Direct and Large-Eddy Simulation

DIRECT AND LARGE-EDDY SIMULATION VI ERCOFTAC SERIES VOLUME 10 Series Editors R.V.A. Oliemans, Chairman ERCOFTAC, Del...

Author: E. Lamballais | Rainer Friedrich | Bernard J. Geurts | Olivier Métais

44 downloads 1631 Views 24MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

DIRECT AND LARGE-EDDY SIMULATION VI

ERCOFTAC SERIES VOLUME 10

Series Editors R.V.A. Oliemans, Chairman ERCOFTAC, Delft University of Technology, Delft, The Netherlands W. Rodi, Deputy Chairman ERCOFTAC, Universität Karlsruhe, Karlsruhe, Germany

Aims and Scope of the Series ERCOFTAC (European Research Community on Flow, Turbulence and Combustion) was founded as an international association with scientific objectives in 1988. ERCOFTAC strongly promotes joint efforts of European research institutes and industries that are active in the field of flow, turbulence and combustion, in order to enhance the exchange of technical and scientific information on fundamental and applied research and design. Each year, ERCOFTAC organizes several meetings in the form of workshops, conferences and summerschools, where ERCOFTAC members and other researchers meet and exchange information. The ERCOFTAC Series will publish the proceedings of ERCOFTAC meetings, which cover all aspects of fluid mechanics. The series will comprise proceedings of conferences and workshops, and of textbooks presenting the material taught at summerschools. The series covers the entire domain of fluid mechanics, which includes physical modelling, computational fluid dynamics including grid generation and turbulence modelling, measuring-techniques, flow visualization as applied to industrial flows, aerodynamics, combustion, geophysical and environmental flows, hydraulics, multiphase flows, non-Newtonian flows, astrophysical flows, laminar, turbulent and transitional flows.

The titles published in this series are listed at the end of this volume.

Direct and Large-Eddy Simulation VI Proceedings of the Sixth International ERCOFTAC Workshop on Direct and Large-Eddy Simulation, held at the University of Poitiers, September 12–14, 2005

Edited by

ERIC LAMBALLAIS University of Poitiers, Laboratoire d’Études Aérodynamiques, Futuroscope Chasseneuil Cedex, France

RAINER FRIEDRICH Technische Universität München, München, Germany

BERNARD J. GEURTS University of Twente, Enschede, The Netherlands and

OLIVIER MÉTAIS ENSHMG/INPG, St Martin d‘Hères, France

123

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 1-4020-4909-9 (HB) ISBN-13 978-1-4020-4909-5 (HB) ISBN-10 1-4020-5152-2 (HB) ISBN-13 978-1-4020-5152-2 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

Printed on acid-free paper

All Rights Reserved © 2006 Springer 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.

To Frans T.M. Nieuwstadt

Frans T.M. Nieuwstadt, 1946-2005 (Photo: Nout Steenkamp/FMAX/FOM)

Obituary Prof. Frans T.M. Nieuwstadt

Laboratory for Aero and Hydrodynamics, Leeghwaterstraat 21, 2628 CA Delft, The Netherlands

Frans Nieuwstadt has been a professor of fluid mechanics since 1986. He died under rather tragic circumstances on May 18, 2005. He made several profound contributions to research in turbulent shear flows, first in meteorology and later in engineering. Frans was born in 1946. After secondary school he studied aerospace engineering at the Technische Hogeschool in Delft (now known as TU-Delft). After his university education he spent two years at the California Institute of Technology, studying computational fluid mechanics. He very much enjoyed this period, especially because of the informal relation between students and professors. At that time the student-professor relation was still very formal in the Netherlands. After leaving Caltech, he accepted a position at the Royal Dutch Institute for Meteorology (KNMI). Here he worked on atmospheric boundary layers. In 1981 he got his PhD, with Henk Tennekes as PhD adviser, on the behavior of the nocturnal boundary layer. This work initiated his international reputation. Best known is his model of the stable boundary layer that (with a few modifications) has since been regarded as the standard model. In 1986 he was appointed Professor of fluid mechanics at TU-Delft. Among his predecessors were renowned persons such as J.M. Burgers and J.O. Hinze. Due to various circumstances, the fluid mechanics Chair at Delft had been vacant for more than two years and a whole new group needed to be formed. Frans proved to be a worthy successor and successfully took up this challenge.

X


During his time at KNMI he had been working on Large Eddy Simulation of the atmospheric boundary layer. He saw the merit of this type of techniques for engineering flows and initiated various projects on the Large Eddy Simulation of pipe and jet flows. He also noticed that this type of work could not be performed without the backup by accurate experimental data. So an extensive experimental program, using Laser Doppler Anemometry and later Particle Image Velocimetry (PIV) was also developed in his laboratory. The combination of the numerical and experimental research turned out to be very successful and formed the backbone of the laboratory. Frans believed strongly in this combination of approaches which forms one of his lasting legacies to us. He noticed that it became more and more difficult to get funding for pure (fundamental) turbulence research. He developed a program he called ‘turbulence plus’, i.e., a turbulent flow plus for instance chemical reactions, suspended particles or polymers. This turbulence-plus program resulted in several very successful PhD projects and many publications in the Journal of Fluid Mechanics and Physics of Fluids. Frans was one of the founders of the JM-Burgerscentre, a national research school for fluid mechanics and related phenomena in the Netherlands. He initiated a number of activities that contributed directly to the excellent collaboration among the various groups in this research school. In this respect, particular mention may be made of the bi-annual meetings of the Dutch turbulence interest group. As a result of his motivating energy, fluid mechanics is a flourishing research-field in the Netherlands, with more than 200 enlisted PhD students. Internationally he was also very active. He saw the value to his laboratory of the new networks that have sprung up in Europe including Euromech, ERCOFTAC, and the European Turbulence Conferences (which he chaired 1990-1996). He also supported the conference of the International Union of Theoretical and Applied Mechanics. Frans was a marvelous companion to everyone in his laboratory and took great pride in having a family atmosphere, and dealing gently but firmly with the managerial and personal difficulties that arise in any organization. Among the students in Delft he was well known for his inspiring lectures, an honor much appreciated by Frans. The door of Frans his office was always open. If one of the employees of the department, or one of his (former) students had a problem, concerning work as well as personal problems, Frans was always there to solve the problem. Also on a national level he was a key player in many scientific organizations, among others he was the president of the Foundation for Fundamental Research of Matter (FOM) until his untimely death on May 18th, 2005. We cherish this image of Frans as initiator, motivator and personal friend, and will miss him dearly.

Colleagues of the laboratory for Aero and Hydrodynamics, June 2005.


XI

A few characteristic publications of Frans Nieuwstadt i. Nieuwstadt, F. T. M.; Mason, P. J.; Moeng, C. H.; Schumann, U., 1991, Large-eddy simulation of the convective boundary layer - A comparison of four computer codes. IN: Symposium on Turbulent Shear Flows, 8th, Munich, Federal Republic of Germany, Sept. 9-11, 1991, Proceedings. Vol. 1 (A92-40051 16-34). University Park, PA, Pennsylvania State University, 1991, p. 1-4-1 to 1-4-6. ii. Eggels, J., Unger, F., Weiss, M., Westerweel, J., Adrian, R., Friedrich, R. & F.T.M. Nieuwstadt, 1994, Fully developed turbulent pipe flow: a comparison between direct numerical simulation and experiment, J. Fluid Mech., 268, 175-209. iii. Draad, A.A., & F.T.M. Nieuwstadt, 1998, The Earth’s rotation and laminar pipe flow, J. Fluid Mech., 361, 297-308. iv. Brethouwer, G., J.C.R. Hunt, & F.T.M. Nieuwstadt, 2003, Micro-structure and Lagrangian statistics of the scalar field with a mean gradient in isotropic turbulence, J. of Fluid Mech., 474, 193-225. v. B. Hof, C. W. H. van Doorne, J. Westerweel, F. T. M. Nieuwstadt, H. Faisst, B. Eckhardt, H. Wedin, R. Kerswell, F. Waleffe, 2004, Experimental Observation of Nonlinear Traveling Waves in Turbulent Pipe Flow, Science, 305, Issue 5690, 1594-1598 .

Preface

The sixth ERCOFTAC Workshop on ’Direct and Large-Eddy Simulation’ (DLES-6) was held at the University of Poitiers from September 12-14, 2005. Following the tradition of previous workshops in the DLES-series, this edition has reflected the state of the art of numerical simulation of transitional and turbulent flows and provided an active forum for discussion of recent developments in simulation techniques and understanding of flow physics. At a fundamental level this workshop has addressed numerous theoretical and physical aspects of transitional and turbulent flows. At an applied level it has contributed to the solution of problems related to energy production, transportation and the environment. Since the prediction and analysis of fluid turbulence and transition continues to challenge engineers, mathematicians and physicists, DLES-6 has covered a large range of topics from more technical ones like numerical methods, initial and inflow conditions, the coupling of RANS and LES zones, subgrid and wall modelling to topics with a stronger focus on flow physics such as aero-acoustics, compressible and geophysical flows, flow control, multiphase flow and turbulent combustion, to quote only a few. In total 141 participants from 16 countries registered for this workshop. The ERCOFTAC support stimulated the organization in a number of essential ways. The specific ERCOFTAC grant allowed easier participation at this workshop for the 42 PhD students. The 33 ERCOFTAC members present in DLES-6 could also benefit from the sponsoring. The present proceedings contain the written versions of 7 invited lectures and 82 selected and reviewed contributions which are organized in 16 parts entitled ’Turbulent Mixing and Combustion’, ’Subgrid Modelling’, ’Flows involving Curvature, Rotation and Swirl’, ’Free Turbulent Flows’, ’Multiphase Flows’, ’Wall Models for LES’, ’Complex Geometries and Boundary Conditions’, ’Flow Control’, ’Heat Transfer’, ’Aeroacoustics’, ’Variable Density Flows’, ’Inflow/Initial conditions’, ’Separated/Reattached Flows’, ’Hybrid RANS-LES Approach’, ’Compressible Flows’, and ’Numerical Techniques’.

XIV

Preface

The workshop was financially supported by the Association Fran¸caise de Mécanique, Centre National de la Recherche Scientifique (SPI), Ecole Nationale Supérieure de Mécanique et d’Aérotechnique, European Research Community On Flow, Turbulence and Combustion, J.M. Burgers Centre, Laboratoire d’Etudes Aérodynamiques-UMR6609, Ministère de l’Education Nationale de l’Enseignement Supérieur et de la Recherche, Région PoitouCharentes (Programme Com’Sciences), Université de Poitiers (Faculté des Sciences), Ville de Poitiers, Communauté d’Agglomération de Poitiers. The organizers of this workshop express their sincere gratitude to these sponsors.

Poitiers, January 2006

Eric Lamballais Rainer Friedrich Bernard J. Geurts Olivier Métais

Contents

Part I Invited Lectures Direct Numerical Simulation of Turbulence and Scalar Exchange at Gas-Liquid Interfaces Sanjoy Banerjee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Large Eddy Simulation of Premixed Turbulent Combustion: FSD-PDF modeling Luc Vervisch, Pascale Domingo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Industrial LES with Unstructured Finite Volumes D. Laurence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Current understanding of jet noise-generation mechanisms from compressible large-eddy-simulations Christophe Bailly, Christophe Bogey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Stable stratified, wall bounded, turbulent flows Vincenzo Armenio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Physics and Control of Wall Turbulence John Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 The physics of turbulent mixing and clustering J.C. Vassilicos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Part II Turbulent Mixing and Combustion LES of Premixed Flame Longitudinal Wave Interactions Christer Fureby, Christophe Duwig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

XVI

Contents

Direct Numerical Simulation of Reacting Turbulent Multi-Species Channel Flow L. Artal, F. Nicoud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 DNS/MILES of Reacting Air/H2 Diffusion Jets L. Gougeon, I. Fedioun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Direct Numerical Simulation of turbulent reacting flows involving dilute particles G´ abor Janiga, Dominique Thévenin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Dissipation of Active Scalars in Turbulent Temporally Evolving Shear Layers with Density Gradients Caused by Multiple Species Inga Mahle, J¨ orn Sesterhenn, Rainer Friedrich . . . . . . . . . . . . . . . . . . . . . . 109

Part III Subgrid Modelling Symmetry invariant subgrid models Dina Razafindralandy, Aziz Hamdouni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Formal properties of the additive RANS/DNS filter Massimo Germano, Pierre Sagaut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Analysis of the SGS energy budget for deconvolution- and relaxation-based models in channel flow Philipp Schlatter, Steffen Stolz, Leonhard Kleiser . . . . . . . . . . . . . . . . . . . . . 135 Symmetry-preserving regularization of turbulent channel flow Roel Verstappen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Large-eddy simulations of channel flows with variable filter-width-to-grid-size ratios Ana Cubero, Ugo Piomelli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 LES of Transition to Turbulence in the Taylor Green Vortex Dimitris Drikakis, Christer Fureby, Fernando F. Grinstein, Marco Hahn, David Youngs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Stochastic SGS modelling in homogeneous shear flow with passive scalars Linus Marstorp, Geert Brethouwer, Arne Johansson . . . . . . . . . . . . . . . . . . 167 Towards Lagrangian dynamic SGS model for SCALES of isotropic turbulence Giuliano De Stefano, Daniel E. Goldstein, Oleg V. Vasilyev, Nicholas K.-R. Kevlahan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Contents

XVII

The sampling-based dynamic procedure for LES: investigations using finite differences G. Winckelmans, L. Bricteux, L. Georges, G. Daeninck, H. Jeanmart . . 183 On the Evolution of the Subgrid-Scale Energy and Scalar Variance: Effect of the Reynolds and Schmidt numbers C. B. da Silva, J.C.F. Pereira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Part IV Flows involving Curvature, Rotation and Swirl Large Eddy Simulation of Flow Instabilities in Co-Annular Swirling Jets M. Garcia-Villalba, J. Fr¨ ohlich, W. Rodi, O. Petsch and H. B¨ uchner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Large Eddy Simulations of the turbulent flow in curved ducts: influence of the curvature radius Cécile M¨ unch, Olivier Métais . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Large Eddy Simulation of Transitional Rotor-Stator Flows using a Spectral Vanishing Viscosity Technique Eric Séverac, Eric Serre, Patrick Bontoux, Brian Launder . . . . . . . . . . . . 217 DNS of Rotating Homogeneous Shear Flow and Scalar Mixing G. Brethouwer, Y. Matsuo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Direct Numerical Simulation of Turbulent Rotating Rayleigh–B´ enard Convection R.P.J. Kunnen, B.J. Geurts, H.J.H. Clercx . . . . . . . . . . . . . . . . . . . . . . . . . 233 DNS of a Turbulent Channel Flow with Streamwise Rotation - Investigation on the Cross Flow Phenomena Tanja Weller, Martin Oberlack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Dynamic structuring and mixing efficiency in rotating shear layers Bernard J. Geurts, Darryl D. Holm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 DNS of turbulent heat transfer in pipe flow with respect to rotation rate and Prandtl number effects L. Redjem Saad, M. Ould-Rouis, A. A. Feiz, G. Lauriat . . . . . . . . . . . . . . 257 DNS of the turbulent Ekman layer at Re=2000 G. N. Coleman, R. Johnstone, M. Ashworth . . . . . . . . . . . . . . . . . . . . . . . . . 267

XVIII Contents

Part V Free Turbulent Flows Large-Eddy Simulation of Coaxial Jets: Coherent Structures and Mixing Properties Guillaume Balarac, Mohamed Si-Ameur, Olivier Métais, Marcel Lesieur 277 Computation of the Self-Similarity Region of a Turbulent Round Jet Using Large-Eddy Simulation Christophe Bogey, Christophe Bailly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 The dependence on the energy ratio of the shear-free interaction between two isotropic turbulences D. Tordella, M. Iovieno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Part VI Multiphase Flows Orientation of elongated particles within turbulent flow J. J. J. Gillissen, B. J. Boersma, F. T. M. Nieuwstadt . . . . . . . . . . . . . . . 303 On the closure of particle motion equations in large-eddy simulation Maria Vittoria Salvetti, Cristian Marchioli, Alfredo Soldati . . . . . . . . . . . . 311 Dispersion of Circular, Non-Circular, and Swirling Spray Jets in Crossflow Mirko Salewski, Laszlo Fuchs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Direct Numerical Simulation of Droplet Impact on a Liquid-Liquid Interface Using a Level-Set/Volume-Of-Fluid Method with Multiple Interface Marker Functions E.R.A. Coyajee, R. Delfos, H.J. Slot, B.J. Boersma . . . . . . . . . . . . . . . . . . 329 Direct numerical simulation of particle statistics and turbulence modification in vertical turbulent channel flow A. Kubik, L. Kleiser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Large-eddy simulation of particle-laden channel flow J.G.M. Kuerten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 High Resolution Simulations of Particle-Driven Gravity Currents Eckart Meiburg, F. Blanchette, M. Strauss, B. Kneller, M.E. Glinsky, F. Necker, C. H¨ artel,, L. Kleiser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

Contents

XIX

Implicitly-coupled finite difference schemes for fictitious domain simulation of solid-liquid flow; marker, volumetric, and hybrid forcing Hung V. Truong, John C. Wells, Gretar Tryggvason . . . . . . . . . . . . . . . . 363

Part VII Wall Models for LES Development of Wall Models for LES of Separated Flows Michael Breuer, Boris Kniazev, Markus Abel . . . . . . . . . . . . . . . . . . . . . . . . 373 Anisothermal Wall Functions for RANS and LES of Turbulent Flows With Strong Heat Transfer Antoine Devesa, Franck Nicoud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Wall Layer Investigations of Channel Flow with Periodic Hill Constrictions Nikolaus Peller, Christophe Brun and Michael Manhart . . . . . . . . . . . . . . . 389 A Multi-Scale, Multi-Domain Approach for LES of High Reynolds Number Wall-Bounded Turbulent Flows M. U. Haliloglu, R. Akhavan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

Part VIII Complex Geometries and Boundary Conditions Modeling turbulence in complex domains using explicit multi-scale forcing A. K. Kuczaj, B. J. Geurts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 Direct and Large-Eddy Simulations of a Turbulent Flow with Effusion S. Mendez, F. Nicoud, P. Miron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Nasal Airflow in a Realistic Anatomic Geometry R. van der Leeden, E. Avital, G. Kenyon . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Large-eddy simulation of a purely oscillating turbulent boundary layer S. Salon, V. Armenio, A. Crise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 Large Eddy Simulation of a Turbulent Channel Flow With Roughness S. Leonardi, F. Tessicini, P. Orlandi, R.A. Antonia . . . . . . . . . . . . . . . . . 439

XX

Contents

Part IX Flow Control Bimodal control of three-dimensional wakes Philippe Poncet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 DNS/LES of Active Separation Control Julien Dandois, Eric Garnier, Pierre Sagaut . . . . . . . . . . . . . . . . . . . . . . . . 459 Direct Numerical Simulation of a Spatially Evolving Flow from an Asymmetric Wake to a Mixing Layer Sylvain Laizet, Eric Lamballais . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

Part X Heat Transfer LES of Flow and Heat Transfer in a Round Impinging Jet M. Hadˇziabdić, K. Hanjalić . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Direct Simulations of a Transitional Unsteady Impinging Hot Jet X. Jiang, H. Zhao, K. H. Luo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

Part XI Aeroacoustics Basic Sound Radiation from Low Speed Coaxial Jets Mikel Alonso, Eldad J. Avital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 Use of surface integral methods in the computation of the acoustic far field of a turbulent jet. P. Moore, B.J. Boersma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 Helmholtz decomposition of velocity field of a mixing layer: Application to the analysis of acoustic sources M. Cabana, V. Fortuné, P. Jordan, F. Golanski, E. Lamballais, Y. Gervais . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 Large-Eddy Simulation of acoustic propagation in a turbulent channel flow Pierre Comte, Marie Haberkorn, Gilles Bouchet, Vincent Pagneux, Yves Aurégan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 Numerical Methodology for the Computation of the Sound Generated by a Non-Isothermal Mixing Layer at Low Mach Number F. Golanski, C. Moser, L. Nadal, C. Prax, E. Lamballais . . . . . . . . . . . . . 529

Contents

XXI

A Hybrid LES-Acoustic Analogy Method for Computational Aeroacoustics K. H. Luo, H. Lai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Numerical simulation of wind turbine noise generation and propagation Drago¸s Moroianu, Laszlo Fuchs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 Large Eddy Simulation for Computation of Aeroacoustic Sources in 2D-Expansion Chambers Gustavo Rubio, Wim De Roeck, Wim Desmet and Martine Baelmans . . . 555

Part XII Variable Density Flows High resolution simulation of particle-driven lock-exchange flow for non-Boussinesq conditions James E. Martin, Eckart Meiburg, Vineet K. Birman . . . . . . . . . . . . . . . . . 567 LES of the jet in low Mach variable density conditions Artur Tyliszczak, Andrzej Boguslawski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 LES of Spatially-developing Stably Stratified Turbulent Boundary Layers Tetsuro Tamura and Kohei Mori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 Numerical Investigation on the Formation of Streamwise Vortices in a Stably Stratified Temporal Mixing Layer Denise Maria Varella Martinez, Edith Beatriz Camano Schettini, Jorge Hugo Silvestrini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

Part XIII Inflow/Initial conditions Interfacing Stereoscopic PIV measurements to Large Eddy Simulations via Low Order Dynamical Systems L. Perret, J. Delville, R. Manceau, J.P. Bonnet . . . . . . . . . . . . . . . . . . . . . . 601 LES of background fluctuations interacting with periodically incoming wakes in a turbine cascade Jan Wissink, Wolfgang Rodi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 Large Eddy Simulations Of Spatially Developing Flows Using Inlet Conditions Dimokratis G.E. Grigoriadis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

XXII

Contents

Impact of Initial Flow Parameters on a Temporal Mixing Layer Evolution M. Fathali, J. Meyers, C. Lacor, M. Baelmans . . . . . . . . . . . . . . . . . . . . . . . 625

Part XIV Separated/Reattached Flows A Comparative Study of the Turbulent Flow Over a Periodic Arrangement of Smoothly Contoured Hills Michael Breuer, Benoit Jaffrézic, Nikolaus Peller, Michael Manhart, Jochen Fr¨ ohlich, Christoph Hinterberger, Wolfgang Rodi, Ganbo Deng, Oussama Chikhaoui, Sanjin uSarić, Suad Jakirlić . . . . . . . . . . . . . . . . . . . . 635 Large-Eddy Simulation of a Subsonic Cavity Flow Including Asymmetric Three-Dimensional Effects Lionel Larchevêque, Odile Labbé, Pierre Sagaut . . . . . . . . . . . . . . . . . . . . . . 643 Numerical Simulation of the Flow Around a Sphere Using the Immersed Boundary Method for Low Reynolds Numbers Campregher, R., Mansur, S.S., Silveira-Neto, A. . . . . . . . . . . . . . . . . . . . . . 651 LES studies on the correspondence between the interaction of shear layers in post-reattachment recovery and in a plane turbulent wall jet A. Dejoan and M. A. Leschziner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 Three Dimensional Wake Structure of Free Planar Shear Flow Around Horizontal Cylinder Marcelo A. Vitola, Edith B. C. Schettini, Jorge H. Silvestrini . . . . . . . . . . 669

Part XV Hybrid RANS-LES Approach Coupling from LES to RANS using eddy-viscosity models G. Nolin, I. Mary, L. Ta-Phuoc, C. Hinterberger, J. Fr¨ ohlich . . . . . . . . . . 679 SAS Turbulence Modelling of Technical Flows F.R. Menter, Y. Egorov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 Zonal LES/RANS modelling of separated flow around a three-dimensional hill F. Tessicini, N. Li and M. A. Leschziner . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 Progress In Subgrid-Scale Transport Modeling Using Partial Integration Method For LES Of Developing Turbulent Flows Bruno Chaouat, Roland Schiestel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703

Contents XXIII

Part XVI Compressible Flows Towards Large Eddy Simulations of Scramjet Flows Christer Fureby, Magnus Berglund . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 DNS and LES of compressible turbulent pipe flow with isothermal wall S. Ghosh, J. Sesterhenn, R. Friedrich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 Large-eddy simulation of separated flow along a compression ramp at high Reynolds number M. S. Loginov, N. A. Adams, A. A. Zheltovodov . . . . . . . . . . . . . . . . . . . . . 729 DNS of Transitional Transonic Flow about a Supercritical BAC3-11 Airfoil using High-Order Shock Capturing Schemes Igor Klioutchnikov and Josef Ballmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737

Part XVII Numerical Techniques and POD A Preconditioned LES Method for Nearly Incompressible Flows N. A. Alkishriwi, M. Meinke,, W. Schr¨ oder . . . . . . . . . . . . . . . . . . . . . . . . . 747 Particle dispersion in turbulent flows by POD low order model using LES snapshots C. Allery, C. Béghein, A. Hamdouni and N. Verdon . . . . . . . . . . . . . . . . . . 755 On the influence of domain size on POD modes in turbulent plane Couette flow Anders Holstad, Peter S. Johansson, Helge I. Andersson and Bjørnar Pettersen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 Implicit Time Integration Method for LES of Complex Flows Frédéric Daude, Ivan Mary, Pierre Comte . . . . . . . . . . . . . . . . . . . . . . . . . . 771

Part I

Invited Lectures

Direct Numerical Simulation of Turbulence and Scalar Exchange at Gas-Liquid Interfaces Sanjoy Banerjee Department of Chemical Engineering, Department of Mechanical Engineering, Bren School of Environmental Science and Management University of California, Santa Barbara, CA 93106

1 Introduction Exchange of heat and mass across deformable fluid-fluid interfaces is central to many industrial and environmental processes but is still poorly understood, despite the considerable effort that has gone into its study. For example, such processes are important in condensers, evaporators, and absorbers, as well as in many environmental problems, like transfer of greenhouse gases at the airsea interface. The poor state of understanding is illustrated in Figure 1-left, where the liquid-side heat transfer coefficient measured for steam condensation on a subcooled water stream is compared with computer code predictions, giving differences of one to three orders of magnitude. 15

Mono lake Crowley lake Rockland lake Lake 302N Pyramid lake K=0.45 U^1.6 Liss & Merlivat (1986)

104 K(600) [cm h−1]

Intf. Head Transfer Coeff. (W/m2-C)

105

1000

10

5

100 Data RELAPS 10 0

0.05

0.1 0.15 Void Fraction

0.2

0.25

0

0

1

2

3 4 5 U(10) [ms−1]

6

7

8

Fig. 1. Left: Predicted (RELAP5 code) and measured transfer coefficients for steam condensation on a subcooled stratified water stream (Kelly, USNRC 1997). Right: Gas transfer coefficients vs. wind speed from lakes (MacIntyre et al. 1995).

Turning to environmental applications, Figure 1-right compares experimental air-water gas transfer coefficients, versus the wind speed 10 meters above the water surface, U10 , to the widely used parameterization by Liss and

4

Sanjoy Banerjee

Table 1. Global oceanic CO2 uptake estimates using different gas exchange-wind speed relationships (Donelan & Wanninkhof 2002).

Merlivat (1986). The scatter in the data is due to several reasons discussed later. The effect of such uncertainties is shown in Table 1, where estimates of global carbon uptake by the oceans vary by factors of 3. The low quality of predictions arises from lack of understanding of fluid motion near deforming interfaces, which usually controls scalar exchange. This, in turn, arises from difficulties in measuring and numerically simulating turbulent velocity fields in the vicinity of moving surfaces. Nonetheless, progress is being made in this difficult area largely due to relatively recent developments of nonintrusive measurement techniques such as digital particle imaging velocimetry (DPIV) and direct numerical simulation (DNS) of turbulent flows over complex boundaries. The objective of this paper is to review recent results obtained, particularly from DNS, focusing on behavior at gas-liquid interfaces. The simplest case is when the gas is not moving and turbulence is generated somewhere else, perhaps at the bottom of a channel or in a shear layer, and then impinges on the interface, on which there is no wind shear imposed. A more complex situation arises when the gas phase is moving, in which case the interface develops waves due to wind forcing, and these waves affect interfacial transport processes. We will discuss in brief what can be learned from DNS for both these situations and induce from these results a model for liquid-side controlled mass and heat exchange processes, e.g. interphase transfer of a sparingly soluble gas. Some experiments results will also be presented to indicate the validity of the direct numerical simulations and the proposed model. Of necessity, this review is rather brief and focuses on work in our own laboratory, but the work in Prof. Komori’s laboratory at Kyoto University indicates similar results (Komori et al. 1993, Komori 1999, and Zhao et al. 2003).

2 The Canonical Problem and Turbulence Phenomena Consider Figure 2-left, which sketches the problem in general terms. A liquid layer flows over a bottom boundary, and the gas flow can either be cocurrent, countercurrent, or not flow at all. This is the situation that is to be

DNS of Turbulence and Scalar Exchange at Gas-Liquid Interfaces

5

simulated, and the gas-liquid interface can develop waves, which interact with the flows, which themselves can be turbulent. If one looks down at the free surface from the top when the flowing liquid is turbulent, then, in the absence of gas shear, the structures shown in Figure 2-right are obtained. These consist of upwellings, which are essentially bursts emanating from the bottom boundary, vortices associated with these upwellings, and downdrafts, which are not visible clearly in this figure but are nonetheless present (see Banerjee 1994). The vortical structures can merge, if they have a like rotation direction, into larger vortices. Sometimes these structures can annihilated by an upwelling. In the next section, we will discuss these vortices based on what is learned from DNS studies and also present some interesting findings regarding the structure of turbulence close to such unsheared free surfaces. It is worth noting at this point that, as shown in Figure 3, the bursts that are generated at the boundary. Upwellings arise from these bursts. Turning now to turbulence phenomena that occur when wind shear is imposed, Figure 4 shows that the free surface region also generates burst-like structures. The top three panels in Figure 4 are for countercurrent gas flow and the bottom three for cocurrent gas flow (Rashidi and Banerjee, 1990). The interesting finding here is that, in spite of the boundary condition at the gasliquid interface being quite different from that at the wall, the qualitative

Fig. 2. Left: Schematic of canonical problem studied here. Gas can flow co- or countercurrent to liquid stream, or not at all. Right: Plan view of free surface structures. Dark areas are upwellings from bottom boundary. Note associated vortices.

Fig. 3. Burst emanating from the bottom boundary (Fig. 2). The free surface is at top of each panel and is unsheared (no gas flow). View looks sideways at the flow.

6

Sanjoy Banerjee

turbulence structure under shear conditions are rather similar, though of course different in detail. If the flow is visualized from the top, then streaky structures, similar to those in turbulent shear flow near solid walls, are also seen at the interface with increasing wind shear. We turn now to discussion of direct numerical simulations of such phenomena.

Fig. 4. Bursts generated both at the top of the panels where the free surface is located and at the bottom boundary. The gas flow is countercurrent for the top three panels and cocurrent for the bottom three.

3 DNS - No Interfacial Shear Consider the situation where the liquid flows in the channel with a free surface and there is no wind stress. Turbulence is generated at the bottom boundary and impinges on the free surface, causing upwellings and associated vortical structures. Much can be learned about the nature of these structures by assuming the interface to be a flat slip surface. This was done in studies by Lam and Banerjee (1992), and Pan and Banerjee (1995). The DNS were essentially similar to those done for channel flows by a number of investigators, except that one of the walls was rigid. There are, of course, small deformations of this surface, but while these are perhaps important for gas transport, in this early work they were not considered. Nonetheless, some interesting results were obtained from these DNS, as illustrated in the following figures. Figure 5-left indicates the structures that form with an upwelling. It appears that vortices, which form rings in the shape of something like a distorted donut around the upwelling as it nears the interface, break and attach to the interface so that the axes of the vortices become normal. Thus, usually two vortices form, associated with each upwelling. These vortices then merge, if there are of like sign and given a sufficient length of time, or counter rotate if they are unlike


7

sign, as indicated in Figure 5-right. As a consequence, these vortical structures are very long lived, dissipate slowly and are essentially two-dimensional, sort of hanging from the interface, as illustrated in Figure 6. Furthermore, as smaller vortices merge into larger vortices, there is an upscattering of energy or a reverse cascade over some part of the energy spectrum. This is also seen in experiments (Kumar et al., 1995) and in DNS (Pan and Banerjee, 1995).

Fig. 5. Left: DNS of an upwelling interacting with an unsheared free surface. The vortical structures associated with the upwelling connect to the free surface and form a pair of counter-rotating vortices. Right: DNS of free-surface turbulence showing vortical structures merging when of like sign. (from Pan and Banerjee, 1995)

Fig. 6. left: Vortical structures attached to an unsheared free surface, on which the instantaneous streamlines are also shown from the free-surface DNS of Pan and Banerjee, 1995. right: Streaky structures from a sheared free surface, top panel just above upwellings; bottom panel just below (De Angelis et al., 1999)

8

Sanjoy Banerjee

If the forcing due to the upwellings is stopped by removing the bottom boundary and letting the turbulence near the free surface decay, then the large hanging vortices approximate a two-dimensional turbulence field and the finer 3-D structure decay away fairly rapidly. However, the larger vortices persist for considerable times and can trap particles or bubbles, if these are available. It is thought that this is the reason why ship wakes are discernable over considerable distances in satellite images of sea glint, as the large-scale vortices formed attach to the free surface as shown in Figure 6-left and persist for long times. They trap microbubbles in the wake, and associated surfactants, causing a change in the capillary wave structure, which can be remotely sensed. We now consider the effects of wind shear.

4 DNS - Sheared Interfaces When wind stresses are imposed on the liquid surface, then the situation becomes more complex and interfacial waves start to have a significant effect. If the waves are not of high steepness and are in the capillary or capillary gravity range, then DNS can be conducted by using a fractional time-step method in which each of the liquid and gas domains are treated separately and coupled to interfacial stress and velocity continuity boundary conditions. In addition, at each time step the domains must be mapped from a physical coordinate system into a computational system, which is rectangular. While the procedure is somewhat laborious, De Angelis et al. (1999) have done extensive testing validating convergence and accuracy, provided the steepness is small enough that the interface does not break. Such simulations are valid for gas-side frictional velocities less than about 0.1ms−1 . If one considers typical wind conditions, frictional velocities of about 0.1ms−1 over open water corresponds to a 10m wind speed of about 3.5ms−1 . Above these velocities, the interface starts to break and the DNS conducted in the manner described become inaccurate or infeasible. However, something can be learned from these DNS, and the typically structures seen in the vicinity of the interface are shown in Figure 6-right. The streaky structures (high speed/low speed regions) seen in the experiments are reproduced here and therefore both on the gas and liquid sides one sees qualitatively similar behavior to that near a wall. However, the details of the intensities are different. If one looks on the gas side, as shown in Figure 7-left, the near-interface intensities look something like a channel flow, with the intensities becoming rather small at the interface when scaled with the gas-side friction velocity. On the other hand, on the liquid side one sees that the intensities peak in the streamwise and spanwise directions and, of course, go to zero in the interface-normal direction, provided that the coordinate system is set relative to interfacial position. If scalar exchange such as gas transfer is studied with DNS, then, if the phenomenon is gas-side controlled, then the mass (or heat flux) is highly correlated with the interfacial shear stress. However, on the liquid side it is


9

not. The DNS indicates that ejections and sweeps, which can be discerned by a quadrant analysis, correlate with the shear stress on the gas side but do not on the liquid side, as shown in Figure 8.

Fig. 7. Turbulence intensities near the gas-liquid interface from De Angelis et al. (1999). Left: gas side. Right: liquid side.

Fig. 8. Left: Gas-side probability density of sweeps and ejections vs. interfacial shear. Center: Liquid-side distribution of ejections and sweeps vs. interfacial shear. (De Angelis et al., 1999). Right: Quadrant II events (ejections) take low-speed fluid away from the interface and quadrant IV events take high-speed fluid towards it.

Note here that the ejections on the gas side arise over the low shear regions, whereas the sweeps, which bring high-speed fluid towards the interface, are associated with high-shear stress regions. This is to be expected, but the same phenomenon does not appear on the liquid side. This is due to the gas driving the shear-stress pattern. Therefore, on the gas side the high-shear stress regions may be expected to correspond to regions of high mass or heat transfer and the low-shear stress regions to low scalar transfer. However, on the liquid side the shear stress patterns are not an indicator of mass transfer rates, nor are they an indicator of ejections or sweeps. Therefore, one has to look at the DNS directly to determine what controls scalar exchange rates when it is liquid-side dominated. De Angelis et al. showed that they were also dominated by the sweeps, with high exchange rates occurring with the sweep, whereas low exchange rates occur with ejections. Since the frequency of sweeps and

10

Sanjoy Banerjee

ejections scale with the frictional velocity on each side of the interface, De Angelis et al. were able to develop a relationship between frictional velocity and scalar exchange rates, as shown in Equations 1 and 2 below. These correspond very well with experimental data at low wind velocities, i.e. u10 < 3.5 ms−1 (more details in their paper). However, there is another way to interpret the DNS, as discussed in the next section, which has a better chance of working under breaking conditions. ∗ βw Sc0.5 w /uf rict,w ∼ 0.1

(1)

with the subscript ‘w’ denoting the liquid side and u∗f rict ∼ (Df /ρ) where ρ is fluid density and Df is the frictional drag, Sc = ν/D is the Schmidt number, with ν being the kinematic viscosity. An equivalent expression was derived for the gas side, with subscript ‘a’ denoting the gas side, in the form ∗ βa Sc2/3 a /uf rict,a ∼ 0.07

(2)

5 The Surface Divergence Model and Microbreaking Regions of high divergence occur in the upwellings which bring fresh bulk liquid to the interface, and regions of convergence form at the downdrafts, which are not apparent in the Figure 2-right, but can be seen clearly in direct numerical simulation (Pan & Banerjee 1995). When the interface is sheared, bursts consisting of ejections (events which take interfacial fluid away) and sweeps (events which bring bulk fluid to the interface) as shown in Figure 4, much like in wall turbulence. Sweeps form regions of high surface divergence, and ejections, of high convergence. The surface divergence model is based on the realization that the region near the interface that controls scalar exchange is of very small interface-normal dimensions. Typically, the thickness of the “film” over which the resistance to transfer lies, is ∼ O(100 μm) for transfer of sparingly soluble gases such as CO2 and methane (De Angelis et al. 1999). For the gas-side controlled processes it is ∼ O(1 mm). It is the motion in these very thin regions that dominate the transport processes. For a gas-liquid interface, the turbulent fluctuations parallel to the interface proceed relatively unimpeded (in the absence of surfactants), a fact validated by direct numerical simulations with the exact stress boundary conditions at a deformable interface (De Angelis et al. 1999; Fulgosi et al. 2003). For example, even with mean shear imposed on the liquid by gas motion, the surface-parallel turbulent intensities peak at the interface (Lombardi et al. (1996), whereas in the gas it peaks some small distance away, as in wall turbulence. Therefore, to a first approximation, the liquid-side interface-normal velocity, ‘w’, can be written w ∼ ∂w /∂z|zint + HOT

(3)

with z being the surface-normal coordinate. This can be related to the divergence of the interface-parallel motions at the surface, as


∂w ∂z

int

∂v ∂u + ≡γ ∂x ∂y int

11

(4)

where the quantity in parentheses is the surface divergence of the surface velocity field fluctuations, γ. x and y are streamwise and spanwise coordinates tangential to the moving interface, and z is normal. In a fixed coordinate system, Eq. (3) requires an additional term due to surface dilation as discussed by Chan & Scriven (1970) and Banerjee et al. (2004). For stagnation flows, interface-parallel motions and diffusion may be neglected, as pointed out by Chan & Scriven (1970), who showed a similarity solution existed in this case. Clearly, a solution of the problem of turbulence giving rise to the surface divergence field, in which γ would vary as a function of position and time, is not possible analytically, but McCready et al. (1986) in a landmark paper showed that the root mean square (rms) surface divergence was nonetheless approximately related to the average transfer coefficients by 2 1/4 ∂v β ∂u −1/2 1/2 −1/2 ∼ Sc + Ret or in dimensional term, β ∼ [Dγ] u ∂x ∂y int (5) from numerical simulations using typical postulated turbulence energy spectra. Here is the rms γ. These ideas were further developed by Banerjee (1990), who related the surface divergence field to the bulk turbulence scales using Hunt and Graham’s (1978) blocking theory, leading to the expression 1/4 β −1/2 3/4 2/3 ≈ Sc−1/2 Ret 0.3 2.83 Ret − 2.14 Ret u

(6)

Ret is the turbulent Reynolds number based on far-field integral length scale, Λ, and velocity scale, u. Note that this is essentially a theoretical expression for γ, and the constants arise by fitting a much more complex expression to this simplified form (Banerjee et al. 2004). Hunt and Graham’s theory connects bulk turbulence parameters to those near the interface by superposing an image turbulence field on the other side, which impedes surface normal motions, redistributing the kinetic energy to surface parallel motions, which are enhanced. The predictions have been well verified in experiments (Banerjee, 1990). The theory also allows a connection to be made between the bulk turbulence parameters and surface divergence. This last model has been used by researchers conducting gas transfer field experiments to correlate data and it applicability has been reviewed by Banerjee and MacIntyre (2004), where it is called the SD model. Forms of the surface divergence model were also tested for gas transfer at the surface of stirred vessels by McKenna and McGillis (2004), as well as recently in direct simulations by Banerjee et al. (2004) − in both cases, with success. Most recently, Turney et al. (2005) tested a form of the model for gas transfer across interfaces, with wind forcing sufficiently high to induce microbreaking, and found that it agreed with data in both microbreaking and non-microbreaking conditions.

12

Sanjoy Banerjee

We will now consider the accuracy of Eqs. 5 and 6 against both experiments and DNS. First, turning to experiments. A typical measurement of the surface divergence field made by top-view DPIV of glass micro balloons floating at the liquid surface is shown in Figure 9. The wind conditions are indicated by u∗ in the figure, and one can see the surface divergence patterns becoming more pronounced as the wind velocity is increased. Strong convergence zones form just upstream of microbreaking wave crests, and divergence zones form behind. Surface topography measured by a shadow-graph method (Turney et al.), from which wave slopes can be calculated, and the surface is seen to roughen substantially as wind velocity is increased, coinciding with the onset of microbreaking − the definition of which is the movement of surface water at the same speed as the wave crest. For example, particles scattered in a liquid surface converge at a microbreaking wave crest and are swept along with it. A typical profile of a microbreaking wave is shown in Figure 10, together with a top down view showing particle accumulation just ahead of the wave crests.

Fig. 9. Plan-view image of the air-water interface with increasing wind speed − u∗ is the wind side friction velocity. Grayscale represents velocity divergence. White dashed lines are wave crests. When microbreaking commences convergence is seen ahead of wave crests and divergence behind. Figure from Turney et al. (2005).

The surface divergence model presented in Eq. (5) appears promising from the comparison in microbreaking conditions with measured gas transfer rates by Turney et al. (2005). The connection between surface divergence and bulk turbulence parameters shown in Eq. (6) has been tested in DNS of windforced flows (without breaking) (Banerjee et al., 2004). While the model was expected to be applicable primarily to unsheared interfaces, it appears to work surprisingly well for wind-forced flows as well. The relationship between the rms surface divergence and the mass transfer coefficient is shown in Figure 11a), from Turney et al. (2005). In Figure 11b) we also show the relationship between surface divergence and mean surface wave slope, which may be measured by satellite remote sensing methods, as discussed in Turney et al.


13

Fig. 10. a) Side-view of a microbreaking wave, showing typical scale. b) Plan-view of a microbreaking wave, visualized with floating particles. Particles accumulate in the convergence zones ahead of the wave crests.

Fig. 11. a) Mass transfer coefficient vs. surface divergence b) surface divergence vs. mean-square wave slope.

In Figure 12, some results of a direct numerical simulation are shown for wind forcing without breaking. The connection between bulk turbulence parameters and surface divergence given in Equation (6) is shown to be valid.

6 Conclusions This review has covered flat interface DNS, in which quasi 2-D structures were seen near the gas-liquid interface in the absence of wind shear. These structures lead to turbulent energy upscattering and give rise to persistent interface-normal vortices that may explain why ship wakes are observed to persist over long distances. A second DNS method was then introduced which was based on boundary fitting and which successfully captured the behavior of low-steepness interfacial waves and the interaction with the underlying

14

Sanjoy Banerjee

Fig. 12. a) Interface configuration for the coupled gas-liquid DNS of Banerjee et al. (2004). b) The surface divergence calculated from the DNS of Banerjee et al. (2004) vs. the prediction in eqn. (6) (shown as the ordinate).

turbulence. The method worked at relatively low wind stress imposed on the free surface, corresponding to u10 < 3.5ms−1 . Above these wind speeds, the DNS indicated interface breaking, which it could not capture. This also corresponds to what is observed in field experiments. Nonetheless, the DNS elucidated the structures that control scalar exchange rates on the gas and liquid side, indicating that sweeps were important in both. This ultimately led to a parameterization that appears to be accurate at low wind speeds. When wind speeds exceeded these conditions, i.e. led to gas-side friction velocities of ∼0.1ms−1 on the surface, waves of lengths ∼10cm, and amplitudes ∼1cm started to “microbreak”, leading to regions of convergence at the wave crests and divergence behind − as if the surface fluid was being sucked under the waves, rolling over the surface. These waves led to qualitative increases in scalar exchange rates on the liquid side, e.g. adsorption of gases such as CO2 . For conditions that did not cause microbreaking, simple parameterizations for scalar exchange rates in terms of friction velocity were derived. from the DNS. These applied only at low wind speed conditions. A more universal approach was needed, and the surface divergence model was assessed, both against DNS and experimental results, which correlated a broad range of effects, including those due to microbreaking. In addition the surface divergence was found to be related to the mean square wave slope which might serve as a surrogate for gas transfer measurements, at least at typical wind speeds of 7.5ms−1 , which is the average for the oceans, and well beyond the limit where surfactants might be expected to affect transfer rates. Should this be proven in field experiments then it would be possible to obtain reliable regional and global estimates of carbon dioxide uptake by the oceans based on remote sensing of wave slope for lengths < O(1m). Thus the DNS, which is done at scales


15

of a few cm, may contribute in this instance to better estimates of what might happen at scales of many hundreds of kilometers.

References 1. Banerjee, S. “Turbulence structure and transport mechanisms at interfaces” In Ninth international heat transfer conference, keynote lectures 1, p. 395. New York, Hemisphere Press, 1990 2. Banerjee, S. “Upwellings, downdrafts, and whirlpools: Dominant structures in free surface turbulence” In Mechanics USA 47 (ed. A. S. Kobayashi). Seattle, Applied Mechanics Reviews, 1994 3. Banerjee, S., D. Lakehal & M. Fulgosi. “Surface divergence models for scalar exchange between turbulent streams” Int. J. Multphase. Flow 30, 963. 2004 4. Banerjee, S. & S. MacIntyre. “The air-water interface: Turbulence and scalar exchange” In Advances in coastal and ocean engineering 9 (ed. P. L. F. Liu), p. 181, World Scientific, 2004 5. Chan, W. C. and L. E. Scriven. “Absorption into irrotational stagnation flow. A case study in convective diffusion theory” Ind. Eng. Chem. Fund. 9, 114. 1970 6. De Angelis, V., P. Lombardi, P. Andreussi & S. Banerjee. “Microphysics of scalar transfer at air-water interfaces” In Proceedings of the ima conference: Wind-over-wave couplings, perspectives and prospects (ed. S. G. Sajjadi, J. C. R. Hunt & N. H. Thomas), p. 257, Oxford Univ. Press, 1999 7. Donelan, M. A. & R. H. Wanninkhof. “Gas transfer at water surfaces concepts and issues” In Gas transfer at water surfaces (ed. M. A. Donelan et al.), p. 1. Washington D.C., American Geophysical Union, 2002 8. Fulgosi, M., D. Lakehal, S. Banerjee & V. De Angelis. “Direct numerical simulation of turbulence in a sheared air-water flow with a deformable interface” Journal of Fluid Mechanics 482, 319. 2003 9. Hunt, J. C. R. “Vorticity dynamics in the water below steep and breaking waves” In Wind over waves (ed. S. G. Sajjadi & J. C. R. Hunt). London, Horwood Publ. Ltd., 2003 10. Hunt, J. C. R. & J. M. R. Graham. “Free-stream turbulence near plane boundaries” J. Fluid Mech. 84, 209. 1978 11. Kelly, J. “Thermal-hydraulic modelling needs for advanced light water reactors with various safety systems” OECD/CSNI Specialists’ Meeting in Adv. Instrumentation and Meas. Techniques, Santa Barbara, 1997 12. Komori, S. “Turbulence structure and mass transfer at a wind-driven airwater interface” In Wind-over-wave couplings: Perspectives and prospects (ed. S. G. Sajjadi et al.), p. 69. New York, Oxford University Press, 1999 13. Komori, S., R. Nagaosa & Y. Murukami. “Turbulence structure and mass transfer across a sheared air-water interface in wind-driven turbulence” J. Fluid Mech. 249, 161. 1993

16

Sanjoy Banerjee

14. Kumar, S. & S. Banerjee. “Development and application of a hierarchical system for digital particle image velocimetry to free-surface turbulence” Phys. Fluids 10, 160. 1998 15. Lakehal, D., M. Fulgosi, G. Yadigaroglu & S. Banerjee. “Direct numerical simulation of turbulent heat transfer across a mobile, sheared gas-liquid interface” J. Heat Trans-T ASME 125, 1129. 2003 16. K. Lam and S. Banerjee, 1992, “On the Conditions of Streak Formation in Bounded Flows,” Phys. Fluids , A4, 306-320. 17. Liss, P. S. & L. Merlivat. “Air-sea gas exchange rates: Introduction and synthesis” In The role of air-sea exchange in geochemical cycles (ed. P. Buat-Menard), p. 113, D. Reidel Publishing Company, 1986 18. Lombardi, P., V. DeAngelis & S. Banerjee. “DNS of near-interface turbulence in coupled gas-liquid flow” Phys. Fluids 8, 1643. 1996 19. MacIntyre, S., R. H. Wanninkhof & J. P. Chanton. “Trace gas exchange across the air-water interface in freshwater and coastal marine environments” In Methods in ecology-biogenic trace gases (ed. P. A. Matson & R. C. Harris), p. 52. New York, Blackwell Science, 1995 20. McCready, M. J., E. Vassiliadou & T. J. Hanratty. “Computer-simulation of turbulent mass-transfer at a mobile interface” AIChE J. 32, 1108. 1986 21. McKenna, S.P. & W.R. McGillis. “The role of free-surface turbulence and surfactants in air-water gas transfer” Int. J. Heat Mass Tran. 47, 539. 2004 22. Nightingale, P. D., G. Malin, C. S. Law, A. J. Watson et al. “In situ evaluation of air-sea gas exchange parameterizations using novel conservative and volatile tracers” Global Biogeochemical Cycles 14, 373. 2000 23. Pan, Y. & S. Banerjee. “A numerical study of free-surface turbulence in channel flow” Phys. Fluids 7, 1649. 1995 24. M. Rashidi and S. Banerjee. “Streak Characteristics and Behavior Near Wall and Interface in Open Channel Flows,” J. Fluids Eng,. 112, 164. 1990 25. Turney, D., W. C. Smith & S. Banerjee. “A measure of near-surface turbulence that predicts air-water gas transfer in a wide range of conditions,” Geophys. Res. Lett. 32, LO4607, 2005. 26. Wanninkhof, R. H. “Relationship between wind-speed and gas-exchange over the ocean” J. Geophys. Res.-Oceans 97, 7373. 1992 27. Wanninkhof, R. H. & W. R. McGillis. “A cubic relationship between airsea CO2 exchange and wind speed” Geophys. Res. Lett. 26, 1889. 1999 28. Zhao, D., Y. Toba, Y. Suzuki & S. Komori. “Effect of wind waves on airsea gas exchange: Proposal of an overall CO2 transfer velocity formula as a function of breaking-wave parameter” Tellus Series B 55, 478. 2003

Large Eddy Simulation of Premixed Turbulent Combustion: FSD-PDF modeling Luc Vervisch1 and Pascale Domingo1 INSA and Université de Rouen, UMR-CNRS-6614-CORIA Campus du Madrillet, Avenue de l’Université, BP 8, 76801 Saint Etienne du Rouvray Cedex, France [email protected], [email protected]

1 Introduction Large Eddy Simulation (LES) of premixed turbulent combustion is discussed. In LES, spatial filtering introduces SubGrid Scale (SGS) fluctuations associated to SGS energies of velocity and scalars that are unresolved by the grid. Numerous questions can be raised concerning LES, as listed by Pope [1]. In the present paper, the Lagrangian Dynamic SGS model by Meneveau et al [2] is used for turbulent transport, and a new formulation for presuming the Probability Density Function (PDF) of a reaction progress variable is discussed. The PDF is used to account for the impact of SGS fluctuations on chemistry, when it is tabulated with laminar flamelets as in FPI or FGM methods [3, 4]. The ratio Δ/δL , between the filter size and the characteristic thickness of the premixed flame, which evolves within the subgrid, is one of the important ingredients of LES of premixed turbulent combustion. In other words, it is needed to introduce in the presumed PDF the influence of the spatially filtered thin reaction zone evolving within the subgrid. In the novel closure, this is achieved via the exact relation between the PDF and the Flame Surface Density (FSD). This relation involves the conditional filtered average of the magnitude of the gradient of the progress variable. Because a laminar premixed flame filtered at the size Δ contains some scaling information on flame filtering, the conditional filtered mean, relating PDF and FSD, is approximated from the filtered gradient of the progress variable of the laminar flame used for chemistry tabulation. Balance equations providing mean and variance of the progress variable, together with the measure of the filtered gradient are used to presume the PDF. A priori test from V-Flame DNS are first performed. Then, a threedimensional flow configuration (ORACLES experiment) is computed with FSD-PDF and the results are compared with measurements.

18

L. Vervisch & P. Domingo

2 A Flame Surface Density-Probability Density Function SGS closure It is proposed to capture the SGS behaviour of the thin premixed flame from three control parameters: A filtered reaction progress variable c (c = 0 in fresh − c c, and, gases and c = 1 in fully burnt products), its SGS variance cv = cc GΔ (c), a filtered reference distribution of spatial gradient of that progress variable. These three quantities are used to presume P(c∗ ; x, t), the PDF of the progress variable. Once the PDF is approximated, all SGS terms related to chemistry (filtered species and filtered chemical sources) may be computed as: 1 (1) Φ(x, t) = ΦF P I (c∗ )P(c∗ ; x, t)dc∗ 0 FPI

∗

where Φ (c ) is the response of the quantity Φ in the FPI [3] tabulation of detailed chemistry with the progress variable c. Because in LES Δ/δL >> 1, the flame is weakly resolved and the PDF features a shape close to its Bi-Modal-Limit, as in BML RANS modeling [5]: P(c∗ ; x, t) = α(x, t)δ(c∗ ) + β(x, t)δ(1 − c∗ ) + F (c∗ ; x, t)

(2)

where the function F (c∗ ; x, t) depends on the SGS statistical properties of the iso-c∗ surfaces evolving between c∗ = 0 and c∗ = 1. The accuracy of the prediction of the PDF in this internal part of the flame is crucial since intermediate species present in detailed chemistry feature strong variations in this zone. The Flame Surface Density Σ(x, t) has been used for LES of premixed turbulent combustion by Hawkes and Cant [6] and Charlette et al [7]. Σ(c∗ ; x, t) is related to the SGS PDF by [8]: Σ(c∗ ; x, t) P(c∗ ; x, t) = |∇c||c∗

(3)

In this last expression, Σ(c∗ ; x, t) the FSD depends on the iso-surface c∗ . In premixed combustion, because the flame is locally thin, it is usually assumed that within the range 0 < c∗ < 1, species iso-surfaces stay mostly parallel to each other, a behaviour confirmed by DNS [9]. Accordingly, Σ(c∗ , x, t) weakly varies in the internal part of the diffusive-reactive layer and one may write Σ(c∗ , x, t) ≈ σ(x, t), where σ(x, t) does not depends on c∗ . σ(x, t) characterises the amplitude of the FSD, which strongly depends on the magnitude of subgrid scale flame wrinkling and on the level of flame resolution. Combining those observations with Eq. 3, it is proposed to approximate the function F (c∗ ; x, t) of the PDF (Eq. 2) in the form: F (c∗ ; x, t) =

σ(x, t) H(c∗ )H(1 − c∗ ) GΔ (c∗ )

(4)

FSD-PDF for LES of Premixed Turbulent Combustion

19

The Heaviside function is introduced to limit the contribution of F (c∗ ; x, t) to ∗ ∗ the internal part of the composition space (0 < c < 1). GΔ (c ) approximates |∇c||c∗ as a gradient distribution measured from the FPI reference planar and unstrained flame that has been filtered at the level Δ: +∞ GΔ (c(x)) = |∇c|F P I GΔ (x − x )dx (5) −∞

Gradient of progress variable

1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Progress variable

Fig. 1. Normalized gradient of progress variable, propane/air mixture at equivalence ratio 0.75. Line: |∇c|F P I . Dash line: GΔ (Eq. 5) Δ/δf l = 5, Dotted line: Δ/δf l = 10, Dot-dash line: Δ/δf l = 20 Double dot-dash line: Δ/δf l = 50.

Figure 1 shows GΔ (c) for increasing values of Δ/δf l . The maximum of GΔ decreases when the filter size increases, up to a point where the distribution of the filtered gradient becomes a plateau, a response typical of a flame whose progress of reaction mainly occurs within the subgrid. Here, as in the ORACLES experiment, propane/air combustion was considered at equivalence ratio 0.75. The GRI mechanism [10] optimised for propane air combustion according to Qin et al [11] is used with the PREMIX [12] complex chemistry flame solver for simulating the FPI reference flame. c = Yc /YcEq , the reaction progress variable was defined after normalising Yc = YCO + YCO2 by YcEq its equilibrium value measured in fully burnt products. Equation 4 therefore appears as a possible candidate to inform the PDF on the scale at which the flame is seen by the coarse LES grid. Combining Eq. 2 with Eq. 4, the SGS progress variable FSD-PDF reads: P(c∗ ; x, t) = α(x, t)δ(c∗ ) + β(x, t)δ(1 − c∗ ) +

σ(x, t) H(c∗ )H(1 − c∗ ) GΔ (c∗ )

(6)

The presumed PDF relies on three parameters, α, β and σ. They may be determined at every location and instant in time from three constraints that

20


the PDF must fulfil: 1

P(c∗ )dc∗ = 1 ;

0

1

c∗ P(c∗ )dc∗ = c;

0

1

c∗2 P(c∗ )dc∗ = c2

(7)

0

These three equations bring the relations: α + β + aσ = 1 ; β + bσ = c ; β + dσ = S c(1 − c) + c2

(8)

c(1 − c) represents the unmixedness. The coefficients a, b and where S = cv / d are fully determined from: 1−

a=

1 dc∗ ; b = GΔ (c∗ )

1−

c∗ dc∗ ; d = GΔ (c∗ )

1−

c∗2 dc∗ GΔ (c∗ )

(9)

is a small parameter used to define the inner zone of the premixed flamelet in composition space, = 0.001 was retained in this study. Tests have been performed where this parameter was slightly varied, LES results are not strongly affected as long as < 0.01. After straightforward manipulations, the coefficients of the presumed PDF are determined from the filtered progress variable and corresponding SGS variance expressed in terms of unmixedness: a−b (1 − S) c(1 − c) α = (1 − c) − b−d b (1 − S) c(1 − c) β= c− b−d 1 (1 − S) c(1 − c) σ= b−d These relations fail for negative values of α or β. Leading to the additional constraints: b−d 1 b−d 1 S >1− ; S >1− (10) a−b c b 1− c Consequently, the SGS presumed PDF proposed in Eq. 6 cannot be used for low values of the unmixedness (i.e. small levels of cv ). For moderate levels of SGS fluctuations, the shape of the SGS PDF is expected to be much less crucial, as it is used to average over values that are close to the filtered level c. Among the available options, a Beta-PDF [13] is retained to deal with points having low level of SGS variance.

3 A-priori test of FSD-PDF from V-Flame DNS To assess the validity of FSD-PDF (Eq. 6) combined with FPI chemistry tabulation, preliminary a-priori tests from a V-Flame DNS database described

FSD-PDF for LES of Premixed Turbulent Combustion 5

0.002

4

PDF

0.0015 Modeling

21

0.001

3 2

0.0005 1 0

0

0.0005

0.001

0.0015

0.002

0

Filtered DNS

(a)

0

0.2

0.4 0.6 Progress variable

0.8

1

(b)

Fig. 2. A priori DNS test of the FSD-PDF closure. (a): Predicted OH mass fraction DN S ). Line: Exact prediction. Square: FSD-PDF. Cir(YOH ) versus filtered DNS (YOH = (0.60,0.74). Line: FSD-PDF. cle: Beta-PDF. (b): PDF of progress variable at ( c, S) Dash-Line: Beta-PDF.

elswhere [14] are first performed. The predictions obtained with both FSDPDF and a Beta-PDF are compared with DNS. Those DNS fields are filtered cDN S , with the characteristic length Δ = 30δL . The filtering operation brings DN S and Y DN S . The first and second moments of the progress variable c2 i

DN S ) are control parameters of the modeling, they enter the pre( cDN S , c2 sumed PDF to provide Yi from Eq. 1. This approximation of the filtered mass fraction can then be compared with YiDN S , the DNS reference. Figure 2(a) shows a comparison between the predictions obtained with the FSD-PDF and a usual Beta-PDF. An exact prediction would correspond in the graph to the line Yi = YiDN S . The prediction of OH is strongly sensitive to the shape of the PDF in the internal part of the flame (i.e. 0 < c < 1). This is visible in Fig. 2(a), where the FSD-PDF provides a OH response that is very close to the DNS value, while the Beta-PDF tends to overestimate OH mass fraction. Similar departures between DNS and Beta-PDF are observed for other species. This observation is further analysed in Fig. 2(b) showing both Betaand FSD-PDF for a representative point ( c, S) = (0.60,0.74). The Beta-PDF features a double delta shape with weak curvature close to the peaks. The plateau of the double delta shape is wider with the FSD-PDF, which is sensitive to the thin flame within the subgrid, and the curvature close to the peaks is much greater than with the Beta-PDF. The FSD-PDF is also non-fully symmetric. All these features are direct results of the effect of GΔ , the filtered distribution of |∇c| in FPI (Fig. 1), which is embedded in the FSD-PDF, to account for the existence of thin reacting layers within the subgrid.

4 Modeled balance equations When this SGS closure is introduced in a LES solver, multiple choices are possible to get the variance from a balance equation. To insure a generic

22


character to the modeling, specifically when coarse grids are used, the variance may be reorganized in: c c= c(1 − c) − ϕ c cv = c c −

(11)

ϕ c is the difference between the variance and its maximum value, it may also be cast in: ρϕc = ρc(1 − c) = ρ c(1 − c) (1 − S) (12) The introduction of ϕ c simplifies numerical treatments linked to the variance c(1 − c). For c = 0 and c = 1 and when that is a bounded quantity 0 < cv < cv reaches its highest theoretical value c(1 − c) (and ϕ c → 0, it is insured that c(1 − c) (and S = 0), S = 1). When ϕ c increases up to its maximum value the SGS variance vanishes. Moreover, the balance equation for ϕ c does not c ≈ 2ρ(νT /ScT )|∇ c|2 contain the modeled production source term −2τ c · ∇ found in the equation for cv . This term is proportional to the gradient of the resolved progress variable field. In the case of large values of Δ/δL (coarse grids), this gradient may not be accurate enough and can easily perturb the solving of the balance equation for cv . With usual notations, the equations used in FSD-PDF then read: c ∂ρ ˙ c + ∇ · (ρ u c) = ∇ · (ρ(D + (νT /ScT ))∇ c) + ρω (13) ∂t c ∂ρϕ + ∇ · (ρ uϕ c ) = ∇ · (ρ(D + (νT /ScT ))∇ϕ c ) ∂t νt cv c|2 + CD + 2ρ D|∇ ScT Δ2 ˙ c − 2ω ˙ cc (14) +ρ ω In the equation for ϕ c , the dissipation rate is a positive source (second term of c|2 , proportional to laminar diffusion, RHS of Eq. 14). Its resolved part, ρD|∇ is generally smaller than its modeled SGS part that is proportional to νT /Δ2 , when it is modeled by a linear relaxation closure as in Eq. 14. Notice however that the diffusive flux proportional to D cannot be dropped. Indeed when cv → 0, the simulation of a laminar flame with tabulated chemistry on a fine grid should be recovered. This is the case with the SGS modeling discussed here. CD is a constant for the scalar dissipation rate, CD = 1 is used for the simulations reported. The various parameters of SGS dynamic modeling closing the Navier-Stokes and progress variable equations (νT eddy viscosity, ScT turbulent Schmidt number) are evaluated from the Lagrangian dynamic procedure developed by Meneveau et al [2]. The chemical sources of Eq. 13 and ˙ c and ω ˙ c c, together with the source for the energy equation, are extracted 14, ω from the FPI table averaged with the PDF via Eq 1. Those terms are precomputed to generate a table of filtered sources whose input are c and S. The introduction of chemistry in the LES solver is then reduced to interpolation into the filtered FPI table.


23

Fig. 3. Instantaneous and time-averaged progress variable iso-contours. Flood: Twodimensional cut of the instantaneous field. White line: Time average.

5 LES of the ORACLES experiment The ORACLES experiment was specifically designed for LES validation by Nguyen and Bruel [15] at CNRS in Poitiers. It consists of two channels of fully premixed mixtures separated by a splitter plate and exposed to a sudden expansion. The channels are 3 meters long to obtain a fully developed turbulent channel flow. The combustion chamber is located after the sudden expansion and is 2 meters long with insulated walls. The outlet was carefully set to monitor the outflow rate of the rig (3.5 m3 /s). The bulk velocity is of the order of Ud = 11.0 m/s. The Reynolds number estimated from Ud , the height of the inlet channel H = 30.4 mm and the viscosity of fresh mixture at the reference temperature T0 = 276K, is of the order of 25 000. In the transverse direction, the depth of the channel is 150.5 mm. The flame is stabilised in the shear layers that develop after the sudden expansion and transverse distributions of mean velocity measurements are available at various streamwise locations. The reported simulation focusses on the case where the equivalence ratio is 0.75. in both streams. The fully compressible set of Navier-Stokes equations together with the balance equations for c and ϕ v (Eq. 13) are solved using a fourth order finite volume skew-symmetric-like scheme proposed by Ducros et al [16] for the spatial derivatives. This scheme was specifically developed and tested for LES, it is combined with a second order Runge Kutta explicit time stepping. Because the equations are solved in their fully compressible form, the time step limitation includes acoustic waves and Navier-Stokes Characteristic Boundary Conditions [17] are retained. The unsteady behavior of the ducted flame was carefully identified and calibrated by time series measurements [15]. For the case studied, a specific acoustic mode with large scale fluctuations at a characteristic frequency of the order of 50 Hz was found. To avoid simulating the full system, preliminary simulations of the channel flows, upstream of the main combustion chamber, were performed to generate proper inlet conditions. They are then perturbed

24

L. Vervisch & P. Domingo 4

y/h

3 2 1 0

01 .5 3 01 .5 3 01 .5 3 01 .5 3 01 .5 3 01 .5 3 x/h = 2 x/h = 4 x/h = 7 x/h = 8 x/h = 9 x/h = 10

4

4

4

4

4

3

3

3

3

3

3

2

2

2

2

2

2

1

1

1

1

1

1

0

0

0

0

0

0

0 0.1 −0.1 0 0.1 −0.1 0 0.1 −0.1 0 0.1 −0.1 0 0.1 −0.1 0 0.1

4

−0.1

y/h

(a)

x/h = 2

x/h = 4 x/h = 7

x/h = 8 x/h = 9 x/h = 10

(b) Fig. 4. Transverse distribution of time averaged velocity component. (a): Streamwise, (b): Spanwise. Symbol: ORACLES experiment. Line: LES.

according to the spectrum experimentally measured in the inlet plane of the combustion chamber [15], right after the splitter plate. The measured power density spectrum of the longitudinal velocity component exhibits a peak at 50 Hz. In the simulations, a sinusoidal forcing at this specific frequency is added to the velocity inlet to mimic the corresponding acoustic mode. Following this procedure, only 450 mm long of the facility is computed, just after the sudden expansion with a spanwise length of 70 mm. The three-dimensional non-uniform grid is composed of 128×96×32 nodes. In the region where the flame develops, the grid is such that Δ/δL ≈ 20 and this value is retained in the FSD-PDF for the filtered gradient GΔ (Eq. 5 and Fig. 1). To construct mean values, instantaneous LES fields are cumulated over twice the time elapsed when a fluid particle travels from inlet to outlet. Figure 3 presents an instantaneous field of c˜, the iso-lines of the time averaged filtered progress variable are also displayed for the values 0.5 and 0.8. Figures 4 and 5 show distributions of time average filtered streamwise, transverse, and RMS velocity. Most of the reacting flow structure is well reproduced, especially the streamwise

4

3

3

3

3

3

2

2

2

2

2

1

1

1

1

1

0

0

0

0

0

x/h = 4

x/h = 7

x/h = 8

x/h = 9

25

0 0.2 0.4 0.6

4

0 0.2 0.4

4

0 0.2 0.4

4

0 0.2 0.4

4

0 0.2 0.4

y/h


x/h = 10

Fig. 5. Transverse distribution of time averaged RMS velocity. Symbol: ORACLES experiment. Line: LES.

velocity that is correctly estimated. The quality of the prediction is slightly reduced in the spanwise direction at the streamwise location x/h = 10, but the overall distribution follows the measurements. The level of RMS velocity is also in agreement with experiment. This preliminary test of FSD-PDF is thus encouraging.

6 Summary A novel presumed probability density function strategy is proposed for LES of premixed turbulent combustion, in which the basic relation between the PDF and the Flame Surface Density is used. The objective is to introduce into the presumed PDF some information on the characteristic premixed flame that is filtered by the LES grid. To validate the subgrid scale modeling, three dimensional LES of a given case of the ORACLES experiment (flames stabilized in a sudden expension) is computed and results are compared with available measurements. The material presented in this paper is part of a manuscript prepared in celebration of Prof. R. W. Bilger’s 70th birthday.

References 1. S B. Pope. Ten questions concerning the large-eddy simulation of turbulent flows. N. J. Physics, 6, 2004. 2. C. Meneveau, T. S. Lund, and W. Cabot. A lagrangian dynamic subgrid-scale model of turbulence. J. Fluid Mech., pages 353–386, 1996. 3. B. Fiorina, R. Baron, O. Gicquel, D. Thevenin, S. Carpentier, and N Darabiha. Modelling non-adiabatic partially premixed flames using flame-prolongation of ildm. Combust. Theory and Modeling, 7(3):449–470, 2003.

26


4. J. A. van Oijen, F. A. Lammers, and L. P. H. de Goey. Modeling of complex premixed burner systems by using flamelet-generated manifolds. Combust. Flame, 127(3):2124–2134, 2001. 5. K. N. C. Bray. The challenge of turbulent combustion. Proc. Combust. Inst., 26:1–26, 1996. 6. E. R. Hawkes and R. S. Cant. Implications of a flame surface density approach to large eddy simulation of premixed turbulent combustion. Combust. Flame, 126(3):1617–1629, 2001. 7. F. Charlette, C. Meneveau, and D. Veynante. A power-law flame wrinkling model for les of premixed turbulent combustion part i: dynamic formulation. Combust. Flame, 131(1/2):181–197, 2002. 8. L. Vervisch, E. Bidaux, K. N. C. Bray, and W. Kollmann. Surface density function in premixed turbulent combustion modeling, similarities between probability density function and flame surface approaches. Phys. Fluids, 7(10):2496– 2503, 1995. 9. L. Vervisch, W. Kollmann, and K.N.C. Bray. Dynamics of iso-concentration surfaces in premixed turbulent flames. In Tenth Symposium on Turbulent Shear Flows, number 22-1, 1995. 10. C. T. Bowman, R. K. Hanson, W. C. Gardiner, V. Lissianski, M. Frenklach, M. Goldenberg, G. P. Smith, D. R. Crosley, and D. M. Golden. An optimized detailed chemical reaction mechanism for methane combustion and no formation and reburning. Technical report, Gas Research Institute, Chicago, IL, 1997. Report No. GRI-97/0020. 11. Z. Qin, V. V. Lissianski, H. Yang, W. C. Gardiner, S. G. Davis, and H. Wang. Combustion chemistry of propane: a case study of detailed reaction mechanism optimization. Proc. Combust. Inst., 28:1663–1669, 2000. 12. R. J. Kee, J. F. Grcar, M. D. Smooke, and J. A. Miller. A fortran program for modeling steady laminar one-dimensional premixed flames. Technical report, Sandia National Laboratories, 1992. 13. P. A. Libby and F. A. Williams. Turbulent combustion: Fundamental aspects and a review. In P.A. Libby and F.A. Williams, editors, Turbulent Reacting Flows, pages 2–61. Academic Press London, 1994. 14. L. Vervisch, R. Hauguel, P. Domingo, and M. Rullaud. Three facets of turbulent combustion modelling: Dns of premixed v-flame, les of lifted nonpremixed flame and rans of jet-flame. J. of Turbulence, 5(4):1–36, 2004. 15. P. D. Nguyen and P. Bruel. Turbulent reacting flow in a dump combustor: experimental determination of the influence of the inlet equivalence ratio difference on the contribution of the coherent and stochastic motions to the velocity field dynamics. In Paper 2003-0958, 41st Aerospace Sciences Meeting and Exhibit, Reno, USA, January 2003. AIAA, 2003. 16. F. Ducros, F. Laporte, T. Soulères, V. Guinot, P Moinat, and B. Caruelle. Highorder fluxes for conservative skew-symmetric-like schemes in structured meshes: application to compressible flows. J. Comput. Phys., 161:114–139, 2000. 17. T. Poinsot and S. K. Lele. Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys., 1(101):104–129, 1992.

Industrial LES with Unstructured Finite Volumes D. Laurence EDF R&D, MFTT, 78400 Chatou, FR. The Uni. of Manchester/MACE/Fluids, M60 1QD, UK.

Summary. Conservation of kinetic energy, while solving only for mass and momentum of incompressible flow, is first discussed in relation with the unstructured finite volume discretisations used in industrial and commercial software. LES applications are shown for a hot wall jet, a fan blade, U-bend pipe, a tube bundle and a T-pipe -junction, using local and systematic embedded grid-refinements or polyhedral cells. The conclusion suggests that ability to locally adapt the grids to the highly variable large eddy scales in complex geometries supersedes the need for higher order schemes or even elaborate subgrid-models.

1 Introduction At EDF R&D LES investigations started 20 years ago, motivated by the flow complexities in power plants (heat transfer, buoyancy, rotation, large and non-streamlined complex geometry), availability of experimental data, and because traditional turbulence models could not provide detailed knowledge of, e.g. extreme or cyclic thermal loading. These early simulations used the staggered pressure-velocity arrangement, which has the well known property of conserving both momentum and kinetic energy in a discrete sense on regular Cartesian grids. After these early Cartesian codes, production runs with RANS for complex geometries had naturally shifted to finite elements, for their unstructured griding flexibility, and LES attempts followed. However, as reported in RolletMiet et al. (1999), it was soon realised that the traditional P1-P0 tetrahedral element of EDF’s N3S FE code (linear velocity and constant pressure per element) was unsuitable for LES, because pressure actually requires more accuracy at high wave-numbers than velocity. A simple illustration of this is the classical Taylor vortex array test-case where the pressure wavenumber is double that of the velocity, and indeed Rollet-Miet et al. only successfully reproduced this vortex array on coarse grids using a collocated element (P1-P1) thus departing from FE practice. Finally the collocated finite volume approach

28

D. Laurence

is now used quite successfully. The presentation will tend to show that for industrial LES applications, the issue of numerical methods is perhaps more important than elaborate SGS models. Indeed, whenever increased computational resources become available, they are more often used to tackle even more complex problems, than to refine grids for academic testcases, hence only a small fraction of the energy spectrum is captured and sustaining resolved turbulence on coarse unstructured grid poses a challenge.

2 Energy conservation with collocated finite volumes One of the issues in discrete kinetic energy conservation is that continuous formulas such as div(pV )=p div(V ) + V grad(p), where p and V stand for the pressure and the velocity respectively, are not true in the discrete sense for most numerical schemes. Ferziger & Peric (1999) show that the fully staggered arrangement on Cartesian regular grids conserves energy. Energy conservation is also studied by Verstappen and Veldman (2003), while Ham and Iaccarino (2002) and Hicken et al. (2005) focussed on the collocated arrangement. Perot (2000) proposed a staggered discretization using tetrahedral elements which conserves global kinetic energy by using the normal components of the velocity located at the centre of faces and the pressure at the circumcentre. Simultaneously, Benhamadouche et al. (2002) introduced a staggered discretisation for tetrahedral elements and a rotational formulation of the non-linear term, rotV × V + 1/2.grad(V 2 ), where by ensuring that V.(rotV × V) = 0 in the discretised sense, and using a proper time discretisation, energy could be strictly conserved. However these methods are limited to tetrahedral meshes. This type of unstructured capability can be misleading because most numerical tests are usually performed on quasi equilateral cells, whereas most mesh generators, when applied to complex geometries, easily generate distorted cells degrading the accuracy of the scheme. It can be shown that a desired property of a triangle (whether with finite elements or finite volumes) is that its circumcentre lies close to the orthocentre, or at least inside the triangle, a feature which is lost as soon as one angle is larger or equal to 90˚, what can happen randomly, or systematically when e.g. elongated quadrangles are introduced in boundary layers, then split in 2 triangles. Nowadays, the general unstructured finite volume method is very popular in complex RANS production studies, thanks to its virtually unlimited meshing flexibility and is in fact quite appropriate for LES. At EDF R&D the Code Saturne software (Archambeau et al. 2004) was developed from 1998 and definitely replaced the finite element code N3S as of 2000. It uses a collocated finite volume approach - all variables are located at the centre of gravity of the cells (which can be of any shape). The generic convection-diffusion-source equation is written:

Industrial LES with Unstructured Finite Volumes

∂ ∂t

φdΩ +

ΩI

φ(u.n)dS =

∂ΩI

φ(u.n)dS = ∂ΩI

Γ (∇φ.n)dS + ∂ΩI

29

QdΩ

(1)

φIJ (uIJ .nIJ )SIJ

(2)

ΩI

J=neighbours

where ΩI is the considered CV, of surface ∂ΩI , J any neighbour CV of I, and IJ subscripts indicate values taken at the centre of the face separating cells I and J. The momentum equations are solved by considering an explicit mass flux (the three components of the velocity are thus uncoupled). The gradients at the cell centres are obtained by the Gauss theorem, but interpolating them to the faces leads to a very large stencil for the Laplacian. Instead cell-face gradients are obtained by exhibiting the finite difference operator along the normal to the face (see figure 1), then adding a correction: ∇φIJ .nIJ

φJ − φ I I I φ J − φI J J + ∇φ − ) ( IJ I J I J I J I J

Fig. 1. Unstructured collocated FV discretisation.

This relation is implicit as it uses the gradient on the r.h.s. but this correction being small is actually taken as explicit. That is, when a non-orthogonal grid is used, the matrix (collecting the implicit part) contains the orthogonal contribution only, and the non-orthogonal part is cast in the right hand side of the equation using ”previous” values. This is known as the deferred correction. However, one can iterate on the system to make it implicit (as in Code Saturne). Velocity and pressure coupling is ensured by a prediction/correction method with a SIMPLEC algorithm (Ferziger & Peric). The Poisson equation is solved with a conjugate gradient method. A second order centred scheme (in space and time) is used (Crank-Nicolson in general and Adams-Bashforth for mass fluxes). Cells can have any number of faces and any shape; hence non conforming local refinement can be used in boundary layers. In this case, no special

30

D. Laurence

treatment is needed for the apparent “hanging nodes”; the cell just has a larger number of faces, some of which happen to be in the same plane, as shown in figure 2. Note that, at variance with some recent “industrial LES” papers which tend to omit the factor 2 in the filter-width and rate of strain (thereby hiding unusually low values of CS , to compensate numerical diffusion), the traditional notations are used herein, e.g. for the Smagorinsky model: 1 τij − τkk δij = −2νt S ij = −2(CS Δ)2 S S ij 3 with S = 2S ij S ij , and Δ the filter length defined as Δ = 2Ω 1/3 , where Ω is the volume of a cell. The Smagorinsky constant, CS , is set to 0.065 for wall bounded flows and 0.17 for homogenous turbulence. A Dynamic model is also available in Code Saturne, but did not exhibit any advantages with the grids and applications shown in this paper.

Fig. 2. Non-conforming cells.

To illustrate some issues concerning energy conservation, consider the convection of a scalar φ, and the effects of the collocated FV scheme on conservation of its second moment φ2 . The discrete energy (in this case second moment) equation is obtained by multiplying the transport equation of φ, by φ at the appropriate time, for instance φn+1/2 when the Crank Nicolson time discretisation is used on the right hand side. If only convection is considered, as well as a periodic domain, the integral of φ2 over the whole domain should be conserved with time. For cell I in figure 1, the rate of change is: n+1/2

φI

.∂φI /∂t.ΩI = 12 (φn+1 + φnI ).(φn+1 − φnI ).ΩI /Δt I I = 12 ((φn+1 )2 − (φn+1 )2 ).ΩI /Δt I I

Let mIJ = SIJ u.n dS be the mass flux across the face between cells I and J, and φIJ = αIJ φ∗I + (1 − αIJ )φ∗J the interpolation of φI on that face.


31

φ∗I contains the non-orthogonality correction in interpolating from node I to I . αIJ is the interpolation weighing. If there is no grid stretching, i.e. I F = F J , then αIJ = 1/2 . The FV integrated convection term for cell I is: CI = ΣφIJ mIJ where the summation is carried out over all cells J sharing a face with cell I. Then the convection fluxes become (for cells I then J): n+1/2 n+1/2 ∗ n+1/2 ∗ CI = (φI φI αIJ mIJ + φI φJ (1 − αIJ )mIJ ) φI J=neighbours n+1/2

φJ

CJ =

n+1/2 ∗ φJ (1

(φJ

n+1/2 ∗ φI αIJ (−mIJ ))

− αIJ )(−mIJ ) + φJ

I=neighbours

Here we used the fact that αJI = 1 − αIJ and mJI = −mIJ . If the mass flux across the boundaries of cell I are such that incompressibility is en n+1/2 ∗ φI αIJ mIJ (as well as sured, then: mIJ = 0. Thus the sum of terms φI J

(φJ φ∗J ) will cancel locally for arbitrary values of the scalar if αIJ is constant, e.g. on constant mesh step: αIJ =1/2. Next when summing the second n+1/2 ∗ φJ from cell I moment over all cells in the domain, the contribution φI n+1/2 ∗ n+1/2 ∗ will cancel globally with term φJ φI from cell J only if φI = φI (and αIJ =1/2). This means that to prove energy conservation, the grid must be orthogonal (no correction from I to I ), with a constant step (αIJ =1/2) and the convected quantity at the face must be second order in time. Note that the mass flux mIJ may be explicit. Ham et al. (2002) suggested new interpolations on variable size Cartesian cells, then Ham et al. (2004) found an energy conservation formulation for unstructured grids, but used an extended neighbourhood. When φ corresponds to successive velocity components, pressure must be considered in addition. The pressure correction which is added after the convection-diffusion step affects energy conservation, but conservation is recovered by global iterations on pressure and velocity (Benhamadouche 2006). This also enables to update the mass flux to reach second order in time on the non-linearity (but this is not a necessary condition for energy conservation). A further complication arises from the Rhie and Chow correction to the pressure gradient, specific to collocated schemes and introduced to avoid checkerboard oscillations on structured grids. This is shown to introduce a systematic but small energy loss. It is noticeable on inviscid test cases, and although it can be omitted on truly unstructured grid, it makes little difference on real (viscous) LES test cases such as channel flow. n+1/2

3 HIT and Taylor array of vortices. Code Saturne has been intensively tested against Decaying Homogenous Isotropic Turbulence (DHIT) and the Taylor vortex array case (Benhamadouche

32

D. Laurence

2002 and 2006). On Cartesian meshes, the kinetic energy is exactly conserved when the viscosity is set to zero. Similarly for HIT, when starting from a k−5/3 spectrum the results tend toward a k2 spectrum as expected in the inviscid case, while in the standard DHIT case the Dynamic model returns the theoretical value of the Smagorinsky constant around 0.17-0.18. This shows that the present collocated FV discretisation conserves energy just as the staggered arrangement, but only on Cartesian, constant step grids. Addad (2005) performed the same tests on Cartesian grids with the StarCD code, which uses a similar discretisation, and showed that even with a commercial code a k2 spectrum can be recovered and stability maintained with full CDS. The time discretisation however is different and does not iterate on pressure velocity, therefore the recommended Smagorinsky constant in DHIT in that code is somewhat lower.

Fig. 3. Kinetic energy conservations for inviscid Taylor vortices case, with hexa (solid), polyhedral (dot) and tetra (chain line) cells in StarCCM.

An interesting development available in the new Star-CCM code is the polyhedral cells feature (Peric 2004). This is similar to the dual mesh that can be obtained from tetrahedral cells, but contains additional mesh smoothing. Results for the inviscid Taylor vortex array are shown in fig. 3 and although the polyhedral cell grid contains half the number of elements of the original tetra grid, the energy is almost perfectly conserved, as with the Cartesian mesh. Note that exactly the same algorithm is use on all 3 grids with StarCCM, therefore the numerical dissipation exhibited on the tetra grid can only be attributed to the non-orthogonality correction discussed above, whereas on


33

Fig. 4. Forward-backward step with non-conforming local refinement (bottom), pressure field (top), and iso-Q (middle).

the polyhedral grid-lines connecting cell centres remain almost perpendicular to the faces.

4 Applications Addad et al. (2003) first used non-conforming hexa grid refinement in combination with LES in the Star-CD code for the flow over a forward-backward facing step at Reh =1.7×105 , and resulting aero-acoustic noise. Using “hanging nodes” type grid expansion in all 3 directions resulted in only 260 000 cells. A Cartesian mesh with the same near wall resolution would have resulted in 7 Million cells. Star-CD results compared favourably with experiments, including for Re stresses, whereas the previous attempts using the old FE code and tetra grids failed to reproduce correctly the separation bubble on the leading corner. Reproducing the correct level of TKE is most certainly thanks to the conservation properties of regular Cartesian cells, while errors due to non orthogonality are very localised, and do no affect the statistical results. Using a precursor k-epsilon simulation to the provide an estimated integral lenghtscale, a similar bloc-Cartesian meshing strategy was used for the case of a downward hot wall jet (fig. 5 & Addad et al. 2004). Both Saturne and

34

D. Laurence

Star CD results compared well with experiments and completed the database down to the wall. This data was then used for improving wall function RANS modelling. Although these “hanging nodes” introduce successive jumps of the filter width by a factor of two (violating filtering/differencing commutation assumptions in deriving the LES equations), mean and rms values remained smooth across interfaces. For the flow around a cambered fan blade at Re=150 000, with leading and trailing edge separations, a 2D mesh was used in the far field (1 cell in

Fig. 5. Hot wall jet in cold counter flow. Temperature field (middle).

Fig. 6. Fan blade. 1/2 refinement & MARS scheme (top), Iso Q levels show delayed transition at leading edge compared to 2/3 refinement & CDS (bottom).


35

spanwise direction) then successive 3D non-conforming refinements allowed quasi DNS resolution in the near wall layer. However, in this case the pure central differencing scheme produced numerical oscillations upstream of the leading edge. Moreau et al. (2005) then continued the Star-CD LES simulation using conditional upwinding (MARS scheme), but this led to delayed turbulent transition and a too large laminar leading edge separation (Fig. 6). The difference with the previous cases is probably in the steady state potential flow upstream the leading edge, whereas the previous applications were fully turbulent from the inlet and this large scale mixing presumably prevented the appearance of checkerboard numerical oscillations, which in the fan blade case seemed to emanate from the 2 cell to 1 coarsening interfaces. Further tests in channel flow showed similar oscillations when the “hanging node” interface was placed perpendicular to the mean flow. The same oscillations were observed in Code Saturne and Star-CCM on frontal mesh discontinuities in channel flow, though less severe than in the potential flow upstream the fan blade. Pending a satisfactory improvement of the numerical scheme, these oscillations are currently circumvented by non-integer coarsening ratio such as 3 cells facing to 2 cells. The fan blade, recomputed on such a grid with full CDS, is now producing a proper development of the turbulence from the leading edge (fig 6.). As mentioned previously in Section 2, polyhedral cells have excellent energy conservation properties. Moulinec et al. (2005) used this feature in

Fig. 7. Polyhedral cells LES of pipe with U bend (r.h.s. zoom: wall cells).

36

D. Laurence

Star-CCM for pipe flow in a U bend at Re 54 700 with good results (Fig. 7). The near-wall layer was well captured by flattened polyhedral cells, while the quasi homogeneous size of cells in the core was appropriate for this type of flow (note that a structured curvilinear grid would have led to elongated cells along the outer boundary of the bend). While the previous Cartesian embedded refinements allow a volumetric control of the cell size in accordance with integral scale variations, this sort of control is more difficult in Voronoi based mesh generators. This feature is however under development in the CDAdapco group and would in time resolve the “refinement discontinuity” issue discussed previously. Using LES in Saturne for tube bundles, a database of lift and drag coefficients for various positions of a displaced tube is constituted, this to feed into flow induced vibrations solid/simplified-fluid software. In the case of a densely packed inline tube array (pitch/diameter = 1.5), an asymmetric mean flow solution was found, even for a nil displacement of the central tube. The asymmetric pressure signal is confirmed by experiments, and a second LES on a different grid and with Star-CCM (Fig. 8 & Benhamadouche et al. 2005).

Fig. 8. Inline tube bundle, asymmetric mean flow and pressure coefficent.

LES is now being used as a generic thermal hydraulics tool in particular for turbulence induced thermal stresses, in collaboration with materials experts to predict and extend the lifespan of power plants. Fig 9 shows hot/cold fluid mixing after a T junction and penetration of temperature fluctuations inside the steel walls (Pasutto et al. 2005).

5 Conclusions While much research has been devoted to SGS modelling, grids used in industrial applications are perhaps too coarse to show benefits over the standard Smagorinsky model. For success of such applications, the primary concern is the fact that grids should be locally adapted to the highly variable integral


37

Fig. 9. T junction. Temperature signal on inner and outer steel wall in elbow.

turbulent scale (in complex flows). But this can be easily satisfied by commercial/industrial unstructured finite volumes codes. It makes sense to tackle the multiscale problem of LES with the extreme meshing flexibility offered by professional software, and this might show that many LES papers based on extrapolations to high Re numbers while assuming structured grids were overly pessimistic. There is scope for further progress on energy conserving schemes, yet this can be achieved today by locally regular grids, and extension to polyhedral grids are promising. LES-specific grid generators present much potential (see also Iaccarino et al. 2005). This combination is of major interest to Industry for future use of LES, while local LES investigations of real engineering problems are already a reality. The community is also actively extending DES and Hybrid RANS-LES approaches, eventually embedding local LES into a general RANS simulation using with synthetic turbulence at interfaces (see Jarrin et al. 2005) and Chimera grids. The overly pessimistic predictions on the expansion of LES applications (including Laurence 2002) probably overlooked the latent progress in numerical method & grid generation, and their potentially huge benefits for LES.

References 1. Addad Y., Laurence D., and Benhamadouche S. (2004). The Negative Buoyant Wall Jet: LES Results, I.J. Heat and Fluid Flow, 25, 795-808 2. Addad Y., Laurence, D., Talotte, C., and Jacob, M.C. (2003). Large Eddy Simulation of a Forward-Backward Facing Step for Acoustic Source Identification. Intl. Jnl. of Heat & Fluid Flow, 24, 562-571 3. Benhamadouche S. (2006) Large Eddy Simulations with the unstructured collocated finite volume method. U. of Manchester, EPS/MACE PhD thesis, February 2006 4. Benhamadouche S., Laurence D., Jarrin N., Afgan I., Moulinec C. (2005) Large Eddy Simulation of flow across in-line tube bundles. NURETH-11 (Nuclear Reactor Thermal-Hyd.), 405, Avignon FR, Oct. 2005

38

D. Laurence

5. Benhamadouche, S., Laurence, D. (2002) Global Kinetic Energy Conservation with Unstructured Meshes. Intl. Jnl. of Numerical Methods & Fluids, 40, 561-272, 6. Ferziger J.H. and Peric M. (1999). Computational Methods for Fluid Dynamics. Springer, second edition. 7. Ham F., Iaccarino G. (2004) Energy conservation in collocated discretization schemes on unstructured meshes. Annual Res. Briefs, Center for Turbulence Research, Stanford U. 8. Ham, F. Lien and A. Strong (2002). A fully conservative second order finite difference scheme for incompressible flow on nonuniform grids. J. Comput. Phys., 177:117–133. 9. Hicken J., F. Ham, J. Militzer and M. Koksal (2005). A shift transformation for fully conservative methods: turbulence simulation on complex, unstructured grids. J. Comput. Phys., 208:704–734. 10. Jarrin N., S. Benhamadouche, D. Laurence (2005). Inflow conditions for Large-Eddy Simulation using a new synthetic vortex method, TSFP4, Williamsburg, June 27-29 2005. 11. Laurence, D. (2002). Large Eddy Simulation of Industrial Flows? In Closure Strategies for Turbulent and Transitional Flows, B. Launder and N. Sandham Eds, Cambridge U.P. 2002, 392-406 12. Moreau S., Mendonca F., Qazi O., Prosser R., Laurence D. (2005). Influence of Turbulence Modeling on Airfoil Unsteady Simulations of Broadband Noise Sources, 11th AIAA/CEAS Aeroacoustics Conference, May 23-25, 2005/Monterey, California, AIAA 2005-2916 13. Moulinec C., Benhamadouche S., Laurence D., Peric M. (2005). LES in a U-bend pipe meshed by polyhedral cells. ERCOFTAC ETMM-6 conference, Sardinia, Elsevier. 14. Pasutto T., Peniguel C, Sakiz M, Stéphan J.M. (2005) Unsteady Fluid/Solid simulations to evaluate thermal stresses in PWR, ASME PVP Denver, 2005 15. Peric, M. (2004). Flow Simulation Using Control Volumes of Arbitrary Polyhedral Shape, ERCOFTAC bulletin No. 62, Page 25-29, Sept. 2004 16. Perot B. (2000). Conservation properties of unstructured staggered mesh schemes. J. Comput. Phys., 159:58–89. 17. Rollet-Miet P., Laurence D., Ferziger J. (1999). LES and RANS of Turbulent Flow in Tube Bundles, I. J. Heat & Fluid Flow, 20, 241-254. 18. Talotte, C., Addad, Y., Laurence, D., Jacob, M., Giardi, H., Crouzet, F. (2002) Comparison of Large Eddy Simulation and experimental results of the flow around a forward-backward facing step. Proc. FEDSM2002-31337. ASME Fluids Eng. Meet. Montreal. 19. Verstappen R., and A. Veldman (2003). Symmetry-preserving discretization of turbulent flow. J. Comput. Phys., 187:343–368.

Current understanding of jet noise-generation mechanisms from compressible large-eddy-simulations Christophe Bailly1 and Christophe Bogey2 1

2

Laboratoire de Mécanique des Fluides et d’Acoustique Ecole Centrale de Lyon & UMR CNRS 5509 69134 Ecully cedex, France [email protected] Same address [email protected]

Summary. In this paper, noise-generation mechanisms of subsonic round jets are investigated numerically. Compressible LES based on explicit filtering are carried out with the aim of computing directly aerodynamic noise. Both the aerodynamic and the acoustic fields are obtained for different Reynolds numbers. The LES procedure as well as comparisons of results with experimental data are described. Noisegeneration mechanisms are then discussed in the light of simulations. Two noise contributions are identified, in agreement with the description of turbulent flows in terms of coherent structures and fine-scale turbulence.

1 Motivations Prediction of the noise generated by a subsonic jet remains a difficult problem. One of the fundamental reason is the real complexity of the developing turbulent flow including the mixing between the jet exiting from a nozzle and the ambient medium. A numerical simulation must be capable, for instance, of relating subtle changes of the flow at the nozzle exit to the radiated noise with the aim of noise reduction. The involved noise-generation mechanisms, on the other hand, are not well understood and still debated in the recent literature. Research efforts to identify noise sources have remained mostly theoretical and experimental. Theoretical approaches are generally based on overly simplifications of the turbulent jet flow, and measurements provide only a limited amount of information on the turbulence. A number of recent technical reviews of jet noise modelling [1, 2, 3, 4] are available. The present study focuses on the application of compressible large-eddy simulations (LES) to compute directly both the aerodynamic turbulent field and the corresponding radiated

40

Bailly and Bogey

acoustic field. With the direct noise computation (DNC), the investigation of sound-generation mechanisms takes the advantage that any turbulent quantity required for the analysis of the acoustic field is available. However, to have confidence in the DNC, serious numerical issues must be addressed before [5, 6]. LES enables to deal with more realistic jets and to study Reynolds number effects on the flow and its acoustics. Furthermore, the rapid development of computational aeroacoustics will allow us to take into account a part of the nozzle geometry in the simulation. The present work is also motivated and guided by the following remarks. Turbulence and aerodynamic noise are intrinsically linked, and a direct identification of sound sources from only the radiated acoustic field is undoubtedly an intricate and ill-posed problem. In addition, methods for predicting the far-field noise from an accurate knowledge of the turbulent field or the nearpressure field are now well established. Before anything else, the challenge is to reproduce a high-fidelity DNC simulation of the flow including the thin turbulent shear layer of the exit boundary layer at the nozzle or the Reynoldsnumber effects for instance. The present paper is organized as follows. In section 2, the numerical procedure used for the compressible LES is detailed. DNC results of round subsonic jets are presented in section 3 and compared with experimental data. Section 4 is devoted to the investigation of noise-generation mechanisms from the DNC data. Finally, concluding remarks are given in section 5.

2 Compressible LES based on an explicit filtering Following the works of Vreman [7] et al., the filtered compressible NavierStokes equations can be recasted as follows: ρu ˜j ) = 0 ∂t ρ¯ + ∂j (¯ ρu ˜i ) + ∂j (¯ ρu ˜i u ˜j + p¯δij − τ˜ij ) = σisgs ∂t (¯ ∂t (¯ ρe˘t ) + ∂j ((˘ et + p¯)˜ uj + q˜j − τ˜ij u ˜j ) =

(1)

σesgs

where ρ represents the density, ui the velocity, p the pressure, τij the viscous tensor and qj the heat flux. The overbar denotes a filtered quantity, and the filtering is assumed to commute with the time and spatial derivatives. The ρ. The variable tilde denotes the Favre (density-weighted) filtering u ˜i = ρui /¯ e˘t is defined as the total energy density of the filtered variables, i.e. ρ¯e˘t ≡ ˜i /2 for a perfect gas, where γ is the specific heat ratio. The p¯/(γ − 1) + ρ¯u ˜i u terms σisgs and σesgs in the right-hand side of (1) are the so-called subgrid-scale (SGS) terms. A detailed definition of each of the other terms of equation (1) is given in references [7, 8]. Low-pass filtering applied to any nonlinear problem introduces unknown terms which represent the interaction between the resolved scales and the non-resolved scales. Since the nineties, considerable efforts have led to clever

Current understanding of jet noise from LES

41

SGS models, see for instance the recent review of Meneveau and Katz [9], and to a better understanding of the interactions with the numerical algorithm solving the governing equations. In parallel, several studies have also pointed out some difficulties to reproduce correctly the behavior of high-Reynolds number flows [10] or of transitional shear flows [11, 12]. This is especially the case when the SGS model is based on a turbulent viscosity which has the same functional form as the molecular viscosity. An alternative to the modelling of the SGS terms is to recover the unfiltered variables appearing in the SGS terms. These deconvolution or more generally defiltering procedures are used to directly compute the SGS terms involving nonlinear interaction between the scales supported by the numerical grid but not accurately resolved by the algorithm. However, the energy transfert between the resolved and the non-resolved scales of the grid need also to be modelled. In the Approximate Deconvolution Model (ADM) introduced by Stolz [13] et al. for instance, a relaxation term draining the energy to nonresolved scales is introduced in the equations to take into account the scales not represented by the numerical grid, and thus to provide a sufficient SGS dissipation. A review of these approaches has been written by Domaradzki and Adams [14]. A highly accurate algorithm has been developed in our works for solving the compressible Navier-Stokes equations [8, 15, 16] in this framework, but also to perform a direct computation of the noise generated by turbulent flows. The discretization of the governing equations is performed with an optimized thirteen-point stencil finite-difference scheme for the spatial derivation. The modified wavenumber of the scheme is plotted in figure 1 for a uniform mesh of grid spacing Δx. Wavenumbers kΔx ≤ kcs 1.83 are accurately discretized without significant dispersion [15]. The cutoff or Nyquist wavenumber of the grid is given by kcg Δx = π, and the corresponding grid-to-grid oscillations are not resolved. They are removed by a high-selective filtering, which is also used as to model the dissipative effects of the SGS. The filter coefficients have been optimized in the Fourier space [15]. The first kth moments of G are zero (1 ≤ k ≤ 3) among the different properties of this class of filters [17]. ˆ of the filter is shown in figure 2. Wavenumbers such The transfer function G f that kΔx ≤ kc Δx = π/2 are not affected by the filtering, and are also well represented by the numerical grid. As shown in the same figure 2, the present filter is very close to the secondary filter proposed by Stolz [12, 13] and coworkers in the ADM. The different scales involved in the numerical resolution are collected in figure 3. To summarize, the following equations are therefore solved in the present LES procedure: ∂t U + ∇.F(U) = −(σd /Δt)(1 − G) ∗ (U − U ) ˜ , ρ¯e˘t ), the vector F is given by the left-hand side of (1), and where U = (¯ ρ, ρ¯u · represents a statistical averaging. All the non-linear terms are computed from the filtered quantities as for a no-model procedure [12, 19, 20]. Note

42

Bailly and Bogey

modified wavenumber

π 3π/4 π/2 π/4 0

0

π/4

π/2 3π/4 wavenumber

π

Fig. 1. Plot of modified or effective wavenumber of the scheme versus exact wavenumber: optimized 13-points finite difference scheme of Bogey and Bailly [15] , 6th-order compact scheme of Lele [18] and 2nd, 4th, 6th, 8th and 10th. Wavenumbers up to kΔx 1.83 or λ/Δx ≥ 3.5 are order central differences accurately resolved by the 13-pts optimized scheme.

transfert function

1.0 0.8 0.6 0.4 0.2 0.0

0

π/4

π/2 wavenumber

3π/4

π

Fig. 2. Transfer function of the optimized 13-points filter of Bogey and Bailly [15] ˆ = 1−G ˆ ˆ , and of D . The tranfer functions involved in the approximate G ˆ i , and implicit primary filter G deconvolution model [12] are also plotted : ˆ ˆ ˆ secondary filter HN = 1 − QN · Gi (N = 5 and kc Δx = 1/2).

Fig. 3. Scales involved in the numerical algorithm for LES. The cutoff frequency of the grid is given by kcg Δx = π but only the scales k ≤ kcf are accurately resolved. The scales with kcf < k < kcg are filtered. Note that the accuracy limit of the spatial numerical derivation is such that kcs ≥ kcf .


43

also that the independence of the results from the filtering has been studied recently [21]. A six-stage low-storage Runge-Kutta algorithm [15] ensures the time integration. Specific non-reflecting boundary conditions are implemented to preserve the acoustic field generated by the turbulent flow. For further details concerning the implementation of the filtering and of the boundary conditions, the reader is referred to [8, 15].

3 Direct noise computation of round subsonic jets The feasibility of the direct computation of noise via compressible LES was demonstrated by Bogey & Bailly [22] for a jet at near sonic conditions, with successful comparisons between predictions and measurements for the flow and for the acoustic. This earlier work was based on the Smagorinsky SGS model. All the results reported subsequently are obtained with the LES procedure described in section 2, with focus on isothermal jets at Mach number M = uj /c∞ = 0.9 and at different Reynolds numbers ReD = uj D/ν where uj is the jet exit velocity, D = 2r0 the jet diameter, ν the kinematic viscosity and c∞ the ambient speed of sound. The mean velocity profile at the inflow is defined by a hyperbolic-tangent profile with a ratio between the shearlayer momentum thickness and the jet radius of δθ /D = 2.5 × 10−2 . Small random vortical perturbations are added to the mean velocity profile in the initial shear-layer zone to seed the turbulence [8, 22]. The influence of these inflow conditions on the flow development as well as on the sound field is investigated in a recent paper [16]. The computational domain is discretized by a 12.5 million point Cartesian grid with 15 points in the jet radius. The flow is calculated up to 25r0 in the axial direction, and up to 15r0 in the transverse directions including a portion of the radiated sound field. Sound waves are accurately resolved up to a Strouhal number St = f D/uj ≤ 2. Snapshots of jets for Reynolds numbers varying from ReD = 1.7 × 103 to 4 × 105 are presented in figure 4. These pictures clearly show the strong modification of the radiated pressure, with the emergence of high-frequency waves in the sound field at 90o to the jet axis as ReD is increased. For the lowest Reynolds number, the radiation pattern seems very similar to the radiation of instability waves in supersonic flows. The same behaviors have been observed for Mach number M = 0.6 jets [23]. The influence of the Reynolds number is also clearly visible on the turbulent flow itself with a larger range of vortical scales when the Reynolds number increases. The ratio δθ /D being kept constant in all the simulations, the decrease of the initial momentum thickness δθ with ReD leads to a stronger viscous diffusion and a larger length of the potential core, from xc 5D to 7D. For ReD = 1.7 × 103 , the development of the turbulent flow occurs even notably later downstream, which prevents vortex pairing.

44

Bailly and Bogey

−

−

−

−

−

−

−

−

−

−

Fig. 4. Jets at Mach number M = 0.9. Snapshots of the vorticity norm in the flow and of the fluctuating pressure outside, in the plane z = 0, for Reynolds number ReD = 1.7 × 103 , 2.5 × 103 , 5 × 103 , 1 × 104 and 4 × 105 . The pressure color scale is [−70, 70] Pa for all the simulations.

Figure 5 shows the evolution of the mean axial velocity uc /uj in the downstream direction for different Reynolds numbers. A good agreement is observed between numerical results and experimental data. The mean velocity decay is more rapid for low Reynolds-number flows. Axial profiles of the turbulence intensity urms /uj are also reported. For low Reynolds numbers, transition to turbulence occurs later as mentioned before. Moreover, turbulence intensity reach higher values in agreement with measurements. All these effects are accurately reproduced with the present LES procedure [11, 23].

4 Jet noise mechanisms The understanding of aerodynamic noise is intrinsically linked to the description of the turbulence. Turbulent flows contain a broad range of scales which generally belong to one of the two following classes. The first one consists of fine-scale turbulence, associated with random motions in turbulent flows, and ranged in size from the larger scale given by the size of the flow, i.e. the nozzle diameter for a jet, to the smallest one, namely the Kolmogorov scale lη . The other class contains coherent structures or wave-packets. These large scales dominate the flow, are organized, and are often reminiscent of instability waves. To avoid some confusions, note that large scales in LES contain

1

0.18

0.9

0.15

0.8

0.12

urms/uj

0.7

′

uc/uj


0.09 0.06

0.6 0.5

45

0.03

5

10

15 x/r0

20

25

0

0

3

6

9

12 x/r0

15

18

21

Fig. 5. Jets at Mach number M = 0.9. Left, axial profile of the mean velocity uc /uj , and right, axial profile of the rms fluctuating axial velocity urms /uj . Simulations: ReD = 4×105 , ReD = 104 , ReD = 5×103 and ReD = 2.5×103 . 3 Measurements: Stromberg [24] et al. (M = 0.9, ReD = 3.6 × 10 ), ◦ Arakeri [25] et al. (M = 0.9, ReD = 5×105 ). DNS of Freund [26] (M = 0.9, ReD = 3.6×103 ). Note that all the profiles are shifted in the axial direction to display the same potential core length.

both fine-scale turbulence and coherent structures. Large scales in this context mean scales resolved by the computational grid, and not only coherent structures. ¿From an experimental point of view, most of the noise originates from near the end of the potential core [27], and seems to be associated with the breakdown of the coherent structures [28]. Two-point azimuthal correlations of the acoustic pressure display high levels in the main emission direction at shallow angles [29, 30]. Lower correlation-levels are measured for angles θ 90o from the jet axis and they appear enhanced as ReD decreases. In the classical framework, this change of the acoustic field with the angle is attributed to mean flow effects on sound propagation [31]. The present numerical works bring support to the conjecture of two distinct noise components as proposed by Tam [32]. The structure of the sound fields have been investigated numerically [23] for the two observation positions θ 30o and θ 90o , and two acoustic radiations have been identified: •

a component nearly independent from the Reynolds number, which dominates the sound field in the downstream direction with a low-frequency spectrum, and high levels of azimuthal correlation. A Strouhal scaling is observed for the spectral peak with a u9j power law. The noise mechanism involved appears to be linked to the periodic intrusion of vortical coherent structures into the jet [22]. • a component closely dependent on the Reynolds number, which vanishes as ReD decreases. A Strouhal scaling is found for this broadband acoustic power law. This radiation, with a weak azimuthal correlation, and a u7.5 j component is responsible for the noise emitted in the sideline direction at high-Reynolds number whereas its contribution at lower angles is masked

46

Bailly and Bogey

by the previous component. It is mainly generated by the transional flow in the shear-layer developping from the nozzle exit to the end of the potential core, and is therefore direcly connected to ReD . Among the different results obtained, the scaling of the peak of the acoustic spectra in the downstream and sideline directions versus the Reynolds number is reported in figure 6. The two noise contributions are distinguished with a fairly constant Strouhal number for the first one, and a Strouhal number decreasing at lower Reynolds numbers for the second one. 0.8

Stpeak

0.6 0.4 0.2 0.0 3 10

4

5

10

6

10

10

ReD Fig. 6. Peak Strouhal number versus Reynolds number obtained for the + Mach 0.9 and × Mach 0.6 jets in the downstream direction θ 30o , and for • Mach 0.9 and Mach 0.6 in the sideline direction, θ 90o .

Noise-source mechanisms can be identified by establishing direct links between turbulent flow events and emitted sound waves. In particular, the causality method can be applied to LES data [33] by calculating the following normalized cross-correlation function: Cf p (x1 , x2 , t) =

f (x1 , t0 ) p (x2 , t0 + t) 1/2

f 2 (x1 , t0 )

1/2

p2 (x2 , t0 )

between two points x1 and x2 , at two times seperated by t. A review from the experimental point of view can be found in reference [34]. Since all the turbulent quantities are available in simulations, variables such as f = u , v , w , u u , v v , w w , k (kinetic energy) or ω (norm of the vorticity vector) can be correlated to the acoustic pressure p . As an illustration, figure 7 shows the correlation obtained between the centerline vorticity and the pressure signal at θ = 40o . A significant correlation level is found near the end of the potential core, and this result still holds when the Reynolds number varies [33], which corroborates the presence of the coherent-noise component. In the same way, the correlations between the flow, along the shear-layer as


47

15

y/r

0

10 5

30

0 −5

0

5

10

x/r0

15

20

25

Fig. 7. Identification of noise sources by cross-correlations between vorticity along the jet axis and pressure at the point •, located at θ = 40o in the acoustic field for the ReD = 4 × 105 jet. The color scale is defined from -0.14 to 0.14, with white in the range [-0.035 0.035]. The solid line represents the acoustic propagation time between the centerline points + and the observer point •. The dotted line shows the end of the potential core.

well as along the jet axis, with the acoustic pressure are found to be very weak or insignificant.

5 Perspectives The present direct noise computations, based on compressible LES with explicit filtering, support the presence of two distinct jet noise-generation mechanisms, corresponding to a decomposition of the turbulent field into coherent structures and fine-scale turbulence. These two noise contributions have specific characteristics which can be identified. In particular, the Reynolds number dependence is well reproduced by the LES for the sound field in the sideline direction. One of the next step is to include a part of the nozzle to better simulate the incoming transitional turbulent flow. This work is in progress and should allow to study noise reduction concepts involving chevrons or beveled nozzles.

Acknowledgments The first author is grateful to the organizers of the DLES-6 for their invitation to present this talk. Computing time and technical assistance provided by the Institut du Développement et des Ressources en Informatique Scientifique (IDRIS - CNRS) are deeply appreciated.

48

Bailly and Bogey

References 1. Huff, D. (compiler) (2001) Proceedings of the jet noise workshop, CP-2002211152, 1071 pages, [www http://gltrs.grc.nasa.gov/] 2. Morris P.J. & Farassat F. (2002) AIAA Journal 40(4):671–680. 3. Bailly C., Bogey, C. (2004) Int. J. Comput. Fluid Dynamics 18(6):481–491. 4. Lele S. (2005) In proceedings of Turbulence and Shear Flow Phenomena-4, Williamsburg, VA, USA, 27-29 june, (3):889–898 5. Tam C.K.W. (1995) AIAA Journal 33(10):1788–1796. 6. Lele S.K. (1997) AIAA Paper 97-0018. 7. Vreman B., Geurts B., Kuerten H. (1995) Applied Scientific Research 54:191– 203. 8. Bogey C., Bailly C. (2005) Computers and Fluids in press :1–15. 9. Meneveau C., Katz J. (2000) Annu. Rev. Fluid Mech. 32:1–32. 10. Porter D.H., Woodwrad P.R., Pouquet A. (1998) Phys. Fluids 10(1):237–245. 11. Bogey C., Bailly C. (2005) AIAA Journal, 43(2):437–439. 12. Schlatter P., Stolz S., Kleiser L. (2004) Int. J. Heat and Fluid Flow 25:549–558. 13. Stolz S., Adams N.A., Kleiser L. (2001) Phys. Fluids 13(4):997–1015. 14. Domaradzki J.A., Adams N.A. (2002) Journal of Turbulence 3(024):1–19. 15. Bogey C., Bailly C. (2004) J. Comput. Phys. 194(1):194–214 16. Bogey C., Bailly C. (2005) AIAA Journal 43(5):1000–1007. 17. Vasilyev O.V., Lund T.S., Moin P. (1998) J. Comput. Phys. 146:82–104. 18. Lele S.K. (1992) J. Comput. Physics 103(1):16–42. 19. Rizzetta D.P., Visbal M.R., Blaisdell G.A. (2003) Int. J. Numer. Meth. Fluids 42:665–693. 20. Mathew J., Lechner R., Foysi H., Sesterhenn J., Friedrich R. (2003) Phys. Fluids 15(8):2279–2289 21. Bogey C., Bailly C. (2005) In proceedings of Turbulence and Shear Flow Phenomena-4, Williamsburg, VA, USA, 27-29 june, (2):817–822, accepted in the International Journal of Heat and Fluid Flow. 22. Bogey C., Bailly C., Juvé D. (2003) Theoret. Comput. Fluid Dynamics 16(4):273–297. 23. Bogey C., Bailly C. (2004) AIAA Paper 2004-3023:1–15. 24. Stromberg J.L., McLaughlin D.K., Troutt, T.R. (1980) J. Sound Vib. 72(2):159– 176. 25. Arakeri V.H., Krothapalli A., Siddavaram V., Alkistar M.B., Lourenco L.(2003) J. Fluid Mech. 490:75–98. 26. Freund J.B. (2001) J. Fluid Mech., 438:277–305. 27. Juvé D., Sunyach M., Comte-Bellot G. (1980) J. Sound Vib. 71(3):319–332. 28. Hussain A.K.M.F. (1986) J. Fluid Mech. 173:303–356. 29. Maestrello L. (1976) NASA TMX-72835. 30. Juvé D., Sunyach, M. (1978) C. R. Acad. Sci. Paris, B, 287:187–190. 31. Goldstein M.E., Leib S.J. (2005) Journal Fluid Mech. 525:37–72. 32. Tam C.K.W. (1998) Theoret. Comput. Fluid Dynamics 10:393–405. 33. Bogey C., Bailly C. (2005) AIAA Paper 2005-2885:1–18. 34. Panda J. (2005) AIAA Paper 2005-2844.

Stable stratified, wall bounded, turbulent flows Vincenzo Armenio Dipartimento di Ingegneria Civile e Ambientale, Universit´ a di Trieste, Piazzale Europa 1, 34127 Trieste, Italy [email protected]

In the present paper we discuss results of recent large-eddy simulation studies of stable stratified wall bounded flows. Three different cases are presented and discussed. In the first two cases the mean shear and the density gradient are aligned and differences come from the use of different boundary conditions (BCs) for both the thermal and the velocity fields. Specifically the first case uses BCs archetypal of a stable stratified atmospheric boundary layer, whereas the second one is relevant for shallow-water applications. In the third case the effect of misalignment between the mean density gradient and the mean shear is discussed.

1 Introduction Due to the many practical applications, stably stratified flows are of great importance in environmental fluid mechanics. As an example, thermal inversion in the low atmosphere causes the stagnation of pollutants and particulates that degrade the quality of air, whereas in the oceans, stable stratification suppresses vertical mixing of chemical species and nutrients. In general, stable stratification can strongly influence the dynamic and the anisotropic characters of turbulence, leading to qualitative and quantitative changes in the small-scale mixing of momentum, of mass and of a dispersed phase. Most literature numerical and experimental studies dealing with the interaction between a mean shear, that is the source of turbulent mixing, and a mean, stable, density gradient acting toward the suppression of turbulence, have been carried out in the very simple case of unbounded homogeneous turbulence, and considering the density gradient aligned with the mean shear. Few investigations have been devoted to the study of stable stratification in wall-bounded turbulence, where with wall we intend either an interface or a solid wall. In this case additional complications arise. Indeed inhomogeneity in the wall-normal direction makes the key parameters (like the gradient Richardson number) to be a function of the distance from the wall; the choice

50

Vincenzo Armenio

of boundary conditions for temperature and velocity at the walls, dramatically affects the wall-normal characteristics of the turbulent field. In the present paper we discuss some recent results of numerical studies of wall bounded stably stratified turbulence. First we discuss the effect of inhomogeneity in the turbulent field in the case of constant-temperature, horizontal solid walls. This flow is archetypal of a stably stratified atmospheric boundary layer (SABL) where the ground surface cools the bottom layers of air. Further we consider a different case, namely a free-surface channel flow with adiabatic bottom wall and constant positive heat flux at the free-surface. This flow is archetypal of a shallow-water basin heated from the top in absence of wind induced mixing and surface waves. Finally a case in which the mean shear supplied by vertical walls is orthogonal to the mean density gradient is discussed. This flow is archetypal of a canyon-like geometry with stable stratification. In all cases the Boussinesq form of the Navier-Stokes equations was employed under the hypothesis that the density variations are small when compared to the bulk density of the fluid. The studies were carried out by large eddy simulation. A dynamic mixed model was used for the SGS stresses, whereas a dynamic eddy viscosity model was employed for the SGS density fluxes. Plane averaging of the model constant was performed in all cases. The governing equations were solved using a second order accurate finite difference scheme. Details are in [1, 2, 3, 8].

2 Key parameters in stably stratified flows It is well established that the Richardson number, which is a measure of the relative importance of the potential energy associated to the presence of the gravitational field with respect to the kinetic energy in the flow field, is the parameter that rules the response of the turbulent field to stable stratification. Among the others, the following forms of the Richardson number are often considered: Rig =

− ρg0 d N2 gΔρh Bk Rig dz = ; Riτ = ; Rif = = d 2 S2 ρ0 u2τ Pk P rT dz

which are respectively the gradient, the friction and the flux Richardson number. In the above given definitions, N and S respectively are the Brunt-Vaisala frequency and the mean shear, z is the vertical direction, Δρ is a bulk density gap, ρ0 is the bulk fluid density, < u > is the mean velocity and g the gravitational acceleration. Bk and Pk are respectively the buoyancy destruction and the turbulent production of the turbulent kinetic energy and P rT = νT /kT is the turbulent Prandtl number. Henceforth we use density gap and temperature gap interchangeably taking advantage of Δρ/ρ0 = αΔT with α the coefficient of thermal expansion. In homogeneous turbulence Rig is constant in space and in time. Recent studies (see among the others [4]) have shown

Stable stratified, wall bounded, turbulent flows

51

that a critical value 0.18 < Rig < 0.25 delimits the level of stratification below which sustained turbulence is observed, from that where turbulence suppression occurs. In wall bounded turbulence, due to the inhomogeneity in the wall normal direction, the gradient Richardson, as well as the flux Richardson number are a function of the distance from the wall, whereas the friction Richardson number is known a priori, once an appropriate density scale has been chosen. The friction Richardson number thus gives an indication of the overall level of stratification in the flow field. Now a question arises: what parameter is suited for the characterization of the flow field? As discussed in the following sections, results from our studies show that in wall bounded stable stratified turbulence the answer is not unique, rather it depends on the way the stratification affects the flow field. Table 1. Parameters of the simulations of Armenio and Sarkar, 2002 together with important bulk quantities Case

Reτ

Riτ

Reb

Rib

103 cf

C0 C1 C2 C3 C4 C5

180 180 180 180 180 180

0 18 60 120 240 480

2800 3100 3660 4150 4570 5120

0. 0.032 0.068 0.112 0.188 0.297

8.18 6.73 4.99 3.71 3.14 2.40

3 Channel flow with horizontal isothermal walls A plane channel flow was studied subjected to a wide range of levels of stratification. The Prandtl number was set equal to 0.71, representing thermally stratified air. The main parameters of the simulations together with important bulk quantities are reported in Table 1. The results of the simulations showed that the increase of stratification suppresses the Reynolds shear stress < u w > and thus causes the increase of the molecular counterpart νd < u > /dz. As a consequence the bulk velocity ub , and thus Reb , (see Table 1) increase with Riτ . The bulk Richardson number Rib = Riτ cf /2 varies from case to case and increases less than Riτ due to the decrease of the friction coefficient cf . The wall normal distribution of the gradient Richardson number (Fig. 1) changes with Riτ although qualitative differences are not detected from case to case. Since Rig |wall ∼ 1/(Re2τ ), in the near wall region the gradient Richardson number is very small; it linearly increases with z in the log-region up to a critical value approximatively Rigc ∼ 0.2 − 0.25 and beyond this point it strongly increases. Interestingly, the point where the critical value

52

Vincenzo Armenio

is reached moves toward the wall with increased Riτ . The analysis of the turbulent field has shown that the channel flow appears split into two different regions, that were defined respectively as a buoyancy affected (BA) near-wall region and a buoyancy dominated (BD) outer region. The border between the two regions is characterized by a value of Rig ∼ 0.2 − 0.25. The stability limit given by the linear theory of [7], namely Rigc > 0.25 for the linear stability to infinitesimal perturbations, that has already been found to approximatively hold in homogeneous turbulence, still holds in inhomogeneous turbulence. In 0.4

Rig

0.3 0.2

C2, Rib=0.137 C3, Rib=0.225 C4, Rib=0.377 C5, Rib=0.593

0.1 0

50

z+

100

150

Fig. 1. Wall normal behavior of the Gradient Richardson number for different levels of stratification. From Armenio and Sarkar 2002.

the near-wall, BA region, turbulence is sustained by the mean shear and the effect of stratification is weak, since an energetic turbulent state persists even in case of strong stratification. In the outer, BD region, turbulent production is very small and the destruction terms prevail. This region was observed to be characterized by the presence of countergradient density fluxes and by the presence of internal waves. The results obtained were consistent with those of the experimental study of [5]. The correlation coefficients of momentum (Cu w ) and density (Cρ w ) fluxes, plotted against Rig are shown in Fig. 2. The coefficients are nearly constant with Rig in the buoyancy-affected region, whereas an abrupt drop is observed for Rig > 0.2 − 0.25, namely in the BD region. It is noteworthy that the correlation coefficients also depend on the overall level of stratification, quantified by Riτ . Figure 2 indeed shows that a reduction of about a factor 2 and 1.5 are observable respectively in Cu w and Cρ w in the BA regime. Finally, the vertical distribution of the turbulent Prandtl number plotted against Rig (Fig. 3) exhibits a very interesting behavior. In the BA region, P rT is nearly independent on the overall level of stratifications and it is nearly equal to 1. This indicates that when active turbulence is present in the flow field, the Reynolds analogy holds, and turbulent transport of mass occurs at the same rate as that of momentum. In the BD region, P rT dramatically increases with Rig and its value also depends on Riτ , thus showing that when


53

< u’ w’ > /urms wrms

1

< ρ’ w’ > /ρrms wrms

1 0.8 0.6 0.4 0.2 0 10

a)

C1, Rib=0.064 C2, Rib=0.137 C3, Rib=0.225 C4, Rib=0.377 C5, Rib=0.593

−2

10

−1

C1, Rib=0.064 C2, Rib=0.137 C3, Rib=0.225 C4, Rib=0.377 C5, Rib=0.593

0.8 0.6 0.4 0.2 10 −2

10

10 −2

100

Rig

b)

Rig

Fig. 2. Correlation coefficients of a) density flux, b) momentum flux, as a function of the gradient Richardson number. From Armenio and Sarkar 2002.

6 C1 , Rib=0.064 C2 , Rib=0.137 C 4, Rib=0.377 C5 , Rib=0.593

5

Prt

4 3 2 1 0

0

0.2

Rig

0.4

0.6

Fig. 3. Turbulent Prandtl number as a function of the gradient Richardson number for the cases studied in Armenio and Sarkar 2002.

turbulent transport is strongly inhibited, suppression of vertical transport of mass occurs at a much higher rate than that of momentum.

4 Free surface channel with given heat fluxes It is well established that in marine applications, the bottom surface is adiabatic and most of the energy exchange between the ocean and the atmosphere occurs at the free surface where heat fluxes are always present. In the present section we discuss the results of a very recent research by [8] aimed at the analysis of the response of a free surface channel flow with adiabatic bottom wall and with positive heat flux coming from the free surface. In spite of the simplifications (absence of rotation, of free surface waves and of the top mixed layer) the present flow field is archetypal of a shallow-water basin heated from the top. The free surface is considered to be shear-free, thus miming absence of wind. The numerical simulations were carried out at Reτ = 400 and considering thermal stratified water (P r = 5). In the present problem the friction Richardson number is defined using, as a density scale, the quantity Δρ = hdρ/dz|f s (with h the channel height and dρ/dz|f s the free surface density gradient associated to the incoming heat flux.

54

Vincenzo Armenio

Substantial differences were observed when comparing the results of the present simulations with those of [2] discussed in the previous section. The main difference comes from the fact that, in [2], the presence of fixed temperature at the walls produces a near-wall density gradient, which directly acts toward the suppression of turbulence in the region where the latter is produced. Conversely, in the present case, the region where turbulence is produced, close to the bottom wall, and the region where the stratification primarily acts, namely the free surface, are far from each other. As a result, the imposition of a fixed heat flux produces a sharp density gradient that thickens with increasing Riτ (Fig. 4a). Consequently the BruntVaisala frequency increases in the free-surface region (Fig. 4b) where weak turbulence is present, and it is nearly zero going down toward the adiabatic wall. The two separate regions already discussed in [2] are present in this case as well. The analysis of the statistics and of the coherent structures has shown that presence of a sharp thermocline in the free surface region inhibits transport of mass and momentum from the deep region of the flow, where turbulence is well sustained toward the free surface. A very interesting aspect of the case herein discussed is that, unlike the case with constant wall temperature of [2], there is not a universal behavior of the relevant turbulent statistics (not shown here) when plotted against Rig . For example there is not evidence of the sharp decrease of the correlation coefficients and the sharp increase of P rT for a certain critical value of Rig . The vertical Froude number F r = wrms /N LE where LE = ρrms /d < ρ > /dz is the Ellison scale, was found to be a parameter better suited to characterize the modification of turbulence under stratification in the present problem. 1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.5 Ri =0 τ Riτ=100 Riτ=250 Riτ=400 Ri =500

0.4 0.3 0.2

a)

0.5 0.4

τ

0.3 0.2

τ

0.1 0 1

Ri =25 τ Riτ=100 Riτ=250 Riτ=400 Ri =500

0.6 z/h

z/h

0.6

0.1

Laminar Solution 0.8

0.6 0.4 0.2 (ρ - ρ )/ Δ ρ z=0

0

0

b)

0

10

N

20

30

Fig. 4. Vertical distribution of a) Density b) Brunt-Vaisala frequency for the cases studied in Taylor et al. 2005.


55

5 Vertical channel with constant wall temperature Many practical applications are characterized by the fact that the mean shear S, responsible of the turbulence production, is not aligned with the mean vertical density gradient Nz which causes the suppression of turbulent motion. The effect of this misalignment was studied by [6] in homogeneous turbulence, considering a wide range of angles between S and Nz , from 0 to 90, which respectively correspond to the cases of parallelism and orthogonality between S and Nz . The authors showed that defining the gradient Richardson number as usual, namely the ratio N 2 /S 2 , irrespective of the angle of inclination of S with respect to Nz , the value of Rigc progressively increases with the angle of inclination, and it gets of the order O(1) when S and Nz are orthogonal to each other. The effect of vertical stratification on the horizontally sheared wall bounded flow was investigated by [3] (see Fig. 5). In this study the friction Reynolds number and the Prandtl number were respectively set equal 390 and 5. In order to have a direct comparison with a flow field where S and Nz be aligned, corresponding simulations were carried out considering a channel with horizontal walls. The parameters of the simulations are reported in Table 2. Similarly to the case where S is aligned with Nz , studied in [2], the increase of stratification causes a weak reduction of the turbulent intensities and a strong

Fig. 5. Schematic of the vertical channel flow with horizontal shear and vertical stable stratification. From Armenio and Sarkar 2004. Table 2. Parameters of the simulations of Armenio and Sarkar, 2004 together with important bulk quantities. Cases labeled CXV refer to S parallel to Nz . Case

Reτ

Riτ

Reb

Rib

103 cf

C0-AS04 C1-AS04 C2-AS04 C3-AS04 C1V-AS04 C2V-AS04

390 390 390 390 390 390

0 15 100 500 100 200

7320 7530 9320 11470 8700 9380

0. 0.041 0.200 0.590 0.210 0.360

6.4 5.5 4.1 2.4 4.2 3.6

56

Vincenzo Armenio

decrease of vertical mixing of mass and of the level of density fluctuations. As a consequence, the turbulent Prandtl number P rT increases with stratification. Interestingly, the universal behavior observed in case of S aligned with Nz , namely P rT ∼ 1 for Rig < 0.25 does not hold when S is orthogonal to Nz (Fig. 6). Indeed, it clearly appears that in cases C1-AS04 to C3-AS04, the turbulent Prandtl number monotonically increases with Rig without showing any universal behavior. The mixing efficiency Bk /k and the Ellison scale behave differently in the two configurations analyzed. Note that, in the outer region where k ∼ Pk the mixing efficiency is nearly equivalent to the flux Richardson number defined above. Again, a somewhat universal behavior can be detected in the cases with S aligned with Nz (see Fig. 7). The mixing efficiency and the Ellison scale peak

10

PrT

8 6 C1-AS04 C2-AS04 C3-AS04 C1V-AS04 C2V-AS04

4 2 0 −2 10

10

−1

10

Rig

0

10

1

Fig. 6. Turbulent Prandtl number plotted against the gradient Richardson number for the cases of Armenio and Sarkar 2004. Note that the vertical lines correspond to the change of sign of P rT due to the presence of countergradient buoyancy fluxes, often observed in the buoyancy dominated region when S is aligned with Nz . In this case the concept of turbulent Prandtl number ceases to hold.

0.4

C1-AS04 C2-AS04 C3-AS04 C1V-AS04 C2V-AS04


0.3

LE

B/ε

0.2

0.4

0

0.2 0.1

−0.2 0

a)

0.5

1

Rig

1.5

0 -2 10

2

b)

10

-1

Rig

10

0

10

1

Fig. 7. a) Mixing efficiency and b) Ellison scale plotted against the gradient Richardson number for the cases of Armenio and Sarkar 2004.


57

/ρrms wrms

in the buoyancy affected regions (Rig < 0.2) and rapidly decay in the buoyancy dominated regime. Negative values of the mixing efficiency in the BD region are due to the presence of countergradient buoyancy fluxes (CGBF) that are often observed in this region. In the horizontal shear case the mixing efficiency first increases in the inner region up to a nearly constant value, and then slowly decreases in the outer region. As expected, a general reduction of Bk /k occurs throughout the wall normal direction with increasing Riτ , and values of the order 0.15 are recorded even in cases of strong stratification. Interestingly, negative values are never recorded, even in case of very strong level of stratification (case C3 − AS04), meaning that CGBF are not observed when the mean shear is orthogonal to Nz . Finally, in Fig. 8 we show the correlation coefficient Cρ =< ρ w > /ρrms wrms plotted against Rig for all the simulations of [3]. As regards the case of S aligned with Nz , as already observed in [2] a rapid decrease is observed for Rig ∼ 0.2. Note that qualitative differences between cases C1 to C5 of [2] and cases C1V-AS04, C2V-AS04 are not observed in spite of the fact that the Reynolds number and the Prandtl number were significantly different in the two mentioned studies. In case of horizontal shear, for a given level of stratification, a rapid decrease of Cρ is observed for Rig ∼ O(1), thus one order of magnitude larger that the value observed when S is aligned with Nz . This result is consistent with the findings of [6] in homogeneous turbulence. To summarize, the analysis has clearly


0.4

0.2

0 10−2

10−1

Rig

100

101

Fig. 8. Correlation coefficient of buoyancy fluxes against the gradient Richardson number for the cases of Armenio and Sarkar 2004.

shown that stable stratification is much more effective in suppressing vertical mixing when turbulence is produced by vertical shear, when compared to an equivalent case with horizontal shear. This has to be attributed to the fact that the mechanism by which stable stratification affects the turbulent field depends on the geometrical configuration: in case of vertical shear, buoyancy directly acts toward destruction of the Reynolds shear stress by means of the term −g/ρ0 < ρ u > and of the vertical turbulent kinetic energy by means of the term −g/ρ0 < ρ w >, which, in turn, affects the other components of the turbulent kinetic energy by pressure strain correlation; in case of horizontal shear, there is not a direct action over the Reynolds shear stress, and buoyancy

58

Vincenzo Armenio

destruction only acts over the vertical turbulent kinetic energy indirectly affecting the other two components by means of pressure strain correlation.

6 Concluding Remarks In the present paper we have shown results of different researches of stably stratified wall bounded turbulence. The differences arise from the boundary conditions employed and the orientation of the mean shear with respect to that of the mean density gradient. The studies have shown that the gradient Richardson number is a key parameter for the characterization of the flow field when the mean shear is aligned with the mean density gradient and constant temperature are imposed at the walls. Even in case of parallelism between S and Nz , the change in the boundary conditions (imposed heat fluxes instead of temperature), dramatically affects the turbulent field and Rig does not appear to be a good parameter for collapsing the turbulent statistics. From a practical point of view, this also means that information from the dynamics of the atmospheric boundary layer cannot be simply extrapolated for modeling stable stratified flow for marine application where heat fluxes are prescribed at the wall rather than temperature. Finally, the analysis of a turbulent wallbounded field with the mean shear orthogonal to Nz has shown that the critical value of Rig for observing a rapid decay of the correlation coefficients increases by more than one order of magnitude when compared to a case with S parallel to Nz . Moreover, the collapse of other relevant turbulent quantities (like the turbulent Prandtl number) with Rig is not observed when S is not aligned with Nz .

References 1. Armenio V, Piomelli U (2000) A Algrangian mixed subgrid-scale model in generalized coordinates. Flow Turbulence and Combustion, 65:51 2. Armenio V, Sarkar S (2002) An investigation of stably stratified turbulent channel flow using large eddy simulation. J. Fluid Mech., 459:1 3. Armenio V, Sarkar S (2004) Mixing in a stably stratified medium by horizontal shear near vertical walls. Theor. and Comp. Fluid Dyn., 17:331 4. Gerz T, Schumann U, Elgobashi S E (1989) Direct numerical simulation of stratified homogeneous turbulent shear flows J. Fluid Mech., 200;563 5. Komori S, Ueda H, Ogino F, Mizushina T (1983) Turbulent structure in a stably stratified open-channel flow J. Fluid Mech., 130;13 6. Jacobitz F G, Sarkar S (1999) The effect of not vertical shear on turbulence in a stably stratified medium Phys. Fluids, 10:1158 7. Miles J W (1961) On the stability of heterogeneous shear flows J. Fluid Mech., 10;496 8. Taylor J, Sarkar S, Armenio V, (2005) Large eddy simulation of stably stratified open channel flow Phys. Fluids, accepted

Physics and Control of Wall Turbulence John Kim Department of Mechanical and Aerospace Engineering University of California Los Angeles, CA 90095-1597 USA [email protected]

1 Introduction The control of turbulent boundary layers (TBLs) requires a thorough understanding of the underlying physics of TBLs and an efficient control algorithm, both of which have been less than satisfactory despite the great interest they have garnered over the years. Great strides on both fronts have been made recently through advancements in computational fluid dynamics and control theories. The availability of accurate time history of full three-dimensional velocity and pressure fields has led to improved understanding of the underlying physics of turbulent flows. Numerical simulations have resolved many existing controversies by putting together bits of information collected by different experimental techniques. Numerical simulations have been also extremely useful in testing various hypotheses by conducting cleverly designed numerical experiments, in which modified Navier-Stokes equations were solved in order to examine the role of certain turbulence mechanisms [1, 2]. On the control front, new approaches to controller design that are significantly different from existing approaches have emerged. In contrast to most existing approaches, which were largely based on the investigator’s physical insight into the flow, new approaches incorporate modern control theories into the controller design. Some of these approaches explicitly exploit certain linear mechanisms present in TBLs, and their success suggests the importance of linear mechanisms in turbulent (and hence, nonlinear) flows. In this paper, I shall review1 the underlying physics of turbulence that are responsible for high skin-friction drag in TBLs, and then discuss a linear systems approach to boundary-layer control. This linear systems approach is based on the recognition that much of of the underlying physics of turbulence responsible for large skin-friction 1

This is a condensed version of a similar review I gave at the 2nd International Symposium on Seawater Drag Reduction, which was held in May 23-26, 2005, in Busan, Korea.

60

John Kim

drag in turbulent boundary layers is a linear process. It is worth mentioning in passing that in spite of the apparent differences in the outer region, near-wall turbulence structures as well as near-wall turbulence statistics are almost identical for turbulent channels and turbulent boundary layers. Since the major concern of this paper is related to near-wall turbulence, no effort is made to differentiate between the two flows, and the term turbulent boundary layers is used throughout this paper. In this paper u, v, w denote the velocity in the streamwise (x), wall-normal (y), and spanwise (z) directions, respectively.

2 A Self-Sustaining Process Discovery of well-organized turbulence structures and the recognition that these structures play important roles in the wall-layer dynamics are among the major advances in turbulent boundary layer research during the past several decades. The ubiquitous structural features in the near-wall region of turbulent boundary layers are low- and high-speed “streaks,” which consist mostly of a spanwise modulation of the streamwise velocity. These streaks are created by streamwise vortices, which are roughly aligned in the streamwise direction. It has now been recognized, in large part due to numerical investigations, that streamwise vortices are also responsible for the high skinfriction drag. There is strong evidence that most high skin-friction regions in turbulent boundary layers are induced by nearby streamwise vortices [3, 4]. It has been also reported that streamwise vortices in drag-reduced TBLs are either eliminated or weakened significantly. Fig. 1 shows a few examples of such flows, where the skin-friction drag has been reduced by various methods. A common feature of these drag-reduced flows, regardless how the drag was reduced, is debilitated near-wall streamwise vortices. The strength of streaks in drag-reduced flows is also significantly reduced and their average spanwise spacing is increased. Streamwise vortices are formed and maintained autonomously (i.e., independent of the outer layer) by a self-sustaining process, which involves the wall-layer streaks and instabilities associated with them [7, 8, 9, 2]. There are some differences in details on the self-sustaining (or regeneration) process (see the references mentioned above for further details), but it is generally accepted that this process is independent of (at least in the first order) the outer part of the boundary layer, and that the presence of wall itself does not play a role in the process other than setting up the mean shear through the no-slip boundary condition on the streamwise velocity. A generally accepted regeneration cycle, except for some details, is shown in Fig. 2. One can start from any place in the cycle, but let’s start from the first leg of the cycle, which involves interactions between streamwise vortices and mean shear. Streamwise vortices (sometimes referred to as streamwise rolls) primarily consist of the wallnormal and spanwise velocities independent of the streamwise direction, i.e.,

Physics and Control of Wall Turbulence

(a)

61

1

y

−1

(b)

0

z

4π/3

0

z

4π/3

0

z

4π/3

−1 0

z

4π/3

1

y

−1

(c)

1

y

−1

(d)

1

y

Fig. 1. Contours of streamwise vorticity in y-z plane: (a) regular channel; (b) channel with an LQR controller on the bottom wall (Min 2005, unpublished work); (c) channel with a hydrophobic surface on both walls [5]; (d) channel with a polymer [6]. Contour levels are from -1 to 1 in increments of 0.1.

v(z) and w(z) respectively. These vortices create streaks, i.e., u(z), through interaction with the mean shear, dU/dy. This process is sometimes referred to as lift-up (of low-momentum fluid). It is also related to the so-called transient growth of disturbances – streamwise disturbances in particular – due to non-normality (or non-self-adjoint) of linearized Navier-Stokes equations. The streaks can also be identified by the presence of strong wall-normal vorticity, ωy (z), at both edges of streaks. Note that this lift-up process (first leg

62

John Kim

Streamwise vortices: v(z), w(z) Vortex formation: ∂ ωy ∂u v ∂ y , ωx ∂ x

Streamwise-dependent disturbances: u(x,z), v(x,z), w(x,z), ωx(x,z), ωy(x,z)

st

rd

1 leg

3 leg

nd

2 leg Breakdown: normal-modeinstability, transientgrowth duetonon-normality

Streak formation: , ∂ v dU v dU dy ∂ z dy

Streaks: u(z),ωy(z)

Fig. 2. Schematic illustration of a self-sustaining process of near-wall turbulence structures.

in the regeneration cycle) is linear. The strength of streaks can grow indefinitely as long as the strength of streamwise vortices and the mean shear are maintained. However, these streamwise-independent vortices will ultimately decay unless they are strengthened through a nonlinear mechanism involving streamwise-dependent disturbances, as shown in the third leg of the regeneration cycle. Hamilton et al. [7] presented a vortex-formation mechanism, in x which an advection term, v ∂ω ∂y , played a dominant role, whereas Schoppa & Hussain [10] showed that a vortex stretching term, ωx ∂u ∂x , was responsible for creating streamwise vortices. It is worth mentioning that both mechanisms do not require the presence of existing vortices for vortex formation in contrast to other generation mechanisms, which require strong pre-existing vortices. Note also that both mechanisms involve streamwise-dependent disturbances, and that they are nonlinear. Streaks created by the linear mechanism (first leg in the regeneration cycle) are unstable to small disturbances, i.e., linearly unstable. They are unstable to the classical normal-mode type disturbances, i.e., there are unstable eigenmodes associated with spanwise-varying mean velocity profiles, U (y, z) [7, 8, 9]. In addition to this normal-mode instability, Schoppa & Hussain [10] showed that streaks are also subject to non normal-mode instability (referred to as streak transient growth), due to non-self-adjoint nature of linearized Navier-Stokes equations [1]. They further illustrated that this transient-growth instability is much stronger than the normal-mode instability, the latter of which requires a rather strong spanwise gradient of the mean velocity, and has limited amplification. Two important points are worth mentioning, especially for the present discussion: both mechanisms are linear, and the streamwise-dependent disturbances that were necessary to form streamwise-independent vortices grow due to these instabilities. The regeneration cycle described above is self-sustaining as it does not require an explicit interaction with the outer part of a boundary layer. In a


63

numerical experiment conducted, in which modified Navier-Stokes equations were solved in order to represent a turbulent channel flow without large-scale motions in the outer part, Jimenez & Pinelli [2] observed no discernible differences in the behavior of the inner part (i.e., near-wall region), thus demonstrating that the inner part of boundary layers could be maintained autonomously by the self-sustaining process. This self-sustaining process of near-wall turbulence structures is a starting point of our discussion of controller design for drag reduction in turbulent boundary layers. Streamwise vortices are responsible for the high skin-friction drag in turbulent boundary layers and they are maintained by the self-sustaining process. Our approach to reduce the skinfriction drag in turbulent boundary layers is therefore to develop a controller, which can disrupt the above self-generating process.

3 Controlling a Linear Mechanism Two key mechanisms in the self-sustaining process described above are linear. In this section, I shall discuss a numerical experiment, which will illustrate that one of these linear mechanisms indeed plays a key role in maintaining near-wall turbulence. It will be also shown that a controller designed to weaken this linear mechanism is indeed effective in reducing the skin-friction drag in fully turbulent (and hence nonlinear) boundary layers, thus validating the notion that an effective approach to achieve a skin-friction drag reduction is through controlling the linear mechanisms in the self-sustaining process. The streak generation mechanism (first leg in the regeneration cycle) is a simple advection of low-speed fluids in the wall region away from the wall by streamwise vortices. Recall that streamwise vortices consist of wall-normal velocity that varies in the spanwise direction. This mechanism can be viewed from a slightly different perspective by considering the linearized NavierStokes equations in the following form: ∂ vˆ vˆ = [A] , (1) ˆy ω ˆy ∂t ω where

Los 0 . [A] = Lc Lsq

(2)

The overhat in Eqn. (1) denotes a Fourier-transformed quantity, and Los , Lsq and Lc in Eqn. (2) represent, respectively, Orr-Sommerfeld, Squire, and linear coupling operators (see [1] for definition). The linear operator A in Eqn. (1) is non-normal (i.e., not self-adjoint), and hence, its eigenmodes are non-orthogonal to each other, which allows transient growth of disturbance energy even if all individual eigenmodes are stable and decay asymptotically. More specifically, it has been shown that, due to this non-normality of the linearized Navier-Stokes system, the so-called optimal

64

John Kim

disturbance can grow up to O(Re2 ) in the transient time period, which is proportional to O(Re), possibly triggering nonlinear transition even below the critical Reynolds number predicted by classical linear stability theory [11, 12, 13]. Butler & Farrell [11] illustrated that the shape of the optimal disturbance resembled near-wall streamwise vortices in turbulent boundary layers. They subsequently showed that when a proper turbulence time scale was imposed in their analysis, the spacing between high- and low-speed streaks induced by this optimal disturbance was close to that commonly observed in turbulent boundary layers [14]. The major contribution to this non-normality comes ˆ in Fourier space from the linear coupling term, Lc , in Eqn. (2). This, dU dy ikz v dU ∂v ˆ y , and or dy ∂z in physical space, is a source term for wall-normal vorticity, ω this coupling between vˆ and ω ˆ y is related to the streak creation mechanism in the self-sustaining process (first leg in the regeneration cycle). In order to investigate the role of the linear coupling term in nonlinear flows, Kim & Lim [1] considered the following modified Navier-Stokes equations: ∂ vˆ vˆ Nv = A˜ + , (3) ˆy ω ˆy Nωy ∂t ω where

L 0 A˜ = os , 0 Lsq

(4)

and Nv and Nωy denote the nonlinear terms in the Navier-Stokes equations. ˜ which Note that the linear coupling term is missing in the modified operator A, is still non-normal, since Los is non-normal, but its non-normality is much reduced as the operator is now bloc symmetrical. Kim & Lim [1] referred this modified system as a virtual flow, which contains all nonlinearity of turbulent flows but contains no coupling between wall-normal velocity and wall-normal vorticity. It can also be viewed as a turbulent flow with perfect control by which the coupling term was completely suppressed. Starting from an initial field obtained from a regular turbulent channel simulation, the above modified nonlinear system was integrated in time, and was compared with a nonlinear simulation with the coupling term. It was found that without the coupling term the near-wall structures first disappeared, and then turbulence intensities were reduced significantly (close to a laminar-like state, but this return to a laminar-like state may be due to the extremely low Reynolds number of their numerical experiment), thus demonstrating that the linear coupling term plays an essential role in maintaining turbulence in nonlinear flows. Motivated by the above results, Lim [15] designed a controller using a linear quadratic regulator (LQR) synthesis, the objective of which was to minimize the linear coupling term. Note that this controller could reduce the coupling term but could not completely eliminate it, in contrast to the virtual flow above where the coupling term was artificially removed from the nonlinear calculations. Despite the fact that the controller was designed based on the linearized system in Eqn. (1), the magnitude of coupling term in the


65

LQR-controlled flow (nonlinear) was substantially reduced, and the strength of near-wall turbulence substantially weakened, resulting in about a 20% drag reduction [15].

4 SVD Analysis Examples in the previous section illustrated that a linear mechanism plays an important role in turbulent boundary layers. Other examples, especially in conjunction with closed-loop flow control, leading to the same conclusion have been reported in [16, 17]. This recognition that a linear mechanism plays a significant role in turbulent boundary layers led Lim & Kim [18] to examine turbulent boundary layers from a linear systems perspective. A brief description of their singular value decomposition (SVD) analysis is given below, but the interested reader is referred to Lim & Kim [18] for further details. Eqn. (1) with control input can be written in the following state-space representation: dx = Ax + Bu, dt u = −Kx,

(5) (6)

where vectors x and u represent, respectively, a ‘state’ of the system and ‘control’ (blowing and suction at the wall in the present study), and matrix K represent a control gain to be determined. Equation (5) represents a state equation inside the flow domain, which is being forced by the control input, u, at the boundary of the domain. The system matrix A is related to the system operator A in Eqn. (1), and the input matrix B depends on a particular method of control input. In linear optimal control theory, the gain matrix K is obtained such that the a certain control objective is minimized. By combining Eqs. (5) and (6), the system equation is given as dx/dt = (A − BK)x. For uncontrolled cases, K is zero and the system equation simply becomes dx/dt = Ax. The traditional eigenvalue analysis, which predicts whether a linear system is stable or unstable based on the eigenvalues of the system, is inadequate in explaining transient growth of kinetic energy of certain disturbances in an otherwise stable system. Instead, transient growth can be analyzed by applying the SVD analysis to the system operator, by which the amplification factor of initial disturbances can be determined. To analyze transient energy growth, we consider the ratio of kinetic energy of a disturbance at a given time (τ ) to that at t = 0, G(τ ) =

||x(τ )||2 . 2 x(·,0)=0 ||x(0)|| sup

The quantity ||x||2 represents kinetic energy of x and can be expressed as

(7)

66

John Kim

||x(t)||2 = x∗ (t)Q x(t) = x∗ (t)F∗ Fx(t) = ||Fx(t)||22 = ||F exp[(A − BK)t ]x(0)||22 ,

(8)

where ||·||2 represents the 2-norm (Euclidian Norm). By substituting Eqn. (8) into (7), it can be shown that G(τ ) is equivalent to 2-norm of matrix F exp[(A − BK)τ ]F−1 , which is equal to the largest singular value of the matrix. The singular values (σ) of F exp[(A − BK)τ ]F−1 represent the amplification of the kinetic energy of initial disturbances over time τ . In naturally evolving turbulent channel flows, only a few singular values are larger than one, as shown in Fig. 3, implying that only a few particular disturbances can have the transient growth. The largest σ represents the maximum energy growth ratio at τ , and the corresponding initial disturbance (first right singular vector) is the so-called optimal disturbance. For further discussion on singular vectors and optimal disturbances, the interested reader is referred to Lim & Kim [18]. Singular values in a turbulent channel flow with various different control schemes have been examined, and the results are shown in Fig. 3. In addition to the channel flow with a linear optimal controller (i.e., the control gain matrix K determined by an LQR synthesis), results from the opposition control [19] (see [18] for the structure of K corresponding to the opposition control) and those from Kim & Lim’s [1] virtual flow (i.e., the operator A with Lc = 0) are also shown in Fig. 3. From the distribution of large singular values corresponding to different control approaches, the efficacy of each controller can be predicted, assuming that the present SVD analysis is still valid for the actual nonlinear system (see below). Note that no singular values corresponding to the virtual flow is larger than one, indicating that there would be no transient growth for all disturbances in this case. The above SVD analysis shows how various controllers are effective in reducing the singular values associated with a linear system. Note that the reduction of singular value is related to the reduction of non-normality of the flow system, which is partially responsible for sustaining near-wall turbulence structures (which are in turn responsible for high skin-friction drag in turbulent boundary layers). There is no guarantee, however, that these controllers will be equally effective in nonlinear flows. The effects of reduced singular values in a turbulent channel flow (i.e., nonlinear system) were examined in Lim & Kim [18]. These nonlinear results were consistent with the SVD analysis, demonstrating that the SVD analysis is indeed a viable tool in predicting the performance of a controller in the nonlinear turbulent channel flow.

Physics and Control of Wall Turbulence 10

si

10

10

10

10

67

2

1

0

−1

−2

0

5

10

15

Singular value index, i

Fig. 3. Singular values in a turbulent channel with different controllers: ◦ , no control; •, opposition control; ×, LQR control; , virtual flow. This is for the case of (kx = 0, kz = 6.0), corresponding to λ+ z ≈ 100, and Reτ =100. A similar trend can also be observed in other wavenumbers. Reproduced from Lim & Kim [18].

5 Concluding Remarks It is shown that boundary layers can be analyzed from a linear system perspective. The SVD analysis can provide useful information regarding the controller’s capability of attenuating the transient growth of disturbances in turbulent boundary layers. It should be noted, however, that linearized Navier-Stokes equations are not sufficient in general to describe many features of turbulent boundary layers, including the self-sustaining mechanism of nearwall turbulence, in which a nonlinear mechanism plays an essential role. This limitation notwithstanding, it is shown here that much can be learned from a proper linear analysis of nonlinear flows, especially for the wall-bounded turbulent shear flows in which a linear mechanism plays an important, if not dominant, role. In this regard, and to some extent paradoxically, wall-bounded turbulent shear flows are more amenable to a theoretical analysis than, say, homogeneous isotropic turbulent flows, where no dominant linear mechanism is present.

Acknowledgments I am grateful to many current and former colleagues at UCLA, NASA Ames Research Center and Stanford University, who have contributed to the results reviewed in this paper. The financial support I have received from Air Force Office of Scientific Research (AFOSR) and Office of Naval Research (ONR) during the course of this work is also gratefully acknowledged. The computer time has been provided by National Science Foundation (NSF) through TeraGrid resources.

68

John Kim

References 1. Kim, J. & Lim, J. (2000) A linear process in wall-bounded turbulent shear flows. Phys. Fluids, 12(8): 1885–1888. 2. Jimenez, J & Pinelli, A. (1999) The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389: 335–359. 3. Kravchenko, A. G., Choi, H. & Moin, P. (1993) On the relation of near-wall streamwise vortices to wall skin friction in turbulent boundary layers. Phys. Fluids A, 5(12): 3307–3309. 4. Choi, H., Moin, P. & Kim, J. (1993) Direct numerical simulation of turbulent flow over riblets. J. Fluid Mech., 255: 503–539. 5. Min, T. & Kim, J. (2004) Effects of hydrophobic surface on skin-friction drag. Phys. Fluids, 16(7): L55–L58. 6. Min, T, Yoo, J. Y., Choi, H. & Joseph, D. D. (2003) Drag reduction by polymer additives in a turbulent channel flow. J. Fluid Mech., 486: 213–238. 7. Hamilton, J. M., Kim, J. & Waleffe, F. (1995) Regeneration mechanisms of near wall turbulence structures. J. Fluid Mech. 287: 317–348. 8. Waleffe, F. & Kim, J. (1997) How streamwise rolls and streaks self-sustain in a shear flow. In Self-Sustaining Mechanisms of Wall Turbulence (ed. Panton), Computational Mechanics Publications. 9. Schoppa, W. & Hussain, F. (1997) Genesis and dynamics of coherent structures in near-wall turbulence. In Self-Sustaining Mechanisms of Wall Turbulence (ed. Panton), Computational Mechanics Publications. 10. Schoppa, W. & Hussain, F. (2002) Coherent structure generation in near-wall turbulence. J. Fluid Mech., 453: 57–108. 11. Butler, K. M. & Farrell, B. F. (1992) Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4(8): 1637–1650. 12. Reddy, S. C. & Henningson, D. S. (1993) Energy growth in viscous channel flows. J. Fluid Mech., 252: 209–238. 13. Bamieh, B & Dahleh, M. (2001) Energy amplification in channel flows with stochastic excitation. Phys. Fluids 13(11): 3258–3269. 14. Butler, K. M. & Farrell, B. F. (1993) Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids A, 5(3): 774–777. 15. Lim, J. (2003) A Linear Control in Turbulent Boundary Layers. Ph.D. dissertation, University of California at Los Angeles. 16. Lee, C., Kim, J., Babcock, D. & Goodman, R. (1997) Application of neural networks to turbulence control for drag reduction. Phys. Fluids, 9(6): 1740– 1747. 17. Lee, C., Kim, J. & Choi, H. (1998) Suboptimal control of turbulent channel flow for drag reduction. J. Fluid Mech., 358: 245–258. 18. Lim, J. & Kim, J. (2004) A singular value analysis of boundary layer control. Phys. Fluids, 16(6): 1980–1987. 19. Choi, H., Moin, P. & Kim, J. (1994) Active turbulence control for drag reduction in wall-bounded flows. J. Fluid Mech., 262: 75–110.

The physics of turbulent mixing and clustering J.C. Vassilicos Department of Aeronautics and Institute for Mathematical Sciences, Imperial College London, SW7 2AZ, London, U.K. [email protected]

1 Introduction In many practical applications, calculations and predictions of concentration fluctuation statistics such as variances < θ2 > − < θ >2 of a scalar field θ (e.g. contaminants, chemicals and other passive or active scalars) are required and can be even more important than predictions of mean concentration values. There may also be a need to know potential peak concentrations and their probabilities. Instead of a scalar field, the problem may involve aerosols and/or droplets in a turbulent air flow or mud/clay particles in a turbulent water flow in which case what may be required is an understanding of the way inertial particles cluster and agglomerate rather than mix. Applications abound: air pollution (e.g. in urban environments); moisture, pollutants, heat, salinity and biologically active micro-organisms in the atmosphere, the earth’s boundary layer and oceans; mixing and reaction rates in chemical reactors; generation of powders in chemical and pharmaceutical industries via particle agglomeration; large variations in efficiency of various industrial processes caused by fluctuations in particle concentrations; clustering, collision and coalescence of water droplets in warm clouds (such small-scale processes play an important role in determining macroscopic properties of clouds such as precipitation efficiency and optical albedo properties) and many more. This paper briefly summarises recent research concerned, firstly, with turbulent fluctuation variances of passive scalars and, secondly, with turbulence clustering of inertial particles. The turbulence considered here is idealised as it is statistically stationary, homogeneous and isotropic. However, the aim is to use such idealised turbulence to understand some fundamental physics of turbulent mixing and clustering in the absence of other physics imposed by external forces and boundary conditions. It may be worth mentioning that the initial steps in the evolution of a scalar field released from an elevated source in a turbulent boundary layer proceed as in stationary homogeneous turbulence, in which case conclusions may be directly applicable.

70

J.C. Vassilicos

In this paper we report results obtained from theoretical analysis and Direct Numerical Simulations (DNS) of two- and three-dimensional turbulence and by Large Eddy Simulations (LES) and Kinematic Simulations (KS) of three-dimensional turbulence.

2 Turbulent mixing calculations by Lagrangian statistics Turbulent mixing, as opposed to stirring, result from the interaction of turbulent eddy motions with molecular diffusion. Scalar variances are good measures of mixing as they vanish when the scalar is fully uniform in space and therefore mixed. As noted by Durbin [6] scalar variances can be obtained from integrations over Lagrangian path probabilities of fluid element pairs and are therefore highly sensitive to relative turbulent dispersion. Calculations of pair statistics and their evolution require some understanding and modelling of turbulent motions over the entire inertial range of scales, in particular of the subgrid turbulent motions which are absent in a LES where only their effect on turbulent motions of size greater than the grid size is modelled.

3 KS as subgrid scalar model for LES Flohr & Vassilicos [8] developed a KS subgrid model which obeys Kolmogorov’s inertial-range scaling, is incompressible and incorporates turbulentlike small-scale eddy topology which is coherent in the sense that it is persistent in time. They described the scalar field by puffs of tracer fluid elements and used the modelled subgrid KS velocity field coupled to an LES velocity field along with a Lagrangian integration method to calculate the path probabilities of fluid element pairs. They then used Durbin’s approach to integrate the scalar variance field from these probabilities. Specifically, they decomposed the velocity field into a grid LES component uLES (calculated using a pseudo-spectral method with a sharp Fourier cutoff and Chollet & Lesieur’s [3] form of the eddy viscosity) and a subgrid component uKS given by the KS velocity synthesis which contains two-point turbulence structure information (energy spectrum) as well as small-scale persistent streamline topology. The coupling between these two velocity fields was effected at the LES cutoff wavenumber by setting the LES kinetic energy dissipation rate per unit mass to be equal to its virtual value in the KS field. This LES-KS method makes it possible to study the scalar variance field with inertial range effects explicitly resolved by the KS subgrid field while the LES determines the value of the Lagrangian integral time scale TL . This calculation method gave good agreement with experimental data by Fackrell & Robins [7] for the scalar variance field generated by an instantaneous release from a line source without the need to adjust any free model parameter. The calculations also produced convincing highly non-gaussian Probability

The physics of turbulent mixing and clustering

71

Density Functions (PDF) of pair separations Δ in the inertial range of times and, as expected, gaussian such PDFs for times longer than TL . Richardson’s law < Δ2 >∼ t3 [14] was also recovered as has recently been confirmed by Osborne et al [13] using very high (effective) Reynolds number KS.

4 Theoretical model of turbulent pair diffusion It is important to obtain some physical understanding and/or develop physical models of the specific mechanisms dominating turbulent pair diffusion, at the very least for the purposes of alleviating as much as possible the calculations necessary to predict scalar fluctuation statistics. The LES-KS approach mentioned above, whilst not requiring “fudge” parameters, remains computationally prohibitive for engineering practice. In a recent series of papers [9], [5], [11], [10], [13] a new model of turbulent pair diffusion has been developed where pairs travel together for long stretches of time till they encounter a persistent straining stagnation point (defined in the frame where the mean flow is zero, which is the frame where the persistence of these points is maximised –the characteristic velocity with which these topologically important points move is minimised in that frame and in fact tends to zero with increasing Reynolds number; furthermore the average life time of these points can be estimated to be of the order of TL ) and undergo a sudden straining action which increases their separation by a factor eα where α > 0. They then travel again at that new separation for a long time till they encounter another persistent stagnation point with a larger length-scale of influence and separate further in a sudden burst by another factor eα . This process is repeated for as long as the extent of the inertial range allows. This mechanistic model of turbulent pair diffusion can be theoretically shown to lead to the following equation for the PDF of Δ, P (Δ, t): ∂ ∂ Bα2 ∂ ∂P = −Bα (Δ1−Ds /d P ) + [Δ (Δ1−Ds /d P )] (1) ∂t ∂Δ 2 ∂Δ ∂Δ where d is the dimensionality of space (d = 2 or 3 in two- or three-dimensional 1/2 turbulence), B ∼ Cs u L−1/3 where u is the rms turbulence velocity and L the turbulence integral length-scale and where Cs and Ds are dimensionless numbers characterising the number density ns of stagnation points and its Reynolds number dependence: ns (L/η) =

Cs (L/η)Ds Ld

(2)

where η is an inner length-scale (the Kolmogorov scale in the case of threedimensional turbulence). The exponent Ds is a scale-similarity exponent or fractal dimension. We may call Cs the “Avogadro number of turbulence” with the proviso that it is not necessarily universal because one can imagine changes

72

J.C. Vassilicos

in large-scale eddies implying changes in the value of Cs . The Richardson constant GΔ in < Δ2 >≈ GΔ t3 can be shown to be an increasing power-law function of Cs and therefore potentially non-universal as well. This constant is very important for the calculation of scalar fluctuation intensities in turbulent flows. It may be valuable to point out that this PDF equation is not a FokkerPlanck type equation: no Wiener processes are involved at any of the stages of the derivation. Instead, this PDF equation takes into account the persistent (coherent) multi-scale streamline topology of inertial range scales. It may be possible to use and generalise our rational method for obtaining such a PDF equation in the context of non-isotropic turbulence (such as stratified and/or rotating turbulence) and non-homogeneous turbulence (e.g. turbulent boundary layers) and efforts are currently under way in this direction. The approach relies on the persistence of stagnation points, which can be measured and tested in terms of statistics of stagnation point velocities Vs and on their number densities and the Reynolds number scalings of these densities. Comparing (1) with the PDF equation originally obtained by Richardson [14] by simply fitting field observation data, namely ∂ ∂ ∂P = [K(Δ)Δd−1 (Δ1−d P )] ∂t ∂Δ ∂Δ

(3)

where the effective turbulent diffusivity K(Δ) ∼ Δ4/3 , we find that the two PDF equations are identical provided that Ds = 2 and α = 6/7 in threedimensional turbulence and Ds = 4/3 and α = 3/2 in two-dimensional inverse energy cascading turbulence where the energy spectrum has a −5/3 powerlaw shape. These values of Ds and α are corroborated by DNS of two- and three-dimensional isotropic homogeneous turbulence! Recently, in a private communication to the author, Dr D.J. Thomson of the UK Met Office pointed out that applying the well-mixed condition P ∼ Δd−1 to the stationary form ( ∂P ∂t = 0) of PDF equation (1) leads to 2d α = d2 −D which gives α = 6/7 for D s = 2, d = 3 and α = 3/2 for Ds = 4/3, s d = 2 in full agreement with the DNS observations and arguments mentioned above.

5 Turbulent clustering of inertial particles and stagnation points In turbulence, inertial particles and zero-acceleration points cluster together. This statement summarises the main result reported in what may be the first attempt to understand the effect of the inertial range on clustering [2] (most analyses to date concentrate on the effects on clustering of the smooth flow below the Kolmogorov scale or the random flow which inertial particles with particle relaxation times larger than TL effectively see). The clustering

The physics of turbulent mixing and clustering

73

of inertial particles reflects that of zero-acceleration points (acceleration stagnation points as opposed to the aforementioned velocity stagnation points) for all particle relaxation times smaller that the integral time-scale of the turbulence. Because of computer limitations this point has been made using DNS of two-dimensional statistically stationary, homogeneous and isotropic turbulence with a well-defined −5/3 power-law energy spectrum over nearly two decades (40962 grid points). The inertial particle trajectories were calculated by integrating the equation of a small rigid sphere’s motion [12] where only the linear Stokes drag was kept because the limit considered was the one where the particles are spherical and of radius much smaller than η, where their mass density is much larger than that of the carrier fluid and where the particle Reynolds number is significantly smaller than 1. Particles were assumed to be dilute enough for particle-particle interactions to be neglected. Being so small, the particles were also effectively assumed not to affect the fluid turbulence. The main conclusions of this study are the following: (i) In turbulence, zero-acceleration points cluster. (ii) On average, the acceleration field is swept by the velocity field more and more accurately as the Reynolds number increases. This is a quantitative (see [2]) reformulation of the Tennekes [15] sweeping hypothesis which stipulates that large turbulence eddies advect small ones. (iii) Inertial particles and zero-acceleration points are swept and stay together for long. Non-zero-acceleration points and inertial particles move apart. (iv) As a result, inertial particles cluster in a way which reflects the zeroacceleration point clustering and which is therefore very similar for different Stokes numbers St (defined as the ratio of the particle relaxation time to a time scale of the turbulence). As St increases, this inertial particle clustering becomes more pronounced because inertial particles are markedly different from fluid elements in more of the space. (See also [1], [4] where inertial particle clustering in two-dimensional and three-dimensional DNS isotropic homogeneous turbulence has already been reported.)

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Boffetta G, De Lillo F, Gamba A (2004) Phys Fluids 16(4):L20 Chen L, Goto S, Vassilicos JC (2006) J Fluid Mech (to appear) Chollet JP, Lesieur M (1981) J Atmos Sci 38:2747-2757 Collins LR, Keswani A (2004) New J Phys 6:119 Davila J, Vassilicos JC (2003) Phys Rev Lett 91:144501 Durbin PA (1980) J Fluid Mech 100:279–302 Fackrell JE, Robins AG (1982) J Fluid Mech 117:1-26 Flohr P, Vassilicos JC (2000) J Fluid Mech 407:315-349 Fung JCH, Vassilicos JC (1998) Phys Rev E 57:1677-1690 Goto S, Osborne DR, Vassilicos JC, Haigh JD (2005) Phys Rev E 71:15301(R) Goto S, Vassilicos JC (2004) New J Phys 6:65

74 12. 13. 14. 15.

J.C. Vassilicos Maxey MR, Riley JJ (1983) Phys Fluids 26(4):883-897 Osborne DR, Vassilicos JC, Sung K-S, Haigh JD (2006) Phys Rev E (sub judice) Richardson LF (1926) Proc R Soc Lond A 110:709-737 Tennekes H (1975) J Fluid Mech 67: 561-570

Part II

Turbulent Mixing and Combustion

LES of Premixed Flame Longitudinal Wave Interactions Christer Fureby1 and Christophe Duwig2 1

2

FOI, Swedish Defence Research Agency, SE-164 90 Stockholm, Sweden [email protected] Division of Fluid Mechanics, Lund Institute of Technology, SE-221 00 Lund, Sweden [email protected]

Summary. Here we study combustion instabilities in a simplified model of a jet engine afterburner at different operating conditions using reacting Large Eddy Simulations (LES). Two LES models, employing the filtered flamelet approach but implemented in two different codes, are used. Comparisons are made between LES and experimental data to clarify the usefulness and limitations of LES. Reacting cases not experiencing thermoacoustic instabilities are generally well predicted, whereas reacting cases with thermoacoustic instabilities are more difficult to predict. Some types of instability (e.g. screech) can be predicted, whereas other types of instabilities (e.g. longitudinal waves) need particular (forced) inflow boundary conditions, suggesting that supergrid boundary conditions is an issue. A parametric study is carried out to quantify the response to the forcing frequency and amplitude.

1 Introduction The demand for ever decreasing emission levels of turbofan and turbojet engines has led to the development of the lean-premixed (prevaporized) combustor technology. Here, low emissions can be achieved by perfecting the premixing of the fuel-air mixture and by operating at fuel-lean (i.e. low temperature) conditions. As a consequence, the combustion is often accompanied by instabilities (or even lean blow-out) of various kinds, which may deteriorate the combustion process or even reduce the combustor life. Although the mechanisms leading to combustion instabilities are not yet fully understood, many such instabilities are the result of the coupling between the unsteady combustion process and acoustic waves in the combustor, including its supply and exhaust systems. The presence of acoustic waves in a combustor can modify the flame significantly due to the scales of the acoustic perturbations and their energy content (comparable to that of the turbulence).

78

Fureby and Duwig

Moreover, the acoustic perturbations often have a spatio-temporal coherence that can amplify their effects in the entire combustion system. Analytical and semi-analytical techniques may be used to gain some insight into the acoustic eigenmodes [2], whereas Reynolds Averaged Navier Stokes (RANS) models in conjunction with numerical solution linearized Euler equations [14] can identify the acoustic eigenmodes but not predict whether they will be amplified or damped. The weakest part of these models is the description of the flame response to the acoustic perturbations. Improved predictions are offered by Large Eddy Simulations (LES) [2]. The fact that the large energy containing structures are resolved, whereas only the smaller structures are modelled gives LES higher generality and accuracy than RANS. To study combustion instabilities in an aircraft engine by means of LES will ideally require a complete geometric model (including compressor and turbine) together with inflow/outflow boundary conditions with proper impedances. This is very expensive, and also subject to many uncertainties, and instead other strategies are recommended. One such strategy is to examine the response of a combustor to a certain forcing of e.g. the inflow velocity or the inflow fuel mass fraction. In this paper this strategy will be used to investigate the experimentally observed instability patterns of a bluff-body stabilized flame, emulating a jet engine afterburner [4, 5, 6].

2 Theoretical Formulation and Numerical Procedures The equations governing turbulent reacting flows are the reactive Navier Stokes Equations (NSE), i.e. the balance equations of mass, species, momentum and energy. Due to the range of scales and the large number of species involved these equations usually must be simplified. In LES we may choose between (i) reduced or global reaction mechanisms with Arrhenius chemistry using Implicit LES (ILES) [1]; (ii) flamelet models in which reaction is confined to a thin layer [2]. (iii) Linear Eddy Models, using a grid-within-the-grid approach to solve 1D species equations with full resolution [3]; (iv) presumed (or transported) Probability Density Function (PDF) methods and the Thickened Flame Approach [2]. Here, two filtered flamelet models are used. The chemical reaction is described by a reactive scalar ζ, such that ζ = 1 in the reactants and ζ = 0 in the products. Thus, the governing equations consist of the low-pass filtered continuity, momentum and energy equations together with the Favré-filtered ζ equation, ˜ + ∇ · (¯ ˜ = ∇ · (D∇ζ) − ∇ · bζ − w ¯˙ ˜ ζ) ρζ) ρv ∂t (¯

(1)

−v ˜ the ˜ ζ) where ρ is the density, v the velocity, D the diffusivity, bζ = ρ¯(vζ ¯ subgrid transport term and w˙ the reaction rate. In the first filtered flamelet model the terms on the r.h.s of (1) are regrouped as a filtered flame front displacement term, −ρSu |∇ζ|, describing the

LES of Premixed Flame Longitudinal Wave Interactions

79

propagation of ζ at the laminar flame speed Su . This term is then expanded ¯ where Ξ is the flame-wrinkling. as ρu Su Ξ|∇ζ|, The second filtered flamelet model consists of splitting the terms on the right hand side of (1) between diffusive and non-diffusive terms. The diffusive transport is modelled using a Fickian scheme while the rest is gathered into a production term Π. Π is a function of ζ and a non-dimensional number a ∼ Δ/δ, where δ is the laminar flame thickness [10]. Here, we took a ∼ 4, and the closed transport equation for ζ reads, − 1 Π(1 − ζ, a)] ˜ + ∇ · (¯ ˜ = ρu Su Ξ[∇ · ( Δ ∇ζ) ˜ ζ) ρζ) ρv ∂t (¯ a Δ

(2)

Both closures require a model for the subgrid flame wrinkling. Following [9] we here apply a fractal model for Ξ = (Γ v /Su )D−2 , where v , D and Γ are respectively the subgrid velocity fluctuation, the fractal dimension and Poinsot’s efficiency function [2] [9]. In the first CFD code the governing equations are discretized using an unstructured finite-volume method. Second order schemes are used in space and time; central differencing for velocity, a monotone scheme for scalars and a Crank-Nicholson time-integration scheme. To decouple the pressure-velocity system, a PISO-type procedure is adopted. The equations are solved sequentially with iteration over the explicit coupling terms with a Courant number of less than 0.3. The One-Equation Eddy-Viscosity Model (OEEVM) [11], is used for closing the filtered equations. The second CFD code used is a Cartesian Finite-Differences code. The code is implicit with multigrid acceleration. Centered fourth order scheme is used for the spatial derivatives except for the convective terms while a second order upwind scheme is used for time discretization. The convective terms are discretized using a 5th order WENO scheme [13]. The Filtered Structure Function (FSF) model [12] is used for closing the filtered equations.

3 Configuration: Jet Engine Afterburner A case for which combustion instabilities, and in particular longitudinal wave interactions, have been observed and documented is the Validation Rig [4]. The rig involves a rectilinear combustor of length L = 25h, height 3h and width 6h in which a triangularly shaped flameholder, with height h = 0.04m is mounted at x/h = 7.95. Optical access is provided by quartz windows on the sides whereas the upper and lower walls are water cooled. The computational domain covers only the combustor and employs 1 or 2 million grid points. At the inlet, Dirichlet conditions are used for all variables besides the pressure for which zero Neumann conditions are used. At the outlet, wave-transmissive conditions are used. For the flameholder and tunnel walls no-slip conditions for the flow. The computational domain is periodic in the spanwise direction, with a width of 3h.

80

Fureby and Duwig

The Validation Rig was operated under a variety of conditions and several non-reacting and reacting cases were studied [4, 5, 6], see Table 1. Previous studies, e.g. [7, 8] focused on the non-reacting cases (Cases I and II) and on the reacting cases in which acoustic effects were deemed small, i.e. Cases III and IV. Here, we will focus on the cases under which the acoustic effects are important, i.e. Cases V and VI, and in particular Case VI. Table 1. Validation Rig: Test cases and operating conditions Case I or II III IV V VI

vin (m/s)

Tin (k)

φ

comments

17 or 34 17 34 34 17

288 288 600 288 288

0 0.61 0.58 0.72 0.95

non-reacting symmetric shedding mixed shedding screech mode longitudinal low frequency mode

4 Cases I to IV: Summary of Results and Discussions Figure 1 shows perspectives of Cases III and IV in terms of iso-surfaces of the flame, vortex cores and superimposed contours of the temperature together with selected profiles of the axial velocity for Cases I to IV and the temperature for Cases III and IV at x/h = 11.5. For the non-reacting cases (Cases I and II) the agreement with measurement data for the axial velocity and its rms-fluctuations is good. Spectral analysis of the experimental and computed vertical velocity component in the wake of the flameholder reveal a shedding frequency of 105 and 103Hz, respectively. For the reacting cases (Case III and IV) the flame anchors behind the flameholder due primarily to the recirculation of hot combustion products in the wake behind the bluff-body, and spanwise vortices are shed off the upper and lower edges of the prism. The symmetric shedding and roll-up (Case III) results in longitudinal vortices stretched between succeeding spanwise vortices on either side of the centreline in the near-wake, whereas the mixed shedding and roll-up (Case IV) results in vortex-dynamics similar to Case III close to the flame holder, whereas further downstream an asymmetric pattern appears similar to the non-reacting case with spanwise vortices stretched between succeeding longitudinal vortices of alternate sign. As the flame advects downstream it propagates normal to itself, causing negatively curved wrinkles to contract and positively curved wrinkles to expand. The high-temperature region is associated with exothermicity, and burning occurs at the fuel-rich side of the shear layers, where also most of the mixing between cold reactants and hot products takes place. The predicted dynamics of Case III is well documented experimentally with high-speed video photography, CARS


81

data and flash schlieren, e.g. figure 9 in [6] and the agreement between LES and data is very good, whereas Case IV is less well documented. As observed in figure 1 the agreement between LES and data is good for both cases. The most noteworthy feature is that LES can capture the experimentally observed and measured differences between cases III and IV. This is emphasized e.g. by comparing measured and simulated Probability Density Functions (PDFs) of the temperature which show good qualitative agreement. x2/h

x2/h

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5 0

0

2

/v 4 1 0

0 0

1000

2000

Fig. 1. Perspective views of Case III (top) and IV (bottom) together with the time averaged axial velocity (middle) and the time averaged temperature (right). Symbols denote measurement data and lines denote LES results using the fractal flame wrinkling model. Gray lines and symbols refer to non-reacting situations whereas black lines refer to the reacting cases III and IV.

5 Cases V and VI: Results and Discussions As opposed to Cases III and IV, Cases V and VI were experimentally observed to experience strong combustion instabilities. For further experimental details and images of the different modes of operation we refer to [6]. The screech mode (Case V) occurs when the equivalence ratio is increased from 0.61 to about 0.72 at an inflow velocity of 34m/s and a temperature of 288K, whereas the longitudinal low frequency oscillation mode occurs when the equivalence ratio was raised to about 0.95 from Case III conditions. Reacting LES of Case V are quite successful in reproducing the observed screech mode, cf. figure 2 and figure 14 in [6]. This suggests that screech is rather insensitive to the details of the inflow/outflow boundary conditions, and is a mode that is determined by the local coupling of the flow and the thermodynamics. An interesting feature observed in the LES, and supported by the experiments, is that the flame is less wrinkled in the spanwise direction (as compared to Cases III and IV), resulting in a synchronized roll-up of the shear layer across the span, and a highly periodic and symmetric heat release. According to the FFT analysis of the pressure signal in the wake in figure 3 this case results in high-frequency pressure oscillations at about 1380 Hz. This

82

Fureby and Duwig

Fig. 2. Numerical schlieren images of Case III (top), Case V (middle) and a time series for Case VI (bottom four pictures). The right column denotes matching experimental schlieren images from the experiments at Volvo Aero Corp. [6].

agrees well with the experimentally observed behavior, with high-frequency pressure oscillations occurring at about 1400 Hz. Reacting LES of Case VI is, however, not capable of reproducing the observed longitudinal low frequency oscillation mode occurring at about 100 Hz. The reacting LES results in a flame similar to that of Case V but with lower frequencies involved. In order to mimic these low frequency longitudinal oscillations in the reacting LES, we performed a sequence of simulations with forced inflow conditions. The forcing is applied as a sinusoidal perturbation of the axial velocity component, with a non-dimensional amplitude ranging from A = 0.05 to A = 0.20 (in steps of 0.05) and frequencies of 50Hz, 100Hz, 200Hz and 400Hz, at the inflow. An equidistant time-sequence of one such simulation is presented in figure 2, cf. figure 13 in [6]. It is found that by modulating the inflow velocity in this way longitudinal low frequency oscillations occur also in the LES calculations. Hence, in contrast to screech, longitudinal low frequency oscillations appears to be more strongly related to the inflow/outflow conditions of the LES. This also implies that the computational domain selected is probably too small and that the upstream and downstream parts of the laboratory combustor should be taken into account. The influence of amplitude and frequency of the inflow velocity perturbations have been systematically investigated, and were found to significantly modify the behavior of the flame. The volumetrically integrated heat release vs time (together with the forcing at 100Hz) in figure 3 suggests that (i) as


83

the amplitude of the velocity perturbations increases the total heat release increases (at a fixed frequency); (ii) the velocity perturbations, irrespectively of A, increase the temporal coherence of the heat release, resulting in clearly separated ‘puffs’ of exothermicity; (iii) the forcing at 200Hz and 400Hz results in less coherent ‘puffs’ suggesting that at these frequencies the correlation with the flow (i.e. the pressure) is less evident. In particular, we notice that peaks at 400 Hz are close to the inertial range of the spectrum (figure 3). The FFT analysis of the heat-release shows that the excitation at 400Hz locks the coherent structures arising in the shear-layer to a harmonic (800Hz and a very strong peak at 1200Hz). On the contrary, excitation at 50Hz or 100Hz (not shown here) are superposed to the turbulent fluctuations. The time lag, τ , between the heat release and the forcing is found to vary substantially (between π/5 and π) between forcing cycles, emphasizing the non-linearity of the reacting flow problem. Best agreement with the experimental schlieren images are obtained at 100Hz at A = 0.10. reaction rate [kg/m /s]

8

15

20

15

x 10

Power Spectrals Density of p case V case VI no forcing case VI A=0.10, f=100 Hz case VI A=0.05, f=100 Hz case VI A=0.20, f=100 Hz case VI A=0.05, f=200 Hz case VI A=0.05, f=400 Hz

10

10

forcing case VI no forcing case VI A=0.10, f=100 Hz case VI A=0.05, f=100 Hz case VI A=0.20, f=100 Hz case VI A=0.05, f=200 Hz case VI A=0.05, f=400 Hz

5

0

0

0.01

0.02

0.03

0.04

5

0

0.05

200

400

600

t [s]

800 1000 1200 1400 1600 f [Hz]

Power Density Spectra of the total reaction rate

60

Power Density Spectra of the turbulent kinetic energy

case VI A=0.1 f=50Hz case VI A=0.1 f=400Hz 6

50

10

5

10

40

4

10

30 3

10

20 2

10

10

0 0

1

10

case VI A=0.1 f=50Hz case VI A=0.1 f=400Hz case VI A=0.1 f=100Hz

0

200

400

600

800 f [Hz]

1000

1200

1400

10 1 10

2

10

3

10 f [Hz]

4

10

Fig. 3. Volumetrically integrated heat release (VIHR) vs time together with the forcing (up-left), FFT analysis of the pressure (up-right). FFT analysis of the VIHR (down-right) and of the turbulent kinetic energy (down-left).

84

Fureby and Duwig

6 Conclusions In this paper we have examined different types of combustion instabilities (screech and longitudinal waves) in a model afterburner using reactive LES. Good agreement between the different LES models was observed both qualitatively and quantitatively. Moreover, good agreement between the LES predictions and experimental data is obtained for the baseline case (no thermoacoustic instabilities). For the cases experiencing combustion instabilities the situation is more involved: The screech mode is reasonably well predicted without additional forcing, whereas the low frequency instability mode needs additional forcing of the inflow. This is further investigated by systematically varying the frequency and amplitude of the inflow forcing. Our results suggest that the screech mode is less sensitive to open inflow/outflow boundary conditions, whereas the longitudinal mode is in fact driven by these. Hence, in order to simulate these instabilities the intake and exhaust systems must be included in the LES model. Acknowledgment H.G. Weller is acknowledged for the development of the C++ class library FOAM (Field Operation And Manipulation), version 1.9.2β, partly used in this study. The authors want to thank A. Sjunnesson and L. Andersson (Vovlo Aero Corp.) for providing the Schlieren pictures. This work was partially financed by the Swedish Gas Turbine Center (GTC). The computations were partially run on HPC2N facilities with the program SNAC.

References 1. Grinstein F.F., Kailasanath K.K (1994) Comb. Flame, 100, p 2. 2. Poinsot T., Veynante D., Theoretical and Numerical Combustion, RT. Edwards, Philadelphia, USA 3. Menon S. (2000) in Advances in LES of Complex Flows, Eds. Friedrich R., Rodi W., p 329, Kluwer, The Netherlands. 4. Sjunneson A., Olovsson S., Sj¨ oblom B. (1991) Presented at the ISABE conference, Nottingham, UK. 5. Sjunnesson A., Nelson C., Max E. (1991) VOLVO Flygmotor AB, S-461 81, Trollh¨ attan, Sweden. 6. Sjunnesson A, Henriksson P, L¨ ofstr¨ om C. (1992) AIAA Paper No 92-3650. 7. Fureby C. (2000) 28th Int. Symp. on Comb, p 783. 8. Fureby C. (2000) Comb. Sci and Tech, 161, p 213. 9. Fureby C. (2004) Proc. of the 30th Int Symp on Comb, p 593. 10. Duwig C., Fuchs L. (2005) Comb. Sci. and Tech., in press. 11. Schumann U. (1975) J. Comp. Phys., 18, p 376. 12. Durcos F., Comte P., Lesieur M. (1996) J. Fluid Mech., 326, p 1. 13. Jiang G.S., Shu C.W. (1996) J. Comp. Phys., 126, p 202. 14. Eriksson L.E.E, Andersson L., Lindblad K., Andersson N. (2003) Proceedings of the AIAA-ISAB conf.

Direct Numerical Simulation of Reacting Turbulent Multi-Species Channel Flow L. Artal1,2 and F. Nicoud2 1

2

CERFACS, CFD Team, 42 avenue Gaspard Coriolis, 31057 Toulouse Cedex 1, France [email protected] University Montpellier II - CNRS UMR 5149, I3M - CC 51, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France [email protected]

1 Introduction Rocket engines eject their gases through propulsion nozzles which are subjected to considerable thermal fluxes. However they must stay as light as possible. This requires precise calculations of wall heat flux in order to optimize the design of insulating material. Industrial design codes use ReynoldsAveraged Navier-Stokes (RANS) methods. High-Reynolds number approaches are often preferred for wall-bounded flows because they require less mesh refinement than their low-Reynolds number counterpart and they are usually more stable. This means using wall functions to assess the momentum/energy fluxes at the solid boundaries knowing the outer flow condition at the first off-wall grid points. The classic logarithmic law is implemented in most of the RANS codes and provides reasonably good results for simple incompressible flows. The trend today is to generalize this wall function approach to account for more physics [1], [2]. Development of wall functions that take into account strong changes in the density due to strong temperature gradient and fluid non-homogeneity is necessary to simulate parietal heat transfers with good accuracy and at moderate cost by using RANS-based design codes. Because the wall heat flux depends on the details of the turbulent flow in the near-wall region, it is essential to analyze detailed relevant data to support the development of such wall models. Classic experimental techniques cannot provide the required space resolution and improved wall functions can hardly be designed or validated due to the lack of relevant data. Direct Numerical Simulations (DNS) can then be used to generate precise and detailed data sets of generic turbulent flows [3], [4] under realistic operating conditions. The first objective of this paper is to describe a DNS of anisothermal multi-species channel flow. The second objective is to illustrate how it can be used to improve the existing wall functions and account for more physics in the heat flux assessment.

86

L. Artal and F. Nicoud

2 Configuration of the DNS: fluid flow, thermochemistry and solved equations A sketch of the computational domain and the coordinate system is shown in figure 1. A periodic boundary condition is used in the streamwise and spanwise directions, while a no-slip isothermal condition is applied to both walls. In order to save CPU time, the DNS configuration is inspired by the minimal channel flow configuration presented in [5] for instance. The domain dimensions in + + terms of wall units are L+ x 560, Ly 360 and Lz 160 in the streamwise, + normal and spanwise directions respectively (Li = Li uτ /νw where uτ denotes the friction velocity and νw the kinematic viscosity at the wall). The mesh contains 17 × 130 × 33 nodes. The associated resolution in x and z which is Δx+ 35 and Δz + 5, allows to capture the elongated turbulence structure. Uniform in the streamwise and spanwise directions, the grid spacing of the parallelepipedic mesh is refined with a hyperbolic tangent law in the direction normal to the flow, in order to be able to solve the viscous sublayer. This results in Δy + 0.9 at the wall and Δy + 5 near the centerline. The Mach number Ma is about 0.2 so that the effects of compressibility can be neglected and the CFL condition is not too restrictive. For the friction Reynolds number Reτ , the classic value 180 encountered in the literature of turbulent channels with inert walls ([6] for instance) is chosen. Table 1 gathers a few data representative of the flow configuration. This computation is initialized thanks to an already turbulent solution coming from a DNS performed on the same configuration but without chemistry, and having reached a converged state.

Lx = 3.14 h

Lz = 0.94 h y, v

Ly = 2 h FLOW

x, u z, w

Fig. 1. Geometry of computational domain and coordinate system

P (MPa) Twall (K) Tmean (K) h (mm) Mamax Reτ = uτ h/νw Re = Umax hρ/μ 10 2700 3000 0.115 0.2 180 3400 Table 1. Data related to the DNS configuration

DNS of Reacting Turbulent Multi-Species Channel Flow

87

The ejected gases from propulsion nozzles form a reactive3 mixture made up with about a hundred gaseous species. The DNS is performed with a gaseous mixture representative of these gases. The transport properties of this equivalent mixture are obtained thanks to the EGLIB library [7]. The DNS is realized with a Schmidt number per species indicated in table 2, and a Prandtl number for the mixture equal to 0.47. Because the numerical code used does not assume chemical equilibrium, a kinetic scheme which reproduces concentration changes in the equivalent mixture through seven chemical reactions has been tuned, thanks to the GRI-Mech data [8]. In report [9], it has been verified that this kinetic scheme gives the right compositions at equilibrium. Species H2 N2 CO H2 O H CO2 OH Sc 0.20 0.87 0.86 0.65 0.15 0.98 0.53 Table 2. Schmidt number Sc of each gaseous species at P = 10 MPa

We consider the three-dimensional compressible unsteady Navier-Stokes equations with chemistry and spatially constant source terms. Because of periodicity in the streamwise direction, a source term Sqdm that acts like a pressure gradient must be added to the streamwise component of the momentum conservation equation. Moreover, it is necessary to add a source term Se to the total energy conservation equation in order to compensate for the thermal losses occuring at the isothermal wall. This source term is constant in space so that it does not modify the temperature profile and drives the averaged temperature in the channel towards the target value Tmean . Tmean is fixed so that the temperature gradient in the boundary layer of a propulsion nozzle is reproduced. The computation is performed with the AVBP parallel code developed at CERFACS and dedicated to DNS/LES of reacting flows on unstructured and hybrid meshes. This code has been thoroughly validated [10].

3 Results of the DNS: chemical equilibrium, shear stress and heat flux Because of very high pressure/temperature values and extreme thinness of the boundary layer, no detailed experimental data are available for typical flows in nozzles of rocket motors, and virtually no model exists for the assessment of the corresponding wall fluxes. Consequently, the present DNS results are analyzed in terms of wall function: the ability of existing laws-of-the-wall to 3

In addition, they chemically react with the carbon-coated walls of the nozzles through an ablation reaction. However, this topic will not be dealt with in this paper.

88


reproduce the results is first tested; then, these laws-of-the-wall are enriched thanks to the data generated by the DNS. Knowing the mean temperature profile, the profiles of mass fractions at equilibrium can be obtained thanks to the equilibrium curves of the equivalent mixture and compared to the profiles obtained by post-processing the fields of mass fractions from the DNS. Such a comparison is made in figure 2 which shows that the hypothesis of chemical equilibrium is valid in the DNS. In terms of wall functions, it means that a law for temperature and velocity is enough to determine the wall heat flux and the shear stress. 0.0396

0.425

0.38 0.3795

0.0394 0.0393

Y_CO

0.4245 Y_N2

Y_H2

0.0395

0.424 0.4235

0.0392 0.0391 −1

0 y/h

0.423

1

−1

0.377 −1

1

0.032

0.0004 0.0003

−1

0.002

0 y/h

1

0 y/h

1

0.0002

0 y/h

1

0 y/h

1

0.0315 Y_CO2

Y_H

Y_H2O

0.125

0 y/h

0.0005

0.126

0.378 0.3775

0.128 0.127

0.379 0.3785

0.031 0.0305 0.03 0.0295

−1

1

0 y/h

0.029 −1

Y_OH

0.0015 0.001 0.0005 0

−1

Fig. 2. Chemical equilibrium - Symbols: Mean mass fractions from the DNS; Continuous line: Mass fractions at equilibrium from the DNS temperature profile.

The derivation of logarithmic wall functions usually relies on the existence of a zone with constant friction/heat flux. For the shear stress and the heat flux in the turbulent wall region, such an assumption leads to: τw = −ρ u v ≈ μt

du dy

(1)

when the Boussinesq hypothesis is applied, and to: qw = ρ C p v T ≈ −

μt Cp ∂T P rt ∂y

(2)

When the chemical composition evolves, the mixture features change, a heat flux related to species diffusion exists and the energy equation admits a chemical source term.

DNS of Reacting Turbulent Multi-Species Channel Flow 1e+07

89

q_w

8e+06 6e+06

W / m2

4e+06 2e+06 0

−2e+06 −4e+06 −6e+06 −8e+06 q_w −1e+07

−1

−0.75

−0.5

-0.25

0 y/h

0.25

0.5

0.75

1

Fig. 3. Heat flux in the turbulent zone - Continuous line: Total heat flux; Dashed line: Sum of the turbulent heat flux and of the turbulent species diffusion term; Dashes and points: Turbulent heat flux.

Figure 3 points out that in the turbulent zone the total heat flux can be reasonably approximated by the sum of the turbulent heat flux ρ C p v T and ns of the turbulent species diffusion term k=1 ρ v Yk Δh0f,k . The wall heat flux can then be obtained by extrapolating the curves up to the wall. Because of the low Reynolds number of this DNS, the Fourier term and the turbulent species diffusion term are found to be of the same order of magnitude in the turbulent zone. This explains the discrepancy noticed between the total heat flux and its approximation. However, we anticipate the Fourier term to become truly negligible when the turbulent Reynolds number is large enough. Therefore, at high Reynolds number, and when there is no forcing term, the wall turbulent zone should be characterized by: ρ C p v T +

ns

ρ v Yk Δh0f,k ≈ qw

(3)

k=1

which differs from equation (2). The classic logarithmic law does not represent the present results, as shown in figure 4 which compares the non-dimensional temperature profile T + given by various laws (the logarithmic law, the Kader correlation [11] and the new wall function proposed in this paper) to the one coming from DNS. T + is defined by T + = (Tw − T )/Tτ , where Tτ = qw /(ρw Cp uτ ) denotes the friction temperature, qw the wall heat flux and Cp the mixture mass heat capacity at constant pressure.

90

L. Artal and F. Nicoud 20

T+

15

10

5

0

1

10

100 y+

Fig. 4. T + profiles - Continuous line: DNS results; Dashed line: New wall function; Dashes and points: Kader correlation; Points: Logarithmic law T + = (0.9/0.4)lny + + 3.9.

Two main reasons have been identified for this disagreement: a) on the contrary to the Kader correlation, the log-law does not account for variations in the molecular Prandtl number and is made for cases with molecular Prandtl number close to unity (it is 0.47 in the present study), b) even if the mixture is always close to chemical equilibrium, the heat release due to chemical reactions is sufficiently large and non uniform over space to prevent the classic constant turbulent heat flux assumption from holding and equation (3) should be used instead of (2). Although in better agreement with the DNS data than the logarithmic law, the Kader correlation does not reflect DNS because of chemistry influence. In order to account for the chemistry effects in the wall region description, a new wall function is derived from equation (3). The turbulent heat flux is again modeled by the classic formula: ρ C p v T ≈ −

μt Cp ∂T P rt ∂y

(4)

The turbulent species diffusion terms must be modeled too. For each species, one makes use of the following approximation, valid under the chemical equilibrium assumption: ρ v Yk ≈ −

μt Wk dX k ∂T μt Wk ∂X k ≈− Sck,t W ∂y Sck,t W dT ∂y

(5)

DNS of Reacting Turbulent Multi-Species Channel Flow

91

Writing equation (1) in terms of non-dimensional variables and integrating it, the following logarithmic approximation is found for the Van Driest transformation [12], [13]: u+ VD

U+

= 0

1 ρ du+ = ln y + + C ρw κ

(6)

Dividing equation (3) by equation (1) and injecting equations (4) and (5) yields: T = C1 − αU + (7) Tw with α=

Cp P rt

+

1 W

and Bq =

ns

Cp Bq

dX k 1 0 k=1 Sck,t Wk dT Δhf,k

qw Tτ =− ρw Cp uτ Tw Tw

(8)

(9)

Assuming that the thermodynamic pressure is constant through the boundary layer, combination of equations (6) and (7) leads to the following coupled wall function including chemistry and sensivity to the molecular Prandtl number: √ √ 2 C1 − C1 − αU + = κ1 ln y + + C α (10) w T + = T −T = Bαq U + + K(P r) Tτ At this point, we still have the liberty to fix the constant of integration C1 . To do so, we impose the wall function (10) to be coherent with the Kader correlation in the limiting case of a passive scalar. This implies C1 = 1 − Bq K(P r), with K(P r) = β(P r)−P rt C+(2.12−(P rt /κ))ln(100) and β(P r) = 2 (3.85 P r1/3 − 1.3) + 2.12 ln(P r) when we impose our wall function to give the same T + as the Kader correlation at y + = 100. Note that this definition makes C1 close to unity, which is consistent with the experimental data [14].

4 Conclusion The DNS of reacting turbulent multi-species wall-bounded flow presented in this paper provided accurate data, which highlighted the chemistry influence on the heat flux and the known sensitivity of the heat flux to the molecular Prandtl number. Thanks to these results, a new wall function which takes into account chemistry and molecular Prandtl number was derived and successfully a priori tested.

92


5 Acknowledgements The authors are grateful to Snecma Propulsion Solide (Groupe SAFRAN) for supporting this work, and to the CINES (Centre Informatique National pour l’Enseignement Supérieur) for the access to supercomputer facilities.

References 1. T. J. Craft, A. V. Gerasimov, H. Iacovides, and B. E. Launder. Progress in the generalization of wall-function treatments. International Journal of Heat and Fluid Flow, pages 148–160, 2002. 2. R. H. Nichols and C. C. Nelson. Wall function boundary conditions including heat transfer and compressibility. AIAA Journal, 42(6):1107–1114, June 2004. 3. P. Moin and K. Mahesh. DIRECT NUMERICAL SIMULATION: A Tool in Turbulence Research. Annual Review of Fluid Mechanics, pages 539–578, 1998. 4. N. Kasagi and O. Iida. Progress in direct numerical simulation of turbulent heat transfer. Proceedings of the 5th ASME/JSME Joint Thermal Engineering Conference, March 15-19 1999. 5. J. Jiménez and P. Moin. The minimal flow unit in near-wall turbulence. Journal of Fluid Mechanics, 225:213–240, 1991. 6. J. Kim, P. Moin, and R. Moser. Turbulence statistics in fully developed channel flow at low Reynolds number. Journal of Fluid Mechanics, 177:133–166, 1987. 7. A. Ern and V. Giovangigli. EGLIB: A General-purpose FORTRAN Library For Multicomponent Transport Property Evaluation. CERMICS and CMAP, 2001. 8. http://www.me.berkeley.edu/gri mech. 9. L. Artal. Modélisation des flux de chaleur stationnaires pour un mélange multiespèce avec transfert de masse ` a la paroi, Rapport d’avancement Année 2. Technical Report CR/CFD/04/113, SNECMA-CERFACS-UMII, 2004. 10. http://www.cerfacs.fr/cfd/avbp code.php. 11. B. A. Kader. Temperature and Concentration Profiles In Fully Turbulent Boundary Layers. International Journal of Heat and Mass Transfer, 24(9):1541– 1544, 1981. 12. E. R. Van Driest. Turbulent boundary layers in compressible fluids. J. Aero. Sci., 18(3):145–160, 1951. 13. P. G. Huang and G. N. Coleman. Van Driest transformation and compressible wall-bounded flows. AIAA Journal, 32(10), 1994. 14. P. Bradshaw. Compressible turbulent shear layers. Ann. Rev. Fluid Mech., 9:33–54, 1977.

DNS/MILES of Reacting Air/H2 Diffusion Jets L. Gougeon1 and I. Fedioun2 1

2

¨ ` Laboratoire de Combustion et SystEmes REactifs, C.N.R.S. - E.P.E.E. 1C, avenue de la Recherche Scientifique, ` F-45072 OrlEans cedex 2, France [email protected] Same address [email protected]

1 Introduction In the general framework of airbreathing hypersonic propulsion, numerical simulation of scramjet combustion chambers is a challenging task. At the moment, Reynolds Averaged calculations (RANS) are the standard approach to this problem but considerable modeling effort is needed to extend standard turbulence models to highly compressible flows and large heat release (e.g. simulation of the LAERTRE chamber, Davidenko & al. [1]). In such complicated situations, turbulence models are to be tuned to reproduce experimental results which serve as reference, and extrapolation to different flows or geometries is uncertain. Hence the Large Eddy Simulation (LES) is a promising tool and has yet produced some valuable results in industrial configurations (e.g. [2]), although, as often claimed by Denis Veynante (plenary lecture, ECM2005) there is a lot of ”dust under the carpet”. Our conviction is that in industrial LES of reacting flows, there is so much ”dust under the carpet” that the physically sound part of the modeling may be completely polluted by the neglected (negligible?) subgrid terms and by numerical errors, mainly the numerical diffusion needed for stable calculations. What is needed is enough viscosity, either subgrid scale or numerical or both for the calculation to be stably resolved on coarse enough, practicable grids. Going to the limit, why not suppress completely the subgrid model and ask the numerical scheme to do the job? This is the Monotone Integrated LES (MILES) approach. From our knowledge, the first MILES simulation of a reacting mixing layer was done by Stoukov [3] using a second order TVD scheme, that was too dissipative. A systematic evaluation of the effects of numerical diffusion on the DNS and LES of free flows and compressible turbulence (non-reacting) was done by Garnier [4][5]. Among all the shock-capturing schemes tested, a 5th order WENO scheme [6] was found to give the best results. In this paper, this scheme is tested in the context of the MILES approach of an air/H2 turbulent diffusion flame and comparison is done with the DNS of the same test case.

94

L. Gougeon and I. Fedioun

2 Description of the codes 2.1 The DNS code As a preliminary to LES or MILES, the development of a non-dissipative DNS code is mandatory for validation purpose such as subgrid model evaluation by a priori tests (e.g. [7][8]) on academic configurations, or other fundamental studies [9]. The DNS code solves the compressible multi-species Navier-Stokes equations in convective form for primitive variables because aliasing errors are reduced, leading to higher stability [10]. The energy equation is written for the sensible internal energy of a perfect gas. The variables advanced in time are (ρ, u, v, T, Yα ). The spatial discretisation uses 6th order compact centered finite differences [11], and time stepping is achieved by a fully explicit thirdorder low-storage Runge-Kutta scheme. Non-reflecting conditions are applied at open boundaries [12]. Careful vector optimization allows reaching high performance (≈ 4 Gflops) on the NEC-SX5 supercomputer. Of major importance (both from physical and computational efficiency prospective) are the thermodynamic basis and the multicomponent transport model used for molecular processes [13]. In the equation for species: ∂Jαj ∂Yα ∂Yα + (uj + Vjc ) =− + ω˙α (1) ρ ∂t ∂xj ∂xj a Fickian approximation is used for the mass flux Jαj = −ρDαm

Yα ∂Xα Xα ∂xj

(2)

and a correction velocity is introduced to ensure mass conservation: Vjc

=

Nsp β=1

Dβm

Yβ ∂Xβ Xβ ∂xj

(3)

The diffusion coefficient for species α in the mixture is approximated with the zero-th order Hirschfelder-Curtiss model, which is a good compromise between accuracy and CPU cost: 1 − Yα (4) Dαm = Xβ β=α Dαβ

In (4), the binary diffusion coefficient Dαβ between species α and β is fitted as a power function of temperature from the CHEMKIN values [14]. This avoids ”CALLs” to CHEMKIN subroutines which would cancel vectorization. The formula (4) is singular in the single species limit. In case of non-premixed combustion, a small amount of each species is added in the pure fuel regions: minα Yα → . The code is stable for ≈ 10−20 . The viscosity and thermal conductivity of the mixture are evaluated with Wilke’s mixing formula in

DNS/MILES of Reacting Air/H2 Diffusion Jets

95

which pure species properties are fitted from the CHEMKIN values with 2nd order polynomials accurate in the range 200K − 4000K. Thermodynamics are evaluated using JANAF polynomial database. Detailed chemistry is implemented as a set of Nreac reversible reactions j = 1, ..., Nreac involving Nsp species α = 1, ..., Nsp with symbol Aα : Nsp

ναj Aα

α=1

Nsp

ναj Aα

j = 1, ..., Nreac

(5)

α=1

The production of species α is ⎡ ⎤ Nsp Nsp N reac ρYβ νβj ρYβ ν”βj ⎣Kf j ⎦ ναj − ναj − Krj ω˙α = Mα Mβ Mβ j=1 β=1

β=1

(6) where forward Kf j and backward Krj rates of reaction j are given by the ` ArrhEnius law and are related by the equilibrium constant of the reaction. The heat release to be computed in the sensible energy equation is T Nsp R 0 Δhf,α + Cpα (θ)dθ − T ω˙α (7) ω˙ T ” = − Mα T0 α=1 A straightforward implementation of (6)-(7) leads to prohibitive CPU cost and careful scalar optimization is needed. The chemical scheme used for air/H2 combustion is the one of Dagaut [15] which involves 9 species (H, H2 , O, O2 , OH, H2 O, HO2 , H2 O2 , N2 ) and 34 elementary reactions (17 reversible). Nitrogen serves as a third-body in 8 elementary reactions. This chemical scheme is comparable to the classic one of Mass and Warnatz (9 species, 37 elementary reactions) [16] except for the auto-ignition delay at initial temperatures lower than 1000K. For a stoichiometric mixture, using a 3rd order RK scheme, numerical stability is obtained for Δt ≤ 1.10−9 s (0D calculation). Advancing 9 species with 34 chemical reactions, the DNS code performs at 1.84 10−6 s per grid node per RK sub-step without any I/Os on one processor NEC SX5. 2.2 The MILES code The MILES code uses a fifth-order characteristic-wise finite difference WENO scheme for the hyperbolic part of the Navier-Stokes equations written in conservation form for the conservative variables (ρ, ρu, ρv, ρEtot , ρYα ), and 4th order central finite differences for viscous terms. Etot is the total energy (internal sensible + chemical + kinetic)[17] per unit mass. As recommended in [6], a 3rd order TVD Runge-Kutta time stepping is used. The WENO reconstruction is performed in the characteristic basis and to ensure stability,

96


upwinding is achieved using the scalar Lax-Friedrichs flux-splitting. Hence, the change from conservative to characteristic variables and vice-versa is done at each grid point, each RK sub-step. This is the more CPU consuming task but this is worthwhile to avoid numerical oscillations in case of higher order methods [18]. The code was first developed by N. Lardjane [7] in the binary non-reacting case and has recently been extended to multi-species reacting flows. Thermodynamics, chemistry and transport routines are the same as in the DNS code. As an illustration of the ability of the WENO scheme to catch simultaneously sharp discontinuities and broad band physics like turbulence, a supersonic non-reacting H2 jet (2000 m/s) impacted by an oblique shock wave generated by a compression ramp in the oxygen co-flow (500 m/s) is simulated. The Reynolds number based on the jet width, the average velocity and diffusivity and is 10, 000. The resolution is 512*512 grid nodes. Figures 1 and 2 show the density and the H2 mass fraction respectively. This calculation is not feasible with the non-dissipative DNS code.

Fig. 1. Density

Fig. 2. H2 mass fraction

3 Comparison MILES/DNS 3.1 Premixed laminar flame front and flame speed We first check the ability of the WENO scheme to capture a sharp flame front without excessive smearing and to predict accurately flame speeds. A premixed 1D air/H2 mixture, at various equivalence ratios, is initially at T=300K


97

in a 1cm long domain, and is ignited in the center at T=1500K. The compact scheme needs between N=512 and N=1024 grid points depending on the equivalence ratio to be stable, whereas the WENO scheme is stable down to N=32 grid points. Figure 3 shows that the chemical scheme [15] associated with the multicomponent transport model described in sec. 2.1 makes the DNS code predict accurately the flame speed over the full range of equivalence ratios. The numerical diffusion of the WENO scheme added to the molecular transport produces a slightly higher flame speed, but still within experimental values. As the resolution is decreased, the computed flame speed is increased (fig. 4) whatever the equivalence ratio. Above N≈ 128, the predicted values do not change any more. The structure of the flame front is observed via the

φ φ φ

Fig. 3. Air/H2 flame speed. Compact (N=1024) and WENO (N=256). Experimental results taken from [19]

Fig. 5. 1D stoichiometric air/H2 flame front. Comparison compact/WENO, t=0, 0.05, 0.2 and 0.4 ms

Fig. 4. Air/H2 flame speed. WENO code at different resolutions. Dashed lines are DNS values

Fig. 6. Same case as in fig. 5: MILES code N=256 with and without viscous terms

98


temperature profiles at various times in figure 5. It is clearly seen that the WENO scheme produces results very similar to those of the compact scheme at high resolution, and smears the temperature gradient at low resolution. In figure 6, the viscous terms have been removed from the calculation, leaving only the numerical diffusion. Results speak by themselves: in case of laminar flames, the molecular transport is absolutely needed to recover the physics. 3.2 Subsonic 2D turbulent reacting jet A turbulent subsonic diffusion flame in an idealized air/H2 reacting jet is simulated. Hydrogen is injected from a pitot tube. The Reynolds number based on reference values is 100 (see table 1 for physical and numerical parameters). The Mach number is ≈ 0.5 in the H2 flow and in the coflowing air. The initial temperature is uniform at 1500K in all the field. The physical domain is 7.1*17.8 mm and the jet dynamic width is 0.12 mm (reference length). The grid is 512*572 nodes for the DNS code and 192*192 for the MILES code. The time step is such that the CFL number is ≈ 0.6. A random noise built Table 1. Physical and numerical parameters for the reacting jet H2 Velocity (m/s) Air Velocity (m/s) H2 Tstat./tot. (K) Air Tstat./tot. (K) H2 Mach number Air Mach number Vref (m/s) Lref (m) μref (kg/m.s) ρref (kg/m3 ) Lx * Ly (ref. units)

1500 375 1500/1570 1500/1558 0.52 0.50 937.5 1.184 10−4 3.353 10−5 3.022 10−2 60*150

m/s 15 00

WENO Compact

1000

500

0 −3

−2

−

0

1

2

x/Lref

H2

on 10 harmonics of the computational box width with an amplitude of 10% of the reference velocity is superimposed on the transverse velocity at the jet outlet. In the picture above, one can see that the inlet profile is highly underresolved in the MILES simulation, which is nevertheless stable. This produces strong numerical diffusion in the laminar core of the jet which mixes reactants and leads to too early ignition compared to the DNS case as observed on figures 7 and 9. This effect is close to the pre-heating one by numerical diffusion of temperature gradient in the 1D premixed case (sec. 3.1). The axial distribution of temperature follows the production of water and higher temperatures are found in the MILES case than in the DNS (figure 8). When the combustion is complete, final temperatures only depending on thermodynamics are


99

Fig. 7. DNS/MILES of the reacting air/H2 jet: instantaneous axial distributions of H2 O mass fraction

Fig. 8. same as fig. 7: temperature

Fig. 9. DNS/MILES of the reacting air/H2 jet: instantaneous contours of H2 O mass fraction. Top: DNS, bottom: MILES

Fig. 10. same as fig. 7: axial velocity

the same. The most interesting feature is the very good agreement concerning the decrease of the axial velocity (figure 10). This is encouraging for further investigations on the MILES approach at higher Reynolds numbers.

Conclusions This first work has demonstrated the ability of WENO schemes to perform MILES simulations of turbulent reacting jets. At high enough resolution, both methods give comparable results. In case of too low resolution, the numerical

100


diffusion of the dissipative WENO scheme increases the flame velocity and smears the gradients in the flame front. A compromise must then be found to achieve the balance between accuracy and computational cost.

Acknowledgements We thank the CNRS supercomputing center I.D.R.I.S. for providing computational resources on NEC-SX5 vector computer. L. Gougeon is supported by ` a joint grant from the CNRS and the Conseil REgion Centre.

References 1. Davidenko D, Gˆkalp I, Dufour E, Magre P (2005) AIAA 2005-3237 2. Selle L, Lartigue G, Poinsot T, Koch R, Schildmacher KU, Krebs W, Prade B, Kaufmann P, Veynante D (2004) Comb. & Flame 137:489–505 ` ` ` 3. Stoukov A (1996) Etude numErique de la couche de mElange rEactive supersonique. PhD thesis, University of Rouen, France ` ` 4. Garnier E (2000) Simulation des grandes Echelles en rEgime transsonique. PhD ` Paris XI Orsay, France thesis, UniversitE 5. Garnier E, Mossi M, Sagaut P, Comte P, Deville M (1999) J. Comp. Physics 153:273–311 6. Liu XD, Osher S, Chan T (1994) J. Comp. Physics 115:200–212 ` ` ` ` 7. Lardjane N (2002) Etude thEorique et numErique des Ecoulements cisaillEs ` libres ‡ masse volumique fortement variable. PhD thesis, University of OrlEans, France 8. Vreman B (1995) Direct and large-eddy simulation of the compressible turbulent mixing layer. PhD thesis, Dept. of Applied Math., University of Twente, The Netherlands 9. Vervisch L, Poinsot T (1998) Annu. Rev. Fluid Mech. 30:655–691 10. Fedioun I, Lardjane N, Gˆkalp I (2001) J. Comp. Physics 174(2):816–851 11. Lele SK (1992) J. Comp. Physics 103:16–42 ` 12. Baum M, Poinsot T, ThEvenin D (1994) J. Comp. Physics 116:247–261 ` 13. Hilbert R, Tap F, El-Rabii H, ThEvenin D (2004) Progress in Energy and Combustion Science. 30:61-117 14. Kee RJ & al. (1999) CHEMKIN collection, release 3.5. Sandia National Laboratories, San Diego, CA 15. Dagaut P (2002) Phys. Chem. Chem. Phys. 4:2079–2094 16. Maas U, Warnatz J (1988)Comb. & Flame 74:549–576 17. Poinsot T, Veynante D (2001) Theoretical and numerical combustion. Edwards Inc. 18. Shu CW (1997) Essentially non oscillatory and weighted essentially non oscillatory schemes for hyperbolic conservation laws. NASA/CR-97-206253, ICASE report 97-65 19. Marinov N, Westbrook CK, Pitz WJ (1996) Detailed and global chemical kinetics model for hydrogen. In: Chan SH (ed) Transport Phenomena in Combustion, Volume 1. Talyor and Francis, Washington DC

Direct Numerical Simulation of turbulent reacting flows involving dilute particles G´ abor Janiga and Dominique Thévenin Laboratory of Fluid Dynamics and Technical Flows, Institute of Fluid Dynamics and Thermodynamics, University of Magdeburg “Otto von Guericke”, Universit¨ atsplatz 2, 39106 Magdeburg, GERMANY [email protected], [email protected] Summary. Multiphase flows play an increasingly important role for many practical applications, for example for energy generation, chemical engineering or environmental problems. All these configurations generally correspond to a turbulent regime. In order to better understand the properties of turbulent multi-phase flows, Direct Numerical Simulations (DNS) play a major role as a complementary tool to experimental measurements, since they give access to the full details of all coupling processes taken into account in the simulation. DNS are nevertheless limited to relatively low Reynolds numbers and simple configurations. In this work the extension of an existing single-phase DNS code toward multi-phase flows is described, in particular concerning particles interacting with a turbulent reacting flow. First results are shown to illustrate the present capabilities of the resulting numerical tool.

1 Introduction and state of the art In order to improve our understanding of the phenomena controlling turbulent flows, numerical simulations are essential as a complementary tool to theory and experimental measurements. Direct numerical simulations are ideally suited for such studies since they do not rely on any hypothesis concerning the turbulence, but they are restricted to simple geometries and relatively low Reynolds numbers. Due to the very high numerical cost of single-phase, non-reacting DNS, the extension of such simulations toward multi-phase resp. reacting flows is still fairly recent. Due to lack of space, an exhaustive literature review cannot be proposed here. We thus refer the interesting reader for example to [1] concerning the beginning of DNS for (single-phase) combustion resp. to [2] for (non-reacting) multi-phase DNS, in particular concerning particles. Concerning studies of non-reacting multi-phase flows relying on DNS, a rapid progress can be observed, but many questions are still open [3]. Turbulent flows with bubbles have been considered for example in [4]. The problem

102

G´ abor Janiga and Dominique Thévenin

of particles in a turbulent flow, already considered by [2], has been further investigated for example in [5] to improve Reynolds-Stress and algebraic turbulence models by comparison with DNS, in [6] to identify particle-fluid interaction processes near the wall, in [7, 8] to investigate heat exchange in a particle-laden turbulent flow, in [9] for particle dispersion in stirred vessels and in [10] to study the impact of the particle Stokes number. The specific question of the non-Newtonian behavior of the resulting suspension has been extensively considered in [11, 12, 13], in order to understand drag reduction processes. First publications considering simultaneously two-phase flows and simple chemical reactions begin to be found [14]. For combustion applications, where the prediction of pollutant emissions or of ignition/extinction limits is of major importance, reliable models must of course be included in the DNS to describe the chemical and diffusion processes as well as the thermodynamical parameters [15]. In many practical combustion flows the fuel is injected in a liquid (e.g. internal combustion engines, cryogenic rocket engines) or a solid (power-plant burning coal or wood chips, fluidized bed combustion) phase. In many other cases, reacting flows have been investigated by adding tracer particles to visualize key properties. Of course, two-phase flows will lead to a different burning behavior and to modifications of the coupling processes between mean flow, turbulence and reaction fronts. Turbulent spray flames have been for example considered using DNS in [16, 17]. Two-phase turbulent flows involving chemical reactions are also found very frequently in the field of chemical engineering, for example for precipitation, crystallization, fluidized beds or particle production (e.g. [18]). Many more important applications are omitted here due to lack of space. In order to be able to investigate numerically with a high accuracy turbulent reacting multi-phase flows, an existing in-house DNS code [19, 20, 21] relying on a low-Mach number formulation and already able to describe with a high accuracy turbulent reacting flows in the gaseous phase has been recently modified to take into account the presence of particles. Those particles are considered as a second, dilute phase overlaid on top of the main flow. Due to the high dilution, the global flow and turbulence properties are not directly impacted by the presence of these particles. But, if these particles are considered combustible, they will burn and release heat when encountering favorable conditions, thus indirectly impacting the main flow and its characteristics, in particular through temperature/viscosity modifications and resulting enhanced burning rates in the gas phase. In a first step, only heat transfer to the particles and devolatilization are considered here. Both two- and three-dimensional Direct Numerical Simulations have been carried out, for non-reacting as well as for reacting flows. The individual particles are tracked individually in the DNS using a Lagrangian approach. Due to the high dilution level, particle-particle interactions are neglected in these first computations. Moreover, the considered geometry does not involve walls, so that wall-particle interactions are not relevant either for our problem.

DNS of turbulent reacting flows with particles

103

In what follows, the employed numerical tools are first briefly described. The physical models employed to describe the reacting gas phase as well as the particles are then introduced. Finally, first results are presented to illustrate the present capabilities of the obtained simulation tool.

2 Numerical methods and physical models All computations shown here have been carried out with the three-dimensional Direct Numerical Simulation code called π 3 . The single-phase version of this code has been described in detail in various previous publications [19, 20, 21], where more information can be found concerning the generic features of the solver. As a summary, π 3 relies on a low-Mach number approximation to describe the turbulent flow. Several tests [22] have proved that this is an excellent alternative to investigate turbulent combustion. Moreover, chemical processes are described using a pre-computed look-up table, corresponding to the method called FPI (see for example [23, 24]). Discretization is of sixth order in space and of fourth order in time. The second phase, consisting of solid particles, is described using a Lagrangian approach. Due to the low volume fraction of this second phase, well below 1% for all cases presented here, only a one-way coupling is considered in the present work. Clearly, this should be modified in future versions if highly loaded flows must be investigated. Considering the small size of the particles (always below 10 μm) and the large density difference between the particles (here ρp = 850 kg.m−3 , corresponding for example to the density of anthracite) and the main phase (gas phase with ρ below 1.1 kg.m−3 ), the drag force is clearly the most important force to predict particle trajectories. As a supplement, the gravity force is also implemented, leading for the particles to the following set of equations (bold symbols denote vectors, the index p is used for the particles, variables without index denote fluid values): dxp = up dt dup = FD + Fg mp dt

(1) (2)

The gravity force Fg = mp g is computed in a standard manner without including buoyancy, while the drag force FD is modeled using [25, 26]: FD =

3 ρ mp CD u − up (u − up ) 8 ρp Rp

(3)

with the drag coefficient CD determined as: 24 1 + 0.15 Rep 0.687 for Rep ≤ 1000, or Rep = 0.44 for Rep > 1000

CD =

(4)

CD

(5)

104


where Rep is the particle Reynolds number defined as Rep =

2ρRp u − up μ

(6)

All further notations used in these equations are standard (ρ, ρp : fluid resp. particle density; u, up : fluid resp. particle velocity; μ: fluid viscosity; mp : particle mass). The correlation used for the drag coefficient reduces as expected to 24/Rep at low Reynolds numbers and is continuous over the whole range of Rep . Concerning heat transfer to the particles, only the convective heat flux φc is taken into account, leading for the particle temperature Tp to: mp cP p

dTp = φc dt

(7)

Since particles are small enough to encounter a locally uniform fluid temperature T , the convective heat flux is computed through: φc = 4πRp2 h(T − Tp )

(8)

In this formula the local fluid temperature is obtained by solving a conservation equation for the enthalpy. The value of the heat transfer coefficient h between flow and particle is obtained through the Nusselt number Nu =

2hRp λ

(9)

involving the thermal conductivity of the fluid, λ. Finally, the Nusselt number itself is determined using the classical correlation [27]: Nu = 2 + 0.6 Pr1/3 Re1/2 p

(10)

involving the particle Reynolds number Rep and the flow Prandtl number Pr=μcP /λ. Onset of particle devolatilization is considered to occur at Tp = 600 K, followed by constant-rate devolatilization [28]. This simple approach is known to often deliver poor results, so that better models will be implemented in the future. Lift force, thermophoretic force and particle rotation are neglected in this first study. Moreover, due to the large mean distance between particles and considering that the geometry does not involve any walls, collisions are not accounted for. All particles are considered to be perfectly spherical (radius Rp ).

3 Configuration, results and discussion All results presented here correspond to the same three-dimensional configuration, of dimension (6.5 mm)3 , discretized using 1203 = 1.728 million grid


105

points, uniformly distributed. An initial homogeneous isotropic turbulence is generated within this domain using a von K´ armán spectrum with Pao correction for near dissipation scales. Resulting Reynolds number based on the integral scale is Ret = 125 with an integral length scale lt = 1.3 mm and velocity fluctuation u = 1.8 m/s. All computations have been carried out on a single Linux Pentium PC with a 2.7 GHz clock and 2 GB RAM. The computational overhead associated to the Lagrangian description of the particles has been checked, without any particular optimization of the corresponding computational procedure. Up to 105 particles, this overhead is fully negligible. Using up to 10 million particles (the highest value tested up to now) leads to a computational time increase of maximum 6.2 %, which is still very small. This is due to the very complex and therefore numerically demanding computations involved by the accurate description of the gaseous reacting flow. First two-phase computations have been carried out in a non-reacting turbulent flow. In Fig.1 the trajectories of 100 randomly chosen particles along with their final position (symbol ∗) are represented at the end of the DNS computation. The total number of particles within the box is kept constant by considering periodic boundary conditions.

Fig. 1. Example of 100 individual particle trajectories (lines) together with the final position of the particles (symbol ∗) obtained in a three-dimensional DNS for a homogeneous isotropic turbulent non-reacting flow.

We then use this same configuration to investigate the growth of a fully premixed methane/air flame, initially perfectly spherical, similarly to the results presented in [19, 20, 21], but simultaneously introducing particles in this flow. The heat exchange between flow and particles is described by Eqs. (7) to (10). Three hundred randomly chosen particle trajectories are shown in

106


Fig.2, together with their temperature. Resulting computation time is about 65 hours.

Fig. 2. Left: example of the temperature evolution (in K) of some particles versus time (in s). Right: 300 individual particle trajectories (lines) are shown together with the present position of the particles (symbol ∗) and possibly the onset of devolatilization (symbol +). Light gray: temperature between 300 and 600 K. Black: temperature above 600 K. Both results are obtained from a three-dimensional DNS for a homogeneous isotropic turbulent flow involving a fully premixed methane/air flame (see also Fig.3.

The instantaneous flame (defined as the isosurface Y (CO2 ) = 0.03 [19]) and an isosurface of the vorticity are shown in Fig.3.

Fig. 3. Instantaneous gaseous phase results at the end of the simulation (same time as for Fig.2). Left: position of the flame surface, defined as Y (CO2 ) = 0.03. Right: isosurface of vorticity.


107

4 Conclusions An existing three-dimensional single-phase DNS code has been successfully extended to deal with a second, dilute phase. Taking into account drag and gravity for heavy, spherical particles using a Lagrangian description does not lead to a large increase of the needed computational time. In this way, particle trajectories and velocities can be determined in a turbulent flow. In a second step, heat transfer to the particles has been implemented, again without impacting considerably the resulting computing time. In the presence of an expanding premixed flame, particles encounter considerable convective heat fluxes, leading to a large temperature increase and finally to devolatilization, depending of course on their initial position and resulting trajectories. Improving the devolatilization model and taking into account the resulting volatile particle material during the combustion process is the subject of our present work.

References 1. T. Poinsot, S. Candel, and A. Trouvé. Applications of direct numerical simulation to premixed turbulent combustion. Prog. Energy Combust. Sci., 21:531–576, 1996. 2. J. B. McLaughlin. Numerical computation of particles-turbulence interaction. Int. J. Multiphase Flow, 20:211–232, 1994. 3. F. Mashayek and R. V. R. Pandya. Analytical description of particle/dropletladen turbulent flows. Prog. Energy Combust. Sci., 29:329–378, 2003. 4. G. Tryggvason, B. Bunner, M. F. G¨ oz, and M. Sommerfeld. Direct numerical simulations of multiphase flows. In Direct and Large-Eddy Simulation IV, pages 517–526. (Geurts, B., Friedrich, R. and Métais, O., Eds.), Kluwer Academic Publishers, 2001. 5. F. Mashayek and D. B. Taulbee. Turbulent gas-solid flows. Num. Heat Transf. B, 41:1–29, 2002. 6. L. M. Portela and R. V. A. Oliemans. Eulerian-lagrangian DNS/LES of particleturbulence interactions in wall bounded flows. Int. J. Num. Methods Fluids, 43:1045–1065, 2003. 7. J. Réveillon and K. Cannevière. Visualization of the effect of dispersed particles on heat transfer from an impinging jet. J. Flow Vis. Image Proc., 9:11–24, 2002. 8. R. V. R. Pandya and F. Mashayek. Non-isothermal dispersed phases of particles in turbulent flow. J. Fluid Mech., 475:205–245, 2003. 9. M. Sommerfeld and S. Decker. State of the art and future trends in CFD simulation of stirred vessel hydrodynamics. Chem. Eng. Technol., 27:215–224, 2004. 10. W. Li, G. Hu, Z. Zhou, J. Fan, and K. Cen. Direct numerical simulation of gas-solid two-phase mixing layer. J. Thermal Sci., 14:41–47, 2005. 11. M. Manhart. Rheology of suspensions of rigid-rod like particles in turbulent channel flow. J. Non-Newt. Fluid Mech., 112:269–293, 2003. 12. M. Manhart. Visco-elastic behaviour of suspensions of rigid-rod like particles in turbulent channel flow. Eur. J. Mech. B, 23(3):461–474, 2004.

108


13. M. Manhart and R. Friedrich. Turbulent channel flow of a suspension of small fibrous particles in a newtonian solvent. In Direct and Large-Eddy Simulation V, pages 287–296. (Friedrich, R., Geurts, B., and Métais, O., Eds.), Kluwer Academic Publishers, 2004. 14. T. Michioka, R. Kurose, K. Sada, and H. Makino. Direct numerical simulation of a particle-laden mixing layer with a chemical reaction. Int. J. Multiphase Flow, 31:843–866, 2005. 15. R. Hilbert, F. Tap, H. Elrabii, and D. Thévenin. Impact of detailed chemistry and transport models on turbulent combustion simulations. Prog. Energy Combust. Sci., 30:61–117, 2004. 16. J. Réveillon, P. Domingo, and L. Vervisch. Structure of flames stabilized on evaporating sprays. In European Combustion Meeting ECM03, Orléans, October 25-28, pages 1–6, 2003. 17. J. Réveillon, C. Péra, M. Massot, and R. Knikker. Eulerian analysis of the dispersion of evaporating polydispersed sprays in a statistically stationary turbulent flow. J. Turb., 1:1–27, 2004. 18. E. G. Moody and L. R. Collins. Effect of mixing on the nucleation and growth of titania particles. Aerosol Sci. Tech., 37:403–424, 2003. 19. D. Thévenin, O. Gicquel, J. de Charentenay, R. Hilbert, and D. Veynante. Twoversus three-dimensional direct simulations of turbulent methane flame kernels using realistic chemistry. Proc. Combust. Inst., 29:2031–2039, 2002. 20. D. Thévenin. Simulation of three-dimensional turbulent flames. In Direct and Large-Eddy Simulation V, pages 335–342. (Friedrich, R., Geurts, B.J. and Métais, O., Eds.), Kluwer Academic Publishers, 2004. 21. D. Thévenin. Three-dimensional direct simulations and structure of expanding turbulent methane flames. Proc. Combust. Inst., 30:629–637, 2005. 22. J. de Charentenay, D. Thévenin, and B. Zamuner. Comparison of direct simulations of turbulent flames using compressible or low-Mach number simulations. Int. J. Numer. Meth. Fluids, 39:497–516, 2002. 23. O. Gicquel, N. Darabiha, and D. Thévenin. Laminar premixed hydrogen/air counterflow flame simulations using Flame Prolongation of ILDM with differential diffusion. Proc. Combust. Inst., 28:1901–1908, 2000. 24. B. Fiorina, R. Baron, O. Gicquel, D. Thévenin, S. Carpentier, and N. Darabiha. Modelling non-adiabatic partially premixed flames using Flame-Prolongation of ILDM. Combust. Theory Modelling, 7:449–470, 2003. 25. R. Clift, J. R. Grace, and M. E. Weber. Bubbles, drops and particles. Academic Press, New York, 1978. 26. M. Sommerfeld. Analysis of collision effects for turbulent gas-particle flow in a horizontal channel. Int. J. Multiphase Flow, 29:675–699, 2003. 27. W. E. Ranz and W. R. Marshall. Evaporation from drops. Chem. Eng. Prog., 48:141–180, 1952. 28. M. M. Baum and P. J. Street. Predicting the combustion behavior of coal particles. Combust. Sci. Tech., 3:231–243, 1971.

Dissipation of Active Scalars in Turbulent Temporally Evolving Shear Layers with Density Gradients Caused by Multiple Species Inga Mahle, J¨ orn Sesterhenn, Rainer Friedrich Fachgebiet Str¨ omungsmechanik, Technische Universit¨ at M¨ unchen, Boltzmannstrasse 15, 85748 Garching, Germany [email protected] Summary. Direct Numerical Siumlations of turbulent temporal shear layers with density gradients caused by multiple species are performed up to the convergence towards a self-similar state. The influence of the density gradient on the scalar gradients, the scalar dissipation rate and its conditional expectation are investigated. First, the investigations are performed in the center of the shear layer, then in various other transverse planes.

1 Introduction An important quantity in mixing and combustion is the conditional expectation of the scalar dissipation rate. It appears not only in the transport equation for the pdf of active and passive scalars but also in models for the reaction rate where a passive scalar, namely the mixture fraction, is used. To investigate the influence of a mean density gradient on the turbulent mixing of active scalars and in particular on some features of the scalar dissipation rate, Direct numerical simulation (DNS) of temporally evolving, turbulent compressible shear layers are performed. The non-reacting case studied here comprises effects that are caused by density gradients due to different mass fractions of the species. It contrasts the case of density gradients that are caused by heat release and hence forms a useful reference. The first section of the paper describes the shear layer configuration and the initial parameters of the simulations. In the next part, a self-similar state of the turbulent mixing layer is found. Samples of this state are used to build statistics in the following sections. Various quantities related to the scalar dissipation rate and its conditional expectation are investigated, first in the centre of the shear layer, then in other planes.

110

Inga Mahle, J¨ orn Sesterhenn, Rainer Friedrich

−

−

−

Fig. 2. Profiles of scalar dissipation rate of H2 at different times (time advances from top to bottom).

Fig. 1. The initial configuration of the shear layer.

2 Initial Configuration and Numerical Parameters Figure 1 shows the initial configuration of the shear layers. All test cases are 3D with x and y denoting the homogeneous streamwise and spanwise directions and z denoting the transverse direction. The upper stream is pure oxygen while the lower stream is either pure nitrogen or a mixture of hydrogen and oxygen. The first case (O2 -N2 ) has a nearly constant density due to similar molecular weight of oxygen and nitrogen. In the second case (O2 -H2 -4), the mass fractions of oxygen and hydrogen in the lower stream are chosen in such a way that the density ratio between this stream and the upper one is 1:4. Temperature and pressure are initially constant. The species, which are also called active scalars, do not only influence the flowfield by the density due to their different molecular weights, but also by the transport properties, like the viscosity and the diffusion coefficients which are computed depending on the local temperature, pressure and species’ composition [2]. The convective Mach number M ac =

Δu c1 + c2

(1)

is 0.15 for all test cases reflecting the low Mach number situation in most combustion chambers. The velocity difference between the two streams is denoted by Δu, and c1 and c2 are their sonic speeds. The Reynolds number at the beginning of the computations Reω,0 =

Δu ρ0 δω,0 μ0

(2)

is 640. It is based on the initial vorticity thickness δω,0 , on the density ρ0 = (ρ1 + ρ2 ) /2 and on the viscosity μ0 = (μ1 + μ2 ) /2. The flow is initialized by a hyperbolic-tangent profile for the mean streamwise velocity and density. In order to accelerate the transition to turbulence, broadband fluctuations in the velocity components are used.

Dissipation of Active Scalars in Shear Layers with Density Gradients

111

Table 1. Geometrical parameters of the simulations. The computational domain has the dimensions Lx , Ly and Lz with Nx , Ny and Nz grid points, respectively. Lx /δω,0 Ly /δω,0 Lz /δω,0 Nx Ny Nz O2 -N2 129.38 O2 -H2 -4 86.25

32.25 21.50

92.23 43.16

512 128 512 512 128 256

Grid sizes and domain sizes are given in table 1. The grid-spacing of all test cases is constant in the x- and y-directions. In the z-direction, the inner third of the domain has a constant grid-spacing with Δzmin = 0.126 δω,0 while the stretching in the external parts is 3%. The compressible Navier-Stokes and species transport equations are solved in a pressure–velocity–entropy–species formulation. The integration of equations is performed using sixth order compact central schemes for the spatial derivatives and a third order low-storage Runge-Kutta scheme in time. The primitive variables are filtered every 20th time step to prevent spurious accumulation of energy in the highest wavenumbers using a sixth order compact filter. The code is parallelized and the simulations were done on up to 192 processors of the Hitachi SR8000-F1 at the Leibniz–Rechenzentrum in Munich (Germany).

3 Self-similar State All computations are performed during time-intervals that are long enough to reach a self-similar state. For this state, a constant growth rate of the momentum thickness δθ is established. The growth rate is reduced by a factor of 0.449 in the case with a mean density gradient which confirms the result of [6]. The Reynolds numbers of the self-similar state range from 10100 to 13800 for O2 -N2 and from 8000 to 8300 for O2 -H2 -4. Reference [6] suggests a non-dimensionalization by the variables δθ and Δu. This is done in fig. 2 for the scalar dissipation rate Y = −ρY Vi

∂Y . ∂xi

(3)

after dividing it by the mean density ρ. Y and Vi are the mass fraction and the diffusion velocity of the respective species. Quantities with an overbar are Reynolds averaged, quantities with a tilde are Favre averaged. Primes and double primes indicate the corresponding fluctuations. The relaxation to a self-similar state is visible. The profiles shown in fig. 2 are shifted towards the side of the lower density because the dividing streamline has moved there which can be explained by the mean-momentum conservation.

112


Fig. 3. Instantaneous magnitude of the scalar gradient of H2 at Reω = 8238 normalized with δω,0 and Ymax , values range from 0 to 1.88.

Fig. 4. Pdfs of the scalar derivatives of H2 normalized by their variance σ, ∂Y /∂x (+), ∂Y /∂y (×), ∂Y /∂z (∗).

4 Scalar Gradients The scalar dissipation rate (3), which contains the scalar gradients, is an important quantity related to mixing as it controls this process at the level of the small scales. Figure 4 shows the pdfs of the scalar derivatives of hydrogen. The samples are taken from the centre of the mixing layer O2 -H2 -4 (six planes around Y /Ymax = 0.5). The exponential tails, which appear in the logarithmic plot as nearly straight lines, are sign of high intermittency. This intermittency can also be seen from the instantaneous scalar gradient in fig. 3: High values do occur but are very rare. The imposed density gradient reduces the intermittency of the scalar derivatives in all directions which is obvious by comparing their flatnesses: The effect is especially strong in the transverse direction z (flatness of 16.16 for O2 -N2 vs. 10.13 for O2 -H2 -4). The skewness of the pdf of the transverse derivatives can be related to sharp scalar fronts. These ramp– cliff events result from the transit of fluid lumps from the low-scalar regions towards the high-scalar regions and vice versa along the imposed gradient [4]. Figure 5 shows that strong scalar gradient fluctuations are aligned with the imposed mean scalar gradient. g denotes the norm of the scalar gradient fluctuations, G the norm of the mean scalar gradient. This direct coupling between large and small scales is influenced by the presence of a density gradient as the curves in fig. 5 have a different slope: For high scalar gradient fluctuations the alignment in the direction of the principal strain rate is less for the test case with no density gradient. For small scalar gradient fluctuations, the alignment can be even in the opposite direction in both test cases, but it is stronger in the case with density gradient.


113

5 Scalar Dissipation Rate and its Statistical Dependence on the Scalar The scalar fluctuations are dissipated at the level of small scales by the scalar dissipation. Like in many turbulent flows, the pdf of ln (Y ) is approximately Gaussian with a slightly negative skewness (fig. 6). The result is nearly independent of a mean density gradient (case O2 -N2 not shown). It would simplify the modeling of the conditional expectation of the scalar dissipation rate if the scalar and its dissipation rate were statistically independent [1]. Then, the conditional expectation of the scalar dissipation rate would be equal to its unconditional mean for all scalar values Y |Y = Y . A first check of such an independence is to compute the correlation coefficient of Y 2 and Y [5]. The result in the centre of the shear layer is −0.01 for O2 -N2 and 0.04 for O2 -H2 -4. Both values are small and seem to suggest statistical independence. A more stringent proof is to show that P DF (Y, Y ) = P DF (Y ) × P DF (Y ). It means comparing the joint pdf P DF (Y, Y ) with the product of the two marginal pdfs. This is done in fig. 7 for O2 -H2 -4. The result for O2 -N2 (not shown) is qualitatively the same. One can see that for the frequent combination of small scalar dissipation rates and scalar values around the mean the contour lines of the joint pdf are nearly identical to those of the product of the marginal pdfs. However, at higher values of the scalar dissipation rate the contour lines differ significantly. Moreover, the product of the two marginal pdfs is predicting combinations of extreme scalar values and high scalar dissipation rate that do not exist in reality due to the fact that in the two free streams the scalar dissipation rate is nearly zero. Fluid that is engulfed from these streams and corresponds to extreme scalar values has therefore small dissipation, too. Apparently, the consequence is P DF (Y, Y ) = P DF (Y ) × P DF (Y ). However, the question arises why the correlation coefficient is indicating independence of the two quantities. The answer is that by computing a correlation coefficient, less frequent values

− −

Fig. 5. Conditional expectation of the cosine between the fluctuation of the scalar gradient and the mean scalar gradient, O2 -N2 (×), O2 -H2 -4 (+).

−

−

−

−

−

Fig. 6. Pdf of the logarithm of the scalar dissipation rate normalized by its variance σ, O2 -H2 -4 (+), Gaussian fit (no symbols).

114


Fig. 7. Joint pdf of YH2 and Y (left) and product of the marginal pdfs (right), horizontal axis: YH2 , vertical axis: Y , axes normalized by maxima, logarithmic spacing of isolines.

Fig. 8. Conditional expectation of Y on Y , O2 -N2 (×), O2 -H2 -4 (+),

Y |Y = Y (horizontal line).

are given a smaller importance than more frequent values (means are taken). So the good agreement between the central parts of the figures is reflected in the correlation coefficient and can lead to a wrong conclusion.

6 Conditional Expectation of the Scalar Dissipation Rate The objection that the rare values which are wrongly predicted by the product of the marginal pdfs in fig. 7 are not that important can be met by looking directly at the conditional expectation of Y . Statistical independence corresponds to the horizontal line in fig. 8 and is different from the actual conditional expectation which has an inverted u-shape reminding of the conditional scalar dissipation rate for a laminar one-dimensional diffusion flame [8]. A density gradient leads to an asymmetry of the conditional expectation which is complicating the modeling task. The scalar dissipation rate ‘that is missing for the extreme values to lift the curve up to the mean’ is the part that is present in the product of the marginal pdfs, but is missing in the joint pdf in fig. 7. It is obvious that this part, though belonging to rare events, plays in its total a quite important role and that it is necessary to retain a certain dependency for the conditional expectation of the scalar itself which means a function Y |Y = Y (Y ). The situation in the interior of the mixing layer can be approximated by homogeneous sheared turbulence with a constant mean gradient. Experimental and numerical investigations of this type of flow found the conditional scalar dissipation rate to be either nearly independent of the scalar value [3] or positively correlated [5] which corresponds to a u-shape behaviour. As our results in fig. 8 indicate an inverse u-shape, one could argue that the inner part of our mixing layers is not yet fully developed showing a very high degree of intermittency. The conditional expectation of [7] for the mixing layer without heat release would then represent a further stage of development as it displays two humps between which the curve has a slight u-shape. However, comparing the final Reynolds number of 5700 [7] with our Reynolds numbers of the self-similar state (sect. 3) this should not be the case. Moreover, there


Fig. 9. Conditional expectation of Y on YN2 , planes: 0.1 (+), 0.3 (×), 0.5 (∗), 0.7 ( ), 0.9 ().

115

Fig. 10. Conditional expectation of Y on YH2 , planes: 0.1 (+), 0.3 (×), 0.5 (∗), 0.7 ( ), 0.9 (), whole domain (no symbols).

is an important difference between the sampling technique of [7] and the one we use: The authors of [7] neglect transverse changes and take samples of the whole domain while we are taking samples from a slice with a small extent in the transverse direction. In the next section, it is shown that the transverse effects have an important impact on the shape of the conditional expectation. So the double hump found in [7] might be due to their larger sampling domain and is not supported by the present results.

7 Transverse Non-homogeneities In the previous sections, pdfs constructed from samples in the centre of the mixing layer are looked at. In this section, pdfs of samples from further transverse positions are compared with each other. The samples for the statistics were chosen by searching an x-y-plane for each self-similar state for which the Favre averaged mass fraction of nitrogen or hydrogen normalized by its largest plane average has a certain value (0.1; 0.3; 0.5; 0.7 and 0.9). From these planes and the two adjacent planes in positive and negative z-direction the pdfs were built. The differences in the mean mass fraction within the planes used for each pdf vary by less than 1%. In order to see the influence of just constructing one pdf for the whole computational domain, this pdf for O2 -H2 -4 is also presented. The pdfs of the conditional scalar dissipation rate are shown in figs. 9 and 10. For O2 -N2 the curves are nearly symmetrical in a way that the curves for the outmost planes (0.1 and 0.9) are approximately the mirror-inverted equivalent of each other. This is not the case for O2 -H2 -4. The differences between the curves from a domain within a certain transverse extent (here: 0.3 to 0.7-plane) are small which would justify a bigger sampling domain. However, taking the whole domain as a sampling area as done in [7], is changing the statistics considerably: The corresponding curve in fig. 10 has a flat part for moderate Y which would lead to the conclusion that for these scalar values Y and Y are independent. This wrong conclusion is due to the fact that

116


sampling over the whole domain includes conditional expectations like those of the 0.1- and 0.9-planes which have a quite different shape: They show big deviations from the unconditional mean for rare but yet present events.

8 Conclusions DNS of turbulent temporal shear layers with density gradients caused by multiple species were performed. After an initial transient, a self-similar state was established and for this state statistics were taken. The density gradient is found to decrease the intermittency of the scalar gradients and of the scalar dissipation rate Y . As for other turbulent flows, the strongest scalar gradient fluctuations in the centre of the shear layer are aligned with the mean scalar gradient. We come to the conclusion that for a shear layer the Y -dependence of the conditional expectation of the scalar dissipation rate should be retained as the newly engulfed fluid from the free streams is always associated with low values of Y . This becomes important for combustion cases where the stoichiometric mixture usually contains small mass fractions of fuel for which the conditional scalar dissipation rate is expected to be below its unconditional mean. The influence of the density gradient on the conditional expectation of the scalar dissipation rate was observed to be significant as it caused asymmetry. Finally, our results indicate that the non-homogeneity of the shear layer in the transverse z-direction has to be taken into account when building statistics. It is not recommendable to sample over the whole computational domain. Taking slices somewhat bigger than ours has probably little influence on some statistics as little differences were observed for the planes 0.3 to 0.7 for Y (not shown) and its conditional expectation. However, differences were present and differently pronounced whether there was a density gradient or not. Other pdfs like the one for the scalar itself (not shown) have strong differences between all investigated planes therefore requiring much more caution when enlarging the sampling slices.

References 1. Bilger R W (1976) Progress in Energy and Combustion Science 1:87–109 2. Ern A, Giovangigli V (1995) Journal of Computational Physics 120:105–116 3. Ferchichi M, Tavoularis S (2002) Journal of Fluid Mechanics 461:155–182 4. Gonzalez M (2000) Physics of Fluids 12:2302–2310 5. Jayesh, Warhaft Z (1992) Physics of Fluids A 4:2292–2301 6. Pantano C, Sarkar S (2002) Journal of Fluid Mechanics 451:329–371 7. Pantano C, Sarkar S, Williams F A (2003) Journal of Fluid Mechanics 481:291–328 8. Peters N (2000) Turbulent Combustion. Cambridge University Press

Part III

Subgrid Modelling

Symmetry invariant subgrid models Dina Razafindralandy1 and Aziz Hamdouni2 Université de La Rochelle, Av. Michel Crépeau, 17042 La Rochelle Cedex 01 1 [email protected] 2 [email protected]

Summary. Navier–Stokes equations have fundamental properties such as the invariance under some transformations, called symmetries, which play an important role in the description of the physics of the equations (conservation laws, wall laws, . . . ). It is essential that turbulent models respect these properties. Unfortunately, the analysis reveals that it is not the case of most of LES models. A new way of deriving a class of symmetry consistent models is then proposed. This class is refined such that the models also conform to the second law of thermodynamics. Finally, a simple model of the class is numerically tested to the configuration of a ventilated room. It gives better results than those provided by Smagorinsky and dynamic models.

Key words: Turbulence modeling, Large-eddy simulation, Symmetry group

1 Introduction Currently, there exists an important number of turbulence models, based on various mathematical or physical hypothesis. While these models give encouraging results, most of them fail to meet natural and fundamental properties of Navier–Stokes equations (NS), as we will see later. Galilean invariance is one of these fundamental properties. Speziale ([8]) was the first who required this invariance to the models. Another property is the material indifference, in the limit of 2D flows. In fact, Galilean invariance and 2D material indifference are two of the symmetry properties of NS, and NS have other transformations, called symmetry transformations, which leave the set of solutions unchanged. These transformations play an important role in the description of the physics of the equations. Indeed, according to Nœther’s theorem ([5]), each symmetry of a Lagrangian corresponds to a conservation law. Notice that, even if NS do not derive from a Lagrangian, it is possible to apply Nœther’s theorem to them (see [6]). Next, scaling transformations, which are other symmetries of ¨ NS, led Oberlack ([4]) to derive wall laws. They also permitted Unal ([9]) to

120

Dina Razafindralandy and Aziz Hamdouni

show that NS admit solutions having the Kolmogorov spectrum. The theory of symmetry groups even permits to calculate exact solutions of NS ([2]). Finally, notice that self similar solutions give an information on the behaviour of the solutions at a very large time (see [1]). To retrieve the properties of NS equations (conservation laws, exact solutions, . . . ) when approximating the solutions by LES approach, the symmetries of NS should be respected by the model. Yet, it is not the case of most of existing models. Another property that models should have is the consistence with the second law of thermodynamics. Unfortunately, many common models, such as the popular dynamic model, violate this principle, since they induce a negative dissipation. Though, this principle leads to the stability of the model ([6]). The aim of this communication is then to present an analysis of existing models under the symmetry consideration and to propose a new way of deriving subgrid models which, in one hand, respect the symmetries of NS and, in the other hand, are conform to the second law of thermodynamics. The paper will be structured as follows. In section 2, the symmetries of NS will be reminded. It will be followed in section 3 by an analysis of some common subgrid models using the symmetry approach. In section 4, a new class of LES models verifying the symmetry and thermodynamic properties will be developed. Finally, a simple model of the class is numerically tested in section 5.

2 The symmetry group of Navier–Stokes equations Consider a 3D incompressible newtonian fluid, with density ρ and kinematic viscosity ν. The motion of this fluid is governed by Navier–Stokes equations (NS). Let t and x be the time and space variables, u and p respectively the velocity and pressure fields and y = (t, x, u, p). Let Ta be a one-parameter transformation, i.e. a transformation Ta : y → y = y(y, a) which depends continuously on the parameter a. Ta is called a symmetry of NS if, to a solution of NS, it corresponds another solution, or, equivalentely, if it remains the set of the solutions unchanged. The set of the symmetries of NS constitutes a group called the symmetry group of NS. Lie’s theory ([5]) permits to calculate the symmetry group of NS which is spanned by the following symmetries: • • • • •

the time translation: (t, x, u, p) → (t + a, x, u, p), the pressure translation: (t, x, u, p) → (t, x, u, p + ζ(t)), the rotation: (t, x, u, p) → (t, Rx, Ru, p), the generalized Galilean transformation: ˙ ¨ (t, x, u, p) → (t, x + α(t), u + α(t), p − ρ x α(t)) , and the scaling transformation: (t, x, u, p) → (e2a t, ea x, e−a u, e−2a p).

In these expressions, ζ and α are arbitrary functions, the symbol dot ( ˙ ) stands for derivative and R is a rotation matrix, i.e. R TR = Id and det R = 1,

Symmetry invariant subgrid models

121

Id being the identity matrix. Note that the classical Galilean transformation can be obtained by choosing α linear. NS admit other known symmetries which are not one-parameter symmetries and which could not then be calculated by Lie’s theory. They are: • the reflection: (t, x, u, p) → (t, Λx, Λu, p), where Λ is a diagonal matrix of which the diagonal elements are ±1, • and the material indifference in the limit of a 2D flow in a simply connected domain: (t, x, u, p) → (t, x, u, p), ˙ with x = R(t) x, u = R(t) u + R(t) x, p = p − 3ωϕ + 12 ω 2 x 2 , R(t) being a 2D rotation matrix with angle ωt, ω an arbitrary real constant, ϕ the usual 2D stream function defined by u = curl(ϕe3 ), e3 the unit vector perpendicular to the plane of the flow and x the Euclidian norm of x. Symmetries have an important role, as seen in the introduction. In some extent, they contain the physics of the equations. So, turbulent models have to respect them. In the next section, we analyze some standard LES models according to the compatibility with the symmetries.

3 Model analysis LES consists in reducing the computation time by dropping the small scales of the unknowns. This is done by using a filter, symbolized here by the bar ( ) and having the width δ. (u, p) is then approximated by the filtered couple (u, p). To obtain (u, p), one applies the filter to NS. It gives: 1 ∂u + div(u ⊗ u) + ∇p = div(T + Ts ), ∂t ρ

divu = 0.

(1)

In these equations, T is the tensor such that ρT is the viscous constraint tensor. It can be linked to the strain rate tensor S according to the relation: T = ∂ψ/∂S, ψ being the positive and convex “potential” ψ = ν tr S2 . Ts = u ⊗ u − u ⊗ u is the subgrid constraint tensor, which must be modeled in order to close the equations. Currently, a large number of models exists (see [7, 3]). Some of the most common ones are reminded below. • Smagorinsky model: Tds = (Cδ)2 |S|S where the superscript d represents the deviatoric part of a tensor, 2 • the dynamic model: Tds = Cd δ |S|S 2 2 ⊗u −u ⊗ u, M = δ |S|S − δ | S|S, where Cd = [ tr(LM)]/[ tr M2 ], L = u • the structure function model: Tds = CSF δ F 2 (δ) S u(x) − u(x + z) 2 dz dx, where F 2 (δ) = z=δ

•

the gradient model:

2

δ ∇u T∇u, Ts = − 12

122

•

Dina Razafindralandy and Aziz Hamdouni 2

Taylor model:

2

δ Ts = − 12 ∇u T∇u + Cδ |S|S, 2

•

2

δ the rational model: Ts = − 12 G ∗ [∇u T∇u] + Cδ |S|S where G is the kernel of the Gaussian filter, ⊗u −u ⊗ u, • the similarity model: Ts = u

•

2

C4 δ (S W − W S) + C5 δ •

2

and Kosovic model:

2

2

2

2

−Tds = C1 δ |S|S + C2 δ (S )d + C3 δ (W )d +

Lund–Novikov model: 2

1

2

2

(S W − S W ) |S|

2 −Tds = (Cδ)2 2|S|S+C1 (S )d +C2 (S W−W S) .

δ, W is the Here, the symbol tilde () represents a test filter, with a width vorticity tensor and C and Ci are real constants. The set of the solutions (u, p) of NS is preserved by each of their symmetries. We then require that so is the set of the solutions (u, p) of (1) too, since (u, p) is expected to be a good approximation of (u, p). More clearly, if a transformation T : (t, x, u, p) → (t, x, u, p) is a symmetry of NS, we require that the model is such that the same transformation, applied to the filtered quantities, T : (t, x, u, p) → (t, x, u, p) is a symmetry of (1). When it is the case, the model will be said invariant under the symmetry. It will be assumed that the test filter does not destroy the symmetry properties. If the model is invariant under all the symmetries, we can expect to retrieve the properties of NS, such as conservation laws, wall laws, exact solutions, spectrum properties, . . . when approximating (u, p) by (u, p). The above presented models will be analyzed according to their invariance under each of the symmetries of NS. All the above models are invariant under the time and pressure translation because neither t nor p is explicitly present in their expressions. It is easy to show that these models are also invariant under the generalized Galilean transformation. Next, a model is invariant under the rotation and the reflection if and only if T varies as follows: Ts = Υ Ts TΥ, where Υ is a (constant) rotation or reflection matrix. It is the case of all the models since ∇u, S and T vary in the same way. A model is invariant under the scaling transformation if and only if Ts = e−2a Ts . Simple calculations show that among the presented models, the lone two which respect this condition are the dynamic and the similarity models. Though, the scaling transformation has a particular importance. Oberlack ¨ ([4]) used it to derive scaling laws and Unal ([9]) to prove the existence of NS solutions having Kolmogorov spectrum. The last symmetry of NS is the 2D material indifference, which corresponds to a time-dependent 2D rotation of matrix R (the dependence of R on t will not be written) with a compensation in the pressure term. The objectivity of S (i.e. S = RS TR) directly leads to the invariance of Smagorinsky model. For the similarity model, we have:


123

˙ xu TR + RM ˙ xx TR˙ Ts = RTs TR + RMux TR + +RM ⊗x and Mxx = x , Mxu = x ⊗u ⊗x . ⊗ x−u ⊗ u−x ⊗ x−x where Mux = u Then the similarity model is invariant if the test filter is such that Mux , Mxu and Mxx vanish, which is not always the case. Under the same conditions on the test filter, the dynamic model is also invariant. The function F 2 is not objective. Indeed, 3 F 2 = F 2 + 2πω 2 δ − 2ω (u − u(x + z)) (e3 × z) dz z=δ

Consequently, the structure function model is not invariant. By the same way, the non-objectivity of the velocity gradient (∇u = R ∇u TR + R˙ TR) leads to the non-invariance of the gradient, Taylor and rational models, and the nonobjectivity of the vorticity tensor (W = R W TR + R˙ TR) to the non-invariance of Lund–Novikov and Kosovic models. This ends the analysis. The above results are summarized in the table 1. It can be seen on it that only two models among the nine, the dynamic and the similarity models, are invariant under the symmetry group of NS. These two models need a test filter and are then concerned by the condition that Mux , Mxu , and Mxx vanish. In addition, the dynamic model is not conform to the second law of thermodynamics since it can induce a negative dissipation. Face to these inconsistencies of existing models, we propose in the next section a new way of deriving models which, in one hand, respect the symmetry group of NS and, in the other hand, are compatible with the second law of thermodynamics. Table 1. Results of the model analysis. Y=invariant, N=not invariant, Y∗ =invariant if Mux = Mxu = Mxx = 0. Translations=time translation, pressure translation, generalized Galilean transformation. Translations Rotation, Scaling Material reflection transformations indifference Smagorinsky

Y

Y

N

Y

Dynamic

Y

Y

Y

Y∗

Structure function

Y

Y

N

N

Gradient

Y

Y

N

N

Taylor

Y

Y

N

N

Rational

Y

Y

N

N

Similarity

Y

Y

Y

Y∗

Lund

Y

Y

N

N

Kosovic

Y

Y

N

N

124


4 A new class of symmetry invariant models Suppose that S = 0. Assume that Ts is an analytic function of S: Ts = F(S). By this way, the invariance under the time, pressure and generalized Galilean translations and the reflection is guaranteed. From that, Cayley–Hamilton theorem and the invariance under the rotation lead to: Tds = A(χ, ζ) S + B(χ, ζ) Adjd S 2

where χ = tr S and ζ = det S are the invariants of S, Adj stands for the operator defined by ( Adj S)S = (det S)Id , and A and B are arbitrary scalar functions. Lastly, Ts is invariant under the scaling transformation if Ts = e−2a Ts . Rewritten for A and B, this condition becomes: A(e−4a χ, e−6a ζ) = A(χ, ζ),

B(e−4a χ, e−6a ζ) = e2a B(χ, ζ).

To satisfy these equalities, one can take A(χ, ζ) = A1(v) and B(χ, ζ) = χ−1/2 B1(v) where A1 and B1 are arbitrary functions and v = ζ/χ3/2 . Thus, 1 Tds = A1(v) S + √ B1(v) Adjd S. χ

(2)

The relation (2) defines a class of subgrid models which are invariant under all the symmetries of NS. Let us now refine this class such that the models are also conform to the second law of thermodynamics. Ts represents the energy exchange between resolved and subgrid scales. It generates certain dissipation, which may take a negative value (backscatter). But to respect the second law of thermodynamics, we must ensure that the total dissipation (sum of molecular and subgrid dissipations) remains positive. At molecular scale, the viscous constraint is ρT = ρ ∂ψ/∂S. The “potential” ψ = ν tr S2 is convex and positive; this ensures that the molecular dissipation Φ = tr(TS) is positive. The tensor ρTs can be considered as a subgrid constraint generating a dissipation Φs = tr(Ts S). To remain compatible with NS, we assume that Ts has the same form as T, i.e.: Ts =

∂ψs ∂S

.

(3)

where ψs is a “potential” depending on the invariants χ and ζ. This hypothesis refines class (2) in the following way. One deduces from (3) that ∂ψs ∂ψs Adjd S. S+ ∂χ ∂ζ 1 ∂ ∂ 1 A1 (v) = This form is compatible with (2) only if √ B1 (v) . If ∂ζ 2 ∂χ χ g is a primitive of B1 , a solution of this equation is Tds = 2


A1 (v) = 2g(v) − 3v g(v) ˙

and

125

B1 (v) = g(v). ˙

Then, hypothesis (3) involves the existence of a scalar function g such that:

Tds

1 ˙ Adjd S. = 2g(v) − 3v g(v) ˙ S + √ g(v) χ

(4)

Now, using (2) and (4), it can easily be shown that the total dissipation ΦT = tr[(T + Ts )S] is positive if and only if ν + g(v) ≥ 0. In summary, a model belonging to class (4) with a continuous function g such that ν + g ≥ 0 is a model respecting the symmetry group of NS and conform to the second law of thermodynamics. We will end this communication by a numerical test of a model of (4).

5 Numerical test The aim of this test is only to show that the symmetry approach can lead to numerically efficient models. We then settle for a simple, linear, expression for g: g(v) = ν(Cd)2 v, where C is the model constant, d = δ/ and a length scale related to the size of the domain. In this case, one has det S 1 d d 2 S+ Adj S τs = ν(Cd) − ||S||3 ||S|| 2

where ||S|| = ( tr S )1/2 . This model, that we will call invariant model, is used to simulate the air flow within a ventilated room (see figure 1) which interests us for applications in building domain. is set to 1m.

Figure 1: Geometry of the ventilated room.

Figure 2: Mean velocity profiles.

A comparison with a dynamical calculation allows us to give to C the value of the Smagorinsky constant (see [7]). Figure 2 shows then the horizontal

126


velocity profiles, along the vertical line defined by (x1 = 2L/3, x3 = W/2), given by Smagorinsky, dynamic and invariant models. It can be deduced from it that the invariant model gives a result in good agreement with experiments, except near the floor, but in all cases better than those provided by the two other models. Notice that no wall law was used.

6 Conclusion We showed that many existing models do not respect the symmetries of NS. It is prejudicial for these models because they are not able to capture all the information contained in the equations, (conservation laws, . . . ). We then proposed a new way of deriving LES models which are consistent with the symmetries of NS and are conform to the second law of thermodynamics. The generality of the method is such that other quantities, such as the dissipation rate, can be introduced and models for subgrid flux of temperature or mass could be derived. The numerical test gave encouraging results. However, deeper studies on g should be carried out (sensitivity of the model on g, ...). Spectral, correlation or other physical approach could be used to optimize the choice of g.

References 1. Cannone M (1995) Ondelettes, paraproduits et Navier-Stokes. Diderot Editeur, Arts et Sciences, Paris 2. Fushchych W, Popowych R (1994) Symmetry reduction and exact solutions of the Navier-Stokes equations. J Nonlin Math Phys, 1(1):75–113 3. John V (2004) Large Eddy Simulation of Turbulent Incompressible Flows. Analytical and Numerical Results for a Class of LES Models. Springer 4. Oberlack M (1999) Symmetries, invariance and scaling-laws in inhomogeneous turbulent shear flows. Flow Turb Comb, 62(2):111–135 5. Olver P (1986) Applications of Lie groups to differential equations. Graduate texts in mathematics. Springer-Verlag, New-York 6. Razafindralandy D (2005) Contribution ` a l’étude mathématique et numérique de la simulation des grandes échelles. Phd, Université de La Rochelle 7. Sagaut P (2004) Large eddy simulation for incompressible flows. An introduction. Scientific Computation. Springer 8. Speziale C (1985) Galilean invariance of sub-grid scale stress models in the large eddy simulation of turbulence. J Fluid Mech, 156:55 ¨ 9. Unal G (1994) Application of equivalence transformations to inertial subrange of turbulence. Lie Group Appl, 1(1):232–240

Formal properties of the additive RANS/DNS filter Massimo Germano1 and Pierre Sagaut2 1

2

Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy, [email protected] Laboratoire de Modélisation en Mécanique, Université Pierre et Marie Curie - Paris 6, 4 place Jussieu, case 162, 75252 Paris Cedex 5, France, [email protected]

Summary. A new formulation of LES based on a mixed RANS/DNS additive filter is presented and some suggestions about its possible practical implementation are proposed.

1 Introduction The Large Eddy Simulation, LES (see for detailed presentations [1, 2, 3]), of turbulent flows should provide more details than the standard informations produced by the solution of the Reynolds Averaged Navier Stokes, RANS, equations. The computed values are in the RANS case limited to the mean values and the Reynolds stresses, while a large eddy simulation should provide more details on the motion of the large scales, the large eddies, and should capture a significant portion of the turbulent kinetic energy in terms of computed fluctuations in time. We recall that the standard LES formulation is based on a convolutional smoothing filter provided with a characteristic filter length of the order of the computational grid length Δf . Obviously when Δf goes to zero we recover a DNS solution, but it is not so clear how to relate this formulation to the RANS representation of turbulent flow, largely used in many engineering computations. The need for a different LES formulation is not only theoretical but also very important for practical use : the interest for hybrid methods that integrate different codes with different resolutions in complex engineering flows are more and more required [1]. In this paper a new formulation of a LES filter is proposed. It is based on an additive RANS/DNS filter [4] and its main peculiarity is that it reduces to RANS when a mixing parameter α goes to zero and to DNS when it goes to one. This mixing parameter α is related to the ratio of the resolved turbulent kinetic energy Kf captured directly by the simulation with the total kinetic energy per unit mass K

128

Massimo Germano and Pierre Sagaut

vi vi Kf = 2K K where vi is the statistical fluctuation of the computed field vi α2 =

vi = vi − v¯i

(1)

(2)

and where we indicate with an overline the statistical values and with an apex the statistical fluctuations. We remark again that the characteristic feature of this new RANS/DNS filter compared with the standard LES convolution filter is the parameter α. Formally the square of α represents the ratio of the resolved turbulent kinetic energy while the resolution parameter of a standard LES is the convolution length. In the following we will relate α with an equivalent resolution length based on classical scaling considerations.

2 The additive RANS/DNS solution of the Navier Stokes equations. Let us consider the Navier Stokes equations for the incompressible flow and the related Reynolds averaged equations ∂j uj = 0 ∂t ui + ∂j (ui uj ) = −∂i p + ν∇2 ui

(3)

∂j u ¯j = 0 ¯i + ∂j (¯ ui u ¯j ) = −∂i p¯ + ν∇2 u ¯i − ∂j τij ∂t u

(4)

¯i u ¯j = ui uj τij = ui uj − u

(5)

where are the Reynolds turbulent stresses. We consider an additive combination of the unaveraged and the averaged field produced by the application of the additive filter F F = αI + βE (6) where I is the identity operator, E the statistical average and α and β two constants. We remark that only in the case in which α+β =1

(7)

the additive RANS/DNS filter is constant preserving. In any case we have a filtered velocity field vi and a filtered pressure q given by ¯i vi = αui + β u q = αp + β p¯

(8)

and we can write the balance equations for the new quantities vi and q ∂j vj = 0 1 ∂t vi + ∂j (vi vj ) = −∂i q + ν∇2 vi − ∂j ϑij α+β

(9) (10)

Formal properties of the additive RANS/DNS filter where ϑij = βτij +

β vi vj α(α + β)

129

(11)

We remark that formally the original quantities ui and p can be exactly recovered only if α = 0 α+β = 0 (12) and in particular ui =

vi α

u ¯i =

v¯i α+β

(13)

so that we have

vi vi Kf = (14) 2K K where K = ui ui /2 is the turbulent kinetic energy per unit mass and Kf = vi vi /2 the resolved turbulent kinetic energy directly captured by the simulation. α2 =

3 Possible implementation coupled with the standard K − ε RANS model Let us now examine a possible implementation of this new LES approach for an incompressible flow. We will consider the constant preserving case where β =1−α

(15)

and a standard K − ε RANS model for the turbulent stresses τij . We remark that in the Very Large Eddy Simulations, VLES, of turbulent flows the interest for subgrid models that reduce for very coarse grids to a standard K−ε RANS model is presently very high [8, 9]. We can write ∂j vj = 0 ∂t vi + ∂j (vi vj ) = −∂i q + ν∇2 vi − ∂j ϑij ϑij = (1 − α)τij + τij =

1−α vi vj α

2 Kδij − νe (∂j v¯i + ∂i v¯j ) 3 K2 νe = cμ ε

(17) (18) (19)

νe ∂j K + P − ε σK νe ε ∂t ε + ∂j (ε¯ vj ) = ∂j ν+ ∂j ε + cε1 P − cε2 ε σε K

∂t K + ∂j (K v¯j ) = ∂j

(16)

ν+

(20)

where ε is the turbulent dissipation and P the production of turbulent kinetic energy

130

Massimo Germano and Pierre Sagaut P = −τij ∂j v¯i

(21)

and the statistical values and the fluctuations of the original turbulent fields u ¯i , ui , p¯, p of velocity and pressure can be recovered as follows ui =

u ¯i = v¯i

vi α

(22)

q (23) α and we remark that following the definitions of Sagaut [1] this new hybrid RANS/LES model is composed by two terms, the first related to the functional properties of the RANS model and the second to the structural properties of the approximate reconstruction. All that is obviously purely formal. In the reality we have to cope with a grid length Δf related to the numerical resolution, and we can reasonably recover the information only at this level. An estimation of the maximum reasonable value for α can be derived as follows. Let us introduce the subgrid, unresolved turbulent kinetic energy Ks Ks = K − K f (24) p =

p¯ = q¯

and let us assume that in the inertial range the dissipation ε is nearly constant. We can write 3/2 Ks K 3/2 = (25) ε= Δe Δf where Δe is an integral scale given by

Δe = π

2 3CK

3/2

K 3/2 ε

(26)

where CK is the Kolmogorov constant, and from (14) and (24) we obtain Ks = 1 − α2 = K

Δ 2/3 f

Δe

(27)

From this relation we can derive a first starting value of α consistent with the disposable numerical resolution. In the development of the computation we can compare this value of α with the computed value α2 =

vi vi Kf = 2K K

(28)

and a hopefully convergent iteration can be conducted. Let us finally consider some limiting situations. We remark that when α → 1 we should recover the DNS formulation and when α → 0 we should have the RANS solution. It is interesting to examine the limiting form of the model ϑij in these two cases. In the DNS limit we can write α = 1 − η with η → 0. We have

ϑij → η τij + vi vj

vi → ui vi vi → 2K

(29)

Formal properties of the additive RANS/DNS filter

131

In the RANS limit we can write α = η with η → 0. We have ϑij → τij +

vi vj η

vi → ηui vi vi → 2η 2 K

(30)

where we remark that when η → 0 also the velocity fluctuations vi disappear.

4 Analysis of the spatial commutation error We now address the issue of the commutation error of the hybrid filter with the spaces derivatives. Such an error will theoretically arise if the weighting parameter α which appears in the hybrid filter definition is not uniform in space, i.e. if α = α(x). This case is realistic, since α is directly tied to the ratio of the turbulent kinetic energy and the resolved turbulent kinetic energy. It is assumed in this section that β = 1 − α for the sake of simplicity. Introducing the commutation error operator: [f, g] (u) ≡ f ◦ g(u) − g ◦ f (u)

(31)

F(∂j ui ) = α∂j ui + (1 − α)∂j ui

(32)

∂j (F(ui )) = α∂j ui + ui ∂j α + (1 − α)∂j ui − ui ∂j α

(33)

and noting that

and

one obtains the following expression for the commutation error with first-order spatial derivatives: [F, ∂j ] (ui ) = −(ui − ui )∂j α = −ui ∂j α

(34)

An interesting result is that the commutation error is directly proportional to the gradient of α and the fluctuation ui . Since α is defined as the ratio of two statistical moments, it is invariant with respect to the statistical averaging operator: α = α, and therefore [F, ∂j ] (ui ) = −ui ∂j α = 0

(35)

showing that the commutation error has a zero mean value. This property is interesting, since it makes it possible to use the computed data for postprocessing purpose without spurious contribution of the commutation error (for first-order statistical moments only). Simple algebra yields the following expression for the dispersion of the commutation error (without summation over the repeated indices): ([F, ∂j ] (ui )) ([F, ∂j ] (ui )) = ui ui (∂j α)2

(36)

Looking at these expressions for the moments of the commutation error, it is seen that large errors will arise in regions with large turbulent kinetic energy and nonvanishing gradient of α. The equation (36) can also be used to derive some practical

132

Massimo Germano and Pierre Sagaut

a priori error estimates for the commutation error. Assuming ui ui 23 K and retrieving K from the K − ε mode introduced in the preceding section, one obtains a useful error estimate, which can also be used to correct the computed data during the postprocessing step. These expressions for the commutation error and its statistical moments are straightforwardly extended to the case of the commutation error which arise from the non-linear convection term. We can write [F, ∂j ] (ui uj ) = − (ui uj ) ∂j α

(37)

[F, ∂j ] (ui uj ) = 0

(38)

and one obtains

We remark finally that the analysis of nonuniform filters and of the commutation error is presently object of intensive studies, see [5, 6, 7], and our relations may have connections with different results peculiar of standard LES filtering. This important point will be developed in more detail in future works.

5 Sensitivity with respect to the weighting parameter α Using the hybrid RANS/DNS method introduced in the present paper, one has to face the question of the sensitivity of the computed solution with respect to the parameter α. Also in this section we assume that β = 1 − α, and we introduce the sensitivity variables v and p [3]: ∂vi ∂p , p≡ (39) ∂α ∂α Evolution equations for these new variables are derived differentiating Eqs. (9) and (10) with respect to α, yielding vi ≡

∂j vj = 0

(40)

∂t vi + ∂j (vi vj + vi vj ) = −∂i p + ν∇ vi − ∂j ϑij

(41)

ϑij ≡ ∂α ϑij

(42)

2

where

All terms but the last one in the right hand side of Eq. (41) are linear with respect to v and p. This generic equation can be easily rewritten in the present case to get a direct insight into the dynamics of the sensitivity since the definition of the additive filter yields vi = ∂α (αui + (1 − α)ui ) = ui − ui = ui

(43)

The sensitivity variables are identical to the turbulent fluctuations and are therefore solutions of the same equations and have the same dynamics (including inter scale transfers, forward and backward cascade, ...). A first trivial result is that v = 0, which is coherent with the analysis conducted above. A second one is that results will be sensitive to α in regions where the kinetic energy K is large. This diagnostic can be easily performed in practical simulation looking at K.

Formal properties of the additive RANS/DNS filter

133

As quoted by Berselli, Iliescu and Layton [3], the sensitivity variables can also be used to improve the results during the postprocessing step and to obtain results which are closer to the ideal DNS results. Let us assume that we are interested in computing a functional J (e.g. drag or lift of an immerged solid body) from the hybrid RANS/DNS results. The value obtained from the results of the simulation is J(v(α)), where v(α) denotes the solution computed with a given (non zero) value of α. The ultimate value for J is the DNS value J(v(1)), which can be estimated correcting the computed value as follows J(v(1)) J(v(α)) + (1 − α)J (v(α)) · v

(44)

were J is the gradient of the functional J.

6 Conclusions A new additive RANS/DNS filter is presented and some suggestions about its possible practical implementation are proposed. We remark that apart from the possible applications as an hybrid filter to mix different computational zones, the proposed RANS/DNS filter can be seen as a new formulation of LES. It reduces to the RANS formulation when the parameter α goes to zero and to the DNS formulation when α = 1, and the parameter α can be related both to the ratio of resolved and the total turbulent kinetic energy and to the ratio of an equivalent resolution length to the integral length. As regards the implementation we stress again that the computation produces a filtered quantity vi vi = v¯i + vi (45) and the statistical values and the statistical fluctuations of the turbulent field ui ui = u ¯i + ui

(46)

can be formally recovered as u ¯i = v¯i

ui =

vi α

(47)

In the filtered equations that give the filtered field vi the related subgrid scale stresses, see (17), can be written as ϑij = (1 − α)τij +

1−α vi vj α

(48)

and two possible strategies can be conceived. The first approach, that we will call the hybrid RANS/LES approach and that we have exposed in the present paper, is to assume that the Reynolds stresses τij are given by some statistical RANS model M (ui , uj ) (49) τij ∼ M (ui , uj ) while the second, the direct approach, is to calculate directly τij during the computation as vi vj (50) τij = ui uj = 2 α where the fluctuations vi = vi − v¯i are given by

134

Massimo Germano and Pierre Sagaut vi = vi −

1 t

t

vi dt

(51)

0

and if the numerical resolution is sufficient we can read this approach as a DNS computation based on an approximate reconstruction of the statistical values.

References 1. Sagaut P (2004) Large Eddy Simulation for Incompressible Flows. Second revised edition, Springer 2. Geurts BJ (2003) Elements of Direct and Large Eddy Simulation. Edwards 3. Berselli L, Iliescu T, Layton W (2005) Mathematics of Large Eddy Simulation of Turbulent Flows. Springer 4. Germano M (2004) Theoret. Comput. Fluid Dynamics 17:225–231 5. Vreman AW (2004) Physics of Fluids 16:2012–2022 6. van der Bos F, Geurts BJ (2005) Physics of Fluids 17:035108 7. van der Bos F, Geurts BJ (2005) Physics of Fluids 17:075101 8. Magnient JC (2001) Simulation des grandes échelles (SGE) d’écoulements de fluides quasi incompressibles. PhD thesis, Université Paris Sud 9. Schiestel R, Dejoan A (2005) Theoret. Comput. Fluid Dynamics 18:443–468

Analysis of the SGS energy budget for deconvolution- and relaxation-based models in channel flow Philipp Schlatter, Steffen Stolz, and Leonhard Kleiser Institute of Fluid Dynamics, ETH Z¨ urich, Switzerland [email protected] Summary. The energy budget of subgrid-scale (SGS) models of deconvolution and relaxation type, e.g. the approximate deconvolution model (ADM), is analysed. Energy transfer properties of the deconvolution and the relaxation regularisation are computed individually. An alternative formulation of ADM based on one filter operation only is analysed and compared to a scale-similarity model with additional relaxation. Depending on the primary LES filter used, the model dissipation caused by the modification of the nonlinear terms drops to roughly one third of the total model dissipation, rendering the relaxation the dominant source of energy dissipation. Consequently, models based on a relaxation regularisation alone perform equally well. All models are exhibiting backscatter in the vicinity of the walls.

Large-eddy simulation (LES) and the development of subgrid-scale (SGS) models is an active field in turbulence research. Understanding the main mechanisms of the SGS energy transfer is a key issue in improving LES strategies [4]. In this contribution we focus on SGS models of deconvolution type [2], and their relation to the scale-similarity model (SSM, [1]). As main example, the approximate deconvolution model (ADM), introduced in [8] and since applied to turbulent channel flow [9] and a variety of other flows, is chosen.

1 Subgrid-Scale Models The governing LES equations are obtained from the Navier-Stokes equations by applying a primary low-pass filter GP yielding equations for the evolution of the filtered quantities u ˜i := GP ∗ ui , p˜ := GP ∗ p. The incompressible LES equations in non-dimensional form thus are ∂u ˜i u ∂ p˜ 1 ∂2u ∂τij ˜j ˜i ∂u ˜i + =− + − , ∂t ∂xj ∂xi Re ∂xj ∂xj ∂xj

(1)

supplemented with the incompressibility constraint ∂ u ˜i /∂xi = 0. The SGS term mi := ∂τij /∂xj is unclosed and needs to be modelled.

136

Philipp Schlatter, Steffen Stolz, and Leonhard Kleiser

With the approximate deconvolution model, the subgrid-scale term is [9] ∂ui uj ∂ui uj ∂τij ADM = mi = − + χ · (I − QN ∗ G) ∗ ui . (2) ! "# $ ∂xj ∂xj ∂xj relaxation ! "# $ “deconvolution”

χ is the model coefficient, a star (·) represents the approximately deconvolved quantities, ui := QN ∗ ui , ui = G ∗ ui with GP = G being the discrete primary N low-pass filter with approximate inverse QN = ν=0 (I − G)ν ≈ G−1 and the deconvolution order set to N = 5. The definition of G is given in Ref. [9]. As shown in equation (2) ADM consists of two independent parts: Firstly, the modification of the nonlinear terms by introducing deconvolved (“defiltered”) quantities, which is supposed to provide an improved prediction of the unclosed advection terms [8]. Secondly, the relaxation term (RT) is used to provide an appropriate amount of SGS dissipation. Similar to the SSM, ADM without relaxation is clearly not dissipative enough [4, 8]. An alternative formulation of ADM (termed ADM ) can be found after convolution of equations (1) and (2) with QN and neglecting commutation errors. For constant χ one obtains (see [5]) ∂ u i uj ∂τij ∂ u% i uj ADM = mi = − + χ(ui − ui ) . (3) ! "# $ ∂xj ∂xj ∂xj relaxation "# $ ! “deconvolution”

& denotes the new primary LES filter, GP = QN ∗ G, i.e. ui = The hat (·) QN ∗ G ∗ ui = ui . Note that only one filter is employed, i.e. the combined filter QN ∗ G. An actual implementation of ADM in the form (3) consists solely of lowpass-filtering the nonlinear terms plus an added relaxation. It should be noted, however, that both models (2) and (3) involve a small frameindifference error [5] of the same order as QN ∗ G ∗ ui . From the classical decomposition of τij into Leonard, cross and Reynolds stresses it can be seen that the ADM model terms arising from the deconvolution operation are basically the (closed) Leonard stresses τij = Lij = u% i uj − ui uj ,

(4)

which makes plausible the high correlation of ADM in a-priori evaluations. As recently shown in [3], a modified decomposition with individually frameindifferent terms leads to modified Leonard stresses, i.e. ˜ ij = u% τij = L i uj − ui uj ,

(5)

which is in fact the (frame-indifferent) scale-similarity model (SSM) [1, 3]. Since it is known that the SSM does not provide enough dissipation, a new

Analysis of SGS energy budget

137

“mixed” formulation (termed mixed-RT) is proposed by combining the SSM with the relaxation-term model (ADM-RT, see below) ∂ ui uj ∂τij ∂ u% i uj mixed-RT = mi = − + χ(ui − ui ) . (6) ! "# $ ∂xj ∂xj ∂xj relaxation "# $ ! SSM

In the relaxation-term model (ADM-RT) [6, 7], the nonlinear terms are not modified and the model dissipation is provided by a relaxation term, ∂τij = mRT = χ(I − QN ∗ G) ∗ u = χ(ui − ui ) . i ∂xj

(7)

Using spectral numerical methods, very accurate results have been obtained using ADM-RT for turbulent and transitional channel flow, see e.g. [6, 5]. The effect of the relaxation term (for constant χ) can also be achieved approximately by explicitly filtering the solution ui every 1/(χΔt) time steps with the filter I − QN ∗ G [9] (termed EXFILT herein). Note that this filtering (and in particular the resulting model) is not equivalent to the one performed in equation (3) where only the nonlinear terms are filtered every time step.

2 Results Turbulent incompressible channel flow at a Reynolds number based on the friction velocity of Reτ ≈ 180 and Reτ ≈ 590 is considered. The numerical code is based on a standard Fourier-Chebyshev method with periodic boundary conditions in the wall-parallel directions. The nonlinear terms are computed with dealiasing in all directions for both DNS and LES model terms. Statistical quantities have been averaged in time and over wall-parallel (x, y) directions. The simulations at Reτ ≈ 590 have been performed on a 3842 × 257 grid for the DNS and on a 642 × 65 grid for the LES [5]. Results for channel flow at Reτ ≈ 180 are all qualitatively very similar to those obtained at Reτ ≈ 590 and are thus not shown. As the sensitivity of the results to the choice of the model coefficient χ is low [9, 5], this parameter was fixed in space and time. A specific value χ = 6 = 3/(20Δt) was chosen which is in agreement with previous investigations [5]. ui x,y,t + u ˜i , the total By introducing the Reynolds decomposition u ˜i = ˜ viscous dissipation can be split into εvisc,mean = −(2/Re) S˜ij S˜ij ,

˜ Sij , εvisc,fluct = −(2/Re) S˜ij

(8)

where Sij = 12 (∂ui /∂xj +∂uj /∂xi ) denotes the strain rate. εvisc,mean describes the dissipation due to the mean flow whereas εvisc,fluct is caused by the fluctuating field. Similarly, the SGS energy transfer is given as [5] ui ∂τij /∂xj = − ˜ ui mi , ε∗SGS,mean = − ˜

ε∗SGS,fluct = − ˜ ui mi . (9)

138

Philipp Schlatter, Steffen Stolz, and Leonhard Kleiser

If τij can be evaluated explicitly (i.e. for the deconvolution operation), the SGS dissipation is given as [5] εSGS,mean = τij S˜ij ,

˜ εSGS,fluct = τij Sij .

(10)

The SGS energy transfer terms can further be split into the two contributions resulting from the modification of the nonlinear terms (e.g. deconvolution, see equations (2), (3), (6)) and from the relaxation term. A first series of simulations was performed to evaluate the two related ADM formulations, equations (2) and (3). The different cases are summarised in table 1. Note that ADM and ADM use different primary LES filters, therefore a direct comparison is only possible between ADM and G∗ADM or QN ∗ADM and ADM . Table 1. Summary of the different DNS and LES. Case

Reτ

Reτ

Comment

ADM QN ∗ADM ADM G∗ADM

176.1 182.0 181.6 179.7

577.7 592.4 583.8 578.0

standard ADM [9] ADM results, deconvolved alternative ADM, equation (3) alternative ADM, results filtered

mixed-RT ADM-RT ADM-RT EXFILT

180.3 176.9 178.6 178.8

582.8 579.1 574.7 578.0

SSM with ADM-RT, equation (6) relaxation-term model, equation (7) relaxation-term model, equation (7), χ = 1/Δt explicit filtering with QN ∗ G every time step

no-model

202.7

628.4

no-model LES

DNS G∗DNS

178.9 176.8

588.2 581.5

interpolated onto respective LES grid DNS filtered with G

Figure 1 shows important flow statistics. Clearly the use of SGS models improves the simulation results. In particular, the Reynolds stresses and the streamwise velocity profile are predicted more accurately. The Chebyshev spectrum (and other spectra, not shown) clearly show the effect of the two different primary LES filters, i.e. G and QN ∗ G. By comparing the respective curves of ADM and G∗ADM or QN ∗ADM and ADM it can be concluded that the alternative formulation ADM is indeed a valid reformulation of the original ADM under the given assumptions. An evaluation of the SGS energy transfer is depicted in figure 2. Figures 2a) and 2b) show the energy dissipation and energy transfer of the deconvolution operation (modification of the nonlinear terms). It can be seen that ADM features significant mean SGS dissipation which is mainly attributed to the lower filter cutoff wavenumber and lower filter order of the explicit filter G. For ADM the influence of the “deconvolution” on the mean flow is negligible.


139

3

Reynolds stresses

20

a)

x,y,t

2

+

1 0

15 10 5

−1

10

0

10

1

10

+

z

2

b)

0

10

0

10

1

10

1

z+

10

2

0.2

10

10

c)

Energy budget

<E3(kz)>x,y,t

0 5

−0.2 −0.4 −0.6 −0.8 −1

10

10

0

kz

10

1

d)

10

0

+

10

2

z

Fig. 1. Comparison of ADM and ADM for Reτ ≈ 590. a) Reynolds stresses, b) mean velocity profile, c) wall-normal Chebyshev spectrum, d) energy budget. ADM, QN ∗ADM, G∗ADM , ADM , • DNS (interpolated no-model LES. onto LES grid), ◦ G∗DNS,

It is further evident that for ADM the energy dissipation due to the deconvolution is significantly higher (by a factor of 8 if integrated in the wall-normal direction) than for ADM . Approximately 90% of the fluctuating energy dissipation of ADM is due to deconvolution. Consequently the energy dissipation caused by the relaxation term is lower for ADM than for ADM-RT and ADM (figure 2c)), whereas the total fluctuating SGS dissipation is comparable for all three models (figure 2d)). However, ADM exhibits a considerable energy transport from the buffer layer towards the wall. In a second series of simulations various models are compared, see table 1 for an overview. Flow statistics are shown in figure 3. The results are very similar for both the Reynolds stresses and the streamwise velocity profile with all models showing an improvement over the no-model calculation. The spectra show distinct differences of the models at higher wavenumbers. In particular, the EXFILT and the ADM-RT with χ = 1/Δt closely follow each other, as to be expected. Similarly, the mixed-RT and ADM can hardly be distinguished. This of course follows from the similarity of equations (3) and (6), in particular for high filter orders as chosen here. Moreover, the ADM-RT model also closely follows ADM and mixed-RT indicating that for the present flow case the scale-similarity term does not further improve the results (i.e. a

140

Philipp Schlatter, Steffen Stolz, and Leonhard Kleiser 0.02

0.15

0

0.1 0.05

−0.04

−ui mi

τ S ij ij

−0.02

−0.06

0.05

−0.08

a)

−0.1

10

0

10

1

z

+

10

2

b)

0

10

1 +

10

2

i

0

mu mm

−0.01

i

−ui mmi

10

0.02

0

−0.02

−0.02 −0.04

−0.03 −0.04

0.1

z

0.01

c)

0

10

0

10

1

z

+

10

2

d)

−0.06

0

10

10

1

2

z+

10

Fig. 2. Comparison of ADM and ADM for Reτ ≈ 590. a) Energy dissipation, fluctuating and mean part), b) energy transfer due to “deconvolution” ( c) fluctuating energy transfer due to relaxation term, d) total fluctuating SGS energy ADM, ADM , ADM-RT. transfer.

modification of the nonlinear terms is not necessary, see also the discussion in [7] and [5]). Figure 4 presents the fluctuating SGS energy budget for the different models. Their influence on the mean flow is at least two orders of magnitude smaller and is therefore not shown. As depicted in figures 4a) and 4b) for both ADM and mixed-RT the effect of the “deconvolution” is similar. Both models clearly exhibit backscatter (i.e. a transfer from the non-resolved scales to the resolved ones) close to the wall. Integrated in the wall-normal direction, the mixed-RT model dissipates approximately twice as much energy as ADM . From the comparison of figures 4a) and 4b) it can also be stated that the “deconvolution” does not contribute to a substantial wall-normal redistribution of energy, it is mainly dissipative. Figure 4c) shows the energy transfer induced by the relaxation term. For ADM and mixed-RT the integrated energy dissipation of the RT is roughly twice as high as the one caused by the “deconvolution”. Moreover, the dissipation peak for both RT and deconvolution is at the same wall-normal distance, z + ≈ 15. As can be seen from the total fluctuating energy transfer due to the SGS model, figure 4d), all models exhibit essentially the same energy transfer, with the relaxation term being the dominant source of dissipation. A comparison with figure 2 yields that lower filter order and lower cutoff wavenumber leads to increased SGS activity.


141

a)

20 2

x,y,t

x,y,t

3

1 0 −1

15 10 5

10

0

10

1

10

+

z

2

b)

0

10

0

10

1

10

2

+

10

2

+

z

0 10

Energy budget

<E3(kz)>x,y,t

0.2

5

− 0.2 − 0.4 − 0.6 − 0.8

10

c)

10

10

0

1

kz

10

d)

1 10

0

10

1

z

Fig. 3. Results for various models at Reτ ≈ 590. a) Reynolds stresses, b) mean veADM , locity profile, c) wall-normal Chebyshev spectrum, d) energy budget. EXFILT, mixed-RT, ADM-RT, ADM-RT with χ = 1/Δt, • DNS (interpolated onto LES grid), ◦ no-model LES.

3 Conclusions Large-eddy simulations of incompressible turbulent channel flow at Reτ ≈ 180 and Reτ ≈ 590 have been performed. The energy transfer of different subgridscale models based on a modification of the nonlinear convection terms in combination with a relaxation have been analysed, e.g. the approximate deconvolution model (ADM), the scale-similarity model (SSM) and the relaxationterm model (ADM-RT). An alternative formulation of ADM, ADM , simplifying rationale and implementation has been derived and shown to be consistent with ADM for the given case. The results show that the treatment of the nonlinear term in ADM is closely related to the scale-similarity ansatz. A new model, mixed-RT, combining the scale-similarity ansatz with the ADM-RT model, provides equally accurate results as ADM. It is further shown that with the alternative formulation ADM and the mixed model one third of the SGS dissipation stems from the modification of the nonlinear term, whereas two thirds are caused by the relaxation rendering it the dominant source of energy dissipation. Accordingly, the results obtained by relaxation regularisation alone are at least as accurate as the ones obtained by the mixed or ADM formulation. Acknowledgements. Calculations have been performed at the Swiss National Supercomputing Centre (CSCS), Manno.

142

Philipp Schlatter, Steffen Stolz, and Leonhard Kleiser 0.005

0.005

0

− 0.005

− 0.005

τ S

ij ij

−u m i i

0

− 0.01

− 0.015

− 0.02

− 0.02

a) − 0.025

− 0.01

− 0.015

10

0

10

1

z+

10

b)− 0.025

2

10

1 +

10

2

0.01

0

0

− 0.01

0.01 0.02

i

−ui mi

−u m

i

− 0.02 − 0.03

c)

0

z

0.01

0.03

− 0.04

0.04

− 0.05

0.05

− 0.06

10

10

0

10

1

z+

10

2

d)

0.06 10

0

1

10

2

z+

10

Fig. 4. Results for various models at Reτ ≈ 590. Fluctuating a) energy dissipation, energy transfer b) due to “deconvolution”, c) due to relaxation term, and d) total fluctuating SGS energy transfer. Line caption see figure 3.

References 1. J. Bardina, J. H. Ferziger, and W. C. Reynolds. Improved subgrid models for large-eddy simulation. AIAA Paper, 1980-1357, 1980. 2. J. A. Domaradzki and N. A. Adams. Direct modelling of subgrid scales of turbulence in large eddy simulations. J. Turbulence, 3, 2002. 3. C. Fureby, R. Bensow, and T. Persson. Scale similarity revisited in LES. In J. A. C. Humphrey, T. B. Gatski, J. K. Eaton, R. Friedrich, N. Kasagi, and M. A. Leschziner, editors, Turbulence and Shear Flow Phenomena 4, pages 1077–1082, 2005. 4. C. Meneveau and J. Katz. Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid Mech., 32:1–32, 2000. 5. P. Schlatter. Large-eddy simulation of transition and turbulence in wall-bounded shear flow. PhD thesis, ETH Z¨ urich, Switzerland, Diss. ETH No. 16000, 2005. 6. P. Schlatter, S. Stolz, and L. Kleiser. LES of transitional flows using the approximate deconvolution model. Int. J. Heat Fluid Flow, 25(3):549–558, 2004. 7. P. Schlatter, S. Stolz, and L. Kleiser. Relaxation-term models for LES of transitional/turbulent flows and the effect of aliasing errors. In R. Friedrich, B. J. Geurts, and O. Métais, editors, Direct and Large-Eddy Simulation V, pages 65– 72. Kluwer, Dordrecht, The Netherlands, 2004. 8. S. Stolz and N. A. Adams. An approximate deconvolution procedure for largeeddy simulation. Phys. Fluids, 11(7):1699–1701, 1999. 9. S. Stolz, N. A. Adams, and L. Kleiser. An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows. Phys. Fluids, 13(4):997–1015, 2001.

Symmetry-preserving regularization of turbulent channel flow Roel Verstappen Research Institute of Mathematics and Computer Science, University of Groningen P.O.Box 800, 9700 AV Groningen, The Netherlands. [email protected] Summary. We propose to regularize the convective term in the Navier-Stokes equations in such a manner that the symmetries that correspond to the invariance of the energy, the enstrophy (in 2D) and helicity are preserved. The underlying idea is to restrain the convective production of smaller and smaller scales of motion by means of vortex stretching in an unconditional stable manner, meaning that the solution can not blow up (in the energy-norm). The numerical algorithm used to solve the governing equations preserves the symmetry properties too. The simulation shortcut is successfully tested for a turbulent channel flow (Reτ = 180).

1 Introduction Most turbulent flows can not be computed directly from the (incompressible) Navier-Stokes equations, ∂t u + C(u, u) + D(u) + ∇p = 0,

(1)

because they possess far too many scales of motion. The computationally almost numberless small scales result from the nonlinear convective term C(u, v) = (u · ∇)v which allows for the transfer of energy from scales as large as the flow domain to the smallest scales that can survive viscous dissipation. As the full energy cascade can not be computed, a dynamically less complex mathematical formulation is sought. In the quest for such a formulation, we consider smooth approximations (regularizations) of the convective term: , u ) + D(u ) + ∇p = 0, ∂t u + C(u

(2)

where the variable name is changed from u to u to stress that the solution of (2) differs from that of (1). Notice that by filtering (2) and comparing the result term-by-term with the equation governing the dynamics of the filtered velocity u in LES, we may identify the closure model, see also [1]: , u ). closure model(u ) = C(u , u ) − C(u

144

Roel Verstappen

Here, we want to smooth the convective term directly to set bounds to the creation of smaller and smaller scales of motion and thus to confine the cascade of energy. That is, the low modes of the solution u of (2) should approximate the corresponding low modes of the solution u of the Navier-Stokes equations (1), whereas the high modes of u should vanish much faster than those of u. In that case, Eq. (2) provides a basis for a simulation shortcut. The first outstanding approach in this direction goes back to Leray [2], who u) = C(u, u) and proved that a moderate filtering of the convective took C(u, velocity is sufficient to regularize a turbulent flow. Cheskidov et al. [3] have analyzed Leray’s approximation for a Helmholtz filter. They show that the complexity of the 3D Leray model lies between that of the 2D and 3D NavierStokes equations. The Navier-Stokes-alpha-model forms another example of regularization modeling, see for instance [4],[5]. In this model, the convective term becomes Cr (u, u) = Cr (u, u), where Cr denotes the convective operator in rotational form: Cr (u, v) = (∇ × u) × v. The regularization method basically alters the nonlinearity to control the convective energetic exchanges. In doing so, one can preserve certain fundamental properties of (the convective operator in) the Navier-Stokes equations exactly, e.g., symmetries, conservation properties, transformation properties, Kelvin’s circulation theorem, Bernouilli’s theorem, Karman-Howarth theorem, etc. [6]. In this paper, we propose to smooth C in such a manner that the symmetry properties that yield the invariance of the energy, the enstrophy (in 2D) and helicity are preserved. The underlying idea is to restrain the convective production of smaller and smaller scales of motion by means of vortex stretching, while ensuring that the solution does not blow up (in the energy-norm; in 2D also: enstrophy-norm). We anticipate that the unconditional stability enhances the accuracy at coarse resolutions. Here, it may be pointed out that the unconditional stability of C allows for simulations at arbitrary coarse grids, provided the discretization of C preserves the symmetry too, see for instance [7].

2 Symmetry-preserving smoothers Approximations of particular interest are the ones that conserve the energy, the enstrophy (in 2D) and the helicity in the absence of viscous dissipation, among others because they are intrinsically stable (in the energy-norm; in 2D: enstrophy-norm). Note: the Leray model conserves the energy, but not the enstrophy or helicity, whereas the Navier-Stokes-alpha model conserves the enstrophy and helicity, yet not the energy. The evolution of the energy follows from differentiating (u, u) with respect to time and rewriting ∂t u with the help of (1). In this way, we get a convective contribution given by the trilinear form (C(u, u), u). Since this form is skewsymmetric with respect to the last two arguments, that is

Symmetry-preserving regularization of turbulent channel flow

(C(u, v), w) = −(v, C(u, w))

145

(3)

(see e.g. [8]), we have (C(u, v), v) = 0 for any pair u, v, which implies that the convective contribution (C(u, u), u) cancels from the energy equation; hence the energy is conserved in the absence of viscous dissipation (D = 0). Similarly it may be shown that (3) yields helicity-conservation. The evolution of the enstrophy is obtained by taking the inner product of the Navier-Stokes equations with the vector field −Δu. The resulting convective contribution vanishes in two spatial dimensions [8]: (C(u, u), Δu) = 0 Actually, an even stronger form of enstrophy invariance holds [9]: (C(u, v), Δv) = (u, C(Δv, v)).

(4)

Since the conservation of energy, enstrophy (in 2D) and helicity are intimately tied up with the symmetry properties (3) and (4) of the convective operator C, we propose to approximate C in such manner that (3) and (4) are preserved exactly. This criterion yields the following class of approximations ∂t u + Cn (u , u ) + D(u ) + ∇p = 0,

(5)

(n = 2, 4, 6) in which the convective term is smoothened according to C2 (u, v) = C(u, v) C4 (u, v) = C(u, v) + C(u, v ) + C(u , v) C6 (u, v) = C(u, v) + C(u, v ) + C(u , v) + C(u , v )

(6) (7) (8)

Here a bar denotes a filtered quantity and a prime indicates the residual. The three approximations Cn (u, u) are consistent with C(u, u), where the error is of the order of n with n = 2, 4, 6 for symmetric filters with a filter length . Both the Leray model and the alpha model are second-order accurate in terms of . The approximations (6)-(8) inherit the skew-symmetry of C by construction. That is, for any filter satisfying (u, v) = (u, v), we have (Cn (u, v), w) = −(v, Cn (u, w)). They inherit the enstrophy invariance in 2D, (Cn (u, v), Δv) = (u, Cn (Δv, v)), provided the filter commutes with the Laplacian. Consequently, Eq. (5) conserves the energy, the enstrophy (in 2D) and the helicity if the filter satisfies both (u, v) = (u, v) and Δu = Δu.

3 Vortex stretching The evolution of the vorticity ω = ∇ × u of any solution u of Eq. (5), ∂t ω + Cn (u , ω ) + D(ω ) = Cn (ω , u ),

(9)

resembles that of the Navier-Stokes equations: the only difference is that C is replaced by the approximation Cn . The Navier-Stokes equations give the source term

146

Roel Verstappen

C(ω, u) = Sω = Sω + Sω + S ω + S ω where S = 12 (∇u term Cn (ω , u ) in

(10)

+ ∇u ) is the deformation tensor. The vortex stretching Eq. (9) reads T

C2 (ω, u) = Sω

(11)

C4 (ω, u) = Sω + Sω + S ω C6 (ω, u) = Sω + Sω + S ω + S ω

(12) (13)

Qualitatively, vortex stretching leads to the production of smaller and smaller scales; hence to a continuous, local increase of both S and ω . Consequently, at the positions where vortex stretching occurs, the terms with S and ω will eventually amount considerably to the right-hand side of (10). Since these terms are diminished in (11)-(13), the conservative smoothing of the convective term counteracts the production of smaller and smaller scales by means of vortex stretching and may eventually stop the continuation of the vortex stretching process. So, in conclusion, the approximations Cn (u, u) restrain the convective production of smaller and smaller scales of motion by means of vortex stretching, while ensuring that the solution can not blow up (in the energy-norm; 2D: enstrophy-norm).

4 Triadic interactions To study the interscale interactions in more detail, we continue in the spectral space, where we will restrict ourselves to the approximation C4 ; a similar analysis may be performed for C2 or C6 . Taking the Fourier transform of (5)+(7) yields the evolution of each Fourier-mode uk (t) of u : |k|2 d + uk +Ck (Gu, Gu)+ Gk Ck (Gu, (I − G)u)+ Gk Ck ((I − G)u, Gu) = 0, dt Re where Ck (u, u) denotes the spectral representation of the convective term in the Navier-Stokes equations and G represents the Fourier transform of our filter. The Navier-Stokes dynamics is obtained by taking G = I. The mode uk (t) interacts only with those modes whose wavevectors p and q form a trangle with the vector k. The local interactions between large scales (meaning that |k| < 1 and |k| ∼ |p| ∼ |q|) approximate the Navier-Stokes dynamics up to O 4 , i.e., the interactions between large scales are only slightly altered by the approximation C4 . In order to investigate interactions involving longer wave-vectors (smaller scales of motion), the filter need be specified further. Since a Helmholtz filter enables a plain analysis of the interactions, we consider Gk = (1 + α2 |k|2 )−1 with α2 = 2 /24. For this filter, the spectral representation of C4 becomes iΠ(k)

p+q=k

up q vq

1 + α2 (|k|2 + |p|2 + |q|2 ) (1 + α2 |k|2 )(1 + α2 |p|2 )(1 + α2 |q|2 )


147

where Π(k) = I − kk T /|k|2 denotes the projector onto divergence free velocity fields in the spectral space. By comparing this expression with the Navier-Stokes interactions, we see that all triad interactions are reduced by the application of the Helmholtz filter. The amount by which the triadic interactions are lessened depends on the length of the legs of the triangle k = p+q. The reduction is the largest for triangles with three long legs, i.e. α|k| > 1, α|p| > 1 and α|q| > 1. In general, we see that the approximation C4 (strongly) attenuates all interactions for which at least two legs of the triangle k = p + q are (much) longer than 1/α, whereas all possible triadic interactions for which at least two legs are (much) shorter than 1/α are reduced to a small degree. Since in the latter case the longest leg is always shorter than 2/α, we may conclude that the approximation C4 confines the dynamics for the greatest part to scales whose wavevector-length is smaller than 2/α. In this way, the resolution requirements resulting from the convective nonlinearity are reduced.

5 Results for a turbulent channel flow As a first step in the application of symmetry-preserving regularization, the approximation C4 is tested for a turbulent channel flow by means of a comparison with the direct numerical simulations performed by Kim et al. [10]. Based on the channel half-width and the friction velocity the Reynolds number is 180. The smooth approximations Cn given by Eqs. (6)-(8) are constructed such that fundamental properties (3) and (4) are preserved. Of coarse, the same should hold for the numerical approximations that are used to discretize Cn . Therefore, Eq. (5) is discretized as in Ref. [7]. We consider two, coarse, computational grids consisting of 16×16×8 and 32×32×16 grid points, respectively. The filtering is based upon the Helmholtz operator, where the boundary conditions that supplement the Navier-Stokes equations are applied to the filter too. Since solving the Helmholtz equation for u is rather expensive, we do not fully solve this equation, but choose to perform just one Jacobi iteration with u = u as initial guess. The least to be expected from a simulation shortcut is a good prediction of the mean flow. As can be seen in Fig. 1, the approximation C4 satisfies that minimal requirement already at the very coarse 16×16×8 grid, provided the filter length is taken equal to two-to-four times the grid width h. Yet, the turbulence intensities do not agree so well with the reference data if only 2048 gridpoints are used. Here, it may be noted that the root-mean-square velocity fluctuations are the least worse (in comparison to the DNS of Kim et al. [10]) if the results are extrapolated linearly to = 0. Overall good agreement between the C4 -calculation at the 32×32×16 grid and the DNS is observed for both the first- and second-order statistics, see Fig. 2. Heuristic arguments as well as computational results (Fig. 3) show that the energy spectrum of the solution of (5)+(7) follows the DNS for large scales of motion, whereas a much steeper (numerically speaking: more gentle) power law is found for small scales.

148

Roel Verstappen

20

filter length 0 h 2h 3h 4h

C4 (16x16x8)

DNS KMM [9]

10

0

1

10

100

urms

3

C4 (16x16X8) DNS KMM [9]

2 wrms 1 vrms 0

0

50

100

150

200

Fig. 1. The upper figure shows the mean velocity (in wall coordinates) for the 16×16×8 simulations with C4 . The filter length varies from zero (no regularization) to the four times the grid width h. The lower figure displays the root-mean-square of the fluctuating velocities. The open boxes and circles correspond with = 2.5h; the filled boxes and circles represent data that is extrapolated linearly to = 0.

The first results shown here illustrate the potential of symmetry-preserving smoothing as a new simulation shortcut for turbulent channel flow. Yet, given the inherent difficulty of turbulence modeling, more thorough investigations and comparisons need be carried out to clarify the pros and cons.


20

149

C4 (32x32X16) DNS KMM [9]

10

0

3

1

10

100

u rms

C4 (32x32X16) DNS KMM [9]

2

wrms

1 vrms 0

0

50

100

150

200

Fig. 2. Results of C4 with 32×32×16 grid points and = 1.5h.

References 1. Geurts BJ, Holm DD (2003) Phys Fluids 15:L13–16 2. Leray J (1934) Acta Mathematica 63:193–248 3. Cheskidov A, Holm DD, Olson E, Titi ES (2005) Proc Roy Soc London A: Math, Phys and Engng Sc 461:629–649 4. Holm DD, Marsden JE, Ratiu TS (1998) Adv in Math 137:1–81 5. Foias C, Holm DD, Titi ES (2001) Physica D 152:505–519

150

Roel Verstappen 0 C4 (16x16X8)

filter length 0 filter length 3h DNS

0.1

0.01

0.001

0.0001

1

10

0 C4 (32x32X16)

filter length 0 filter length 1.5h DNS

0.1

0.01

0.001

0.0001

1

10

Fig. 3. One-dimensional (streamwise) energy spectra at y + ≈ 5.

6. Geurts BJ, Holm DD (2004) In: Friedrich R et al. (eds.) Direct and Large-Eddy Simulation V, Kluwer, pp. 5–14 7. Verstappen RWCP, Veldman AEP (2003) J Comp Phys 187:343–368 8. Foias C, Manley O, Rosa R, Temam R (2001) Navier-Stokes Equations and Turbulence. Cambridge University Press 9. Vukadinovic J (2004) Nonlinearity 17:953–974 10. Kim J, Moin P, Moser R (1987) J Fluid Mech 177:133–166

Large-eddy simulations of channel flows with variable filter-width-to-grid-size ratios Ana Cubero1 and Ugo Piomelli2 1

2

Fluid Mechanics group, University of Zaragoza, Spain [email protected] Dept. of Mechanical Engineering, University of Maryland, College Park, MD, USA [email protected]

Summary. In this paper, decoupling the filter-width from the grid size is proposed in order to reduce the numerical errors arising in LES on complex geometries. A preliminary investigation of this approach is presented by simulations of spatially developing channel flows.

1 Introduction Most commonly, in large-eddy simulations (LES), the turbulent flow-field is filtered implicitly by the discrete grid and the filter width Δ to be included in the sub-grid scale (SGS) model is taken to be proportional to the grid size. In principle, however, the filter width is totally independent of the grid size: Δ represents the length-scale of the smallest eddy resolved, and the grid must be sufficiently fine to resolve this scale. This requirement results in a filter width that is larger than the grid size, at least by a factor of two. The larger this factor, the better the numerical resolution of the subgrid eddies (but the higher the cost of the calculation). Discussions of the relationship between filter width and grid size can be found in [1, 2, 3, 4, 5, 6] As LES is being applied to more complex geometries, more sophisticated numerical techniques are being used; these include embedded meshes, unstructured grids, or block-structured meshes with different resolutions across blocks. In all these cases, the choice of a filter-width strictly related to the grid size may result in numerical instabilities or unphysical behaviors, as the eddy viscosity becomes discontinuous across the interface between a coarse-grid and a finer-grid region [7, 8]. Numerical errors may arise if the grid is suddenly coarsened: the coarse mesh cannot support the smaller eddies convected into it from the fine-grid size, and aliasing errors arise; since the increased eddy viscosity due to the increased grid acts mostly on the smallest scales, while the errors are distributed throughout the spectrum, these errors will propagate into the flow, corrupting the results. Unphysical behaviors may also

152

Ana Cubero and Ugo Piomelli

occur when the flow goes from a coarse-grid region to a finer one: in this case a certain distance will be required to generate the small-scale eddies that the finer grid can support; in this transition region the eddy viscosity has decreased, but the Reynolds-stress supporting eddies have not yet been generated; the momentum balance, therefore, becomes incorrect and errors are again introduced. Several authors [9, 10, 11] have investigated the effect on the flow of the numerical errors introduced in the standard large-eddy equations when a variable filter width is used. First, Ghosal and Moin [9] studied this problem in bounded flows, where the grid must be refined close to walls in order to capture the small turbulent scales generated there. They showed that the effect on the flow of a hyperbolic tangent stretching is dissipative and no larger than the discretization error using a second-order finite difference. More recently, van der Bos and Geurts [11] extended this work to asymmetric filters and found that their effect is also dissipative. Furthermore, they show that commutator errors scale with the gradient of the filter width and can become comparable in magnitude with the SGS stress. These authors conclude that explicit modeling of the closure terms is unavoidable when LES with skewed filters and sharp variations is intended. The lagrangian dynamics of commutator errors are discussed by the same authors in a later paper [12]. One possible way to remove the problems described above is to decouple the filter width from the grid size. If the filter width is maintained constant across a discontinuity in the grid size (at a level consistent with the finer-grid resolution capabilities), for instance, the velocity discontinuity is removed at the cost of an over-resolution of the smallest resolved eddy. The filter-width can then be decreased or increased smoothly to a value consistent with the finer grid. Thus, the numerical errors due to variations in the filter-width can be controlled. A sharp jump in the resolution of the smallest scales resolved will, however, be introduced, which may result in numerical errors which will not be studied here. We report here the initial results of an investigation of this approach. We have performed simulations of a spatially developing channel flow on a structured grid, with filter width that increased or decreased in the streamwise direction, while the grid remained constant. In this way, we study uniquely the effects of using a filter-width decoupled from the cell size as a preliminary stage before applying the proposed approach to grids including sharp variations of the cell size. In the following, we will first present the formulation of the problem, and describe the numerical experiments carried out. We will then show and discuss the simulation results, and finish with some concluding remarks.

LES with variable filter-width-to-grid-size ratios

153

2 Numerical formulation We carry out LES of spatially developing channel flow at Reτ = 400, using a standard 2nd-order finite-difference scheme. The SGS stresses are parametrized using the Lagrangian Dynamic eddy-viscosity model [13]. A grid with 480 × 80 × 120 points in the streamwise (x), spanwise (y) and wall-normal (z) directions is used to discretize a computational domain of dimensions 8πδ × πδ × 2δ. Grid stretching is applied in the normal direction by a hyperbolic tangent function. The resultant cell sizes in wall units are: Δx+ = 19, Δy + = 14 and 0.29 ≤ Δz + ≤ 7.3. Periodic boundary conditions are used in the spanwise directions, no-slip conditions at the walls, at the outflow a convective condition is used [14]. At the inflow, planes of data from a separate calculation of fully developed channel flow (with periodic boundary conditions in x) are read at each timestep. To determine the effect of filter width decoupled from the grid size on the turbulent eddies and on the statistical data, a short distance downstream of the inflow the filter width is gradually increased or decreased in the streamwise direction. Note that the calculation of the Smagorinsky constant by the dynamic model requires two filtering levels, the test filter, Δ, and the grid filter, Δ. The filter-width enters the calculation only through the definition of the eddy viscosity 2 (1) νT = CΔ |S| (where |S| is the magnitude of the strain-rate tensor). On the other hand, the velocity field is explicitly filtered over the test-filter size to apply the Germano identity [15]. A test-filtered variable is defined as:

v(x, y, z) =

1 Δx Δy Δz

x+Δx /2

y+Δy /2

z+Δz /2

v(ξ, η, ζ)dξdηdζ; x−Δx /2

y−Δy /2

(2)

z−Δz /2

where the test-filter width in each direction varies with x according to Δi = 2f (x)Δxi (Δxi is the grid size), resulting in a conformal transformation of the filtering volume (see Fig 1). The integral is discretized and calculated by the trapezoidal rule. Linear interpolation of variables is used when the integration limits do not match with any grid node. In all the results presented in this paper, f (x/δ) is given by a hyperbolic tangent that ranges between 1 and 2: f (x/δ) = 0.5 tanh(x/δ − 11) + 1.5, for the increasing and f (x/δ) = 3 − 0.5 tanh(x/δ − 11) + 1.5), for the decreasing filter-grid ratio. The filter width for discrete filters was calculated according √ √ to [16]. Thus, the effective test-filter width ranges from 6Δg to 18Δg (where Δg = (Δx Δy Δz )1/3 ) The grid filter-width used in the eddy viscosity √ calculation is fixed at Δ = Δ/ 6; consequently, Δ varies between Δg and √ 3Δg .

154


22 21 20 19 18 17 16 15 14 13 12

5

10

15

20

x = 8.3 x = 10.9 x = 13.5 x = 15.6 Δ = Δg Δ=31/2Δg

Fig. 1. Illustration of the filter variation in the streamwise-normal plane. Increasing filter-width case.

100 z+

200

300 400

22 21 20 19 18 17 16 15 14 13 12

5

10

15

20

x = 8.3 x = 10.9 x = 13.5 x = 15.6 Δ = Δg Δ = 31/2Δg

z+

100

200

300 400

Fig. 2. Mean velocity profiles at various locations for increasing (left) and decreasing (right) filter-width.

3 Results In this section, results obtained from the variable filter approach described above are presented. The results will be compared with those obtained from two periodic calculations that use constant filter-widths equal respectively to the fine and coarse one. Figure 2 shows profiles of the mean velocity at several sections of the channel. For the increasing case (left) one observes a smooth transition from the constant-filter solution obtained with the coarser filter-width, Δ = Δg to that √ corresponding to 3Δg . The intercept of the logarithmic layer increases from the standard value of 5.2 to approximately 6.2. This behavior reflects the expected thickening of the near-wall layer associated with the larger filter width. Figure 3 supports this reasoning by presenting the streamwise development of the wall stress, τw , which decreases by approximately 3% as the filter-grid ratio increases (left). Inverse trends are observed when a decreasing filter width is applied. For both increasing and decreasing cases, the transition of the mean velocity is quite rapid, taking place over a length of roughly 5 channel half-

LES with variable filter-width-to-grid-size ratios

155

0.965 0.965 0.960 0.960 0.955

0.955

0.950 0.945

0.945 0.940

0.940

0.935

0.935 0.930

0.950

0.930 5

10

15

5

10

15

20

20

x/δ

x/δ

Fig. 3. Streamwise development of the wall stress. −

−

−

−

−

−

−

−

−

−

−

−

−

−

Fig. 4. Streamwise development of the SGS stress with an increasing (left) and decreasing (right) filter width in z + = 20, 58 and 199 planes.

heights. The wall stress develops more slowly and presents an undershoot or overshoot in the transition to the downstream equilibrium. The values of τw at the inlet and outlet are consistent with those of periodic calculations with the coarse and fine filter-widths, respectively. The streamwise development of the SGS shear stress, τ13 , at z + = 20, 58, and 199 is shown in Fig. 4. As expected, the absolute value of τ13 increases (or decreases) as the filter-width-to-grid ratio increases (or decreases), since larger (or smaller) scales have to be modeled. The effect of the variable filter width is more significant in the channel core than in the near-wall region. For an increasing filter-width in the downstream equilibrium region the SGS stress is increased by 50%, 100% and 150% in the z + = 20, 58 and 199 planes, respectively. Additionally, an overshoot is observed close to the wall (z + = 20) at the end of the region where the filter-width changes significantly. We conjecture that this overshoot may be due to insufficient damping of the scales of size comparable to Δg as the coarser-filter region is approached. According to the studies by van der Bos and Geurts [11], lengthening the region over which the transition takes place may decrease this phenomenon. Similar results are obtained for the decreasing filter-width case; a slight undershoot is observed at the location closest to the wall, somewhat less significant than in the increasing-filter case. Contours of the instantaneous velocity fluctuations in a plane close to the wall are shown in Fig. 5. One can observe a smoother flow, with fewer

156


Fig. 5. Contours of streamwise velocity fluctuations in the z + = 20 plane. z+ = 20

1.250 L y,uu

1.200 1.150

1.050

λ y,uu

1.100 1.050 1.000 0.950

2

4

6

8

10

12 x/δ

14

16

18

20

22

24

Integral and Taylor scales

Integral and Taylor scales

1.300

z += 20 1.000 0.950 0.900

λ y,uu

0.850 0.800 0.750

L y,uu 2

4

6

8

10

12 14 16 18 20 22

24

x/δ

Fig. 6. Development of the spanwise integral scale, Ly,uu , and Taylor micro-scale, λy,uu , of the streamwise velocity fluctuations in the z + = 20 (bottom) and z + = 58 (top) planes. Both length-scales have been normalized by their value at the inlet.

small-scale fluctuations in the regions of larger filter-width. The changes in scale of the turbulent eddies with the filter-width changes are an expected result, which is further illustrated in Fig. 6. This figure shows the evolution along the channel of the spanwise integral scale and Taylor micro-scale of the streamwise velocity fluctuations, Ly,uu and λy,uu . The integral scale has been defined as the spanwise distance where the autocorrelation, Ruu (r) = u (x, y, z)u (x, y + r, z) / u2 (where · denotes time-averaging), is reduced to 0.2. The transition to the downstream equilibrium is quite slow and the integral scale continues changing even at the end of the channel. Nevertheless, the change in the largest scales size is small, less than 25% in the cases shown. A similar change is observed in the Taylor micro-scale. Another way to judge the effect of the variable filter-width on the turbulent eddies is by considering the spectra. In Fig. 7 we show the development of the

1.0 0.8

k = 20

0.6

k = 35

0.4

<Euu/Euu(inlet)>

k=5

5

10

x/δ

15

20

+ 1.2 z = 58

k=5 1.0 0.8

k = 20

0.6 0.4

k = 35 5

10

15 x/δ

20

<Euu/Euu(inlet)>

<Euu/Euu(inlet)>

+ 1.2 z = 2 0

<Euu/Euu(inlet)>

LES with variable filter-width-to-grid-size ratios 2.4 + 2.2 z = 20 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 5

2.2 z+ = 58 2.0 1.8 1.6 1.4 1.2 1.0 0.8 5

157

k = 35 k = 20 k=5 10

x/δ

15

20

k = 35

k = 20

k=5 10

x/δ

15

20

Fig. 7. Development of the spanwise 1-D spectral coefficient of the streamwise velocity fluctuations Euu (normalized by the value at the inlet) for three representative wavenumber: k = 5 (large scales), k = 20 (intermediate scales) and k = 35 (small scales) in the z + = 20 (top) and z + = 58 (bottom) planes.

spanwise spectra of streamwise velocity fluctuations Euu for three wavenumbers which correspond to large, intermediate and small scales. As the filter width increases (or decreases), the dissipation introduced by the SGS model increases (or decreases) and the small and intermediate resolved scales contain less (or more) energy. The energy contained by largest scales, instead, is unaffected by the filter-grid ratio. A delay in the response to the variable filter-grid ratio is observed, which indicates that some history effects are felt by the smaller eddies. As discussed above regarding the overshoot in the development of the SGS stress, we expect that this effect could be reduced when the filter is spread over a wider region.

4 Conclusions In this work we have reported initial tests of LES that use a filter-width decoupled from the cell size, with the eventual aim to reduce the numerical errors that may arise close to grid discontinuities. This is an alternative approach to the explicit modelling of the additional closure terms that emerge in the standard LES equations when a variable (and, in particular, discontinuous) filter-width is used. We have performed a priori -like test calculations on a spatially developed channel. Information about the effect of using a variable filter-grid ratio on statistics and structures, in the near-wall region and in the channel core has been provided.

158


Both increasing and decreasing variable filter widths have been used. As expected, the wall stress and the magnitude of the SGS stress decreases, the smaller scales contain less energy and the size of the large scales is essentially unaffected as the filter width is increased. Inverse trends are observed for the decreasing filter-grid ratio. We also observed that the transition from one filter-width to the other often included overshoots or undershoots in some variables in the near-wall region. This effect may be related to the additional closure terms on the standard LES equations. According to the investigations of van der Bos and Geurts [11], this undesirable phenomenon may be decreased by reducing the gradient of the filter width. To conclude, a posteriori tests, including calculations in which grid discontinuities exist, are underway. Their results will be reported in a forthcoming paper.

Acknowledgments The stay of AC at Maryland was funded by the Ibercaja International Program of Researching Stays (Grant No. 282-116). UP was supported by the Office of Naval Research under Grant No. N00014-03-1-0491, monitored by Dr. Ronald D. Joslin.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Mason PJ, Callen NS (1986) J. Fluid Mech. 162:439–462 Gullbrand J (2002) CTR Res. Briefs 2002. CTR, Standford University. Chow FK, Moin P (2003) J. Comput. Phys. 184:366-380 Lund TS (2003) Comput. Math. with Appl 46(4):603-616 Meyers J, Geurts BJ, Baelmans M (2003) Phys. Fluids 15(9): 2740-2755 Meyers J, Geurts BJ, Baelmans M (2005) Phys. Fluids 17: 045108 Benhamadouche S., Uribe J, Jarrin N, Laurence D (2005) Fourth International Symposium on Turbulence and Shear Flow Phenomena. Camarri S, Salveti MV, Koobus B, Dervieux A (2002) Int. J. Numer. Meth. Fluids 40:1431-1460 Ghosal S, Moin P (1995) J. of Comput. Phys. 118:24-37 Iovieno M, Tordella D (2003) Phys. Fluids 15:1926-1937 van der Bos F, Geurts B (2005) Physics of Fluids 17:035108-035128 van der Bos F, Geurts B (2005) Physics of Fluids 17:075101-2005 Meneveau C, Lund TS, and Cabot WH (1996) J. Fluid Mech. 319:353-385 Orlanski I (1976) J. Comput. Phys 21:251-269 Germano M (1992) J. Fluid Mech. 238:325-336 Lund TS (1997) CTR Annual Res. Briefs 1997: 83-95. CTR, Standford University

LES of Transition to Turbulence in the Taylor Green Vortex Dimitris Drikakis1 , Christer Fureby2 , Fernando F. Grinstein3 , Marco Hahn1 , and David Youngs4 1

2

3

4

School of Engineering, Aerospace Sciences, Fluid Mechanics and Computational Science Group, Cranfield University, Bedfordshire, MK43 0AL, UK [email protected] FOI, Swedish Defence Research Agency, SE-164 90 Stockholm, Sweden [email protected] Naval Research Laboratory, Washington DC, USA; now at Los Alamos National Laboratory, MS-F699, Los Alamos, NM 87545, USA [email protected] AWE, Atomic Weapons Establishment, Aldermaston, Reading, Berkeshire, RG7 4PR, UK [email protected]

Summary. The paper presents an investigation of different conventional Large Eddy Simulation (LES) and Implicit LES approaches - Monotone Integrated LES (MILES) in particular, for the Taylor-Green Vortex (TGV), which features transition to turbulence. The LES models are examined with respect to their ability in allowing to simulate the two basic empirical laws of turbulence, namely the existence of an inertial subrange on the energy spectra for sufficiently high Reynolds number (Re) and the finite (viscosity-independent) energy dissipation limit law.

1 Introduction A grand challenge for Computational Fluid Dynamics (CFD) is the modeling and simulation of the time evolution of the fully non-linear turbulent flow in and around realistic engineering applications. For such flows it is unlikely that we will have in the foreseeable future a really-deterministic predictive framework based on CFD, because of the inherent difficulty in modeling and validating all the relevant physical sub-processes, and acquiring all the necessary and relevant initial and boundary conditions information. The modeling challenge is to develop computational models that although may not be explicitly incorporating all dynamic eddy scales of the flow will still give accurate and reliable results for at least the large energy-containing scales of motion. 4

The names of the authors appear in alphabetical order.

160

Drikakis et al.

The computational challenge is thus to solve the equations of the computational model as accurately as possible. The current drive is towards Large Eddy Simulations (LES) in which the large energy containing structures are resolved, whereas the smaller, more isotropic, structures are filtered out and, therefore, their effects need to be modeled. This gives LES a much higher generality than Reynolds Averaged Navier Stokes (RANS) models, where the entire turbulent spectrum is modeled. Different approaches are available for deriving the LES equations and the associated Sub Grid Scale (SGS) models required to handle the effects of the unresolved flow physics. In conventional LES the Navier-Stokes (NS) equations are filtered by convolving all dependent variables with a predefined filter in order to extract the large scale components, see e.g. [11] for a recent survey. In the MILES approach (see [12, 13], for recent surveys), the effects of the SGS physics on the resolved scales are incorporated in the functional reconstruction of the convective fluxes using high-resolution methods (as defined by Harten [14]). In practice, this amounts to using either monotonic or total variation diminishing (TVD) numerical methods, and the Monotone Integrated LES (MILES) technique based on monotone high-resolution methods provides an efficient implicit LES framework. Modified equation analysis indicates that the leading truncation-error terms introduced by such methods provide implicit SGS models of mixed anisotropic type [15, 16]. Major properties of the implicit SGS model are related to: (i) the choice of highand low-order schemes - where the former is well-behaved in smooth flow regions, and the latter is well-behaved near sharp gradients; (ii) the choice of flux-limiter which determines how these schemes should be blended locally, depending on prescribed characterization of the flow smoothness; (iii) the balance of the dissipation and dispersion contributions to the numerical solution, which strongly depends on the design details of each numerical method. The Taylor-Green Vortex (TGV) is a fundamental case that has been traditionally used as prototype for vortex stretching and the consequent production of small-scale eddies, to investigate the basic dynamics of transition to turbulence [9]. In what follows, we report on the performance of MILES based on various limiting schemes to examine the space/time development of the TGV flow problem [8, 9]. Comparisons are established with a conventional LES method and available DNS data.

2 Theoretical Formulation and Numerical Procedures The mathematical model consists of the conservation and balance equations of mass, momentum, and energy for a nominally-inviscid or linear viscous fluid (i.e., based on Euler or NS equations, respectively). As already alluded to, both MILES and conventional LES models have been used to investigate the TGV problem. In the conventional LES model the SGS modeling is performed by means of a Mixed Model (MM) [24]. This means that the subgrid stress

LES of Transition to Turbulence in the Taylor Green Vortex

161

tensor is decomposed into Leonard, cross and Reynolds stresss tensors, of which the Leonard stress tensor does not need closure modeling, and that the combined effects of the cross and Reynolds stress tensors are modeled using a conventional eddy viscosity model. Here, the One Equation Eddy Viscosity Model (OEEVM) of Schumann [1] was chosen. For further details on this model and some applications we refer to [2]. MILES was tested on this TGV case using various limiting algorithms, including Flux Corrected Transport (FCT), Characteristics-Based (CB) Godunov, and a Lagrange Remap (LR) method. The FCT schemes considered involved the standard 4th order FCT algorithm [6] and the 4th order 3D monotone limiter FCT [17]. Furthermore, a 2nd order scheme using hybridization of first-order upwind and second order central differences, was employed; depending on the choice of limiter, different algorithms were tested in this context, including, the Gamma limiter [3], the minmod limiter [4], the vanAlbada limiter [5] and the FCT limiter [6]. The flow code using the 4th order FCT is a compressible Euler-based code whereas the code using the 2nd order schemes is NS-based and the same as used for the conventional LES calculations described above. The other two compressible codes employed here are Euler -based and use the CB and LR methods. CB schemes [18, 19] are used in the context of Godunov-type schemes [21, 20] and hybrid TVD schemes [19]. The fluxes are discretized at the cell faces using the values of the conservative variables along the characteristics. Third-order variants of the fluxes can be obtained through flux limiting, based on the squares of second-order pressure or energy derivatives. Examples of flux limiting in connection with the CB scheme can be found in [18, 20]. The LR method [22] uses a non-dissipative finite difference method plus quadratic artificial viscosity in the Lagrange phase, and a 3rd order van Leer monotonic advection method [23] in the remap phase.

3 Problem Specification The TGV configuration considered here involves triple-periodic boundary conditions enforced on a cubical domain with box side length 2π cm using 643 or 1283 evenly spaced computational cells (only the 1283 results are included here, due to space constraints). Note that the same grid was used with all computer codes employed in this study. The flow is initialized with the solenoidal velocity components, u = uo sin(x)cos(y)cos(z), v = −uo cos(x)sin(y)cos(z) and w = 0, and the pressure given by a solution of the Poisson equation for the above given velocity field, i.e., p = po +(ρu2o /16)[2+cos(2z)][cos(2x)+cos(2y)], where we further select po = 1.0 bar, mass density, ρ = 1.178kg/m3 , uo = 100m/sec (corresponding to a Mach number of M = 0.28), and an ideal gas equation of state for air. Using truncated series analysis techniques, Morf et al. [7], an inviscid instability for the TGV was identified with estimated onset at a non-dimensional time t∗ = kuo t = 5.2, where the wavenumber k is

162

Drikakis et al.

unity here. These results were later questioned in Brachet et al. [9], where it was pointed out that accuracy in the analytic continuation procedure used by Morf et al. deteriorates too quickly to lead to a definite conclusion regarding their early prediction. Further estimates of t∗ based on the DNS of Brachet et al. [9], and Brachet [8], reported a fairly consistent dissipation peak at t∗ ∼ 9, for Re = 800, 1600, 3000, and 5000, where Re = kuo /ν (= 1/ν in [9]). The almost indistinguishable results for Re = 3000 and 5000 (e.g., Figure 1), suggested that they may be close to a viscosity independent limit. [10] 0.02 0.018 0.016

−dK/dt

0.014

DNS Re = 400 DNS Re = 800 DNS Re = 1600 DNS Re = 3000 DNS Re = 5000

0.012 0.01 0.008 0.006 0.004 0.002 0

2

4

6

8

10

t*

Fig. 1. DNS results for the kinetic energy dissipation at different Reynolds numbers [8, 9].

4 Results and Discussion By design, MILES emulates the dynamics of convectively-dominated flows characterized by high (but finite) Re ultimately determined by the nonvanishing residual dissipation of the numerical algorithms. The focus here is on comparing the trends of the MILES and LES predictions with those from available DNS. The evolution in time of the kinetic energy K (normalized with its value at t = 0) and kinetic energy dissipation, −dK/dt (m2 /s3 ), where K = 1/2 < u2 > and denotes mean (volumetric average), is demonstrated in Figure 2 for the 1283 resolution, for FCT, CB, LR, and the conventional LES, using the mixed model of Bardina et al. [24]. Fastest K decay at the dissipation peak (and peak mean enstrophy, not shown here) corresponds to the onset of the inviscid TG instability at t∗ ∼ 9. The observed agreement between MILES, Mixed-Model LES, and DNS is quite good in predicting both the time and height of the dissipation peak for the large-Re limit in Fig. 1 suggested by the DNS results for Re= 1600, 3000, and 5000. While some sort of Re-independent regime seems to be asymptotically attained with MILES with increasing grid resolution (not shown due to

LES of Transition to Turbulence in the Taylor Green Vortex

163

Fig. 2. Kinetic energy (top) and Kinetic energy dissipation (bottom) using conventional LES and MILES models (see text for description).

space constraints), actual values of Re characterizing the flow at the smallest resolved scales are not a priori available in MILES as in well-resolved DNS. Lower observed characteristic t* at dissipation peaks (as well as wider peaks) are predicted by coarser-grid (643 ) MILES ; this trend is consistently exhibited by the DNS results as Re is lowered (cf. Fig. 1. This suggests that it might be possible to parametrize MILES in terms of a characteristic effective Re that increases with grid resolution. This possibility will be addressed in future work. Some obvious differences between the present MILES and conventional LES compared to the DNS, such as the additional structure near t∗ ∼ 5 − 6 in Figure 2 not present in the DNS results, and other observed finer structure in the details of the results are attributed to specifics of the various limiting algorithms and/or their implementation, and a more systematic analysis of these features will be reported separately. For example, the double-peaked structure of the dissipation near t∗ ∼ 9 predicted by the LR method is likely due to the dispersive properties of this scheme compared to the less dispersive FCT and CB schemes. On the other hand, the LR method is also the least dissipative of the compared methods, especially at the early stages of the flow development; exhibiting relatively slower decay rate of K. The slight (but unphysical) kinetic energy increase of the early-times MILES CB is due to the particular conditions used in the present set of CB runs, i.e., high value of the Courant number in conjunction with the dual time-stepping explicit algorithm. The fluid dynamics underlying the dissipation results shown above is analyzed in what follows. Figure 3 compares instantaneous flow visualizations ranging from the initial TGV at t∗ = 0, the transition to increasingly smallerscale (but organized) vortices (top row), to the fully developed (disorganized) decaying worm-vortex dominated flow regime (bottom row), as characteristic of developed turbulence. The particular simulation was the one run using the 4th order 3D monotone FCT on the 1283 grid. The flow visualizations are based on (ray tracing) volume renderings of the second-largest eigenvalue of

164

Drikakis et al.

the velocity gradient tensor [25]), where hue and opacity maps were chosen the same for all times, except for peak magnitude values (normalized by value at t∗ = 0), indicated at the lower right of each frame of Figure 3. As the size of the smallest scale structures approaches the cutoff resolution, kinetic energy is removed at the grid level through numerical dissipation.

Fig. 3. Instantaneous flow visualizations ranging from the initial TGV at t∗ = 0, the transition to increasingly smaller-scale (but organized) vortices (top row), to the fully developed (disorganized) decaying worm-vortex dominated flow regime (bottom row), as characteristic of developed turbulence.

Corresponding 3D velocity spectra are shown in Figure 4. The peak in the velocity spectra around the k = 3 wave-number shell reflects the imprint leftover by the chosen initial (TGV) conditions (t∗ = 0). Higher wave-number modes are populated in time (t∗ = 2.2), through the virtually inviscid cascading process. As the size of the smallest scale structures approaches the cutoff resolution, kinetic energy is removed at the grid level through numerical dissipation. Figure 4 shows that the spectra consistently emulate a (-5/3) power law inertial subrange and self-similar decay for the times t∗ > 6. As the Kolmogorov spectra becomes more established in time (t∗ > 15 in Figure 4), it is associated with the (more disorganized) worm-vortex dominated flow regime (bottom row of Figure 3) - as characteristic of developed turbulence (e.g., Jimenez et al. [27]). The depicted self-similar decay in Figure 4 suggests that the removal of kinetic energy by numerical dissipation may occur at a physically suitable rate. The decay rates were examined with more detail based on selected representative 1283 data and with respect to slopes corresponding to power laws with exponents −1.2 and −2 through the mean value of K at the observed dissipation peaks (see Figure 2). Despite the unavoidable degree of subjectivity introduced by choice of origin times when such power-law fits are attempted,

LES of Transition to Turbulence in the Taylor Green Vortex time

165

t* = 0.0 t* = 2.2

3D Velocity Spectra

1E+08

t* = 6.7

k −5/3

t* = 8.9 t* = 14.7

1E+07

t* = 36.3 t* = 49.4 t* = 62.8

1E+06

1

10

3 1/2 k

Fig. 4. Evolution in time of the 3D velocity spectra (units on vertical axis are the same for all spectra, but are otherwise arbitrary).

all compared methods showed decay rates consistent with each other and with the −1.2 law for times immediately after that of the dissipation peak at t∗ ∼ 9, and with the −2 exponent for much later times. The power law with -1.2 exponent, is the one generally accepted as characteristic of decaying turbulence, whereas the later −2 exponent can be understood in terms of the expected saturation of the energy containing length scales reflecting that eddies larger than the simulation box side length cannot exist [26]. Differences between the LR results and the other schemes employed were observed and these were attributed to the Lagrange step that tends to be less dissipative and more dispersive than the FCT and Godunov-type schemes.

5 Conclusions We have examined the behavior of different LES and MILES approaches in the context of transition and decaying turbulence in the TGV. The results show that all the high-resolution schemes employed here can provide stable and acceptable (in terms of accuracy) solutions without resorting to an explicit SGS model using relatively coarse grids. The results also show that the kinetic energy dissipation rate does depend on the details of the numerical scheme employed (and its particular associated implicit SGS model). Therefore, even though implicit LES based on high-resolution methods provides a fairly robust computational framework for LES, there is plenty of room to achieve improvements on MILES performance based on better understanding of the specific dissipation and dispersion properties of the different high-resolution schemes. In particular, further such investigations are clearly warranted in order to gain better insights into the accuracy (and computational behavior in general) of MILES in relation to LES.

166

Drikakis et al.

Acknowledgment H.G. Weller is acknowledged for the development of the C++ class library FOAM (Field Operation And Manipulation), version 1.9.2β, partly used in this study. Support of one of us (FFG) from ONR through NRL is greatly appreciated.

References 1. Schumann, U., (1975) J. Comput. Phys. 18, 376. 2. Fureby C., Bensow R., Persson T. (2005) Turbulent shear Flow Phenomena IV, p 1077. 3. Jasak H., Weller H.G., Gosman A.D. (1999), Int. J. Numer. Meth. Fluids, 31, 431. 4. Roe P.L. (1985), Lectures in Applied Mathematics, 22, 163. 5. van Albada, G.D., van Leer, B., Roberts, W.W. (1982) Astron. Astrophys. 108, 76. 6. Boris J.P., Book D.L. (1973) J. Comp. Phys, 11, 38. 7. Morf R.H., Orszag S.A., Frisch U. (1980) Phys. Rev. 44, 572. 8. Brachet M.E. (1991) Fluid Dynamics Research, 8, 1. 9. Brachet M.E., Meiron D.I., Orszag S.A., Nickel B.G., Morg, R.H., Frisch, U., (1983) J. Fluid Mech., 130, 411. 10. Frisch, U., Turbulence, Cambridge University Press, Cambridge, 1995, Chapter 5, p 57. 11. Sagaut, P. (2001) Large Eddy Simulation for Incompressible Flows, Springer Verlag. 12. Drikakis, D. (2003) Progress in Aerospace Science, 39, 405-424. 13. Grinstein, F.F. and Fureby, C. (2004) Computing in Science and Engineering, 6: 37-49. 14. Harten, A. (1983) High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 1983; 49:357–93. 15. Fureby C. and Grinstein F.F. (1999) AIAA. J., 37, p 544. 16. Fureby C. and Grinstein F.F. (2002) J. Comp. Phys. 181, p 68. 17. DeVore C. R. (1998) , Naval Research Lab. Report NRL-MR-6440-98-8330. 18. Zoltak J., Drikakis D. (1998) Comp. Meth. in Appl. Mech. and Eng., 162, 165. 19. Bagabir A., Drikakis D. (2004) Comp. Meth. in Appl. Mech. and Eng, 193, 4675. 20. Drikakis D., Rider W. (2004) High-Resolution Methods for Incompressible and Low-Speed Flows, Springer Verlag. 21. Eberle A. (1987) VKI Lecture Series, Computational Fluid Dynamics, Report 1987-04. 22. Youngs, D.L. (1991) Phys. Fluids A3, 1312. 23. van Leer, B.(1977) J. Comp. Phys, 23, 276. 24. Bardina J., Ferziger J.H., Reynolds W.C. (1980) AIAA Paper No. 80-1357. 25. Jeong J., Hussain F. (1995) J. Fluid Mech., 285, 69. 26. Skrbek, L.. Stalp, S.R. (2000) Phys. Fluids 12, p 1997. 27. Jimenez J., Wray A., Saffman P., Rogallo R. (1993) J. Fluid Mech., 255, 65.

Stochastic SGS modelling in homogeneous shear flow with passive scalars Linus Marstorp, Geert Brethouwer and Arne Johansson Department of Mechanics, Royal Institute of Technology, KTH [email protected], [email protected], [email protected]

Summary. A new stochastic Smagorinsky model for the subgrid stress and subgid scalar flux is proposed. The new model is applied in LES of rotating homogeneous shear flow, which is an excellent case for developing and testing subgrid scale models. The proposed model provides for backscatter of energy and scalar variance, and reduces the length scale of the subgrid dissipation compared to the standard Smagorinsky model. At the same time, the flatness factor of the subgrid dissipation obtained from the stochastic model is of the same order of magnitude as for the Smagorinsky model.

1 Introduction Accurate descriptions of turbulent flows including passive scalar mixing are desirable in many engineering applications and geophysical situations. A passive scalar is mixed by the flow field, but it has no effect on the flow. It can represent a pollutant carried by the flow or small temperature fluctuations. Many approaches to develop subgrid models in LES have been proposed in the past. The use of stochastic processes in subgrid modelling in LES has been treated by several authors, e.g. Leith [6] and Schumann [9] who both focused on how to obtain a correct description of the backscatter of turbulent kinetic energy. Alvelius and Johansson [1] showed that the Smagorinsky model predicts a too large length scale of the subgrid dissipation and proposed a new stochastic model that solved this problem. The purpose of this study is to follow the approach of Alvelius and Johansson to develop a new stochastic model for the subgrid scales of the velocity field and validate this new model. At the same time, we apply the idea of stochastic modelling also to the subgrid scales of a passive scalar field.

168

Linus Marstorp, Geert Brethouwer and Arne Johansson

2 Stochastic subgrid modelling 2.1 Filtered equations and existing models The filtered incompressible Navier-Stokes and passive scalar equations in a rotating frame reads (ijk is the permutation tensor) 1 ∂ p˜ ∂τij ∂u ˜i ∂u ˜i +u ˜j =− + ν∇2 u ˜i − 2ijk Ωj u ˜k − ∂t ∂xj ρ ∂xi ∂xj ˜ ˜ ν 2 ˜ ∂qj ∂u ˜i ∂θ ∂θ +u ˜j ∇ θ− = , =0 ∂t ∂xj Pr ∂xj ∂xi where u ˜i and θ˜ denotes the filtered velocity and passive scalar respectively, and where p˜ includes both the pressure and the centrifugal force. Ωi is the system rotation vector and P r is the Prandtl number. The widely known Smagorinsky model for the unclosed subgrid stress, τij , and subgrid flux, qi , has several known drawbacks. Two examples are that it does not provide for backscatter, neither of turbulent kinetic energy, nor of scalar variance, and that it over-predicts the subgrid dissipation length scale. In the dynamic Smagorinsky model, due to Germano [5], the model constant is determined from local flow conditions. The dynamic model provides for backscatter but the model constant yields too large fluctuations and it can easily become unstable. Applying averaging in homogeneous directions to obtain the constant eliminates the stability problem but the model looses generality and the ability to account for backscatter. 2.2 A stochastic model The idea behind the present stochastic subgrid model is to model the random behaviour of the subgrid stress and flux by a stochastic process, which can decrease the correlation length scale of the smallest resolved scales, and provide for backscatter. The new stochastic model is based on the widely known Smagorinsky model for the unclosed subgrid-scale stress tensor, j − θ˜ ˜uj , which reads ˜i u ˜j , and subgrid-scale flux, qj = θu τij = u i uj − u τij = −2νT S˜ij ,

qj = −

νT ∂ θ˜ P rT ∂xj

where the turbulent Prandtl number, P rT , is a constant and νT = (Cs Δ)2 |S˜ij | is the eddy viscosity. Cs is the Smagorinsky constant, Δ is the filter width, and S˜ij denotes the filtered, or resolved, rate of strain. Similar to Alvelius and Johansson [1], we model the eddy viscosity as the sum of the Smagorinsky model and a stochastic term νT = Cs2 (1 + X(t))Δ2 |S˜ij |

Stochastic SGS modelling in homogeneous shear flow with passive scalars

169

The part corresponding to the Smagorinsky model generates the right amount of mean dissipation whereas the stochastic part creates realistic SGS fluctuations. X(t) is obtained from independent Ornstein-Uhlebeck processes at each spatial point dX(t) = aX(t)dt + bdW √ where a and b are constants and dW has the normal distribution N (0, dt). X(t) has zero mean, E[X(t)] = 0, and constant variance, V [X(t)] = b2 /2a. The time scale of the process can be characterised by the decay rate of the correlation E[X(t)X(t − t )]/VX = exp(−at ). It follows that the time scale τ sgs = 1/a decreases with increasing values of a. The length scale, Δ, of the subgrid field is known, and it can be used to estimate τ sgs . From dimensional arguments we have 1/3 τ sgs = C Δ2 / Π where C is a constant of order 1 and Π = −τij S˜ij is the subgrid dissipation. Hence, stochastic noise which has a correlation time about as long as the time scale of the subgrid velocity field, and which is uncorrelated in space, enters both the subgrid stress and the subgrid flux through the eddy viscosity. The eddy viscosity incorporated in the eddy diffusivity model for the subgrid scalar flux is the same as that for the subgrid stress.

3 Simulations The performance of the new model was tested in rotating homogeneous shear flow, which is an excellent case for developing and testing subgrid scale models. The simple geometry of the flow enables the use of the accurate spectral methods which makes it easy to separate SGS model features from the numerical errors. In addition the recent 1536 × 1280 × 1024 grid-point DNS by Brethouwer and Matsuo [2] provides for very good reference data with the opportunity to perform a priori tests and validation of the simulations in the future. The filtered incompressible Navier-Stokes equations with a constant uniform shear, Ui = Sx3 δi1 , and the passive scalar equation with a mean scalar gradient, Gi = δi3 , were solved using a pseudo spectral code, with a third order Runge-Kutta method for time advancement. Rogallo’s method has been used to simulate homogeneous shear flow, i.e. the grid moves along with the mean flow to enable the use of periodic boundary conditions, and it is re-meshed periodically. Some information is lost during the re-meshing. However, the losses are not significant for the evolution of the flow. LES with 1283 gridpoints were performed in a periodic box with the dimensions 4π × 3π × 2π with three different subgrid models; the standard Smagorinsky model, the dynamic model as defined by Lilly [7] (both with model constant averaging in all homogeneous directions, and with clipping of large negative values), and

170


the new stochastic model described above. The model parameters were chosen as a = 1/τ sgs , b = 4, and C = 1.0. These parameters may not be optimal and they can be improved by a priori and a posteriori test with DNS data, in the future. The parameter C should, however, be of order one to assure that 1/a represents a time scale of the SGS velocity field. Fully developed turbulence obtained from isotropic decay was used as initial condition and the flow is rotating about the spanwise direction, Ωi = Ωδi2 , at the non-dimensional rotation numbers R = 2Ω/S = 0, −1/2, and −1. The initial non-dimensional ˜ = 3.38, and the turbulent Reynolds number shear rate was chosen as S K/˜ 2 ˜ ν) = 33000. The turbulent length scales grow rapidly and was ReT = K /(˜ therefore the initial integral length scales must be small. The relatively high resolution (1283 ) was needed in order to resolve the initial turbulence.

4 Results 4.1 Large scale statistics ˜ is Figure 1 shows the time development of the turbulent kinetic energy. K ˜ strongly destabilised at R = −1/2. At R = −1 K still grows but at much slower rate than at R = 0. These observations agree well with the DNS data by Brethouwer and Matsuo [2] and show that all models produce the right amount of mean dissipation. The growth has a pronounced exponential part, ˜ =K ˜ 0 eαSt at R = 0 and R = −1/2. At R = 0 the exponential growth rate K is in agreement with the experiment by Tavolaris and Corrsin [10]. Cs = 0.10 is used in the standard Smagorinsky and stochastic model, which is equal to the value Cs ≈ 0.10 predicted by the dynamic model. This value is also close to the value Cs = 0.11 suggested by Canuto et al. [3], for homogeneous shear ˜ (and the Reynolds stresses) are very similar flow. The time development of K for the LES with the stochastic model and the dynamic model according to the figure. The results for the Smagorinsky model and stochastic model were in fact the same. Next, we turn our attention to the scalar mixing. The direction of the turbulent flux is defined as ˜3 / θ˜ u ˜1 αf = atan θ˜ u where u ˜i and θ˜ denote the fluctuating velocity and scalar components, and ˜1 and θ˜ u ˜3 are the mean turbulent scalar fluxes in the x1 and where θ˜ u x3 direction, respectively. In figure 2 the development of αf is seen to depend on the rotation number. The angle αf in the LES with the standard, dynamic, and stochastic Smagorinsky model approached the same equilibrium values and the results were found to yield good agreement with DNS results of Brethouwer and Matsuo, and Rogers et al. [8]. Also here, the results of the


171

stochastic model are very close to those of the Smagorinsky model. The ratio of the mean turbulent diffusivity to the molecular diffusivity is about 500. 0

R = 1/2 101

R=0 30

R=0

αf

K/K0

R = 1/2 60

R=1 100

R=1 90 0

5

St

10

15

Fig. 1. The time history of the turbulent kinetic energy. Dynamic model, dashed line; stochastic and Smagorinsky model, solid line.

0

5

10

15

St

Fig. 2. The direction of turbulent scalar flux. Dynamic model, dashed line; stochastic and Smagorinsky model, solid line.

4.2 Backscatter Despite the small differences in large scale statistics in the LES with the standard Smagorinsky and stochastic model, there are significant differences at the smaller scales. The stochastic process adds a stochastic part to the subgrid dissipation, which allows for backscatter Πsto = XCs2 Δ2 |S˜ij |3 Backscatter is local energy transfer from the SGS into the resolved scales, see Cerutti and Meneveau [4]. In the present model backscatter is represented by intermittently negative eddy viscosity. This is of course only a very simple model for the real backscatter phenomena, assuming that the SGS stress is aligned with the resolved rate of strain. In figure 3 we see that the PDF of the stochastic subgrid dissipation is symmetric about its mean value Π = 0. The backscatter variance can be adjusted by adjusting b. A priori and a posteriori tests can be used to find the proper value. The Smagorinsky part of Π is without any backscatter of turbulent kinetic energy and has a positive mean value. The combination of the stochastic part and the Smagorinsky part thus accounts for a mean energy flux and at the same time backscatter. The PDF of the total subgrid dissipation in figure 4 is non-symmetric about zero, which is its most probable value. The shape of the PDF resembles of that of Cerutti and Meneveau [4], who extracted the PDF of the exact subgrid dissipation from DNS data of isotropic turbulence. The stochastic model accounts also for

172


backscatter of scalar variance in the same manner as for the turbulent kinetic energy. This is shown in figure 4, where the PDF of the total scalar variance ˜ dissipation, Q = −2qi ∂ θ/∂x i is plotted. Negative eddy viscosity has to be treated with care. The stochastic model predicts locally negative total viscosity (νT + ν) and can be unstable under some circumstances. A large time scale, τsgs , and a large variance of the stochastic process increases the probability for numerical instability. For the present choice of parameters and initial conditions it was not necessary to clip negative values of subgrid dissipation due to the short time scale of the backscatter. Locally negative total viscosity is not dangerous as long as it only occurs during short periods of time. 100 PDF of Π / and Q/

PDF of Π/

100 101 102 103 104 10

5

0

5 10 Π/

15

20

Fig. 3. PDF of the subgrid dissipation according to the stochastic Smagorinsky model. Smagorinsky part, dashed line; stochastic part, solid line.

101 102 103 104 0

5

10 15 20 Π/ and Q/

25

Fig. 4. PDF of the total subgrid dissipation, solid line. PDF of the total scalar variance dissipation, dasheddotted line.

4.3 Subgrid dissipation length scale We chose to study the length scale of the subgrid dissipation at the most destabilised rotation number R = −1/2 where resolved length scales grow very fast. The length scale of the subgrid dissipation is computed from the correlation lx LΠ = Π (x0 ), Π (x0 + x) dx 0

where lx is half box length in the the streamwise direction, and Π = Π− Π is the fluctuating part of the subgrid dissipation. It can be seen from figure 5 that the stochastic part of the subgrid dissipation decreases the length scale. For the present choice of parameters the the length scale is reduced by approximately 20% compared to the Smagorinsky model. The reduction is similar for the scalar variance dissipation.


173

4.4 Intermittency of SGS dissipation The flatness factor of the subgrid dissipation F =

(Π − Π )4 (Π − Π )2 2

is a measure of the intermittency. Large values indicate high intermittency. Cerutti and Meneveau [4] compared the flatness factor of the subgrid dissipation predicted by various subgrid stress models from a velocity field obtained from DNS. They found that the dynamic model without spatial averaging is too intermittent and that the Smagorinsky model is about as intermittent as the real subgrid dissipation. The intermittency, at R = 0, extracted from the LES of the stochastic model, the clipped dynamic model (Cs2 > −0.01), and the standard Smagorinsky model are plotted in figure 6. One can see that the flatness of the stochastic model is of the same order of magnitude as for the standard Smagorinsky model whereas the intermittency of the clipped dynamic model is much too large. Hence, the stochastic Smagorinsky model provides for backscatter without being too intermittent. Both the dynamic model and the stochastic model allow locally negative total viscosity. The reason why the result of the dynamic model is more intermittent is the time scale of the backscatter rather than the negative viscosity itself. 105 3 104

F

LΠ / Δx

2.5

2

103

1.5

102

1 1

2

3

4

5

6 St

7

8

9

10 11

Fig. 5. The time development of the length scale of the subgrid dissipation. Stochastic model, solid line; standard Smagorinsky model, dash-dotted line.

101

2

4

6

8

10

12

14

St

Fig. 6. The flatness of the subrid dissipation. Clipped dynamic model, dashed line; stochastic model, solid line; standard Smagorinsky model, dash-dotted line.

5 Conclusions LES of rotating homogeneous shear flow with a passive scalar was performed. Three different subgrid models were used: the standard Smagorinsky model,

174


the dynamic Smagorinsky model, and a newly developed stochastic model. The subgrid models had a small influence on the large scale velocity and scalar statistics, but a large effect on the smaller scales. The proposed stochastic Smagorinsky model was shown to reduce the length scale of the subgrid energy and scalar variance dissipation and provide for backscatter, which may be a promising feature for the development of improved subgrid scale models for reacting flows. In that case the properties of the subgrid scales are very important. The LES with the dynamical model using spatial averaging did not provide for backscatter. With clipping of large negative values of the dynamic model constant instead of spatial averaging the dynamic model accounts for backscatter, but due to long correlation time of the negative eddy viscosity the intermittency of the dissipation was very high. The time scale of the stochastic backscatter of the proposed model is adjustable and for the present choice of parameters the intermittency of the subgrid dissipation was of the same order of magnitude as for the standard Smagorinsky model and realistic. However, it is necessary to tune the model parameters and determine their dependence on the filter length scale. The DNS by Brethouwer and Matsuo can be used for a priori and a posteriori test.

References 1. Alvelius K, Johansson A V (1999) Stochastic modelling in LES of a tubulent channel flow with and without system rotation. Doctoral Thesis, Department of Mechanics KTH Sweden 2. Brethouwer G, Matsuo Y (2005) DNS of rotating homogeneous shear flow and scalar mixing. Proc. 4th Int. Symp. on Turbulence and Shear Flow Phenomena (TSFP4), Williamsburg, USA. Editors: J A C Humphrey et al. 3. Canuto V M, Cheng Y (1997) Determination of the Smagorinsky-Lilly constant Cs Phys. of Fluids 9(5) pp. 1368-1378 4. Cerutti S, Meneveau C (2001) Intermittency and relative scaling of subgridscale energy dissipation in isotropic turbulence. Phys. Fluids 10(4) pp. 928-937 5. Germano M, Piomelli U, Cabot H (1991) A dynamic subgrid-scale eddy viscosity model. Phys. of Fluids A 3(7):1760-1765 6. Leith C E (1990) Stochastic backscatter in a subgrid-scale model: Plane shear mixing layer. Phys. of Fluids A 2(3):297-299 7. Lilly D (1992) A proposed modification of the Germano subgrid scale closure method. Phys. of Fluids A 4:633-635 8. Rogers M M, Mansour N N, Reynolds W C (1989) An algebraic model for the turbulent flux of a passive scalar. J. Fluid Mech. 203:77-101 9. Schumann U (1995) Stochastic backscatter of turbulence energies and scalar variance by random subgrid-scale fluxes. Proceedings of the Royal Society of London, Series A, Phys. 451:293-318 10. Tavoularis S, Corrsin S (1981) Experiments in nearly homogeneous turbulent shear flow with a uniform mean temperature gradient. Part 2. The fine structure, J. Fluid Mech. 104:349 - 367

Towards Lagrangian dynamic SGS model for SCALES of isotropic turbulence Giuliano De Stefano1 , Daniel E. Goldstein2 , Oleg V. Vasilyev2 , and Nicholas K.-R. Kevlahan3 1

2

3

Dipartimento di Ingegneria Aerospaziale e Meccanica, Seconda Universit` a di Napoli, 81031 Aversa, Italy ([email protected]) Department of Mechanical Engineering, University of Colorado at Boulder, 427 UCB, Boulder CO, USA ([email protected], [email protected]) Department of Mathematics & Statistics, McMaster University, Hamilton, ON, Canada L8S 4K1 ([email protected])

1 Introduction Although turbulence is common in engineering applications, a solution to the fundamental equations that govern turbulent flow still eludes the scientific community. Due to the prohibitively large disparity of spatial and temporal scales, direct numerical simulation (DNS) of turbulent flows of practical engineering interest are impossible, even on the fastest supercomputers that exist or will be available in the foreseeable future. Large eddy simulation (LES) is often viewed as a feasible alternative for turbulent flow modelling, e.g., [1]. The main idea behind LES is to solve only large-scale motions, while modelling the effect of the unresolved subgrid scale (SGS) eddies. When dealing with complex turbulent flows, current LES methods rely on, at best, a zonal grid adaptation strategy to attempt to minimize the computational cost. While an improvement over the use of regular grids, these methods fail to resolve the high wavenumber components of spatially intermittent coherent eddies that typify turbulent flows, thus, neglecting valuable physical information. At the same time, the flow is over-resolved in regions between the coherent eddies, consequently wasting computational resources. Another important drawback of LES, which is often overlooked, is that a priori decided grid resolution distorts the spectral content of any vortical structure by not supporting its small-scale contribution. Recently, a novel approach to turbulent complex flow simulation, called stochastic coherent adaptive large eddy simulation (SCALES) has been introduced [2, 3]. This method addresses the above mentioned shortcomings of LES by using a wavelet thresholding filter to dynamically resolve and “track” the most energetic coherent structures during the simulation. The less energetic

176

De Stefano, Goldstein, Vasilyev and Kevlahan

unresolved modes, the effect of which must be modeled, have been shown to be composed of a minority of coherent modes that dominate the total SGS dissipation and a majority of incoherent modes that, due to their decorrelation with the resolved modes, add little to the total SGS dissipation [2, 4]. The physical coherent/incoherent composition of the SGS modes is reflected in the naming of the SCALES methodology, yet as pointed out in [4] this physical coherent/incoherent composition of the SGS modes is also present in classical LES implementations. For this work, as in much of classical LES research, only the coherent part of the SGS modes will be modeled using a deterministic SGS stress model. The use of a stochastic model to capture the effect of the incoherent SGS modes will be the subject of future work. The first step towards the construction of SGS models for SCALES was undertaken in [3], wherein a dynamic eddy viscosity model based on Germano’s classical dynamic procedure redefined in terms of two wavelet thresholding filters was developed. The main drawback of this formulation is the use of a global (spatially non-variable) Smagorinsky model coefficient. The use a global dynamic model unnecessarily limits the SCALES approach to flows with at least one homogeneous direction. This is unfortunate since the dynamic adaptability of SCALES is ideally suited to fully non-homogeneous flows. In this paper a localized dynamic model is developed to allow the application of the SCALES methodology to inhomogeneous flows. The proposed model is based on the Lagrangian formulation introduced in [5].

2 Stochastic coherent adaptive large eddy simulation 2.1 Wavelet thresholding filter Let us very briefly outline the main features of the wavelet thresholding filter. More details can be found, for instance, in [6]. A velocity field ui (x) can be represented in terms of wavelet basis functions as ui (x) =

l∈L0

+∞ 2 −1 n

c0l φ0l (x)

+

j=0

μ=1

μ,j dμ,j k ψk (x) ,

(1)

k∈Kμ,j

where φ0k (x) and ψlμ,j are n-dimensional scaling functions and wavelets of different families and levels of resolution, indexed with μ and j, respectively. One may think of a wavelet decomposition as a multilevel or multiresolution representation of ui , where each level of resolution j (except the coarsest one) consists of a family of wavelets ψlμ,j having the same scale but located at different positions. Scaling function coefficients represent the averaged values of the field, while the wavelet coefficients represent the details of the field at different scales. Wavelet filtering is performed in wavelet space using wavelet coefficient thresholding, which can be considered as a nonlinear filter that depends on each flow realization. The wavelet thresholding filter is defined by,

Lagrangian dynamic SGS model for SCALES

ui > (x) =

l∈L0

+∞ 2 −1 n

c0l φ0l (x) +

j=0

μ=1

k∈K

μ,j dμ,j k ψk (x) ,

177

(2)

μ,j

|dμ,j | > Ui k

where > 0 stands for the non-dimensional (relative) threshold value, Ui being the (absolute) dimensional velocity scale. The latter can be specified, for instance, as the norm Ui = u 2 . 2.2 Wavelet-filtered Navier-Stokes equations When applying the wavelet thresholding filter to the Navier-Stokes equations, each variable should be filtered, according to Eq. (2), with a corresponding absolute scale. However, this would lead to numerical complications due to the one-to-one correspondence between wavelet locations and grid points. In particular, each variable would be solved on a different numerical grid. In order to avoid this difficulty, in the present study, the coupled wavelet thresholding strategy is used. Namely, after constructing the masks of significant wavelet coefficients for each primary variable, the union of these masks results in a global thresholding mask that is used for filtering each term. Note that other additional variables, like vorticity or strain rate, can be used for constructing the global mask. Once the global mask is constructed, one can view the wavelet thresholding as a local low-pass filtering, where the high frequencies are removed according to the global mask. Such interpretation of wavelet threshold filtering highlights the similarity between SCALES and classical LES approaches. However, the wavelet filter is drastically different from the LES filters, primarily because it changes in time following the evolution of the solution, which, in turn, results in an adaptive computational grid that tracks the areas of locally significant energy in physical space. Therefore, the SCALES equations for incompressible flow, which describe the evolution of the most energetic coherent vortices in the flow field, can be formally obtained by applying the wavelet thresholding filter to the incompressible Navier-Stokes equations: ∂ui > =0, ∂xi

(3)

∂(ui > uj > ) 1 ∂p> ∂ 2 u i > ∂τij ∂ui > + =− +ν − , ∂t ∂xj ρ ∂xi ∂xj ∂xj ∂xj

(4)

where ρ, ν are the constant density and kinematic viscosity, and p stands for the pressure. As a result of the filtering process the unresolved quantities > > τij = ui u> j − ui uj ,

(5)

commonly referred to as SGS stresses, are introduced. They represent the effect of unresolved (less energetic) coherent and incoherent eddies on the

178


resolved (energetic) coherent vortices. As usual in a LES approach, in order to close equations (4), a SGS model is needed to express the unknown stresses in terms of the resolved field. From a numerical viewpoint, the SCALES methodology is implemented using the dynamically adaptive wavelet collocation (DAWC) method, e.g., [7]. The DAWC method is ideal for the actual approach as it combines the resolution of the energetic coherent modes in a turbulent flow with the simulation of their temporal evolution. The wavelet collocation method employs wavelet compression as an integral part of the numerical algorithm such that the solution is obtained with the minimum number of grid points for a given accuracy.

3 Lagrangian dynamic SGS model The primary objective of the current work is to develop a local SGS model for SCALES of inhomogeneous turbulent flows. In previous work a dynamic Smagorinsky model with a global (spatially non-variable) coefficient has been developed and successfully tested for decaying isotropic turbulence [3]. In this work this idea is further extended by exploring the use of a local Lagrangian dynamic model [5]. Following [3], where it was shown that when a wavelet thresholding filter is applied to the velocity field, the resulting SGS stresses scale like 2 , the following Smagorinsky-type eddy viscosity model is used for simulating the deviatoric part (hereafter noted with a star) of the SGS stress tensor (5): > > ∗ ∼ (6) τij = −2CS Δ2 2 S Sij , > > ∂uj > i is the resolved rate-of-strain tensor and = 12 ∂u where Sij ∂xj + ∂xi Δ(x, t) is the local characteristic vortical lengthscale implicitly defined by wavelet thresholding filter. Note that Δ is distinctively different from the classical LES, where the local filter width is used instead. Also note that despite its implicit definition, Δ can be extracted from the actual thresholding mask during the simulation. Following the modified Germano’s dynamic procedure redefined in terms of two wavelet thresholding filters, originally introduced in [3], the SGS stress corresponding to the wavelet test filter at twice the threshold, noted (·) defined as Tij = ui uj >2 − ui >2 uj >2 .

>2

, is (7)

Note that, the wavelet filter being a projection operator, by definition, it holds >

>2

>2

(·) ≡ (·) . Filtering (5) at the test filter level and combining with (7) results in the following modified Germano identity for the Leonard stresses: Lij ≡ Tij − τij >2 = ui > uj >

>2

− ui >2 uj >2 .

(8)


179

Exploiting the model (6) and the analogous relation for the test filtered SGS stresses >2 2 >2 , (9) Tij∗ ∼ = −2CS Δ2 (2) S Sij one obtains >2 >2 > > 2 >2 2CS Δ2 2 S Sij − 2CS Δ2 (2) S Sij = L∗ij .

(10)

A least square solution to (10) leads to the following local Smagorinsky model coefficient definition: L∗ij Mij , (11) CS (x, t)2 = Mhk Mhk where Mhk

>2 > >2 > >2 . ≡ 2Δ S Shk − 4S Shk 2

(12)

The coefficient CS can be actually positive or negative, that allows for local backscatter of energy from unresolved to resolved modes. However, it has been found that negative values of CS cause numerical instabilities. To avoid this fact, for homogeneous flow, one can introduce an average over homogeneous directions. This procedure results in the global dynamic model proposed in [3]. In this study we follow a Lagrangian dynamic model formulation [5] and take the following statistical averages over the trajectory of a fluid particle: 1 t τ −t e T Lij (x (τ ) , τ ) Mij (x (τ ) , τ ) dτ , (13) ILM (x, t) = T −∞ 1 t τ −t e T Mhk (x (τ ) , τ ) Mhk (x (τ ) , τ ) dτ , (14) IM M (x, t) = T −∞ which leads to the following local Smagorinsky model coefficient CS (x, t)2 =

ILM . IM M

(15)

To avoid the computationally expensive procedure of Lagrangian pathline averaging, following [5], Eqs. (13) and (14) are differentiated with respect to time leading to the following evolution equations for ILM and IM M : 1 ∂ILM ∂ILM + u> = (Lij Mij − ILM ), l ∂t ∂xl T 1 ∂IM M ∂IM M + u> = (Mhk Mhk − IM M ). l ∂t ∂xl T

(16) (17)

As in [5] the relaxation time scale T is defined as T (x, t) = θΔ (ILM IM M )−1/8 , θ being a dimensionless parameter of order unity.

180


1

500

0.95 compression

energy

400

00

200

0.85

100

0

0.9

0

0.1

time

0.2

0.3

0.8 0

0.1

time

0.2

0.3

Fig. 1. Kinetic energy decay (left) and grid compression (right) for GDM (dashed line) and LDM (solid line). The reference energy decay for DNS is also reported (dotted line).

The equations (16) and (17) should be solved together with the SCALES equations, (3) and (4). It should be noticed that both ILM and IM M have higher frequency content when compared to the velocity field. This is due to two main factors: the quartic character of nonlinearity of ILM and IM M with respect to velocity and the creation of small scales due to chaotic convective mixing. Thus, in order to adequately resolve both ILM and IM M , one needs to have a substantially finer computational mesh than the one required by the velocity field, which is impractical. To by-pass this problem, an artificial diffusion term is added to Eqs. (16) and (17): 1 ∂ 2 ILM ∂ILM ∂ILM + u> = (Lij Mij − ILM ) + DI , l ∂t ∂xl T ∂xl ∂xl

(18)

1 ∂ 2 IM M ∂IM M ∂IM M + u> = (Mhk Mhk − IM M ) + DI . l ∂t ∂xl T ∂xl ∂xl

(19)

To avoid the creation of small scales, the diffusion time scale, Δ2 /DI , should > −1 be smaller than the convective time scale associated with local strain, S , > which results in DI = CI Δ2 S , where CI is a dimensionless parameter of order unity.

4 Results In this paper, the preliminary results of the application of the SCALES method together with Lagrangian dynamic modeling (for discussion: LDM)


181

to incompressible isotropic decaying turbulence simulation are presented. The LDM solution is compared to SCALES with a global dynamic model (for discussion: GDM) [3]. The initial velocity field is a realization of a statistically stationary turbulent flow at Reλ = 48, as provided by a pseudo-spectral DNS database, e.g. [4]. In both SCALES cases the wavelet thresholding parameter is set to = 0.5. For a detailed discussion on the SCALES formulation we refer to [2]. The additional SGS modeling variables are initialized as IM M = Mhk Mhk and ILM = C¯s 2 IM M , C¯s being the volume averaged value. For the time relaxation scale definition, the suggested value θ = 1.5 is chosen. For a discussion of the model sensitivity to this parameter, one can see the original work [5]. As to the artificial diffusion coefficient, several experiments have been performed, leading to the choice of CI = 5 for this preliminary test, in order to have a stable solution. In Figure 1 the kinetic energy decay and grid compression for LDM are compared to GDM. The energy decay for a pseudo-spectral DNS solution is also reported for reference. The compression is always evaluated with respect to the maximum field resolution, that is 1283 for both SCALES cases. The LDM case appears initially slightly over dissipative in comparison to DNS. Though both SCALES use the same relative , yet the compression for the LDM run is slightly better. In fact, a very interesting aspect of the SCALES methodology is that the dynamic grid evolution is closely coupled to the flow physics and is therefore affected by the SGS stress model forcing. Figure 2 shows the energy density spectra at a given time instant, that is t = 0.104. The spectral DNS and wavelet-filtered DNS solutions are also shown for reference. It can be seen that, at this point in the decay, both the LDM and the GDM models show excess energy in the small scales, leading to the conclusion that the model is either not damping out small scales or is itself introducing excess small scale motions. This again highlights the strong coupling between the dynamically adapting grid and the flow physics. In conclusion, we want to emphasize that the work on local Lagrangian model is ongoing. However, from these limited initial results, one can conclude that the local model works as well as the global one. Further work will pursue a more computationally efficient formulation as well as improve the model behavior at small scales level. It is worth reporting that SCALES with GDM at higher Reynolds number have provided better agreement with a DNS solution [3]. The same good results are expected for the LDM case. Moreover, once a cost effective model implementation is developed, the LDM approach will allow the study of non-homogeneous flows.

Acknowledgements This work was supported by the Department of Energy (DOE) under Grant No. DE-FG02-05ER25667, the National Science Foundation (NSF) under grants No. EAR-0327269 and ACI-0242457, and the National Aeronautics and

182

De Stefano, Goldstein, Vasilyev and Kevlahan 102 101

energy density

100 10−1 10−2 10−3 10−4 10−5

16 wavenumber

32 48 64

Fig. 2. Energy density spectra at a given time instant (t = 0.104) for DNS (dotted line), wavelet filtered DNS (dash-dotted line), GDM (dashed line) and LDM (solid line).

Space Administration (NASA) under grant No. NAG-1-02116. In addition, G. De Stefano was partially supported by Regione Campania (LR 28/5/02 n.5), D. E. Goldstein by the Minnesota Supercomputing Institute Research Scholarship and N. K.-R. Kevlahan by the Natural Sciences and Engineering Research Council of Canada. The authors thank Prof.s Charles Meneveau and Thomas S. Lund for helpful suggestions.

References 1. P. Moin. Advances in large eddy simulation methodology of complex flows. Int. J. Heat Fluid Flow, 23:710–720, 2002. 2. D. E. Goldstein and O. V. Vasilyev. Stochastic coherent adaptive large eddy simulation method. Phys. Fluids, 16(7):2497–2513, 2004. 3. D.E. Goldstein, O.V. Vasilyev, and N.K.-R. Kevlahan. CVS and SCALES simulation of 3D isotropic turbulence. To appear on J. of Turbulence, 2005. 4. G. De Stefano, D. E. Goldstein, and O. V. Vasilyev. On the role of sub-grid scale coherent modes in large eddy simulation. Journal of Fluid Mechanics, 525:263– 274, 2005. 5. C. Meneveau, T. S. Lund, and W. H Cabot. A Lagrangian dynamic subgrid-scale model of turbulence. J. Fluid Mech., 319:353–385, 1996. 6. I. Daubechies. Ten Lectures on Wavelets. Number 61 in CBMS-NSF Series in Applied Mathematics. SIAM, Philadelphia, 1992. 7. O. V. Vasilyev. Solving multi-dimensional evolution problems with localized structures using second generation wavelets. Int. J. Comp. Fluid Dyn., Special issue on High-resolution methods in Computational Fluid Dynamics, 17(2):151– 168, 2003.

The sampling-based dynamic procedure for LES: investigations using finite differences G. Winckelmans1 , L. Bricteux1 , L. Georges2 , G. Daeninck1 , H. Jeanmart1 1

2

Université catholique de Louvain (UCL), Mechanical Engineering Department, 1348 Louvain-la-Neuve, Belgium, [email protected], Center for Research in Aeronautics (CENAERO), 6041 Gosselies, Belgium

Summary. The dynamic procedure for LES performed solely in physical space (i.e., no Fourier transform) is considered. It amounts to a procedure working at the force (vector) level that is natural and quite general: it only requires a numerical tool for restriction of the discrete LES field and forces to a coarser level. It is here investigated using finite differences and with restriction done by sampling. It gives good results on flows with homogenous directions: Burger’s turbulence and homogeneous isotropic turbulence. Preliminary results on the turbulent channel flow are also presented: they are encouraging but not yet satisfactory (velocity profile underpredicted). The obtained profile of CΔ2 is found to have the proper near and far wall behaviors, but with too low amplitude. Further improvements are required: they might include some filtering (using tensor-product stencil-3 discrete filters, also iterated) prior to the sampling, to mitigate the aliasing effects due to the sampling; they might also require to modify the procedure itself, following what was done by others when using the classical procedure expressed at the force level.

1 Introduction Practical LES mainly consists of a “restriction”: a truncation to much less information than that required to numerically capture the complete field, u(x), as in DNS. Considering a grid of size h, the discrete LES-restricted field is here written uh = Rh (u(x)). Notice that, contrary to regular filtering, restriction is such that applying it twice is the same as applying it once (idempotent): Rh (uh ) = uh . The time variation of the discrete field is evaluated using the discrete “forces” (convection term, subgrid-scale (SGS) term, viscous term). This is here written as a discrete numerical operator acting on the discrete field: def

= Qh (uh ) . (1) ∂t uh + ∇h Ph = −∇h · (uh uh ) + ∇h · (2 ν Sh ) + ∇h · τ sgs h Notice that the pressure correction step (the projection to ensure incompressibility) is here seen as part of the proper time integration.

184

G. Winckelmans, L. Bricteux, L. Georges, G. Daeninck, H. Jeanmart

We here present and investigate a procedure for dynamic LES performed solely in physical space (i.e., no recourse to spectral space accessible or allowed): finite differences (FD), etc. For simplicity, we only consider LES without explicit filtering; the approach being easily extended to LES with added explicit filtering (e.g., as those in [6]), when using discrete filters. We here investigate the case with FD (staggered approach), and with sampling used as the basic restriction operator: either “pure sampling” (thus with some aliasing) or “regularized sampling” (using discrete filtering to mitigate the aliasing effects). Results were already presented at [7]. Similar work for such “LES without filtering”, and the related dynamic procedure, was proposed in [2]: they investigated it using a pseudo-spectral code (thus collocated approach) and mimicking FD behavior. The procedure is really quite simple, and is designed to be self-consistent with the numerics (whatever those are). In the case of FD, it is: 1. Further restrict the LES field, to obtain the discrete field on the coarser grid: (2) u2h = R2h (uh ) 2. Evaluate the discrete forces at both levels and using the same numerics: Qh (uh )

and

Q2h (u2h )

(3)

3. The equivalent of a “Germano identity” involving the forces is then obtained; it corresponds to the following statement: the restriction of the discrete forces (evaluated using the discrete field) should be equivalent to the discrete forces evaluated using the restricted field: R2h (Qh (uh )) = Q2h (u2h ) .

(4)

4. To obtain the best coefficient(s) in the SGS model, the square of the error on this identity is averaged over the homogeneous direction(s) and is minimized. Consider, for instance, using the Smagorinsky SGS model: 2 ∇h · τ sgs h = ∇h · 2 C h |Sh | Sh .

(5)

The error vector on the identity is then: e = [R2h (∇h · (uh uh )) − ∇2h · (u2h u2h )] − [R2h (∇h · (2 ν Sh )) − ∇2h · (2 ν S∗2h )] −(C h2 ) [R2h (∇h · (2 |Sh | Sh )) − ∇2h · (8 |S∗2h | S∗2h )] def

= a − (C h2 ) b .

(6)

Averaging the error squared over the homogeneous direction(s) and minimiza·b ing gives (C h2 ) = b·b . Notice that, since (to make it simpler and cheaper)

Sampling-based dynamic procedure: investigations using finite differences

185

we do not enforce that u2h be divergence free (i.e., we avoid solving the Poisson equation for P2h required to project it), we have to use the deviatoric part of the tensor: S∗2h . In most LES, one can also neglect the molecular viscous term contribution in the error minimization, as it is small compared to the SGS contribution. So far, this is the approach followed here. Recall also that this term is linear and thus has no contribution in the classical spectral-based dynamic procedure. When using the staggered approach for incompressible flows (as here, with FD), an intermediate interpolation is required as part of the “global restriction step”, see Section 3. For the present validations, the procedure is investigated using FD and the Smagorinsky SGS model. Three cases are investigated: (1) LES of forced “Burgers turbulence”, using second order FD; (2) LES of decaying incompressible isotropic turbulence, using the staggered approach and fourth order FD; (3) LES of channel flow, using the same code (preliminary results). Reference solutions are those obtained using a pseudo-spectral method (DNS and LES with the spectral-based dynamic procedure: with restriction to grid 2h done using a sharp Fourier cutoff).

2 “Burgers turbulence”, using 2nd order FD The continuous equation is : ∂t u = −∂x

uu 2

+ ν ∂x2 u + F

(7)

where F is a white noise forcing (here applied using Fourier space). The discrete LES equation is: u u h h + ∂h Ch2 |∂h uh | ∂h uh + ν ∂h2 uh + Fh (8) ∂t uh = −∂h 2 where we have used the 1-D equivalent of the Smagorinsky model. Using 2nd order FD on a grid of size h, one has C (|ui+1 − ui |(ui+1 − ui ) − |ui − ui−1 |(ui − ui−1 )) h (ui+1 − 2ui + ui−1 ) +ν + Fi h2

∂t ui = −Hi +

(9)

where Hi is the discretized convective term. The divergence form (Hd2), Hi =

(u2i+1 − u2i−1 ) , 4h

(10)

dissipates energy. The advective form (Ha2), Hi = u i

(ui+1 − ui−1 ) , 2h

(11)

186


produces energy (and often leads to blow up). The skew-symmetric form (Hs), Hi =

(ui+1 + ui + ui−1 ) (ui+1 − ui−1 ) , 3 2h

(12)

conserves energy (thus best for DNS and LES; it is also equal to (2 Hd2+Ha2)/3). The “other” divergence form (Hd1), 2 2 (ui+1 + ui ) − (ui + ui−1 ) , (13) Hi = 8h also produces energy, yet significantly less than Ha2 (as is it equal to (Hd2+Ha2)/2). As to the “other” advective form (Ha1), it is the same as Hd2.

10−1 10−2

10−4

10−3

10−5

10−5

E

E

10−4 10−6

10−7

10−7

10−8

10−8 10−9 0 10

10−6

101

102

k

103

10−9 102

k

103

Fig. 1. Burgers DNS spectra on a 2048 point grid (left) and zoom on the dissipation range (right): spectral (solid), Hs (dash), Hd2 (dash-dot), Hd1 (dot)

We first present, in Fig. 1, the energy spectra for a 2048 points DNS. The reference spectrum is obtained using a de-aliased pseudo-spectral method (energy conserving and without dispersion). It is verified that Hs performs best. It is also seen that Hd1 produces a small amount of energy (spectrum is above reference). It is also seen that Hd2 dissipates energy (spectrum too low). We consider next a challenging LES: on 128 points. The restriction from grid h to grid 2h is done using pure sampling (thus with aliasing). The obtained spectra and dynamic C coefficient (averaged in time) are presented in Fig. 2. We notice that the C obtained using the best scheme (Hs) is close to that obtained using the classical dynamic spectral method; the obtained spectrum is also quite good: in fact better than that obtained using the dynamic spectral method. The spectra obtained using the energy producing schemes (Ha2 and Hd1) are found to be close to each other, and not as good as those obtained using Hs. The proposed dynamic procedure thus appears to work properly, the best results being obtained when using the best scheme.


187

0.25 10

−1

10−2

C

E

0.2

10−3

10−4 0 10

0.15

0.1

101

K

102

0.05

0

2

4

6

8

10 12 14 16 18 20

t

Fig. 2. Burgers LES on a 128 point grid: spectra (left) and history of the dynamic coefficient (right): spectral (solid-thin), Hs (dash), Ha2 (dash-dot), Hd1 (dot). Also shown is DNS (solid-thick),

3 Decaying isotropic turbulence, using 4th order FD We consider next LES of decaying incompressible isotropic turbulence. Our code simulates unsteady incompressible flows, using the staggered approach (MAC), 4th order FD, the convection scheme of Vasilyev [4] (which conserves energy), the fractional step method of Choi and Moin, explicit convection and SGS terms (Adams-Bashforth), and the choice between explicit or implicit molecular diffusion (in case of implicit, Crank-Nicolson with ADI). It is multiblock and parallel, and it uses a Multigrid Poisson solver.

Fig. 3. Schematic of the restriction operation on a 3D MAC cell.

Due to the staggered approach, the restriction operation from grid h to grid 2h requires an intermediate interpolation step, see Fig. 3. For instance, to obtain the u2h , the interpolation only uses the uh values that lie in the same plane as that of grid 2h; the values on the planes at −h and h are not used. This ensures that the global step (interpolation + sampling) is indeed a good restriction. We consider a 2563 de-aliased pseudo-spectral DNS, projected to 643 (easy LES) and 483 (more challenging), and then run further in time using the

188


100

0.025

DNS 3 DYNAMIC LES FD 64 3 DYNAMIC LES SPECTRAL 64

90 80

0.02

70 60

0.015

50

C

dE/dt

DYNAMIC LES FD 643 3 DYNAMIC LES SPECTRAL 64

40

0.01

30 20

0.005

10 0

0

1

2

3

4

5

0

6

t 100

1

2

3 t

0.025

DNS 3 DYNAMIC LES FD 48 3 DYNAMIC LES SPECTRAL 48

90

0

80

4

5

6

DYNAMIC LES FD 483 DYNAMIC LES SPECTRAL 483

0.02

60

0.015

50

C

dE/dt

70

40

0.01

30

0.005

20 10 0

0 0

1

2

3

t

4

5

6

0

1

2

3 t

4

5

6

Fig. 4. Evolution of total dissipation (left) and of the dynamic coefficient (right) for 643 (top) and 483 (bottom) LES of decaying isotropic turbulence (reference DNS is spectral 2563 ).

Smagorinsky model (same cases as those considered in [6]). Comparisons are made with dynamic LES using the de-aliased pseudo-spectral method and with the dynamic procedure based on sharp spectral cutoff. It is seen, on Fig. 4, that both dynamic LES have too much dissipation initially: a long recognized weakness of the dynamic procedure when LES is started from restricted DNS data. Apart from that, the dissipation histories of the two LES compare well with each other. The sampling based dynamic procedure also performs well. The only noticeable difference in the results is the more abrupt variations observed in the histories (dissipation dE dt and dynamic C) in the case of FD: this was expected as the present procedure is theoretically not as pure as the spectral-based procedure. Of course, the history of the resolved energy (E(t), not shown) is smooth in both cases.

4 Turbulent channel flow, using 4th order FD It remains to test the proposed approach on a flow with walls. LES of the channel flow at Reτ = uτνH = 395 is considered next, the reference being the DNS of [3]. The computational domain is 2πH × 2H × πH in streamwise,


189

normal, and spanwise directions. The LES grid is quite coarse: 64 × 48 × 48, thus Δx+ = 38.8 and Δz + = 25.8. The stretching used in y is: tanh(α(ζ − 1)) y =1+ H tanh(α)

(14)

with ζ = y = 0 at the lower wall and ζ = y/H = 2 at the upper wall, and + with stretching typical for LES: α = 2.75 (which gives Δyw = 0.8 at the wall + and Δyc = 45.4 at the center). 7

7 20

6 5 CΔ

2

15 u+

x 10

10

4 3 2

5

1 0

100

101 y+

2

10

0

0

50

100 150 200 250 300 350 y+

Fig. 5. LES of channel flow at Reτ = 395: averaged velocity profile (left) and averaged dynamic coefficient C Δ2 (right). DNS profile shown in bullets.

Dynamic LES was performed and some results are presented in Fig. 5. So far, the velocity profile is underpredicted. The dynamic CΔ2 curve appears to have the proper near and far wall behaviors, yet its amplitude is too low when compared to that obtained in dynamic and spectral LES on the same grid [1]. Notice that clipping was used (whenever the value produced by the dynamic procedure is negative, it is set to zero); however, on average, there isn’t much clipping. We also notice that Morinishi and Vasilyev [5] tested the classical dynamic procedure (filter-based), and with the Germano identity expressed at the force level. In their case, they observed excessive near-wall clipping behavior. They were able to cure this problem, and thus improve the results, by better taking into account the y variation of CΔ2 in the minimization procedure itself. We will also investigate whether their improved methodology can be used with the present approach and improve the results.

5 Conclusion The sampling-based dynamic procedure for LES performed solely in physical space (i.e., no Fourier transform) was considered. It amounts to a procedure at the force (vector) level that is natural and quite general: it only requires a tool

190


for restriction of the discrete LES field and forces to a coarser level. It was here investigated using FD and restriction done by sampling. It gave good results on flows with homogenous directions: Burger’s turbulence and homogeneous isotropic turbulence. Preliminary results on the turbulent channel flow were also presented, with modest, yet encouraging, results: the obtained CΔ2 profile has the correct near and far wall behaviors but its amplitude is too low; hence the velocity profile is underpredicted. Future work will focus on improving the procedure for such wall bounded flows, e.g.: (a) include some filtering (based on tensor-product stencil-3 discrete filters, eventually iterated) prior to the sampling of uh , to mitigate the aliasing effects, (b) take the molecular viscous term into account in the minimization, (c) modify the procedure itself, as was done by others when working with the classical procedure when expressed at the force level. The present dynamic procedure can likely be applied generally, as it only requires the proper “tool for restriction to a coarser level”. It could also be used with finite volumes (then with an implicit top-hat filter also implied), for unstructured grids, etc. Of course, a proper restriction step must be used in all cases.

References 1. Jeanmart H, Winckelmans G (2002) Comparison of recent dynamic subgridscale models in the case of the turbulent channel flow, Proc. Summer Program 2002, Center for Turbulence Research, Stanford University & NASA Ames: 105–116. 2. Knaepen B, Debliquy O, Carati D (2005) Large-eddy simulation without filter. J. Comp. Phys., 205: 98–107. 3. Moser R, Kim J, Mansour N (1999) Direct numerical simulation of turbulent channel flow up to Reτ = 590. Phys. Fluids11: 943–945. 4. Vasilyev O (2000) High order finite differences schemes on non-uniform meshes with good conservation properties. J. Comp. Phys., 157(2): 746–761. 5. Morinishi Y, Vasilyev O (2002) Vector level identity for dynamic subgrid scale modeling in large eddy simulation. Phys. Fluids 14: 3616–3623. 6. Winckelmans G, Wray A, Vasilyev O, Jeanmart H (2001) Explicit-filtering large-eddy simulation using the tensor-diffusivity model supplemented by a dynamic Smagorinsky term. Phys. Fluids 13(5): 1385–1403. 7. Winckelmans G, Georges L, Bricteux L, Jeanmart H (2004) Sampling-based dynamic procedure for LES in physical space. 57th Annual Meeting of APS-DFD, Seattle, WA, Nov. 21–24, Bulletin of the American Physical Society 49(9).

On the Evolution of the Subgrid-Scale Energy and Scalar Variance: Effect of the Reynolds and Schmidt numbers C. B. da Silva and J.C.F. Pereira Instituto Superior Técnico, Pav. M´ aquinas I, 1o andar/LASEF, Av. Rovisco Pais, 1049-001 Lisboa, Portugal csilva,[email protected]

A promising trend in subgrid-scale modeling consists in using transport equations for the subgrid-scale kinetic energy as e.g. in Ghosal et al. [1]. In the present work Direct Numerical Simulations (DNS) of forced homogeneous isotropic turbulence are used to analyze some of the assumptions often used in these models, and their dependence on the Reynolds number and implicit filter size. In particular, three key issues are analyzed: (a) the relative importance between production and diffusion of SGS kinetic energy; (b) the modeling of the viscous SGS dissipation and; (c) the acceleration statistics of the local and convective terms. Finally, the same analysis is applied to an equation for the evolution of the SGS scalar variance.

1 Introduction In the context of Large-Eddy Simulations (LES), the hypothesis of (a) equilibrium and (b) self-similarity, were often invoked in the past, and are still often used today, in order to develop and improve new subgrid-scale models. Indeed, most subgrid-scale models today use one of these assumptions, either to derive mathematical expressions or to compute model constants. However, it has been recognized that, particularly the equilibrium hypothesis, does not work very well. As shown by da Silva and Métais [2], and da Silva and Pereira [3], even for statistically stationary turbulence the large and small scales of the velocity field are only weakly correlated. The same occurs for any passive scalar field. This problem is even more important when considering flows that are statistically unsteady at the large scales e.g. in periodically forced jets or in unsteady boundary layers. For these cases, the ”past history” of the flow field has to be taken into consideration if one is to compute accurately the local turbulence characteristics. One interesting trend in Large-Eddy Simulations that overcomes these difficulties by dropping out the equilibrium and self similarity assumptions, is

192

C. B. da Silva and J.C.F. Pereira

to consider simplified transport equations for the subgrid-scale stresses or the subgrid-scale kinetic energy. Examples of this approach include the works of Ghosal et al. [1], Wong [4], Debliquy et al. [5], Schiestel and Dejoan [6] and Chaouat and Schiestel [7]. The estimation of the subgrid-scale kinetic energy is interesting in other situations as for instance the evaluation of the Reynolds stresses in LES [8, 9]. Also, in numerical simulations of turbulent reacting flows, knowledge of the SGS scalar variance might be useful. For these reasons it is important to analyze the dynamics of the SGS kinetic energy and SGS scalar variance, and hopefully to try to develop simple or approximated expressions for their evolution. The present work uses Direct Numerical Simulations (DNS) of statistically stationary (forced) homogeneous isotropic turbulence to analyze three key issues related to the modeled or simplified expressions for the evolution of the subgrid-scale kinetic energy: (a) the relative importance between production and diffusion of SGS kinetic energy; (b) the modeling of the viscous SGS dissipation and; (c) the acceleration statistics of the local and convective acceleration terms. Finally, a similar analysis is applied to an equation for the evolution of the SGS scalar variance.

2 Transport equations: exact and simplified forms 2.1 Evolution of the SGS kinetic energy The classical filtering operation used to obtain the Filtered Navier-Stokes equations, is defined by, φ(x )GΔ (x − x )dx , (1) φ< (x) = Ω

where φ is a given flow variable, and GΔ (x) is the filter kernel. When applied to the Navier-Stokes equations the filtering operation yields the subgrid-stress < tensor, τij = (ui uj )< −u< i uj . The exact equation for the subgrid-scale kinetic energy, τii , reads, ' ' <

Direct and Large-Eddy Simulation

Direct and Large-Eddy Simulation VIII

Industrial and Environmental Applications of Direct and Large-Eddy Simulation