visscher, J., Pettersen, B. and Andersson, H. I., PIV study on the turbulent wake behind tapered cylinders. In: Advances in Turbulence XI, Eds. J. M. L. M. Palma, A. Silva Lopes, Springer, Portugal, Porto (2007), 254–256. 7. Williamson, C. H. K., The natural and forced formation of spot-like vortex dislocations in the transition of a wake. J. Fluid Mech. 243 (1992), 393–441. 8. Williamson, C. H. K., Vortex dynamics in the cylinder wake. Annu. Rev. Fluid. Mech. 28 (1996), 477–539. 9. Peller, N., Le Duc, A., Tremblay, F. and Manhart M., High-order stable interpolations for immersed boundary methods. Int. J. Num. Meth. Fluids. 52 (2006), 1175–1193. 10. Manhart M., A zonal grid algorithm for DNS of turbulent boundary layers. Comp. & Fluids, 33 (2004), 435–461. 11. Zdravkovich, M. M., Flow Around Circular Cylinders: Volume 1, Oxford University. Press (1997). 12. Jeong, J. and Hussain, F., On the identification of a vortex. J. Fluid Mech. 285 (1995), 69–94. 13. Barkley, D. and Henderson, R. D., Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322 (1996), 215–241.


Dynamics of Oblate Freely Rising Bodies P.C. Fernandes, P. Ern, F. Risso, and J. Magnaudet

Abstract We have investigated the characteristics and the causes of the zigzag periodic motions followed by solid bodies rising freely under the effect of buoyancy in liquid otherwise at rest. The frequencies, amplitudes and relative phases of the body velocity and orientation have been determined for a large range of parameters. Thanks to the determination of the body kinematics and using the force and torque balances provided by the generalized Kirchhoff equations, we have analyzed the dynamics of the body periodic motion. Keywords Wake instability Periodic motion Freely moving body

1 Introduction We have investigated the zigzag periodic motion followed by a solid body rising freely under the effect of buoyancy in a liquid otherwise at rest. The central difficulty is tied to the intrinsic coupling between the fluid and body motions, the body displacement inducing a disturbance in the fluid which in turn imposes loads that govern the body motion. Also the governing equations and the interpretation of the hydrodynamic couplings between the various degrees of freedom become significantly more complex as soon as the body exhibits some geometrical anisotropy and starts rotating. Predicting the kinematics and the hydrodynamic loads on such freely-moving bodies as a function of the characteristic parameters of the fluid/body system is however of primary importance in many applications ranging from aerodynamics [14], meteorology [13], sedimentology [20] to biomechanics [21] and dispersed two-phase flows [15]. Freely rising or falling bodies in a fluid at rest can display various types of path and a number of investigations has thus been devoted to identify these paths and the range of parameters in which they occur. The canonical cases of a thin disk [9, 22] P.C. Fernandes, P. Ern (), F. Risso, and J. Magnaudet Institut de Mécanique des Fluides de Toulouse, UMR CNRS/INP/UPS 5502, Allée du Prof. C. Soula, 31400 Toulouse, France e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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Fig. 1 Sketch of the body with the rotating axes (x,y,z) and definition of the components of the velocity U and rotation rate of the body

and that of a two-dimensional flat plate [1, 12] have been worked out in detail. In both cases, it turns out that, depending on the value of the control parameters, the motion can be rectilinear, planar time-periodic either without lateral drift or mean rotation (i.e. fluttering or zigzag) or with both of them (tumbling), quasi-periodic or even fully chaotic. Most of these regimes were also identified and analyzed numerically in the case of a solid sphere [11]. The kinematics and the dynamics of oblate gas bubbles as well as the motion induced in the liquid have also been investigated in detail experimentally [2, 3, 19] and by three-dimensional direct numerical simulations [17, 18]. The bodies considered here are short-length axisymmetric solid cylinders (Fig. 1) of diameter d , thickness h and density s chosen very close to the density f of the liquid, s =f ' 1. The rising motion of the bodies was followed by two travelling cameras; the time evolution of the position and orientation of the body was then determined by image processing. The characteristics of the body translation and rotation were investigated for aspect ratios, 2 < D d= h < 20, and Reynolds numbers, 100 < Re D V d= < 300, V being the vertical mean rise velocity of the body and the kinematic viscosity of the liquid. For some representative situations, the liquid motion induced by the body was characterized by Particle Image Velocimetry (PIV) and dye visualization (Fig. 2). We here summarize the main results of this study which are reported comprehensively in Refs. [4–8].

2 Onset of the Periodic Path The transition from the rectilinear rise to the oscillatory motion occurs for a critical Reynolds number Rec that depends on the body aspect ratio . As shown in Fig. 3 (symbols), Rec first decreases when increases, but then increases for 5 < < 10 and eventually becomes nearly independent of for thin bodies ( > 10). Since the periodic path is synchronized with the shedding of vortices, we have investigated the role of the wake instability on the body motion. We have therefore performed direct numerical simulations of the flow around fixed bodies

Dynamics of Oblate Freely Rising Bodies

215

Fig. 2 Velocity field for a steady case ( D 10, Re D 100): (a) freely rising body (rectilinear path; obtained by PIV), (b) fixed body (by DNS). Three dye-visualizations of the body wake: (c) rectilinear path ( D 10), (d) and (e) periodic path of a thick ( D 2) and a thin ( D 10) body, respectively 300 Rec Rec1 Rec2

Re

250

200

150

100

1 2

4

6

10

χ

15

20

Fig. 3 Onset of the oscillatory motion for a freely-rising body compared to the thresholds of the two successive wake instabilities of the same body when it is held fixed

216


of same shape and PIV measurements of the flow surrounding the freely-moving bodies. For all aspect ratios, the wake of the fixed body looses its axial symmetry above a critical Reynolds number Rec1 (plain line in Fig. 3) and becomes unsteady at Rec2 > Rec1 (dotted line). Both Rec1 and Rec2 decrease when the aspect ratio increases. For a Reynolds number Re smaller than Rec1 , the numerical result for the flow around a fixed body corresponds to the PIV measurements of the flow around the rectilinearly-rising body; in particular, the maximum of the reverse velocity Vt in the recirculation region, which characterizes the strength of wake effects, is the same. This velocity was shown to be an increasing function of the aspect ratio. Introducing the Reynolds number, Re D Vt d=, which is related to Re through the empirical relation 0:62 Re, the two wake bifurcations of the fixed body then corre1C spond to constant values of Re whatever the aspect ratio, Rec1 ' 72 for the loss of axial symmetry and Rec2 ' 78 for the loss of stationarity. Figure 3 shows that for thick bodies ( < 5), the onset of path oscillation coincides with the first destabilization of the fixed body wake (Rec Rec1 ), the wake instability causing a lift force and a torque able to induce the oscillatory motion. For thin bodies, on the contrary, the oscillatory motion appears at a Reynolds numbers higher than Rec2 . For Rec2 < Re < Rec , PIV measurements have shown that the wake of the freely-rising thin body is still stationary. These results are presented in detail in Ref. [8].

3 Characteristics of the Oscillatory Motion For Re > Rec , after a short transient following the release of the body from rest, the body exhibits a planar periodic motion, called zigzag or flutter. This motion is characterized by horizontal and vertical oscillations of constant amplitude superimposed to the body mean vertical rise, as well as oscillations of the body symmetry axis about the vertical direction. In all cases, the orientation and the horizontal velocity oscillate at frequency f , whereas the vertical velocity oscillates at 2f . We have shown that the body kinematics exhibit interesting properties in the system of axes sketched in Fig. 1, where x is directed along the body symmetry axis, y and z along two perpendicular radial directions, (x,y) defining the plane of motion: the component of the body velocity parallel to its axis is constant along the path u D u and the oscillatory behaviour of the velocity is thus restricted to the transverse component v along the y direction. Figure 4 shows that the evolution with Re of the axial velocity u compared to the gravitational velocity uo D ..1 s =f / g h/1=2 (g denoting gravity) can be conveniently modeled by Eq. (1). Similarly, scaling the transverse velocity v by f d , allowed us to gather all the results along a unique curve given by (2), the amplitude of which depends only on Re , as shown in Fig. 5 (left). The measurements of the amplitude of the body inclination (i.e. is the angle between the body symmetry axis x and the vertical) also gather along a master curve given by (3), the amplitude of which depends only on Re (Fig. 5, right). The Strouhal number, S t D f d=V , is approximately independent of Re but varies strongly with the aspect ratio, increasing from approximately 0:1 to 0:25 when varies from 2 to 10. This contrasts with the behaviour of the Strouhal number associated to


217

1.5 1.25

_ u / uo

1 0.75 χ=2 χ=3 χ=6 χ=8 χ = 10 eq. (1)

0.5 0.25 0

60

72 78

100

150

125

175

Re*

Fig. 4 Mean axial velocity u normalized by uo as a function of Re and . The fit corresponds to Eq. (1)

0.5

v / (2 π f d)

0.4

40

χ=2 χ=3 χ=6 χ=8 χ = 10 eq. (2)

χ=2 χ=3 χ=6 χ=8

30

θ (degrees)

0.6

0.3

χ = 10 eq. (3) 20

0.2 10

0.1 0

60 72 78

100

125 Re*

150

175

0

60

72 78

100

125

150

175

Re*

Fig. 5 On the left: Evolution of the amplitude of the transverse velocity v scaled by 2f d with Re and . On the right: Evolution of the amplitude of the body inclination as a function of Re and

the vortex shedding behind the bodies held fixed, since numerical simulations for Re just above Rec2 provided values nearly independent of the body aspect ratio: S t 0:12 for all . We have shown however that the Strouhal number S t built with the gravitational velocity uo follows a simple evolution in 1=2 (Eq. (4), Fig. 6, left). Though the body velocity and orientation oscillate at the same frequency, their phase difference depends strongly on the aspect ratio and is nearly independent of the Reynolds number: when varies from 2 to 10, the phase lag increases continuously from approximately 0ı to 110ı and then remains nearly constant for > 10. The phase difference ˆ between the oscillations of and v also depends on the aspect ratio though it evolves in a shorter range, as can be seen in Fig. 6 (right). For further details, the reader is referred to Refs. [5–8].

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P.C. Fernandes et al. 225

0.15

0.1

Φθ (degrees)

f . d / (uo χ1/2 )

210

0.05

0

125 Re*

100

150

180

χ=2 χ=3

χ=2 χ=3 χ=6 χ=8 χ = 10

60 72 78

195

χ=4

165

χ=6 χ=8 χ = 10

150

175

60 72 78

100

125 Re*

150

175

Fig. 6 On the left: Scaling of the Strouhal number St with the aspect ratio. On the right: Evolution of the phase difference ˆ between the oscillations of the body orientation and of the transverse velocity v as a function of Re and

u=uo 1:35 3:5 103 .Re Rec1 /;

v=.f d / St

1=2 / sin.2 S t t/; 9 10 .Re Rec1 1=2 / sin.2 S t t C 5:8 102 .Re Rec1 1=2

D f d=uo 0:1

2

(1)

ˆ /;

:

(2) (3) (4)

4 Forces and Torques Acting on the Freely-Moving Body The motion of a non-deformable body through an unbounded viscous fluid at rest at infinity is governed by the generalized Kirchhoff equations [10, 16], which express the linear and angular momentum balances for the complete fluid/body system. These equations are commonly written in the system of axes used previously with an origin fixed with respect to the observer and axes rotating with the body (Fig. 1). For uniform fluid and body densities, they read

s s du CA C B v r D Fx! C cos f dt f

(5)

s s dv C CB C A u r D Fy! sin f dt f

(6)

s dr . A B / u v D !z ; J2 C Q f dt

(7)


219

where r D d dt is the body rotation rate about the z axis. Equations (5)–(7) are written in dimensionless form, using the scales lo D d , uo D ..1 s =f / g h/1=2 , to D lo =uo , and Fo D .f s /# g for the length, velocity, time and force, respectively, # being the body volume. The left-hand side of (5)–(7) contains the proper- and added-inertia force and torque, the coefficients A; B; J2 and Q being known functions of the aspect ratio [5]. The right-hand side of (5)–(7) contains the projections of the buoyancy force, which are harmonic functions of the inclination angle , and the components Fx! , Fy! and !z of the force and torque resulting from the vorticity produced at the body surface. For the final periodic path on which all forces and torques evolve periodically, the inertia and buoyancy contributions can be determined using the kinematic results (1)–(4) and the vortical loads deduced from the balances (5)–(7). This led to the following conclusions regarding the main features of the body dynamics (more details can be found in Ref. [7]). The force balance in the axial direction is dominated by the steady components of drag and buoyancy, Fx! 1, resulting in a constant axial velocity u D u. In contrast the oscillatory contributions to the transverse force and torque balances are of the same order of magnitude as the mean drag. The transverse vortical force Fy! and the vortical torque !z exhibit harmonic oscillations at frequency S t . Their amplitudes are plotted in Fig. 7 as a function of Re and their phase difference ˆ (the force Fy! is in all cases ahead of the torque !z ) corresponds to the dimensionless time delay td D ˆ=.2 S t /, displayed in Fig. 9. Since td 0:8 whatever Re and , the time delay between the transverse vortical force and the vortical torque is of the order of the characteristic time of the mean rise, i.e. to D d=uo. This indicates that the problem is governed by two independent time scales. On the one hand, the body motion and the vortex shedding process have a periodicity of f 1 / to 1=2 . On the other hand, the evolution of the vortical structures in the body wake is governed (at leading order) by the time scale to . 0.1

0.8

0.08

Γ ω / χ3/4

0.4

0.06

r

y

Fω

0.6

χ=2 χ=3 χ=4 χ=6 χ=8 χ = 10

0.04 χ χ χ χ χ χ

0.2

0 50

72 78

100

150

Re*

=2 =3 =4 =6 =8 = 10

0.02

200

0 50

72 78

100

150

200

Re*

Fig. 7 On the left: Evolution of the amplitude of the transverse vortical force Fy! with Re and . On the right: Evolution of the amplitude of the vortical torque !z with Re and . The fits correspond to square root evolutions

220


0.75

0.75

0.5

0.5

0.25

0.25

0

0

−0.25

−0.25

−0.5

−0.5

−0.75

20

25

30

35

−0.75 58

60

t / to

62

64

66

t / to

Fig. 8 Time evolution of Fyur ( ), Fyg ( ), Fy! ( ), !z ( ) and v (); is superimposed to Fyg ; On the left: thick body with D 3, Re D 180, St D 0:125 and u D 1:23; On the right: thin body with D 10, Re D 240, St D 0:275 and u D 1:17

1.5

td

1

χ=2 χ=3 χ=4 χ=6 χ=8 χ = 10

0.5

0

150

200

250

300

Re Fig. 9 Dimensionless time delay between the oscillation of the transverse vortical force and that of the vortical torque as a function of Re and

The determination of the different terms of the Kirchhoff Eqs. (5)–(7) is also instructive on how the loads combine to support the body periodic motion. Figure 8 presents for a thick ( D 3) and a thin ( D 10) body the time evolution of the predominant periodic loads governing the zigzag motion. Along the transverse direction, it turns out that the vortical force Fy! is mostly balanced by the sum of the inertia term proportional to the body rotation r, say Fyur , and the transverse component of buoyancy, say Fyg D sin ; the transverse linear acceleration of the body having a smaller contribution. Similarly, the inertia torque related to the angular acceleration of the body is negligible and the vortical torque !z is mostly balanced


221

by the restoring added-mass torque associated with the transverse velocity component v. In particular, the stronger vortical torque observed for larger values of corresponds to a larger transverse velocity v, i.e. a larger angle between the symmetry axis and the instantaneous velocity of the body. It can therefore be noted that, on the one hand, the transverse vortical force Fy! is mainly balanced by two forces that depend on the body inclination and are =2 out-of-phase, whereas the vortical torque is linked to the transverse velocity v. The chronology of the loads acting on the body can then be summarized as follows (the description is detailed in Ref. [4] together with the evolution of the liquid motion obtained by PIV). Consider a body that is horizontal ( D 0) with a transverse vortical force acting on it, say Fy! < 0. To balance this force, the body is rotating clockwise (r < 0), generating the inertia force Fyur . While the body rotates, a transverse component of the buoyancy force is generated, compensating partly for Fy! and allowing the body to decrease its rotation rate. Since the rotation also induces a rearrangement of the wake, the vortical force saturates and starts decreasing. This in turn allows the body to reach a maximum inclination, for which the buoyancy force and the transverse vortical force balance each other, which happens after a time period of f 1 =4. In parallel, the wake also generates a torque !z on the body, which is now negative, corresponding to a positive transverse velocity v, i.e. the velocity vector is to the left of the body axis. Since the maxima of the vortical force and torque are separated by a time period set by the mean rise motion to , when the body reaches its maximum inclination, the vortical torque either continues to grow ( 3) or has just started decreasing ( D 2). Since the body axis starts now to rotate counter-clockwise, this results in very different constraints on the velocity vector, leading to the different types of body motion described in Ref. [6]: the thick body continues straight ahead and broadside on (v decreases in magnitude) whereas the velocity vector of the thin body has to rotate faster counter-clockwise (v increases in magnitude) so that the body leaves sideways. The next step, for which a numerical approach might be better suited, could be to determine quantitatively the rate at which vorticity is produced along the body surface and shed into the wake and to understand how this rules the behavior of the vortical force and torque. Acknowledgements We are very grateful to D. Fabre, H. Ayroles, S. Cazin, E. Cid, C. Trupin, J.-P. Escafit, J.-J. Huc and O. Eiff for technical assistance and helpful discussions.

References 1. A. Andersen, U. Pesavento, and Z.J. Wang. Unsteady aerodynamics of fluttering and tumbling plates. J. Fluid. Mech., 541:65–90, 2005. 2. C. Brücker. Structure and dynamics of the wake of bubbles and its relevance for bubble interaction. Phys. Fluids, 11:1781–1796, 1999. 3. K. Ellingsen and F. Risso. On the rise of an ellipsoidal bubble in water: oscillatory paths and liquid - induced velocity. J. Fluid. Mech., 440:235–268, 2001.

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4. P. Ern, P.C. Fernandes, F. Risso, and J. Magnaudet. Evolution of wake structure and wakeinduced loads along the path of freely rising axisymmetric bodies. Phys. Fluids, 19:113302, 2007. 5. P. C. Fernandes. Etude expérimentale de la dynamique de corps mobiles en ascension dans un fluide peu visqueux. Thèse de Doctorat, Institut National Polytechnique de Toulouse, France, 2302, 2005. 6. P.C. Fernandes, P. Ern, F. Risso, and J. Magnaudet. On the zigzag dynamics of freely moving axisymmetric bodies. Phys. Fluids, 17:098107, 2005. 7. P.C. Fernandes, P. Ern, F. Risso, and J. Magnaudet. Dynamics of axisymmetric bodies rising along a zigzag path. J. Fluid Mech., 606:209–223, 2008. 8. P.C. Fernandes, F. Risso, P. Ern, and J. Magnaudet. Oscillatory motion and wake instability of freely-rising axisymmetric bodies. J. Fluid Mech., 573:479–502, 2007. 9. S. Field, M. Klaus, M. Moore, and F. Nori. Chaotic dynamics of falling disks. Nature, 388: 252–254, 1997. 10. M. Howe. On the force and moment on a body in an incompressible fluid, with application to rigid bodies and bubbles at low and high Reynolds numbers. Q. J. Mech. Appl. Math., 48:401–426, 1995. 11. M. Jenny, J. Dusek, and G. Bouchet. Instabilities and transition of a sphere falling or ascending in a newtonian fluid. J. Fluid. Mech., 508:201–239, 2004. 12. M.A. Jones and M.J. Shelley. Falling cards. J. Fluid. Mech., 540:393–425, 2005. 13. P. Kry and R. List. Angular motions of freely falling spheroidal hailstone models. Phys. Fluids, 17(6):1093–1102, 1974. 14. H.J. Lugt. Autorotation. Ann. Rev. Fluid. Mech., 15:123–147, 1983. 15. J. Magnaudet and I. Eames. The motion of high-Reynolds-number bubbles in inhomogeneous flows. Ann. Rev. Fluid Mech., 32:659–708, 2000. 16. G. Mougin and J. Magnaudet. The generalized Kirchhoff equations and their application to the interaction between a rigid body and an arbitrary time-dependent viscous flow. Int. J. Multiphase Flow, 28:1837–1851, 2002. 17. G. Mougin and J. Magnaudet. Path instability of a rising bubble. Phys. Rev. Lett., 88(1):14502, 2002. 18. G. Mougin and J. Magnaudet. Wake-induced forces and torques on a zigzagging/spiralling bubble. J. Fluid Mech., 567:185–194, 2006. 19. W. Shew, S. Poncet, and J.-F. Pinton. Force measurements on rising bubbles. J. Fluid Mech., 569:51–60, 2006. 20. G.E. Stringham, D.B. Simons, and H.P. Guy. The behavior of large particles falling in quiescent liquids. US Geol. Surv. Prof. Pap., 562-C:C1–C36, 1969. 21. Z.J. Wang. Dissecting insect flight. Ann. Rev. Fluid Mech., 37:183–210, 2005. 22. W. Willmarth, N. Hawk, and R. Harvey. Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids, 7(2):197–208, 1964.

Parametric Study of Two Degree-of-Freedom Vortex-Induced Vibrations of a Cylinder in a Two-Dimensional Flow D. Lucor and M.S. Triantafyllou

Abstract We derive accurate, continuous response surfaces of two degree-offreedom vortex- induced vibrations (VIV) of flexibly mounted cylinders, for a wide range of transverse and in-line natural frequencies, to identify the parametric sensitivity of the VIV response. The flow is assumed to be two-dimensional and the Reynolds number equal to 1,000; the structure has the same low damping for the inline and transverse motions, while the transverse and in-line mass ratios are equal. The VIV response is studied within a wide range of the transverse natural frequency around the synchronization region. The variation of the in-line natural frequency is chosen to be larger than for the transverse natural frequency, in order to study multimodal response. The numerical technique uses a stochastic generalized Polynomial Chaos representation coupled to a spectral element based deterministic solver; hence the response is obtained as a continuous function of the parameters. Keywords Two degree-of-freedom motion VIV Surface response Sensitivity analysis

1 Introduction In vortex-induced vibration (VIV) studies of flexibly-mounted bluff bodies in crossflow, often only the transverse motion of the body is considered, in order to simplify the problem. This is supported by the fact that the amplitude of vibrations along the in-line direction is generally much smaller than along the transverse direction, when the in-line and transverse natural frequencies are equal. Recently, it was reported that the effect of the in-line motion on the transverse motion can be significant when D. Lucor () Institut Jean Le Rond d’Alembert, Unité Mixte CNRS-UPMC 7190, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France M.S. Triantafyllou Department of Ocean Engineering, 77 Massachusetts Avenue, Massachusetts Institute of Technology, Cambridge, MA 02139, USA M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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the natural frequency ratio fn X/fnY departs from one. The presence of the in-line X-motion can cause a significant change in the flow pattern behind the cylinder and may enhance the transverse Y-motion. The purpose of the present study is to systematically explore the effects of the coupling of the two motions as function of the oscillators natural frequencies. Instead of testing the response of the system for a finite set of values of the parameters, we use a method that provides us with a continuous representation of the response as a function of the variable parameters. We treat the natural frequencies of the oscillator as random quantities, in the sense that the frequencies are uncertain within specified ranges. This is often the case in complex operating environments when the specific parameters of the system are known only approximately; hence, it is important to find the response as well as its sensitivity to parametric changes. The numerical technique we chose couples a stochastic generalized Polynomial Chaos (gPC) representation to a spectral element based deterministic solver. The power of the gPC representation resides in its ability to assign a given probability distribution to the parameters (the natural frequencies), and to propagate its effects through the model to the numerical solution (the VIV responses). The gPC model then provides fast and efficient approximations of the response for any set of natural frequencies within the study interval. The main advantage of the method from a numerical point of view is to significantly reduce the computational cost compared to, for example, the Monte-Carlo method.

2 Numerical Method We will first introduce the general framework of the stochastic collocation method. Then we will apply it to our fluid-structure interaction problem and briefly present the deterministic solver on which it relies.

2.1 Stochastic Collocation Method The generalized Polynomial Chaos (gPC) method is a non-statistical method used to solve stochastic differential (SDE) and partial equations (SPDE) [20] and has been used for numerous applications [5–7, 14, 24]. It is a spectral representation of a random process in terms of orthogonal basis functions; the spatial and temporal evolutions of the basis coefficients providing quantitative estimates of the modeled random process solution. It is a means of representing second-order stochastic processes X./ parametrically through a set of independent random variables fj ./gN j D1 ; N 2 N; through the events of a random event space . The approach is very similar to the variational finite elements formulation for deterministic mechanical problems [8].

Parametric Study of Two Degree-of-Freedom Vortex-Induced Vibrations of a Cylinder

225

The representation has the advantage of separating the stochastic variables (only present in the polynomial basis) from the deterministic ones (modal coefficients): X./ D

1 X

Xk ˆk ..//:

(1)

kD0

Here fˆj ..//g are orthogonal polynomials in terms of a zero-mean random vec2 tor WD fj ./gN j D1 , satisfying the orthogonality relation hî ˆj i D hî iıij . For our application, we will only keep a finite set of random variables, i.e. fj gN j D1 with N < 3, and a M finite-term truncation of (1). Due to its tensorstructure form, a complete basis has M D .N C P /Š=N ŠP Š terms with P being the highest polynomial order in the expansion. We will drop the -dependence of in the following for notation simplicity. We have: X./ D

M 1 X

Xk ˆk ./:

(2)

kD0

The efficiency of the representation depends on the choice of the appropriate parametric family of random variables. There exists in fact, in (1), a one-to-one correspondence between the type of the orthogonal polynomials fˆg and the probabilistic law of the random variables . More details are given in [18, 25]. After solving for the deterministic coefficients Xk , we hold a representation which can be apprehended as a response surface providing with the sensitivity of the solution to the variability of the different parameters. It is then possible to perform a number of analytical operations onto this explicit representation. Moments as well as probability density function (pdf) of the solution can be evaluated. Due to the orthogonality of the modes, the moments can be easily computed. The mean solution is contained in the expansion term with zero-index. The second moment, i.e., the covariance function is given by a linear combination of the modal fluctuations [8]. The ‘non-intrusive’ approach or stochastic collocation method [21] of the gPC application is used in this study. It does not require any substantial modifications to the existing deterministic solver, and consists of projecting the stochastic solution onto the orthogonal basis spanning the random space. The Xk random coefficients can be directly computed as they take the following form: 8k 2 f0; : : : ; M 1g Xk D

< X./ ˆk ./ > : < ˆ2k ./ >

(3)

The inner product is based on the measure ./ of the random variables: Z

Z hf ./g./i D 2

f ./g./d P./ D

f ./g././d ;

(4)

226


with ./ denoting the density of the law d P./ with respect to the Lebesgue measure d and with integration taken over a suitable domain , determined by the range of . We recall that D 0 for k > 0 and the denominator can be tabulated prior to the projection. The evaluation of (3) is equivalent to computing multi-dimensional integrals over the domain . Different ways of dealing with high-dimensional integrations can be considered depending on the prevalence of accuracy versus efficiency [9]. A convenient approximation through numerical quadrature consists in replacing the integral by a finite weighted sum of the integrand values taken at some chosen points. When the number of grid points in multi-dimensions N becomes too large, one should not use a grid based on the full tensor product of one-dimensional grids. An alternative is to use sparse quadratures [15, 16] which require less quadrature points. For instance, the sparse quadrature based on Smolyak algorithm [19] has the advantage of remaining accurate with a convergence rate depending weakly on the number of dimensions. In this study, we use numerical quadratures of Gauss– and Gauss-Lobatto–type by full tensor products as our number of random dimensions is small. We insist on the fact that the deterministic solver will compute/provide X at those known quadrature points and not at randomly selected locations. The number of quadrature points nq to use depends on the regularity of the function to integrate. If it is well-known that nq points are enough to integrate exactly a polynomial function of leading order P D .2 nq 1/, there is no way of knowing a priori how smooth the solution X will be. Indeed, the knowledge of the ˆk ’s is not sufficient to foresee the regularity of the integrand. Therefore, we choose nq > P as our lower bound for this study. The minimum number of sampling of the solution (or minimum number of calls to the deterministic solver) for the computation of the kth coefficient of (3), when the leading order of the ˆk polynomial is p, is: Nq D .nq /N

with nq D p C 1:

(5)

This number becomes Nq D .P C 1/N for the estimation of the M th coefficient. Therefore, the minimum number of samplings to compute all M coefficients Xk is .P C 1/N .

2.2 Two Degree-of-Freedom Structural Model The non-dimensional equations of motions, based on a reference length D (cylinder diameter) and a reference velocity U (inflow velocity), that are numerically solved are: @X 1 CDrag .t/ XR C 2 X !X ./ C !X2 ./X D @t 2 mX @Y 1 CLift .t/ C !Y2 ./Y D YR C 2 Y !Y ./ ; @t 2 mY

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where !X ./ D 2fnX ./ and !Y ./ D 2fnY ./ represent the natural frequencies of the oscillator along the X – and Y –direction respectively. The -dependency indicates that the natural frequencies are considered uncertain within the intervals study. The forcing involves the non-dimensional time-dependent drag CDrag .t/ and lift CLift .t/ coefficients, computed iteratively by the flow solver. The hydrodynamic forcing will act as the coupling between these two equations. For all examples considered in this paper, the mass ratios of the structure are: mX D mY D m D s =f D 2 D 2, (s is the structural linear density and f is the fluid volumic density) and the damping ratios are: X D Y D with D 0 or D 3% depending on the case under consideration. Dimensional frequency values fOn may be computed with a proper scaling: fOn D .fn U /=D. The reduced velocity of the oscillator is Un D U=.fOnD/. The hydrodynamic loads in (7) are computed by a two-dimensional Navier– Stokes direct numerical simulation (DNS) solver, N T ˛r, based on the spectral/hp element method [9]. It was used previously for various VIV applications [3, 11–13]. A 2D rectangular grid of size Œ.22DI 55D/ .22DI 22D/ in the (x, y)–plane and made of 708 triangular elements [10] with Jacobi polynomial order p D 11 is used. This spatial resolution insures the presence of (at least) four computational nodes within the flow boundary layer developing at the wall at Re D 1;000. Our chosen non-dimensional temporal resolution requires at least 10;000 time iterations per period of oscillation. Time-statistics are collected over approximately 500–1,000 (when necessary) non-dimensional time units. In the following, we focus on the representation of natural frequencies following uniform distributions, so that only bounded variability ranges are considered and no physical values are ‘favored’ within each interval. Legendre polynomials are chosen to represent the response. For stochastic processes that require more than one random dimension to be represented, multi-dimensional Legendre polynomials are built in a tensor-like form. Two generic cases are considered in this study corresponding to different parametric ranges. In the first case (Case-A), we take fnX D fnY D fn C where follows a uniform distribution with zero mean and unit variance. The parameters fn and are constant parameters referring to the mean value and half the width of the support of the natural frequency respectively, and are chosen in such a way that fnX and fnY are uniformly distributed in Œ0:1114I 0:3024. This implies that only one uncertain parameter is considered and fnX =fnY D 1 always. The damping factor is D 0. This case will be used in the following as our reference case. In the second case (Case-B), we take fnX D fn1 C 1 1 and fnY D fn2 C 2 2 , where 12 are chosen to be finite independent identically-distributed (iid) uniform random variables with zero mean and unit variance. The parameters fn12 and 12 are constant and chosen in such way that fnX and fnY are uniformly distributed in Œ0:1273I 0:3820 and Œ0:1273I 0:1910 respectively. The fnY range is narrower than Case-A due to resolution requirements. This setup implies that the natural frequencies can both vary independently and that fnX =fnY 2 Œ0:5I 3. A more realistic damping factor of D 3% is used for this case.

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The output of the DNS simulations can be treated as a random field and decomposed onto the gPC basis (2). Herein, we focus mainly on the cylinder response statistics and show the drag force distribution; other physical quantities of interest can be evaluated similarly. It is worth mentioning that gPC representations of complex and highly nonlinear processes are sometimes inefficient in capturing the correct behavior of the system; particularly for long-time integration of stochastic systems characterised by a limit cycle oscillation response. For these cases, it was found that a spectral decomposition of the solution in terms of global basis exhibits severe limitations [1, 23]. Herein, we circumvent this problem by considering the statistical moments of the simulated response and loads as functions of the uncertain parameters, instead of decomposing the time-dependent turbulent pressure and velocity fields.

3 Results Continuous gPC representation of the cylinder response and loads are constructed with Legendre polynomials. To this end, the collocation procedure described in Section 2.1 needs the deterministic solution of our VIV problem at some discrete quadrature points in the parametric domain of interest. The deterministic DNS solver is therefore called for each known sample point, corresponding to a specific chosen pair of natural frequencies and time-statistics are collected for each case.

3.1 Case-A Here, a multi-element gPC approach is employed along the lines introduced by Wan [22]. This allows to decompose the parametric domain into smaller subdomains and then use the gPC representation in each element. Local refinements in chosen elements are possible when local gradients are large and better accuracy becomes critical. Here, we use seven Gauss-Lobatto–quadrature points in each sub-element and up to P D 5th order Legendre polynomials to reconstruct the response. Figure 1 presents the time traces of the cylinder vertical displacement for different natural frequency pairs. Those signals are quite representative of the possible scenarios that can be encountered in our computations (both Case-A and Case-B). In the first case, the motion eventually settles down to some regular single pattern after some time. This reflects on the cylinder trajectory as well. In the second case, there is an alternance of regions with small or large amplitude response that correspond to multiple trajectory patterns. The frequency of occurrence of these regions can be low and/or very irregular. Those cases are the most difficult to simulate and require long time integration. Finally, the last signal is quite irregular at a small time scale but exhibits some stationarity for a long time domain. This diversity implies that the temporal statistics collected for each case will not bear the same level of


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regularity and confidence. In other words, the temporal statistics might not be fully converged for some cases. This will obviously affect the overall accuracy of our study. Figure 2 presents the gPC response of the average 10% highest amplitude of the X- and Y-motion against the reduced velocity. The symbols correspond to the DNS deterministic samplings. The dotted lines delimit the computational sub-elements. The thin dashed curve with circles shows the one degree-of-freedom (d-o-f) response of the same system and is used as a reference. We emphasize that a maximum amplitude of 0:6D is what is usually reported in the literature and commonly accepted for 2D numerical simulations of one degree-of-freedom VIV [4]. We notice that allowing in-line motion enhances the maximum transverse cylinder motion relative to transverse-response only, with a 33% increase of the highest amplitude from around 0:6D–0:8D. Moreover the maximum transverse amplitude of the two degree-of-freedom case does not coincide with the one of the one degree-offreedom. We also notice that the gPC representation is more continuous and accurate for the Y- than the X-motion, in particular for the UnY Œ5I 7 range. Interestingly, the distribution of the in-line amplitude closely mimics the same peaks than the transverse amplitude.

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eight Gauss–quadrature points along each direction, which totals 64 DNS simulations. The deterministic solver is called for each quadrature point on the map, corresponding to a specific chosen set of parameters (represented by white dots in the figure). The negative values are due to some end-effects oscillation and only affect very small regions of the domain. A measure of the error of the representation is obtained by comparing the exact DNS solution and the gPC reconstructed solution at the 64 sampling points. In this case, the L2 norm of the error is within 12% accuracy. The first finding is the increase of the transverse amplitude response compared to the case with no X -motion. Indeed, the maximum amplitude reaches above 90% of the cylinder diameter D for certain combinations of the natural frequencies. Nevertheless, this result is still below 3D experimental amplitude results [2]. Figure 4b shows the same result presented differently in the light of the experimental work of Dahl et al. [2]. The isocontours are plotted vs. the reduced velocity (based on fnY ) and the natural frequency ratio fnX =fnY . Despite the somewhat reduced parametric ranges, it is clear that increasing the in-line to transverse frequency ratio caused a shift in the peak amplitude response to increasingly higher reduced velocities. This is in agreement with recent 3D experimental results [2]. Moreover, we notice at a frequency ratio between Œ1:5I 2:0 that two distinct response peaks appear. One centered around UnY 5 (and somewhat incomplete due to the range limitation) and one centered around UnY 6:25. This is in qualitative agreement with earlier experiments [2, 17]. Interestingly, there exists another peak centered around UnY 7:35 for higher in-line to transverse frequency ratios. Figures 4 and 5 show the average 10% X -motion highest amplitude and the drag coefficient distribution respectively. The gPC representation of integrated quantity such as the drag force on the cylinder is very accurate with a L2 norm of the error within 5% accuracy. As seen on the plots, there exists a strong relationship between the drag forces and the X -motion. The mean drag force globally increases for increasing fnX frequency. a

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4 Conclusions This is a first application of recently developed numerical stochastic collocation techniques to obtain accurate, continuous response surfaces in two degree-offreedom vortex-induced vibrations (VIV) of flexibly mounted cylinders; and to capture the sensitivity of the response to the change in both the transverse and in-line natural frequencies of the structure. The system was studied for a wide range of transverse natural frequency around the synchronization region using a stochastic generalized Polynomial Chaos representation coupled to a DNS flow-structure interaction deterministic solver. Although the results are obtained assuming twodimensional flow, the parametric dependence of the response is qualitatively close to observed results in experiments where the flow is three-dimensional: the in-line motion causes the maximum transverse response to increase by about 33%, but the peak responses occur at different reduced velocities and at multiple points. These results will guide us to explore a computationally more costly three-dimensional study of the same phenomenon.

References 1. P.S. Beran, C.L. Pettit, and D.R. Millman. Uncertainty quantification of limit-cycle oscillations. Journal of Computational Physics, 217(1):217–247, 2006. 2. J. Dahl, F.S. Hover, and M.S. Triantafyllou. Two-degree-of-freedom vortex-induced vibrations using a force assisted apparatus. Journal of Fluids and Structures, 22:807–818, 2006. 3. S. Dong and G.E. Karniadakis. DNS of flow past a stationary and oscillating cylinder at Re D 10000. Journal of Fluids and Structures, 20:519–531, 2005. 4. C. Evangelinos. Parallel Simulations of Vortex-Induced Vibrations in Turbulent Flow: Linear and Non-Linear Models. Ph.D. thesis, Division of Applied Mathematics, Brown University, 1999.


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5. R. Ghanem and B. Hayek. Probabilistic modeling of flow over rough terrain. J. Fluids Eng., 124(1):42–50, 2002. 6. R.G. Ghanem. Ingredients for a general purpose stochastic finite element formulation. Computer Methods in Applied Mechanics and Engineering, 168:19–34, 1999. 7. R.G. Ghanem and J. Red-Horse. Propagation of uncertainty in complex physical systems using a stochastic finite elements approach. Physica D, 133:137–144, 1999. 8. R.G. Ghanem and P. Spanos. Stochastic Finite Elements: A Spectral Approach. Springer, New York, 1991. 9. Andreas Keese. Numerical solution of systems with stochastic uncertainties: a general purpose framework for stochastic finite elements. Ph.D. thesis, Technische Universität Braunschweig, Mechanik-Zentrum, 2005. 10. D. Lucor. Generalized Polynomial Chaos: Applications to Random Oscillators and FlowStructure Interactions. Ph.D. thesis, Brown University, Rhode Island, 2004. 11. D. Lucor, J. Foo, and G.E. Karniadakis. Vortex mode selection of a rigid cylinder subject to VIV at low mass-damping. Journal of Fluids and Structures, 20:483–503, 2005. 12. D. Lucor, L. Imas, and G.E. Karniadakis. Vortex dislocations and force distribution of long flexible cylinders subjected to sheared flows. Journal of Fluids and Structures, 15:641–650, 2001. 13. D. Lucor and G.E. Karniadakis. Effects of oblique inflow in vortex induced vibrations. Flow, Turbulence and Combustion, 71:375–389, 2003. 14. D. Lucor and G.E. Karniadakis. Noisy Inflows Cause a Shedding-Mode Switching in Flow past an Oscillating Cylinder. Physics Review Letters, 92(15):154501–1; 154501–4, 2004. 15. Erich Novak and Klaus Ritter. High dimensional integration of smooth functions over cubes. Numerische Mathematik, 75:79–97, 1996. 16. Erich Novak and Klaus Ritter. Simple cubature formulas with high polynomial exactness. Constructive Approximation, 15:499–522, 1999. 17. T. Sarpkaya. Hydrodynamic damping, flow-induced oscillations, and biharmonic response. ASME Journal of Offshore Mechanics and Arctic Engineering, 117:232–238, 1995. 18. W. Schoutens. Stochastic Processes and orthogonal polynomials. Springer, New York, 2000. 19. S.A. Smolyak. Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Mathematics, Doklady, 4:240–243, 1963. 20. P. Spanos and R.G. Ghanem. Stochastic finite element expansion for random media. ASCE Journal of Engineering Mechanics, 115(5):1035–1053, 1989. 21. M. Tatang, W. Pan, R. Prinn, and G. McRae. An efficient method for parametric uncertainty analysis of numerical geophysical models. Journal of Geophysical Research, 102:21925– 21932, 1997. 22. X. Wan and G.E. Karniadakis. An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. Journal of Computational Physics, 209:617–642, 2005. 23. X. Wan and G.E. Karniadakis. Long-term behavior of polynomial chaos in stochastic flow simulations. Computer Methods in Applied Mechanics and Engineering, 195:5582–5596, 2006. 24. X. Wan and G.E. Karniadakis. Stochastic heat transfer enhancement in a grooved channel. Journal of Fluid Mechanics, 565:255–278, 2006. 25. D. Xiu and G.E. Karniadakis. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM Journal on Scientific Computing, 24(2):619–644, 2002.


Vortex Dynamics Associated with the Impact of a Cylinder with a Wall L. Schouveiler, M.C. Thompson, T. Leweke, and K. Hourigan

Abstract The flow resulting from the oblique collision without rebound of a circular cylinder with wall in a still viscous fluid is investigated both computationally and experimentally. We focus on the dynamics of the different vortex systems that form during such a motion and on their dependance on the impact angle. For this purpose, dye visualizations and numerical simulations using a spectral-element method are performed. Keywords Cylinder impact Vortex dynamics DNS Spectral-element method Flow visualization

1 Introduction A solid body colliding with a solid surface is a common situation in everyday life, involving complex energy exchanges. We focus here on the hydrodynamic effects resulting from such a collision. Their understanding is of interest for industrial applications, such as heat exchangers using fluidized-bed technology, where the collisions of particles with surfaces create strong currents allowing to enhance heat exchanges due to the resulting forced convection. A study like the present one, of the impact of an individual particle can also help understand particle-laden flows. Moreover, when the collision occurs on a dusty surface, the resulting flow gives rise to dust resuspension (see, e.g., [1]), an issue of importance to environmental and climate studies.

L. Schouveiler () and T. Leweke ´ Institut de Recherche sur les Phénomènes Hors Equilibre (IRPHE), CNRS/Universités Aix-Marseille, 49 rue Joliot-Curie, B.P. 146, F-13384 Marseille Cedex 13, France e-mail: [email protected] M.C. Thompson and K. Hourigan Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical Engineering, Monash University, Victoria 3800, Australia M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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2 Methodology The flow resulting from the normal or oblique collision without rebound of a circular cylinder (diameter D) on a wall in a still viscous fluid is investigated. For this purpose, the cylinder is impulsively started from rest, travels a distance L through the fluid at a constant velocity U towards a wall, then stops at the moment of contact with the wall. The parameters of this configuration (see Fig. 1) are the nondimensional travel distance L=D, the Reynolds number Re D UD=, where the fluid’s kinematic viscosity, and the impact angle ˛, i.e., the angle between the body trajectory and the normal to the wall. In the present paper we mainly focus on the dependance of the vortex dynamics on the impact angle ˛. For this purpose visualizations are performed for five values of ˛, namely 0ı , corresponding to the normal impact, 21ı , 33ı , 47ı and 62ı . Experiments were performed at a Reynolds number Re D 200, for which the flow stays two-dimensional, except near the cylinder ends. This Reynolds number is above the critical value of around 50 for the onset of vortex shedding behind the cylinder. Because we wish to limit the present investigation to stationary initial states, with two symmetrical vortices of opposite sign attached behind the cylinder, the approaching distance L=D is chosen short enough to avoid the development of vortex shedding, but large enough for the development of well-defined vortices. These conditions are met for the distance L=D D 4 considered here.

Fig. 1 Problem definition and parameters

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2.1 Numerical Method The numerical approach for investigating the hydrodynamics associated with the impact of a particle on a wall is based on the spectral-element method incorporating a deforming mesh. More details on the method can be found in Thompson et al. [2].

2.2 Experimental Details Experiments are conducted in a 600 mm high glass tank with a square horizontal cross-section of 500 500 mm2 , filled with water. An aluminium cylinder of length 368 mm and diameter D D 15.91 mm is suspended horizontally using two inelastic threads (see Fig. 2). The threads are attached to a computer controlled stepper motor, allowing to lower the body at a specified velocity U , and to impulsively stop the cylinder motion at the position of contact with the wall. An inclined wall is placed at the bottom of the tank. It consists in a plate of plexiglas attached to a second plate (using a swivel articulation), itself placed at the bottom of the tank. This allows for a continuous variation of the inclination angle. The cylinder axis is perpendicular to the wall slope in such a way that the contact between the cylinder and the wall after impact is a line perpendicular to the wall slope.

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For the visualizations, the cylinder is coated with a solution of green fluorescent dye prior to lowering it into the water. Dye patterns created by the flow entrainment are illuminated by a vertical light sheet perpendicular to the cylinder axis and passing by its center. An argon laser beam is spread out by a cylindrical lens for generating the light sheet. Visualizations are recorded with a digital video camera operating at 25 frames per second and aligned with the cylinder axis. The movies are then analyzed frame by frame to extract the vortex trajectories. We use the coordinate system shown in Fig. 1, and time t is non-dimensionalized using the advection time scale: D t=.D=U /. The origin D 0 corresponds to the instant of impact.

3 Results 3.1 Normal Collision Figure 3 shows, for a normal impact (˛ D 0) at Re D 200, dye visualizations at the instant of the impact ( D 0) and three different times after the impact corresponding to D 2:1, 4.9 and 15.1, as well as the corresponding evolution of the vorticity field deduced from the numerical predictions. While the cylinder approaches wall, the dye is entrained into the two longitudinal vortices that form in its wake, as can be seen at the instant of the impact D 0. After the impact, these primary vortices continue to travel towards the wall because of the inertia, thus going round the cylinder. Then they impact the wall and are convected away from the cylinder. This motion induces secondary vorticity on the cylinder surface that rolls up to form a secondary longitudinal vortex of opposite sign on each side of the cylinder ( D 2:1). Simulations show another source of secondary vorticity of same sign in the viscous boundary layer at the wall, that can not be exhibited by the visualizations because the dye is on the sphere surface. Secondary vorticity encircles the primary vortex when time increases. Both dye visualizations and numerical simulations show that the planar symmetry with respect to the plane x D 0 is preserved during the normal impact for this value of the Reynolds number. Experimental trajectories of the primary vortex centers and of the centers of the secondary vortices that are induced at the sphere surface are plotted in Fig. 4. Experimental vortex centers are defined as being the center of the dye spiral that appear on the visualizations. Centers of the secondary vortices can be followed for a short time between D 0:7 and D 2:8 because of the fast diffusion of the vorticity. In contrast with secondary vortices, it is possible to follow the primary vorticity centers for a long time. Their trajectories are nevertheless plotted for D 0–20.3 only, because it has been previously shown (Thompson et al. [3]) that, due to the large difference between the diffusivities of dye and vorticity, the dye centers are a good markers for the maximum of the vorticity for a limited time only. Figure 4 shows that primary wake vortices are advected radially outwards with an additional vertical “rebound” component due to the secondary vorticity.


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Fig. 3 Flow visualizations (left) and simulations (right) of the cylinder impact for Re D 200, L D 4D and ˛ D 0; D 0; 2:1; 4:9; 15:1 from top to bottom

3.2 Oblique Collision For investigating the effect of the impact angle ˛ between the normal to the wall and the cylinder trajectory, experiments have been performed for four values of ˛, namely 21ı ; 33ı ; 47ı and 62ı , in addition of the case of the normal impact ˛ D 0 described in the previous section. Figure 5 compares dye visualizations and vorticity fields as deduced from the numerical simulations for the intermediate angle ˛ D 33ı and Fig. 6 for the maximum angle investigated ˛ D 62ı . We first note on Figs. 5 and 6 that for non-normal impacts, the planar symmetry is not preserved even at the initial instant D 0.

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Fig. 4 Trajectories of vortex centers for Re D 200; L D 4D, ˛ D 0. Primary wake vortices () and secondary vortices from sphere surface (ı)

Fig. 5 Flow visualizations (left) and simulations (right) of the cylinder impact for Re D 200, L D 4D and ˛ D 33ı ; D 0; 2:1; 4:9; 15:1 from top to bottom


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Fig. 6 Flow visualizations (left) and simulations (right) of the cylinder impact for Re D 200, L D 4D and ˛ D 62ı ; D 0; 2:1; 4:9; 15:1 from top to bottom

On the far side of the impact line (x < 0) the wake vortex and the secondary one combine together into a pair of vortices of opposite sign. Such a dipole structure has a self-induced velocity, which is responsible for its ejection, as seen on Figs. 5 and 6. The trajectories of the centers of these two vortices appear curved towards the primary vortex (see Fig. 7), because this vortex is stronger than the secondary one. Comparison of Fig. 7(a)–(d) show that the radius of curvature tends to increases with ˛, because the relative difference in strength of the two vortices tends to diminish with ˛. On the other side of the contact line, the primary vortex travels on smaller circles because the circulation of the secondary vortex is much lower than the one of the primary vortex.

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Fig. 7 Trajectories of vortex centers for Re D 200; L D 4D. (a) ˛ D 21ı , (b) ˛ D 33ı , (c) ˛ D 47ı , (d) ˛ D 62ı . Primary wake vortices () and secondary vortices from sphere surface (ı)

4 Conclusion Numerical simulations and experimental flow visualizations have been performed for investigating the hydrodynamics of a cylinder impacting a wall, these two approaches show good agreement. Non-normal collisions give rise to strong vortex dipoles that are advected over long distance from the impact area. This flow leads to an increase of the mass transfer that can result in an enhancement of mixing and heat transfer at the wall.

References 1. Eames I., Dalziel S.B., Dust resuspension by the flow around an impacting sphere, J. Fluid Mech. 403, 305–328, 2000. 2. Thompson M.C., Hourigan K., Cheung A., Leweke T., Hydrodynamics of a particle impact on a wall, Appl. Math. Model. 30, 1356–1369, 2006. 3. Thompson M.C., Leweke T., Hourigan K., Sphere-wall collision: vortex dynamics and stability, J. Fluid Mech. 575, 121–148, 2007.

Modification of the Flow Structures in a Swirling Jet K. Atvars, M. Thompson, and K. Hourigan

Abstract The effect of a downstream bluff body on upstream vortex breakdown has been investigated in an open-flow, motivated by the potential for tissue culture in swirling flow bioreactors. A sphere was placed on the central axis of a swirling jet issuing into a tank of stagnant water. The position of the stagnation point of a vortex breakdown was tracked as a result of varying the azimuthal Reynolds number. While it is known that an increase in azimuthal Reynolds number leads to the upstream movement of the vortex breakdown stagnation point, this investigation focuses on the position of the vortex breakdown as a function of sphere size, axial Reynolds numbers, and sphere position. It was found that the distance from the jet exit to the stagnation point scaled with axial Reynolds number to the half-power, and that distinctive flow topologies of closed and open recirculation zones were found that were analogous to the cone and bubble forms previously found. Varying the size of the sphere was found to affect the flow only for sphere diameters comparable or larger than that of the nozzle – a smaller sphere size was found not to modify the flow significantly from the no-sphere case. Finally, the distance of the sphere from the nozzle affected the stagnation point location in the near-nozzle region. A correlation was also found between the swirl setting for various sphere distances at which the vortex breakdown changed form, with the swirl setting that gave the same stagnation point position with the no-sphere case. Keywords Vortex breakdown Bioreactors Stagnation point Bluff-bodies Incompressible flow

K. Atvars (), M. Thompson, and K. Hourigan Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical & Aerospace Engineering, Monash University 3800, Melbourne, Australia M. Thompson and K. Hourigan Division of Biological Engineering, Monash University 3800, Melbourne, Australia e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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1 Introduction Tissue engineering typically involves cells being seeded through convection on scaffolds in a bioreactor in order to establish a 3D culture. There is a requirement for the tissue cells to be cultured under well-controlled operating conditions and in an optimum environment. The process of tissue culture depends crucially on the supply of soluble nutrients and oxygen. Furthermore, in the case whereby the engineered tissues would need to possess biomechanical properties that are critical to their function in vivo (relevant to musculoskeletal tissue engineering), the application of mechanical forces acts as an important modulator of cell physiology. Different forms of bioreactors for which we have investigated the flow fields include the closed type with a spinning lid plus either a fixed lid [10] or a free surface [4, 5], or an open type with a swirling jet injection below a free surface and an outflow at the bottom [7]. In all these situations and for a range of conditions, a recirculation region, or vortex breakdown bubble, has been observed to occur along the centreline axis. Whether the presence of vortex breakdown in a bioreactor is beneficial or detrimental to tissue engineering is unclear, but certainly optimization of tissue growth requires the control of vortex breakdown. Many control methods have been tried in an attempt to influence the position and occurrence of vortex breakdown. As described in the review by Mitchell & Delery [9], vortex breakdown generally can be catagorised as either pneumatically or mechanically based. In seeking to understand the underlying physics of the open flow vortex breakdown, Billant et al. [2] performed an experiment on a swirling jet issuing into an open tank. This minimised the effects of confinement that could, in addition to the effect of changing flow conditions, influence vortex breakdown onset significantly [11]. Using a rotating honeycomb section in an axial flow, Billant et al. [2] were able to independently vary the rotational and axial velocity of the vortex core. They found, in addition to the known bubble state of vortex breakdown, a cone state, which has no closed recirculation zone. Furthermore, they confirmed the existence of a critical swirl number that determines the onset of vortex breakdown, with the swirl number defined as a ratio of maximum azimuthal and axial components of velocity upstream of breakdown. Interestingly, vortex breakdown was also found to be relatively independent of the axial Reynolds number. Investigations on pneumatic control of vortex breakdown in an open tank are limited: Khalil et al. [7] used a sinusoidal variation in the axial flow rate to alter the shedding frequency of shear-layer vortices, and Gallaire et al. [6] directed radial jets to act on the perimeter of the swirling jet to excite particular mode shapes in the wake of a swirling jet. However, unlike Khalil et al. [7], Gallaire et al. [6] were unable to affect the position or occurrence of breakdown with their shear-layer manipulation. It was suggested that this was because the mechanisms leading to breakdown act on the central region of the jet [12]. In primarily looking at blockage effects in pipe flows, Mattner et al. [8] studied a mechanical method of vortex interaction, by placing a sphere on the central axis of a swirling pipe flow. It was found that for particular swirl settings, a steady recirculation zone could be formed upstream of the sphere surface, which was enclosed

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by the vortex breakdown shear-layer reattaching to the sphere surface. It was noted that the swirl required to form this upstream stagnation point was significantly less than that required to form breakdown without a sphere. This was not unexpected in light of the work of [3], who theorised that the onset of breakdown was caused by the appearance of negative upstream azimuthal vorticity, the existence of which was confirmed in [8] by interpolating LDV measurements of the flow. The current investigation aims to extend research into mechanical means of interacting with a vortex core by placing a sphere in an open tank swirling jet. It is intended that some insight can be inferred from these experiments into the behaviour and possibly the mechanism of open-flow vortex breakdown, as well as the modification of bioreactor flows by the presence of tissue culture scaffolds.

2 Experimental Setup In this investigation, experiments were performed in an open-tank flow apparatus, as shown in the schematic diagram of Fig. 1 and which is similar to that designed and used by Billant et al. [2]. A large perspex tank of rectangular cross section (570 600 1540 mm) is filled with filtered tap-water. A head unit issues a swirling

Motor, Gearbox

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Head Unit

Outlet Nozzle Sphere

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Multiple outlet holes Axial Flow Pump

Fig. 1 Schematic diagram of experimental apparatus. A sphere is placed under the contraction nozzle with a sting attached to the side of the tank. Both motor and pump are controlled separately by computer to generate axial and azimuthal velocity components. Tank, sphere and head unit dimensions are to scale

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jet of water through an exit nozzle of diameter DON D 39:5 mm, held well beneath the water level in the tank. The axial component of the jet is generated using a viscous-disk pump, and monitored for consistency via a feedback mechanism from a magnetic flow meter. The rotational or swirling component of the jet is generated by the rotation of a honeycomb section inside the head unit, and a contraction nozzle is then used to reduce to the size of the jet. Both the axial-flow pump and the motor rotating the honeycomb are controlled by individual motors and motor controllers, allowing independent variation of the axial and rotational velocity components. Previous studies by Khalil et al. [7] on this particular apparatus have confirmed the axial and azimuthal velocity profiles are comparable with those of Billant et al. [2]. Buoyancy effects can be non-negligible for the flow velocities involved here, as noted by [2], so an attempt was made to reduce thermal effects by allowing the entire rig to equalise in temperature over a period of days by running both motors at speed before experiments commenced. During experiments, the temperature was constantly monitored to ensure flow (and thermal) conditions remained constant. A short distance downstream of the nozzle outlet, the swirling jet impinges on a sphere of diameter DOS , placed on the central axis of the nozzle at a distance away from the nozzle tip of xOS . Various sphere sizes were used, and each was held in place on the central axis using a 0.127DON diameter sting of a length greater than 7DON and held against the side wall of the tank with a perpendicular arm downstream of the sphere. This allowed the sphere to be held rigidly in position, with no observable movement or vibration when the swirling jet was switched on. Accurate placement under the nozzle was achieved by imaging the sphere-nozzle arrangement from perpendicular positions. The sphere position was measured to be horizontally within 1% of the central axis, and vertically within the measurement uncertainties (typically 3%) of the required distance from the nozzle. Visualisation was achieved by the use of long exposure photography with a small quantity of particles in the water, giving streak-line images to depict the flow structures. The flow was illuminated simultaneously from both sides of the tank by a 3 mm thick light-sheet made with two apertured 1,200 W stage-lamps. This method of visualisation ensured only a relatively small number of particles was needed to visualise the flow over the course of experimenting. This had the advantage of not interfering with steady-state flow conditions by having to regularly inject flowvisualisation liquids into the system, or needing to keep filtering the fluid between experiments, such as is required with PLIF. Images were taken using a single 1.3 MB CCD Pixelfly camera set perpendicular to both the light sheets and the flow axis, and focused with a 28 mm Nikon lens to image only the region around the upstream side of the sphere and the outlet part of the contraction nozzle. Throughout this investigation, the characteristic length used for scaling was chosen to be the nozzle diameter, because only a single nozzle diameter was used and it is a representative length of the jet. The axial Reynolds number is used as the fundamental flow descriptor of the jet, and defined here in the downstream, or axial direction x, in terms of dimensional parameters of the volumetric axial flow rate Q, and the nozzle exit diameter DON :


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4Q ; DON

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where is the kinematic viscosity of the fluid. The amount of swirl imparted on the jet is defined in terms of the rotation speed of the honeycomb section inside the head unit, !M , and takes the form of a rotational Reynolds number Re! : Re! D

2 2!M DON :

(2)

A measure of the system response was given by the stagnation point location P D x= O DON , and a key output from this investigation would be a quantitative relationship between this observable and the control variables.

2.1 Stagnation Point Movement with Axial Reynolds Number A sphere of size DS D 1:47 ˙ 0:02 was placed at a distance of xS D 1:99 ˙ 0:03 downstream of the nozzle exit, where DS and xS are non-dimensionalised by nozzle diameter. Four axial Reynolds numbers were chosen, increasing in steps of 150 from Rex D 450 to Rex D 900, with an uncertainty of ˙20. At each swirl setting, the position of the stagnation point was determined. It was found that by plotting the position against a modified non-dimensional swirl parameter, defined as a ratio of the azimuthal Reynolds number to the axial Reynolds number taken to the half power, that the data collapse to a single curve, as shown in Fig. 2a. A smoothed 0.0

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Fig. 2 (a) Stagnation point location P plotted against a rescaled swirl parameter of SK , defined as SK D Re! =Rex0:5 . Axial Reynolds numbers plotted are .C/Rex D 450, .4/Rex D 600, . /Rex D 750 and .ı/Rex D 900. (b) Stagnation point location for Rex D 750 plotted against rotational Reynolds number, demonstrating the apparent hysteresis in stagnation point location by the deviation away from an averaged spline curve. Arrows indicate direction of increasing and decreasing swirl

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least-squares spline has also been fitted to the data, to show the trend of the general collapse, and to separate this general collapse from any hysteresis in the stagnation point position, which was evident for the Rex D 600 and Rex D 750 cases. The quality of the collapse suggests that the assumption of a power-law scaling for the stagnation point position seems appropriate, and we can conclude that the stagnation 1=2 point location is a function of Rex to good accuracy. It is not yet clear what the physical significance of this scaling is; however, it is noted that the boundary layer 1=2 thickness also scales with Rex . The data for Rex D 750 have been replotted separately in Fig. 2b to show the hysteretic behaviour of the stagnation point location as the swirl is increased or decreased. In the figure, increasing swirl corresponds to the points below the average line, and decreasing swirl to those above the line. One reason for this hysteresis in stagnation point position may be found by looking at the flow topologies in the hysteretic region, which are shown in Fig. 3. For increasing swirl, the recirculation zone is upstream of the sphere surface, and the shear-layer reattaches to the sphere surface. But when decreasing the swirl, the stagnation point is closer to the nozzle, and the shear-layer is clear of the sphere. It would appear from this observation that the reason for the observed hysteresis effect in the stagnation point position is that both forms of the vortex breakdown in the near-nozzle and the near-sphere regions are stable states for the same swirl ratio. A further point is that Billant et al. [2] found hysteresis in the swirl number required for the appearance and disappearance of the vortex breakdown for a similar range of axial Reynolds number, Rex D 626 839. Although in the present study the hysteresis of breakdown appearance was not accurately determined, the trace of the stagnation point that displayed hysteresis-like behavior occurred in the same Rex range as for Billant et al. [2].

:

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a Closed state, at Reω = 253

b Open state, at Reω = 257

Fig. 3 Comparison between (a) a closed recirculation zone between the sphere surface and the stagnation point, with the vortex breakdown shear-layer re-attaching to the sphere surface, and (b) an open form of breakdown, with shear-layer clear of sphere surface. Images are taken for Rex D 750, and DS D 1:47 ˙ 0:02, xS D 1:99 ˙ 0:03


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In order to further understand the physics of this flow, we can look at the curve displayed in Fig. 2a as consisting of three parts corresponding to distinct flow behaviours. The first, indicated with the letter A, corresponds to when the breakdown is close to the sphere. Here, on increasing the swirl, an upstream stagnation point first occurs, followed by the formation of an upstream recirculation zone between the stagnation point and the sphere surface (as depicted in Fig. 3). This is identical to the flow topology found by Mattner et al. [8], and because the recirculation zone is closed to the outside stagnant flow, it can be said that the vortex breakdown is in a closed form, analogous to the bubble state of breakdown seen by Billant et al. [2]. Increasing rescaled swirl up to SK D 9:0, the stagnation point moves steadily upstream in a relatively linear fashion, with the re-attachment point moving downstream of the sphere. The second region of interest corresponds to when the stagnation point is close to the nozzle exit and well upstream of the sphere, indicated in Fig. 2a by the letter B. As mentioned earlier, the vortex breakdown in this higher swirl condition is of an open form, with the shear-layer completely detached from the sphere surface and open to the surrounding flow. The open topology of the flow (depicted in Fig. 3b) seems to be very much analogous to the cone-form of breakdown seen by Billant et al. [2], and indeed, a time sequence of this flow shows that the stationary recirculating zones of the closed form of breakdown are here replaced with traveling vortices advecting along and shedding from the shear layer. With the rescaled swirl increasing from SK D 9:5 to SK D 10:5, the stagnation point moves upstream, in much the same manner as with the near-sphere region. The region C, between regions A and B, corresponds to flow states between these two extremes. At around SK D 9:2, flow visualisation shows that the shear-layer region enclosing the recirculation zone has a width comparable to that of the sphere. Small increases in Re! in this region result in large movements of the stagnation point upstream. It is in this transition region where a unique topology could not be determined because of hysteresis. Of note, Billant et al. [2] found that for a given swirl, the bubble form of breakdown first appeared at a higher swirl ratio than the cone breakdown.

2.2 The Effect of Sphere Size on Stagnation Point Location For this set of experiments, the axial Reynolds number was kept constant at Rex D 600 ˙ 20, and two sphere sizes were tested along with a third no-sphere case. The spheres were placed at approximately the same axial distance downstream of the nozzle exit. The smaller sphere diameter was measured to be DS D 0:62 ˙ 0:05 and placed at xS D 0:99 ˙ 0:03. The larger sphere diameter was measured to be DS D 0:99 ˙ 0:02 and placed at a measured distance of xS D 1:00 ˙ 0:02. The tracking of the stagnation point with swirl has been plotted in Fig. 4, and can be seen to comprise two distinctive regions. The first, marked by the letter A, is where the onset of the upstream stagnation point occurs. For the case of the smaller

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ı

(a) No Sphere

(b) DS D 0:62

(c) DS D 0:97

Fig. 5 Flow visualisation of the same stagnation point location for the cases of (a) no-sphere at SK D 9:6, (b) DS D 0:62 at SK D 9:7 and (c) DS D 0:99 at SK D 10:3. The shear-layer of the vortex breakdown region for the larger sphere is attached to the sphere surface but open for the smaller sphere, as for the no-sphere case. Images taken at Rex D 600

sphere, the stagnation point moves rapidly upstream as swirl is increased to SK D 10:0, where it reaches a position (scaled on the nozzle diameter) P D 0:5 away from the nozzle. Increasing the swirl further beyond this position moves the stagnation point upstream in a fairly steady and linear fashion. These general trends are shown also with the no-sphere case, which we note is hysteretic, but a linear fit of the nearnozzle region gives a gradient of 0:23DN =SK . The close matching of these two cases can be seen by the flow topologies, which are shown in Fig. 5a and b. It can be seen that the shear-layer for the small-sphere case is clear of the sphere surface, and has the same shape as for the no-sphere case. Even when the stagnation point is


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close to the small-sphere surface, the shear-layer simply continues undisturbed past the sphere into the general flow, maintaining the same cone-angle as the stagnation point moves upstream. The flow visualisation for the larger sphere case in Fig. 5c shows that for sufficient sphere size, the flow structure can be modified. Here, the stagnation point movement in the low-swirl section is significantly different to the previous two cases, in that the breakdown is in a closed form. From the data of Fig. 4, as swirl is increased for the larger-sphere case, the upstream recirculation region grows in size and the stagnation point moves upstream, keeping the shear layer attached to the sphere surface. This is much the same type of motion observed in the previous section (Section 2.1) Only once the stagnation point reaches a height of around P D 0:5 does the shear layer detach from the sphere, and the stagnation point moves upstream at the same rate as the small-sphere and no-sphere cases. These separate paths of stagnation point movement for different sphere sizes suggest that there exists a critical sphere size, of between DS D 0:62 and DS D 0:99, that determines whether the breakdown in the near-sphere region will be an open or closed form, and so whether the stagnation point will move upstream at a fast or slow rate with increasing swirl. Significantly, however, is the near-nozzle region indicated in Fig. 4 by the letter B and deals with the steady upstream movement of the stagnation point in the upperswirl range starting from around SK D 10:4. Here, the plot shows the same linear movement of the stagnation point towards the nozzle with increasing swirl as seen in Section 2.1. Also, the breakdown in this region is in an open form, with the shearlayer clear of any contact with the sphere surface.

2.3 Sphere Position Effect on Stagnation Point Location The single sphere of size DS D 0:99 ˙ 0:02 was placed at distances of xS D 0:75, 1:00, 1:50, 2:00 and 2:50 away from the nozzle outlet. The axial Reynolds number was once again held constant at Rex D 600 ˙ 20. The results of tracking the stagnation point for all five sphere positions are shown in Fig. 6, together with the data of the no-sphere case of Section 2.2. Plotted is the distance from the nozzle, measured in nozzle diameters, against the modified swirl parameter SK . Once again, the stagnation point data of the measured stagnation point position are represented by a least-square spline of the averaged data. The first feature of note in Fig. 6 is that towards the higher swirl setting, (>SK D 10:7), the stagnation point location paths for all sphere positions seem to converge in a similar upward trend, much as in the near-nozzle regions of the previous sections. The second feature of this plot is that the paths traced out for the individual sphere locations eventually cross over that of the no-sphere case. The reasons for this are not entirely clear, but a clue as to its significance can be found from looking at the flow visualisations. For each sphere location case, the swirl setting for which

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Fig. 6 Least-squares spline fits of averaged stagnation point location against the swirl parameter of Section 2.2 for various sphere positions: (4)xS D 0:75, ( )xS D 1:0, ()xS D 1:5, (C)xS D 2:0, ( )xS D 2:5. Dark solid line represents data of the no-sphere case

0.0 0.5 1.0

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Table 1 Comparison of the swirl setting for complete detachment of the shear-layer from the surface of the sphere. Re! is measured from Fig. 6 where the stagnation point location path of each sphere crosses that of the no-sphere case, and Re! is determined from observing flow visualisations

xS

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the path of each sphere location curve in Fig. 6 crosses that of the no-sphere case is summarised in Table 1 as Re! . The swirl value at which the shear-layer can be seen to detach from the surface of the sphere is indicated in the same table as Re! , and is in very good agreement with Re! , to well within 10%. This analysis indicates that, although the presence of the sphere is able to keep the breakdown in a closed form close to the sphere surface, once the stagnation point reaches a height that it would ordinarily be at in a no-sphere case, the shear-layer detaches from the sphere surface. It can also be seen in the plot of Fig. 6 that there appears to be an overshoot of the stagnation point location above the no-sphere position once the shear-layer has detached. This is a significant finding, since if the criticality theory developed by [1] were to hold, it would be expected that varying the downstream sphere position should not alter the upstream flow conditions. However, it is not immediately obvious why the no-sphere case position gives an indication of when the shear layer detaches from the sphere surface.

3 Conclusions A novel and comprehensive empirical investigation was performed into the effects of a bluff body on the vortex breakdown in a swirling jet in an open flow. While systematically changing flow conditions and bluff body parameters, the stagnation


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point of the vortex breakdown was monitored as a measure of the system response. It was shown that with increasing rotational Reynolds number (Re! ), the stagnation point location moved upstream linearly in the near-nozzle and near-sphere regions of the flow topology. By scaling Re! inversely by axial Reynolds number (Rex ) to the power of 1=2, all data collapsed to a single, well-defined curve. It was also shown that, although sphere size appeared not to affect the behaviour of the stagnation point in the near-nozzle region of the flow, there did appear to be a critical sphere size in the range 0:62 < DS < 0:99 that determined whether for a particular Re! the breakdown upstream of the sphere surface would be in an open or closed state. When the sphere location was varied, it was shown that a good correlation existed between the stagnation point position at which the vortex breakdown shear-layer detached from the sphere surface, and that of the no-sphere case at the same Re! . Further increasing Re! beyond this point resulted in a stagnation point location moving closer to the nozzle than for the same Re! of the no-sphere case. Acknowledgements This research was partly supported under Australian Research Council’s Discovery Projects funding scheme (project number DP0452664).

References 1. T.B. Benjamin. Theory of the vortex breakdown. J. Fluid Mech., 14:593–629, 1962. 2. P. Billant, J.-M. Chomaz, and P. Huerre. Experimental study of vortex breakdown in swirling jets. J. Fluid Mech., 376:183–219, 1998. 3. G.L. Brown and J.M. Lopez. Axisymmetric vortex breakdown. Part 2. Physical mechanisms. J. Fluid Mech., 221:553–576, 1990. 4. J. Dusting, J. Sheridan, and K. Hourigan. A fluid dynamics approach to bioreactor design for cell and tissue culture. Biotechnol. Bioeng., 94(6):1196–1208, DOI: 10.1002/bit.20960, 2006. 5. A. Fouras, J. Dusting, and K. Hourigan. A simple in-situ calibration technique for stereoscopic particle image velocimetry. Exp. Fluids, 42(5):799–810, 2007. 6. F. Gallaire, S. Rott, and J.-M. Chomaz. Experimental study of a free and forced swirling jet. Phys. Fluids, 16(8):2907–2917, 2004. 7. S. Khalil, M.C. Thompson, and K. Hourigan. Response of unconfined vortex breakdown to axial pulsing. Phys. Fluids, 18(3):38102–1–4, March 2006. 8. T.W. Mattner, P.N. Joubert, and M.S. Chong. Vortical flow. Part 2. Flow past a sphere in a constant-diameter pipe. J. Fluid Mech., 481:1–36, 2003. 9. A.M. Mitchell and J. Delery. Research into vortex breakdown control. Prog. Aerospace Sci., 37:385–418, 2001. 10. L. Mununga, K. Hourigan, M.C. Thompson, and T. Leweke. Confined flow vortex breakdown control using a small disk. Phys. Fluids, 16(12):4750–4753, 2004. 11. T. Sarpkaya. On stationary and travelling vortex breakdown. J. Fluid Mech., 45(3):545–559, 1971. 12. S. Wang and Z. Rusak. The dynamics of a swirling flow in a pipe and transition to axisymmetric vortex breakdown. J. Fluid Mech., 340:177–223, 1997.


Thickness Effect of NACA Symmetric Hydrofoils on Hydrodynamic Behaviour and Boundary Layer States H. Djeridi, C. Sarraf, and J.Y. Billard

Abstract The present study investigates experimentally the hydrodynamic behavior of 2D NACA (15%, 25%, 35%) symmetric hydrofoils at Reynolds number range 0:5 105 to 1:3 106 . A particular attention is paid on the hysteretic behavior at static stall angle and a detailed cartography of boundary layer structures (integral quantities and velocity profiles) is given in order to put in evidence the mechanism of the detachment and the onset of von Karman instability. Keywords Lift and drag Thick foils Boundary layer Von Karman street

1 Introduction In the context of podded propulsion, thick foils are used in naval shipyards to develop fins, rudders or POD Struts. One of the main interest of these thick hydrofoils is the stall delay that such profiles can provide when used at high angle of attack. In spite of their increasing use their behavior is not clearly understood and very few data are available concerning the hydrodynamic behavior at low and high incidences and also concerning the turbulent boundary layer structures, performance control, unsteady separated flows and vibrations. The effect of the thickness that leads to a surprising increase of the lift coefficient has been studied by Thwaites [1]. This effect can be observed for moderate thicknesses and the threshold value of the thickness above what this effect is observed is not clear. Some studies have been devoted to thickness effects on global parameters but few of them have investigated the detail of the flow in the boundary layer with adverse pressure gradient. The present work intends to clarify this point. H. Djeridi () Laboratoire de Physique des Océans (LPO, UMR 6523), UFR Sciences et Techniques, 6 Avenue le Gorgeu, C.S. 93837, 29238 Brest Cedex 3, France e-mail: [email protected] C. Sarraf and J.Y. Billard Institut de Recherche de l’Ecole Navale (IRENav, EA 3634), Ecole Navale BP 600, Lanvéoc Poulmic, 29240 BREST ARMEES France M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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If numerous works have focus on flat plates boundary layers with or without adverse pressure gradients [2–4] very few have been presented concerning boundary layers on foils and particularly the links between the boundary layer pattern with the performances of the foil remains unclear. Nevertheless, an increasing interest on boundary layer at high angle of attack can be noticed in order to predict the type of hysteretic loop that may occur [5] or to quantify the unsteady oscillatory flow in the near wake that produces pressure fluctuations as sources of hydroacoustic noise [6]. In addition, experiments have been performed to explore unsteady separating boundary layer [7] or the influence or laminar separation and transition on hysteresis on particular airfoil [8] but in these few references no systematic study of the effect of thickness is performed. The present work proposes such an attempt and with this aim, flows on three NACA symmetric foils (15%, 25% and 35% of relative thickness) are presented and detailed. The present study is a preliminary work of a major global VIV project in order to quantify the thickness effects of foils on the topology of the vortex shedding characterized by a predominant frequency persisting at high Reynolds.

2 Experimental Set Up and Measurements The experiments have been conducted in the hydrodynamic tunnel of the French Naval Academy. This facility is fitted with a 1 m long and 0:192 0:192 m2 square cross test section, in which a maximum velocity of 15 m s1 can be achieved. The turbulence intensity upstream at the entrance of the test section is 2%. The three designed foils are two-dimension symmetric profiles with relative maximum thickness of 15%, 25%, 35% located at 25% from the leading edge. The experiments have been performed on 100 mm chord length and 192 mm span length hydrofoils to a range of Reynolds number based on the chord length of 0:5 106 < Re < 1:3 106 . The blockage ratios defined as t/H (H is the height of the test section, and t the maximum thickness of the foil are respectively 0.078 for the NACA0015 and 1.92 for the NACA0035. The draft of the experimental configuration is shown on Fig. 1.

(d) (c)

(b) (a)

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Fig. 1 Experimental set up

(a) Back wall of the vein (b) Force balance (c) Downstream pressure tap (d) Upstream pressure taps (e) Acoustical pressure sensor (f) Profile L1 = L2 = 192mm L3 = 625mm L4 = 100mm L5 = 225mm

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Lift and drag measurements have been performed using a resistive gauge hydrodynamic balance calibrated in our laboratory. The mean and rms values have been performed from 30 s test measurements realized at 1,830 Hz. The determination of hydrodynamic parameters has been performed for 0ı < a < 35ı and the maximum range for the balance is 0–180 daN for the lift and 0–17 daN for the drag force. To characterize the structure of the boundary layer that develop on the hydrofoils, detailed velocity measurements have been performed using a refined spatial grid. Two components, three beams, LDV Dantec Dynamics System was used to measure normal and tangential velocity components in water seeded with micron size particles of Iriodine. The dimensions of the probe volume are 0.8 mm in spanwise direction and 0.05 mm in vertical and longitudinal directions. Velocity measurements are performed in the boundary layer from the leading to the trailing edge .0 < x=c < 1/ and a remote mechanical positioning system with a minimum translation step of 10 m was used to obtained a cartography around the hydrofoil along a curvilinear map (x,y). For the boundary layer measurements the time histories were registered with 20,000 to 8,000 samples acquired in the range of 20 to 60 s corresponding to the mean data rates of 1,000 to 130 Hz. These parameters are sufficient to obtain the mean and rms values of the velocity with an uncertainty of 1% and 1.5% respectively. The plan of measurements is located at a quarter the spanwise length and at each station x/c measurements are performed on the wall normal from the outer region of the boundary layer. The grid, refined in the near wall region (50 m between grid points) allows the determination of the mean velocity gradient with a good accuracy. The grid represents 15 normal lines from the leading to the trailing edge with about 70 points per line. To locate the laminar/turbulent transition with an accuracy of 2% of x/c, the grid has been refined locally in x direction. The closest point corresponds to as mean distance to the wall of y=c D 5 104 . In the chord wise direction, measurements are provided every x=c D 0:1 for every y=c D 5 104 . The typical grid is shown Fig. 2. 0.5 3%

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Concerning unsteady separated states, a spectral analysis has been conducted in the shear layer downstream and in the wake of the foils using a maximum data rate of 2,000 Hz for 400 s samples leading to a spectral resolution of 0.25 Hz. Spectral analysis of the vertical velocity component are provided by the Fast Fourier Transform method on the randomly sampled signals.

3 Thickness Effects on Global Parameters 3.1 Lift and Drag Measurements Classic behaviour of both lift and drag coefficients is observed in Fig. 3 for low angles of attack. An abrupt loss of lift, characteristic of the stall, is observed on the NACA 0015 and on NACA 0025 for higher angles. For the two thinnest profiles the lift behaviour is linear for small values of the angle of attack. For the thickest one, a screen effect delays the establishment of the lift, leading to a non linear behaviour for small angles of attack indicated by (a) on Fig. 3. It can be noticed that the linear range of angles of attack increases with the thickness of the profile. Thus angles limiting the linear behaviour are 7ı ; 13ı and 16ı respectively on the three profiles. Stall appears for angles of 21ı ; 33ı and 40ı . The evolution of the drag coefficient remains classical: for low incidences the effect of thickness is visible by an increase of the drag coefficient with increasing thickness. On these curves the stall is linked with a violent increase of the drag coefficient except for the NACA0035 for which stall appears at larger incidences. The effect of thickness introduces a decrease the slop of the lift coefficient with increasing thickness. This effect is characteristic of thick profiles.

3.2 Hysteretic Behaviour and Fluctuating Efforts For the NACA0015 and NACA0025 profiles an abrupt loss of lift, characteristic of the stall, is observed for an angle of attack of 21ı and 33ı respectively. dC /2πdα =0.81

1.2

(a)

L

L

0.8

dC /2πdα= 0.43

dC /2πdα= 0.58

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= 0.009°

CDα0 = 0.014°

CDα0 = 0.023°

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Fig. 3 Lift and drag coefficients for the three profiles

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10 α [°]

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10 α [°]

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Fig. 4a Zoom of the hysteretic behaviour of lift (left) and drag (right) coefficients for NACA0015 (a) and NACA0025 (b)

(b)

(a)

10−1

std(Cl)

Fig. 4b Zoom of the hysteretic behaviour of fluctuating efforts lift (left) and drag (right) for NACA0015 (a) and NACA0025 (b)

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For flow reattachment the angle of attack must be drastically reduced and are respectively 15ı and 23ı . This behaviour is illustrated in Figs. 4a and 4b where the hysteresis loops have been underlined by arrows following the cycle of the lift coefficient. In fact, the cycle provides an upper and lower branches characterized by two different states of the flow. The starting point for the flows along the increasing angle branch is an attached flow (called State I), whereas it is a massively separated flow for the flows along the branch corresponding to the decreasing angle (called state II). Corresponding boundary layer pattern has been observed and described in detail in Section 4.2. This phenomenon is not observed on the thickest profile in the range of our investigation. We were obliged to increase the angle of attack up to 40ı in order to observe a similar phenomenon. The rms values of lift and drag coefficients at the stall angle show that the hysteretic behaviour is associated with an abrupt increase of fluctuations of forces. This increase of fluctuating part is associated with the unsteady component of hydrodynamic coefficients.

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3.3 Turbulent Boundary Layer States For each point of measure 20,000 samples have been validated during a maximum of 60 s (corresponding to the near wall locations). This has been proven sufficient after having performed successive tests to measure the mean and the rms values of the u and v velocity components with a very good convergence and repetitiveness of the processing. According to the previous refined grid, a survey of the distribution of mean tangential and normal velocity components is presented. Firstly, our goal was to characterise the boundary shape parameters on a Naca0015, 25 and 35 hydrofoils. Velocity profiles were numerically integrated to compute the displacement, ı1 , momentum, ı2 and energy, ı3 , thicknesses from which shape factors H12 and H23 are deduced. The following formulas are applied: Z• Z• Z• u u u2 u u 1 dy; •2 D 1 1 2 dy; •3 D dy •1 D Ue Ue Ue U Ue e 0

H12

0

0

•1 •2 D ; H23 D •2 •3

Ue represents the external velocity on the normal line. In our case this velocity is the maximum velocity measured at the location. Shape factors are presented Fig. 5 versus normalized chord length. It can be observed that the thickness effect is associated with an increase of the length of the laminar region near the leading edge which grows from less than 10% on the NACA0015 to 30% on the NACA0035. At the end of this area the shape factor increases to reach a value of 3 to 4 just before the occurrence of the transition. After the transition, the value of H12 remains constant and equal to 1.6. This value increases near the trailing edge when the flow reaches conditions of separation. The

5

H1 2

4 3 2 1

0

0

0.2

0.4

0.6

0.8

1

x/c Fig. 5 Shape factors for the three profiles, Re D 5105 ; ’ D 10ı ; NACA0035

ı NACA0015, } NACA0025,


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constant value of H12 is larger than the value generally observed for turbulent boundary layers (1.4). It must be noticed that this value depends rigorously of the Reynolds number [9]. Near the trailing edge the shape factor is lower than 3 on the two thinner profiles and is greater than 4 on the third where separation is observed. These values are consistent with the threshold values proposed by Bradshaw [10]. To check the self similarity and to quantify the thickness effect of the foils on the boundary layer, mean velocity profiles with inner variables are presented. Inner variables are classically defined as yC for the normal position and uC for the tangential component of the velocity respectively given by: q q £w uC D A log yC C B with yC D yu$ and u D Ue C2f D ¡ , where £w is the wall shear stress. To determine the value of the shear stress velocity u , an efficient estimation of the skin friction is required. The friction has been determined using an experimental model based on the value of the integral quantities proposed by Ludwieg and Tillman in 1950 [11]: Re

0:268

•2 Cf D 0:246 100:678 H12 , the values obtained with the model are in a good agreement with numerical results (boundary layer code 3C3D) and with the near wall velocity gradients. Figure 6 the velocity profiles using scaling law are reported for the three foils at different locations on x/c. It can be noticed that the slopes of the profiles in the logarithmic area are quite far from the classical value 5.75 obtained for turbulent boundary layers on flat plates. This is due to the non equilibrium boundary layer condition due to the low Reynolds number effects. Moving downstream, the velocity profiles in wall coordinates deviate significantly from the standard log-law and it can be noticed that the thickness effect pronounce this trend. Despite the Reynolds effect, the thickness effect is only characterized here by the shape of the wake region. For the locations x/c near the trailing edge, the wake area can be described by a second law proposed by Coles [12] for yC > 100 W uC D f yC C …› ¨ y• where • is the boundary layer thickness. With the non equilibrium layers the wake law can be used according the evolution of … with x/c and the different value of › [13]. The hypothesis of a universal wake function is reported Fig. 6 for the three profiles at different station x/c. The thickness effect is then characterized by a deviation of the wake law accentuated for the two thicker profiles.

4 Unsteady Separated Flow 4.1 Strouhal Number When incidence is increased over 20ı , an organized motion due to the regular vortex shedding appears. This shedding is characterized by a predominant frequency f. The evolution of von Karman instability is shown for the three profiles on Fig. 7. The establishment of the instability is associated with an increase of maximum amplitude and a decrease of the frequency. For NACA0015 the frequency evolves linearly but for the two thicker profiles if the behaviour remains linear, two different slopes can

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a 60 50

U+

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U+ +10

30 20 10

+

U -Π i /κ *i ω(y/δc)

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0 100

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+ U +10

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Fig. 6 Velocity profiles with inner coordinates and associated wake laws, ’‚ D 10ı ; 5 m s1 , (a) NACA0015, (b) NACA0025, (c) NACA0035


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NACA0015 spectre 6ms, Vz, a =17° à 30°

30

x²/Hz

102 10 10 10

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a(°) 15 60

80

100

Fréquence [Hz]

Fig. 7 Frequency spectrum of the vertical velocity in the shear layer for the three profiles 0.4 0.35 (b) 0.3 St

(c)

0.25

(a)

0.2 0.15 15

20

25 α [°]

30

35

Fig. 8 Strouhal number versus incidence for the three profiles

be observed. These different evolutions are characterized by two values of Strouhal number, based on the projected area of the foil defined as St D c sin ’ f=U. This number is reported versus the angle of attack for each profile on Fig. 8. Two states of the flow which characterized by two values of St (0.3 in state I and 0.2 in state II) are shown. The transition between the two states is characterized by a

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NACA0015, α=20°

NACA0025, α=33°

NACA0035, α=40°

NACA0015, α=21°

NACA0025, α=34°

NACA0035, α=40°

Fig. 9 Flow visualization of the two states I and II for the three profiles

jump of the Strouhal value. In state I, the flow is partially attached in the vicinity of the leading edge when in state II, the flow detaches at the leading edge (stall mechanism). This sketch is illustrated Fig. 9. This figure shows the flow patterns that characterize the two branches of hysteresis loop. This behaviour is observed on the three profiles but experimental range is not large enough to measure precisely the value on NACA0035. When the incidence angle of the foil is increased, the separation point appears near the trailing edge and moves progressively towards the leading edge (steady state configuration). A vortex shedding appears as it is observed through the spectral analysis, and the shear layer at separated point is affected by Kelvin Helmotz instability. This instability, forced by the periodic vortex shedding, lead to an oscillation of the stagnation point and further an oscillation of the separation point. The non linear interaction between these two instabilities seems to be responsible of the coexistence of the state I and II [13]. A second mechanism that can lead to the static stall (transition from state I to state II) is linked with the value of the maximum speed near the leading edge. A comparison between the flow patterns on the three profiles shows that: (i) For the same incidence angle the maximum velocity observed on the three profiles is a decreasing function of the relative thickness. (ii) Just before the transition the maximum velocity observed on the three profiles are equal. Thus it seems that for a fixed Reynolds number a velocity threshold that can’t be overcome by the flow exists. The measurements realized for the two flow states are presented in the next paragraph.


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4.2 Boundary Layer State During Hysteresis On the Naca0015 boundary layer measurements have been performed in the range of angle where hysteretic behaviour is observed at the same incidence angle for the two flow states I and II. The chosen incidence angles are 16ı and 20ı . The same measurements have been performed on NACA0025 for the two specific states at ’ D 25ı . When the incidence angle of the foil is increased two phenomena occur simultaneously on the foil. The transition point and the detachment points move progressively toward the leading edge and the distance between the two points decreases. The detachment point behaves more and more unsteady. As it is observed, Figs. 10a and 10b, for the thinner profile the transition to the state II is characterized by an abrupt modification of the localisation of the detached point x=c D 0:45 for state I and x=c D 0:023 for state II. For NACA0025 profile, the magnitude of the displacement of the detached point is less pronounced (x=c D 0:35 for state I and x=c D 0:043 for state II). To conclude, the thickness of the profile tends to increase the thickness of the boundary layer at low angle of attack instead of at stall, the magnitude of the detached point raising is attenuated leading to a less decrease of the lift. This effect corroborates the well known stall mechanism of the thick profile. On Fig. 11 the local velocity on the suction side is presented versus x coordinate 2 e as a Cp value: Cp D 1 UUref . It can be seen that for the two conditions that prevail just before stall occurrence the maxima of the velocity (corresponding to the

20

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Fig. 10a Velocity profile for NACA0015 at 16ı corresponding to state I (left) and state II (right)

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Fig. 10b Velocity profile for NACA0025 at 25ı corresponding to state I (left) and state II (right)

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1-(Ue /Uref)2

−1 −2 −3 −4 −50

0.2

0.4

0.6

0.8

1

x /c Fig. 11 Velocity on the two profiles (suction side) just before (open symbols) and just after (solid symbols) stall occurrence : NACA0015 ’ D 16ı }: NACA0025 ’ D 25ı

ı

Cp minimum) are equal for the two profiles. After stall occurrence (corresponding to the state II), the thickness effect induces a larger maximum of the velocity on the thicker profile.

5 Conclusions This experimental work realized on three symmetrical NACA profiles of 15%, 25% and 35% relative thickness have shown that: Above about 10% thickness the lift coefficient of the profile decreases when the thickness is increased. The hysteretic behavior of the profile at stall is delayed when the thickness is increased, this phenomenon is correlated with a modification of the flow state linked with the position of the detachment point which is forced by the establishment of Karman instabilities. Concerning the unsteady separated flow at high angle of attack, the thickness effect is associated with a modification of the establishment of the Karman street. Indeed, as it is observed through the spectral analysis for the thinner profile, the frequency evolves linearly but for the two thicker profiles if the behaviour remains linear, two different slopes can be observed. The thickness of the profile tends to increase the thickness of the boundary layer at low angle of attack instead of at stall, the magnitude of the detached point raising is attenuated leading to a less decrease of the lift. The boundary layers that develop on the three profiles are turbulent even if they have not reach their equilibrium state with establishment of the self similarity. The effect of the adverse pressure gradient implies the use of two velocity laws (near wall logarithmic law and wake law) that can be unified using the Coles law for the velocity profiles close to the detachment point. The thickness effect is then characterized by a deviation of the wake law accentuated for the two thicker profiles.


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References 1. Thwaites, B., Incompressible aerodynamics, an account of the theory and observation of the steady flow of incompressible fluid past aerofoils, wings and other bodies. Dover, 1960. 2. Na, Y. and Moin, P., Direct numerical simulation of a separated turbulent boundary layer, J. Fluid Mech., 370, 1998. 3. Yang, Z. and Voke, P., Large eddy simulation of boundary layer separation and transition at a change of surface curvature, J. Fluid Mech., 439, 2001. 4. Marusic, I. and Perry, E., A wall-wake model for the turbulence structure of boundary layers, J. Fluid Mech., 298, 1995. 5. Mittal, S. and Saxena, P., Prediction of hysteresis associated with the static stall of an airfoil, AIAA J., 38(5), 2000. 6. Bourgoyne, D.A., Ceccio, S., Dowling, D., Jessup, S., Park, J., Brewer, W., and Pankajakshan, R., Hydrofoil turbulent boundary layer separation at high Reynolds numbers, 23rd Symposium on Naval Hydrodynamice, Val de Reuil, France 2000. 7. Lurie, E.A., Keenan, D.P., and Kerwin, J.E., Experimental study of an unsteady separating boundary layer, AIAA J., 36(4), 1998. 8. Mueller, T., The influence of laminar separation and transition on low Reynolds number airfoil hysteresis, J. Aircraft, 22(9), 1985. 9. Cousteix, J., “Turbulence et couche limite” Aérodynamique, Editions Cepadues, 1989. 10. Bradshaw P., “The turbulence structure of equilibrium boudary layers”, J. Fluid Mech., 29:625–645, 1967. 11. Ludwieg, H., and Tillman, W., Investigation of the wall-shearing stress in turbulent boundary layer, NACA Technical Memorandum, 1285, 1950. 12. Coles, D., “The law of the wake”, J. Fluid Mech., 1:191–226, 1956. 13. Hoarau Y., Braza M., Ventikos Y., Faghani D., and Tzabiras G.: “Organized modes and the three dimensional transition to turbulence in the incompressible flow around a NACA0012 wing”, J. Fluid Mech., 496:63–72, 2003. 14. Hoarau, Y. and Braza, M., Simulation et contrôle d’un e´ coulement fortement décollé autour d’un profil d’aile, 39 e` me colloque d’aérodynamique appiqué AAAF 2004-04-08.


Quasi-Steady Self-Excited Angular Oscillation of Equilateral Triangular Cylinder in 2-D Separated Flow S. Srigrarom

Abstract This paper studies the flow field of a particular fluid–structure interaction phenomenon – the continuous angular oscillation of a centrally-pivoted equilateral triangular cylinder, under uniform two dimensional incompressible flow. Dye flow visualization of a 30 cm long and 10 cm wide cylinder in the two-dimensional water tunnel was conducted. Under a uniform incoming flow of 7.5 cm/s, after an initial perturbation, the cylinder oscillated continuously. On the windward side of the cylinder, a vortex was formed at the sharp edges of the cylinder during the initial phase, whereas on the leeward side, the flow stayed attached. The phase-averaged Particle Image Velocimetry (PIV) measurements are also presented. PIV results show the interchange of flow patterns from that over a flat plate to flow past a sharp edge and vice versa. Together with flow visualization, we recorded the angular position, the hydrodynamic force and moment acting on this cylinder with time. We found the dominant oscillating frequency is about 0.1 Hz. Hence, we are able to analyze the cylinder’s oscillating dynamic motion. From the recorded data, we propose a dynamic model based on sinusoidal moment. The sinusoidal moment model provides decent angular position, which is analogous to the oscillation of the pendulum of large amplitude. Keywords Dye flow visualization Fluid–structure interaction Triangular cylinder Strouhal number Force/moment measurement PIV Oscillation dynamic model

1 Introduction To date, this coupled fluid–structure interaction has not yet been well studied nor simulated. This is due to the unsteadiness surrounding flow. The available prior research works were only dealt with the stationary cylinder at different position S. Srigrarom Nanyang Technological University, School of Mechanical and Aerospace Engineering, 50, Nanyang Avenue, Singapore, 639798 e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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(Nagashima [1]), or with other geometries (Hu et al. [2], Sakamoto et al. [3], Naudascher and Wang [4], Nakamura and Nakashima [5]). Prior experiments (Srigrarom [6]) had been conducted to investigate such behaviour by means of Food coloring dye and Laser Induced Fluorescence flow visualization. The oscillating frequencies appeared to be governed by Strouhal numbers, which fell within limited range of 0:13 < S t r < 0:18. Beyond this range, the cylinder was either stationary or rotates only in one direction. In this experimental study, we examine the unsteady flow pattern in the water tunnel, by means of direct dye injection flow visualization to get the flow pattern, and Particle image velocimetry to get the velocity field.

2 Oscillation Model The explanation of the self-excited oscillation behaviour was elaborately discussed in Srigrarom [7]. The key features are excerpted here. Consider the flow pattern around a symmetrical triangular cylinder rotating in the clockwise direction (Fig. 1). The upstream flow is uniform at zero angle of attack, and is brought to stagnation at the front of the cylinder. The flow is divided into two identical zones, upper and lower. The divided stream varies smoothly from a 90ı turn at S to a stream merging with the free stream at edge A or B. At A or B, the flow separates from the cylinder at both tips or shape edges of the cylinder, creating back flow or eddies on both lateral sides. At the trailing edge C , the flow pattern changes according to the free stream Reynolds number. At low Reynolds number .104 < Re < 105/, the two divided streams create alternating vortices, shed downstream. At higher Reynolds number .Re > 105/, the two streams join together at alternating streamwise locations creating a turbulent wake downstream of the cylinder (Luo et al. [8]). When the front face of the cylinder AB is inclined with the free stream, the flow separates at both A and B (the pattern is asymmetrical). The radius of curvature of the separated streamline at A; rA , is smaller than the radius of curvature at B; rB . As a result of conservation of angular momentum, the velocity at A is higher than at B; therefore, the pressure at A is lower than at B .PAC < PBC /. Hence, the cylinder rotates clockwise about the pivot, and the frontal surface AB becomes more aligned with the free stream.

Fig. 1 Sequence of flow past the equilateral triangular cylinder, rotating clockwise

Quasi-Steady Self-Excited Angular Oscillation of Equilateral Triangular Cylinder

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With the continuous motion, the triangle will arrive at a position where AC is parallel to the free stream. The flow separates at the upper tip of the cylinder .B/, but the flow in the lower part separates only at the lower tip of the cylinder .A/, before reattaching to the lower lateral face (AC). The flow is then similar to that over a flat plate. Because of the difference between the two flow patterns, the local pressures differ at the upper and lower parts of the cylinder. The upper part, with the existence of a large eddy, has lower pressure, compared with the free stream; whereas at the lower part the pressure is equal to the free stream pressure. The pressure in the lower part is now greater than the upper part, .PAC > PBC / and the cylinder tends to rotate back to its original position. As a consequence of the above, the resultant pressure forces the cylinder to rotate counter-clockwise back to its original position (under the assumption that the cylinder starts rotating in a clockwise direction, as described in the previous step). Due to the inertia of the cylinder, the motion does not stop when it returns to the original symmetric position as shown in the left sequence of Fig. 1. Instead, the cylinder continues to swing in the counterclockwise rotation. The flow patterns are just flipped ones from those in Fig. 1. The overall phenomenon can be viewed as the interchange of the flow patterns, from the flow past the sharp edge to flow over the flat plate, and vice versa.

3 Direct Dye Injection Flow Visualization The flow was visualized using direct dye flow injection and particle image velocimetry (PIV). This was elaborately discussed in Srigrarom and Koh [9]. The key features are similarly excerpted here. The experiment was conducted in the 45 45 100 cm water tunnel facility at Nanyang Technological University, Singapore. The free stream velocity, U1 was 7.5 cm/s. The cylinder was made from Delrin plastic with density 1;400 kg=m3 . The cylinder’s width, W was 10 cm. This corresponds to Reynolds number, ReW , of 7,500 based on the cylinder’s width. We put the plate at the end of the cylinder, such that, the flow surrounding this cylinder was essentially two-dimensional. The cylinder geometry and the experimental setup are shown in Fig. 2. Firstly, direct dye injection technique was used to observe the flow pattern. The food coloring dye was released upstream of the cylinder. The images were captured by digital camera are shown in Figs. 3 and 4. The oscillation frequency, f was 0.13 Hz, observed from the side-force (cross-stream direction) measurement, corresponding to a Strouhal number .S t f W=U1 / of 0.17. The overall phenomenon of the triangular cylinder oscillating in a to-and-fro manner can be clearly observed. As seen from the dye, there is an interchange of flow patterns from the flow over the flat plate (AC) to flow past sharp edge .A/, resulting in a clockwise rotation of the cylinder.

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U•

Triangular wedge End plate to make flow 2D

Fig. 2 The geometry of the cylinder (wedge), with the built-in bearings and the mounted force transducer (left). All dimensions are in millimeters (mm). Experimental setup (only the test section in the water tunnel shown) (right)

A

C

A

A C B

B

B

C

Fig. 3 Cylinder’s oscillation in clockwise motion

A

A

A B

C

C

C B

B

Fig. 4 Cylinders’ oscillation in counter-clockwise motion

4 Particle Image Velocimetry (PIV) Particle Image Velocimetry (PIV) measurements of the velocity field was conducted, with the same flow conditions as in the previous direct dye injection experiment, i.e. U1 D 7:5 cm=s; ReW D 7;500. The camera was mounted on the top of the water tunnel to capture the cylinder’s side view flow image, whereas the laser was


273

Image capturing area PIV camera

Cylinder

A

+q

B PIV laser

Fig. 5 Experiment setup for PIV (left) and image capturing area (right)

positioned at the side to create planar laser sheet. The setup is shown in Fig. 5. The 0:1 m nylon particles for PIV were released upstream of the triangular cylinder, at a time synchronizes to the laser firing and camera capturing time. Since the oscillation frequency was consistently at 0.13 Hz .S t D 0:1733/, we could do phase-average PIV, i.e. capturing and processing the PIV images and data at the same cylinder’s angular position and the rotational motion direction, but from different cylinder’s oscillation cycles. The process was done by (1) recording the cylinder’s motion by high speed camera for extended period of time to get the sufficient oscillation history (approx. 20 cycles), (2) synchronizing the laser firing time with the cylinder motion, (3) capturing PIV images at the prescribed cylinder’s angular position. In our PIV experiment, due to position and orientation of the camera, the free stream flow in the image came from right to left. When the cylinder was swinging following the flow (counter-clockwise motion), the PIV camera images and the processed velocity flow field are shown in Fig. 6a–c. Figure 6 shows the time-sequence PIV results when the cylinder moves in a clockwise direction. The camera images are shown on the left and the corresponding velocity fields are on the right. Note that, in these plots, the free stream direction is from the right to left. In Fig. 6a at D C120ı , the flow separates from the sharp edge at B. There is reversed flow on the observed surface AB as indicated by the velocity vectors. In Fig. 6b at D C72ı , the cylinder moves and the fluid adjacent to the surface AB is moved by the cylinder. The velocity vectors appear to point upwards and to the left with the cylinder’s clockwise motion. In Fig. 6c at D C45ı , the velocity vectors on the right side on the observed surface AB are attached to the cylinder’s surface. At lower left C , there is jet efflux, caused by the cylinder’s clockwise motion. Overall, Fig. 6 shows the change of flow pattern from the flow past sharp edge like (Fig. 6a) to the flow over flat plate like on the observed surface AB

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B

Cylinder’s

A Cylinder at q = +120°

A Cylinder’s motion

B

Cylinder at q = +72°

A

Cylinder’s motion

B C Cylinder at q = +45°

Fig. 6 Sequence of captured camera images (left) and the corresponding velocity fields (middle) and vorticity contours (right). The cylinder was swinging against the flow (clockwise)

(Fig. 6c); which is in agreement with the dye flow visualization discussed in previous sections. When the cylinder moves in the counter-clockwise direction similar agreement was observed.

5 Side Force Comparison Between Oscillating and Stationary Cylinder In this section, we compare the side force acting on the oscillating and stationary cylinder. If we fix the cylinder in place (referred as “stationary cylinder”), such that, the cylinder cannot rotate freely, there will be vortex shedding on the cylinder from the two sharp edges on both sides. This vortex shedding will generate side force


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acting on the cylinder. The actual side force, Fy reading for the stationary cylinder at U1 D 7:5 cm=s has been recorded as shown in Fig. 7 below. The corresponding power spectral density is shown in Fig. 8.

Fig. 7 Experimental reading of Fy , on the stationary cylinder, at U1 D 7:5 cm=s

Fig. 8 Power spectral density of Fy , on the stationary cylinder, at U1 D 7:5 cm=s

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Fig. 9 Experimental reading of Fy , on the oscillating cylinder

Fig. 10 Power spectral density of Fy , on the oscillating cylinder

In comparison, the actual side force reading for the self-excited rotational oscillating triangular cylinder at U1 D 7:5 cm=s has been recorded as shown in Fig. 9. The corresponding power spectral density is shown in Fig. 10. For comparison, Fig. 11 shows the zoomed in of Fig. 7, for the shorter period of time .0 < t < 100 s/. This is to compare with Fig. 12 (reprint of Fig. 9). As shown


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Fig. 11 Reading of Fy on the stationary cylinder, at U1 D 7:5 cm=s, for 0 < t < 100 s.

Fig. 12 Reading of Fy acting on the self-excited oscillating cylinder, at U1 D 7:5 cm=s. for 0 < t < 100 s (Reprint of Fig. 23, with the sinusoidal approximation)

in the figures, there are distinctive differences between the stationary and oscillating cylinders, in terms of amplitudes and frequencies. The stationary cylinder has smaller side force magnitude (amplitudes) in the order of 0.02 N, and the dominate frequency in the order of 0:01 Hz (see Fig. 8). Whereas the oscillating cylinder has bigger side force magnitude around 0.05 N, and the dominate frequency around

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0.13 Hz (see Fig. 10). Therefore, it is obvious that this self-excited rotational oscillating motion is different from the classical vortex shedding one for stationary cylinder. With the side force recording on the cylinder, we note the dominate frequency of 0.13 Hz for the self-excited oscillating cylinder at U D 7:5 cm=s. We can approximate the force and moment with sinusoidal function, based on experimental data, as the followings. Fy .t/ D F0 sin .!t/ where ! D 2f; F0 D 0:05 N and f D 0:13 Hz This formula will be used in cylinder’s oscillation dynamics analysis in the following section.

6 Cylinder’s Oscillation Dynamics Model In this section, we examine the cylinder motion under hydrodynamic force from the incoming flow, based on rigid body dynamics. From observation, the motion could be considered quasi-steady, i.e. the movement of the cylinder is comparatively slow, and that the flow settled faster than the motion of the cylinder. The buoyancy force is also assumed negligible. By applying these assumptions, we can obtain relatively simple curve-fitting mathematical models, which explain the cylinder oscillation well. Consider the side view of the equilateral triangular cylinder, of length L, and width W , hinged at its geometric center O, as shown in Fig. 13. At far upstream the uniform flow velocity is U1 . At one instance, the cylinder is inclined at angular position , from the balance position.

y

U• 30

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+q

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+M

q


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From the conservation of angular momentum, the mathematical description of the rotational oscillation can be formulated in terms of the equation: X

M DJ

d 2 dt 2

(1)

where J D Polar moment of inertia about the axis of rotational oscillation .z/ D Angular position of the triangular cylinder, measured relative to the free stream flow (™ D 0ı is the equilibrium position and at horizontal plane). The only moment acting on the cylinder caused by the hydrodynamic force, M ./ is approximated to be sinusoidal: M ./ D M0 sin

(2)

where M0 is the amplitude of the moment. This sinusoidal approximation is based on observed experimental data, as discussed in the previous section. The minus sign is because the moment is always opposite to the cylinder’s angular position. Substitute Equation (2) into (1), we get the non-linear differential equation: d 2 M0 sin D 0 C dt 2 J

(3)

This problem is analogous to the large amplitude oscillation of the pendulum (Davis [10]). The solution of this equation can be achieved by means of elliptic integrals and expressed in terms of elliptic functions. "

r

.t/ D 2 arcsin k sn t

M0 ;k J

!# (4)

where

k D sin !2 is the transformation ! max is the maximum displacement of cylinder from its equilibrium q M0 sn t J ; k is the Jacobian elliptic function.

The period of the rotational oscillation motion is defined to be the time required to make a complete oscillation between positions of maximum displacement. Let P .k/ to be the period of the oscillating cylinder, we get s P .k/ D 4

J M0

Z

=2

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p D4 1 k 2 sin2

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where K .k/ is the complete elliptic integral of the first kind.

(5)

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Here, we select the case of the oscillating cylinder, submerged in the free stream flow at U1 D 7:5 cm=s. The maximum displacement is ! max D 30ı . This gives ! 30 D sin D 1:6906 (6) k D sin 2 2 We observe the oscillation period from the experimental reading to be approximately 7.8 s. (The corresponding Strouhal number, Str, is 0.17, in the range of 0:13 < S t r < 0:18 as reported earlier). Thus s P D 2

s J D 7:8 or M0

J D 1:1534 M0

(7)

Therefore, from Equation (5) we get the angular position, defined as elliptic .t/

elliptic .t/ D 2 arcsin 1:6906 sn

t ; 1:6906 1:1534

(8)

This elliptic .t/ is plotted in comparison to the actual angular position reading from the experiment as shown in Fig. 14. It appears that, this obtained angular position, established the lower limit of amplitude of oscillation as compared to the actual

Fig. 14 Cylinder’s angular position comparison between models and experiment readings at U1 D 7:5 cm=s ( D 0ı is the equilibrium position and at horizontal plane)


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experimental reading. This sinusoidal moment approximation model appears to provide decent angular position prediction. This also means the self-excited oscillation of the triangular cylinder is closely similar to the oscillation of the pendulum of large amplitude.

7 Conclusions The present study of a simplified quasi-steady flow past an oscillating triangular cylinder found an interesting self-excited oscillation of the cylinder as observed in experimental investigation before. Under the simple uniform incoming flow, the interchange flow pattern, at certain flow velocity, was found. The results from direct dye injection flow visualization as well as Particle Image Velocimetry (PIV) appeared to agree with the proposed physical explanation of the phenomena. It is the unbalance force acting on the cylinder’s side faces that causes such movement. On one side, the flow will be flow-past-flat-plate like, whereas the other side will be flow-past-sharp-edge like. When the cylinder rotates, these mechanisms switch side interchangeably, and bring the cylinder to continuous oscillation. Together with experiment, the simple sinusoidal approximations of acting hydrodynamic force and moment on the cylinder give simple, yet fairly accurate, explanation of the cylinder dynamics. The qualitative agreement between the theory, simulation and the experiments are shown.

References 1. Nagashima, T. and Hirose, T., “Potential Flow around Two Dimensional Isosceles Triangular Cylinder Subjected to Uniform Flow from Base Surface”, Journal of Japan National Defense Agency, 1992 (in Japanese). 2. Hu, C.C., Miau, J.J. and Chou, J.H., “Instantaneous Vortex-Shedding Behaviour in Periodically Varying Flow”, Proceedings of the Royal Society of London Series A, Vol. 458, pp. 911–932, 2002. 3. Sakamoto, H., Takai, K., Alam, M.M. and Moriya, M., “Suppression and Characteristics of Flow Induced Vibration of Rectangular Prisms with Various Width-to-Height Ratios”, in “Fluid Structure Interaction” (eds.) Chakrabarti, S.K. and Brebbia, C.A., pp. 67–76. WIT Press, Southampton, 2001. 4. Naudascher, E. and Wang, Y., “Flow-Induced Vibrations of Prismatic Bodies and Grids of Prisms”, Journal of Fluids Structures, Vol. 7, pp. 341–373, 1993. 5. Nakamura, Y. and Nakashima, M., “Vortex Excitation of Prisms with Elongated Rectangular, H and ? Cross-Sections”, Journal of Fluid Mechanics, Vol. 163, pp. 149–169. 1986. 6. Srigrarom, S., “Self-Excited Oscillation of Triangular Cylinder”, Master thesis, Department of Aeronautics and Astronautics, University of Washington, Washington, DC, 1998. 7. Srigrarom, S., “Self-Excited Oscillation of Equilateral Triangular Cylinder”, Proceedings of the IUTAM Symposium on Fluid-Structure Interactions, Vol. 1, New Brunswick, NJ, June l2–16, 2003.

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8. Luo, S.C., Yazdani, M.G., Lee, T.S. and Chew, Y.T., “Aerodynamic Stability of Square, Trapezoidal and Triangular Cylinders”, Proceedings of the Third International Offshore and Polar Engineering Conference, Singapore, pp. 709–714, 1993. 9. Srigrarom. S., Koh, A.K.G., Flow Field of Self-excited Rotationally Oscillating Equilateral Triangular Wedge, Journal of Fluids and Structures, doi:10.1015/j.fluidstructs.2008.10.015, 2007. 10. Davis, H.T. “Introduction to Nonlinear Differential and Integral Equations”, Dover, New York, 1962.

Part V

Compressibility Effects Related to Unsteady Separation


Compressibility Effects on Turbulent Separated Flow in a Streamwise-Periodic Hill Channel – Part 1 J. Ziefle and L. Kleiser

Abstract We present large-eddy simulation (LES) results of the streamwiseperiodic hill channel configuration, which is a standard test case for massively separated flows. The Reynolds number (computed with the hill height, the bulk mass flux through the cross-section above the hill crest and the dynamic viscosity at the wall) is chosen as 2,800, in accordance with recent DNS data from Peller & Manhart [4] for incompressible flow. The Mach number was varied between Ma D 0:2 and Ma D 2:5. The numerical simulation code NSMB discretises the compressible Navier-Stokes equations with the finite-volume method on a deliberately-chosen coarse structured mesh. The subgrid-scales are accounted for by the well-proven approximate deconvolution model (ADM). This investigation is an extension of our previous work on this configuration, which focused on the validation of our simulation approach at nearly incompressible flow conditions. In this first part of a two-part contribution [8], the scope lies primarily on the effect of compressibility, especially on the separation characteristics and the flow conditions at the walls. To this end, we introduce a new measure that quantifies reverse flow at the walls and study the distributions of the friction and pressure coefficients. Unlike other investigations of this flow case, we also include the upper wall in this study. Furthermore, we describe the dependence of the turbulence and separation characteristics on the Mach number. Keywords LES Separation Compressibility Streamwise-periodic hill channel Reverse flow

1 Introduction The streamwise-periodic hill channel is a canonical test case for separated flow from curved surfaces. It is based on experiments of Almeida et al. [1], with some slight geometric modifications. Figure 1a shows a lateral view of the configuration. The J. Ziefle () and L. Kleiser Institute of Fluid Dynamics, ETH Zürich, Sonneggstrasse 3, 8092 Zürich, Switzerland M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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Fig. 1 (a) Sketch of the periodic-hill channel and (b) computational mesh used for the LES. The dotted line in (a) denotes the edge of the mean flow recirculation zone with clockwise orientation

bottom and the top of the channel are constrained by solid walls. Periodic boundary conditions are employed in the streamwise (x) and spanwise (y) directions, while at the top and bottom walls no-slip boundary conditions are enforced. All lengths are made dimensionless with the hill height h, henceforth omitted for brevity. The computational domain extends over Lx D 9 in the streamwise direction, Lz D 3:036 in the vertical direction, and Ly D 4:5 in the spanwise direction. Although the structure of the mean flow field depends on the specific flow conditions, the flow typically separates near the hill top and reattaches somewhere in the flat region between the hills. Due to the highly unsteady character of the separation process, the resulting separation bubble can be recognised only in the mean flow field [9]. An animation of the instantaneous flow fields reveals a periodic but irregular shedding of smaller vortices that are convected downstream. These highly unsteady flow properties lead to long sampling times (about 40–50 flow-through times) in order to obtain sufficiently converged statistics. The first incompressible DNS of the configuration at the present Reynolds number of 2,800 was conducted recently by Peller and Manhart [4]. An extensive study of the flow physics in the streamwise-periodic hill channel was presented in the LES-based study [2]. This publication, as well as the preceding work that it summarises [5, 6], is based on an approximately four times higher Reynolds number of 10,595. In our previous paper on the periodic-hill channel [9], we compared results of our low-Mach number LES to the above-mentioned incompressible reference data at both Reynolds numbers. The work was aimed at the assessment and validation of our simulation approach using the approximate-deconvolution subgrid-scale model (ADM) for unsteady separated flows. Despite our deliberately-chosen coarse

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Table 1 Calculation parameters, mesh dimensions, and separation/reattachment locations of the present and reference simulations. The cross-sectional Reynolds number above the hill crest is Re D 2;800 in all cases Simulation Peller and Manhart (DNS) [4] 8 (a) ˆ ˆ ˆ < (b) Present LES (c) ˆ ˆ ˆ : (d) (e)

Ma

n/a 0:2 0:7 1:0 1:5 2:5

Nx Ny Nz 464 304 338 128 72 69

Mio. Cells 47:68 .D100 O %/ 0:64 .D1:33 O %/

xsep 0:21 0:21 0:28 0:33 0:39 0:46

xreatt 5:41 5:14 5:40 6:06 5:33 7:58

resolution and our compressible simulation method, we found our results to be in good agreement with the comparison data for most quantities. Most notably, the hard-to-predict separation and reattachment locations of the mean flow and the mean velocity profiles were found to be in excellent agreement. In the present work, we extend the previous study to higher Mach numbers up to the supersonic regime. For reasons of computational economy, we restricted our computations to the lower of the two Reynolds numbers (Re D 2;800). The focus lies on the physical aspects of the flow, especially the Mach number dependence of the separation characteristics and the flow properties at both walls. The parameter study was conducted by performing simulations with successively increasing Mach number, thereby using an instantaneous flow field of one simulation as an initial condition for the simulation at the next higher Mach number. Details about the present computational setup, which is the same as in our previous study, can be found in [9]. Table 1 summarises the key parameters and the mean separation and reattachment locations of the present simulations and the reference DNS.

2 Results In the leftmost picture of Fig. 2a, the mean (i.e., spanwise- and time-averaged) streamwise velocity profiles at Ma D 0:2 are compared to incompressible DNS data of Peller and Manhart [4]. Despite of the present coarse resolution, excellent agreement is achieved. In our previous publication [9], this validation was extended, with similar results, to other statistical quantities and important flow features such as separation and reattachment locations. The left column of Fig. 2 also shows the contours of the mean streamfunction for all five investigated Mach numbers. At Ma D 0:2 this is characterised by a large recirculation region downstream of the hill, and a small separation bubble at its windward foot. Additionally, at the leeward hill face a small secondary separation bubble with opposite orientation is contained in the primary recirculation zone. With increasing Mach numbers, the separation bubbles grow and the reattachment point moves farther downstream. They merge for Ma > 1:5, and one large recirculation region covers the lower wall between the hills. Additionally,

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Fig. 2 First column: contours of the mean streamfunction h iy;t ( positive values, negative values, h iy;t h wall iy;t ) and, in plot (a), mean streamwise velocity profiles, present LES results, incompressible DNS data [4]. Second column: fraction of the spanwise-averaged backflow hriy at lower wall and upper wall. Boundary of mean-flow separation region (shaded in grey), domain boundaries. Third column: contours of the mean Lumley flatness parameter hF iy;t . (a) Ma D 0:2, (b) Ma D 0:7, (c) Ma D 1:0, (d) Ma D 1:5, (e) Ma D 2:5

at the two supersonic Mach numbers the flow also separates at the upper wall due to the presence of an extended region with a large adverse pressure gradient. The primary reason for the growth and the downstream movement of the recirculation zone with increasing Mach number is the inverse relationship between the Mach number and the heat transfer coefficient, k 1=Ma2 . With rising Mach number, the heat transfer coefficient decreases, thus the increasing amount of heat generated by dissipation in the turbulent flow is less easily transported out of the fluid body through the isothermal walls. As a consequence, the fluid temperature rises. Since the dynamic viscosity is directly dependent on the temperature, T 0:7 , the increased fluid temperature comes along with a higher fluid viscosity. The effect is similar to reducing the Reynolds number of the flow, which is generally known to result in a downstream movement of the separation and reattachment locations [9]. As apparent from visualisations of the instantaneous flow field, there is considerable backflow along the walls at all times. To better quantify this phenomenon, we analysed the time fraction at which the wall-parallel velocity just above the wall R t C T u t wall is negative, i.e., r.x/ D t00 Œ1 H.u.x; t/ t wall / dt= T (with the wall-tangent vector in positive x direction t wall and the Heaviside step function H ). The result is displayed in the second column of Fig. 2. At the lower wall, the flow


289

acceleration and the resulting very high velocities above the hill crest prevent backflow, thus the backflow fraction is almost zero. Just downstream of the hill crest, the flow separates and vortices are shed, yielding a sharply rising backflow fraction with a peak of more than 70% backflow. This location lies just downstream of the mean-flow separation point and exhibits the second-largest magnitude of backflow along the bottom wall. Due to the secondary recirculation zone with opposite orientation, which is embedded in the main separation bubble, the fraction of backflow is decreasing rapidly by a considerable amount. The backflow rates grow again at a slower rate downstream of the secondary separation region. For most Mach numbers, a little local maximum appears in the strongly-curved region at the leeside foot of the hill. Downstream of that dent, the fraction of backflow is increasing again to a global maximum, which is located near the streamwise coordinate of the centre of the primary mean-flow separation bubble. Downstream of this maximum, the backflow ratio decreases moderately, until is reaches a plateau around the mean-flow reattachment location. It remains between 40% and 50% in the flow recovery region, until it grows again somewhat due to the small separation bubble at the windward hill foot. Farther downstream, the flow accelerates strongly, and the ratio of backflow sinks fast. Along the second half of the windward hill face (z > 0:5), the backflow rate is negligible. An exception occurs for the two Mach numbers Ma D 0:7 and 1:0, where the backflow ratio rises again upstream of the hill crest. Also note the kink of the backflow ratio graph at the end of the curved region of hill foot (x 8), which is potentially due to wall curvature effects. At the upper wall, the backflow rates are negligible for the simulations without the presence of a detached boundary layer, i.e., up to Ma D 1:0. Significant backflow is only occurring in the range 3 x 6, which coincides with the location of the mean-flow recirculation region for the two highest Mach numbers. Note that the backflow decreases from Ma D 0:2–0:7, while it rises considerably by a factor of about two to the next higher Mach number, Ma D 1:0. Additionally the curve changes its shape slightly to a dual-peak form (with maxima at x 3 and 5, respectively) that gets more distinct for the higher Mach numbers exhibiting a detached boundary layer. For the two highest Mach numbers Ma D 1:5 and Ma D 2:5, a strong adverse pressure gradient causes the upper-wall boundary layer to detach, and a large recirculation zone develops. This separation bubble leads to very high backflow rates of 70–80%, which are of the same magnitude as the values observed in the recirculation region above the lower wall. The large backflow rates are observed primarily in the streamwise range of the mean-flow separation region 3 x 7. However, while the curve rises sharply from almost zero at the upstream edge of the recirculation region, its decay back to very low values at the downstream end of the separation bubble occurs much more gradually until above the upper half of the windward hill side (8 x 9). Within the range of the recirculation region, the graph exhibits the two-peak shape mentioned above. After a local maximum at the upstream edge of the separation bubble, the backflow rates decrease somewhat, until they rise again to a higher, global maximum in the second half of the detached boundary layer. Since increasing Mach numbers lead to stronger pressure gradients,

290


the boundary-layer detachment and thus the upper-wall recirculation region intensifies, resulting in generally higher backflow rates. It is surprising, however, that significant backflow is already occurring in this region at very low Mach numbers such as Ma D 0:2. This leads to the conclusion that already at this Mach number, the adverse pressure gradient exerts a strong effect on the upper-wall boundary layer. Lumley’s flatness parameter F [3], displayed in the third column of Fig. 2, allows for an analysis of the turbulence characteristics of the flow. While the observed values of F reach levels that are very close to zero, i.e., two-dimensional turbulence, the maxima of F (about 0.9 throughout all Mach numbers) lie considerably below the value for three-dimensional turbulence, F D 1. This result is typical for channel flow and can be explained by the nature of the flow configuration and the low Reynolds number. The specific geometry, including the separated boundarylayer downstream of the hill, as well as the influence of the upper and lower walls, does not allow for the development of truly three-dimensional isotropic turbulence. On the other hand, two-dimensional turbulence characteristics are enforced in wall-bounded turbulence by the presence of solid walls. For the three lower Mach numbers Ma D 0:2, 0:7 and 1:0, high values of F are present in the whole upper channel part between the upper-wall boundary layer and the shear layer (z > 1), and in the lower channel part (z < 1) between the lower-wall boundary layer and the shear layer. The highest flatness values with almost three-dimensional turbulence occur in distinct elongated patches in the high-velocity region above the leeward hill face, close to the upper wall. With increasing Mach number, this area grows in streamwise size and thickens somewhat, but in the major part of the remaining channel, the level of F is considerably lower. This is in contrast to the result for the two highest Mach numbers Ma D 1:5 and 2:5, where the area with the highest values of F (and thus almost three-dimensional turbulence structure) covers almost the complete cross-section between the shear layer and the upper-wall boundary layer, and is only interrupted by its separation and the upward-extending influence of the shear layer in the middle of the channel (at x 4:5). As expected, the values of F are very small in the boundary layers at the lower and upper walls for all Mach numbers. In Fig. 2a–c, the thickening of the upper-wall boundary layer due to the adverse pressure gradient can be well observed. For the two highest Mach numbers Ma D 1:5 and 2:5 the detached upper-wall boundary layer is evident from the considerably thicker region with two-dimensional turbulence for 3 x 7. Near the lower wall, two-dimensional turbulence is prevalent in the shear layer downstream of the hill crest, whose characteristics are only gradually displaced by that of the surrounding flow. With increasing Mach number, this region of two-dimensional turbulence extends farther downstream and gains in thickness. At the highest Mach number, the turbulence is primarily two-dimensional in the major part of the domain between the hills (z < 1), an exception being the core of the mean-flow recirculation region above the leeward hill foot (x 1:75). In Fig. 3 we present the streamwise development of the mean pressure and friction coefficients along both walls for the five investigated Mach numbers. Additionally, at the lower wall DNS data from [4] is displayed for comparison. (For the upper wall, no data is available from literature.) Note that the pressure coefficient


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has been centred by subtracting its streamwise mean value, so that the mean pressure coefficient vanish. This allows for a comparison with the incompressible reference data, where the pressure coefficient is usually presented in this manner due to the lack of a reference pressure. For the mean pressure coefficient cp at the lower wall in Fig. 3c, our Ma D 0:2 simulation lies close to the incompressible reference data. The small deviations can be attributed to effects of compressibility in conjunction with the normalisation method [9]. Slightly upstream of the hill crest (x 8:75), the pressure reaches a global minimum due to the very high velocities in this narrow cross-section. After crossing the peak of the hill, the flow decelerates and the pressure rises. At x 0:25, which roughly marks the beginning of the primary recirculation zone, the pressure stagnates at a low level along the leeward hill face. At the beginning of the flat region between the hills, the pressure rises again. In the first half of the plane region, the increase is somewhat stronger. Around the mean reattachment location, the slope of the curve gets flatter, and the pressure coefficient grows with an almost constant gradient in the recovery region until the stagnation zone around x 7 at the windward hill foot. Here the pressure reaches its global maximum and falls quickly along the windward hill face, where the flow is subject to strong acceleration. Within the high subsonic and transonic regimes, i.e., for Ma D 0:7 and 1:0, the development of the pressure coefficient is very similar to the one at Ma D 0:2. Most notably, the pressure gradient between the hills is larger, leading to more distinct peaks. In the range 0 x 4:5, the development of the gradient of the

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pressure coefficient is approximately the same for all three subsonic Mach numbers. (Differences in magnitude of cp are primarily due to the chosen normalisation.) In the downstream half of the channel (x 4:5), the curves for the two higher Mach numbers deviate notably from Ma D 0:2 by rising stronger. Furthermore, the maximum pressure coefficient is reached slightly farther downstream, which coincides with the downstream-movement of the stagnation region for increasing Mach numbers. Between Ma D 0:7 and 1:0 the differences are marginal. A drastic change in form and magnitude of the cp distribution can be observed in the supersonic regime. Here the mean separation point lies considerably farther downstream. Furthermore, the velocity maximum is not reached slightly upstream of the hill crest but near the mean separation point. Consequently, the pressure minimum also occurs downstream of the hill crest. Slightly farther down the hill, the wall is covered by the relatively slowly moving fluid of the recirculation region, thus the pressure coefficient rises sharply. For Ma D 1:5 a short plateau is recognisable which reaches roughly to the hill foot. At Ma D 2:5, the pressure coefficient begins to rise immediately, albeit at a weaker rate along the windward hill face. Downstream of x 2, the pressure coefficient curves for both supersonic Mach numbers rise roughly parallel to each other at a considerably steeper slope than for the three lower Mach numbers. As a result of the strong influence of compressibility, the pressure maxima in the stagnation region above at the windward foot of the hill lie much higher than in the subsonic and transonic regimes. As expected, the pressure coefficient maximum at Ma D 2:5 considerably surpasses the one at Ma D 1:5 and occurs slightly farther downstream. The pressure coefficient cp at the flat upper wall of the configuration, depicted in Fig. 3a, generally exhibits smaller variations than at the lower wall, which are however in the same order of magnitude. For the three lower Mach numbers its distribution is quite regular. The appearance resembles a sinusoidal shape, with the lower part of the curve occurring in the first channel half (0 x 4), and the positive semi-oscillation appearing in the second channel half. The physical explanation of this pattern lies in the effect of the contour of the lower wall on the flow. The strong acceleration caused by the hill contraction causes a pressure drop above the hill (i.e., for 7D O 2 x 2), with a minimum appearing approximately at the location of the maximum velocity, e.g., at x 1 for Ma D 0:2. Downstream of the hill the channel cross-section expands and the flow decelerates, resulting in rising pressures. The maximum pressure is reached roughly vertically above the stagnation zone at the foot of the hill. The following contraction of the channel yields increasing velocities and falling pressures. While the curves of all three Mach numbers Ma D 0:2, 0:7 and 1:0 exhibit very similar shape, there are some minor differences. Of course, there are generally larger pressure differences within the flow for the higher Mach numbers due to compressibility. Therefore, their extremal values are higher. Additionally, the locations of the maximum as well as the minimum values of cp move downstream with rising Mach numbers. This is in agreement with our above findings, where all characteristic locations such as separation and reattachment location move downstream with increasing Mach number. It is however striking that all three curves intersect in the same streamwise location of x 4:25,


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at cp 0 in the chosen normalisation. Note that the pressure drop takes place over a considerably shorter streamwise length than the following pressure increase. This again correlates with the contour of the lower wall, where the hill causing the flow acceleration extends over a much shorter streamwise stretch than the flat region between the hills, where the flow decelerates. In the supersonic range, the separation of the boundary layer at the top wall yields a more complex pressure coefficient distribution. Here also the previously observed oscillating pattern appears, but the amplitudes of the minimum and maximum values are even higher and about of the same magnitude for both Ma D 1:5 and 2:5. Additionally, the pressure maxima occur farther downstream and well above the windward hill face, at x 8 for Ma D 1:5 and at x 8:5 for Ma D 2:5, i.e., shortly above the hill crest. The pressure drop following the maximum extends along the whole leeward hill face and is much steeper in the supersonic range than observed for the three lower Mach numbers. Note that for Ma D 2:5, the slope of the curve exhibits a kink at x 2, where the hill face bends into the flat region, and the decrease of cp continues at a slightly steeper slope. This slight curvature effect can be also observed for other mean quantities. In both supersonic cases, the separation of the boundary layer is preceded by a sharp pressure rise approximately 0:5 hill heights upstream of the separation point, which is located at x 2:7 for Ma D 1:5 and at x 3:25 for Ma D 2:5. Within the recirculation region, the pressure first continues to rise at a lower rate. Roughly at the thickest part of the separation bubble, the cp distribution steepens and rises until the maximum above the hill is reached. For Ma D 2:5, the pressure maximum appears quite distinct and exhibits small curvature, whereas for Ma D 1:5 and the lower Mach numbers the curve around the maximum is rounder. Strong compressibility effects, especially at the hill crest, are evident from the mean friction coefficient cf at the lower wall in Fig. 3d. Again our lowest Mach number case reproduces the DNS data very well. After a sharp peak at x 8:5, i.e., still at the windward side of the hill, where very high velocities occur, the friction coefficient drops steeply until the hill crest. Here cf reaches a short plateau until the mean separation point, where it drops slightly into the negative range. Shortly thereafter it changes its sign again, revealing a confined secondary recirculation region with opposite orientation embedded into the primary separation bubble. Downstream thereof, cf remains quite constant at negative values close to zero until the foot of the hill at x 2. At this point the friction coefficient drops further to its global minimum at x 2:8, where it begins to rise slowly, leading to its sign reversal, which marks the mean reattachment point at x 4:7. In the post-reattachment zone, cf exhibits only small positive values, until it experiences a short dip into the negative range at the windward hill foot (x 7:2). Here the flow stagnates, and a small recirculation zone appears. The following strong flow acceleration can be witnessed by the steep increase of the friction coefficient along the windward hill face. The cf distribution does not change considerably over the whole range of Mach numbers. With increasing Mach number, the peak of the friction coefficient rises somewhat and wanders downstream. The short region with constant cf after the hill crest occurs at a much higher level for the higher Mach numbers. At Ma D 1:5 and

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2:5, a second short peak of cf at slightly lower values than their global maxima is visible, before the friction coefficient drops strongly into the negative range. The secondary separation bubble is also affected by the Mach number. It grows and moves downstream towards the foot of the hill for higher Mach numbers. In the supersonic range, it appears much flatter but considerably more elongated, and ranges over a large part of the lower hill face until it bends towards the flat region. The dip of cf to lower negative values in the flat region downstream of the hill foot gets less pronounced with higher Mach numbers. Instead, the reattachment points move farther downstream. For Ma D 2:5, the friction coefficient remains negative throughout the flat region (although it runs very close to zero at x 6), and the primary recirculation zone covers the bottom walls between the hills. This results in a more pronounced local minimum of cf at the stagnation region near the windward hill foot. In the other simulations, there is a small separation bubble which grows from Ma D 0:2–1:0 and appears again smaller for Ma D 1:5. However, the negative values of cf in this region are quite similar for all those simulations. The magnitude of the friction coefficient cf at the upper wall, see Fig. 3(b), lies in the same range as on the lower wall. While for Mach numbers up to Ma D 1:0 the pressure coefficient distribution exhibited only minor differences, the friction coefficient is subject to larger variations along the upper domain boundary. For Ma D 0:2, cf is highest above the hill crest, where very high velocities occur. The flow decelerates and cf sinks, as the channel and its “virtual contour” (the cross-section bounded by the upper wall and the boundary of the mean-flow recirculation zone) expand. The minimum friction coefficient, less than half of its maximum, is reached roughly vertically above the mean reattachment location. In the following flow acceleration, cf rises at approximately the same rate as it decreases before. The friction coefficient at Ma D 0:7 follows the same general distribution as for Ma D 0:2, but at a higher level. While the minimum values are similar, the maximum friction coefficient lies considerably higher at Ma D 0:7. At Ma D 1:0 the peak is located even higher and downstream of the hill crest, at x 0:5. The friction coefficient then falls quite steep to a minimum at x 3, which lies significantly farther upstream than for the lower Mach numbers, but still at a comparable magnitude. Farther downstream cf is subject to a quite steady rise which steepens at the windward hill foot. For the two supersonic Mach numbers, the adverse pressure gradient at the upper wall is so large that the boundary layer separates. This results in a major change of appearance in the distribution of the friction coefficient. At Ma D 1:5, its maximum is reached approximately at the same position and magnitude as at Ma D 1:0, slightly downstream of the hill crest. However, the immediate strong drop occurring there is delayed until x 2. This coincides with the location where the pressure coefficient reaches its minimum and is approximately 0:5 hill heights upstream of the separation point, where the friction coefficient changes sign. Within the recirculation zone, cf first remains close to zero and experiences a dip at x 4, where the pressure gradient increases. The minimum friction coefficient is located at x 5, at less than one fourth of the magnitude of the maximum. Downstream of this location cf rises at relatively constant slope, except for a kink at x 7, which is also visible for the lower Mach numbers. It is an effect of the beginning cross-sectional


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contraction caused by the contour of the lower wall. The general appearance of the upper-wall friction coefficient at Ma D 2:5 is similar to Ma D 1:5. However, its maximum reaches only 3=4 of the previous value. On the other hand, the plateau of the maximum is wider, so that the drop in cf preceding the separation region and the recirculation zone itself occur about 0:5 farther downstream. Additionally the extent of the recirculation zone is considerably longer and reaches until x 7:5.

3 Conclusions We extended our previous study of the streamwise-periodic hill channel configuration at nearly incompressible flow conditions to Mach numbers up to Ma D 2:5. The present series of large-eddy simulations was performed on a deliberately-chosen coarse grid using the well-proven approximate deconvolution subgrid-scale model with the finite-volume flow solver NSMB. In this first paper of a two-part contribution [8], we investigate the Mach-number dependence of the separation characteristics with an analysis of the mean streamfunction, and study the flow conditions at the walls in detail using distributions of the mean pressure and friction coefficients. In contrast to previous works, we also consider the flow behaviour at the top wall. We introduce a new quantity, the backflow fraction r, to determine the time fraction of instantaneous backflow along the walls. The turbulence structure is analysed using Lumley’s flatness parameter. We are currently extending our research to more complex separated flows by studying two configurations of a jet in crossflow, a generic case [7] and a case related to film-cooling. Acknowledgements All simulations were performed at the Swiss National Supercomputing Centre (CSCS). We would like to thank Jan Vos for fruitful discussions and technical support.

References 1. G. P. Almeida, D. F. G. Durão, and M. V. Heitor. Wake flows behind two-dimensional model hills. Experim. Therm. Fluid Sci., 7:87–101, 1993. 2. J. Fröhlich, C. P. Mellen, W. Rodi, L. Temmerman, and M. A. Leschziner. Highly resolved largeeddy simulation of separated flow in a channel with streamwise periodic constrictions. J. Fluid Mech., 526:19–66, 2005. 3. J. L. Lumley. Computational modeling of turbulent flows. In C.-S. Yih, editor, Advances in Applied Mechanics, volume 18, pages 123–176. Academic Press, 1978. 4. N. Peller and M. Manhart. DNS einer Kanalströmung mit periodisch angeordneten Hügeln. In STAB (Arbeitsgemeinschaft “Strömung mit Ablösung”) Jahresbericht 2004, pages 178 f., 14. DGLR/STAB Fachsymposium, November 16–18, 2004, Bremen, Germany, 2004. 5. L. Temmerman and M. A. Leschziner. Large eddy simulation of separated flow in a streamwise periodic channel constriction. In E. Lindborg, A. Johansson, J. Eaton, J. Humphrey, N. Kasagi, M. Leschziner, and M. Sommerfeld, editors, Turbulence and Shear Flow Phenomena 2, volume 3, pages 399–404, 2001.

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6. L. Temmerman, M. A. Leschziner, C. P. Mellen, and J. Fröhlich. Investigation of wall-function approximations and subgrid-scale models in large eddy simulation of separated flow in a channel with streamwise periodic constrictions. Int. J. Heat Fluid Flow, 24(2):157–180, 2003. 7. J. Ziefle and L. Kleiser. Large-eddy simulation of a round jet in crossflow. In 36th AIAA Fluid Dynamics Conference, San Francisco, CA, USA, June 5–8 2006, 2006. AIAA Paper 2006-3370. 8. J. Ziefle and L. Kleiser. Compressibility effects on turbulent separated flow in a streamwiseperiodic hill channel — part 2. In Notes on Numerical Fluid Mechanics and Multidisciplinary Design. Second DESider Symposium on Hybrid RANS-LES Methods, Corfu, Greece, June 17/18 2007, Springer, 2007. 9. J. Ziefle, S. Stolz, and L. Kleiser. Large-eddy simulation of separated flow in a channel with streamwise-periodic constrictions. In 17th AIAA Computational Fluid Dynamics Conference, Toronto, Canada, June 6–9, 2005. AIAA Paper 2005-5353.

Global Structure of Buffeting Flow on Transonic Airfoils J.D. Crouch, A. Garbaruk, D. Magidov, and L. Jacquin

Abstract The flow field associated with transonic airfoil buffet is investigated using a combination of global-stability theory and experimental data. The theory is based on perturbing a steady flow field obtained from the Reynolds-averaged Navier– Stokes equations. Linearized perturbations are described by an eigenvalue problem, with the frequency and growth rate given by the eigenvalue and global-flow structure provided by the eigenfunction. The experiments provide both steady and unsteady information on the airfoil surface and in the flow downstream of the shock. The theory and experiment show good agreement for the buffet onset conditions – including the critical angle of attack and the buffet-onset frequency. The post-buffet flow structure is also in good agreement, and shows a shock oscillation phase locked to an oscillating shear layer downstream of the shock. Keywords Buffet Global instability Transonic flow RANS Unsteady flow

1 Introduction The upper surface flow over transonic airfoils is characterized by a supersonic zone followed by a shockwave and a subsonic pressure recovery. As the airfoil lift is increased with angle of attack, the shock becomes stronger. At some angle of attack, the flow separates – either from the trailing edge, or locally as a bubble at the foot of the shock. Further increase in the angle of attack results in an onset J.D. Crouch () Boeing Commercial Airplanes, Seattle, USA e-mail: [email protected] A. Garbaruk and D. Magidov Saint-Petersburg Polytechnic University, St. Petersburg, Russia e-mail: [email protected], [email protected] L. Jacquin ONERA-Fundamental/Experimental Aerodynamics Dept, Meudon, France e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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of large-scale unsteadiness, leading to large oscillations in the sectional lift. This unsteadiness has been shown to result from global flow instability [1]. The unsteadiness is characterized by phased-locked oscillations of the shock and the separated shear layer. Results from the global-instability analysis are in good agreement with earlier experiments on a NACA 0012 airfoil [2]. However, these earlier experiments do not provide details about the flow structure or spectral information at the onset conditions. More recent experiments on a modern supercritical airfoil have captured the details of the buffeting flow field in the neighbourhood of the buffet onset [3]. Subsequent results from unsteady Reynolds-averaged Navier–Stokes (URANS) equations show good agreement with the experiments [4, 5]. The URANS results show a dependence on the turbulence model used, as well as a potential influence from the tunnel walls. Here we combine the global-instability theory [1] and the detailed experiments [3] to provide a description of the origins and flow structure of airfoil buffeting.

2 Problem Formulation 2.1 Global Stability Analysis Theoretical predictions are based on a global stability analysis of a steady-state solution obtained from the Reynolds-averaged Navier–Stokes (RANS) equations – see [1] for expanded details. The compressible form of the S-A turbulence model is used [6], including the compressibility correction [7]. The total flow is described by the state vector q D f; u; v; T; Q g, where is the density, u, v are the velocities in the x-, y-directions, T is the temperature, and Q is a scaled form of the eddy viscosity. The total flow solution is split into a steady-state solution q.x; N y/ and an unsteady perturbation q 0 .x; y; t/; q D qN C q 0 . Substituting this into the RANS equations, removing terms governing q, N and linearizing with respect to q 0 provides a set of 0 equations governing q . The unsteady perturbation is then expanded in modal form, q 0 .x; y; t/ D q.x; O y/ exp.i !t/. This leads to an eigenvalue problem governing the complex frequency ! and the unsteady-disturbance mode shape q. O The governing equations are discretized using a finite-different approximation. The steady flow is obtained using the NTS code [8]. The perturbation equations are then solved on the same grid using a hybrid scheme, which blends a forthorder centered scheme with a third-order upwind scheme [1]. The size of the final eigenvalue problem is O.106 /. This is solved using the implicitly restarted Arnoldi method [9]. Using the shift-invert mode, a small number of eigenvalues can be calculated in the neighborhood of a prescribed frequency.

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In the limit of an incompressible laminar basic flow, the analysis simplifies to the approach used by Jackson [10] and Zebib [11] for the analysis of vortex shedding behind cylinders. Results from the current approach for the onset of vortex shedding behind cylinders are shown to be in very good agreement with experiments [1]. For high-Reynolds-number transonic flows, a turbulence model is required to obtain the basic-state flow. In addition, shock smoothing is used to prevent “ringing” in the eigenmode response. The shock smoothing (in conjunction with the local grid spacing) affects the shock thickness; coarser grids and increased smoothing yield a thicker shock, which enhances stability of the steady base flow. Physically, the thicker shock is representative of the time-averaged flow observed once shock buffeting has occurred. Small levels of smoothing on a fine grid do not substantially alter the results.

2.2 Experimental Setup and Signal Processing The experiment considered here is that conducted by Jacquin et al. [3] in the continuous closed-circuit transonic wind tunnel S3Ch of the Fundamental and Experimental Aerodynamics Department of ONERA. The model was a supercritical airfoil OAT15A characterized by a 12.3% thickness-to-chord ratio, a 230 mm chord length and a 780 mm span (aspect ratio AR 3:4). Average Reynolds number was Rec D 3.10/6 based on the chord length. Laminar-turbulent transition was fixed on the model using a Carborundum strip located at x=c D 7% from the leading edge. The Mach number M was varied between 0.70 and 0.75 and the flow incidence ˛, controlled by means of adaptable walls, between 2:5ı to 3:91ı . The measurements comprised surface flow visualisations by oil and sublimating products, steady and TM unsteady pressure by 68 static pressure taps and 36 unsteady Kulite pressure transducers (in the central section of the wing), flow field characterizations by Shlieren films, and velocity fields by means of a two component laser-Doppler velocimeter. At M D 0:73, the value considered herein, unsteady pressure signals revealed perturbation onset at ˛ D 3:1ı , and perfectly periodic signals indicating buffet were obtained for incidences comprised between ˛ D 3:25ı and 3:9ı . The value ˛ D 3:5ı was selected for a full characterisation of the unsteady velocity field with the laser-Doppler velocimeter. As shown in Fig. 1, the meshing used consisted in a series of vertical lines located in the airfoil symmetry plane with variable meshes with separations varying from 2 to 0.5 mm. The phase-averaged technique introduced by Hussain and Reynolds [12] has been applied to this data, following the method described in Forestier et al. [13]. The technique can separate the ‘coherent’ motion, related to the periodic excitation, from the random fluctuating part. A component of the velocity, u.x; t/ for instance, is decomposed into three contributions, u.x; t/ D u.x/ C uQ .x; t/ C u0 .x; t/, where u.x/ is the ensembleaverage, uQ .x; t/ the cyclic component and u0 .x; t/ the fluctuating component. The phase-averaged velocity is defined as < u.x; t/ >D u.x/ C uQ .x; t/. The remaining fluctuating component should be seen as a residue characterising events which are

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Fig. 1 LDV probing of the flow: definition of the meshing (the shaded region corresponds to a separation between two windows)

not in phase with the reference signal. In the present case, the reference signal has TM been chosen as the pressure signal measured by the Kulite transducer located in the mean shock location. The global oscillation of the flow, which can be compared to the global mode provided by the global stability analysis previously described, is characterized by the cyclic component uQ .x; t/. Phase averages of the two-components of the velocity .u; v/ were determined TM using the following procedure: (1) the pressure signal measured by the Kulite transducer located in the mean shock location is used to synchronize the acquisition of LDV signals; (2) the flow period is determined from the low-pass filtered pressure signal and the period is segmented into 20 bins in which data are stored; (3) in each bin, ensemble averages of the velocity (phase averages) and moments of the differences with respect to this phase averages (random fluctuations) are computed.

3 Results and Discussion 3.1 Steady Flow Field We first consider the steady flow field as characterized by the surface pressure distribution. Figure 2 shows a comparison at the sub-critical (steady) condition M D 0:73 and ˛e D 3:0 (where the subscript e signifies the experimental value), with calculations at ˛ D 3:0 and ˛ D 3:5. The overall agreement is good, but the RANS solution yields a shock position farther downstream than the experiment. However, as the angle of attack is increased the RANS shows a forward movement of the shock – roughly matching the sub-critical experimental location .˛e < 3:2/ at ˛ D 3:5. At a fixed Mach number, a higher angle of attack in the RANS results in a slight increase in the roof-top pressure and increased levels of separated flow (with greater unsteadiness, as shown below). This suggests the need for a “Mach correction” in order to match the shock position, the roof-top level and the trailing-edge pressure.

Global Structure of Buffeting Flow on Transonic Airfoils Fig. 2 Surface pressure coefficient at M D 0:73 from theory .˛ D 3:0; ˛ D 3:5/ and experiment .˛e D 3:0/

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These features can be expected to influence the specific point of buffet onset. However, for examining the global flow structure, small differences in the Mach number or angle of attack are not considered significant.

3.2 Buffet-Onset Conditions The global-stability analysis shows the buffet onset results from a Hopf bifurcation. The stability boundary (separating the steady- and unsteady-flow regimes) is show in Fig. 3, along with the critical-mode frequency. Experimental values for the buffet boundary are given for two of the Mach numbers. The results show that the critical angle of attack for buffet onset decreases with increasing Mach number – consistent

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with earlier observations. The frequency at buffet onset increases with increasing Mach number. The experimental frequencies plotted in the figure are for a fixed angle of attack, which is above the critical value. The comparisons show a good agreement between the theory and experiment – especially considering the offset in shock position. An earlier study using URANS showed that the predicted buffet-onset conditions are influenced by the turbulence model and the treatment of the wind-tunnel walls [5]. Inclusion of the wind-tunnel walls reduced the frequency in the URANS results by 3 Hz. This is comparable to the level of frequency offset seen in Fig. 3.

3.3 Unsteady Flow Field The structure of the unsteady flow field is given by the eigenfunction of the global instability. Figure 4 shows the streamwise-velocity and pressure perturbations at four different phases in the oscillation for M D 0:73; ˛ D 3:5ı as predicted by the stability theory. The velocity perturbation is concentrated at the shock and in the shear layer downstream of the shock. This velocity perturbation corresponds to an oscillation of the shock coupled with the modulation of the downstream flow – the velocity in the neighbourhood of the shock increases (or decreases) in phase with the velocity variations in the shear layer. Thus, as the shock moves downstream, the shear layer moves toward the airfoil surface. The velocity perturbations extend into the wake, where the downstream evolution shows a periodic increase and decrease in streamwise velocity. The unsteady pressure is also shown in Fig. 4. The pressure fluctuations appear to originate near the foot of the shock. The disturbance then propagates up away for the surface along the shock. At the same time, the pressure fluctuation (with lower level) propagates downstream in the shear layer. Near the trailing edge the pressure fluctuation intensifies. After rounding the trailing edge, the pressure disturbance propagates upstream along the lower surface. Figure 5 shows the vertical-velocity fluctuation for conditions similar to Fig. 4. Here the contours are chosen to allow a direct comparison to the flow-field fluctuations measured in the experiments. The figure shows ten phases in the oscillation. The theory and experiment show the same overall structure for the oscillating flow. This supports the global-mode description for the buffeting flow, as given above. A quantitative look at the unsteady flow is given in Fig. 6, which shows the rms of the surface pressure. The unsteady pressure fluctuations are largest in the neighborhood of the shock, similar to the field perturbation shown in Fig. 4. The pressure fluctuations downstream of the shock are roughly a factor of 5 smaller than the values near the shock. Upstream of the shock, the fluctuations are negligible. The global mode from the stability theory shows the same form for the unsteady surface pressure. In the stability results, the ratio of the pressure fluctuations at, and downstream of, the shock depends on the effective shock thickness.

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4 Conclusions The combination of stability theory and experiment shows the onset of transonic buffeting flow results from global instability. The global mode is characterized by an oscillating shock phased-locked with an oscillating shear layer downstream. The conditions for the onset of buffeting flow, and the buffeting-flow structure, are well predicted by the global-stability theory. Acknowledgements Some of this work was done while the first author was visiting ONERA/DAFE in Meudon, France.

References 1. Crouch, J.D., Garbaruk, A. and Magidov, D., Predicting the onset of flow unsteadiness based on global instability. J. Comp. Phys. (2007) doi:10.1016/j.jcp.2006.10.035. 2. McDevitt, J.B. and Okuno, A.F., Static and dynamic pressure measurements on a NACA0012 airfoil in the Ames high Reynolds number facility. NASA Tech. Paper No. 2485 (1985). 3. Jacquin, L., Molton, S., Deck, S., Maury, B. and Soulevant, D., An experimental study of shock oscillation over a transonic supercritical profile. AIAA Paper No. 2005–4902 (2005). 4. Deck, S., Detached-eddy simulation of transonic buffet over a supercritical airfoil. AIAA Paper No. 2004–5378 (2004). 5. Mylène, T. and Coustols, E., Numerical prediction of shock induced oscillations over a 2D airfoil: Influence of turbulence modelling and test section walls. Int. J. Heat Fluid Flow 27 (2006) 661–670. 6. Spalart, P.R. and Allmaras, S.R., A one-equation turbulence model for aerodynamic flows. La Recherche Aérospatiale 1 (1994) 5–21. 7. Spalart, P.R., Trends in turbulence treatments. AIAA Paper No. 2000–2306 (2000). 8. Strelets, M., Detached-eddy simulation of massively separated flows. AIAA Paper No. 2001– 0879 (2001). 9. Lehoucq, R.B., Sorensen, D.C. and Yang, C., ARPACK User’s Guide, SIAM Publications (1998).

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10. Jackson, C.P., A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182 (1987) 23–45. 11. Zebib, A., Stability of viscous flow past a circular cylinder. J. Eng. Math. 21 (1987) 55–165. 12. Hussain, F. and Reynolds, W., The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech. 41 (1970) 241–258. 13. Forestier, N., Jacquin, L. and Geffroy, P., The mixing layer over a deep cavity in a transonic regime. J. Fluid Mech., 475 (2003) 101–145.

Low-Order Modeling for Unsteady Separated Compressible Flows by POD-Galerkin Approach R. Bourguet, M. Braza, G. Harran, and A. Dervieux

Abstract A low-dimensional model is developed on the basis of the unsteady compressible Navier–Stokes equations by means of POD-Galerkin methodology in the perspective of physical analysis and computational savings. This approach consists in projecting the complex physical model onto a subspace determined to reach an optimal statistical content conservation. This leads to a drastic reduction of the number of degrees of freedom while preserving the main flow dynamics. The high-order system formulation is modified and an inner product which couples the contributions of both kinematic and thermodynamic state variables is selected. The associated reduced order model is a quadratic polynomial ordinary differential equation system which presents an inherent sensitivity to POD basis truncation for long-term prediction. A calibration process based on the minimisation of the prediction error with respect to reference dynamics is implemented. The predictive capacities of the low-order approach are evaluated by comparison with results issued from the 2D Navier–Stokes simulation of a transonic flow around a NACA0012 airfoil, at zero angle of incidence. This configuration is characterised by a complex unsteadiness caused by a von Kármán instability mode induced by shock/vortex interaction, and a low frequency buffeting mode. Keywords Low-order modeling POD-Galerkin approach Navier–Stokes equations Compressibility effects

R. Bourguet (), M. Braza, and G. Harran Institut de Mécanique des Fluides de Toulouse, UMR 5502 CNRS-INPT/UPS, Allée du Prof. Camille Soula, 31400 Toulouse, France e-mail: [email protected] Present address: Massachusetts Institute of Technology, Department of Mechanical Engineering, 77 Massachusetts Avenue, Cambridge, MA 02139, USA A. Dervieux Institut National de Recherche en Informatique et en Automatique, B.P. 93, 06902 Sophia-Antipolis, France M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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1 Introduction The purpose of the present study is to develop a Reduced-Order Model (ROM) for the prediction of the complex wall-flow features induced by compressibilily effects at high transonic regimes. From a general point of view, the objective of surrogate modeling is to mimic a realistic and complex physical model by a local but faithful approximation involving lower calculation costs. In the context of mutli-disciplinary optimisation or fluid/structure interaction, where complex physical simulations are integrated into iterative processes, low-dimensional models can allow drastic gains of computational ressources. Moreover, such approaches represent relevant tools for physical investigations owing to their inherent mathematical simplicity, especially in the transitional case. To elaborate reduced-order models, meta-modeling like polynomial interpolations, neural networks or response surfaces are often developed on the basis of a pure data-driven approach which involves many high-order computations. The physicdriven methodology presented in this paper consists in a Galerkin projection of the complex model onto a finite-dimensional basis determined to reach optimal energy reconstruction. The physical model is the compressible Navier–Stokes system. The basis is issued from a separable Proper Orthogonal Decomposition (POD, [1], also known as Karhunen-Loève expansion [2]) of the flow variables, which extracts the main fluid energetic properties ([3], among others). The corresponding low-order model is an ordinary differential equation system of considerably reduced dimension compared to the high-order one. This ROM enables the prediction of the main flow dynamics. Various low-order dynamical models were derived from the incompressible Navier–Stokes system. In 2D, the laminar flow past a circular cylinder [4] and transitional cavity flows [5] were efficiently predicted by POD-Galerkin approach. The relevance of this methodolody was illustrated in the 3D laminar case [6, 7] with databases issued from Direct Numerical Simulations. The inherent instability of POD-Galerkin systems was studied and calibration processes lead to significant improvments, based on the addition of an artificial dissipation [8], on data-driven optimisations in the laminar case [9], in the transitional/turbulent case [10], and on the introduction of additional instability or “shift” modes in the low-order basis [11]. POD-Galerkin models were integrated into optimal control processes [12] and error estimates were obtained in this context [13]. Theoretical extensions were developed to increase the robustness of the empirical basis with respect to changes in flow configuration [14]-[17] and to adapt the POD basis to domain deformations [18] in the perspective of design optimisation. The case of reacting flows involving dilatation effects was investigated in [19] by means of a coupling between POD modes, “shift” and “expansion” modes. For compressible flows, the coupling of kinematic and thermodynamic variables in the state system induces specific difficulties concerning the state variable formulation and the inner product invovled in the POD. In [20] a general framework was provided to derive low-order models based on the inviscid Euler equations, via POD-Galerkin approach, among others. Promising results are reported in [21],

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considering an approximation of the compressible Navier–Stokes system valid for moderate Mach numbers and cold flows, by means of an isentropic inner product. As discussed in the present paper, a specific variable change considerably simplifies the ODE system obtained after projection of the complex physical model onto the empirical basis [22, 23]. Investigations of stability properties lead to relevant ROM for moderate Mach numbers and short time predictions, in laminar and transitional cases [24]. In the present paper, an accurate POD-Galerkin model is derived from the compressible Navier–Stokes system expressed in term of modified state variables. The classical spatial inner product is extended to the compressible case and the corresponding quadratic polynomial ODE system enables flow dynamic reconstructions including all state variable contributions. Taking into account of the inherent instability of this low-order model, a stabilisation strategy is applied. Optimal constant and linear terms are added to the dynamical system in order to minimize a specific prediction error with respect to the initial reference dataset. The reliability of the calibrated low-order model is examined in the high transonic regime, in the two-dimensional case at first. The unsteady compressible flow past a NACA0012 airfoil at moderate Reynolds number is considered. The database is issued from a direct numerical simulation via ICARE/IMFT compressible solver [25]. The complex unsteadiness of this configuration is induced by compressibility effects and especially by shock/vortex interaction, as detailled in the first section. In the following, the extraction of the empirical POD basis, the development of the Navier– Stokes ROM and the calibration process are described. The predictive capacities of the present low-order approach are quantified in the last section.

2 Physical Context 2.1 Transition Features in the Compressible Flow Past an Airfoil The transonic flow past a NACA0012 airfoil at zero angle of incidence develops an inherent unsteadiness due to compressible effects during the transition to turbulence. At moderate Reynolds number 0:51 104, the flow is steady at the incompressible regime. At Mach number higher or equal to 0:3 the onset of a von Kármán instability occurs in the wake (mode I). In the Mach number interval Œ0:5; 0:7, this mode I becomes more pronounced and the periodic alternating vortex pattern is clearly developed. The near-region is progressively contaminated by the instability developing in the wake. At Mach number 0:75 a lower frequency mode induced by the oscillation of the supersonic pockets is observed (buffeting) and this mode II has disappeared at Mach number 0:85. Thus, the configuration studied in the present paper (Mach number 0:80, Reynolds number 104 ) is characterized by a complex unsteadiness caused by the two instability mode interaction (Fig. 1). More details about the flow physics of this specific two-dimensional configuration can be found in [25–28].

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a

b Mach: 0.40 0.57 0.74 0.91 1.08

Mach : 0.40 0.57 0.74 0.91 1.08

Fig. 1 (a) Re D 5;000 and M D 0:85: Mode I. (b) Re D 10; 000 and M D 0:8: Mode I and Mode II (present study)

2.2 The Compressible Navier-Stokes System The high-order model to approximate is the unsteady compressible Navier–Stokes system. If no external effort and no external heat flux are imposed on the fluid flow, the governing equations expressed in term of conservative variables can be formulated as follows, in two dimensions: 2

3

2

ui

3

2

0

6 u 7 6 u u C pı 7 6

1i 7 6 17 6 i 1 6 1i vi s U;t C Fi;i D Fi;i with U D 6 7 ; Fi D 6 7 ; Fivi s D 6 4 u2 5 4 ui u2 C pı2i 5 4 2i e

ui e C pui

3 7 7 7; 5

ij uj qi

where is the volumic mass, ui are the components of the velocity, p is the thermodynamic pressure which satisfies the ideal gas lawp D RTp where Tp is the temperature and R is the ideal gas constant. ij D ui;j C uj;i 2=3uk;k ıij is the viscous effort tensor where is the fluid viscosity, qi are the components of the heat flux (qi D CT Tp;i , with the conductivity coefficient CT ) and ıij is Kronecker symbol. e is the total energy defined by: e D Cv Tp C

u21 C u22 ; 2

where Cv is the specific heat coefficient. :;t and :;i denotes respectively the time and space derivatives. Concerning the boundary conditions at the frontiers of the physical domain, only time-independent relations are prescribed: constant values on inflow frontier and free-stream conditions on the outlet. No-slip condition is imposed on the airfoil. The numerical dataset is issued from a calculation via ICARE/IMFT compressible solver. This is a structured finite volume code in which Roe’s upwind spatial scheme is implemented with MUSCL approach for the convective part and a centered second order scheme for the diffusive term. The temporal integration is ensured by an explicit four-stage Runge-Kutta scheme of fourth order accuracy.

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The C-type meshgrid (Nx D 369 89 nodes) used was validated in the present flow configuration, the corresponding pressure and aerodynamic coefficients are reported in [25].

3 The POD-Galerkin Model 3.1 Reduced Order Basis Extraction via POD The Proper Orthogonal Decomposition is a Singular Value Decomposition which consists in expanding each physical variable as a linear combination of specific eigenfonctions. The POD of a time/space-dependent function v can be written as follows: v.x; t/

NX POD

ai î .x; t/; or v.x; t/

NX POD

i D1

yi .t/î .x/

(1)

i D1

assuming time/space separation. yi are time-dependent functions and î stationnary spatial modes determined as successive solutions of the following constrained optimisation problem: î C1 .x/ D arg max

2L2 ./

D

E .v …i v; /2 subject to kk2 D 1;

(2)

where represents a temporal averaging, is the spatial domain, .:; :/ is an inner product which has to be defined on L2 ./ and …i is the projector onto fˆ1 ; :::; î g. Finding î in (2) is equivalent to find the orthonormalised eigenvectors of a state variable spatial correlation matrix. After discretisation, Nx is the number of space discretisation points and Nt the number of flow snapshots collected. If the first NPOD spatial modes are taken into account with NPOD Nx and NPOD Nt , the expression (1) provides a low-order approximation of v which is optimal in the sense of energy reconstruction. The “snapshot-POD” technic [29] consists in finding the eigenvectors of the temporal correlation matrix which reduces considerably the size of the problem in the case of numerical simulations where Nx Nt . The spatial inner product involved in (2) is a crucial point in case of multiple state variables (vi ; i D 1; :::; 4). In the present study the classical choice is adopted by considering an addition of the each state variable contribution as in [22, 24]: .s1 ; s2 / D

4 Z X i D1

vi1 vi2 dx:

This approach allows an important reduction of the number of degrees of freedom in the state system, from 4 Nx to NPOD . Only time-independent boundary

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conditions are prescribed and the POD has to be calculated on time-centered snapshots (vi .x; t/ vi .x/) because spatial modes can only respect homogeneous boundary conditions. The expansion of each state variable is then: vi .x; t/ vO i .x; t/ D vi .x/ C

NX POD

yj .t/îj .x/; with vi .x/ D

j D1

1 T

Z vi .x; t/dt: T

(3)

3.2 Projection onto the POD Basis and Stabilsation Strategy A direct use of the variable expansions (3) in the previously defined compressible system leads to fractional expressions which do not allow trivial Galerkin projection onto the POD basis. An alternative is suggested by [22] to derive a quadratic polynomial ODE system by considering a modified formulation of the state vector U ! UQ D Œ1= u1 u2 pt . The corresponding low-order model is then, for i D 1; :::; NPOD : 8 N PC1 POD C C NP ˆ 0:5 [shown in green-blue]), above a layer of relatively low-velocity close to the wall (typically u=U1 < 0:5 [shown in red-orange]). The term layer is used here to emphasize that while they are defined instantaneously, they typically extend in the streamwise direction across the measurement domain. The reader should note, however, that this is somewhat of a subjective characterization, given the complex nature of the interaction, and is done so for the purposes of simplifying the conceptual interpretation of the results. The two layers are typically separated by a thin region of relatively high shear, indicated by the yellow contour. This interface has an irregular and intermittent nature, which is a particularly dominant feature of the redeveloping boundary layer. Figure 2b, c, and e show that this interface typically has a downstream-sloping pattern, and appears to be more distinct in the first part of the interaction region, than farther downstream. Spatiotemporal studies of an incident SWTBLI at Mach 2.3 [12] have determined that frequencies tend to decrease along the separated flow region, noting that in subsonic recirculating flows, such frequency evolutions are associated with large-scale structures convected in a mixing layer, which develops downstream of flow detachment. It therefore appears possible that vortical structures develop within the separated shear layer. Another feature, shown in Fig. 2a–c, is that while the reattachment process takes place within a relatively short streamwise extent, the overall velocity deficit within the inner layer persists much farther downstream, and it is clear the redeveloping boundary layer does not fully recover to its incoming conditions within the present measurement domain.

3.2 Identification of Vortical Structures To complement the discussion on the instantaneous velocity fields, the presence of vortical (or coherent) structures within the interaction are visualized. Following the method proposed by Hunt et al. [13], vortical structures are identified as any contiguous region of flow where the second invariant Q of the velocity gradient tensor is greater than zero. In the present study, it is assumed that the criterion Q > 0 is valid in compressible flow, since the divergence of velocity field was found to be relatively small, and because results were obtained that were very similar to the use of other (incompressible) criterion, such as the 2 criterion [14]. Note that in two-dimensional measurements there are missing components of the velocity gradient tensor, and the following determined patterns are only a section of the complete three-dimensional structure from the x–y plane, therefore representing only a footprint of the spanwise vortical structure of the interaction. Figure 3 shows contours of Q > 0, with a small non-zero threshold to distinguish from measurement noise. Note the nonuniform axes. Under-sampled velocity vectors, displaying 1 in 5 in the streamwise direction are also shown, with a convective velocity of Uc D 0:7U1 . (Note that each figure part corresponds to the preceding figure parts of Fig. 2.) The results reveal numerous distinct regions of Q > 0, which are interpreted to be sections through vortical structures. Structures

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Fig. 3 Vortical structures within the interaction using the Q criterion. Note the nonuniform axes. Contours of Q > 0 are shown, along with convective velocity vectors Uc D 0:7U1 . Vectors are displayed 1 in 5 in the streamwise direction for clarity

can be distinguished relatively close to the wall, within the incoming boundary layer, although the present resolution is considered insufficient to properly resolve the swirling motion in this region, and they are most likely numerical artefacts. Figure 3b, c, and e show that as the vortical structures approach the interaction, they are lifted away from the wall, upwards into the shear layer in the detachment region. They appear to turn around the separation bubble, where they propagate mainly through the incident shock wave’s tip. This is consistent with the direct numerical simulation (DNS) of an incident SWTBLI by Pirozzoli and Grasso [6], who found that the oscillatory motion of the incident shock wave occurs mainly at its tip. Farther downstream, the vortical structures return to the proximity of the wall in the reattachment region. Notably, very few vortical structures can be observed within the separated flow region itself. Rather, vortical structures typically occur along the interface between our two layers. It is interesting to observe that a similar lifting of the vortical structures occurs even when the boundary layer remains fully attached throughout the interaction, such as observed in Fig. 3f for instance. It therefore appears that the vortical structures respond to a strongly retarded (yet downstream moving) inner layer flow, in much the same way as they do to the reversed-flow. Accordingly, when there is no significant inner velocity deficit, then there is no observed lifting of the vortical structures, as demonstrated in Fig. 3d for instance. This is especially important in

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the present interaction, because it means that the interaction can instantaneously exhibit many of the features of a large-scale separated interaction, while the boundary layer in fact remains fully attached. The convective velocity vectors show that fluid is drawn in and ingested inbetween the vortical structures, as they propagate throughout the interaction, resulting in an intermittent fluid exchange between the two layers described above. This is in fact not dissimilar to the scenario that has been described in low-speed separated flows by Simpson et al. [15], who discusses how movies of laser-illuminated smoke and turbulence energy results reveal how large-scale eddy structures supply the near wall separated flow. It therefore appears that the present interaction shares similarities with incompressible separated flows. Farther downstream, vortical structures become much more broadly distributed normal to the wall, as observed in Fig. 3e for example, and as hinted at by the frequency evolutions mentioned earlier. Wall-normal turbulence intensity profiles of the present interaction have been found to spread more broadly over the vertical height of the interaction than the streamwise intensity [8], and it is now clear that, in contrast to what occurs within the incoming boundary layer, a significant wall-normal fluid exchange takes place away from the wall within the redeveloping boundary layer.

3.3 POD Analysis We now decompose the fluctuating part of the flowfield into a series of eigenmodes using POD, to construct a simplified description of the unsteady behaviour of the interaction. Since the POD is based on ensemble-averaged correlations, the eigenmodes represent only statistical information about the flow features they represent. A conceptually appealing interpretation is to view the eigenmodes as being perturbations on the mean flowfield. As the motivating prelude, the modal energy and cumulative modal energy distributions of the POD eigenmodes are shown in Fig. 4. The POD eigenspectrum reveals that the first eigenmode is by far the most dominant, capturing almost 20% of the total energy; four times the amount of the second eigenmode. A mode that contributes a substantial proportion of the total energy of the flow is considered to play a more significant role in the flow dynamics than a mode that contains less energy, and it therefore seems that this first mode represents a configuration that playsPa significant P role in the interaction’s dynamics. The cumulative ratio i D1 K i = i D1 M i gives an indication of the energy convergence, by showing the amount of energy contained in the M element system, contained in the first K eigenmodes. The cumulative energy distribution shows that over 70 modes are required to capture 75% (say) of the total energy. In comparison, other POD studies that have used planar PIV data, such as the one by Patte-Rouland et al. [16], report about 60 modes in their study of an annular jet, for the same amount of energy. Only a limited number (say 2%), and we therefore restrict our attention to discussing these eigenmodes.


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Fig. 4 Eigenspectrum of the POD analysis. Modal energy content shown for the kth eigenmode and cumulative modal energy shown for the Kth cumulative sum. Arrows assign data with appropriate axis

To visualize the eigenmodes, we first create a 1D phase space using the temporal coefficients from the analysis. Recall that one can arbitrarily choose a finite number K of the most energetic modes, to form a subspace spanned by the first K eigenmodes. Similarly, subspaces can be formulated based upon a single eigenmode, by first ordering the temporal coefficients of all M observations, such that n k D fak .tn / ak .tnC1 / : : : ak .tM /g. An eigenmode then yields M subspaces, the nth subspace of the kth eigenmode un k .x; y/ given by ukn .x; y/ D ak .tn /

k

.x; y/ ;

n D 1; : : : ; M

(3)

These subspaces provide a convenient method to analyze the behaviour of the kth eigenmode. For notational convenience, we first define as ( nk

D

0 when ak D .ak /min I n D 1 1 when ak D .ak /max I n D M 2

(4)

Using the above procedure, eigenmodes 1, 2, and 4 are shown in Fig. 5, with the streamwise and wall-normal velocity components on the left and right, respectively. Recall that the first eigenmode is by far the most energetic. Figure 5a shows that this first mode consists of relatively large streamwise velocity fluctuations within the incoming boundary layer, separated flow region, and redeveloping boundary layer. In contrast, the associated wall-normal velocity component in Fig. 5b shows fluctuations across the reflected shock wave. When viewed as a perturbation on the mean flow, this eigenmode contributes to the unsteadiness observed in the instantaneous

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Fig. 5 POD eigenmodes for u0 =U1 (left) and v0 =U1 (right). Note that modes 2 and 4 show 2u0 =U1

realizations, through an energetic association between fluctuations within the incoming boundary layer, separated flow region, and reflected shock wave. The reader may note the lack of velocity fluctuations along the incident shock wave, indicating that it is a steady feature. Higher-order eigenmodes portray a more intricate structure. Figure 5c and e reveal that subspace bifurcations take place within the separated flow region and redeveloping boundary layer. The term bifurcation refers here to the qualitative changes that take place. Such bifurcations are attributed to the higher-order harmonics required to properly represent the high-dimensional phase space of the data, and when used in combination with other modes, separate the different temporalscales within the flow. In particular, Fig. 5e and f show that coherent flow features bifurcate to become smaller-scale features within the redeveloping boundary layer. The subspace arrangement of these features suggests that they are associated with


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the propagation of perturbations within the redeveloping boundary layer. Overall, it appears that the subspace features contained within the eigenmodes represent the phenomenology observed in the instantaneous realizations.

4 Conclusions The unsteady flow organization of an incident SWTBLI has been investigated using PIV and POD. Generally, the interaction instantaneously exhibits a two-layer structure, characterized by a relatively high-velocity outer layer and a low-velocity inner layer. Discrete vortical structures are prevalent at their interface, which appear to play a role in the interaction between the two layers. Low-order eigenmodes from the POD analysis show an energetic association between the incoming boundary layer, separated flow region, and the reflected shock wave. Higher-order eigenmodes show subspace bifurcations leading to smaller-scale features, which are required to properly represent the flow. The subspace features contained in the eigenmodes appear to represent the phenomenology observed in the instantaneous realizations and provide a convenient synthesis of the results. Acknowledgments This work is supported by the Dutch Technology Foundation STW under the VIDI – Innovation Impulse program, grant DLR.6198.

References 1. Dolling, D.S., Fifty years of shock wave/boundary layer interaction research: what next? AIAA J. 39 (2001) 1517–1531. 2. Knight, D.D. and Degrez, G., Shock wave boundary layer interactions in high speed flows. A critical survey of current numerical prediction capabilities, Advisory Rept. 319, AGARD 2 (1998) pp. 1.1–1.35. 3. Erengil, M.E. and Dolling, D.S., Physical causes of separation shock unsteadiness in shock wave/turbulent boundary layer interactions, AIAA Paper 93–3134 (1993). 4. Andreopoulos, J. and Muck, K.C., Some new aspects of the shock-wave/boundary-layer interaction in compression flows, J. Fluid Mech. 180 (1987) 405–428. 5. Dolling, D.S. and Murphy, M.T., Unsteadiness of the separation shock wave structure in a supersonic compression ramp flowfield, AIAA J. 21 (1983) 1628–1634. 6. Pirozzoli, S. and Grasso, F., Direct numerical simulation of impinging shock wave/turbulent boundary layer interaction at M D 2:25, Phys. Fluids 18 (2006) 065113. 7. Holmes, P., Lumley, J.L., and Berkooz, G., Turbulence, coherent structures, dynamical systems and symmetry, Cambridge University Press, Cambridge (1996). 8. Humble, R.A., Scarano, F., and van Oudheusden, B.W., Experimental study of an incident shock wave/turbulent boundary layer interaction using PIV, 36th AIAA Fluid Dynamics Conference & Exhibit, San Francisco, CA (2006). 9. Scarano, F., Iterative image deformation methods in PIV, Meas. Sci. Tech. 13 (2002) R1–R19. 10. Sirovich, L., Turbulence and the dynamics of coherent structures, Q. Appl. Math. XLV (1987) 561–590.

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11. Rowley, C.W., Colonius, T., and Murray, R.M., Model reduction for compressible flows using POD and Galerkin projection, Physica D 189 (2004) 115–129. 12. Dupont, P., Haddad, C., and Debiève, J.F., Space and time organization in a shock-induced separated boundary layer, J. Fluid Mech. 559 (2006) 255–277. 13. Hunt, J.C.R., Wray, A.A., and Moin, P., Eddies, streams, and convergence zones in turbulent flows, Center for Turbulence Research Report, CTR-S88 (1988) 193–208. 14. Jeong, J., Hussain, F., Schoppa, W., and Kim, J., Coherent structures near the wall in a turbulent channel flow, J. Fluid Mech. 332 (1997) 185–214. 15. Simpson, R.L., Chew, Y.-T., and Shivaprasad, B.G., The structure of a separating turbulent boundary layer. Part 2. Higher-order turbulence results, J. Fluid Mech. 113 (1981) 53–73. 16. Patte-Rouland, B., Lalizel, G., Moreau, J., and Rouland, E., Flow analysis of an annular jet by particle image velocimetry and proper orthogonal decomposition, Meas. Sci. Technol. 12 (2001) 1404–1412.

Dependence Between Shock and Separation Bubble in a Shock Wave Boundary Layer Interaction J.F. Debiève and P. Dupont

Abstract We present experimental results obtained in a turbulent boundary layer at a Mach number of 2.3 impinged by an oblique shock wave. Strong unsteadinesses are developed in the interaction, involving several frequency ranges which can extend over two orders of magnitude. In this paper, attention is focused on the links between the low frequencies shock motions and the separation bubble. An interpretation based on a simple scheme of the longitudinal evolution of the instantaneous pressure is proposed. As it is mainly based on the pressure signals properties inside the region of the shock oscillation, it may be expected that it will be still relevant for different configurations of shock induced separation as compression ramp, blunt bodies or over expanded nozzles. Keywords Shock wave Boundary layer separation Unsteadiness

1 Introduction Shock wave boundary layer interactions occur in various aeronautical applications: for example, in inlets of supersonic aircrafts or in over expanded nozzles. Depending on the geometry of the problem, different kind of interactions can be found. An important family of shock wave interaction is the case where the boundary layer separates because of the adverse pressure gradient produced by the shock wave and reattaches downstream. In these cases, a limited region of the flow becomes subsonic downstream the shock wave, allowing spatial couplings through the pressure between different regions of the interaction. Such a behaviour involves, for example, flows as compression ramps, incident shock waves, blunt fins or over expanded nozzles in restricted separation cases. In all these separated cases, new unsteadinesses are developed. They involve a large number of time scales, which can be at J.F. Debiève () and P. Dupont Institut Universitaire des Systèmes Thermiques Industriels, Université de Provence and UMR CNRS 6595, 5 rue Enrico Fermi, 13 Marseille Cedex 13, France e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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least one or two orders of magnitude larger than the energetic time scales of the upstream boundary layer. Several pioneering works, [1, 2], and more recent experimental studies [3, 4] have shown that these flows exhibit very low frequency shock motions as well as the development of new large scales downstream the separation point. Recent works [5, 6] have shown that, independently of the particular spatial organisation of these interactions (compression ramp, incident shock, blunt fin...) a typical dimensionless frequency, or Strouhal number, can be associated with these low frequency shock movements. Nevertheless, the origin of these unsteadinesses are not yet well understood. Two scenarii can be considered: The shock motions can be driven by some upstream low frequency events. Or they can be driven by some low frequency downstream events from the subsonic separated region. Both scenarii can be expected, with possibly some couplings between them. In that respect, evidences of upstream or downstream dependencies have been obtained in compression ramps [2, 7–9] and in shock wave reflection [6, 10]. Some of these experimental results have shown statistical links between signals associated to the shock motions and the separated bubble. The aim of this paper is to examine the spatial and temporal dependencies between the shock movements and the separated bubble. From the different experimental observations, we developed a simple scheme able to reproduce the spatial and temporal wall pressure evolutions inside the interaction. To develop this scheme, we will consider unsteady wall pressure measurements together with instantaneous velocity fields obtained recently by PIV measurements in the case of an incident shock wave. The output of the scheme will be compared with experimental observations such as longitudinal evolution of mean wall pressure, standard deviation and phase shift between pressure signals recorded in different region of the interaction. As it is mainly based on the description of wall pressure variation in the vicinity of the foot of the unsteady detached shock, some generality can be expected for different cases such as compression ramp or blunt fins.

2 Experimental Set Up and Description of the Flow Unsteadiness The experiment was carried out in the hypo-turbulent supersonic wind tunnel at IUSTI. It is a continuous facility with a closed-loop circuit. The nominal conditions of the interaction are summarised in Table 1. A shock generator is fixed on the ceiling of the wind tunnel. The flow deviation can be set to 8ı and 9.5ı . The length of interaction L is defined as the distance between the foot of the reflected shock Table 1 Aerodynamic parameters of the flow upstream of the interaction

M

U1

ı0

Re

Cf

Tt

2.28

550m=s

11mm

6:9 103

2 103

300 K

Dependence Between Shock and Separation Bubble

333

Fig. 1 Spark Schlieren visualisation of the interaction, flow deviation of 8ı

(X0 ) and the extrapolation down to the wall of the incident shock. The dimensionless coordinate is therefore X D .x X0 /=L, and the interaction extends from X D 0 to 1. The global organisation of the incident shock wave boundary layer interaction obtained by spark Schlieren visualisation is presented in Fig. 1. The flow deviation due to the incident shock is 8ı and the pressure gradient is strong enough for the layer to separate. In this case, the reflected shock which originates upstream of the recirculating zone is known to be strongly unsteady with low frequencies oscillations of several hundreds of Hertz [4, 6, 11]. A dimensionless frequency, or Strouhal number, defined by SL D f L=U1 , where U1 is the external velocity, has been used to compare the different shock induced separated flows [5, 6]. In all cases, very low Strouhal numbers are obtained: SL is of the order of 0.03. PIV measurements [10] highlighted the development, inside the recirculating zone, of high energetic structures. These structures have been shown to be generated in the mixing layer which develops from the separation point. They correspond to frequencies one order of magnitude lower than the energetic eddies in the upstream boundary layer, as in subsonic separations. Superimposed on these ones, frequencies of one order of magnitude below (in the same range as the characteristic frequencies associated with the reflected shock motions), have been identified in the interaction zone. These low frequencies correspond to a Srouhal number SL ' 0:03, therefore they cannot be compared directly with low frequencies observed in subsonic detached flows, generally associated to some “flapping” phenomena of the mixing layer, involving Strouhal number of about 0.12 [12]. Hereafter only the low passed signal associated with this range of frequencies is taken into account. These low frequencies are found in almost linear dependence with the reflected shock signal: the coherence function between the pressure in the vicinity of the unsteady reflected shock and pressure in the separation where 0:5 < X < 1 keep a significant value close to one (see Fig. 2). For sake of comparison, we have also shown in this figure the low level of coherence between the shock and the initial boundary layer for the case D 8ı .

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Coherence

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −3 10

−2 10 Strouhal (θ = 8°)

−1

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Fig. 2 Coherence between the reflected shock and the downstream flow (solid lines) and the upstream boundary layer (dashed line), D 8ı 0.3 0.2 0.1

Correlation

0 −0.1 −0.2

x2* = 0.57 x2* = 0.67 x2* = 0.76 x2* = 0.86 x2* = 0.95

−0.3 −0.4 −0.5 −0.6 −0.7 −6

−4

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Fig. 3 Intercorrelation function between wall pressure signals near the reflected shock and in the recirculating zone (X 1), where the intercorrelation vanishes progressively.


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0.3 0.25

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0.2 0.15 0.1 0.05 0 −0.05 0.5

1

1.5

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Fig. 4 Ratio of the optimal delay time to the characteristic time of the reflected () D 8ı , (ı) D 9:5ı

Moreover, one may remark that the optimal delay time of the intercorrelation function is nearly equal to zero for X < 1, which means that the instantaneous pressure variations due to the shock motions are simultaneous with the pressure variations in the recirculating bubble. More precisely, the ratio of this delay time to the characteristic time scale for the phenomenon variation which is the time scale of the shock excursion (i.e. the inverse of the shock dominant frequency), is presented on Fig. 4. This ratio is negligible in the recirculating bubble (X < 1). For the strongest shock D 9:5ı , some similar results are obtained. Signals are found in phase opposition with zero delay time near the reattachment point (X D 1), see Fig. 4. So, as in the 8ı case, the link between the reattachment region and the reflected shock movements is quasi instantaneous, even compared to an acoustic time defined as L=a or a convection time L=U1 . This means that the instantaneous pressure variations due to the shock movements are simultaneous with the pressure variations at these points and are not related through any propagative or convective scheme along the longitudinal direction. On the other hand, for the case D 9:5ı , non-zero, negative optimal delay times can be observed in the recirculating bubble (see Fig. 4). Therefore, it seems that, in this case, some propagative or convective scales are overimposed to the global pulsation of the recirculating bubble. In both cases, this strong link becomes weaker downstream in the relaxation zone.

3 Links Between Shock Motions and the Separated Region Previous works have already put in evidence strong statistical links between the low frequency shock movements and the flow which develops downstream in various shock induced configurations. Most of these results have been derived from

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unsteady wall measurements. For example, in a compression ramp at M D 1:5, Thomas et al. [2] finds a strong coherence between the wall pressure fluctuations recorded near the mean position of the foot of the separation shock and recorded near the reattachment point. Moreover, he observed that signals where in phase opposition. Similar results were obtained in a M D 5 compression ramp [9]. Nevertheless, these finding were only deduced from wall pressure measurements, so that the aerodynamic interpretation are rather difficult. More recently, [8, 10] PIV measurements in compression ramp and incident shock wave have confirmed these results. In the case of the incident shock wave, evidences of dependence between the shock movements, with large amplitude vertical displacement of the recirculating bubble have been given. A model based on such behaviour has been proposed by Thomas et al. to explain the phase opposition in the case of ramp compression. The main point of the model is the observation of a specific pattern on the instantaneous pressure repartitions at the reattachment point producing some unsteady signals when the recirculating bubble is contracting or dilating. Nevertheless in our case of incident shock wave, no particular trend on the pressure fluctuation is observed near the reattachment point, and the scheme cannot be directly used. Moreover, it is difficult with Thomas’ interpretation to explain the phase opposition between the shock signal and any point inside the recirculating bubble. In the next part, we modify this starting point, and we propose a simple model describing the longitudinal instantaneous wall pressure repartitions.

3.1 Instantaneous Wall Pressure Scheme Our objective is to define the simplest model for the longitudinal instantaneous wall pressure repartition and its variations when the reflected shock moves, compatible with the experimental observations. As the delay time of influence is negligible in respect of the time scale of the phenomenon, and as signals are nearly linearly related (C oh ' 1) the instantaneous longitudinal pressure repartitions can be expressed in a one-to-one correspondence with the instantaneous movements of the reflected shock around its mean position, i.e. a function of the position of the focused shock. The time history of the wall pressure near the foot of the unsteady reflected shock is reported Fig. 5 for two different positions inside the region of shock oscillations. The upstream shock displacements correspond to the positive rises and the downstream ones are associated with the negative steps. In this figure, the upstream wall pressure and the pressure step given by the inviscid theory is also given for the considered case ( D 9:5ı ). It is clear that, at least near the wall, a model based on a single focused shock wave cannot be sufficient to describe the pressure signals. Only about the half of the pressure increase can be associated with a shock-like increase, when the rest corresponds to a progressive longitudinal increase of pressure. From these observations, we define our model with the following characteristics: Firstly, the step of pressure through the focused shock is constant but less than the theoretical total increase of pressure.


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70 65

p(mmHg)

60 55 50 45 40 35 30 35

40

45

50

55

t(msec) Fig. 5 Wall pressure signals for different positions near the mean position of the reflected shock

Secondly, the levels of pressure at the point of the focused shock depends on its position. Then, a crude scheme describing these signals can be as follows: A constant pressure p0 upstream the region of shock oscillations. A longitudinal pressure gradient in the region of oscillation, defined from the offset of the signals plotted Fig. 5 A constant step of pressure starting from the current pressure upstream of the shock position. This upstream pressure depends on the instantaneous position of the shock wave. This step of pressure is about the half of the total increase pressure downstream of the shock wave with respect to the infinite upstream pressure p0 . Downstream the instantaneous shock wave, the recirculating zone is developing: it is mimicked by a constant adverse pressure gradient deduced, for example, from the mean pressure measurements. This leads to the global scheme which is detailed on the Fig. 6, with four main parameters:

Lex : length of the region of shock oscillations. p: step of pressure through the focused wave. @p1 =@x: longitudinal pressure gradient in the region of shock oscillations. @p2 =@x: longitudinal pressure gradient in the recirculating zone. Finally, the probability density function of the shock presence on the excursion length Lex is supposed to be Gaussian.

The different parameters are deduced from experimental signals. The length of oscillations Lex is derived from the longitudinal evolution of the standard deviation of the wall pressure fluctuations [11]: Lex ' 0:3.

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J.F. Debiève and P. Dupont unsteady shock shema (downstream and upstream shock motion). Longitudinal pressure distribution

P(t)

60 40 20

upstream BL

shock intermitency

0 −0.5

separation

0

0.5

0 X*

0.5

60

P(t)

Reflected shock 40 20 0 −0.5

Fig. 6 Unsteady shock scheme: shock motions in upstream and downstream directions

The residual adverse pressure gradient is defined as a linear increase of the pressure immediately upstream of the shock wave, starting at the beginning of the oscillation zone of the shock. The step of pressure p and the residual pressure gradient @p1 =@x are deduced from the experimental signals ( p=Dp D 0:53, where Dp is the incident shock pressure step) and the pressure gradient such as the mean pressure reaches the right level at the end of the intermittent shock region. Finally, the pressure gradient inside the recirculating zone, @p2 =@x, is deduced from the mean wall pressure slope in this region. With these four parameters, the model is fully defined. In the next section, comparisons with experimental results are presented.

3.2 Comparison with Experimental Results From this scheme we can obtain statistical informations on the pressure repartitions, which depend only on the position of the moving shock. A random simulation with a Gaussian law leads to the mean, rms, phase of the pressure. We show in Fig. 7 that the simulated mean pressure reproduces conveniently the experiments. The instantaneous repartitions are smoothed by the weighting action of the shock intermittency. In Fig. 7, we check, for the case D 9:5ı , that the simulated rms pressure level reproduces in a satisfactory way the rms value p of low passed pressure signals. We will detail now some results which can be found in some particular points and the comparison with the equivalent experimental results.


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unsteady shock shema 60

P mean

50 40 30 20 −0.5

0

0.5

p’ RMS

6

4

2

0 −0.5

0

0.5

X* Fig. 7 Mean and standard deviation: — instantaneous pressure (scheme); experiments (passband on low frequencies shock motion); scheme results

1. The maximum level of pressure fluctuations The maximum of the rms pressure is located near of the median position of the shock (X D 0), where the intermittency coefficient D 1=2 and its value is in our case p 0 D p = p ' 0:43, which can be compared to the experimental value (0.34). With this model, it may be retained that the maximum of pressure fluctuations, in the intermittency zone, is proportional to the step of pressure for the focused part of the reflected shock. 2. Level of rms pressure downstream the shock oscillations Downstream of the intermittent zone the model leads to a fluctuation level such as: p D j@p1 =@x @p2 =@xjc (1) where c is the rms values for the shock position excursions. From this expression or from the simulations (Fig. 7), typical values compared to the incident shock pressure step are p =Dp D 10% (case D 9:5ı ). This model indicates also that the pressure fluctuation level for the low frequency range, in the recirculating bubble, is proportional to the slope difference (@p1 =@x @p2 =@x) of the unfocused compressions and is also proportional to the length scale of the reflected shock motion.

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70 60 50

P(t)

40 30 20 P(t): downstream

10

shock

0 −0.5 −0.4 −0.3 −0.2 −0.1

0 X*

0.1

0.2

0.3

0.4

0.5

Fig. 8 Instantaneous phase opposition between signals recorded at the foot of the shock (X D 0) and in the recirculating bubble (X D 0:22)

3. Phase shift between the reflected shock and the recirculating bubble The phase shift between the pressure signals at the foot of the shock and in the recirculating bubble can be deduced from the model. The fluctuations correspond to the difference between the instantaneous pressure (thick line in Fig. 8) and the mean pressure (thin line). In the simplified case of a periodic shock motion, a reconstruction of the temporal signal (in arbitrary scales) for two points placed respectively at the mean position of the reflected shock and in the recirculating bubble, is presented Fig. 8. They are in phase opposition, which is in agreement with the experimental results. We can notice that the two cases of in-phase or out-of- phase signals are defined by the sign of the slope difference (@p1 =@x @p2 =@x). Therefore, a shock model as an Heaviside distribution with constant plateau downstream the shock (@p1 =@x @p2 =@x D 0) does not leads to a phase opposition between the two signals. The proposed scheme for the low passed wall pressure behaviour in the interaction synthesizes the different experimental observations. However, it is only descriptive, it does not explain the shock decomposition in two part in the intermittent zone: about one half for a focused shock, and another half with an continuous compression. This hypothesis is the basic point for the explanation of the phase opposition phenomena or for the low passed residual pressure fluctuation level in the recirculating bubble.


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4 Conclusions The present results give a preliminary overview on unsteadiness for an incident shock wave induced separation, with different shock intensities. We propose for the wall pressure repartition, in a low frequency range, a very simplified scheme for the wall pressure repartition able to synthesize the various experimental observations. Experiments show that the link between the low frequency shock motions and the vicinity of the reattachment is quasi instantaneous, quicker than an acoustic propagation or convection in a streamwise direction. This leaves the possibility of a global perturbed motion in the other directions. So, the analysis of the stability of such a system would be of great interest. The possibility of an excitation of the system by perturbations coming from the upstream turbulent boundary layer could also be considered. Acknowledegments Part of this work was carried out with the support of the Research Pole CNES/ONERA Aérodynamique des Tuyères et Arrière-Corps (ATAC) and with a grant of the European STREP UFAST (DGXII). Their support is gratefully acknowledged. Comments of Dr. J.P. Dussauge are also gratefully acknowledged.

References 1. D. Dolling, AIAA Journal, 39, 8, 1517 (2001) 2. F. Thomas, C. Putman, H. Chu, Experiments in Fluids, 18, 69 (1994) 3. A. Nguyen, H. Deniau, S. Girard, T.A. de Roquefort, in 38th AIAA Joint Propulsion Conference, Indianapolis, AIAA Paper, 02-4001, Indianapolis, Indiana, USA (2002) 4. C. Haddad, Instationnarités, mouvements d’onde de choc et tourbillons a` grande e´ chelles dans une interaction onde de choc/couche limite avec décollement. Thèse de doctorat, Université de Provence Aix-Marseille I (2005) 5. J. Dussauge, P. Dupont, J. Debiève, Aerospace Science and Technology, 10, 85 (2006) 6. P. Dupont, C. Haddad, J. Debiève, J. Fluid Mech., 559, 255 (2006) ¨ Unalmis, ¨ 7. O. D. Dolling, AIAA Journal, 36, 3, 371 (1998) 8. B. Ganapathisubramani, N. Clemens, D. Dolling, in 45th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 8–11 January (2007) 9. M. Erengil, D. Dolling, AIAA Journal, 29, 5, 728 (1991) 10. P. Dupont, S. Piponniau, A. Sidorenko, J. Debiève, AIAA Journal, 46, 6, 1365–1370 (2008) 11. P. Dupont, C. Haddad, J. Ardissone, J. Debiève, Aerospace Science and Technology, 9, 7, 561 (2005) 12. M. Kiya, K. Sasaki, Journal of Fluid Mechanics 137, 83 (1983)


Large Eddy Simulation of a Supersonic Turbulent Boundary Layer at M D 2 :25 A. Hadjadj and S. Dubos

Abstract This work deals with numerical simulation of a spatially-developing supersonic turbulent boundary layer at a free-stream Mach number of M D 2:25 and a Reynolds number of Re D 5;000 with respect to free-stream quantities and momentum thickness at inflow. Since a shock-capturing scheme is used, a hybrid numerical scheme has been developed to reduce its dissipative properties. The issue of the generation of coherent turbulent boundary conditions is also addressed. A method originally developed by Lund, based on a rescaling technique, has been modified by adjusting the scaling coefficient to provide smooth transition between the inner and the outer parts of the boundary layer. This modification is essential for avoiding the drift previously observed in the mean streamwise velocity profile. The obtained results are analysed and discussed in terms of mean and turbulent quantities. Excellent agreement between LES, DNS and experimental data is obtained. The validity of the assumptions of the strong Reynolds analogy (SRA) is also addressed. Keywords Unsteady turbulent supersonic flows Large eddy simulation Turbulent boundary conditions Strong Reynolds analogy Shock-capturing schemes

1 Introduction Eddy structures and internal dynamics of compressible supersonic turbulent boundary layers may play an important role in aerospace applications, specifically when surface heat transfer on high-speed vehicles or unsteadiness in shock/turbulent boundary layer interactions are of concern. The purpose of this paper is to develop reliable CFD tools and estimate the area of their applicability for complex compressible flows situations, including shocks, boundary layer, acoustics, compressibility

A. Hadjadj () and S. Dubos Institut National des Sciences Appliquées de Rouen, CORIA – Unité Mixte de Recherche C.N.R.S. 6614. Avenue de l’Université, 76801 Saint Etienne du Rouvray, France e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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effects... The primary focus of the present contribution is the study of a spatiallydeveloping turbulent boundary layer at Mach number 2.5 over an adiabatic flat plate using LES method. This test-case provides the (unsteady) inflow conditions to the shock-boundary layer interaction problem studied experimentally by Deleuze and Laurent [6, 8]. In addition, analyses of the turbulent structures may significantly contribute to the understanding of the turbulence behaviour of supersonic boundary layers as well as the development of improved compressible turbulence models.

2 Numerical Procedure and LES Methodology In this study, a fifth-order WENO scheme [2] combined with a centred fourth-order scheme is used to calculate the convective fluxes. Using a selective Ducros’ sensor, it was possible to confine the use of the WENO scheme to the portions of the flow that contain discontinuities (shocks). This technique contributes to reduce significantly the dissipation of the numerical scheme. Viscous terms are discritized using a centred fourth-order accurate, while an explicit third-order Runge-Kutta of Shu and Osher [3] is used for time integration. For numerical stability reasons, the minimum value of ˆ (where ˆ is the Ducros’ sensor defined in Fig. 1), for which the centred scheme is selected, is fixed to ˆc D 0:035. The computed mean value of ˆ shows that the centred scheme is mainly used within the boundary layer, since ˆ < ˆc for y=ı < 0:96 (see Fig. 1 – left). The advantage of using a hybrid scheme is evC ident from Fig. 2 (right), where the normalised mean velocity profile Uvd (with Z U Cq C Uvd D =p d U C .y C / D ln.y C /= C C ) exhibits a better behaviour. In 0

particular, if only the WENO scheme is used, the value of the skin-friction velocity, u , is underestimated by approximately 30% compared to experimental data. However, this value is reduced to 10% with the hybrid scheme. As previously

Fig. 1 Mean distribution of Ducros’ sensor in the boundary layer (left). Influence of numerical scheme on the longitudinal velocity profile (right)

Large Eddy Simulation of a Supersonic Turbulent Boundary Layer at M D 2:25

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Fig. 2 Numerical flow visualization. Instantaneous density field with iso-vorticity contours

reported [9, 10], this kind of underestimation is customary for compressible LES. Concerning the inflow boundary conditions, an existing method of generation of unsteady compressible turbulent boundary layers [5, 11, 13] has been modified to avoid the drift of the mean velocity profile, observed in supersonic boundary layer simulations. The modification was achieved through an appropriate adjustment of the scaling coefficient to provide smooth transition between the inner and the outer parts of the boundary layer [1]. Doing so, the new recycling and rescaling method becomes robust and relaxes faster towards the target experimental values (mainly 1=2 the skin-friction velocity, u D w , where w D .@u=@y/jw and the boundary-layer thickness ı).

3 Results and Discussion A supersonic incoming boundary layer at M1 D 2:3 and Re D 5;000 (in the absence of interacting shock) are reported here after. The size of the computational domain is: Lx 15 ı, Ly 6:5 ı and Lz 0:6 ı, where ı D 10:83 mm is the incoming boundary-layer thinckness. Notice that the spanwise length of the computational domain represents 1/10th of the experimental wind tunnel extent. The two-point autocorrelation coefficients in the homogeneous direction (z), for both the turbulent velocity and thermal variables, are examined. Results (not presented here for concision) show that the decorrelation of velocity fluctuations is achieved over a distance of Lz =2, indicating that the computational domain is chosen large enough to not inhibit the turbulence dynamics. The mesh has about 2:4 106 grid C points, distributed in wall units as: x C D 40, zC D 7 and ymi n D 1, where

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y C D yu =w , with w and w the kinematic viscosity and the density at the wall, respectively. These computations were performed on a parallel IBM-SP Power4 using 40 processors and required 140 h of CPU time. An unsteady view of the supersonic flow is presented in Fig. 3. The examination of the instantaneous three-dimensional iso-vorticity field shows that the boundary layer is fully developed and self preserving. Also, the simulation reveals the appearance of large-scale motion in the outer region of the boundary layer, dominated by the entrainment process. These large-scale structures are particularly active near the boundary-layer edge, where they remain coherent long enough and are strongly responsible for the intermittency of the boundary layer its growth rate. Near-wall streaks can be visualized by contours of the streamwise velocity fluctuation, which is shown in Fig. 3 in a wall-parallel plane at y C 10. It is obvious from Fig. 3 that the computational domain contains several streaks (more than five) in the spanwise direction, spaced by about LC z D 455 wall units, which is four times larger than the “Minimal Flow Unit”recommended by Jimenez and Moin [12]. The reported turbulence statistics are examined to evaluate their consistency with both DNS [4] and experimental measurements [6, 8]. They are based on timeaveraging of the instantaneous three-dimensional fields that were extracted from a time series covering 160 characteristic times m D ıi =U1 , where ıi is the incoming boundary-layer thikness evolving at a free-stream velocity, U1 . As shown in Figs. 4 and 5, simulations match well with experimental results (for other parameters of interest see the reference [1]).

Fig. 3 Instantaneous longitudinal velocity fluctuations in a wall-parallel plane at y C 10

Fig. 4 Distributions of normalized mean flow variables (left) and subgrid turbulent viscosity as function of wall-normal distance (right)

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Fig. 5 Distributions of normalized Reynolds shear stresses (left) and r.m.s values (right)

4 Conclusions In this paper, a new approach, based on the use of a combined filter and discontinuity sensor for monitoring the flow solution, is developed and validated for the simulation of supersonic turbulent flows containing shocks with fine scale flow structures. The current research is motivated by the desire to construct reliable compressible Navier–Stokes solvers with accurate numerical tools for predicting complex supersonic aerodynamics in real applications. The numerical procedure, developed in this study (a 3D compressible LES solver with improved inflow-data generation method) has been used to analyse the spatial evolution of a supersonic turbulent boundary layer at M D 2:25. This test-case provides the (unsteady) inflow conditions to the shock-boundary layer interaction problem studied experimentally by Deleuze and Laurent [6, 8]. Distributions of mean and turbulent flow quantities are analysed and compared to experimental measurements and DNS data. Very interesting results are obtained. In particular, it is found that the LES accurately predicts the mean temperature and density profiles, skin friction, root mean square of velocity, temperature fluctuations and Reynolds shear stress profiles. In agreement with DNS [4, 7], this study shows that the u velocity component and temperature are weakly anti-correlated (RuT is approximately 0.5). Experimental evidence, however, suggests a higher value of the correlation coefficient than was found in this simulation. Finally, fluctuations of the total temperature are not negligible and the strong Reynolds analogy (SRA) is not valid. Acknowledegments Part of this work has been carried out within the research activities of the ATAC group (Aérodynamique des Tuyères et Arrière-Corps) supported by CNES and ONERA. Computational facilities were provided by CNRS – IDRIS (Institut du Développement et des Ressources en Informatique Scientifique, Paris) and CRIHAN (Center de Resources Informatiques de HAute Normandie, Rouen).

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References 1. Dubos S. (2005) Simulation des grandes e´ chelles d’écoulements turbulents supersoniques, Ph.D. thesis, INSA of Rouen. 2. Jiang G. S. and Shu C. W. (1996) Efficient implementation of weighted ENO schemes, J. Comp. Phys. 126: 202–228. 3. Shu C. W. and Osher S. (1988) Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comp. Phys. 77: 439–471. 4. Pirozzoli S., Grasso F. and Gatski T. B. (2004) Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at M D 2:25, Phys. Fluids 16: 530–545. 5. Lund T. S., Wu X. and Squires K. D. (1998) Generation of turbulent inflow data for spatiallydevelopping boundary layer simulations, J. Comput. Phys. 140: 233–258. 6. Deleuze L. (1995) Structure d’une couche limite turbulente soumise a` une onde de choc incidente, Ph.D. thesis, Université Aix-Marseille II, France. 7. Guarini S., Moser R., Shariff K. and Wray A. (2000) Direct numerical simulation of a supersonic turbulent boundary layer at Mach 2.5, J. Fluid Mech. 414: 1–33. 8. Laurent H. (1996) Turbulence d’une interaction onde de choc/couche limite sur une paroi plane adiabatique ou chauffée, Ph.D. thesis, Université Aix-Marseille II, France. 9. Spyropoulos E. T. and Blaisdell G. A. (1998) Large-eddy simulation of a spatially evolving supersonic turbulent boundary-layer flow, AIAA J. 36: 1983–1990. 10. Sagaut P., Garnier E., Tromeur E., Larcheveque L. and Labourasse E. (2004) Turbulent inflow conditions for large-eddy simulation of compressible wall-bounded flows, AIAA J. 42: 469– 477. 11. Stolz S. and Adams N. A. (2003) Large-eddy simulation of high-Reynolds number supersonic boundary layers using the approximate deconvolution model and a rescaling and recycling technique, Phys. Fluids 15: 2398–2412. 12. Jimenéz J. and Moin P. (1991) The minimal flow unit in near-wall turbulence, J. Fluid Mech. 225: 213–240. 13. Urbin G. and Knight D. (1999) Compressible large eddy simulation using unstructured grid: supersonic boundary layer. Second AFOSR Conference on DNS/LES, Kluwer Academic Publishers, June 7–9 Rutgers University: 443–458.

Film Cooling Mass Flow Rate Influence on a Separation Shock in an Axisymmetric Nozzle P. Reijasse and L. Boccaletto

Abstract The influence of the mass flow rate of a film cooling upon the free shockinduced separation inside an over-expanded axisymmetric slot nozzle has been experimentally investigated in the Onera R2Ch wind tunnel. The characteristics of the shock wave – turbulent boundary layer interaction are differently affected depending on whether the film mass flow rate is greater or smaller than a critical value. Above this critical value the laminar nature of the film governs the interaction, which leads to a more precocious flow separation in the nozzle. Beneath the critical value, the interaction properties are weakly affected by the presence of the film. Even, for very weak film mass flow rates, one can observe a slight favourable influence upon the separation position. One suggests that the film – not energetic enough – breaks up into the turbulent boundary layer and reinforces it by adding momentum. Keywords Flow separation Shock wave-boundary layer interaction Film cooling Nozzle

1 Introduction The film cooling effectiveness has been largely investigated in supersonic and hypersonic regimes for application to gas turbine blades or scramjet combustors. From this abundant literature some studies have focused onto the film cooling influence either in the presence of bump-induced adverse pressure gradients as seen by Teekaram et al. [8] and by Zakkay et al. [9], or with a fixed incident shock induced by a shock generator as seen by Juhany and Hunt [3] and by Kanda and Ono [4].

P. Reijasse () ONERA, Experimental and Fundamental Aerodynamics Department, 8 rue des Vertugadins, 92190 Meudon, France e-mail: [email protected] L. Boccaletto CNES, Launchers Directorate, Centre Spatial d’Evry, Rond Point de l’Espace, 91023, Evry Cedex M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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Wall film cooling is also a technology used in rocket engine nozzles to prevent intense convective heat fluxes. In addition to thermal protection, film cooling may improve engine performance by reducing skin friction and providing thrust in the high Mach number domain. At ignition and at low altitudes rocket engine nozzles run in over-expansion regime. This leads to the presence of an oblique shock interfering with the nozzle boundary layer and the wall cooling film. In this paper, the effect of the film cooling mass flow rate upon the free shock-induced separation has been investigated in a subscale axisymmetric over-expanded slot nozzle through the analysis of wall pressure properties.

2 Test Set-Up and Model The test campaign has been executed in the ONERA R2Ch blowdown wind tunnel of Meudon Center [1]. The nozzle model is shown in Fig. 1. The nozzle throat and exit diameters are d D 20 mm and D D 112:9 mm, respectively. The throat-to-exit distance is L D 125 mm. The nozzle set-up allows to independently feed the main nozzle and its annular injection slot (Fig. 2). A calibrating throat mounted upstream of the film injection device determines the mass flow rate qfilm to be symmetrically distributed into four pipes. The pipes thus feed an annular settling chamber upstream of the wall film injection slot. The ratio of the nozzle jet stagnation pressure over the ambient pressure is close to 47. This nozzle pressure ratio (NPR) value leads to the overexpansion of the supersonic nozzle flow with an extended shock-induced separation zone. The film-to-nozzle jet stagnation pressure ratio pst;film =pst can vary

Fig. 1 Mounting in the R2Ch wind tunnel

Film Cooling Mass Flow Rate Influence on a Separation Shock

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Fig. 2 Principles of the nozzle set-up with the file injection slot device

from 1% to 8%. As the nozzle jet conditions (NPR) are fixed, the variation of the ratio pst;film =pst is equivalent to the variation of the film mass flow rate relatively to that one of the nozzle jet. The nozzle jet Mach number at the slot is close to M D 3:5. The nominal wall film Mach number at the slot injection exit is Mfilm D 1:88 (Fig. 3b). The height of the sonic throat of the injection slot is 0.58 mm. The film Reynolds number based on the injection slot throat conditions varies from 3:34 103 to 2:41 104. For a fixed film injection mass flow rate of 8%, former theoretical studies [6, 7] of the flow separation in this nozzle have shown that the film is laminar.

3 Experimental Data Analysis 3.1 Film Adaptation The co-flowing of the two supersonic flows downstream of the injection slot base geometry (Fig. 3a) leads the film to adapt to the main flow static pressure along a certain distance. As shown in Fig. 4 the adaptation is obtained at X=L D 0:45 whatever the level of the film stagnation pressure pst;film is.

3.2 Wall Pressure Distributions The wall pressure distributions are severely affected by the film mass flow rate qfilm (Fig. 5). The two characteristics which are notably changed are (a) the pressure slope p= X induced by the pressure jump by crossing the shock foot and (b) the streamwise position X0 of the pressure curve.

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Fig. 3 Injection slot (a) pressure taps (b) local Mach numbers

0,022 0,020 0,018 0,016

1,05% 1,42% 1,85% 2,23% 2,66% 3,03% 3,42% 4,24% 5,02% 5,84% 6,50% 7,45% 0,19%

X/L=0.3516 at the exit of injection slot X/L=0.3432 upstream of injection slot

p/pst

0,014 0,012 0,010

X/L=0.3732 downstream of injection slot

X/L=0.3532 at the base of injection lip

0,008 0,006 0,004 0,002 0,000 0,34

0,345

0,35

0,355

0,36

X/ L

0,365

0,37

0,375

0,38

0,4

0,45

0,5

0,55

0,6

0,65

X/L

Fig. 4 Wall pressures in the vicinity of the injection slot for different film stagnation pressures

One can distinguish two types of pressure slope values as shown in Fig. 6. Sharp rises of wall pressures are seen for small film mass flow rates as for no-film configurations. For qfilm values above 1% the maximum values of the pressure slopes decrease. This can reveal whether the film exercises an influence on the shock foot by spreading it or not. The streamwise position X0 of the pressure curves has been roughly evaluated as shown in Fig. 5. Then one defines a distance DX which is the difference between the X0;film values obtained in the presence of the film relatively to the X0 values obtained without film. The distance DX=X0 is plotted versus the film mass flow rate ratio (Fig. 7). The value DX D 0 is thus a reference to discuss the influence of the film. Three types of pressure curve positions versus film mass flow rate ratios are observed. Positive values of DX which indicate a less extended separation zone


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0,020 0,018 0,016

p/pt

0,014 0,012 0,010

1,05% 1,42% 1,85% 2,23% 2,66% 3,03% 3,42% 4,24% 5,02% 5,84% 6,50% 7,45% sans film sans film

0,008 0,006 0,004 0,5

0,6

X0

0,7

0,8

0,9

1

X/L Fig. 5 Wall pressure distributions .NPR D 47/ for different film mass flow rate ratios Fig. 6 Pressure slopes .NPR D 47/ for different film mass flow rate ratios

9000 Meridian line 0° Meridian line 120° Meridian line 240°

8000 7000

(Dp/Dx)m

6000 5000 4000 3000 2000 1000 0 0%

2%

4%

6%

8%

pt,film /pt

in the nozzle are obtained for the smallest film mass flow rate ratios (qfilm less than 2%). For qfilm in the range 2–4%, there is no influence upon the flow separation position. For qfilm greater than 4%, DX is negative and the separation region is thus more extended than in the case without film. The behaviour (DX < 0 when qfilm greater than 4%) is coherent with the fact that the film is laminar, thus less resistant to an adverse pressure gradient. An explanation for DX values greater than 0 is that the film – weakly energetic – should break up into the nozzle boundary layer. Thus its non-zero momentum adds to the one of the nozzle turbulent boundary layer which becomes thus more resistant. In the intermediate range where DX is close to 0 one can suggest that it results from the balance between the two antagonistic facts relatively to the separation position: the film momentum contribution and the laminar nature of the film sub-layer.

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P. Reijasse and L. Boccaletto 0,1 Generating line 0⬚ Generating line 120⬚ Generating line 240⬚

DX=(X0,film-X0)/X0

0,05 0 −0,05 −0,1 −0,15 −0,2 −0,25 0%

1%

2%

3%

4% pt,film/pt

5%

6%

7%

8%

Fig. 7 Pressure curves positions versus film mass flow rate ratios .NPR D 47/

3.3 Pressure Fluctuations Levels The pressure fluctuations levels normalized by the local wall pressure p0 have been plotted versus the normalized streamwise coordinate .X X0 /=L (Fig. 9). One clearly observes the three zones which characterize a shock wave – boundary layer interaction: The attached flow region where prms is small and does not exceed 1% of the mean local wall pressure p0 . The interaction region where prms rapidly increases and reaches the highest values, representing 35% of the mean local wall pressure p0 . This level is about twice the maximum of prms values measured in supersonic compression ramps at Mach 3 [2, 5]. The fully separated flow region where prms =p0 values are nearly constant at a level smaller than 10%, a level which is comparable to the values measured on compression ramps. Two classes of pressure rms distributions have been distinguished (Fig. 9a and b) according to the film mass flow rate ratio qfilm . For qfilm smaller than 4% the pressure rms distributions obtained are better correlated than those for qfilm greater than 4%. An order of magnitude of the interaction length is about 10% to 20% of the nozzle length L, which represents 6 to 12 times the nozzle boundary layer height δ evaluated by Pitot probing at about 2 mm (Figs. 8 and 9).


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10 dZ (mm)

X = 47mm

8

6 Pi/Pc ~ 84

4 δ≅2mm

2 2508

0 0

0,04

0,08

0,12

0,16

P'i/Pi

0,2

Fig. 8 Pitot pressure probings of the flows at 2 mm downstream of the injection slot (the height of the injection slot is 1.8 mm)

b

0.4 0.35

Test No.

0.3

3262 3263 3264 3263 3262 3262 3263 3263 3263

0.25 0.2 0.15 0.1

x +

pst_inj/pst

+ x

0% 0% 0% 1.02% 1.84% 2.29% 2.70% 3.11% 3.53%

x+ x +

prms/p0

prms/p0

a

x + x +

0.4 0.35

Test No.

0.3

3263 3264 3264 3264 3264

0.25 0.2

p

st_inj

/p

st

4.38% 5.21% 6.08% 6.89% 8.53%

0.15 0.1

+ + x+ x x x+

0.05 x+ ++x x+ x+ x 0 xxx+++ −0.4 −0.3 −0.2 −0.1

0.05

x+

+ + x+ x x +

0

0.1

0.2

(X-X0)/L film mass flow rates < 4%

0.3

0.4

0 −0.4 −0.3 −0.2 −0.1

0

0.1

0.2

0.3

0.4

(X-X0)/L film mass flow rates > 4%

Fig. 9 Pressure fluctuations levels normalized by the local wall pressure p0 versus the normalized streamwise coordinate

4 Conclusions The effect of the film cooling mass flow rate upon the characteristics of a free oblique shock wave – boundary layer interaction has been investigated in a subscale axisymmetric over-expanded slot nozzle through the analysis of wall pressure properties. The test campaign has been executed in the ONERA R2Ch blowdown wind tunnel of Meudon Center. The film-to-nozzle jet stagnation pressure ratio pst;film =pst can vary from 1% to 8%. The characteristics of the interaction are differently affected depending on whether the film mass flow rate is greater or smaller than a critical value. In the present study, the critical value is qfilm 4%. For qfilm > 4% the film keeps its laminar nature all along the nozzle wall which induces the expected changes (pressure slope more gentle, separation more precocious, interaction length larger) relatively

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to the case of the nozzle turbulent boundary layer without film. For qfilm < 4% the interaction properties are weakly affected by the presence of the film, and for very weak values of qfilm .< 1%/ one can observe a slight favourable influence upon the separation position. One suggests that, in the latter case, the film is not energetic enough and breaks up into the turbulent boundary layer, but reinforces it by adding momentum. Acknowledegments This work has been carried out within the CNES-ONERA ATAC (Aérodynamique des Tuyères et des Arrière-Corps) research program. The authors would thank the collaboration efforts of G. Rancarani, D. Coponet, J.-M. Luyssen and J.-C. Lorier from the Fundamental and Experimental Aerodynamics Department of ONERA.

References 1. Coponet, D., Influence of a film cooling on flow separation and side loads in propulsive nozzles. R2Ch wind tunnel. (translation of the French title). Onera report, 2/10515 DAFE, October 2006. 2. Dolling, D.S. and Or, C.T., Unsteadiness of the shock wave structure in attached and separated compression ramp flow fields, AIAA 83–1715, July 1983. 3. Juhany, K.A. and Hunt, M.L., Flow-field measurements in supersonic film cooling including the effect of shock wave interaction. AIAA 92–2950 (1992). 4. Kanda, T. and Ono, F., Experimental studies of supersonic film cooling with shock wave interaction (II). J. Thermophys., 11(4) (1997) 590–593. 5. Muck, K., Dussauge, J.-P. and Bogdonoff, M., Structure of the wall pressure fluctuations in a shock-induced separated turbulent flow. AIAA 85–0179, January 1999. 6. Reijasse, P., Morzenski, L., Blacodon, D. and Birkemeyer, J., Flow separation experimental analysis in overexpanded subscale rocket-nozzles. AIAA Paper 2001–3556. 7. Reijasse, P., Aerodynamics of overexpanded propulsive nozzles: free separation and side loads in stabilized regime. Ph.D. thesis, University of Pierre-et-Marie-Curie (Paris 6-Jussieu, Paris, France) (2005). 8. Teekaram, A.J.H., Forth, C.J.P. and Jones, T.V., Film cooling in the presence of mainstream pressure gradients. ASME 90-GT-334 (1990). 9. Zakkay, V., Wang, C.R. and Miyazawa, M., Effect of adverse pressure gradient on film cooling effectiveness. AIAA-73-0697 (1973).

Detached Eddy Simulation of a Nose Landing-Gear Cavity R. Langtry and P. Spalart

Abstract Some aircraft have exhibited a noticeable vibration and aero-acoustic phenomenon inside the nose landing gear cavity. The goal of the present study was to determine whether unsteady CFD using either unsteady RANS or detached eddy simulation (DES) could predict the cavity oscillation that was measured in a Boeing wind tunnel test. In general the agreement between the experiment and CFD was good. The CFD predicted an aircraft scale cavity tone frequency of 17 Hz compared to the measured value of 15 Hz. As well, the CFD predicted sound pressure level of the tone was within 4 dB of the measurements. From the present results it would appear that CFD can be used as a tool to investigate and possibly mitigate nose gear cavity tone mechanisms on new aircraft designs. Keywords DES URANS Cavity Noise Aircraft Landing gear

1 Introduction During early flight tests aircraft have occasionally exhibited a noticeable vibration and aero-acoustic phenomenon inside their nose-gear cavity. An oscillation of interest occurred when the gear was partially retracted (wheels up, rear doors closed, front doors open, see Fig. 1). These oscillations can often be eliminated through the introduction of various fixes during the development or flight test program. One such fix that proved very effective was to introduce baffles behind the wheels in order to damp out the pressure oscillations. These baffles were designed based on a series of wind-tunnel experiments, and the problem does not occur on production airplanes. Currently, Computational Fluid Dynamics (CFD) has progressed to a point where it is used routinely at Boeing for investigating steady flows. For industrial purposes, unsteady flows are usually only investigated on an experimental basis. The goal of the present study is thus to determine whether unsteady CFD can R. Langtry () and P. Spalart Boeing Commercial Airplane Group, Seattle, WA, 98124 e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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R. Langtry and P. Spalart

Shear layer oscillation

Rear bay doors (closed)

Fig. 1 Flow over the nose-gear cavity when the gear is partially retracted with the rear doors closed and the front doors open. Contour plot of Mach number

predict the magnitude and frequency of the pressure oscillations that were measured in a nose gear cavity during flight and wind tunnel testing. A suitable level of accuracy in terms of the oscillation magnitude would be on the order of ˙25% of the pressure change during one oscillation period. The problem of intense nose-gear cavity oscillations, among many others, must be avoided in new aircraft designs and CFD is likely to become an essential tool for doing this. It is expected that sooner or later during the early development phase of a landing gear the design will start being checked using CFD. This would include situations where the unsteady loads could potentially be strong enough to raise the noise and vibration contribution from the landing gear to unacceptable levels. The ultimate goal would thus be to rely on unsteady CFD simulations to check the design and reduce the likelihood of having to develop a fix for the cavity oscillation during the flight test program. Very large calculations will probably be needed, however these are expected to be possible in an industrial context within a few years as cluster computing becomes more prevalent. The CFD technique of choice for the flow physics involved in cavities is Detached Eddy Simulation [1]. The main advantage of DES over either Unsteady Reynolds Averaged Navier-Stokes (URANS) or Large Eddy Simulation (LES) is that the unsteady geometry-specific three-dimensional eddies which constitute turbulence can be resolved, but only where directed by the user through the use of grid density. Consequently, the external fuselage boundary layers can be modeled using a relatively coarse RANS grid (and thus be nearly steady) while the complex flow field inside the cavity can be modeled using Large Eddy Simulation (LES),

Detached Eddy Simulation of a Nose Landing-Gear Cavity

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provided the grid inside the cavity is fine enough. Similar applications have been done for generic cavities and DES has been rather successful, at least for empty cavities but also including the effect of various types of bay doors [2, 3]. The geometry that will be included in the simulation consists of the fuselage from the nose to the beginning of the wing, the cavity, the bay doors, wheels and the main landing-gear strut. It should be noted that a full LES that attempts to resolve the turbulence eddies in the fuselage boundary layers is not currently feasible. This is because the grid and timescale requirements for resolving the turbulent boundary layer eddies would be far too costly on present day computing clusters.

2 Experiment The wind tunnel test was done in Boeing’s Low Speed Aero-acoustic Facility (LSAF). The tunnel consists of a 2.5 by 3.5 m free jet in a large anechoic test chamber which is 20 m long, 23 m wide and 9 m high. The wind tunnel model scale is approximately 1/16th of a typical large passenger aircraft assuming a fuselage diameter 6 m (0.375 m for the wind tunnel model). Noise measurements for frequencies between of 200 Hz and 80 kHz (12.5 Hz and 5 kHz aircraft scale) are possible with the existing foam wedges present in the tunnel [4]. A representative fuselage model was mounted upside down in the test section (see Fig. 2). The lifting surfaces were not present in the experiment and their effect on the cavity oscillation is thought to be minimal as the upwash near the nose of the aircraft is only on the order of 0:15ı . The nose gear cavity is shown in Fig. 2 and includes the rear bay doors (which are closed) the front bay doors (open) and a simplified nose landing gear geometry. The free jet for this case had a Mach number of 0.25 and the fuselage angle of attack was 3ı . The noise measurements consisted of a number of kulite microphones positioned inside the nose gear cavity (see Fig. 2). The frequency and tonal noise levels measured inside the cavity were found to be relatively insensitive to the kulite location.

3 Numerical Method The CFD code used for this study is the commercial code CFD C C of Metacomp Technologies Inc. [5,6]. The code can handle both structured and unstructured grids (hexahedra were used for this study) and has a number of algorithms available for solving steady and transient incompressible and compressible flows. The algorithm selected within CFDCC for this study was the double precision unsteady compressible formulation. This algorithm is a dual time-stepping, second-order backward block-implicit scheme, which uses multi-grid acceleration to converge the linearized systems within each pseudo time iteration step. The time step was selected to give approximately 200 time steps per oscillation period and 15 inner-iterations (i.e. pseudo time steps) were used to converge the equation residuals by two orders of

360 Farfield microphone boom

R. Langtry and P. Spalart Kulite microphone #1 Rear nose gear doors (closed)

Kulite microphone #2

Front nose gear doors (open)

Fig. 2 Fuselage model with nose gear cavity, bay doors (front open, rear retracted) and representative landing gear (retracted inside the cavity) installed in LSAF for the noise test (left). Close up view of the nose cavity region (right)

magnitude before moving on to the next time step. The inviscid flux discretization was based on a second order multi-dimensional TVD scheme employing the HLLC Riemann solver to determine the wave interactions at cell faces. Low-speed preconditioning was not used for the unsteady computations. The computations for this study were run on the NASA Columbia supercomputer which is a 10,240 processor system composed of twenty 512 node Itanium 1.6 GHz processors. Between 100 and 200 nodes were used for each run. The coarse and fine grids were half-models (in order to reduce the grid size) of the wind tunnel geometry and were generated in ICEM Hexa as a multi-block hex mesh of 5 and 12 million nodes respectively. Based on an oscillation frequency of 15 Hz inside the cavity the coarse grid had approximately 30 grid points per wavelength whereas the fine grid had approximately 58 grid points per wavelength. A symmetry plane was imposed along the centreline of the geometry. The fine grid that was used for the DES runs is shown in Fig. 3. The grid near the walls was fine enough to give a yC of one or less. The far field boundaries were located 20 fuselage diameters away from the cavity and are not shown in Fig. 3.


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Fig. 3 3D half-model view of the nose gear cavity surface grid (top) including the fuselage, cavity walls, landing gear, front nose gear door (open) and rear nose gear door (closed). Volume grid (bottom) near the plane of symmetry

4 Results The experimental and CFD predicted sound pressure level versus the normalized frequency at the back of the nose gear cavity near the closed bay doors (i.e. kulite microphone #1 in the experiment) are shown in Fig. 4. The unsteady Reynolds-Averaged Navier Stokes (URANS) computations were done with the SST turbulence model while the DES results were obtained with the S-A DES model [1]. Both the DES and URANS results shown in Fig. 4 were obtained on the 5 million node coarse mesh. The coarse grid DES results (as well as URANS) are in relatively good agreement with the LSAF experimental data. In terms of the peak SPL the CFD results differ from the experiment by only about 4 dB. This is considered an acceptable amount of error given the fact that the CFD geometry is considerably simplified (particularly the landing gear) compared to the actual wind tunnel model. The effect of grid refinement on the DES results for the 5 and 12 million node grids are shown in Fig. 5. The peak sound pressure levels are virtually identically between the two different grids.

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Fig. 4 Experimental and predicted sound pressure level (SPL) versus normalized frequency inside the cavity at the kulite microphone #1 position

Fig. 5 Effect of grid refinement on the predicted sound pressure level (SPL) versus normalized frequency inside the cavity at the kulite microphone #1 position

In terms of the tonal frequency there is a small shift in the frequency predicted by the CFD (17 Hz) compared with the measured tone frequency (15 Hz). The measured cavity tone frequency of 15 Hz is very close to the expected frequency of a Rossiter type [8] cavity flow when the length scale (L) is taken as the distance


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fL/U

Present case: Mach 0.25, S = 0.39 corresponds to a Mode 1 frequency of 14.7 Hz

Fig. 6 Non-dimensional Rossiter frequencies (fL/U) of various cavity modes as a function of Mach number (Reproduced from Ref. [7])

between the start of the cavity and the leading edge of the closed rear bay doors. This is illustrated in the plot shown in Fig. 6 for a number of different cavity experiments [7]. The solid lines in Fig. 6 correspond to the semi-empirical equation of Heller et al. [9]. Sm D

fm L m˛

q U1 2 C 1=k M= 1 C 1 M v 2

(1)

where m is the mode number (equal to 1, 2, 3, : : :) and γ D 1:4 is the ratio of specific heats. The quantities α and kv are empirical constants that were set to α D 0:25 and kv D 0:57 in order to match the experimental data in Fig. 6 [7]. Assuming a Mach 0.25, Mode 1 type oscillation (i.e. the shear layer consists of only one oscillating wave) corresponds to a Strouhal number of 0.39 in Fig. 6. When converted to the normalized length and velocity scales of the experiment this gives a tone frequency of 14.7 Hz. This is very close to the measured value of 15 Hz and would seem to confirm that the nose gear cavity oscillation is a Rossiter type cavity flow with the leading edge of the rear doors dominating the interaction with the shear layer. The predicted Mach number contours and instantaneous pressure waves (Pressure – Time averaged pressure) at five points in time during one cavity oscillation period (T) from the URANS results are shown in Fig. 7. The contour plots are located in the span wise plane of the landing gear wheel and are useful for illustrating the cavity oscillation physics. At the first point in time .T D 0/ the cavity shear layer is pointing towards the inner part of the rear bay door leading

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Fig. 7 Mach number contours (left) and acoustic pressure contours (right) for five points in time during one cavity oscillation period (T). The contour plots are located in the span wise plane of the landing gear wheel. Present case: Mach 0.25, S D 0:39 corresponds to a Mode 1 frequency of 14.7 Hz


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edge and consequently freestream air is entering the cavity. This raises the pressure inside the cavity and a high pressure wave propagates upstream around the landing gear towards the front of the cavity .T D 0:2/. Once the high pressure wave reaches the front of the cavity it begins to deflect the shear layer away from the cavity .T D 0:4/. As the shear layer moves to the outside of the rear bay door leading edge, the air begins to empty out of the cavity .T D 0:6/. As the air leaves the cavity the cavity pressure is reduced and eventually it is low enough that the shear layer is sucked back towards the inside of the cavity .T D 0:8/ at which point the whole cycle repeats. From the CFD flow visualization there appeared to be only one wave present in the shear layer (i.e. its oscillation to and away from the cavity) and this confirms that the nose gear cavity oscillation was a mode 1 Rossiter type cavity flow. Another point to note is that the magnitude of the cavity tone is expected to be a strong function of the amount of obstructions (i.e. pressure damping) inside the cavity. Certainly the landing gear would act as a partial barrier to the high pressure wave that must propagate towards the front of the cavity in order to trigger the shear layer oscillation. This could also explain how baffles could be used to reduce (or even eliminate) the cavity tone by decreasing the magnitude of the high pressure wave as it propagates towards the front of the cavity and shutting down the mechanism that reinforces the oscillation. Another promising approach to mitigating the cavity tone would be to actually use CFD during the early design phase to determine cavity and bay door shapes that would completely eliminate the cavity oscillation. From the encouraging results obtained in the present study this would appear to be a viable approach.

5 Conclusions In this study the nose landing gear cavity oscillation measured in the Boeing LSAF tunnel has been computed using both URANS and DES CFD simulations. In general the agreement between the experiment and CFD was good. The CFD predicted a cavity tone frequency of 17 Hz compared to the measured value of 15 Hz. As well, the CFD predicted sound pressure level of the tone was within 4 dB of the measurements. From the present results it would appear that CFD can be used as a tool to investigate and possibly mitigate nose gear cavity tone mechanisms on new aircraft designs. Future work will involve more in depth studies on the effect of grid refinement, time step, Mach number as well as potential modifications to the cavity geometry that would eliminate the oscillation. Acknowledegments The authors would like to thank the NASA Advanced Supercomputing Division (NAS) for donating the computing time on the Columbia supercomputer. As well we would like to thank Metacomp Technologies Inc. for all of their help with the CFD C C setup on Columbia. Finally the authors would like to thank Cyrille Breard, John Rose, Srini Bhat and Bill Simmons of the Boeing Company for all of their help with the problem setup and experimental data.

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References 1. Spalart, P.R., Jou, W.-H., Strelets, M., and Allmaras, S.R., Comments on the Feasibility of LES for Wings and on Hybrid RANS/LES Approach, Advances in DNS/LES, Proceedings of the First AFOSR International Conference on DNS/LES, Rouston, Louisiana, USA, 1997. 2. Nayyar, P., Barakos, G., Badcock, K., and Kirkham, D., Analysis and Control of Transonic Cavity Flow Using DES and LES, AIAA-2005-5267 35th AIAA Fluid Dynamics Conference and Exhibit, Toronto, Ontario, June 6–9, 2005. 3. Peng, S., Simulation of Flow Past a Rectangular Open Cavity Using DES and Unsteady RANS, AIAA-2006-2827 24t D.M. D.M.h AIAA Applied Aerodynamics Conference, San Francisco, CA, June 5–8, 2006. 4. Guo, Y., Yamamoto, K.J., and Stoker, R.W., Experimental Study on Aircraft Landing Gear Noise, Journal of Aircraft, Paper 0021-8669, Vol. 43, No. 2, pp. 306–317, 2006. 5. Batten, P., Leschziner, M.A., and Goldberg, U.C., Average State Jacobians and Implicit Methods for Compressible Viscous and Turbulent Flows, JCP, Vol. 137, pp. 38–78, 1997. 6. Batten, P., Clarke, N., Lambert, C., and Causon, D.M., On the Choice of Wave Speeds for the HLLC Riemann Solver, SIAM J. Sci. Stat. Comp., Vol. 18, No. 6, pp. 1553–1570, 1997. 7. Bliss, D.B. and Hayden, R.E., Landing Gear and Cavity Noise Prediction, NASA Contractor Report CR-2714, 1976. 8. Rossiter, J.E., Wind Tunnel Experiments on the Flow Over Rectangular Cavities at Subsonic and Transonic Speeds, Royal Aircraft Establishment ARC R&M 3438, 1966. 9. Heller, H.H., Holmes, G., and Covert, E.E., Flow-Induced Pressure Oscillations in Shallow Cavities, Journal of Sound and Vibration, Vol. 18, No. 4, pp. 545–553, 1971.

Experimental and Numerical Study of Unsteady Wakes Behind an Oscillating Car Model E. Guilmineau and F. Chometon

Abstract This research focuses on the analysis of the instability of passenger vehicles associated with transient crosswind gusts. A new vehicle model, created to analyze the behavior of unsteady wakes on bluff bodies, is proposed. This test model called Willy is designed using the following criteria: the geometry is realistic compared to a real vehicle, the model’s plane under-body surface is parallel to the ground, and the separations are limited to the region of the base for a moderated yaw angle. In the present paper, the tests are performed on the model animated by an oscillating yaw angle at a frequency of 2 Hz in a steady wind. Experiments are carried out at Reynolds number of 0.9 106 at the Conservatoire National des Arts et Métiers and computations are performed at the Ecole Centrale de Nantes. The numerical results are compared with experimental data. Keywords Automotive flow Crosswind Transient gust Incompressible flow

1 Introduction The aerodynamic characteristics of a vehicle and their evolution as a function of situations such as passing, crossing, the presence of an unsteady gust of wind or of a non uniform turbulent atmospheric flow are factors in the vehicle’s on-road stability and of the safe manoeuvrability appreciated by the driver [7]. The simplest way to define the stability of a vehicle is to measure the steady forces and moments in a wind tunnel as a function of the yaw angle ˇ. The knowledge of the lift or weight forces at the level of the wheels axes and of the forces and moments on the body allows to define the position of the lateral thrust center in comparison with the center E. Guilmineau () Laboratoire de Mécanique des Fluides, CNRS UMR 6598, Ecole Centrale de Nantes BP 92101, 44321 Nantes Cedex 3, France F. Chometon Laboratoire d’Aérodynamique, Conservatoire National des Arts et Métiers 15 rue Marat, 78210 Saint Cyr l’Ecole, France M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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of gravity. This result can be improved by introducing these steady aerodynamic data in a dynamic model that includes the dynamic of the suspension, springing and tyres. This approach postulates that there is no phenomena of phase shifting or hysteresis and actually a dynamic approach is necessary in order to prove that these phenomena do not exist or remain weak compared to others. Experimental studies in a wind tunnel or on the road of the dynamic behaviour of vehicles are in counterpart very complex and much analysis must be performed on models. Several techniques allow the reproduction of a side gust of wind in a wind tunnel, assimilated to a pulse of velocity. For Macklin et al. [12], it was obtained by propelling the model on a rail crossing the test section of the wind tunnel. Ryan and Duming [15], used a technique where the side wind was produced by a cross jet. Another technique, more easily set up, consists of submitting the model to a periodic movement in a steady wind. This approach does not actually simulates the situation of passing but it does bring into evidence the phenomena of phase shifting or hysteresis and allows their analysis [4]. This last approach was retained here [1]. More complicated problems like vehicle passing or crossing were correctly analysed on 1=5 scale models propelled on a rail [14]. The transient aerodynamic effects experienced by every member of a platoon during passing manoeuvres were also analysed by Tsuei and Savas [17] on 1=20 scale models. The same techniques are also applied to the analysis of cross winds effects on high-speed trains [16]. To date, accuracy of numerical simulation has been improved and simulation is now widely used in the automotive industry for exploration of flow physics or for effort prediction on full scale vehicles. Many of the CFD tests are done at at zero yaw angle for steady flows and use simplified bluff bodies [10] or the reference Ahmed car model [9, 11]. More recently a new model (the Willy car model) was proposed and experimented for steady and unsteady conditions. The unsteady results showed phenomena of phase shift and hysteresis [3]. A detailed comparison between numerics and experiments were done on this model at large yaw angles up to ˇ D 30ı for steady flows [5, 6]. Comparisons show a good agreement between numerics and experiments. Experimental and numerical data presented in this paper are concerned with the analysis of unsteady flows around the Willy model. The tests are performed on the model animated by an oscillating yaw angle of amplitude ˇ D 10ı in a steady wind at a Reynolds number of 0.9 106 . The experimental unsteady wall pressures are compared to the numerical results for a frequency f = 2 Hz. The post-processing of other numerical data allows the understanding of the physics of phase shifting phenomena observed on wall pressures.

2 Test Model Experimental and numerical tests are performed on the squareback Willy test model, which is realistic, compared to a van-type vehicle. A complete definition of the model is given in reference [2]. The main characteristics are as follows, see Fig. 1:

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Fig. 1 Model definition

(a) Side view

(b) Bottom view

Table 1 Dimensions of the test model (data in mm and mm2 )

L1 675

B 140

D 550

E 118

F 140

G 29

H 345

1 20

Sref 41791

The geometry is realistic, compared to a real van-type vehicle. The model’s plane underbody surface is parallel to the ground. The separations are limited to the region of the base for a moderated yaw angle, i.e. ˇ D 10ı . The digital definition of the model allows the modification of the shape through only four parameters given in [2]. The overall length of the model is L1 D 675 mm, the width 240 mm, the maximum height 192 mm and its surface reference is the maximum cross section Sref D 41;791 mm2 . The ground clearance is G D 29 mm and the diameter of the four feet (f) which are used to secure the model to the floor of the wind tunnel is 1 D 20 mm. All dimensions are defined in the Table 1. In previous steady analysis [6], a cylinder of diameter 40 mm located under the body was used to protect the pressure tubes passing from the pressure taps to the multi-manometer. This tube modifies strongly the flow around the model and was removed for the present unsteady experimental and numerical tests. The reference axis of the model is the Eiffel axis where the axis is parallel to the upstream velocity V0 . The origin of axis lies at the point O located on the floor of the model, see Fig. 1. This point O is the centre of rotation of the model when submitted to a harmonic oscillation.

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3 Experimental Set-Up Wall pressures measurements are performed in a Prandtl-type wind tunnel of the CNAM. The semi-open test section has a cross section of 1.45 by 1.45 m and the ground of the wind tunnel is fixed. The wind tunnel is equipped with a six components balance. The value of the yaw angle ˇ is positive when the right side of the car model is windward. The turbulence level at the centre of the test section is 2%. For an upstream velocity V0 D 20 m=s, the thickness of the turbulent boundary layer at a point located on the floor of the wind tunnel, at x D 670 mm, i.e. upstream the body, and y D 10 mm is ı D 40 mm with a shape factor H D 1:29. This data is obtained with the model at a yaw angle of ˇ D 30ı . The experimental results given in the paper are obtained at a Reynolds number of 0:9 106 based on the velocity V0 and the length L1 of the model. The set-up allowing the model to be put into movement at the frequency f is described in Fig. 2. The model is attached to a turntable (T) oscillating around its z axis (A). A rod (CD) of length l is attached to the disc of a step motor (Mo). The set-up is conceived so that the rod was parallel to axis y when ˇ D ˇ; in these conditions, r D R sin ˇ and l D d . With r D 0:0255 m and l D d D 0:3264 m, the movement ˇ D f .˛/ is close to the sine function. For an angle ˛ D !t, where

Fig. 2 Oscillating bench


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! is a constant angular velocity and t the time, the actual movement ˇ D f .˛) is the physical root of: ˇ ˇ a tan2 C b tan Cc D0 (1) 2 2 where:

cos ˛ 2l sin ˇ bDB 2 sin ˇ sin ˛ (2) a D A C B cos ˇ C sin ˇ r c D A B .cos ˇ C sin ˇ cos ˛/ (3) 2r 2 (4) A D B .1 C 0:5 sin 2 ˇ cos ˛/ 2rl sin ˛ BD sin2 ˇ

Wall pressures are measured along the curve (A) of the model, see Fig. 1. The internal diameter of the pressure taps is 1.5 mm. Pressure coefficients Cp are defined by: Cp D

p p0 1 2 2 V0

(5)

where p0 and p are respectively the upstream static pressure and the local pressure. Steady and unsteady pressure measurements are performed with dynamic differential pressure sensors Honeywell DC010BDC4 with pressure range ˙10 mb. Sensors are mounted inside the model and the electric wires pass through a foot (f). The frequency response of this device was tested in a shock tube which was modified in order to provide weak shock waves suitable with the pressure range of the sensors. The natural frequency of the actual measurement device is 520 Hz. The bandwidth frequency at which the error is 5% is about 30 Hz. Concerning the post processing of unsteady pressures, the phase shifting introduced by the mechanisms and the angle triggering is calculated on quasi-steady reference results obtained at a low frequency f D 0:1 Hz. It was admitted that this angle adjustment is independent of the frequency and applied to all tests.

4 Numerics 4.1 Flow Solver The ISIS-CFD flow solver, developed by the EMN (Equipe Modélisation Numérique) of the Fluid Mechanics Laboratory of the Ecole Centrale of Nantes, uses the incompressible unsteady Reynolds-averaged Navier–Stokes equations (RANSE). The solver is based on the finite volume method to build a spatial discretization of the transport equations. The face-based method is generalized to two-dimensional or three-dimensional unstructured meshes for which non-overlapping control volumes are bounded by

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an arbitrary number of constitutive faces. The velocity field is obtained from the momentum conservation equations and the pressure field is extracted from the mass conservation constraint, or continuity equation, transformed into a pressureequation. A second-order accurate three-level fully implicit time discretization is used and surface and volume integrals are evaluated using second-order accurate approximation [8]. In the case of turbulent flows, additional transport equations for modeled variables are solved in a form similar to that of the momentum equations and they can be discretized and solved using the same principles. In this study, the turbulence model used is the K! SST model of Menter [13]. To take account of the oscillation of the model, all mesh points move with the same velocity than the one of the car model. However, this approach does not represent the experimental set-up because in experiments, the model is attached to a turntable and in this case, it is not all the ground that oscillates. We suppose that this different approach in numerics does not induce some secondary effects.

4.2 Mesh The computational domain starts 3.5 L1 in front of the model and extends 4 L1 behind the model. The width of the domain extends from C1;000 mm (1.48 L1) to 1;000 mm and the height is 1;050 mm (1.55 L1). The mesh is generated by using HEXPRESSTM , an automatic unstructured mesh generator. This software generates meshes only containing hexaedrals. The mesh is composed to about 6.3 millions of points with approximatively 93;000 points on the model and 48;100 points on the wind tunnel floor. For the model, we use the near-wall low-Reynolds number turbulence model, the distance between the body and the first fluid points is fixed to 0.006 mm, i.e. yC D 0:5. For the wind tunnel floor, we use a wall function, and the distance between the floor and the first fluid points is fixed to 0.6 mm.

5 Results Experimental and numerical results given in this paper are obtained for ˇ D 10ı and a oscillation frequency f D 2 Hz i.e. a Strouhal number S t D 0:07 calculated with the upstream velocity V0 and the lenght L1 of the model. For the simulation, the time step used is t D 0:001 s.

5.1 Unsteady Forces and Moments The numerical and experimental forces Cx0 and Cy0 drawn in Figs. 3 and 4 are respectively the drag and the side forces coefficients in the Eiffel axes linked to upstream velocity Vo. The labels [1] to [10] drawn in the unsteady curves indicate


373

Fig. 3 Cx0 D f .ˇ/

Fig. 4 Cy0 D f .ˇ/

the evolution of the angle ˇ along a complete cycle. The experimental static values of Cx0 D f.ˇ/ and Cy0 D f.ˇ/ are also given as well as the numerical static values calculated at ˇ D 0ı and ˇ D 10ı . These results show that, for a low value of the Strouhal number St D 0:07, the unsteadiness of the flow introduces a phase shift effect for Cx0 and Cy0 and increases the value of the drag, particularly around ˇ D 0ı . In counterpart the value of d(Cy0)=dˇ does not change. In the same way, the sign of d(CN)=dˇ, where CN is the yaw coefficient, does not change, and the unsteadiness does not modify the stability of the model. The interest of the polar curve Cy0 D f.Cx0/ drawn in Fig. 6 is to show that the dynamic thickness " D Cy0=Cx0 of the model, presented in Fig. 7, is maximum, about " D 1:27 for ˇ D 6:95ı located between the labels [9] and [10]. At the same yaw angle ˇ D 6:95ı the static value of the thickness is about " D 0:78.

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5.2 Wall Pressures The comparison between the experimental and calculated wall pressure are given in Fig. 8 for two pressure taps noted P1, P2 located on the curve (A) of the model, and a third tap P3 located on its base, see Fig. 1. The agreement is very good and confirms the results obtained for the forces and moments. The three results show clearly a phase shift phenomena with no non-linear effects. At location P1 which is at the level of the nose of the model, the phase shift is very low, close to zero but increases along the line (A). At location P2 the shape of the curve obtained for the unsteady pressure is identical to the curve for the unsteady side force Cy0. In the same way the loop observed for the drag Cx0 is also observed for the base pressure.

5.3 Unsteadiness Effects in the Wake For cross flows, the rotation of vortices is defined for an observer looking the model backward. For each frame the number into the brackets corresponds to the label drawn in Figs. 3 to 7. The sense of the loop is also given. The unsteadiness and temporal variation of the wake along a cycle from ˇ D 10ı to ˇ D C10ı and then from ˇ D C10ı to ˇ D 10ı is described in Fig. 9 for several yaw angles. The flow is analysed through cross flow velocities drawn at X=L1 D 0:70. The reading of these results must take into account that the vortices which appear at X=L1 D 0:70 are shed at the level of the base located at X=L1 D 0:5. This figure shows that the pair of vortices observed on frames [1], [2] and [3] vanish in frame [4] and are replaced in frame [5] by a new pair of symmetrical vortices. The phenomena is the same for the frames evoluting from [6] to [10]. At ˇ D C4:56ı and C4:59ı which are roughly at half-cycle, the results described in Fig. 9 show that no vortex are visible in the cross section. The detail of the modification of the wake is described in Fig. 10. In this figure the first line of

Fig. 5 CN D f .ˇ/


375

Fig. 6 Cy0 D f .Cx0/

Fig. 7 " D f .ˇ/

frames must be read from left to right, the second line from right to left, and the last from left to right. At ˇ D C2:18ı , the vortex V1 is visible but the vortex V2 is weak; at ˇ D C2:79ı only the clockwise vortex V1 is present. At ˇ D C3:39ı no vortex appear in the cross flow. At ˇ D C5:11ı , a counterclock vortex V3 appears and grows up as seen in the following frames. At angle ˇ D C6:65ı a new vortex V4 is rolling up, and form a vortex pair with the vortex V3 at ˇ D C7:12ı . A comparison of the cross flow velocity vectors drawn at X=L1 D 0:70 at ˇ D 0ı in static and in dynamic is given in Fig. 11 for steady and unsteady flows. In static there are two symmetrical vortices V1 and V2, see the central frame . These two vortices are due to the facts that the base of the model is not circular and that the wake interacts with the fixed ground. In dynamic, the angular velocity of the model is maximum at ˇ D 0ı , and only the clockwise vortex V1 appears when it moves from 10ı to C10ı , see the left frame. When the model is moving back

376


a Pressure tap P1

b Pressure tap P2

c Pressure tap P3 Fig. 8 Pressure coefficient CP D f .ˇ/

from ˇ D C10ı to ˇ D 10ı , only the counter-clockwise vortex V2 appears, see the right frame. The two wakes are symmetrical about the (x,z) plane and the drags Cxo are the same for the two dynamic situations, see Fig. 3.

6 Conclusions This paper presents the simulations and experiments of the flow around the Willy car model submitted to a harmonic oscillation at a Strouhal number S t D 0:07 and an amplitude ˇ D 10ı . Unsteady experimental wall pressures, used as reference, has been carried out at the CNAM. For the numerics, the ECN ISIS-CFD is used with K! SST turbulence model. The results confirms that even at low frequency and low amplitude, there are phase shift effect phenomena associated with an increase of the drag, particularly at ˇ D 0ı where the angular velocity of the model


[1]: ¯ = −8:83◦ , from−10◦ to +10◦

[10]: ¯ = −8:85◦ , from+10◦ to −10◦

[2]: ¯ = −4:50◦ , from−10◦ to +10◦

[9]: ¯ = −4:59◦ , from+10◦ to −10◦

[3]: ¯ = +0:31◦ , from−10◦ to +10◦

[8]: ¯ = +0:19◦ , from+10◦ to −10◦

[4]: ¯ = +4:56◦ , from−10◦ to +10◦

[7]: ¯ = +4:46◦ , from+10◦ to −10◦

[5]: ¯ = +8:98◦ , from−10◦ to +10◦

[6]: ¯ = +8:97◦ , from+10◦ to −10◦

Fig. 9 Unsteady cross flows at X=L1 D 0:70 for a full cycle

377

378


¯ = 2:18◦

⇓

¯ = 6:16◦

¯ = 6:65◦

¯ = 2:79◦

⇒

⇐

⇒

¯ = 5:65◦

¯ = 3:39◦

⇒

⇓

¯ = 5:11◦

⇐

¯ = 7:12◦

Fig. 10 Cross flows at X=L1 D 0:70, detail of the wake

Unsteady from−10◦ to +10◦

Static

Unsteady from+10◦ to −10◦

Fig. 11 Steady and unsteady cross flows at ˇ D 0ı and at X=L1 D 0:70

is maximum. Comparisons show a good agreement between numerics and experiments, and confirm the capability of the ECN ISIS-CFD code to catch hysteresis or phase shift phenomena for 3D flows. Moreover, this approach shows that, after a validation of numerical results by experiments, the post-processing of numerical results gives access to a large lot of data allowing a fructuous analysis of flow physics. This project is performed in the framework of a collaboration between the Ecole Centrale de Nantes, the Conservatoire National des Arts et Métiers and the Ecole Nationale Supérieure de l’Aéronautique et de l’Espace (SUPAERO).


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Acknowledegments Experiments were performed by Mr. Jean-Luc Alleau in the framework of preparation for an engineering degree from the CNAM. We would like to thank Mr. E. A¨ıssaoui for his help and cooperation. The authors gratefully acknowledge the Scientific Committee of IDRIS (project 0129) and of CINES (project dmn2049) for the attribution of CPU time.

References 1. F. Chometon and P. Gilliéron. Analysis of unsteady wakes by images processing in automotive aerodynamics. In Congress FLUCOME, Tokyo, Japan, 1997. 2. F. Chometon, A. Strzelecki, V. Ferrand, H. Dechipre, P.C. Dufour, M. Gohlke, and V. Herbert. Experimental study of unsteady wakes behind an osci. SAE Technical Paper 2005-01-0604, 2005. 3. F. Chometon, A. Strzelecki, J. Laurent, and E. A¨ıssaoui. Experimental analysis of unsteady wakes on a new simple car model. In Fifth International Colloquium on Bluff Body Aerodynamics and Applications, pp. 545–548, Ottawa, Ontario, Canada, 2004. 4. K.P.K. Garry and K.R. Cooper. Comparison of quasi-static and dynamic wind tunnel measurements on simplified tractir-trailer models. Journal of Wind Engineering and Industrial Aerodynamics, 22:185–194, 1986. 5. E. Guilmineau. Numerical simulation of wakes behind a car model. In FISITA World Automotive Congress, Yokohama, Japan, 2006. 6. E. Guilmineau and F. Chometon. Experimental and numerical analysis of the effect of side wind on a simplified car model. SAE Techincal Paper 2007-01-0108, 2007. 7. W.H. Hucho. Aerodynamics of road vehicles. SAE International, 1998. 8. H. Jasak, H.G. Weller, and A.D. Gosman. High resolution nvd differencing scheme for arbitrarily unstructured meshes. International Journal for Numerical Methods in Fluids, 31:431–449, 1999. 9. S. Kapadia, S. Roy, and K. Wurtzler. Detached eddy simulation over a reference Ahmed car model. In 41st Aerospace Sciences Meeting and Exhibit, AIAA Paper 2003-0857, Reno, NE, January 2003. 10. B. Khalighi, S. Zang, C. Koromilas, S.R. Balkany, L.P. Bernal, G. Laccarino, and P. Moin. Experimental and computational study of unsteady wake flow behind a bluff body with drag reduction device. SAE Technical Paper 2001-01B-207, 2001. 11. S. Krajnovic and L. Davidson. Flow around a simplified car – part 1: Large eddy simulation. Journal of Fluids Engineering, 127:907–918, 2005. 12. A.R. Macklin, K.P. Garry, and J.P. Howell. Comparing static and dynamic testing techniques for the crosswind sensitivity of road vehicles. SAE Technical Paper 960674, 1996. 13. F.R. Menter. Zonal two-equation k! turbulence models for aerodynamic flows. In AIAA 24th Fluid Dynamics Conference, AIAA Paper 93-2906, Orlando, FL, July 1993. 14. C. Noger, C. Regardin, and E. Szechenyi. Investigation of the transient aerodynamic phenomena associated with passing manoeuvres. Journal of Fluids and Structures, 21:231–241, 2005. 15. A. Ryan and R.G. Dominy. The aerodynamic forces induced on a passenger vehicle in response to a transient cross-wind gust at a relative incidence of 30ı . SAE Technical Paper 980392, 1998. 16. S. Sanquer, C. Barré, M. Dufresne de Virel, and L.M. Cléon. Effect of cross winds on highspeed trains: development of a new experimental methodology. Journal of Wind Engineering and Industrial Aerodynamics, 92:535–545, 2004. ¨ Savas. A wind tunnel investigation of the transient aerodynamic effects on a 17. L. Tsuei and O. four-car platoon during passing manoeuvers. SAE Paper 2000-01-0875, 2000.


Physical Analysis of an Anisotropic Eddy-Viscosity Concept for Strongly Detached Turbulent Unsteady Flows R. Bourguet, M. Braza, R. Perrin, and G. Harran

Abstract A tensorial eddy-viscosity turbulence model is developed in order to take into account of the structural anisotropy appearing between the mean strain-rate tensor and the Reynolds turbulent stresses in strongly detached high Reynolds number flows. In the framework of the Organised Eddy Simulation, a physical investigation of the misalignment of these two tensor principal directions is performed by means of phase-averaged 3C-PIV measurements in the near-wake of a circular cylinder at Reynolds number 1:4 105 . Considering the stress–strain misalignment as a local sign of the turbulence non-equilibrium, anisotropic criteria are derived. This leads to a tensorial eddy-viscosity concept which is introduced in the turbulent stress constitutive law. Additional transport equations for the misalignment criteria are derived from a degenerated SSG second order closure scheme. A two-dimensional version of the present model is implemented in the NSMB solver on the basis of a twoequation k" isotropic OES model. Numerical simulation results are compared to an experimental dataset concerning the incompressible flow past a NACA0012 airfoil at 20 degrees of incidence and Reynolds number 105 . Keywords Turbulence modeling Advanced URANS methods Anisotropic Organised Eddy Simulation

1 Introduction In the context of high-Reynolds number turbulence modeling and especially in the case of parietal flows, recent advances like Large Eddy Simulation (LES) and hybrid methods (Detached Eddy Simulation, DES) have considerably improved the physical relevance of the numerical simulation. However, the LES approach is still R. Bourguet (), M. Braza, R. Perrin, and G. Harran Institut de Mécanique des Fluides de Toulouse, UMR 5502 CNRS-INPT/UPS, Allée du Prof. Camille Soula, 31400 Toulouse, France e-mail: [email protected] Present address: Massachusetts Institute of Technology, Department of Mechanical Engineering, 77 Massachusetts Avenue, Cambridge, MA 02139, USA M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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limited to the low Reynolds number range concerning wall flows and the Unsteady Reynolds Averaged Navier–Stokes (URANS) approach remains a widespread and robust methodology for complex flow computation particularly in the near-wall region. Second-order closure schemes (Differential Reynolds Stress Modeling, DRSM) can provide an efficient simulation of turbulent stresses. Nonetheless, from a computational point of view, the main drawbacks of such approaches are a higher cost for unsteady and three-dimensional configurations and above all, numerical instabilities which imply the addition artificial dissipation terms. The present study is founded on the Organised Eddy Simulation (OES) methodology [1–3] which consists in distinguishing the flow structures to model according to their coherent or chaotic aspect instead of their size as in LES. The improvment of the advanced first order statistical approaches in the context of OES, especially in the sense of a realistic simulation of the anisotropy tensor for non-equilibrium flows, represents one of the main objectives of the present development. Concerning the first order statistical turbulence modeling, the linear eddyviscosity models utilise the Boussinesq approximation [4] which establishes a linear relation between the Reynolds stresses and the strain-rate by means of a scalar eddy-viscosity concept. The Boussinesq law can be written as follows under the incompressibility assumption:

ui uj 2 t C ıij D aij D 2 Sij ; k 3 k

where ui ui are the turbulent stresses, k is the turbulent kinetic energy (k D 12 ui ui ), ıij is Kronecker symbol and S the mean strain-rate tensor, defined by Sij D 1 2

@Ui @xj

C

@Uj @xi

. Ui is the mean flow velocity. t is the scalar eddy-viscosity that is

often expressed by means of the turbulence length and time scales as t D C k 2 =", where " is turbulence dissipation rate. The Boussinesq approximation assumes, among others, that the principal directions of the two tensors a and S always remain collinear. This leads to an over-production of turbulent kinetic energy [5] especially in flow regions upstream of the detachment, where the strain-rate is high and the flow is laminar [6, 7]. The Non-Linear Eddy-Viscosity Models (NLEVM) provides modified behaviour laws which attempt to overcome these limitations. The associated constitutive laws are derived from a complete tensorial basis of the turbulent stresses [8, 9] involving quadratic or cubic combinations of the strain and vorticity tensors. The Explicit Algebraic Stress Models which are derived from algebraic forms of the turbulent stresses issued from the DRSM [10–13] provide improved results for nonequilibrium flows but imply significant calibration processes according to the flow configuration of interest [14, 15]. In the framework of OES methods, an alternative to NLEVM is suggested to derive a tensorial eddy-viscosity model sensitised for non-equilibrium turbulence [16]. As discussed in the present paper, the non-equilibrium can be illustrated by means of stress-strain misalignment [17], among other concepts, as well as by the ratio of the mean flow time-scale over the turbulence time-scale [11]. A selective reduction

Physical Analysis of an Anisotropic Eddy-Viscosity Concept

383

of the eddy-diffusion coefficient, varying according to the non-equilibrium flow regions and the coherent flow structures, to reach an improved prediction of the turbulence production in respect of the flow physics, is expected. As presented in the first section, the analysis of the stress-strain behaviour is based on a detailed high-Reynolds PIV experiment concerning the incompressible flow past a circular cylinder at Reynolds number 1:4 105 in high blockage and aspect ratios [18]. The phase-averaged turbulence properties are considered, allowing distinction of the organised coherent physical process from the random turbulence. Furthermore, anisotropic misalignment criteria are investigated and a tensorial definition of the eddy-viscosity is put forward, leading to a new Reynolds stress constitutive law. Transport equations for these criteria are derived from the Speziale et al. second order closure scheme [19]. The predictive capacities of this anisotropy resolving approach are examined in the last section by comparison of two-dimensional numerical simulation results issued from NSMB solver with an experimental dataset concerning the incompressible flow around a NACA0012 airfoil at 20 degrees of incidence and Reynolds number 105 .

2 Stress–Strain Anisotropy as a Non-equilibrium Criterion 2.1 The Organised Eddy Simulation Framework The OES methodology consists in a separation of the turbulent kinetic energy spectrum into a resolved part corresponding to organised flow structures and a modeled part associated with chaotic fluctuations. Experimental studies emphasised a modification the spectrum to be modeled in the inertial region where coherent structures and random turbulence interact [2, 3]. This modification, which illustrates the nonequilibrium compared to the equilibrium turbulence described by Kolmogorov’s statistical theory, implies a recalibration of the turbulence time and lengh scales in URANS methodology. The OES approach proposed a modification of the diffusivity constant C in two-equation closure schemes, using an isotropic Boussinesq law as a first step, and this methodology reached an efficient prediction of massively detached unsteady turbulent flows around bodies. From a physical point of view, a consequence of the non-equilibrium is the misalignment observed between the Reynolds stress and mean strain-rate tensor. In the present study, the structural properties of this misalignment are used to reach a more relevant prediction of the non-equilibrium turbulence physics.

2.2 An Investigation of the Stress-Strain Misalignment via 3C-PIV in the Cylinder Wake The experiment has been carried out in the wind tunnel S1 of IMFT. The channel has a 670 670 mm2 cross section. The cylinder spans the width of the channel without

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endplates and has a diameter D of 140 mm, giving an aspect ratio L=D D 4:8 and a blockage coefficient D=H D 0:208. The upstream velocity U0 at the centre of the channel is 15 m/s, therefore the Reynolds number based on the upstream velocity and the cylinder diameter D is 1:4 105 . The free stream turbulence intensity, measured by hot wire technique in the inlet was found 1:5%. The three-component measurements have been performed by means of stereoscopic PIV. The procedure used is reported in [18]. In the present study, the median plan has been considered at half distance spanwise and located in the near-wake region (Fig. 1). The near periodic nature of the flow, due to the von Kármán vortices, allows the definition of a phase. In the following all quantities are phase-averaged. Angles between the principal directions of the strain-rate and turbulence anisotropy tensors are quantified. The main coherent vortex regions are delimited by the Q criterion [20]. The first principal directions of each tensor are represented in Fig. 2. In specific flow regions their misalignment becomes predominant. The largest misalignment is observed near the vortex center (x1 =D D 1, x2 =D D 0:03) in Fig. 2a for instance. The best alignment is reached in free shear flow regions. Fig. 1 Flow configuration

a

Q: -20.0 -15.8 -11.5 -7.2 -3.0

1.2

5.5

b

9.8 14.0

Q: -20.0 -15.8 -11.5 -7.2 -3.0

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-5.8

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-4.4

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x1/D

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-.2

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Fig. 2 a (dashed) and S (solid) first principal directions and Q criterion iso-contours at phases (a) ' D 500 and (b) ' D 1400


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angle(Va1,Vs1) (degree)

50

40

30

20

10

1

1.5

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x1 / D

Fig. 3 Angle variation between a and S first principal directions along the three lines in bold in Fig. 2a

In Fig. 3 the angle between the directions of va1 and vS1 is represented for given ordinates (cf. bold lines in Fig. 2a). In spite of the measurement noise induced by PIV technic, the solid and dashed-dotted curves (x2 =D D 0:21 and x2 =D D 0:06, respectively) confirm the misalignment peak near the vortex center (up to 50ı around x1 =D D 1) whereas the dashed curve (x2 =D D 0:39) demonstrates a quasialignment near the saddle point and in free flow regions (beyond x1 =D D 1:5).

2.3 An Anisotropic Misalignment Criterion The analysis of the high misalignment zones allows to locate precisely the validity regions of the Boussinesq isotropic law. As a consequence, in the perspective of an improvment of the Reynolds tensor constitutive law, it seems judicious to take into account of these effects. However, a direct monitoring of the misalignment between the three principal directions of the two tensors implies an assumed knowledge about these tensors which does not make sense since the stress tensor is derived from the constitutive law. In this context, a misalignment criterion is defined as the correlation rate between the projection of the anisotropy tensor onto the eigen basis of the strain-rate tensor and the corresponding eigenvector of S . Without any estimation of the eigenvectors of a, this directional criterion can provide sufficient information about the alignment between the principal directions of a and vSi , in each space direction: aj k vSi k vSi j S Ci D for i D 1; 2; 3; where k:k is the euclidian norm. av i

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Fig. 4 Ci anisotropic criterion discriminates (a) planar and (b) global misalignments

The Ci coefficients gives an anisotropic knowledge which enables to describe locally the distortion between the stress and strain-rate tensors, allowing a distinction between the global and planar misalignment as presented schematically in Fig. 4. As can be shown in the present experiment, the criterion decreases in highlystrained shear flow regions and especially near the vortex center whereas it remains maximum when the two principal tensorial directions are aligned. Moreover, this directional criterion is “advectable” through specific transport equations that can be derived from DRSM as suggested in the next section.

3 An Anisotropic First Order Eddy-Viscosity Model 3.1 The Tensorial Eddy-Viscosity Concept The previous analysis concerning the specific decorrelations between Reynolds stress and mean strain-rate tensors in each space direction demonstrates the relevance of a constitutive law taking account of the individual contribution of each element of a spectral decomposition which is applied to the strain-rate tensor. The following definition of an anisotropic eddy-diffusion coefficient can be suggested by an extension of the scalar C definition, for i D 1; 2; 3:

C i D

ˇ ˇˇ ˇ ˇaj k vSi k vSi j ˇ i

aj k vSi k vSi j " ˇ Sˇ : D jCV i j where CV i D ˇ ˇ k i

k jS j k i D " i is a vectorial version of D kkS mean flow/turbulent time scale " rate which emphasises the non-equilibrium turbulence regions [11]. Whenever is higher than 3:3 approximately, the non-equilibrium turbulence becomes predominant.


387

Therefore a consistent definition of the eddy-viscosity as a symmetric tensor tt is suggested on the basis of a positive directional eddy-viscosity td : .tt /ij D .td /k vSk i vSk j with .td /i D jCV i j k:

(1)

Expression (1) leads to a weighted summation of S spectral decomposition: Si k .tt /kj D .td /l Sl vSl i vSl j D .td /l .Sl /ij ;

(2)

and thus, the linear EVM behaviour law can be generalised as: 2 2 ui uj C kıij D 2Si k .tt /kj Rıij ; 3 3

(3)

where R D .td /i Si is the trace of Si k .tt /kj . From expression (2), the symmetry property of the turbulence anisotropy tensor is ensured. Expression (3) leads to the following generalization of averaged Navier–Stokes momentum equations: DUi @ D Dt @xj

@Uj @Ui @Ui @Uk 2 1 @P C C : C .tt /kj .k C R/ ıij @xj @xi @xk @xi 3 @xi

The tensorial definition enables a selective reduction of the effect of one (or more) elements of the strain-rate tensor spectral decomposition with respect to the corresponding physical alignment (or misalignment) between the associated principal directions. Moreover, if a perfect alignment is observed in an equilibrium and isotropic strain region the tensorial expression degenerates into a classical Boussinesq-like scalar model.

3.2 “Experimental” Validation Comparison between normal and shear Reynolds stresses evaluated from the PIV experiment and from modelling via (3) and measured stress tensor at two location points and over a period of vortex shedding is presented in Fig. 5 and can be regarded as an “experimental” validation. The modelled quantities present a good match with the experiment for both normal and shear Reynolds stresses. This is verified when examining the complete fields at a given phase angle (Fig. 6): despite slight differences in shear flow region, the predictive capacities of the tensorial constitutive law are confirmed.

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u1u2_exp, u1u2_mod, u1u2_mod

0.04 0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1 −0.12 −0.14 0

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Fig. 5 Comparison between measured (solid) and modeled (dashed) Reynolds stresses: shear layer (red) and wake (black)

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Fig. 6 Comparison between phase-averaged Reynolds stresses ui uj obtained directly from the PIV experiment (a) shear and (c) normal, and those evaluated via Equation (3) and experimental strain-rate tensor (b) shear and (d) normal at phase ' D 50ı


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3.3 An Anisotropic First Order Closure Scheme From a degeneration of the Speziale et al. second order closure scheme [19], three advection equations are derived to transport the CV i coefficients as state variables of the physical system. For q D 1; 2; 3: 0

D Vq ij

Daij DCV q 1 D ˇ ˇ @ Vq ij C aij ˇ ˇ Dt Dt ˇSq ˇ vSq ; Vq ij D vSq i

Dt

ˇ ˇ1 ˇ ˇ D ˇSq ˇ A with C CV q Dt

j

which leads by introducing the SSG modeling for the pressure/strain correlation in a similar way as [21] for a non-directional misalignment: DCV q D Dt

V S 1 Vq ij ai k Sj k q ij ij c2 4 ˇ ˇ C .2 2c4 / ˇ ˇ Vq ij ai k akj C c3 IIa2 c3 ˇ Sˇ ˇ Sˇ 3 q ˇq ˇ ˇq ˇ Vq ij ai k j k " c2 IIa ˇ ˇ C .2 2c5 / C .1 c1 / CV q C 1 C c1 CV q aij Sij C ˇ Sˇ k 3q ˇq ˇ ˇ ˇ 0 ˇ ˇ1 D ˇSq ˇ 1 @ D Vq ij 2 .c4 1/ aij Sij A C D CV q ˇ ˇ ˇ ˇ aij C C CV q ˇ Sˇ ˇ Sˇ 3 Dt Dt ˇ ˇ ˇ ˇ

q

q

where D CV q, the diffusion term can be approximated by:

D

CV q

@ D @xi

C

.tt /ij

!

CV q

@CV q @xj

! :

IIa D aij aij , and the seven constants ci and ci are reported in Table 1. Assuming a similarity with the diffusion term of k transport equation, CV q coefficient can be set, firstly, to the value of one.

Table 1 SSG second order closure scheme constants [19]

c1

c1

c2

c3

c3

c4

c5

1.7

0.90

1.05

0.8

0.65

0.625

0.2

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4 Numerical Results 4.1 Implementation in the Navier–Stokes Multi–Block Solver On the basis of the k–" OES turbulence model, the previous transport equations were implemented in the Navier–Stokes Multi-Block (NSMB) code. The NSMB solver is constructed on a finite volume formulation of the fully compressible Navier–Stokes governing equations. In the present study, spatial discretization is ensured by a second order central scheme and temporal intergration by a second order backward scheme based on a dual time stepping method with constant CFL parameters. More details about NSMB numerical issues can be found in [22] and validation results concerning the C type meshgrid (256 81 nodes) used in the present configuration are reported in [23]. The isotropic OES version of the k" two-equation closure scheme is founded on Chien’s low Reynolds number model [24] where eddy-diffusivity coefficient and damping function were reconsidered to take into account of the turbulent kinetic energy spectrum modification induced by the extraction of phase-avared quantities in non-equilibrium turbulent configurations. In the present development, the scalar C parameter is replaced by the tensorial one and the following isotropic OES damping function is considered, leading to a reduction of the eddy-viscosity closer to the wall than using Chien’s function: f y C D 1 exp 0:0002 y C 0:000065 y C2 ; where y C is the non-dimensional wall distance.

4.2 Detached Turbulent Flow Around a NACA0012 Airfoil The predictive capacities of the present anisotropic turbulence model are analysed on a well-documented two-dimensional test-case, at first. The incompressible unsteady flow past au NACA0012 airfoil at 20 degrees of incidence is simulated by means of the present model. The Reynolds number based on the chord length and the free-stream velocity is equal to 105 . The numerical results are compared to an experimental dataset [25]. As presented in Fig. 7, the CV i criteria transported by the additional equations allow a local modulation of the eddy-diffusion coefficient, leading to specific reductions in highly sheared region and in the near-wake coherent structures. In the far-wake where a certain equilibrium is reached, a homogenisation of the criterion is observed. A comparison between the experimental and computed aerodynamic efforts emphasises the quality of the anisotropic turbulence model in this two-dimensionnal context (Fig. 8). Numerical values are slightly higher than the experimental ones. Relative errors are 2:4/. A similar study of the local flow field of a small-aspect-ratio stack .AR D 8:3/ [8] identified three zones within the jet: zone 1, immediately above the stack exit, where the jet dominates the flow; zone 2, where the jet begins to bend and the jet flow and cross-flow have the same velocity; and zone 3, further downstream, where the cross-flow dominates the flow. Depending on the value of R and the corresponding flow regime, one or more of these zones may be absent. In a more recent study of a small-aspect-ratio stack .AR D 9/ partially immersed in a flat-plate turbulent boundary layer [2], wake velocity measurements were made in cross-stream (y–z) planes for jet-to-cross-flow velocity ratios ranging from R D 0 to 3. The near-field behaviour of the stack was broadly classified into three flow regimes; adopting the terminology used by [5, 6], these flow regimes were classified as the downwash .R < 0:7/, cross-wind-dominated .0:7 R < 1:5/, and jet-dominated .R 1:5/ flow regimes. Each flow regime had a distinct structure to the mean velocity field, turbulence intensity field, and Reynolds stresses [2]. The influence of the jet-to-cross-flow velocity ratio, R, on the streamwise vortex structures has not been extensively investigated. These structures are known to have

Vortex Structures in the Wake of a Stack

479

Fig. 1 Stack of uniform circular cross-section (of diameter, D, and height, H ) mounted normal to a ground plane and partially immersed in a turbulent flat-plate boundary layer (of thickness, •, and freestream velocity, U1 ): (a) top view; (b) side view

an important role in the problem of stack downwash [7], which occurs at low jetto-cross-flow velocity ratios. In the present study, the behaviour of the streamwise vortex structures in the near-field of a small-aspect-ratio .AR D 9/ stack is investigated for a range of jet-to-cross-flow velocity ratios.

2 Experimental Approach The experimental set-up was similar to that adopted by [2]. The experiments were conducted in a low-speed, closed-return wind tunnel with a test section of 0.91 (height) 1.13 (width) 1.96 m (length). The longitudinal freestream turbulence intensity was less than 0.6% and the velocity non-uniformity outside the test section wall boundary layers was less than 0.5%.

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The test section floor was fitted with a ground plane. A rough strip located about 200 mm from leading edge of the ground plane was used to produce a turbulent flat-plate boundary layer on the ground plane at the location of the stack.

2.1 Experimental Apparatus A cylindrical stack of H D 171:5 mm; D D 19:1 mm; d=D D 0:67 (where d is the internal diameter, see Fig. 1), and AR D 9, was used in the present study. The experiments were conducted at a freestream velocity of U1 D 20 m=s, giving a Reynolds number, based on the stack external diameter, of ReD D 2:3 104 . The solid blockage ratio was 0.3% and no wall interference corrections were made. The stack was located 700 mm downstream of the rough strip on the ground plane. At the location of the stack, the boundary layer thickness was ı D 83 mm, the boundary layer shape factor was 1.3, and the Reynolds number based on momentum thickness, , was Re D 0:86 104 . This boundary layer provided a thickness-to-height ratio of ı=H 0:5 at the location of the stack. A pair of screens was installed at the stack supply inlet to ensure the flow exiting the stack was turbulent. The mean axial velocity profiles at the stack jet exit were similar to the typical velocity profile for a turbulent pipe flow. The exhaust velocity of the non-buoyant stack jet was varied with two MKS 1559A-200L mass flow controllers arranged in parallel. The jet-to-cross-flow velocity ratio was varied from R D 0 (no jet exiting the stack) to R D 3, corresponding to momentum flux ratios of Rm D 0 to 9. The jet Reynolds number for the minimum exit velocity, when R D 0:5, was Red D 7:6 103 (based on the internal diameter of the stack, d , and the average jet exit velocity, Ue ). The jet Reynolds number for the maximum exit velocity, when R D 3, was Red D 4:7 104 .

2.2 Measurement Instrumentation The wind tunnel data were acquired with a personal computer, a National Instruments PCI-6031E 16-bit data acquisition board, and LabVIEW software. The freestream conditions were obtained with a Pitot-static probe (United Sensor, 3.2-mm diameter), Datametrics Barocell absolute and differential pressure transducers, and an Analog Devices AD590 integrated circuit temperature transducer. Wake measurements were made with a seven-hole pressure probe (3.45-mm diameter, 30ı cone angle). The seven pressures were measured with a Scanivalve ZOC-17 pressure scanner. The probe was calibrated in situ using an automated variable-angle calibrator, with a calibration grid spacing of 8:1ı over a flow angle range of ˙72:9ı . A direct-interpolation calibration data-reduction method was used [10]. The measurement uncertainty was estimated to be less than 3ı for the flow angle and less than 5% for the velocity magnitude.


481

The seven-hole probe was mounted in a three-axis, computer-controlled traversing system. For each value R, the time-averaged wake velocity field (u, v, w components) in the cross-stream (y–z) plane was measured over a 5-mm uniform grid at streamwise locations from x=D D 6 to 10 downstream of the stack. From each velocity field, the time-averaged streamwise vorticity field, !x .y; z/, was determined. In addition, the wake velocity field was measured in a vertical (x–z) plane parallel to the test-section centreline (at y=D D 0).

3 Results and Discussion Results are presented for three values of the jet-to-cross-flow velocity ratio, R, each value representing one of the three flow regimes identified by [2]: R D 0:5 (downwash flow regime), R D 1 (cross-wind-dominated flow regime) and R D 2:5 (jet-dominated flow regime). Similar to the definition of [4], the stack wake is defined as the region below the free end, where 0 < z=H 1.0 < z=D 9/, and the jet wake is defined as the region above the free end, where z=H > 1.z=D > 9/.

3.1 Time-Averaged Velocity Fields on the Wake Centreline Time-averaged velocity vector fields in the vertical (x–z) plane along the wake centreline (at y=D D 0) are shown in Fig. 2 for three values of R. In each case, immediately behind the stack is a region of highly angled or recirculating flow, indicated by an absence of velocity vectors. The absence of vectors in this region 14

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Fig. 2 The time-averaged velocity vector field along the wake centreline .y=D D 0/ for the (a) downwash flow regime, R D 0:5; (b) cross-wind-dominated flow regime, R D 1; and (c) jetdominated flow regime, R D 2:5

482


indicates that the local flow angle exceeded the angular range of the seven-hole probe, which was estimated at ˙70ı [10, 11]. For each value of R, the streamwise extent of this recirculation region varies along the stack height. For R D 0:5 (Fig. 2a), which represents the downwash flow regime, the downwash flow (downward-directed velocity) is observed within the stack wake close to the stack’s free end. This downwash flow persists in the streamwise direction and descends from the free end towards the mid-height of the stack. An upwash flow (upward-directed velocity) is observed near the base of the stack and within the flatplate boundary layer on the ground plane. These downwash and upwash flows are similar to the case of the finite circular cylinder [11, 12], which corresponds to the case of R D 0. However, with the presence of the weak jet flow exiting the stack at R D 0:5, the strength of the downwash flow, and the streamwise and vertical extents of the upwash flow, are reduced compared to the finite circular cylinder. In addition, the maximum streamwise extent of the recirculation region is smaller, while closer to the stack free end the length of the recirculation region is greater, compared to finite-cylinder .R D 0/ case. For R D 1 (Fig. 2b), which corresponds to the cross-wind-dominated flow regime, the downwash flow is absent in the near-wake region of the stack. Compared to R D 0:5 (Fig. 2a), the streamwise extent of the upwash flow is reduced but it now extends to a greater vertical distance above the ground plane. In addition, for R D 1 (Fig. 2b), the maximum streamwise extent of the recirculation region within the stack wake is smaller, but a more sizeable recirculation region is now found in the vicinity of the stack free end. This is caused by the stronger jet flow, which now behaves more like a bluff body. Another recirculation region is also observed above the free end of the stack and within the jet wake. This region may contain the vortex formed within the jet wake as observed by [5] for 0:95 < R < 2:4, which is due to the presence of the jet flow. For R D 2:5 (Fig. 2c), which corresponds to the jet-dominated flow regime, the stronger momentum of the jet flow allows it to penetrate the cross-flow, as shown by the upward-directed velocity vectors above the free end of the stack. With the increased jet rise, the size of the recirculation region within the jet wake is now larger compared to the cross-wind-dominated flow regime at R D 1 (Fig. 2b). A strong recirculation region within the jet wake was also observed by [8]. There is also a corresponding reduction in the size of the recirculation region within the stack wake, and the streamwise extent of this region becomes more uniform along the stack height.

3.2 Time-Averaged Cross-Stream Velocity Fields Time-averaged cross-stream velocity vector fields (v, w velocity components measured in the y–z plane) at x=D D 6 are presented in Fig. 3 for the same three values of R as in Fig. 2. For the downwash flow regime (Fig. 3a, R D 0:5) and


483

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Fig. 3 The time-averaged velocity vector field downstream of the stack at x=D D 6 for the (a) downwash flow regime, R D 0:5; (b) cross-wind-dominated flow regime, R D 1; and (c) jetdominated flow regime, R D 2:5

the cross-wind-dominated flow regime (Fig. 3b, R D 1), the vector fields show two pairs of counter-rotating velocity fields within the stack wake, one closer to the free end and the other closer to the ground plane. For the jet-dominated flow regime (Fig. 3c, R D 2:5), an additional pair is observed in the jet wake region. For all of the three flow regimes, the upwash flow is observed within the flat-plate boundary layer on the ground plane near the stack base. The strength of this upwash flow is strongest within the boundary layer and reduces towards the mid-height of the stack. For R D 0:5 (Fig. 3a), there is a presence of strong a downwash flow within the stack wake and below the stack free end, which is the main characteristic of the downwash flow regime. The strength of the downwash flow reduces along the height of the stack when moving away from the free end and towards the base, similar to the case of the finite circular cylinder [11]. For R D 1 (Fig. 3b), representing the cross-wind-dominated flow regime, the downwash velocity field is located further away from the ground plane and closer to the stack’s free end. The downwash for R D 1 is weaker than for R D 0:5 (Fig. 3a). In addition, for the same streamwise position, the upper pair of two the counter-rotating velocity fields now occurs closer to the free end of the stack. For R D 2:5 (Fig. 3c), a second and much stronger upwash flow occurs above the free end of the stack within the jet wake region. This upwash extends more than four stack diameters above the free end, and is associated with the third pair of the counter-rotating velocity fields. These features characterize the jet-dominated flow regime and result from the stronger jet flow exiting the stack.

484


3.3 Time-Averaged Streamwise Vorticity Fields The time-averaged streamwise vorticity (!x D !x D=U1 , where !x is the streamwise vorticity component) fields in the y–z plane at x=D D 6 are shown in Fig. 4 for the same three values of R as in Figs. 2 and 3. For each of the three flow regimes, the streamwise vorticity field shows two counter-rotating vortex pairs within the stack wake: one pair near the stack free end, referred to as the “tip vortex pair”, and another of opposite sense of rotation closer to the base of the stack, referred to as the “base vortex pair.” These two pairs of vortex structures were evident in the mean cross-stream velocity vector fields, shown in Fig. 3, and are also observed in the wake of a finite circular cylinder [11, 12]. The tip vortex structures are stronger than the base vortex structures, consistent with [11, 12]. In the downwash flow regime (Fig. 4a, R D 0:5), the tip vortex pair at x=D D 6 is located below the free end, similar to the finite cylinder [11, 12]. In the cross-winddominated flow regime (Fig. 4b, R D 1), the tip vortex pair now extends above the free end. In addition, the base vortex structures are stretched upward along the sides of the stack towards the mid-height position. In the jet-dominated flow regime (Fig. 4c, R D 2:5), the tip vortex pair has weakened considerably compared to the downwash and cross-wind-dominated flow regimes (Fig. 4a and b, respectively). In addition to the two pairs of streamwise vortex structures found within the stack wake region, a third counter-rotating vortex pair is now found in the jet wake region. Evidence of this vortex pair was seen in

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Fig. 4 The time-averaged streamwise vorticity field downstream of the stack at x=D D 6 for the (a) downwash flow regime, R D 0:5; (b) cross-wind-dominated flow regime, R D 1; and (c) jetdominated flow regime, R D 2:5. Iso-vorticity contour lines of !x . Solid lines represent positive (CCW) vorticity; dashed contour lines represent negative (CW) vorticity; minimum vorticity contour of !x D ˙0:04; contour increment of !x D 0:04


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Fig. 5 Streamwise development of the time-averaged streamwise vorticity field behind the stack for R D 2:5: (a) x=D D 6; (b) x=D D 8; and (c) x=D D 10. Contour lines as in Fig. 4

the velocity vector field (Fig. 3c). This third vortex pair, referred to as the “jet-wake vortex pair”, is located above the free end of the stack and is similar to the kidneytype vortex pair observed for a bent jet in cross-flow. The sense of rotation of the jet-wake vortex pair is the same as the base vortex pair but opposite to the tip vortex pair. The jet-wake vortex structures are associated with the jet rise and the strong upwash velocity field above the free end of the stack (Fig. 3c), and are stronger than the tip vortex structures. The streamwise development of the vortex structures in the jet-dominated flow regime is presented in Fig. 5, for x=D D 6 to 10. The size and strength of the tip and jet-wake vortex pairs decrease with increasing distance from the stack. The behaviour of the weaker base vortex pair with x=D is more irregular.

4 Conclusions In the present study, a seven-hole pressure probe was used to measure the timeaveraged velocity and streamwise vorticity fields in the wake of a stack operating at jet-to-cross-flow velocity ratios from R D 0 to 3. This range of velocity ratio covered the three main flow regimes describing the fluid behaviour close to the stack, namely the downwash, cross-wind-dominated, and jet-dominated flow regimes. As R is varied, marked changes occur in the downwash and upwash velocity fields, the location, strength, and number of streamwise vortex structures, and the size and shape of the recirculation zone in the near wake of the stack. In the downwash flow regime, the flow is similar to that of a finite circular cylinder, with two pairs of counter-rotating streamwise vortex structures, each of opposite

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sign, in the stack wake. The tip vortex pair is found closer to the free end of the stack and is associated with a strong downwash velocity field immediately behind the stack. The weaker base vortex pair is found within the flat-plate boundary layer on the ground plane and is associated with an upwash velocity field directed away from the ground plane. In the cross-wind-dominated flow regime, the tip vortex pair extends just above the free end of the stack and base vortex pair is stretched vertically towards the midheight of the stack. The downwash and upwash velocity fields, and the vortex pairs, are weakened compared to the downwash flow regime. For the jet-dominated flow regime, three pairs of counter-rotating streamwise vortex structures are observed. In addition to the tip vortex and base vortex pairs found at lower velocity ratios, the jet-wake vortex pair appears in the jet wake region above the free end of the stack. The jet-wake vortex pair is associated with the jet rise and a strong upwash velocity on the jet wake centreline. Acknowledgments The authors acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canada Foundation for Innovation (CFI), the Innovation and Science Fund of Saskatchewan, and the Division of Environmental Engineering. The assistance of D.M. Deutscher and Engineering Shops is appreciated.

References 1. Adaramola, M.S., Akinlade, O.G., Sumner, D., Bergstrom, D.J., and Schenstead, A.J., Turbulent wake of a finite circular cylinder of small aspect ratio. J. Fluid Struct. 22 (2006) 919–928. 2. Adaramola, M.S., Sumner, D., and Bergstrom, D.J., Turbulent wake of a stack and the influence of velocity ratio. Proceedings of the 6th International Symposium on Fluid-Structure Interactions, Aeroelasticity, Flow-Induced Vibration & Noise, Vancouver, Canada, ASME PVP Division, July 23–27, 2006, Paper No. PVP2006-ICPVT11-93629 (2006) 1–9. 3. Eiff, O.S., Kawall, J.G., and Keffer, J.F., Lock-in of vortices in the wake of an elevated round turbulent jet in a crossflow. Exp. Fluids 19 (1995) 203–213. 4. Eiff, O.S. and Keffer, J.F., Parametric investigation of the wake-vortex lock-in for the turbulent jet discharging from a stack. Exp. Therm. Fluid Sci. 19 (1999) 57–66. 5. Huang, R.F. and Hsieh, R.H., An experimental study of elevated round jets deflected in a crossflow. Exp. Therm. Fluid Sci. 27 (2002) 77–86. 6. Huang, R.F. and Hsieh, R.H., Sectional flow structures in near wake of elevated jet in a crossflow. AIAA J. 41 (2003) 1490–1499. 7. Johnston, C.R. and Wilson, D.J., A vortex pair model for plume downwash into stack wakes. Atmos. Environ. 31 (1997) 13–20. 8. Mahjoub Sa¨ıd, N., Mhiri, H., Le Palec, G., and Bournot, P., Experimental and numerical analysis of pollutant dispersion from a chimney. Atmos. Environ. 39 (2005) 1727–1738. 9. Okamoto, S., Flow past circular cylinder of finite length placed on ground plane. Trans. Jpn. Soc. Aeronaut. Space Sci. 33 (1991) 234–246. 10. Sumner, D., A comparison of data-reduction methods for a seven-hole probe. J. Fluid Eng.-T. ASME 124 (2002) 523–527. 11. Sumner, D., Heseltine, J.L., and Dansereau, O.J.P., Wake structure of a finite circular cylinder of small aspect ratio. Exp. Fluids 37 (2004) 720–730. 12. Tanaka, S. and Murata, S., An investigation of the wake structure and aerodynamic characteristics of a finite circular cylinder. JSME Int. J. B-Fluid T. 42 (1999) 178–187.

Flow Separation of a Rotating Cylinder S.C. Luo, T.T.L. Duong, and Y.T. Chew

Abstract The effects of rotating a circular cylinder on the suppression of flow separation were investigated experimentally. Flow separation and vortex shedding were studied by flow visualization to guide the hot-film anemometry measurements that follow. The experiments were conducted for flow past a circular cylinder with an aspect ratio of 16 in a recirculating water channel at Reynolds numbers in the range of 140 to 1,000. The rotational to translational speed ratio, ’, varied from 0 to 5. The present results show the existence of a critical ’ value of about 2.3 at which the vortex shedding is suppressed. Below this critical value of ’, the Kármán vortex street and separation points are observed. Vortex shedding is deflected and separation points are displaced more and more towards the rotation direction of the cylinder as ’ increases. Above the critical ’ value, vortex shedding disappears. The two separation points on the cylinder surface seem to move very close to each other at ’ > 2:3. This issue will also be discussed by analyzing the flow pictures obtained from flow visualization. The flow regime close to the cylinder surface is analyzed at different Reynolds numbers and different values of ’ to study how the cylinder’s rotation affects the flow separation. The effects of rotating circular cylinder to vortex shedding frequency as well as the suppression of vortex shedding and flow separation are studied in order to understand the moving-wall effects in flow separation control. Keywords Flow separation Vortex shedding suppression Rotating circular cylinder Magnus effect

1 Introduction When a viscous fluid flows over a bluff body, flow separation can occur under certain conditions resulting in phenomena such as vortex-induced oscillation, drag increase, lift decrease, wake buffeting, etc. The phenomenon of flow separation has important S.C. Luo (), T.T.L. Duong, and Y.T. Chew Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 e-mail: [email protected]; [email protected]; [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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effects to the bluff body which are associated with the formation of vortices and energy loss in the wake behind the body. The shedding of vortices from the sides of the bluff body then causes fluctuating forces to exert on the body. It is known that to reduce or to eliminate these unfavorable effects, the flow separation should be delayed or even suppressed. In order to effectively control flow separation, different techniques had been introduced to add momentum to the near-wall flow field either passively (based on the design of geometrical configuration, for example disposing slats around the structure [8]) or actively (uses of external power to counteract the adverse pressure gradient, for example using an additional rotating cylinder [9]). In the present study, the influence of the moving-wall effects to the flow separation from a bluff body is investigated when a fluid flows past a rotating circular cylinder.

2 The Experiment Figure 1 shows the present experimental set-up. A circular cylinder with diameter D D 25 mm and aspect ratio of 16 rotating at angular speed is subjected to an oncoming flow at a free-stream velocity U1 . Flow visualization is carried out by injecting dye from within the cylinder at Reynolds numbers in the range of 140 < Re < 1;000, and the cylinder rotation rate ’ (cylinder rotational to free stream speed ratio D D=2U1 ) of 0 ’ 5. Vortex shedding frequency f and hence Strouhal number .St; D fD=U1 / are obtained by hot-film measurement in the cylinder wake.

3 Results and Discussion The fundamental data in St–Re relationship compared with literatures will be presented first, both as a calibration of the present experimental set-up, and to examine how the cylinder rotation affects the frequency of vortex shedding. Results of stationary circular cylinder .’ D 0/ are first studied before the more complex rotating

Fig. 1 Definition sketch of flow past a rotating circular cylinder

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cylinder .’ ¤ 0/ cases are investigated. After that, some typical pictures obtained from flow visualization will be analyzed to understand the cylinder rotation’s effects to vortex shedding process and flow separation.

3.1 Vortex Shedding Frequency For the stationary cylinder, St–Re relationship in mode A .Re < 190/ and mode B .Re > 260/ regimes (Fig. 2) is achieved. Good agreement is obtained between the present data and the highly cited experimental St–Re curve reported in Williamson [6]. Figure 3 also presents the comparison between the present experimental results and the results obtained from the universal St–Re relation proposed in Williamson and Brown [7] and Fey et al. [2]. The data show a rather good agreement, except for the transition regime from mode A to mode B regimes where the vortex shedding process is really complicated. The agreement obtained in the comparison presented in Figs. 1 and 2 gives some indication of the quality of the present rotating circular cylinder data.

Fig. 2 St–Re of stationary cylinder compared with Williamson’s St–Re curve [5]

Fig. 3 St–Re of stationary cylinder

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Fig. 4 St–Re at different ’

Figure 4 shows an increase in vortex shedding frequency when rotation rate increases. In the present study, from Reynolds number of 600 up to 1,000, the Strouhal number seems to reach a constant value at higher Re which is higher for greater rotation rate. This confirms that changing the rotation rate can influence or even control the frequency at which the vortices are shed.

3.2 Vortex Shedding Suppression Karman vortex street is observed for stationary cylinder (Fig. 5) and the flow pictures match well with those reported in Van Dyke’s album [5]. The periodic vortex shedding is also clearly observed for rotating circular cylinder (Fig. 6) up to a critical value of rotation rate, ’crit ; .’ < 2:3/. Above the critical ’ .’ > 2:3/, regular vortex street is no longer seen, the vortex shedding ceases gradually and disappears as shown in Fig. 7. The existence of a critical ’ is also reported in the literatures (’crit D 2 in the computation of Chew et al. [1]; ’crit is of 2 and 2.2 at Re of 200 and 1,000, respectively, by Ling and Shih [3]). The suppression of vortex shedding at high rotation rate shows that cylinder rotation plays a very important role in controlling the flow process behind the cylinder. Even when the vortex street vanishes, the dye trace in the wake still continues to increase in inclination in the rotation direction as ’ increases, showing the effect of cylinder rotation on its wake.

3.3 Effect on Flow Separation When the cylinder is non-rotating .’ D 0/, flow separation occurs symmetrically on both sides of the cylinder causing a large wake behind the cylinder (Fig. 8a). For rotating cylinder when the vortex shedding still exists, the two separation points are shifted on the cylinder surface due to the rotation and seem to move closer to each other as ’ increases (Fig. 8). On the side of the cylinder where the surface moves

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a

b

Fig. 5 Karman vortex street. (a) at Re D 140 [5]. (b) at Re D 141 (present study) a

d

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Fig. 6 Vortex shedding at Re D 141. (a) ’ D 0:4. (b) ’ D 1:4. (c) ’ D 1:8. (d) ’ D 2:3 a

c

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Fig. 7 Suppression of vortex shedding at Re D 226. (a) ’ D 2:5. (b) ’ D 3. (c) ’ D 4. (d) ’D5 a

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Fig. 8 Flow separation (close up view) at Re D 164. (a) ’ D 0. (b) ’ D 0:6. (c) ’ D 1:2. (d) ’ D 2:5

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Fig. 9 Flow separation at Re D 114. (a) ’ D 2:6. (b) ’ D 3:2. (c) ’ D 4

in the same direction with the free stream flow (lower side in Fig. 8), additional momentum is injected into the near-surface fluid region, resulting in the delay of boundary separation which can be seen by the displacement of the separation point in the cylinder rotation direction. On the other side of the cylinder, the increase in relative motion between the moving surface and the free stream flow shifts the separation point further upstream. This upstream movement increases with increasing ’. Above ’crit .’ > 2:3/, the wake appears to become narrower as ’ increases (Figs. 7 and 9) and this could result in a decrease in the pressure drag as well as an

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increase in the lift force. On the side where the cylinder surface moves in the same direction with free stream flow, the separation seems to be suppressed at high value of ’ (’ D 4 in Fig. 9) as a large additional momentum is injected to the close-wall flow field to overcome the effects of the adverse pressure gradient. Prandtl, in his experiment [4], also noted that flow separation on one side of the rotating cylinder is completely eliminated when the rotational speed is high.

4 Conclusion The present study shows that rotating a circular cylinder can result in the suppression of the vortex shedding and elimination of boundary layer separation on one side of the cylinder at high rotation rate. It confirms the existence of a critical ’ value of about 2.3 at which the vortex shedding is suppressed. Below this critical value of ’, the Kármán vortex street and separation points are observed. Vortex shedding is deflected and separation points are displaced more and more towards the rotation direction of the cylinder as ’ increases. Above the critical ’ value, vortex shedding disappears. Acknowledgements One of the authors (T.T.L. Duong) is a Ph.D. student and recipient of a graduate scholarship from the National University of Singapore. This scholarship is gratefully acknowledged.

References 1. Chew, Y. T., Cheng, M. and Luo, S. C., A numerical study of flow past a rotating circular cylinder using a hybrid vortex scheme. Journal of Fluid Mechanics 299 (1995) 35–71. 2. Fey, U., König, M. and Eckelmann, H., A new Strouhal-Reynolds number relationship for the circular cylinder in the range 47 < Re < 2 105 . Physics of Fluids 10 (1998) 1547–1549. 3. Ling, G. P. and Shih, T. M., Numerical study on the vortex motion patterns around a rotating circular cylinder and their critical characters. International Journal of Numerical Methods in Fluids 29 (1999) 229–248. 4. Prandtl, L., Application of the “Magnus Effect” to the wind propulsion of ships. N.A.C.A – Technical memorandum 367 (1926) 1–45. 5. Van Dyke, M., An Album of Fluid Motion, The Parabolic Press, Stanford, CA (1982). 6. Williamson, C. H. K., Vortex dynamics in the cylinder wake. Annual Review of Fluid Mechanics 28 (1996) 477–539. p 7. Williamson, C. H. K. and Brown, G. L., A series in 1= Re to represent the Strouhal-Reynolds number relationship of the cylinder wake. Journal of Fluids and Structures 12 (1998) 1073– 1085. 8. Wong, H. Y., A mean of controlling bluff body flow separation. Journal of Industrial Aerodynamics 4 (1979) 183–201. 9. Yoshinoby, K., Eijirou, Y., Vinod, J. M., Eiki, Y., Kusuo, K. and Shin-ichi, K., Control of flow separation from leading edge of a shallow rectangular cylinder through momentum injection. Journal of Wind Engineering and Industrial Aerodynamics 83 (1999) 503–514.

Part VII

Theoretical/Industrial Aspects of Unsteady Separated Flow Control


The Wake Dynamics of a Cylinder Moving Along a Plane Wall with Rotation and Translation B. Stewart, K. Hourigan, M. Thompson, and T. Leweke

Abstract A numerical investigation examined the two-dimensional flow structures forming around a cylinder moving along a wall with different combinations of rolling and sliding, with Reynolds numbers, Re, ranging from 20 to 500. Past research has shown that both wall effects and rotation can suppress the mechanisms which bring about unsteady flow. Knowing this, it is natural to question how the flow will be affected when both these properties are present, as is the case for the cylinder rolling along a wall. Results indicate that the transition from steady to timevarying flow is strongly influenced by the rate of rotation of the body, and for the case of reversed rolling, the onset of unsteady flow is delayed until Reynolds numbers above 400. Keywords Wake dynamics Flow control Rolling cylinder

1 Introduction The current study utilised a two-dimensional, spectral element method to examine the flows occurring around a cylinder moving along a wall with several different combinations of rolling and sliding. Past research has shown that the flow structures forming around a stationary cylinder change dramatically when the body is in close proximity to a wall, and vortex shedding may be suppressed [3, 4]. Furthermore, the unsteady flow around an isolated cylinder may be suppressed if the cylinder is rotating at a sufficient speed, relative to the free-stream velocity [5]. The rate of rotation at which vortex shedding is suppressed is a function of the Reynolds B. Stewart (), K. Hourigan, and M. Thompson Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical Engineering and Division of Biological Engineering, Monash University, Clayton, 3800, Australia B. Stewart and T. Leweke Institut de Recherche sur les Phénomènes Hors Equilibre, CNRS/Universités Aix-Marseille, Technopôle de Château-Gombert, 49 rue Frédéric Joliot-Curie, BP 146, F-13384 Marseille Cedex, France M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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ω

Fig. 1 Problem configuration

U

D G

number, Re, and results in a fully attached boundary layer and the formation of closed streamlines around the body [6]. Knowing that both wall effects and rotation can suppress the mechanisms which bring about unsteady flow, it is natural to question how the flow will be affected when both these properties are present, as is the case for the cylinder rolling along the wall. This configuration has the potential to control the formation of flow structures which may enhance particle deposition and mixing in both biological and industrial flows. The non-slip rolling cylinder has been reported briefly in [7] and the current study aims to extend this work. The problem geometry is shown in Fig. 1. The rotation rate of the cylinder, ˛, is defined as the ratio of rotational to translational cylinder velocity. That is, ˛ D !D=2U , where ! is the angular rotation, D is the cylinder diameter and U is the translational velocity of the cylinder. Simulations were carried out for ˛ D 1; 0:5; 0; 0:5 and 1 to include a range of ‘reversed’ rolling when ! is negative (opposite in direction to that shown in Fig. 1). The Reynolds number of the flow is based on the cylinder diameter and ranges from 20 to 500.

2 Wake Dynamics Behind the Rolling and Sliding Cylinder 2.1 Numerical Method and Resolution The numerical code used a Galerkin method with seventh-order Lagrangian polynomials fitted to each macro-element of the mesh to solve for the relevant variables. A more complete description of the numerical method can be found in [8]. The mesh consisted of approximately 550 macro-elements, with dimensions of 100D to the upstream and downstream boundaries and 150D to the cross-stream boundary. This resulted in a blockage ratio of less than 1%. Further increases in polynomial order or domain size gave a variation in the Strouhal number and the mean lift and drag within 0:1% for Re D 200 and ˛ D 1. To overcome numerical constraints, a small distance, G, was introduced between the cylinder and the wall. Increasing this distance from 0:004D to 0:01D allowed a small amount of fluid to pass between the body and the wall, however, the streamlines in the wake displayed no discernible change for G < 0:01D. G D 0:005 was used throughout the present study.

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From this point forward, results are presented in the frame of reference attached to the cylinder centre, with the lower wall and the free-stream flow moving with the same velocity relative to the fixed, rotating cylinder.

2.2 Steady Flow For each of the five cylinder rotation rates examined, the flow structures behind the body remain steady at the lower limit of Reynolds numbers studied (Re D 20). This flow is characterised by either 0, 1 or 2 closed areas of recirculating fluid. These recirculation regions are defined by the closed streamlines of the flow, and the upper and lower recirculation regions (labeled 1 and 2, respectively) are shown in Fig. 2. For cylinder motions with a positive rotation (˛ D 1; 0:5) the upper recirculation region (1), is always present and encircling the body. This occurs as the cylinder motion entrains a fluid layer and moves it toward the contact region. The fluid being forced into the constriction then reverses direction and passes back over the top of the body. As the fluid moves downstream past the cylinder, it moves into the low pressure region behind the body where it is then sucked into the constriction at the rear of the body and re-entrained by the cylinder motion (see Fig. 2a). When the cylinder is sliding along the wall with no rotation present (Fig. 2b), the topology of the flow is altered, and the upper recirculation region no longer encircles the cylinder. Instead, there exists points of flow separation and reattachment on the

1

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α = 1, Re = 50

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α = 0, Re = 100

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1 2

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α = −1, Re = 300

Fig. 2 Upper and lower recirculation regions

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surface of the cylinder. The position of these separation and reattachment points remains relatively stable as Re is increased in the steady flow regime. For both the sliding cylinder and the cylinder with positive rotation, streamlines near the wall only form a closed recirculation region, (2), when Re is increased beyond a critical value. For Re below this value a smaller region of upstream flow is observed but with no closed streamlines present. When the rotation of the cylinder has a negative value it is possible for the flow near the body and the wall to remain fully attached. When negative rotation is present, the flow entering the constriction near the wall is then able to reverse direction and be transported around the body in the cylinder boundary layer. The increased flow attachment for negative ˛ has a stablising effect on the wake which prevents the formation of recirculation regions 1 and 2 for Reynolds number less than Re 70 for ˛ D 0:5 and Re 115 for ˛ D 1. When the upper recirculation region (1) is present, it no longer encircles the body but is instead separated from the cylinder by the attached boundary layer flow. The effect of increasing Re and varying the rotation rate is shown below in Fig. 3a–c. Increasing Re results in a near linear increase in the length of the steady recirculation zones. This is in agreement with observations of the steady flow around wall-mounted bodies and the backward facing step [2, 9]. As the rotation rate is varied between rolling (Fig. 3a), sliding (Fig. 3b) and reversed rolling (Fig. 3c), the wake narrows and moves closer to the wall. This is particularly apparent for the cases of reversed rotation. Another change in flow topology, which becomes apparent for negative ˛, is that associated with the formation of the upper and lower recirculation regions. For the cases of ˛ D 1; 0:5; 0 and 0:5 the upper recirculation region forms from the shear layer at lower Reynolds numbers, before recirculation region 2. However, as ˛ becomes more negative, the recirculation region nearest the wall forms first. For ˛ D 1, even after appearance of recirculation region 1, the recirculating flow near the wall remains the larger of the two, which is not the case for the other values of ˛.

a Re=60, α = 1

b Re=100, α = 0

c Re=300, α =

− 1

Fig. 3 Streamlines and vorticity contours for steady flow


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This difference in the relative size of the two areas of recirculating fluid is shown in Fig. 4 for the two cases of negative cylinder rotation studied. For increasing Re (until the limit of steady flow), the filled curves in Fig. 4 indicate the length of the recirculation zones measured from the upstream to downstream-most points. Zones 1 and 2 are as defined in Fig. 2 and x* is the stream wise distance scaled by D. For ˛ D 0:5 the size of the two recirculation zones is comparable but with zone 1 occurring first, at lower Re. As described above, as ˛ became more negative the formation of the recirculation regions was suppressed and there was a larger range of Re for which no closed regions of recirculating flow was observed. During the steady flow regime the drag and lift forces on the body decrease smoothly according to a power law of the form CD / Re b , where 0:74 < b < 0:80. The results for the drag coefficient at ˛ D 1; 0 and 1 are shown in Fig. 5. As the rotation rate is varied the dominant force acting on the body changes from being a lift force at ˛ D 1 to a drag dominated system when the cylinder is sliding along the wall or moving with a positive rotation. This is shown in Fig. 6 for Re D 80. As the cylinder undergoes larger magnitude negative rates of rotation, the

Fig. 4 Filled curves indicate the regions in the x direction occupied by the two steady recirculation zones. For ˛ D 0:5 the upper recirculation region (1) appears at lower Re, while for ˛ D 1 the recirculation region nearest the wall (2) forms first

Fig. 5 Coefficient of drag for varying Re and ˛

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Fig. 6 Coefficients of lift and drag for Re D 80

Fig. 7 Transition from steady to unsteady flow for varying ˛

high pressures in the contact region and the lower pressures over the top of the body create a net lift force. In addition, the attached flow around the body acts to reduce the pressure drag.

2.3 Unsteady Flow For each of the five rotation rates under examination, the flow undergoes a transition from steady flow to periodic vortex shedding as Re increases. No hysteresis was observed for this transition and the critical Re dividing these two regimes is shown in Fig. 7. As described above for the development of the steady recirculation zones, negative values of ˛ tend to enhance the stability of the flow, and for ˛ D 1 the flow does not undergo the transition to time-varying flow until Re > 400. Once the flow becomes unsteady, the wake is characterised by vorticity from the upper shear layer inducing opposite-sign vorticity at the wall. These regions of opposite sign vorticity then move away from the wall and form a rotating vortex pair


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Fig. 8 Vorticity contours for the cylinder with ˛ D 0 at Re D 200 over a single shedding cycle

similar to those observed by Arnal et al. [1] for the square cylinder sliding along a wall. A typical shedding period is shown in Fig. 8 at Re D 200 for the sliding cylinder with ˛ D 0. The movement downstream of the shed vortex pair is observed (from top to bottom) with the next image in the sequence starting again from the top. A clockwise rotation of the vortex pair takes place due to the higher strength of the vortex which is shed from the cylinder shear layer. Increasing ˛ from 0 to 1 at Re D 200 results in a considerably shorter formation length and reduces the frequency of vortex shedding. This can be seen in Fig. 9, which shows the three rotation rates at approximately the same moment in the shedding cycle (taken at the moment of maximum lift). The increasing spacial separation of the shed vortex pair is a reflection of the change in shedding frequency and Strouhal number, S t. The strength of the shed vorticity also increases with the rotation rate, resulting in the shed vortex pair being propelled further from the wall. The same trends are observed in Fig. 10, as ˛ is increased from negative rotation to sliding (˛ D 0). For each cylinder rotation the frequency of vortex shedding appeared to be largely insensitive to to Re, with the exception of the case of ˛ D 1. These trends are shown in Fig. 11 and at Re 450 the shedding in the reversed rolling case

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Fig. 9 Vorticity contours for unsteady flow at Re D 200 for ˛ D 0, ˛ D 0:5 and ˛ D 1 (from top)

Fig. 10 Vorticity contours for unsteady flow at Re D 450 for ˛ D 1, ˛ D 0:5 and ˛ D 0 (from top)


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Fig. 11 Variation of Strouhal number with cylinder rotation rate

experiences a sudden decrease in the Strouhal number. However, from Fig. 10 it is observed that the strength of the shed vorticity is also much less and the vortex pairs have all but disappeared by a distance of 15D downstream. From these observations it is apparent that the imposed rotation of the cylinder is capable of either delaying or enhancing the onset of unsteady flow, depending on the magnitude and direction of rotation.

3 Conclusions When the cylinder has a negative imposed rotation as it moves along the wall the stability of the wake is enhanced and the wake region narrows and moves towards the wall. For ˛ D 1 the wake is highly stable, and the transition to unsteady flow is not reached until Re 420. As the rotation rate is varied from reversed rolling to sliding (˛ D 0), the steady wake region increases in both length and distance from the wall. At Re D 200, the sliding cylinder wake is unsteady and the flow is characterised by vorticity from the upper shear layer inducing opposite sign vorticity at the wall. These regions of opposite sign vorticity then move away from the wall and form a rotating vortex pair. Increasing ˛ from 0 to 1 at constant Re results in a considerably shorter formation length and increases the strength of the shed vorticity in the wake. The frequency of vortex shedding appeared to be insensitive to the Reynolds number of the flow, except for the case of ˛ D 1 which experienced a rapid decrease in S t at Re 450. Acknowledgments This research was made possible via the financial support of an Australian Postgraduate Award.

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References 1. M. P. Arnal, D. J. Goering, and J. A. C. Humphrey. Vortex shedding from a bluff body adjacent to a plane sliding wall. Transactions of the American Society of Mechanical Engineers, 113:384– 398, 1991. 2. S. C. R. Dennis and Gau-Zu Chang. Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100. Journal of Fluid Mechanics, 42:471–489, 1970. 3. A. Dipankar and T. K. Sengupta. Flow past a circular cylinder in the vicinity of a plane wall. Journal of Fluids and Structures, 20(3):403–423, April 2005. 4. C. Lei, L. Cheng, S. W. Armfield, and K. Kavanagh. Vortex shedding suppression for flow over a circular cylinder near a plane boundary. Ocean Engineering, 27:1109–1127, 2000. 5. S. Mittal and B. Kumar. Flow past a rotating cylinder. Journal of Fluid Mechanics, 476:303– 334, 2003. 6. J. C. Padrino and D. D. Joseph. Numerical study of the steady-state uniform flow past a rotating cylinder. Journal of Fluid Mechanics, 557:191–223, 2006. 7. B. E. Stewart, K. Hourigan, M. C. Thompson, and T. Leweke. Flow dynamics and forces associated with a cylinder rolling along a wall. Physics of Fluids, 18:111701–1–111701–4, 2006. 8. M. Thompson, K. Hourigan, and J. Sheridan. Three-dimensional instabilities in the wake of a circular cylinder. Experimental Thermal and Fluid Science, 12:190–196, 1996. 9. P. T. Williams and A. J. Baker. Numerical simulations of laminar flow over a 3D backwardfacing step. International Journal for Numerical Methods in Fluids, 24:1159–1183, 1997.

Simulation Study of the Robust Closed-Loop Control of a 2D High-Lift Configuration B. Gunther, ¨ A. Carnarius, F. Thiele, R. Becker, and R. King

Abstract The investigation focuses on the closed-loop separation control of a two dimensional high-lift configuration in a numerical simulation study. The lift is to be controlled by adjusting the non-dimensional intensity of the harmonic excitation near the leading edge of the single slotted flap. Since control laws based on a high-dimensional discretisation or low-dimensional description of the Navier– Stokes equations are not applicable in real-time, this investigation presents a fast and efficient controller synthesis methodology employing robust methods. This offers real-time capability for future experimental implementations. In spite of the nonlinear and infinite-dimensional Navier–Stokes equations, it is surprising to observe that the dynamic behaviour appears very simple. This input–output behaviour in the vicinity of set points can be empirically approximated by stable linear blackbox models of second order. Based on these, a simple robust controller is synthesised that autonomously adjusts the excitation such that a desired lift is obtained. Keywords High-lift Active flow control RANS Incompressible flow Robust Closed-loop

1 Introduction The wings of commercial aircraft must generate a tremendous amount of lift during take-off and landing in order to reduce ground speeds and runway lengths. Instead of incorporating complex, heavy and expensive multi-element high lift devices, single B. Günther (), A. Carnarius, and F. Thiele Institute of Fluid Mechanics and Engineering Acoustics, Berlin University of Technology, MüllerBreslau-Str. 8, 10623 Berlin, Germany e-mail: [email protected] R. Becker and R. King Department of Process and Plant Technology, Berlin University of Technology, Hardenbergstr. 36a, 10623 Berlin, Germany e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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flaps without slats are desirable. Such flaps can however only be applied if flow separation at high flap angles can be avoided. Experimental investigations [1] as well as numerical simulations [2] have shown that flap separation can be significantly delayed by periodic excitation near the flap leading edge in the case of low and high Reynolds numbers [3] and the lift can be enhanced. In further investigations [4, 5] oscillatory suction and blowing was found to be much more efficient than steady blowing with respect to lift. The process becomes very efficient if the excitation frequencies correspond to the most unstable frequencies of the free shear layer, generating arrays of spanwise vortices that are convected downstream and continue to mix across the shear layer. In order to create an effective and efficient control method, previous studies have primarily focused on the parameters of the excitation apparatus itself. Overviews are given by Wygnanski and Gad-el-Hak [6, 7]. The present investigation focuses on lift control of a two dimensional, three element high-lift configuration in a numerical simulation study. The lift is to be controlled by adjusting the amplitude of the harmonic excitation near the leading edge of the single slotted flap. Open-loop control, however, suffers from two severe drawbacks related to uncertainty. First, it has to be based on a profound knowledge of the flow system considered in order to correctly invert the input-output relation from the actuator to the quantity being controlled. If this relation is not fully known, openloop performance will deteriorate if not fail. A second source of uncertainty will always be given by disturbances, especially when leaving the well-defined conditions of a laboratory wind tunnel. To overcome these problems, closed-loop control is employed here to control the high-lift configurations. Keeping the requirement of a real time capability in mind, thus excluding feedback control based on a numerical solution of the Navier–Stokes equations, only three approaches seem to be promising: 1. The best results will be obtained by nonlinear controllers exploiting lowdimensional models that describe the nonlinear physics, i.e. Galerkin or vortex models. However, these models are presently restricted to rather simple flow configurations and will still rely on a rather detailed knowledge of the system. 2. In adaptive control [8], models are identified online. This information is then used to adapt the controller in real time. As an alternative, model-free methods [9] can also be applied in adaptive control, such as extremum seeking control. 3. A fast and cost-effective controller synthesis employing robust methods is possible as a third alternative based on a family of linear or nonlinear black-box models [8, 10] identified offline. This latter approach will be used here. The paper is organised as follows: A description of the configuration is given in Section 2. Section 3 describes controller design and simulation results. Finally, the results are summarized.

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2 Flow Configuration 2.1 General Description The numerical test model represents the practically-relevant SCCH (Swept Constant Chord Half-model) high-lift configuration, which has already been used for several experimental studies targeting passive/active flow and noise control concepts [11– 13]. In the experiments, the three-dimensional wing has a sweep angle of ˆ D 30ı and a constant chord length in the spanwise direction. The numerical investigation is mainly focused on a 2D wing profile in order to reduce the computational costs. The typical three-component setup consists of the main airfoil equipped with an extended slat of 0:158 c relative chord length and an extended flap which has a relative chord length of ck D 0:254 c (Fig. 1). All profiles have blunt trailing edges. The flap is situated at a fixed position underneath the trailing edge, forming a gap of 0:0202 c and an overlap of 0:0075 c. The flap deflection angle was increased from the base value (32ı ) to a angle of 37ı in order to maximise the recirculation above the flap. The angle of attack is fixed at 6ı for the whole configuration. In addition, the flow over the flap is detached whereas the flow over the slat and the main wing are still fully attached. In all numerical investigations the freestream velocity corresponds to a Reynolds number of Rec D 106 based on the chord of the clean configuration (with retracted high-lift devices). This high Reynolds number is chosen to demonstrate the relevance to industrial applications.

2.2 Numerical Method All computational investigations have been carried out using the numerical code ELAN was developed at the Institute of Fluid Mechanics and Engineering Acoustics of the TU Berlin. The numerical method is based on a three-dimensional incompressible finite-volume scheme for solution of the Reynolds-averaged Navier–Stokes equations [14]. All URANS simulations are based on the statistical

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turbulence model LLR k-! by Rung and Thiek [15], which represents an improved two-equation eddy viscosity model formulated with special respect to the realisability conditions. The computational c-type mesh consists of 90,000 cells in total with 15 chords upstream, above and below the configuration and 25 chords downstream. The nondimensional wall distance of the first cell centre remains below Y C D 1 over the entire surface. A separate study of the influence of time step size indicated that a typical time step of t D 2:1 103 c=u1 is sufficient to resolve the important flow structures. All computations presented here are based on t D 1:26 103 c=u1 , which allows a resolution of 630 time steps per period of vortex-shedding in the unexcited case and 336 time steps per oscillation cycle for a non-dimensional oscillating frequency of F C D 0:6. To model the excitation apparatus, a periodic suction/blowing type boundary condition is used. The perturbation to the flow field is introduced through the inlet velocity on a small wall section representing the excitation slot:

u1 C t uexc .t/ D ua si n 2 F ck

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where F C is the non-dimensional perturbation frequency, C is the nondimensional steady momentum blowing coefficient, H is the slot width (H D 0:001238 ck ) and ua is the amplitude velocity of the perturbation oscillation. The oscillating jet is emitted perpendicular to the wall segment of the excitation slot, and is located at 4.1% chord behind the flap leading edge.

2.3 Unexcited Flow As a first step, two-dimensional, unsteady investigations without excitation have been carried out. The flow field of the SCCH-configuration without excitation is characterised by massive separation above the upper surface of the flap. The mean separation point is located at 4.1% chord behind the flap leading edge, and downstream a large recirculation region occurs. The unsteady behaviour of separated flow is mainly governed by large vortices shed from the flap trailing edge that interact with the vortices generated in the shear-layer between the recirculation region and the flow passing through the slot between main airfoil and flap nose (Fig. 2a). The spectrum of the lift coefficient in Fig. 2b shows a dominant amplitude for a Strouhal k number of the unexcited flow formed with the flap chord of S tu D f uc1 D 0:32, mainly produced by this vortex shedding. The base flow configuration involving massive separation over the flap described above is undesirable, and the goal is to improve this using active flow control.


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2.4 Excited Flow After the base flow investigations, flow control mechanisms are applied. All flow control computations use the baseline case solutions as initial flow conditions. In order to find an optimum excitation, simulations with different variable excitation parameters, such as excitation frequency, excitation intensity, blow out angle or duty cycle are performed (Fig. 3). The numerical investigations employed sinusoidal perturbation and periodic blowing in comparison to the experimental excitation mode [11]. The studies with variable excitation frequencies and intensities (Figs. 3a and 4b) show that active flow control by periodic blowing could not achieve the gain in lift seen for excitation by sinusoidal suction and blowing.

2.4.1 Periodic Excitation with Different Intensity For the excited flow case periodic perturbations with an intensity of between C D 10105 and C D 500105 have been introduced. This excitation parameter has been selected as the control signal in order to design a robust controller. Compared

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to the detached, unexcited flow, the excited lift could be continuously increased with growing intensity. The maximum of lift can be identified for a harmonic excitation of C D 400 105 . In this case the lift coefficient can be enhanced by 19% compared to the baseline simulation (Fig. 4b). The gain in lift by the excitation is mainly based on a change of flow direction at the trailing edge of the main airfoil. With the excited condition of flow above the flap the trailing edge departure angle is increased (compare Figs. 2a and 4a) and the pressure distribution above the main airfoil is enhanced. The natural flow above the flap is mainly governed by large-scale vortex shedding from the flap trailing edge, which is nearly eliminated in the optimal excited flow (see Figs. 2a and 4a). Smallsized vortices generated by periodic suction and blowing above the perturbation slit enable the transport of energy from the main flow to the recirculation near the wall. Thereby the time-average detachment position moves from less than 5% chord downstream to more than 12%, the recirculation area is reduced and the downflow condition is modified.

3 Robust Control Figure 4b illustrates the benefits of periodic excitation (sinusoidal perturbation). The plot displays the time-averaged, steady-state input–output map of the configuration, i.e. the lift coefficient cl versus excitation intensity C . Excitation with low intensity (C D 10 to 100105) leads to a strong increase in lift (max. 17%). If C becomes larger the lift shows a saturation behaviour. The controller computes the non-dimensional intensity C . / of the harmonic excitation in order to track a desired lift coefficient cl;desired . / referred to as the reference command in the following. Here, is the dimensionless time given in convective units. In control notation C . / is taken as the control signal and cl . / as the system output to be controlled. Therefore, a classical closed-loop feedback control system as shown in Fig. 5 is used. The controller calculates the control signal C . / as function of the difference between cl;desired . / and cl . /.


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3.1 Dynamic System Behaviour In spite of the nonlinear and infinite dimensional Navier–Stokes equations, it is surprising that the dynamic behaviour of the lift cl . / as a function of time looks very simple in step experiments (see Fig. 6). The plots on the left-hand and right-hand side display step responses for switching C . / on and off, respectively. All system responses can be approximated by stable linear black-box models GP .s/ of first or second order [10]. Due to the nonlinear characteristics of the investigated system, large ranges for the identified model parameters have to be accepted for various step heights and initial values of the control signal C . /. To avoid a significant detuning of the controller C.s/, it is desirable to reduce the model uncertainty. As the steady-state gain KP D lims!0 GP .s/ of all identified models GP .s/ shows a highly nonlinear dependence on the size of the control signal C , i.e. the forcing amplitude, the inverse f 1 of the steady-state input-output map cl D KP .C / C D f .C / (C D const:, ! 1, see Fig. 4b) can be used to compensate for this nonlinearity as displayed in Fig. 5. Here, a simple exponential relation is fitted between the data points for sinusoidal perturbation in Fig. 4b. In order to describe the nonlinear behaviour of the real process, a family … of linear black-box models GP .s/ 2 … is identified from representative step experiments as illustrated in Fig. 6 for all relevant operating points. The black lines in Fig. 7 display the Bode magnitude (top) and phase (bottom) plots of the models GP .s/ 2 ….

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Due to the compensation f 1 , the steady-state gain of all models is about one. The spread of the cut-off frequencies is caused by the different dynamics of the experiments with the varying step heights for switching on (positive) and switching off (negative) cases. However, positive steps correspond to faster dynamics resulting in higher cut-off frequencies. The flow system description by continuous black-box models is only valid for significantly lower frequencies than the excitation frequency, as illustrated in Fig. 7. Otherwise, the sinusoidal excitation signal will be distorted and the active flow excitation mechanism does not work properly. This upper bound for validity is illustrated by the dotted model frequency responses and the vertical dashed line for the excitation frequency.

3.2 Determination of the Controller Design Model A linear nominal model Gn .s/ with a multiplicative uncertainty description ˇ ˇ ˇ GP .j!/ Gn .j!/ ˇ ˇ lM .!/ D max ˇˇ ˇ GP 2… G .j!/

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speaking, lM .!/ gives the validity range of Gn .j!/. Gn .s/ and lM .!/ are plotted in the Bode diagram in Fig. 7 with a thick solid and a dash-dotted line, respectively. By searching for a nominal model Gn .s/ with the smallest uncertainty radius, it turns out that a simple second-order transfer function Gn .s/ D K=.1 C sT1 C s 2 T2 / can be used [10]. The parameters T1 and T2 are identified by solving an optimization problem such that the least mean square uncertainty radius is obtained.

3.3 Controller Design Due to the uncertainty of the identified models, robustly designed controllers are required to maintain closed-loop stability for all operating points. As a PI-controller CPI .s/ D K.1 C 1=sTI / is the most common controller for such systems, its application was tested first. Here, the PI-break frequency ! D 1=TI was chosen near to the nominal model cut-off frequency as displayed in the Bode magnitude plot in Fig. 8. The thin solid line indicates the nominal model and the bold dash-dotted line indicates the PI-controller. In order not to amplify oscillations, only a moderate gain K was chosen for the first closed-loop studies in order to obtain high phase and gain margins. Additionally, as described below, the PI-structure was simplified to an I-structure CI .s/ D 1=sTI as illustrated in Fig. 8 by the bold solid line. Both controllers meet the robust stability requirement.

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3.4 Results for Command Tracking Performance In the left-hand time plot of Fig. 9, which shows the closed-loop command tracking performance of the PI-controller, it is seen that the controlled system output cl . / follows the step input reference command cl;desi red . / although with strong oscillations. Despite a closed-loop design that does not amplify oscillations in the case of linear systems due to the high gain and phase margins, the flow system still shows a strong nonlinear behaviour with oscillating dynamics leading to this poor closedloop performance. Even the variation of the controller parameters toward smaller gains K and break frequencies ! D 1=TI does not reject or damp the oscillations. Corresponding results are not shown here. Since the robust PI-controller does not reject the oscillations, which are inherent to the flow system, a modified controller structure with low frequency response magnitude in the range of the oscillations is chosen next. The robust I-controller described above is the simplest and most obvious candidate for this purpose, see the tracking response in the right-hand time plot of Fig. 9. However, the controlled lift follows the reference command signal thereby damping the oscillations to low acceptable level. In order to achieve faster closed-loop tracking responses, simulation studies with higher I-controller crossover frequencies ! D 1=TI have been carried out. Again, non-acceptable oscillations occured in these cases with smaller gain and phase margins. Again, corresponding results are not shown in this paper. Obviously, a sufficient low frequency response magnitude is needed to damp the inherent oscillations in the range of the excitation frequency and this is not given for higher I-controller crossover frequencies.

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4 Conclusions This paper describes a successful approach for closed-loop separation control by active means on a high-lift configuration, i.e. a two dimensional wing with a slat and a single slotted trailing edge flap. Well-known local periodic forcing near the leading edge of the flap is used to excite the flow in order to control flow separation and, hence, the corresponding lift. Since closed-loop flow control suffers from the lack of sufficiently simple, low-dimensional controller design models based on the Navier– Stokes equations, an empirical family of linear black-box models is identified from representative simulation studies, which are then used for robust controller design. Loop shaping yields a linear controller with I-structure and this offers real-time capability for future experimental implementations. However, the inherent non-linear oscillating flow system behaviour imposes a limitation on the achievable performance, i.e. the closed-loop crossover frequency needs to be sufficiently smaller than the oscillations in the range of the excitation frequency. Acknowledgements The research project is funded by Deutsche Forschungsgemeinschaft (German Research Foundation) as part of the Collaborative Research Centre 557 Complex turbulent shear flows at TU Berlin.

References 1. F.H. Tinapp. Aktive Kontrolle der Strömungsablösung an einer Hochauftriebs-Konfiguration. Ph.D. thesis, Technische Universität Berlin, 2001. 2. M. Schatz and F. Thiele. Numerical study of high-lift flow with separation control by periodic excitation. AIAA Paper 2001-0296, 2001. 3. M. Schatz, F. Thiele, R. Petz, and W. Nitsche. Separation control by periodic excitation and its application to a high lift configuration. AIAA Paper 2004-2507, 2004. 4. S.S. Ravindran. Active control of flow separation over an airfoil. TM-1999-209838, NASA, Langley, 1999. 5. J.F. Donovan, L.D. Kral, and A.W. Cary. Active flow control applied to an airfoil. AIAA Paper 98-0210, 1998. 6. I. Wygnanski. The variables affecting the control separation by periodic excitation. AIAA Paper 2004-2505, 2004. 7. M. Gad-el Hak. Flow control: The future. Journal of Aircraft, 38(3), 402–418, 2001. 8. R. King, R. Becker, and M. Garwon. Robust and adaptive closed-loop control of separated shear flows. AIAA-Paper 2004-2519, 2004. 9. R. Becker, R. King, R. Petz, and W. Nitsche. Adaptive closed-loop separation control on a high-lift configuration using extremum seeking. AIAA-Paper 2006-3493, 2006. 10. R. Becker, M. Garwon, C. Gutknecht, G. Bärwolff, and R. King. Robust control of separated shear flows in simulation and experiment. Journal of Process Control, 15, 691–700, 2005. 11. B. Günther, F. Thiele, R. Petz, W. Nitsche, J. Sahner, T. Weinkauf, and H.-C. Hege. Control of separation on the flap of a three-element high-lift configuration. AIAA Paper 2007-265, 2007. 12. M. Schatz, B. Günther, and F. Thiele. Computational investigation of separation control for high-lift airfoil flows. In R. King, ed., ‘Active Flow Control’, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Vol. 95. Berlin, Springer, 2007.

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13. K. Kaepernick, L. Koop, and K. Ehrenfried. Investigation of the unsteady flow field inside a leading edge slat cove. In 11th AIAA/CEAS Aeroacoustics Conference (26th Aeroacoustics Conference), Monterey, CA, 2005. 14. L. Xue. Entwicklung eines effizienten parallelen Lösungsalgorithmus zur dreidimensionalen Simulation komplexer turbulenter Strömungen. Ph.D. thesis, Technische Universität Berlin, 1998. 15. T. Rung and F. Thiele. Computational modelling of complex boundary-layer flows. In 9th International Sympolium on Transport Phenomena in Thermal-Fluid Engineering, Singapore, 1996.

Large-Eddy Simulation of Passively-Controlled Transonic Cavity Flow Pierre Comte, F. Daude, and I. Mary

Abstract A 30 dB reduction of the peak pressure tone and a reduction by 6 dB of the background pressure found in an experiment of high-subsonic cavity flow controlled by a spanwise rod is retrieved numerically. The injection of deterministic upstream fluctuations in the LES domain is found to be of crucial importance, in contrast with the baseflow case. Reduction of the vortex impingement onto the aft edge of the cavity is confirmed, together with reduction of mass flow rate breathing through the grazing plane. Visual evidence of merging between the Kelvin-Helmholtz-type vortices shed downstream of the fore edge of the cavity and the von Kármán vortices shed behind the cylinder is provided. Shocklets downstream of the cylinder are also observed. Keywords Transonic cavity flow LES Rossiter modes

1 Introduction The high levels of pressure fluctuations caused by compressible flows over open cavities have motivated considerable efforts, in order to understand the underlying physical mechanisms and develop control strategies which are effective not only for a specific design point but also for a sufficiently wide range of parameters around it to be of practical interest. It is well established that grazing flows over open cavities, namely, cavities too short for the recirculation zone past the upstream edge to close, develop unsteadiness due to some coupling between the reattachement region near the aft edge and the region where the incoming boundary layer detaches, past the upstream edge. P. Comte () Laboratoire d’Etudes Aérodynamiques, Unité Mixte C.N.R.S. - Université de Poitiers - ENSMA nı 6609, CEAT, 43 avenue de l’Aérodrome, F-86036 Poitiers Cedex, France e-mail: [email protected] F. Daude and I. Mary ONERA, 29, avenue de la Division Leclerc, 92322 Châtillon Cedex, France M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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Conceptual models developing self-sustained oscillations exist in the incompressible limit [6], featuring Biot-Savart-type instantaneous interaction. Pressure tones are found, the frequencies of which scale on the inverse of the cavity length L, and which are enharmonic up to an end correction as customary in impinging flows [16, 17]. These models can be extended to weakly compressible flows, as in [3]. At higher Mach number, the propagative nature of the coupling has to be taken into account. The interaction is referred to as “fluid-acoustic mode” in [17], in contrast with the “fluid–fluid mode” that prevails at vanishing Mach number. Assuming that the frequency of shedding of the Kelvin-Helmholz vortices matches that of acoustic waves propagating upstream within the cavity yields the Rossiter model fn D UL1 Mn , in which M D U1 =a denotes the external Mach number C1

and D Uc =U1 involves an average convection speed of the Kelvin-Helmholtz vortices [18]. A broad set of experimental configurations provided values for both constants and as a function of the length-to-depth aspect ratio L=D of the cavity, and the model has proved to match quite well frequencies of the pressure tones, up to some ad-hoc tuning of both parameters. However, attempts to educe a universal behaviour of cavity flows have not been successful, because of the observed influence of the following parameters on the receptivity of the detached mixing layer: L=D, but also L=W , L= and the flow parameters Re , M1 , the shape factor of the incoming boundary layer H D ı = and the overall pressure level prms =q1 within the cavity [2]. This sensitivity not only to the level of the tones but also to the broad-band noise makes the prediction and the optimization of control strategies particularly challenging (see [2, 19] for a review). The motivation here is focussed on the assessment of the needs of numerical insight to reproduce quantitatively the unsteadiness reduction effects of a simple passive actuator, in the simplest possible high-subsonic cavity configuration: such a low aspect ratio as L=D D 0:42 is considered, as in [5, 12] in order to minimize large-scale three-dimensional effects. The control device considered is a spanwise cylinder placed in the upstream boundary layer, as proposed in [15]. This was proved to reduce significantly both the tone and broadband pressure levels, provided the diameter d and its height1 y are suitably chosen. One of the key enablers for this so-called High-Frequency Tone Generator is the Reynolds number independence of the cylinder’s wake Strouhal number S t D f d=U1 0:2. A detailed experimental investigation [7, 8] performed at ONERA for different L=D, boundary layer thicknesses, and Mach numbers ranging from 0:6 to 0:78, concluded that the main parameter is the ratio y=d , with an optimum at y=d D 1:2 yielding a reduction by 30 and 6 dB of the peak tone and the overall pressure rms, respectively, and this despite the added noise of the cylinder. In contrast with the baseline case, it was found that prms did not increase with M , hence a highest efficiency found at the highest M possible with the experimental facility before the onset of sonic choking effects, viz., M D 0:78.

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Here, two calculations will be compared, differing essentially in the treatment of the incoming boundary layer. The numerical details are given in the next section. The results of both calculations are compared in Section 3, in which pressure and velocity statistics are presented together with visualisations. The contribution of this investigation to the current understanding of this intriguing feedback loop will eventually be summarized.

2 Computational Setup The numerical methodology employed here has been adapted from [12] (see also [10,11] for other aspect ratios), for which very good agreement with the experimental counterparts was found, in particular regarding the pressure levels and the dynamics of the phase-averaged coherent structures. In this L=D D 0:42 case, the overall large-scale two-dimensionality of the flow makes it possible to use 2D URANS in the portions of the flow where the boundary layer is attached, and devote most of the computing power to the matched Large-Eddy Simulation of a streamwise portion of the flow, in a 3D domain of span Wnum D L with periodic boundary conditions, whereas the span W D D of the experimental test section is 2:4 times as large. The Reynolds number based on the length L of the cavity is 8:21 105 , as in the experiment. The major difficulty on the numerical side is to account for the cylinder (Fig. 1), which is of diameter d D 2:5 mm, whereas the thickness of the incoming boundary layer is ı99% D ı D 9:8 mm. The thickness of the boundary layer cylinder is smaller than 0.1 mm. Turbulent boundary layers have to be gridded with a wall-normal resC olution of about one wall unit. In fact, ymi n 2 proved to be sufficient, either in URANS or LES. The block-structured grid shown in Fig. 2 has about 20 106 grid points, and meets the LES requirements x C 50, y C 2, zC 20. The cavity length is discretized over about 200 meshes, and there are 256 points in the

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spanwise direction, with Wnum D 20 d , that is, 5 times as much as in the incompressible cylinder wake simulations [9]. In the LES, the 3D zone is preceded by a box of width Wnum =5 D 4d replicated 5 times in the spanwise direction, and in which the incoming boundary layer and its fluctuations are generated by means of a recyling method inspired by [14] and adapted to compressible RANS-LES interfacing. We are aware that this spanwise replication, motivated by CPU time considerations, could be critized because it deprives the upstream forcing of spanwise scales significantly larger than the spanwise integral scale of the turbulence generated by a canonical wake in a turbulence-free environment. Nevertheless, Wnum=5 D 4d is large enough for the dominating instabilities of the wake to develop, as in [9]. The SubGrid-Scale model used in the LES portion (in pale in the left plot of Fig. 2) is the Selective Mixed-Scale model proposed in [13]. The dark regions are treated in 2D URANS, with the Spalart-Allmaras model, as in [12]. In that paper, the injection of realistic upstream fluctuations was not found to be needed for recovery of correct results in the baseline configuration. A repetition of it with the cylinder has thus been undertaken, with the same grid as described above, except that the region upstream of the cylinder is treated in 2D URANS. Because it switches from URANS to LES, this calculation will be hereafter referred to as DES, although the switching is monitored by the multi-block decomposition and not by comparison between the mesh size and the local turbulence integral scale. Implicit time integration is used, as in [11], but with a block-local determination of the number of iterations of the Newton-type inner process designed in such a way that the balance of the convergence errors is ensured despite the high gradient of CFL number near the cylinder. Consistency with results obtained with time-explicit schemes has been assessed in the case of the linear advection of a 2D vortex and in the case of the low Mach number flow on an airfoil at moderate angle of attack, near the recirculation bubble on the leeside [4]. Table 1 highlights the computational effort and the mesh size discrepancies. The CPU times mentioned correspond in each case to 50 periods of the fundamental Rossiter mode (i.e. 0.025 s), computed on a NEC SX6 with an average speed of four Gflops per processor.


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Table 1 Mesh and computational parameters for cavity simulations Number of cells ( 106 ) LES without control [11] LES with control DES with control

CFL 16 1.6 20 17

CFL 700 / 0.5 0.5

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3 Results Analysis Figure 3 shows the organized vortices educed by means of a positive Q surface. Both the DES and the LES develop Kelvin-Helmholtz and von-Kármán-type vortices in between which streamwise vortices are stretched. The LES shows a higher level of small-scale turbulence. The corresponding movies show less large-scale unsteadiness in LES than in DES, with low-frequency flapping of the mixing layer dramatically reduced with respect to the baseline configuration, without the cylinder, which is consistent with [8] and other experimental references. In particular, the impingement of spanwise-organized large scale vortices on the aft edge of the cavity and the trapping events are visibly inhibited by the presence of the cylinder. The pressure spectra (Fig. 4) recorded on the rear wall of the cavity at y=D D 0:08 show more differences between the LES and the DES than the visualizations: the LES reproduces satisfactorily the reduction of the first Rossiter mode at 2 kHz and that of the next ones observed in [8]. In contrast, the DES shows much less reduction of the tone levels, which shows the influence of the upstream boundary layer fluctuations. However, both calculations underestimate the peak at 20 kHz due to the wake of the cylinder, by almost 10 dB. They also underpredict the width of this peak, which is significantly wider than that of the Rossiter tones, but one should keep in mind that the pressure signal is recorded 20 diameters downstream of the cylinder, which is quite demanding in terms of resolution and numerical dissipation. Note also that the experimental pressure signal has been low-pass filtered at 30 kHz, which prevents the assessment of the numerical prediction of the high-frequency background noise. However, the latter is higher in LES than in DES, which is in agreement with the visualisations. Regarding the recycling method, the distance between the re-injection and the extraction planes would correspond, assuming advection at U1 , to a frequency of the order of 12 kHz, which does not show up on the spectra, either in LES or DES. This confirms that the recycling technique has been applied sufficiently upstream, so that the spurious correlation it introduces has enough room to decrease before it reaches the cylinder. The effect of these upstream fluctuations is highly visible on the mean flow (Fig. 5): the recirculation length in LES is reminiscent of that of a freestream turbulent wake, whereas that in DES is about 3 times as large. It is well known that this recirculation length strongly depends on the turbulence level in the boundary layer of the cylinder, which determines the position of the separation points. As the Reynolds number, based on the cylinder diameter, is close to 4 104 , the wake

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Fig. 3 Isosurfaces of Q D 2 .U1 =d /2 coloured by the streamwise velocity. DES (left), LES (right) 160

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would be in the subcritical regime in free stream. However, the LES yields a drag coefficient of 0:48, rather reminiscent of the supercritical regime. We however cannot be conclusive regarding the accuracy of the treatment of the boundary layer of the cylinder, and can only notice the dramatic (and beneficial) effect of the injection of deterministic upstream fluctuations. The mean streamlines show a much less

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pronounced upward deviation of the meanflow in LES than in DES, with about the same additional thickening of the mixing layer due to the cylinder. This was not expected a priori, since the deviation and thickening of the mixing layer are considered as one of the possible explanations for the tone reduction caused by the cylinder [21]. Notice also that the baseline configuration (right plot of 5) exhibits a small recirculation bubble, analogous to that observed at higher aspect ratio by LarchevOeque et al. [11], who emphasized its possible importance in the feedback process. Although such a bubble is not visible in either the DES or the LES, the latter shows more attached mean streamlines around the upstream edge of the cavity, which is in favour of the argument in [11]. The beneficial influence of the deterministic upstream perturbations is not outstandingly visible on the velocity statistics (Fig. 6). The mean velocity profiles are prescribed at x D 50 mm in both the DES and the LES. At x D 0 mm, above the fore edge of the cavity, that is, 5 mm downstream of the cylinder’s centerline, the turbulent kinetic energy and the Reynolds stress u0 v0 (not shown here) in LES are correct, but the width of the wake is ever so slightly overestimated. The DES cannot build up the right turbulence level in such a short distance downstream of the cylinder. This was also the case in the DES in [1]. Farther downstream (x D 30 and 40 mm), the differences between DES and LES are less visible, but, with respect to the LDV measurements of [8], the LES tends to underestimate the wake’s diffusion, whereas the DES overpredicts it. The presence of locally supersonic regions, too mild to educe experimentally, was suspected in [7, 8]. This is confirmed, not only by the mean Mach lines shown above, but also by instantaneous snapshots and movies, as in Fig. 7. This shows,

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Fig. 7 Isosurfaces of Q D 2 .U1 =d /2 coloured by the streamwise vorticity (blue/red), and of j@x jjdivuj D 1:3 d 2 =.1 U1 (green), at two instants of the cycle, in LES

in addition to Mach contours that indicate local Mach numbers beyond 1:8, and a positive Q surface, a Schlieren-type representation of density gradient conditioned by a high value of the dilatation, in order to educe the shocklets. The upstream part of the supersonic region is found to be relatively stable, whereas its downstream part, in which the flow decelerates causing the shocklets, oscillates at the wake’s shedding frequency. There is also visual evidence of merging between the Kelvin-Helmholtz-type vortices shed downstream of the fore cavity edge and the lower row of von Kármán-type vortices shed behind the cylinder. This interaction takes place either immediately around x D 0, as in the bottom row of Fig. 7, or farther downstream during the other alternance of the wake shedding sequence, in which case the two vortical systems can clearly be distinguished before they merge. One of the conjectures about the tone reduction is that the cylinder’s wake reduces the ‘breathing’ of the cavity, namely, the variations of mass flow rate through its grazing plane, which is difficult to measure experimentally. This is shown here in Fig. 8 from LES, in .x; t/ evolution after spanwise averaging (top row) and in time evolution only, after streamwise averaging (bottom row), over three periods of the fundamental Rossiter mode. In the baseline configuration, one can see about ten in-out alternances per period near the fore edge, reduced to about three in the downstream quarter of the cavity, in a consistent fashion with the phase-averaged visualizations in [12], in which the classical ‘escaped, cut or trapped’ sequence was educed with outstanding agreement between numerical results and PIV measurements. With the cylinder, the breathing amplitude is reduced by a factor of about 6, with more high frequencies and more alternances in the near fore edge region, close to the cylinder.


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Fig. 8 Evolution of the mass flow rate through the grazing plane, Baseline configuration (left), LES with cylinder (right)

4 Conclusion Two hydrid RANS=LES calculations of the transonic flow over a cavity passively controlled by means of a spanwise rod are presented, and compared with the baseline configuration. The results are assessed with respect to the experimental measurements of [8]. With the cylinder dimensioned and position for maximal pressure tone reduction, a low L=D aspect ratio is considered in order to minimize the complexity of the physics involved: in particular, the large-scale structure of the flow is quasi two dimensional, which makes it possible to use spanwise periodic boundary conditions. The simulations remain nonetheless quite computationally extensive ( 2,000 CPU hours per run), despite the adaptation of the numerics to the grid size variations required to capture both the incoming boundary layer and that which develops on the cylinder. In contrast with the baseline case, strong sensitivity of the nature of the upstream boundary layer fluctuations is found: indeed, in the absence of deterministic forcing, the wake of the cylinder is not turbulent enough to reduce the pressure tones to the experimental level, whereas an analogous simulation with

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deterministic fluctuations generated by recycling method is proved to be successful. The visualisations confirm that the impingement of Kelvin-Helmholtz-type vortices onto the aft edge of the cavity is indeed reduced, in agreement with [21]. However, the two simulations do not show dramatic differences in the mean flow properties, and the mean upward deflection of the mean flow does not seem to be significant. Reduction of the amplitude of the mass flow rate ‘breathing’ through the grazing plane by a factor of 6 is observed w.r.t. the baseline case, together with the enrichment of the frequency content, due to additional small scales injected in the vicinity of the wake near the upstream edge of the cavity. This is in a sense consistent with the argument of [20], although analogous tone reduction effects were observed by [7] at lower frequency ratio between the fundamental Rossiter mode and the wake shedding mode, here equal to 10. Finally, evidence of shocklets is provided, which was conjectured experimentally but difficult to measure.

References 1. S. Arunajatesan, J. D. Shipman, and N. Sinha. Hybrid RANS-LES simulation of cavity flow fields with control. AIAA Paper 2002-1130, 2002. 2. L. Cattafesta, D. Williams, C. W. Rowley, and F. Alvi. Review of active control of flow-induced cavity resonance. AIAA Paper 2003-3567, 2003. 3. L. Chatellier, J. Laumonier, and Y. Gervais. Theoretical and experimental investigation of low mach number cavity flow. Exp. Fluids, 36:728–740, 2004. 4. F. Daude, I. Mary, and P. Comte. Improvement of a Newton-based iteration strategy for the large-eddy simulation of compressible flows. submitted to J. Comp. Phys. 5. N. Forestier, L. Jacquin, and P. Geffroy. The mixing layer over a deep cavity at high-subsonic speed. J. Fluid Mech, 475:101–145, 2003. 6. M. Howe. Edge, cavity and aperture tones at very low mach numbers. J. Fluid Mech., 330:61– 84, 1997. 7. H. Illy. Contrôle de l’écoulement au-dessus d’une cavité en régime transsonique. Ph.D. thesis, ´ Ecole Centrale de Lyon, 2005. 8. H. Illy, P. Geffroy, and L. Jacquin. Control of cavity flow by means of a spanwise cylinder. 21st ICTAM, Warsaw, 2004. 9. A. G. Kravchenko and P. Moin. Numerical studies of flow over a circular cylinder at ReD D 3900. Phys. Fluids, 12(2):403–417, 2000. ´ 10. L. Larchevêque. Simulation des Grandes Echelles de l’écoulement au-dessus d’une cavité. Ph.D. thesis, Université de Paris VI-Pierre et Marie Curie, 2003. 11. L. Larchevêque, P. Sagaut, T.-H. Lê, and P. Comte. Large-eddy simulation of a compressible flow in three-dimensional open cavity at high Reynolds number. J. Fluid Mech., 516:265–301, 2004. 12. L. Larchevêque, P. Sagaut, I. Mary, O. Labbe, and P. Comte. Large-eddy simulation of a compressible flow past a deep cavity. Phys. Fluids, 15(1):193–210, 2003. 13. E. Lenormand, P. Sagaut, L. Ta Phuoc, and P. Comte. Subgrid-scale models for large-eddy simulation of compressible wall bounded flows. AIAA J., 38(8):1340–1350, 2000. 14. T. S. Lund, X. Wu, and K. D. Squires. Generation of turbulent inflow data for spatiallydeveloping boundary layer simulations. J. Comp. Phys, 140(2):233–258, 1998. 15. S. F. McGrath and L. L. Shaw. Active control of shallow cavity acoustic resonance. AIAA Paper 96-1949, 1996. 16. A. Powell. On the edgetone. J. Acoust. Soc. Am., 33:395–409, 1961.


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17. D. Rockwell and E. Naudascher. Review: Self-sustained oscillations of flow past cavities. ASME J. Fluids Eng, 100:152–165, 1978. 18. J. E. Rossiter. Wind-tunnel experiments on the flow over rectangular cavities at subsonic and transonic speeds. Aeronautical Research Council Reports and Memoranda, 3438, 1964. 19. C. W. Rowley and D. R. Williams. Dynamics and control of high Reynolds-number flow over open cavities. Ann. Rev. Fluid Mech., 38:251–276, 2006. 20. M. J. Stanek, G. Raman, V. Kibens, J. A. Ross, J. Odedra, and J. W. Peto. Control of cavity resonance through very high frequency forcing. AIAA Paper 2000-1905, 2000. 21. L. S. Ukeiley, M. K. Ponton, J. M. Seiner, and B. Jansen. Suppression of pressure loads in cavity flows. AIAA Paper 2002-0661, 2002.


Passive Drag Control of a Turbulent Wake by Local Disturbances O. Cadot, B. Thiria, and J.-F. Beaudoin

Abstract Global properties such as recirculation bubble, shedding frequency, base pressure and global mode amplitude of a turbulent wake produced by a “D” shape cylinder are studied when local stationary disturbances are placed downstream in the wake (one or two small control cylinders). The control cylinders have for main effect to push away further downstream the maxima of the global mode amplitude (or to increase the formation length). On the other hand and in accordance to the cavity models, it is found that the larger the drag reduction, the larger the size of the recirculation bubble of the “D” cylinder. These results suggest that the efficiency of this passive drag control depends crucially on the role of the global mode instability in the mechanism of the bubble closure. Keywords Global instability Bluff-bodies Drag Incompressible flow

1 Introduction The turbulent drag of 2D bluff bodies in incompressible flow at large Reynolds numbers is one of the oldest problem in fluid mechanics that remains one of the most important, for practical reasons and theoretical interest [1]. The underlying physics can be approached by three major ingredients, the steady potential flow with cavity models (see [2] and references therein), the momentum turbulent diffusion (see [3] and references therein) and the global mode instability (referred later as the BvK instability) corresponding to the well known Kármán street (see [4] and references therein). The connection between these ingredients has always been one of the best hope to edify a complete theory for bluff body wakes. In the case of O. Cadot () and B. Thiria Unité de Mécanique de l’Ecole Nationale Supérieure de Techniques Avancées, Chemin de la Hunière, 91761 Palaiseau cedex, France e-mail: [email protected] J.-F. Beaudoin Department of Research and Innovation, PSA Peugeot-Citroën, 2 route de Gisy, 78943 VélizyVillacoublay, France M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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natural wakes of different cylinder shape, some empirical [5] and semi empirical [6] fruitful approaches allowed to relate global properties of the wake such as the frequency of the vortex shedding (a major ingredient for the BvK instability) to the characteristic size and the pressure of the region of slow fluid motion (major ingredients of the cavity models). Generally, it is found for natural wakes that the lower the drag (or equivalently the higher the base pressure) the lower the shedding frequency and the larger the region of slow fluid motion. However, when wakes are disturbed by the presence of a splitter plate or small secondary cylinder, the opposite can be observed : drag reduction can be associated with an increase of the shedding frequency [7, 8]. In the present work, we study the wake properties of a “D” shape cylinder with one and too local disturbances. The disturbances we consider are small circular cylinders whose positions in space have been optimized to obtain the higher base pressure coefficient of the “D” cylinder (i.e. lower drag). The configuration with one cylinder is similar to cases studied by [9–11].

2 Experimental Set-Up and Specific Arrangement of Control Cylinders The Eiffel type wind tunnel is an open loop air flow facility. The turbulent intensity is less than 0:3% and the homogeneity of the velocity over the 400 400 mm blowing section is ˙0:4%. The wake is produced by a symmetric “D” shape cylinder (see Fig. 1) with a leading face profiled as a semi-circle, and a flat trailing plane at right angles to the flow. The width of the trailing vertical plane is D D 25 mm. The main flow velocity is U0 D 22 m=s, and the Reynolds number of the wake, defined as Re D U0 D= 36;600. The three different flow configurations we studied are shown in Fig. 1. The first one, denoted ]N , is the natural wake. The two others have been obtained after an optimization of the base pressure coefficient Cpb of the “D” cylinder. For the natural wake, the base pressure coefficient is 0:56. For the second configuration [12] (denoted ]1), a small circular cylinder of diameter d=D D 0:12 is placed downstream. This control cylinder is a local and steady disturbance to the natural wake of the “D” cylinder. We measured the base pressure versus the position of the control cylinder. The contour map of the isolines of Cpb is reported in Fig. 2(a). Whatever the position, the base pressure is always increased (which is equivalent to a drag reduction), we also find that the wake frequency is always increased. The maxima of base pressure corresponds to maxima of frequency. For configuration ]1 we chose to fix the control cylinder at xC=D D 0:5, yC=D D 0:6 where the base pressure coefficient is high, about 0:33. For the third configuration, another small circular cylinder (identical to the previous one) is placed downstream the previous one. Now the second control cylinder is a local disturbance of the previous configuration ]1. We measured again the base pressure of the “D” cylinder versus the position of the control cylinder. The contour map of the isolines of Cpb is reported in Fig. 2(b). Again, we

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Fig. 1 The three different flow configurations. ]N is the natural wake, ]1 is with one control cylinder and ]2 with two control cylinders. The control cylinder labelled (1) is located at xC=D D 0:5, yC=D D 0:6. The control cylinder labelled (2) is located at xC=D D 1:28, yC=D D 0:45. Arrows represent the mean velocity field, the background grayscale is the x-component of the mean velocity field. The white regions correspond to a zero of , they enclose regions of negative representative to the back flow. The crosses indicate the positions (xG=D, yG=D) of the global mode maxima measured from the Fig. 4

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find that whatever the disturbance position, the base pressure is always increased, but now the frequency is always decreased compared to that of configuration ]1. The maxima of the base pressure corresponds to minima of frequency. We chose to place the second control cylinder at the position xC =D D 1:28, yC =D D 0:45 for which a large base pressure coefficient of Cpb D 0:23 is obtained. It is worth noticing that if there is only one control cylinder in the wake at this location (see Fig. 2(a)), the Cpb would have increased only to 0:51. The experimental arrangement for the configurations ]N; ]1; ]2 and dimensions are depicted in Fig. 1. The pressure distribution around the body is given for the three configurations in Fig. 3(a). For the Reynolds number considered, the viscous stresses are negligible compared to that of the pressure. Since the pressure distribution are equivalent for the three configu-


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Fig. 3 (a): pressure coefficient distribution around the main body for the three configurations. (b): amplitude spectrum of the velocity measured with the hot wire probe for the three configurations. Each spectrum corresponds to the locations (measured from Fig. 4 and localized by crosses in Fig. 1) in the wake where the amplitude at the fundamental frequency of the BvK instability is maximum Table 1 Main characteristics of the wake for the three configurations Config. ]N ]1 ]2

Cpb (˙0.02) 0.56 0.33 0.23

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rations at the front of the body, the larger the pressure deficit at the rear of the body, the larger the drag exerted on the bluff body. We can then attest for a significant drag reduction, the drag in ]2 is lower than ]1 that is lower than the natural case ]N . The main wake characteristics for the three configurations are listed in Table 1. The velocity field is investigated with a PIV set-up and a hot wire probe. In each case, measurements are performed in the plane xOy (see Fig. 1 and concern the u and v components of the velocity field only. For the hot wire measurements, the wire is orientedpsuch a way to be essentially sensitive to the modulus u2 C v2 . We will call U D u2 C v2 , the velocity measured by the wire probe. At each point in space (the resolution is ıx=D D 0:24 and ıy=D D 0:16), the probe signal is recorded during 120 s at a sampling frequency of 5 kHz.

3 Results The mean flows, obtained by averaging in time the PIV velocity fields, are shown in Fig. 1. Comparing the vector fields, We can see that the mean momentum deficit in the wake is increased by the presence of the control cylinders. From the pressure distribution measurements for each configuration (Fig. 3b, we observe that the larger the deficit momentum the lower Cpb (or equivalently the drag). In the close wake,

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the drag is related to the strong pressure gradient and not to the momentum deficit as it is the case in the far wake [3]. The characteristic length of the recirculation bubble, revealed by the region of negative component in Fig. 1 is increased as the drag is reduced. Both observations about the momentum deficit and the bubble length are qualitatively consistent with the cavity models [2], [13]. In this model, the larger the dead zone region (i.e. the region of the slow fluid motion), the lower the Cpb or equivalently the drag. The recirculation bubble that becomes asymmetric with one control cylinder split into two parts with two control cylinders. Spectral measurements in the wake indicate that the flow is synchronized for the three configurations. Whatever the position in the wake, we find a clear global frequency whose magnitudes are 211 Hz for the natural wake, 220 Hz with one control cylinder and 186 Hz for two control cylinders (see spectra in Fig. 3b). The corresponding Strouhal numbers based on the characteristic length D are displayed in Table 1. We extracted the amplitude of the peak corresponding to the fundamental frequency (mode 1) and to the harmonic (mode 2). This peak’s amplitude measured in space, say Af0 .x; y/, defines the global mode envelope of the mode 1 [14]. This envelope is characteristic to the underlying global instability [4]. The global mode envelope for the fundamental frequency and its harmonic are shown in Fig. 4. ]N For the natural wake ]N , two maxima of magnitudes about Amax D 2:86 m=s are found very close to the rear of the bluff-body at xG =D D 0:6D and symmetrically located at yG =D D ˙0:42D (Fig. 4a). We can see in Fig. 1 that the global mode maxima (found in Fig. 4a and displayed by the crosses in Fig. 1 for configuration ]N ) are situated upstream the closure of the re-circulation bubble. For the controlled wake in configuration ]1, the maxima are significantly damped by a fac]1 tor 2, their magnitudes are now about Amax D 1:36 m=s and located much further downstream in the wake at xG =D D 2:8. Now, the maxima are situated downstream the closure of the re-circulation bubble (see Fig. 1). Their positions are still rather symmetrically located as for the natural wake at positions yG =D D ˙0:42D. With the second control cylinder, the maxima are again shifted downstream at about xG =D D 4:25 and further downstream the closure of the recirculation bubbles (see Fig. 1). The positions are still rather symmetrically located but their separating distance is increased compared to cases ]N and ]1 since now yG =D D ˙0:5D. The magnitude of the maxima is increased compared to that of the case with one ]2 cylinder, Amax D 1:78 m=s but remains lower than the natural case. The study of the global mode envelope at the harmonic frequency in Fig. 4b leads to same conclusions.

4 Discussions and Conclusion There are only few attempts [9, 10] to understand the effect of drag reduction due to a small control cylinder placed in the wake of a larger cylinder. From our point of view, the first control cylinder in configuration ]1 “forces” the shear layer to roll up further downstream. This is a similar effect to that observed in the case of “wake in-

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Fig. 4 Pressure coefficient distribution around the main body for the three configurations

terference” in [6]. The consequence is an increase of the recirculation bubble and, in accordance to the cavity model [13], a higher base pressure (or equivalently a lower drag). With the second cylinder, the recirculation bubble split into two parts, but the region of slow fluid motion is larger than the previous cases. The drag reduction mechanism seems to be again in agreement with the cavity model. Although the relationship between the size of the slow fluid region and the base pressure coefficient seems rather clear, the corresponding modification of the Bvk instability deserves attention. As the region of slow fluid motion increases, the maxima of the global mode are shifted downstream (it is similar to an increase of the

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formation length as defined in [5]). Both effects are consistent and related to a delay in the instability. However the frequency modification are opposite (see Table 1). There is then no direct relation between the base pressure and the Bvk instability. It is worth noticing that a Bvk global mode instability coupling too shear layers can be obtained without any low pressure between them [15]. However, it seems obvious that the global mode plays an important role in the closure of the region of slow fluid motion (and then to the drag) for the natural wake. Actually, in that case the maxima of the global mode occur before the closure (see Fig. 1). For the two other configurations, with passive control, the maxima occur much further downstream the closure of the recirculation bubble (see Fig. 1) and one may wonder whether the global mode is the only relevant mechanism for the closure. Actually, the opposite shears produced by the “D” cylinder are strongly disturbed by the control cylinders implying flow deviations and increase of the turbulent mixing. Both effect could be responsible for the closure. In conclusion, since the main effect of this passive control with small cylinders is to shift downstream the global mode envelop, our results suggest that its efficiency to reduce drag depends on the role of the global mode in the closure mechanism of the recirculation bubble. Another property that we would like to study is the reversed behavior between the wake frequency and the global mode amplitude. We observed that an increase of the frequency is always associated to a decrease of the amplitude and vice versa with no exception (whatever the position and the number of disturbances in the wake). The effect is certainly a manifestation of the budget equation of the flux of circulation per unit of length primarily created by the “D” shape cylinder.

References 1. ROSHKO , A. 1993 Perspectives on bluff body aerodynamics. J. Fluid. Wind Eng. Ind. Aero. 49, 79–100. 2. W U Y.T. 1978 Cavity and wake flows. Ann. Rev. Fluid. Mech. 4, 243–284. 3. P OPE S.B. 2000 Turbulen flows. Cambridge University Press Cambridge. 4. C HOMAZ , J.-M. 2005 Global instabilities in spatially developing flows: non normality and nonlinearity. Ann. Rev. Fluid. Mech. 37, 357–392. 5. G ERRARD , J.H. 1966 The mechanics of the formation region of vortices behind bluff bodies J. Fluid. Mech. 25, 401–413. 6. ROSHKO , A. 1954 On the drag and shedding frequency of 2D bluff bodies. NACA Tech. Note 3169. 7. B EARMAN P.W., 1965 Investigation of the flow behind a two-dimensionnal model with a blunt trailing edge and fitted splitter plates J. Fluid. Mech. 21, 241. 8. S AKAMOTO H., TAN K. & H ANIU , H. 1991 An optimum suppression of fluid forces by controlling a shear layer separated from a square prism. J. Fluid. Eng. 113, 183–189. 9. S REENIVASAN , K. R. & S TRYKOWSKI , P.J. 1990 On the formation and suppression of vortex shedding at low Reynolds number. J. Fluid. Mech. 218, 71–108. 10. M ITTAL , S. & R AGHUVANSHI , A. 2001 Control of vortex shedding behind circular cylinder for flows at low Reynolds numbers. Int. J. Numer. Mech. Fluids 35, 421–447. 11. DALTON , C., X U , Y. & OWEN , J.C. 2001 The suppression of lift on a circular cylinder due to vortex shedding at moderate Reynolds numbers. J. Fluids Struc. 15, 617–628.


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12. T HIRIA B., B EAUDOIN J.-F., AND C ADOT O. 2009 Passive drag control of a blunt trailing edge cylinder. Journal of Fluids and Structures, doi:10.1016/j.j?uidstructs.2008.07.008 13. R IABOUCHINSKY D. 1920 On steady fluid motions with free surface. Proc. London Math. Soc. Ser. 2 19, 206–215. 14. Z IELINSKA , B. & W ESFREID , J.E. 1995 On the spatial structure of global modes in wake flow. Phys. Fluids 7, 1418. 15. A BERNNATHY, F. H. & K RONAUER , R.E. 1962 The formation of vortex streets. J. Fluid. Mech. 13, 1–20.


Active Control of a Cylinder Wake Using Surface Plasma T. Jukes and K.-S. Choi

Abstract An experimental investigation has been undertaken using high-speed Particle Image Velocimetry to study the possibility of controlling the global flow field in the near wake of a circular cylinder at Re D 6;500. Surface plasma actuators were mounted at strategic locations around the cylinder (both fore and aft of the separation point) and used for flow control by producing a body force close to the wall. It was found that the plasma can significantly alter the vortex shedding in the wake of the cylinder, with effectiveness depending upon the actuator location and forcing frequency. The most dramatic effects were observed when the plasma was located very close to the natural laminar separation point. Here, amplification of the shedding was observed when the plasma was excited at the natural vortex shedding frequency .Stf 0:2; StK D 0:206/. This was accompanied by periodic flow reattachment to at least the rearward stagnation point. At higher forcing frequency .Stf 0:8/, the plasma completely suppressed the vortex shedding process which lead to a short and narrow wake, reduced turbulence intensity, and 60% reduction in the wake momentum thickness. At still higher frequency .Stf 2:0; StSL D 1:7/, only the shear layers were excited and the vortex street remained unaltered. Keywords Flow control Surface plasma Circular cylinder PIV

1 Introduction Flow around circular cylinders is one of the oldest problems in fluid mechanics and has been extensively studied since the late nineteenth century due to its engineering significance and relative simplicity in experiment. Many review papers have been written on the subject (see, for example, Williamson [17]). The focus of these experiments are within the Transition-in-Shear-Layer (TrSL or subcritical)

T. Jukes () and K.-S. Choi University of Nottingham, Nottingham, NG7 2RD, UK e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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state of flow [18], which was first examined in detail by Bloor and Gerrard [2]. Within this flow regime, the shear layers initially develop 2D transition waves .400 < Re < 1–2k/, which then roll-up into discrete vortices that feed into the Kármán vortex street .2k < Re < 20–40k/. Here Re D U1 d=$ is the Reynolds number, where U1 is the free-stream velocity, d is the cylinder diameter and $ is the kinematic viscosity of the fluid. Prasad and Williamson [13] showed that the transition vortices occur at a frequency (normalised with the primary vortex shedding frequency) that varies with Re0:67 . An effective method for controlling the cylinder wake is by rotational oscillation of the cylinder about its axis. Tokumaru and Dimotakis [16] studied the flow at Re D 1:5 104 using a maximum rotational speed, ˝ D Vr;max=U1 , with a forcing Strouhal number, Stf D f d=U1 , in the range 0 ˝ 16 and 0:17 Stf 3:3, where f is the oscillation frequency. Four vortex shedding modes were identified for different values of Stf . When the forcing frequency was similar to the natural shedding frequency .Stf D 0:2/, two vortices of the same sign were released per half cycle. At higher Stf (0.2–1), the wake structure became synchronised to the rotational oscillation frequency. At 1:1 Stf 1:5, the near-wake structure was synchronised but became unstable and developed into a vortex street with lower spatial frequency downstream. For Stf > 2, the oscillation only affected the shear layers near the cylinder, with a largely undisturbed vortex street. A broad minimum in drag occurred when Stf > 0:8, due to a delay in mean separation and a thinning of the wake. Similar results were reported using DNS [4, 5], although it was noted that the range for eddy shedding lock-on became narrower as ˝ was reduced. Smallamplitude oscillations have also been used .0:005 ˝ 0:03/, which were very effective at exciting the shear layer instability for Stf > 0:6 [8]. Surface plasma is an emerging technique for flow control due to its unique ability to create a body force close to the wall in atmospheric pressure air. The actuators are simple, lightweight, require no moving parts and are extremely fast acting. Plasma actuators have been observed to create laminar wall jets accompanied by initiation vortices [9], and have been used to reduce the skin friction drag in turbulent boundary layers by up to 45% [10]. Enloe et al. [7] provide details of the plasma formation and Corke and Post [6] presented details of the body force produced by the plasma. Plasma actuators have proved successful in locking-on the vortex shedding to the forcing frequency over a circular cylinder [11, 12], and synchronising the shedding from two side-by-side cylinders [1]. In addition, Thomas et al. [15] observed that the vortex street could be suppressed when Stf 1. The wake turbulence was dramatically reduced and a very short, tapered wake region occurred which was much thinner than the canonical case. The aim of this study is to explore the potential of plasma actuators for dynamic separation control over a circular cylinder. High frame-rate PIV is used to study the dynamics of the near-wake at Re D 6;500. Actuators are placed at several azimuthal locations both fore and aft of the separation point .˙70ı ; 89ı ; 100ı ; 130ı / with periodic forcing in the range 0:2 Stf 4:0. Top and bottom actuators are operated in unison and alternately, with both constant forcing duration and constant duty cycle throughout the Stf range.

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2 Experimental Arrangement Experiments were conducted in a low-speed open return wind tunnel with test section of dimensions 1:5 0:3 0:3 m. The circular cylinder consisted of a hollow acrylic tube .d D 50 mm/ spanning the test section 0.4 m downstream of the contraction. Endplates were fitted to the cylinder which extended 5d in the streamwise direction .1:5 < x=d < 3:5/ and 3d in the cross-flow direction .1:5 < y=d < 1:5/, where the origin is located at the centre of the cylinder. All experiments were conducted at U1 D 2 m=s .Re D 6;500/. The plasma actuators consisted of 17 m thick copper electrodes separated by 250 m thick Mylar dielectric (dielectric constant, " D 3:1). Several actuators were photochemcially etched onto a single sheet, which was wrapped around the acrylic tube and bonded in place so that they were located at 30ı intervals (Fig. 1). Wires were soldered to the electrodes and fed outside the wind tunnel so that the actuators could be used independently or in pairs. In order to create the plasma, a bipolar square waveform was delivered to the upper electrodes with frequency, fplasma D 25 kHz, 35% duty cycle, and voltage, Eplasma D ˙3:5 kV. This locally ionized the ambient gas around it, causing plasma to spread out over the surface of the cylinder. This appeared as a light purple glow extending for around 3 mm to the side of the electrode under which the lower electrode was placed. In these experiments, plasma was formed only on the downstream edges of the electrodes which caused a body force to be directed in the downstream direction. Details of the plasma characteristics and power supply can be found in [9]. Measurements of the flow field in the near wake were made using a High Frame-Rate Particle Image Velocimetry (PIV) System from TSI. The system consisted of a PowerView HS-3000 high-speed camera, New Wave Research Pegasus PIV laser (45 W Nd:YLF), TSI 9307-6 Oil Droplet Generator and a dedicated

Fig. 1 Circular cylinder cross-section. All dimensions in mm. Note the height of the upper electrodes, k, is exaggerated (roughness ratio, k=d D 3 104 )

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PC. The laser sheet was aligned along the streamwise direction and at the centreline of the wind tunnel. Olive oil was used to seed the flow with 1 m diameter droplets. Generally, the camera was set to view the cylinder wake in the region 0 < x=d < 2:5; 1 < y=d < 1. Image pairs were then taken at a frame rate of 250 Hz ( t D 300 s between pairs, 10 ns laser pulse). Velocity vectors were computed on a 16 16 pixel grid using a recursive cross-correlation technique.

3 Results 3.1 Effect of Plasma in Still Air The effect of activating a single plasma actuator in still air is shown in Fig. 2. Plasma was activated for a total of 33.2 ms, corresponding to one quarter of the Kármán shedding period, TK , when the flow is present. The plasma creates a starting vortex which travels along the cylinder. Taking the characteristic plasma velocity, Uplasma , as the time-averaged peak velocity in the vortex after initiation .7 t 33 ms/ yields Uplasma =U1 D 0:73 (Fig. 3). Figure 4 shows the total momentum, M , added to the flow by the surface plasma with time. There is a reasonably linear increase, indicating that the plasma produces a constant body force whilst it is present. After the plasma switches off, there is a slow decrease due to viscous dissipation and transport out of the measurement area. Note that a significant portion of the induced flow is outside of the PIV image for t > 24 ms, thus causing the non-linear increase after this time. The slope of the linear region yields the force produced by the plasma per unit width, Fplasma D 4:7 mN=m. Expressed as a ratio of the dynamic force on the cylinder, Cplasma D Fplasma =1=2U1 2 d D 0:041.

Fig. 2 Instantaneous velocity magnitude induced by plasma in still air. Plasma activated for 33 ms at fplasma D 25 kHz; Eplasma D ˙3:5 kV . (a): t D 8 ms. (b): t D 24 ms. (c): t D 56 ms after plasma initiated. Plasma on during frames (a) and (b) with location as depicted in purple


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Fig. 3 Maximum velocity magnitude induced by plasma in still air with time

Fig. 4 Total plasma-induced momentum in measurement region with time

3.2 Flow Field Without Plasma The velocity profile in the wake of the circular cylinder without plasma is shown in Fig. 5. Six seconds of data were taken at 250 Hz (1,500 PIV image pairs), corresponding to approximately 47 vortex shedding cycles. The time-averaged profiles ((a) and (b)) exhibit symmetry along the horizontal axis, showing that the vortex shedding is equal from the top and bottom of the cylinder as expected. The wake region shows two mean recirculating cells centred at x=d D 1:25; y=d D ˙0:25. Downstream of this region there is a sharp increase in velocity fluctuations and mean velocity, indicating the onset of the Kármán vortex street. The position of maximum velocity fluctuations on the streamwise axis marks the length of the eddy formation region, Lf =d D 1:92 [2].

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Fig. 5 Flow field around the circular cylinder without plasma. (a) Time-averaged velocity magnitude over 47 shedding cycles. (b) Total turbulence profile. (c) Instantaneous vorticity profile. (d) Power spectrum based on measurements at 0:95 x=d 1:05; 0:7 y=d 0:5

The roll-up of the upper shear layer into a large-scale Kármán vortex can be seen in the instantaneous vorticity profile in (c). In addition, small-scale roll-ups relating to the shear layer instability can be observed in the lower part of frame with spacing of x=d 0:3. At this Reynolds number .Red D 6;500/, the boundary layers remain laminar up to the separation point but the shear layers roll-up into transition eddies shortly after separation and before forming the Kármán vortex street [18]. Detailed study of the vortex shedding process was presented by Cantwell and Coles [3]. The Kármán shedding frequency and shear layer roll-up frequency were measured by taking the average velocity in the region 0:95 x=d 1:05; 0:5 y=d 0:7 (depicted in (c)). The Kármán shedding frequency, fK , was clearly shown in the U -component time-trace since the shear layers waver back and forth throughout this region during the shedding cycle. The shear layer instability frequency, fSL , was better shown in the V -component trace because the roll-ups rapidly switch from positive to negative V as the vortices pass through. The energy spectra for the two signals are shown in (d). Both show a peak at fK D 7:8 Hz, corresponding to StK D 0:206. This compares very well to the data set of Roshko [14]. The V -velocity component spectrum shows a peak corresponding to fSL D 64:9 Hz .StSL D 1:71/, such that fSL =fK D 8:3, agreeing well with Prasad and Williamson [13].


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3.3 Effect of Plasma on the Cylinder Wake The effect of Stf on the wake behaviour was studied using two plasma actuators at a circumferential angle, D ˙89ı . The plasma extended for an angle of approximately 7ı , such that forcing occurs in the region 89ı 96ı . The profile of the body force throughout this region is not known, although it is expected to be strongest at the edge of the exposed electrode and decrease with . The two plasma actuators were either operated in phase (simultaneously) or 180ı out of phase (oscillatory). In the oscillatory mode, the forcing frequency is defined as the reciprocal of the time between two occurrences of an electrode firing. Hence, exactly the same energy is expended for oscillatory and simultaneous modes at the same Stf . The pulsing was performed with both fixed plasma duration of 3 ms .0:025TK / and fixed duty cycle of 25%. Figure 6 shows the velocity magnitude, turbulence level, and instantaneous vorticity profile at values of Stf for which four different wake regimes were observed. Firstly (Case A), the plasma causes two large-scale vortices to form on the top and bottom sides of the cylinder at the same time. This only occurred when the plasma actuators are fired together at Stf D 0:2 (Fig. 6a). The two vortices are released together from the back of the cylinder and travel downstream as a counter-rotating vortex pair. This process occurs once every time the plasma fires (i.e. at the Kármán shedding frequency) and note that the vortices form much closer to the cylinder than without plasma (Fig. 5c). The vortex pairs interact and mix chaotically downstream. A quite different mode of shedding was observed when the plasma actuators were energised simultaneously at Stf D 0:4, or when oscillated at Stf D 0:2. In this second mode, Case B, the wake becomes hugely amplified with a vortex street that wavers dramatically back and forth. The amplification can be clearly seen in the diverging mean velocity field and increased velocity fluctuations in Fig. 6b. As with Case A, large scale vortices are formed quite close to the rear of the body, such that the formation region virtually vanishes. However, the vortices are now shed alternately from the lower and upper surfaces. Such a vortex can be seen in the instantaneous vorticity field, where a large-scale vortex shed from the top surface protrudes right across the wake. These vortices protrude to at least y=d D 1 within the measurement region, whilst it was rare to see a vortex cross beyond y=d D 0:5 without plasma. Also in Fig. 6b, it can be seen that the shear layer from the lower surface remains attached until quite close to the separation point on the opposite side of the cylinder. This amplified wake is expected to lead to rapidly fluctuating aerodynamic forces. Similar observation have been made by rotating a cylinder at Stf D 0:2 [8]. For 0:8 Stf 2:0, a third mode of shedding is observed (Case C). In this regime, the Kármán vortex street is completely suppressed and a vortex is released from the top and bottom of the cylinder each time the plasma is activated. Thus, the frequency of these eddies increases with fplasma . The eddies are smaller scale than the Kármán vortices and it is therefore likely that the plasma is triggering rollup of the shear layers. The train of vortices travel downstream at a shallow angle towards the centreline and combine after a short distance .x=d 2/. They then

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Fig. 6 Flow field around the cylinder with plasma actuators at D ˙89ı . Time-averaged velocity magnitude after t U1 =d D 80 (t D 2 s, left column), total turbulence profile (middle) and instantaneous vorticity profile at t U1 =d D 40 (right). (a) Electrodes actuated simultaneously at Stf D 0:2 (3 ms, 2% duty). (b) Oscillatory actuation at Stf D 0:2 (3 ms, 2% duty). (c) Simultaneous at Stf D 1:0 (3 ms, 11% duty). (d) Simultaneous at Stf D 2:3 (3 ms, 26% duty)

undergo mutual annihilation. This can be clearly seen in Fig. 6c, where two vortices of opposite sign appear on either side of the wake at the same distance downstream (simultaneous mode). Similar behaviour occurs with oscillatory plasma except that the chain of vortices is staggered. The velocity fluctuations are significantly reduced in this regime and the mean velocity shows a very short, thin and tapered wake region, as also observed by Thomas et al. [15]. Our observations show that the wake becomes less tapered as Stf increases so that the wake is shortest when Stf D 0:8–1:0 (c). It is likely that the drag is at a minimum at these frequencies. At still higher forcing frequency .Stf > 2:0/, the flow enters another flow regime: Case D. Here, the shear layers still roll-up at the plasma forcing frequency and separation is delayed, leading to a thinner wake (Fig. 6d). However, the


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Fig. 7 Wake velocity profile at x=d D 2 with plasma at D ˙89ı for various Stf

shear layers initially travel parallel to the free stream and start to form large-scale structures further downstream with similarity to the natural Kármán vortex street. The formation length is much longer than the canonical case so that the large-scale structures form quite close to the downstream edge of the measurement region. Hence, there is some uncertainty about the exact nature of this regime at present. It seems that at high Stf , the plasma only excites the shear layers near the cylinder with a largely undisturbed vortex street downstream. In order to give a quantitative measure of the effectiveness of the plasma forcing, the momentum thickness of the cylinder wake, wake , was calculated from the timeaveraged streamwise velocity, U , at x=d D 2 (Fig. 7): Z wake D

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Figure 8 shows that wake increased by over 20% in the Case B flow regime .Stf 0:4/. In contrast, a broad plateau was observed within the Case C and D regimes, where wake reduced by over 60% .Stf 0:8/. These are similar findings to those of Tokumaru and Dimotakis [16] for a rotationally oscillating cylinder, although note that the plasma induced velocity is lower than in their study (Uplasma =U1 D 0:73, as compared to Vr;max =U1 D 2) and the plasma only produces an effect in the region 89ı 96ı . The wake behaviour in each regime was very similar regardless of whether the plasma was activated with constant duration or constant duty cycle. This would suggest that the shear layers are very sensitive to small disturbances at this actuator location, so that the flow control can be achieved with low power. This has huge implications for energy saving. For example, at Stf D 0:2 the power consumption is reduced by a factor of 10 between the two forcing modes (2.5% and 25% duty), yet the wake profile and momentum thickness is very similar (Fig. 8).

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Fig. 8 Momentum thickness at x=d D 2 for various Stf ; , and actuation mode

The effect of actuator location is also shown in Fig. 8. When the plasma was located upstream of the natural separation point . D ˙70ı /, Case C behaviour could not be observed at any Stf . At high forcing frequency .Stf > 1/ the shear layers did roll-up at fplasma , but the Kármán vortex street and the wake profile remained similar to those without plasma. However at low forcing frequency, the amplification regime (Case B) was extended to Stf 0:8, reflected by the increase in wake . Thus the plasma can only act to amplify the Kármán vortex shedding when applied upstream of the separation point. At D ˙100ı , the behaviour was similar to that at D ˙89ı except that there was less reduction in wake momentum thickness .Stf 0:8/, suggesting that the plasma was not as effective for flow control at this location. When the actuators were placed still further downstream . D ˙130ı /, very little effect was observed. The plasma only weakly altered the recirculation in the wake region and did not affect the shear layers or the Kármán vortex shedding. The momentum thickness shows canonical values throughout the entire Stf range.

4 Conclusions The near wake of a circular cylinder has been investigated using high-speed PIV in the subcritical regime .Re D 6;500/. The flow was actively controlled using surface plasma actuators at D ˙70–130ı from the front stagnation point. Pulsed actuation was applied in the range 0:2 Stf 4:0, with plasma forcing such that Uplasma =U1 D 0:73 and Cplasma D 0:041. Four flow behaviours were observed, depending on Stf : Case A – Two large scale vortices are shed simultaneously from the upper and lower sides of the cylinder. Stf D 0:2 (actuated simultaneously at D ˙89ı , only). Case B – Amplification of the Kármán vortex street. Stf 0:4.


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Case C – Suppression of the Kármán vortex street. Shear layers roll-up at Stf and merge at x=d 2: 0:8 Stf 2:0. Case D – Shear layers excited at Stf but form large-scale vortex street downstream. Stf > 2:0. The wake became much wider in the Case B regime, with nearly 30% increase in turbulence intensity and 20% increase in wake momentum thickness. In contrast, the wake became much thinner in the Case C regime, where the turbulence intensity was reduced by 50% and the momentum thickness was reduced by over 60%. Flow control was most effective when D ˙89ı and we expect drag to be a minimum when Stf 0:8–1:0. Future work will consist of direct measurement of the dynamic lift and drag forces experienced by the cylinder. Acknowledgements EPSRC Research Grant (EP/D500850/1) and the use of the PIV system from the EPSRC Engineering Instrument Loan Pool are gratefully acknowledged.

References 1. Asghar, A. and Jumper, E. J., Phase Synchronization of Vortex Shedding from Multiple Cylinders Using Plasma Actuators, In: 41st Aerospace Sciences Meeting and Exhibit, Reno, NV (2003) AIAA 2003–1028. 2. Bloor, M. S. and Gerrard, J. H., Measurements of Turbulent Vortices in a Cylinder Wake, Proc. Roy. Soc. A294 (1966) 319–342. 3. Cantwell, B. and Coles, D., An Experimental Study of Entrainment and Transport in the Turbulent Near Wake of a Circular Cylinder, J. Fluid Mech., 136 (1983) 321–374. 4. Cheng, M., Chew, Y. T., and Luo, S. C., Numerical Investigation of a Rotationally Oscillating Cylinder in Mean Flow, J. Fluid Struct., 15 (2001) 981–1007. 5. Choi, S., Choi, H., and Kang, S., Characteristics of Flow Over a Rotationally Oscillating Cylinder at Low Reynolds Number, Phys. Fluids, 14 (8) (2002) 2767–2777. 6. Corke, T. C. and Post, M. L., Overview of Plasma Flow Control: Concepts, Optimization, and Applications, In: 43rd Aerospace Sciences Meeting, Reno, NV (2005) AIAA 2005-0563. 7. Enloe, C. L., McLaughlin, T. E., VanDyken, R. D., Kachner, K. D., Jumper, E. J., and Corke, T. C., Mechanisms and Responses of a Single Dielectric Barrier Plasma Actuator: Plasma Morphology, AIAA J., 42 (3) (2004) 589–594. 8. Filler, J. R., Marston, P. L., and Mih, W. C., Response of the Shear Layers Separating from a Circular Cylinder to Small-Amplitude Rotational Oscillations, J. Fluid Mech., 231 (1991) 481–499. 9. Jukes, T. N., Choi, K.-S., Johnson, G. A., and Scott, S. J., Characterisation of Surface PlasmaInduced Wall Flows Through Velocity and Temperature Measurement, AIAA J., 44 (2006) 794–771. 10. Jukes, T. N., Choi, K.-S., Johnson, G. A., and Scott, S. J., Turbulent Drag Reduction by Surface Plasma Through Spanwise Flow Oscillation, In: 3rd AIAA Flow Control Conference, San Francisco, CA (2006) AIAA 2006–3693. 11. McLaughlin, T. E., Munska, M. D., Vaeth, J. P., Dauwalter, T. E., Goode, J. R., and Siegel, S. G., Plasma-Based Actuators for Cylinder Wake Vortex Control, In: 2nd AIAA Flow Control Conference, Portland, OR (2004) AIAA 2004–2129. 12. Munska, M. D. and McLaughlin, T. E., Circular Cylinder Flow Control Using Plasma Actuators, In: 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV (2005) AIAA 2005–2141.

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13. Prasad, A. and Williamson, C. H. K., The Instability of the Shear Layer Separating from a Bluff Body, J. Fluid Mech., 333 (1997) 375–402. 14. Roshko, A., On the Development of Turbulent Wakes from Vortex Streets, NACA Report 1191 (1954). 15. Thomas, F. O., Kozlov, A., and Corke, T. C., Plasma Actuators for Bluff Body Flow Control, In: 3rd AIAA Flow Control Conference, San Francisco, CA (2006) AIAA 2006–2845. 16. Tokumaru, P. T. and Dimotakis, P. E., Rotary Oscillation Control of a Cylinder Wake, J. Fluid Mech., 224 (1991) 77–90. 17. Williamson, C. H. K., Vortex Dynamics in the Cylinder Wake, Ann. Rev. Fluid. Mech., 28 (1996) 447–539. 18. Zdravkovich, M. M., Flow Around Circular Cylinders. Vol. 1: Fundamentals, Oxford University Press, Oxford (1997).

Active Control of Flow Separation Over an Airfoil Using Synthetic Jets D. You and P. Moin

Abstract We perform large-eddy simulation of turbulent flow separation over an airfoil and evaluate the effectiveness of synthetic jets as a separation control technique. The flow configuration consists of flow over a NACA 0015 airfoil at Reynolds number of 896,000 based on the airfoil chord length and freestream velocity. A small slot across the entire span connected to a cavity inside the airfoil is employed to produce oscillatory synthetic jets. Detailed flow structures inside the synthetic-jet actuator and the synthetic jet/cross-flow interaction are simulated using an unstructured-grid finite-volume large-eddy simulation solver. Simulation results are compared with the experimental data of Gilarranz et al. (J. Fluids Eng. 127, pp. 377–387 (2005)), and qualitative and quantitative agreements are obtained for both uncontrolled and controlled cases. As in the experiment, the present largeeddy simulation confirms that synthetic-jet actuation effectively delays the onset of flow separation and causes a significant increase in the lift coefficient. Modification of the blade boundary layer due to oscillatory blowing and suction and its role in separation control is discussed. Keywords Flow separation Synthetic jets Flow control Large eddy simulation (LES) Airfoil

1 Introduction The performance of an airplane wing has a significant impact on the runway distance, approach speed, climb rate, payload capacity, and operation range, but also on the community noise and emission level as an efficient lift system also reduces thrust requirements (e.g., Ref. [12]). The performance of an airplane wing is often D. You () Department of Mechanical Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA P. Moin Center for Turbulence Research, Stanford University, 488 Escondido Mall, Building 500, Stanford, CA 94305, USA M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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degraded by flow separation. Flow separation on an airfoil surface is related to the aerodynamic design of the airfoil profile. However, non-aerodynamic constraints such as material property, manufacturability, and stealth capability in military applications often conflict with the aerodynamic constraints, and either passive or active flow control is required to overcome the difficulty. Passive control devices, for example, vortex generators [5], have proven to be quite effective in delaying flow separation under some conditions. However, they can introduce a drag penalty when the flow does not separate. Over the past several decades various active flow control concepts have been proposed and evaluated to improve the efficiency and stability of lift systems by controlling flow separation. Many of these techniques involve continuous blowing or suction, which can produce effective control but is difficult to apply in real applications. In recent years, control devices involving zero-net-mass-flux oscillatory jets or synthetic jets have shown good feasibility for industrial applications and effectiveness in controlling flow separation (e.g., Refs. [3, 10, 13]). The application of synthetic jets to flow separation control is based on their ability to stabilize the boundary layer by adding/removing momentum to/from the boundary layer with the formation of vortical structures. The vortical structures in turn promote boundary layer mixing and hence momentum exchange between the outer and inner parts of the boundary layer. The control performance of the synthetic jets greatly relies on parameters such as the amplitude, frequency, and location of the actuation. Therefore an extensive parametric study is necessary for optimizing the control parameters. For numerical simulations, an accurate prediction, not to mention control, of the flow over an airfoil at a practical Reynolds number is a challenging task. The flow over an airfoil is inherently complex and exhibits a variety of physical phenomena including strong pressure gradients, flow separation, and confluence of boundary layers and wakes (e.g., Refs. [6, 7, 9, 14]). The complex unsteady flow is difficult to compute by traditional computational fluid dynamics (CFD) techniques based on Reynolds-Averaged Navier-Stokes (RANS) equations [11]. For prediction of such unsteady flows, large-eddy simulation (LES) offers the best promise in the foreseeable future because it provides detailed spatial and temporal information regarding a wide range of turbulence scales, which is precisely what is needed to gain better insight into the flow physics of this configuration. Recently, Gilarranz et al. [2] performed an experimental study of flow separation over a NACA 0015 airfoil with synthetic jet control. They reported the flow visualization, mean pressure coefficients, and wake profiles in both controlled and uncontrolled cases. However, the mechanism for separation control and how the boundary layer is modified by the control have not been clearly identified. In the present study we address the issues using large-eddy simulation. An understanding of the control mechanisms is valuable in reducing the effort for optimizing the control parameters. In this study we employ an unstructured-grid LES solver, CDP, to predict turbulent flow separation over an airfoil and its control by synthetic jets, and to understand the control mechanism for separation control. The unstructured-grid capability of the solver allows us to effectively handle the complex flow configuration involving an embedded synthetic-jet actuator and wind-tunnel

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walls. The present LES results are compared to the experimental data [2] in both controlled and uncontrolled cases. The effects of flow control on the boundary layer properties, flow separation, and lift enhancement are discussed.

2 Computational Methodology 2.1 Numerical Method The numerical algorithm and solution methods are described in detail in Refs. [4,8]; the main features of the methodology are summarized here. The spatially filtered incompressible Navier-Stokes equations for resolved scales in LES are @ ij @ @ui @p 1 @ @ui C ui uj D C ; @t @xj @xi Re @xj @xj @xj @ui D 0; @xi

(1) (2)

where ij is the subgrid-scale (SGS) stress tensor modeled by the dynamic Smagorinsky closure [1]. All the coordinate variables, velocity components, and pressure are non-dimensionalized by the airfoil chord length C , the inflow 2 freestream velocity U1 , and U1 , respectively. The time is normalized by C=U1 . The Cartesian velocity components and pressure are stored at the center of the computational elements. A numerical method that emphasizes discrete energy conservation was developed for the above equations on unstructured grids with hybrid, arbitrary elements. Controlling aliasing errors using kinetic energy conservation instead of employing numerical dissipation or filtering has been shown to provide good predictive capability for successful LES [15]. The temporal integration method used to solve the governing equations is based on a fully-implicit fractional-step method that avoids the severe time-step restriction that would occur in the synthetic jet orifice region with an explicit scheme. All terms in (1) and (2) are advanced using a second-order accurate fully-implicit method in time, and are discretized by the second-order central difference in space. A biconjugate gradient stabilized method (BCGSTAB) is used to solve the discretized nonlinear equations. The Poisson equation is solved by an algebraic multigrid method.

2.2 Flow Configuration The flow configuration is shown in Fig. 1. This configuration was experimentally studied by a team at Texas A&M [2]. In the experiment, a NACA 0015 airfoil with a chord length of 375 mm was installed in a wind tunnel. The slot of the actuator

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slip BC 1:37C

y

U∞

x

uniform inflow

convective outflow BC

−1:07C

−3C

3C

CL

Fig. 1 Flow configuration for LES of flow over a NACA 0015 airfoil with synthetic-jet control

® (degrees) Fig. 2 Lift coefficient as a function of angle of attack (˛) measured by Gilarranz et al. [2]. ı, controlled case (f D 1:2U1 =C ); , uncontrolled case

had a width of 2 mm across the entire length of the span and was placed at 12% of the chord measured from the leading edge on the suction side of the airfoil. This location was selected to provide sufficient volume to accommodate the synthetic-jet actuator inside the airfoil. Figure 2 shows the maximum lift coefficient measured in the experiment [2] as a function of angle of attack (˛) in both the uncontrolled and controlled cases. The use of the synthetic-jet actuator causes a dramatic increase in the maximum lift coefficient when the baseline (uncontrolled) flow separates. In the experiment, it was found that the angle of attack for which stall occurs is increased from 12ı for an uncontrolled airfoil to approximately 18ı for the controlled case. For the synthetic-jet actuation, the frequency of the actuation in the range of 60 130 Hz (or fC =U1 D 0:65 1:40) does not seem to have a significant effect on the


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maximum lift coefficient. Figure 2 indicates that the uncontrolled airfoil first suffers from a docile stall, which is also referred to as a trailing-edge stall when the angle of attack reaches approximately 12ı . The separation point gradually moves upstream as the angle of attack increases. The leading-edge stall at approximately 19ı produces an abrupt change in the lift coefficient. With the synthetic-jet actuation, the docile stall is effectively controlled and produces further enhanced lift coefficient up to the attack angle of approximately 18ı . For an angle of attack greater than 18ı , the controlled airfoil also suffers from a sharp drop of the lift coefficient due to the leading-edge stall, which is characterized by the formation of a separation bubble near the leading edge. Even after the massive stall (leading-edge stall) occurs, the synthetic-jet actuation increases the maximum lift coefficient compared to the uncontrolled case, but the amount of the lift augmentation is relatively small. The present study focuses on cases with the angle of attack of 16:6ı , where flow separates from the mid-chord location of the airfoil in the uncontrolled case, and the control effect is most remarkable. For this angle of attack, experimental data such as the mean surface pressure coefficients and wake profiles are available for comparison [2]. The computational domain is of size Lx Ly Lz D 6C 2:44C 0:2C . In the present LES, a smaller domain size than that in the experiment is employed in the spanwise direction to reduce the computational cost. The Reynolds number of this flow is 8:96 105 , based on the airfoil chord and inflow freestream velocity. In this study, it is important to precisely predict the flow through the synthetic-jet actuator because the directional variation of the jets during the oscillatory period greatly affects the boundary layer. Therefore, in the present study, the flow inside the actuator and resulting synthetic jets are simulated along with the external flowfield using an unstructured-grid capability of the present LES solver. Figure 3a shows the synthetic-jet actuator modeled with an unstructured mesh. In the experiment, a piston engine is utilized to generate a sinusoidal mass flux and generates synthetic jets through the spanwise cavity slot. To mimic the oscillatory motion of a piston engine in the experiment, we apply sinusoidal velocity boundary conditions to a a

b

velocity BC

Fig. 3 (a) Computational mesh and (b) instantaneous spanwise vorticity contours inside and around the synthetic-jet actuator. Twenty contour levels in the range of 50 60 are shown

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cavity side wall as shown in Fig. 3b. Figure 3b shows the spanwise vorticity contours representing flow inside the cavity and the interaction between synthetic jets and boundary layer flow. The frequency of the oscillation of the cavity side wall is f D 1:284U1 =C , which corresponds to 120 Hz in the experiment of Gilarranz et al. [2]; the peak bulk jet velocity at the cavity exit nozzle is Umax D 2:14U1 . The same momentum coefficient as in the experiment is produced as: C D

2 h.Umax / sin j D 1:23 102 ; 2 / C.U1

(3)

where h and j .D30:2ı / are the width of the cavity nozzle exit and the jet angle with respect to the airfoil surface. No-stress boundary conditions are applied along the top and bottom of the wind tunnel, and no-slip boundary conditions are applied on the airfoil surface and cavity wall. Periodic boundary conditions are used along the spanwise (z) direction. At the exit boundary, the convective boundary condition is applied, with the convection speed determined by the streamwise velocity averaged across the exit plane. Two different mesh sizes of approximately 8 and 15 million cells have been employed while the results obtained with 8 million cells are presented in this paper. A total of 24 mesh points are allocated along the cavity slot. The grid spacings are distributed such that the resolution in the streamwise, wall-normal, and spanwise directions is less than 60, 1.2 and 16.2 wall-units, respectively. The simulation is advanced in time with a maximum Courant-Friedrichs-Lewy (CFL) number equal to 3.5, which corresponds to tU1 =C 1:7 104 , and each time step requires a wallclock time of approximately 15 s when 128 CPUs of the ASC Linux Cluster (2.4 GHz Intel Pentium 4 Prestonia) are used. The present results are obtained by integrating the governing equations over an interval of approximately 20C =U1.

3 Results and Discussion Gross features of the flow over uncontrolled and controlled airfoils are revealed in Fig. 4, showing iso-surfaces of the instantaneous vorticity magnitude overlapped with pressure contours predicted by the present LES. The vortical structures present over the suction surface qualitatively indicate the degree of flow separation. In the uncontrolled case (Fig. 4a), flow massively separates from the half aft portion of the suction surface while the flow separation is dramatically prevented with the synthetic-jet actuation in the controlled case (Fig. 4b). Qualitatively, these features are consistent with the change in the experimentally measured maximum lift coefficient [2] with flow control (see Fig. 2). The pressure distributions over the airfoil surfaces in both uncontrolled and controlled cases are compared with the experimental data in Fig. 5. In general, the present LES shows favorable agreement with experimental measurements in both


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a

b

−Cp

Fig. 4 Iso-surfaces of the instantaneous vorticity magnitude (jjC=U1 ) of 40 overlapped with the pressure contours. (a) Uncontrolled case; (b) controlled case

x=C Fig. 5 Mean pressure distribution over the airfoil surface. Solid line, controlled case; dashed line, uncontrolled case; symbols, experimental data [2]

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cases. The pressure distribution directly indicates the effect of synthetic jets on flow separation. As seen in Fig. 5, most of the lift enhancement is achieved in the upstream portion of the airfoil suction surface, while the control effect of synthetic jets on the pressure distribution in the pressure surface is negligible. The lift and drag coefficients predicted by the present LES in the uncontrolled and controlled cases are in excellent agreement with the experimental data [2] as shown in Table 1. The present synthetic-jet actuation with the momentum coefficient of 1.23% produces more than a 70% increase in the lift coefficient. The drag coefficient is found to decrease approximately 15% 18% with the synthetic-jet actuation. The drag reduction due to the synthetic-jet actuation is also indicated by the wake profiles. Figure 6 shows the mean streamwise velocity profiles in the uncontrolled ( ) and controlled ( ) cases in a downstream location at x=C D 1:2. The width of the wake and the peak magnitude of velocity deficit decrease with synthetic jet control. The present wake profiles are in favorable agreement with experimental data [2] in both uncontrolled and controlled cases. Both the suction and blowing phases modify the boundary layer on the suction surface of the airfoil. The synthetic-jet actuation not only stabilizes the boundary layer either by adding/removing the momentum to/from the boundary layer, but also Table 1 Summary of lift and drag coefficients

Case

Controlled CL CD 1:43 0:23 1:41 0:22

U=U∞

Present LES Experiment [2]

Uncontrolled CL CD 0:83 0:28 0:82 0:26

y=C Fig. 6 Mean streamwise velocity profiles at x=C D 1:2. Solid line, controlled case; dashed line, uncontrolled case; symbols, experimental data [2]


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a

b

c Fig. 7 Mean streamlines overlapped by the mean pressure contours. (a) .1=4/T (suction phase); (b) .2=4/T ; (c) .3=4/T (blowing phase), where T denotes the period of synthetic-jet actuation

enhances mixing between inner and outer parts of the boundary layer. The change of the blade boundary layer during a period of synthetic-jet actuation is shown in Fig. 7 in terms of the phase-averaged streamlines. In the suction phase (Fig. 7a) the low momentum flow in the upstream boundary layer is removed by the suction and prevents downstream flow separation. On the other hand, synthetic-jet blowing (Fig. 7c) energizes the downstream boundary layer and prevents downstream flow separation. The modification of the boundary layer in the upstream (x=C D 0:11) and downstream (x=C D 0:16) proximity to the exit slot of the synthetic-jet actuator (x=C D 0:12) is shown in Fig. 8. Compared to the velocity profile in the uncontrolled case (ı), in the blowing phase (Fig. 8a), the downstream velocity profile becomes fuller due to additional momentum while the modification of the upstream velocity profile is not noticeable. On the other hand, in the suction phase (Fig. 8b), the thickness of the downstream boundary layer is significantly thinned. Therefore, the downstream flow separation is effectively prevented by the favorable modification of the blade boundary layer in both the blowing and suction phases.

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ujet

blowing

22:5 67:5 112:5 157:5 202:5 247:5 292:5 337:5

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Fig. 8 Profiles of the phase-averaged streamwise velocity. (a) blowing phase: B2; , B3; , B4; (b) suction phase: , S1; , S2; ı, uncontrolled case. The cavity slot is located at x=C D 0:12

, B1; , S3;

, , S4.

4 Conclusions We have performed large-eddy simulation of turbulent flow separation over an airfoil with synthetic-jet control. Detailed flow structures inside a synthetic-jet actuator and the synthetic jet/cross-flow interaction have been simulated using an unstructured-grid finite-volume large-eddy simulation solver. Simulation results show favorable agreements with experimental data in terms of mean pressure co-


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efficients and wake profiles for both uncontrolled and controlled cases. For a docile stall, synthetic-jet actuation has been found to stabilize the blade boundary layer and effectively delay the onset of flow separation and cause a significant increase in the lift coefficient. Acknowledgements The authors gratefully acknowledge support from Boeing company and valuable discussions with Dr. Arvin Shmilovich. The authors are also grateful to Dr. Frank Ham for his help with the unstructured LES solver CDP.

References 1. M. Germano, Ugo Piomelli, Parviz Moin, and W. H. Cabot. A dynamic subgrid-scale eddyviscosity model. Physics of Fluids (A), 3(7):1760–1765, 1991. 2. J. L. Gilarranz, L. W. Traub, and O. K. Rediniotis. A new class of synthetic jet actuators - part II: application to flow separation control. Journal of Fluids Engineering, 127:377–387, 2005. 3. Ari Glezer and Michael Amitay. Synthetic jets. Annual Review of Fluid Mechanics, 34:503– 529, 2002. 4. F. Ham and G. Iaccarino. Energy conservation in collocated discretization schemes on unstructured meshes. Annual Research Briefs, 3-14, Center for Turbulence Research, Stanford, California, 2004. 5. A. Jirasek. A vortex generator model and its application to flow control. AIAA Paper 20044965, 2004. 6. M. R. Khorrami, M. E. Berkman, and M. Choudhari. Unsteady flow computations of a slat with blunt trailing edge. AIAA Journal, 38(11):2050–2058, 2000. 7. M. R. Khorrami, B. A. Singer, and R. H. Radeztsky. Reynolds-averaged navier-stokes computations of a flap-side-edge flowfield. AIAA Journal, 37(1):14–22, 1999. 8. K. Mahesh, G. Constantinescu, S. Apte, G. Iaccarino, F. Ham, and P. Moin. Large-eddy simulation of reacting turbulent flows in complex geometries. Journal of Applied Mechanics, 73:374–381, 2006. 9. D. L. Mathias, K. R. Roth, J. C. Ross, S. E. Rogers, and R. M. Cummings. Navier-Stokes analysis of the flow about a flap edge. Journal of Aircraft, 36(6):833–838, 1999. 10. C. L. Rumsey, T. B. Gatski, W. L. Sellers III, V. N. Vatsa, and S. A. Viken. Summary of the 2004 CFD validation workshop on synthetic jets and turbulent separation control. AIAA Paper 2004-2217, June 2004. 11. C. L. Rumsey and S. X. Ying. Prediction of high lift: Review of present CFD capability. Progress in Aerospace Sciences, 38:145–180, 2002. 12. J. J. Thibert, J. Reneaux, F. Moens, and J. Priest. ONERA activities on high lift devices for transport aircraft. Aeronautical Journal, 99:395–411, 1995. 13. I. Wygnanski. The variables affecting the control of separation by periodic excitation. AIAA Paper 2004-2505, 2004. 14. S. X. Ying, F. W. Spaid, C. B. McGinley, and C. L. Rumsey. Investigation of confluent boundary layers in high-lift flows. Journal of Aircraft, 35(3):550–562, 1998. 15. D. You, M. Wang, and P. Moin. Large-eddy simulations of flow over a wall-mounted hump with separation control. AIAA Journal, 44(11):2571–2577, 2006.


Electromagnetic Control of Separation at Hydrofoils G. Mutschke, T. Weier, T. Albrecht, G. Gerbeth, and R. Grundmann

Abstract Lorentz forces originating from surface-mounted actuators of permanent magnets and electrodes in weakly conducting fluids like seawater provide a convenient tool for separation control at hydrofoils. A well-known actuator design is considered which creates a mainly streamwise Lorentz force that is exponentially decaying in wall-normal direction. Separation control by steady forcing at the suction side and by oscillatory forcing near the leading edge of a symmetric foil is investigated numerically, mostly in the post-stall regime. Direct numerical simulations are performed in the laminar flow regime in order to reveal basic control phenomena as well as simulations using turbulence modelling at higher Reynolds numbers which are closer to possible naval applications. Strong enough steady control s capable of suppressing separation completely, and the scaling behaviour of the maximum lift gain CLmax in the turbulent regime is found to agree nicely with experimental results. As oscillatory forcing always has to compete with natural shedding, lock-in behavior is detected, and lift-optimum control at strong control is found in a frequency band around the natural shedding frequency. In terms of the momentum coefficient describing the control effort, appropriate excitation allows for a more effective lift control than steady forcing for small lift gains; for large lift enhancement the effort seems to approach the level of steady control. Keywords Electromagnetic flow control Separation control Wings Numerical simulation Incompressible flow

1 Introduction Separation control is an important issue in many industrial, aviation and marine applications, and a large variety of different control methods does exist [1]. Apart G. Mutschke (), T. Weier, and G. Gerbeth Forschungszentrum Dresden-Rossendorf, MHD Dept., PO Box 510119, 01314 Dresden, Germany T. Albrecht and R. Grundmann Dresden University of Technology, Inst. Aero. Eng., 01062 Dresden, Germany M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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from steady control schemes, active control allows for more distinct benefits in certain flow configurations, as, e.g., proper time-periodic blowing and suction is known to enhance the lift of airfoils quite effectively as compared to steady blowing. Greenblatt and Wygnanski [2] attribute this effect to the periodic excitation of the separating shear-layer, thereby using far-field momentum to advantageously reorganize the vortex shedding process at the suction side. Optimum control frequencies with respect to lift enhancement found there are typically of O.1/ based on chord length and free stream velocity. Wu et al. [3] have performed two-dimensional RANS simulations of the turbulent flow over an airfoil at post-stall angles of attack when periodic blowing-suction near the leading edge is applied. Interestingly, in certain control parameter ranges, the still separated flow became periodic or quasiperiodic, associated with significant lift enhancement. The physical mechanisms are attributed to the non-linear mode competition of the two basic constituents of the flow, the leading-edge shear layer and the vortex shedding from the trailing edge. The present paper is concerned with separation control by applying Lorentz forces into the near-wall region of an hydrofoil by surface setups of electrodes and magnets. We have in mind saltwater or electrolyte flows of weak electric conductivity ( O.10/S=m) where induction effects can be neglected and, besides externally applied magnetic fields, electric currents are fed to the fluid in order to generate Lorentz forces large enough for achieving control. Due to the momentum modification of the near-wall flow, a similarity to suction or blowing does exist. However, an obvious advantage of the Lorentz force approach is that its amplitude is easily adjustable in time by applying alternating currents up to high frequencies. First efforts in applying Lorentz forces to weakly-conducting fluids were undertaken more than 40 years ago [4, 5]. Meanwhile, control of transition in a flat-plate boundary layer [6], turbulent boundary layer control with respect to drag reduction [9] and control of the flow around a circular cylinder [10, 11] have been discussed extensively. Hoarau et al. [16] have investigated the 3-D transition around a NACA-0012 airfoil, and recently, first results on the separation control of flow around hydrofoils were published [12–14]. Although most potential control problems experience turbulent flow conditions, there is also particular interest in the transitional and lowReynolds-number range for, e.g. RPV’s and UAV’s in aviation or for, e.g., active hydrofoils in naval applications. In this paper, we first present DNS results which are limited to relatively low chord Reynolds numbers but aimed to understand basic control effects. Second, results of turbulent simulations are compared with experimental data of [12]. Steady and oscillatory control are investigated in a first step of 2-D simulations as the surface tangential forcing weakens possible 3-D side effects in both laminar and turbulent flows. The basic mechanisms of time-periodic control were recently proven to work in the transitional range as control was achieved by conventional blowing/suction [15] as well as by applying Lorentz forces in experiments [12, 13].

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2 Problem Definition 2.1 Hydrofoil and Actuator p A particular profile named PTL-4 (characteristic polynom: p.x/ D d.a1 xCa2 xC a3 x 2 C a4 x 3 C a5 x 4 , 0 x 1, coefficients: d D 0:1676154, a1 D 1:26854, a2 D 0:292071, a3 D 1:34964, a4 D 0:478002, a5 D 0:104831, ) was chosen to allow later for comparison with experimental results. As can be seen from the left side of Fig. 1, its shape is rather close to that of a standard NACA-0017 profile. Lorentz forces fL result from the cross product of current density in moving media j D .E C u B/. and the magnetic induction B. Neglecting induction and currents due to motion, only the externally applied current density j0 D E0 and the applied magnetic field B0 contribute to fL . A cross-cut sketch of the actuator is shown in the right side of Fig. 1. It consists of an alternating arrangement of stripes of electrodes of varying polarity and permanent magnets of varying magnetization direction (black arrows). The width of both electrodes and permanent magnets is assumed to be equal to a. Surface flush mounted, due to the crossing electric (dashed) and magnetic (solid) field lines, it creates a mainly streamwise (x) Lorentz volume force in the fluid which, when averaging over the spanwise direction z, reads fL D

j0 M0 e a y ex : 8

(1)

Hereby, M0 denotes the magnetization of the magnet, and larger values of the “penetration depth” a lead to Lorentz forces acting deeper inside the fluid. For more details we refer to [6, 17]. Figure 2 illustrates the actuator at the suction side of the hydrofoil for steady (left) and time-periodic (right) separation control.

2.2 Equations and Parameter Based on chord length c of the hydrofoil and freestream velocity U0 , the 2-D NavierStokes equation for an incompressible fluid (r u D 0) reads in dimensionless form @u 1 c C .u r/u D rp C u C N g.y / e a y et : @t Re

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Hereby, y measures the local wall-normal distance, and et denotes the corresponding tangential direction vector along the foil. The shape function g.y / is zero everywhere except above the active actuator range x at the suction side where g.y / D 1 holds; end effects are approximately taken into account by a linear growth or decay of the force amplitude function g.y / in a small transition range at the actuator ends while maintaining the total momentum input. In case of time-periodic forcing, by introducing a nondimensional frequency f based on chord length c and freestream velocity U0 , the shape function above the actuator Q reads g.y ; t/ D cos .2 f t/, f D fUc0 . The two dimensionless characteristic parameters of the problem are the Reynolds number Re D U0 c and the interaction 0 B0 c parameter N D 4 jU 2 describing the ratio of electromagnetic to inertial forces. 0 Here, denotes the kinematic viscosity of the fluid, and M0 D 2 B0 is used assuming infinitely long magnets. In analogy to conventional control by blowing, a momentum coefficient may be introduced which describes the ratio of the total momentum added by the Lorentz force to the dynamic pressure. In case of steady control (where x 1 holds) and oscillatory control (based on the rms value of N ) the momentum coefficient reads 2a a j0 B0 x C D D N; c c 2U02

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Forces acting on the hydrofoil are due to friction and pressure but additionally due to action of the Lorentz force. Based on dynamic pressure and chord length, the F dimensionless total drag and lift coefficients are defined as CD D FUx2 c , CL D Uy2 c 2 0 2 0 where Fx and Fy denote the total force component per spanwise length unit in streamwise and normal direction, respectively. The non-dimensional force input due to the Lorentz force follows from integration over the area above the active part of the actuator (for more details see [14]).


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3 DNS 3.1 Simulation Details A well-established spectral element code was used for the DNS simulations [21] which has already been successfully applied to EMHD problems [7, 8, 10]. More simulation details can be found in [14]. The computational domain with respect to width and position of the hydrofoil was chosen to resemble the test section geometry of an existing experimental facility. Typical grids used consist of about 200 spectral elements and ensure sufficient resolution at boundary and shear-layer regions in the near wake while trying to avoid computational overhead in low-shear regions. The rectangular domain ranges from 1 x 7 and 1:5 y 1:5; the center of gravity of the hydrofoil is located at (x D 0:3, y D 0). The boundary conditions applied in the simulations are no-slip (u D 0, v D 0) at the hydrofoil, freestream (u D 1, v D 0) at inlet, top and bottom of the computational domain and an outflow condition at the outlet. At validation runs, for Reynolds numbers up to Re D 600 under investigation here, the final choice of 9 9 inner element resolution ensured lift and drag accuracy of about ˙1%.

3.2 Stationary Forces Figure 3 shows snapshots of the uncontrolled flow around the hydrofoil at an angle of attack of 30ı and the controlled flow under action of a momentum coefficient of C D 1:61 and a penetration depth of a=c D 0:1265 at a Reynolds number of Re D 500. The flow without control clearly experiences separation, which can be completely suppressed when applying control. Furthermore, at this large value of the momentum coefficient, the controlled flow is almost steady, and a jet on the suction side can already be detected. The left side of Fig. 4 shows the behavior

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of the time-averaged lift coefficient as a function of the momentum coefficient at Re D 600, ˛ D 30ı . Hereby, in the force balance, the momentum input due to the Lorentz force CM grows only linearly with C whereas the total lift coefficient CL seems to grow stronger than linearly. For small values of C , a quadratic dependence due to the separation delay is expected, until the flow is completely attached. However, in our numerical investigation, this might be disturbed by the finite channel width which influences the pressure field. The variation of the penetration depth a=c offers a possiblity for reducing the energy effort needed to achieve some fixed lift gain. The right side of Fig. 4 shows the ratio of the lift CL .C / at momentum coefficient C divided by the uncontrolled lift CL0 versus momentum coefficient C for different values of the penetration depth a=c. For large values of the momentum coefficient, the largest chosen penetration depth performs best. At low control amplitudes, the smallest value of the penetration depth a=c D 0:03163 gives only weak lift enhancement. In general, although the lift-optimum penetration depth depends on the details of the flow configuration, it seems to be advantageous to choose a=c not smaller than a characteristic boundary p layer thickness of the flow which is ılam 1= Re 0:044 in the considered case.

3.3 Oscillatory Forcing Oscillatory actuation was applied at the front part of the hydrofoil, whereby the optimum position of the actuator depends on flow details as, e.g., the position of the separation point. The following results were obtained at a Reynolds number of Re D 500 and an angle of attack of ˛ D 30ı where the uncontrolled flow is already separated (see Fig. 3). Until noted otherwise, an active actuator range of 0:05 x=c 0:15 was chosen, and the penetration depth of the Lorentz force is a=c D 0:1265. Figure 5 summarizes the behaviour of the mean lift and drag coefficient versus actuation frequency f at different values of the momentum coefficient c0 . For the weakest control of c0 D 2:8% applied, there exist distinct maxima for

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both lift and drag coefficient where a weak control influence is sufficient to advantageously modify the process of vortex shedding. Stronger control in general leads to larger gains in lift but also to larger drag penalties. Interestingly, the control frequencies of the lift maxima are close to the shedding frequency of the uncontrolled flow f0 D 0:233 and its harmonics. With growing control amplitude, largest lift and, at the same time, largest drag, is observed more and more in a broad band of frequencies around the natural shedding frequency, and the importance of higher excitation frequencies decays. The lift gain obtained in the broad frequency band mentioned above corresponds to only a modification of the natural vortex shedding by a strong external excitation. Lock-in behavior of the lift coefficient with the external excitation, well-known from oscillatory cylinder control [18–20, 23] occurs in certain frequency ranges near the natural shedding frequency f0 and its harmonics as shown in Fig. 6 for two values of the momentum coefficient. Figure 7 shows streamtraces of the time-averaged flow in case without control and with control at

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c0 D 0:11 for two selected frequencies. As might be also deduced from the size of the separation bubble, control at f D 0:235 gives strongly enhanced lift, although the flow does not attach completely, whereas at f D 0:53 lift enhancement is already smaller. Although lift enhancement is usually coupled to drag penalty, as can be seen from the left side in Fig. 8, the lift to drag ratio can clearly be enhanced by oscillatory control.

3.4 Comparison with Steady Control Figure 8 compares on the right side the lift enhancement obtained at the optimum frequency of f D 0:235 with results obtained by steady control at different values of the penetration depth a=c. As mentioned above, the larger value of the penetration depth a=c is preferable in case of steady control. But, at small values of the momentum coefficient, oscillatory control is clearly more effective than steady control which here achieves only a small amount of separation delay, whereas oscillatory control is already able to reorganize vortex shedding due to a large receptivity of the uncontrolled flow to optimum control. For larger values of the momentum coefficient, steady forcing might perform better as oscillatory forcing can not completely suppress separation.


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4 Turbulent Simulations 4.1 Simulation Details First turbulent simulations were performed by using the commercial finite element code FIDAP [22] and applying the extended k model by Chen and Kim [25]. Geometry details of the rectangular testsection where also a part of the experimental work was performed can be found in [13]; typical grids consist of about 10,000 linear elements with near-wall resolution C 1 for U-RANS simulations. Top and bottom of the domain are no-slip walls; at inflow, freestream of 2% turbulence level was used to force early transition. Although validation runs for a NACA-0015 profile resulted in a 10–15% underestimation of the maximum lift at critical angle CLmax .˛c / due to obvious lacks in turbulence modelling, as will be seen later, there is good quantitative agreement of the scaling behavior of CLmax with experimental results.

4.2 Steady Control Figure 9 shows left lift enhancement and stall delay due to steady control at Re D 800;000 for different values of C . Separation control leads to considerably larger maximum lift values CLmax at larger angles of attack before stall occurs. As for certain applications the maximum lift gain CLmax D CLmax .C / CLmax .C D 0/ as function of C is of importance, on the right side of Fig. 9 the scaling behavior of

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CLmax versus C is shown in comparison to experimental results [12]. Apart from the nice agreement, the slight over–estimation of CLmax obtained numerically as compared to experimental data can be attributed to electrolytic bubble production in the experiment, the simple turbulence model applied and possible 3-D effects. However, the expected strong effect of separation control can not be found as the 1

overall scaling of CLmax is only like C2 . As the electrical power density can j2

4 C2 for steady forcing, the effort for CLmax be estimated as pE 0 U1 4 finally scales as pE U1 . CLmax /4 which makes applications at large Reynolds numbers energetically expensive. However, larger magnetic fields applied would strongly reduce the energy consumption as pE B02 which gives some future perspective.

4.3 Oscillatory Control First results for oscillatory control are shown at the left side of Fig. 10 for an angle of attack ˛ D 17ı and an active actuator range of 0:07 x=c 0:12 at Re D 800;000. As expected, lift-optimum control corresponds to a frequency of f 1. Interestingly, lock-in phenomena as described by Wu [3] were also found. The r.h.s. of Fig. 10 shows first results of the comparison of steady and periodic control where additionally the case ˛ D 21ı with two different actuator locations was investigated for oscillatory control. As can be seen, in case of ˛ D 21ı , for small values of the momentum coefficient C , oscillatory control can be more efficient than steady control, but larger lift gains seem to require similar energy efforts as steady control.

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5 Discussion Control by steady forcing can achieve full reattachement at strong enough control amplitudes but at high cost. Oscillatory control, properly designed with respect to location, frequency and amplitude, can be more efficient for small values of lift enhancement than steady control which can be achieved at small values of the momentum coefficient. Interesting lock-in phenomena were found in both the laminar and the turbulent flow regime which deserve further investigation. Lift-optimum control frequencies as recently noted also in the literature [3,24] were found near the natural shedding frequency of the uncontrolled flow which holds in the laminar and in the turbulent flow regime as oscillatory control always has to struggle with vortex shedding. Currently, more turbulent simulations, also with different modelling, are underway, combined with new PIV measurements of the flow. Acknowledgements We gratefully acknowledge the code PRISM from G.E. Karniadakis group at Brown University and financial support from DFG in frame of Sonderforschungsbereich 609.

References 1. M. Gad-el-Hak, Flow Control, Cambridge University Press, Cambridge, 2000. 2. D. Greenblatt, I.J. Wygnanski, Prog. Aero. Sci. 36 (2000) 487–545. 3. J.-Z. Wu, X.-Y. Lu, A.G. Denny, M. Fan and J.-M. Wu, J. Fluid Mech. 371 (1998) 21–58. 4. E.L. Resler & W.R. Sears, J. Aero. Sci. 25 (1958) 235–245. 5. A. Gailitis, O. Lielausis, Appl. Magnetohydrodyn. Rep. Riga Inst. Phys. 12 (1961) 143–146. 6. T. Albrecht, R. Grundmann, G. Mutschke, G. Gerbeth, Phys. Fluids 18 (2006) 098103. 7. Y. Du and G.E. Karniadakis, Science 288 (2000) 1230–1234. 8. Y. Du, V. Symeonidis and G.E. Karniadakis, J. Fluid Mech. 457 (2002) 1–34. 9. V. Shatrov, G. Gerbeth: Phys. Fluids 19 (2007) 035109 10. O. Posdziech and R. Grundmann, Eur. J. Mech.-B/Fluids 20 (2001) 255–274. 11. S.-J. Kim and C.M. Lee, Exp. Fluids 28 (2000) 252–260. 12. T. Weier, G. Gerbeth, G. Mutschke, O. Lielausis and G. Lammers, Flow, Turbul. Combust. 71 (2003) 5–17. 13. T. Weier and G. Gerbeth, Eur. J. Mech./B - Fluids 23 (2004) 835–849. 14. G. Mutschke, G. Gerbeth, T. Albrecht, R. Grundmann, Eur. J. Mech./B - Fluids 25 (2006) 137–152. 15. D. Greenblatt, I. Wygnanski, J. Aircraft 38 (2001) 190–192. 16. Y. Hoarau, M. Braza, Y. Ventikos, D. Faghani, G. Tzabiras, J. Fluid Mech. 496 (2003) 63–72. 17. T. Weier, G. Gerbeth, G. Mutschke, E. Platacis and O. Lielausis, Exp. Therm. Fluid Sci. 16 (1998) 84–91. 18. G.E. Karniadakis and G.S. Triantafyllou, J. Fluid Mech. 199 (1989) 441–469. 19. S. Choi, H. Choi and S. Kang, Phys. Fluids 14 no. 8 (2002) 2767–2777. 20. P.T. Tokumaru and P.E. Dimotakis, J. Fluid Mech. 224 (1991) 77–90. 21. R.D. Henderson and G.E. Karniadakis, J. Comp. Phys. 122 (1995) 191–217. 22. FIDAP 8.0; Fluent Inc., 2002. 23. S. Mittal, J. Fluids Struct. 15 (2001) 291–326. 24. A. Darabi and I. Wygnanski, J. Fluid Mech. 510 (2004) 105–129. 25. Y.S. Chen, S.W. Kim, NASA CR 179204, 1987.


Active Control of Flows with Trapped Vortices R.M. Kerimbekov and O.R. Tutty

Abstract An approach to developing active control strategies for high Reynolds number flows with trapped vortices is presented. The problem considered is the stabilisation of the vortex in a special cavity on the airfoil using suction as an actuator. The flow dynamics are modelled by the parallel discrete vortex method capable of handling wall irregularities, arbitrary boundary conditions, and turbulence. System identification is performed based on the open-loop analysis with the constant flow rate suction. Feedback control results show that a properly designed linear PI controller prevents the large-scale vortex shedding from the cavity region and reduces considerably flow unsteadiness in the downstream boundary layer. Keywords Trapped vortex Discrete vortex method Feedback control design

1 Introduction Large vortices forming in separated flows over bluff bodies tend to be shed downstream, with new vortices arising in their stead. This results in the increased drag, unsteady loads on the body, and produces an unsteady wake. An alternative flow pattern involves ‘trapped’ vortices which are permanently kept near the body surface. Vortices can be trapped in vortex cells that are special cavities on the airfoil, as shown in the picture. In essence, a trapped vortex reproduces the effect of moving wall, resulting in the postponing or even eliminating flow separation. The idea of trapping a vortex was first suggested (and implemented in flight experiments) by Witold Kasper in the early 1960s. However, soon it became obvious that a proper flow control is required to ensure that the vortex remains stably trapped. For example, in the aircraft EKIP designed by Lev Schukin in 1980–1996, vortices were stabilised with the help of central bodies in the cells and a constant flow R.M. Kerimbekov () and O.R. Tutty AFM Research Group, School of Engineering Sciences, University of Southampton, Southampton, SO17 1BJ, United Kingdom e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

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rate suction (see website http://www.ekip-aviation-concern.com). A more reliable and significantly less power consuming system involving active feedback control was recently proposed by [6]. Somewhat earlier, [1] showed that a simple linear PI controller and backside suction as an actuator can be used in order to stabilise the vortices behind a flat plate oriented perpendicular to the free-stream velocity. This paper is aimed at designing a feedback control system for stabilising real flows with trapped vortices. The flow geometry has been chosen according to the experimental setup provided by the Centro Italiano Ricerche Aerospaziali [5]. We shall consider an airfoil with a cavity mounted on the lower wall of a straight channel, as shown in Fig. 1. The shape of the cavity has been computed at the Politecnico di Torino by [10]. The actuator comprises three suction slots on the upstream cavity wall, and the suction velocity is assumed uniform and normal to the body surface. The flow state is monitored by a sensor situated on the airfoil near the cavity exit (see Fig. 1). The paper is organised as follows. In Section 2 the parallel discrete vortex method (DVM) is introduced as a numerical tool for modelling the flow dynamics. Section 3 performs system identification and explains the procedure for designing a linear PI controller. The results of our work are summarised in Section 4.

2 Numerical Method In order to calculate the flow past an airfoil with a cavity, the two-dimensional discrete vortex method is employed. According to the DVM approach, the Navier– Stokes equations are written in the vorticity/stream function form. The flow field is partitioned into a large number of blobs having a Gaussian distribution of vorticity. The solution is discretised in time, and for each time step the convection and diffusion processes are treated independently. Such an operator-splitting technique was

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first introduced by [2] and is commonly used in the viscous discrete vortex methods. The convection step is governed by the kinematic relation xP j D uŒxj .t/; t, where xj is the position vector of the j -th vortex, t is time and u is the velocity field calculated from the known vorticity field using the Biot–Savart law. The diffusive part of the Navier–Stokes equations is solved for each vortex by computing the fraction of vorticity to be distributed amongst neighbouring vortices (for details, see [8]). In our DVM code, the boundary conditions on the body surface are satisfied approximately with the aid of novel vortex-source panel method, in which the vortex and source panels are located just outside and just underneath the wall, respectively. This approach allows us to account for eventual suction and/or blowing through the surface. The panel elements are taken in the form of curved segments with a linear distribution of vorticity. Previously, [3] showed that such elements can significantly reduce the boundary leakage as compared to the standard textbook panels. Upon completing each time step, the vortex panels are transformed into vortex blobs and released in the flow, thus imitating the effect of vorticity diffusion in the boundary layer. It is worth noting that the blob radius has the same order of magnitude as the panel length. Thus, for an adequate resolution of the boundary layer, the number of panels must be kept proportional to the square root of the Reynolds number. As panel methods normally break down near the sharp corners, we find it convenient to transform physical coordinates in such a way that in the new variables the flow domain turns into a straight channel of unit width. The mapping is based on the generalised Schwarz–Christoffel formula for channels with curved walls [4, 9]. It is provided in the form of grid-to-grid transformation with a bilinear interpolation between the mesh lines. Taking into account that the Navier–Stokes equations are invariant under conformal mappings, the flow field can be computed by supplying the Jacobian of the transformation to the standard DVM solver and mapping the results back to the physical plane when the calculation is complete. A special care is still required though for the vortices that come close to the cusp point, where the mapping is singular. This problem is handled by imposing a lower boundary on the value of the Jacobian. The DVM code has been validated against various published results on laminar and turbulent flows past a circular cylinder and a flat plate at zero incidence. In particular, a good agreement has been observed in predicting the drag crisis for a cylinder flow and the structure of the flat-plate boundary layer up to the Reynolds number of 107 . On average, the number of vortices used in our calculations was 5 105 , with the number of vortex panels being 1200. For further details of the discrete vortex method, see [7].

3 Feedback Control Design The numerical results presented in this section are obtained for a channel flow past the test-bed airfoil of Fig. 1, with the Reynolds number being Re D 2:1 106 per unit length. As shown in Figs. 2a and b, the uncontrolled flow is characterised by

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the large-scale vortex shedding from the cavity region, which results in the increased unsteady drag force. The control objective is to reduce this drag force by trapping the vortex in the cavity using suction as an actuator. The non-dimensional rate of creation of vorticity at the sensor point may conveniently be used as an output parameter for monitoring the flow state. In the open-loop tests with constant suction, we discovered that the vortex remains stably trapped if the suction velocity (non-dimensionalised by the inlet velocity U1 ) reaches the level S0 D 0:04. In this case, the time average value of the output signal becomes hi D 0:038, and a sharp drop in the variance of is observed. The instantaneous vorticity field and streamline pattern for such flow are displayed in Fig. 2c and d. However, in practice the amount of suction required to capture the vortex is not known a priori, therefore the feedback control strategy capable of computing the appropriate suction is desired. Although the system dynamics proves to be highly nonlinear, we have found that a linear PI controller can be used to stabilise the vortex. The control law in this case is given by the equation Z

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Fig. 3 Time histories of (a) output error e, and (b) suction strength S. The PI controller is activated at t D 5

suction strength fluctuates about S D 0:44 after some overshoot. Thus, the linear PI controller is able to achieve the target output measurement and to inhibit the vortex shedding process, but the required average suction is approximately 10% higher than the value of S0 obtained in the open-loop analysis.

4 Conclusions The paper develops an active control strategy for stabilising high Reynolds number flows with trapped vortices using suction as an actuator. The flow dynamics are modelled by the parallel discrete vortex method capable of handling wall irregularities, arbitrary boundary conditions, and turbulence. We find it convenient to accept the rate of creation of vorticity at the wall near the cavity exit as an input control parameter, since it can easily be linked to the values observable in experiments (e.g. pressure, wall shear stress). The open-loop analysis with constant suction reveals a strongly nonlinear behaviour of the system and determines the level of the actuation required for stabilising the flow. Feedback control results show that a properly designed linear PI controller prevents the large-scale vortex shedding from the cavity region and reduces considerably flow unsteadiness in the downstream boundary layer. Acknowledgements This work has been completed as an integral part of the research project Fundamentals of Actively Controlled Flows with Trapped Vortices, funded by the European Commission within its FP6 Program, Contract No: AST4-CT-2005-012139. The project particulars can be found on the website http://www.vortexcell2050.org.

580

R.M. Kerimbekov and O.R. Tutty

References 1. ANDERSON, C. R., CHEN, Y.-C. AND GIBSON, J. S.: Control and identification of vortex wakes. Trans. ASME J. Dyn. Sys. Meas. Contr. 122 (2000) 298–305. 2. CHORIN, A. J.: Numerical study of slightly viscous flow. J. Fluid Mech. 57 (1973) 785–796. 3. CLARKE, N. R. AND TUTTY, O. R.: Construction and validation of a discrete vortex method for two-dimensional incompressible Navier–Stokes equations. Comput. Fluids 23 (1994) 751–783. 4. DAVIS, R. T.: Numerical methods for coordinate generation based on the Schwarz–Christoffel transformation. AIAA Paper 79-1463 (1979) 4th Computational Fluid Dynamics Conference, July 1979. 5. DONELLI, R.: Mechanical design of the drawer equipped with cavity and instrumentation for trapped vortex equipment. Tech. Rep. AST4–CT–2005–012139–D3.3 (2007) Centro Italiano Ricerche Aerospaziali, Italy. 6. IOLLO, A. AND ZANNETTI, L.: Trapped vortex optimal control by suction and blowing at the wall. Eur. J. Mech. B–Fluids 20 (2001) 7–24. 7. KERIMBEKOV, R. M. AND TUTTY, O. R.: Discrete vortex method code. Tech. Rep. AST4–CT–2005–012139–D2.1 (2006) University of Southampton, UK. 8. SHANKAR, S. AND VAN DOMMELEN, L. L.: A new diffusion procedure for vortex methods. J. Comput. Phys. 127 (1996) 88–109. 9. SRIDHAR, K. P. AND DAVIS, R. T.: A Schwarz–Christoffel method for generating twodimensional flow grids. Trans. ASME J. Fluids Eng. 107 (1985) 330–337. 10. ZANNETTI, L.: Report on geometry for test bed experiments. Tech. Rep. AST4–CT–2005– 012139–D4.6 (2007) Politecnico di Torino, Italy.

A Three-Dimensional Numerical Study into Non-Axisymmetric Perturbations of the Hole-Tone Feedback Cycle M.A. Langthjem and M. Nakano

Abstract This paper is concerned with the hole-tone feedback cycle problem, also known as Rayleigh’s bird-call. A simulation method for analyzing the influence of non-axisymmetric perturbations of the jet on the sound generation is described. In planned experiments these perturbations will be applied at the jet nozzle via piezoelectric or electro-mechanical actuators, placed circumferentially inside the nozzle at its exit. The mathematical model is based on a three-dimensional vortex method. The nozzle and the holed end-plate are represented by quadrilateral vortex panels, while the shear layer of the jet is represented by vortex rings, composed of vortex filaments. The sound generation is described mathematically using the Powell-Howe theory of vortex sound. The aim of the work is to understand the effects of a variety of flow perturbations, in order to control the flow and the accompanying sound generation. Keywords Aeroacoustics Self-sustained flow oscillations Three-dimensional vortex method

1 Introduction Self-sustained fluid oscillations can occur in a variety of practical applications where a shear layer impinges upon a solid structure [1]. The oscillations are the cause of sound generation, which typically is powerful. In cases of music instruments (flutes, etc.) and whistles, sound generation is, of course, the aim. By engineering applications however, the sound generation is, in most cases, an unwanted, annoying side effect. M.A. Langthjem () Graduate School of Science and Engineering, Yamagata University, Yonezawa-shi, 992-8510 Japan e-mail: [email protected] M. Nakano Institute of Fluid Science, Tohoku University, Sendai-shi, 980-8577 Japan e-mail: [email protected] M. Braza and K. Hourigan (eds.), IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries 14, c Springer Science+Business Media B.V. 2009

581

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M.A. Langthjem and M. Nakano

Fig. 1 Left: Geometry and physical features of the hole-tone problem. Right: Flow visualization of the vortex roll-up [4]

The present paper is concerned with the so-called hole-tone problem [2, 3]. The common teakettle whistle is an example of utilization of the sound generation in this system. The steam jet, issuing from a nozzle, passes through a similar hole in a plate, placed a little downstream from the nozzle. The shear layer of the jet is unstable and rolls up into a large, coherent vortex (‘smoke-ring’). This large vortex cannot pass through the hole in the plate and hits the edge of the hole, where it creates a pressure disturbance. This disturbance is thrown back (with the speed of sound) to the nozzle, where it disturbs the shear layer. This initiates the roll-up of a new coherent vortex. In this way an acoustic feedback loop is formed. Figure 1a illustrates the principle of the hole-tone phenomenon. Figure 1b shows an experimental realization, with the vortex roll-up visualized by the smoke wire technique [4]. The basic dynamics of the hole-tone feedback system was studied numerically in [5], using an axisymmetric discrete vortex method, combined with an aeroacoustic model based on Curle’s theory [7]. This approach could predict the fundamental characteristics of the problem quite well, in particular the fluid-dynamic characteristics. The acoustic model gave qualitative correct results but overestimated the sound pressure levels. An improved acoustic model which gives much better results has been published recently [6]. The hole-tone system is a part of many engineering systems, where sound generation is unwanted. Examples include automobile intake- and exhaust systems, gas/steam distribution systems (bellows, valves, etc.), and solid-propellant rocket motors. In these cases, if a geometry which avoids the sound-generation cannot easily be obtained, a control method which can eliminate, or at least suppress, the sound generation is desirable. Nakano et al. [4] studied experimentally a forced excitation strategy to eliminate the hole-tone feedback cycle in the system depicted in Fig. 1. The shear layer near the nozzle exit was acoustically excited by means of an excitation chamber equipped with six loudspeakers, placed equidistantly around the circumference. By harmonic excitation at frequencies away from the fundamental frequency f0 , noise level reductions (at f0 ) of up to 6 dB were achieved.

The Hole-Tone Feedback Cycle

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The aim of the present work is to develop a numerical method for simulating the hole-tone problem with the jet subjected to non-axisymmetric ‘mechanical’ (‘nonacoustic’) perturbations, by piezoelectric or electro-mechanical actuators mounted at the nozzle exit, similar to the concept of Kasagi [8]. This is expected to be more efficient than acoustic perturbations [4], and the aim is to study/verify this carefully before experimental verification, using a three-dimensional vortex method.

2 Flow Model The shear layer of the jet issuing from the nozzle is represented in a lumped form, by a ‘necklace’ of discrete vortex rings. These rings are disturbed mechanically at the nozzle exit such that they loose their natural axisymmetric form, and are thus represented by three-dimensional vortex filaments. The induced velocity ui D .u1 ; u2 ; u3 /i , at position xi D .x1 ; x2 ; x3 /i and time t, from J vortex rings represented by the space curves rj .; t/, is given by ui .xi ; t/ D

Z J X j fxi .t/ rj .; t/g @rj =@ d; 3 4 fjxi .t/ rj .; t/j2 C ˛j2 .; t/g 2 j D1

(1)

where j is the strength (circulation) of the j ’th vortex, is a material (vortex) coordinate, and j .; t/ is the core radius. The parameter ˛ represents the vorticity distribution within the core; for a Gaussian distribution, ˛ 0:413. The space curves rj .; t/ are discretized by employing K marker points on each curve (vortex ring), connected via cubic splines or, optionally, via straight segments. The integration in (1) is carried out using Gauss-Legendre quadrature when splines are used, and analytically when straight segments are used. A vortex ring is released from the nozzle at each time step in the simulation. [Earlier studies [5] have shown that the vortex shedding from the edge of the hole in the end plate is insignificant.] The strength of the vortex ring to be released is dictated by the Kutta condition. The convection velocity of a shed vortex ring is dictated by the induced velocities from all other vortex rings, plus the self-induced velocity, as indicated by (1). The positions ri of the shed vortex filament ring marker points are updated by solving numerically the system of ordinary differential equations dri .t/=dt D ui .ri ; t/. The solid surfaces are represented by quadrilateral vortex panels, made up of four straight vortex filaments, as indicated in Fig. 2. The inviscid boundary condition of zero normal velocity is imposed at control points in the center of these panels. The mean jet flow is provided by a number of panels placed on the ‘back’ of the nozzle tube; see again Fig. 2. The strengths of the bound vortex panels are dictated by the boundary conditions and by the mean jet velocity. The mechanical/piezoelectric actuator system is simulated by periodical ‘edge wave’ deformations of the nozzle end section, as illustrated also by Fig. 2.

584


Z

Y

Y

Y 3 2 1 0 −1 −2 −3

3 2 1 0 −1 −2 −3

3 2 1 0 −1 −2 −3

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Fig. 2 Illustration of the perturbation mechanism (actuator model) [For purpose of illustration the amplitude is exaggerated]

3 Aeroacoustic Model The theory of vortex sound is applied to compute the sound generation. For low Mach-number flows it is described by the inhomogeneous wave equation [7] c02 @2 p=@t 2 r 2 p D 0 r L;

(2)

where p.x; t/ is the acoustic pressure, L D ! u is the Lamb vector, with the vorticity ! given by r u; 0 is the mean air density, and c0 the speed of sound. In the present work (2) is solved in two different ways: (i) by using the compact Green’s function approach; (ii) by using the boundary element method. A detailed description of approach (i) can be found in e.g. [7]; accordingly only approach (ii) will be described here. Using the free space Green’s function G.r; t / D ı.t r=c0 /=4 r, where r D jx yj and ı is the Dirac delta function, the solution to (2) can be expressed as p.x; t/ D pv .x; t/ C

X“ e

1 4 rxˇ

1 rxˇ

pQˇe

t

1 C c0

@pQˇe @t

! cos.rxˇ ; nˇe /d 2 yˇe ; t

(3) where pQˇe is the pressure difference across the end plate at the boundary element control point yˇe , rxˇ D jxyˇe j is the distance between the observation point x and yˇe , and nˇe is the normal vector at yˇe . Square brackets with P subscript t indicate evaluation at the retarded time t D t jx yˇe j=c0 , and e indicates summation over all boundary elements. The source term pv .x; t/ is given by • pv .x; t/ D

4 r

!

xj yj 3 1 @Lj 1 Lj t C d y; r c0 @t t r

(4)

The Hole-Tone Feedback Cycle

585

where summation over repeated subscript j ’s applies. The vector of pressure differences across the end plate control points, pQ ˇ , can be determined by taking the normal derivative of (3); see [10]. Evaluation at the point x˛ on the end plate gives @pv .x˛ ; t/ X C @n˛ e

“

1 2 4 r˛ˇ

1 1 pQˇe t C r˛ˇ c0

@pQˇe @t

! d 2 yˇe D 0:

(5)

t

4 Numerical Example and Concluding Remarks Computations have been carried out for data corresponding to an experimental rig with nozzle and end plate hole diameter d0 D 2r0 D 50 mm [4]. The outer diameter of the end plate is 250 mm. The gap length ` is 50 mm, e.g., equal to d0 . The mean velocity u0 of the air-jet is 10 m/s. At 20ı C this corresponds to a Reynolds number Re D u0 d0 = 3:3 104 and a Mach number M D u0 =c0 0:03, where the speed of sound c0 = 340 m/s and the kinematic viscosity D 1:5 105 m2 /s. The initial vortex core radii j D 0:275r0. A number of side view ‘snapshots’ of the jet during one period of the oscillations are shown in Fig. 3. The computed fundamental frequency f0 190 Hz. The experimentally observed value is 196 Hz. The influence of nozzle-oscillations is exemplified by Fig. 4 which shows the sound pressure levels (in dB) midway between nozzle exit and end plate, five nozzle diameters away from the central axis. Part (a) is for the case without nozzle

X2

X1

X3

X3

t D0

t D 1=.17f0 /

X2

X1

t D 9=.17f0 /

t D 11=.17f0 /

t D 13=.17f0 /

Fig. 3 Side view of the jet during one period of oscillation

t D 7=.17f0 /

X2

X1

X1 X3

X3

t D 5=.17f0 /

X2

X1 X3

X1 X3

t D 3=.17f0 /

X2

X1 X3

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X1

X1 X3

X2

X2

X2

X2

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t D 15=.17f0 /

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t D 17=.17f0 /

586

a


b

60

55

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55 50 45 40

60

50 45

0

200

400

600

Frequency [Hz]

800

1000

40

0

200

400

600

800

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Fig. 4 Sound pressure levels (a) without and (b) with nozzle oscillations

oscillations. Part (b) is for a case where the radius of the nozzle, in polar coordinates, is given by r.; z/ D r0 .1 C sin.N / cos.2fd t/.1 C z=r0 //, 0 2, r0 z 0 (with nozzle exit at z D 0), N D 3, D r0 =150, and fd D 200 3 Hz. It is seen that the nozzle oscillations suppress the fundamental tone by approximately 3 dB. From the appearance of the jet (not shown here due to lack of space) it can be seen that the oscillating nozzle destroys the coherence of the ‘smoke rings’ and thus limits the noise generation. Acknowledgement The support of the present project through a JSPS Grant-in-Aid for Scientific Research (No. 18560152) is gratefully acknowledged.

References 1. D. Rockwell and E. Naudascher. Self-sustained oscillations of impinging free shear layers. Annu. Rev. Fluid Mech. 11:67–94, 1979. 2. Lord Rayleigh. The Theory of Sound, Vol. II. Dover Publications, New York, 1945. 3. R.C. Chanaud and A. Powell. Some experiments concerning the hole and ring tone. J. Acoust. Soc. Am. 37:902–911, 1965. 4. M. Nakano, D. Tsuchidoi, K. Kohiyama, A. Rinoshika, and K. Shirono. Wavelet analysis on behavior of hole-tone self-sustained oscillation of impinging circular air jet subjected to acoustic excitation (in Japanese). Kashikajouhou 24:87–90, 2004. 5. M.A. Langthjem and M. Nakano. A numerical simulation of the hole-tone feedback cycle based on an axisymmetric discrete vortex method and Curle’s equation. J. Sound Vib. 288:133–176, 2005. 6. M.A. Langthjem and M. Nakano. Numerical study of the hole-tone feedback cycle based on an axisymmetric formulation. Fluid Dyn. Res., 2009 (to be published). 7. M. S. Howe. Theory of Vortex Sound. Cambridge University Press, Cambridge, 2003. 8. N. Kasagi. Toward smart control of turbulent jet mixing and combustion. JSME Int., J. Ser. B 49:941–950, 2006. 9. G.-H. Cottet and P.D. Koumoutsakos. Vortex Methods: Theory and Practice. Cambridge University Press, Cambridge, 2000. 10. Y. Kawai and T. Terai. A numerical method for the calculation of transient acoustic scattering from thin rigid plates. J. Sound Vib. 141:83–96, 1990.

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