Progress in Wall Turbulence: Understanding and Modeling: Proceedings of the WALLTURB International Workshop held in Lille, France, April 21-23, 2009 (ERCOFTAC Series)

Progress in Wall Turbulence: Understanding and Modeling ERCOFTAC SERIES VOLUME 14 Series Editors R.V.A. Oliemans, Cha...

Author: Michel Stanislas | Javier Jimenez | Ivan Marusic

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Progress in Wall Turbulence: Understanding and Modeling

ERCOFTAC SERIES VOLUME 14 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.

For other titles published in this series, go to www.springer.com/series/5934

Progress in Wall Turbulence: Understanding and Modeling Proceedings of the WALLTURB International Workshop held in Lille, France, April 21–23, 2009 Edited by Michel Stanislas Université Lille Nord de France, LML, UMR CNRS 8107, France

Javier Jimenez Universidad Politécnica de Madrid, Spain

and Ivan Marusic The University of Melbourne, Australia

Foreword by Rémy Denos European Commission DGXII

Editors Prof. Michel Stanislas Université Lille Nord de France LML, UMR CNRS 8107 Cité Scientifique boulevard Paul Langevin 59655 Villeneuve d’Ascq CX France [email protected]

Ivan Marusic The University of Melbourne Department of Mechanical Engineering 3010 Melbourne, Victoria Australia [email protected]

Javier Jimenez Universidad Politécnica de Madrid School of Aeronautics Pz. Cardenal Cisneros 3 28040 Madrid Spain [email protected]

Additional material to this book can be downloaded from http://extra.springer.com ISSN 1382-4309 ISBN 978-90-481-9602-9 e-ISBN 978-90-481-9603-6 DOI 10.1007/978-90-481-9603-6 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2010937478 © Springer Science+Business Media B.V. 2011 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. Cover design: VTEX, Vilnius Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Foreword

Over the years, the issue of turbulence has attracted the attention and skills of many researchers, maybe because of the complexity of turbulent flow and the challenge that its understanding represents for scientists. Between the von Karman analytical expression for turbulent boundary layers profiles and the analysis of the results of Direct Numerical Simulation (DNS), this understanding has considerably progressed. Among the different regions of the boundary layer, the wall region is particularly challenging because the dimensions are getting so small that it is difficult to investigate it experimentally. Wallturb, a European Synergy for the Assessment of Wall turbulence, is the first European Commission funded collaborative project that addresses near wall turbulence. During more than 4 years, 16 European partners have collaborated on this topic. The project generated considerable amount of new data, from both experiments and DNS computations. Improved measurement techniques (e.g. high resolution Particle Image Velocimetry), high density time resolved instrumentation and scaled up experiments gave researchers access to a level of detail never achieved before. A number of test cases were investigated, including the presence of adverse pressure gradients and separated flows. Thanks to the availability of large computing resources, DNS computations could be carried out at levels of Reynolds number never computed before. All the data generated was made available for researchers in the form of databases. In parallel, other groups were working on the turbulence modelling aspects including Reynolds Average Navier–Stokes methods (RANS), Large Eddy Simulation (LES) as well as an innovative Low Order Dynamical Systems method (LODS). Comparison between experiments and computations highlighted strengths and weaknesses of the different approaches. But what is the rational for a better understanding and for improved prediction methods if these are not used for practical applications? This is where the two industrial members of the consortium brought their needs and a challenging application: the prediction of the friction coefficient, which is directly dependent on the velocity profile at the wall, around the wing of an aircraft. Accurate prediction of the friction coefficient means accurate prediction of drag and thus of fuel consumption. v

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Foreword

Wallturb has demonstrated that the (sometimes turbulent) mixing of academia, research centres and industry, and, the associated diversity of views and approaches, resulted in the creation of new knowledge and a realisation that the fertility of applied research is dependent on knowledge created by more upstream research. The project has clearly contributed to pushing the boundaries of understanding closer to the wall, at higher levels of Reynolds numbers and under conditions representative of aeronautical applications. This has lead to a better accuracy of the different prediction methods, ranging from those used in academia to those used in industry. The final workshop was the occasion to share this new knowledge developed in the frame of European cooperation with other parts of the world, whose representatives acknowledged the high quality and innovativeness of the work performed. These proceedings of the final workshop attest to the quality of the work and advances that have been achieved during the project. Brussels

Rémy Denos

Preface

This book is gathering the contributions of most of the participants to the WALLTURB workshop on “Understanding and modelling of wall turbulence” held in Lille (France) on April 21st to 23rd 2009. This workshop was organized by the WALLTURB consortium, as a final open workshop of the WALLTURB project, in order to present to the relevant scientific community the main results of the project and to stimulate scientific discussions around the subject of wall turbulence. The workshop assembled 105 participants from all over the world, 7 invited lecturers and 46 contributions from which 27 where presented by the WALLTURB consortium as an output of the WALLTURB project. This workshop was an opportunity to review the recent progress in theoretical, experimental and numerical approaches to wall turbulence. The problems of zero pressure gradient, adverse pressure gradient and separating turbulent boundary layers were addressed in detail with the three approaches, using most advanced tools in each of them. This book is gathering papers from most of the contributors to the workshop. It is aimed as being a milestone in the research field, thanks to the high level of the invited speakers and the involvement of the contributors. It is also aimed at being a testimony of the achievement of the WALLTURB project. Lille, Madrid, Melbourne

Michel Stanislas Javier Jimenez Ivan Marusic

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Acknowledgements

As WALLTURB coordinator, and in the name of the consortium, I would like to acknowledge the funding of the European Community, under the 6th framework program, through the contract no AST4-CT-2005-516008. This project lasted from April 1st 2005 to June 30st 2009 and allowed the 16 partners of the consortium, from universities, research organisation and industry, to join their research effort to contribute to the understanding and modelling of wall turbulence. Cooperative research project at the European level and at such a scale would not be possible without this European funding, which makes a significant contribution to the building of the European research community. Lille

Michel Stanislas

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Contents

The WALLTURB Project . . . . . . . . . . . . . . . . . . . . . . . . . . . Michel Stanislas, Javier Jimenez, and Ivan Marusic

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Invited Speakers The Law of the Wall. Indications from DNS, and Opinion . . . . . . . . . Philippe R. Spalart A Web-Services Accessible Turbulence Database and Application to A-Priori Testing of a Matrix Exponential Subgrid Model . . . . . Charles Meneveau Modeling Multi-point Correlations in Wall-Bounded Turbulence . . . . . Robert D. Moser, Amitabh Bhattacharya, and Nicholas Malaya Theoretical Prediction of Turbulent Skin Friction on Geometrically Complex Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pierre Sagaut and Yulia Peet Scaling Turbulent Fluctuations in Wall Layers . . . . . . . . . . . . . . . Ronald L. Panton

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Session 1: The WALLTURB LML Experiment The WALLTURB Joined Experiment to Assess the Large Scale Structures in a High Reynolds Number Turbulent Boundary Layer . Joel Delville, Patrick Braud, Sebastien Coudert, Jean-Marc Foucaut, Carine Fourment, W.K. George, Peter B.V. Johansson, Jim Kostas, Fahrid Mehdi, A. Royer, Michel Stanislas, and Murat Tutkun

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Calibration of the WALLTURB Experiment Hot Wire Rake with Help of PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michel Stanislas, Jean-Marc Foucaut, Sebastien Coudert, Murat Tutkun, William K. George, and Joel Delville Spatial Correlation from the SPIV Database of the WALLTURB Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Marc Foucaut, Sebastien Coudert, Michel Stanislas, Joel Delville, Murat Tutkun, and William K. George Two-Point Correlations and POD Analysis of the WALLTURB Experiment Using the Hot-Wire Rake Database . . . . . . . . . . . . Murat Tutkun, William K. George, Michel Stanislas, Joel Delville, Jean-Marc Foucaut, and Sebastien Coudert

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Session 2: Experiments in Flat Plate Boundary Layers Reynolds Number Dependence of the Amplitude Modulated Near-Wall Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Ivan Marusic, Romain Mathis, and Nicholas Hutchins Tomographic Particle Image Velocimetry Measurements of a High Reynolds Number Turbulent Boundary Layer . . . . . . . . . . . . . 113 Callum Atkinson, Sebastien Coudert, Jean-Marc Foucaut, Michel Stanislas, and Julio Soria Study of Vortical Structures in Turbulent Near-Wall Flows . . . . . . . . 121 Sophie Herpin, Sebastien Coudert, Jean-Marc Foucaut, Julio Soria, and Michel Stanislas Session 3: Experiments in Adverse Pressure Gradient Boundary Layers Two-Point Near-Wall Measurements of Velocity and Wall Shear Stress Beneath a Separating Turbulent Boundary Layer . . . . . . . . . . . 135 Paul Nathan and Philip E. Hancock Experimental Analysis of Turbulent Boundary Layer with Adverse Pressure Gradient Corresponding to Turbomachinery Conditions . . 143 Stanislaw Drobniak, Witold Elsner, Artur Drozdz, and Magdalena Materny Near Wall Measurements in a Separating Turbulent Boundary Layer with and without Passive Flow Control . . . . . . . . . . . . . . . . . 151 Davide Lengani, Daniele Simoni, Marina Ubaldi, Pietro Zunino, and Francesco Bertini

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Session 4: Boundary Layer Structure and Scaling On the Relationship Between Vortex Tubes and Sheets in Wall-Bounded Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Sergio Pirozzoli Spanwise Characteristics of Hairpin Packets in a Turbulent Boundary Layer Under a Strong Adverse Pressure Gradient . . . . . . . . . . . 173 S. Rahgozar and Y. Maciel The Mesolayer and Reynolds Number Dependencies of Boundary Layer Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 William K. George and Murat Tutkun A New Wall Function for Near Wall Mixing Length Models Based on a Universal Representation of Near Wall Turbulence . . . . . . . 191 Michel Stanislas Session 5: DNS and LES Direct Numerical Simulations of Converging–Diverging Channel Flow . . 203 Jean-Philippe Laval and Matthieu Marquillie Corrections to Taylor’s Approximation from Computed Turbulent Convection Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Javier Jiménez and Juan C. del Álamo A Multi-scale & Dynamic Method for Spatially Evolving Flows . . . . . . 219 Guillermo Araya, Luciano Castillo, Charles Meneveau, and Kenneth Jansen Statistics and Flow Structures in Couette–Poiseuille Flows . . . . . . . . . 229 Matteo Bernardini, Paolo Orlandi, Sergio Pirozzoli, and Fabrizio Fabiani Session 6: Theory LES-Langevin Approach for Turbulent Channel Flow . . . . . . . . . . . 239 Rostislav Dolganov, Bérengère Dubrulle, and Jean-Philippe Laval A Scale-Entropy Diffusion Equation for Wall Turbulence . . . . . . . . . 249 Hassan Kassem and Diogo Queiros-Conde A Specific Behaviour of Adverse Pressure Gradient Near Wall Flows . . . 257 Syed-Imran Shah, Jean-Philippe Laval, and Michel Stanislas

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Session 7: RANS Modelling A Nonlinear Eddy-Viscosity Model for Near-Wall Turbulence . . . . . . . 269 B. Anders Pettersson Reif and Mikael Mortensen ASBM-BSL: An Easy Access to the Structure Based Model Technology . 277 Bertrand Aupoix, Stavros C. Kassinos, and Carlos A. Langer Introduction of Wall Effects into Explicit Algebraic Stress Models Through Elliptic Blending . . . . . . . . . . . . . . . . . . . . . . . . 287 Abdou G. Oceni, Rémi Manceau, and Thomas B. Gatski Session 8: Dynamical Systems POD Based Reduced-Order Model for Prescribing Turbulent Near Wall Unsteady Boundary Condition . . . . . . . . . . . . . . . . . . . . . 301 Guillaume Lehnasch, Julien Jouanguy, Jean-Philippe Laval, and Joel Delville A POD-Based Model for the Turbulent Wall Layer . . . . . . . . . . . . . 309 Bérengère Podvin HR SPIV for Dynamical System Construction . . . . . . . . . . . . . . . 317 Jean-Marc Foucaut, Sébastien Coudert, and Michel Stanislas The Stagnation Point Structure of Wall-Turbulence and the Law of the Wall in Turbulent Channel Flow . . . . . . . . . . . . . . . . . 327 Vassilios Dallas and J. Christos Vassilicos Session 9: Large Eddy Simulation Wall Modelling for Implicit Large Eddy Simulation of Favourable and Adverse Pressure Gradient Flows . . . . . . . . . . . . . . . . . 337 ZhenLi Chen, Antoine Devesa, Michael Meyer, Eric Lauer, Stefan Hickel, Christian Stemmer, and Nikolaus A. Adams LES of Turbulent Channel Flow with Pressure Gradient Corresponding to Turbomachinery Conditions . . . . . . . . . . . . . . . . . . . . . 347 Witold Elsner, Lukasz Kuban, and Artur Tyliszczak LES Modeling of Converging Diverging Turbulent Channel Flow . . . . . 355 Jean-Philippe Laval, Witold Elsner, Lukasz Kuban, and Matthieu Marquillie Large-Scale Organized Motion in Turbulent Pipe Flow . . . . . . . . . . 365 Sebastian Große, Dirk Jan Kuik, and Jerry Westerweel

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Session 10: Skin Friction Near-Wall Measurements and Wall Shear Stress . . . . . . . . . . . . . . 377 T. Gunnar Johansson Measurements of Near Wall Velocity and Wall Stress in a Wall-Bounded Turbulent Flow Using Digital Holographic Microscopic PIV and Shear Stress Sensitive Film . . . . . . . . . . . . . . . . . . . . . 385 Omid Amili and Julio Soria Friction Measurement in Zero and Adverse Pressure Gradient Boundary Layer Using Oil Droplet Interferometric Method . . . . . 393 Philippe Barricau, Guy Pailhas, Yann Touvet, and Laurent Perret Session 11: Modified Wall Flow Scaling of Turbulence Structures in Very-Rough-Wall Channel Flow . . . 405 David M. Birch and Jonathan F. Morrison Characterizing a Boundary Layer Flow for Bubble Drag Reduction . . . 413 Marc Harleman, René Delfos, Jerry Westerweel, and Thomas J.C. van Terwisga Direct and Large Eddy Numerical Simulations of Turbulent Viscoelastic Drag Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Laurent Thais, Andres E. Tejada-Martínez, Thomas B. Gatski, Gilmar Mompean, and Hassan Naji DNS of Supercritical Carbon Dioxide Turbulent Channel Flow . . . . . . 429 Mamoru Tanahashi, Yasuhiro Tominaga, Masayasu Shimura, Katsumi Hashimoto, and Toshio Miyauchi Session 12: Industrial Modeling Evaluation of v 2 –f and ASBM Turbulence Models for Transonic Aerofoil RAE2822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 Jeremy J. Benton Turbulence Modelling Applied to Aerodynamic Design . . . . . . . . . . 451 Vincent Levasseur, Sylvain Joly, and Jean-Claude Courty

Contributors

Nikolaus A. Adams Omid Amili LTRAC, Department of Mechanical and Aerospace Engineering, Monash University, VIC 3800, Melbourne, Australia, [email protected] Guillermo Araya Swansea University, Swansea, SA2 8PP, UK, [email protected] Callum Atkinson Laboratoire de Mécanique de Lille, Ecole Centrale de Lille, Bd Paul Langevin, Cite Scientifique, 59655 Villeneuve d’Ascq cedex, France and Laboratory for Turbulence Research in Aerospace and Combustion, Department Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia, [email protected] Bertrand Aupoix Aerodynamics and Energetics Department, ONERA, BP 74025, 2 Avenue E. Belin, 31055 Toulouse Cedex 4, France, [email protected] Philippe Barricau Aerodynamics and Energetics Department, ONERA, BP 74025, 2 Avenue E. Belin, 31055 Toulouse Cedex 4, France, [email protected] Jeremy J. Benton Aerodynamics Design and Data, Building 09B, Airbus, Filton, Bristol, BS99 7AR, UK, [email protected] Matteo Bernardini Dipartimento di Meccanica e Aeronautica, Università La Sapienza, Via Eudossiana 16, 00184 Roma, Italy Francesco Bertini AVIO S.p.A., Turin, Italy, [email protected] Amitabh Bhattacharya University of Pittsburgh, Pittsburgh, PA, USA, [email protected] David M. Birch University of Surrey, Guildford, UK GU2 7XH, [email protected] Patrick Braud LEA UMR CNRS 6609, Poitiers, France, [email protected] xvii

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Contributors

Luciano Castillo Rensselaer Polytechnic Institute, 110 8th St., Troy, NY 12180, USA, [email protected] ZhenLi Chen Institute of Aerodynamics, Technische Universität München, 85748 Garching, Germany, [email protected] Sebastien Coudert LML UMR CNRS 8107, Bd Paul Langevin, Cite Scientifique, 59655 Villeneuve d’Ascq cedex, France, [email protected] Jean-Claude Courty Dassault Aviation — Advanced Aerodynamics and Aeroacoustics, 78 quai Marcel Dassault, 92552 Saint-Cloud CEDEX, France, [email protected] Vassilios Dallas Institute for Mathematical Sciences & Department of Aeronautics, Imperial College London, London SW7 2PG, UK, [email protected] Juan C. del Álamo School of Aeronautics, U. Politécnica, 28040 Madrid, Spain and MAE Department, U. of California San Diego, La Jolla, CA 92093, USA, [email protected] René Delfos Joel Delville LEA/CEAT UMR CNRS 6609, Université de Poitiers-ENSMA, 86036 Poitiers, France, [email protected] Antoine Devesa Institute of Aerodynamics, Technische Universität München, 85748 Garching, Germany, [email protected] Rostislav Dolganov Laboratoire de Mécanique de Lille, CNRS, 59655 Villeneuve d’Ascq, France, [email protected] Stanislaw Drobniak Czestochowa Univ. of Technology, Armii Krajowej 21, Czestochowa, Poland, [email protected] Artur Drozdz Czestochowa Univ. of Technology, Armii Krajowej 21, Czestochowa, Poland, [email protected] Bérengère Dubrulle Groupe Instabilités et Turbulence, SPEC-CNRS, 91191 Gif sur Yvette, France, [email protected] Witold Elsner Czestochowa Univ. of Technology, Armii Krajowej 21, Czestochowa, Poland, [email protected] Fabrizio Fabiani Dipartimento di Meccanica e Aeronautica, Università La Sapienza, Via Eudossiana 16, 00184 Roma, Italy Jean-Marc Foucaut Laboratoire de Mecanique Lille, Ecole Centrale de Lille, Bd Paul Langevin, Cite Scientifique, 59655 Villeneuve d’Ascq cedex, France, [email protected] Carine Fourment LEA UMR CNRS 6609, Poitiers, France, [email protected]

Contributors

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Thomas B. Gatski Laboratoire d’études aérodynamiques (LEA), Université de Poitiers, ENSMA, CNRS, Bd Marie et Pierre Curie, BP 30179, 86962 Futuroscope Chasseneuil Cedex, France and Center for Coastal Physical Oceanography and Ocean, Earth and Atmospheric Sciences, Old Dominion University, Norfolk, VA, USA, [email protected] William K. George Chalmers University of Technology, Dept. of Applied Mechanics, 41296 Gothenburg, Sweden, [email protected] Sebastian Große Laboratory for Aero- and Hydrodynamics, Delft University of Technology, Leeghwaterstraat 21, 2628 CA Delft, The Netherlands, [email protected] Philip E. Hancock University of Surrey, Guildford, UK, GU2 7XH, [email protected] Marc Harleman Delft University of Technology, Laboratory of Aero and Hydrodynamics, Leeghwaterstraat 21, 2628 CA Delft, Netherlands, [email protected] Katsumi Hashimoto Central Research Institute of Electric Power Industry, 2-6-1 Nagasaka, Yokosuka-shi, Kanagawa, 240-0196, Japan Sophie Herpin Laboratoire de Mecanique Lille, Ecole Centrale de Lille, Bd Paul Langevin, Cite Scientifique, 59655 Villeneuve d’Ascq cedex, France and Laboratory for Turbulence Research in Aerospace and Combustion (LTRAC), Monash University, VIC 3800, Melbourne, Australia, [email protected] Stefan Hickel Nicholas Hutchins Department of Mechanical Engineering, University of Melbourne, Melbourne, Victoria 3010, Australia, [email protected] Kenneth Jansen Rensselaer Polytechnic Institute, 110 8th St., Troy, NY 12180, USA, [email protected] Javier Jimenez Universidad Politécnica de Madrid, School of Aeronautics, Pz. Cardenal Cisneros 3, 28040 Madrid, Spain, [email protected] and Center for Turbulence Research, Stanford U., Stanford, CA 94305, USA Peter B.V. Johansson TRL, Chalmers University of Technology, Gothenburg, Sweden, [email protected] T. Gunnar Johansson Chalmers University of Technology, 41296 Gothenburg, Sweden, [email protected] Sylvain Joly Dassault Aviation — Advanced Aerodynamics and Aeroacoustics, 78 quai Marcel Dassault, 92552 Saint-Cloud CEDEX, France, [email protected] Julien Jouanguy LML UMR CNRS 8107, Ecole Centrale de Lille, 59655 Villeneuve d’Ascq, France, [email protected]

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Contributors

Hassan Kassem ENSTA-ParisTech, Unite Chimie et Procédés, 32 Bb Victor, 75015 Paris, France, [email protected] Stavros C. Kassinos Computational Sciences Laboratory — UCY-CompSci, Department of Mechanical & Manufacturing Engineering, University of Cyprus, 75 Kallipoleos, Nicosia 1678, Cyprus, [email protected] Jim Kostas LML UMR CNRS 8107, Villeneuve d’Ascq, France, [email protected] Lukasz Kuban Czestochowa University of Technology, Institute of Thermal Machinery, Czestochowa, Poland, [email protected] Dirk Jan Kuik Carlos A. Langer Computational Sciences Laboratory — UCY-CompSci, Department of Mechanical & Manufacturing Engineering, University of Cyprus, 75 Kallipoleos, Nicosia 1678, Cyprus, [email protected] Eric Lauer Jean-Philippe Laval Laboratoire de Mécanique de Lille, CNRS, 59655 Villeneuve d’Ascq, France, [email protected] Guillaume Lehnasch LEA/CEAT UMR CNRS 6609, Université de PoitiersENSMA, 86036 Poitiers, France, [email protected] Davide Lengani Dipartimento di Macchine, Sistemi Energetici e Trasporti, Università di Genova, Via Montallegro 1, 16145 Genoa, Italy, [email protected] Vincent Levasseur Dassault Aviation — Advanced Aerodynamics and Aeroacoustics, 78 quai Marcel Dassault, 92552 Saint-Cloud CEDEX, France, [email protected] Y. Maciel Dept of Mech Eng, Laval University, Quebec City, Canada, [email protected] Nicholas Malaya University of Texas at Austin, Austin, TX, USA, [email protected] Rémi Manceau Laboratoire d’études aérodynamiques (LEA), Université de Poitiers, ENSMA, CNRS, Bd Marie et Pierre Curie, BP 30179, 86962 Futuroscope Chasseneuil Cedex, France, [email protected] Matthieu Marquillie Laboratoire de Mécanique de Lille, CNRS, 59655 Villeneuve d’Ascq, France, [email protected] Ivan Marusic Department of Mechanical Engineering, University of Melbourne, Melbourne, Victoria 3010, Australia, [email protected] Magdalena Materny Czestochowa Univ. of Technology, Armii Krajowej 21, Czestochowa, Poland, [email protected]

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Romain Mathis Department of Mechanical Engineering, University of Melbourne, Melbourne, Victoria 3010, Australia, [email protected] Fahrid Mehdi TRL, Chalmers University of Technology, Gothenburg, Sweden, [email protected] Charles Meneveau Johns Hopkins University, Baltimore, MD 21218, USA, [email protected] Michael Meyer Toshio Miyauchi Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan Gilmar Mompean Laboratoire de Mécanique de Lille, CNRS-UMR 8107, Université Lille I-Sciences et Technologies, Polytech’Lille, Villeneuve d’Ascq, France, [email protected] Jonathan F. Morrison Imperial College London, London, UK SW7 2AZ Mikael Mortensen Norwegian Defence Research Establishment (FFI), Kjeller, Norway Robert D. Moser University of Texas at Austin, Austin, TX, USA, [email protected] Hassan Naji Laboratoire de Mécanique de Lille, CNRS-UMR 8107, Université Lille I-Sciences et Technologies, Polytech’Lille, Villeneuve d’Ascq, France, [email protected] Paul Nathan University of Surrey, Guildford, UK, GU2 7XH Abdou G. Oceni Laboratoire d’études aérodynamiques (LEA), Université de Poitiers, ENSMA, CNRS, Bd Marie et Pierre Curie, BP 30179, 86962 Futuroscope Chasseneuil Cedex, France Paolo Orlandi Dipartimento di Meccanica e Aeronautica, Università La Sapienza, Via Eudossiana 16, 00184 Roma, Italy Guy Pailhas Aerodynamics and Energetics Department, ONERA, BP 74025, 2 Avenue E. Belin, 31055 Toulouse Cedex 4, France, [email protected] Ronald L. Panton University of Texas, Austin, USA, [email protected] Yulia Peet Institut Jean Le Rond d’Alembert, Université Pierre et Marie Curie, 4 place Jussieu — case 162, 75252 Paris cedex 5, France, [email protected] Laurent Perret Laboratoire de Mécanique de Lille, Villeneuve d’Ascq, France and École Centrale de Nantes, Nantes, France, [email protected] B. Anders Pettersson Reif Norwegian Defence Research Establishment (FFI), Kjeller, Norway, [email protected] Sergio Pirozzoli Dipartimento di Meccanica e Aeronautica, Università La Sapienza, Via Eudossiana 16, 00184 Roma, Italy

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Contributors

Bérengère Podvin LIMSI-CNRS, Université Paris-Sud, 91403 Orsay Cedex, France, [email protected] Diogo Queiros-Conde ENSTA-ParisTech, Unite Chimie et Procédés, 32 Bb Victor, 75015 Paris, France, [email protected] S. Rahgozar Dept of Mech Eng, Laval University, Quebec City, Canada, [email protected] A. Royer LEA UMR CNRS 6609, Poitiers, France, [email protected] Pierre Sagaut Institut Jean Le Rond d’Alembert, Université Pierre et Marie Curie, 4 place Jussieu — case 162, 75252 Paris cedex 5, France, [email protected] Syed-Imran Shah Laboratoire de Mécanique de Lille, CNRS, 59655 Villeneuve d’Ascq, France, [email protected] Masayasu Shimura Tokyo Institute of Technology, 2-12-1 Ookayama, Meguroku, Tokyo 152-8550, Japan Daniele Simoni Dipartimento di Macchine, Sistemi Energetici e Trasporti, Università di Genova, Via Montallegro 1, 16145 Genoa, Italy, [email protected] Julio Soria LTRAC, Department of Mechanical and Aerospace Engineering, Monash University, VIC 3800, Melbourne, Australia, [email protected] Philippe R. Spalart Boeing Commercial Airplanes, PO Box 3707, Seattle, WA 98124, USA, [email protected] Michel Stanislas Université Lille Nord de France, LML, UMR CNRS 8107, Cité Scientifique, boulevard Paul Langevin, 59655 Villeneuve d’Ascq CX, France, [email protected] Christian Stemmer Institute of Aerodynamics, Technische Universität München, 85748, Garching, Germany, [email protected] Mamoru Tanahashi Tokyo Institute of Technology, 2-12-1 Ookayama, Meguroku, Tokyo 152-8550, Japan, [email protected] Andres E. Tejada-Martínez Department of Civil and Environmental Engineering, University of South Florida, Tampa, FL, USA, [email protected] Laurent Thais Laboratoire de Mécanique de Lille, CNRS-UMR 8107, Université Lille I-Sciences et Technologies, Polytech’Lille, Villeneuve d’Ascq, France, [email protected] Yasuhiro Tominaga Tokyo Institute of Technology, 2-12-1 Ookayama, Meguroku, Tokyo 152-8550, Japan Yann Touvet Aerodynamics and Energetics Department, ONERA, BP 74025, 2 Avenue E. Belin, 31055 Toulouse Cedex 4, France, [email protected]

Contributors

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Murat Tutkun Norwegian Defence Research Establishment, P.O. Box 25, 2027 Kjeller, Norway, [email protected] and TRL, Chalmers University of Technology, Gothenburg, Sweden, [email protected] Artur Tyliszczak Czestochowa University of Technology, Czestochowa, Poland, [email protected] Marina Ubaldi Dipartimento di Macchine, Sistemi Energetici e Trasporti, Università di Genova, Via Montallegro 1, 16145 Genoa, Italy, [email protected] Thomas J.C. van Terwisga J. Christos Vassilicos Institute for Mathematical Sciences & Department of Aeronautics, Imperial College London, London SW7 2PG, UK, [email protected] Jerry Westerweel Delft University of Technology, Delft, The Netherlands, [email protected] Pietro Zunino Dipartimento di Macchine, Sistemi Energetici e Trasporti, Università di Genova, Via Montallegro 1, 16145 Genoa, Italy, [email protected]

The WALLTURB Project Michel Stanislas, Javier Jimenez, and Ivan Marusic

The WALLTURB project was a challenging research program, in line with the objectives of the EC FP6 program in Aeronautics and of strong industrial interest at intermediate and long term. The WALLTURB consortium was a combination of 16 research teams from University, research organizations and aeronautical industry shaped to tackle the challenges of the project: WALLTURB Consortium LML UMR CNRS 8107 ONERA LEA UMR CNRS 6609 LIMSI UPR CNRS 3251 Chalmers University of Technology ENSTA/ARMINES CNRS SPEC/CEA Saclay University of Cyprus University of Rome la Sapienza University of Surrey Polytechnic University of Madrid

F F F F SE F F CY IT UK SP

M. Stanislas () Université Lille Nord de France, LML UMR CNRS 8107, Cité Scientifique, boulevard Paul Langevin, 59655 Villeneuve d’Ascq CX, France e-mail: [email protected] J. Jimenez Universidad Politécnica de Madrid, School of Aeronautics, Pz. Cardenal Cisneros 3, 28040 Madrid, Spain e-mail: [email protected] I. Marusic The University of Melbourne, Department of Mechanical Engineering, 3010 Melbourne, Victoria, Australia e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_1, © Springer Science+Business Media B.V. 2011

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Technische Universität München Technical University of Czestochowa Norwegian Defence Research Establishment AIRBUS DASSAULT AVIATION

G PL NO UK F

The global aim of WALLTURB was to bring in four years a significant progress in the understanding and modelling of near wall turbulence in boundary layers. This went through: • • • • •

generating and analyzing new data on near wall turbulence, extracting physical understanding from these data, putting more physics in the near wall RANS models, developing better LES models near the wall, investigating alternative models based on Low Order Dynamical Systems (LODS).

To reach these objectives, the WALLTURB Consortium took advantage of the recent progress in the experimental and numerical approaches of turbulence and the complementary skills of some of the leading teams in Europe working on turbulence. The WALLTURB project has generated a large and original database, with new and relevant data about near wall turbulence (from both experiments and DNS). This database is giving new insights into both Zero and Adverse Pressure Gradient Turbulent Boundary Layer physics, with and without separation. It was shared by the partners, during the course of the project to extract relevant physical data. It will be made available to the scientific community. This database was used by some partners to improve RANS, and LES near wall turbulence models and to develop a LODS/LES coupling approach near the wall. The work performed has allowed further development and improvement of recent turbulence models, based on a detailed physical characterization, and the assessment of the relative merits and drawbacks of these models. To reach its objectives, the work in WALLTURB was organized in the following way: a/ A shared database on near wall turbulence At the start of the project, large amounts of recently generated data (both experimental and from DNS) was already existing at some partners on both low and high Reynolds number near wall turbulence. With the efficiency of the web and some portable data formats available, it was possible to share these data in a network between the different partners. Each partner could thus process them in order to extract relevant physical content for modelling purposes. b/ Adding new data on APG TBL Although this type of flow is of particular interest to the aeronautical industry, very little is known about the turbulence physics in adverse pressure gradient

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boundary layers, either in the near wall region or further away from the wall. As all the necessary skills were available in the consortium, it was possible to perform state-of-the-art experiments and DNS and to exploit these data to allow a better understanding of the flow physics for the APG TBL. c/ Improving near wall turbulence models RANS modeling In spite of their limitations, many consider two-equation turbulence models as a good industrial compromise between complexity and generality. It is well-known that the main weakness of these models is the length scale equation, especially near the wall. To improve such models, due to the anisotropy near the wall, it is necessary to have representative scales in different spatial directions. This information is provided by the double spatial correlation tensor, which can be fully extracted from DNS at low and moderate Reynolds number. It can also be obtained in considerable detail by PIV at high Reynolds numbers. One aim of the project was to take advantage of the shared database to extract these scales and use them to improve the models. Classical two-equations and Reynolds stress models were examined, along with advanced models such as elliptic relaxation and Algebraic Structure-Based Models. Tests of these improved models were also performed by Airbus in one of their internal codes. LES modeling Large Eddy Simulation is a prediction method which offers an alternative in some situations of strong industrial interest, especially where the RANS approach fails due to its basic hypothesis. It has been demonstrated that the standard LES models fail near the wall, unless they are turned locally into a DNS in order to represent correctly the near wall turbulence structure. This is very memory and time consuming, and limits presently the extensive use of this approach in industry. The objective of WALLTURB was here again to take advantage of the network database to quantify the accuracy of standard LES models near the wall and develop alternative approaches near the wall. LODS/LES coupling By introducing the Proper Orthogonal Decomposition (POD) and the Low Order Dynamical System (LODS) approach, Lumley and his co-workers (a number of them partners in this proposal) have opened an entirely different approach to near wall turbulence modelling. This is easily demonstrated by the number of teams working on this approach around the world. LODS is an elegant way to cope with the representation of the relevant coherent structures in turbulence, and it has already shown its ability to represent the very near wall structure and dynamics. Although it is not suited to represent a full industrial flow, this approach is a very good alternative, solidly grounded in physics, to adapt LES near the wall. The detailed DNS/PIV/HW data that was maid available allowed to develop such models, and test them and their underlying hypotheses on real near wall flows.

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Fig. 1 Workprogram structure

For management purposes, the work program was divided into 6 work packages which formed coherent groups of similar tasks. The diagram in Fig. 1 makes clear important links between them. As can be seen, the program was organized in three levels. At level 1, WP1 covered all the management and exploitation activity needed for smooth operation of the project. The WALLTURB web site (http://wallturb. univ-lille1/fr) was an important tool of this work package. At level 2, the objective of the consortium was to generate a full database of new results on wall turbulence. These data came from both experiments and DNS. The most advanced tools were used to generate them. They organized as a network shared database. This level was structured in two workpackages: WP2 was focussed on the experiments and DNS performed during the project. It was divided in 4 tasks in order to clearly identify the type of experiments and DNS performed. WP3 was the kernel of the project as it was responsible for the management and processing of the different databases (a total of 8) that were be put in common by the partners. These databases were fed both by existing data and by the experiments performed in WP2. The workpackage was organized in three tasks which group the databases in a coherent manner with respect to the type of flow. Level 3 was divided into three work packages which were devoted to the development, improvement and validation of three different kinds of modelling approaches: WP4 was concerned mostly with the classical and industrial RANS approach and had the aim of improving the physical content of the models. WP5 was devoted to the improvement of LES modelling near the wall, and especially the investigation of new models for this region. WP6 did investigate the possibilities of the fairly recent Low Order Dynamical Systems approach, and of its coupling with LES in the near wall region.

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Of course, one of the main interest of the project was the direct exchange which existed during all the course of the program between levels 2 and 3. The aim was to make the best use of the database generated at level 2 to help building and validating the models developed at level 3. A good deal of the results obtained during this project are presented as papers in the present book. To give a general overview of the output of the project, an extended summary is available on the CD attached to the present book.

Invited Speakers

• The Law of the Wall. Indications from DNS, and Opinion P. Spalart • A Web-Services Accessible Turbulence Database and Application to A-Priori Testing of a Matrix Exponential Subgrid Model C. Meneveau • Modeling Multi-point Correlations in Wall-Bounded Turbulence R.D. Moser, A. Bhattacharya, and N. Malaya • Regularization Modeling for LES of Turbulent Flow Separation B. Geurts (no paper) • Reduced-, Low- and Least-Order Models of Turbulence: Insights from Nonlinear Dynamics, Statistical Physics and Turbulence Theories B. Noack (no paper) • Theoretical Prediction of Turbulent Skin Friction on Geometrically Complex Surfaces P. Sagaut and Y. Peet • Scaling Turbulent Fluctuations in Wall Layers R.L. Panton

The Law of the Wall. Indications from DNS, and Opinion Philippe R. Spalart

Abstract The law of the wall and its companion the logarithmic law are at the center of our knowledge of wall turbulence, and after almost a century still the subject of intense speculation and proposed amendments, based on experimental and simulation results, as well as putative theories. In particular, the consensus over the value of the Karman constant κ has been lost. However, compensation by the additive constant C is such that the two main contenders cross at y + = 510, and differ by 1 unit in U + only when y + = 40,000; thus, the effect is modest in practice, and any errors in skin-friction measurement are magnified. It has further been suggested that κ has different values in different flows, such as boundary layers and pipes, and in different pressure gradients. This seeming amendment amounts to abandoning the concept of a law of the wall, which would be very damaging to our prediction power in wall-bounded flows. At the same time, it is noted again that the law of the wall fails even for the simplest quantities other than the mean velocity. A model problem is described which generates log laws with artificial values of κ, both for the true flow and for RANS CFD, but only as long as the flows in question are unsettled. Results of Direct Numerical Simulations (DNS) with different enough numerics are found to be consistent with uniqueness of the laws, but these results are still narrow-based and unable to indicate the value of κ, even after the very consequent increase in Reynolds number since the 1980s. They have firmly established the “DNS answer” for the law of the wall well beyond 100 wall units, but not up to 500, and there is now agreement that the log law is entered only near 300 at best, and is entered from above contrary to previous expectations. The general message is to acknowledge the weakness of theory and DNS but to reason, without a proof, that the credibility of the twin laws for mean velocity is intact, and in particular that a true and unique value of κ exists.

P.R. Spalart () Boeing Commercial Airplanes, PO Box 3707, Seattle, WA 98124, USA e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_2, © Springer Science+Business Media B.V. 2011

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1 Classical Position The readers of this volume do not need background. This position therefore will be presented without justification, for clarity and notation; a good source, in addition to numerous textbooks, is Ref. [1]. The interaction between inner and outer layer consists in the friction velocity uτ , which the dynamics of the two layers determine together. The wall distance is y, and the thickness of the entire turbulent layer called δ, even if it is the pipe radius R or channel half-width h; the kinematic viscosity is ν. The flow is in the x direction. Wall units denote quantities non-dimensionalized with uτ and ν, for instance, y + ≡ yuτ /ν. It is assumed that “history effects” are weak in such a wall-bounded shear flow, thus allowing the inner/outer-layer interaction to reach an established and potentially universal state dominated by uτ . Logically, this implies ∂x 1/δ, and an equivalent limit of the intensity of time variations: ∂t uτ /δ. At the least, it implies considering y small enough to ensure ∂x 1/y and ∂t uτ /y. The pressure gradient in wall units satisfies p + 1, with typical values well below 0.01.

1.1 Mean Velocity The law of the wall states that, for y δ, U + (y) = f (y + ),

(1)

the short-hand “f ” meaning that U + is a unique function of y + . It is recognized that the range of applicability, that is, the value of y/δ up to which the law applies, is flow-dependent. A typical value is 0.1. The logarithmic law states that, if further y + 1, 1 log(y + ) + C, (2) κ where κ is the Karman constant and C the intercept. Here, the lower limit on y + has been thought to be of the order of 100. As a result, a log region would barely begin to form when δ + exceeds 1,000. This lower limit most probably needs to be revised upwards, from values accepted in the 20th century. The fit of Chauhan et al. implies that it is near y + = 300 [2] while Superpipe data analysis suggests values as high as 600 [3]. This is related to George’s concept of the Mesolayer, discussed in this volume. Furthermore, a consensus now exists that the velocity profile blends into the log law from above; in other words there is an overshoot (U + > log(y + )/κ + C), which peaks around y + = 40. The Kolmogorov spectrum is similarly thought to overshoot the k −5/3 law before plunging in the viscous range. This invites confusion, but is not at all in conflict with theory. A consequence is that the local “Karman Measure” (KM) d(log(y + ))/dU + (the only impartial way to exhibit logarithmic behavior and truly reveal κ) also overshoots its final value, the overshoot of the KM peaking closer to y + = 70 at about 0.44. DNS and experiments agree on this. This makes it easy U+ =

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to extract an erroneous and excessive value from results at an insufficient Reynolds number. In simplistic terms, to mistake an inflection point for a straight region in the velocity profile, or to mistake an extremum for a plateau in the KM. However, it is not the case that 0.44 is currently proposed as the true value of κ for that reason; the Superpipe data was first thought to indicate 0.436, but was definitely not limited to low Reynolds numbers. This new situation places the direct finding of a log law well out of reach of DNS for the time being, since the best value of δ + reached is only 2,000 [4]. We may plausibly define such a direct finding as having the Karman Measure substantially constant over a factor of 2 in y + , demanding δ + in the 6,000 range at the least (assuming an upper limit at δ/10).

1.2 Other Quantities The reasoning applied to the mean velocity, logically extended, predicts for a Reynolds stress, for instance the streamwise intensity, u +2 (y) = f (y + ).

(3)

Wall quantities are also predicted to be universal constants, for instance, the pressure + rms at the wall is pwall = f . Similarly, these quantities take constant values in the + log layer, i.e., if y 1 and y/δ 1, u+2 (y + ) = f.

(4)

Now it has long been observed that these properties are incorrect, at least in the Reynolds-number range of DNS or even almost all experiments. The shear stress u v has a dependence directly attributed to that of the total shear stress, due to the pressure gradient which follows the inverse of the Reynolds number δ + in channel flow. The wall-normal stress v 2 also appears well-behaved, but the other Reynolds stresses at fixed y + depend on Reynolds number, pressure gradient, and flow type in ways which have not been convincingly linked to quantities such as δ + . In channel flow and zero-pressure-gradient they do so mildly, but unquestionably, and are definitely not uniform in the log layer, in conflict with (4). For instance the peak of u v is well-understood in terms of location and value, but that of u 2 is not. The same applies to the wall pressure rms, the wall dissipation, and so on. Physical justifications invoking “inactive motion,” that is, eddies with wave+ lengths λ y, are very plausible and supported by data. For instance, pwall is predicted by this theory to increase like the logarithm of the Reynolds number δ + ; experiments come close to it. In the same framework the diagonal Reynolds stresses probably follow log(y/δ) in the log layer, rather than (4); both properties, or rather conjectures, derive from a spectrum in 1/k which is fully in line with law-of-thewall thinking. Nevertheless, these variations raise legitimate suspicion over the law of the wall. It is possible that such variations do saturate, but only at surprisingly large values of the Reynolds number; a suggestion made by Hunt years ago. Some superpipe findings are consistent with this idea, but not confirmed yet.

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1.3 Behavior of Turbulence Models Near a wall, turbulence models are driven by the mean shear dU/dy and influenced by wall proximity, either directly in some terms or via diffusion terms and boundary conditions. They incorporate the viscosity. As expected, they all reproduce the behavior outlined in Sect. 1.1 for mean velocity, both the response to y δ and that to y + 1. None have intentional overshoots of U + over the log law, but this has little impact. Models have not been created outside the classical position. More irritating is the fact that the models naturally also follow the “predictions” of Sect. 1.2. They return the law of the wall for all Reynolds stresses, and in particular constant values in the log layer. The prototype for this is in the k–ε model: the √ turbulent kinetic energy (TKE) k satisfies k + = 1/ cμ where cμ is a key constant in the model. This has little or no influence on the mean flow, consistently with the concept of Inactive Motion, but it will plague all attempts to make models correctly reproduce quantities besides mean velocity and shear stress. Sadly, this is precisely where DNS could have best guided the calibration of Reynolds-stress transport models: to begin with, over the anisotropy of the tensors. This problem is absent only in one-equation models. It can be argued with good reason as Wilcox did that the models need only account for the “Active Motion,” but a precise quantitative definition of it is not in hand. The content of a turbulence model as applied to wall-bounded flows is usefully divided into its “law of the wall” and its reactions in the outer part of the flow. The first is summarized well with the values of κ and C, and is not very different between models. The second is much more involved, full of variety between models, and not as deliberately “designed” by the modeler. It results from a few simple and arbitrary terms, heavily constrained by invariance and dimensional analysis, and is also a compromise between desired performance in wall-bounded and in free shear flows. Going back 40 years to the Stanford conferences, testing has aimed at determining which models have better “luck” in the outer layer, with pressure gradients. Adverse gradients make the outer region occupy more of the layer, and account for more of the velocity difference. These regions are the most important in practice.

1.4 Alternative Analytical Proposals The principal contender of this type has been the power law, which had high empirical status in the early 20th century, and still does in some literature on atmospheric boundary layers, as opposed to engineering. It had a resurgence around this turn of centuries, this time based on theoretical arguments, or rather mathematical ones. In the elementary arguments used to motivate the laws, the root of the difference between power law and log law comes from letting the velocity U (y) itself enter the inner-outer matching process, rather than only its derivative dU/dy. In other words, it is a difference of position on Galilean invariance. Most unfortunately, none of the power-law literature brings this out, thus missing a key chance to debate the physics.

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For this simple reason, all modern turbulence models, based on invariant equations (i.e., sensitized only to the local gradient of velocity), choose the log law over the power law. Another drawback of power laws is that their generalization to rough walls, riblets, flows over belts (non-zero slip velocity), and so on, is very uncertain whereas the same task for the log law is conceptually limpid. For instance, roughness and riblets reduce and increase the C constant, respectively, leaving κ exactly the same, and the laws in the outer layer unchanged. Again, the convenience of a conjecture does not imply its correctness, but clarity and well-understood generalization to other situations are always desirable. Note that the argument for Galilean invariance in what may be labeled turbulence semi-theories and in turbulence models is not a rigorous one (in contrast with arguments over the acceleration, made in Ref. [10]). We all recognize that every region of the entire turbulent flow is interacting with every other region and every boundary condition, so that the no-slip condition U = 0 at y = 0 is in fact information pertinent to the turbulence at y = 0. Turbulence is not local, even if models are. In other words, this invariance argument is a conscious simplification, but it is a clear and meaningful one. It can be argued that the log law is the envelope of power laws, with the power tending to 0 from its historical value near 1/7. This could bridge the two schools of thought. However, the power must then depend on Reynolds number, which conflicts with the law of the wall (1). This power could then be adjusted to better fit data towards lower Reynolds numbers, but only at the cost of making the theoretical setting even more complex. Another point which bears repeating is that higher-order extensions of knowledge such as the law of the wall has always been beset by the choice of the small parameter “ε” in the powers of which to make an expansion. The law of the wall itself is not a rigorous first-order expansion of a know set of equations, the way the boundarylayer equations are. As mentioned, it is a simplification with much support in ex+ periments and DNS. In wall-bounded flows, ε could be any inverse power of δ , or the skin-friction coefficient Cf , or Cf . Thus, no matter how much math follows, these attempts begin with an arbitrary decision. Curiously, when this is contemplated for a RANS model, we do have a well-defined set of differential equations. However, these equations are a greatly stripped-down description of turbulence, and what they predict has little status in theory. Recall their “naïve” behavior for the diagonal Reynolds stresses and for wall values, discussed above.

2 Conflicting Experiments Recent experiments in pipes and boundary layers conflict with the previously accepted range of values for κ, roughly [0.40, 0.41], but also with each other. The difference is nearly 10%, which appears very large; however, the differences in C bring the laws back to close agreement in the y + range of a few hundred (and both teams believe the region near 100 and 200 is below the log law), so that very high

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Reynolds numbers are needed for the conflict to become visible, and the impact in practice is certainly not near 10% on key quantities such as the skin friction. To determine this, the author has shown using calculations with a modified Spalart–Allmaras model that changing κ from 0.41 to 0.436 only altered the skin friction coefficient Cf by about 2% even at the highest Reynolds numbers reached over large vehicles [9]. The model was trained to the new log law in its inner layer, and the outer layer re-adjusted to the same Cf at Rθ = 104 , following the original calibration. An extended exchange with Prof. Nishioka in 2008 showed that this procedure was subtly erroneous, because there is no reason to retain the same outer-layer formula in the model. Recall how arbitrary these formulas are. In the S-A model, it is plain that the fw function has an arbitrary shape, and only its slope at r = 1 is adjusted to match data, at Rθ = 104 only. There is no systematic process to improve the rest of the curve. The cb constants which control free shear flows are also arbitrary. A start, however, would be to demand a match of the correct skinfriction coefficient at two Reynolds numbers rather than only one. Matching three values may guide both the Karman constant and the outer-layer response. On the other hand, devoting so much attention to the flow without pressure gradient would ignore the higher priorities. To summarize, this exercise in 2000 with the S-A model is less quantitative than it appeared, but the point that κ could be changed by 10% without sending a “tsunami” across the world of RANS CFD is absolutely correct. Consistently, users and developers of RANS models have shown very little interest in the Karman-constant controversy, let alone power laws. The author has a version of the S-A model with each “new” κ, but has refrained from spreading them, following a much appreciated policy of minimizing the number of versions of the model. Returning to the experiments, both derive their importance from having very high Reynolds numbers and independent measurement of skin friction (by Oil Film Interferometry for boundary layers) to give uτ ; Clauser plots are avoided. Because of the weak sensitivity discussed above, the requirement on Cf is below 1%, which is a tall challenge. The experimental findings also depend on corrections for probe size, strong because the range of scales in the facility is so wide; this difficulty is analogous to that in DNS. This author is not qualified to judge on experimental techniques, but the low “gain” from (κ, C) to Cf does help make sense of this conflict.

3 Proposals of Non-uniqueness 3.1 Essence of the Proposals The controversial recent idea is that different flows would contain log layers at sufficient Reynolds numbers, but with different κ values. It is most clearly advocated by Nagib & Chauhan [7]. The initial driving force was the conflict between pipe and boundary-layer results, but the idea was extended to boundary layers with different

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pressure gradients, and channel flow was also attributed a value, incidentally not one close to that for the pipe. This is noted because two internal flows with favorable pressure gradient could be expected to deviate from boundary layers in similar ways. Logically, Couette flow would have its own value. In any case, in this setting, the teams studying different flows would declare that they are all correct instead of challenging each other’s κ value. As of today, the Superpipe results are isolated in terms of κ, but are also the ones with the highest Reynolds numbers, an absence of side-walls, and the best handle on uτ . They have also been scrutinized and revisited for years, giving them much weight.

3.2 Conceptual Consequences The existence of different κ values runs against the law of the wall (1). It denies the argument that when y δ, the type of outer flow is immaterial. Therefore, the proposals for different values represent a frontal attack on the classical thinking, rather than a refinement of that thinking. This is not addressed in the papers, and must be viewed as a most controversial aspect of them. The consequences for the predictability of turbulence are considerable. As a example, the flow along a long cylinder (such as a submarine or airplane body) could not be accurately predicted even very near the wall, given any amount of knowledge of other boundary layers. It is a different canonical flow. By the same logic, the flow past a cone may have its own Karman coefficient, as may the flow in a diffuser, and the Ekman layer. The all-important flow over a wing would be in a class of its own. A practical consequence would be to make turbulence modeling unmanageable. Current models strongly conform with the expectation of a universal near-wall behavior, and a precise Karman constant can be deduced analytically from the equations, by assembling a log law with all the turbulence variables obeying dimensional analysis, and applying the momentum and model equations. This value comes out in numerical solutions at sufficient Reynolds numbers. Note that this sufficient value is quite different in different models; it is particularly high in the k–ω and therefore the SST model. It is of course conceivable that the conception of the models has been marked by a narrow-mindedness driven by the classical thinking. Some observers believe a radically different and superior theory of turbulence exists, and will be discovered.

3.3 A Situation with Log Laws and Erratic κ Values During the Wallturb meeting, Prof. Nagib presented CFD results in which different κ values were returned by the same turbulence model in different flows. This was a surprise to the author, since as mentioned each model has a “hard-coded” value of κ, for instance 0.41 for Spalart–Allmaras and 0.4082 for k–ω and SST. Dr. Aupoix in

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private cited similar findings in appreciable pressure gradients. This suggested that even a (pseudo-physical) system with a unique internal Karman constant could give the appearance of non-uniqueness, and led to the following construct. Consider a steady channel flow with an established log layer and the Karman constant κ0 , the friction velocity being uτ 0 . Now obtain a sudden acceleration or deceleration via a strong pressure gradient dP /dx, applied for a short time. It alters the channel centerline velocity significantly. Except near the wall, vorticity is conserved through the sudden stimulation by pressure and therefore dU/dy is unchanged. An internal boundary layer forms at the wall, and the friction velocity uτ (t) varies widely. When the velocity profiles are plotted in “current” wall units at different values of t, a log layer will be present at the same y levels as before the change. Its Karman Measure will be given by uτ (t)/(ydU/dy), and dU/dy is still given by the undisturbed profile: dU/dy = uτ 0 /(κ0 y). In the end, the Karman “constant” exhibited at time t in that region will be: uτ (t) κ0 . (5) uτ 0 This reasoning applies equally well to the true flow and to its rendition in RANS CFD. Both will conserve vorticity and create the internal boundary layer under an unchanged region (in terms of dU/dy). Therefore, this unsettled situation will create log layers which can remain long and therefore have the appearance of a standard situation, but the κ value will be controlled by the evolution of uτ (t) as seen in (5), and therefore vary freely, and also be different between the true flow and the models. Decelerated flows have low apparent Karman constants (because uτ (t) < uτ 0 ), and accelerating flows high ones. The brief pressure gradient does not alter the fluctuating vorticity field in the bulk of the flow any more than it does the mean vorticity, so that the turbulence is not altered. Since good turbulence models are not sensitized to pressure (directly, in their formulation), they correctly reproduce this behavior. Eddy viscosity and shear stress are unchanged. After the sudden alteration of the flow, it will relax towards a new equilibrium in a new mass flow, with the internal boundary layer gradually invading the log layer from below until the “correct” Karman constant κ0 comes through again. This situation, therefore, can easily make different κ values result from the same RANS model, as observed by Nagib and Aupoix although not in this exact flow. Figure 1 presents a calculation by the group of Dr. Strelets, and matches the expectations and (5). The channel starts at Reτ = 106 . It is then decelerated to reduce the centerline velocity by 50% within a time t = 0.05h/uτ 0 , after which the sustained pressure gradient is reduced by 75%, so as to also reduce uτ in the final equilibrium by 50%. If the sustained pressure gradient had been returned to its original value, the flow would have taken longer to return to equilibrium, but the qualitative behavior would have been the same. The profiles at different times clearly show the “fossil” log layer with very low κ (high slope of U + versus y + ), and the “live” log layer rising from the wall. Spatially-developing flows do not obey as simple an equation as this channel, because the streamlines along which vorticity is conserved do not remain at a conκ(t) =

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Fig. 1 Velocity profiles in channel flow after sudden deceleration

stant y, but it is very plausible that similar “non-conforming” log layers can be observed both in reality and in RANS. Some measurements in Ref. [8] could also be interpreted as having, in a single profile, two log layers with different Karman Measures. Naturally, the boundary between “settled” and “un-settled” flows is a vague one.

4 DNS Evidence 4.1 Logarithmic Law The DNS evidence in the debate over κ values is unfortunately non-existent, an observation which was less clearly formulated when the title of this article was chosen than it is now. The primary reason is the need, discussed above, for Reynolds numbers far above even what record-setting computing efforts can reach today [4]. A secondary reason is that a close examination of Ekman-layer DNS results the author co-wrote [12] and was counting on to support the log law showed that the accuracy was unfortunately not sufficient by modern standards. The published results were consistent with κ near 0.38, if an origin shift was allowed, i.e., using log(y + + 7.5) in place of log(y + ). The Karman Measure was then fairly convincing, although not enough to remove the word “cautious” from the title. However, subsequent comparisons with channel-flow DNS [4] were not favorable even in the buffer layer, prompting a grid resolution study. Simulations under way with finer grids in preparation for a correction now compare well with other DNS but will not reach as high a Reynolds number; in addition, they appear to demand a much larger origin shift, which becomes difficult to reconcile with the thickness of the buffer layer. The failure of the Karman Measure (without shift) to show a plateau in the

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few-hundred range of y + , which was a major disappointment in channel flow, essentially is also present in the Ekman layer. This would not have been expected in the early days of DNS, when the value of δ + was only 1/10th of current ones.

4.2 Law of the Wall In contrast to the log-law matter, the DNS evidence in favor of the Law of the Wall (1) is unanimous; with fine enough grids, simulations with rather different numerics agree very closely up to y + of a few hundred. This includes pipe flow [13]. When the profiles eventually differ, they do so in a manner consistent with the difference in flow types, once the condition y δ is not satisfied any more. An odd event is that the published Ekman-layer profiles agreed to perfection with the analytical fit to experiments made by Chauhan et al., but the improved Ekman results do not. The disagreement is best illustrated near y + = 30: experiment as fitted gives about 13 for U + , and DNS about 13.3. This needs to be resolved in the coming years; however, it is difficult to imagine all the DNS results moving to 13. The buffer layer is a strength area for DNS, not for high-Reynolds-number experiments. It is also difficult to imagine the DNS results confirming κ values in the region of 0.42; the current trends, however vague, suggest values below 0.40. The DNS evidence against (3) and (4), a law of the wall for diagonal Reynolds stresses, is just as unanimous even where U + displays excellent agreement. As mentioned above, it remains possible that this is a “low-Reynolds-number effect,” but it will be late in the century before DNS can make claims on this precise question, and the conjecture of Inactive Motion is quite a credible one; there is little reason why it should stop adding turbulent energy at higher Reynolds number.

4.3 Response to Pressure Gradients The inconclusive findings led the author and his partners away from a pure Reynolds-number quest towards other effects, again pressure gradients [5]. This follows studies in strained channels, conducted in very much the same spirit [6]. The article will not be repeated here, only its premise. DNS was conducted of a Couette–Poiseuille flow to have one wall with favorable pressure gradient, and the other with adverse gradient. The question is then which of the three standard behaviors of a length scale in a zero-pressure-gradient flow survives, if any. The first is a logarithmic velocity law; the second, the mixing length being equal to κy; and the third, the eddy viscosity being equal to κyuτ . In the author’s view the three “laws” are equally plausible, so that only simulations and measurements can provide any answer. It will be more useful to RANS modelling than to theory, since this is not an asymptotic result, with y δ; the interest is at finite values of y/δ. There could also be a lesson in turbulence physics. Near both walls, the outcome is that the velocity

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law is much more closely satisfied than the other two. This is not new with favorable gradient, Poiseuille flow having that behavior. It is of interest in the adverse gradient. As of writing, this manuscript is being revised, and may stimulate other work in Couette–Poiseuille flows, including experiments. DNS will be more helpful than experiments in studies of coherent structures, which take advantage of complete solution fields. This kind of knowledge should stimulate inventions towards drag reduction. However, the most promising concept, namely riblets, is not traced to DNS. Nowadays, riblets could be convincingly tested and refined by DNS, and Large-Eddy-Break-Up devices have been tested by DNS, unfortunately without leading to applications [11].

5 Highlights The place of invited lecturer was an encouragement to freely express opinion, and this has been the case here, especially since the DNS evidence appeared weaker than expected at the outset. Note that the claim itself that the answer to a recognized question is a matter of opinion can be non-trivial, especially if the claim is that it will remain so; in other words, that the question will never receive a rigorous, conclusive, mathematical answer. Unfortunately, several claims of this nature are made here, which does not contribute to the beauty of turbulence research (recall words such as “semi-theory”), but could steer effort away from wasteful projects. With any modesty, it is hard to imagine a “clean” solution appearing now to a conceptual problem which occupied some excellent minds for almost a century, before the computer age and its distractions. The rather weak sensitivity of turbulence predictions to the disputed values of the Karman constant κ (coordinated with changes in C) was noted, both to put the practical importance of the debate in perspective and to better understand how a 10% uncertainty in κ could materialize suddenly. The recent literature has shown that, unfortunately, pursuing DNS solely for the purpose of a Reynolds-number “record” is not very rewarding. In particular, extracting a κ value will be out of reach for many years. This brings up an area of new consensus between DNS and experiments: the log layer is truly entered only at higher values of y + than was previously believed, namely at several hundred, and is entered from above. The lack of reward in Reynolds-number endeavors by DNS leads to introducing other effects such as pressure gradients and strains; a Couette–Poiseuille flow which is promising in this area has been submitted for publication. It was shown using a simple unsteady channel-flow problem that rapid perturbations can create log layers with artificial Karman constants, both in reality and in RANS calculations. However, these artificial κ values vary continuously in time, and it would not be correct to describe such flows as canonical. Most important is the point that introducing flow-dependent log laws amounts to a negation of the Law of the Wall itself. The internal logic of such a step is in fact very debatable. Separately in each type of flow, it makes the argument that the

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value of δ loses its influence on (1) and (2) at height y once δ y, but it denies the argument that the type of outer flow also loses its influence.

References 1. Bradshaw, P., Huang, G.P.: The law of the wall in turbulent flow. Proc. R. Soc. Lond. A 451, 165–188 (1995) 2. Chauhan, K., Nagib, H., Monkewitz, P.A.: On the composite logarithmic profile in zero pressure gradient turbulent boundary layers. AIAA-2007-532 3. Coleman, G.N., Kim, J., Spalart, P.R.: A numerical study of strained three-dimensional wallbounded turbulence. J. Fluid Mech. 416, 75–116 (2000) 4. Hoyas, S., Jiménez, J.: Scaling of velocity fluctuations in turbulent channels up to Reτ = 2003. Phys. Fluids 18, 011702 (2006) 5. Johnstone, R., Coleman, G.N., Spalart, P.R.: The resilience of the logarithmic law to pressure gradients: evidence from DNS. J. Fluid Mech. 639, 443–478 (2009) 6. McKeon, B.J., Li, J., Jiang, W., Morrison, J.F., Smits, A.J.: Further observations on the mean velocity distribution in fully developed pipe flow. J. Fluid Mech. 501, 135–147 (2004) 7. Nagib, H.M., Chauhan, K.A.: Variations of von Karman coefficient in canonical flows. Phys. Fluids 20, 101518 (2008) 8. Nagib, H.M., Christophorou, C., Monkewitz, P.A.: High Reynolds number turbulent boundary layers subjected to various pressure gradient conditions. In: IUTAM 2004: One Hundred Years of Boundary Layer Research, Aug. 12–14, 2004, Göttingen, Germany 9. Spalart, P.R.: Trends in turbulence treatments. AIAA-2000-2306 10. Spalart, P.R., Speziale, C.G.: A note on constraints in turbulence modelling. J. Fluid Mech. 391, 373–376 (1999) 11. Spalart, P.R., Strelets, M., Travin, A.: Direct numerical simulation of large-eddy-break-up devices in a boundary layer. Int. J. Heat Fluid Flow 27(5), 902–910 (2006) 12. Spalart, P.R., Coleman, G.N., Johnstone, R.: Direct numerical simulation of the Ekman layer: a step in Reynolds number, and cautious support for a log law with a shifted origin. Phys. Fluids 20, 101507 (2008) 13. Wu, X., Moin, P.: A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow. J. Fluid Mech. 608, 81–112 (2008)

A Web-Services Accessible Turbulence Database and Application to A-Priori Testing of a Matrix Exponential Subgrid Model Charles Meneveau

Abstract We review a new public database of isotropic turbulence and methods of accessing it. In particular, a Web-Services Method is used that enables analysis programs running on user’s local computer to access the database with subroutine-like calls to the data over the Internet. The data are from DNS of isotropic turbulence at Rλ ∼ 430 and contain 1 turn-over time evolution of a 10243 simulation, with 1024 stored time-steps. User-defined functions are implemented as stored procedures on the database servers, and include functions such as differentiation, interpolation, box-filtering as well as calculation of subgrid-scale stresses. These functions are used to evaluate a-priori a new subgrid scale model based on matrix exponentials proposed in Li et al. (J. Turbul. 9(31):1–29, 2008). It is found that correlation coefficients for this model are significantly higher than for eddy-viscosity closures. However, expansions of the matrix exponential can be used to relate the model to prior nonlinear models, which suggests that the increased correlation may not be dynamically significant. The need for additional public databases for wall-bounded flows, such as channel flow, is highlighted.

1 Introduction: The Web-Accessible Public Turbulence Database The scale of scientific computing in fluid dynamics, and turbulence in particular, continues to grow. The number of grid-points now often exceeds three orders of magnitude in each of the three space directions. However, only a relatively small fraction of research groups in turbulence and fluid dynamics have convenient access to the considerable cyber infrastructure required to analyze and process the resulting data sets. In order to address this important challenge, a new approach has been proposed and implemented [2]. The approach is based on a public database system which archives a direct numerical simulation (DNS) data set of isotropic, forced C. Meneveau () Johns Hopkins University, Baltimore, MD 21218, USA e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_3, © Springer Science+Business Media B.V. 2011

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turbulence. The data set consists of the DNS output on 10243 grid points and 1024 time-steps that span roughly one large-scale turn-over timescale. Therefore, a complete 10244 space-time history of turbulence, comprising 27 Terabytes, is easily accessible to remote users. The users may access the data remotely through an interface that is based on the Web-services model. As explained in detail in Ref. [2], users may write and execute analysis programs on their host computers, and the programs can make subroutine-like calls requesting desired parts of the data over the Internet. Using regular platforms such as laptops, remote users are thus able to perform numerical experiments by accessing quite easily desired parts of the 27 Terabytes of DNS data. The database is available from its portal at http://turbulence.pha.jhu.edu. Access to data in this fashion allows analysis tasks that are difficult if not impossible to perform during DNS as the simulation proceeds. For example, a user may inquire about the pre-history of particular fluid elements in the flow. As an example, Fig. 1 shows a snippet of a fortran program which accesses the database using the function GetVelocity inside a loop that advects fluid particles backwards using a simple 1st order Euler integration in time (for purposes of illustration). The equation being integrated backward is dxp /dt = −u(xp , t), using time-step dt. The vector function u(xp , t) is the velocity field from the DNS evaluated at time t at the particle position xp . Since the data are stored only on grid points and discrete times, the database function returns interpolated values of the velocity. As explained in detail in Ref. [2], in space we use Lagrange polynomial interpolation. In time, piecewise cubic Hermite polynomial interpolation is used. In the illustrative example shown here, a set of 1000 particles are selected along a square (250 particles to each side), and the backward time-evolution is shown in Fig. 2. Thus a remote user may analyze the geometric features of a loop and its time-evolution given that at the final time it will become the specified square. Analysis and visualization may also be done extracting samples of the database. For instance, a single command may extract all 9 elements of the velocity gradient tensor Aij = ∂ui /∂xj on a grid of points in a plane inside the data. If one is interested in the structure of the dissipation field (where the dissipation rate is defined as ε(x) = 2νSij Sij , Sij = (Aij + Aj i )/2 is the strain-rate tensor, and ν is the fluid viscosity) one may use the command GetVelocityGradient for (say) a square of 256 × 256 points on a subplane. Figure 3 displays the dissipation thus plotted as a 3D plot and as a contour plot at the bottom of the figure. The well-known intermittent structure [1] of the dissipation field is immediately visible. For more details about the architecture of the database, usage statistics, speed and performance measures, as well as the locally defined functions, such as differentiation and interpolation, see Ref. [2].

2 The Matrix Exponential Subgrid Model for LES We first describe the model and in the next section use the database to perform some a-priori tests of the model. The general background describing the matrix exponen-

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Fig. 1 Snippet of FORTRAN program that assigns values to 1000 points along a square loop, then enters an iteration that advects each point by calling the database to obtain fluid velocities at the points using the function GetVelocity

tial model as derived from the stress transport equation, or alternatively based on a multiscale expansion, has been presented in [3]. Here we present the model based on a simple argument for the time-evolution of a velocity fluctuation. The basic turbulence modeling problem for Large Eddy Simulations (LES) is to model the subgridscale stress tensor in terms of the large-scale flow. Using a scale-decomposition of the velocity as u = u˜ + u where u˜ is the large-scale component and u the small scale one (see e.g. [5]), the stress tensor containing only small-scale quantities is u . As is usually done, a transport equation can be written for the given by τij = u i j fluctuation velocity, starting from the Navier–Stokes equations. It contains an advective term, a linear stretching-rotation term by large scale gradients, a pressure term, a viscous term and a nonlinear interaction term. In the ‘linear’ approximation (reminiscent of Rapid Distortion Theory), and neglecting any pressure and viscous effect, the highly simplified equation for the fluctuation reads du ˜ ≈ −u · A, dt

(1)

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Fig. 2 Loop of fluid particles being advected backwards in time, integrated on a user-side host computer but using the public Web-accessible turbulence database to obtain the interpolated fluid velocities from a 10244 space–time dataset

Fig. 3 Snapshot of intermittent dissipation field on a plane (256 × 256 points) of data obtained from the public Web-accessible turbulence database, using the function GetVelocityGradient and evaluating the dissipation based on the symmetric part of the velocity gradient tensor at each point

where d/dt denotes the Lagrangian material derivative. Next, we integrate this equation formally along a fluid particle path between some time t0 and t0 + ta , ˜ = ∇ u˜ does not change during this time interval ta . This assumes and assume that A a separation in time-scales, which as is well known does not hold in turbulence, but will be used here nonetheless as a strong approximation. The solution to Eq. 1 can be written in terms of the matrix exponential function as follows ˜ · u (t0 ). u (t0 + ta ) ≈ exp(−ta A) {u (u ) },

(2)

The stress tensor involves the product appropriately averaged or filtered, where the superscript means transpose of the matrix. Using conditional averaging, ˜ (that is fixed and thus conditioned on a fixed given large-scale velocity gradient A can be taken out of the conditional averaging), we obtain ˜ ˜ · exp(−ta A ˜ · u (u ) |A ˜ ). u (u ) |A = exp(−ta A) (3) t +t t 0

a

0

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The modeling also assumes that the term on the left side of the above equation is spatially filtered at scale Δ in order to define the subgrid stress tensor τ . Applying such a filter on both sides of the equation and neglecting “turbulent” diffusion effects, one may express the subgrid stress at some time t0 + ta as a function of the stress at a previous time t0 according to ˜ · τ (t0 ) · exp(−ta A ˜ ). τ (t0 + ta ) ≈ exp(−ta A)

(4)

Following the reasonings proposed earlier [3], we assume that the “upstream condition” τ (t0 ) is an isotropic tensor due to loss of directional information caused by diffusion and randomization processes. Using a Smagorinsky-like modeling for the magnitude of this isotropic tensor yields the proposed form of the matrixexponential closure for the subgrid stress tensor ˜ 2 exp(−ta A) ˜ · exp(−ta A ˜ ), τ ≈ Ce Δ|S| (5) where Ce is a dimensionless model coefficient and ta is a characteristic time-scale. Its order of magnitude is dictated by the following opposing assumptions made in ˜ is constant to the derivation: ta must be sufficiently short for the assumption that A hold, but ta must be sufficiently long so that the assumption of “upstream isotropy” of the unknown initial-condition stress tensor is reasonable. In Ref. [3], analysis of DNS was used to provide some empirical justification to these assumptions. Finally, the behavior of this closure when ta is small enough [3] so that the norm ˜ ≈I− of ta A is much smaller than unity can be explored by expanding exp(−ta A) 2 ˜ ˜ ta A + (1/2)(ta A) + · · · . Up to second order one obtains for the trace-free part (superscript d) of the tensor

2 d ˜ 2 −2ta S˜ + ta2 A ˜ )2 ˜ + (A ˜A ˜+ 1 A τ d ≈ Ce Δ2 |S| + ··· . (6) 2 As can be seen, the first term has the form of eddy viscosity, indeed not surprising from an approach solving the linearized equations for fluctuations for short time periods. It is interesting that the second-order terms are reminiscent of “nonlinear models”, although with slight differences (see discussion in Ref. [3]). Initial a-posteriori tests of this model [3] have shown good performance in simulations of isotropic turbulence, for values near Ce = 0.1 and ta = (A˜ ij A˜ ij )−1/2 . In the next section, the model is compared with directly measured subgrid-stresses with data obtained from the turbulence database.

3 Database-Enabled A-Priori Tests A user-defined function for box-filtering of the velocity gradients has been implemented and compiled on the database (these functions are not yet publicly available to users but can be implemented on the user side). Another function that has been implemented computes the subgrid-stress tensor τ directly from its definition by filtering velocity products according to τij = u ˜ i u˜ j . The modeled stress tensor i uj − u

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Fig. 4 Tensor correlation coefficient between the real subgrid tensor and predictions from two models, the standard Smagorinsky model (stars) and the matrix exponential model (empty squares). Spatial filtering uses a box filter of size Δ and the data is obtained from the public Web-accessible turbulence database exp

τij is evaluated based on the matrix exponential function as in Eq. 5 with Ce = 0.1 and ta = (A˜ ij A˜ ij )−1/2 . The tensor-level correlation coefficient between two tensors is defined according aij bij − aij bij ρ[a, b] = . (7) (aij aij − aij 2 )1/2 (bij bij − bij 2 )1/2 The correlation ρ[τ , τ exp ] is computed based on a statistically meaningfully sample of locations in the database, for 8 different filter sizes ranging from 5 gridspacings (i.e. Δ ∼ 10η, where η is the Kolmogorov scale) up to 18 grid-spacings (i.e. about Δ = 36η). Resulting correlation coefficients are plotted in Fig. 4 as function of filter scale. As is clearly visible the matrix exponential model displays higher correlation than the Smagorinsky model. However, the interpretation of such correlation coefficients has to be done with great caution. For instance, it has been verified that when the SGS stress is computed using a de-aliased spectral filtering operation, the correlation of the matrix exponential model drops down to values close to those of the Smagorinsky model (Yi Li, personal communication, 2008). The situation is very similar to what happens to the nonlinear and similarity models (see discussion in Ref. [4]). Therefore, the results do not allow to conclude unequivocally that the model is superior. However, coupled with the good performance in a-posteriori tests [3] and the physically-based arguments used in the derivation of the model, suggests that it is worth to be tested more fully in the future.

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4 Conclusions In this paper we have illustrated how the database approach enables users to access freely a large DNS data set in a user-friendly fashion and address issues of relevance to turbulence research. Illustrative examples tracking particles backward in time, and visualizing a 2-D subset of the dissipation field, have been presented. Also, initial results from a-priori testing of a new subgrid model have been described. Further developments of the database approach will need to focus on several open issues such as: automated data ingestion algorithms, more analysis functions, and improvements in access speed. Very importantly, it will be of great interest to generate a database of wall-bounded turbulent flow such as channel flow. Acknowledgements The database has been built as a multidisciplinary effort involving Dr. Yi Li, Mr. Eric Perlman, Dr. Minping Wan, Mr. Yunke Yang, Prof. Randal Burns, Prof. Alex Szalay, Prof. Shiyi Chen, Prof. Gregory Eyink, and the author. Discussions with Prof. Ethan Vishniac are also gratefully acknowledged. The research is supported by the National Science Foundation through the ITR program, grant AST-0428325. The DNS was performed on a cluster supported by NSF through MRI grant CTS-0320907. Additional hardware support was provided by the Moore Foundation. The author also thanks Tamas Budavari, Ani Thakar, Jan Vandenberg, and Alainna Wonders for their valuable help at various stages of development and maintenance of the database cluster.

References 1. Frisch, U.: Turbulence. Cambridge University Press, Cambridge (1995) 2. Li, Y., Perlman, E., Wan, M.P., Yang, Y., Meneveau, C., Burns, R., Chen, S., Szalay, A., Eyink, G.: A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. J. Turbul. 9(31), 1–29 (2008) 3. Li, Y., Chevillard, L., Eyink, G., Meneveau, C.: Matrix exponential-based closures for the turbulent subgrid-scale stress tensor. Phys. Rev. E 79, 016305 (2009) 4. Meneveau, C., Katz, J.: Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid Mech. 32, 1–32 (2000) 5. Pope, S.: Turbulent Flows. Cambridge University Press, Cambridge (2000)

Modeling Multi-point Correlations in Wall-Bounded Turbulence Robert D. Moser, Amitabh Bhattacharya, and Nicholas Malaya

Abstract In large eddy simulation (LES), one is generally not interested in the large-scale or filtered quantities computed in the simulation, but rather the corresponding characteristics of the underlying real turbulence. One approach to reconstructing the statistics of turbulence from the filtered statistics of an LES is to employ models for the small separation multi-point velocity correlations, which can be parameterized using the statistics of the LES. This has been employed to good effect in isotropic turbulence, but to employ this technique for near-wall turbulent shear flows requires a model for the anisotropy and inhomogeneity in the correlations. Here we explore the use of multi-point correlation models in LES modeling and reconstruction, and propose a anisotropy/inhomogeneity model for the two-point second-order correlation.

1 Introduction The statistical characteristics of turbulence have long been described through the multi-point velocity correlations. Indeed, the structure functions of arbitrary order are the subjects of the well-known Kolmogorov inertial-range theory [12], and in particular, the second-order correlations in homogeneous turbulence are commonly described and modeled through their Fourier transform, the spectrum tensor (see for example [10, 17]). Multi-point velocity correlations are of particular interest in the context of Large Eddy Simulation (LES) because an LES filter can be applied R.D. Moser () · N. Malaya University of Texas at Austin, Austin, TX, USA e-mail: [email protected] N. Malaya e-mail: [email protected] A. Bhattacharya University of Pittsburgh, Pittsburgh, PA, USA e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_4, © Springer Science+Business Media B.V. 2011

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directly to multi-point correlations to obtain correlations of LES (i.e. filtered) quantities. Indeed a correlation representation with a finite number of parameters can be filtered and then fit to the correlations of an LES. The resulting parameters then yield a model for the underlying unfiltered correlations. This ability to reconstruct the correlations of u from the statistics of the filtered fields is fundamental to LES; it forms the basis of subgrid modeling, and is necessary if one is to use the results of LES for analysis of turbulent flows. It is this need to represent multi-point velocity correlations in turbulent flows with complex geometries that motivates the current work. Because filtered fields simulated in an LES are missing information about the small scales of turbulence, representing the small-separation multi-point correlations is of particular interest here. In high Reynolds number isotropic turbulence, or when the small-scales of interest can be considered to be isotropic, the Kolmogorov inertial range assumptions provide a very rich description of the low-order correlations. In particular, for separations in the inertial range, the second and third-order two-point correlations are determined in terms of the velocity variance (or equivalently the turbulent kinetic energy) and the dissipation rate [8, 10]. Furthermore, a simple modeling ansatz introduced by Chang & Moser [8] leads to an expression for the three-point third-order correlation, and by invoking the quasi-normal approximation, the four-point fourthorder correlation is determined. These theoretical and modeling considerations then allow any statistical correlation up to fourth order to be reconstructed from statistics of a filtered turbulence, provided the assumption of small scale isotropy is valid. This has been used successfully to formulate LES models for isotropic turbulence [16]. However, in some situations, such as near walls, isotropy of the sub-filter scale turbulence is not a good approximation. In this case, reconstruction of turbulence statistics from statistics of filtered turbulence requires a model for the anisotropic multi-point correlations. This is the motivation for the research discussed in this paper. In particular, here we focus on representing anisotropy of the two-point secondorder correlation tensor. Unfortunately, the general description of anisotropy in the two-point correlation requires an infinite number of parameters. Indeed, Arad et al. [4] introduced a formal expansion of the two-point correlation in a basis built to span an infinite sequence of rotationally invariant subspaces of possible correlations (the SO(3) decomposition). A power law form was proposed for the radial dependence of the correlations (consistent with inertial range ideas), and it was speculated that the sequence of rotationally invariant spaces formed a hierarchy in the power law exponent so that for small separations, the lowest order spaces would dominate, yielding a natural finite dimensional truncation. However, no clear evidence of this hierarchy was found in DNS of wall-bounded turbulence [6], consistent with our own DNS observations (not shown here). It thus appears, that obtaining a formulation for the anisotropy of the twopoint correlation is not a matter of formal approximation, but rather of modeling. The modeling ansatz employed here is that the anisotropy of the correlation can be parameterized in terms of the anisotropy of the single point structure tensors introduced by Kassinos & Reynolds [11]. This single-point parameterization of

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anisotropy is related to a formulation introduced by Cambon & Rubinstein [7], who also identified the current formulation as a particular truncation of the SO(3) expansion. It was found [5] that this parameterization yielded good fits to many features of the correlations in wall-bounded turbulence, but that a generalization was needed to account for the effects of inhomogeneity on the anisotropy. Such a generalization is pursued here. In the following, we start with a brief overview of the role of multi-point correlation models in LES reconstruction and modeling, with examples from isotropic turbulence (Sect. 2). The structure tensor based anisotropy representation for wallbounded turbulence in the log-layer is presented in Sect. 3, and further challenges in anisotropy modeling for wall-bounded turbulence are discussed in Sect. 4.

2 Multi-point Correlations and LES In large eddy simulation, one generally is solving for the large-scale velocity field w, which is commonly defined in terms of the turbulent velocity field u through a linear filter operator (call it L ), that is w = L u. For example, L could be the average over a discrete set of volumes to produce a “finite volume” filter. Since L is linear, it is straight-forward to determine the relationship between the multi-point correlations of the filtered fields w and those of turbulent fields u. For example the two-point correlation tensor Rijw (x, y) = wi (x)wj (y) of w is written in terms of the two-point turbulent velocity correlation Rij as Rijw = L x L y Rij .

(1)

Note that even the single-point statistics of w are determined in terms of the multipoint correlations of u, due to the filtering. In performing an LES, one is generally seeking information about the statistics of turbulence, such as the multi-point velocity correlations, rather than statistics of the filtered or large-scale velocity that is produced by an LES. To make use of LES in this way, a model that allows the turbulent velocity correlations to be reconstructed from the filtered correlations is needed. In some LES model formulations, such as Pullin’s spiral vortex model [15, 20], such a reconstruction is natural for a limited set of statistics. To reconstruct velocity correlations, and related statistics, a more general approach is possible by positing a model for small-separation multi-point correlations. The correlations of the filtered velocities from an LES, along with relationships such as (1) between the filtered and unfiltered correlations are then used to set the parameters of the model. The model of the small separation correlation can be combined with the correlations of filtered velocities to form a composite representation of the correlation over all scales. As an example, this approach has been used to reconstruct the two-point correlation computed from filtered DNS of isotropic turbulence. The small-scale correlation model is simply the Kolmogorov expression: Rij (r) = u2 δij +

C 2/3 −4/3 (ri rj − 4r 2 δij ) ε r 6

(2)

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Fig. 1 Reconstruction of turbulence statistics from filtered statistics in isotropic turbulence. Shown in (a) are the reconstructed contracted two-point correlation along with the filtered and DNS statistics. In (b) the relative error in reconstructing the dissipation from the third-order structure function is shown as a function of separation for different filtered grids

and the values of u2 and ε were determined from the second- and third-order structure functions, respectively, of the finite-volume filtered velocities. In addition, to account for the fact that the Reynolds number of the turbulence being reconstructed is finite, a viscous correction like that used by Sirovich et al. [18] was employed. The reconstructed longitudinal correlation function, along with the filtered and DNS correlations are shown in Fig. 1. Also as an example, consider the dissipation rate ε; it can be determined from ε=

SΔ 3 (r) ˆ S Δ (r)

(3)

3

where S Δ 3 is the longitudinal third-order “structure function” of the filtered field with filter width Δ. Also, SˆΔ is the same filtered structure function computed from 3

the model of the three-point third-order correlation of a turbulence with unity ε (the model correlation is proportional to dissipation) [8]. The relative error in the value of ε estimated in this way is also shown in Fig. 1. Reconstructing the small-scale velocity correlations from filtered correlations is also useful in modeling LES evolution. In particular, in the optimal LES modeling approach [13, 21, 22], in which stochastic estimation [1–3] is used to estimate the model terms in an LES, the model is usually formulated in terms of surface and volume integrals of multi-point correlations up to fourth order. As described in Sect. 1, in isotropic turbulence, the Kolmogorov inertial range theory for two-point correlations as in (2), the model of [8] and the quasi-normal approximation are sufficient to express the multi-point correlations up to fourth order (see [16] for details). Thus, once ε and u2 are determined using the process described above, a complete LES evolution model can be defined [16]. Three-dimensional energy spectra from LES of infinite Reynolds number isotropic turbulence performed with such a model are shown in Fig. 2. These spectra are precisely what is expected, that is they are a good approximation to the spectrum of a filtered infinite Reynolds number inertial range.

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Fig. 2 Three dimensional spectra from LES of forced isotropic turbulence with optimal LES subgrid models [13, 21, 22] constructed using the models discussed here for the second, third and fourth-order velocity correlations, and reconstructed values for u2 and ε, which appear as parameters in the models (see [16]). Shown are results from LES at a range of resolutions

The examples of LES statistical reconstruction given above are based on isotropic models of the small-scale multi-point correlations. Such models can be applied in many inhomogeneous and anisotropic turbulent flows, provided the LES filter scale is sufficiently small for an assumption of small-scale isotropy to be valid. This is expected to be possible in many flows. An important flow situation in which such assumptions are certainly not valid is turbulent shear flow near a solid wall. The modeling of anisotropy in wall-bounded turbulent flows is thus of great interest. An approach to modeling anisotropy in the two-point second-order correlation is described in the following section.

3 Modeling Anisotropy in Wall-Bounded Turbulence In developing anisotropy models for multi-point correlations, we begin first with models for the two-point second-order correlations. As proposed by [5, 7], we consider parameterizations of the anisotropy in the two-point correlation in terms of the anisotropy of a small number of single-point tensors. In particular, the structure tensors introduced by Kassinos et al. [11] are used as in [5]. Kassinos & Reynolds introduced several structure tensors, but two in particular were found to be of primary importance, the componentality tensor and the dimensionality tensor. The componentality tensor is simply the Reynolds stress, and anisotropy of this tensor represents anisotropy in the magnitude of velocity fluctuations. In contrast, anisotropy in the dimensionality tensor represents anisotropy in the correlation length [5]. Modeling the two-point correlation in terms of these two tensors is a particular truncation of an SO(3) expansion [7]. This approximation is attractive because it is a simple way to include anisotropy of components and length-scales, and because it appears that the structure tensors provide a good characterization of anisotropy in the RANS modeling context [11]. It was found in [5] that when fit to DNS data for a turbulent channel [9], a representation in terms of componentality and dimensionality tensors produces correlation models that agree reasonably well with the DNS data, with one significant

34

R.D. Moser et al.

exception. The models produced by [5] exhibit the symmetry Rij (r) = Rij (−r), which is not satisfied by the real turbulence. This symmetry is present because the representation is for a homogeneous non-helical turbulence, and of course the real turbulence is not homogeneous near the wall. It appears then that a treatment of the anisotropy introduced by the inhomogeneity is needed. In addition to componentality and dimensionality, Kassinos & Reynolds [11] introduced an inhomogeneity tensor to represent the affect of inhomogeneity on anisotropy. This representation can account for a wide range of inhomogeneities and is apparently overly general for out purposes in wall-bounded turbulence. Such a representation also adds significant complexity to the already complex anisotropy representation. Instead, we have sought a more targeted treatment of inhomogeneity of the sort encountered in wall-bounded turbulence. In particular, in the log region of a wall-bounded turbulent shear flow, we expect that Rij = Rij − Bij , where Bij is the componentality or Reynolds stress tensor, will depend only on r/y, where y is the distance to the wall. That is, the inhomogeneity is one of spatial variation of scale, not variation of the anisotropy. To accommodate such an inhomogeneity, a generalization of the formulation used by Bhattacharya et al. [5] is used. In this formulation, the correlation is additively decomposed into four components Rijα , which include an isotropic part (RijI ), a component anisotropy part (Rijb ), a dimension anisotropy part (Rijd ), and a Reynolds stress part (RijR ). Each of these parts is expressed as a function of the separation r and the location of the midpoint y between the two correlation points in the following form: + − α α ∂l c (y)r pα +2 Hkn (ˆr, tα ) (4) Rijα = εilk εj mn ∂m where the operators ∂i± are given by ∂i±

=

1 ∂ ∂ . ± 2 ∂yi ∂ri

(5)

α are rotationally invariant linear functions of a tensor argument The generators Hkn α t which can be either the isotropic tensor (for the isotropic part), the component anisotropy tensor (b), the dimensional anisotropy tensor (d) or the Reynolds stress tensor B. Each Hα is written using tensor invariant theory and to satisfy a consistency constraint as in [5], so that one part (e.g. Rijb ) does not contribute to the structure tensor associated with another part (e.g. the dimensional anisotropy). See [5] for details. In (4), rˆ is the normalized separation vector r/r, and a power law dependence on the separation magnitude r is assumed, with exponents given by pI = 2/3, pb = 1.4, pd = 1 and pB = 0 as determined in [5]. Finally, the cα ’s represent the only y dependence, which subsumes the r/y dependence discussed above as a special case. Further, cB (y) is simply q 2 (y) = Bii (y). To evaluate the utility of this anisotropy representation, the free parameters in the formulation are fit to correlations computed in a turbulent channel flow at Reτ = 940 [9]. There are 15 parameters: 5 components of b, 5 components of d, q 2 and the four gradients of cα . Of these, six (q 2 and the components of b) are determined directly from the Reynolds stress. The results of this fit at y + = 115,

Modeling Multi-point Correlations in Wall-Bounded Turbulence

35

Fig. 3 Contours of the two-point correlation in the x–y plane of a turbulent channel flow at Reτ = 940 with the mid-point between the correlation points located at y + = 115 and no spanwise separation (x is streamwise and y is wall-normal direction). Shown are correlations computed directly from the DNS of [9] and from the parameterized model discussed here

along with the corresponding correlations computed from the DNS are shown in Fig. 3. The model form does a reasonably good job of representing the anisotropy of the two-point correlation, and in particular, the inhomogeneity form captures the small-separation asymmetry in the R12 component, though it does not do as well at larger separations (r/ h ∼ 0.1 where h is the channel half-width). To use this formulation in LES reconstruction, the filtered model form is fit to correlations of filtered velocities. This has been tested using finite-volume filtered DNS correlations, and the results (not shown) are consistent with those in Fig. 3.

4 Discussion The model developed here of anisotropy in the two-point correlation does a reasonably good job of representing anisotropy in near-wall turbulence (i.e. in the loglayer). While the fit is far from perfect, it appears to be sufficient for use in the sort of LES reconstruction for which it is being developed. In particular, the model presented here can be used to reconstruct second-order turbulence statistics from LES statistics in wall-bounded turbulence. However, there are three important developments that are needed before this general approach can be widely deployed for LES modeling and reconstruction in wall-bounded turbulence: 1. This model formulation is overly complex with more fitting parameters (15) than would be preferred. The difficulty with having a large number of fitting parameters is that it increases the amount of statistical information that must be gathered from an LES to perform the reconstruction. Thus, if one is reconstructing

36

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the turbulent kinetic energy, for example, LES correlations with many different separations will need to be evaluated for each point at which the energy is required. Further modeling to reduce the number of independent parameters will be useful. 2. To reconstruct higher order statistics from LES of wall-bounded turbulence, models for anisotropy of higher order correlations are needed. In particular, correlations up to fourth order are needed to formulate optimal LES evolution models. Fortunately, the quasi-normal approximation can be used to model the fourthorder correlations in terms of the second order model developed here [19], but no ready generalization for the three-point third-order correlations is available. A formulation is needed that produces an anisotropic third-order correlation that is consistent with the anisotropy in the second-order correlations. 3. Recent analysis of LES models in wall-bounded turbulence suggest that refinements are needed to the treatment of pressure in wall-bounded turbulent flows. Langford & Moser [14] found in isotropic turbulence that a simple treatment of pressure and continuity consistent with numerical methods commonly employed in finite volume schemes was near-optimal, and subsequent LES studies [16, 22] found that such treatments yielded excellent LES results. However, in near-wall turbulence, the pressure plays a much more critical role of energy redistribution among components, and recent analysis suggests that the treatment proposed by Langford & Moser is inadequate in this case. To address this issue, models for anisotropy and inhomogeneity in the multi-point pressure–velocity correlations in near wall turbulence may also be needed. With these generalizations and refinements, models of anisotropic multi-point correlations will enable the deployment of robust reliable LES models. Acknowledgements The financial support of the National Science Foundation, the Air Force Office of Scientific Research and the National Aeronautics and Space Administration are gratefully acknowledged. In addition, we would like to thank Profs. Javier Jimenez and Ron Adrian for many helpful discussions of near-wall turbulence.

References 1. Adrian, R.: On the role of conditional averages in turbulence theory. In: Zakin, J., Patterson, G. (eds.) Turbulence in Liquids, pp. 323–332. Science Press, Princeton (1977) 2. Adrian, R.: Stochastic estimation of sub-grid scale motions. Appl. Mech. Rev. 43(5), 214–218 (1990) 3. Adrian, R., Jones, B., Chung, M., Hassan, Y., Nithianandan, C., Tung, A.: Approximation of turbulent conditional averages by stochastic estimation. Phys. Fluids 1(6), 992–998 (1989) 4. Arad, I., L’vov, V.S., Procaccia, I.: Correlation functions in isotropic and anisotropic turbulence: the role of the symmetry group. Phys. Rev. E 59(6), 6753–6765 (1999). doi:10.1103/ PhysRevE.59.6753 5. Bhattacharya, A., Kassinos, S.C., Moser, R.D.: Representing anisotropy of two-point secondorder turbulence velocity correlations using structure tensors. Phys. Fluids 20(10) (2008). doi:10.1063/1.3005818 6. Biferale, L., Lohse, D., Mazzitelli, I., Toschi, F.: Probing structures in channel flow through SO(3) and SO(2) decomposition. J. Fluid Mech. 452, 39–59 (2002)

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7. Cambon, C., Rubinstein, R.: Anisotropic developments for homogeneous shear flows. Phys. Fluids 18(8) (2006) 8. Chang, H., Moser, R.D.: An inertial range model for the three-point third-order velocity correlation. Phys. Fluids 19, 105,111 (2007) 9. Del Álamo, J., Jiménez, J., Zandonade, P., Moser, R.: Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135–144 (2004) 10. Frisch, U.: Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press, Cambridge (1995) 11. Kassinos, S., Reynolds, W., Rogers, M.: One-point turbulence structure tensors. J. Fluid Mech. 428, 213–248 (2001) 12. Kolmogorov, A.N.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. Acad. Sci. USSR 30, 301 (1941) 13. Langford, J.A., Moser, R.D.: Optimal LES formulations for isotropic turbulence. J. Fluid Mech. 398, 321–346 (1999) 14. Langford, J.A., Moser, R.D.: Breakdown of continuity in large-eddy simulation. Phys. Fluids 11, 943–945 (2001) 15. Misra, A., Pullin, D.I.: A vortex-based subgrid model for large-eddy simulation. Phys. Fluids 9, 2443–2454 (1997) 16. Moser, R.D., Zandonade, P.S., Vedula, P., Malaya, N., Chang, H., Bhattacharya, A., Haselbacher, A.: Theoretically based optimal large-eddy simulation. Phys. Fluids 21(10) (2009) 17. Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000) 18. Sirovich, L., Smith, L., Yakhot, V.: Energy spectrum of homogeneous and isotropic turbulence in the far dissipation range. Phys. Rev. Lett. 72, 344–347 (1994) 19. Vedula, P., Moser, R.D., Zandonade, P.S.: On the validity of quasi-normal approximation in turbulent channel flow. Phys. Fluids 17, 055,106 (2005) 20. Voelkl, T., Pullin, D.I., Chan, D.C.: A physical-space version of the stretched-vortex subgridstress model for large-eddy simulation. Phys. Fluids 13, 1810–1825 (2000) 21. Volker, S., Venugopal, P., Moser, R.D.: Optimal large eddy simulation of turbulent channel flow based on direct numerical simulation statistical data. Phys. Fluids 14, 3675 (2002) 22. Zandonade, P.S., Langford, J.A., Moser, R.D.: Finite volume optimal large-eddy simulation of isotropic turbulence. Phys. Fluids 16, 2255–2271 (2004)

Theoretical Prediction of Turbulent Skin Friction on Geometrically Complex Surfaces Pierre Sagaut and Yulia Peet

Abstract This article can be considered as an extension of the paper of Fukagata et al. (Phys. Fluids 14:L73, 2002) who derived an analytical expression for the componential contributions into skin friction in a turbulent channel, pipe and plane boundary layer flows. In this paper, we extend theoretical analysis of Fukagata et al. limited to canonical cases with two-dimensional mean flow to a fully three-dimensional situation allowing complex wall shapes. We start our analysis by considering arbitrarily-shaped surfaces and then formulate a restriction on a surface shape for which the current analysis is valid. Theoretical formula for skin friction coefficient is thus given for streamwise and spanwise homogeneous surfaces of any shape, as well as some more complex configurations, including spanwise-periodic wavy patterns. Current theoretical analysis is validated using the results of Large Eddy Simulations of a turbulent flow over straight and wavy riblets with triangular and knife-blade cross-sections. Decomposition of skin friction into different componential contributions allows to analyze the influence of different dynamical effects on a drag modification by riblet-covered surfaces.

1 Introduction Accurate estimation of skin friction coefficient is required for determining skin friction drag on a body moving relative to a fluid. Negative impact of skin friction drag on performance and efficiency of practical engineering devices is responsible for the past and present quest for drag reduction methods. As a result, a bulk of experimental and computational data have been generated over the last several decades P. Sagaut () · Y. Peet Institut Jean Le Rond d’Alembert, Université Pierre et Marie Curie, 4 place Jussieu — case 162, 75252 Paris cedex 5, France e-mail: [email protected] Y. Peet e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_5, © Springer Science+Business Media B.V. 2011

39

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P. Sagaut and Y. Peet

concerning skin friction and the ways to reduce it, and a number of various drag reduction methods have been proposed [10]. It is now understood that turbulent flows have higher skin friction drag than their laminar counterparts due to the interaction of coherent near-wall turbulent structures, or quasi-streamwise vortices, with the surface [16]. Quasi-streamwise vortices pump a high-speed fluid towards the wall during the turbulent sweep events, thus increasing the local shear rate and, ultimately, the drag [4]. Reduction of this interaction would generally lead to a lower drag, and various passive and active drag reduction methods are attempting to alter the near-wall turbulence in one way or another in order to achieve this reduction. Wall blowing and suction, or opposition control of turbulence, prevents the downwash of high-speed fluid toward the wall during sweep events [6, 15]; riblets [2, 17] and microbubbles [7] displace the quasi-streamwise vortices away from the wall, so that the vortex-surface interaction is reduced; spanwise-wall oscillation [5, 12] breaks the coherence between the streamwise vortices and the low-speed streaks, thus weakening the near-wall burst activity and reducing the drag. However, in spite of an extensive amount of literature on skin friction drag reduction, most of the conclusions are drawn based on hypothetical arguments relating the measured/calculated drag to the observed flow features. A clear understanding of the contribution of different dynamical effects into the skin friction coefficient, based on theoretical analysis, is lacking. Fukagata et al. [8] derived an analytical expression relating local skin friction coefficient to the properties of the flow above the surface for the canonical cases of plane turbulent channel flow, pipe flow and flat plate boundary layer. They used their analysis to explain drag modification by the opposition control and uniform wall blowing/suction. Their approach was subsequently extended to analyze friction drag reduction by near-wall turbulence manipulation at high Reynolds numbers [11], and by superhydrophobic surfaces [9]. In the present paper, we extend theoretical analysis of Fukagata et al. [8], limited to canonical cases with two-dimensional mean flow, to a fully three-dimensional situation of complex wall shapes. The generalized approach would allow to treat analytically configurations involving geometrical surface modification, as occurs with the use of riblets [2, 17] or wall deforming actuators in active flow control [13, 15]. In Sect. 2, we present the details of the derivation of the closed-form expression relating skin friction coefficient to the statistical information of the flow in a case of complex three-dimensional wall shapes. We start our analysis by considering arbitrarily-shaped surfaces and then formulate a restriction on a surface shape for which the current analysis is valid. In Sect. 3, we use the derived expression to analyze the modification of drag by straight [2, 17] and wavy [3] riblets with two different types of cross-section, triangular and knife-blade. Statistical flow information needed for the analytical expression is obtained from Large Eddy Simulations (LES). Skin friction coefficient obtained from the analytical expression is compared with the computed LES value. The derived expression allows to decompose skin friction coefficient into a sum of bulk, convective and turbulent components and thus provides better understanding of the mechanism of drag modification by straight and wavy riblets.

Prediction of Skin Friction on Complex Surfaces

41

Fig. 1 Schematics of a flow over an arbitrarily-shaped surface (left) and schematics of a cross-section Σ at a streamwise location x (right)

2 Mathematical Formulation 2.1 Skin Friction Coefficient Consider a flow over an arbitrarily-shaped surface as depicted in Fig. 1. Cartesian coordinate system (x, y, z) is introduced, where x axis is aligned with the direction of incoming, y axis is parallel to the mean surface normal and z axis is in spanwise direction. Local force that acts on a surface at a particular point consists of shear and pressure forces (1) F = − μ ∇v + (∇v)T w · n − Pw · n , where μ is the fluid viscosity, v = (u, v, w) is the velocity vector, P is the pressure, n is the outward pointing local surface normal of the unit length, and the subscript “w” stands for the quantities evaluated at the wall. Local skin friction drag can be written as ∂u , (2) Fd = −μ ∂n w

and its non-dimensional counterpart, skin friction coefficient defined as Cf = is therefore expressed as Cf = −

Fd , 1/2ρUr2

2μ ∂u , ρUr2 ∂n w

(3)

(4)

where ρ is the density and Ur is the reference velocity. In the following, bar over variables denotes filtered quantities, and prime — the difference between local and filtered quantities. Velocities are normalized by the characteristic velocity U , spatial coordinates are normalized by the characteristic

42

P. Sagaut and Y. Peet

length L, Re = U L/ν is the Reynolds number of the flow, and ν is the kinematic viscosity. Note that with the above normalization and filtering equation (4) for skinfriction coefficient will read ∂u 2 Cf = − . (5) Re Ur2 ∂n w We now propose an extension of the triple integration procedure proposed by Fukagata et al. [8] to get a componential decomposition of the mean skin friction. To this end, we integratethe mean streamwise momentum equation at each streamwise location x over the cross-section Σ shown in Fig. 1. Cross-section Σ is y–z crosssection of the flow volume at a given streamwise location at the wall, and represents an area occupied by the flow between the wall and an imaginary flat surface, which is perpendicular to the vertical axis and located far enough from the wall, so that the flow at that location can be considered uniform. Such an imaginary surface would, for example, correspond to an edge of a boundary layer or to a channel center-plane. Side boundaries of the cross-section Σ are chosen to be parallel to y axis. The lateral spacing between the side boundaries is denoted by s. Please, note that in a general case both Σ and s are functions of streamwise coordinate x. To obtain the relation for the componential contributions of different dynamical effects to the skin friction coefficient, Fukagata et al. [8] in their analysis of a homogeneous case of a plane channel flow applied triple integration of the form 1 y y 0 dy 0 dy 0 dy to the expression for skin-friction coefficient (in our case y = 0 and y = 1 signified the bottom wall and the channel center-plane, respectively, in their normalization). With the three-dimensional shape of the wall, the situation is more complicated, and a line integration dy should be replaced by a surface inte gration dy dz over some elementary surfaces within Σ . To ensure orthogonality, an integration has to be performed along the gridlines corresponding to an orthogonal body-fitted curvilinear coordinate system (η, ζ ). Notice that we do not switch coordinate systems, Navier–Stokes equations are not modified, all distances are calculated in the Cartesian coordinate system, and functions η(y, z), ζ (y, z) are merely used to define elementary surfaces of integration. Note that the wall is represented by η = 0, top imaginary surface — by η = ηtop , and side boundaries — by ζ = ζ1 and ζ = ζ2 , respectively. As a result, one gets the following formula for spanwise-averaged skin friction coefficient on any three-dimensional surface which satisfies ad hoc symmetries Cf z (x, t) =

1 (Tb + Tc + Tp + Ttr + Tt ) Ur2 A s cos β

(6)

where A , Tb , (Tc + Tp + Ttr ) and Tt are defined by (9), (10), (11) and (12) of the Appendix, respectively. A in the normalization term is a function of a crosssectional shape and a spanwise skin-friction distribution w(γ ). Note that all terms in (6) except for Ur are functions of a streamwise coordinate x in a general case. Tb corresponds to the bulk contribution, Tc — convective contribution, Tp — pressure contribution, Ttr — transient contribution, and Tt — turbulent contribution into the skin friction.

Prediction of Skin Friction on Complex Surfaces

43

3 Application of the Formula to Surface Riblets In this section, we will show some examples of application of the derived expression for skin friction coefficient, (6), to realistic flows. This will serve a dual purpose of validating the formula and demonstrating its usefulness in extracting new information about the flow. We look at riblets with triangular and knife-blade cross-sections. For both triangular and knife-blade riblets, analytical formulas for the transformation from a Cartesian coordinate system (y, z) into a curvilinear orthogonal system (η, ζ ) are provided by Bechert & Bartenwerfer [1] and represent a series of conformal mappings from a surface inside a rectangular channel onto a surface inside a polygon. We investigate both straight and wavy riblets. Straight riblets represent straight lines if viewed from above, while wavy riblets represent sinusoidal waves characterized by the function 2π

z(x) = a sin x , (7) λ where z(x) is the deviation of a spanwise coordinate of the wavy riblet surface from the corresponding spanwise coordinate of the straight riblet surface, a is the oscillation amplitude, and λ is the oscillation wavelength. Note that dependence of cross-sectional geometry on streamwise coordinate is absent for straight and wavy riblets. An angle β(x) defined as d z β(x) = arctan (8) dx is the same angle which enters (6) for skin-friction coefficient and represents an angle between the local surface normal and its orthogonal projection to y–z plane. A set of LES of channel flows with different types of riblets was carried out investigate the componential generation of turbulent friction. Physical and numerical parameters are displayed in Tables 1 and 2.

3.1 Componential Contributions It is observed (see Tables 3 and 4) that skin friction for riblet surfaces is smaller than skin friction for a flat surface for all the calculated cases, proving that riblets are an effective method of skin friction drag reduction provided that riblet geometry parameters are chosen correctly. Wavy riblets were introduced in an attempt to achieve additional drag reduction benefits relative to straight riblets [3, 14]. Results shows that those additional benefits are achieved for wavy riblets with longer wavelength but aren’t achieved for short wavy riblets. Current theoretical approach allows to give more insight into drag reduction mechanism for riblet-covered surfaces and explain the difference in drag reduction properties of short and long wavy

44

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Table 1 Geometrical parameters for the calculated cases Case

α

Straight T

60°

Short waves T

60°

Long waves T

60°

Straight B

90°

Short waves B Long waves B

βmax

s+

h+

a+

λ+

s/δ

h/s

a/δ

λ/δ

0°

0.1164

0.866

0

0

21

18

0

0

11.3°

0.1164

0.866

0.1

3.22

21

18

18

580

11.3°

0.1164

0.866

0.19

6

21

18

34

1080

0°

0.0908

0.5

0

0

16

8

0

0

90°

9.6°

0.0908

0.5

0.085

3.22

16

8

15

580

90°

9.6°

0.0908

0.5

0.16

6

16

8

28

1080

Table 2 Numerical grid parameters for the calculated cases Case

Lx /δ

Lz /δ

Straight T

3.22

0.9312

L+ x 580

L+ z

Nx × Ny × Nz

x + × y + × z+

168

16 × 64 × 128

36 × (0.4–14) × 1.32

Short waves T

3.22

0.9312

580

168

16 × 64 × 128

36 × (0.4–14) × 1.32

Long waves T

6

0.9312

1080

168

32 × 64 × 128

33 × (0.4–14) × 1.32

Straight B

3.22

0.7264

580

131

16 × 64 × 128

36 × (0.4–14) × 1.03

Short waves B

3.22

0.7264

580

131

16 × 64 × 128

36 × (0.4–14) × 1.03

Long waves B

6

0.7264

1080

131

32 × 64 × 128

33 × (0.4–14) × 1.03

riblets by looking at quantitative contribution of different dynamical effects in the total skin friction coefficient. Componential contribution of different dynamical effects into skin friction is shown in Table 3 and represents the value of each individual term (bulk Cf b , convective Cf c , turbulent Cf t , and total Cf — their sum, where the double bar denotes streamwise integration). In order to better understand the relative effect of each term, we also look at the ratio of each component (and their sum) to the total skin friction coefficient of a flat surface Cf f in Table 4. In the rest of this section, we analyze each term contributing to skin friction coefficient in more details in order to understand the differences in drag reduction properties between the different surfaces.

3.2 Drag Reduction Drag reduction value defined as RD = (Cf f − Cf r )/Cf f , predicted by the theoretical formula and by LES, is cited in Table 5 for the six calculated cases. Here Cf f is the skin friction coefficient of a flat surface, and Cf r is the skin friction coefficient of a riblet surface. Decomposition of skin-friction coefficient into componential contributions discussed above allows to explain drag reduction mechanism of straight and wavy riblets and clarify why short and long wavy riblets have so different drag reduction properties.

Prediction of Skin Friction on Complex Surfaces

45

Table 3 Componential contribution of different dynamical effects into skin friction Surface

Bulk (Cf b )

Convective (Cf c )

Turbulent (Cf t )

Total (Cf )

Flat

2.20 × 10−3

0.008 × 10−3

6.39 × 10−3

8.59 × 10−3

Straight T

2.41 × 10−3

0.008 × 10−3

5.66 × 10−3

8.08 × 10−3

Short waves T

2.43 × 10−3

0.49 × 10−3

5.51 × 10−3

8.43 × 10−3

Long waves T

2.43 × 10−3

0.10 × 10−3

5.49 × 10−3

8.02 × 10−3

Straight B

2.36 × 10−3

0.03 × 10−3

5.24 × 10−3

7.63 × 10−3

Short waves B

2.38 × 10−3

0.11 × 10−3

5.34 × 10−3

7.83 × 10−3

Long waves B

2.38 × 10−3

0.006 × 10−3

5.07 × 10−3

7.46 × 10−3

Table 4 Ratio of each componential contribution to the total flat plate skin friction coefficient Surface

Bulk f (Cf b /Cf )

Convective f (Cf c /Cf )

Turbulent f (Cf t /Cf )

Total f (Cf /Cf )

Flat

0.26

0

0.74

1

Straight T

0.28

0

0.66

0.94

Short waves T

0.28

0.06

0.64

0.98

Long waves T

0.28

0.01

0.64

0.93

Straight B

0.28

0

0.61

0.89

Short waves B

0.28

0.01

0.62

0.91

Long waves B

0.28

0

0.59

0.87

Table 5 Drag reduction value, RD

Surface

Theory

LES

Straight T

6%

5%

Short waves T

2%

3%

Long waves T

7%

6%

11%

11%

Straight B Short waves B

9%

9%

Long waves B

13%

14%

Straight Riblets For the straight riblets (cf. Table 4), two terms come into play: bulk term and turbulent term. Although bulk term is increased slightly from 26% to 28% for straight riblets over a flat surface, significant reduction in turbulent term (from 74% to 66% for triangular riblets and to 61% for knife-blade riblets) overcome 2% increase in a bulk term resulting in 6% drag reduction for triangular riblets and 11% drag reduction for knife-blade riblets. Those values are in good agreement with the drag reduction values published in the open literature for those configurations. Further-

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more, the accepted hypothesis that drag on a straight riblet surface is reduced due to a favorable interaction of riblets with the near-wall turbulence is confirmed, as well as the opinion that knife-blade riblets possess smaller drag than riblets with other cross-sectional shapes because they are more effective in reducing the near-wall turbulence.

Wavy Riblets For the wavy riblets, the situation is slightly more complicated, since a delicate balance between small changes in all the three terms determine positive or negative increment of skin friction with respect to straight riblets. Indeed, bulk contribution is slightly increased for wavy riblets due to the bulk flow oscillations and non-unity cos β; convective contribution is also slightly increased for all the cases except for the “Long waves B” case; turbulent contribution is slightly decreased for all the cases except for the “Short waves B” case. As a result, long wavy riblets exhibit smaller Cf values than their straight riblet counterparts, while short wavy riblets exhibit larger Cf values. For long wavy riblets, drag increase due to a convective term is very small (1% for triangular, 0% for knife-blade), whereas decrease in turbulence intensity by spanwise oscillations is 2% with respect to straight riblets, resulting in a positive effect of riblet oscillations on drag reduction, especially pronounced for knife-blade riblets. However, if the wavelength of the oscillations is too small, smooth transition of the flow between different phases of oscillation no longer occurs, local turbulence does not have enough time to adjust to a change in a flow direction and is constantly in excited and locally non-equilibrium state, leading to asymmetries in a streamwise distribution of a convective term and large streamwise fluctuations of a turbulent term. For the “Short waves T” case, tremendous increase of a convective term by 6% does not overcome 2% decrease in turbulent term, resulting a drag reduction increase of 4% for the “Short waves T” compared the straight riblet case. For the “Short waves B” case, although convective term increase is not so dramatic, of only 1%, the turbulent term is also increased by 1% in this case, still resulting in a drag reduction loss over straight riblets. To summarize, introducing spanwise oscillations to the conventional (straight) riblet shape does produce additional drag reduction benefits, provided the wavelength of oscillations is not too small. It can therefore be a promising method of achieving even lower drag by simple and inexpensive geometrical modifications. Benefits might be even larger for the optimized configuration in terms of oscillation parameters, and for higher Reynolds numbers relevant to practical applications.

4 Conclusions An analytical formula is derived for a turbulent skin friction coefficient on geometrically complex surfaces, whose shape satisfies certain conditions. Analytical expression for skin friction coefficient is thus given for streamwise and spanwise

Prediction of Skin Friction on Complex Surfaces

47

homogeneous surfaces of any shape, as well as some more complex configurations including wavy patterns. The derivation consists of integrating governing equations of motion and leads to a closed-form expression between a skin friction and statistical information in the flow above the surface. The expression shows that contribution into a skin friction can be decomposed into several terms: bulk, convective, pressure, transient and turbulent. The derived expression can be particularly useful for analyzing skin friction drag reduction properties of various surfaces. Analytical formula is validated for a flat plate and for a surface covered with straight and wavy riblets of triangular and knife-blade cross-section by comparing LES results with the prediction of the formula. A difference of no more than 4% is reported for all the simulated cases. Analysis of different dynamical effects contributing to a total skin friction value for riblet-covered surfaces shows that: 1. Bulk contribution is slightly larger for straight riblet surfaces than for flat surfaces, and even larger for wavy riblet surfaces. 2. Convective contribution is zero for a flat plate and for straight riblets, but not zero for wavy riblets in both laminar and turbulent cases. Convective contribution is generally larger for short riblets versus long riblets and for triangular riblets versus knife-blade riblets. In a turbulent case, the maximum value is 6% for short triangular riblets, and the minimum value is 0% for long knife-blade riblets. Percentage signifies the fraction of each particular term with respect to the total f

skin friction of a flat surface, Cf . 3. Turbulent contribution is reduced by 8% for straight triangular riblets as compared to a flat surface, with even further reduction of up to 13% for knife-blade riblets. Turbulent term exhibits large fluctuations with respect a streamwise coordinate for short wavy riblets, and those fluctuations are much reduced for long wavy riblets. Turbulent term is consistently smaller by about 2% for long wavy riblets than for straight riblets, and no consistency is observed for short wavy riblets. Drag reduction values of 6% and 11% for straight triangular and knife-blade riblets, respectively, are documented and are in agreement with the published literature. It is found that further drag reduction benefits are achieved for wavy riblets with large oscillation wavelength due to additional turbulence suppression by spanwise motion, probably by tilting streamwise vortices and violating their spatial coherence with respect to the low-speed streaks, similar to the effects found in a spanwiseoscillating wall [5, 12]. If the oscillation wavelength is too small, however, drag is increased with respect to the straight riblets due to non-equilibrium effects leading to large streamwise fluctuations of a turbulent term and increase in a convective term. Wavy riblets represent a promising way of improving drag reduction properties of riblet-covered surfaces at no cost, which might work even better for higher Reynolds numbers. Wavelength of the oscillations should be carefully chosen, however, and should not be made too small. Knife-blade riblets are confirmed to be more efficient in drag reduction than triangular riblets for a straight riblet configuration. Knifeblade riblets are also more efficient in bringing out the benefits from a spanwise

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shape oscillation: they show larger values of drag reduction increase with respect to the straight riblets, when a spanwise shape oscillation of the long wavelength is applied. Acknowledgements Dr. Yves Charon (IFP, France) is gratefully acknowledged for many enlightening discussions. This project was supported by ANR as project ANR-PANH-READY.

Appendix In this appendix, we spell out the formulas obtained for the components of friction ηtop η after the double integration 0 dγ 0 dγ and summation over dζ → 0 is applied to this equation. As before, γ denotes the local variable along the integration contour, and dγ — its differential. We will also be using the notations γ (η) and γ (ζ ) to distinguish the integration along η or ζ lines, and dγ (η), dγ (ζ ), referring to their differentials. The transformation of the four terms can be written as follows: η

ηtop dγ (ζ ) dAσ (η, ζ ) lim w γ (ζ ) − dγ (η) dγ (η) dζ →0 s AΣ 0 ∂σw 0 dζ

ζ2

=

ζ1

ηtop

γ (ηtop ) − γ (η)

0

dγ (ζw ) dAσ (η, ζ ) dγ (η) = A , − × w γ (ζw ) s AΣ

(9)

η

2 ηtop ∂u dγ (η) dγ (η) dγ (ζ ) dζ →0 Re 0 ∂σt ∂nσ 0 lim

dζ

2 ηtop 2Ub AΣ = Tb , dγ (η) u(η) − u(0) dγ (ζ ) = dζ →0 Re 0 Re ∂σt

= lim

(10)

dζ

lim

dζ →0

dζ

=

ηtop

dγ (η)

0

dγ (ζ )

dγ (η) 0

ζ2

ζ1

η

dγ (η) 0

ηtop

ηtop

(−2F ) dy dz

σ

η

dγ (η) 0

η

(−2F ) dγ (η)

0

2 1 γ (ηtop ) − γ (η) (−2F ) dγ (ζ ) dγ (η) 2 ζ1 0 ζ2 ηtop 2 ∂P ∂u top − dγ (ζ ) dγ (η) = γ (η ) − γ (η) −Ix − ∂x ∂t ζ1 0 = Tc + Tp + Ttr , (11) =

ζ2

Prediction of Skin Friction on Complex Surfaces

lim

dζ →0

dζ

=

ζ2 ζ1

ηtop

dγ (η)

0

ηtop

η

dγ (η) 0

0

49

∂σt

(−2u vn σ ) dγ (ζ )

γ (ηtop ) − γ (η) (−2u vη ) dγ (ζ ) dγ (η) = Tt .

(12)

Integration by parts was used to transform multiple integrations over η to a single integration in the derivation of the formulas (9), (11) and (12). Ub in (10) is the bulk velocity.

References 1. Bechert, D.W., Bartenwerfer, M.: The viscous flow on surfaces with longitudinal ribs. J. Fluid Mech. 206, 105–129 (1989) 2. Bechert, D.W., Bruse, M., Hage, W., Van Der Hoeven, J.G.T., Hoppe, G.: Experiments on drag-reducing surfaces and their optimization with an adjustable geometry. J. Fluid Mech. 338, 59–87 (1997) 3. Charron, Y., Lepesan, E., Surface structurée tri dimensionnelle à onde transverse en vue d’une réduction de la traînée aérodynamique. Patent FR 2899 945, 2007 4. Choi, K.-S.: Near-wall structure of a turbulent boundary layer with riblets. J. Fluid Mech. 208, 417–458 (1989) 5. Choi, K.-S.: Near-wall structure of turbulent boundary layer with spanwise-wall oscillation. Phys. Fluids 14(7), 2530–2542 (2002) 6. Choi, H., Moin, P., Kim, J.: Active turbulence control for drag reduction in wall-bounded flows. J. Fluid Mech. 262, 75–110 (1994) 7. Ferrante, A., Elghobashi, S.: On the physical mechanisms of drag reduction in a spatially developing turbulent boundary layer laden with microbubbles. J. Fluid Mech. 503, 345–355 (2004) 8. Fukagata, K., Iwamoto, K., Kasagi, N.: Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows. Phys. Fluids 14, L73 (2002) 9. Fukagata, K., Kasagi, N., Koumoutsakos, P.: A theoretical prediction of friction drag reduction in turbulent flow by superhydrophobic surfaces. Phys. Fluids 18, 051703 (2006) 10. Gad-El-Hak, M.: Flow Control. Cambridge University Press, Cambridge (2000) 11. Iwamoto, K., Fukagata, K., Kasagi, N., Suzuki, Y.: Friction drag reduction achievable by nearwall turbulence manipulation at high Reynolds numbers. Phys. Fluids 17, 011702 (2005) 12. Laadhari, F., Skandaji, L., Morel, R.: Turbulence reduction in a boundary layer by a local spanwise oscillating surface. Phys. Fluids A6(10), 3218–3220 (1994) 13. Lumley, J., Blossey, P.: Control of turbulence. Annu. Rev. Fluid Mech. 30, 311–327 (1998) 14. Peet, Y., Sagaut, P., Charron, Y.: Turbulent drag reduction using sinusoidal riblets with triangular cross-section. In: 38th AIAA Fluid Dynamics Conference and Exhibit. AIAA Paper 2008-3745 (2008) 15. Rebeck, H., Choi, K.-S.: A wind-tunnel experiment on real-time opposition control of turbulence. Phys. Fluids 18, 035103 (2006) 16. Robinson, S.K.: Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601 (1991) 17. Walsh, M.J.: Riblets, In: Bushnell, D.M., Heffner, J.N. (eds.) Viscous Drag Reduction in Boundary Layers, pp. 203–261. AIAA, Washington (1990)

Scaling Turbulent Fluctuations in Wall Layers Ronald L. Panton

Abstract The correlation of Reynolds number effects on profiles of turbulent fluctuations in wall layers is considered. It is proposed that, for some quantities, asymptotic expansions with two terms are needed to represent profiles. The two terms inherently have different scalings. Fluctuation profiles of all three vorticities and all three velocities are discussed.

1 Introduction Nomenclature in this article is: x, streamwise distance, y, wall normal distance and z, spanwise distance. Corresponding velocity fluctuations are u, v, w, and the free stream or centerline mean velocity is Uo . Non-dimensional distances are y + = yu∗ /ν where u∗ is the friction velocity and ν the viscosity, and Y = y/ h where h is layer thickness. The Reynolds number is Re∗ = u∗ h/ν. Townsend [9, 10], conjectured that there are two categories of fluctuations; active fluctuations make essential contributions to the Reynolds shear stress, while inactive motions do not. More recently, many, but not all, have accepted the conclusion of [2] that uu does not scale with u2∗ , as does uv, but with the mixed scale u∗ Uo . Moreover, although the Reynolds numbers are low, the DNS of [5] also confirm a mixed scaling. Thus, it is reasonable to conjecture that u (and w) fluctuations have two velocity scales. Modifying Townsend’s definition slightly, let us call motions that scale with u∗ , the same scaling as the Reynolds shear √ stress, active. Motions that scale with any other scale, such as the mixed scale u∗ Uo , will be called inactive. When both active and inactive motions are present, a quantity needs to be represented by a two term asymptotic expansion. Here, the DNS channel data [3–5] is used to test the scaling ideas. A similar analysis for boundary layers is given by [6]. R.L. Panton () University of Texas, Austin, USA e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_6, © Springer Science+Business Media B.V. 2011

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2 Composite Expansions The Reynolds number dependence of fluctuations profiles is displayed when the profiles are expressed as composite asymptotic expansions. An additive composite expansion has the form of an inner function fi (y + ) plus an outer function Fo (Y ) minus the common part. The common part is Fcp (Y ) = fcp (y + ) = f (y + → ∞) = F (Y → 0). f (y + , Re∗ ) = fi (y + ) + Fo (Y → y + /Re∗ ) − Fcp (Y → y + /Re∗ )

(1)

Matching is required between the inner and outer functions and determines the common part. It is well known that the outer regions boundary layers, pipes, and channels are different, while the inner regions are remarkably similar.

3 Reynolds Shear Stress By definition the Reynolds shear stress motions uv are entirely active. A composite expansion has the form: uv − 2 = g(y + ) + G(Y → y + /Re∗ ) − 1 (2) u∗ Here the outer function is known from pipe and channel theory to be G = 1 − Y and the common part is 1. Channel flow DNS data can be used to find g(y + ) in Eq. 2 by inserting data for uv+ and the known G − 1 = Y = y + /Re∗ . The results show an excellent correlation in Fig. 1.

4 Vorticity Fluctuations There has been some question about the proper scale for fluctuating vorticity. Most experimenters use the velocity scale u∗ and the viscous length scale ν/u∗ . However, our experience with streamwise fluctuations u provides two scales: an active motion scaled by √ u∗ (active quantities will be given a + superscript) and inactive motions scaled by u∗ Uo (given a # superscript). With different velocity scaling, the nondimensional vorticity forms are: ωω u4∗ /ν 2 u∗ ωω = ωω+ ωω# = 3 2 Uo Uo u∗ /ν

ωω+ =

(3) (4)

The outer representation, say for the y-direction component, has a two term asymptotic expansion: u∗ ωy ωy # (Y, Re∗ ) ∼ ωy ωy #o_0 (Y ) + ωy ωy #o_1 (Y ) (Re∗ ) + · · · (5) Uo

Scaling Turbulent Fluctuations in Wall Layers

53

Fig. 1 Inner Reynolds shear stress function

Zeroth and first-order subscripts 0 and 1 stand for the active and inactive parts, while subscripts o and i stand for the outer and inner regions. Similarly, for the inner region: ωy ωy # (y + , Re∗ ) ∼ ωy ωy #i _0 (y + ) + ωy ωy #i _1 (y + )

u∗ (Re∗ ) + · · · Uo

(6)

Although measurements are not accurate enough to establish Reynolds number trends, DNS provides precise values to do so. The highlights of a detailed analysis, Panton [8], will be given here.

4.1 Outer Vorticity If one considers any vorticity component in the outer region in the inactive scaling, say ωx ωx # (Y, Re∗ ), for fixed Y , as Re∗ → ∞ it is observed that ωx ωx # → 0. Furthermore, if one considers the any active scaling component such as ωx ωx + (Y, Re∗ ), for fixed Y , as Re∗ → ∞, it is again observed that ωx ωx + (Y, Re∗ ) → 0. The conclusion is that neither u4∗ /ν 2 nor Uo u3∗ /ν 2 scales vorticity fluctuations in the outer region. Later, Sect. 4.5, we will find that outer vorticity fluctuations are of a higher order than assumed in Eq. 5.

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Fig. 2 Active motion of the vertical vorticity fluctuation

4.2 Inner Vertical Vorticity It is convenient to first consider the vertical vorticity. If one plots the inactive scaling ωy ωy # (y + , Re∗ ), it is found that for fixed y + as Re∗ → ∞, ωy ωy # → 0. The inactive inner motion is zero. However, plotting the active scaling (see Fig. 2), ωy ωy # (y + , Re∗ ), reveals that curves for all Reynolds numbers, especially the highest three values, collapse together nicely. The vorticity rises slowly from zero to a maximum in the near wall region at about y + = 14 and then falls to become negligible around y + = 300 to 400. It is in this region that viscous events associated with the production of Reynolds shear stress occur. The vertical vorticity fluctuations are entirely active.

4.3 Inner Spanwise Vorticity The fluctuating spanwise vorticity ωz ωz # (y + ) is plotted in inactive scaling for the inner region in Fig. 3. Curves for different Reynolds numbers are close together near the wall. The finite wall value is directly related to the fluctuating wall shear stress. However, above y + = 10 the curves separate in a bump that is more pronounced as the Reynolds number becomes lower. This bump is located in the same region as the active motion found for the vertical vorticity ωy ωy + . It is assumed that active motions cause the bumps, and a line is drawn on Fig. 3 conjectured to be the inactive component ωz ωz #i _0 (y + ). This would be the ωz ωz # curve when Re∗ → ∞ (or u∗ /Uo → 0).

Scaling Turbulent Fluctuations in Wall Layers

55

Fig. 3 Inactive scaling of spanwise vorticity fluctuations

Fig. 4 Active component of spanwise vorticity fluctuations

The idea that the bump is active motions can be tested. Using Eq. 6, the data and the inactive curve can be subtracted and the result divided by u∗ /Uo to obtain: u∗ + # # (7) / (y ) = ω ω − ω ω ωz ωz + z z z z i _0 i _1 Uo The result should be the active part, and should correlate in the active scaling. The scaled data are shown in Fig. 4. The reasonable correlation supports the idea that the bumps are the result of active motions.

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Fig. 5 Inactive scaling of streamwise vorticity fluctuations

Fig. 6 Active component of streamwise vorticity fluctuations

4.4 Inner Streamwise Vorticity The streamwise vorticity ωx ωx (y + ) has a similar pattern to the spanwise vorticity. Figure 5 displays the data in the inactive scaling. The active motion bumps are more pronounced and further out in y. Processing the data to extract the active part, with an equation similar to Eq. 6, results in Fig. 6. The correlation is reasonably good as the Reynolds number increases.

Scaling Turbulent Fluctuations in Wall Layers

57

Fig. 7 Outer vorticity in Kolmogorov scaling

4.5 Outer Vorticity and Dissipation The vorticity in the outer region is of a higher-order than the terms in Eq. 5. It turns out that the non-dimensional form that is of order one is (see Fig. 7): ωi ωi ⊗ (Y ) ≡

ωi ωi = ωi ωi + Re∗ u3∗ /(νh)

(8)

The scaling in Eq. 8 has √ the physical interpretation of Kolmogorov time scale (reciprocal squared), τo = ν/ε = (νh/u3∗ /)1/2 . All the other vorticity components have a similar behavior in the outer region, and there is a tendency toward isotropy.

5 Normal Reynolds Stresses Townsend also proposed an “attached eddy” model for the inactive motions and used a single velocity scale u∞ for all fluctuations. This model predicts that, in the overlap region, the velocities scale as uu/u2∗ ∼ ln(1/Y ), vv/u2∗ ∼ C and ww/u2∗ ∼ ln(1/Y ), and the Reynolds shear stress is constant: uv/u2∗ ∼ C. Townsend attributed all of the u and w to inactive motions.

5.1 Vertical Velocity Fluctuations Townsend considered the vertical velocity fluctuations entirely active, and thus the should scale on u∗ . Channel flow DNS for vv is displayed in Fig. 8. Here there

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Fig. 8 Vertical velocity in active scaling

is a slight trend with Reynolds number. Whether this is inactive motion, or just a low Reynolds number effect, is not clear. Also note that there is a finite value at the centerline. Here the Reynolds shear stress is zero so the motion is inactive.

5.2 Streamwise Velocity Fluctuations In the outer region the streamwise stress with two scalings has the form: uu# (Y, Re∗ ) ∼ uu#o_0 (Y ) + uu#o_1 (Y )

u∗ (Re∗ ) + · · · Uo

(9)

Here the non-dimensional forms are: uu+ =

uu u2∗

uu# =

u∗ uu = uu+ Uo u∗ Uo

(10)

Experimental and DNS data is not refined enough to determine the two terms in Eq. 9 by limits. + Temptative and provisional assumptions, [7], are that uu+ o_1 and uui _1 are + proportional to the Reynolds shear stress −uv . The rational behind this assumption is that active motions are responsible for all of these quantities. This may be a crude approximation, but it is better than ignoring the active motion entirely. With

Scaling Turbulent Fluctuations in Wall Layers

59

Fig. 9 Streamwise velocity in inactive scaling and inactive component

this assumption Eq. 9 can be solved for the inactive correlation in terms of the data for uu and uv to obtain: u∗ uu#o_0 = uu# + −uv+ (11) Uo The question is does the data, processed according Eq. 11, correlate into an outer function uu#o (Y ) and an inner function uu#i (y + ) that match? Analysis of DNS data leads to Fig. 9 where the outer correlation is shown. The collapse is good. Experimental data is given in [7]. Consider that an additive composite expansion, of the form of Eq. 2, can be solved for the inner function. y+ uu#i _0 (y + ) = uu# (y + ) − uu#o_0 Y → + Ccp (12) Re∗ DNS data substituted into Eq. 12 is shown in Fig. 10. The correlation is reasonable but not perfect. Perhaps a better assumption is required for the active component.

5.3 Spanwise Velocity Fluctuations For the ww fluctuations, the approach of Sect. 5.2, which assumes that the inactive motions are proportional to the Reynolds number, has been formulated by [1]. The resulting correlations are not strong. There are some issues that need to be resolved to successfully correlate the spanwise fluctuation data.

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Fig. 10 Inner streamwise component

Fig. 11 Spanwise velocity in inactive scaling

Figure 11 displays the outer DNS data for ww# for all Reynolds numbers. The trend at the centerline appears to be a constant, however, for smaller Y the trend is definitely decreasing. Trends are revealed by plotting the values at Y = 1, Y = 0.4 and Y = 0.2 as a function of u∗ /Uo . Extrapolating to u∗ /Uo = 0, it is not unreasonable that ww# (Y → y + /Re∗ ) = constant. This would be the inactive outer function.

Scaling Turbulent Fluctuations in Wall Layers

61

Fig. 12 Spanwise velocity in inactive scaling

Another issue that needs to be resolved is the common part function in the overlap region. If one considers the active scaling ww+ as Re∗ → ∞, the common part appears to be a logarithmic function (the behavior predicted by Townsend’s model). Logarithmic overlap behavior complicates the inner region expression, and on occasion Poincare expansions cannot be used. Figure 12 gives the data in the inner region scaled as ww# = ww/(u∗ Uo ). The peak does not reach a limit at these Reynolds numbers. However, if one considers ww/(Uo2 ), the peak correlates nicely. Since the continuity equation restricts the velocity components, their scalings should be related.

6 Summary Several researchers have noted that uu scales better with the mixed scale u∗ Uo than with the u2∗ . On the other hand, uv definitely scales only with u2∗ . Townsend recognized that some motions contributed to the Reynolds shear stress while others do not. In light of these facts, profiles of fluctuations are proposed to, in general, have two terms with different scalings. Terms that scale with u∗ only, as does the Reynolds shear stress, are called active, while terms with Uo are called inactive. In the inner region vorticity fluctuation components in the streamwise and spanwise directions have both active and inactive parts while the normal component has only an active part. In the outer region vorticity fluctuations scale with the Kolmogorov time scale, and tend to be isotropic. Streamwise velocity fluctuations have both active and inactive parts. Townsend proposed that the normal velocity is completely active, however, the data is not clear on this point. The spanwise velocity fluctuation has some interesting trends that need to be considered if a successful correlation is obtained in the future [8].

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References 1. Buschmann, M.H., Gad-el-Hak, M.: Composite expansion for cross-flow stress. In: BAIL Conference, University of Limerick, Limerick, Ireland (2008) 2. DeGraaff, D.B., Eaton, J.K.: Reynolds number scaling of the flat plate turbulent boundary layer. J. Fluid Mech. 422, 319–346 (2000) 3. Del Alamo, J.C., Jimenez, J.: Spectra of very large anisotropic scales in turbulent channels. Phys. Fluids 15, L41–L44 (2003) 4. Del Alamo, J.C., Jimenez, J., Zandonade, P., Moser, R.D.: Scaling the energy spectra in turbulent channels. J. Fluid Mech. 500, 135–144 (2004) 5. Jimenez, J., Hoyas, S.: Turbulent fluctuations above the buffer layer of wall-bounded flows. J. Fluid Mech. 611, 215–236 (2008) 6. McKee, R.J.: Composite expansions for active and inactive motions in the streamwise Reynolds stress of turbulent boundary layers. Ph.D. Dissertation, University of Texas (2008) 7. Panton, R.L.: Composite expansions and scaling wall turbulence. Philos. Trans. R. Soc. A 365, 733–754 (2007) 8. Panton, R.L.: Scaling and correlations of vorticity fluctuations in turbulent channels. Phys. Fluids 21(11) (2009) 9. Townsend, A.A.: Equilibrium layers and wall turbulence. J. Fluid Mech. 11, 97–120 (1961) 10. Townsend, A.A.: The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press, Cambridge (1976)

Session 1: The WALLTURB LML Experiment

• The WALLTURB Joined Experiment to Assess the Large Scale Structures in a High Reynolds Number Turbulent Boundary Layer J. Delville, P. Braud, S. Coudert, J.-M. Foucaut, C. Fourment, W.K. George, P.B.V. Johansson, J. Kostas, F. Mehdi, A. Royer, M. Stanislas, and M. Tutkun • Calibration of the WALLTURB Experiment Hot Wire Rake with Help of PIV M. Stanislas, J.-M. Foucaut, S. Coudert, M. Tutkun, W.K. George, and J. Delville • Spatial Correlation from the SPIV Database of the WALLTURB Experiment J.-M. Foucaut, S. Coudert, M. Stanislas, J. Delville, M. Tutkun, and W.K. George • Two-Point Correlations and POD Analysis of the WALLTURB Experiment Using the Hot-Wire Rake Database M. Tutkun, W.K. George, M. Stanislas, J. Delville, J.-M. Foucaut, and S. Coudert

The WALLTURB Joined Experiment to Assess the Large Scale Structures in a High Reynolds Number Turbulent Boundary Layer Joel Delville, Patrick Braud, Sebastien Coudert, Jean-Marc Foucaut, Carine Fourment, W.K. George, Peter B.V. Johansson, Jim Kostas, Fahrid Mehdi, A. Royer, Michel Stanislas, and Murat Tutkun

Abstract Experiments, involving the joint effort of three European teams and aiming at using the state-of-the-art techniques to study the dynamics of the high Reynolds turbulent boundary layer, have been performed in June 2006 in the LML large wind tunnel. A set of four stereoscopic PIV systems and a rake of 143 hot wires were used to provide synchronised measurements. This paper summarises these exJ. Delville () · P. Braud · C. Fourment · A. Royer LEA UMR CNRS 6609, Poitiers, France e-mail: [email protected] P. Braud e-mail: [email protected] C. Fourment e-mail: [email protected] A. Royer e-mail: [email protected] S. Coudert · J.-M. Foucaut · J. Kostas · M. Stanislas LML UMR CNRS 8107, Villeneuve d’Ascq, France e-mail: [email protected] J.-M. Foucaut e-mail: [email protected] J. Kostas e-mail: [email protected] M. Stanislas e-mail: [email protected] W.K. George · P.B.V. Johansson · F. Mehdi · M. Tutkun TRL, Chalmers University of Technology, Gothenburg, Sweden e-mail: [email protected] P.B.V. Johansson e-mail: [email protected] F. Mehdi e-mail: [email protected] M. Tutkun e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_7, © Springer Science+Business Media B.V. 2011

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periments, presents sample results of both PIV and hot-wire rake data and illustrates the complementarity of such a coupled approach that combines the advantages of each technique.

1 Introduction The turbulence structure near a wall is a key to the understanding (and hopefully the reduction) of skin friction. From an experimental point of view, many researchers have worked on this subject since the fifties. They used different experimental approaches ranging from visualisation to hot wire anemometry and, more recently, to Particle Image Velocimetry. The initial goal was to measure statistics of the boundary layer turbulence. In parallel, studies on the organisation were performed to try to understand the process of auto generation of turbulence in the near wall region. Based also on numerical simulation, different models of this organisation have been published [5]. Modern optical measurement techniques (especially stereo PIV) have opened new opportunities for obtaining spatial information about turbulent flows. However, in spite of the rapid advance of optical measurement techniques, the hotwire anemometer is still the first choice of researchers when high frequency response and temporal information on the flow are needed, especially when the turbulence intensities are not too high (of the order of 30% typically). The primary interest in WALLTURB was to combine multi-plane stereo PIV with extensive rakes of hot-wire probes so that temporally resolved hot-wire data can be used to augment and even animate the more slowly sampled spatial and multi-component information from the PIV. This work implied a deep collaboration of the involved teams (Fig. 1).

2 Experimental Setup The experiments were performed in the large Laboratoire de Mécanique de Lille (LML) wind tunnel which test section is 20 m long and 1 × 2 m2 in cross section. The whole test section and the measurement section can be seen in Fig. 2. Two values of the Reynolds number Reθ , based on momentum thickness, were tested: 9,800 and 19,800. The freestream velocity was 5 m/s and 10 m/s respectively. The boundary layer thickness at the measurement location was about 30 cm. The hot-wire rake (HWR) designed and manufactured by the Laboratoire d’Etudes Aérodynamiques (LEA) (Fig. 3), was made of 143 single wire probes distributed in a (Y Z) plane1 normal to the flow. The HWR was used for all measurements in order to provide both spatial and temporal information simultaneously. The 1 The

following coordinates system was used: X, Y and Z correspond respectively to the streamwise, wall normal and spanwise directions.

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Fig. 1 One part of the Wallturb team during the experiment in Lille, June 2006. From left to right: M. Tutkun, M. Stanislas, J.-M. Foucaut, J. Delville, W.K. George, J. Kostas, P.B.V. Johansson, S. Coudert and F. Mehdi

Fig. 2 The LML wind tunnel (left) and the measurement section (right)

143 probes are distributed on an array such that 11 probes were placed logarithmically in wall-normal direction on each vertical comb and 13 combs were staggered in spanwise direction. The first row of probes was at 0.3 mm from the wall, corresponding to y + = 7.5 at Reθ = 9,800. The last probe is located at y = 306.9 mm, that is approximately δ99 . The rake is 280 mm in width and the vertical combs are also distributed logarithmically, symmetrically around the one placed at z = 0. The sensing wires of the probes were 0.5 mm in length l and 0.25 µm in diameter d, corresponding to wire length and diameter in wall units of: l + and d + of 11.8 and 0.006 for Reθ of 19,800, and 6.1 and 0.003 for Reθ of 9,800 respectively.2 The anemometry system was designed, manufactured and tested in the facilities of the Turbulence Research Laboratory (TRL) in its previous incarnation at the State University of New York at Buffalo (see Woodward [10]) and had been previously used and tested extensively (e.g., [1, 4]). The anemometer system went through a complete overhaul and modification after moving to Sweden with TRL in 2001. 2 Note

that l + = luτ /ν and d + = duτ /ν.

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Fig. 3 The 143 hot wires rake built by LEA (left); Zoom on the closest to the wall probes of one individual comb of the rake (centre-top) and view of rake connectors (centre-bottom); The Chalmers constant temperature anemometers system (right)

Data were sampled using a fast A/D converter with an on-board processor and a buffer to store the readings before spooling to the computer disk. A Microstar Laboratories DAP 5400a processor and additional extensions boards were used as the data acquisition (see [9] for a more complete description). With this architecture, it was possible to connect and simultaneously sample up to 192 channels at a rate of 52 kHz. The actual hot-wire sampling frequency was set to fs = 30 kHz for each experiment, corresponding to a sampling interval Δ+ = u2τ /(fs ν) ≈ 0.27. Sets of 2,200 blocks of 6 s of data were collected in total. Special connectors were used between the combs and 5 m long coaxial cables, which then connected to the hot-wire anemometers. Particular care was given to the comb end of the coaxial cables to prevent any interference with each other. To complete the spatial information on the flow, stereo PIV systems were used in combination with the HWR. To be able to use in a same experiment two stereoscopic PIV systems at the same time, and to avoid spurious illumination of one system by the light sheet of the other one, polarisation filters were used. Using the HWR implies some technical aspects to be considered. The PIV and HWR were synchronised using an external master clock system. The pollution of the wires by the seeding was found to be negligible concerning its frequency response. This pollution only introduces a drift in the calibration law. This consideration was even more drastic while very long experiments, several tens of hours, were involved. Due to limitations imposed by the mechanical design of the hot-wire rake and the support systems, it was not possible to perform a conventional calibration of the probes in the laminar flow. To solve these problems a specific procedure was developed for which the rake was permanently recalibrated in situ all along the experiments using the turbulent statistics of the flow at each actual probe location. Endly the blockage effect due to the rake intrusion within the boundary layer was addressed (see [6]). Two setups were retained that are described in the following. The main features of the databases created during these experiments are summarised in Table 1.

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Table 1 Data base for the zero pressure gradient turbulent boundary layer Ue [m/s]

Reθ

Configuration

No of HWR blocks (6 s)

No of PIV records

10

20,000

HWR + XY & Y Z

600

9,600

10

20,000

HWR + XZ

1,100

1,100 × 40

10

20,000

HWR + XZ

1 block of 2.29 s

6,880

10

20,000

HWR

613

0

Total: 2,314

Total: 60,480 9,600

5

10,000

HWR + XY & Y Z

600

5

10,000

HWR + XZ

1,100

1,100 × 40

5

10,000

HWR + XZ

1 block of 1.96 s

2,943

5

10,000

HWR

620

0

Total: 2,321

Total: 56,543

Fig. 4 Setup 1. Orthogonal dual planes stereoscopic PIV system and HWR configuration

Setup 1: HWR + XY & Y Z In this first HWR + PIV arrangement, described in Fig. 4, two stereo PIV systems were used to record a 30 × 30 cm2 (Y Z) plane located 1 cm upstream of the HWR plane (D). These two systems used a BMI 2 × 150 mJ dual cavity Yag Laser and four LaVision Image Intense PIV cameras (B and C) with a CCD of 1,376 × 1,040 pixels and a sampling rate of 4 VF/s. A third stereo PIV system was used to record a streamwise–wallnormal (XY ) plane in the plane of symmetry (Z = 0) and with a field of view of 10 × 15 cm2 . It used a BMI 2 × 150 mJ dual cavity Yag Laser and 2 LaVision Flowmaster PIV cameras (A) with a CCD of 1,280 × 1,024 and a sampling rate of 4 VF/s. These three SPIV systems were synchronised together and with the HWR data acquisition system. The first 600 blocks of data of the HWR system were acquired together with this SPIV arrangement.

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Fig. 5 Setup 2. High repetition rate SPIV system and HWR configuration

Setup 2: HWR + XZ In a second setup (Fig. 5), combined with the HWR, one high repetition rate SPIV system was used in the streamwise–spanwise (XZ) plane to get the spatial and temporal information in the near-wall region. This system was based on a Quantronix dual cavity 2 × 20 mJ YLF laser and two Phantom V9 cameras. This system recorded PIV planes parallel to the wall, at y + = 50 with sampling frequency of 3,000 VF/s for Reθ = 20,000 and y + = 100 with sampling frequency of 1,500 VF/s for Reθ = 10,000. The fields of view were 4.5 × 4.5 cm2 and 6.5 × 6.5 cm2 respectively. A total of 1,000 blocks of HWR data were collected synchronised with this high repetition SPIV.

3 Samples results We present in this section some illustrative examples of typical results that can be obtained by using the joined HWR–SPIV experimental arrangement of the Wallturb programme. Each of these means has its own advantages and drawbacks. Using these systems all together was expected to provide some information combining space and time description over long times and over the full boundary layer extent while remaining locally resolved both in time or in space. A compromise has generally to be found between the spatial description and resolution that can be provided by the stereoscopic PIV systems. Multiplane PIV can increase the spatial extent of the plane of analysis and even bring some quasi 3D information by using cross planes however the samples are obtained only at uncorrelated times. The excellent correspondence and continuity obtained by combining the information arising from the 3 stereoscopic PIV systems of setup 1 is illustrated by the instantaneous snapshots of the streamwise velocity component shown of Fig. 6 for the two values of Ue . The two SPIV systems (Y Z plane and cameras B and C of Fig. 4) lead to a very smooth reconstruction of the velocity in this plane, with a spatial resolution of 2 mm that is of the order of h+ = 40 and 20 for Reθ = 19,800 and

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Fig. 6 Instantaneous snapshots of velocity obtained from setup 1; Ue = 5 m/s (left) ; Ue = 10 m/s (right)

Fig. 7 Successive snapshots of velocity obtained from setup 2

9,800 respectively. The same excellent connexion of this plane with the XY plane is obtained. Note that these figures are obtained in an experimental configuration with the rake present in the boundary layer. The blockage effect due to the HWR is noticeable in the symmetry plane (XY ) where the velocity close to the wall seems smaller than in the (Y Z) plane. This effect is however less pronounced in the highest external velocity case (Ue = 10 m/s). High Repetition Rate PIV can combine PIV and HWR advantages by bringing some temporal description of the flow field for time records which remain limited in size (Fig. 7). The hot wire rake, by its limited discrete distribution, is sparse. However it can provide time resolved signals for very long periods and for frequencies only limited by the size of the wire. Using rakes of hot-wire can bring a limited local spatial resolution, however on a wide spatial extent. A very large field of view can then be obtained, by using Taylor’s hypothesis. As an example, Fig. 8 gives (for FP 10 m/s)

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Fig. 8 Spatial correlations obtained from the HWR by using Taylor’s hypothesis (contour lines) compared to the spatial correlations obtained directly from the PIV experiment (colored maps). X–Z plane (top) and X–Y plane (bottom)

the spatial correlations for a reference probe located y ∼ 0.015δ99 . The color plots corresponds to the PIV data and the solid lines to HWR data. This figure clearly illustrates the advantages of each technique.

4 Conclusions This complicate experiment was dreamed and performed in the framework of the WALLTURB programme, combining: Stereoscopic PIV; High Repetition PIV; Rake of 143 hot wires. The combination of time resolution (HW & HR PIV) and spatial resolution (PIV) made it possible to cover at the same time, small inner scales and large outer scales and to covers very long temporal integral scales. An equivalent experiment has been performed for an Adverse Pressure Gradient boundary layer configuration in the same wind tunnel and the same set-up. The pressure gradient was generated by a bump set on the wind tunnel floor. The shape of the bump was designed by Dassault Aviation to generate APG representative of airfoil at high angle of attack. A well documented data base has been created and all raw data are now available for the FP and APGFP configurations at: http://lmlm6-62.univ-lille1.fr/db/. Some non exhaustive results have been presented during the Wallturb final workshop in Lille (among them [2, 3, 8]) and can be retrieved in [7]. Acknowledgements This work has been performed under the WALLTURB project.WALLTURB (A European synergy for the assessment of wall turbulence) is funded by the CEC under the 6th framework program (CONTRACT No: AST4-CT-2005-516008).

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References 1. Citriniti, J.H., George, W.K.: Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition. J. Fluid Mech. 418, 137–166 (2000) 2. Foucaut, J.-M., Coudert, S., Stanislas, M.: HR SPIV for dynamical system construction. In: Stanislas, M., Jimenez, J., Marusic, I. (eds.) Progress in Wall Turbulence: Understanding and Modeling. Proceedings of the WALLTURB International Workshop Held in Lille, April 21– 23, 2009. ERCOFTAC Series, vol. 14. Springer, Dordrecht (2011) 3. Foucaut, J.-M., Coudert, S., Stanislas, M., Delville, J., Tutkun, M., George, W.K.: Spatial correlation from the SPIV database of the WALLTURB experiment. In: Stanislas, M., Jimenez, J., Marusic, I. (eds.) Progress in Wall Turbulence: Understanding and Modeling. Proceedings of the WALLTURB International Workshop Held in Lille, April 21–23, 2009. ERCOFTAC Series, vol. 14. Springer, Dordrecht (2011) 4. Jung, D., Gamard, S., George, W.K.: Downstream evolution of the most energetic modes in a turbulent axisymmetric jet at high Reynolds number. Part 1. The near-field region. J. Fluid Mech. 514, 173–204 (2004) 5. Panton, R.L.: Self-Sustaining Mechanisms of Wall Turbulence. Computational Mechanics Publications, Halifax (1997) 6. Stanislas, M., Foucaut, J.-M., Coudert, S., Tutkun, M., George, W.K., Delville, J.: Calibration of the WALLTURB experiment hot wire rake with help of PIV. In: Stanislas, M., Jimenez, J., Marusic, I. (eds.) Progress in Wall Turbulence: Understanding and Modeling. Proceedings of the WALLTURB International Workshop Held in Lille, April 21–23, 2009. ERCOFTAC Series, vol. 14. Springer, Dordrecht (2011) 7. Stanislas, M., Jimenez, J., Marusic, I. (eds.) Progress in Wall Turbulence: Understanding and Modeling. Proceedings of the WALLTURB International Workshop Held in Lille, April 21– 23, 2009. ERCOFTAC Series, vol. 14. Springer, Dordrecht (2011) 8. Tutkun, M., George, W.K., Stanislas, M., Delville, J., Foucaut, J.-M., Coudert, S.: Twopoint correlations and POD analysis of the WALLTURB experiment using the hot-wire rake database. In: Stanislas, M., Jimenez, J., Marusic, I. (eds.) Progress in Wall Turbulence: Understanding and Modeling. Proceedings of the WALLTURB International Workshop Held in Lille, April 21–23, 2009. ERCOFTAC Series, vol. 14. Springer, Dordrecht (2011) 9. Tutkun, M., George, W.K., Delville, J., Foucaut, J.-M., Coudert, S., Stanislas, M.: Space–time correlations from a 143 hot-wire rake in a high Reynolds number turbulent boundary layer. In: 5th AIAA Theoretical Fluid Mechanics Conference, 23–26 June 2008, Seattle, Washington. AIAA 2008-4239 10. Woodward, S.H.: Progress toward massively parallel thermal anemometry system. M.Sc. Thesis, State University of New York at Buffalo (2001)

Calibration of the WALLTURB Experiment Hot Wire Rake with Help of PIV Michel Stanislas, Jean-Marc Foucaut, Sebastien Coudert, Murat Tutkun, William K. George, and Joel Delville

Abstract In the WALLTURB LML experiment presented in Delville et al. (Stanislas, M., Jimenez, J., Marusic, I. (eds.) Progress in Wall Turbulence: Understanding and Modeling. Proceedings of the WALLTURB International Workshop Held in Lille, France, April 21–23, 2009. ERCOFTAC Series, vol. 14. Springer, Dordrecht, 2011), the calibration of the hot wire rake had to be performed on site, due to the large number of wires on the rake (143). For that purpose, the position of the wires with respect to the wall was carefully measured using two complementary techniques. The blockage effect due to the rake was characterized using StereoPIV and modeled using a simple potential flow model. The results show that the blockage affects mostly the mean flow and not the turbulence itself. Finally, the calibration data were built using the StereoPIV data in a plane located 1 cm upstream of the wires plane. The mean velocity, the second and third moments were provided. The calibration was checked on the fourth moment of the fluctuating velocity. M. Stanislas () · J.-M. Foucaut · S. Coudert LML UMR CNRS 8107, Villeneuve d’Ascq, France e-mail: [email protected] J.-M. Foucaut e-mail: [email protected] S. Coudert e-mail: [email protected] M. Tutkun FFI, Kjeller, Norway e-mail: [email protected] W.K. George Chalmers University of Technology, Gothenburg, Sweden e-mail: [email protected] J. Delville LEA UMR CNRS, Poitiers, France e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_8, © Springer Science+Business Media B.V. 2011

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Table 1 Theoretical wires position with respect to the wall Number

1

2

3

4

5

y (mm)

0.30

0.9

2.1

4.5

9.3

Number

7

8

9

10

11

y (mm)

38.1

76.5

153.3

230.1

306.9

6 18.9

1 Introduction In Delville et al. [1], a joint StereoPIV/Hot Wire Rake experiment was presented which was performed in the frame of the WALLTURB EC project. Due to the large number of wires on the rake, it was not possible to use a standard calibration procedure of the hot wire anemometers by putting the rake in a calibration wind tunnel. It was also not possible to make this calibration in the middle of the wind tunnel (outside the boundary layer) as the rake was not movable. Consequently, only an in situ calibration was possible. For this calibration, it was also not possible to vary the wind tunnel velocity, as is usually practiced, as this would have been too time consuming. Consequently, a specific calibration procedure had to be developed to perform the calibration from the measurements themselves. This was made possible thanks to the set-up used, with a stereo PIV plane 1 cm upstream of the hot wires plane, allowing a good degree of accuracy and convergence the mean streamwise velocity in front of each wire together with the second and third moments of the streamwise velocity fluctuations. Based on these data and thanks to the very large number of samples recorded with the Hot Wire Rake, a successful calibration could be performed. Before performing this calibration, it was necessary to determine accurately the actual wires’ position and to investigate the blockage effect of the rake on the incoming turbulent boundary layer.

2 Wires Location The wires’ position with respect to the wall were optimized based on the mean velocity and turbulence intensity profiles at four Reynolds numbers (Reθ = 7600, 10100, 13400, 19800) obtained previously from single hot wire traverses. The final theoretical position of the wires is given in Table 1 and compared in Fig. 1 to the mean velocity and streamwise turbulence intensity profiles. Due to the manufacturing procedure used, the actual position of each wire was slightly different from the theoretical one. Consequently it was necessary to measure this actual position of the wires used for the measurement. This was done immediately after the test campaign in a two step procedure. First, the wires were illuminated with an HeNe light sheet, generated with a 10 mW laser. This light sheet was introduced through the ceiling window of the wind tunnel, in the plane of the wire, as illustrated in Fig. 2. The image of the illuminated wires was recorded with the PIV stereoscopic set-up used for the Y Z plane,

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Fig. 1 Wall normal position of the rake hot wires, compared to the mean velocity (left) and streamwise turbulence intensity (right) distributions obtained from single hot wire traverses at four different external velocities

Fig. 2 Photograph of the hot wire rake with the wires illuminated by an HeNe laser light sheet

thanks to the depth of field of the set up and to the fact that it was calibrated enough in depth to reconstruct in the plane of the wires (which was 1 cm downstream of the PIV plane), it was possible to reconstruct the images of the wire and to measure their position with respect to the wall. Based on this calibration, the uncertainty on the position of the wires could be estimated at ±0.13 mm with 95% confidence. Such an uncertainty was acceptable only for the wires far away from the wall that is above 10 mm. This method was thus used for the wires in positions 6 to 13 leading to a relative error less than ±0.7%.

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Fig. 3 Map of the mean streamwise component U of the velocity in the two SPIV planes. Left: without rake. Right: with rake

For the wires nearer to the wall, the rake was dismounted from the wind tunnel, set on a table with a proper illumination and the wires’ position was measured three times successively. The mean value of the three measurements was taken as the wire actual position and the span was used to estimate the measurement uncertainty which, in this case is ±0.015 mm. This gives in the worst case (for wire n° 1) a relative uncertainty of ±5% on wire location. The actual position of the 143 wires of the rake is not reproduced here but is available in the database (http://lmlm6-62.univ-lille1.fr/db/). For the wires nearer to the wall, the rake was dismounted from the wind tunnel, set on a table with a proper illumination and the wires position was measured three times successively. The mean value of the three measurements was taken as the wire actual position and the span was used to estimate the measurement uncertainty which, in this case is ±0.015 mm. This gives in the worse case (for wire n° 1) a relative uncertainty of ±5% on wire location. The actual position of the 143 wires of the rake is not reproduced here but is available in the database (http://lmlm6-62.univ-lille1.fr/db/).

3 Blockage Effect One important concern with such a dense rake of probes is the blockage effect on the incoming boundary layer. This effect was foreseen and SPIV measurements were performed in the XY and Y Z set-ups without the rake, just before the measurements with the rake. 1600 samples were recorded in each plane, allowing the computation of the mean velocity and compare it to that with the rake installed. This comparison

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Fig. 4 Sketch of the rake, showing the position of the SPIV plane 10 mm upstream of the wires and of the potential flow model plane (parallel to the wall)

is given in Fig. 3 for the streamwise component of the mean velocity. As can be seen, there is a significant blockage effect. It was inferred from preliminary estimations that this blockage would be mainly a potential flow effect, not affecting strongly the boundary layer turbulence structure. To quantify this better, a simple potential flow model was developed based on the complex potential theory. In an XZ plane parallel to the wall (see Fig. 4), the flow was modeled with a classical expression of a uniform flow combined with a source: m m f (z ) = φ + iψ = Uo .z + . ln(z ) − i (1) 2π 2 where f (z ) is the complex potential with z = x + iz, Uo is the mean velocity without rake at the y location of the plane of interest and m is the intensity of the source. The origin is at the leading edge of the comb which is in the symmetry plane of both the flow and the rake. The sources are distributed in span at x = 0 and at the leading edge of each comb supporting the hot wire probes (that is 34 mm downstream of the wires plane). Using the superposition principle of complex potential, a model with N = 13 combs can easily be built. The streamwise velocity component is given by: N x m Ub = Uo (y) + du + 2π x 2 + (z − ai )2

(2)

i=1

and the spanwise component by: Wb =

N z m 2 2π x + (z − ai )2

(3)

i=1

where the ai parameters are the leading edge coordinates of each comb in the spanwise direction, m is the intensity of each source, Uo (y) is the velocity of the unperturbed flow and du is a constant allowing to adjust slightly the upstream velocity. As the mean velocity is not uniform along y, Uo (y) is set equal to the local mean streamwise velocity obtained by averaging along Z the velocity field without the rake. The values of du and m which are the same for all y and for all combs, are adjusted by a least square fit over 13 profiles extracted from the PIV measurement at each comb location. This fit was computed for all y between 1 cm and 30 cm.

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Fig. 5 Mean streamwise velocity distribution in the SPIV planes computed by use of the potential flow model

Fig. 6 Mean streamwise velocity spanwise profile at y = 10 cm, computed by use of the potential flow model in the wires plane, in the SPIV plane and compared to the actual SPIV data at the two Reynolds numbers

By moving the position of the plane with respect to the wall, it is possible to compute a 2D blockage effect (not taking into account the displacement of the streamlines above the rake). The result of such a computation for the streamwise velocity component is given in Fig. 5 in the XY an Y Z SPIV planes and can be compared to Fig. 3 (left). The agreement is fairly good. This is confirmed by the data in Fig. 6 which gives a spanwise profile at y = 10 cm of the streamwise mean velocity given by the model, in the plane of the wires and in the Y Z PIV plane. The results of the model are compared to the same profile from the SPIV measurements at the two Reynolds numbers. Two conclusions can be drawn from these figures: first, the model is in fairly good agreement with the data, supporting the idea that the blockage is mostly potential and second that the streamwise velocity component does not change between the SPIV plane and the wires plane which are 1 cm apart. Consequently, the mean velocity measured by SPIV in the Y Z plane can be used for an in situ calibration of the hot wires. Of course, the potential flow model is not realistic very near the wall, where the wall normal velocity gradient is strong and near the top of the rake where a wall

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normal mean velocity is induced by the blockage. But the fairly good agreement between observed above strongly supports the idea that the blockage is mostly potential. This conclusion is further supported by the comparison of the turbulence intensities (not shown here) which evidence a very good agreement between the SPIV measurements without and with rake (this will be confirmed further downstream).

4 Calibration The calibration method used to link the velocities to the wires’ voltages is detailed in [2]. A polynomial function of second order was used to link the instantaneous velocity to the instantaneous voltage: u˜ = a0 + a1 e˜ + a2 e˜2

(4)

Based on this relation, the mean streamwise velocity, second and third order moments of the streamwise fluctuations could be expressed: U = a0 φ0 + a1 φ1 + a2 φ2 u = a12 φ3 u3 = a13 φ6 2

+ 2a1 a2 φ4 + a22 φ5 + 3a12 a2 φ7 + 3a1 a22 φ8

(5) (6) + a23 φ9

(7)

where the φi are combinations of the voltage moments up to the 6th order (see [2] for details). Consequently, for each wire of the rake, to determine the three coefficients a0 , a1 and a2 , it is necessary to provide the mean velocity U and the two moments u2 and u3 , but also, the mean voltage E and the moments of the fluctuating voltage en up to the 6th order. Thanks to the large number of samples recorded (see [1]), it was possible to properly converge the moments of the fluctuating voltage up to order 6 [2]. For the velocity, as the blockage was demonstrated to be potential and as it was shown to vary negligibly between the SPIV Y Z plane and the wires plane, the SPIV data were interpolated at each Y Z wire position to provide both the mean velocity and the fluctuating velocity moments. This was done of course with the rake in place and here again, thanks to the large number of samples (see [1]), a good convergence could be achieved. Equations 5, 6 and 7 form a nonlinear system of equations which can be solved for a0 , a1 and a2 with a set of initial conditions for the coefficients. A nonlinear solver working in a least-squares sense was used to compute the calibration coefficients. A Matlab function called lsqnonlin with Levenberg–Marquardt method was implemented in our analysis. Selection of initial conditions, or guesses, for the coefficients to initiate the computation is very important for the convergence and speed of convergence in this kind of nonlinear system of equations, therefore they should be chosen properly.

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Fig. 7 Mean streamwise velocity spanwise profiles deduced from the SPIV data at each comb spanwise location and used for the wires calibration

The spatial resolution of the PIV data is uniform and was chosen as a compromise with the field of view, taking into account the size of the CCD cameras used (25+ at Re = 9800 and 50+ at Re = 19800). If this resolution is good away from the wall, it is much lower than that of the hot wire rake near the wall. In order to improve the spatial resolution very near the wall, a blending was performed between the actual SPIV data and the mean velocity profile provided by separate single hot wire traverses performed without the rake, before the joint experiment. This was done using the following formula: y (8) U (y) = UHWA (y) + UPIV (yo ) − UHWA (yo ) 1 − exp A. yo for y < yo , where yo = 6 mm is the first wall distance where the SPIV profiles are validated and A = 5 was chosen to make a smooth connection to the viscous sublayer. Based on this approach, the mean streamwise velocity profiles shown in Fig. 7 where obtained in the different XY planes corresponding to each comb. As can be observed, the effect of the blockage on the mean velocity is significant. It is maximum in the XY plane of symmetry of the rake (z = 0), around which the combs are close to each other. It is negligible on the sides of the rake (z = ±140 mm) where the agreement with the single hot wire anemometry traverse without rake is very good. As far as the turbulence statistics are concerned, as said previously, it was observed from the comparison of the SPIV data without and with the HWR in place that they were not affected by the rake blockage. Consequently, the second and third moments of the streamwise velocity fluctuations obtained from SPIV in the Y Z plane were averaged along z, to increase the convergence. They were extended close to the wall, using the same kind of coupling as for the mean profiles. Figure 8 shows the profile of respectively the second (top) and third (bottom) moments compared to the HWA data obtained from the single wire traverse without rake. As can be seen, the agreement is quite satisfactory. In order to verify the calibration performed, the fourth order moment profile of the streamwise fluctuations was also extracted from the SPIV data. After calibration, this profile could also be reconstructed independently from the Hot Wire Rake

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Fig. 8 Profiles of top: second order and bottom: third order moments of the streamwise velocity fluctuations deduced from the SPIV data, compared to single hot wire measurements without rake and used for the rake wires calibration

Fig. 9 Profiles of the fourth order moment of the streamwise velocity fluctuations deduced from the SPIV data and compared to the rake wires data after calibration

data. Figure 9 gives the comparison of these two profiles for the two Reynolds numbers under investigation. As can be seen, the agreement is fairly good, giving some confidence in the calibration procedure.

5 Conclusion An in-situ calibration of the Hot Wire Rake used in the Wallturb joint experiment [1] was performed successfully. For that purpose, it was necessary first to measure accurately the actual position of each of the 143 wires of the rake. It was then necessary to characterize the blockage effect. This was done using a simple potential flow model (2D) and it was demonstrated that the blockage was affecting only the mean velocity but not the turbulence structure. Based on these observations, an orig-

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inal calibration procedure was developed, based on a polynomial relation between the instantaneous streamwise velocity component and anemometer voltage. This calibration procedure needs the mean value of both, together with the second and third moments of the velocity fluctuations and the moments up to the sixth order of the voltage fluctuations to determine the three calibration coefficients, using a nonlinear least square fit. The velocity data were provided by a blending of SPIV data measured in a plane 1 cm upstream of the rake wires’ plane and single hot wire anemometry profiles measured previous to the joint experiment. The calibration was validated looking at the fourth order moment of the streamwise velocity fluctuations. To the authors knowledge, it is the first time that a hot wire anemometer is calibrated with the help of PIV data. Acknowledgements The authors would like to acknowledge P.B.V. Johansson who did a significant contribution to the success of the experiment itself. This work has been performed under the WALLTURB project. WALLTURB (A European synergy for the assessment of wall turbulence) is funded by the CEC under the 6th framework program (CONTRACT No: AST4-CT-2005-516008).

References 1. Delville, J., Braud, P., Coudert, S., Foucaut, J.-M., Fourment, C., George, W.K., Johansson, P.B.V., Kostas, J., Mehdi, F., Royer, A., Stanislas, M., Tutkun, M.: The WALLTURB joined experiment to assess the large scale structures in a high Reynolds number turbulent boundary layer. In: Stanislas, M., Jimenez, J., Marusic, I. (eds.) Progress in Wall Turbulence: Understanding and Modeling. Proceedings of the WALLTURB International Workshop Held in Lille, France, April 21–23, 2009. ERCOFTAC Series, vol. 14. Springer, Dordrecht (2011) 2. Tutkun, M., George, W.K., Foucaut, J.M., Coudert, S., Stanislas, M., Delville, J.: In situ calibration of hot wire probes in turbulent flows. Exp. Fluids 46(4), 617–629 (2009)

Spatial Correlation from the SPIV Database of the WALLTURB Experiment Jean-Marc Foucaut, Sebastien Coudert, Michel Stanislas, Joel Delville, Murat Tutkun, and William K. George

Abstract An original experiment has been performed in the frame of the WALLTURB EC project. In this experiment a specific set-up of SPIV allows to compute the full 3D tensor of velocity spatial correlation by using the homogeneity of the flow. The two-point correlations are tools to study the coherence of a flow. Stanislas et al. (C. R. Acad. Sci. Paris 2b 327:55–61, 1999) and Kahler (Exp. Fluids 36:114–130, 2004) showed that double spatial correlations, computed from Particle Image Velocimetry (PIV) fields, allow a better understanding of the turbulent flow organization.

1 Introduction The turbulent boundary layer is one of the most difficult problems in fluid mechanics. Since the seventies, studies on wall turbulence organization were performed to J.-M. Foucaut () · S. Coudert · M. Stanislas LML UMR CNRS 8107, Villeneuve d’Ascq, France e-mail: [email protected] S. Coudert e-mail: [email protected] M. Stanislas e-mail: [email protected] J. Delville LEA UMR CNRS 6609, Poitiers, France e-mail: [email protected] M. Tutkun FFI, Kjeller, Norway e-mail: [email protected] W.K. George Chalmers University of Technology, Gothenburg, Sweden e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_9, © Springer Science+Business Media B.V. 2011

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try to understand the process of auto generation in the near wall region of the flow. Based on both Direct Numerical Simulation and experiments, different models of this organization have been published (e.g. [1, 2, 4]) which are still the subject of intensive investigations in several research groups. A first approach to the organization study consists in computing the spatial and/or temporal two-point velocity correlation tensor. This approach has been used with both hot wire rakes, which give poor spatial resolution, and with PIV, mainly for the spatial correlations. The first investigation of this type was performed by Favre et al. [6, 7] who studied the spatio-temporal structure of the streamwise velocity component by using a pair of spatially separated hot-wire probes. Stanislas et al. [14] showed that double spatial correlations, computed from PIV fields, allow a better understanding of the turbulent flow organization. They demonstrated the richness of such a tool when it is coupled with conditional sampling. Kahler [11] characterized the structure size and organization of the buffer region of a turbulent boundary layer using such an approach, based also on PIV data. For the last 10 years, Stereoscopic PIV has been developed and characterized in detail [13, 16]. Such a method allows us to study the organization of near wall flows [15]. An improvement of the technique is to use the light polarization to record velocity fields at the same time in two different planes. If both planes are parallel, this method is called dual plane stereoscopic PIV [12]. The dual plane technique allows measurement of 2 velocity fields with an adjustable time delay or spatial separation between them. By varying this delay, the space–time correlation of the velocity field can be built. By varying the separation, the gradient tensor or the 3D spatial correlation can be built. Ganapathisubramani et al. [8] used the dual plane technique to get the full gradient tensor and to study the near wall flow structure. If both planes are perpendicular, it can give some information about the spatial properties and the organization of the flow [9, 10]. The LML wind tunnel allows us to obtain high Reynolds number boundary layers with a thickness of about 30 cm. In this wind tunnel, an original experiment for the characterization of a fully turbulent boundary layer was performed in the frame of the WALLTURB European project [5]. This experiment is concerned with the simultaneous measurement of 2 perpendicular SPIV planes (i.e. both planes are normal to the wall, one is streamwise and the other spanwise) and of 143 single hot wires in a plane normal to both the wall and the flow. From this experiment an original method to obtain the 3D correlation tensor will be proposed.

2 Experimental Setup As presented in Delville et al. [5], the experiment was carried out in a turbulent boundary layer wind tunnel. The test section is 1 m high, 2 m wide and 20 m long to allow the development of the boundary layer. The present experiment was carried out at two Reynolds numbers Rθ 9800 and 19800, which correspond to velocities of 5 and 10 m/s.

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Fig. 1 Top view of the experimental setup Fig. 2 Front view of the experimental setup

2.1 SPIV System Figures 1 and 2 show a top and a front view of the set-up, respectively. The first PIV plane (normal to the flow) is imaged with two Stereoscopic PIV systems in order to enlarge the field of view. Each system is based on Imager Intense cameras, from LaVision and a 50 mm lens. Both systems are adjusted with a small overlap region in order to obtain, after merging, a final field of view of about 30 × 32 cm2 . These dimensions are comparable to the boundary layer thickness. In that plane, the spatial resolution is 2 mm which corresponds to 40 wall units for the higher Reynolds

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Table 1 PIV analysis characteristics Rθ = 9800

Rθ = 19800

Characteristics

Units

External velocity

m/s

5

Friction velocity

m/s

0.185

Spatial resolution YZ

mm

2

2

Spatial resolution YZ

wua

25

50

Spatial resolution XY

mm

1

1

Spatial resolution XY

wua

12

25

Interrogation window size YZ

mm

6

6

Interrogation window size YZ

wua

75

150

Interrogation window size XY

mm

3

3

Interrogation window size XY

wua

36

75

a Wall

10 0.35

units

number and about 20 wall units for the smallest. The second PIV system (plane parallel to the flow) field of view is 10 cm long and 15 cm high. It is composed of two PCO Sensicam cameras and two 100 mm lenses. The spatial resolution in this plane is 1 mm, which is half that of the previous plane. The x and z axes are in streamwise and spanwise respectively. The y axis is normal to the wall. The laser used was a BMI YAG system with 4 cavities, which can deliver 4 beams two by two recombined and orthogonally polarized. Each camera lens was suited with a polarizing filter. Consequently, for each camera, only the corresponding light sheet is recorded on the images as for the dual plane technique. Moreover each SPIV system was adjusted with respect to the Scheimpflug conditions. A total of 9600 velocity fields per Reynolds number were recorded. Figure 6 in [5] gives an example of the instantaneous streamwise velocity component u in the two planes for both Reynolds numbers.

2.2 PIV Analysis The images from both cameras were processed with a standard multi-grid algorithm with discrete window offset. The analysis was made by the classical FFT-based cross-correlation method with integer shift of both windows. A 1D Gaussian peak fitting algorithm was used for the sub-pixel displacement determination. The final interrogation window size was 30 × 42 pixels and 26 × 38 pixels respectively for the plane normal and parallel to the flow. The PIV analysis characteristics are given in Table 1. The interrogation window size can be slightly large which is not an issue because the focus is on the large scale motions in this experiment. The spatial resolution given in Table 1 corresponds to a 70% mean overlap. The Soloff method using 3 calibration planes was used to reconstruct the three velocity components in

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Fig. 3 2D correlation R11 in the YZ plane

the plane of measurement. This was performed using the PIVlml software developed at LML. The calibration was done with different targets using crosses. From the set of recorded calibration planes and SPIV images, the misalignment between the light sheet and the calibration plane was corrected (Coudert and Schon [3]).

3 Spatial Correlation 3.1 2D Correlations The double spatial correlations can be computed independently from the SPIV data in both planes. Stanislas et al. [14] and Kahler [11] have detailed the properties of these correlations for standard PIV fields obtained in the same boundary layer. They showed that the spatial correlation shape allows quantitative information about coherent structures like both shape and size. SPIV gives the full tensor of correlation in a plane. As an example, R11 is shown in Fig. 3 for the lowest Reynolds number. The fixed point is at y = 25 mm corresponding to 0.08δ (i.e. 300 wall units). This correlation map corresponds to a standard turbulent boundary layer: a single peak is located at the fixed point. The size of the peak is indicating the size of a coherent region of the flow. This size is about 0.2δ wide and 0.25δ high. It gives an idea of the size of the large scale motion of the boundary layer relative to this wall distance. Two negative peaks are located on each side of the main peak at about 0.4. This large scale organization is in quite good agreement with the results of Ganapathisubramani et al. [9].

3.2 3D Correlations To compute the 3D correlation a volume measurement is usually necessary. However, if the flow is homogeneous along the x and z directions, the 3D spatial corre-

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Fig. 4 Comparison of 2D and 3D correlation

Fig. 5 Comparison of 2D and 3D correlation

lation can be computed from the present experiment using data of the two perpendicular SPIV planes: Rij (y, δx, δy, δz) = u1i (x0 − δx, y, 0)u2j (x0 , y + δy, δz) (1) N

Rij (y, δx, δy, δz) =

u2i (x0 , y, −δz)u1j (x0 + δx, y + δy, 0)

(2)

N

where x = x0 and z = 0 are the coordinates of the two planes intersection, superscripts 1 and 2 correspond to the normalized fluctuation components of the planes z = 0 and x = x0 respectively and N is the number of independent realizations (here about 10000). In the present case, as the mean flow is 2D, the symmetry can be enforced in order to improve the convergence. The correlation Rij must be symmetrical if i = j or i = j = 3 in the other cases the correlation must be anti-symmetrical. It is interesting to note that the 3D correlations were obtained by a direct computation. The FFT, which is much faster, has not been chosen to limit the effect of eventual outliers. The computational time, which would be a few months, was reduced to a few hours by a full parallelization of software (about 200 of CPU were used). Figures 4 and 5 show a comparison of the 2D and the 3D correlation as a cut along x and z. As can be seen both correlations are in quite good agreement. Close

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Fig. 6 Iso-contour at 0.6 of 3D correlation R11

Fig. 7 Iso-contour at 0.6 of 3D correlation R22

to 0 the 3D correlation does not reach 1. This is due to the fact that this correlation is computed from two different PIV systems. A correlation of 1 would show a perfect measurement with no difference between both systems. The errors between both PIV systems are not correlated. A strong difference of convergence level is also shown. In the 2D case the homogeneity is used to increase the convergence (about 100000 samples) and in the 3D case the homogeneity is used to build the 3D (only 9600 samples). The lack of convergence is more critical when the correlation level decreases (as, for example, when δz is far from 0). Figures 6, 7 and 8 show the 3D isovalue of the correlations R11 , R22 and R33 corresponding to a correlation level of 60%. R11 shows a very elongated region of correlation due to the streaky shape on the streamwise velocity component. This correlation looks like a long tube of 0.85δ with a diameter of 0.08δ which is inclined at about 5° to the x axis. R22 looks like a sphere (diameter 0.06δ) which confirms the isotropic behavior of the normal velocity fluctuation. R33 is also elongated in the streamwise direction but less than R11 and also more inclined (15°). The length of R33 is of the order of 0.17δ.

4 Conclusion A method to measure the 3D correlation from a specific PIV experiment in a turbulent boundary layer is detailed. It is based on the fact that there exists two ho-

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Fig. 8 Iso-contour at 0.6 of 3D correlation R33

mogeneous directions in the flow. From this method, by means of two stereoscopic PIV fields recorded in perpendicular planes at the same time, the full tensor of 3D correlation can be computed. Cuts of the 3D correlation match the 2D ones. Between the YZ PIV plane and one wire of the comb, the spatio-temporal correlation can also be computed from this method. These results should allow to validate the Taylor hypothesis and to characterize the organization of the flow as far as the large scale motions are concerned. To complete the database a Linear Stochastic Estimation method can be envisaged based on the 3D and spatio-temporal correlations. Acknowledgements The authors would like to acknowledge F. Benyoucef and D. Krolak who did a significant contribution to the development of the correlation computation software. This work has been performed under the WALLTURB project. WALLTURB (A European synergy for the assessment of wall turbulence) is funded by the CEC under the 6th framework program (CONTRACT No: AST4-CT-2005-516008).

References 1. Adrian, R.J., Meinhart, C.D., Tomkins, C.D.: Vortex organisation in the outer region of the turbulent boundary layer. J. Fluid Mech. 422(1), 1–54 (2000) 2. Carlier, J., Stanislas, M.: Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry. J. Fluid Mech. 535, 143–188 (2005) 3. Coudert, S., Schon, J.P.: Back projection algorithm with misalignment corrections for 2D3C stereoscopic PIV. Meas. Sci. Technol. 12, 1371–1381 (2001) 4. Del Alamo, J.C., Jimenez, J., Zandonade, P., Moser, R.D.: Particle imaging techniques for experimental fluid mechanics. J. Fluid Mech. 561, 329–358 (2006) 5. Delville, J., Braud, P., Coudert, S., Foucaut, J.-M., Fourment, C., George, W.K., Johansson, P.B.V., Kostas, J., Mehdi, F., Royer, A., Stanislas, M., Tutkun, M.: The WALLTURB joined experiment to assess the large scale structures in a high Reynolds number turbulent boundary layer. In: Stanislas, M., Jimenez, J., Marusic, I. (eds.) Progress in Wall Turbulence: Understanding and Modeling. Proceedings of the WALLTURB International Workshop Held in Lille, France, April 21–23, 2009. ERCOFTAC Series, vol. 14. Springer, Dordrecht (2011) 6. Favre, A., Faviglio, J., Dumas, R.: Space–time double correlations and spectra in a turbulent boundary layer. J. Fluid Mech. 2, 313–342 (1957) 7. Favre, A., Faviglio, J., Dumas, R.: Further space–time correlations of velocity in a turbulent boundary layer. J. Fluid Mech. 3, 344–356 (1958)

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8. Ganapathisubramani, B., Hutchins, N., Hambleton, W.T., Longmire, E.K., Marusic, I.: Investigation of large-scale coherence in a turbulent boundary layer using two-point correlations. J. Fluid Mech. 524, 57–80 (2006) 9. Ganapathisubramani, B., Longmire, E.K., Marusic, I.: Experimental investigation of vortex properties in a turbulent boundary layer. Phys. Fluids 18, 055105 (2006) 10. Hambleton, W.T., Hutchins, N., Marusic, I.: Simultaneous orthogonalplane particle image velocimetry measurements in a turbulent boundary layer. J. Fluid Mech. 560, 53–64 (2006) 11. Kahler, C.J.: Investigation of the spatio-temporal flow structure in the buffer region of a turbulent boundary layer by means of multiplane stereo PIV. Exp. Fluids 36, 114–130 (2004) 12. Kahler, C.J., Kompenhans, J.: Fundamentals of multiple plane stereo PIV. Exp. Fluids 29(Suppl.), S70–S77 (2000) 13. Soloff, S., Adrian, R., Liu, Z.C.: Distortion compensation for generalized stereoscopic particle image velocimetry. Meas. Sci. Technol. 8, 1441–1454 (1997) 14. Stanislas, M., Carlier, J., Foucaut, J.M., Dupont, P.: Double spatial correlations, a new experimental insight into wall turbulence. C. R. Acad. Sci. Paris 2b 327, 55–61 (1999) 15. Stanislas, M., Perret, L., Foucaut, J.M.: Vortical structures in the turbulent boundary layer: a possible route to a universal representation. J. Fluid Mech. 602, 327–382 (2005) 16. Willert, C.: Stereoscopic digital particle image velocimetry for applications in wind tunnel flows. Meas. Sci. Technol. 8, 1465–1479 (1997)

Two-Point Correlations and POD Analysis of the WALLTURB Experiment Using the Hot-Wire Rake Database Murat Tutkun, William K. George, Michel Stanislas, Joel Delville, Jean-Marc Foucaut, and Sebastien Coudert

Abstract We use the novel hot-wire rake of 143 single wire probes to measure 30 cm thick boundary layer of the large LML wind tunnel at Reynolds number based on momentum thickness of 9800 and 19 100. Multiple-point cross-correlation analysis using the data from the hot-wire rake show that the physical length of the correlation contour in the streamwise direction is about 7 boundary layer thickness. In addition, the shape of the correlations maps obtained from two-point correlations on the streamwise–wall-normal plane retains its shape approximately throughout the entire boundary layer. The data is also used for POD analysis of the boundary layer using only the streamwise turbulent fluctuations. The normalized eigenvalue distribution shows that the first POD mode has more than 40% of the turbulence kinetic energy, while the second one has about 20% of the total turbulence kinetic energy for both of the Reynolds number tested here. It is possible to recover about 90% of the kinetic energy only using the first four POD modes. The reconstructed velocity fluctuations on the spanwise–wall-normal plane show how organized motions of M. Tutkun () Norwegian Defence Research Establishment, P.O. Box 25, 2027 Kjeller, Norway e-mail: [email protected] W.K. George Chalmers University of Technology, Dept. of Applied Mechanics, 41296 Gothenburg, Sweden e-mail: [email protected] M. Stanislas · J.-M. Foucaut · S. Coudert Laboratoire de Mécanique de Lille, UMR CNRS 8107, 59655 Villeneuve d’Ascq, France e-mail: [email protected] J.-M. Foucaut e-mail: [email protected] S. Coudert e-mail: [email protected] J. Delville Laboratoire d’Etudes Aérodynamiques, UMR CNRS 6609, ENSMA, 86036 Poitiers, France e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_10, © Springer Science+Business Media B.V. 2011

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turbulence with significant amounts of energy interact with each other across the boundary layer. It is also possible to observe the strength of the interaction between the inner and outer layer using these reconstructed velocity fields.

1 Two-Point Correlations of WALLTURB Experiments The large and very large scales of motions in a high Reynolds number turbulent boundary layer have been analyzed and documented by Tutkun et al. [5] using the novel hot-wire rake of 143 single wire probes shown in Fig. 1. The large LML wind tunnel provides a very thick boundary layer of about 30 cm with resolvable small scales. The hot-wire rake with many probes distributed on an array enables us to look at both spatial and temporal characteristics of the turbulent boundary layers by means of multiple-point cross-correlation analysis. The results for the twopoint cross-correlations observed in a turbulent boundary layer at Reynolds numbers based on momentum thickness, Reθ , of 19 100 and 9800 can be summarized from [5] as follows: (i) the maximum extension of the correlations maps on the streamwise–spanwise plane is bounded between ±3.5δ, (ii) the shape of the correlations maps obtained from two-point correlations on the streamwise–wall-normal plane retains its shape approximately throughout the entire boundary layer, (iii) the streamwise–wall-normal plane correlation maps show that extent of the most elongated correlation contour lines in the streamwise direction is about equal to or smaller than 8δ. See [5] for more details and thorough discussions on the two-point correlations, their Reynolds number dependence and structural implications.

2 Proper Orthogonal Decomposition The full four dimensional representation of the POD integral in Cartesian coordinate system is given by: Ri,j (x, x , y, y , z, z , t, t )φj(n) (x , y , z , t ) dx dy dz dt D

(n)

= λ(n) φi (x, y, z, t)

(1)

where x, y, z and t denote coordinates in streamwise, wall-normal and spanwise directions and time respectively. represents different position in space and time. Because the turbulent boundary layer is stationary in time and homogeneous in the spanwise direction, the two-point cross-correlation tensor, Ri,j (x, x , y, y , z, z , t, t ) = ui (x, y, z, t)uj (x , y , z , t ) is only function of separations in space and time. Since the POD reduces to the harmonic decomposition in the homogeneous and stationary directions, these directions can be removed by taking the Fourier transform of the two-point cross-correlation tensor. Therefore, Fourier transformations in time, t, and homogeneous direction, z, are performed to obtain Fourier coefficients of the two-point cross-spectral tensor:

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Fig. 1 Hot-wire rake in place in the LML wind tunnel and close-up of one of the combs

Si,j (x, x ; y, y ; k; f ) ∞ ∞ i,j (x, x ; y, y , z, τ )e−i(2πf τ +kz) dτ d(z) = R −∞ −∞

(2)

where f is the frequency corresponding to τ and k is the spanwise Fourier mode number corresponding to z. If only one downstream location is considered, streamwise, x, dependence of the two-point cross-spectral tensor in (2) can be treated as a parameter. The resulting POD integral equation is therefore called the slice-POD (cf., [1, 2, 4]). As detailed by Krogstad et al. [3] utilizing the Taylor hypothesis in connection with the convection velocities across the boundary layer is a reasonable assumption to convert spatial dependence to the time dependence. Therefore, x dependence is essentially the same as time dependence, and the slice POD equation can be written as: ∗(n) (n) Si,j (y, y ; k; f )φj (y ; k; f ) dy = λ(n) (k; f )φi (y; k; f ) (3) y

(n)

where λ(n) (k; f ) and φi (y; k; f ) represent the eigenspectra and eigenfunctions for each spanwise mode and frequency, respectively. Since the integration is performed over the wall-normal coordinate direction which is an inhomogeneous direction, the domain is of finite total energy and Hilbert–Schmidt theory applies. We consider therefore the domain to be bounded by the boundary layer thickness, δ, since the energy and correlations beyond it make a negligible contribution.

2.1 Eigenvalue Distribution over POD Modes First, the eigenvalue distribution is presented in Fig. 2 to show how POD is efficient and optimal in terms of capturing the largest amount of turbulence kinetic energy with the fewest modes. The high and low Reynolds number cases are given in Figs. 2(a) and 2(b) respectively. Each of the bars in these figures represents the

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Fig. 2 Normalized eigenvalue distribution, ζ n , for each POD mode, n

normalized eigenvalues integrated over frequency and summed over the spanwise Fourier modes. The distribution of eigenvalues is computed using (4) as follows: M ∞ (n) λ (k, f ) df k=1 (n) (4) ζ = N M−∞ ∞ (n) n=1 k=1 −∞ λ (k, f ) df where the denominator is the total turbulence kinetic energy at the plane normal to the streamwise direction. These results are based on a one-component scalar proper orthogonal decomposition of the streamwise turbulence fluctuations, since only the one component of the velocity was measured in the experiments. Each bar in Fig. 2 indicates the contribution of the corresponding POD modes to the total kinetic energy of the domain. We observe slightly higher (∼0.4%) energy captured by the first POD mode in the high Reynolds number case, whereas the second POD mode of the low Reynolds number case is found to be approximately the same amount higher than that of the high Reynolds number case. The results show that the first six POD modes contain more than 97% of the total energy. If the energy content of the first four POD modes is investigated, we see that approximately 90% of the total energy is carried by these three modes.

2.2 Eigenvalue Distribution over POD and Spanwise Fourier Modes The eigenspectra of the POD modes are integrated over frequency to investigate the kinetic energy distribution only over azimuthal Fourier modes. These results are presented in normalized form in Fig. 3. Each bar denotes the contribution to the turbulence kinetic energy of the POD mode at the spanwise Fourier mode shown by the abscissa of the plots. The ∗ signs in the figures indicate the total contribution of each spanwise Fourier mode to the total turbulence kinetic energy at each spanwise Fourier mode and are computed by summing all the POD modes for each spanwise

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Fig. 3 Normalized eigenvalue distribution, ξ n , for POD modes, n, and spanwise Fourier modes, k

mode. Figures 3(a) and 3(b) show the normalized eigenvalue distribution for high and low Reynolds numbers respectively. There are some common features in both of these figures; namely: (i) Most of the energy is found at spanwise Fourier mode-1 and mode-2; (ii) Spanwise Fourier mode-2 is slightly larger than spanwise Fourier mode-1; (iii) The first POD modes of the first and second spanwise Fourier modes of the high Reynolds number case are larger than those of the low Reynolds number case, whereas the second POD modes of the first and second Fourier modes of the high Reynolds number case are smaller than those of the low Reynolds number case.

2.3 Reconstruction of Velocity Field Reconstruction of the instantaneous velocity field of the flow is performed using the deterministic POD eigenmodes together with their the random coefficients, which are obtained by projecting the velocity field onto the deterministic eigenmodes. Since the kernel, S1,1 (y, y ; k; f ), is written as a function of both spanwise Fourier mode number, k, and frequency, f , the resulting eigenfunctions and eigenvalues are also functions of these two parameters as described in the previous sections. Therefore, reconstruction of velocity field begins by finding the random coefficients, an (k, f ), by projecting the eigenfunction onto the double Fourier transformed velocity fluctuations as follows: ∞ ˆˆ k, f )φ (n)∗ (y, k, f ) dy (5) u(y, an (k, f ) = 0

where the integration is performed in the inhomogeneous wall-normal direction using the trapezoidal rule. The upper limit of integration is actually replaced by the boundary layer thickness, δ. The doubly Fourier transformed streamwise compo-

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nent of fluctuating velocities is obtained as a linear combination of eigenfunction using the random coefficients obtained by (5): uˆˆ rec (y, k, f ) =

N

a (n) (k, f )φ (n)∗ (y, k, f )

(6)

n=1

where subscript rec stands for “reconstructed”. The reconstructed velocity expressed in terms of spanwise Fourier modes and frequency can be mapped back into real space to get urec (y, z, t) in two steps: (i) inverse Fourier transformation in frequency, (ii) inverse Fourier transformation in spanwise Fourier index. Since large scale features of turbulent boundary layers are of much interest, lowpass filtering of the data to remove the contributions due to high frequency (or wavenumber) (i.e., small scale features of turbulence) is performed by setting the random coefficient, an (k, f ), to zero for those high frequencies. The location of the cut-off frequency in the spectrum was decided to be 100 Hz for the high Reynolds number case and 49.4 Hz for the low Reynolds number case. It has been observed that the filtering in our case only removes the high frequency and small scale contributions, and does not affect the large scale features of turbulence. Figure 4 shows the reconstructed velocity for Reynolds numbers of 19 100 using only one POD mode in the spanwise–wall-normal (YZ) plane. Figure 4(a) presents the reconstruction of velocity fluctuations using only the first spanwise Fourier mode. Reconstructed velocity fluctuations using only the second spanwise Fourier mode are presented in Fig. 4(b). The variation in velocity fluctuations across the spanwise direction is clear this time, because of the second spanwise Fourier mode. Notice that two lobes fill almost the entire extent of the view in the spanwise direction, which is about one boundary layer thickness. We observe strong positive and negative momentum sources in the outer layer as shown in Fig. 4(b), especially between y/δ of 0.1 and 0.8. These are one of the most dominant features if only the reconstruction based on spanwise Fourier mode 2 is considered. The time-resolved reconstruction of this POD and spanwise Fourier mode show that once there is large enough momentum developed in the inner layer including both log layer and near wall layers, then the high momentum sources located in the outer layer and inner layer merge. This initially results in very large scales of motion occupying the entire outer and ‘log’ layers. Reconstructed velocities for high Reynolds number experiment using spanwise Fourier mode-3 together with only the first POD mode are plotted in Fig. 4(c). First of all, there are more variations in velocity profiles and zero crossings along the spanwise direction. Figure 4(c) shows that the organized motions extend from halfway in the outer region to the end of the buffer layer. Figure 4(d) shows the velocity contour lines of the reconstructed turbulent fluctuations using the fourth spanwise Fourier mode. The positive and negative momentum sources using only one POD mode based reconstructions appear to be fixed in space and they do not change their location substantially.

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Fig. 4 Reconstructed velocity fluctuations using only the first POD and first four spanwise Fourier modes at Reθ = 19 100. k denotes spanwise Fourier mode number

3 Discussion and Summary The normalized eigenvalue distribution shows that the first POD mode has more than 40% of the turbulence kinetic energy, while the second one has about 20% of the total turbulence kinetic energy. It is possible to recover about 90% of the kinetic energy only using the first four POD modes. Similar figures are obtained at both Reynolds number with no significant indication of Reynolds number dependence. The eigenspectra always peak near zero frequency and most of the large scale features are found below 100 Hz and 50 Hz for the high and low Reynolds number cases studied here respectively. The kinetic energy distribution is maximum at spanwise Fourier mode-2, while there is a slight difference between spanwise Fourier mode-1 and -2. The normalized eigenvalue distribution obtained from two different Reynolds number have almost the same distribution and features.

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The low-pass filtering of turbulence kinetic energy by means of the random coefficients of POD is very effective with no significant change in the large scale features of the turbulent boundary layer. The reconstructed velocity fluctuations on the spanwise–wall-normal plane show how organized motions of turbulence with significant amounts of energy interact with each other across the boundary layer. It is also possible to observe the strength of the interaction between the inner and outer layer using these reconstructed velocity fields. Acknowledgements This work has been performed under the WALLTURB project. WALLTURB (A European synergy for the assessment of wall turbulence) is funded by the CEC under the 6th framework program (CONTRACT No: AST4-CT-2005-516008). M. Tutkun acknowledges the partial support by the Center of Excellence grant from the Norwegian Research Council to the Center for Biomedical Computing at Simula Research Laboratory.

References 1. Citriniti, J.H., George, W.K.: Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition. J. Fluid Mech. 418, 137–166 (2000) 2. Johansson, P.B.V., George, W.K.: The far downstream evolution of the high Reynolds number axisymmetric wake behind a disk. Part 2. Slice proper orthogonal decomposition. J. Fluid Mech. 555, 387–408 (2006) 3. Krogstad, P., Kaspersen, J.H., Rimestad, S.: Convection velocities in a turbulent boundary layer. Phys. Fluids 10(4), 949–957 (1998) 4. Tutkun, M., Johansson, P.B., George, W.K.: Three-component vectorial proper orthogonal decomposition of axisymmetric wake behind a disk. AIAA J. 46(5), 1118–1134 (2008) 5. Tutkun, M., George, W.K., Delville, J., Stanislas, M., Johansson, P.B.V., Foucaut, J.M., Coudert, S.: Two-point correlations in high Reynolds number flat plate turbulent boundary layers. J. Turbul. 10(21), 1–23 (2009)

Session 2: Experiments in Flat Plate Boundary Layers

• Reynolds Number Dependence of the Amplitude Modulated Near-Wall Cycle I. Marusic, R. Mathis, and N. Hutchins • Lagrangian and Eulerian Aspects of a Turbulent Boundary Layer Flow. An Investigation Using Time-Resolved Tomographic PIV A. Schröder, R. Geisler, K. Staack, A. Henning, B. Wieneke, G.E. Elsinga, F. Scarano, C. Poelma, and J. Westerweel (no paper) • Tomographic Particle Image Velocimetry Measurements of a High Reynolds Number Turbulent Boundary Layer C.H. Atkinson, S. Coudert, J.-M. Foucaut, M. Stanislas, and J. Soria • Study of the Vortical Structures in Turbulent Near-Wall Flows S. Herpin, S. Coudert, J.-M. Foucaut, J. Soria, and M. Stanislas

Reynolds Number Dependence of the Amplitude Modulated Near-Wall Cycle Ivan Marusic, Romain Mathis, and Nicholas Hutchins

Abstract The interaction in turbulent boundary layers between very large scale motions centred nominally in the log region (termed superstructures) and the small scale motions is investigated across the boundary layer. This analysis is performed using tools based on Hilbert transforms. The results, across a large Reynolds number range, show that in addition to the large-scale log region structures superimposing a footprint (or mean shift) on to the near-wall fluctuations, the small-scale structures are also subject to a high degree of amplitude modulation due to the large structures. The amplitude modulation effect is seen to become progressively stronger as the Reynolds number increases.

1 Introduction Advances in numerical simulation, measurement techniques and high Reynolds number facilities have provided the opportunity in recent years to study in greater detail the relationship between eddying motions of different length scales in wallbounded flows. The near-wall cycle, related to the near-wall streaks described by Kline et al. [11], has been largely viewed as depending only on global viscous wall units. The study by Jimenez & Pinelli [7] has shown that this region can self-sustain in the absence of an outer region and it is therefore often referred to as being autonomous. This autonomous view was based largely on an understanding of low Reynolds number flows, which by definition have a limited range of scales of motions. More recently, studies at higher Reynolds number (with high-fidelity measurement or simulation) have shown that the near-wall cycle is affected by the outer I. Marusic () · R. Mathis · N. Hutchins Department of Mechanical Engineering, University of Melbourne, Melbourne, Victoria 3010, Australia e-mail: [email protected] R. Mathis e-mail: [email protected] N. Hutchins e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_11, © Springer Science+Business Media B.V. 2011

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Fig. 1 Pre-multiplied energy spectrogram of streamwise velocity fluctuation kx φuu /Uτ2 at Reτ = 7300 across the turbulent boundary layer

flow region [1–3, 6, 8, 14, 16] and hence perhaps should not be considered as purely “autonomous”. Hutchins & Marusic [4] described the large-scale motions responsible for this as “superstructures”, with their origin nominally in the logarithmic region of the boundary layer. These observations came from experiments conducted at a range of Reynolds numbers, and Hutchins & Marusic [4] found that the strength (and influence) of the superstructures increased with increasing Reynolds number. Furthermore, they observed that low-wavenumber energy associated with these very large scale motions is not merely superimposed on the near-wall streamwise fluctuations, but seem to “amplitude modulate” the small-scale fluctuations [5]. This paper is concerned with this amplitude modulation interaction between the large and small (near-wall) scales in turbulent boundary layers. In the remainder of the paper the amplitude modulation effect will be quantified using a correlation coefficient and the effects of increasing Reynolds number will be considered. (A fuller discussion of these effects is given in a paper by Mathis et al. [15].)

2 Quantifying Amplitude Modulation Hutchins & Marusic [4] showed that, at sufficiently high Reynolds numbers, two distinctive peaks appear in the pre-multiplied spectrogram of the fluctuating streamwise velocity, an example of which is shown in Fig. 1. Here the coordinate system, x, y and z, refers to the streamwise, spanwise and wall-normal directions. The spectral density function of the streamwise velocity fluctuation is described by φuu

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Fig. 2 Example of small-scale decomposition on fluctuating u+ velocity signal at z+ = 15; (a) the + raw signal u+ ; (b) the large-scale signal u+ L ; (c) the small-scale signal uS ; (d) its envelope; (e) filtered envelope (solid line) against the large-scale component (dot-dashed). For comparison, the mean of the filtered envelope has been adjusted to zero

and the streamwise wavenumber and wavelength are denoted by kx and λx respectively (where λx = 2π/kx ). The superscript “+” is used to denote viscous scaling (z+ = zUτ /ν, u+ = u/Uτ , etc.). The outer peak is related to superstructures. Hutchins & Marusic [4] showed that a very high level of correlation was found between the filtered (long wavelength) signatures of u simultaneously measured at the locations of the inner and outer peaks. This was understood to mean that the large-structures superimpose their “footprint” near the wall. To quantify the interaction we begin by decomposing the signal from a wall normal location corresponding to the inner peak site (z+ = 15). Figure 2 shows a sample of the u signal at z+ = 15 for Reτ = 7300, as well as its large (uL ) and small (uS ) scale components. Here ‘large’ corresponds to a long-wavelength filtered signal (λx > δ retained) and ‘small’ refers to the remainder (λx < δ). In order to determine the relationship between the large- and smallscale structures contained in any velocity signal, the small-scale component of the signal (u+ S ) is analysed using the Hilbert transformation. This allows us to extract an envelope (E(u+ S )), representative of any modulating effect (assumed here to be the large-scale component u+ L ). The envelope returned by the Hilbert transformation will track not only the large-scale modulation due to the log region events, but also the small-scale variation in the ‘carrier’ signal. To remove this effect, we filter the envelope at the same cut-off as the large-scale signal (λx /δ > 1). Hence, a filtered envelope (EL (u+ S )) describing the modulation of small-scale structures is obtained.

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It is now possible to compute a meaningful correlation coefficient, R, of the filtered envelope with the large-scale velocity fluctuation u+ L R=

+ u+ L EL (uS ) 2 2 u+ EL (u+ L S)

(1)

where u2 denotes the r.m.s. value of the signal u. For the Reynolds number and signal shown in Fig. 2, R is found to be 0.33, which is a significant correlation.

3 Experiments The experiments for this study were conducted in two facilities. The first being the High Reynolds Number Boundary Layer Wind-Tunnel (HRNBLWT) at the University of Melbourne with a working section 27 × 2 × 1 m. Full details of the facility are available in Ref. [17]. Measurements consist of velocity measurements using a single-normal hot-wire probe across the entire boundary layer, and close enough to the wall to measure at the location of the inner peak (z+ = 15). The probe is made using a Wollaston platinum wire sensing element, operated in constant temperature mode using an AA Lab Systems AN-1003 with overheat ratio set to 1.8. For each Reynolds number, the diameter d and length l of the sensing element was adjusted to give a constant viscous scaled length of l + = lUτ /ν = 22 with l/d = 200, to allow comparison without any spatial resolution influences [6]. Measurements were made at five separate Reynolds numbers, namely: Reτ = 2800, 3900, 7300, 13600, and 19000. The second facility is from very high Reynolds number measurements in the atmospheric surface layer at the SLTEST facility, located at the Great Salt Lake Desert in Western Utah. Full details of the facility are available in Refs. [9, 12, 16]. The unique geography of this site allows us to obtain measurements in extremely high Reynolds number turbulent boundary layers (Reτ ∼ O(106 )). The boundary layer develops naturally over 100 km of salt playa, which is remarkably flat and has a low surface roughness. Measurements were performed using an array of 9 logarithmically spaced wall-normal sonic anemometers (Campbell Scientific CSAT3) from z = 1.4 to 25.7 m. We will consider here one hour of data taken from a period of prolonged neutral buoyancy and steady wind conditions. Mean statistics were found to compare well with canonical turbulent boundary layers from laboratory facilities [4, 13]. The estimated Reynolds number was Reτ = 650000.

4 Variations with Reynolds Number Figure 3 presents the pre-multiplied energy spectra maps, kx Φuu /Uτ2 for all sets of measurements, in the same manner as presented in Fig. 1, but here simply shown with contour levels. The inner-peak is marked by the “+” symbols and is seen to

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Fig. 3 Reynolds number effect — Iso-contours of the pre-multiplied energy spectra of streamwise velocity fluctuation kx φuu /Uτ2 ; (a) Reτ = 2800; (b) Reτ = 3900; (c) Reτ = 7300; (d) Reτ = 13600; (e) Reτ = 19000; (f) Reτ = 650000; Contour levels are from 0.2 to 1.8 in steps of 0.2; The large “+” marks the inner-peak location (z+ = 15, λ+ x = 1000); The vertical dot-dashed line indicates the mid-point of the log-layer

scale well in viscous units (z+ = 15; λ+ x = 1000) for all cases. The location of the outer peak was found by Hutchins & Marusic [4] to be at z/δ = 0.06 and λx /δ = 6, but this was based on a study over a limited Reynolds number range. Here, the data in Fig. 3, which cover a larger Reynolds number range, shows that the location of the outer peak appears to correspond well with the geometric centre of the logarithmic region (on a log plot), which is indicated by the vertical dashed lines in the figure. It should be noted that the data from SLTEST are considerably less reliable that the laboratory data, and here are used only as a guide. For the SLTEST data, fluctuating signals (and energy spectra) are only available at 9 locations within the log region.

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Fig. 4 Wall-normal variation of the degree R of modulation for several Reynolds numbers; Reτ = 2800, 3900, 7350, 13600 & 19000 from laboratory facility (HRNBLWT); Reτ = 6.4 × 105 from atmospheric surface layer (SLTEST)

Figure 4 shows the wall-normal evolution of the degree of amplitude modulation R for all six Reynolds number considered. The global shape of each curve is seen to be the same for all Reτ . A feature of interest is the wall-normal location at which the degree of amplitude modulation crosses zero. This point delineates the wall-normal position where R(z+ ) changes sign, or to be more precise the location at which the amplitude of the small-scale fluctuations is completely uncorrelated with the large-scale envelope (i.e. a position with no amplitude modulation), and possibly may be interpreted as the centre of the “source” location of the main modulating motion. The wall-normal position where R = 0 (marked with the vertical dashed line) is seen to correspond well with the location of the outer-peak which, as seen in Fig. 3, agrees well with the nominal mid-point of the log region. Figure 5 shows the location of the zero amplitude modulation for all Reynolds numbers considered here. (It is important to note that the Utah results have large error-bars given the large experimental uncertainties associated with those measurements.) There are considerable differences in the literature as to what constitutes the bounds of the log law. Here, our attention is on a nominal location (rather than a precise one). However, for comparison two lines are shown in Fig. 5 as estimates of the centre of the logarithmic region (on a log plot). The solid line corresponds to a definition of 1/2 + the log region as 100 < z+ < 0.15Reτ giving zM = 3.9Reτ , while the dotted line 1/2 + = 0.39Re3/4 , from suggestions by is based on KReτ < z+ < 0.15Reτ giving zM Klewicki et al. [10] and others that the lower bound of the log region is Reynolds number dependent. Overall, the results in Figs. 3 & 5 support the notion that the outer-peak in the streamwise energy spectra is coincident with the wall-normal location at which the degree of amplitude modulation crosses zero, and that this nominally agrees with the

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Fig. 5 Wall-normal location where the degree of amplitude modulation reaches zero (R = 0) versus Reynolds number. Lines represent estimates for the location of the geometric middle of the log-layer, corresponding to (solid) 100 < z+ < 0.15Reτ , and (dashed) based on Reynolds number 1/2 dependant boundaries KReτ < z+ < 0.15Reτ (here K = 1.2)

centre of the log region (all of which are Reynolds number dependent). It is difficult to interpret this result within a mechanistic description of the boundary layer. Close to the wall, the positive values of correlation are as expected. A large-scale deviation in the local streamwise velocity, will produce a corresponding change in the local velocity gradient at the wall, altering the turbulence production (or input of vorticity) in the near-wall region. Thus, in this region the magnitude (or envelope) of the smallscale fluctuations would be expected to follow the sign of the large-scale fluctuations (hence positive values of R). Based on the hairpin packet paradigm, one would expect opposite behaviour to occur at some point within the log region. A regime of hairpin packets would imply that most of the small-scale vortical activity would be located within or about the large-scale regions of negative velocity fluctuation (hence the negative values of R in the log region). However, further study is required before a firm conclusion can be made as to why the reversal in sign between these two regions should correspond so well to the location of the ‘outer site’ in the energy spectra. Acknowledgements We gratefully acknowledge the financial support of the Australian Research Council through grants DP0663499, FF0668703, and DP0984577.

References 1. Abe, H., Kawamura, H., Choi, H.: Very large-scale structures and their effects on the wall shear-stress fluctuations in a turbulent channel flow up to Reτ = 640. Trans. ASME: J. Fluid Eng. 126, 835–843 (2004)

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2. DeGraaff, D.B., Eaton, J.K.: Reynolds number scaling of the flat-plate turbulent boundary layer. J. Fluid Mech. 422, 319–346 (2000) 3. Hoyas, S., Jiménez, J.: Scaling of the velocity fluctuations in turbulent channels up to Reτ = 2003. Phys. Fluids 18, 011702 (2006) 4. Hutchins, N., Marusic, I.: Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28 (2007) 5. Hutchins, N., Marusic, I.: Large-scale influences in near-wall turbulence. Philos. Trans. R. Soc. Lond. A 365, 647–664 (2007) 6. Hutchins, N., Nickels, T., Marusic, I., Chong, M.S.: Spatial resolution issues in hot-wire anemometry. J. Fluid Mech. 635, 103–136 (2009) 7. Jiménez, J., Pinelli, A.: The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359 (1999) 8. Klewicki, J.C., Falco, R.E.: On accurately measuring statistics associated with small-scale structure in turbulent boundary layers using hot-wire probes. J. Fluid Mech. 219, 119–142 (1990) 9. Klewicki, J.C., Metzger, M.M., Kelner, E., Thurlow, E.M.: Viscous sublayer flow visualizations at Reθ = 1500000. Phys. Fluids 7, 857–963 (1995) 10. Klewicki, J., Fife, P., Wei, T., McMurty, P.: A physical model of the turbulent boundary layer consonant with mean momentum balance structure. Philos. Trans. R. Soc. Lond. A 365, 823– 840 (2007) 11. Kline, S.J., Reynolds, W.C., Schraub, F.A., Rundstadler, P.W.: The structure of turbulent boundary layers. J. Fluid Mech. 30, 741–773 (1967) 12. Kunkel, G.J., Marusic, I.: Study of the near-wall-turbulent region of the high-Reynoldsnumber boundary layer using atmospheric flow. J. Fluid Mech. 548, 375–402 (2006) 13. Marusic, I., Hutchins, N.: Study of the log-layer structure in wall turbulence over a very large range of Reynolds number. Flow Turbul. Combust. 81, 115–130 (2008) 14. Marusic, I., Kunkel, G. J.: Streamwise turbulence intensity formulation for flat-plate boundary layers. Phys. Fluids 15, 2461–2464 (2003) 15. Mathis, R., Hutchins, N., Marusic, I.: Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J. Fluid Mech. 628, 311–337 (2009) 16. Metzger, M.M., Klewicki, J.C.: A comparative study of near-wall turbulence in high and low Reynolds number boundary layers. Phys. Fluids 13, 692–701 (2001) 17. Nickels, T.B., Marusic, I., Hafez, S., Chong, M. S.: Evidence of the k1−1 law in high-Reynolds number turbulent boundary layer. Phys. Rev. Lett. 95, 074501 (2005)

Tomographic Particle Image Velocimetry Measurements of a High Reynolds Number Turbulent Boundary Layer Callum Atkinson, Sebastien Coudert, Jean-Marc Foucaut, Michel Stanislas, and Julio Soria

Abstract Streamwise/wall-parallel volumes of a turbulent boundary layer at Reθ = 7800 and 11800 are measured using a 4 camera (2048 × 2048 px) tomographic particle image velocimetry (Tomo-PIV) system in the turbulent boundary layer wind tunnel at the Laboratoire de Mécanique de Lille (LML). A measurement volume of 1200 × 180 × 1200 voxels was achieved, the large boundary layer provided by this tunnel (δ ≈ 0.3 m) resulting in measurements volumes of 470+ × 70+ × 470+ and 920+ × 140+ × 920+ , respectively. Volumes were reconstructed using both the standard MART algorithm and the accelerated MLOS-SMART approach (Atkinson and Soria in Exp. Fluids 47(4–5):553–568, 2009). The quality of the data acquired by these two techniques is assessed based on the mean velocity profiles, velocity fluctuations and velocity power spectra.

1 Introduction Aerodynamic applications from automotive to aircraft industries require the robust and accurate prediction of drag and the separation of turbulent flows. While the governing equations for these flows can be solved using direct numerical simulation (DNS) the need to resolve the Kolmogorov scale and simultaneously examine a large region of the flow, limits these computations to low Reynolds numbers (Re). Presently the highest Reynolds numbers simulated in a boundary layer have been on the order of Reθ = 103 , well below the Reθ ≈ 105 –106 that are commonly encountered over aircraft. For information on flow behaviour at higher Reynolds C. Atkinson · S. Coudert · J.-M. Foucaut · M. Stanislas Laboratoire de Mécanique de Lille, Ecole Centrale de Lille, Bd Paul Langevin, Cite Scientifique, 59655 Villeneuve d’Ascq cedex, France C. Atkinson () · J. Soria Laboratory for Turbulence Research in Aerospace and Combustion, Department Mechanical and Aerospace Engineering, Monash University, Melbourne, Victoria 3800, Australia e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_12, © Springer Science+Business Media B.V. 2011

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Table 1 Parameters of recent Tomo-PIV turbulent boundary layer studies References

Reθ

Elsinga et al. (2007) [8]

34000a

Elsinga (2008) [6]

δ+

Volume size IW length xi yrange min, max (δ) Sx , Sy , Sz (δ) (px [δ]) (δ) 0.15, 0.47

1900 816 0.04, 0.5

3.5, 0.32, 1.8 40 [0.1]

0.025 0.05

1.4, 0.46, 1.1 50 [0.07]

0.018 0.05

0.3, 0.8

1.4, 0.5, 1.1

42 [0.06]

0.016 0.05

0, 1.2

1.8, 1.3, 0.42 42 [0.06]

0.020 0.05

1.7, 0.39, 1.8 32 [0.07]

0.018 0.05

Schröder et al. (2008) [13] 2460 832 0, 0.39 a Flow

Seeding (ppp)

was supersonic Mach 2.1

numbers it is necessary to perform experimental measurements, ideally providing time-resolved volumetric velocity fields with a high spatial dynamic range (SDR). Until recently such all-inclusive measurements were not possible. Hot-wire measurements can provide excellent temporal resolution, but are limited to point measurements, while particle image velocimetry (PIV) can be used to provide snapshot spatial measurements in a plane through the velocity field. Fortunately PIV has evolved significantly over the past 20 years to include the out-of-plane velocity component via stereoscopic PIV (Stereo-PIV) [18], temporally resolved velocity fields via high-speed cameras and lasers [3] and volumetric velocity measurements via digital holographic PIV (DHPIV) [11], tomographic PIV (Tomo-PIV) [7] or a combination of the two [15]. These advances have brought us much closer to capturing the three-dimensional, chaotic, multi-scale nature of turbulence. Tomo-PIV utilises the same equipment as Stereo-PIV meaning that images and their associated reconstructions can be acquired over similar large measurement domains, magnifications and optical resolutions. This provides an advantage over lens-less DHPIV, where small CCD arrays currently limit measurements to very small flow regions ≈1 mm3 . Tomo-PIV involves reconstructing the intensity distribution in a seeded flow volume based on a series of images taken simultaneously from different observation angles [7]. Small interrogation volumes (IW) taken from these intensity volumes are then three-dimensionally cross-correlated to provide three-component three-dimensional velocity field data. Tomo-PIV measurements have been performed on low and high speed turbulent boundary layers by Elsinga et al. [6, 8] and time-resolved Tomo-PIV by Schröder et al. [13], as summarised in Table 1. These experiments have provided interesting visualisations of the three-dimensional low-speed streaks and large scale vortex structures, however this is after smoothing the velocity fields. Little information about the true noise limited spatial resolution of the technique is given in any of these studies. In this paper we present a preliminary investigation on the application of TomoPIV to a high Reynolds number turbulent boundary layer. The effect of the reconstruction method and the noise limited spatial resolution are examined in terms of the measured velocity power spectrum. The mean profiles and fluctuating components are also examined.

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Table 2 Properties of the turbulent boundary layer under investigation U∞ (m/s)

Reθ

δ (m)

uτ (m/s)

ν/uτ (µm)

δ+

3

7800

0.33

0.115

125

2650

6a

11800

0.29

0.223

654

4510

a Parameters

are interpolated from data at 5 and 7 m/s

Fig. 1 Left: Schematic of Tomo-PIV setup. Right: Top view showing camera arrangement. Dashed region indicates the measurement volume

2 Experimental Procedure Turbulent boundary layer measurements were performed in the 1 × 2 m2 test-section of the Laboratoire de Mécanique de Lille boundary layer wind tunnel, approximately 18 m downstream of the initial boundary layer development zone. The long development generates a boundary layer up to a momentum based Reynolds number of Reθ = 20600, while maintaining a thickness of approximately δ ≈ 0.3 m and a freestream turbulence intensity of 0.3%. An extensive description of the facility can be found in Carlier and Stanislas [5]. The boundary layer properties for the two Reynolds numbers considered are given in Table 2. Tomo-PIV was setup to measure a volume with major dimensions parallel to the wall (see Fig. 1). A 300 mJ Nd:YAG laser was used to illuminate the measurement volume at wavelength of 532 nm. A cylindrical diverging lens was used to shape the beam into a 10 mm thick light sheet, which was then passed through a slit in order to trim lower intensity regions of the laser sheet profile and provide a more even illumination through the volume. This resulted in a 8 mm light sheet with no noticeable variation across the measurement volume. In order to reduce scattering bias between the cameras a mirror was used to reflect the light-sheet back through the measurement volume in the opposite direction. This mirror was adjusted such that the reflected light sheet was effectively parallel to the incident sheet throughout the measurement volume. Four 12-bit 2048 × 2048 pixel Hamamatsu CCD array cameras were situated beneath the tunnel, with each rotated about the y-axis such that they were situated at

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Table 3 Parameters of the Tomo-PIV setup Magnification

M = 0.15

Resolution

20 px/mm

Reconstructed field of view

[Sx ; Sy ; Sz ] = [60; 9; 60 mm] = [1200; 180; 1200 px]

Lens aperture

f# = 8

Particle image diameter

dp = 1.6 px

Depth of field

z = 9.2 mm

Separation time Reθ = 7800, 11800

t = 400, 200 µs

45 deg angles in the x–z plane. The Scheimpflug condition was set in each camera’s horizontal axis to maintain focus throughout the measurement region. Rotating the cameras in this way required only one Scheimpflug angle for each camera and provided each with the same angle relative to forward and backward scattering. The cameras were fitted with 105 mm Micro Nikkor lenses resulting in an average magnification of M = 0.15 and a common field of view of [Sx , Sy , Sz ] = [60, 8, 60] mm. The aperture size was set at f# = 8 in order to provide an estimated depth of field (z ≈ 9.2 mm) greater than the light sheet thickness. The flow was seeded using particles produced by a poly-ethylene-glycol smoke generator with a nominal particle size of 1 µm. At the selected lens aperture this corresponds to a diffraction limited particle image diameter of dp = 1.6 px. Two cavities of the laser were pulsed at a frequency of 1 Hz, using a separation time t = 400 and 200 µs for Reθ = 7800 and 11800, respectively. This corresponded to a maximum displacements on the order of 15 px for each case. A summary of these parameters can be found in Table 3. The Tomo-PIV measurement volume was calibrated using the Soloff method [14]. A calibration plate was initially placed on the wall and then translated in the wall normal direction from y = 12 mm to y = 0 mm with steps of 500 µm, using a microcontrol with an accuracy of 5 µm. Images of the calibration target were simultaneously recorded on each camera. A volume self calibration analogous to that of Wieneke [17] showed an initial calibration error on the order of 1 px. After self calibration correction this was reduced to 0.1 px.

3 Volume Reconstruction and PIV Processing Tomographic reconstruction of the measurement volumes and 3D cross-correlation were performed using in-house software developed by the author at the Laboratory for Turbulence Research in Aerospace and Combustion (LTRAC) at Monash University [1]. Reconstructions were made using both the standard 5 iterations of the MART algorithm and 10 iterations of the accelerated MLOS-SMART algorithm [1, 2]. For a seeding density of 0.03 particles per pixel (ppp) MLOS-SMART reconstruction of a single volume required only 11 minutes on a single processor, while MART required just over 2 hours. Images were preprocessed using histogram normalisation, background subtraction and 3 × 3 Gaussian filtering. In each case 100

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Table 4 Parameters of the present Tomo-PIV turbulent boundary layer studies δ+

yrange min, max (δ)

Volume size Sx , Sy , Sz (δ)

IW length (px [δ])

xi (δ)

Seeding (ppp)

7800

2650

0, 0.027

0.18, 0.03, 0.18

64 [0.010]

0.002

0.04

11800

4510

0, 0.031

0.21, 0.03, 0.21

64 [0.011]

0.003

0.06

Reθ

Fig. 2 Mean and fluctuating velocity profiles produced by Tomo-PIV at Reθ = 7800. Results for volume reconstruction based on both 5 MART iterations and 10 MLOS-SMART iterations are shown

volumes pairs were reconstructed and cross-correlated using two passes, with a 3D integer shift. A constant interrogation volume or 3D interrogation window IW size was maintained, with a 75% overlap and normalised mean vector validation [16].

4 Results To assess the effectiveness of Tomo-PIV for turbulent boundary layer measurements, two Reynolds numbers were examined with seeding densities and IW sizes listed in Table 4. Results were obtained by averaging over 100 velocity fields in each case and along both the streamwise (x) and spanwise (z) directions, both assumed to be homogeneous. This resulted in averaging over 500000 valid vectors for IW = 643 px, with approximately 5% of vectors rejected. The longitudinal mean and fluctuating velocity profiles computed for Reθ = 7800 are presented in Fig. 2, the measurement region extends from the top of the viscous sub-layer (y + ≈ 8) to the top on the buffer layer (y + ≈ 50). Hot wire anemometry results (HWA) taken at U∞ = 3 m/s [4] and the following universal laws are plotted for comparison: • Linear law: y+ < 5, U + = y + ; y+ • Van Driest profile: 5 < y + < 50, U + = 0 • Log law: 50 < y + < 200, U + =

1 K

√

1+

ln(y + ) + C.

2dy

1+4(Ky (1−exp(−y /A+ ))2

;

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Fig. 3 Longitudinal velocity power spectra and probability density function at Reθ = 7800, y + = 53 produced by Tomo-PIV. Results for volume reconstruction based on both 5 MART iterations and 10 MLOS-SMART iterations are shown. Stereo-PIV IW = 322 px or 16.3+ with 75% window overlap

Constants K, C and A+ are set at 0.41, 5.3 and 26, respectively. Near the wall the strong velocity gradient results in separation of the profiles as commonly observed in planar and Stereo-PIV measurements. Above this a good collapse with the HWA and universal laws is observed. A slight underestimation of the turbulent fluctuations is present, which can also occur for standard PIV this close to the wall [5]. Longitudinal velocity spectra are shown in Fig. 3, using inner scaling [12]. Spectra obtained by HWA and Stereo-PIV [10] are shown for comparison. In all cases the influence of measurement noise causes the spectra to peel away from the true turbulent flow spectrum. This typically occurs at lower wavenumbers for PIV compared to HWA. The nature of this spectral noise and the effective resolution of PIV was examined by Foucaut et al. [9], defining an effective cut-off wavenumber kmax for PIV as the wavenumber at which the signal to noise ratio SNR = 1. In this case Tomo-PIV is generating significantly more noise than it’s Stereo-PIV counterpart with kmax y = 4.6. This corresponds to a minimum spatial variation of 2πy +

PIV SV + min = kmax y = 68 wall units or 10 vectors, in this case. The maximum spatial + variation is limited by the dimensions of the interrogation volume SV + max = Sx ,

Sy+ , Sz+ such that the effective spatial dynamic range SDR =

SV + max SV + min

of the present

measurement is SDRx,z = 7.0 and SDRy = 1.2. The probability density function (PDF) of the u component of velocity shows the presence of the peak-locking effect, spacing between these peaks corresponding to 1 pixel. All plots show an excellent collapse between the data obtained by MART and MLOS-SMART reconstruction techniques. The mean and fluctuating velocity profiles for Reθ = 11800 are presented in Fig. 4, ranging from y + ≈ 16 to y + ≈ 100. Behaviour is similar to that at the lower Reynolds number, with a better estimate of the turbulent fluctuations further from the wall. The spectra and PDF also exhibit similar behaviour (see Fig. 5). In this case kmax y = 2.2, with a minimum noise limited spatial resolution of SV + min = 143 wall units or 12 vectors. This corresponds to SDRx,z = 6.6 and SDRy = 1.0.

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Fig. 4 Mean and fluctuating velocity profiles produced by Tomo-PIV at Reθ = 11800. Results for 10 MLOS-SMART iterations are shown

Fig. 5 Longitudinal velocity power spectra and probability density function at Reθ = 11800, y + = 53 produced by Tomo-PIV. Results for 10 MLOS-SMART iterations are shown

5 Conclusion Tomo-PIV was performed on a turbulent boundary layer at Reθ = 7800 and 11800. The MLOS-SMART reconstruction technique was found to perform tomographic reconstructions almost 11 times faster than the standard MART, with negligible difference in the calculated turbulence statistics. The velocity power spectra produced by Tomo-PIV showed the presence of significant noise when compared to HWA and Stereo-PIV. At best a SDR = 7.0 was achieved with a minimum resolvable vortex diameter of SV + min = 68 at a SNR = 1. It is hoped that with further optimisation this can be improved. Acknowledgements C. Atkinson was supported by an Eiffel Fellowship and an Australian Postgraduate Scholarship while undertaking this research. The support of the Australian Research Council is gratefully acknowledged.

References 1. Atkinson, C., Soria, J.: An efficient simultaneous reconstruction techniques for tomographic particle image velocimetry. Exp. Fluids 47(4–5), 553–568 (2009)

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2. Atkinson, C.H., Dillon-Gibbons, C.J., Herpin, S., Soria, J.: Reconstruction techniques for tomographic PIV (Tomo-PIV) of a turbulent boundary layer. In: 14th Int. Symp. on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal (2008) 3. Baur, T., Koengeter, J.: High-speed PIV and the post-processing of time-series results. In: Euromech 411 (2000) 4. Carlier, J.: Etude des structures coherentes d’une couche limite turbulente à grand nombre de Reynolds. Ph.D. thesis, Univ. de Lille, France (2001) 5. Carlier, J., Stanislas, M.: Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry. J. Fluid Mech. 535, 143–188 (2005) 6. Elsinga, G.E.: Tomographic particle image velocimetry and its application to turbulent boundary layers. Ph.D. thesis, Technische Universiteit Delft (2008) 7. Elsinga, G.E., Scarano, F., Wieneke, B., van Oudheusden, B.W.: Tomographic particle image velocimetry. Exp. Fluids 41, 933–947 (2006) 8. Elsinga, G.E., Adrian, R.J., van Oudheusden, B.W., Scarano, F.: Tomographic-PIV investigation of a high Reynolds number turbulent boundary layer. In: Proc. 7th Int. Symp. on Particle Image Velocimetry, Roma, Italy (2007) 9. Foucaut, J.M., Carlier, J., Stanislas, M.: PIV optimization for the study of turbulent flow using spectral analysis. Meas. Sci. Technol. 15, 1046–1058 (2004) 10. Herpin, S.: Study of the influence of the Reynolds number on the organization of wall-bounded turbulence. Ph.D. thesis, Ecole Centrale de Lille and Monash University (2009) 11. Meng, H., Pan, G., Pu, Y., Woodward, S.H.: Holographic particle image velocimetry: from film to digital recording. Meas. Sci. Technol. 15, 673–685 (2004) 12. Perry, A.E., Henbest, S., Chong, M.S.: A theoretical and experimental study of wall turbulence. J. Fluid Mech. 165, 163–199 (1986) 13. Schröder, A., Geisler, R., Staack, K., Wieneke, B., Elsinga, G.E., Scarano, F., Henning, A.: Lagrangian and Eulerian views into a turbulent boundary layer flow using time-resolved tomographic PIV. In: 14th Int. Symp. on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal (2008) 14. Soloff, S.M., Adrian, R.J., Liu, Z.C.: Distortion compensation for generalized stereoscopic particle image velocimetry. Meas. Sci. Technol. 8, 144–1454 (1997) 15. Soria, J., Atkinson, C.: Towards 3c-3d digital holographic fluid velocity vector field measurement — tomographic digital holographic PIV (Tomo-HPIV). Meas. Sci. Technol. 19, 1–12 (2008) 16. Westerweel, J., Scarano, F.: Universal outlier detection for PIV data. Exp. Fluids 39, 1096– 1100 (2005) 17. Wieneke, B.: Volume self-calibration for 3d particle image velocimetry. Exp. Fluids 45(4), 549–556 (2008) 18. Willert, C.: Stereoscopic digital particle image velocimetry for application in wind tunnel flows. Meas. Sci. Technol. 8, 1465–1479 (1997)

Study of Vortical Structures in Turbulent Near-Wall Flows Sophie Herpin, Sebastien Coudert, Jean-Marc Foucaut, Julio Soria, and Michel Stanislas

Abstract Streamwise and spanwise vortices are investigated in a database of nearwall turbulence constituted of SPIV data of boundary layer covering a large range of Reynolds numbers (Reθ ∈ [1300; 18950]).The detection algorithm is based on a fit of the velocity field surrounding extrema of swirling strength to an Oseen vortex. Some statistical results on the characteristics of the vortices (radius, vorticity, convection velocity, density) are investigated, giving some new insight into the organization of near-wall turbulence.

1 Introduction Near-wall turbulence is organized into coherent structures which are believed to play a key role in the maintenance of turbulence. In particular, the streamwise and spanwise oriented vortices, through their ability to transport mass and momentum across the mean velocity gradient, are a fundamental feature of near-wall turbulence. The description, scaling laws and generation mechanism of these structures have been the focus of many studies, but still remain unclear. The present study aims at providing new insight into these outstanding issues through the analysis of a SPIV database of turbulent boundary layer covering a large range of Reynolds numbers (Reθ ∈ [1300; 18950]).

S. Herpin () · S. Coudert · J.-M. Foucaut · M. Stanislas Laboratoire de Mecanique Lille, Ecole Centrale de Lille, Bd Paul Langevin, Cite Scientifique, 59655 Villeneuve d’Ascq cedex, France e-mail: [email protected] S. Herpin · J. Soria Laboratory for Turbulence Research in Aerospace and Combustion (LTRAC), Monash University, VIC 3800, Melbourne, Australia M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_13, © Springer Science+Business Media B.V. 2011

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Table 1 Characteristics of the LML and LTRAC database Facility Plane (1–2)

Reθ

δ+ Reτ

Domain S1 , S2

820 4δ, 1.4δ

LIW

Mesh 1st point step i

n° records

15.2+

4.7+

y + = 15

605

LTRAC XY

1300

LTRAC XY

2200 1390 2.6δ, 0.75δ

14.2+

4.3+

y + = 15

LML

XY/YZ

7630 2590 0.15δ, 0.6δ

16.8+

3.9+

y+

LML

XY/YZ 10140 3620 0.08δ, 0.28δ 11.2+

3.2+

y + = 13 / y + = 17 4096/4096

LML

XY/YZ 13420 5020 0.08δ, 0.28δ 15.4+

4.5+

y + = 15 / y + = 27 4096/4352

LML

XY/YZ 18950 6860 0.08δ, 0.28δ 20.6+

6.0+

y + = 20 / y + = 33 3840/4608

= 24 /

1815 y+

= 34 3840/2048

2 Description of the Database The boundary layer data was acquired by means of stereoscopic particle image velocimetry in two complementary flow facilities: the LTRAC water-tunnel in Melbourne (Australia) and the LML wind-tunnel in Lille (France). The characteristics of the database (size of the field of views, interrogation window, and mesh spacing) are summarized in Table 1. A brief summary is given below. The measurements in the LTRAC water-tunnel were realized in a streamwise/wall-normal (XY) plane of a turbulent boundary layer at moderate Reynolds numbers (Reθ = 1300 and 2200). The free-stream turbulence intensity in the watertunnel is relatively high, on the order of 2.6%U∞ and 5.4%U∞ for the measurements at Reθ = 1300 and Reθ = 2200 respectively. Extensive details on the experimental procedure and qualification of the data can be found in [6, 7]. The data feature low measurement uncertainty (0.75% of the free stream velocity on the inplane velocity components, and 1.5% on the out-of plane component), large spatial dynamic range (thanks to the large CCD array of the PCO 4000 camera), and high spatial resolution (with an interrogation window size of 15+ and a mesh spacing of 4.5+ on average). The measurements in the LML wind-tunnel were realized both in a streamwise/wall-normal (XY) plane and in a spanwise/wall-normal (YZ) plane of a turbulent boundary layer at high Reynolds numbers (Reθ = 7800, 10140, 13420 and 18950). The description of the setup as well as the qualification of the data can be found in [6]. At all Reynolds numbers and in both planes, the measurement feature low uncertainty (0.7%U∞ for all velocity components) and high spatial resolution (with an IW size varying from 11.2+ to 20.6+ and a mesh spacing from 3.2+ to 6+ depending on the Reynolds number).

3 Average Properties of the Database The wall normal evolution of the mean streamwise velocity was computed by averaging the data over the number of sample acquired and over the homogeneous directions (x for the XY plane, z for the YZ plane). Only the profiles obtained in

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Fig. 1 Wall-normal evolution of mean longitudinal velocity

the XY plane are shown; the profiles in the YZ plane show the same overall behavior (except for the position of the first valid point in the wall-normal direction, see Table 1). It is plotted in external units (non dimensionalized with the free velocity U∞ and the momentum thickness θ ) in Fig. 1(a) and in wall units with a logarithmic scale in the near-wall region (scaled with the friction velocity Uτ and the viscosity ν) in Fig. 1(b). In the outer region (y/δ ≥ 0.2), the boundary layer data display a good collapse in external scaling (Fig. 1(a)). In the inner region (y/δ ≤ 0.2), all the data display a very good collapse in wall-units, and is in excellent agreement with the law of the wall defined by the Van Driest law (for 0 < y + < 55) and the logarithmic law (for 55 < y + and y/δ < 0.2). Spectral analysis is a tool of special interest for the analysis of turbulence which is, in essence, a multi-scale phenomenon. Here we consider only the onedimensional power spectra of the streamwise velocity E11 (k) along the streamwise and spanwise direction, at y + = 100. It is computed as the product of the Fourier transform of the u velocity (after a periodization of the field for the SPIV data) times its complex conjugate, divided by the length of the field, and averaged over the number of samples and the homogeneous direction. The streamwise spectra E11 (kx ) are shown in Fig. 2(a), and the spanwise spectra E11 (kz ) in Fig. 2(b), in an inner scaling (scaled with the distance to the wall y and the friction velocity Uτ , see [9]). The SPIV spectra are compared with a spectrum computed on the DNS of channel flow from [3] at Reτ = 2000. As it can be seen, the SPIV spectra and the DNS spectra are in excellent agreement in the low (k¯ < 1) and intermediate (1 < k¯ < 10) wavenumbers range. In particular, all spectra tend to a ‘−5/3’ power law in the inertial subrange. In the high wavenumber domain, a spurious liftup of the PIV spectra with respect to the DNS spectra is visible: it indicates that the effect of measurement noise dominates the effect of spatial averaging over the interrogation window [5]. Taking the DNS spectrum as the reference spectrum of the flow, it is possible to compute the wavenumber k¯SNR=1 at a signal-to-noise-ratio of 1 (and the associated structure size, e.g. radius of a vortex that would be re-

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Fig. 2 Spectra of u velocity at y + = 100 1 2Πy + 2 k¯SNR=1 ) [7]. The cut-off wavenumbers are reported + = 25) for the LTRAC read: k¯SNR=1 = 12.7 (rSNR=1

+ solved with SNR = 1, rSNR=1 =

on the spectra. Their values + datasets, k¯SNR=1 = 15.2 (rSNR=1 = 21) for the LML datasets in the XY plane, and + k¯SNR=1 = 16.6 (rSNR=1 = 19) for the LML dataset in the YZ plane.

4 Detection Technique ∂ui Local detection techniques based on the 3D velocity gradient tensor Aij = ∂x (such j as the criterion [2]) present the advantage of being Galilean invariant. They have been successfully used to detect vortices in turbulent boundary layer. Because the database used in the present contribution is planar, the detection technique employed is based on the 2D velocity gradient tensor. This tensor is computed using a second order least-square derivative scheme that minimizes the propagation of measurement noise [4]. When complex eigenvalues of the tensor exist, their imaginary part, called the swirling strength, is used as a detection function of vortex cores. This function is first normalized by the wall-normal profile of its standard deviation (as suggested by DelAlamo et al. [3] and Wu and Christensen [12]), and then smoothed using a 3 × 3 sliding average to remove the remaining noise. Extrema exceeding a fixed threshold (λci (x1/3 , x2 ) > 1.5λci,RMS (x2 )) are retained as center of vortex cores. The velocity fields surrounding extrema of the detection function are then fitted to a model vortex with a non-linear least square algorithm (Levenberg–Marquardt). The model is an Oseen vortex, defined in Eq. 1. 2 r Γ 1 u(r, θ ) = uc + (1) eθ 1 − exp − 2Π r r0

This procedure has already been employed by Carlier and Stanislas [1] and Stanislas et al. [11]. It validates that the structure detected is indeed a vortex, and allows the

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Fig. 3 Wall normal evolution of vortex densities

retrieval of the vortex characteristics (radius, circulation, convection velocity, subgrid position of the center) through the fitted parameters of the model.

5 Results: Characteristics of the Vortices The detection technique described above is employed to detect streamwise vortices (in the YZ planes) and spanwise vortices (in the XY planes) in the SPIV boundary layer database. Some statistics are computed on the characteristics of the detected vortices. The wall-normal evolution of the mean radius, vorticity and density are obtained by taking into account eddies contained in layers of 25+ in height.

5.1 Density of the Vortices The vortex densities are normalized such that they represent the densities per wallunit square. At given orientation of the measurement plane, all datasets of the LML database show the same trend with wall-normal distance, and the only difference in the vortex densities is a shift in absolute value due to the effect of spatial resolution. Here, only the densities obtained in both planes in the boundary layer data at Reθ = 10140 (featuring the highest spatial resolution) and Reθ = 18950 (with the lowest spatial resolution) are shown in Figs. 3(a) and 3(b) respectively. The total density of spanwise vortices (detected in the XY plane), the density of prograde (spanwise vortices with rotation in the same senses as the mean shear w0 < 0) and retrograde (w0 > 0) vortices, and the total density of streamwise vortices (detected in the YZ plane) are represented. In the region y + < 150, the streamwise oriented vortices (detected in the YZ plane) are more numerous than the spanwise oriented vortices (detected in the XY plane). The fact that quasi-streamwise vortices are the major constituent of the nearwall region was also observed using a λ2 criterion by Jeong et al. [8] in a DNS of

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Fig. 4 Mean radius in wall-units

channel flow at Reτ = 180 and by Sheng et al. [10] in holographic 3D velocity field of a turbulent boundary layer at Reτ = 1400. The wall-normal location where the density of the streamwise vortices reaches a maximum is obtained at y + ≈ 60. After this maximum is reached, the density of streamwise vortices decreases rapidly with increasing wall-normal distance. This behavior of the streamwise vortices density in the near-wall region is in good agreement with the findings of Stanislas et al. [11] in SPIV measurements of a turbulent boundary layer at Reθ = 7800. The density of the spanwise vortices, in contrast, continuously increases from the wall up to y + ≈ 150 where a maximum is reached. Among the spanwise vortices, the predominance of the prograde vortices is overwhelming. Both prograde and retrograde forms increase with increasing wall-normal distance, but at very different rates: the increase is very fast for the prograde vortices, and rather slow for the retrograde ones. This suggests that different mechanisms are responsible for their formation. In the outer region (y + > 150), the vortex densities behave differently. First, it is of interest to note that the vortices population is now almost equally constituted of streamwise and spanwise vortices. The density of both vortices decreases at a medium rate with increasing wall-normal distance. Again, the prograde and the retrograde vortices follow different evolutions: the density of the prograde vortices continuously decreases with increasing wall-normal distance, while the density of the retrograde vortices stabilizes until y/δ = 0.5.

5.2 Radius of the Vortices The wall-normal evolution of the mean radius for all Reynolds numbers is plotted in Fig. 4(a) for the XY plane, and in Fig. 4(b) for the YZ plane, in wall units. It is of interest to note that, within each measurement plane, the wall-normal evolution of mean radius in wall units of the datasets at Reθ = 1300, 2200, 7630 and 13420 (which feature the same spatial resolution, see Table 1) are in excellent agreement. On a different note, the dataset at Reθ = 10140 (with the highest spatial resolution)

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Fig. 5 Mean radius in Kolmogorov units

displays smaller values of mean radius, and the dataset at Reθ = 18950 (with the lowest spatial resolution) higher values of mean radius. In the XY plane, the radius appears to be slowly increasing with wall normal distance over the whole field for all Reynolds numbers, except for the dataset at Reθ = 1300 where a steep increase of mean radius is observed for y + > 400 ⇔ y/δ > 0.5. This may be due to intermittency with the free stream; in any case the spanwise vortex density in this region is quite low, cf. previous paragraph. Close to the wall (y + ≈ 50), the mean radius is about 20+ on the dataset with the highest spatial resolution (Reθ = 10140) which is comparable to the findings of [1] at Reθ = 7500. In the YZ plane, the behavior of the mean radius is quite different. Two regions must be distinguished: the region y + < 100, where the radius increases strongly with wall-normal distance, and the region y + > 100, where the radius increases slowly with wall-normal distance. This composite behavior in the YZ plane was also observed in [11] at Reθ = 7800 and Reθ = 13420. Close to the wall (y + ≈ 50), the radius is as small as 15+ on the dataset with the highest spatial resolution (Reθ = 10140), which is in excellent agreement with the estimation of [10] for the radius of the streamwise vortices in the upper buffer layer. Therefore the streamwise vortices (detected in the YZ plane) are found to be smaller than the spanwise vortices (detected in the XY plane) in the near wall region. This may indicate that, being inclined to the wall, the streamwise vortices are stretched by the mean velocity gradient. In the region y + > 100, the mean radius of the streamwise and spanwise vortices are comparable and slowly increase with increasing wall-normal distance toward a value of 25+ . The wall-normal evolution of the mean radius scaled with the Kolmogorov length scale is plotted in Fig. 5(a) for the XY plane and in Fig. 5(b) for the YZ plane. The influence of the spatial resolution is less noticeable in this representation, and the mean radius at all Reynolds numbers is in quite good agreement. It is roughly independent of wall-normal distance for y + > 150 and equal to 8η, except for the channel flow data. In the YZ plane, the behavior is the same except that the mean radius is found to be y independent over the full region of investigation (and not only for y + > 150).

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Fig. 6 PDF of radius in Kolmogorov units, superimposed for different wall-normal distances in the log region

Fig. 7 Mean vorticity of the vortices, in wall-units

It is also of interest to examine the full distribution of the vortex radius. The radius PDFs in Kolmogorov scaling computed at different y in the logarithmic region are shown in Fig. 6(a) for the spanwise vortices (XY plane) and in Fig. 6(b) for the streamwise vortices (YZ plane). It can be seen that the distribution of radius is also universal in both Re and y in Kolmogorov scaling.

5.3 Vorticity of the Vortices The wall normal evolution of the absolute mean value of vorticity at the center of the vortices is represented in Fig. 7(a) for the XY plane and in Fig. 7(b) for the YZ plane, in wall units. As it can be seen, in both planes, the vorticity decreases exponentially with the wall normal distance. In the near wall region (y + < 250) a good collapse of the vorticity in all datasets and in both planes is observed. The peak of vorticity at the wall is slightly higher in the YZ plane than in the XY plane. This may indicate that the streamwise vortices (detected in the YZ plane) are more

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Fig. 8 Mean vorticity of the vortices, in Kolmogorov units

Fig. 9 PDF of vorticity in Kolmogorov units

intensified by the mean velocity gradient than the spanwise vortices (detected in the XY plane). A good universality is observed for the streamwise vortices detected in the YZ plane, but some differences are visible for the spanwise vortices detected in the outer region of the XY plane, maybe owing to Reynolds number effects. The wall-normal evolution of the mean vorticity scaled with the inverse of the Kolmogorov time scale is plotted in Fig. 8(a) for the XY plane and in Fig. 8(b) for the YZ plane. The behavior is similar in the two planes. The quantity w0 τ appears to be quite constant in the upper buffer layer and in the logarithmic region of the SPIV datasets (w0 τ ≈ 1.4). It is also of interest to examine the full distribution of the vortex vorticity. The vorticity PDFs in Kolmogorov scaling computed at different y in the logarithmic region are shown in Fig. 9(a) for the spanwise vortices (XY plane) and in Fig. 9(b) for the streamwise vortices (YZ plane). It can be seen that the distribution of vorticity is also universal in both Re and y in Kolmogorov scaling.

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6 Conclusion A coherent structure detection was undertaken on the SPIV database of boundary layer flow at Reθ ∈ [1300; 18950]. At all Reynolds numbers, the near-wall region is the most densely populated region, predominantly with streamwise vortices that are on average smaller and more intense than spanwise vortices. In contrast, the logarithmic region is equally constituted of streamwise and spanwise vortices having equivalent characteristics. In the outer region, some differences between the datasets are observed depending on the scaling employed: the wall-units scaling or the Kolmogorov scaling. In wall-unit scaling, a good universality in Reynolds numbers is observed in the near-wall and logarithmic region: the vorticity is found to be maximum at the wall, decreasing first rapidly and then slowly with increasing wall-normal distance; the radius is increasing slowly with wall-normal distance in both regions, except for the streamwise vortices for which a sharp increase in radius is observed in the near-wall region. The wall-units scaling is found to be deficient in the outer region, where Reynolds number effects are observed. In contrast, the Kolmogorov scaling appears to be universal both in Reynolds number and wall-normal distance across the three regions investigated, with a mean radius on the order of 8η and a mean vorticity on the order of 1.5τ −1 . A good universality of the PDF of the vortex radius and vorticity is also observed in Kolmogorov scaling in the logarithmic region.

References 1. Carlier, J., Stanislas, M.: Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry. J. Fluid Mech. 535, 143–188 (2005) 2. Chong, M.S., Perry, A.E.: A general classification of three-dimensional flow fields. Phys. Fluids 5, 765–777 (1990) 3. DelAlamo, J.C., Jiménez, J., Zandonade, P., Moser, R.D.: Self-similar vortex clusters in the turbulent logarithmic region. J. Fluid Mech. 561, 329–358 (2006) 4. Foucaut, J.M., Stanislas, M.: Some considerations on the accuracy and frequency response of some derivative filters applied to particle image velocimetry vector fields. Meas. Sci. Technol. 13, 1058–1071 (2002) 5. Foucaut, J.M., Carlier, J., Stanislas, M.: PIV optimization for the study of turbulent flow using spectral analysis. Meas. Sci. Technol. 15, 1046–1058 (2004) 6. Herpin, S.: Study of the influence of the Reynolds number on the organization of wall-bounded turbulence. Ph.D. thesis, Ecole Centrale de Lille and Monash University (2009) 7. Herpin, S., Wong, C.Y., Stanislas, M., Soria, J.: Stereoscopic PIV measurements of a turbulent boundary layer with a large spatial dynamic range. Exp. Fluids 45, 745–763 (2008) 8. Jeong, J., Hussain, F., Schoppa, W., Kim, J.: Coherent structures near the wall in a turbulent channel flow. J. Fluid Mech. 332, 185–214 (1997) 9. Perry, A.E., Henbest, S., Chong, M.S.: A theoretical and experimental study of wall turbulence. J. Fluid Mech. 165, 163–199 (1986) 10. Sheng, J., Malkiel, E., Katz, J.: Using digital holographic microscopy for simultaneous measurements of 3d near wall velocity and wall shear stress in a turbulent boundary layer. Exp. Fluids 45, 1023–1035 (2008)

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11. Stanislas, M., Perret, L., Foucaut, J.M.: Vortical structures in the turbulent boundary layer: a possible route to a universal representation. J. Fluid Mech. 602, 327–342 (2008) 12. Wu, Y., Christensen, K.T.: Population trends of spanwise vortices in wall turbulence. J. Fluid Mech. 568, 55–76 (2006)

Session 3: Experiments in Adverse Pressure Gradient Boundary Layers

• Two-Point Near-Wall Measurements of Velocity and Wall Shear Stress Beneath a Separating Turbulent Boundary Layer P. Nathan and P.E. Hancock • Experiments and Modeling of Boundary Layers Subjected to Various Pressure Gradients H. Nagib (no paper) • Experimental Analysis of Turbulent Boundary Layer with Adverse Pressure Gradient Corresponding to Turbomachinery Conditions S. Drobniak, W. Elsner, A. Drozdz, and M. Materny • Near Wall Measurements in a Separating Turbulent Boundary Layer with and without Passive Flow Control D. Lengani, D. Simoni, M. Ubaldi, P. Zunino, and F. Bertini

Two-Point Near-Wall Measurements of Velocity and Wall Shear Stress Beneath a Separating Turbulent Boundary Layer Paul Nathan and Philip E. Hancock

Abstract Two-point measurements of tangential velocity and wall shear stress have been made using LDA and pulsed-wire anemometry in the near-wall layer beneath a turbulent boundary layer passing through separation. The zero-timelag correlation shows a dramatic decrease in the viscous sub-layer, followed by a roughly constant level in the remainder of the near-wall layer. A qualitatively similar behaviour is seen analytically by supposing an oscillating pressure gradient. This analysis shows that in the viscous sub-layer the velocity and wall shear stress are out of phase, except in the limit at the wall. Thus, an assumption that the very-near-wall flow behaves simply in a quasi-steady manner in response to a fluctuating pressure field imposed by large-scale outer-layer motions would not be correct. The two-point time-lag correlation shows the wall shear stress to be most strongly correlated with the tangential velocity at a point almost directly above; the correlation contours are almost perpendicular to the surface. The normal velocity, in contrast, shows no more than a weak correlation with wall shear stress.

1 Introduction The present paper is an investigation of the relationship between the instantaneous velocity near the surface and the instantaneous wall shear stress beneath, in a boundary layer passing through separation. This is of interest in itself, and of relevance to near-wall models for high Reynolds number Large Eddy Simulations, where a full spatial resolution is not practical. Near the surface, the fluctuations in an adverse pressure gradient boundary layer are large enough for instantaneous flow reversal, with attendant reversal of wall shear stress. Very near the surface, in the limit, the wall shear stress fluctuations must be entirely correlated with the tangential velocity P. Nathan · P.E. Hancock () University of Surrey, Guildford, UK, GU2 7XH e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_14, © Springer Science+Business Media B.V. 2011

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Fig. 1 LDA probe supported beneath the surface, and mirror

fluctuations. Earlier work [2, 3] showed that the relationship between velocity and wall-normal distance beneath reattachment is quadratic rather than linear. Clearly, the mean wall shear stress cannot be used to provide a meaningful velocity scale where the mean wall shear stress passes through zero. However, the fluctuations in wall shear stress do not become zero in magnitude, but in fact change fairly slowly through separation, and so may provide the basis for a meaningful velocity scale. In simple terms one may think of the larger-scale motions sloshing’ the near-wall flow back and forth.

2 Measurement Techniques The velocity measurements were made with a Dantec two-component LDA system, and the wall shear stress by means of pulsed-wire anemometry. While both techniques provide instantaneous measurements, neither was sufficiently fast to provide time-resolved measurements. Indeed, the pulsed-wire technique data rate was constrained to 10 Hz. Because the measurements from the LDA are random in time, special software needed to be written to obtain coincident or verynearly-coincident samples of velocity and wall shear stress. This condition of verynearly-coincident was necessary because very few samples would have coincided in the typically 10 minute sampling times at each point. The velocity from nearly coincident samples, before and after the wall shear stress sample, was interpolated linearly to give a coincident sample. By varying the width of an acceptance window the effect of non-coincidence was found to be negligible, within the limits used here. This approach was applied to both the zero-time-lag correlations and the time-lag correlations. As far as the authors are aware this is the first time coupled LDA and pulsed-wire wall shear stress measurements have been made. The LDA probe was held beneath the surface and mounted on a traversing mechanism, which also supported a mirror above the surface. The beams passing through an optical glass window flush with the surface were reflected by the high-quality front-silvered mirror such that one beam was parallel to the surface, as shown in Fig. 1. This allowed measurements to be made as close as 0.5 mm from the surface. The focal point of the beams was above the pulsed wire probe, 80 mm from the mir-

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Fig. 2 Rear section of bump. Origin at start of bump. Dimensions in mm. x local to flat section

ror centre, and far enough for no more than a small interference effect caused by the presence of the mirror. (Ideally, a long-focus probe would have been held outside the working section, but none was available.) The slight angle of the beams means that the normal’ velocity measurement contains a contribution from the lateral fluctuation but, because the flow is two-dimensional in the mean, a simple correction can be applied to give v 2 and uv correctly. Time-weighted averaging was applied to the velocity measurements. The pulsed-wire probe was calibrated against a Preston tube in a nearly-zero pressure gradient two-dimensional boundary layer. The probe, once calibrated, provides a direct measurement of wall shear stress no longer constrained by the limitations of the Preston tube. The measurements themselves were made on the rear flat section of a bump, as illustrated in Fig. 2. In this figure, the horizontal and vertical axes give the distance from the beginning of the bump, and above the reference height, respectively. The reference height is the surface of the flat plate that extends 7.75 m upstream of the bump leading edge, and downstream of the (internal) corner formed with the flat section at the end of the bump. The separation bubble sat above the corner in Fig. 2, where x in the figure is the distance from a datum on the flat surface, 274.5 mm from the corner. Upstream of the bump crest, and for part of the way downstream, the bump profile was the same as that used in the WALLTURB experiments at Lille (see also [1]), but at half scale. Further downstream, the present bump was steeper, leading to a separation bubble. The mean wall shear stress on the flat surface was measured by means of oil-film interferometry. Necessarily, the flow converges towards a line of separation and, in the case as here of a flow that is two-dimensional in the time mean, would cause the oil to accumulate near separation. Therefore, the plate and bump over which the flow was developed was mounted vertically so that excess oil was drained gravitationally. The associated analysis of an oil film undergoing surface shear stress and gravitational body force beneath a boundary layer in an adverse pressure gradient will be reported separately.

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Fig. 3 Mean shear stress through separation

Fig. 4 Example of simultaneous oil films upstream, at and downstream of separation, after 120 minutes. Free-stream flow right to left

3 Results 3.1 Mean Wall Shear Stress, Mean Velocity and Reynolds Stresses Figure 3 shows the mean wall shear stress along the flat section as measured by means by the oil film technique, and Fig. 4 shows an example of the interference pattern fringes. The latter figures shows the flow to be closely two-dimensional. Profiles of mean velocity and three Reynolds stresses are shown in Fig. 5, in dimensional form. From these it can be seen that the mean separation was between x = 140 and 162 mm. (Oil film measurements gave separation at x = 139 mm, in the absence of the mirror, so the small discrepancy here might be from a residual interference effect of the mirror and its support.) The overall height of the shear layer was about 130 mm above the bump at the separation position.

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Fig. 5 Mean velocity and Reynolds stress profiles. y is perpendicular to x

3.2 Velocity-Wall-Shear-Stress Correlation The correlation coefficient is shown in Fig. 6, against distance from the surface, where this distance has been normalised by uτ /ν. uτ is the velocity scale formed from the r.m.s. of the wall shear stress fluctuations measured by the pulsed-wire probe. (In a zero pressure gradient boundary layer yuτ /ν ≈ 0.64yuτ /ν.) Surprisingly, the correlation decreases rapidly even inside yuτ /ν of about 7. It is then relatively constant further out, with most of the profiles bunched together. This dramatic decrease was not anticipated. However, support for the observed behaviour can be demonstrated analytically, at least with qualitative agreement. By neglecting the convective terms the boundary layer equations near a surface can be written as ∂u/∂t = −∂p/∂x + ν∂ 2 u/∂y 2 ,

(1)

where p is the kinematic pressure. If, now, the pressure gradient is supposed to oscillate sinusoidally with frequency ω and magnitude k, i.e. ∂p/∂x = k exp(iωt) we get, after time averaging, y y y μk 2 1 cos − sin , τw u = 2 1 − exp − δ δ δ δω 2

(2)

(3)

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Fig. 6 Zero time lag correlation coefficient, τw u/τw u vs yuτ /ν

τw2 = (kμ/ωδ)2 and

u2 = (k/ω)2

1 1 y y y + exp −2 − exp − cos , 2 2 δ δ δ

(4)

(5)

where δ 2 = 2ν/ω , from which it is straightforward to obtain an expression for the correlation coefficient, τw u/τw u , where τw 2 = τw2 and u 2 = u2 . Asymptotically, the ensuing expression becomes, as y tends to zero and infinity, respectively, 1y 1y / 1− (6) τw u/τw u = 1 − 2δ 6δ √ and 1/ 2. The fact that at any instant the pressure gradient is independent of x implies that the associated length scale driving the fluctuations is infinite; that is, that the pressure gradient is being driven by large-scale outer-layer motions. τw u/τw u as given by the above is illustrated in Fig. 6. Altering ω changes the steepness of the initial decrease, but cannot have any affect on the asymptotic level. Clearly, although the level does not fall to that observed experimentally, the behaviour is qualitatively similar. The figure also shows the form given by (6). It does not appear that a superposition of modes would lead to a greater concurrence with the measurements. It also turns out that the phase between velocity and wall shear stress changes from zero at the surface to 45° further out. Thus, it is quite wrong to think of the very near wall viscous layer, and its response to large outer-layer motions, as quasi steady (i.e. ignoring ∂u/∂t in (1)). Rather, an instantaneously imposed pressure gradient will lead to a velocity profile that is out of phase. It appears a phase lag of

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Fig. 7 Time ‘lag’ correlation coefficient τw (t)u(t + t)/τw u . Bottom axis is t . x = 0 (left), 140 mm (right)

about 60° is needed to give the observed decrease in the coefficient. Leaving out the convective terms in (1) in effect leaves out any representation of structures, which would plausibly give rise to a reduced correlation between u(t) and τw (t).

3.3 Time-Lag Correlations Figure 7 shows the cross correlation at two streamwise positions, x = 0 and 140 mm, that is, separated a distance slightly larger than the overall shear layer thickness. A very distinctive feature is that the contour lines are nearly vertical and symmetrical about t = 0, in both cases. This implies a strong correlation between the wall shear stress fluctuation and the tangential velocity fluctuation directly above, to a distance well above the surface. Note, the top of these figures correspond to the points to the furthest right in Fig. 6. The bottom of these figures are at yuτ /ν = 7.4 and 5.6 (left and right, respectively). The fact that the contours are not precisely vertical means that the maximum correlation does not occur at precisely zero time delay. Were the data of Fig. 6 to be replaced with the maxima there would be an up-shift of not more than about 0.03, slightly reducing the gradient. The mean convection velocity at x = 0 does not cause a noticeable displacement of the contours to one side, compared with those at x = 140 mm, but, not surprisingly, the time interval for a given change in correlation level is reduced. The patterns seen in Fig. 7 are consistent with the motion being determined largely by motion outside the field represented, that is from above, through the imposed pressure field. If Taylor’s hypothesis is supposed then the length scale over which the correlation goes to zero is comparable with the overall shear layer thickness, reinforcing the inference that the near-wall motion is dominated by large-scale outer-layer motion. The correlation coefficient for the normal velocity and wall shear stress fluctuations, τw (t)v(t + t)/τw v (not shown), where v is the r.m.s. of v(t), is an order of

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magnitude smaller than τw (t)u(t + t)/τw u and exhibits no such correlation pattern. From Fig. 5, v , though smaller, is comparable with u . Therefore, there is no more than a weak relationship between v(t) and τw (t). This is consistent with the motions being predominantly associated with large-scale motions. Acknowledgements This work is part of that performed by the University of Surrey under the WALLTURB project. WALLTURB (A European synergy for the assessment of wall turbulence) is funded by the EC under the 6th framework program (CONTRACT N: AST4-CT-2005-516008).

References 1. Bernard, A., Foucaut, P., Dupont, P., Stanislas, M.: Decelerating boundary layer: a new scaling and mixing length model. AIAA J. 41, 248–255 (2003) 2. Hancock, P. E.: Velocity scales in the near-wall layer beneath reattaching turbulent separated and boundary layer flows. Eur. J. Mech. B, Fluids 24, 425–438 (2005) 3. Hancock, P.E.: Scaling of the near-wall layer beneath turbulent separated flow. Eur. J. Mech. B, Fluids 26, 271–283 (2007)

Experimental Analysis of Turbulent Boundary Layer with Adverse Pressure Gradient Corresponding to Turbomachinery Conditions Stanislaw Drobniak, Witold Elsner, Artur Drozdz, and Magdalena Materny

Abstract The paper deals with the experimental analysis of turbulent boundary layer at the flat plate for Reynolds number Reθ ≈ 3000. The adverse pressure gradient generated by curvature of the upper wall corresponded to the case of pressure distribution in axial compressor. The fully developed turbulence structure was achieved by proper triggering of the boundary layer. The mean and turbulent flowfields were investigated with the use of hot-wire technique. A substantial effort has been devoted to the precise definition of inlet boundary conditions as well as consistency of measurements obtained with different HWA sensors. The analysis in the paper was concentrated on the problem of scaling of turbulent boundary layer and on the physical background behind scaling laws being compared.

1 Introduction The research on turbulent boundary layers (TBL hereinafter) is being carried out for more than a century and as it has been pointed out in recent review by W.K. George [5]: . . . a little more than a decade ago the basic characteristics of turbulent boundary layer. . . were widely believed to be well understood. . . and it bothered only a few. . . that real shear stress measurements. . . differed consistently from (theoretical) results. . . and instead of reexamination of theory it became common wisdom that there was something wrong with the experimental techniques. . . S. Drobniak () · W. Elsner · A. Drozdz · M. Materny Czestochowa Univ. of Technology, Armii Krajowej 21, Czestochowa, Poland e-mail: [email protected] W. Elsner e-mail: [email protected] A. Drozdz e-mail: [email protected] M. Materny e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_15, © Springer Science+Business Media B.V. 2011

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During the last decade this problem started to be re-examined again, the more so that the practical motivation for better understanding of TBL has appeared from the aeronautical industry, which needs more accurate modelling tools for the design of aircraft body. The modelling of TBL is of particular importance as more than a half of drag of modern aircraft originates from wall friction, which is a phenomenon directly governed by the behavior of TBL. However, the current CFD tools reflect the physics of TBL to the extent which is presently understood and the need to improve the knowledge of TBL physics was behind the establishing the joint EU project WALLTURB [9], which is aimed at generating and analyzing new data on near wall turbulence and extracting physical understanding of TBL from the new experiments. This in turn should enable to improve modelling of boundary layers, especially in near-wall regions. Among the crucial issues of the subject is the problem of TBL scaling or rather the physical background behind these scaling laws. The search for scaling laws, which were a valuable analytical tool for decades, today leads to improvement of turbulence closure models which reflect the current understanding of TBL physics. This issue becomes more complex when one considers the TBL at large Reynolds numbers with the presence of pressure gradient, where the lack of reliable data is particularly evident. As it has been pointed out by Stanislas [9], the Adverse Pressure Gradient (APG) TBL is particularly problematical, because of the lack of proper experimental data for sufficiently large Re numbers and for experimental conditions corresponding to real engineering applications. That is why the experimental analysis of APG TBL was selected as the aim of the present research. The experiment has been performed for the conditions representative for practical turbomachinery flows that determined not only the distribution of pressure but also required a sufficiently high value of Reynolds number (Reθ ≈ 3000).

2 Experimental Setup and Measuring Techniques The experiment was performed in an open-circuit wind tunnel with outlet test section shown in Fig. 1, where the turbulent boundary layer developed along the flat plate (manufactured of 10 mm thick Plexiglas), which was 2807 mm long and 250 mm wide. In order to achieve the proper structure of turbulent boundary layer the tripping was applied, which consisted of cylindrical wire of 2 mm placed at the distance x = 210 mm from the elliptical leading edge and followed by strip of controlled roughness. For that location laminar boundary layer thickness was about 2.6 mm so d/δ parameter was equal 0.76 and the relative distance from the leading edge was x/d = 105. To accelerate the breakdown of the large-scale vortex structures, the strip of coarse-grained sandpaper (of grain size g ∼ = 0.1 µm was additionally located 3 mm downstream of the cylindrical wire. Tripping of boundary layer allowed to obtain a relatively high value of characteristic Reynolds number equal Reθ ≈ 3000. The free-stream mean velocity at the inlet plane was U0 = 15 ms−1 and the turbulence intensity of the undisturbed flow was equal Tu ≈ 0.4%.

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Fig. 1 The shape of the upper wall of the wind tunnel and the corresponding pressure distribution along the flat plate Table 1 Positions of measuring planes No

Distance from inlet xs (mm)

Relative distance Sg

No

Distance from inlet xs (mm)

Relative distance Sg

1

0

0.000

10

667

0.625

2

427

0.400

11

697

0.653

3

457

0.428

12

727

0.681

4

487

0.456

13

757

0.709

4

487

0.456

14

787

0.737

5

517

0.484

15

817

0.765

6

547

0.512

16

847

0.793

7

577

0.540

17

877

0.822

18

907

0.850

8

607

0.569

9

637

0.597

The upper wall was shaped according to the assumed distribution of pressure gradient shown in Fig. 1b, which corresponded to the conditions encountered in stator passages of turbomachinery. The static pressure distribution was measured at the flat plate with DATA INSTRUMENTS DCXL01DN pressure transducer connected to KULITE D486 amplifier. The measuring accuracy of pressure is documented by error bars shown along the pressure distribution at Fig. 1b, the mean relative error of pressure measurements was equal 2.16% for ZPG area and 2.67% for APG. Flow-field measurements were made at two upstream planes (Sg = 0.0 and Sg = 0.197) and 17 planes at APG region. The distances of measuring planes from the first plane and their corresponding non-dimensional coordinates Sg are given in Table 1, while their locations are shown by thick lines at Fig. 1. Velocity profiles were measured with single (1.25 mm wire sensor) and X-wire HWA probes

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Fig. 2 Comparison of the mean and fluctuating streamwise velocities obtained from different probes

(1.25 mm wire sensors), the latter one had to be specially calibrated due to high turbulence intensity in TBL (for details of calibration procedures see [8]). Application of two different HWA sensors made it necessary to verify the consistency of their readings, which was done for streamwise velocity component. The results presented in Fig. 2 show the sample comparison of streamwise velocity and velocity fluctuation profiles measured at the inlet measuring section Sg = 0.00. It must be noticed, that due to larger size the X-wire could not penetrate the boundary layer as close as the single probe. With single wire probe it was possible to get as close to the wall as y + = 2 while for X-wire the minimum value was y + ≈ 8. The comparison of the results presented in Fig. 2 reveals a good consistency in both mean and fluctuating components at sample coordinate Sg = 0.00 and similar conclusions were found for other coordinates. The distribution of mean and fluctuating velocity confirmed that the turbulent boundary layer was fully developed. In particular it could be noticed, that the distribution of u revealed a single peak located at y + ≈ 20, which is typical for turbulent boundary layers at ZPG conditions. The comparison of results obtained with single and X-wire probes performed in all cross-sections allowed to estimate the uncertainty of velocity measurements to be of the order of 0.6% for mean velocity and 3–4% for fluctuating components. The detailed measuring error analysis given in [7] confirms the quality and accuracy of measurements reported here.

3 Experimental Results and Scaling of TBL The main task of the investigations reported here was the analysis of the turbulent boundary layer under the influence of adverse pressure gradient. Figure 3a presents the downstream evolution of shape factor parameter H which is typical of TBL approaching separation under the APG conditions, however its behavior reveals that the TBL analyzed has not been separated yet. This conclusion is confirmed by the analysis of Fig. 3b, where distribution of wall shear stresses τw is given. Apart from the data calculated from velocity profiles (according to Clauser method [7]) the results from oil-film interferometry technique [3] as well as from RANS computations

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Fig. 3 Distribution of the shape parameter (a), and wall shear stress (b)

with SST turbulence model are presented for comparison. The results agree fairly well except for a region downstream Sg = 0.6 plane, where HWA data are slightly underestimated, what is rather typical defect of this technique in decelerating boundary layers. The more detailed insight into the TBL structure near the wall may be obtained from Fig. 4a, which presents mean velocity profiles in consecutive measuring planes. All these profiles were presented in universal coordinates u+ ; y + , which is the inner region Prandtl’s scaling. One may notice the viscous sublayer (y + = 2–6) which then is transformed into the buffer zone (y + = 6–30) and then the area of log-law which extends up to y + ≈ 300 for initial cross sections. The wake-law region visible at Fig. 4a as the deflection of velocity profile from log-law distribution accompanies the appearance of APG. One may notice that in the cross sections most distant from inlet plane the deviation of velocity profile from loglaw appears as early as y + < 100 (see Sg = 0.850 at Fig. 4a), which indicates the pronounced increase of wake-law area and a shortening of log-law region. The analysis of TBL outer scaling laws began from transformation of mean velocity profiles with von Karman scaling proposal [4] and the results are presented in Fig. 4b. The Δ in this figure is a Clauser–Rotta length scale defined as δ ∗ U∞ /uτ . One may notice that measuring points do not collapse on the common curve that proves that friction velocity uτ , which is an inner velocity scale, does not perform satisfactorily in outer TBL region. The explanation for failure of inner scaling is the distribution of Clauser parameter β given by: β=

δ ∗ dP∞ uτ dx

(1)

which is by no means constant [7]. One may conclude that it agrees with statement of Castillo and George [6] that equilibrium boundary layers, which would fulfill the Clauser’s requirements are nearly impossible to generate and maintain, that in turn explains the failure of von Karman scaling. The next attempt was directed towards the verification of George–Castillo outer scaling law given in [6] that is summarized in Fig. 4c. One may notice that also in this case measuring points do not collapse on the common curve, however the behavior of particular velocity profiles measured at consecutive cross-sections is

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Fig. 4 Different scaling for TBL: (a) inner Prandtl’s scaling; (b) outer von Karman scaling; (c) outer Castillo–George scaling; (d) outer Zagarola–Smits scaling

more consistent than in the case of von Karman scaling. To explain the failure of this concept one should analyze the behavior of Castillo–George pressure parameters Λ and Λθ : Λ= Λθ =

δ dP∞ 2 dδ/dx dx ρU∞

θ 2 dθ/dx ρU∞

θ dU∞ dP∞ = dx U∞ dθ/dx dx

(2) (3)

which according to [1] should be constant to enable the successful application of this scaling. These parameters (what is not shown here) are unfortunately not constant along the measuring region what indicates that it is a case of nonequilibrium flow and explains the failure of George–Castillo outer scaling. The most satisfactory results were obtained with the scaling proposed by Zagarola and Smits [2]. One can observe at Fig. 4d, that measuring points collapse at the common curve for all the measuring cross-sections. The success of this approach is because of applied scaling factor, which reflects the variation of upstream conditions and provides means to remove this influence. The other reason could be, according to Castillo and Wagan [1], that despite of nonequilibrium conditions defined as Λθ = const the boundary layer could remain in equilibrium but only locally. One may conclude therefore, that the outer scaling originally proposed by Zagarola and Smits is in agreement with the two-layer approach and that Zagarola–Smits scaling is the most suitable for the mean-velocity profile even for very strong APGs. This conclusion is in agreement with the findings of Indinger et al. [4] who found also, that scaling proposed by Castillo and George fails very close to separation, due to the effect of backflow.

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Fig. 5 Reynolds stresses in xy plane

The controversy on the applicability of inner and outer scaling is in fact a controversy on more fundamental issue concerning the physics of TBL and in particular contribution of particular zones to downstream growth of TBL thickness. As it was pointed out by George [3]: it is commonly (and erroneously) believed that because the main contribution to θ comes from near the wall, then the main contribution to dθ/dx must also come from near the wall.

In fact, the near wall region grows much more slowly than the outer part of the boundary layer that in turn implies, that most of the contribution to dθ/dx originates from areas located far from the wall. It may be assessed from the distribution of intensity of velocity fluctuations, which may be treated as the indirect indication of intensity of turbulent transport. Analysis of Reynolds normal u u ; v v and shear stresses u v distributions, shown at Fig. 5 in inner scaling, reveal in the initial region (Sg = 0.4 to 0.5) a single maximum located in the immediate vicinity of the wall (y + ≈ 20), which gradually decays in the downstream area. At the same time the outer maximum located at y + ≈ 300 appears and becomes the more pronounced, the further downstream is located the measuring cross-section (arrows at Fig. 5 denote the decay and appearance of inner and outer maxima respectively). It means therefore, that in the presence of APG the appearance of second peak of turbulent velocity fluctuations confirms the more pronounced contribution of outer region to the downstream development of TBL.

4 Conclusion The results obtained suggest, that turbulent boundary layer at APG conditions requires two velocity scales, i.e. inner (imposed by inner boundary condition from constant shear stress layer) and outer (imposed by outer layer) velocity scales. Among the scaling proposals published in literature so far the Zagarola–Smits scaling seems to be the most suitable for the mean-velocity profile even for very strong APGs, bearing in mind the experimental data obtained during the present research. The results concerning the turbulent velocity fluctuations confirm the basic physics

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behind the idea of outer scaling, in particular the appearance of second peak of velocity fluctuations confirms the more pronounced contribution of outer region to the downstream development of TBL, that in turn suggests that physical reasoning advocated by George [5] seems to be in agreement with experimental evidence. The uncertainty analysis and comparison with available literature data confirm the good quality of experimental results and support the validity and trustworthiness of conclusions presented in the paper. Acknowledgements This work has been performed under the WALLTURB project and SPB WALLTURB funded by MNiSzW. WALLTURB (A European synergy for the assessment of wall turbulence) is founded by CEC under the 6th framework program (Contract #: AST4-CT-2005516008).

References 1. Castillo, L., George, W.K.: Similarity analysis for turbulent boundary layer with pressure gradient: outer flow. AIAA J. 39, 41–47 (2001) 2. Castillo, L., Wang, X.: Similarity analysis for nonequilibrium turbulent boundary layers. TransASME J. Fluid Eng. 126, 827–834 (2004) 3. Clauser, F.H.: The turbulent boundary layer. AIAM 4, 1–51 (1956) 4. Drozdz, A., Drobniak, S., Elsner, W.: Application of oil-fringe interferometry for measurements of wall shear stresses in turbulent boundary layer. Turbomach., Tech. Univ. Lodz 133, 103–110 (2008) 5. George, W.K.: Recent advancements toward the understanding of turbulent boundary layers. AIAA J. 44(11), 2435–2449 (2006) 6. Indinger, T., Buschmann, M.T., Gad-el-Hak, M.: Mean-velocity profile of turbulent boundary layers approaching separation. AIAA J. 44(11), 2465–2474 (2006) 7. Materny, M., Drozdz, A., Drobniak, S., Elsner, W.: Experimental analysis of turbulent boundary layer under the influence of APG. Arch. Mech. 60, 1–18 (2008) 8. Materny, M., Drozdz, A., Drobniak, S., Elsner, W.: The structure of turbulent boundary layer with adverse pressure gradient corresponding to turbomachinery condition. Turbomach., Tech. Univ. Lodz 133, 221–228 (2008) 9. Stanislas M.: WALLTURB: a European synergy for the assessment of wall turbulence. In: Proc. AIAA Conf., Reno (2008)

Near Wall Measurements in a Separating Turbulent Boundary Layer with and without Passive Flow Control Davide Lengani, Daniele Simoni, Marina Ubaldi, Pietro Zunino, and Francesco Bertini

Abstract The present paper reports the results of a detailed experimental study on low profile vortex generators used to control a turbulent boundary layer separation on a large-scale flat plate with prescribed adverse pressure gradient, which conditions are representative of aggressive turbine intermediate ducts. Laser Doppler Velocimetry has been employed to investigate the velocity fields in different measurement planes to characterize the turbulent boundary layer at separation conditions, without and with control devices application. The detail of the performed measurements allows the evaluation of the deformation works in the test section symmetry plane; normal and shear contributions of viscous and turbulent deformation works have been distinctly analyzed. Furthermore the analysis on the flow generated by VGs has been extended evaluating velocity and turbulence profiles in a cross stream plane immediately downstream of the control device.

1 Introduction In the last thirty years many efforts have been done to apply flow control devices inside a real environment in a reliable and efficient way. In particular inside a modern D. Lengani () · D. Simoni · M. Ubaldi · P. Zunino Dipartimento di Macchine, Sistemi Energetici e Trasporti, Università di Genova, Via Montallegro 1, 16145 Genoa, Italy e-mail: [email protected] D. Simoni e-mail: [email protected] M. Ubaldi e-mail: [email protected] P. Zunino e-mail: [email protected] F. Bertini AVIO S.p.A., Turin, Italy e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_16, © Springer Science+Business Media B.V. 2011

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aeroengine one of the most interesting applications of boundary layer control may be the prevention of flow separation. The modern tendency in gas turbines design leads to the reduction of blades count and ducts length, which may be introduced without the decay of aerodynamic performance suppressing or delaying separation. Low profile vortex generators (VGs) seem up to now one of the most appealing flow control device to be employed in a modern aeroengine (e.g. [1]). For this reason in the last years CFD works (e.g. [2, 3]) and tough experimental parametric analysis ([4] among others) introduced a large amount of data presenting global performance evaluation parameter (as also reviewed in [5]) which may allow a relatively easy and efficient implementation of VGs in common applications. Although for the design and prediction of flow control in more complex environment more data may be provided; loss generation mechanism and turbulence characteristics may be also analyzed via different tools, performing both numerical and experimental investigations. In the present work, that is part of an European research project named AIDA “Aggressive Intermediate Duct Aerodynamic for Competitive and Environmentally Friendly Jet Engines”, mean velocity, turbulence characteristic and local deformation works are analyzed with and without VGs in a linear diffuser, which pressure gradient is typical of newly designed aeroengine intermediate duct. Furthermore in aeroengine intermediate ducts, the boundary layer is fully turbulent and prone to separation, the same condition has been imposed at the inlet of the diffuser’s test section [6]. The data presented in this paper extended the analysis on VGs physical mechanism of interaction carried out in previous works [6, 7] introducing the evaluation of viscous and turbulent contributions to the work of deformation and their relative importance. Since the flow modified by VGs is three-dimensional the present discussion focus also on the flow characteristic in a cross-stream plane downstream of VGs.

2 Experimental Apparatus and Methodology The facility (described in details in [6]) consists of an open-loop wind tunnel in which air is blown by a centrifugal fan installed in the Aerodynamics and Turbomachinery Laboratory of the University of Genova. The test section (sketched in Fig. 1) used for this investigation was designed to provide several adverse pressure gradients typical of aeroengine diffusers. The boundary layer flow develops on a large-scale flat plate 1700 mm long and 400 mm wide, mounted parallel to the test section floor, with the leading edge located about 600 mm upstream of the test section inlet. The inlet test section height H0 is 196 mm. For the experiment discussed in this paper the test section top wall was regulated with an inclination of 16 deg; the imposed adverse pressure gradient is here reported for completeness in Fig. 2 (to the left) in which it may be observed the increase of the static pressure from the test section inlet up to separation. The test section inlet boundary layer (x = 0 mm) is also depicted in Fig. 2 (to the right), while more data on the inlet flow condition are provided in [6, 7].

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Fig. 1 Test section scheme (left) and measurement mesh (right)

Fig. 2 Static pressure coefficient distribution (left) and boundary layer velocity profiles (right)

In order to control the boundary layer separation, low-profile vortex generators have been employed. They consist in small wings of trapezoidal shape 64 mm long and 16 mm high (0.8 and 0.2 times the inlet section boundary layer thickness respectively). A more comprehensive explanation and description of their parameters choice may be found in [6]. VGs have been arranged in a co-rotating pattern at an angle of 23 deg to the incoming flow and the tunnel centerline provides a plane of symmetry for the configuration, such that the two vortex generators near the symmetry plane are arranged in a counter rotating manner. In a previous work [6] the VGs axial location most effective in delaying separation was established. For the top wall inclination studied in the present work, it resulted to be 95 mm (x/H0 = 0.48) upstream of the detachment point, that was detected at x = 385 mm (x/H0 = 1.93).

2.1 Measurement Technique and Experiment Organization A four beams two colours Laser Doppler Velocimeter (Dantec Fiber Flow), in backward scatter configuration, was employed for the velocity measurements and it was largely described in [7, 8]. The probe consists of an optical transducer head of

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60 mm diameter, with a focal length of 300 mm and a beam separation of 38 mm, connected to the emitting optics and to the photomultipliers by means of optic fibres. The probe volume is 0.09 mm × 0.09 mm × 1.4 mm. Flow was seeded with mineral oil droplets with a mean diameter of 1.5 m. The experimental uncertainty was evaluated [8–10] to be less than 1 percent of the mean velocity. Statistical moments were weight-averaged with transit time to avoid statistical bias. Thanks to the large number of samples acquired (30000) the statistical uncertainty on the averaged velocity due to finite number of samples was estimated better than 4% for a probability of 95% and a local turbulence intensity of 100%, which condition may occur in the near wall region. The boundary layer developing over the flat plate was surveyed by means of traverses normal to the flat plate. For the case without VGs installed traverses were performed only in the symmetry plane while for the case with VGs installed traverses were performed in the symmetry and cross-stream plane as depicted in Fig. 1. Each boundary layer traverse was constituted by 103 measuring points, with the first point at a distance of 50 µm from the wall and the distance between adjacent points has progressively increased in the outer part.

3 Results and Discussion The baseline flow evolution is represented in Fig. 2 (to the right); the flow is decelerating and separating at x equal to 385 mm [6, 7], as shown by the velocity profiles. The boundary layer profile at x = 385 mm (x/H0 = 1.93) is in fact characterized by a zero mean velocity gradient in y direction at the wall. When applying VGs, the boundary layer separation is delayed. In the near wall region downstream of VGs position (x = 350 mm) the most relevant differences with respect to the baseline case are observable. The flow is accelerated near the wall since streamwise momentum is transferred from outer to inner region of the boundary layer. This acceleration of the flow leads to a significant reduction of the near wall momentum deficit with respect to the baseline case but even with respect to the boundary layer upstream of the VGs location. As consequence the static pressure recovery in presence of VGs results around 15% greater than in the case with no flow control (Fig. 2 to the left) as described in [6].

3.1 Dissipation Mechanism The energy dissipation mechanism responsible of losses appears in the transport equation of the mean flow kinetic energy [11] as the work of deformation D of the mean motion performed by the stress tensor τij : D = τij · Sij . The stress tensor τij includes both the viscous and the turbulent components: τij = 2μSij − ρu v . The ∂u ∂ui + ∂xji ). rate of strain of the mean flow Sij is instead defined as follows: Sij = 12 ( ∂x j

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Fig. 3 Deformation work operated by Reynolds shear and normal stresses: baseline case (left) and with boundary layer control device (right)

The viscous dissipation terms μ(∂u/∂x)2 , μ(∂u/∂y)2 , μ(∂v/∂y)2 and μ(∂v/∂x)2 resulted to be not negligible only upstream of the separation onset in the very near to the wall region (y < 5 mm) where strongest normal to the wall velocity gradient may be observed. As the flow velocity is reduced by the channel divergence, the viscous terms start to be negligible since the near wall velocity gradient becomes small and consequently are not shown here. The other contributions to the dissipation of the mean flow energy are due to the turbulent dissipation: energy is exchanged from the mean flow to the turbulence, increasing turbulent kinetic energy [11]. The terms more relevant in the work of deformation of the mean motion due to turbulent action have been found to be the deformation work operated by the Reynolds normal and shear stress along the x direction ((Dn )T x = −ρu 2 ∂u/∂x and (Dt )T x = −ρu v ∂u/∂y respectively) and the deformation work operated by Reynolds normal stress along the y direction ((Dn )T y = −ρv 2 ∂v/∂y). In the uncontrolled case, up to x = 200 mm, the near wall losses are increased mainly by the term (Dt )T x (Fig. 3 to the left). The deformation work operated by the Reynolds normal stress starts, instead, to generate losses at x = 200 mm where the test section channel divergence begins to reduce the mainstream velocity (Fig. 2); upstream of this position this contribution is, in fact, negligible. Close to separation (x = 350 mm) the term (Dn )T x becomes the main responsible of energy dissipation. This effect is observable in Fig. 3 comparing the distributions of the term (Dt )T x to the one operated by (Dn )T x respectively for the two axial position in analysis. Consequently the classical assumption that the normal components are negligible can not be done for a separating turbulent boundary layer. On the contrary in this condition there are the shear terms to be negligible (as also observed by [12]). The only non-negligible contribution to the mean energy dissipation rate along the y direction is the deformation work operated by the Reynolds normal stress (Dn )T y . This term is due to the channel divergence, by which fluid is moving away from the wall, and to the wall normal velocity fluctuations. The distribution of this term resembles the distribution of the term (Dn )T x and starts to have a significant contribution as separation is approached. But in this case, since the gradient of the

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velocity normal to the wall is positive along y direction, the term (Dn )T y tends to increase the average flow energy subtracting it to the turbulence. This contribution to the deformation work results in fact negative in this flow region (Fig. 3). The same contributions to loss generation have been evaluated with VGs installed in the test section (Fig. 3 to the right). The viscous terms resulted not negligible downstream of the VGs due to the strong normal velocity gradients induced by the momentum transferred towards the wall. Nevertheless their effects are confined in the very near to the wall region (y < 5 mm) and consequently are not shown here. Downstream of vortex generators, at x = 350 mm, since the flow accelerates close to the wall due to the control device action, the term (Dt )T x results greater than his homologous in the baseline flow in the near wall region y < 5 mm (compare left and right diagram of Fig. 3). Nevertheless the term (Dt )T x is far lower than the deformation work operated by Reynolds normal stress (Dn )T x even in presence of VGs. This latter term results reduced by the presence of VGs with respect to the baseline case since separation is postponed. The presence of VGs tends also to modify substantially the shape of (Dn )T y . In fact the VGs induce near to the wall a negative average normal velocity [6, 7], with flow moving towards the wall. In this case the term (Dn )T y becomes positive, and the energy dissipation is increased near the wall, on the contrary above y = 18 mm the term (Dn )T y returns negative as the mean v component follows again the distribution imposed by the channel divergence. This consideration results in agreement with what observed in [7]: the flow in presence of VGs is accelerated at the wall by y advection of high momentum fluid towards the wall, which mechanism requires energy as (Dn )T y is positive for y < 18 mm. This peculiar aspect, which is one of the main positive effects of VGs in controlling separation, may be more carefully analyzed since the presence of VGs induces a 3D flow. In fact VGs are generating a vortex that around its core transports fluid creating downwash (as presented above) but also upwash effects. As it will be shown on the following section the process of momentum transport and turbulent mixing introduced in this way modifies considerably the mean flow characteristic and may not be neglected in the analysis of the controlled flow.

3.2 Three-Dimensional Effect of VGs The three-dimensional effects of the vortex generated by the VGs on velocity and rms(u ) are shown in Fig. 4. Considering at first the contour plot of the mean streamwise velocity component (Fig. 4 to the left) it may be observed that VGs introduce a relevant modification on the flow field. The trace of the generated vortex, which center may be identified at around y = 18 mm and z = 25 mm, is clearly visible in the cross-stream plane y–z. Near this core low velocities are observed as a trace of the VG’s wake. Nevertheless, in the upward motion generated to the right with respect to the vortex center, the vortex rotation is transporting momentum to low energy portion of flow leading to an almost separated boundary layer in a small and

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Fig. 4 Average mainstream velocity (left) and rms of velocity contour plots: VGs effect in the cross stream plane at x = 350 mm

confined region close to the wall at z = 40 mm. The disposition of the VGs adopted is in fact generating vortex rotating counter clockwise. The region of downwash, near the symmetry axis, is instead presenting for all his extension higher streamwise velocity. From z equal to 0 till 20 mm the flow is in fact highly accelerated even close to the vortex center position while from 20 to 50 mm a low momentum region is visible near the center in an asymmetric disposition. The peculiar shape of the vortex is even more clearly visible looking at the rms of velocity (Fig. 4 to the right). The rms(u ) present higher values in the right side with respect to the vortex center. High values of the rms(u ) may be found in the downward motion in particular in the region of space between y equal to 20 till 40 mm and z equal to 10 till 40 mm. In this area, momentum is subtracted to high energy fluid and moved down by y advection but consequently also z advection. Closer to the wall between z equal to 5 till 20 mm this high momentum flow is transferred near the wall. Low fluctuations of velocity are instead observable all along the upward part of the vortex motion and in the vortex center; momentum is only in small part transferred in this area leading to small rms(u ). This process is then acting in the way to preserve the global turbulence intensity level inside the boundary layer. The processes of turbulent mixing and the y and z advection momentum transfer operated by the VGs induce high values of the Reynolds shear stress u v that are addressing the downward motion of the vortex in the same area of high values of rms(u ) (u v at z = 10 and 25 mm of Fig. 5). High Reynolds shear stress u v seems typical of flow controlled by vortex generation, as also found by [13] in which pulsed-jet vortex generators were applied. The stretching of the flow due to the control device is leading to the formation of high velocity gradients in both y and z directions where the y gradient of velocity induces the shear stress u v . The change in the slope of this quantity, observable in particular for the profile at z = 25 mm, suggest two opposite dynamic effects for the flow mixing: a positive ∂u v /∂y produces a local deceleration, while

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Fig. 5 Reynolds stress profiles at x = 350 mm; comparison of baseline (no VGs) and controlled cases for different transversal positions

for ∂u v /∂y < 0 the opposite, as suggested by a classical analysis of the Navier– Stokes equation. Away from the vortex center this high shear stress vanishes and the distribution of u v for the controlled flow (z = 0 and 70 mm of Fig. 5) tends to the shape of u v in the baseline flow (also shown in Fig. 5).

4 Conclusion A detailed experimental study on low profile vortex generators has been carried out in a turbulent boundary layer developing on a large-scale flat plate with a prescribed adverse pressure gradient. The boundary layer with and without VGs has been investigated by the use of LDV in the meridional and cross-stream planes. The modification of the mean flow applied by VGs have been discussed computing the terms of the global dissipation rate in the symmetry plane by means of LDV measurements. Mainly three terms, in a two-dimensional domain, resulted relevant for both controlled and baseline case: the shear turbulent dissipation terms acting in the mainstream direction and the normal turbulent dissipation term acting both in the mainstream and normal direction. In particular it has been shown that close to separation, for the uncontrolled case, the normal dissipation term acting along the main direction may not be neglected (as typically assumed for an attached boundary layer), besides it resulted the greater one. For the uncontrolled case the normal dissipation term acting in the normal direction is negative, due to the channel divergence, and consequently tends to redistribute mechanical energy from turbulence to the mean flow. The dissipation terms acting along the mainstream direction show that the energy dissipation rate for the controlled case is lower than the baseline case one. Only near the wall this trend is not respected since the controlled flow results fully attached and high velocity gradients are present. Furthermore one of the main mechanism by which VGs work, y advection of high velocity flow portion [7], tends to dissipate energy since the normal dissipation term acting in the normal direction results positive, and not negligible, near the wall. To complete this analysis the mean flow characteristics in a cross-stream plane just downstream of VGs trailing edge have been

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shown, the control device introduces in the separating flow high non-uniformities. In the symmetry axis of the test section the separation is avoided by streamwise momentum enhancement. Nevertheless very near to the wall, in the region of upwash, the controlled boundary layer results even poorer of near wall momentum with respect to the baseline one in a limited portion of the investigated cross stream plane. High velocity gradients and consequently high shear stresses characterize the flow downstream of VGs; this peculiar aspect may be detrimental and turbulence characteristics should be simulated properly for the design and prediction of flow control using VGs taking into account the resulting 3D flow. Particular care should be also taken to evaluate the performance of this control device, for this reason the authors are carrying on a loss investigation in the measurement planes here described and in downstream planes. Acknowledgements The authors gratefully acknowledge the financial support of the European Commission as part of the research project AIDA “Aggressive Intermediate Duct Aerodynamic for Competitive and Environmentally Friendly Jet Engines”, contract n° AST3-CT-2003-502836.

References 1. Gad-el-Hak, M., Bushnell, D.M.: Separation control: review. J. Fluids Eng. 113, 5–29 (1991) 2. Waithe, K.A.: Source term model for vortex generator vanes in a Navier–Stokes computer code. In: 42nd AIAA Aerospace Sciences Meeting and Exhibit. AIAA paper 2004-1236 (2004) 3. Wallin, F., Eriksson, L.-E.: Tuning-free body-force vortex generator model. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 9–12 January 2006. AIAA paper 2006-873 (2006) 4. Wendt, B.J.: Initial circulation and peak vorticity behavior of vortices shed from airfoil vortex generators. NASA-CR-2001-211144 (2001) 5. Lin, J.C.: Review of research on low-profile vortex generators to control boundary-layer separation. Prog. Aerosp. Sci. 38, 389–420 (2002) 6. Canepa, E., Lengani, D., Satta, F., Spano, E., Ubaldi, M., Zunino, P.: Boundary layer separation on a flat plate with adverse pressure gradients using vortex generators. ASME Paper GT-2006-90809 (2006) 7. Satta, F., Simoni, D., Ubaldi, M., Zunino, P., Bertini, F., Spano, E.: Velocity and turbulence measurements in a separating boundary layer with and without passive flow control. Proc. IMechE, Part A: J. Power Energy 221, 815–823 (2007) 8. Satta, F., Simoni, D., Ubaldi, M., Zunino, P.: Experimental difficulties in measuring separating boundary layers with the LDV technique. In: XVIII Symposium on Measuring Techniques in Turbomachinery Transonic and Supersonic Flow in Cascades and Turbomachines, Thessaloniki, Greece (2006) 9. Modarress, D., Tan, H., Nakayama, A.: Evaluation of signal processing techniques in laser anemometry. In: Fourth International Symposium on Application of Laser Anemometry to Fluid Dynamics, Lisbon (1988) 10. George, W.K.: Processing of random signals. In: Proceedings of Dynamic Flow Conference, pp. 757–800 (1978) 11. Tennekes, H., Lumley, J.L.: A First Course in Turbulence. MIT Press, Cambridge (1972) 12. Simpson, R.L.: Turbulent boundary layer separation. Annu. Rev. Fluid Mech. 21, 205–234 (1989) 13. Kostas, J., Foucaut, J.M., Stanislas, M.: The flow structure produced by pulsed-jet vortex generators in a turbulent boundary layer in an adverse pressure gradient. Flow Turbul. Combust. 78, 331–363 (2007)

Session 4: Boundary Layer Structure and Scaling

• On the Relationship Between Vortex Tubes and Sheets in Wall-Bounded Flows S. Pirozzoli • Spanwise Characteristics of Hairpin Packets in a Turbulent Boundary Layer Under a Strong Adverse Pressure Gradient S. Rahgozar and Y. Maciel • The Mesolayer and Reynolds Number Dependencies of Boundary Layer Turbulence W.K. George and M. Tutkun • A New Wall Function for Near Wall Mixing Length Models Based on a Universal Representation of Near Wall Turbulence M. Stanislas

On the Relationship Between Vortex Tubes and Sheets in Wall-Bounded Flows Sergio Pirozzoli

Abstract The statistical relationship between vortex tubes and vortex sheets in turbulent wall-bounded flow is analyzed by means of conditional averaging of DNS fields. The results support strong association between the two types of coherent structures, and indicate that vortex tubes are produced upon roll-up of vortex sheets (as in the hairpin vortex paradigm), or interact causing the ejection of near-wall vorticity, or generate sheets of streamwise vorticity through a rubbing effect caused by the no-slip condition.

1 Introduction Coherent eddy structures are believed to play a major role in the dynamics of turbulent flows. It is well known that zones of intense vorticity in isotropic turbulence have either tube- or sheet-like shape [18, 19]. In wall-bounded flows, coherent structures are regarded to be responsible for transport of low-momentum fluid and for Reynolds stress production, and are found to be associated with intense events, such as ejections and sweeps [20]. A wide body of experimental and numerical works has confirmed that boundary layers are indeed populated by tubular hairpin vortices inclined at a positive angle with respect to the wall [4], either alone or arranged in packets [1]. Such arrangement would explain the dominance of Q2 and Q4 events (i.e. Reynolds shear stress production), and the occurrence of streamwise-elongated streaks [10]. Shear layers in wall-bounded turbulent flows have received comparatively much less attention, with some notable exceptions. Johansson et al. [8] studied the evolution and dynamics of shear layers, and found that they are responsible for intense events in the near-wall region, and provide an important contribution to the turbulence production. Liu et al. [11] observed the presence of shear layers which proS. Pirozzoli Dipartimento di Meccanica e Aeronautica, Università La Sapienza, Via Eudossiana 16, 00184, Roma, Italy M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_17, © Springer Science+Business Media B.V. 2011

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trude into the downstream flow at an angle less than 45°, having relatively sharp boundaries, and being spatially associated with regions of large Reynolds stress. The relationship between vortex sheets and vortex tubes in channel flow was studied by Jiménez et al. [7]. Those authors observed that the viscous layer is dominated by intense three-dimensional shear layers whose prevalent vorticity component is spanwise, and suggested similarity with the behavior of nonlinear Tollmien– Schlichting waves in a two-dimensional channel, whereby vorticity is spontaneously ejected from the wall and then stretched by the mean flow into long thin shear layers. Orlandi and Jiménez [14] analyzed the formation of streaky velocity structures in the near wall region of turbulent boundary layers through a simplified two-dimensional computational model, and showed that compact streamwise vortices form naturally from general sheet-like vorticity distributions. Klewicki and Hirschi [9] performed a conditional analysis of boundary layer flow in the proximity of shear layers, and observed strong spatial correlation of the shear layer motions with clusters of spanwise vortices, identifying two types of patterns, whereby: (i) a clockwise vortex forms upon roll-up of a shear layer, inducing clock-wise vorticity at the wall (referred to as NUPL configuration); and (ii) an outer layer, counter-clock-wise vortex causes the ejection of a shear layer (PUNL configuration). The objective of the present paper is to shed additional light on the statistical relationship between vortex tubes and vortex sheets. For this purpose we interrogate a turbulent supersonic flat plate boundary layer database, and apply state-of-the-art eduction criteria to extract vortex sheets and tubes.

2 Statistical Analysis The direct numerical simulation (DNS) database of supersonic turbulent boundary layer in zero-pressure-gradient at M = 2, Reθ ≈ 1350 of Pirozzoli et al. [16] is used for the statistical analysis. Under the flow conditions of that study the turbulent Mach number never exceeds 0.3, and we therefore expect that the analysis to follow also applies to low-speed boundary layers. To fully resolve the vortical structures in the boundary layer, the grid spacings of the simulation (in terms of wall units) are x + = z+ = 4.10 in the streamwise and the spanwise direction, respectively, and y + = 0.71 for the first point off the wall. The data for statistical analysis have been collected in a small portion of the entire domain where the momentum thickness Reynolds number (Reθ ) varies between 1340 and 1370. Statistics have then been obtained by averaging in time (70 samples are considered, with spacing t + = 1.50), and in the streamwise and spanwise directions. Further details are reported in [16]. To identify vortex tubes and vortex sheets we introduce two “vorticity variables”, designed so as to reduce to the local vorticity modulus in the limit cases of idealized tube-like and sheet-like vorticity distributions. For vortex tubes, we extend the incompressible swirling strength criterion of Zhou et al. [21] by considering the deviatoric part of the velocity gradient tensor A∗ . The local swirling strength λci is then defined as the imaginary part of the complex conjugate eigenvalue pair of A∗ .

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It is a simple matter to show that, for solid-body rotation, λci = ω/2. We then introduce the vorticity-like variable ωt = 2λci ,

(1)

and define vortex tubes as those regions where ωt exceeds a physically relevant threshold value (ε). With regard to vortex sheets, we extend the algorithm originally proposed by Horiuti and Takagi [5], based on a (suitably selected) eigenvalue (λL ) of the tensor ∗ W + S ∗ W . For two-dimensional parallel flow it turns out that λ = L∗ij = Sik kj L j k ki ω2 /2. We therefore introduce the vorticity-like variable ωs = 2λL , (2) and define vortex sheets as those regions where ωs exceeds a physically relevant threshold (ε). A physically relevant value for ε is selected to be proportional to the local value of the r.m.s. vorticity, and accordingly we set ε(y) = αω (y), where α is a suitable nondimensional constant (α ≥ 1). For eduction purposes, the mean shear is subtracted out [16, 17], and the vorticity variables ωs and ωt are defined in terms of the fluctuating velocity field. Inclusion of the mean shear has the main effect of making the near-wall shear layers stronger, but the qualitative nature of the results does not change.

3 Conditional Expected Fields To extract statistically significant information from the DNS database we conditionally average the flow samples based on the basis of the occurrence of strong vortex tubes oriented along the coordinate axes. For that purpose, as done by Hutchins et al. [6], we define the signed tube strength in the generic i-direction by: (i) computing the local swirl strength from the reduced velocity gradient tensor in the plane normal to the i-direction (say ωt,i ); (ii) multiplying the absolute swirl strength by the sign of the i-th vorticity component, i.e. ω˜ t,i = ωt,i sign(ωi ). Cross-stream vortex cores are then identified by the condition |ω˜ t,z | ≥ αω (y), whereas streamwise cores are defined by the condition |ω˜ t,x | ≥ αω (y), where α = 2 is used for all the forthcoming results. Analogously, we define a local signed vortex sheet strength as follows ω˜ s,i = ωs sign(ωi ). The distributions of the average signed tube- and sheet-strength conditioned to the presence of a spanwise, clock-wise (i.e. ω˜ t,z < 0) core at several distances from the wall is reported in Fig. 1 in the x–y plane through the center of the distribution. At the nearest wall-normal location (y + = 16) the flow pattern consists of a shear layer fed by the wall vorticity, and having shallow slope with respect to the wall. The shear layer is observed to roll up, forming a vortex tube near the origin of the reference system, where the conditioning event is placed, in a fashion that very closely resembles the Kelvin–Helmholtz mechanism of vorticity collapse [2, 15]. A secondary roll-up is observed approximately 100 wall units downstream, suggesting

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Fig. 1 Conditional expected fields for cross-stream, clock-wise vortex cores. Flood: contours of + + . Solid lines indicate ; Lines: contours of signed vortex sheet strength ω˜ s,z signed swirl strength ω˜ t,z negative values, dashed lines indicate positive values

that multiple vortices can be produced by the same shear-layer event, as observed by Hambleton et al. [3]. Further away from the wall, the association between vortex core and generating shear layer (attached to the wall) is still obvious, even though looser, and the occurrence of a positively signed sheet of vorticity is more clearly observed near the wall. The observed conditional events are therefore presumably associated with local decrease of the skin friction. The same data are reported in

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Fig. 2 Conditional expected fields for cross-stream, counter-clock-wise vortex cores. Flood: con+ + . Solid lines ; Lines: contours of signed vortex sheet strength ω˜ s,z tours of signed swirl strength ω˜ t,z indicate negative values, dashed lines indicate positive values

Fig. 2, under the condition of occurrence of a spanwise counter-clock-wise, vortex core (i.e. ω˜ t,z > 0). The figure shows quite a different mechanism: contrary to the previous case, the core vorticity has opposite sign with respect to the one observed in the associated shear layer, and the Kelvin–Helmholtz mechanism is not active. It seems that in this case the vortex is impinging onto the near-wall shear layer, stimulating its ejection, and causing it to wrap around. Finally, in Fig. 3 we show

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Fig. 3 Conditional expected fields for stream-wise vortex cores (only clock-wise cores only are + ; Lines: contours of signed vortex sheet shown). Flood: contours of signed swirl strength ω˜ t,x + . Solid lines indicate negative values, dashed lines indicate positive values strength ω˜ s,x

the average fields obtained by conditioning on the occurrence of strong clock-wise swirl in the longitudinal direction (note that the conditional fields for counter-clockwise cores are exactly symmetrical). Again, we observe the roll-up of an oppositely signed vorticity sheet around the vortex core, thus indicating the occurrence of a mechanism of near-wall vorticity stripping. As explained by Orlandi [13], a vortex impinging on a wall causes the formation of oppositely-signed vorticity at the wall

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Fig. 4 Conditional expected fields associated with: (a) cross-stream, clock-wise vortex cores; (b) cross-stream, counter-clock-wise vortex cores; (c) stream-wise vortex cores. Light gray: iso-surfaces of sheet strength ωs ; Orange: iso-surfaces of vortex tubes strength ωt

due to the no-slip condition, which then rolls-up around the interacting vortex, giving rise (in a two-dimensional environment) to a vortex dipole. The formation of tertiary (clock-wise) vorticity at the left of the interacting vortex is also visible, as also observed by Orlandi [13]. To give a three-dimensional perception of the structure of the conditional educed structures, in Fig. 4 we report iso-surfaces of the tubes and sheets strength (note that the same threshold is used for both ωt and ωs ).

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The figure shows that, in the case of clock-wise, cross-stream cores, a pattern very similar to the canonical hairpin vortex occurs, whereby spanwise vorticity bends towards the backward direction, forming a shallow angle with respect to the horizontal. Figure 4(a) clearly shows that such pattern is associated with the presence of a three-dimensional shear layer ejecting from the wall, whose tip rolls-up to form the hairpin vortex. Secondary shear layers are also observed at the left and right side of the primary one, presumably associated with the induction of the legs of the hairpin. The case of counter-clock-wise, cross-stream cores, reported in Fig. 4(b), shows that the conditionally-averaged vortex structure is in this case an inverted hairpinlike vortex, stripping a shear layer from the bottom wall (the asymmetry being due to lack of statistical convergence of the results). The conditional average structure for streamwise cores is a quasi-longitudinal vortex (having shallow inclination with respect to the horizontal), surrounded by a vortex sheet.

4 Conclusions The analysis of the conditional average fields reported in the present paper suggests a strong correlation between tube-like and sheet-like vorticity distributions in a turbulent boundary layer. The conditional fields associated with spanwise, clock-wise vortex cores highlight a scenario that is roughly consistent with the hairpin vortex paradigm, and with the NUPL pattern [9]. On the basis of the present data, it seems that hairpin vortices are associated with the three-dimensional roll-up of near-wall shear layers. The results related to counter-clockwise, spanwise vortices suggest a scenario similar to the PUNL pattern of Klewicki and Hirschi [9], whereby the ejection of a near-wall shear layer is promoted by the presence of a nearby cylindrical vortex. However, our data indicate that the shear layer actually rolls-up around the interacting vortex, and we do not find substantial traces of an associated clockwise core, that could be evidence of the occurrence of a vortex ring, or a severely deformed omega-shaped vortex [12]. With regard to streamwise vortex cores, the typical flow pattern is very similar to the one observed by Orlandi [13], whereby vorticity is induced by a quasi-cylindrical vortex because of the wall no-slip condition, and subsequently rolls-up around the primary vortex. Contrary to the findings of Hutchins et al. [6] we do not observe any oppositely-signed vortex in the proximity of the generating one, which could be the trace of the other leg in the case of an hairpin vortex. Further efforts will be directed to improve the statistical convergence of the results, and to establish the association of the typical educed vortical structures with Reynolds stress and skin friction generation. Acknowledgements The support of the CASPUR computing consortium through a 2009 Standard HPC Grant is gratefully acknowledged.

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References 1. Adrian, R.J., Meinhart, C.D., Tomkins, C.D.: Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54 (2000) 2. Baker, G.R., Shelley, M.J.: On the connection between thin vortex layers and vortex sheets. J. Fluid Mech. 215, 161–194 (1990) 3. Hambleton, W.T., Hutchins, N., Marusic, I.: Simultaneous orthogonal-plane particle image velocimetry measurements in a turbulent boundary layer. J. Fluid Mech. 560, 53–64 (2006) 4. Head, M., Bandyopadhyay, P.: New aspects of turbulent boundary-layer structure. J. Fluid Mech. 107, 297–338 (1981) 5. Horiuti, K., Takagi, Y.: Identification method for vortex sheet structures in turbulent flows. Phys. Fluids 17, 121703 (2005) 6. Hutchins, N., Hambleton, W.T., Marusic, I.: Inclined cross-stream stereo particle image velocimetry measurements in turbulent boundary layers. J. Fluid Mech. 541, 21–54 (2005) 7. Jiménez, J., Moin, P., Moser, R., Keefe, L.: Ejection mechanisms in the sublayer of a turbulent channel. Phys. Fluids 31, 1311–1313 (1988) 8. Johannson, A.V., Alfredsson, P.H., Kim, J.: Evolution and dynamics of shear-layer structures in near-wall turbulence. J. Fluid Mech. 224, 579–599 (1991) 9. Klewicki, J.C., Hirschi, C.R.: Flow field properties local to near-wall shear layers in a low Reynolds number turbulent boundary layer. Phys. Fluids 16, 4163–4176 (2004) 10. Kline, S.J., Reynolds, W.C., Schraub, W.C., Runstadler, F.A.: The structure of turbulent boundary layers. J. Fluid Mech. 30, 741–773 (1967) 11. Liu, Z.C., Landreth, C.C., Adrian, R.J., Hanratty, T.J.: High resolution measurement of turbulent structure in a channel with particle image velocimetry. Exp. Fluids 10, 301–312 (1991) 12. Natrajan, V.K., Wu, Y., Christensen, K.T.: Spatial signatures of retrograde spanwise vortices in wall turbulence. J. Fluid Mech. 574, 155–167 (2007) 13. Orlandi, P.: Vortex dipole rebound from a wall. Phys. Fluids A 2, 1429–1436 (1990) 14. Orlandi, P., Jiménez, J.: On the generation of turbulent wall friction. Phys. Fluids 6, 634–641 (1994) 15. Passot, T., Politano, H., Sulem, P.L., Angilella, J.R., Meneguzzi, M.: Instability of strained vortex layers and vortex tube formation in homogeneous turbulence. J. Fluid Mech. 282, 313– 338 (1995) 16. Pirozzoli, S., Bernardini, M., Grasso, F.: Characterization of coherent vortical structures in a supersonic turbulent boundary layer. J. Fluid Mech. 613, 205–231 (2008) 17. Robinson, S.K.: Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639 (1991) 18. Ruetsch, G.R., Maxey, M.R.: The evolution of small-scale structures in homogeneous isotropic turbulence. Phys. Fluids A 4, 2747–2760 (1992) 19. She, Z.S., Jackson, E., Orszag, S.A.: Intermittent vortex structures in homogeneous isotropic turbulence. Nature 344, 226 (1990) 20. Wark, C.E., Nagib, H.M.: Experimental investigation of coherent structures in turbulent boundary layers. J. Fluid Mech. 230, 183–208 (1991) 21. Zhou, J., Adrian, R.J., Balachandar, S., Kendall, T.M.: Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353–396 (1999)

Spanwise Characteristics of Hairpin Packets in a Turbulent Boundary Layer Under a Strong Adverse Pressure Gradient S. Rahgozar and Y. Maciel

Abstract Hairpin structures in the outer region of a turbulent boundary layer subjected to a strong adverse pressure gradient have been studied using PIV with inclined spanwise planes. This work is a follow-up to the study of Maciel and Shafiei Mayam (iTi Conference on Turbulence III, Bertinoro, Italy, October 12–15, 2008) done in the same flow but with streamwise/wall-normal planes. In agreement with Maciel and Shafiei Mayam, it is found that the gross features of the hairpin vortices and packets in the outer region above 0.2δ do not differ significantly from those found in a zero-pressure-gradient turbulent boundary layer at a comparable Reynolds number. In both flows, two-legged hairpins although asymmetric occur more frequently than one-legged ones and the outer-scaled hairpin width distributions are similar with a mean width of 0.15δ. The results suggest however finer legs of higher swirl in the zero-pressure-gradient case. In accord with Maciel and Shafiei Mayam, the streamwise separation between hairpins is smaller in the present decelerated flow.

1 Introduction In canonical turbulent wall flows, several studies support the existence of hairpin structures in the outer region and conclude on their important dynamical roles. Moreover, many of these studies have shown that the hairpins tend to occur in groups, usually termed packets, coherently aligned in the streamwise direction. The readers are referred to Adrian [1] for a recent comprehensive review on hairpin structures and packets in wall turbulence. In contrast, not much is known about these structures in adverse-pressure-gradient turbulent boundary layers (APG TBL). S. Rahgozar () · Y. Maciel Dept of Mech Eng, Laval University, Quebec City, Canada e-mail: [email protected] Y. Maciel e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_18, © Springer Science+Business Media B.V. 2011

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Fig. 1 (a, b) Respectively top view and side view of the modified test section of the boundary-layer wind tunnel. Dimensions in m. (c) Pressure coefficient distribution along the floor of the test section: ++ Measurement, −− NACA 2412 airfoil (2.5 m, 18°), — Potential flow calculation

DNS studies of TBL with separation bubbles, e.g. [2], have provided some insight into some of the turbulent structures found in these flows. But the focus of these studies was not on characterizing coherent structures and the limited information pertained mostly to near-wall structures. By analyzing the DNS data of Na and Moin [2], Chong et al. [3] suggest that in the APG zone of the flow prior to detachment there are proportionally more eddies which are detached from the wall than in zero-pressure-gradient (ZPG) TBL, where wall-attached eddies are more frequent. Maciel and Shafiei Mayam [4], hereafter referred to as MSM, recently studied the hairpin structures and packets in the outer region of an APG TBL via PIV measurements in xy-planes. The particular flow case studied was a turbulent boundary layer under external flow conditions similar to those found on the suction side of airfoils in trailing-edge stall conditions (Fig. 1). Even if the flow was very different from ZPG TBL, MSM found that the following features of the hairpin vortices and hairpin packets remained similar for y > 0.2δ, even as separation was approached: population density and diameter of hairpin heads. The upper arch part of the hairpin vortices was however slightly more inclined with respect to the wall, and the streamwise separation of the heads was smaller when scaled with the boundary layer thickness. The growth angle of the hairpin packets in the streamwise direction was also found to be more important. All these differences were found to be consistent with the variations of the mean strain rates, in particular rates of streamwise contraction and wall-normal extension. The objective of the present study is to further improve our understanding of hairpins and hairpin packets by analyzing their spanwise characteristics in the same flow as MSM. For the moment, one streamwise location has been studied with the help of inclined planes and comparisons have been made with the xy-plane data of MSM and with inclined plane data of ZPG TBL obtained by Hutchins et al. [5].

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Fig. 2 Position of measurement planes in the test section: rectangles are the xy-planes of MSM, the inclined line is the present 160° inclined plane

Fig. 3 Side view schematic of inclined PIV planes with an idealized hairpin packet representation. (a) Present 160° inclined plane. (b) 135° inclined plane used by Hutchins et al. [5] in ZPG TBL

2 Experimental Procedure As seen in Fig. 1, the turbulent boundary layer studied undergoes an abrupt transition from very strong favorable pressure gradient to very strong APG, leading to a large separation zone. The statistical properties of this flow and details of the experimental set-up and PIV system can be found in Ref. [6]. In the present experiment, 5000 instantaneous velocity fields are acquired by two-component PIV in a spanwise plane inclined in the streamwise direction (inclined at 160° to the streamwise axis), and located at the same streamwise position as the first xy-plane of MSM (Fig. 2). MSM found that the upper neck of the hairpins (or arch region) has an average inclination of ≈70° with respect to the wall. The 160° inclination was therefore chosen in order to reveal sectional cuts of the hairpin vortices in the arch region and the low-momentum regions at the center of these arches. Figure 3a presents a schematic representation of the inclined plane and of an idealized hairpin packet. The dimensions of the measurement plane are 2.6δ in the x direction and 1.1δ in the spanwise z direction (Fig. 2). As was done by MSM, measurements were made simultaneously with two cameras resulting in two overlapping large planes in order to increase the spatial resolution. The spatial resolution of the inclined plane and of the three xy-planes of MSM is identical in outer units, 0.018δ, at the plane’s central x-position. Note that the boundary layer thickness δ grows by more than 20% along the streamwise extent of the plane in this strong APG flow. Since an overlap of 50% of the interrogation windows was used to compute the velocity vectors, the vector spacing is 0.009δ. Table 1 presents the interrogation window width as well as boundary layer parameters for the central positions of the present measurement plane and the xy-planes of MSM. The last station is very close to the position where Cf = 0, which is x = 1615 mm.

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Table 1 PIV interrogation window width (z = x ) and boundary layer parameters for the reference streamwise positions. ZPG TBL data from Ref. [5] for 135° inclined PIV plane. x is the in-plane coordinate aligned in the streamwise direction: x = x135 , x160 and x in ZPG, present experiment and MSM respectively x (mm)

x /δ

x +

δ+

Reθ

H

uτ /Ue

UZS /Ue 0.17

ZPG TBL

–

0.025

69.0

2800

7440

–

0.0352

Present exp.

1156

0.018

13.2

765.9

5329

2.03

0.0250

0.350

MSM xy-plane

1392

0.018

15.2

836.7

8638

2.86

0.0159

0.501

MSM xy-plane

1600

0.018

–

≈0

12095

3.65

≈0

0.535

A coherent structure identification technique based on the low-momentum regions and the swirling rate of vortices is used to infer the presence of hairpins and hairpin packets. First, vortices are detected with an automated procedure based on the out-of-plane component of the swirling rate, λ. The swirling rate is defined as the imaginary part of the eigenvalue of the local velocity gradient tensor [7]. Contrary to the vorticity, the swirling rate does not identify regions of intense shear that have no rotation. Since the swirling rate does not indicate the sense of rotation, it is signed according to the sign of the corresponding vorticity component, ωy160 . To remove noise from the instantaneous swirling rate fields and to identify the boundaries of individual vortices, a threshold on λ has to be used. Wu and Christensen [8] showed that the root-mean-square value of the swirling rate, λrms , is a representative scale of non-zero λ. They found that a threshold of |λ/λrms | ≥ 1.5 defined well the boundaries of the vortex cores while minimizing experimental noise. In our case, the best threshold value was found to be 1.45. Because this criteria does not work well outside the boundary layer where vortices seldom occur, the threshold |λ/λmax | ≥ 0.1 is also used, where λmax is the maximum of λ in the instantaneous field. Finally, only clusters with at least three contiguous points with non-zero λ were considered to form a vortex core. Figure 4 shows an example of the vortices detected via this procedure. Once the vortices have been detected, one needs to discriminate those that may be hairpin legs from the others. To reduce the level of ambiguity, only hairpin-leg candidates within a packet consisting of at least three one-legged and/or two-legged hairpins are considered. The first step in the identification of the hairpins is to inspect the zones with negative u160 since the hairpin legs are expected to be close to the edges of these zones (Fig. 4). Indeed long zones of low streamwise momentum (u < 0) are known to exist within the packets and each hairpin generates a strong Q2 region (u < 0, v > 0) within its upper arch part [1]. As can be deduced from Fig. 3a, a long zone of negative u160 can represent a combination of low streamwise momentum and Q2 events within a packet. If the vortices are signed properly (λ < 0 on starboard side of the low momentum zone and λ > 0 on the port side), then they are hairpin-leg candidates. The second step is to verify that the identified hairpinleg candidates within a group are convected at about the same velocity (u160 within ±20%). At the moment, 5 instantaneous velocity fields have been analyzed with the aforementioned procedure leading to the detection of 49 packets and 291 hairpins.

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Fig. 4 Example of instantaneous flow field in 160° inclined plane in region x = 1.127–1.181 m. (a) Whole field with contours of out-of-plane swirling rate λ and grey shading shows low-momentum regions, u160 < 0. (b) Enlarged view of the packet in the boxed region in (a) with contours of u160 . Contour line: u160 = 0. Circles identify hairpin vortex legs

Hairpin and packet parameters discussed in the next section were then computed. Numerical values of the parameters have to be considered with caution since the detection procedure and the limited spatial resolution can introduce bias errors. We therefore only focus our attention in comparing hairpin and packet parameters obtained by the exact same procedure but in different flow situations. Five inclined-plane instantaneous velocity fields at the highest Re case of the ZPG TBL database of Hutchins et al. [5] were analyzed in the same manner in order to ensure an equivalent basis of comparison. The two sets of measurements are not however completely equivalent. Hutchins et al. [5] used three-component PIV with planes inclined at 135° (see Fig. 3b). The dimensions of the planes are 0.8δ in the x direction and 1.45δ in the z direction. Moreover, their outer scaled interrogation window width is 0.025δ, which means less spatially resolved fields than in the present study. These differences are taken into account when analyzing and comparing the results. As in the present study, Hutchins et al. [5] used 50% overlap of the interrogation windows to compute the velocity vectors. The interrogation

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window width and boundary layer parameters for the database of Hutchins et al. [5] are given in Table 1.

3 Results and Discussion When comparing the inclined-plane results obtained in the present flow and in ZPG TBL, it is found that the gross features of the hairpin vortices and hairpin packets remain similar in the region y > 0.2δ, despite the presence of a very different pressure environment. MSM arrived at the same conclusion when they compared xy-plane results in the present flow with those computed from the database of Adrian et al. [9] in ZPG TBL. Although the different inclinations of the planes (Fig. 3) lead to different in-plane streamwise extent of the low momentum regions, the overall shape of these regions, their spatial distribution and the distribution of vortices at their edges are comparable. An average of 6 hairpins per packet is found in both flows (range of 3 to 12). A smaller number for the ZPG TBL case was expected because of the higher inclination of the plane with respect to the wall, meaning that more hairpins within a packet might be missed in this case (see Fig. 3). Two factors may explain this result. Firstly, near-wall packets are sometimes found to lie underneath larger packets [4, 9]. Secondly, hairpins of two side-by-side packets sometimes connect or merge [1]. In both cases, these neighboring packets are counted as a single packet. Such events seem to occur more frequently in the ZPG case, hence the stronger bias towards more hairpins per packet in the ZPG case. Note also that MSM found an average of 4 hairpins per packet in the xy-planes for both the present flow and ZPG TBL. But an underestimated value was expected in this case (hairpin signature absent in the plane) because packets are not necessarily straight and perfectly aligned in the streamwise direction, as can be appreciated from Fig. 4. Some characteristic features of the hairpins are now discussed for the upper region y > 0.2δ. The limit y > 0.2δ was chosen based on the fact that the focus, like in MSM, is on the outer region of the boundary layer. The near-wall region behaves very differently in both flows and the spatial resolution of the measurements does not permit a detailed study of that region. The proportion of two-legged hairpins is found to be 70% in both flows. Although the two-legged hairpins are dominant, the hairpins are generally asymmetric and not necessarily aligned in the streamwise direction. Figure 5a presents the probability density distributions of the hairpin width of the two-legged hairpins (distance between the legs’ centers) in outer scaling for the present flow and the ZPG TBL. The width distributions are remarkably similar in both flows with an average of about 0.15δ in both cases. Other hairpin parameters of interest are the effective diameter D and the swirl intensity λ¯ of the hairpin legs. The swirl intensity λ¯ was defined by MSM as the nodal arithmetic average of the swirling rate λ per vortex. Contrary to the other parameters presented, these two hairpin parameters have to be considered with caution as they are sensitive to both mesh and spatial resolutions (interrogation window width). However, MSM found that D is mostly sensitive to mesh resolution while

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Fig. 5 (a) Pdf of hairpin width. (b) Pdf of hairpin diameter. Both are in outer units for y > 0.2δ

Fig. 6 Hairpin parameters in outer units for the heads (xy planes) and the legs (inclined planes) for y > 0.2δ: (a) Average effective diameter as a function of mesh width. (b) Average of λ¯ as a function of interrogation window width. Data of all xy planes from MSM. ZPG TBL xy-plane data computed from database of Ref. [9]

λ¯ is mostly sensitive to spatial resolution. They computed the average of these parameters on six streamwise intervals for each xy-plane to show these sensitivities. Representative results expressed in outer units are plotted in Fig. 6, where UZS is the Zagarola–Smits outer velocity scale. The large variations of mesh width and interrogation window width when scaled with δ in the present flow are due to the large variations of δ itself within one measurement plane. When accounting for mesh and spatial resolution effects, MSM found that both the average diameter and the average swirl intensity of the hairpin heads remain constant in the present flow, within experimental uncertainty. The trend shown in Fig. 6a also indicates that the average diameter of the hairpin heads is comparable to its value in the ZPG case, while Fig. 6b reveals that the swirl intensity is much lower in the APG case. Figures 6a and 6b also show respectively average values of D and λ¯ computed from the inclined planes of the present study. Note that these averages are computed

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from realizations that cover the whole measurement plane for y > 0.2δ. The individual realizations of D and λ¯ are therefore not obtained for constant values of outer scaled mesh width and interrogation width. The average values are plotted in these figures at single values of outer scaled mesh width and interrogation width corresponding to those of the plane’s central streamwise position. Since the inclination of the hairpin legs can vary greatly, these average values are only rough estimates of the average effective diameter and average swirl intensity of the hairpin legs. It is also expected that in the case of the present flow, the use of a plane inclined at 160° leads to a slight overestimation of the diameter and slight underestimation of the swirl intensity since the normal to the plane is not aligned with the most probable hairpin inclination, which is known to be around 45° [1]. Moreover, as it is illustrated in Fig. 5b by comparing the pdf of hairpin head’s diameter obtained by MSM with the pdf of hairpin leg’s diameter, both at the same location of the present flow, it can be implied that a small fraction of the leg’s diameter pdf (small diameter extremity) was not captured because of the limitation in spatial resolution; thus, again a very small overestimation of the average diameter of the hairpin legs is expected. With these considerations in mind, and again taking into account mesh and spatial resolution effects, Fig. 6a suggests that in both flows the hairpin legs are finer than the heads, but with a larger difference for the ZPG case. As for the swirl intensity, Fig. 6b suggests that the legs have on average a swirl intensity comparable to, or maybe slightly smaller than, that of the heads in the present flow. In the ZPG case, the legs are found to have a lower swirl intensity than the heads, but this result has to be interpreted with care because of the spatial resolution limitations. Turning our attention now to the features of the packets, let us first note that packets of hairpins are found throughout the boundary layer like in previous works. They are aligned on average with the streamwise direction but their deviation from that direction can be important and is comparable in both flows. MSM found that the average streamwise separation between consecutive hairpin heads is smaller in the present decelerated flow than in ZPG TBL. Since the hairpins are curved and vary greatly in shape and orientation, an assumption had to be used in order to compute the streamwise separation of the legs obtained from inclined planes with different inclinations. We assumed straight vortices inclined at 45° with respect to the wall in order to convert the in-plane streamwise separation into true streamwise separation in the x direction. The average streamwise separation of the legs is found to be 0.122 ± 0.009 for the present flow and 0.152 ± 0.015 for the ZPG TBL, where the random uncertainties are only those due to the limited number of samples. The trend observed by MSM in the case of the heads’ separation is therefore corroborated by these results. The velocity dispersion of hairpins within a packet was also found to be higher in the present flow (10% vs. 7%). Acknowledgements Financial support from NSERC and CFI of Canada is gratefully acknowledged by the authors. They also wish to thank N. Hutchins, W.T. Hambleton and I. Marusic for sharing their data, as well as M.H. Shafiei Mayam and D. Lopez for their help with the measurements and data processing.

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References 1. Adrian, R.J.: Hairpin vortex organization in wall turbulence. Phys. Fluids 19(4) (2007) 2. Na, Y., Moin, P.: Direct numerical simulation of a separated turbulent boundary layer. J. Fluid Mech. 374, 379–405 (1998) 3. Chong, M.S., Soria, J., Perry, A.E., Chacin, J., Cantwell, B.J., Na, Y.: Turbulence structures of wall-bounded shear flows found using DNS data. J. Fluid Mech. 357, 225–247 (1998) 4. Maciel, Y., Shafiei Mayam, M.H.: Hairpin structures in a turbulent boundary layer under stalled-airfoil-type flow conditions. In: iTi Conference on Turbulence III, Bertinoro, Italy, October 12–15, 2008 5. Hutchins, N., Hambleton, W.T., Marusic, I.: Inclined cross-stream stereo particle image velocimetry measurements in turbulent boundary layers. J. Fluid Mech. 541, 21–54 (2005) 6. Maciel, Y., Rossignol, K.S., Lemay, J.: A study of a turbulent boundary layer in stalled-airfoiltype flow conditions. Exp. Fluids 41, 573–590 (2006) 7. Zhou, J., Adrian, R.J., Balachandar, S., Kendall, T.M.: Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353–396 (1999) 8. Wu, Y., Christensen, K.T.: Population trends of spanwise vortices in wall turbulence. J. Fluid Mech. 568, 55–76 (2006) 9. Adrian, R.J., Meinhart, C.D., Tomkins, C.D.: Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54 (2000)

The Mesolayer and Reynolds Number Dependencies of Boundary Layer Turbulence William K. George and Murat Tutkun

Abstract Spectral measurements from the WALLTURB Lille experiment in a flat plate turbulent boundary layer at Rθ = 19,100 are used to evaluate the role of viscosity on the turbulence in different parts of the boundary layer. The measurements support the idea of a mesolayer from 30 ≤ y + ≤ 300 in which viscosity affects all scales of motion. An approximately k−1 range emerges near the outer part of the mesolayer and over inertial sublayer (220 ≤ y + ≤ 890). But in spite of the relatively high value of Rθ , the spectra in neither the inertial sublayer (300 ≤ y + ≤ 0.1δ0.99 ) nor the main part of the boundary layer show a true inertial subrange behavior; i.e., + a k−5/3 spectrum. The spectra outside of y + ≈ 0.12δ99 were shown to be consistent −5/3+μ with a k behavior where μ > 0 and decreases as the inverse of the logarithm of the Reynolds number. An immediate consequence is that the asymptotic state of the boundary layer can be reached only at very high values of Rθ , probably near 105 .

1 Historical Context In the context of turbulent boundary layers near walls, the term mesolayer was originally introduced by Long and Chen [5]. Their theoretical argument was based on the idea that there existed a second boundary layer very near the wall, beneath the ‘log’ layer but outside the buffer layer, whose properties scaled with Reynolds number to the 1/2-power. They found evidence for it by scrutinizing all of the available data at that time. Although their arguments have been largely disregarded (but not refuted) by the community, some at various times have expressed at least moral support for the idea (cf., [7]). W.K. George () Chalmers University of Technology, Dept. of Applied Mechanics, 41296 Gothenburg, Sweden e-mail: [email protected] M. Tutkun Norwegian Defence Research Establishment (FFI), P.O. Box 25, 2027 Kjeller, Norway e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_19, © Springer Science+Business Media B.V. 2011

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An alternative view that perhaps our thinking about this same region of the flow was lacking something important was put forth by George and Castillo [3] (hereafter referred to as GC97). Although they adopted the same term ‘mesolayer’ to describe the region beginning at y + = 30 and ending approximately at y + = 300 or so, their arguments were based not on the mean velocity, but on the dynamics of the turbulence in this region. Before reviewing them, it is useful to summarize the prevailing views at the time, especially since they are still held by many. The boundary layer was generally believed to consist of only four regions: the linear region (y + < 5) in which viscous stresses dominate; the buffer layer (5 ≤ y + ≤ 30), at the outer edge of which the viscous stresses eventually are overwhelmed by the Reynolds stresses; + the inertial (or ‘log’ layer, 30 ≤ y + ≤ 0.1δ0.99 approximately), where the Reynolds shear stress is approximately constant but mean convection terms are still negligible, and the outer boundary layer (y/δ0.99 ≥ 0.1) where the mean convection terms are balanced by the Reynolds shear stress gradient. So entrenched were these ideas and related theoretical arguments up until the 1990s that it was widely accepted that measurements of wall shear stress were unnecessary. The shear stress could simply be evaluated using measurements of the mean velocity at y + = 100, the log velocity profile and the accepted value of the ‘universal’ von Kármán constant (usually taken as κ = 0.41). Moreover, it was widely believed that the ‘universal constants’ could be determined from experiments (or DNS) where Rθ was low as a few thousand or less. GC97 challenged this traditional view on a number of fronts, only a few of which are of interest here. First they argued that there is indeed something different going on above the buffer layer and below the inertial sublayer; namely that the Reynolds stress producing (or energetic) turbulence fluctuations themselves are still directly influenced by viscosity, even though the mean flow is not. In the vocabulary of usual turbulence theory: the dissipative scales and energetic scales of the turbulence overlap, so all scales are more or less influenced by viscosity. GC97 suggested that this layer existed between approximately 30 ≤ y + ≤ 300, at least as long as the mean convection effects could be ignored in this region (i.e., as long as y/δ0.99 0.1). + Thus only when δ0.99 > 3,000 could the beginning of a true inertial layer be ex+ pected, approximately bounded by 300 ≤ y + ≤ 0.1δ0.99 . Second, GC97 also argued, following a suggestion of Long and Chen [5], that the inertial layer itself should have a residual and asymptotically vanishing dependence on Reynolds number. Although the analysis required (Near-Asymptotics) is quite involved, the physical reason is quite simple: the overlap region between a region dominated by viscosity and another region dominated by inertia would not be an inertial region, but surely one containing a mixture of inertial and viscous effects; i.e., at finite Reynolds number it should show a weaker dependence on viscosity than the inner layer, but stronger than the outer. For both the pipe/channel flow [12] and the GC97 boundary layer, all of the important parameters (e.g., the von Kármán ‘constant’, the power and corresponding coefficients) retained a dependence on 1/ ln δ + (1/ ln R + in the case of pipe/channel), and were only asymptotically constant. It was further suggested that these asymptotic values began to be approached slowly only when both conditions for an inertial sublayer could be said to unequivocally be

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Fig. 1 Sketch from GC97 showing various layers in turbulent boundary layer in inner (left) and outer (right) variables

satisfied; i.e., when both y + > 300 and y/δ0.99 < 0.1 or equivalently δ + 3,000. This roughly corresponds to Rθ 10, 000. Moreover the parameters could become only truly constant for much higher values of δ + (or Rθ ) than previously believed. Figure 1 taken from the GC97 paper shows their view of the boundary layer. Support for both their arguments can be found in Fig. 2, which shows running integrals (from the WALLTURB experiment at Rθ = 19,100) of the energy spectra (counting down) and the dissipation spectra (counting up) for three positions in the boundary layer. At y + = 50, a significant fraction of the dissipation has already occurred over the range for which there is still a considerable amount of the energy left. The overlap is reduced by two-thirds or more by y + = 445; but is still not insignificant by

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. Fig. 2 Running integrals of one-dimensional energy and dissipation spectra from the Rθ = 19,100 k k + (δ0.99 = 7,250). Solid lines: 1 − (1/u2 ) 0 1 F11 (k) dk; Dashed lines: (15ν/ε) 0 1 k 2 F11 (k) dk + y + = 3618, which is half way to δ0.99 (i.e., in the main part of the boundary layer). Thus in no way can this data, in spite of the relatively high value of Rθ , be interpreted as support for the classical arguments which require a spectral gap between the energy (nearly dissipationless) range and the viscous (but nearly energy-less) dissipative range. A consequence of the mesolayer is that the inertial profile (be it log or power law) could not be reached until much larger values of y + than previously believed; namely outside of y + > 300. Although some still profess not to accept the GC97 mesolayer arguments, it is worth noting that almost no one now tries to fit a log law below y + of a few hundred. Also, although there has not be widespread support for the Near-Asymptotic analysis of GC97, virtually no one now cites values for the presumably constant von Kármán and other parameters obtained at Reynolds numbers below those suggested by GC97 and Wosnik et al. [12] of Rθ > 10,000 or so.

2 Spectra at Rθ = 19,100 The primary goal of this paper is to move the discussion beyond arguable interpretations of the mean velocity profile data to a discussion of the turbulence itself. The essence of the GC97 argument was that it was the absence of a spectral gap that necessitated the mesolayer, and delayed the inertial layer from reaching its asymptotic character. The arguments applied depended heavily on the classical interpretation of the scale separation between viscous and dissipative scales as set forth by K41 [4], and as summarized quite eloquently by Tennekes and Lumley [11] and many others over the years. Those consequences of most interest here are the region of constant spectral flux (i.e., the inertial subrange), the consequent k−5/3 -law (or corresponding regions for the velocity structure functions), and the inviscid relation for the dissipation, ε ∝ u3 /L where L is assumed to be proportional to the physical integral scale. These can be viewed as the infinite Reynolds number limit of turbulence, and form the basis for most of our attempts to model turbulence. Clearly any finite

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Reynolds number result should take into account the overlap of energetic and dissipative ranges like that noted above. A strong motivation to do so was provided by the grid turbulence experiments of Mydlarski and Warhaft [8], who showed the K41 behavior could at most only be expected at much higher Reynolds numbers than previously believed, if at all. In fact, the Near-Asymptotics methodology and results of Gamard and George [2] showed that it was possible to carry the K41 argument one step forward to examine how the infinite Reynolds number limits are approached. Unfortunately there have been only a very few experiments (cf., [1, 10]) which satisfied the conditions for existence of an inertial layer, and were also able to resolve enough information about the near wall region to evaluate the alleged effects on the turbulence. This was usually because the probes were too large relative to the viscous wall length and/or Kolmogorov microscales. In fact, the lower Reynolds number experiments and DNS barely satisfy the conditions for even a mesolayer to exist free from the mean convection effects of the outer flow, much less an inertial layer. As we shall see below WALLTURB has provided data to evaluate the GC97 hypotheses. Over almost the entire boundary layer, the measuring devices (whether hot-wires or PIV interrogation volumes) were at most a few Kolmogorov length scales, and always small enough to approximate (using local isotropy) the dissipation. 5/3 Figure 3 shows how the one-dimensional velocity spectra multiplied by k1 , 5/3 (1) k1 F1,1 (k1 ), varies across the boundary layer. There are several significant features that one might expect. First, if there were a region where the spectral flux were constant, there should be a level region on each plot. The spectra show nothing that resembles the expected classical inertial layer. Second, between 220 ≤ y + ≤ 890 there is an approximately k−1 region. This can be deduced from purely dimensional considerations and has been discussed extensively elsewhere (e.g., [6, 9]). Interestingly it only seems to dominate in a region corresponding to the approximately to the inertial sublayer. This is contrary to the expectations of GC97 who conjectured that this region would show a k−5/3 -range. Clearly it is the k−1 behavior which characterizes the inertial subrange. The mesolayer as postulated by GC97 is predicated upon the absence of spectral gap — meaning that the dissipative and energy-containing ranges of the spectra overlap so that viscosity affects all the scales of motion directly (even though the direct effect of viscosity on the mean motion is negligible). The justification for their argument, as long recognized by RANS modelers, is that the local turbulence Reynolds number is roughly proportional to distance from the wall. And it is only outside of y + ≈ 300 that the ratio of local integral scale (roughly proportional to y) becomes large enough compared to the Kolmogorov microscale, ηK = (ν 3 /)1/4 , for viscosity to only weakly affect the Reynolds stress. There are many other consequences of this. Turbulence structures in this part of the flow (or at least the part of structures passing through this region) should show a need for viscous scaling that is absent farther out. Easiest to see, however, are the consequences for spectra and structure functions. If there is no significant scale separation (and hence no spectral gap), then none of the conditions for the existence of a k−5/3 range in the

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Fig. 3 Premultiplied spectra at y + = 22, 50, 220, 445, 890 and 3,618. Dashed line and solid line denote k−5/3 and k−1 slopes, respectively

velocity spectra nor the r2/3 in the second-order structure function are satisfied. In fact, as noted above, there is not even a k−1 -range in the spectra over most of this region. The absence of these ranges in data clearly support the GC97 mesolayer argument. By the GC97 arguments, the true inertial sublayer (in y + ) would have to satisfy two conditions: First the effects of mean convection should be negligible throughout the layer; and second, there should be a scale separation between energetic and viscous scales so that the Reynolds-stress-producing scales are nearly inviscid. The first condition means that the Reynolds shear stress should be nearly constant, while the second was taken to imply that Kolmogorov inertial subrange arguments should apply (at least approximately). Interestingly, it is in this range that the k−1 -range appears. Only outside of y + = 890 do the spectra begin to show a limited range 5/3 (1) where k1 F1,1 (k) ≈ constant (with emphasis on the ‘approximate’), and the extent of this range appears to increase with increasing values of y + . All of the spectra, (1) in fact, show a more extensive range where k5/3−μ F1,1 (k) = constant where μ > 0.

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Fig. 4 Plot of 5/3 − μ versus Rλ for present data, grid turbulence and theory

Fig. 5 Spectra at y + = 3618 showing data, composite spectrum and overlapping high and low wavenumber spectral components. Slope of lines = −1.529

Figure 4 plots for each of these spectra the exponent, 5/3 − μ, which is required to achieve the best constant range as a function of the local value Rλ . Also shown on the plot are the values from [8] for grid turbulence and the finite Reynolds number theory of Gamard and George [2] in which the departures from the −5/3 exponent vary as the inverse of the logarithm of the Reynolds number. Clearly the same phenomena are in play; namely the consequences of the overlap between the dissipative and energy-containing ranges and its residual influence on what would otherwise be an inertial range. It also is clear from the logarithmic variation and the plot that the asymptotic −5/3-behavior will only be reached for much higher Reynolds numbers than the current experiment. Figure 5 shows a one-parameter fit of the composite spectrum, including the k−1 bump, of Gamard and George [2] to the spectrum at y + = 3618 for which μ = 0.138. The composite spectrum is obtained by multiplying the high and low wavenumber spectra shown together, and then dividing by the common part. The agreement between the theory and experiment is all the more remarkable since it was developed quite independently of the data (or for that matter, any boundary layers at all).

3 Summary and Conclusions Since the turbulence spectra continue to show a residual dependence on Reynolds number at Rθ ≈ 20,000, then it would seem to be quite unreasonable to expect any scaling law to behave any differently. In fact, this is precisely the source of the residual Reynolds number dependence proposed for the log profile parameters (for channels and pipes, [12]) and the power law parameters (for boundary layers, GC97). And it lends support to arguments that we must go to much higher values

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of Rθ (GC97 suggest 100,000) if we are to find truly asymptotic behavior. Clearly all of these observed Reynolds number dependencies must also have a direct effect on our attempts to model and simulate turbulence. Most importantly, however, it establishes unequivocally that even this relatively high Reynolds number boundary layer is far from reaching anything resembling an asymptotic limit. Acknowledgements This work has been performed under the WALLTURB project. WALLTURB (A European synergy for the assessment of wall turbulence) is funded by the CEC under the 6th framework program (CONTRACT No: AST4-CT-2005-516008).

References 1. Carlier, J., Stanislas, M.: Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry. J. Fluid Mech. 535, 143–188 (2005) 2. Gamard, S., George, W.K.: Reynolds number dependence of energy spectra in the overlap region of isotropic turbulence. Flow Turbul. Combust. 63, 443–477 (2000) 3. George, W.K., Castillo, L.: Zero-pressure-gradient turbulent boundary layer. Appl. Mech. Rev. 50(12), 689–729 (1997) 4. Kolmogorov, A.N.: Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 16–18 (1941) 5. Long, R.R., Chen, T.C.: Experimental evidence for the existence of the ‘mesolayer’ in turbulent systems. J. Fluid Mech. 105, 19–59 (1981) 6. McKeon, B., Morrison, J.: Asymptotic scaling in turbulent pipe flow. Philos. Trans. R. Soc. A 365(1852), 771–787 (2007) 7. McKeon, B.J., Sreenivasan, K.R.: Introduction: scaling and structure in high Reynolds number wall-bounded flows. Philos. Trans. R. Soc. A 365(1852), 635–646 (2007) 8. Mydlarski, L., Warhaft, Z.: On the onset of high-Reynolds-number grid-generated wind tunnel turbulence. J. Fluid Mech. 320, 331–368 (1996) 9. Nickels, T., Marusic, I., Hafez, S., Chong, M.: Evidence of the k−1 1 law in a high-Reynoldsnumber turbulent boundary layer. Phys. Rev. Lett. 95(7), 074501 (2005) 10. Priyadarshana, P.J.A., Klewicki, J.C.: Study of the motions contributing to the Reynolds stress in high and low Reynolds number turbulent boundary layers. Phys. Fluids 16(12), 4586–4600 (2004) 11. Tennekes, H., Lumley, J.L.: A First Course in Turbulence. MIT Press, Cambridge (1972) 12. Wosnik, M., Castillo, L., George, W.K.: A theory for turbulent pipe and channel flows. J. Fluid Mech. 421, 115–145 (2000)

A New Wall Function for Near Wall Mixing Length Models Based on a Universal Representation of Near Wall Turbulence Michel Stanislas

Abstract In a recent paper (Stanislas et al. in J. Fluid Mech. 602:327–382, 2008), the authors proposed a representation of the turbulent boundary layer characteristics based on the analysis of coherent vortices performed from stereo PIV. These results are briefly recalled and a new wall function model is proposed and tested in the case of the turbulent channel.

1 Introduction The universal representation of turbulent boundary layers is a problem addressed since its discovery by Prandtl and is still a subject of strong controversy in the concerned scientific community. It has long been recognized that two sets of parameters are needed to represent the mean velocity profile: (1) an inner set based on the fluid kinematic viscosity ν and the wall friction velocity uτ which is valid near the wall and (2) an outer set based on the free stream velocity Ue , the wall friction velocity uτ and the boundary layer thickness δ which works in the wake part of the boundary layer. Asymptotic expansion between these two regions leads to the well known log law which is also a vivid subject of discussion. In a recent paper, Stanislas et al. [9] have proposed a new set of parameters: the external velocity Ue and the local Kolmogorov length scale η which allow to represent in a single way, both the velocity profile and the turbulent intensity profiles over the whole boundary layer thickness and for a range of Reynolds number Reθ = going from 8,000 to 20,000. This proposal was based on a detailed analysis of the vortical structures embedded in the turbulent boundary layer (TBL) and on the fact that they appear to scale with the Kolmogorov scales. In the present contribution, the arguments of [9] and the proposed scaling are quickly reviewed. Some empirical laws are proposed to fit the universal profiles M. Stanislas () LML UMR CNRS 8107, Villeneuve d’Ascq, France e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_20, © Springer Science+Business Media B.V. 2011

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obtained and a wall function model is proposed, based on these empirical laws. This model is then tested in a channel flow, coupled to a standard mixing length model. The results are discussed and compared to experimental data and to those of the mixing length model without wall function.

2 Vortices Properties in the TBL The vortices properties in turbulent wall flows have been extensively studied through Direct Numerical Simulation by Tanahashi et al. [5, 8, 10]. These authors have found a good scaling of the size and intensity of the vortices with Kolmogorov scales. Recently, Stanislas et al. [9] performed a detailed study of the vortices characteristics in a high Reynolds number flat plate boundary layer. This study was performed on Stereo PIV data recorded by Kähler et al. [7]. Data were available in a plane normal to the wall and to the flow at two Reynolds numbers: Reθ = 7,800 and 15,000. The vortices were detected in the instantaneous PIV maps using the swirling strength criterion, based on the velocity gradient tensor and introduced by Adrian [1]. They were then identified and characterized by non linear fit of an Oseen vortex model, using a procedure very similar to that used by Tanahashi et al. Figure 1 reproduced from [9] gives an example of the probability density function of the vortices’ radii, scaled using the Kolmogorov length scale. The data are given for the two Reynolds numbers and compared to the data of [8]. The universality is evident and the agreement with the DNS data is good. The difference in the larger sizes is attributed to the fact that in the DNS all vortices were extracted while in the PIV, due to the orientation of the plane, only the streamwise vortices could be looked at. The conclusion of this study was that, besides the universality of the pdf, the vortices had the following constant mean characteristics through the BL: ro 6η and ωo 1.6τ . Where ro and ωo are respectively the radius and vorticity of the vortices and η and τ are respectively the Kolmogorov length and velocity scales.

3 Universal Representation Based on this scaling result of the vortices, [9] could revisit the physical interpretation of the TBL and came to the proposal of a new scaling, based on the external velocity Ue and the local Kolmogorov length scale η(y). The basis was that Ue is providing information on the kinetic energy available to feed the turbulence and η is the proper parameter to represent the dissipation of this turbulent kinetic energy (and not the dissipation ε itself). It appeared that this was true as long as η was at least three orders of magnitude smaller than δ the boundary layer thickness. A clipping had thus to be introduced in order to replace η by δ at a certain wall distance. This clipped scale was called η∗. Figures 2 and 3 respectively give the mean velocity and the turbulence intensities profiles for four Reynolds numbers scaled with Ue and η∗. As can be observed, in this range of Reynolds number, a good superposition

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Fig. 1 Pdf of vortex radius scaled with Kolmogorov length scale. Exp. [9] Req = 7,800, ♦ Req = 15,000, — DNS Ret = 1,270 [8] Fig. 2 Wall normal evolution of the mean velocity in the proposed representation

is observed for the different parameters. As the mean velocity is plotted in semilog, a logarithmic region is easily evidenced. A fit to this region is given in Fig. 2 with the following equation: U y y = 0.11 ln + 0.192 with 10 ≤ ≤ 600 (1) Ue η∗ η∗ To obtain such a representation, the Kolmogorov length scale must be estimated in the whole BL thickness. This was done by [9] from the time correlation signal of the streamwise velocity fluctuations measured with a hot wire anemometer.

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Fig. 3 Wall normal evolution of the three components of the turbulence intensity in the proposed representation

Fig. 4 Wall normal evolution of the Kolmogorov length scale in wall units

The Taylor microscale was estimated and an hypothesis of local homogeneity and isotropy of the small scales allowed to estimate η. This parameter is plotted in wall units in Fig. 4 for the four Reynolds numbers. A good universality is observed in this representation, although a slight Reynolds number effect can be evidenced, especially in the outer part. On the same figure, a fit to the highest Reynolds number data is given, with the following equation: 0.25 η+ = κy + + 0.48 with 50 ≤ η+ ≤ 2000 (2) where κ = 0.41 is the Von Karman constant and the 0.25 exponent can be deduced from the mixing length theory. Of interest is the thickness of the layer in which the flow variables scale with Ue and η. The outer limit y∗/δ of this region is given as a function of the Reynolds number in Fig. 5 and compared to the usual limit between the log and outer regions.

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Fig. 5 Evolution with Reynolds number of the region of universality of the Kolmogorov scaling compared to the inner region thickness

As can be seen, if the log/outer limit is relatively constant (of the order of 0.2δ), on the contrary, y∗ increases linearly with Reθ , indicating that above 40,000, η would be the right length scale for the whole BL thickness. This result and its implications are discussed in detail in [9]. The question which follows immediately from the above results is the following: Is it possible to represent the Kolmogorov length scale in external units? A simple similarity analysis leads to an equation of the following form: y U e.δ η =f , y δ ν A fit to the hot wire data of [2] leads to the following equation: −0.8 η U e.δ −0.6 y y = 1.6 with 0.01 ≤ ≤ 0.8 y δ ν δ

(3)

This equation is compared to the data used to define it in Fig. 6. As can be observed, the agreement is fairly good over two decades of y/δ. A slight departure is observed very near the wall, when η/y goes to order 1.

4 Wall Function Model Wall functions is a well known solution to limit the computational effort near a wall. It has been used extensively with RANS models [4, 6]. It is now looked at for LES approaches. Looking at Eqs. 1–3, it appears that it is possible to build a wall function model to be coupled with a mixing length model in the near wall region. Knowing Ue and δ, Eq. 3 allows to compute η for any y in the corresponding range. Knowing y and η, Eq. 2 can be solved iteratively to find uτ (and thus τw ) and Eq. 1 gives directly U . The advantage of the proposed model being that the range of validity of Eqs. 1–3 should allow some freedom on the value of y at which the wall function is coupled to the turbulence model.

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Fig. 6 Evolution of the Kolmogorov length scale with wall distance in outer variables representation

To test the feasibility of such a model, an application was performed in the case of a fully developed turbulent plane channel flow. As is well known, this flow shares a lot of similarity with the turbulent boundary layer, at least in the inner part. To perform the analogy, the boundary layer thickness δ was taken as H the channel half width. The mixing length used was a standard one, coupled with the Van Driest damping function: y 26.ν with A = (4) lm = lo 1 − exp A uτ y 2 y 4 lo = 0.12 − 0.06 1 − 2 − 0.06 1 − 2 (5) H H The momentum equation writes in that case: ∂ 1 ∂P ∂U (ν + νt ) = (6) ∂y ∂y ρ ∂x where νt is the eddy viscosity given by the mixing length model: 2 ∂U ∂U (7) νt = lm ∂y ∂y This set of equations was solved by a finite volume method on a non uniform grid varying exponentially with wall distance.

5 Channel Flow Validation In order to test the proposed model, plane channel flow experiments at high Reynolds number were looked for. There are several experiments available in the

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Table 1 Main characteristics of the channel flow experiment of Comte-Bellot [3] Uq (m/s)

Req

Umax (m/s)

Remax

uτ (m/s)

Reτ

1 ∂P ρ ∂x

9.1

109,200

10.5

126,000

0.39

4,680

−1.78

19.43

233,400

22.2

266,000

0.8

9,600

−7.11

36.83

442,000

42.8

514,000

1.36

16,320

−20.55

Fig. 7 Prediction of the Comte-Bellot [3] channel flow with a standard mixing length model

literature which will not be reviewed here. The highest Reynolds number available is the experiment of Comte-Bellot [3]. The main characteristics of this experiment are given in Table 1. The channel is 0.18 × 2.4 m2 in cross section and 12 m long. The fluid used was air at room temperature giving a kinematic viscosity ν = 1.510−5 SI. Uq is the flow rate velocity, Umax the maximum velocity and uτ the friction velocity which is the same on both walls. The pressure gradient is computed from the friction velocity. Given the channel height, fluid viscosity and pressure gradient, Eqs. 4–6 can be solved on a 1D grid. Figure 7 gives the comparison between the prediction and the experimental results for the three values of the Reynolds number. The agreement is fairly good at the lowest Reynolds number and decreases slightly when the Reynolds number increases. In Fig. 8, the comparison is between the wall function proposed (Eqs. 1–3), coupled to the above mixing length model at y + = 100, and the experimental data. First, it can be observed that the coupling is working, allowing retrieval of the global shape of the velocity profile at the three Reynolds numbers. The second observation is that the agreement is improving when the Reynolds number increases. This is not surprising as Eq. 2 has been fitted to the highest Reynolds number data (there is a slight Reynolds number effect on η+ ). Also, in Eq. 1, η∗ was taken as η which is less and less valid when the Reynolds number decreases. In order to test the sensitivity of the wall function model to the position of the coupling point, this position was varied from 100 to 300 wu for the highest Reynolds number. The result is given in Fig. 9, compared to the experimental data. As can

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Fig. 8 Prediction of the [3] channel flow with a standard mixing length model coupled to the proposed wall function model at y + = 100

Fig. 9 Prediction of the [3] channel flow at Re = 514,000, with the proposed wall function model coupled at different wall distances

be observed, the model is relatively insensitive to this parameter at high Reynolds number. This is not true of course at low Re.

6 Conclusion The analysis of the vortical structures in a fully developed flat plate turbulent boundary layer at high Reynolds number has led Stanislas et al. [9] to propose a new scaling of the flow parameters working on the whole BL thickness. Based on this scaling, empirical laws describing the mean velocity and the Kolmogorov length scale were derived and proposed as a new set of wall functions to be coupled to a mixing length model. In a first step, this coupling was performed on a turbulent channel flow and compared to the mixing length model alone and to the experimental data. The results show that the coupling works, that it is better at high Reynolds number and that it is relatively insensitive to the wall distance of the coupling point (thanks to the range of validity of the empirical laws used). Of course, this is just a preliminary attempt. The main interest of wall functions is that they allow to save grid points (and computer time) in the near wall region. It is clear that the mixing length model is nowadays of limited interest but, the proposed wall functions, together with the estimation of η at the coupling point, also provide the value of ε. As

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the three turbulence intensities are also universal in the proposed representation, it will not be difficult to propose an empirical function for the turbulent kinetic energy, thus also providing the coupling condition for this parameter. As can be seen, the proposed model would then provide clean coupling conditions for a k–ε model. It will then be of interest to test it on the prediction of a turbulent boundary layer. An other point to be addressed is the effect of the pressure gradient. The proposed model has been established from data obtained in a flat plate TBL. It is clear that its validity in BL submitted to favorable or adverse pressure gradient has to be assessed. Acknowledgements This work has been performed under the WALLTURB project. WALLTURB (A European synergy for the assessment of wall turbulence) is funded by the CEC under the 6th framework program (CONTRACT No: AST4-CT-2005-516008).

References 1. Adrian, R.J., Christensen, K.T., Liu, Z.C.: Analysis and interpretation of turbulent velocity fields. Exp. Fluids 29, 275–290 (2000) 2. Carlier, J., Stanislas, M.: Experimental study of eddy structures in a turbulent boundary layer using PIV. J. Fluid Mech. 535, 143–188 (2005) 3. Comte-Bellot, G.: Ecoulement turbulent entre deux parois paralléles. Publ. Sci. Minist. Air, vol. 419 (1965) 4. Craft, T., Gerasimov, A., Iacovides, H., Launder, B.: Progress in the generalization of wall function treatments. Int. J. Heat Fluid Flow 23, 148–160 (2002) 5. Das, S.K., Tanahashi, M., Shoji, K., Miyauchi, T.: Statistical properties of coherent fine eddies in wall-bounded turbulent flows by direct numerical simulation. Theor. Comput. Fluid Dyn. 20, 55–71 (2006) 6. Goncalves, E., Houdeville, R.: Reassessment of the wall functions approach for RANS computations. Aerosp. Sci. Technol. 5(1), 1–14 (2001) 7. Kähler, C., Stanislas, M., Dewhirst, T.P., Carlier, J.: Investigation of the spatio-temporal flow structure in the log-law region of a turbulent boundary layer by means of multi-plane stereo particle image velocimetry. Selected paper presented at the 10th Int. Symp. on Appl. of Laser Techn. to Fluid Mech., Lisbon (Portugal). Springer, Berlin (2000) 8. Kang, S.J., Tanahashi, M., Miyauchi, T.: Dynamics of fine scale eddy clusters in turbulent channel flows. In: Fourth International Symposium on Turbulence and Shear Flow Phenomena, Williamsburg, VA, USA, pp. 183–188 (2005) 9. Stanislas, M., Perret, L., Foucaut, J.: Vortical structures in the turbulent boundary layer: a possible route to a universal representation. J. Fluid Mech. 602, 327–382 (2008) 10. Tanahashi, M., Kang, S.J., Miyamoto, T., Shiokawa, S., Miyauchi, T.: Scaling law of fine scale eddies in turbulent channel flows up to Reτ = 800. Int. J. Heat Fluid Flow 25, 331–340 (2004)

Session 5: DNS and LES

• Direct Numerical Simulations of Converging–Diverging Channel Flow J.-P. Laval and M. Marquillie • Corrections to Taylor’s Approximation from Computed Turbulent Convection Velocities J. Jiménez and J.C. del Álamo • A Multi-scale & Dynamic Method for Spatially Evolving Flows G. Araya, L. Castillo, C. Meneveau, and K. Jansen • Statistics and Flow Structures in Couette–Poiseuille Flows M. Bernardini, P. Orlandi, S. Pirozzoli, and F. Fabiani

Direct Numerical Simulations of Converging–Diverging Channel Flow Jean-Philippe Laval and Matthieu Marquillie

Abstract Two Direct Numerical Simulations (DNS) of a converging–diverging channel flows were performed at Reτ = 395 and Reτ = 617. The present DNS of adverse pressure gradient flow were designed within the WALLTURB project to meet two main objectives. The first one was to gather three-dimensional fully resolved data in order to investigate statistics and coherent structures of turbulence under strong pressure gradient with and without curvature. The second one was to have a reference for the evaluation of RANS and LES models. The flow slightly separates at the lower curved wall and is at the onset of separation at the opposite flat wall. Intense vortices are generated at the location of the minimum friction velocity even without averaged recirculation. The full budget of the Reynolds stresses were computed along the channel at both walls. The occurrence of a well documented secondary peak of velocity fluctuations due to adverse pressure gradient is shown to be the consequence of a very strong production peak.

1 Introduction The realistic flows in engineering or geophysics are still unaffordable by direct numerical simulation but some very large DNS of academic flows like isotropic turbulence (see [2] for a review) or channel flow brought new possibilities to investigate the complex behavior of turbulence as well as its statistics. Only few DNS of wall bounded flows have been performed with adverse pressure gradient and most of them used suction or directly prescribed the adverse pressure gradient at the opposite wall [5, 6, 8]. However, DNS over curved surfaces are important in order to study the curvature effect in addition to the pressure gradient effect. One of the J.-P. Laval () · M. Marquillie Laboratoire de Mécanique de Lille, CNRS, 59655 Villeneuve d’Ascq, France e-mail: [email protected] M. Marquillie e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_21, © Springer Science+Business Media B.V. 2011

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Fig. 1 Computing grid of the DNS at Reτ = 617 in the XY plane (every 24 meshes are plotted in each direction). The flow is coming from the left

WALLTURB project objectives was to gather an important database on wall turbulence with and without pressure gradient. Two DNS of converging–diverging channel flow have been performed within the project taking advantage of the numerical code developed by LML and adapted to this geometry.

2 Description of the DNS Two large DNS of converging–diverging channel were performed using the MFLOPS3D code developed at LML (Lille Mechanics Laboratory) for DNS and LES of wall bounded flow with a curved wall. Instead of writing the Navier–Stokes system in curvilinear coordinates, the wall curvature is obtained by a mathematical mapping of the partial differential operators from physical coordinates to Cartesian ones [4]. The spatial discretization uses Fourier modes in the spanwise direction, a Chebyshev collocation is used in the normal direction and 4th order explicit finite differences are used in the streamwise direction. The objective of the two DNS is to work out a database of turbulent flows with adverse pressure gradient at the highest possible Reynolds number, with a geometry comparable to the experiment which was carried on in the LML wind tunnel [1]. A channel flow configuration was chosen instead of two separate boundary layers. Actually, channel flow inlet conditions are much easier to generate than a configuration with two separated boundary layers. The reason is that it is difficult to define a-priori a simulation which leads to two different boundary layers with statistics comparable to the experiment. Therefore, the inlet for the two DNS were generated either by precursor DNS or LES of flat channel flows at the equivalent Reynolds numbers. The two DNS were simulated on the same domain which is 4π long in streamwise direction, 2 in normal direction and π in spanwise direction. The Reynolds number based on the inlet friction velocity (uτ ) and half the channel width (h) are respectively Reτ = 395 and Reτ = 617 and the spatial resolutions are (1536 × 257 × 384) and (2304 × 385 × 576) respectively. The grid is stretched in the streamwise direction for the highest Reynolds number (see Fig. 1) but is homogeneous for the lower Reynolds number. The two DNS at Reτ = 395 and Reτ = 617 were integrated over 30 and 50 convective times respectively (based on half the channel width and the maximum velocity at the inlet) and a total of 9Tb of velocity and pressure fields were stored in order to compute converged statistics.

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Fig. 2 Spatial resolution of the DNS at Reτ = 617 as compared to the Kolmogorov scale η = ν 3/4 /ε 1/4 . x, y, z are the mesh sizes in streamwise, normal and spanwise direction respectively

The ratio of the Kolmogorov scale η = (ν 3 /ε)1/4 with respect to the maximum mesh size is shown in Fig. 2 for the DNS at the largest Reynolds number. The maximum of this ratio is of the order of 2 to 3 in most of the domain and goes up to 4 in the diverging part and up to a value of 5 very close to the wall (not visible in the figure). However, in order to evaluate the discretization of the near wall region, the mesh size in wall units (x + , y + , z+ ) are more relevant. The maximum values + = 0.02 and z+ = 3.4. The global maximum at the inlet are x + = 5.1, ymin + = values are reached in the converging part of the channel with x + = 10.7, ymin + 0.03 and z = 7.4.

3 Results The pressure coefficient of the two DNS are compared with experiment of Bernard et al. [1] (see Fig. 3). The experimental results are shown in order to compare the behavior and the order of magnitude of the pressure gradient. However the differences between the experiment and the DNS are expected because of different inlet conditions and because the Reynolds number of the experiment is more than one order of magnitude larger. The friction coefficient indicates that the two DNS slightly separates at the lower wall (contrarily to the experiment) but not at the upper wall. However the minimum of Cf at Reτ = 395 is a very low positive value. DNS data are useful to analyze the evolution of coherent turbulent structures. The streaks interpolated at a constant altitude (y + = 20) in wall unit are visualized at the two walls in Fig. 4 by the streamwise velocity fluctuations normalized by its standard deviation (a minimum friction velocity uτ = 1.E–4 was used for the interpolation near the separation and reattachment at the lower wall). The low speed streaks and high speed streaks can be defined as region with u /σu < 1 and u /σu > 1 respectively. One can clearly see that the streaks are wiped out by the adverse pressure gradient. They are first elongated by favorable pressure gradient (−2 < x < 0) and then destroyed at the location of the minimum friction velocity for the upper wall and at the beginning of the recirculation region at the lower wall. The streaks are built up again before the end of the simulation domain x > 4 in the region of weak pressure gradient.

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Fig. 3 Comparison of the pressure coefficient 2 ) Cp = (P − Po )/( 12 ρUmax and skin friction coefficient 2 ) for the Cf = τw /( 12 ρUmax two DNS at Reτ = 395 and Reτ = 617. The friction coefficient of the experiment (at the lower wall) of Bernard et al. [1] is given as a reference, and the Reτ of the experiment is roughly estimated from the boundary layer thickness in front of the profile

Fig. 4 Fluctuating streamwise velocity (normalized by its standard deviation) interpolated at 20 wall units from the two walls of the converging diverging channel. The spanwise coordinates are evaluated in wall unit (z+ ) at the inlet. The low speed streaks (dark) and high speed streaks (light) can be defined as regions where u /σu < 1 and u /σu > 1 respectively

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Fig. 5 Visualization of strong coherent vortices in the DNS at Reτ = 617 using iso-value (300) of Q = Ω 2 − S 2 normalized by the maximum velocity at the inlet (umax 1.0) and half the channel height (h = 1)

The strong coherent vortices are visualized in Fig. 5 with iso-values of the second invariant Q = Ω 2 − S 2 of the velocity gradient tensor. A strong iso-value (Q = 300) was chosen in order to visualized the more intense vortices which are generated near the location of the minimum friction velocity. These intense vortices are not due to the small separation region as the same behavior is obtained at the upper wall without separation. However, the density and the intensity of the vortices are lower at the upper wall. This may be due to the intensity of the adverse pressure gradient as well as the curvature effect. These vortices are likely to be generated from a local instability in space and time due to local inflection points of the streamwise velocity. The detailed mechanism of generation is difficult to address but an analysis of fluctuating vorticity shows that these strong vortices are not necessary the outcome of spanwise vortices breakdown. The profiles of the Reynolds stresses are shown for the lower wall (Fig. 6) at the location where streamwise turbulent stress is maximum just after the separation region. As reported by several authors, for boundary layer flows with adverse pressure gradient [9], the profiles of streamwise turbulent stresses exhibit a secondary maximum. The original peak located near y + 12, characteristic of zero pressure gradient wall turbulence, first decreases and moves toward the wall in physical units. This peak increases under the effect of the adverse pressure gradient and moves away from the wall indicating that the internal layer grows away from the wall. The secondary maximum closed to the wall, more visible in both the streamwise and spanwise turbulent stress, starts to grow near the position of the minimum friction velocity (x = 1.3 at the lower wall). The statistics are comparable at the upper flat wall but the intensity of the secondary peak is less pronounced for the spanwise turbulent stress as compared to the streamwise turbulent stress and the intensity of u u /u2τ (0) reaches a maximum value of 7 instead of 10 at the lower wall. The full budget of the Reynolds stress tensor was computed in the local frame of reference at the two walls. An example of the balance of the kinetic turbulent energy is given at the lower wall (Fig. 7) at the streamwise position of the minimum friction velocity. The ratio of each terms are in accordance to the description given by Marquillie et al. [4] for the DNS at the lowest Reynolds number with a larger recirculation region. The minimum of friction velocity for Reτ = 617 is shifted downstream as compared to the case with Reτ = 395, the cross evolution of

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Fig. 6 Reynolds stresses profiles normal to the lower wall of the DNS at Reτ = 617 (the streamwise position, x = 1.5, is indicated by a dark circle on the curve of Cf in the inset). The plot are normalized with the friction velocity at the submit of the bump uτ (0) (left and lower axis) and with the local friction velocity uτ (x) (right and upper axis)

Fig. 7 DNS at Reτ = 617: Budget of the turbulent kinetic energy normal to the lower wall in the adverse pressure gradient region (the streamwise position x = 1.2 is shown in the inset and corresponds to the minimum of Cf )

each term are also shifted downstream, but the global behavior stay comparable. We retrieve the maximum peak value of the kinetic energy production at this streamwise location, the peak starting to move away from the wall. Part of the production of energy is dissipated but part of it is transported away as shown by the strong peak of advection and the two positive peaks of the turbulent transport on each side of the production peak, phenomenon also reported in the experimental work of Skare & Krogstad [3, 7].

4 Conclusions The DNS of converging–diverging channel flow at significant Reynolds number has been presented. This flow configuration is able to account for an adverse pressure gradient flow with and without curvature and was designed to be a useful engineering test case. The channel flow inlet condition are well documented and easy to model for simulations with a turbulent model like Reynolds Averaged

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Navier–Stokes or LES models. The numerical code used for the present simulation is original as it uses a transformation of coordinates. This numerical code allows us to perform simulations with smooth profiles at reasonable cost keeping the benefit of the efficiency and the accuracy of the spectral solver in spanwise and normal direction. Several turbulent statistics as well as a first investigation of turbulent structures such as coherent vortices were presented. These two DNS of converging–diverging channel flows form a large database which can be used for further detailed investigation of wall turbulent in adverse pressure gradient flows. Acknowledgements This work was supported by WALLTURB (A European synergy for the assessment of wall turbulence) which is funded by the EC under the 6th framework program (CONTRACT: AST4-CT-2005-516008). The largest DNS was performed through two successive DEISA Extreme Computing Initiatives (DEISA is a Distributed European Infrastructure for Supercomputing Applications). The other calculations were done at IDRIS (French CNRS Computing Facilities) and at CRIHAN (Centre de Ressources Informatiques de HAute-Normandie).

References 1. Bernard, A., Foucaut, J.M., Dupont, P., Stanislas, M.: Decelerating boundary layer: a new scaling and mixing length model. AIAA J. 41(2), 248–255 (2003) 2. Ishihara, T., Gotoh, T., Kaneda, Y.: Study of high-Reynolds number isotropic turbulence by direct numerical simulation. Annu. Rev. Fluid Mech. 41, 165–180 (2005) 3. Krogstad, P.Å., Skåre, P.E.: Influence of the strong adverse pressure gradient on the turbulent structure in a boundary layer. Phys. Fluids 7(8), 2014–2024 (1995) 4. Marquillie, M., Laval, J.P., Dolganov, R.: Direct numerical simulation of separated channel flows with a smooth profile. J. Turbul. 9(1), 1–23 (2008) 5. Na, Y., Moin, P.: Direct numerical simulation of a separated turbulent boundary layer. J. Fluid Mech. 374, 379–405 (1998) 6. Na, Y., Moin, P.: The structure of wall-pressure fluctuations in turbulent boundary layers with adverse pressure gradient and separation. J. Fluid Mech. 377, 347–373 (1998) 7. Skåre, P.E., Krogstad, P.Å.: A turbulent boundary layer near separation. J. Fluid Mech. 272, 319–348 (1994) 8. Skote, M., Henningson, D.S.: Direct numerical simulation of separating turbulent boundary layers. J. Fluid Mech. 471, 107–136 (2002) 9. Webster, D.R., Degraaf, D.B., Eaton, J.K.: Turbulence characteristics of a boundary layer over a two-dimensional bump. J. Fluid Mech. 320, 53–69 (1996)

Corrections to Taylor’s Approximation from Computed Turbulent Convection Velocities Javier Jiménez and Juan C. del Álamo

Abstract A new method for estimating the convection velocity of individual Fourier modes in turbulent shear flows is applied to numerical channels in the range Reτ = 180–1880. Smaller scales travel at the local mean velocity, while large ‘global’ modes travel at a more uniform speed proportional to the bulk velocity. The modifications introduced in the energy spectrum by the use of Taylor’s approximation with a wavelength-independent convection velocity are then discussed. Near the wall, it not only displaces the large-scale end of experimental spectra to shorter apparent wavelengths, but it also modifies its shape, giving rise to spurious peaks similar to those observed in some experiments. This result suggests that some of the recent challenges to the kx−1 energy spectrum may have to be reconsidered.

1 Introduction Eddies in turbulent flows with a dominant velocity component, such as jets or boundary layers, propagate downstream at a ‘convection velocity’ that is usually assumed to be close to the average flow velocity [23, Chap. 1.7]. This property is often used in experiments to describe the spatial structure of the flow by applying Taylor’s frozen turbulence approximation [22], and is also important in the analysis of turbulence dynamics [2, 16, 17]. Despite this, a complete characterization of the J. Jiménez () · J.C. del Álamo School of Aeronautics, U. Politécnica, 28040 Madrid, Spain e-mail: [email protected] J. Jiménez Center for Turbulence Research, Stanford U., Stanford, CA 94305, USA J.C. del Álamo MAE Department, U. of California San Diego, La Jolla, CA 92093, USA e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_22, © Springer Science+Business Media B.V. 2011

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turbulent convection velocity is still unavailable for many flows, because its measurement is arduous both in the laboratory and in simulations. Data is especially scarce for the dependence of the convection velocity on eddy size, even if it was shown in [26] that neglecting that dependence leads to large errors in the interpretation of the large-scale structures in jets. The same is presumably true for other flows, and warnings against the uncritical use of Taylor’s approximation appear regularly in the literature. They usually go unheeded, in part for lack of a better alternative. A historical introduction is [26]. The convection velocity of a generic variable q is often derived from its frequency–wavenumber spectrum, or equivalently from its space–time correlation function [8, 24, 25]. Assume x to be the streamwise direction, the flow to be inhomogeneous along y, and homogeneous along x and z (with velocity components u, v ˜ x , kz , y, ω)q˜ ∗ (kx , kz , y, ω), where and w). Consider the ω–kx spectrum Φq = q(k denotes ensemble averaging, ω is the frequency, and kx and kz are wavenumbers. We use ( ˜· ) to denote the Fourier transform with respect to the two homogeneous directions and time, and ( ˆ· ) for spatial Fourier coefficients that have only been transformed with respect to x and z, but that retain an explicit temporal dependence. A common definition for the convection velocity of each Fourier mode, cq,1 (kx , kz , y), is the one maximizing Φq (kx , kz , y, −cq,1 kx ) [25], ∂c Φq (kx , kz , y, −ckx )|c=cq,1 = 0.

(1)

Alternative definitions in terms of the ω–kx spectrum are surveyed in [10, 13], but [3] showed that most of them are roughly equivalent for low turbulence levels. A problem with most of these methods is that the estimation of the ω–kx spectrum requires uniformly-sampled data in both space and time — and, as we will see below, preferably also in z. This makes their computation experimentally impractical in many cases. A new method which only requires Fourier decomposition along the directions in which spectral information is desired was introduced in [9, 15]. A full description can be found in [5], where it is validated against the spectral definitions mentioned above, and used to estimate the corrections required to the frozen-turbulence hypothesis. The present report is an abridged version of that paper, with emphasis on the effects on the spectra. The new method is briefly described in Sect. 2, and related to the older spectral ones. Section 3 applies it to the dependence of the turbulent convection velocities on eddy size, wall-distance, and Reynolds number in channels. Section 4 estimates the effects of Taylor’s approximation, and computes the equivalent uncorrected Taylor one-dimensional (frequency) spectra of the existing simulation data. The paper concludes by summarizing the results, including the possible implications for the recent challenges to the k −1 spectral scaling of wall turbulence.

2 The Estimation of the Convection Velocities Consider the convection velocity of the generic variable q(x, t), which is to be approximated as a frozen wave q(x − ct). A reasonable definition for c is the value cq = −∂t q∂x q/(∂x q)2 ,

(2)

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which minimizes the mean-squared total derivative, (∂t q + c∂x q)2 . When q contains a single Fourier mode, the equation (2) becomes cq (kx , kz , y) = −kx−1 Imqˆ ∗ ∂t q/| ˆ q| ˆ 2 ,

(3)

which can be used to study the spectral dependence of the convection velocities. Equations (2) and (3) also provide natural definitions for the mean convection velocity associated with a general range of wavenumbers [5]. The new definition can be related to methods such as (1), based on the ω–kx spectrum. Consider the temporal Fourier transform q(k ˜ x , kz , y, ω) of the spatial co˜ and it follows from efficient q(k ˆ x , kz , y, t). The Fourier transform of ∂t qˆ is iωq, Parseval’s theorem that ∞ ˆ =i ωΦq (kx , kz , y, ω) dω, (4) qˆ ∗ ∂t q −∞ ∞ qˆ ∗ q ˆ = Φq (kx , kz , y, ω) dω, (5) −∞ ∞ 1 −∞ ωΦq (kx , kz , y, ω) dω ∞ . (6) cq = − kx −∞ Φq (kx , kz , y, ω) dω The new phase velocity is therefore based on the position of the center of gravity of the ω-spectrum at a given wavenumber, instead of on its maximum. Both definitions are equivalent for a frozen monochromatic wave, and differ little if the spectrum is narrow enough to justify a wave representation.

3 Spectral and Spatial Dependence of the Convection Velocity By comparing definitions (1) and (2) in a numerical channel at Reτ = 550, it was shown in [5] that the difference between them is generally smaller than between different spectral definitions of the type mentioned above. The new method was consequently used to compute the spectral distributions of the convection velocity as a function of λx = 2π/kx , λz , y, and Reτ , in numerical channels in the range Reτ = 180–1880 [4, 6, 7]. As shown in Fig. 1(a), the convection velocity cu of the streamwise velocity depends on both wavelengths, especially near the wall, where long-wide structures propagate faster than short-narrow ones. Conversely, larger structures move slower than smaller ones far from the wall. The largest deviations from the local mean velocity are in the region (λx , λz ) (2, 0.4)h, which coincides with the ‘global modes’ identified in [1, 4, 6] as outer layer structures penetrating into the near-wall region. Figures 1(b–c) display wall-normal profiles of cu , respectively averaged over the ‘small’ and ‘global’ wavelength rectangles defined in Fig. 1(a). They show that the convection velocity of the global modes varies relatively little with y, and scales approximately with the bulk velocity Ub , while the small scales follow more closely the mean velocity profile, except very near the wall. The convection velocities for the three velocity components are roughly similar, but the global modes of u and w

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Fig. 1 (a) Contour plot of the two-dimensional spectral distribution of the convection velocity of u, as a function of the streamwise and spanwise wavelengths. Channel with Reτ = 934 [6] at y + = 15 (y = 0.016h, U + = 10.7). The contour lines are, 1, cu+ = 11; e, 13; P, 15; !, 17. The shaded contours are linearly equispaced isolines of the two-dimensional energy density of the streamwise velocity at the same wall distance. The horizontal and vertical dashed lines are respectively λz = 0.4h and λx = 2h. (b) Mean convection velocity profiles of the streamwise velocity cu (y), plotted as a function of the wall distance, for the ‘small’ wavelengths (λx < 2h, λz < 0.4h). · · ·, Reτ = 180; – · –, 550; - - -, 935; —, 1880. The thick solid line is the mean velocity profile from Reτ = 934. (c) For the ‘large’ wavelengths (λx ≥ 2h, λz ≥ 0.4h). (d) Ratio between large- and small-wavelength velocities

tend to move with the local mean velocity above the logarithmic layer, while those of v, which are very coherent in that region, have velocities that correspond to layers much closer to the wall.

4 The Effect of Taylor’s Approximation We now examine the errors introduced in the energy spectra by applying Taylor’s approximation with a wavelength-independent advection velocity. It is clear from Fig. 1(a) that the analysis requires two-dimensional spectral information, because the convection velocity depends on both wavelengths. The one-dimensional

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wavenumber spectrum, which is the expected outcome of most experiments, is derived from the two-dimensional one by integration with respect to kz , E(kx ) = E(kx , kz ) dkz . (7) Experimental results are typically one-dimensional frequency spectra, reduced to ‘Taylor-uncorrected’ wavenumbers defined as kx,T = ω/U (y). The ‘uncorrected’ wavenumber spectrum can be derived from the ω–kz two-dimensional one as, ET (kx ) = U (y)E(ω) = U (y) E(ω, kz ) dkz . (8) The direct ‘Taylor correction’ problem would be to estimate an approximation to E(kx ) starting from E(ω, kz ), but two-dimensional ω–kz spectra are rare, and we will center on the ‘inverse’ problem of estimating the uncorrected spectrum (8) from the two-dimensional wavenumber spectrum E(kx , kz ), available from simulations. The result will then be compared to one-dimensional experimental frequency spectra, to try to separate which of their features correspond to properties of the underlying spatial spectra, and which ones are artifacts of the transformation. From the definition of passing frequency, the ‘corrected’ frequency spectrum is (9) ωE(ω) = kx,T ET (kx ) = kx E[kx (ω, kz ), kz ]J (kx , kz ) dkz , where the integral is taken along paths of constant frequency in the wavenumber plane, and J = (ω/kx )|dkx /dω|

(10)

is a dimensionless version of the Jacobian of the transformation. It ensures that the energy of an eddy is the same in both representations, E(ω)|dω| = E(kx )|dkx |, and can be considerably different from unity if the advection velocity changes rapidly with the wavenumber. This behavior is shown in Fig. 2(a), which displays the same buffer-layer plane used for Fig. 1(a). The general location of the u-spectrum is shown shaded for reference, and the solid lines are contours of constant kx,T . They spread out near the short-wavelength edge of the global modes, because the global wavelengths to the right of that limit are aliased to higher frequencies (or to shorter uncorrected wavelengths) by their higher convection velocities. Where the isofrequency lines diverge, the Jacobian increases, because the energy is crowded into a narrower frequency interval. The overall effect is to damp the frequency spectrum at intermediate and long wavelengths, and to augment it near λx ≈ 3h, where the advection velocity changes from U (y) to Ub . The magnitude of this change depends on Ub /U (y), and is therefore strongest in the buffer and deep logarithmic layers, where it may create artificial maxima or minima in ET (kx ). It can also be expected to increase with increasing Reynolds numbers at a fixed y + , as the two velocities diverge approximately proportionally to log(Reτ ). This is seen in Fig. 1(d), which plots the ratio of the measured large- and small-scale convection velocities as a function of wall distance and Reynolds number. As the latter increases, the ratio also increases near

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Fig. 2 (a) The shaded area is the premultiplied two-dimensional spectrum of u at y + = 15, above 0.2 times its maximum in that plane. Reτ = 934 [6]. Solid lines are constant frequency isocontours, corresponding to uncorrected Taylor wavelengths. The value of each contour is given approximately by the wavelength at which it intersects the horizontal axis. Dashed lines are isocontours of the mapping factor (10), J = 1(0.1)1.4. (b) One-dimensional premultiplied energy spectra, kx Euu (kx ). !, experimental pipe Reτ = 2325 [21], y + = 100. Lines are a numerical channel at Reτ = 2003 [12]. - - -, true wavelengths; —, uncorrected Taylor wavelengths

the wall, leading to stronger corrections, and to an enhancement of the distorting effect. The maximum of J in Fig. 2(a) is 1.5, although its influence on the integral (9) is not as large, because it is located in a region of relatively low energy. Bimodal spectra have often been documented near the wall, perhaps first emphasized by [11] in boundary layers. Using those data, it was noted in [14] that the outer energy peak in experimental boundary layers extends closer to the wall than in numerical channels, although they cautioned that this could in part be due to the difference between wavenumber and frequency spectra. A similar peak near the wall was observed in experimental boundary layers by [18, 19], and [20] have noted recently that a broad long-wavelength peak is present near the wall in experimental channels, but not in numerical ones. That outer structures extend into the buffer layer has been well documented in simulations, independently of Taylor’s approximation [12], and is not in doubt, but the premultiplied long-wave spectra in those cases tend to be flat, rather than bimodal, consistent with the theoretical arguments for a kx−1 spectrum [23, pp. 150–158]. In view of the previous arguments, it is fair to ask whether the bimodal nature of the experimental spectra might be partly an artifact of the Taylor approximation. This possibility is tested in Fig. 2(b), using experimental pipes that also display the peak in question [21]. The figure contains the experimental data reduced to wavelengths using the local mean velocity, the true wavelength spectrum for the channel, and the channel spectrum reduced to uncorrected Taylor wavelengths using (9). The appearance of a long-wavelength peak due to Taylor’s approximation is obvious, and matches quantitatively the uncorrected experimental spectrum in the y + = 100 case.

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5 Conclusions We have presented a new method, based on minimizing the total advective derivative, to compute turbulent convection velocities in shear flows. It does not require knowledge of the frequency–wavenumber spectrum, but can be naturally extended to deal with arbitrary regions of the Fourier plane. In contrast to older methods based on maxima of the frequency–wavenumber spectrum, the new velocity is related to the center of gravity of the frequency spectrum at constant wavenumber, but the spectrum itself is not required. We have shown that both methods agree well in a numerical channel at Reτ = 550. A fuller discussion is found in [5]. We have used the new scheme to determine convection velocities in computational channels with Reynolds numbers 180 ≤ Reτ ≤ 1880, as functions of the streamwise and spanwise wavelength, of Reτ , and of the wall distance. The smallest eddies follow the local mean velocity everywhere except near the wall, but the convection velocity becomes less dependent on y with increasing wavelength, and eddies with λx 2h and λz 0.4h, corresponding to the ‘global’ modes identified in [1, 4, 6], travel at almost uniform speeds, close to the bulk velocity. These results have been used to estimate the errors introduced by applying Taylor’s frozen-turbulence approximation in experimental data with a wavelengthindependent convection velocity. They depend on the two-dimensional spectra, and can be considerable. Not only are the large scales aliased into shorter ones near the wall, but the spectral shape changes, creating spurious spectral peaks near the shortwavelength limit of the global modes, where the energy is squeezed into a local maximum by the differential deformation of the wavenumber range. The effect is largest in the buffer and inner logarithmic layers, and increases with the Reynolds number. Such peaks have been reported in experiments, but a quantitative comparison of the Taylor-corrected channel data with experimental data from pipes suggests that they may be, at least in part, artifacts of the incorrect application of the frozenturbulence approximation. A consequence is that the true wavenumber spectrum of turbulence near walls may be closer to the kx−1 prediction in [23] than concluded in recent experiments. Acknowledgements This work was supported in part by the CICYT grant TRA2006-08226. JCA was partially supported by consecutive FPU and Fulbright fellowships from the Spanish Ministry of Education. The preparation of the report was done in the context of the EU FP6 Wallturb Strep AST4-CT-2005-516008.

References 1. 2. 3. 4. 5. 6.

Bullock, K.J., Cooper, R.E., Abernathy, F.H.: J. Fluid Mech. 88, 585–608 (1978) Choi, H., Moin, P.: Phys. Fluids A 2, 1450–1460 (1990) Comte-Bellot, G., Corrsin, S.: J. Fluid Mech. 48, 273–337 (1971) del Álamo, J.C., Jiménez, J.: Phys. Fluids 15, L41–L44 (2003) del Álamo, J.C., Jiménez, J.: J. Fluid Mech. 640, 5–26 (2009) del Álamo, J.C., Jiménez, J., Zandonade, P., Moser, R.D.: J. Fluid Mech. 500, 135–144 (2004)

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del Álamo, J.C., Jiménez, J., Zandonade, P., Moser, R.D.: J. Fluid Mech. 561, 329–358 (2006) Fisher, M.J., Davies, P.O.A.L.: J. Fluid Mech. 18, 97–116 (1963) Flores, O., Jiménez, J.: J. Fluid Mech. 566, 357–376 (2006) Goldschmidt, V.W., Young, M.F., Ott, E.S.: J. Fluid Mech. 105(APR), 327–345 (1981) Hites, M.: Scaling of high-Reynolds number turbulent boundary layers in the National Diagnostic Facility. Ph.D. thesis, Illinois Inst. of Technology (1997) Hoyas, S., Jiménez, J.: Phys. Fluids 18, 011702 (2006) Hussain, A.K.M.F., Clark, A.R.: AIAA J. 19, 51–55 (1981) Jiménez, J., Hoyas, S.: J. Fluid Mech. 611, 215–236 (2008) Jiménez, J., del Álamo, J.C., Flores, O.: J. Fluid Mech. 505, 179–199 (2004) Kim, J., Hussain, F.: Phys. Fluids 6, 695–706 (1993) Krogstad, P.A., Kaspersen, J.H., Rimestad, S.: Phys. Fluids 10(4), 949–957 (1998) Kunkel, G.J.: An experimental study of the high Reynolds number boundary layer. Ph.D. thesis, Aerospace Engng. and Mech., U. Minnesota (2003) Kunkel, G.J., Marusic, I.: J. Fluid Mech. 548, 375–402 (2006) Monty, J., Chong, M.: In: Proc. of the 61st Ann. Meeting Div. Fluid Dyn., EB-10. Am. Phys. Soc., Providence (2008) Perry, A.E., Abell, C.J.: J. Fluid Mech. 67, 257–271 (1975) Taylor, G.I.: Proc. R. Soc. Lond. 164(919), 476–490 (1938) Townsend, A.A.: The Structure of Turbulent Shear Flows, 2nd edn. Cambridge University Press, Cambridge (1976) Willmarth, W.W., Wooldridge, C.E.: J. Fluid Mech. 14(2), 187–210 (1962) Wills, J.A.B.: J. Fluid Mech. 20, 417–432 (1964) Zaman, K.B.M.Q., Hussain, A.K.M.F.: J. Fluid Mech. 112, 379–396 (1981)

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A Multi-scale & Dynamic Method for Spatially Evolving Flows Guillermo Araya, Luciano Castillo, Charles Meneveau, and Kenneth Jansen

Abstract A method for generating realistic turbulent velocity inlet boundary conditions is presented for simulations of spatially evolving turbulent boundary layers. The approach is based on the rescaling–recycling method proposed by Lund et al. (J. Comput. Phys. 140:233–258, 1998). The standard rescaling process requires prior knowledge about how the appropriate velocity and length scales are related between the inlet and recycle stations (e.g. classic single scaling laws). In the present study the scales for the inner and outer regions are determined from the multi-scale approach based on the original equilibrium similarity method developed by George and Castillo (Appl. Mech. Rev. 50:689–729, 1997) (for ZPG flows) and Castillo and George (AIAA J. 39(1):41–47, 2001) (for PG flows). In addition, a new dynamic approach is proposed in which power law ratios of inner/outer scales are used with scaling exponents that may depend on flow conditions and are deduced dynamically by involving an additional plane, a “test plane”. This plane is located between the inlet and recycle stations of the simulation. This improvement, as well as the use of multiple velocity scales, permits the simulations of turbulent boundary layers subjected to arbitrary pressure gradients. DNS for zero (ZPG), and pressure gradient flows are discussed with special emphasis on adverse pressure gradient flows (APG). The agreement obtained by comparing present results with experimental and numerical data demonstrates the suitability of the proposed method for G. Araya () Swansea University, Swansea, SA2 8PP, UK e-mail: [email protected] L. Castillo · K. Jansen Rensselaer Polytechnic Institute, 110 8th St., Troy, NY 12180, USA e-mail: [email protected] K. Jansen e-mail: [email protected] C. Meneveau The Johns Hopkins University, Baltimore, MD 21218, USA e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_23, © Springer Science+Business Media B.V. 2011

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generating inflow boundary conditions for simulations of developing boundary layer flows including those subjected to pressure gradient.

1 Introduction While Computational Flow Dynamics (CFD) has gone through notable developments in the last few decades, spatially evolving turbulent flows (i.e., boundary layers) are very challenging to compute numerically due to the need to impose time-dependent turbulent inflow conditions. The most straight-forward procedure to prescribe inflow information consists of starting from the laminar regime including small disturbances, simulating transition to turbulence; and, finally, using this information as input to the main domain of interest [6, 13, 14]. This method has the advantage that no turbulent fluctuations are needed at the inlet, but the main drawback is its very high computational cost compounding limitations about the highest Reynolds number that can be simulated. On the other hand, several techniques for generating turbulent inflow conditions have been put in practice with different degrees of success. A comprehensive review can be found in [10] and [11], and more recently in [5]. The simplest approach is to superpose random fluctuations on a desired mean profile. The random fluctuation method has been widely used with different variants [8, 9, 12]. Unfortunately, random fluctuations necessitate long distances (i.e., developing sections) for the development of realistic turbulent flow structures, not to mention the problem of controlling turbulent flow parameters at the inlet. Lund and co-authors [10] proposed a modification to the concept of [15] for accounting for spatial growth in the inflow condition based on the similarity of the velocity profiles at different locations, in a spatially evolving turbulent boundary layer. They considered an auxiliary simulation where the velocity field was extracted from a downstream plane, or recycle plane, rescaled it and reintroduced it as boundary condition at the inlet of the auxiliary zone (see Fig. 1). Subsequently, the instantaneous velocity field on a selected plane of the auxiliary domain was utilized as inflow information for the principal domain. This technique has been successfully implemented in ZPG flows [7] after having been tested by [10]. However, two of the major drawbacks of this method limits its application to simple canonical flows: (i) the assumption of single velocity scaling for the inner and outer regions, and (ii) the use of an empirical correlation in order to connect the friction velocities, uτ , between the inlet and recycle planes. In this paper, we propose to dynamically compute the necessary flow parameters at the inlet plane based on the solution downstream in such a way as to avoid dependency on empirical correlations. This feature plus the use of a multi-scale approach during the rescaling process [1] permits us to extend the original approach of Lund et al. to flows with external pressure gradients.

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Fig. 1 Schematic of the evolving boundary layer

2 Formulation of the Problem and Methodology Figure 1 shows a schematic of the turbulent boundary layer with the inner and outer regions as well as the corresponding inlet and recycle stations, where the flow is 2D in the mean. In this study, x, y and z denote the streamwise, wall-normal, and spanwise Cartesian coordinates, respectively. The basic idea of the rescaling–recycling method consists of expressing the instantaneous velocity, ui , as a contribution of a mean value, U¯ i , plus a fluctuation, ui . Furthermore, by normalizing the velocity components by the appropriate scales in the inner and outer flows of the boundary layer, it is possible to attenuate the nonhomogeneity along the streamwise direction and to apply “quasi” periodic conditions in the new coordinate system.

2.1 The Rescaling–Recycling Method: The Multi-scale Similarity Approach The proper scaling for turbulent flow variables is essential to capture common behaviors, especially in spatially developing flows. This scaling will be needed to connect the inlet and recycle planes. The appropriate scales for the level of fluctuation of any dependent variable can be sought by applying the equilibrium similarity analysis to the boundary layer equations for the fluctuating variables. Subtracting the mean flow in x-momentum equations from those for the instantaneous flow and rearranging terms, the following equation for the u fluctuation in the outer region is obtained, and given here as an example: ∂u ∂u ∂u +U +V ∂t ∂x ∂y ∂u 2 ∂u v ∂u w ∂u v ∂U 1 ∂p − v − − − + =− ρ ∂x ∂y ∂x ∂y ∂z ∂y

(1)

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The same approach is true for the v and w components as well as the mean flow, Reynolds stresses equation and two-point correlations. In this equation the viscous stress was neglected since it is only important in the inner layer. As in the mean flow, the solutions for the fluctuations are sought in the outer layer as product of two functions; one depends on x only and it is the unknown scaling determined from the similarity analysis and the second is the similarity function: u (x, y, z, t) = uso (x)fuo∞ (η, ηz , to , ∗)

(2)

(3)

(4)

(5)

v (x, y, z, t) = vso (x)fvo∞ (η, ηz , to , ∗) w (x, y, z, t) = wso (x)fwo∞ (η, ηz , to , ∗) p (x, y, z, t) = pso (x)fpo∞ (η, ηz , to , ∗)

where η = y/δ(x), ηz = y/φ(x) and t = t/To with To as the time scale which must be determined from the similarity arguments applied to the equations of motion. The functions fuo∞ , fvo∞ , fwo∞ , fpo∞ represent similarity functions in the limit of δ + → ∞. Notice that the scales uso , vso , wso , pso , and To are functions only of x; also the inlet conditions are given by ∗. The outer spanwise length scale, φ(x), is the boundary layer thickness, i.e. φ = δ and the outer characteristic time scale, To = δ/(U∞ dδ/dx) (obtained from similarity, see analysis below) where dδ/dx is the growth rate of the boundary layer. The same scaling functions are obtained when this derivation is carried out for the v and w fluctuation components. If the same analysis is conducted for the inner region, the classical scales are obtained where the inner flow is scaled with the friction velocity, uτ . In addition, the inner characteristic time scale is determined from the kinematic fluid viscosity and the friction velocity, Ti = ν/u2τ . The importance of this procedure is that the scales (velocity, space and temporal) are determined from the similarity analysis of the equations of motions instead of a single scaling as in the traditional approach. Using the product similarity forms in Eq. 5 and the equations for the streamwise fluctuations, Eq. 1, the scales for the fluctuations in the outer region are determined from the constraint that the scales must have the same x-dependence: 2 dδ uso 1 dpso duso uso dδ U∞ uso vso U∞ ∼ U∞ ∼ ∼ vso ∼ ∼ ∼ U∞ To dx δ dx ρ dx δ δ δ dx

(6)

It follows that the similarity scales for the fluctuations are: uso = wso ∼ U∞ ;

vso ∼ U∞

dδ dx

(7)

where the scale for w was obtained by doing the same analysis on the z-momentum equation. Thus, with these similarity scalings the appropriate scale for the velocity at the inlet plane can be specified in terms of known quantities (U∞ , δ, etc.). The inlet velocity components are then evaluated by the following equations in terms of the appropriate ratios (λ’s) connecting recycle (rcy) and inlet planes, outer (u )outer inl = λo,1 (u )rcy (ηinl , z, t)

(8)

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Table 1 Scaling parameters for the velocity field and temperature comparison between (classical approach) and multi-scaling similarity (current method) Parameter

Type

Classical scaling

Proposed scaling

Inner

Outer

Inner

Outer

U

Mean streamwise velocity

uτ

uτ

uτ

U∞

V

Mean wall-normal velocity

U∞

U∞

uτ

u

dδ U∞ dx

Streamwise velocity fluctuations

uτ

uτ

uτ

U∞

v

Wall-normal velocity fluctuations

uτ

uτ

uτ

w

dδ U∞ dx

Spanwise velocity fluctuations

uτ

uτ

uτ

U∞

outer (v )outer inl = λo,2 (v )rcy (ηinl , z, t)

(w )outer inl

= λo,3 (w )outer rcy (ηinl , z, t)

(9) (10)

where λo,1 = λo,3 =

dδ )inl (U∞ dx (U∞ )inl λo,2 = dδ (U∞ )rcy (U∞ dx )rcy

(11)

In the same way, the scale for the fluctuations in the y and z directions can also (uτ )inl which are clearly be obtained. And for the inner flow the λi,1 = λi,2 = λi,3 = (u τ )rcy different than the λ’s for the outer flow unlike in the traditional approach. Finally, a composite profile is formed at the inlet plane by weighting the contributions of the inner and outer mean and fluctuating velocities, respectively: (ui )inl = (Ui )inner + (ui )inner 1 − W (ηinl )t inl inl outer + (Ui )outer W (ηinl ) (12) inl + (ui )inl where W (η) is a weighting function that ranges from 0 to 1. In the classical approach the inner and outer flows are scaled with a single scaling whereas in the multi-scale similarity the inner and outer flows are scaled differently. The scales from similarity shown in Table 1 apply for boundary layers with or without external pressure gradients. Still, the downstream development of the scaling variables must be prescribed. George and Castillo [3] showed that the streamwise inhomogeneity in boundary layers yielded different scalings for the inner and outer flow in boundary layers. And consequently, the ratio of two different scalings yield power law solutions in the overlap region. However, in the case of homogeneous flows as in channels and pipes the inner and outer mean flows are both characterized by a single scaling, and thus logarithmic solutions are obtained in the overlap region but with coefficients that depend on the Reynolds number. Therefore, from the context of the present investigation, the rescaling–recycling approach must be adapted for applicability to more general flows without the need for any empirical correlation and to capture any Reynolds number dependence. A dynamic approach is proposed in the next section.

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2.2 Dynamic Approach In Large Eddy Simulation, the dynamic approach of Germano et al. [4] has provided a highly successful approach by invoking a ‘test-filter’ and interrogating simulated scales to determine unknown model parameters. In the present work, we begin by assuming that a power law variation of the friction velocity as (uτ /U∞ ) ∼ (Reδ )γ , exists where Reδ = δU∞ /ν (see e.g. [3]). By relating the friction velocities at the inlet and recycle stations, we can write (uτ /U∞ )inl Reδinl γ = (13) (uτ /U∞ )rcy Reδrcy where uτ inl and γ are unknowns. In order to determine uτ inl , we must specify γ . Here we propose to introduce a new plane, called “test plane”, located between the inlet and recycle stations. By applying the same expression to these two planes, γ may now be evaluated dynamically ln[(uτ /U∞ )test /(uτ /U∞ )rcy ] (14) γ= ln[Reδtest /Reδrcy ] Using this same γ for the inlet assumes that γ is constant along the entire computational domain. Since the Reynolds number does not vary greatly, and the flow is in reasonable equilibrium during the downstream evolution, this assumption is warranted. Consequently, once the power γ is obtained from Eq. 14, the value of (uτ )inl is calculated from Eq. 13. The freestream velocity, U∞ , is evaluated by averaging the mean streamwise velocity in the region above the boundary layer edge in the entire computational domain. In addition, the ratio of dδ/dx at the inlet plane to the recycle plane is computed by also assuming a power law of the boundary layer thickness with respect to x-coordinate Reynolds number (Rex = xU∞ /ν), i.e. (δ/x) ∼ (Rex )γδ . Again, by relating the inlet to the recycle stations, (δ/x)inl Rexinl γδ = (15) (δ/x)rcy Rexrcy and γδ is calculated by linking the corresponding quantities of the test and recycle planes as follows, ln[(δ/x)test /(δ/x)rcy ] (16) γδ = ln[Rextest /Rexrcy ] Finally, the boundary layer growth ratio is also obtained as, γ

(dδ/dx)inl d(xRexδ )inl /dx = (dδ/dx)rcy d(xReγxδ )rcy /dx

(17)

3 Results and Discussion Direct Numerical Simulations are carried out for a moderate adverse pressure gradient. The full Navier–Stokes equations are solved using a Finite Element (FE) procedure, details can be found in [16]. At the upper surface of the computational box,

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Fig. 2 Variation of U∞ (left) and θ (right) along the streamwise direction

Fig. 3 Mean streamwise velocity (left) in outer units, turbulence intensities and Reynolds shear stresses in inner units (right)

a desired curvature is prescribed in such a way to obtain a power-law variation of the freestream velocity, i.e. U∞ ∼ (x − xo )m with m = −0.17 as seen in Fig. 2 (left). On the other hand, the virtual origin of the boundary layer, xo , can be obtained by extending the linear trend of the momentum thickness, θ , see Fig. 2 (right). The dimensions of the computational domain are selected as follows: 0 < x < 15δinl , 0 < y < 3δinl and 0 < z < π/2δinl , where δinl is the inlet boundary layer thickness. The range for the momentum thickness Reynolds number, Reθ , + + = 6.8 and is 438–633. The mesh resolution is: x + = 20, ymin = 0.2, ymax +

z = 5. In Fig. 3 (left), the mean streamwise velocity profiles are shown in outer units, by using a deficit law with the local freestream velocity, at several downstream positions of the computational domain. A good collapse of all profiles is observed by employing this scale. Furthermore, good agreement with data from Skote [14] at a similar Reθ with m = −0.15 can be observed. It is worth mentioning that in

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Skote [14], the inlet conditions were obtained by simulating the laminar-transitionturbulent stage, which is the most precise technique as an inflow generator but com+ , v + and w + ) and putationally very expensive. The turbulence intensities (urms rms rms + + Reynolds shear stresses (u v ) are shown in Fig. 3 at Reθ = 580. The pressure gradient effects are mostly manifested in the outer region, i.e. y ≈ 100, and represented by peaks or “shoulders” in the profiles. This behavior is particularly more evident in streamwise velocity fluctuations. The same effects can be observed in Skote’s simulations.

4 Conclusions A rescaling–recycling method together with a new multi-scale/dynamic approach has been proposed, and was shown to yield realistic results when applied to simulations of momentum evolving boundary layers under streamwise pressure gradients. The major effect of the adverse pressure gradient on flow parameters has been identified as a local peak, particularly on streamwise velocity fluctuations, in the outer region.

References 1. Araya, G.: DNS of turbulent wall bounded flows with a passive scalar. Ph.D. thesis, Rensselaer Polytechnic Institute, Troy, New York (2008) 2. Castillo, L., George, W.K.: Similarity analysis for turbulent boundary layer with pressure gradient: outer flow. AIAA J. 39(1), 41–47 (2001) 3. George, W.K., Castillo, L.: Zero-pressure-gradient turbulent boundary layer. Appl. Mech. Rev. 50, 689–729 (1997) 4. Germano, M., Piomelli, U., Moin, P., Cabot, W.H.: A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A 3, 1760–1765 (1991) 5. Keating, A., Piomelli, U., Balaras, E., Kaltenbach, H.: A priori and a posteriori tests of inflow conditions for large-eddy simulation. Phys. Fluids 16(12), 4696–4712 (2004) 6. Khujadze, G., Oberlack, M.: DNS and scaling laws from new symmetry groups of ZPG turbulent boundary layer flow. Theor. Comput. Fluid Dyn. 18, 391–411 (2004) 7. Kim, K., Sung, H.J.: Effects of unsteady blowing through a spanwise slot on a turbulent boundary layer. J. Fluid Mech. 557, 423–450 (2006) 8. Le, H., Moin, P.: Direct numerical simulation of turbulent flow over a backward-facing step. Report TF-58. Thermosciences Div., Dept. Mech. Eng. Stanford University, Stanford, CA 94305 (1994) 9. Lee, S., Lele, S., Moin, P.: Simulation of spatially evolving turbulence and the applicability of Taylor’s hypothesis in compressible flow. Phys. Fluids A 4, 1521–1530 (1992) 10. Lund, T.S., Wu, X., Squires, K.D.: Generation of turbulent inflow data for spatially-developing boundary layer simulations. J. Comput. Phys. 140, 233–258 (1998) 11. Moin, P., Mahesh, K.: Direct numerical simulation: a tool in turbulence research. Annu. Rev. Fluid Mech. 30, 539–578 (1998) 12. Park, J., Choi, H.: Effects of uniform blowing or suction from a spanwise slot on a turbulent boundary layer flow. Phys. Fluids 11(10), 3095–3105 (1999) 13. Rai, M., Moin, P.: Direct numerical simulation of transition and turbulence in a spatially evolving boundary layer. J. Comput. Phys. 109, 169–192 (1993)

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14. Skote, M.: Studies of turbulent boundary layer flow through direct numerical simulation. Ph.D. thesis, Royal Institute of Technology, Stockholm, Sweden (2001) 15. Spalart, P.R., Leonard, A.: Direct numerical simulation of equilibrium turbulent boundary layers. In: Proc. 5th Symp. on Turbulent Shear Flows, Pennsylvania State University, University Park, PA, pp. 9.35–9.40 (1985) 16. Whiting, C.H., Jansen, K.E.: A stabilized finite element method for the incompressible Navier–Stokes equations using a hierarchical basis. Int. J. Numer. Methods Fluids 35, 93–116 (2001)

Statistics and Flow Structures in Couette–Poiseuille Flows Matteo Bernardini, Paolo Orlandi, Sergio Pirozzoli, and Fabrizio Fabiani

Abstract Direct Numerical Simulations (DNS) of Couette–Poiseuille turbulent channel flows have been performed to understand whether universality of the flow near the stationary and the moving wall occurs. As in previous experiments, we have observed that the flow near the stationary wall is not affected by the flow in the rest of the channel. Near the moving wall streaky structures form if the normalized shear rate Sk/ε is larger than a threshold value, which is found to agree with that found in other turbulent flows. The statistics in the region near the moving wall are not observed to scale with the friction velocity, and a better collapse of the data is instead obtained in terms of dissipative units evaluated at the wall.

1 Introduction The near-wall region of turbulent flows has been extensively investigated in boundary layers, channels and pipes with smooth stationary wall. Experiments and DNS in a very wide range of Reynolds number have demonstrated satisfactory collapse of mean velocity and velocity fluctuations in wall units. However, careful examination of experimental and numerical data in canonical channel flows for 180 < Reτ < 2000 [1] seems to suggest weak linear variation of the peak longi+ < 3.0. tudinal Reynolds stress with the Reynolds number, with 2.0 < umax In the first experiments in wall-bounded flows the boundary conditions were changed by using a rough surface instead of a smooth one [9]. An infinite number of rough surfaces can be used, leading to very different results, the common feature being that near-wall structures become more isotropic, with a completely different shape from those found in the presence of smooth walls. Rough-walls produce variations in both the mean and the r.m.s. velocity, with a substantial downward shift of the log-layer. M. Bernardini · P. Orlandi · S. Pirozzoli · F. Fabiani Dipartimento di Meccanica e Aeronautica, Università La Sapienza, Via Eudossiana 16, 00184 Roma, Italy M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_24, © Springer Science+Business Media B.V. 2011

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Our interest is to find flows where the boundary conditions change in a more systematic way, thus controlling the (an)isotropy of the near-wall structures. The simplest way to generate such a flow is to let one of the walls to move (with velocity UW ), while keeping the other wall fixed. For that purpose, an experimental apparatus was first designed by El Telbany and Reynolds [2], and subsequently by Nakabayashi et al. [8]. The latter authors reported detailed velocity statistics (scaled by the friction velocity) near the stationary wall, whereas they did not attempt to study the flow near the moving wall owing to difficulties to measure flow properties (and in particular the friction velocity) in the vicinity of the moving belt. DNS does not have this drawback, since all the quantities are accessible. The only DNS study of the Couette–Poiseuille flow (to our knowledge) was performed by Kuroda et al. [5]. Those authors considered a relatively low Re, and found good agreement with the experiments of Nakabayashi et al. [8]. Kuroda et al. [5] presented a detailed statistical analysis for a limited number of flow conditions, including (i) UW = 0 (pure Poiseuille flow); (ii) UW = 0.5 (Poiseuille-like flow); (iii) UW = 0.81 (intermediate type flow); and (iv) UW = 1.0 (Couette-like flow). The present simulations have been performed in a wider range of flow conditions, and the Reynolds number has been increased, since the value used by Kuroda et al. [5] (Reτ = 150) is too low for fully developed turbulent Poiseuille flow. The results obtained from the DNS study can also be of interest for RANS closure validation; such comparison is currently in progress, and the results will be presented elsewhere. In the present simulations the action is on UW , which has the primary effect of controlling the mean shear, and, consequently, the turbulent kinetic energy production and dissipation rate (ε). In wall-bounded flows ε assumes its largest values at the wall and it does not vary substantially in the viscous layer; hence, it is expected that a velocity scale can be constructed based on the value of the dissipation rate at the wall (εw ), which could allow a better scaling than the friction velocity. This scaling could also partially work in the case of shear-less flow, when wall units cannot be consistently defined.

2 Numerical Methodology We have solved the nondimensional Navier–Stokes momentum equations for a divergence-free velocity field ∂Ui Uj ∂Ui ∂P 1 ∂ 2 Ui + =− + + Πδi1 , ∂t ∂xj ∂xi Re ∂xj ∂xj

∂Uj = 0, ∂xj

(1)

where Π is the pressure gradient required to maintain a constant flow rate, Ui is the component of the velocity vector in the i direction and P is the pressure. The Navier–Stokes equations have been discretized in an orthogonal coordinate system (x1 , x2 , x3 ) using staggered central second-order finite-difference approximations. The details of the numerical method can be found in [10], and are not repeated here. The boundary conditions are enforced at the lower (moving) wall, indicated with the

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Fig. 1 Schematic picture of mean velocity profiles in P-like and C-like flows

subscript M, by setting the spanwise (U3 = w) and the wall-normal (U2 = v) velocity components to zero, and U1 = UW . On the opposite (stationary) wall, indicated with the subscript S, the three velocity components are all set to zero. A sketch of the velocity profile in Couette–Poiseuille flow is depicted in Fig. 1. The controlling parameter for the flow under consideration is γ = τM /τS , where τM indicates the shear stress at the moving wall, and τS at the stationary wall. Couette-like flows are recovered for γ > 0, whereas Poiseuille-like flows are found for γ < 0. Specifically, pure Couette and pure Poiseuille flows are recovered for γ = 1 and γ = −1, respectively. Flows with γ ≈ 0 (referred to as ‘shear-less’) are of special interest, as they exhibit nearly-zero mean shear near the moving wall; under such conditions, coherent structures are created near the moving wall which significantly differ from the ones found in canonical channel flows. A series of simulations at different wall speeds have been performed in a computational box of size L1 = 4πh in the streamwise and L3 = πh in the spanwise directions, h being the channel half-width. The computational domain has been discretized with 256 × 128 × 128 cells in the x1 , x2 , x3 directions, respectively. The simulations were initiated with a laminar parabolic Poiseuille velocity profile, with maximum velocity UP at the centerline, and bulk velocity Ub = 2/3UP . The calculations were then time-advanced by forcing constant mass flux through the channel, in such a way that Ub does not change with time. The Reynolds number in Eq. 1 is then conveniently defined in terms of the Poiseuille velocity, and of the channel half-width, i.e. Re = UP h/ν. All the simulations reported in the present study were performed at Re = 7200, the corresponding bulk Reynolds number being Rb = Ub h/ν = 4800. The main flow parameters (UW , γ and the friction Reynolds numbers at the moving and the stationary wall, ReτM , ReτS ) are reported in Table 1. The range of γ covered by the present simulations is quite large, and each run has been labeled with the letter P or C according to the sign of γ (note that, strictly speaking the run C1 is C-like, but it is very close to the shear-less case). The time evolution of the wall friction shows that after an initial transient, it fluctuates around a constant mean value. After the end of the transient (lasting approximately 100 time units), the velocity and pressure fields are saved and processed to evaluate the statistical flow properties. Statistics have been computed on a col-

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Table 1 Flow parameters and symbols used to indicate each run Run

P1

P2

P3

P4

C1

C2

C3

C4

UW

0

0.5

0.6

0.7

0.813

0.9

1

1.6

γ

−1

−0.228

−0.151

−0.081

0.008

0.108

0.239

2.15

ReτS

290

266

262

262

254

253

251

239

Symbol

"

Q

F

2

!

P

E

1

Fig. 2 Distributions of: (a) mean velocity; (b) |U − UW |. The dashed line in (b) denotes the law-of-the-wall for Couette flow (U + = 5.4 + 2.44 log y + )

lection of 120 fields saved every 0.5 time units, sufficient to guarantee statistical convergence.

3 Mean and Fluctuating Properties The mean velocity distributions are reported in Fig. 2(a) in computational units. The perfect symmetry with respect to the centreline of the plane Poiseuille flow is broken when γ = −1, and the moving wall affects the velocity distributions in the entire channel. P-like flows exhibit a maximum which shifts towards the moving wall at lower values of |γ | while in C-like the mean velocity increases monotonically, showing an inflection point whose location depends on γ . Due to the constancy of the flow rate, the velocity at the centreline and the stress at the stationary wall slightly decrease when γ increases. To investigate the occurrence of universality in the region near the moving wall, in Fig. 2(b) we report the distribution of |U − UW | normalized by the friction velocity at the moving wall uτM = (|τM |/ρ)1/2 . Except for the P4 and C1 cases, all profiles show the occurrence of a viscous region up to y + = yuτM /ν ≈ 5, y being the distance from the moving wall. The linear behavior holds up to y + = 2 for P4 run, and it is absent from the shear-less simulation. A log region of limited extent is visible for the P1 and C4 simulations, with an additive constant that is lower for the Couette-like flow C4 (characterized by large γ ), consistent with the findings of Nakabayashi et al. [8]. A small log region is also present for the P-like flow P2.

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Fig. 3 Distribution of r.m.s. velocity fluctuations (urms , vrms , wrms ) and turbulent kinetic energy, scaled by Kolmogorov units

As pointed out by Kuroda et al. [5], the velocity fluctuations near the moving wall do not scale with the friction velocity. Furthermore, wall units cannot be consistently defined for shear-less flow. The velocity distributions can be alternatively scaled [11] with respect to Kolmogorov units taken at the wall (ηw = (ν 3 εw )1/4 , uη w = (νεw )1/4 ). The normalized velocity r.m.s. components and the turbulent kinetic energy are reported in Fig. 3. Compared to standard scaling based on friction quantities (not reported) a better collapse of statistics is found, even though a universal behavior is not achieved.

4 Turbulence Structure near the Moving Wall The effects of the wall speed on the structure of turbulence can be analyzed in terms of the Reynolds stress anisotropy tensor [7], ui uj

1 − δij . (2) 2k 3 The anisotropy of the turbulent stress field can be characterized in terms of its second and third invariants, bij =

III b = bij bj k bki ,

II b = bij bj i .

(3)

All realizable fields must lie in a domain (Lumley’s triangle) whose vertices represent one-, two- and three-components isotropic turbulence.

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Fig. 4 Anisotropy invariant map of the Reynolds stress tensor in the lower part of the channel

The anisotropy invariant map in the lower half of the channel is reported in Fig. 4 for the complete set of current simulations. For flows with large values of |γ | (corresponding to high shear rates) the maps are very similar to a canonical channel [3], the main differences being found in the central region for C-like flows, where turbulence is less isotropic. However, when the mean shear is large, the anisotropy in the wall region is very similar. The blocking effect of the wall (suppressing the wallnormal velocity fluctuations) yields two-component turbulence [4], but the streamwise fluctuating motions are enhanced with respect to the spanwise ones. When |γ | approaches zero in the neighborhood of the wall, the flow is only affected by the blocking effect, and the transfer of energy from the vertical component to the others is more equilibrated. The maps are therefore shifted towards the point corresponding to two-component isotropic turbulence. To analyze the effect of mean shear on structures, we consider the shear parameter S ∗ and the energy parameter K ∗ , defined as S ∗ = 2Sk/ε,

K ∗ = 2u 2 /(v 2 + w 2 ).

(4)

Lee et al. [6] have shown that large values of the shear rate parameter in homogeneous turbulence are typically associated with coherent structures similar to the “streaks” present in the sublayer of wall-bounded turbulent shear flows. Those authors also identified the energy partition parameter K ∗ as a proper quantity to quantify the concentration of turbulent kinetic energy in the streamwise component, and found that K ∗ = 5 is a suitable threshold level to infer the presence of streaky structures. The distributions of S ∗ and K ∗ are reported in Fig. 5 as a function of the distance from the moving wall, and normalized by the dissipative scale at the wall. Inspection of Fig. 5 shows the occurrence of large values of S ∗ for the C1, C2 and P4 simulations, moderate values for C3, P3 and rather small values for the other

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Fig. 5 Distribution of shear parameter S ∗ (a), and energy parameter K ∗ (b) in Kolmogorov units. The dashed line in (b) denotes K ∗ = 5

runs. Assuming K ∗ > 5 as criterion to detect streaky structures, we would expect to find streaks in five of the eight runs here presented (even if for C3 and P3 the range with K ∗ > 5 is very narrow). This inference is confirmed by the visual inspection of the streamwise velocity field (not shown).

5 Conclusions A systematic investigation of Poiseuille–Couette flow has been carried out in a wide range of flow conditions, including Couette-like, Poiseuille-like, and (quasi) shearless flow, at the same Reynolds number. The flow on the moving wall has been investigated in terms of mean and statistical flow properties. The study has shown the occurrence of a narrow logarithmic region for a limited set of conditions. Furthermore, the velocity fluctuations have been found to scale well with the Kolmogorov velocity- and length-scales taken at the wall, rather than in wall units, thus supporting the claims of Stanislas et al. [11]. The analysis of the turbulence structure suggests that the occurrence of streaky patterns in Poiseuille–Couette flow is a function of the parameter γ , and in particular that streaks are absent in the case of shear-less flow over the moving wall. Acknowledgements This study was carried out within the WALLTURB research project, funded by EU. Some of the calculations reported in the paper were performed on clusters of the CASPUR consortium, and some on parallel computers purchased with a PRIN grant of the Italian Ministry of University and Research (MIUR).

References 1. del Álamo, J.C., Jiménez, J.: Spectra of the very large anisotropic scales in turbulent channels. Phys. Fluids 15, L41–L44 (2003) 2. El Telbany, M., Reynolds, A.J.: Velocity distributions in plane turbulent channel flows. J. Fluid Mech. 100, 1–29 (1980)

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3. Frohnapfel, B., Lammers, P., Jovanovic, J., Durst, F.: Interpretation of the mechanism associated with turbulent drag reduction in terms of anisotropy invariants. J. Fluid Mech. 577, 457–466 (2007) 4. Hunt, J., Graham, J.: Free-stream turbulence near plane boundaries. J. Fluid Mech. 84, 209– 235 (1978) 5. Kuroda, A., Kasagi, N., Hirata, M.: Direct numerical simulation of turbulent plane Couette– Poiseuille flows: effect of mean shear on the near-wall turbulence structures. In: Proc. 9th Symp. Turbulent Shear Flows, Kyoto, vol. 1, pp. 8.4.1–8.4.6 (1993) 6. Lee, M.J., Kim, J., Moin, P.: Structure of turbulence at high shear rate. J. Fluid Mech. 216, 561–583 (1990) 7. Lumley, J.: Computational modelling of turbulent flows. Adv. Appl. Mech. 18, 123–176 (1978) 8. Nakabayashi, K., Kitoh, O., Katoh, Y.: Similarity laws of velocity profiles and turbulence characteristics of Couette–Poiseuille turbulent flows. J. Fluid Mech. 507, 43–69 (2004) 9. Nikuradse, J.: Laws of flow in rough pipes. NASA TM 1292 (1950) 10. Orlandi, P.: Fluid Flow Phenomena: A Numerical Toolkit. Kluwer, Dordrecht (2000) 11. Stanislas, M., Perret, L., Foucaut, J.M.: Vortical structures in the turbulent boundary layer: a possible route to a universal representation. J. Fluid Mech. 602, 327–382 (2008)

Session 6: Theory

• LES-Langevin Approach for Turbulent Channel Flow R. Dolganov, B. Dubrulle, and J.-P. Laval • A Scale-Entropy Diffusion Equation for Wall Turbulence H. Kassem and D. Queiros Conde • Direct Simulations for Wall Modeling of Multicomponent Reacting Compressible Turbulent Flows O. Cabrit and F. Nicoud (no paper) • A Specific Behaviour of Adverse Pressure Gradient Near Wall Flows S.I. Shah, J.-P. Laval, and M. Stanislas

LES-Langevin Approach for Turbulent Channel Flow Rostislav Dolganov, Bérengère Dubrulle, and Jean-Philippe Laval

Abstract The LES-Langevin model was studied on turbulent channel flows. The approach is based on the dynamics of the subgrid velocity scales previously studied in the frame of Rapid Distortion Theory. The subgrid stress tensor is modeled through a combination of a turbulent force and an eddy-viscosity. The turbulent force equation is derived from the subgrid scales velocity dynamics within the hypotheses of the Rapid Distortion Theory. The complex nonlinear terms containing the subgrid scale pressure were modeled by a stochastic forcing. The well-known eddy-viscosity closure was chosen to improve the resolved scales dissipation due to subgrid Reynolds stress. The advantage of the model is that the dynamics of turbulent force is prescribed by a dynamical equation derived from the Navier–Stokes equations. This allows a possibility to include all the important physical effects of the subgrid scales like the backward subgrid scale energy transfer. The direct modeling of the gradient of the subgrid scale tensor allows a reduction of the computational time compared to direct numerical simulation, which is the global objective of LES. The work demonstrates a need for further investigation on the equation of the turbulent force in order to better estimate the subgrid scale action.

1 Introduction The LES approach resolves the turbulent motion for the large (resolved) velocity scales which are sensitive to modeled (non-resolved) scales. An LES model provides R. Dolganov () · J.-P. Laval Laboratoire de Mécanique de Lille, CNRS, 59655 Villeneuve d’Ascq, France e-mail: [email protected] J.-P. Laval e-mail: [email protected] B. Dubrulle Groupe Instabilités et Turbulence, SPEC-CNRS, 91191 Gif sur Yvette, France e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_25, © Springer Science+Business Media B.V. 2011

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the dynamics of the large scales resolved in time. An extensive description of the existing models was provided by Sagaut [16]. The main purpose of the LES models is to represent correctly the energy transfer between the resolved and non-resolved scales. The eddy-viscosity models (like Smagorinsky, [17]) produce the dissipation of the resolved scales due to the motion of the non-resolved scales. Kraichnan [9] has shown that the turbulent scales closest to the separation scale are of particularly importance for the dissipation of the resolved scales. Nevertheless, the backward energy transfer from the smallest scales to the largest ones must be taken in consideration for the turbulence near a wall. These interactions are mostly non-local for large Reynolds numbers. Härtel et al. [8] studied a priori the energy transfer of the resolved scales in the presence of a wall. They have shown the importance of the mean and fluctuating parts of the SGS tensor for the energy transfer, as a function of the wall distance and scale separation. The analyses of the strain-rate correlation with the SGS stress have shown the non-trivial structure of the dissipation produced by the subgrid scales. Domaradzki et al. [5] studied the spectral representation of the energy transfer of the near-wall turbulence. The modeling of the backward energy transfer by a random force was studied by Leith [12]. The further development of the LES strategy led to the vast application of the model based on the explicit separation of the mesh-represented scales on the re¯ and subfilter scale velocity (u ). Between others, this is the solved scale velocity (u) case of the variational multiscale model (VMS) [15], the Subgrid Scale Estimation Model (SGEM) [6], the Resolvable Subfilter Scale model (RSFS) [18]. These models reconstruct the subfilter scales by interpolation (SGEM) or by a derived approximated equation (VMS, RSFS). The interaction between the resolved and subfilter ¯ scales are calculated by its definition from the nonlinear term: (u¯ · ∇)u + (u · ∇)u, so the energy transfer between the subfilter and resolved scales is computed exactly. Nevertheless, the backward energy transfer between the smallest turbulent scales and the resolved ones is not represented by these models. The LES-Langevin model consider the effect of the subgrid scales as a sum of a turbulent force and an eddy diffusive term. The diffusive term does not participate in the backscatter. The turbulent force can produce both the effects of dissipation and backscatter, and it is sensitive to the geometry of the flow and to the distance to the wall. The dynamics of this force is derived from the Navier–Stokes equations using the hypothesis of the time scale separation and the phenomenological approach. This approach is not based on the cascade mechanism. Compared to the model of Leith [12], the LES-Langevin method does not assume the correlation between the turbulent force and strain-rate and the turbulent force is not random, but is governed by an dynamical equation.

2 LES-Langevin Model for Wall Turbulence The LES-Langevin model is based on the approved Langevin model of the subgrid velocity of Laval et al. [11]. The approach of Laval et al. [11] resolves the

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filtered incompressible momentum equation for the resolved velocity u¯ and represents the subgrid scale dynamics by an approximate momentum equation for subgrid velocity u . Initially, the LES-Langevin model of Laval et al. [10] studied the isotropic turbulence using the modeling analogy to the Langevin model of turbulence of Laval et al. [11]. The approach consist in modeling of the non-linear velocity resolved-subfilter scales interaction by an acceleration vector l, called turbulent force: li = ∂j Cij = ∂j (u¯ i uj + u¯ j ui ). The subgrid–subgrid scales nonlinear term, represented by subgrid stress Rij = ui uj , is supposed to be effectively modeled by an eddy-viscosity approach: Rij = νt S¯ij , based on the filtered strain-rate: S¯ij = 1/2(∂j u¯ i + ∂i u¯ j ). The modeling of the tensor Rij by a turbulent diffusion is common to different LES models like Mixed model [14], Variational multiscale model [3], Resolved subfilter scale model [18]. The energy transfer produced by the acceleration l is dominant in the total energy transfer produced by the nonlinear term, as it was shown by Laval et al. [11] and Dubrulle et al. [7]. The development of a dynamical equation for the turbulent force l by Laval et al. in [10] leads to a complete LES-Langevin model: ∇ · u¯ = 0, ¯ ∂t u¯ + (u¯ · ∇)u¯ + l = −∇ p¯ + ∇(ν + νt )∇ u,

(1)

∂t l = −(l/τf ) − (u¯ · ∇)l − (l · ∇)u¯ + ∇(ν + νt )∇l + ξ0 + ξ, ¯ the vector f is defined by fi = ∂j (u¯ i u¯ j − u¯ i u¯ j ) and where ξ0 = −(u¯ · ∇)f − (f · ∇)u, τf is a characteristic time used for the friction term l/τf . The two first equations for resolved scales are solved on a coarse grid (x, y, z) and the last equation for the acceleration vector l was derived for all scales. In practice, it has be shown that a grid twice as fine as the one used for the resolved scales is sufficient to solve the equation for the acceleration vector. The vector ξ models the pressure term: ξ = (u¯ · ∇)∇p + [(∇p ) · ∇]u¯

(2)

as a stochastic force with zero mean value and a prescribed correlation function Tij (t − t , x − x ) such as: ξ = 0, ξi (t, x)ξj (t , x ) = Tij (t − t , x − x ).

(3)

In the homogeneous isotropic turbulence, the solution of Navier–Stokes equation is solenoidal and the pressure terms p¯ and p can be eliminated. In this case, the stochastic forcing can be completely eliminate from the model [10]. In the problem with a non-periodic boundary conditions, the modeling of the stochastic forcing should be taken into account explicitly.

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3 Estimation of Stochastic Forcing in the Case of Channel Flow The simulation of the channel flow by the LES-Langevin model requires the computation of the subgrid pressure for the evaluation of the stochastic forcing (2). The analytical derivation of Tij from (3) is difficult and probably needs a further hypothesis on the subgrid scale pressure. Our approach is based on a priori estimation of ξ(x) from the DNS data described in Marquillie et al. [13]. The 3D channel flow has the streamwise (x), wall-normal (y) and spanwise (z) directions. We approximate the vector ξ from (2) by a function, which depends separately on space and time dimensions: ξ0i (t, x, y, z) = Fti (t)Fxi (x)Fyi (y)Fzi (z),

(4)

where index “i” means the components of the vector ξ . The functions Fti (t), Fxi (x), Fyi (y), and Fzi (z) model the time, streamwise, wall normal and spanwise direction dependence respectively. Fyi provides the intensity of the vector ξ as a function of the normal direction to a wall. The intensity of ξ does not change in the homogeneous x and z directions. The functions Fxi and Fzi supply the spectral form of the ξ -vector in these directions. Time function Fti provides a time-dependence of ξ . Without any loss of generality, we can assume: 2 Fxi x = Fzi2 z = Fti2 t = 1

Fyi (y) ≡ ξi2 (x, t)2x,z,t .

where ·x,z,t stands for the averaging of the function in the streamwise direction, spanwise direction and time. The functions Fxi and Fzi are defined from (2) and through the spectral form of the mean functions Fˆxi and Fˆzi . At each time step, a new set of random phases is chosen for Fˆxi and Fˆzi . If we suppose that the typical correlation time of ξi is larger than the time step, a time behavior of ξ can be introduced using a Markovian series [2]: ⎧ n+1 = C1 ξin + C2 ξ0i , ⎪ ⎨ ξi (5) C1 = 1 − dt/τstf , ⎪ ⎩ C2 = 2dt/τstf , where τstf is the expected time correlation. The vector ξ becomes random, if C1 = 0 and C2 = 1.

3.1 A Priori Tests The vectors l and ξ depend on the form and size of the explicit filter used to separate the resolved scales from the subgrid scales and depend on the physics of the flow. The DNS database of channel flow was used to study the sensitivity of these vectors to the filter size, filter shape and to the smallest turbulent scales. The DNS data were generated for the channel flow at Reτ = 600 [13]. The computational domain was 2π × 2 × π with a resolution of 768 × 257 × 384 grid points. The a priori tests were done from 24 fields equally separated with a time interval of T ∗ T0 = 400,

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Table 1 kx , ky and kz are the maximal wavelengths of retained for the subgrid scales u . N1, N2 and N3 correspond to the DNS resolution, others are obtained by a sampling to a coarser mesh. k¯x , k¯y and k¯z are the cutoff size of the explicit filter used for the resolved scales; kx of N6 corresponds to a discrete filter, described in [4] k¯x k¯y k¯z Case kx ky kz N1 N2 N3 N4 N5 N6

384 384 384 160 128 128

257 257 257 129 97 97

384 384 384 192 128 128

64 64 32 64 64 discrete

65 49 33 49 49 49

64 32 16 32 32 32

where characteristic time T0 is based on the half height of the channel and the center channel velocity: T0 ≡ Ly /2/u0 = 1. The tested filters and resolutions are presented in Table 1. As most of the filters used for the tests are spectral, the maximum wavenumbers k i = Li /i are indicated instead of the mesh size.

3.2 The Filter and Spatial Resolution Dependence of the Stochastic Forcing and the Turbulent Force The a priori computation of the l and ξ vectors presented in Fig. 1 has shown that a + ∈ [60; 110]) than the position of maximum is achieved at larger wall distance (ymax + the turbulence kinetic energy maximum (y ≈ 15). This is surprising because the vector l describes most of the resolved-subgrid scale energy transfer which is maximal at the peak of turbulent kinetic energy [16]. The correction can not be supplied by the vector ξ because its maximum is also a priori shifted toward larger y + . The spectral maximum of the vector l is close to the filter size. So, the energy transfer is particularly important for the near cutoff modes. This confirms the statement of Kraichnan [9] that near cutoff turbulent kinetic energy transfer is preponderant for the whole resolved-subgrid energy transfer. Nevertheless, a priori tests shows a large decrease of the vector l in the small resolved scales range, which is similar to the results of Dubrulle et al. [7]. This is crucial for LES-Langevin modeling of the resolved-subfilter energy transfer: the largest modes of the vector l are not represented in the resolved range and the resolved range spectrum has a very small intensity. The forcing ξ has the same spectral property and cannot qualitatively change the spectrum of the vector l during the simulations.

3.3 Time Scale Separation ¯ u and ∇p were calculated from the a priori tests by a The time correlation of u, classical definition: cf = f (t)f (0)/f (0)2 ,

(6)

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Fig. 1 A priori analysis of the stochastic forcing (vector ξy ): mean profile of |ξz | (a) and |lz | (b), kz spectra of |ξz | at y + = 17 (c) and kx spectra of |lz | in the center of channel (y + = 620) (d). Tests from Table 1: —— is N1, · · · · · · is N2, - - - - is N3, –·– is N4, —· · ·— is N5, — — — is N6

where the average · is performed over the time and the homogeneous directions of the system. An example of the correlation function is presented in Fig. 2. The normal large scale velocity is compared to the small scale velocity and to the same component of the small scale pressure gradient. We see that the small scale pressure gradient correlation time is shorter than that of the subgrid velocity and the resolved velocity: τv¯ > τv > τ∂y p τv¯ ≈ 1 τv ≈ 0.3 τ∂y p ≈ 0.03. The derivation of the turbulent force equation requires the hypothesis of rapidness ¯ The shortest correlation of the of the subgrid velocity scales [10]: |∂t u | |∂t u|. subgrid pressure gradient ∇p allows a possibility to model the stochastic forcing ξ by a random function, or by a Markovian series (5) with τstf < τ∂y p .

4 Results and Discussions The LES-Langevin model (cs = 0.1) was compared to the Smagorinsky model (cs = 0.1) with Van-Driest damping, VMS model (small–small formulation of Col¯ lis et al. [3], cs = 0.1) and coarse DNS resolved on u-grid (CDNS-RS) and on

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Fig. 2 Time correlation cf defined by (6) at y + ≈ 17 (a) and in the center of the channel (b) of test N5 of Table 1. —— is the large scale normal velocity, · · · · · · is the small scale normal velocity, - - - - is the normal small scale pressure gradient component Table 2 The error of the friction velocity: δτ = (uτ,LES − uτ,DNS )/uτ,DNS Model DNS

+ x

+ y

+ z

Reτ

δτ

5.0

0.05–7.58

5.0

618

–

CDNS-RS

27.5

0.3–18.4

55.1

561

−0.092

SMAG

28.7

0.3–19.1

57.4

585

−0.0537

LES-L

28.7

0.3–19.1

57.4

584

−0.054

VMS

30.3

0.3–20.2

60.7

618

0.000

CDNS-SS

30.4

0.3–20.3

60.8

620

0.0003

u -grid (CDNS-SS). The resolution of the simulations, Reynolds numbers based on the shear velocity and the error of the friction velocity prediction are presented in Table 2. In Fig. 3 one can see that the VMS model better predicts the mean streamwise velocity and Reynolds stress than the LES-Langevin model. The Smagorinsky model and the LES-Langevin approach lead to similar results for the mean streamwise velocity and slightly different ones of the buffer layer. The discrepancy of the streamwise fluctuating velocity RMS produced by LES-Langevin or Smagorinsky model and reference DNS is due to the coarse LES resolution in spanwise direction, which leads to the production of the spurious structures called “superstreaks” [1]. Dolganov [4] has shown that for + z = 20, LES-Langevin produces better streamwise velocity RMS in the buffer and log regions. The difference in the modeling of the energy transfer between the resolved and subfilter scales brings the difference in the resolved scale velocity statistics presented in Fig. 3. The VMS model computes this ¯ which is different to the energy transfer by its definition: u¯ · [(u¯ · ∇)u + (u · ∇)u], LES-Langevin model: (u¯ · l). Its is clear that for such a rough resolution (presented in Table 2), the modeling of the equation for the turbulent force (1) needs a further refinement. The optimization of the vector l equation does not significantly improve the resolved velocity statistics. Partially, this is due to the fact, that the amplitude

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Fig. 3 Velocity profiles (a) and velocity RMS (b, c, d) of Smagorinsky model with different cs compared to the DNS. All the curves are referenced in Table 2. DNS ——, CDNS-RS · · · · · ·, the Smagorinsky model - - - -, LES-Langevin —·—, VMS —· · ·—, CDNS-SS — — —

of l variations can not reach its a priori values [4] and the feedback of the vector l on the resolved velocity u¯ is weak. Probably, some local mechanisms of the turbulent energy transfer are not represented by actual model, which can be studied in the future. The resolved velocity becomes more sensitive to the modeling of the vector l for the modeling constant cs > 0.1 [4]. This shows that the eddy-viscosity action is stronger than that of the acceleration l. The dominance of the backscatter produced by the vector l is clear from the comparison of the LES-Langevin and Smagorinsky models. We see from Fig. 3, that the modeling of the acceleration l increases the resolved scale energy. The effect of the forward energy transfer can be studied in the future to understand what mechanisms should be produced by the vector l. The comparison of the VMS model with the coarse DNS resolved on the u grid shows that the eddy-viscosity closure helps to improve the Reynolds stress in the center of the channel. The mean velocity profiles are coincident as well as the Reynolds stress in the viscous, buffer and log regions. So, even a coarse DNS resolved up to u is not so bad and produces the reasonable turbulent statistics, which limit the interest of the LES applications to more coarse resolutions.

5 Conclusion The present study shows the complexity of modeling of the near-wall flows in the frame of Langevin approach. The modeled turbulent force does not provide the cor-

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rect energy transfer between the resolved and unresolved scales as it is expected for gradient of the Cross Stress tensor. Further investigations are necessary to improve the RDT hypotheses and the phenomenological determination and stochastic approximations of the subfilter/subgrid velocity and pressure. The work illustrates the sensitivity of a SGS closure to the type of turbulent flow. The LES-Langevin provided good statistics for isotropic turbulence but the modeled turbulence intensity in the center of the channel does not compare well with the DNS reference value. The a priori estimation of the performance of the model to particular boundary conditions and numerical scheme can be an issue for LES in general. Acknowledgements This work was supported by WALLTURB (A European synergy for the assessment of wall turbulence) which is funded by the EC under the 6th framework program (CONTRACT: AST4-CT-2005-516008). The simulations were performed at IDRIS (French CNRS Computing Facilities).

References 1. Bagett, J.S.: The feasibility of merging LES with RANS for the near-wall region of attached turbulent flows. Annual Research Briefs. Center for Turbulent Research, pp. 267–277 (1998) 2. Castronovo, E., Kramer, P.R.: Subdiffusion and superdiffusion in Lagrangian stochastic models of oceanic transport. Monte Carlo Methods Appl. 10(3–4), 245–257 (December 2004) 3. Collis, S.S.: Monitoring unresolved scales in multiscale turbulence modeling. Phys. Fluids 13(6), 1800–1806 (June 2001) 4. Dolganov, R.: Développement d’un modèle LES basé sur la théorie de la distorsion rapide. Ph.D. thesis, Ecole Centrale de Lille (2009) (in English) 5. Domaradzki, J.A., Liu, W., Härtel, C., Kleiser, L.: Energy transfer in numerically simulated wall-bounded turbulent flows. Phys. Fluids 6(4), 1583–1599 (April 1994) 6. Domaradzki, J.A., Loh, K.C., Yee, P.P.: Large eddy simulations using the subgrid-scale estimation model and truncated Navier–Stokes dynamics. Theor. Comput. Fluid Dyn. 15, 421–450 (2002) 7. Dubrulle, B., Laval, J.-P., Sullivan, P., Werne, J.: A new dynamical subgrid model for planetary surface layer. Part I: The model and a priori tests. J. Atmos. Sci. 59(4), 861–876 (February 2002) 8. Härtel, C., Kleiser, L., Unger, F., Friedrich, R.: Subgrid-scale energy transfer in the near-wall region of turbulent flows. Phys. Fluids 6(9), 3130–3143 (September 1994) 9. Kraichnan, R.H.: Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33, 1521 (1976) 10. Laval, J.-P., Dubrulle, B.: A LES-Langevin model for turbulence. Eur. Phys. J. B 49, 471–481 (2006) 11. Laval, J.-P., Dubrulle, B., McWilliams, J.C.: Langevin models of turbulence: renormalization group, distant interaction algorithms or rapid distorsion theory? Phys. Fluids 15(5), 1327– 1339 (May 2003) 12. Leith, C.E.: Stochastic backscatter in a subgrid-scale model — plane shear mixing layer. Phys. Fluids A, Fluid Dyn. 2(3), 297–299 (March 1990) 13. Marquillie, M., Laval, J.-P., Dolganov, R.: Direct numerical simulation of a separated channel flow with a smooth profile. J. Turbul. 9(1), 1–23 (January 2008) 14. Piomelli, U., Moin, P., Ferziger, J.H.: Model consistency in large eddy simulation of turbulent channel flow. Phys. Fluids 31(7), 1884–1891 (July 1988) 15. Ramakrishnan, S., Collis, S.S.: Partition selection in multiscale turbulence modeling. Phys. Fluids 18, 075105 (2006)

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16. Sagaut, P.: Large Eddy Simulations for Incompressible Flows. Springer, Berlin (2002) 17. Smagorinsky, J.: General circulation experiments with the primitive equations. Mon. Weather Rev. 91(3), 99–166 (March 1963) 18. Zhou, Y., Brasseur, J.G., Juneja, A.: A resolvable subfilter-scale model specific to large-eddy simulation of under-resolved turbulence. Phys. Fluids 13(9), 2602–2610 (September 2001)

A Scale-Entropy Diffusion Equation for Wall Turbulence Hassan Kassem and Diogo Queiros-Conde

Abstract We applied on a database of PIV fields obtained at Laboratoire de Mécanique de Lille corresponding to a turbulent boundary layer the statistical and geometrical tools defined in the context of entropic-skins theory. We are interested by the spatial organization of velocity fluctuations. We define the absolute value of velocity fluctuation δV defined relatively to the mean velocity. For given value δVs (the threshold), the set Ω(δVs ) is defined by taking the points on the field where δV ≤ δVs . We thus define a hierarchy of sets for the threshold δVs ranging from the Kolmogorov velocity (the corresponding set is noted ΩK ) to the turbulent intensity U (the corresponding set is noted ΩU ). We then characterize the multi-scale features of the sets Ω(δVs ). It is shown that, between Taylor and integral scale, the set Ω(δVs ) can be considered as self-similar which fractal dimension is noted Ds . We found that fractal dimension varies linearly with logarithm of ratio δVs /U . The relation is Ds = 2 + β ln(δVs /U ) with β ≈ 0.12–0.26: this result is obtained for all the values y + we worked with. We then defined an equivalent dispersion scale le such as N (δVs ) − NK = le2 . It is shown that δVs /U ∼ le1.52 . We thus can write Ds = 2 + β ln(le / l0 ) with β ≈ 0.18–0.39. These results are interpreted in the context of a scale-entropy diffusion equation introduced to characterize multi-scale geometrical features of turbulence.

1 Introduction Wall turbulence displays three important features which strongly suggests to deal with this problem by using topological and multi-scale tools. (i) The level of intermittency increases in the vicinity of the wall: it increases as the distance to the H. Kassem · D. Queiros-Conde () ENSTA-ParisTech, Unite Chimie et Procédés, 32 Bb Victor, 75015 Paris, France e-mail: [email protected] H. Kassem e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_26, © Springer Science+Business Media B.V. 2011

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wall is decreasing. Due to bursting events, highly non-Gaussian statistics characterize wall turbulence and classical tools can not be used. (ii) The scale-range in the direction perpendicular to the wall is progressively reduced as the distance to the wall is reduced. Near to the wall, the scale-range is very small and almost reduced to a single scale. (iii) The need to describe a wall turbulent flow (but also an homogeneous and isotropic turbulence) by a variety of forms such as filaments, sheets or specific vortices. This represents a problem for the models dealing with turbulence since the definition of scales is usually done through Kolmogorov theory i.e. which do not display high intermittency [1, 2]. Other arguments and tools are thus necessary to define the scale-range of a turbulence in the vicinity of a wall. Through a great number of works, it appears more and more clearly that a deep thinking on scale-dynamics is necessary to understand wall turbulence and, more generally, turbulence. We present in this paper a multiscale analysis of PIV fields using an adequate thresholding procedure based on the velocity fluctuation relatively to the mean. We propose to describe the geometry of these sets by using the concept of scale-entropy ruled by a scale-entropy diffusion equation.

2 Scale-Entropy Diffusion Equation Scale entropy [3, 4] noted Sx,t defines at a given scale li defined by x = ln(li / l0 ) (l0 is the integral scale) and time t the level of space-occupation of the system for the scale li at time t: Sx,t = ln(l03 /Vx,t ) where Vx,t is the volume occupied by the set at scale defined by x and at time t. It can be written Vx,t = Ni,0 lid where Ni,0 represents the number of balls Ni,0 of scales li needed to cover a part of the system having a size l0 obtained by a box-counting procedure. It has been shown that this quantity is ruled by a scale-entropy diffusion equation which gives scale-entropy as a function of scale and time: ∂ 2 Sx,t 1 ∂Sx,t . − ω(x, t) = 2 χ ∂t ∂x

(1)

The quantity ω(x, t) represents a scale-entropy flux sink (or source) which is, in the general case, scale and time dependent. The ability to describe scale-dynamics in time if given thanks to the concept of scale-diffusivity. The evolution of multiscale structure can be studied by this equation and compared to experimental measurements. We will consider here only the stationary case. The scale-entropy flux is defined by φx = dSx /dx which follows φx = Δx − d where Δx is the local fractal dimension (fractal dimension is scale-dependent) and d = 3 is the embedding dimension. If we assume the simple case of an uniform scale-entropy sink (ω(x) = β with β ≥ 0) then it can be easily obtained Sx = (β/2)2 + (Δ0 − d)x where β = (Δ0 − Δc )/ ln(l0 / lc ) where Δ0 and Δc are respectively the local fractal dimension associated to integral scale l0 and smallest cutoff scale lc . The fractal dimension depends thus linearly on scale logarithm: Δx = Δ0 + βx . It means that, for an uniform sink, a scale analysis should lead to a parabolic form in logarithmic

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coordinates. Fractality has thus here the status of a very specific case corresponding in scale-space to a stationary regime and no sink of scale-entropy (β = 0). For scales belonging to the range [lc ; l0 ], an usual scale analysis at scale li gives the number of balls Ni,0 needed to cover a part of the system having a size l0 . Parabolic scaling (ω(x) = β) corresponds thus to a local fractal dimension evolving linearly with scale-logarithm.

3 Experimental Measurement of Structure Functions, Scaling Exponents and Intermittency Efficiency We studied the flow corresponding to a boundary layer with a mean velocity UE = 3 m.s−1 and with zero pressure gradient. Its global thickness is δ = 0.35 m; the displacement thickness is δ ∗ = 0.055 m; the momentum thickness is θ = 0.041 m. The von Kármán constant is κ = 0.41. The friction velocity is equal to u∗ = 0.11 m.s−1 . The values of y + belong to the range 14–44. A wall unit (wu) is equal to 0.125 mm. By using classical methods we computed the integral scale and the Taylor scale of the flow on the PIV data. This is done by computing the auto-correlation function of the longitudinal velocity. Let us note Vz (x) and Vz (x + l) the velocities at x and x + l for the transversal coordinate z. The auto-correlation function is ≺

Vz (x)Vz (x + l) . Vz (x)2

(2)

The average is done taking all the lines by varying z the transversal coordinate. In the range of y + available, the integral scale and Taylor scale seem to be constant: l0 ≈ 120 wu and λ = 38 wu i.e. l0 ≈ 15 mm and λ = 4.75 mm.

4 Detection of Structures by a Thresholding Procedure of Velocity Fluctuations We studied the sets obtained by a thresholding procedure on the velocity fluctuation relatively to the mean velocity noted Vmoy . We used the positive velocity fluctuation. Let thus be δVz (x) = |Vz (x) − Vmoy | the absolute value of this fluctuation. We then define a thresholding noted δVs which can vary from the Kolmogorov velocity to several times the value of U . From the PIV fields, we then define the sets Ω(δVs ) such as x ∈ Ω(δVs ) if δVs ≤ δVs . As an illustration, we can see in Fig. 4 three sets extracted by the thresholdind procedure on the fluctuation δVz (x) = |Vz (x) − Vmoy |. We are interested by the space-filling properties of the sets obtained by this way and the fractal dimension the set displays if a fractal character can be evidenced. To do this we used the box-counting method. The set to study is placed on a grid of mesh-size li . The global size of the grid being lT . We then compute the number Ni of meshes touched by the set. If the set is fractal we should have a power law

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Fig. 1 Scale-analyses (ln Ni vs ln li ) for two values of threshold (y + = 44). Each point is an average on 500 images −Δ

Ni ∼ li f over an enough wide scale-range at least for the scales belonging to Taylor scale and integral scale, Δf being the fractal dimension. We are expecting that the values of corresponding to Kolmogorov velocity uK and turbulent intensity U will have a particular behavior. Some theoretical developments of entropic-skins theory showed that for δVs = uK we should obtain the corresponding fractal dimension Δ(uK ) =

2 + ζ1 , 1 + ζ1

(3)

where ζ1 is the scaling exponent corresponding to the first order structure function ≺|δV |l with ≺|δV |l ∼ l ζ1 . This quantity corresponds to the turn-over time of the flow at the scale l. Without intermittency, the Kolmogorov theory predicts ζ1 = 1/3 which would give Δ(uK ) = 7/4. With intermittency, we have ζ1 = 0.36 i.e. Δ(uK ) = 1.735. For thresholdings approaching turbulent intensity we expect to recover all the field i.e. Δ(U ) = 2. We give some examples of scale-analyses in Fig. 1. We determined a fractal dimension on a scale-range from to Taylor scale to integral scale which have been measured directly by classical means. The slope gives the fractal dimension of the corresponding set. Our main result is shown in Fig. 2. The fractal dimension measured noted Ds corresponding to a threshold value is determined. We emphasize the fact that the measurement is an average on 500 images for each case. This means that each point on the curve results from a treatment on 500 images obtained at the same distance from the wall. We found that the fractal dimension Ds is a linear function of the quantity ln(δVs /U ) as it is expressed in Fig. 3 for y + = 44. This behavior is the same for all the values of y + but the slope varies. Two sets of curves can be observed around y + = 30. But the main result is that the behavior is linear. We thus obtain for the fractal dimension the following relation Ds = 2 + β ln(δVs /U )

with β ≈ 0.12–0.26.

(4)

This linear evolution of fractal dimension is particularly noticeable. The slope depends on y + . In the context of the scale-entropy diffusion equation and in the

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Fig. 2 Fractal dimension Ds as a function of threshold δVs . Each point is an average on 500 images Fig. 3 Quantity 2 − Ds vs ln(δVs /U )

specific case of “parabolic scaling” we developed, the fractal dimension is a linear function of scale-logarithm [1, 2]. Can we interpret the ratio δVs /U as a ratio of scales?

5 The Notion of Equivalent Dispersion Scale We propose in this report an “equivalent dispersion scale” defined by the following way. We start from the set ΩK corresponding to the a velocity fluctuation equal to Kolmogorov velocity uK . This set will be taken as a reference. On Fig. 4, we give an example of this procedure. We would like to introduce a characteristic scale which would quantify the “distance” of a thresholded set Ω(δVs ) to the set of reference ΩK obtained for Kolmogorov velocity. To simplify our presentation let us consider the sketches in Fig. 5 where the set ΩK is symbolized in dark and two thresholded sets are indicated in different greys. We thus want to determine a sort of mean distance from the thresholded set Ω(δVs ) to the Kolmogorov set ΩK taken as a reference. A simple way to do this is to determine a scale from the increase of surface brought by the set Ω(δVs ) relatively to the Kolmogorov set. In Fig. 5a, we show the differentiated set obtained by subtracting ΩK to the set of Fig. 5b.

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Fig. 4 The Kolmogorov set ΩK is taken as a reference. We measure a sort of distance to Kolmogorov set

Fig. 5 Sketch explaing the definition of equivalent scale from thresholded sets Ω(δVs ). The set in dark correspond to the set ΩK obtained for Kolmogorov fluctuation which is taken as the reference

Having the increase of surface due to the thresholded set Ω(δVs ) relatively to Kolmogorov set ΩK , we determine an equivalent surface which gives us a characteristic scale r. Let us note N (uK ) the number of pixels occupied by the Kolmogorov set and N (δVs ) the number of pixels occupied by the set Ω(δVs ). The difference N(δVs ) − N(uK ) represents in pixels the differentiated surface. We choose a scale le (in pixels) such as 4le2 = 12 [N (δVs ) − N (uK )]. It thus implies the relation (in pixels) for equivalent scale le = 12 [N (δVs ) − N (uK )]1/2 (see Fig. 6). The measurement of equivalent dispersion scale is done by varying the threshold value and several values of y + . The variation with y + seems to be weak. When the threshold δVs is such as δVs = U , the equivalent scale is close to the integral scale

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Fig. 6 Sketch explaining the definition of equivalent dispersion scale le

Fig. 7 The ratio δVs /U as a function of le / l0 in ln–ln coordinates

measured by using the auto-correlation function of the velocity. We can verify that integral scale l0 is closed to the measured equivalent scale in our range of y + values. Let us come back to the evolution of equivalent dispersion scale with threshold δVs . We express in Fig. 7, in a ln–ln diagram, the quantity δVs /U as a function of le / l0 . A power law links the ratio to the ratio of scales of le / l0 . We thus can write δVs le 1.52 . U = ( l0 ) Since Ds = 2 + β ln(δVs /U ) with β ≈ 0.12–0.26, we thus can write Ds = 2 + β ln(le / l0 )

with β ≈ 0.18–0.39.

(5)

The fractal dimension is thus a linear function of the scale-logarithm based on the equivalent dispersion scale.

6 Conclusion Based on the scale-entropy diffusion equation introduced a few years ago to describe the multi-scale nature of turbulent interfaces and, more generally, scale dynamics in turbulence, we analyzed a database of PIV fields corresponding to a turbulentboundary layer. We mainly showed that, using the absolute fluctuation of velocity as a thresholding quantity, the multi-scale structure of the resulting sets can be considered fractal between Taylor and integral scales. The corresponding fractal dimension varies linearly with the logarithm of ratio δVs /U . We thus introduced an equivalent dispersion scale which follows a power law with ratio δVs /U ; this implies that fractal dimension is a linear function of logarithm of this equivalent scale. This behavior is the one predicted in the context of scale-entropy diffusion for stationary conditions and a constant scale-entropy production through scale-range.

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References 1. Queiros-Conde, D.: Entropic skins model in fully developed turbulence. C. R. Acad. Sci. 328(7), 541–546 (2000) 2. Queiros-Conde, D.: Internal symmetry in the multifractal spectrum of fully developed turbulence. Phys. Rev. E 64, 015301(R) (2001) 3. Queiros-Conde, D.: A diffusion equation to describe scale- and time-dependent dimensions of turbulent interfaces. Proc. R. Soc. Lond. A 459(2040), 3043–3059 (2003) 4. Queiros-Conde, D., Foucher, F., Mounam-Rousselle, C., Kassem, H., Feidt, M.: A scaleentropy diffusion equation to describe the multi-scale features on turbulent flames near a wall. Physica A 387, 6712–6724 (2008)

A Specific Behaviour of Adverse Pressure Gradient Near Wall Flows Syed-Imran Shah, Jean-Philippe Laval, and Michel Stanislas

Abstract Adverse pressure gradient turbulent flows are of prime importance for aeronautics as they are characteristic of the suction side flow along an airfoil at any positive angle of attack. These flows have been known for a long time to pose modelling problems. The turbulence behaviour departs significantly from the standard near wall turbulence (as observed in flat plate boundary layers and channels). In diffusers, as well as in boundary layers, the flow separates from the wall as soon as the adverse pressure gradient is strong enough. In the present contribution, a few selected data from the literature are reviewed and compared together and with a recent DNS of converging–diverging channel flow performed in the frame of the WALLTURB project. This simulation provides data on APG near wall flow, both with and without curvature. The analysis of these data indicates that an instability is developing inside the turbulent near wall flow in both cases. The comparison of turbulent statistics with the data from the literature indicates that this phenomenon is fairly general in APG near wall flows.

1 Introduction Whenever a flow experiences a rising static pressure in the mean flow direction, it is said to undergo an Adverse Pressure Gradient (APG). In this case the flow decelerates, inducing a reduction in the kinetic energy. This has important consequences for a boundary layer flow because the adverse pressure gradient, if strong enough, can cause flow separation which leads to a large loss of total pressure alongwith loss of lift and control. Therefore turbulent boundary layers subjected to adverse pressure S.-I. Shah () · J.-P. Laval · M. Stanislas Laboratoire de Mécanique de Lille, CNRS, 59655 Villeneuve d’Ascq, France e-mail: [email protected] J.-P. Laval e-mail: [email protected] M. Stanislas e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_27, © Springer Science+Business Media B.V. 2011

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gradients is an important phenomenon in engineering and is of great technological interest. It has been the subject of many research projects and still remains an active field of research. The interest in understanding adverse pressure gradient turbulent boundary layer flows also originates from modelling requirements. Modeling of turbulent flows with adverse pressure gradient is still a tough objective. Calibration of turbulence models requires that precise experiments and simulations be conducted. Understanding of adverse pressure gradient flows is also important from the point of view of flow and turbulence control. The research on turbulent boundary layers started with Prandtl’s zero pressure gradient (ZPG) boundary layer along a flat plate and progressed to boundary layer flows with pressure gradients. For research purposes, the flows with adverse pressure gradient were mostly generated in straight- or curved-walled diffusers or other geometries with suction to avoid separation. In practical flows, a favourable pressure gradient (FPG) usually precedes the adverse pressure gradient and thus the converging–diverging geometries were mostly adopted for experiments and numerical simulations. This paper on adverse pressure gradient turbulent boundary layers focuses on the behaviour of streamwise velocity fluctuations.

2 LML Experiment The hot-wire anemometry (HWA) experiments of Bernard et al. (2003) [2], was carried out under the European project AEROMEMS. These hot-wire measurements were taken in the boundary layer wind tunnel of Laboratoire de Mécanique de Lille (LML), France. The tunnel has a length of 20 m and its section is 1 × 2 m2 . The boundary layer under study develops on the lower wall of the tunnel. The maximum external velocity is 10 m/s, resulting in a momentum thickness Reynolds number between 7.5 × 103 and 2 × 104 . Constant-temperature anemometers were used, with platinum-plated tungsten wire of 2.5 micron in diameter and 0.5 mm in length, manufactured by AUSPEX. The probes were mounted on a computer controlled traversing system. APG was created by the second-half of a surface bump. The bump consists of a main convex surface with smaller concave surfaces on the leading and trailing edge sides. It was designed to simulate the conditions of the suction side of an aircraft wing at high angle of attack. The boundary layer studied develops over the bump and thus the flow undergoes both the pressure gradient and curvature. For further details, see [2]. Figure 1 shows the streamwise velocity fluctuations for this data-set for different values of the curvilinear coordinate s (with origin of s at the top of the bump). The reference friction velocity uτ o is the one at the origin.

3 LML Direct Numerical Simulation The direct numerical simulation (DNS) of the turbulent flow in a converging– diverging channel was performed as part of the WALLTURB European project. The

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Fig. 1 Streamwise velocity fluctuations at different positions along the bump for the LML experiment (from [2])

geometry of the lower wall is the same as the one used in the experiment of Bernard et al. (2003) [2]. The DNS code was developed at LML [6]. Details on the numerical schemes used for space and time discretisation are as follows: for space discretisation, fourth-order central finite differences are used for the second derivatives in the streamwise x-direction. All first derivatives of the flow quantities appear explicitly in the time-advancing scheme and the first derivatives in x are discretized using eighth-order finite differences. Chebyshev-collocation is used in the wall-normal y-direction. The transverse direction z is assumed periodic and is discretized using a spectral Fourier expansion with Nz modes, the non-linear coupling terms are computed using the conventional de-aliasing technique using one-third of the largest modes. For time-integration, implicit second-order backward Euler differencing is used; the Cartesian part of the Laplacian is taken implicitly whereas an explicit second-order Adams–Bashforth scheme is used for the operators coming from the mapping as well as for the nonlinear convective terms. The three-dimensional system uncouples into Nz two-dimensional subsystems and the resulting 2D-Poisson equations are solved efficiently using the matrixdiagonalisation technique. In order to ensure incompressibility, a fractional-step method (cf. [3, 4]) has been adapted to the formulation of the Navier–Stokes system. The adverse pressure gradient was obtained by a wall curvature through a mathematical mapping from physical coordinates to Cartesian ones. The code combines the advantage of a good accuracy with a fast integration procedure compared to standard numerical procedures for complex geometries. For more detail about the numerical code, see [6]. Figure 2 shows the streamwise velocity fluctuations at different streamwise locations of the DNS for an upstream channel flow Reynolds number of Reτ = 600.

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Fig. 2 Streamwise velocity fluctuations along the top and bottom walls of the diverging part of the channel

4 Literature Data Here we describe the cases selected from the literature specifying the geometry that produced the APG. The dominant role of APG, the availability of data and the nonequilibrium state were the factors considered for choosing the data-sets. Firstly the cases without curvature are described, e.g., Spalart and Watmuff (1993) [8], Maciel et al. (2006) [5], Materny et al. (2008) [7]. Without curvature means that the measurements are carried out on the bottom flat wall, although the top wall may be curved. All cases pertain to adiabatic incompressible turbulent boundary layers. For the first cases, APG has been created by the divergent portion of a converging–diverging flow geometry. The boundary layer studied is the one that develops on the bottom flat wall of the diverging part. The flow geometry of Maciel et al. (2006) [5] has a very small elongated bump on the floor too. The second group consists of Webster et al. (1996) [9], Baskaran et al. (1987) [1], Bernard et al. (2003) [2] and Laval-DNS (2008). Baskaran et al. (1987) [1] had tested two flow geometries of curved hill and free wing. We have chosen the free wing for our study. For these cases, APG has been created by the second half of a surface bump. This bump consists of a main convex surface with smaller concave surfaces on the leading and trailing edge sides. The boundary layer develops over the bump and thus the flow undergoes both pressure gradient and curvature effects.

5 Discussion Table 1 lists some important parameters, like the nature of the work (experimental or numerical), flow geometry, Reynolds number based on momentum thickness Reθ , inlet velocity, presence or absence of separation, curvature parameter, pressure gradient parameter β and pressure gradient coefficient dCp /ds with s in mm and

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Table 1 Global characteristics and parameters of the selected data-sets Nature Geometry Reθ

U∞ Separation δ/Rc βmax (m/s)

dCp /dsmax

Elsner (2008)

HWA CD

1767–5705

15

No

–

9

0.0019

Maciel (2006)

PIV

CD

3355–14318

9.1

Yes

–

72000 0.1723

Spalart (1993)

DNS

CD

lowest 600

6.5

No

–

2

0.0016

Bernard (2003)

HWA Bump

No

0.16

79

0.0017

∼23

0.0041

Case

7500–32000 10

Webster (1996)

HWA Bump

4030–4890

17

No

0.06

Baskaran (1987)

HWA Bump

500–4600

20

Yes

0.012 ∼6

0.0025

Laval-LW (2008) DNS

Bump

–

–

Yes

0.02

–

0.0027

Laval-UW (2008) DNS

Bump

–

–

No

–

–

0.0007

Fig. 3 Gradient of pressure coefficient for the selected data-sets

dCp /ds being computed from the Cp data through least-squares curve-fitting. In Table 1, DNS stands for Direct Numerical Simulation, HWA for Hot-Wire Anemometry and PIV for Particle Image Velocimetry. The curvature parameter δo /Rc is the boundary layer initial thickness to convex radius of curvature ratio and is used for comparing the curvatures strength. δo is the initial boundary layer thickness. Curvature is considered as weak if the value of δo /Rc ∼ 0.01, moderate around 0.1 and strong if it is around 1. From Table 1, it appears that Bernard et al. (2003) has the strongest curvature followed by Webster et al. (1996) and Laval-LW (2008). In addition to the strength of pressure gradient, dCp /ds gives an idea of the pressure gradient being increasingly or decreasingly adverse for a certain boundary layer. Figure 3 shows dCp /ds plotted for all the data-sets, with s in mm. Looking at the peak values, it is clear that Maciel (2006) has the strongest pressure gradient followed by Webster et al. (1996), Laval-LW (2008), Elsner et al. (2008), Bernard et al. (2003), Spalart (1993) and Laval-UW (2008) respectively. The peak (or max) values of dCp /ds are listed in Table 1. To quantify the strength of the pressure gradient and its local effect on the boundary layer, it is presented in several non-dimensional forms like the afore-mentioned

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Table 2 Parameters used for internal and external scaling Case

δo (mm)

uτ o (m/s)

Uref (m/s)

Elsner (2008)

18

0.77

16.9

1.2*

38

Maciel (2006)

4.5*

Spalart (1993)

23

0.39

7.7

Bernard (2003)

30*

0.59*

14.9

Webster (1998)

25

1.05

21

Baskaran (1987)

12

0.92

27

Laval-LW (2008)

30

0.0262

0.543

Laval-UW (2008)

51.5

0.026

0.463

* Extrapolated

Clauser’s β, P + , Maciel’s βzs , etc. Clauser’s β shows the effect of APG on the outer layer. In this parameter, the pressure gradient dP /dx is normalised by the wall shear stress τw and the displacement thickness δ ∗ . A value of β greater than 3 is generally taken as a strong APG. From the point of view of β, we can see in Table 1 that the strongest pressure gradient is Maciel et al. (2006) followed by Bernard et al. (2003) and Webster et al. (1996) respectively. This is different ranking as compared to dCp /dx itself. It is worth noting that although the flow geometries for both Bernard et al. (2003) and Laval-LW (2008) are the same, the curvature and pressure gradient effects are different because of different Reynold numbers and different upstream conditions. Differences in upstream conditions include the fully-developed channel flow profile at inlet of the DNS as compared to the developping boundary layer flow of the experiment of Bernard et al. (2003). β=

δ ∗ dP δ ∗ dP = 2 τw dx ρuτ dx

(1)

The external velocity Ue , if not given √ by the author, is calculated from Cp using the Bernoulli equation, Ue = Uref ∗ (1 − Cp ) and wall √ friction velocity uτ , if not given by the author, is calculated through uτ = Ue ∗ (Cf /2). Table 2 lists the internal and external scaling parameters. Location for normalisation parameters is s = 0 in all cases. Values of the internal and external parameters (used for non-dimensionalisation) were extrapolated in the cases of Bernard et al. (2003) and Maciel (2006) because these values were not available at s = 0. Traditionally, for the turbulent boundary layers, two length scales and two velocity scales hare proposed: the viscous length scale ν/uτ and friction velocity scale uτ as internal scales on one side, the boundary layer thickness δ and friction velocity scale uτ as external scales (with Uinf − U as mean velocity) on the other side. Both of these scalings are used for the mean velocity and streamwise Reynold stress u 2 . Scaling with the local value of δ and uτ mask partly the changes in the Reynolds stress by trying to collapse the different profiles. To see the absolute evolution of the

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Fig. 4 rms velocity fluctuations for different data-sets

profiles, reference values of δo and uτ o at s = 0 are used for the scaling of turbulence data. When the streamwise rms velocity fluctuations are scaled with uτ o and yo+ = yuτ o /ν (Fig. 4), it can be seen that the near-wall peak rapidly diminishes and a second peak appears around yo+ ∼ 1000. This second peak rapidly grows and moves away from the wall, towards higher yo+ . This phenomenon is common to cases with and without curvature. Thus it may be deduced that it is triggered by the pressure gradient. In terms of y/δ, the second peak reaches y/δ ≥ 0.4, with the exception of DNS of Spalart (1993) that was performed at low-Reynolds numbers and hence may have low-Reynolds number effects. Figure 5 shows the comparison between the peak of the PDF of Q-criterion and the peak of u + for the lower-wall of the converging–diverging channel. The Q-criterion is used to detect the coherent vortices which are usually linked to turbulent stress in the boundary layer. It is clear that the peaks almost coincide with each other, especially in the APG region. Figure 6 shows the same results for the upper-wall with the same behaviour.

6 Conclusion Different experimental and numerical data-sets have been analysed for the flow in a turbulent boundary layer with different adverse pressure gradients. The develop-

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Fig. 5 PDF of Q-criterion overlaid on rms velocity fluctuations for Laval-LW. Solid line represents the u + and dashed line represents PDF of Q-criterion

Fig. 6 PDF of Q-criterion overlaid on rms velocity fluctuations for Laval-UW. For symbols, see Fig. 5

ment of a turbulence peak is evidenced, which is present both with and without wall curvature. This peak is quite different from the one found in ZPG turbulent layers: it is wider, stronger and moving away from the wall. This means that this peak should have a different physical origin from the standard near wall peak. It is proposed here that this peak is triggered by the adverse pressure gradient and has its origin in an instability hidden in the turbulent boundary layer and developing soon after the change of sign of the pressure gradient. This may explain the difficulties encountered up to now in finding a universal scaling for turbulent boundary layers. Acknowledgements We are thankful to Y. Maciel from University of Laval (Canada) and W. Elsner from University of Tceztochowa for sharing their databases. The data for Webster et al. (1998) and Spalart and Watmuff (1994) were taken from the database maintained by J. Jimenez (UPM Madrid) for various flow cases. Part of this work has been performed under the WALLTURB project. WALLTURB (A European synergy for the assessment of wall turbulence) is funded by the CEC under the 6th framework program (CONTRACT No: AST4-CT-2005-516008).

References 1. Baskaran, V., Smits, A., Joubert, P.: A resolvable subfilter-scale model specific to large-eddy simulation of under-resolved turbulence. J. Fluid Mech. 182, 47–83 (1987) 2. Bernard, A., Foucaut, J.M., Dupont, P., Stanislas, M.: Decelerating boundary layer: a new scaling and mixing length model. AIAA J. 41(2), 248–255 (2003)

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3. Karniadakis, G.E., Israeli, M., Orszag, S.A.: High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97, 414–443 (1991) 4. Kim, J., Moin, P.: Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308–323 (1985) 5. Maciel, Y., Rossignol, K.S., Lemay, J.: Similarity in the outer region of adverse-pressuregradient turbulent boundary layers. AIAA J. 44(11), 2450–2464 (2006) 6. Marquillie, M., Laval, J.P., Dolganov, R.: Direct numerical simulation of separated channel flows with a smooth profile. J. Turbul. 9(1), 1–23 (2008) 7. Materny, M., Drozdz, A., Drobniak, S., Elsner, W.: Experimental analysis of turbulent boundary layer under the influence of adverse pressure gradient. Arch. Mech. 60, 449–466 (2008) 8. Spalart, P.R., Watmuff, J.H.: Experimental and numerical investigation of a turbulent boundary layer with pressure gradients. J. Fluid Mech. 249, 337–371 (1993) 9. Webster, D.R., Degraaf, D.B., Eaton, J.K.: Turbulence characteristics of a boundary layer over a two-dimensional bump. J. Fluid Mech. 320, 53–69 (1996)

Session 7: RANS Modelling

• A Nonlinear Eddy-Viscosity Model for Near-Wall Turbulence B. Anders Pettersson Reif and M. Mortensen • ASBM-BSL: An Easy Access to the Structure Based Model Technology B. Aupoix, S.C. Kassinos, and C.A. Langer • On the Role of the Convective Term for Scaling Laws and Improved WallFunctions for Adverse Pressure Gradient Flow T. Knopp (no paper) • Introduction of Wall Effects into Explicit Algebraic Stress Models Through Elliptic Blending A.G. Oceni, R. Manceau, and T.B. Gatski

A Nonlinear Eddy-Viscosity Model for Near-Wall Turbulence B. Anders Pettersson Reif and Mikael Mortensen

Abstract In this work the original V2F model of Durbin (Theor. Comput. Fluid Dyn. 3:1–13, 1991) and the nonlinear V2F model of Pettersson Reif (Flow Turbul. Combust. 76:241–256, 2006) have been assessed for several adverse pressure gradient (APG) turbulent boundary layer flows, designed as benchmark cases within the WALLTURB consortium. The APG flows have been created by putting curved surfaces (bumps) on walls of otherwise plane channels. On the diverging side of the bumps the flows either separate or are close to separation. We have found that even though the NLV2F model accounts for the Reynolds-stress anisotropy, this does not seem to significantly improve the model predictions. It is in fact very interesting to note that without the effect of curvature, the two models respond almost identical to the imposed adverse pressure-gradient. When the flow gets close to separation the differences between the models are on the other hand significant and the models generally perform worse. All in all, the results seem to reveal that it is not the lack of turbulence anisotropy information per se that is the primary source for the long standing problem related to the prediction of adverse pressure-gradient turbulent boundary layers. This is our primary motivation for our long term objective to include turbulence structure information alongside improved turbulence anisotropy predictions in an attempt to improve the modeling of the turbulence time scale (k/ε).

1 Introduction The utilization of the Reynolds-Averaged Navier–Stokes (RANS) approach continues to dominate computational predictions of turbulent flows in a wide range of disciplines. The reason is not so much that of physical attractiveness as it is of convenience: the RANS approach offers a computationally robust and relatively B.A. Pettersson Reif () · M. Mortensen Norwegian Defence Research Establishment (FFI), Kjeller, Norway e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_28, © Springer Science+Business Media B.V. 2011

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inexpensive method. It is based on computing the average over an ensemble of realization of the disorderliness motion of fluid turbulence that produces a smooth reproducible flow field. The outcomes are mean and variance quantities that provide values particularly relevant for engineers. As a rigorous mathematical consequence, the ensemble averaging procedure of the instantaneous Navier–Stokes equations that leads to the RANS equations only one term, the Reynolds stress tensor, is left to represent all effects of turbulence on the mean flow field. This reduction of the number of degrees-of-freedom comes with a price. The second order single-point correlation of fluctuating velocity components ui uj , sometimes also referred to as the kinematic Reynolds stress tensor, inherently only contains information that is loosely related to the turbulence energy. There is no explicit dependence on turbulent length- or times scales. Among all challenges RANS modelers have to face, and there are many, the lack of a mathematically rigorous framework for modeling turbulent length- and time scales is perhaps the most fundamental one. Another very important challenge related to the majority of practical applications is how to treat near-wall effects. Current RANS model development is essentially based on the assumption of locally homogeneous shear flow. At the same time as this is a mathematically convenient choice, it also implies a local treatment of the unclosed terms. This locality assumption is fundamentally flawed in near-wall flows where nonlocal effects dominate. Durbin’s elliptic relaxation approach is today the closest we get to a nonlocal modeling approach. Despite the remaining challenges associated with the implementation of the elliptic relaxation method for industrial use, it remains perhaps as the most physically appealing approach towards improved predictive capabilities of complex industrial flows. The original V2F model by Durbin utilizes the elliptic relaxation approach within the framework of the linear eddy-viscosity formulation. Despite the indirect measure of near-wall turbulence anisotropy, the utilization of a linear eddy-viscosity relation makes the original V2F model incapable of predicting normal stress anisotropy. It should be recalled, however, that the success of the V2F model up to now is closely related to the fact that it is capable of representing effect of solid surfaces on the Reynolds shear stress component. In flows where the mean velocity field predominately depends on the off-diagonal Reynolds component it is therefore not expected the mean flow predictions will be improved by accounting also for the normal stress anisotropy. The objective of the present work is to explore whether or not the inclusion of Reynolds stress anisotropy extend the predictive capability of the linear V2F model in flows affected by adverse pressure-gradients. Predictions of adverse pressuregradient boundary layers, with and without flow separation, considered in WALLTURB are compared with experimental results and DNS data. A more ambitious and long term objective is to explore the use of turbulence structures based tensors in an attempt to systematically include turbulence scale information. The novel Algebraic Structure Based Model (ASBM) serves as a basis for this endeavor which is based on a coupling between the ASBM and the scale-determining equation in the V2F model. It is believed that this is a viable route towards an improved eddy-viscosity formulation for practical fluid flow predictions.

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2 Mathematical Modeling The V2F model by Durbin [2] is a four equation model that utilizes the elliptic relaxation approach to characterize near-wall turbulence anisotropy. Below the V2F model and the nonlinear extension (NLV2F) of Pettersson Reif [4] are described.

2.1 The Linear V2F Model In the linear V2F model the components of the kinematic Reynolds stress tensor, ui uj , are modeled using the traditional linear eddy-viscosity relation 2 ui uj = −2νT Sij + kδij , (1) 3 where νT is the turbulent viscosity, δij is the Kronecker delta and the mean rate of strain tensor Sij = 0.5(∂Ui /∂xj + ∂Uj /∂xi ). The velocity is decomposed as u˜ i = Ui + ui , which represents instantaneous, mean and fluctuating components respectively. The modeled transport equations for k and ε are ∂k νT ∂k ∂k ∂ ν+ + Pk − ε + Uj = (2) ∂t ∂xj ∂xj σk ∂xj and ∂ε ∂ ∂ε + Uj = ∂t ∂xj ∂xj

νT ∂ε Cε1 Pk − Cε2 ε ν+ + , σε ∂xj T

(3)

where σk and σε are model constants. The rate of production of turbulent kinetic energy Pk = −ui uj ∂Ui /∂xj . In order to model the impact on νT in the vicinity of solid surfaces, the V2F model solves two additional scalar equations ∂v 2 νT ∂v 2 ∂v 2 ε ∂ ν+ + kf − v 2 + Uj (4) = ∂t ∂xj ∂xj σv 2 ∂xj k and L2

∂ 2f Pk C1 − 1 v 2 2 − − C2 , =f + ∂xj ∂xj T k 3 k

where T and L are time- and lengthscales determined by 3 3 1 k k2 ν 4 ν T = max , 6 , Cη , and L = CL min ε ε ε ε

(5)

(6)

respectively. Through Eqs. 4 and 5 an additional and more appropriate near-wall 2 velocity scale v is obtained and the turbulent viscosity can now be computed as νT = Cμ1 v 2 T .

(7)

The parameters of the V2F model vary somewhat in the literature, but are most commonly set to Cμ1 = 0.22, Ced = 0.045, Cε2 = 1.9, C1 = 1.4, C2 = 0.3, σk = 1.0,

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σv 2 = 1.0, σε = 1.3, CL = 0.25, and Cη = 80. The last parameter Cε1 is in this work computed with a model that has been shown superior to the original form for separating flows 0.0567|S 2 |k 2 , (8) Cε1 = 1.35 1 + 0.13 2 ε + 0.0323|S 2 |k 2 where |S 2 | = Sij Sj i .

2.2 The Nonlinear V2F Model (NLV2F) In order to remedy the inherent inability of the original model to reproduce the turbulence anisotropy in shear turbulence the NLV2F utilizes a nonlinear constitutive equation. The choice of basis is arbitrary but here only a quadratic expansion in the mean deformation rate has been used aij = −Cμ1 T1 Sij − Cμ2 T 2 (Sik Wkj + Sj k Wki ) 2 2 2 + Cμ3 T Sik Skj − |S |δij (9) 3 where the Reynolds-stress anisotropy tensor aij = 2rij − 2/3δij , where rij = ui uj /2k are the normalized components of the Reynolds-stress tensor. The components of the mean vorticity tensor are in an inertial frame of reference given by Wij = 0.5(∂Ui /∂xj − ∂Uj /∂xi ). The functional form of the remaining coefficients in Eq. 9 was determined directly by imposing realizability. More details can be found in Pettersson Reif [4]. As already alluded to, the choice of basis for the constitutive relation 9 is arbitrary. The particular choice of using only a quadratic formula is based on its close connection to significantly more elaborate differential Reynolds stress models (RSM). As shown by Pope [6], a quadratic relation between the Reynolds stresses and the mean deformation rates is physically fully consistent with the solution to a RSM in the limit of homogeneous turbulence in equilibrium. Although the quadratic expansion only is valid for two-dimensional mean flows, it is important to note that this is the only true physical solution to the RSM. Structural equilibrium implies that both d/dt (Sij k/ε) = 0 = Wij k/ε independently. As shown by Durbin and Pettersson Reif [3] the latter of these requires η3 = Sik Skj Sj i = 0. This limits the solution of RSMs to two-dimensional mean flow fields. Although cubic algebraic formulas for ui uj , corresponding to three-dimensional mean flow fields, mathematically can be shown to satisfy a RSM, the necessary equilibrium condition for the solution to be physical is not met.

3 Results and Discussion Within the framework of the WALLTURB consortium there are several groups working both numerically and experimentally with adverse pressure gradient (APG)

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turbulent boundary layers. The established WALLTURB database containing experimental and numerical results are used in this paper to verify the model predictions. Alltogether three cases are considered, two of which undergoes flow separation and reattachment. A brief description of the cases used in this paper is provided in Sect. 3.1. The models have been implemented in a modified version of the general purpose finite volume code CALC [1]. The code utilizes a mix of second order upwind (convective terms) and second order central differencing (diffusion terms) discretization. A new coupled solver that solves k–ε and v 2 –f systems has been implemented in order to increase the numerical stability of the V2F and NLV2F models. The original V2F model that is used in this work requires a coupled solver because the boundary conditions on ε and v 2 need to be implemented implicitly for stability. Grid independent solutions of all cases considered here have been verified and the location of the first internal node is always closer than y + = 1, also over the bump. The total number of gridpoints used is in the order of 150 × 100 in streamwise and wallnormal directions respectively. The inlet conditions are computed separately from developing boundary layers with flat plates. The inlet profiles are then chosen from the location that best matches the inlet conditions reported in the experiments.

3.1 Experimental and Numerical Reference Data At the Technical University of Czestochowa (TUCz) an experiment was designed where the effect of the APG was measured along a plane wall, whereby curvature effects were avoided, cf. Fig. 1. The diverging side of the bump is rather long and separation has not been observed. Measurements were performed with hot-wires near the flat wall on the opposite side of the bump. Streamwise mean velocity and turbulence intensity data are used here. Measurements were also conducted upstream the APG region to establish details about the incoming boundary layer. At the University of Surrey (USur) they considered a bump geometry similar to LML but with a slight modification in order to promote flow separation to occur downstream the bump crest and from a flat surface. The flow separates on the bump and reattaches on the flat surface further downstream. Measurements upstream the APG region were also here conducted to establish details about the incoming boundary layer for the use by modelers. At Laboratoire de Mécanique de Lille (LML) there has in addition to experimental investigations been conducted DNS of the same experimental configuration but at a lower Reynolds number (Reτ = 400, 600). Contrary to the experimental findings, the numerical results indicated that a mild separation occur just downstream the bump crest. This difference is attributed the lower Reynolds-number in the numerical simulations and it poses a severe challenge for the modeling. The data for Reτ = 600 are still processed by LML and all data are thus not yet available.

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Fig. 1 Profiles of the axial velocity component along the TUCz channel. V2F and NLV2F are represented with solid and dashed lines respectively and the experimental data are represented with open circles. The plots are shown for locations x = 0.2, 0.43, 0.61, 0.76 and 0.91 Fig. 2 The axial component of the Reynolds stress in the TUCz channel is shown in (a)–(d) for the last four positions of Fig. 1 respectively. V2F and NLV2F are represented with solid and dashed lines respectively and the experimental data are represented with open circles

3.2 Model Results and Discussion In Fig. 1 the distribution across the domain of the predicted mean velocity is compared with experimental data for the TUCz case. Comparisons at a position upstream the bump confirms correct upstream conditions. Both models yield very similar results along the flat wall and the correspondence with the experimental data is quite good. Notable difference between the two models can be seen along the curved wall. Although NLV2F seems more prone to predict separation than V2F, the flow remains attached. No experimental data are available along the curved wall. The variation of wall shear stress along the straight wall is shown in Fig. 3. Again no notable differences between the models; both underpredicts the initial acceleration. Although the model predictions agree better further downstream the streamwise slope is underpredicted. The streamwise Reynolds stress component at four different streamwise positions are shown in Fig. 2. The NLV2F consistently overpredicts uu in the inner part of the boundary layer whereas the correspondence is better in the outer part (y + > 500). As expected the NLV2F agrees closer with the experiments although the overprediction is significant. Figure 4 shows the mean velocity predictions for the USur case. There are no notable differences between V2F and NLV2F in the vicinity of the bump crest and as long as the flow is attached there is very good agreement with experiments. Differences between the models can be observed further downstream, where there unfortunately are no experimental data. Note that in the experiments and simulations separation starts very close to the end of the bump and the separation bubble is

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Fig. 3 Plot of the wall shear stress measured on the flat plate of the TUCz channel. V2F and NLV2F are represented with solid and dashed lines respectively and the experimental data are represented with open circles. The x-axis shows the axial distance from the bump crest (x = 0)

Fig. 4 Profiles of the axial velocity component along the USur bump. V2F and NLV2F are represented with solid and dashed lines respectively and the experimental data are represented with open circles

Fig. 5 Profiles of the turbulent kinetic energy along the USur bump. V2F and NLV2F are represented with solid and dashed lines respectively and the experimental data are represented with open circles

small. A small separation bubble is predicted by both V2F and NLV2F, even though the NLV2F shows more tendency to predict separation. From Fig. 5 it can be seen that the NLV2F once again overpredicts the turbulent kinetic energy in the immediate vicinity of the wall. The position of the peak is predicted too close to the wall as compared to the experimental data, yet there is good agreement for both models. Figure 6 compares model predictions with DNS results in the LML case. Data upstream the bump have been confirmed to give the correct inflow condition. Although the experimental results did not reveal flow separation the DNS data do. The DNS results exhibits separation quite close to the bump crest whereas NLV2F models predict attached flow significantly further downstream. V2F does not predict separation of this flow. Note that the models performances are much worse than for the other cases where comparisons were made during attached flow.

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Fig. 6 Profiles of the tangential velocity component along the LML channel. DNS, V2F and NLV2F are represented with solid, dashed and dotted lines respectively

4 Concluding Remarks Accounting for the Reynolds-stress anisotropy does not seem to significantly improve the model predictions using state-of-the-art near-wall modeling. It is in fact very interesting to note that without the effect of curvature, the two models responds almost identical to the imposed adverse pressure-gradient. When the flow gets close to separation the differences between the models are on the other hand significant. All in all, the results seem to reveal that it is not the lack of turbulence anisotropy information per se that is the primary source for the long standing problem related to the prediction of adverse pressure-gradient turbulent boundary layers. This is our primary motivation for our long term objective to include turbulence structure information alongside improved turbulence anisotropy predictions in an attempt to improve the modeling of the turbulence time scale (k/ε). Some progress has already been made within WALLTURB to link the novel Algebraic Structure Based Model with the NLV2F [5]. Acknowledgements This work has been performed under the WALLTURB (A European synergy for the assessment of wall turbulence) project funded by the EC under the 6th European framework programme (CONTRACT N: AST4-CT-2005-516008). The authors also acknowledge support from the Norwegian Research Council through the Norwegian Centre of Excellence for Biomedical Computing.

References 1. Davidson, L., Faranieh, B.: Technical report 95/11, Thermo- and Fluid Dynamics, Chalmers University of Technology (1995) 2. Durbin, P.A.: Theor. Comput. Fluid Dyn. 3, 1–13 (1991) 3. Durbin, P.A., Pettersson Reif, B.A.: Flow Turbul. Combust. 63, 23–37 (1999) 4. Pettersson Reif, B.A.: Flow Turbul. Combust. 76, 241–256 (2006) 5. Pettersson Reif, B.A., Mortensen, M., Langer, C.A.: Flow Turbul. Combust. 83(2), 185–203 (2009) 6. Pope, S.B.: J. Fluid Mech. 72, 331–340 (1975)

ASBM-BSL: An Easy Access to the Structure Based Model Technology Bertrand Aupoix, Stavros C. Kassinos, and Carlos A. Langer

Abstract The Algebraic Structure Based Model (ASBM) offers unique features to represent the Reynolds stress tensor from the underlying turbulent structures. It is usually coupled with a non standard length scale equation. A way of coupling it with the more popular BSL ω equation is proposed here. Only a minor modification of the ω equation is required to obtain a realistic turbulent kinetic energy profile and thus achieve fair predictions.

1 Introduction Present industrial design heavily relies upon CFD, using mainly eddy viscosity models, the two most popular ones being the Spalart–Allmaras one equation model and the Menter SST k–ω model. However, they are known to fail to capture many flow issues, first of all because of the eddy viscosity assumption. Two routes are generally considered to get rid of this assumption. The use of transport equations for the Reynolds stress tensor (DRSM models) allows one to circumvent most of the failures of the eddy viscosity assumption, by accounting for turbulence memory effects, most of the rotation and curvature effects. Explicit Algebraic Reynolds Stress Models (EARSM) only require two transport equations and replace the eddy viscosity assumption by an equilibrium assumpB. Aupoix () Aerodynamics and Energetics Department, ONERA, BP 74025, 2 Avenue E. Belin, 31055 Toulouse Cedex 4, France e-mail: [email protected] S.C. Kassinos · C.A. Langer Computational Sciences Laboratory — UCY-CompSci, Department of Mechanical & Manufacturing Engineering, University of Cyprus, 75 Kallipoleos, Nicosia 1678, Cyprus e-mail: [email protected] C.A. Langer e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_29, © Springer Science+Business Media B.V. 2011

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tion for the anisotropy tensor. They so inherit many properties of the underlying Reynolds stress model. A third modelling route was proposed by Reynolds and Kassinos [2–4, 9] to circumvent most of the failures of DRSM models. While DRSM models assume that all the information about turbulence is in the turbulence length and velocity scales and in the anisotropy tensor, they pointed out the importance of the dimensionality of turbulence. This led them to Structure Based Models (SBM), which are able to better represent rapid distortion, rotation effects and many subtle non-equilibrium effects than current DRSM models. But these models use extra transport equations, compared to DRSM models and are thus less prone to a prompt industrial use. A simplified version, using the SBM approach to directly represent the anisotropy and dimensionality tensors was derived [5, 6]. This Algebraic Structure Based Model (ASBM) only requires two transport equations for turbulence velocity and length scales, and brings some similarities with EARSM models. This approach is very powerful as it allows to sensitize the length scale equation to the dimensionality of turbulence. But the length scale equation is unusual [10] and may require some changes to be implemented in a commercial code. Therefore, a simpler version, coupling the ASBM with a more standard set of length scale equations has been developed, to give an easy access to the ASBM technology.

2 ASBM Modelling Structure Based Models, and especially ASBM models, can be envisioned in two complementary ways. A first way is the use of turbulence structure tensors to characterize the turbulent motion. While DRSM models only deal with the anisotropy tensor, other tensors can be derived from the turbulent stream function vector, the curl of which is the turbulent velocity field. The dimensionality tensor characterizes the changes of the turbulence structure along the various directions; the circulicity tensor characterizes the vorticity field associated with the energy bearing structures, the inhomogeneity tensor the degree of inhomogeneity of the turbulent field. These three tensors and the Reynolds stress tensor are related and, in SBM complete models, transport equations are solved for these tensors to fully describe the turbulent field. A second way is to consider that turbulence can be mimicked with a combination of simplified turbulent structures, i.e. as an ensemble of hypothetical 2D eddies the direction of independence of which is aligned with the eddy-axis. These eddies differ by their componentality and dimensionality and are: • Jetal motions: 2D-1C fields where the motion is only along the eddy axis, • Vortical motions: 2D-2C fields where the motion is around the eddy axis, • Helical eddies: superpositions of the above two motions A last important notion is the flattening of the eddies, which is related to the degree of asymmetry of the turbulent kinetic energy distribution around the hypothetical eddies.

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A complete turbulent field is envisioned as a large ensemble of these eddies, with varying characters and orientations. Averaging over the ensemble produces statistical quantities representative of the field. Decomposing the fluctuating velocity in components aligned and normal to the eddy-axis direction, taking the product ui uj , and then averaging, results in the algebraic constitutive relation for the normalized Reynolds stress tensor (related to statistics of the ensemble) rij =

ui uj

1 = (1 − φ)(δij − aij ) + φaij 2k 2 1 1 + (1 − φ)χ (1 − anm bnm )δij − (1 + anm bnm )aij − bij 2 2 Ωk 1 (ipr apj + jpr api ) 1 − χ(1 − anm bnm ) δkr + ain bnj + aj n bni − γ Ω 2 + χ(bkr − akn bnr ) (1)

where the different symbols are defined below. The eddy-axis tensor aij represents the energy-weighted average direction cosine of the eddy-axis vector. Eddies, like material lines, tend to align with the direction of positive mean strain rate, and are rotated kinematically by the mean rotation rate. To ensure the algebraic model for the eddy-axis tensor is realizable, it is computed via a two-step procedure. Initially a strained eddy axis, aijs , is evaluated based upon the mean strain rate tensor, Sij . aijs =

s + S a s − 2 |Sa s |δ Sik akj j k ki ij δij 3 +τ , 3 a + 2 a2 + τ 2S S as 0

1

(2)

kp kq pq

|Sa s |

s , and {a , a } are “slow” param= Spq aqp where τ is a turbulent time scale, 0 1 eters. Next a rotation operation is applied on aijs , so that the final (strained, then rotated) eddy-axis tensor, aij , is obtained as

aij = Hik Hj m aijs .

(3)

The rotation operator Hij is modeled as Ωij Ωik Ωkj , + h2 Hij = δij + h1 |Ω 2 | |Ω 2 |

(4)

which satisfies the orthonormal conditions Hik Hj k = δij for h1 = 2h2 − h22 /2. h2 is chosen to satisfy theoretical rapid distortion limits for combined homogeneous plane strain and rotation. ⎧ ⎨ 2 − 2 (1 + √1 − r )/2 if r ≤ 1 , (5) h2 = √ ⎩ 2 − 2 (1 − 1 − 1/r )/2 if r ≥ 1 where r = (apq Ωqr Srp )/(Skn Snm amk ). bij , the eddy-flattening tensor, is based upon an ensemble average of the flattening direction of the hypothetical eddies. This flattening direction is found to be

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dependent on the vorticity vector, so that the model for bij takes the simple form bij = Ωi Ωj /|Ω 2 | in a fixed reference frame. φ, the jetal parameter, is representative of the amount of energy in the jetal mode of motion. (1 − φ) is representative of the amount of energy in the vortical mode. In irrotational flows φ = 0, and of course 0 ≤ φ ≤ 1. Mathematically, it represents the average (over all eddies) of the dot product between the eddy-axis vector and the velocity vector. In simple shear flows (this study), it reads φ = 1 × fslow (a 2 ),

a 2 = alm aml

2.5 2 0.5 2 fslow (a 2 ) = 0.35fiso (a ) + (1 − 0.35)fiso (a )

(6)

fiso (a ) = (3/2)(a − 1/3) 2

2

γ , the helical parameter, is representative of the correlation between the jetal and the vortical modes. In the general case,

2φ(1 − φ) γ =β , (7) 1+χ and for simple shear flows β = 1. χ is the flattening parameter, a representative of the average magnitude of the lack of symmetry in the energy distribution around the eddies. In simple shear flows χ = 0.2 × fslow (a 2 ).

(8)

The model is moreover sensitized to the wall blocking with the help of an elliptic relaxation equation for a blocking parameter Φ, from which a partial projection of the eddy axis tensor towards the wall is deduced 1 h h aij = Pik akl Plj , Pik = (δik − Bik ), Da2 = 1 − (2 − Bkk )amn Bnm , (9) Da where Pik is the partial-projection operator, and Da2 is such that the trace of aij remains unity (note the superscript “h” is used to indicate the unblocked tensor). The blockage tensor Bij gives the strength and the direction of the projection. If the wall-normal direction is x2 , then B22 = Φ is the sole non-zero component, and varies between 0 (no blocking) far enough from the wall, to 1 (full blocking) at the wall. The blocking parameter, Φ, is computed by an elliptic relaxation equation

3/2 2Φ 3 ∂ k 4 ν 2 L , Cν (10) = Φ, L = CL max ε ε ∂xl2 ∂Φ with Φ = 1 at solid boundaries, and ∂x = 0 at open boundaries, where xn is the n direction normal to the boundary. Values of the constants are CL = 0.17, Cν = 80. To retrieve the proper near-wall asymptotics for the Reynolds stress tensor, the blocking parameter Φ is also used for blending of φ and γ with their wall values; 1 and 0 respectively,

φ = 1 + (φ h − 1)(1 − Φ)2 ,

γ = γ h (1 − Φ).

At last, δij and εij k are the identity and Ricci tensors, as usual.

(11)

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It must be pointed out that the so obtained Reynolds stress tensor is fully realizable. Model details can be found in Reynolds et al. [9] and Langer and Reynolds [6].

3 Coupling with a k–ω Model Among two-equation models, the k–ω model is known to give fair predictions of pressure gradient effects. However, as the Wilcox [12] model is sensitive to free stream values, Menter’s BSL model [7] has been preferred. Coupling the above ASBM representation of the Reynolds stress tensor with the BSL model however requires two significant changes. To take advantage of the fair representation of the Reynolds stress tensor provided by the ASBM model, the turbulent kinetic energy production must be comi puted with the ASBM model as Pk = −ui uj ∂U ∂xj . This is used for both k and ω equations, i.e. the source term in the ω equation is expressed as αPk ωk . Moreover, the ASBM model has been designed to represent real near wall turbulence while the basic k–ω model yields turbulent kinetic energy profiles which are more similar to v 2 profiles in the wall region and do not exhibit a near-wall peak. Wilcox later proposed to add three wall functions to retrieve the correct wall behaviour. When using the ASBM model, modifications on the eddy viscosity and on the ω production rate are not needed. Only the dissipation rate is altered, using a function adapted from Peng and Davidson [8] as 2 Rt 13 k , Rt = exp − . (12) fw = 1 − ε = βfw ωk, 18 10 νω This form ensures that k ∝ y 2 , ε having a plateau in the wall region (see Fig. 2). At last, the length scale definition in the elliptic wall blocking has been re-tuned, with CL = 0.1285.

4 Validation Results Channel flows are first addressed. Comparisons are performed with DNS data at various Reynolds numbers provided by Madrid Polytechnic University [1]. The transport equations used here are Wilcox’ k–ω ones, which are nearly equivalent to BSL ones for channel flows. They are coupled either with eddy viscosity or with ASBM constitutive relations. Results are given for the largest Reynolds number Rτ = huν τ = 2000, using wall scaling. Figure 1 shows that the model inherits many characteristics from the underlying k–ω model and yields very similar predictions for the mean velocity profile, with some discrepancies with respect to DNS in the buffer and wake regions. The error on the friction coefficient, w.r.t. Prandtl’s correlation, is about −4% for a wide range of Reynolds numbers, while it is about −6% with the BSL model.

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Fig. 1 Channel flow Rτ = 2000 — Mean velocity predictions

Fig. 2 Channel flow Rτ = 2000 — Turbulent kinetic energy and dissipation rate predictions

Figure 2 points out that the model, as any other, is unable to reproduce the growth in the turbulent kinetic energy peak as it does not account for the inactive motions. The improvement brought about by the use of the damping function (12) is apparent on the dissipation profile, which is now in better agreement with DNS data, and also on the appearance of the near-wall turbulent kinetic energy peak, which is not reproduced by Wilcox’ model. The three diagonal stresses are plotted in Fig. 3. “Wilcox” values are obtained assuming u 2 = k, v 2 = 0.4k, w 2 = 0.6k. v 2 profile is well reproduced, w 2 is depleted in the buffer region and of course u 2 and w 2 are underestimated as inactive motions are not accounted for. Profiles of all turbulent quantities are very similar to the ones provided by the k–ω model in the logarithmic region and above.

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Fig. 3 Channel flow Rτ = 2000 — Diagonal stresses predictions

Fig. 4 Relative error on skin friction predictions versus Reynolds number for zero pressure gradient boundary layers (in per cent)

For zero pressure gradient boundary layers, errors on the skin friction coefficient prediction w.r.t. Fernholz’ correlation are plotted in Fig. 4. The error is less than one percent over a wide range of Reynolds numbers Rθ based upon the momentum thickness. Comparisons were also performed with experimental data for high Reynolds number boundary layers, performed in the framework of WALLTURB in Laboratoire de Mécanique de Lille (LML). Nice agreement is obtained on the mean velocity profiles but the turbulent quantities evidence that the model, as others, yields the same solution in the wall region whatever the Reynolds number and hence does not account for inactive motions. At last, several adverse pressure gradient test cases have been considered. Only the most difficult case, i.e. the nearly equilibrium flow subjected to a strong adverse pressure gradient investigated by Skåre and Krogstad [11] is presented. Figures 5

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Fig. 5 Skin friction predictions for Skåre and Krogstad experiment

Fig. 6 Mean velocity profile predictions for Skåre and Krogstad experiment — From right to left stations X = 3, 4.2, 4.6 and 5 m

and 6 show that the model is able to reproduce the very low skin friction levels encountered in the experiment and even does better than the SST model.

5 Conclusions and Perspectives An easy way to use the Algebraic Structure Based Model, coupling it with the widespread BSL model, has been proposed. It only requires slight modifications of the BSL part, besides of course adding an ASBM routine to compute the Reynolds stress tensor. It thus gives access to the much better representation of the Reynolds stress tensor, e.g. for flows with rotation, provided by the ASBM model. The model has been shown to give fair predictions and to rather nicely reproduce the near wall turbulence, what the basic BSL model is not able of.

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Another way to determine the wall function fw , from known near-wall physics, will be investigated and reported later. Author names appear in alphabetical order. Acknowledgements This work has been performed under the WALLTURB project. WALLTURB (A European synergy for the assessment of wall turbulence) is funded by the EC under the 6th framework program (CONTRACT N: AST4-CT-2005-516008).

References 1. Hoyas, S., Jimenez, J.: Scaling of velocity fluctuations in turbulent channels up to Reτ = 2000. Phys. Fluids 18, 011702 (2006) 2. Kassinos, S.C., Reynolds, W.C.: A structure-based model for the rapid distortion of homogeneous turbulence. Technical Report TF-61, Thermosciences Division, Department of Mechanical Engineering, Stanford University (1994) 3. Kassinos, S.C., Langer, C.A., Haire, S.L., Reynolds, W.C.: Structure-based turbulence modelling for wall bounded flows. Int. J. Heat Fluid Flows 21, 599–605 (2000) 4. Kassinos, S.C., Reynolds, W.C., Rogers, M.M.: One-point turbulence structure tensors. J. Fluid Mech. 428, 213–248 (2001) 5. Kassinos, S.C., Langer, C.A., Kalitzin, G., Iaccarino, G.: A simplified structure-based model using standard turbulence scale equations: computation of rotating wall-bounded flows. Int. J. Heat Fluid Flows 27, 653–660 (2006) 6. Langer, C.A., Reynolds, W.C.: A new algebraic structure-based turbulence model for rotating wall-bounded flows. Technical Report TF-85, Mechanical Engineering Department, Stanford University (2003) 7. Menter, F.R.: Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J. 32(8), 1598–1605 (August 1994) 8. Peng, S.H., Davidson, L., Holmberg, S.: A modified low-Reynolds number k–ω model for recirculating flows. J. Fluids Eng. 119, 867–875 (December 1997) 9. Reynolds, W.C., Kassinos, S.C., Langer, C.A., Haire, S.L.: New directions in turbulence modeling. In: Third International Symposium on Turbulence, Heat and Mass Transfer, Nagoya, Japan, April 3–6, 2000 10. Reynolds, W.C., Langer, C.A., Kassinos, S.C.: Structure and scales in turbulence modelling. Phys. Fluids 14(7), 2485–2492 (2002) 11. Skåre, P.E., Krogstad, P.-Å.: A turbulent equilibrium boundary layer near separation. J. Fluid Mech. 272, 319–348 (August 1994) 12. Wilcox, D.C.: Reassessment of the scale-determining equation for advanced turbulence models. AIAA J. 26(11), 1299–1310 (November 1988)

Introduction of Wall Effects into Explicit Algebraic Stress Models Through Elliptic Blending Abdou G. Oceni, Rémi Manceau, and Thomas B. Gatski

Abstract In order to account for the non-local blocking effect of the wall, responsible for the two-component limit of turbulence, in explicit algebraic models, the elliptic blending strategy, a simplification of the elliptic relaxation strategy, is used. The introduction of additional terms, dependent on a tensor built on a pseudo-wallnormal vector, yields an extension of the integrity basis used to derive the analytical solution of the algebraic equation. In order to obtain a tractable model, the extended integrity basis must be truncated, even in 2D plane flows, contrary to standard explicit algebraic models. Four different explicit algebraic Reynolds-stress models are presented, derived using different choices for the truncated basis. They all inherit from their underlying Reynolds-stress model, the Elliptic Blending Model, a correct reproduction of the blocking effect of the wall and, consequently, of the two-component limit of turbulence. The models are satisfactorily validated in plane Poiseuille flows and several configurations of Couette–Poiseuille flows.

1 Introduction Explicit Algebraic Stress Models (EASMs) are a compromise between representation of the physics and numerical robustness. They inherit most of the capacities of the Reynolds stress model (RSM) from which they are derived to account for complex physical mechanisms. A corollary of the previous remark is that the EASMs also inherit some of the shortcomings of their underlying RSM; particularly, the influence of the blocking effect of the wall, which is not taken into account within the usual context of EASMs. The present work aims at incorporating in EASMs the elliptic blending method proposed by Manceau and Hanjali´c [2, 3]. A.G. Oceni · R. Manceau () · T.B. Gatski Laboratoire d’études aérodynamiques (LEA), Université de Poitiers, ENSMA, CNRS, Bd Marie et Pierre Curie, BP 30179, 86962 Futuroscope Chasseneuil Cedex, France e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_30, © Springer Science+Business Media B.V. 2011

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The Elliptic-Blending Reynolds-Stress Model (EB-RSM) aims at reproducing the blocking effect of the wall by enforcing the correct limiting behavior of the difference between the velocity–pressure-gradient and dissipation terms of the Reynolds stress (τij ) transport equation. The EB-RSM model is characterized by a simple blending between two asymptotically-correct forms of the model for φij∗ −εij τij 2 (1) ε + α 2 φijh − εδij . φij∗ − εij = (1 − α 2 ) φijw − k 3 In order to reproduce the nonlocal character of the blocking effect, the blending function α is obtained from the elliptic relaxation equation α − L2 ∇ 2 α = 1,

(2)

with the boundary condition α = 0, such that α goes from 0 at the wall to 1 far from the wall. φijh denotes hereafter the SSG [10] model, valid far from the wall. The analysis of the near-wall asymptotic behavior [3] shows that φijw must be of the form ε 1 (3) φijw = −5 τik nj nk + τj k ni nk − τkl nk nl (ni nj + δij ) , k 2 where n is a pseudo-wall-normal vector defined by n = ∇α/∇α. The present article describes the derivation and validation of explicit algebraic representations based on this Reynolds-stress model.

2 Explicit Algebraic Methodology Using the weak equilibrium assumptions dbij /dt = 0 and Dij /Dkk = τij /τkk , where bij = τij /(2k) − δij /3 and Dij are the anisotropy and the total diffusion of τij , respectively, the following algebraic equation for the Reynolds stress is obtained τij τij ∗ Pij − P + φij − εij − ε = 0. (4) k k Introducing the EB-RSM model into (4) yields, under tensorial form 1 2 − b − a3 bS + Sb − {bS}I + a2 (bW − W b) a4 3 2 1 a5 M , − a5 bM + Mb − {bM}I − {bM}M = a1 S + 3 2 2

(5)

where {.} denotes the trace, and S and W are the mean strain and mean rotation tensors, respectively. In (5) and henceforth, the enclosed terms are the terms due to the introduction of the elliptic blending procedure. These terms vanish far from the wall, where the parameter α goes to one. The ai ’s are given by

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a1 =

289

2 1 g5 2 g4 2 a2 = 1 − α , a3 = 1 − α , − (g3 − g3∗ 1 − α 2 )α 2 , 3 2 2 2 −1 g∗ P 13 13 k g1 1 + 1 α2 − − α2 + −1 , (6) a4 = ε 2 ε 3 2 3 ε a5 = 5 (1 − α 2 ) , k

where the gi ’s are the coefficients of the SSG model. Since the implicit algebraic system (5) is numerically intractable, an explicit solution must be sought. The theory of invariants [9] indicates that the solution of such a relation between tensors is a polynomial function of the tensors involved in the equation, of the form b=

N

(7)

βi T i

i=1

where T i are the tensors of the so-called functional integrity basis, and the βi ’s are polynomial invariant functions.

3 Invariant and Functional Integrity Bases The specificity of the present model lies in the presence of the tensor M in (5). Indeed, in the standard explicit algebraic methodology [7], the relation only involves b, S and W , such that the solution is of the form (7), in which the functional integrity basis consists of the N = 10 terms [7] 1 T 4 = W 2 − {W 2 }I ; 3 2 T 5 = W S 2 − S 2 W ; T 6 = SW 2 + W 2 S − {SW 2 }I ; 3 (8) 2 2 2 T 7 = W SW − W SW ; T 8 = SW S − S 2 W S; 2 T 9 = W 2 S 2 + S 2 W 2 − {S 2 W 2 }I ; T 10 = W S 2 W 2 − W 2 S 2 W . 3

T 1 = S;

T 2 = SW − W S;

1 T 3 = S 2 − {S 2 }I ; 3

The βi ’s are polynomial functions of the terms of the invariant integrity basis η1 = {S 2 };

η2 = {W 2 };

η3 = {S 3 };

η4 = {SW 2 };

η6 = {SW S 2 W 2 }.

η5 = {S 2 W 2 }; (9)

In the present case, the relation (5) involves b, S, W and M, such that the functional integrity basis now contains N = 41 terms and the invariant integrity basis 29 terms [9]. However, using the fact that M 2 = 13 M + 29 I , the functional integrity basis reduces to N = 27 terms, i.e., (8) and the 17 additional terms

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2 T 12 = SM + MS − {SM}I ; T 13 = W M − MW ; 3 2 2 T 14 = MW S − SW M − {MW S}I ; T 15 = S 2 M + MS 2 − {S 2 M}I ; 3 3 2 2 2 2 2 T 16 = MW + W M − {MW }I ; T 17 = W MW − W MW 2 ; 3 2 2 T 18 = W MS − SMW − {W MS}I ; T 19 = W SM − MSW − {W SM}I ; 3 3 2 T 20 = W S 2 M − MS 2 W − {W S 2 M}I ; 3 2 2 2 T 21 = MW S − S W M − {MW S 2 }I ; 3 (10) 2 2 2 T 22 = W MS − S MW − {W MS 2 }I ; 3 2 T 23 = SW S 2 M − MS 2 W S − {SW S 2 M}I ; 3 2 T 24 = SW 2 M + MW 2 S − {SW 2 M}I ; 3 2 2 2 T 25 = W SM + MSW − {W 2 SM}I ; 3 2 2 2 2 2 T 26 = W S M + MS W − {W 2 S 2 M}I ; 3 2 T 27 = W 2 SW M − MW SW 2 − {W 2 SW M}I , 3 and the invariant integrity basis to 16 terms, i.e., (9) and the 10 additional terms T 11 = M;

η7 = {SM};

η8 = {S 2 M};

η11 = {W S M}; 2

η14 = {W S M}; 2 2

η9 = {W 2 M};

η12 = {W S MS}; 2

η15 = {W SW M}; 2

η10 = {W SM};

η13 = {W 2 SM};

(11)

η16 = {W MW S }. 2

2

The solution (7) of (5) can be obtained by performing a Galerkin projection, which leads to a 27 × 27 invertible linear system for the βi functions.

4 Truncated Bases In order to reduce the complexity of the model, the usual approach, for instance followed by [8] for the SSG model, is to consider a 2D plane flow. In this case, it can be shown [7] that the functional integrity basis is reduced to the 3 terms T 1 –T 2 –T 3 , and the invariant integrity basis to η1 –η2 . The expression (7) with N = 3 is the solution of (5) in 2D plane cases only, and can be used as an approximation in 3D. However, in our case, the integrity basis in 2D plane flows contains the 6 terms T 1 –T 2 –T 3 –T 11 –T 12 –T 13 , which leads to an overly complex model. Therefore, in the present paper, only bases consisting of at most 3 terms are used.

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This restriction is not too severe, considering that in a 2D plane flow case, the anisotropy tensor is determined by only 3 independent parameters, b11 , b22 and b12 . However, since such a basis is not an integrity basis, the function (7) obtained by Galerkin projection can be singular at particular locations in the flow domain. The standard choice for the 3-term basis is the 2D plane flow integrity basis T 1 –T 2 –T 3 . The use of this basis leads to a model denoted by EB-EASM #1 (First Elliptic Blending Explicit Algebraic Stress Model). However, basis tensors involving the tensor M are attractive. Indeed, this tensor is independent of the mean field, and tensors such as T 11 , T 12 and T 13 are at most linear in the mean velocity gradient. This is a very desirable property for improving numerical robustness. Moreover, M carries the information about the orientation of the wall, which is crucial in its vicinity to ensure a correct representation of the anisotropy in 3D flows (where a 3-term basis representation of b is incomplete). Another interesting characteristic of M is that it does not vanish where S and W vanish. Several combinations of models based on 2-, 3- and 5-term bases have been analytically investigated. The complexity of the formulation is only dependent on the number of tensors retained in the basis, not on the particular choice of the basis tensors. Using a 5-term basis may be valuable in 3D, complex flows, but at the price of a considerable increase of the complexity of the formulation. Four different attractive choices for the basis have been identified, and the resulting models are • • • •

EB-EASM #1: b = β1 S + β2 (SW − W S) + β3 (S 2 − 13 {S 2 }I ) EB-EASM #2: b = β1 S + β2 M EB-EASM #3: b = β1 S + β2 M + β3 (SM + MS − 23 {SM}I ) EB-EASM #4: b = β1 S + β2 (SW − W S) + β3 M

The reasons for selecting these particular models can be summarized as follows: EB-EASM #1 is the standard choice and can thus be easily compared with standard models, but it is nonlinear in the mean velocity gradient; EB-EASM #2 is the simplest formulation (only 2 basis tensors) that preserves the two-component limit of turbulence at the wall (b22 = −1/3); EB-EASM #3 is linear in the mean velocity gradient, which is desirable for numerical robustness, but degenerates to EB-EASM #2 in 1D flows, since the last two tensors of the basis are linearly dependent in this situation; EB-EASM #4 is not susceptible to this degeneracy, and incorporates the tensor M, such that it does not degenerate where S and W vanish. In the following sections, the focus will be on 1D flows, where EB-EASM #3 and #4 are identical to EB-EASM #2 and #1, respectively. Thus, it will only be necessary to present results given by the models #1 and #2. For a 2D plane flow, the Galerkin projection of (5) onto either one of the 4 bases selected in the previous section provides βi ’s of the form k P (12) βi η, R, P , Q , , , α ε ε √ √ where η = η1 = {S 2 } and R = −η2 /η1 = −W 2 /S 2 are the mean strain parameter and the mean kinematic vorticity number, respectively. The introduction of

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elliptic blending results in the appearance of two additional invariants in the models, P = η7 = {SM} and Q = 2η10 = 2{W SM}, that both characterize the orientation of the velocity gradient in the coordinate system linked to the wall. P is zero in a flow parallel to the wall and maximum at an axisymmetric impingement point, and Q is zero at an impingement point and maximum in a flow parallel to the wall. Therefore, P and Q are called the Impingement invariant and the Boundary layer invariant, respectively. The dependence on k/ε, P /ε and α originates from the variable ai coefficients of (7). k, ε and α are provided by their own differential equations, such that the original second-moment closure is reduced to a 3-equation model. The ratio of production to dissipation, P /ε, appears in (5) via a4 , and is a function of β1 , since P /ε = −2β1 η2 k/ε. Consequently, β1 is the solution of a nonlinear algebraic equation, that is cubic for the model EB-EASM #2, but quartic for the others. It is worth pointing out that this equation is only cubic in standard models using 3-term bases: the increase of the degree of the equation is due to the introduction of the elliptic blending method. This peculiarity does not make the selection of the proper root more problematic in the 1D cases under consideration in the present article, since there is a single physically admissible root (real and negative).

5 Validation of the Models The first validation test is the investigation of the analytical form of the models in a channel flow, in order to check that the models inherit from the EB-RSM the reproduction of the two-component limit of √ turbulence at the wall. In such a 1D flow, the invariants reduce to η = ∂U/∂y/ 2, R = 1, P = 0 and Q = η2 , and it can be shown that the models EASM #1 and #2 yield ⎡ ⎤ √ 2 −β2 η2 + 16 β3 η2 β1 η 0 2 ⎢ ⎥ √ ⎥ and 2 2 + 1 β η2 b=⎢ β η β η 0 1 2 3 ⎣ ⎦ 2 6 1 2 0 0 − 3 β3 η (13) ⎡ ⎤ √ 1 2 − 3 β2 β1 η 0 2 ⎢√ ⎥ ⎥, 2 2 b=⎢ β 0 2 ⎣ 2 β1 η ⎦ 3 1 0 0 − 3 β2 respectively. In both models, −kβ1 plays the role of an eddy-viscosity. In model #1, β2 and β3 drive the anisotropy of the normal stresses, while in model #2, the use of a 2-term basis does not enable the reproduction of the full anisotropy of the normal stresses, leading to b11 = b33 throughout the channel. As the wall is approached (y → 0), it can be shown that β1 → 0. For model #1, β2 → −1/(4η2 ) and β3 → −1/(2η2 ), while for model #2, β2 → −1/2. Therefore, for both models, the original limiting behavior of the EB-RSM (b12 = 0, b22 = −1/3, b11 = b33 = 1/6, b12 = 0) is preserved, and the two-component limit

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293

Fig. 1 EB-EASM #1: Mean velocity profiles in plane Poiseuille flows

Fig. 2 EB-EASM #1: Reynolds-stress profiles in the plane Poiseuille flow at Reτ = 590

of turbulence is correctly enforced (b22 = −1/3). Such a favorable behavior with the linear, 2-term model #2 is noteworthy. Figure 1 shows the mean velocity profiles given by EB-EASM #1 and its underlying RSM, the EB-RSM, in the case of Poiseuille flows at Reynolds numbers ranging from Reτ = 180 to 2000 [1, 5]. In Fig. 2, the corresponding Reynolds stresses at Reτ = 590 are shown. It is seen that the algebraic model gives profiles almost identical to the EB-RSM, except for the anisotropy at the channel center, due to the weak equilibrium hypothesis on the diffusion term. The Reynolds stresses given by the explicit algebraic model of [8] are also shown in Fig. 2. This model is identical to EB-EASM #1 far from the wall (α → 1), as highlighted in (5)–(12). This comparison emphasizes the effect of the introduction of the elliptic blending method in the explicit algebraic formulation. Figures 3 and 4 show the same results for EB-EASM #2, compared to the rescaled-v 2 –f model [4]. As pointed out previously, the reproduction of the anisotropy is not complete since the wall-normal component v 2 and the shear-stress uv are closely approximated, but the model yields exactly u2 = w 2 throughout the

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Fig. 3 EB-EASM #2: Mean velocity profiles in plane Poiseuille flows

Fig. 4 EB-EASM #2: Reynolds-stress profiles in the plane Poiseuille flow at Reτ = 590

channel. This behavior is similar to that of the v 2 –f model, provided that the v 2 component used for comparison is the one given by the additional “v 2 ” equation, not by the Boussinesq relation. Figures 5–8 show the results obtained using EB-EASM #1, EB-EASM #2 and the EB-RSM, in Couette–Poiseuille flows, compared with the DNS data of Orlandi [6]. These 1D flows are generated by imposing a pressure gradient and a moving wall. Three different configurations are studied, distinguished by the velocity gradient at the moving wall: positive for the “Poiseuille-type” flow; negative for the “Couettetype” flow; and nearly zero for the “intermediate-type” flow. Two Reynolds numbers are studied for each configuration. Figure 5 shows the satisfactory reproduction of the mean velocity profiles by all the models, and in Figs. 6, 7, 8 it is shown that the EB-RSM reproduces the Reynolds stress very well for the 3 types of flows. The nonlinear model EB-EASM #1 gives satisfactory results overall, but overestimates the anisotropy in the vicinity of the moving wall, in particular for the Couette-type flow. In such a 1D flow, this discrepancy necessarily comes from the use of the weak equilibrium hypothesis for the

Introduction of Wall Effects into Explicit Algebraic Stress Models Fig. 5 Couette–Poiseuille flows: mean velocity profiles. PT, CT, IT denote Poiseuille-, Couette- and intermediate-type, respectively

Fig. 6 Couette-type flow: Reynolds-stress profiles for Reτ = 207

Fig. 7 Intermediate-type flow: Reynolds-stress profiles for Reτ = 182

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Fig. 8 Poiseuille-type flow: Reynolds-stress profiles for Reτ = 204

diffusion terms: in the region close to the moving wall, the relative weight of these terms is increased due to the reduction of the turbulent level. The results given by EB-EASM #2 are comparable to those shown previously for Poiseuille flows. The crucial components uv and v 2 are correctly reproduced; although, u2 = w 2 is obtained. It is worth pointing out that EB-EASM #2 actually better approximates uv and v 2 than EB-EASM #1 in the intermediate-type flow. This can be traced to the vanishing of the shear component S12 in the vicinity of the moving wall, that leads to the degeneracy towards zero of the 3 basis tensors of EBEASM #1; whereas, in EB-EASM #2, the tensor M is independent on the mean flow.

6 Conclusions The introduction of the elliptic blending strategy into explicit algebraic stress models was presented. The extended integrity basis due to the introduction of the wallnormal-sensitive tensor in the algebraic relation led to a number of possible approximated formulations. The validation of selected models, for several cases of Poiseuille and Couette–Poiseuille flows, has shown satisfactory behavior, and that the main properties of the underlying Elliptic Blending Reynolds-Stress Model are preserved. The possibility of building models linear in the mean velocity gradients but resolving the anisotropy is attractive from a numerical robustness standpoint. The 2term linear model appears as a very acceptable simplified model, with many similarities with the v 2 –f model, but derived from an approach valid in general configurations, and with only 3 differential equations for k, ε and α.

References 1. Hoyas, S., Jimenez, J.: Scaling of velocity fluctuations in turbulent channels up to Reτ = 2000. Phys. Fluids 18(1), 011702 (2006)

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2. Manceau, R.: An improved version of the Elliptic Blending Model. Application to non-rotating and rotating channel flows. In: Proc. 4th Int. Symp. Turb. Shear Flow Phenomena, Williamsburg, VA, USA, 2005 3. Manceau, R., Hanjali´c, K.: Elliptic blending model: a new near-wall Reynolds-stress turbulence closure. Phys. Fluids 14(2), 744–754 (2002) 4. Manceau, R., Carlson, J.R., Gatski, T.B.: A rescaled elliptic relaxation approach: neutralizing the effect on the log layer. Phys. Fluids 14(11), 3868–3879 (2002) 5. Moser, R.D., Kim, J., Mansour, N.N.: Direct numerical simulation of turbulent channel flow up to Reτ = 590. Phys. Fluids 11(4), 943–945 (1999) 6. Orlandi, P.: Database of turbulent channel flow with moving walls. http://dma.ing.uniroma1. it/users/orlandi (2008) 7. Pope, S.B.: A more general effective viscosity hypothesis. J. Fluid Mech. 72, 331–340 (1975) 8. Rumsey, C.L., Gatski, T.B., Morrison, J.H.: Turbulence model predictions of strongly curved flow in a U-duct. AIAA J. 38(8), 1394–1402 (2000) 9. Spencer, A.J.M.: Theory of invariants. In: Eringen, A.C. (ed.) Continuum Physics, vol. 1. Academic Press, New York (1971) 10. Speziale, C.G., Sarkar, S., Gatski, T.B.: Modeling the pressure-strain correlation of turbulence: an invariant dynamical system approach. J. Fluid Mech. 227, 245–272 (1991)

Session 8: Dynamical Systems

• POD Based Reduced-Order Model for Prescribing Turbulent Near Wall Unsteady Boundary Condition G. Lehnasch, J. Jouanguy, J.-P. Laval, and J. Delville • A POD-Based Model for the Turbulent Wall Layer B. Podvin • HR SPIV for Dynamical System Construction J.-M. Foucaut, S. Coudert, and M. Stanislas • The Stagnation Point Structure of Wall-Turbulence and the Law of the Wall in Turbulent Channel Flow V. Dallas and J.C. Vassilicos

POD Based Reduced-Order Model for Prescribing Turbulent Near Wall Unsteady Boundary Condition Guillaume Lehnasch, Julien Jouanguy, Jean-Philippe Laval, and Joel Delville

Abstract Only limited regions of turbulent/unsteady flows can generally be simulated. The physical character of the solution thus depends on the reliability of the boundary conditions. Various methods (synthetic eddies, recycling/rescaling procedures) have already been proposed for inflow conditions of turbulent boundary layers, yielding various degrees of efficiency. But the prescription of realistic conditions at boundaries parallel to the wall is still an open question. The main objective of this study is to evaluate the feasibility of an alternative approach which consists in coupling the simulation with some POD-based reduced order models and/or stochastic models. The specific difficulties encountered in the case of a turbulent boundary layer, are first addressed. Preliminary a priori boundary flow reconstructions, as well as the results of some coupling with a large-eddy simulation are then analysed to determine the best suited approach.

This work has been performed under the WALLTURB project and the CALINS project. WALLTURB (A European Synergy for the assessment of wall turbulence) is funded by the CEC under the 6th framework program (CONTRACT No: AST4-CT-2005-516008). CALINS (Conditions aux limites instationnaires pour Navier–Stokes) is a non-thematic project funded by the French ANR (Agence Nationale de la Recherche) (reference project number 05-BLAN 0311). G. Lehnasch () · J. Delville LEA/CEAT UMR CNRS 6609, Université de Poitiers-ENSMA, 86036 Poitiers, France e-mail: [email protected] J. Delville e-mail: [email protected] J. Jouanguy · J.-P. Laval LML UMR CNRS 8107, Ecole Centrale de Lille, 59655 Villeneuve d’Ascq, France e-mail: [email protected] J.-P. Laval e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_31, © Springer Science+Business Media B.V. 2011

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Table 1 Databases used for this study Author

Reθ /Reτ

+ L+ x × Lz ≡ nx × nz

y+

Channel

UPM/LIMSI

Reτ = 180/550

8π × 4π ≡ 1024 × 1023

100

Channel

LML

Reτ = 180/550

2π × π ≡ 128 × 65

DB

Flow

DNS LES PIV

TBL

LML/TUCh/LEA

Reθ = 8000

(8.4 × 6.65

cm2 ) ≡ 169 × 134

50 100

These databases are extracted from the data provided by UPM (Madrid), LIMSI (Orsay), TUCh + (Gothenburg) and LML (Lille). y + , L+ x and Lz are the height and the sizes in wall units of the extracted planes corresponding to the artificial boundary

1 Introduction The ultimate objective of this study, performed in the framework of the WP6 package of the Wallturb programme, is to couple boundary layer Large Eddy Simulations (LES) to Reduced-Order Models (ROM) at a boundary parallel to the wall. This approach aims at improving the reliability of unsteady near-wall boundary conditions by modelling the incoming or outgoing coherent structures through the boundary of a reduced computational domain. This coupling is typically performed at a distance y + in the range [50 : 100]. In this first attempt, the model provides information to the computational domain with no feed-back from the computation. The approach follows from the recent availability of both temporally and spatially resolved velocity fields (by means of Time Resolved PIV (TR-PIV) or Direct Numerical Simulation (DNS) data, obtained by other partners of Wallturb). We propose a two-fold model where the lowest order POD modes are modeled using a Low Order Dynamical System (LODS) while the highest order modes are modeled stochastically. The features of the databases used in this paper are summarised in Table 1. A POD analysis is first presented in Sect. 2 to highlight the specific difficulties of POD-based ROM encountered when applied in a plane parallel to the wall. The feasibility of the ROM/LES coupling at a virtual wall of a channel flow is then discussed in Sect. 3. The procedures developed to improve the LODS calibration and some preliminary results of the integration of these LODS are described in Sect. 4. Finally, some new perspectives are discussed regarding other possible improvements.

2 POD Analysis and Modelling Strategy In spite of the quasi-homogeneity found in planes parallel to the wall, the spatial POD modes obtained are not simple combinations of Fourier modes. Whatever the database used, the Karhunen Loève dimension is very high and the convergence rate relatively small. The number of POD modes N necessary to represent 99% of the turbulent kinetic energy (TKE) is typically O(1000). The required number of snapshots being at least one order of magnitude greater than N , the eigenvalue problem to be solved is very large. A low-storage Lanczos iterative algorithm [2]

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Fig. 1 Typical POD eigenspectrum (left) and Spectral Power Density function of the POD coefficients an (t) vs frequency and POD mode number (right). Vectorial POD of the PIV database from the Turbulent Boundary Layer; Reθ = 8000

has been implemented and used to overcome this computational difficulty and to extract only the most energetic modes. For all databases, the POD eigenspectrum obtained always yields three typical zones, as illustrated in Fig. 1 for the case of the PIV database: (i) the lowest mode range, where the distribution of energy follows some complex laws depending on the nature of the samples used and on the level of convergence of the turbulence statistics; (ii) a regularly decaying zone (ranging in this example from mode 20 to 200) which yields a typical power law function exhibiting a slope of about −1.1 (close to −11/9) that can be considered the footprint of the Kolmogorov −5/3 law in POD mode space [3]. Consideration of the conventional frequency spectrum of each POD coefficient ai (t) individually also indicates that some general scaling laws can be attempted to take into account the mode hierarchy for the highest order modes; (iii) a final zone, where the decay of energy is more pronounced (yielding a power law in ∼−2.5). For cylinder wake or planar jet flows, the corresponding eigenspectra exhibit a degeneration type behaviour. This is characterised by successive modes which evolve in pairs with a similar energy. In this case, two successive POD modes in quadrature correspond to one single physical mode corresponding to the convection of coherent structures. Identifying the mode hierarchy can thus be a key element in order to correctly take into account the strong relationships between modes in the modelling process, and, eventually to reduce its dimension. This degeneration-like behaviour can also be observed in the case of boundary layers or channel flows when considering the topology of spatial POD modes Φ (i) (x). However, in the case of boundary layers or channel flows, such a strict hierarchy of modes is not observed furthermore. The mode hierarchy can change depending on the database considered (cf. Table 1). In addition, the topological similarities between modes depend on the physical quantity considered. This precludes any unambiguous pairing of these modes. Figure 2 illustrates, for example, that two modes can yield very similar spatial turbulent integral scales if only their spatial autocorrelation is considered, whereas the corresponding structure of their streamlines, analysed here using Line Integral Convolution [1], are in fact very different.

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Fig. 2 Analysis of POD modes topology obtained from PIV database: example of the autocorrelation field ((x , z ) from (0, 0) to (Lx /2, Lz /2)) (left) and field of streamlines (obtained by Line Integral Convolution [1] ((x, z) from (0, 0) to (Lx , Lz ))) (right) of 5th (top) and 9th (bottom) POD modes

According to these observations, the global strategy we use is the following: (i) perform a snapshot POD [4, 7] of the history of the velocity fields in such a way that the velocity field can be expanded, in terms of the mean flow u, POD coefficients and spatial eigenvectors Φ (i) (x); (ii) truncate the POD basis to NT = O(20) POD modes; (iii) calibrate the LODS giving the evolution of the first NT mode amplitudes ai (t) from the available time history of the corresponding POD coefficients; (iv) model in some stochastic way the amplitude evolution bi (t) of the highest modes i > NT ; (v) rebuild the velocity field at the boundary by: u(x, t) ≈ u(x, t) +

NT i=1

ai (t)Φ (i) (x) +

N

bi (t)Φ (i) (x)

i=NT +1

3 Flow Reconstruction and Coupling with LES In order to check the feasibility of coupling a LES with a robust and realistic model it is first necessary to distinguish the error due to the inaccuracy of LODS from the possible inappropriate stochastic modelling of highest order modes. The original simulation combines the use of a finite-difference scheme, Fourier polynomials and a Chebyshev collocation discretisation method, respectively in the streamwise, wall-normal and spanwise directions. A second order Adams– Bashforth is used for time-stepping with the help of a projection method (prediction of the velocity and pressure field, followed by a projection to reimpose the incompressibility condition). More details about the numerical procedure may be found in [5]. In the present study, the subgrid effects are recovered by using a dynamic Smagorinski model. As a preliminary test of coupling, the following strategy is thus followed with the time-resolved LES database. First, the lowest order (most energetic) modes ai (t) (i = 1, . . . , NT ) are exactly retained in order to reimpose an exact coherent information carrying at least 50% of

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Fig. 3 Reconstructions of instantaneous fields of longitudinal velocity (m/s) from LES database at a threshold truncation level NT = 49. From left to right: original snapshot, “truncation” and “model”

the TKE. Second, the highest order modes (bi (t)) are described by new time histories. These are generated from random time series which preserve the original aubi∗ (f ) bi (f ) of the POD coefficients,1 satisfying the statistical tospectra Ei (f ) = independence between modes (bi (t)bj (t) = 0), and preserving the mode hierarchy, i.e.: bi (t) is set such that bi (t)2 = λi where λi is the eigenvalue of mode i. Note that this approach does not preserve the phase between the highest modes. Examples of such a priori reconstructions are illustrated in Fig. 3. It is noteworthy that this procedure preserves the general structure of the original velocity fields up to very small scales. However a more refined analysis involving movies of these reconstructions shows that the lack of phase between the highest modes “smears” the dynamical history. In order to test the coupling at a virtual plane parallel to the wall, a new computational domain is used and the grid distribution near the coupling plane is modified. The coupling of the LES solver with the exact original data at this plane was first applied to check that satisfactory reproduction of the original statistics can be achieved. Then, two different flow reconstructions are reimposed at the virtual wall. In the first case, denoted “truncation case”, only the first POD modes are taken into account to rebuild exactly up to 50% of the TKE (which corresponds to 49 modes). In the second case, denoted “model case”, the contribution of the 851 highest order modes, modelled stochastically, is added. The mean longitudinal velocity and turbulent stresses obtained from the new simulation are shown on Fig. 4 for the two types of flow reconstructions considered at the virtual wall. In both cases, the mean flow appears to be hardly affected by the new virtual boundary condition. This indicates that the main effects of the shear and dissipative mechanisms are globally reproduced. A short adaptation zone above this boundary is noticeable. However, in this zone, the turbulence structure does not fulfil a correct energy balance. The correct evolution of the turbulent stresses is then qualitatively well reproduced when one goes toward the channel centre. The largest error is observed at the channel centre for the u component which is overestimated, while the other v and w components are underestimated. We see how the turbulence does not relax towards the expected equilibrium in the channel. However, it is worth noting that, despite the fact that the phase between the highest order modes is neglected, corresponds to the direct Fourier transform in the time domain, ()∗ corresponds to the () complex conjugate and corresponds to a time average.

1 Here

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Fig. 4 Virtual wall/LES coupling at y + = 50 in a channel flow based on the “truncation” (top) or the “model” (bottom): mean and fluctuating velocity. Line: filtered DNS, + LES, ° coupling

the stochastic modelling of their contribution helps restore a significant part of the spatial correlation above the virtual wall. These results are encouraging and indicate that the main properties of the turbulence structure near the centre of a channel flow can be expected to be globally reproduced even if the coherent information (that are given by a LODS) is only partially supplied. Of course, additional studies would be necessary to improve the modelling of highest order modes and to incorporate at least a partial phase relationship.

4 Low-Order Dynamical Systems The low-dimensional modelling strategy followed consists first in assuming a polynomial representation of the quadratic inter-mode dependency which would be obtained by considering a Galerkin projection of the weak form of the Navier–Stokes equations on the POD basis. Then, the coefficients are calibrated by applying a global minimisation procedure [6]. Thus, for each mode i at each time tn , the following low order modal equation is assumed: T T T dai (tn ) = Ci + Lij aj (tn ) + Qij k aj (tn )ak (tn ) dt

N

N

N

j =1

j =1 k=j

Note that this approach requires to know both the temporal modes and their derivatives. In this case, each set of coefficients can be obtained by solving, in a least square approach, this overdetermined system. A singular value decomposition is used for this purpose, in order to overcome possible ill-conditioning problems.

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Fig. 5 Phase portrait of POD coefficients obtained from DNS database: a2 vs a1 (left); time signals (arbitrary scale) given by a 10 equations ROM. From top to bottom: modes 1, 2, 3, 4 and 10 (right)

One should recall that the number of model coefficients required to calibrate the systems is of order NT2 for each mode. The maximal size of the LODS that can be considered is thus limited by the rank of the corresponding calibration problem, and depends directly on the number of snapshots available. A full exploration of the LODS behaviour has been carried out for truncation NT ranging typically from 4 to a maximal number of modes equal to 24 (depending on the database considered), which represents anyway a quite small portion of the TKE. By integrating the LODS obtained from every possible initial condition corresponding to the known reference POD signals, three typical dynamical regimes are observed: (i) chaotic-like and rapidly diverging; (ii) oscillator-like with increasing or decreasing amplitude; (iii) convergence around a stable limit cycle. The systems of the last class are robust but generally yield poor dynamical behaviour. Most of the systems belong to the first class and yield a richer dynamical behaviour, as illustrated in Fig. 5. But they are not robust enough to be coupled with the LES. We should remark that the features of the database used can have a great impact on the LODS obtained. Use of the largest TR-PIV database (2150 uncorrelated sets of 40 successive fields) led to chaotic-like systems only. However, using a subset of only 6120 uncorrelated snapshots from the same database could lead to a limit cycle. This suggests that the sampling process, the ratio between the number of snapshots and the number of LODS parameters to calibrate and the initial conditions used have a strong influence on the LODS representativity. Note that various approaches have been followed to improve the LODS robustness and accuracy. They include, in particular: noise filtering procedures of time histories of POD coefficients; robust algorithms for time-integration, generally dedicated to stiff problems, and improvement of the problem conditioning. The main conclusion is that the chaotic behaviour of most LODS defined with constant model parameters can only lead to short-term predictions.

5 Conclusions and Perspectives The POD analysis of various databases shared during the Wallturb program has revealed the high complexity of the mode hierarchy which represents the spatial flow

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organisation in planes parallel to the wall within boundary layers. Accordingly, the modelling strategy determined is based both on using LODS for representing the coherent largest scales and a stochastic modelling approach for the highest order modes. Some preliminary virtual wall/LES coupling simulation have been carried out. The results obtained are encouraging as far as this modelling approach is concerned. However the phase information between the highest modes appears to be a necessary ingredient for further improvement. These on-going studies now focus on the evaluation of alternative approaches. These are based on a more robust parametric definition of the LODS parameters in a probabilistic framework and improved generalised scaling laws to incorporate a physically representative phase information between the highest modes.

References 1. Cabral, B., Leedom, C.: Imaging vector fields using line integral convolution. Proc. SIGGRAPH 93, 263–270 (1993) 2. Calvetti, D., Reichel, L., Sorensen, D.C.: An implicitly restarted Lanczos method for large symmetric eigenvalue problems. Electron. Trans. Numer. Anal. 2, 1–21 (1994) 3. Knight, B., Sirovich, L.: Kolmogorov inertial range for inhomogeneous turbulent flows. Phys. Rev. Lett. 65, 1356–1359 (1990) 4. Lumley, J.: The structure of inhomogeneous turbulent flows. In: Atmospheric Turbulence and Wave Propagation, pp. 166–176. Nauka, Moscow (1967) 5. Marquillie, M., Laval, J.-P., Dolganov, R.: Direct simulation of separated channel flows with a smooth profile. J. Turbul. 9, 1–23 (2008) 6. Perret, L., Collin, E., Delville, J.: Polynomial identification of POD based low-order dynamical system. J. Turbul. 7, 1–15 (2006) 7. Sirovich, L.: Turbulence and the dynamics of coherent structures. Q. Appl. Math. XLV(3), 561–590 (1987)

A POD-Based Model for the Turbulent Wall Layer Bérengère Podvin

Abstract A POD-based model is built for the inner wall region (0 < y+ < 70) of a turbulent channel flow at Rτ = 180. The cut-off scales in the streamwise and spanwise directions are respectively 100 and 30 wall units. Only one eigenmode is considered in the wall-normal direction. The modelling hypotheses such as the effect of the mean velocity and the energy loss to higher-order modes are adjusted from the DNS. The statistics of the model are compared with those of the simulation. The time spectra and convection velocity of the modes are found to be in good agreement with the simulation. The solution of the model is found to hover near a cluster of travelling waves on the streamwise invariant subspace. The behavior of the model is found to exhibit some robustness with respect to the modelling assumptions.

1 Introduction POD-based low-dimensional models have been originally derived for wall turbulent flows [2] and have since then been extended to a variety of flows, such as free shear flows [5], separated flows [4], flows with thermal variations [16] and compressible flows [7]. For a number of these flows, especially in closed domains, it seems that a very small number of POD modes can reproduce faithfully the dynamics of the flow. The POD-based model can then be used as a basis for estimation and control [3]. In contrast, wall turbulence is characterized by a very high number of degrees of freedom. Keefe, Moin and Kim [10] have estimated the number of the attractor dimension to be larger than O(500) in channel flow. However, Aubry et al. [2] produced a model whose intermittent dynamics appeared to capture at least qualitatively the bursting process observed in real turbulent flows. A body of work [1, 14, 15, 17] has been constituted over the years, the aim of which is to examine more precisely B. Podvin () LIMSI-CNRS, Université Paris-Sud, 91403 Orsay Cedex, France e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_32, © Springer Science+Business Media B.V. 2011

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the relationship between higher-dimensional models and real-life flows. The present work is part of this effort, as we derive a model for the inner region of the wall layer and proceed to compare its predictions with that of the corresponding DNS. We first present the characteristics of the direct numerical simulation, then the details of the POD-based model. We then proceed with a statistical comparison between the model and the DNS.

2 Characteristics of the Direct Numerical Simulation The code used is similar to that of Kim, Moin and Moser [11] and has been described in [14]. It is based on a Chebyshev expansion in the normal (y) direction and Fourier expansion in the horizontal directions. The velocity components will be denoted indifferently u, v, w or u1 , u2 , u3 . The flow directions will be referred to as x, y, z or x1 , x2 , x3 . The equations were advanced in time using a third-order Runge–Kutta scheme for nonlinear terms and Crank–Nicolson discretization for the linear terms. Periodic boundary conditions were imposed in the horizontal directions x and z. The Reynolds number based on the channel half-height h and centreline velocity U of a laminar profile is 5500. The numerical code is based on the outer units (U, h). The mass flow rate was kept constant. In wall units, which are based on the fluid viscosity ν and friction velocity uτ , the channel half-height is about 180 wall units. The box dimensions are identical to Kim et al.’s (Lx , Lz ) = (4πh, 4πh/3).

3 The Proper Orthogonal Decomposition The Proper Orthogonal Decomposition (see Holmes et al. [6] for an introduction) is applied to the full flow unit, in the region 0 < y+ < 70. We selected 500 flow realizations spanning 2500 outer time units. Due to use of flow symmetries (see [12]) the number of samples was increased four-fold. In the horizontal directions (x and z), the POD modes are simply Fourier modes and the decomposition of the fluctuating velocity field can be written in Fourier space as: uilk (y, t) =

∞

n in alk (t)φlk (y)

(1)

n=0

where • the i-component of the velocity field u is Fourier-transformed in the horizontal directions x and z so that the spatial modes are organized by their wavenumber (l, k) in those directions and the wall-normal index n corresponding to the wallnormal direction y; n refers to the temporal coefficient and φ in corresponds to the associated spatial • alk lk POD mode.

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The POD spectrum and first eigenfunctions were compared to those obtained in the smaller unit of size (4πh/3, 2πh/3) studied in Podvin [14]. No noticeable difference was observed for the most energetic structures when scaled in wall units. The POD modes are obtained by integrating the product of the velocity field with the corresponding eigenfunction over the wall layer. The dominant modes consist of an alternance of low and high-speed streaks, combined with streamwise vortices. A model for the evolution of the POD modes is then obtained by projecting the n , and is of the form equations onto the basis of POD eigenfunctions φlk n nmp dalk p nmp p nm m n alk + Clkl k alm k al−l k−k + Dlkl k |alm k |2 alk + plk = Blk dt

(2)

lk

where the tensor notation is used for the superscripts but not for the indices. The form of the model is further detailed in the next section.

4 Derivation Hypotheses Three main assumptions, based on Aubry et al.’s original derivation, are made to put the model in closed form: • The horizontally averaged mean velocity profile U (y, t) contains a feedback term which represents the energy of the fluctuations. Its expression is based on a stationary balance between the pressure gradient and the total stress. u2 y2 1 U (y, t) = τ y − + uv(y, t) dy (3) ν 2h ν The Reynolds stress term is computed using the POD modes from the truncation. The interaction of the POD modes with the mean velocity profile is therefore composed of a linear source term and cubic terms. • The small-scale Reynolds stress tensor τ< is modelled with a Smagorinsky-like ss in the model, with hypothesis, which leads to an additional linear term blk ss n n = −αlk alk blk

(4)

n representing the action of the pressure at the top of the wall • The pressure term plk layer, was computed and found to be small. It was therefore neglected.

5 Model Validation Since the behavior of the POD modes is inherently chaotic, we have to rely on statistical measures of the dynamics. Specifically, we consider the energy level of each mode. We examine the time histories of the most energetic modes, i.e. the zero streamwise subspace and the first streamwise modes. We then compute the temporal

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Fig. 1 Normalized amplitude of the POD modes 1/2 |alk |/λlk

spectra. We point out that by construction the projected field has the same secondorder spatial statistics as those of the real flow. Finally, we consider the convection velocity of the modes, which is a very important criterion since it characterizes how the structures are transported throughout the flow. • Mode Energy: The total energy captured by the model is equal to the sum of the energy of individual modes in the truncation lk |alk |2 . The model was found to capture about 110% of the energy, i.e. model |2 lk |alk = 1.1 (5) DNS 2 lk |alk | Again, we emphasize that the calibration procedure does not enforce any constraints on the dynamics or the energy of the modes. Figure 1 gives a more precise indication of the energy repartition in the model. It shows the norm of each mode 1/2 integrated in the model relative to its expected value, i.e. |alk |2 1/2 /λlk where λlk is the variance of the POD mode alk . The norm is close to 1 for the most energetic modes. Lower-order modes tend to be slightly underestimated, while higher-order modes tend to be slightly overestimated. One can see that discrepancies are observed for “border” modes of the truncation, i.e. modes that are either associated with a purely streamwise variations (i.e. a zero spanwise wavenumber) or with high wavenumbers in both directions. Each of these modes corresponds to a low eigenvalue, so that the peak in the relative norm of the integrated mode actually results in only a small contribution to the total energy. • Power Spectral Density: Figure 2 compares the model and DNS power density spectra of some spanwise modes with no streamwise variations a0k , for k = 2 and k = 4. This allows us to compare the characteristic time scales of the modes. We can see on the figure that the power spectrum can be decomposed in three parts: a quasi-flat region at low frequencies up to a frequency of 0.05−0.1U/ h, a region of stronger decrease in the range 0.1 − 1U/ h, then a region at higher frequencies dominated by numerical noise. The model and the DNS spectra agree relatively well with each other, in particular for frequencies between 0.1U/ h and U/ h. In that range the spectral energy decreases like f −β with β around 2.2–3.8 in the DNS. The exponent β appeared to depend on the wavenumber and its values

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Fig. 2 Power density spectrum of the POD zero streamwise modes a0k , (a) k = 2, (b) k = 4

were about the same in the model and the DNS, except for the mode a03 (not represented here), where we found a value for β of 4.8 in the model, against a value of 3.8 in the DNS. We point out that the computation of very long time scales is an issue as our time window (over 1000 outer time units or 5000 viscous time units) is relatively short compared to the streamwise length of the box, so it is difficult to interpret the results at these time scales. • Convection Velocity: Next, we determine the speed at which the POD modes, and by extension, the coherent structures of the near-wall layer are convected through the domain. To estimate the convection velocity of the POD modes, we used the same expression as Jimenez (see (4.1) in [9]) clk = −

∗ da /dt] Lx Im[alk lk , ∗ 2πl alk alk

l = 0

(6)

Figure 3 compares the velocity extracted from the DNS with that obtained from the model. The mean convection velocity varies between 10 and 14uτ in both the DNS and the model. The oscillations for the first streamwise wavenumber in Fig. 3b) are likely due to a lack of convergence, since the domain is relatively long. The convection velocity in the DNS tends to decrease with both streamwise and spanwise wavenumbers, in agreement with the observations of Jeon et al. [8]. These trends are reproduced by the model.

6 Influence of the Calibration Procedure In the previous section, we have relied on the DNS to estimate the energy transfer to unresolved modes and calibrate the model. We now examine how the model is affected in the absence of calibration. We therefore integrated the model setting the energy transfer to zero, and computed the various statistics again. • Energy: The total energy was found to be about 95% of that of the DNS, which is quite close to the DNS value. Higher spanwise modes tend to be overesti-

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∗ da /dt] Im[alk lk , ∗ alk alk

(a) in the DNS, (b) in the model

Fig. 4 Statistics of the model with no transfer to the unresolved scales: (a) Power density spectrum of the most energetic POD zero streamwise mode a04 ; (b) Energy of the modes |alk |2

mated, which makes sense since they are no longer losing energy to the unresolved scales. Conversely, modes with a high streamwise wavenumber and a low spanwise wavenumber tend to be underestimated. This agrees with the fact that the sign of the transfer due to the unresolved stress tensor was found to be positive at these wavenumbers in the DNS. See Podvin [15] for more details. • Temporal Spectra: The power density spectra of the modes is represented in Fig. 4. The only discrepancy with the calibrated model is a small underestimation of the low-frequency content of the modes. Clearly the large-scale frequency content of the modes remains essentially unaffected by the amount of energy transfer to the unresolved modes, which means that our truncation seems adequate to reproduce the key features of the dynamics. The horizontal cut-off scales are 100 and about 35 wall units in respectively the streamwise and spanwise directions, which is close to Piomelli and Balaras’s wall resolution requirements [13]. • Convection Velocity: The convection velocity in the model represented in Fig. 5 matched the DNS at low streamwise wavenumbers but was slightly lower than expected at high streamwise wavenumbers. The maximum error was 25% of the DNS value and was observed at the highest streamwise wavenumber of the trun-

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Fig. 5 Convection velocity defined as ∗ Lx Im[alk dalk /dt] clk = − 2π ∗ l alk alk with no transfer to the unresolved scales

cation, which is also the location of the maximum of the imaginary correction term due to the unresolved modes.

7 Conclusion We have presented a model for the wall layer based on the POD, which on is an extension of the original Lumley’s model, developed by Aubry, Holmes, Lumley and Stone [2]. The truncation consisted of one wall-normal eigenfunction and cutoff scales of respectively 100 and 30 wall units in the streamwise and spanwise directions. The energy content of POD modes in the model was found to agree well with that of their DNS counterparts. The convection velocity was correctly estimated in the DNS and in the model. We found that the broad characteristics of the model were relatively insensitive to the accuracy of the unresolved small-scale transfer. However adjusting the closure assumptions resulted in slightly improved statistics for the POD modes. Acknowledgements This work has been performed under the WALLTURB project. WALLTURB (A European synergy for the assessment of wall turbulence) is funded by the EC under the 6th framework program (CONTRACT No : AST4-CT-2005-516008). Most of the computations were carried out at the IDRIS-CNRS center. The author is grateful to Pr. W.K. George, Pr. M. Stanislas and Dr. J. Delville for many stimulating discussions.

References 1. Armbruster, D., Guckenheimer, J., Holmes, P.: Heteroclinic cycles and modulated travelling waves in systems with o(2) symmetry. Physica D 29, 257–282 (1988) 2. Aubry, N., Holmes, P., Lumley, J., Stone, E.: The dynamics of coherent structures in the wall region of the wall boundary layer. J. Fluid Mech. 192, 115–173 (1988) 3. Bergmann, M., Cordier, L., Brancher, J.: Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced-order model. Phys. Fluids 17, 097101 (2005) 4. Couplet, M., Sagaut, P., Basdevant, C.: Intermodal energy transfers in proper-orthogonal decomposition Galerkin representation of a turbulent separated flow. J. Fluid Mech. 491, 275– 284 (2003)

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5. Delville, J., Ukeiley, L., Cordier, L., Bonnet, J., Glauser, M.: Examination of large-scale structures in a turbulent plane mixing layer. Part 1. Proper orthogonal decomposition. J. Fluid Mech. 391, 91–122 (1999) 6. Holmes, P., Lumley, J., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996) 7. Ilak, M., Rowley, C.W.: Reduced-order modeling of channel flow using traveling POD and balanced POD. In: 3rd AIAA Flow Conference (2006) 8. Jeon, S., Choi, H., Yoo, J., Moin, P.: Space–time characteristics of the wall shear-stress fluctuations in a low-Reynolds-number channel flow. Phys. Fluids 11(10), 3084–3094 (1999) 9. Jimenez, J., Alamo, J.D., Flores, O.: The large-scale dynamics of near-wall turbulence. J. Fluid Mech. 505, 179–199 (2004) 10. Keefe, L., Moin, P., Kim, J.: The dimension of attractors underlying periodic turbulent Poiseuille flow. J. Fluid Mech. 242, 1–29 (1992) 11. Kim, J., Moin, P., Moser, R.: Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166 (1986) 12. Moin, P., Moser, R.: Characteristic-eddy decomposition of turbulence in a channel. J. Fluid Mech. 200, 471–509 (1989) 13. Piomelli, U., Balaras, E.: Wall-layer models for large-eddy simulations. Annu. Rev. Fluid Mech. 34, 349–374 (2002) 14. Podvin, B.: On the adequacy of the 10-d model for the wall layer. Phys. Fluids 13, 210–224 (2001) 15. Podvin, B.: A pod-based model for the wall layer of a turbulent channel flow. Phys. Fluids 21(1), 015111 (2009) 16. Podvin, B., Quéré, P.L.: Low-order P.O.D.-based models for the flow in a differentially heated cavity. Phys. Fluids 13, 3204 (2001) 17. Sanghi, S., Aubry, N.: Mode interaction models for near-wall turbulence. J. Fluid Mech. 247, 455–488 (1993)

HR SPIV for Dynamical System Construction Jean-Marc Foucaut, Sébastien Coudert, and Michel Stanislas

Abstract Specific experiments of High Repetition Stereoscopic PIV were carried out in the high Reynolds number flat plate turbulent boundary layer of the Laboratoire de Mécanique de Lille (LML) wind tunnel. This experiment was done in the frame of the WALLTURB European project. The planes under study were parallel to the wall located at 50 and 100 wall units. As the flow is time resolved, space–time correlations are computed.

1 Introduction Stereoscopic Particle Image Velocimetry is now a reliable method to measure quantitative parameters in complex flows [11, 14]. Such a method allows us to obtain the 3 components of the velocity field in a plane. Generally, the cameras and lasers used have of low repetition rate of the order of 10 Hz. This does not allow us to record the time sequence of the flow. Recently the progress of new CMOS cameras and Nd:YLF lasers has allowed us to reach repetition rates higher than 1000 im/s [6]. This new approach, called High-Repetition PIV (HR-PIV), allows us to study the turbulence organization and the flow structure evolution. High-Repetition PIV needs both high energy output of the laser and good sensitivity of the camera [9]. In the present paper, two high speed cameras (kHz) are used in stereoscopic configuration [14]. The wind tunnel of LML allows us to obtain a high Reynolds number boundary layer, with a thickness of about 30 cm. In this wind tunnel, original experiments of characterization of the turbulent boundary layer were performed in the frame of the WALLTURB project. These experiments are concerned with both the J.-M. Foucaut () · S. Coudert · M. Stanislas LML UMR CNRS 8107, Villeneuve d’Ascq, France e-mail: [email protected] S. Coudert e-mail: [email protected] M. Stanislas e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_33, © Springer Science+Business Media B.V. 2011

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Fig. 1 Scheme of the mesh reduction for LES computation using dynamic boundary condition

measurement in a plane parallel to the wall at 100 wall units [6] and the simultaneous measurement of High Rep. Stereoscopic PIV in a plane parallel to the wall and of the signal of 143 single hot wires in a plane normal to the wall and to the flow [4]. In the present contribution, the database will first be validated. Then, space– time correlations will be computed. As shown in Fig. 1, these correlations are the first step to POD decomposition and Low Order Dynamical System generation [10] in order to obtain a dynamic boundary condition near the wall which mimics the unsteady behavior of turbulence. The coupling of this dynamic condition with LES simulation can be envisaged as future work. It should reduce the number of mesh points necessary for a LES computation close to a wall.

2 Experimental Setup The experiment was carried out in the LML turbulent boundary layer wind tunnel [2]. This wind tunnel is 1 m high, 2 m width and 20 m long to allow the development of the boundary layer. The last 5 m of the test section is transparent on all sides to allow the use of optical investigation technique. The turbulent boundary layer is studied on the bottom wall of the wind tunnel test section. This flow presents a tiny longitudinal pressure gradient which is negligible and has no effect on near wall turbulence.

2.1 HR SPIV System The present data base consists of three set-ups as described in Table 1. The PIV records were obtained in a plane parallel to the wall by varying both the Reynolds number and the wall distance. The x and z axis are in streamwise and spanwise directions respectively. The y axes is normal to the wall (Fig. 2). The 2 cameras used are Phantom V9. They are located in symmetrical forward scattering conditions. The H and L parameters defined in Fig. 2 are H = 1.20 m and L = 2 m. The camera chip has a full size of 1600 × 1200 pixels of 11.5 × 11.5 µm2 each. Such a size allows us to record at a rate of 500 image pairs per second, with a total memory of 2.6 GB. The recorded images are then transferred by means of a fast Ethernet connection to a computer. A 200 mm focal lens was used on each camera with f# = 4.

HR SPIV

319

Table 1 Characteristics of the experiments External velocity

m/s

Reynolds number

3

5

10

7500

8900

19800

Wall distance

wua

100

50

100

Rate

kHz

1

1.5

3

Sensor size

pix

1200 × 762

920 × 576

592 × 384

Field size

mm

86 × 70

66 × 60

42 × 40

Field size

wua

680 × 540

800 × 740

960 × 920

2000

1100

1200

Paquet of 40 fields Full memory paquet Full memory time

s

2

1

1

1.8

1.95

2.3

Spatial resolution

mm

0.5

0.5

0.5

Spatial resolution

wua

4

6

12

Interrogation window size

mm

1

1

1

Interrogation window size

wua

8

12

24

a Wall

units

Fig. 2 Front view of the experimental setup

The cameras were set in Scheimpflug conditions [14]. The magnification was 0.12. The light sheet was generated by two ExcelTechnology Darwin pulsed YLF lasers recombined, which are able to provide 2 × 18 mJ of energy at 1 kHz. The sheet thickness in the field of view was about 1 mm. The whole flow was seeded with Poly-Ethylene Glycol. The particle size was of the order of 1 µm. In this configuration, the Airy disk diameter is of the order of 6 µm. It gives a particle image size of the order of 1 pixel [1]. The PIV delay was chosen to obtain a mean displacement on each camera of about 8 pixels.

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Fig. 3 Front view of the experimental setup with the rake [4]

As an example, for the lowest Reynolds number (i.e. Rθ = 7500), the number of active pixels was reduced to 1200 × 672 pixels in order to increase the acquisition rate up to 1000 pairs per second (i.e. 1 kHz) which is higher than the Kolmogoroff frequency (about 400 Hz). At the wall distance under investigation (i.e. 100), the convection velocity was about 2 m/s. The convected structures of the flow move about 1.8 mm (i.e. 25 pixels) between two records. A total of 2000 uncorrelated packets of 40 time resolved fields (i.e. at 1 kHz) were recorded. This significantly improves the statistics convergence. Moreover, two full memory packets of 1800 time resolved velocity fields were also recorded (each corresponding to 1.8 s of the flow). Concerning the 2 other set-ups, the images from the same HR SPIV configuration and the signal of 143 single hot wires were simultaneously recorded [4]. Two Reynolds numbers based on momentum thickness were studied: 9800 and 19100. As illustrated in Fig. 3, the plane was located at a distance of about 4.5 mm, which corresponds to 100 wall units for the highest Reynolds number and 50 wall units for the smallest one. For the highest Reynolds number, it was necessary to increase the acquisition rate to 3000 image pair/s, the image size was consequently reduced to 592 × 384 pixels as given in Table 1. At the wall distance investigated, the convection velocity was about 5.8 m/s. The convective displacement was kept at about 25 pixels between two samples. For the lowest Reynolds number, the acquisition frequency was set at 1500 image pair/s which was also chosen to keep the convective displacement around 25 pixels. This value fulfills the Nyquist criterion as compared to the Kolmogoroff frequency. As can be seen in Fig. 3, the PIV system is located upstream of the hot wire rake. As given in Table 1, a total of 1100 packets of 40 time resolved fields were recorded to be time uncorrelated in order to compute the statistics. One packet was also recorded with the full memory in both cases. This corresponds to 6880 and 2943 time resolved velocity fields respectively for the highest and the lowest Reynolds number.

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321

Fig. 4 Mean velocity contour (a) streamwise and (b) spanwise

2.2 PIV Analysis The PIV image processing was performed using the PIVlml software developed at LML, that have been specially tuned to run on a computer cluster. The images from both cameras were processed with a standard multi-grid algorithm with discrete window offset. The analysis was made by the classical FFT-based cross-correlation method with integer shift of both windows. A 1-D Gaussian peak fitting algorithm was used for the sub-pixel displacement determination. The final interrogation window size was 16 × 24 pixels with a 50% mean overlap. The Soloff method using 3 calibration planes was used to reconstruct the three velocity components in the plane of measurement. This was also performed with in-house software. The calibration was done with a transparent target using crosses separated by 0.5 mm. From the self calibration process, the misalignment between the light sheet and the calibration plane was corrected (Coudert and Schon [3]). Figure 4 gives the mean velocity contours (averaged on the times and realizations) of the streamwise and the spanwise components respectively. A decrease in the streamwise velocity can be observed on the right part and positive and negative velocities are respectively seen on the upper and lower part of this figure. In that kind of flow these two velocities should be constant along x and z. This evidences the blockage effect due to the hot wire rake. This blockage effect is expected to affect only the mean velocity [13]. As the interest of such a study concerns particularly turbulence, Fig. 5 shows the contour of standard deviation of the streamwise, wall normal and the spanwise components and the shear stress. Even if convergence is not fully reached, the blockage effect is less visible in this figure which shows a good homogeneity of the turbulence results. The Reynolds stress tensor elements are then computed by averaging along x and z to increase the convergence. The main elements are presented in Table 2 and are compared with results of a previous hot wire experiment [2]. The standard deviation of each component and the shear stress were computed for the three velocities. The values are in good agreement with the hot wire anemometry.

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Fig. 5 Contour of (a) streamwise, (b) normal, (c) spanwise velocity fluctuations and (d) shear stress

3 Space–Time Correlations The first step for a POD decomposition consists of computing the two points correlations. The correlations of the velocity field can also be used to condense the space and time information and discuss the flow organization. Both Stanislas et al. [12] and Kähler [8] have detailed the properties of the two point spatial correlations for standard PIV fields obtained in the same boundary layer. They showed that the correlation shape provides quantitative information, like shape and size, about the coherent structures. SPIV gives the full two points correlation tensor in a plane and HR SPIV gives access to the space–time two point correlation: Rij (δt, δx, δz). u2i . u2j =

1 Np Nt Nx Nz

p

p

ui (t, y, z)uj (t + δt, x + δx, z + δz)

(1)

Np Nt Nx Nz

where Np is the number of uncorrelated packets (indexed p) of Nt = 40 time fields whose size is Nx times Nz vectors. The correlations are normalized by the standard deviations of ui and uj . The homogeneity along t, x and z are used to increases the convergence of the correlations, this drastically increases the computational time and the parallelization requirement.

HR SPIV

323

Table 2 Comparison of the Reynolds stress between HSSPIV and HWA results External velocity

5 m/s

3 m/s

10 m/s

Reynolds number

9800

7500

19100

50

100

100

Wall

distancea

HSSPIV Mean velocitya

–

16.4

–

Streamwise RMS velocitya

2.265

2.188

2.263

Wall normal RMS velocitya

0.951

1.089

0.946

Spanwise RMS velocitya

1.481

1.482

1.451

−0.774

−0.925

−0.764

Main velocitya

15.0

16.2

16.2

Streamwise RMS velocitya

2.378

2.232

2.232

Wall normal RMS velocitya

1.000

1.198

1.198

Spanwise RMS velocitya

1.481

1.451

1.451

−0.672

−0.829

−0.829

Shear

stressa

HWA

Shear a Wall

stressa

units

Fig. 6 Spatial correlation R11 in the plane (δx+ , δz+ ), (a) Rθ = 7500, (b) Rθ = 19800

Figure 6 shows the spatial correlation R11 of the streamwise velocity in the plane (δx+ , δz+ ) (i.e. δt+ = 0) for the two Reynolds numbers at 100 wall units. This correlation presents an elliptical shape with a maximum centered at zero, indicating a significant spatial coherence in the streamwise direction. Ganapathisubramani et al. [7] did the same computation at 92 wall units. The results they obtained are very similar to the present ones. An evolution of the size of the correlation can be clearly observed in wall units representation, probably due to a Reynolds number effect. Figure 7 shows the spatial correlation R12 in the same plane as Fig. 6 for the two Reynolds numbers at 100 wall units. This correlation is important because it is associated with the production of turbulent kinetic energy. It shows a main negative peak centered at zero corresponding to the Reynolds shear stress and two large

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Fig. 7 Spatial correlation R12 in the plane (δx+ , δz+ ), (a) Rθ = 7500, (b) Rθ = 19800

Fig. 8 Space–time correlations Rii in the plane (δt U + , δx+ ) (i.e. δz+ = 0)

positive regions located symmetrically at about 100 wall units. The main negative peak is the trace of a turbulent energy production region which corresponds to high negative values of the product of u and v (Kähler [8]). As in Kähler [8], this peak is not symmetrical in the x direction, the maximum is located at about δx+ = −12. The Reynolds number influence is less visible in Fig. 7. As far as the space–time correlations are concerned, Favre et al. [5] were the first, in 1957, to measure these correlations. They used two hot wire probes with a separation in space. Recently, Kähler [8] computed such correlations from dual plane SPIV. These correlations give some information about the coherence of the flow and the time scale of this coherence. Figure 8 shows the correlations Rii in the plane (δt U + , δx+ ) for (δz+ = 0) for the three velocity components. These correlations show for each component a single peak which is inclined with respect to the x axis with a slope of 1. This shows that convection of the turbulent structure by the mean flow is important and validates the Taylor hypothesis. Figure 9 shows 5 time steps of the two point correlation R23 in the plane (δx+ , δz+ ). The time steps are chosen in the range of −40 ms < δt < +40 ms (i.e. −38 < δt+ < +38). This correlation presents two main peaks of opposite sign and two secondary peaks. The intensity of these peaks increases with δt up to a region between 5 and 20 ms where there is a maximum. The spanwise distance between the two peaks is quite constant whatever the value of δt , it is of the order of 200 wall units. Such a shape is typical of vortices. This phenomenon of motion of the four peaks is linked to the convection and the maximum of the correlation level for δt > 0 is probably

HR SPIV

325

Fig. 9 Space–time correlation R23 in the plane (δx+ , δz+ ) for δt = −30, −15, 0, 15, 30 ms

linked to the shape of the vortex cut by the plane of measurement. This should be investigated by means of conditional sampling.

4 Conclusion High Repetition Stereoscopic PIV experiments have been performed in a high Reynolds turbulent boundary layer in order to obtain time resolved 3 components velocity fields. A large number of packets of 40 time-resolved fields were recorded in a plane parallel to the wall for 3 Reynolds numbers and 2 wall distances. A few long recordings corresponding to the full memory of the cameras were also obtained. The experimental database has been validated by statistical analysis. The statistics such as mean and RMS (and also PDF and spectra not presented here) have been compared with results of hot wire anemometry. The two-point spatial correlations were computed from this database and are in good agreement with the literature. Space–time correlations have also been computed with 40 time steps, providing the full space–time correlation tensor at the two wall distances under study. The results show that, at these wall distances, turbulence is dominated by the convection of localized turbulent structures at the local mean velocity. The detailed analysis of the space time correlations should give new characteristics about the turbulence structures. The experiments designed to allow us to build a low order dynamical system in a plane parallel to the wall, which will mimic the unsteady behavior of turbulence. The database will be used to compute a POD decomposition. The large number of packets will enable this computation with good convergence. The next step will be to compute the dynamical system and to couple it to a LES simulation. Acknowledgements The authors would like to acknowledge F. Benyoucef and D. Krolak who did a significant contribution to the development of the correlation computation software. This work has been performed under the WALLTURB project. WALLTURB (A European synergy for the assessment of wall turbulence) is funded by the CEC under the 6th framework program (CONTRACT No: AST4-CT-2005-516008). Part of the computations presented in this paper were carried out using the Grid’5000 experimental testbed, an initiative from the French Ministry of Research through the ACI GRID incentive action, INRIA, CNRS and RENATER and other contributing partners (see https://www.grid5000.fr).

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References 1. Adrian, R.J.: Dynamic ranges of velocity and spatial resolution of particle image velocimetry. Meas. Sci. Technol. 8(12), 1393–1398 (1997) 2. Carlier, J., Stanislas, M.: Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry. J. Fluid Mech. 535, 143–188 (2005) 3. Coudert, S., Schon, J.P.: Back projection algorithm with misalignment corrections for 2D3C Stereoscopic PIV. Meas. Sci. Technol. 12, 1371–1381 (2001) 4. Delville, J., Braud, P., Coudert, S., Foucaut, J.-M., Fourment, C., George, W.K., Johansson, P.B.V., Kostas, J., Mehdi, F., Royer, A., Stanislas, M., Tutkun, M.: The WALLTURB joined experiment to assess the large scale structures in a high Reynolds number turbulent boundary layer. In: Stanislas, M., Jimenez, J., Marusic, I. (eds.) Progress in Wall Turbulence: Understanding and Modeling. Proceedings of the WALLTURB International Workshop Held in Lille, France, April 21–23, 2009. ERCOFTAC Series, vol. 14. Springer, Dordrecht (2011) 5. Favre, A., Faviglio, J., Dumas, R.: Space–time double correlations and spectra in a turbulent boundary layer. J. Fluid Mech. 2, 313–342 (1957) 6. Foucaut, J.M., Coudert, S., Stanislas, M.: Unsteady characteristics of near wall turbulence using high repetition stereoscopic PIV. Meas. Sci. Technol. 20(7) (2009) 7. Ganapathisubramani, B., Hutchins, N., Hambleton, W.T., Longmire, E.K., Marusic, I.: Investigation of large-scale coherence in a turbulent boundary layer using two-point correlations. J. Fluid Mech. 524, 57–80 (2006) 8. Kähler, C.J.: Investigation of the spatio-temporal flow structure in the buffer region of a turbulent boundary layer by means of multiplane stereo PIV. Exp. Fluids 36, 114–130 (2004) 9. Okamoto, K.: Dynamic PIV: a strong tool to resolve the unsteady phenomena. In: Stanislas, M., Westerweel, J., Kompenhans, J. (eds.) Particle Image Velocimetry: Recent Improvements. Springer, Berlin (2004) 10. Perret, L., Braud, P., Fourment, C., David, L., Delville, J.: 3 component acceleration field measurement by dualtime stereoscopic particle image velocimetry. Exp. Fluids 40, 813–824 (2006) 11. Soloff, S., Adrian, R., Liu, Z.C.: Distortion compensation for generalized stereoscopic particle image velocimetry. Meas. Sci. Technol. 8, 1441–1454 (1997) 12. Stanislas, M., Carlier, J., Foucaut, J.M., Dupont, P.: Double spatial correlations, a new experimental insight into wall turbulence. C. R. Acad. Sci. Paris 2b 327, 55–61 (1999) 13. Stanislas, M., Foucaut, J.-M., Coudert, S., Tutkun, M., George, W.K., Delville, J.: Calibration of the WALLTURB experiment hot wire rake with help of PIV. In: Stanislas, M., Jimenez, J., Marusic, I. (eds.) Progress in Wall Turbulence: Understanding and Modeling. Proceedings of the WALLTURB International Workshop Held in Lille, France, April 21–23, 2009. ERCOFTAC Series, vol. 14. Springer, Dordrecht (2011) 14. Willert, C.: Stereoscopic digital particle image velocimetry for applications in wind tunnel flows. Meas. Sci. Technol. 8, 1465–1479 (1997)

The Stagnation Point Structure of Wall-Turbulence and the Law of the Wall in Turbulent Channel Flow Vassilios Dallas and J. Christos Vassilicos

Abstract DNS of turbulent channel flows propose the following picture. (a) The Taylor microscale λ(y) is proportional to s (y), the average distance between stagnation points of the fluctuating velocity field, i.e. λ(y) = B1 s (y) where B1 is constant, for δν y < δ. (b) The number density of stagnation points ns varies with −1 3 /δν with Cs constant in the range δν y < δ. (c) In that same height as ns = Cs y+ range, the kinetic energy dissipation rate per unit mass, = 23 E+ u3τ /(κs y) where E+ = E/u2τ is the normalised total kinetic energy per unit mass and κs = B12 /Cs is the stagnation point von Kármán coefficient. (d) For Reτ 1, large enough for the production to balance dissipation locally and for −uv ∼ u2τ in the range δν y δ, du/dy 23 E+ uτ /(κs y) in that same range. (e) The von Kármán coefficient κ is a meaningful and well-defined coefficient and the log-law holds only if E+ is independent of y+ and Reτ in that range, in which case κ ∼ κs . The universality of κs = B12 /Cs depends on the universality of the stagnation point structure of the turbulence via B1 and Cs , which are conceivably not universal.

1 Introduction The universality of the classical log-law theory of von Kármán and Prandtl has been the source of controversy for the last decade [7]. The mean velocity profile’s scaling with Reynolds number is debatable by some who propose power laws [1, 3] and the universality of the so called von Kármán constant, κ, in the case of a logarithmic velocity profile is questioned [9]. Recent estimates [9] imply a dependence of overall flow geometry on κ indicating values away from the most widely accepted value, V. Dallas () · J.C. Vassilicos Institute for Mathematical Sciences & Department of Aeronautics, Imperial College London, London SW7 2PG, UK e-mail: [email protected] J.C. Vassilicos e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_34, © Springer Science+Business Media B.V. 2011

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κ 0.41. Due to this non-universality the von Kármán constant is being renamed von Kármán coefficient [9]. The main aim of the present study is to relate the multiscale structure of the stagnation points of the fluctuating velocity field of turbulent channel flows to mean flow quantities, such as the dissipation rate of turbulent kinetic energy and mean velocity gradient, following the work by [4]. Stagnation points are the most basic aspect of field topology and are a well-defined concept, unlike “eddies”. Our new approach is validated by Direct Numerical Simulations (DNS) of fully developed incompressible turbulent channel flows proposing a resulting new intermediate asymptotic analysis for the mean flow profile of turbulent channel/pipe flows. The Reynolds numbers that we consider in this study are too low for the direct assessment of the log-law but appear to be sufficient for this new approach to turbulent mean flow profiles that we propose here.

2 Conventional Results of DNS of Turbulent Channel Flow We use the code of Laizet & Lamballais [5] to numerically solve the incompressible Navier–Stokes equations (1) in Cartesian coordinates applying periodic boundary conditions in the x (longitudinal) and z (lateral) homogeneous directions and no slip conditions at the y = 0 and y = 2δ walls. 1 1 ∂jj ui ∂i ui = 0 and ∂t ui + [∂j ui uj + uj ∂j ui ] = −∂i p + 2 Rec

(1)

where Rec ≡ Uc δ/ν is the Reynolds number based on Uc ≡ 32 Ub and Ub is the bulk velocity of the flow kept constant in time by adjusting the mean pressure gradient −dp/dx at each time step. We use different near-wall forcings and boundary conditions at the walls so as to demonstrate how our stagnation point approach accounts for the way that different wall-actuations modify the mean flow profile. Specifically, we considered the following three control schemes: (a) u = 0 with forcing f(y) = (A sin(2πy/Λ)H (Λ − y), 0, 0) added to the Navier–Stokes equations [14] where A = 0.16Uc2 /δ 2u2τ /δν and Λ = 11δν (case A1). This scheme corresponds to a steady wall-parallel forcing localised within eleven wall units from the wall and uniform in the direction parallel to it. This force decelerates the flow closest to the wall but accelerates it in the adjacent thin region. (b) u(x, t) = (0, a cos(α(x − ct)), 0) [10] with a/Uc = 0.05, α/δ = 0.5 and c = −2Uc (case A2). This boundary condition corresponds to a blowing-suction travelling wave on the wall. (c) v(x, yd , z, t), 0) replaced by −v(x, yd , z, t), 0) at all (x, z) points on the planes yd = 10δν and yd = 2δ − 10δν (case A3). This boundary condition corresponds to a computational control scheme whereby the normal velocity at a distance yd from the walls is made to change sign at every time step. Table 1 shows the numerical parameters of our computations. The Reynolds numbers of our computations based on the skin friction velocity uτ , the channel half-width δ and the fluid’s kinematic viscosity ν, i.e. Reτ ≡ uντ δ = δδν , where δν ≡ ν/uτ is the wall unit, range between Reτ 110 and 400.

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329

Table 1 Parameters for the DNS of turbulent channel flow Nx × Ny × Nz

Case

Forcing1

A

No

4250

179

4πδ

4πδ/3

200 × 129 × 200

A1

Yes2

4250

114.4

4πδ

4πδ/3

200 × 129 × 200

A2

Yes3

4250

222.3

4πδ

4πδ/3

200 × 129 × 200

A3

Yes

4250

141.6

4πδ

4πδ/3

200 × 129 × 200

Rec

Reτ

Lx

Lz

B

No

2400

109.5

4πδ

2πδ

100 × 65 × 100

C

No

10400

392.6

2πδ

πδ

256 × 257 × 256

1 Refers 2 Xu

to wall or near-wall actuations

et al. [14]

3 Min

et al. [10]

Fig. 1 (left) Profile of the production to dissipation ratio. Note the existence of an approximate equilibrium layer which grows with Reτ and where production approximately balances dissipad tion. (right) The inverse von Kármán coefficient ≡ y dy U+ versus y+ . The effects of the various near-wall actuations are significant

One might expect production to balance the rate of dissipation of turbulent kinetic energy in an intermediate region δν y δ when Reτ 1, i.e. P ≡ d −uv dy u . To check if this balance exists in our simulations we plot in Fig. 1(left) B2 = P /

(2)

This intermediate region, where this approximate balance holds increases as Reτ increases and asymptotically suggests that B2 → 1 as Reτ → ∞. This asymptotic result has been proved recently [2] assuming, however, that the mean velocity profile is logarithmic in this intermediate region δν y δ. The slight discrepancy away from B2 1 at these moderate Reynolds numbers is well known and agrees with other previously published DNS results [8].

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In the case of channel/pipe flows, as δ/y → ∞ and δν /y ≡ 1/y+ → 0, it can be d u − uv = u2τ (1 − y/δ) shown mathematically that the momentum balance ν dy implies −uv u2τ , i.e. B3 = −uv/u2τ → 1

(3)

in the intermediate region δν y δ. This is true assuming, that, in this limit d p d ln y+ U+ (where U+ ≡ u/uτ ) does not increase faster than y with p ≥ 1. Then from the above balance of production–dissipation and with Reτ 1 follows that, in this equilibrium intermediate region,

u3τ κy

implies

du uτ dy κy

(4)

However, at finite Reynolds numbers the equation for the mean velocity gradient B2 uτ should be replaced with du dy B3 κy , since at finite values of Reτ , B2 and B3 are different from 1 and even functions of y. According to the logarithmic scaling of the mean velocity profile U+ = 1 log y+ + B, which is obtained by integrating (4) if 1/κ is independent of y, we κ d U+ against y+ , which usually refers to 1/κ. However, in the plot the coefficient y dy d U+ B2 /(B3 κ) for these finite Reynolds numbers. It is difficult present context y dy to conclude from Fig. 1(right) for the validity of the log-law, which clearly shows a significant dependence on near-wall conditions, Reτ and y+ . It may be that the log-law is not valid at all or it may be that the log-law is not valid unless Reτ is sufficiently high, definitely higher than the Reτ of our simulations.

3 The Stagnation Point Approach Our DNS study for the equation of the mean velocity gradient in (4) does not yield clear results. Therefore, we choose instead to study the validity of the dissipation equation in (4) using the stagnation point approach, which have been shown to be useful in other contexts of turbulence theory [4, 6, 12]. Stagnation points are points where u ≡ u − u = 0, with . denoting temporal and spatial average in the homogeneous directions. A generalised Rice theorem for high Reynolds number Homogeneous Isotropic Turbulence (HIT) was proved recently [4], under two main assumptions: (a) statistical independence between small and large scales and (b) absence of small-scale intermittency effects. The theorem states that the Taylor microscale is proportional to the average distance between neighboring stagnation points. So the question which arises in the context of the present work is: Is there an intermediate region of turbulent channel flow with λ(y) = B1 s (y)

(5)

where B1 is independent of y and Reynolds number for Reτ 1? The definition of the Taylor microscale λ2 ≡ 23 νE relates it to the dissipation (y) = 2νsij sij , where sij is the fluctuating velocity’s strain rate tensor and the turbulent kinetic

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331

Fig. 2 Support for the generalised Rice theorem as a meaningful approximation in turbulent channel flows with various Reτ and wall-actuations (Table 1). B1 as a function of (left) y+ and (right) y/δ

energy E(y) ≡ 12 |u|2 . The average distance between stagnation points is defined as s ≡ LNx Ls z , where Ns (y) is defined to be the total number of stagnation points in a thin slab parallel to and at a distance y from the channel walls with dimensions Lx × δy × Lz with slab thickness δy ∼ δν . Using our DNS data, we are able to locate the stagnation points and finally obtain B1 from (5). Figure 2 gives the answer to our question showing the approximate constancy of B1 over an intermediate range δν y < δ. The small discrepancy away from B1 = const may be accounted to the neglected small-scale intermittency effects, which in the case of high Reynolds number HIT are known to appear as a weak Reynolds number dependence on B1 [6]. For wall-bounded turbulence this effects could therefore manifest themselves as a weak dependence of B1 on local Reynolds number y+ = y/δν . However, this refinement is left for future studies. An interpretation of the constancy of B1 can be given by considering the definition of the eddy turnover time τ ≡ E/ and combining it with λ2 ≡ 23 νE and (5) imply τ ∼ 2s /ν

(6)

Therefore, B1 constant means that in the equilibrium layer, the time it takes for viscous diffusion to spread over neighbouring stagnation points is the same proportion of the eddy turnover time (i.e. the time it takes to cascade the energy to the smallest scales) at all locations and all Reynolds numbers. By rearranging the definition of λ, using (5) and s ≡

2 νE 3 λ2

=

2 νE 3 B 2 2s 1

=

2 νE Ns 3 B 2 Lx Lz 1

=

2 νE 3 B 2 δν n s , 1

Lx Lz Ns ,

we have =

where ns ≡ Ns /(Lx Lz δν ) is the number

d u = B2 , usdensity of stagnation points. Then starting from the balance −uv dy

ing the above relation for the dissipation and follows ns =

Cs −1 y δν3 +

uτ d dy u = κy

with Cs =

B12 κB2 C

E as well as C ≡ − 23 uv , it

(7)

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Fig. 3 (left) Normalised number of turbulent velocity stagnation points for different Reτ and wall-actuation cases (Table 1); (inset) Points where u ≡ u − u = 0 for case C at a given instant it time. (right) C as a function of y+ for different Reynolds numbers and different wall-actuation cases (Table 1)

According to the classical claims [11] κ 0.4, B2 1 and C 2 in the intermediate range 1 y+ Reτ in the limit Reτ → ∞. Hence, provided B1 is constant then Cs should also be constant in that range and limit. As we have seen B1 is not far from being constant in the range δν y < δ, whereas κ and C are significantly far from constant in this range. Nevertheless, our DNS results indicate that Cs tends to a well-defined constant within the range δν y+ < δ as Reτ increases (see Fig. 3). Remarkably, the constancies of Cs and B1 seems to require as little as Reτ exceeding a few hundred showing the persistence of the stagnation point structure. The constancy of Cs entails that in the range δν y+ < δ the number density of stagnation points ns is inverse proportional with distance from the wall y+ (see (7)) and this is also depicted in the superimposed Fig. 3(left, inset), where the stagnation points become increasingly denser as the wall is approached. An interpretation of the constant Cs is that represents the number of turbulent velocity stagnation points within a cube of side-length equal to a few multiples of δν (see (7)) placed where y equals a few multiples of 10δν as seen in Fig. 3(left). This is the lower end of the −1 range where nS ∼ y+ and seems to be where the upper edge of the buffer layer is usually claimed to lie [11]. For an insight into the constancy of Cs one has to notice that (7) and 2s = LNx Ls z = and (5) follows ns δν give 2s = δν y/Cs . Then combining with = 23 νE λ2 =

2 Euτ 3 κs y

with κs ≡

B12 Cs

(8)

Therefore, Cs = const means τ ∼ y/uτ throughout the range where it is constant, given that the eddy turnover time τ = E/, where the constant of proportionality 3κs /2 is the same at all locations and all Reynolds numbers. We refer to κs as the stagnation point von Kármán coefficient, since it is determined by the stagnation point coefficients B1 and Cs and is constant in some intermediate range only if they are constant. In the present context the equation (8) replaces the usual = u3τ /κy

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Fig. 4 (left) Linear-log plots of P y/u3τ versus y+ for different Reτ and wall-actuation cases (Table 1); (right) Linear-log plots of 32 EP+ uy3 versus y+ for different Reτ and wall-actuation cases τ (Table 1)

[11] but they reduce to the same one only if and where E ∼ u2τ independently of Reτ and y+ .

4 Consequences of the Constancies of B1 & Cs u3

For Reτ 1, B1 and Cs constant in the equilibrium layer suggest = 23 E+ κsτy , where E+ = E/u2τ with a constant κs = B12 /Cs in that layer and limit. Moreover, d from (2) we have −uv dy u = B2 in the equilibrium layer. However, as Reτ → 2 ∞ then −uv → uτ in an intermediate layer δν y δ and is expected that u3

B2 → 1, therefore using = 23 E+ κsτy , then we obtain 2 uτ d u E+ dy 3 κs y

(9)

in that same layer and limit. However, at finite Reynolds numbers the equation (9) uτ d 2 should be replaced by dy u 23 B B3 E+ κs y to account for the potentially different rates of convergence of B2 , B3 and κs towards their high Reynolds number asymptotic constant values. The local high Reynolds number balance P in the equilibrium layer is the vital step to derive both (4) and (9). In terms of the classical assumption u3τ /κy, then P = B2 implies Bκ2 = Pu3y , whereas using (8) implies Bκs2 = 32 EP yu3 . In either + τ τ case, both of these equations should be constant in the equilibrium layer due to the balance of P and . It has to be noted that the main difference in the scaling of the dissipation in the present context is the presence of E+ . Figure 4 shows the DNS results for the two cases and clearly the collapse between different Reynolds numbers and wall-actuations is far worse and the y-dependence far stronger for Py B2 B2 3 Py κ = u3 (Fig. 4(left)) than for κs = 2 E u3 (Fig. 4(right)). τ

+ τ

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5 Conclusion The DNS suggest that B1 = λ/s and Cs = ns δν3 y+ are constants in the region δν y < δ and their asymptotic values can be reached for Reτ as low as a few hundred and therefore will be constants as Reτ → ∞. This is equivalent to stating that in 2 τ the limit Reτ → ∞ and in the region δν y < δ, = 23 Eu κs y with κs ≡ B1 /Cs constant in that range and limit. Either of these equivalent statements imply that as d u = E+ κusτy in the equilibrium region δν y δ where we may Reτ → ∞, dy expect production to balance dissipation. According to classical similarity scalings, in the limit Reτ → ∞, E ∼ u2τ independently of y in the equilibrium range δν y δ. If this is true, then the log-law (4) will be recovered from (8) and (9) with κ proportional to κs , which is inversely proportional to Cs , the number of stagnation points within a volume δν3 at the upper edge of the buffer layer. There is no a priori reason to expect Cs to be the same in turbulent channel and pipe flows and hence κ to be independent of overall flow geometry. On the other hand, Townsend’s idea of inactive motions [13] would suggest that E does not scale as u2τ in the equilibrium layer as Reτ → ∞ then (9) does not yield (4) and B2 /κ does not tend to a constant in that limit. Thus, mean flow data fitted by a log-law may yield non-universal κ as a result of κs = B12 /Cs but also as a result d of the small effect that inactive motions have on E, and thereby on dy u = E+ κusτy . In conclusion, whatever the scalings of E+ , one can expect measured values of the von Kármán coefficient to be non-universal. Acknowledgements We are grateful to Dr. Sylvain Laizet for providing the Navier–Stokes solver and to Halliburton for the financial support.

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

Barenblatt, G.I., Chorin, A.J., Prostokishin, V.M.: Proc. Natl. Acad. Sci. 94, 773–776 (1997) Brouwers, J.J.H.: Phys. Fluids 19, 101702 (2007) George, W.K.: Phil. Trans. R. Soc. A 365, 789–806 (2007) Goto, S., Vassilicos, J.C.: Phys. Fluids 21, 035104 (2009) Laizet, S., Lamballais, E.: J. Comp. Phys. 228(16), 5989–6015 (2009) Mazellier, N., Vassilicos, J.C.: Phys. Fluids 20, 015101 (2008) McKeon, B. (ed.): Theme Issue: Scaling and Structure in High Reynolds Number WallBounded Flows. Phil. Trans. R. Soc. A 365 (2007) Moser, R.D., Kim, J., Mansour, N.N.: Phys. Fluids 11, 943 (1999) Nagib, H.M., Chauhan, K.A.: Phys. Fluids 20, 101518 (2008) Min, T., Kang, S.M., Speyer, J.L., Kim, J.: J. Fluid Mech. 558, 309–318 (2006) Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000) Salazar, J.P.L.C., Collins, L.R.: Annu. Rev. Fluid Mech. 41, 405–432 (2009) Townsend, A.A.: The Structure of Turbulent Shear Flow. Cambridge University Press, Cambridge (1976) Xu, J., Dong, S., Maxey, M.R., Karniadakis, G.E.: J. Fluid Mech. 582, 79–101 (2007)

Session 9: Large Eddy Simulation

• Wall Modelling for Implicit Large Eddy Simulation of Favourable and Adverse Pressure Gradient Flows Z.L. Chen, A. Devesa, M. Meyer, E. Lauer, S. Hickel, C. Stemmer, and N.A. Adams • LES of Turbulent Channel Flow with Pressure Gradient Corresponding to Turbomachinery Conditions W. Elsner, L. Kuban, and A. Tyliszczak • LES Modeling of Converging Diverging Turbulent Channel Flow J.-P. Laval, W. Elsner, L. Kuban, and M. Marquillie • Large-Scale Organized Motion in Turbulent Pipe Flow S. Große, D.J. Kuik, and J. Westerweel

Wall Modelling for Implicit Large Eddy Simulation of Favourable and Adverse Pressure Gradient Flows ZhenLi Chen, Antoine Devesa, Michael Meyer, Eric Lauer, Stefan Hickel, Christian Stemmer, and Nikolaus A. Adams

Abstract In order to perform Implicit Large Eddy Simulation (ILES) on complex geometries at high Reynolds numbers, a wall model based on the simplified Thin Boundary Layer Equations (TBLE) is designed in the framework of ILES with a cut-cell finite-volume immersed boundary method. This wall model is validated for turbulent channel flow at friction Reynolds number up to Reτ = 2,000 on very coarse grids. The results compared with DNS and LES without wall model show that the wall model has the potential to improve the mean velocity in the outer flow region well at high Reynolds number. The wall model is applied to a complex converging diverging channel flow at Reynolds number Re = 7,900 on very coarse meshes. Improved mean velocities and Reynolds stresses are obtained, which shows that the wall model has the ability to perform ILES on complex geometries at high Reynolds numbers.

1 Introduction Structured and unstructured body-fitted meshes are extensively used in industrial Computational Fluid Dynamics (CFD) simulations. In complex flows, large computational effort is required to generate good quality meshes, and the accuracy of the simulation is reduced due to the limited grid smoothness and orthogonality. Multiblock technique is used to deal with complex geometry, but the grid smoothness can deteriorate at the interface between subdomains. The grid quality of unstructured meshes also deteriorates with the increased complexity of the geometry especially in the boundary layer. In order to ease on the quality requirements for the preprocessing of the grid generation and to maintain the accuracy of high-order numerical schemes, an alternative approach combining the advantage of Cartesian meshes Z.L. Chen () · A. Devesa Institute of Aerodynamics, Technische Universität München, 85748 Garching, Germany e-mail: [email protected] A. Devesa e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_35, © Springer Science+Business Media B.V. 2011

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with the ability to deal with complex geometries has been developed in recent years [5, 9–11]. Those methods simulating the viscous flow with immersed boundaries on the background of Cartesian grids which do not conform to the boundary of the geometry, are usually named according to the way how to impose the boundary condition. Here, we call them Immersed Boundary (IB) methods in general. The primary advantage of this kind of method is that it separates the grid generation from the complexity of the geometry almost entirely. There are two key issues in IB methods: the way how to impose the boundary condition and the grid refinement near the wall to resolve the near-wall region. In most IB methods, it is difficult to impose the boundary condition sharply and directly as in body-conformal grid method. Forcing functions are included into the governing equations as source terms to take the effect of the boundary on the fluid into account, which categorise as continuous forcing approach and discrete forcing approach. The former uses a smooth distribution function to distribute the sharp boundary force on a discrete mesh, which always spreads the force into several cells near the boundary, and makes the method only first-order accurate. The latter updates the discrete force indirectly through reconstruction of the fluid field near the boundary by interpolation, which always requires the presence of the first grid point in the viscous layer and makes it not suitable for simulation at high Reynolds numbers. The non-body-conformal Cartesian grids need more grid points to get appropriate grid resolution in the boundary layer in the vicinity of the wall than bodyconformal grids. The ratio of grid points for a Cartesian grid to a body-conformal grid scales as (L/δ)3 in outer flow simulation of 3D bodies [8], where L is the characteristic length of the body, and δ is the boundary layer thickness, although the simulations carried out here are not of this type. Adaptive mesh refinement methods can ease this difficulty with local refinement, but it increases the complexity of the algorithm. From the point of view of Large Eddy Simulation (LES), a lot of small structures scaled with wall units are contained in the near-wall viscous and buffer regions and play a significant role in the generation and transport of turbulent kinetic energy and shear stress there. A fully resolved LES must simulate the bulk of these features. This makes the grid resolution of LES comparable to that of Direct Numerical Simulation (DNS) and limits the use of LES to moderate Reynolds numbers. It is more difficult to satisfy the grid-resolution requirement in the framework of IB method and in the industrial flow simulations characterised by high Reynolds numbers. One way to overcome this difficulty is to replace the near-wall region with a wall model which provides the outer LES with approximate wall boundary conditions [1, 12] and imposes the boundary indirectly in the IB method. As sequel of previous work [2], the objective here is to extend the wall modelling based on simplified Thin Boundary Layer Equations (TBLE) for Implicit Large Eddy Simulation (ILES) to complex geometries in the content of cut-cell finite-volume IB method. This wall modelling is validated for Turbulent Channel Flow (TCF) at friction Reynolds numbers up to Reτ = 2,000 by comparisons with DNS results, and is applied to a converging-diverging channel flow (bump flow) with weak separation at Reynolds number Re = hUmax /ν = 7,900, where h is half of the channel width and Umax is the maximum mean velocity at the inlet.

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2 Numerical Method and Wall Modelling The cut-cell finite-volume IB method and the wall model both are implemented in the framework of ILES. In ILES, a finite volume method based on a nonlinear deconvolution operator and a numerical flux function, called Adaptive Local Deconvolution Method (ALDM) is used to formulate the truncation error as an implicit SGS model [4]. An explicit third-order Runge–Kutta scheme is used for time advancement. The time step is dynamically adapted to satisfy a Courant–Friedrichs– Lewy condition. The pressure Poisson equation and diffusive terms are discretized by second-order centred differences, whereas the convective terms are discretized by Simplified Adaptive Local Deconvolution method (SALD) for improved computational efficiency [3]. The Poisson equation is solved at every Runge–Kutta substep. The validation of this numerical methodology has been established for plane channel flow, separated channel flow, passive-scalar mixing, and massively separated turbulent boundary layer under adverse pressure gradient. In the next subsections, we will briefly introduce the cut-cell finite-volume approach and describe the wall models in detail.

2.1 Cut-Cell Finite-Volume IB Method Most of the IB methods could not satisfy the underlying conservation laws for the cells in the vicinity of the IB. Strict global and local conservation of mass and momentum are very important for large-eddy simulations to get reasonable large-scale structures. This requirement can be guaranteed by a cut-cell finite-volume immersed boundary method. A signed level-set function is used to represent the geometry with its zero values. All of the Cartesian grid cells are categorised by level-set values at their nodes into three groups: fluid cell with positive level-set values at all of its nodes, solid cell with negative levelset at all of its nodes, and cut cell with positive and negative values at its nodes. The Navier–Stokes equations are discretized on a staggered Cartesian grid, so the velocity cells in the three directions and pressure cells are treated separately. For a cut-cell, its edge ratio in the fluid part is calculated by level-set values at the two nodes of each edge. The face apertures immersed in the fluid are calculated on its six faces by its edge ratios, and then its volume fraction occupied by the fluid is calculated using face apertures and edge ratios. The face apertures are used to calculate the convective and diffusive fluxes in the cutcells and the velocity boundary conditions for the convective flux calculation can be imposed directly on the wall. The diffusive flux on the wall is supplied by the wall model or is approximated by local velocity and wall distance in LES without wall model. The small cut-cells (say volume fraction less than 0.5) are merged with their neighbours in order to improve the stability constrained by the small cut-cells. The detail of this method are presented in [7].

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Fig. 1 The sketch of interpolation and tangential direction definition

2.2 Wall Model on IB Boundary The simplified Thin Boundary Layer Equation (TBLE) is used as a two-layer model to supply LES with the diffusive flux boundary condition driven from the no-slip boundary condition on the wall. The following simplified TBLE is used ∂utan ∂utan ∂ 1 ∂p (ν + νt ) = , + ∂xn ∂xn ∂t ρ ∂xtan

(1)

in which xn is the wall-normal coodinate, xtan is the wall-tangential coordinate, and utan is the velocity in the wall-tangential direction. The mixing-length eddyviscosity model with damping function is used, which accounts for all near-wall turbulent scales +

νt = κxn uτ (1 − e−xn /A )2 ,

(2)

where κ = 0.4, A = 19.0, and xn+ is the non-dimensional wall-distance scaled by √ wall-units l + = ν/uτ . The friction velocity is defined as uτ = τw /ρ. The upper boundary of the TBLE is interpolated from LES at the points having the same level-set value, that is, having constant wall distance. The velocities there are interpolated by an inverse distance weighting formula on the grid points surrounding the interpolation node. In order to solve the TBLE once for every cut-cell, just one wall-tangential direction is defined by the wall-normal direction and the velocity that is interpolated from LES at the interpolation point. The interpolation and tangential direction definition are shown in Fig. 1. The simplified boundary-layer equations neglecting the convective terms are ordinary differential equations and are solved algebraically every time step on embedded grids in the wall-normal direction to get the wall-shear stress value. However, this wall-shear stress can not be used directly on Cartesian grids in LES. A complex tensor transformation has to be used to transform wall-shear stresses from the local coordinates to Cartesian coordinates. This needs coodinate differential computation and matrix inversion, and bears high computational cost. To avoid this difficulty, the

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Fig. 2 Defect-velocity comparisons in DNS, coarse LES with/without wall model (WM)

wall-shear forces are calculated from the wall-shear stress and the local wall-surface area at every Runge–Kutta substep and are used as source terms in the momentum equations.

3 Validation and Application In order to investigate the effect of this wall model, the simulations of canonical TCF are carried out at several friction Reynolds numbers. The defect velocity and Reynolds stresses are compared with DNS results. Then the wall model is applied to a converging-diverging channel flow with inlet friction Reynolds numbers Reτ = 395. The results are compared with that of DNS at the same flow condition and that of coarse ILES without wall model.

3.1 Validation for Turbulent Channel Flow For all TCF cases, the plane channel (domain size 2πh × 2h × πh) is immersed symmetrically in a domain of size 2πh × 2.5h × πh discretized on 32 × 36 × 32 regular cells in three dimensions. Periodic boundary conditions are applied in streamwise and spanwise directions, and the wall model is used on both walls. The nondimensional defect velocity which is the mean velocity scaled by the centreline mean velocity and the friction velocity is compared in Fig. 2. For all four cases, the wall-modelled LES under predicts the mean velocities around y/ h = 0.1, but improves it considerably compared with LES without wall model in the outer flow

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Fig. 3 Reynolds stresses comparison at Reτ = 395

Fig. 4 Reynolds stresses comparison at Reτ = 2,000

region with increasing friction Reynolds number. The Reynolds stresses are compared in Fig. 3 at Reτ = 395 and Fig. 4 at Reτ = 2,000, which show that LES with wall model over-predicts the Reynolds stresses severely at low friction Reynolds number, but become much better at high friction Reynolds number. Coarse LES without the wall model severely under-predicts the Reynolds stresses at the high friction Reynolds number Reτ = 2,000. The peak values of Reynolds stresses are further away from the wall than DNS, which is related to the under-predicted velocity profile near the wall. In order to know what causes this phenomenon, different ways of cutting and with/without merging of small cut-cell cases are carried out on a 16 × 22 × 16 mesh. In Fig. 5, the defect velocities are compared, which show that

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Fig. 5 Defect velocity comparison of different ways of cutting and with/without merging at Reτ = 395

Fig. 6 Streamwise normal Reynolds stress comparison of different ways of cutting and with/without merging at Reτ = 395

it is not the way of cutting but the merging of small cut-cell that makes the velocity be under-predicted near the wall. This is reflected in Fig. 6 as the peak of uu is located closer to the wall in the cases of LES without cell merging compared to the LES with cell merging. For general use, the small cut-cell merging can not be avoided, but can be improved by increasing grid resolution near the wall. From the current investigation, we can conclude that wall model has the potential to improve the mean velocity and Reynolds stresses at high Reynolds number.

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Fig. 7 Mean streamwise velocity comparison at several streamwise locations Fig. 8 Reynolds stresses comparison at streamwise location x = 500

3.2 Application to Bump Flow The domain size of the converging-diverging channel is 4πh × 2h × πh which is symmetrically immersed in a domain of size 4πh×2.24h×πh discretized on 172× 64 × 32 regular cells in three dimensions. Inflow and outflow boundary conditions are imposed in the streamwise direction. Periodicity is imposed in the spanwise direction and the wall model is used on both walls. The flow is under complex pressure gradient and curvature. In the DNS [6], there is a weak separation bubble at the backward side of the bump. The mean streamwise velocities are compared in Fig. 7, which shows that the wall model has not much effect in front of the apex of the bump, but shortens the length and height of the separation bubble towards the DNS result. Also it improves the velocity gradient near the wall in the separation and reattachment region. Reynolds stresses at streamwise position x = 500 in the separation region are compared in Fig. 8. The Reynolds stresses are improved a

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lot by wall-modelled LES like the trend in TCF, but the peak values of Reynolds stresses are far from the wall due to the small cut-cell merging.

4 Conclusion From the investigation of TCF and bump flow, we observe that the investigated TBLE-based wall model improves the mean velocity and Reynolds stresses at high friction Reynolds number Reτ = 2,000 in the framework of ILES in combination with a cut-cell finite-volume IB method on very coarse grids. On one hand, the wall model approximates the near wall region with wall-shear force and provides the LES with diffusive-flux boundary condition. On the other hand, it imposes the no-slip boundary condition for IB method indirectly. The results show that this wall model has the potential to be used for LES on more complex geometries at high Reynolds numbers. The comparisons of the Reynolds stresses show that the peakstress values occurs further away from the wall in wall-modelled LES and coarse LES than in DNS. This is caused by the small-cell merging algorithm, which can not be avoided due to stability constraints, but can be improved by increasing the near-wall grid resolution. Acknowledgements The authors gratefully acknowledge the members of LES group in the Institute of Aerodynamics of TUM for their valuable discussions and heartful help. This work has been performed under the WALLTURB project. WALLTURB (A European synergy for the assessment of wall turbulence) is funded by the EC under the 6th framework program (CONTRACT No.: AST4-CT-2005-516008).

References 1. Cabot, W., Moin, P.: Approximate wall boundary conditions in the large-eddy simulation of high Reynolds number flow. Flow Turbul. Combust. 63, 269–291 (1999) 2. Chen, Z.L., Devesa, A., Hickel, S., Stemmer, C., Adams, N.A.: A wall model based on simplified thin boundary layer equations for implicit large eddy simulation of turbulent channel flow. In: 16th DGLR Symposium of STAB, RWTH Aachen University (2008) 3. Hickel, S., Adams, N.A.: Implicit LES applied to zero-pressure-gradient and adverse-pressuregradient boundary-layer turbulence. Int. J. Heat Fluid Flow 29(3), 626–639 (2008) 4. Hickel, S., Adams, N.A., Domaradzki, J.A.: An adaptive local deconvolution method for implicit LES. J. Comput. Phys. 213, 413–436 (2006) 5. Lai, M.C., Peskin, C.S.: An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J. Comput. Phys. 160, 705–719 (2000) 6. Marquillie, M., Laval, J.P., Dolganov, R.: Direct Numerical Simulation of a separated channel flow with a smooth profile. J. Turbul. 9(1), 1–23 (2008) 7. Meyer, M., Devesa, A., Hickel, S., Hu, X.Y., Adams, N.A.: An immersed interface method in the framework of implicit large eddy simulation. In: Proceedings DLES 7, Trieste, Italy, 2008 8. Mittal, R., Iaccarino, G.: Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261 (2005) 9. Mohd-Yosuf, J.: Combined immersed boundary/B-spline methods for simulation of flow in complex geometries. Annual Research Briefs. Center for Turbulent Research, pp. 317–328 (1997)

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10. Tessicini, F., Iaccarino, G., Fatica, M., et al.: Wall modeling for large eddy simulation using an immersed boundary method. Annual Research Briefs. Center for Turbulent Research, pp. 81– 187 (2002) 11. Udaykumar, H.S., Mittal, R., Rampunggoon, P., Khanna, A.: A sharp interface Cartesian grid method for simulating flows with complex moving boundaries. J. Comput. Phys. 174, 345–380 (2001) 12. Wang, M., Moin, P.: Dynamic wall modeling for large-eddy simulation of complex turbulent flows. Phys. Fluids 14(7), 2043–2051 (2002)

LES of Turbulent Channel Flow with Pressure Gradient Corresponding to Turbomachinery Conditions Witold Elsner, Lukasz Kuban, and Artur Tyliszczak

Abstract A converging–diverging channel flow with the geometry corresponding to the blade channel of axial compressor has been simulated using Large-EddySimulation (LES) for the inlet Reynolds number based on the friction velocity equal to Reτ = 395. The channel geometry causes the existence of the adverse pressure gradient which induces the boundary separation. Both the influence of the mesh spacing and the subgrid scale models (SGS) have been investigated. Two SGS model have been considered: the Wall Adapting Local Eddy-Viscosity and the Smagorinsky with the dumping function.

1 Introduction Large Eddy Simulation technique is increasingly used as a tool to model various flow including non-equilibrium, threedimensional flows, relaminarizing and retransitioning boundary layers in presence of the low Reynolds number, and massively separated flows. The main limitation of LES appears to be in the application for wall-bounded flow. In case when the near wall region needs to be resolved, the grid must be proportional to the size of inner-layer eddies depending strongly on Reynolds number. A lot of effort has been put into the development of this technique in order to reduce its computational cost. The proper solution of the near wall flow depends among the others on the applied subgrid model. One of the important tasks is to take into account subgrid-scale dissipation at the smallest scales, which leads to significant improvements of isotropic turbulence description, which is assumed to W. Elsner () · L. Kuban · A. Tyliszczak Czestochowa University of Technology, Czestochowa, Poland e-mail: [email protected] L. Kuban e-mail: [email protected] A. Tyliszczak e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_36, © Springer Science+Business Media B.V. 2011

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be present near the wall. Moreover the model has to produce an eddy viscosity with correct near-wall behavior and to be inactive on the well-resolved scales of the flow. In the paper the turbulent flow through the channel with one curved wall has been investigated. Shape of the channel corresponds to the geometry of axial compressor blade channel. The wall curvature and change of cross-section area produces the adverse pressure gradients which creates conditions for boundary layer separation. The test case was experimentally investigated by Materny et al. [6] for Reτ = 950. The objective of research was to apply for this test case popular and commonly used SGS models i.e. the Wall Adapting Local Eddy-Viscosity (WALE) of Nicoud and Ducros [2] and the Smagorinsky model with an explicit near wall dumping function and to compare the accuracy of the solutions obtained on three different meshes.

2 Numerical Procedure For the simulation the in-house code SAILOR-WALL was used. The numerical code is based on the projection method [3] for the pressure-velocity coupling. Time integration is performed by the Adams–Bashforth/Adams–Multon corrector–predictor scheme. Spatial derivatives are approximated by a high-order compact scheme [4] in the wall normal and streamwise direction. In the spanwise direction the Fourier approximation [1] is used. The high-order discretization usually allows for coarsening of the grid and in this way decreasing the computational cost. As it was mentioned before two eddy viscosity based models were considered in the paper. The first one was Smagorinsky model for which the eddy viscosity does not scale correctly close to the wall, causing excess of dissipation. To improve this deficiency, a dumping function D similar to the Van Driest function was applied. + n y . (1) D = 1 − exp 26 The second model was Wall Adapting Local Eddy-Viscosity model abbreviated as WALE. This model is based on the square of the velocity gradient tensor and accounts for the effects of both the strain and the rotation rate to obtain the local eddy-viscosity. It yields proper near-wall scaling for eddy-viscosity without requiring a dynamic procedure. For current test case the universal constants CS = 0.1 of the Smagorinsky model and CW = 0.5 of the WALE model were applied. Because SAILOR is a code with structural mesh it is difficult to perform the calculations of channel with curved walls. To solve this problem we decided to follow the procedure of mathematical mapping of Marquillie et al. [5]. Instead of writing Navier–Stokes system in curvilinear coordinates we transform the partial differential operators using a mapping which has property of bump profile (x) at the lower wall and a flat surface on the upper wall. One of the main problems doing LES simulation of spatially developing boundary layer is the correct prescription of the inflow boundary condition. The most straightforward approach is to simulate instantaneous inflow boundary condition by imposing random fluctuation on the mean velocity profile and then allow natural

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Table 1 The investigated cases Sgs model

Mesh

x +

+ ymin

+ ymax

z+

SMAG SMAG SMAG WALE WALE WALE

128 × 96 × 32 96 × 96 × 32 64 × 96 × 32 128 × 96 × 32 96 × 96 × 32 64 × 96 × 32

38.8 51.7 77.6 38.8 51.7 77.6

0.75 0.75 0.75 0.75 0.75 0.75

18.7 18.7 18.7 18.7 18.7 18.7

38.8 38.8 38.8 38.8 38.8 38.8

Fig. 1 The computational domain together with the mesh structure

transition of turbulence to occur. Such an implementation gives good results but is usually very expensive because natural transition appears far downstream from the inlet. Usually applied solution is to perform an additional calculation in the channel with periodic conditions in the streamwise direction, during which the instantaneous velocity components at the assumed plane can be stored and later used as an inlet condition for the spatially developing boundary channel. Such a procedure has been adopted in current work, where the periodic channel with Smagorinsky subgrid model was simulated. The mesh in the spanwise and wall-normal direction was the same as in the calculations for spatially developing boundary channel flow. Figure 1 shows schematic view of the computational domain as well as one of the mesh used in computations. The domain has dimension 4πh in the streamwise direction, 2h between the walls and πh in the spanwise direction. Three different meshes were tested i.e. 64 × 96 × 32, 96 × 96 × 32, 192 × 96 × 32. Each mesh was refined near the walls by a tangent hyperbolic function and is adjusted to have 7–8 nodes from wall to y + = 10, with the first node located approximately at y + = 0.95. This ensures that mesh in the wall normal direction was fine enough, so the aim of the study was to analyze the influence of density in the streamwise direction only. The number of grid points as well as the resulting resolution for the consecutive cases is given in Table 1. One may see that the resolution in normal and spanwise directions is typical for periodic channel flow benchmark (Reτ = 395). In streamwise direction the near

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Fig. 2 The pressure distribution along the longitudinal dimension of computational domain with pressure distribution messured in test section

wall spacing increments vary from 38.8 to 77.6. In the calculation the characteristic grid spacing is taken as cubic root of the mesh control volumes = V 1/3 with V = xyz.

3 Analysis of the Results In the work three different meshes and two different subgrid models have been analyzed. To shorten the description it was decided to discuss only streamwise and wall-normal velocity fluctuation components and the skin friction coefficient. Besides that, the mean velocity field does not differ significantly in the consecutive cases. The comparison of the velocity fluctuation profiles has been performed on two different cross-section planes. Their location is presented in Fig. 1 by two white lines. The first plane is located at the top of the bump and the second is in the middle of the recirculation zone. For all cases, simulated time needed to collect statistics was chosen such, that the inlet velocity profile was advected five times throughout the domain. The objective of the paper was to check the influence of grid as well as subgridscale model on the solution. The simulations were performed at Reynolds number Reτ = 395 at inlet, which does not however make it directly comparable with the experimental test case (where Reτ = 950 [6]) in terms of mean and fluctuating flow field. Nevertheless, as it is seen from Fig. 2 the pressure distribution predicted in the simulation reproduces the experimental results quite well. The first aim was to investigate the streamwise grid spacing influence. For this purpose the results obtained with Smagorinsky subgrid model for three meshes are presented. In Fig. 3 the skin friction coefficient distribution along the longitudinal direction is given. One can see that in all cases, no difference occurs for the area between inlet plane and the separation point. The separation point is also identically predicted by all cases. The discrepancies are observed further downstream. It is apparent that results obtained on two finest meshes (128 × 96 × 32 and 96 × 96 × 32) are almost identical. The solution on the most coarsed mesh differs slightly from other cases especially in the recirculation zone. The reattachment point for the coarse mesh is located more downstream than for the fine meshes.

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Fig. 3 Comparison of the skin friction parameter distribution along steamwise direction. Data presented for three differently coarsed meshes

Fig. 4 Streamwise and spanwise velocity fluctuation profiles at the top of the bump (x = 4.1)

In Fig. 4 the velocity fluctuation profiles in the streamwise and wall-normal direction at the top of the bump are presented. One may observe that in terms of streamwise velocity fluctuations the results are almost identical for all considered cases. In terms of wall-normal velocity fluctuations some differences may be noticed near walls at the peak values. Again the coarse mesh case differs from the others, showing smaller values (see zoom of the plot). Figure 5 shows the velocity fluctuation profiles in the middle of the recirculation zone are presented. The streamwise velocity fluctuation profiles seem to be independent of the mesh refinement in the streamwise direction. Some discrepancies may only be observed close to the wall, where the coarsest case shows lowest value of fluctuation. The difference is about 2.5 ms−1 . In terms of wall-normal fluctuation profiles similar behavior as for the distributions at the top of the bump is observed, where the coarsest mesh shows the lowest values. One can conclude that the influence of streamwise spacing is not very strong, possible due to low numerical errors introduced by high-order discretization. It is also seen that the mesh of second case (96 × 96 × 32) is fine enough to obtain independent results.

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Fig. 5 Streamwise and spanwise velocity fluctuation profiles in the recirculation zone (x = 5.0) Fig. 6 Comparison of the skin friction parameter distribution along steamwise direction. Data presented for two different SGS models

The next aim of the paper was to investigate the influence of subgrid models. The comparison of results for WALE and Smagorinsky models on the finest mesh (not presented here) has shown that both models predict almost the same distribution of the skin friction coefficient. To observe the influence of the model it was decided to compare the results for the coarse mesh. The results from WALE SGS model on finest mesh has been taken as a reference value. In Fig. 6 the skin friction coefficient for both subgrid models is presented. It might be observed that results obtained for both subgrid models on coarse mesh are almost identical up to the separation point. Some discrepancies are observed in the recirculation zone but further downstream the results are again close to each other. It might be noticed that WALE SGS model predicts the reattachment point slightly closer to the reference results than Smagorinsky model. The velocity fluctuation profiles at the top of the bump are presented in Fig. 7, where no major differences have been noticed. The comparison at the plane located in the recirculation zone (Fig. 8) shows some discrepancies both for streamwise and wall-normal direction. In that region WALE model predicts higher fluctuation values close to the lower wall, which could be due to the differences in the level of the subgridscale viscosity. It is well known that the Smagorin-

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Fig. 7 Comparison of the WALE and Smagorinsky velocity fluctuation profiles at the top of the bump (x = 4.1)

Fig. 8 Comparison of the WALE and Smagorinsky velocity fluctuation profiles in the middle of recirculation zone (x = 5.0)

sky model tends to give higher viscosity values than WALE. Therefore, one could expect that the dumping is higher for the Smagorinsky, leading to lower level of the fluctuations. In the consequence the reattachment point for WALE model is located somewhere closer than for Smagorinsky model.

4 Conclusions The simulations performed in the frame of the paper were done for turbomachinery test case of Materny et al. [6], but for lower Reynolds number. Despite this fact the pressure distribution was predicted successfully. Obtained results have shown very weak influence of the subgrid model. Both the WALE and Smagorinsky models yield very similar results. The velocity profiles, the profiles of velocity fluctuation and the profiles of the skin friction coefficient are almost independent of the SGS

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model applied. The same applies to the location of the separation point. The only noticeable differences are observed for the localization of the reattachment point. This parameter was sensitive both to the SGS used as well as applied mesh. However, it should be stressed that application of high order discretization allowed to obtain almost grid independent results for relatively coarse mesh. Acknowledgements This work has been performed under the WALLTURB project and SPB WALLTURB funded by MNiSzW. WALLTURB (A European synergy for the assessment of wall turbulence) is founded by CEC under the 6th framework program (Contract #: AST4-CT-2005516008).

References 1. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988) 2. Ducros, F., Nicoud, F.: Subgrid-scale stress modeling based on the square of the velocity gradient tensor. Flow Turbul. Combust. 62, 183–200 (1999) 3. Fletcher, C.A.J.: Computational Techniques for Fluid Dynamics. Springer, Berlin (1991) 4. Lele, S.K.: Compact finite difference with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1992) 5. Marquillie, M., Laval, J.P.: Direct numerical simulations of a separated channel flow with a smooth profile. J. Turbul. 9, 1–23 (2008) 6. Materny, M., Drozdz, A., Drobniak, S., Elsner, W.: Experimental analysis of turbulent boundary layer under the influence of APG. Arch. Mech. 60, 1–18 (2008)

LES Modeling of Converging Diverging Turbulent Channel Flow Jean-Philippe Laval, Witold Elsner, Lukasz Kuban, and Matthieu Marquillie

Abstract The paper presents the results of LES application for turbulent channel flow with varying pressure gradient obtained by the adequate curvature of one of the wall. The adverse pressure gradient in the second part of the channel together with a wall curvature results in small separation of the flow at that side. The main objective of the paper was an assessment of various subgrid models implemented in two different codes as well as the sensitivity of the predictive accuracy to grid resolution. The simulations were performed on a few levels of coarsening grids compared to the reference DNS. The influence of the grid refinement and constant parameters for the subgrid-scale models were investigated. It was demonstrated that all subgrid-scale models require a comparable minimum grid refinement in order to capture accurately the recirculation region.

1 Introduction Large Eddy Simulation (LES) is a methodology, where large turbulent eddies proportional to a given mesh are captured and the influence of non-resolved scales onto resolved ones are modeled. As the large eddies contain the major part of energy, it is expected that the proper physical characteristic of the flow are correctly captured by the simulation. Due to growing computational power during last years, LES is J.-P. Laval () · M. Marquillie Laboratoire de Mécanique de Lille, CNRS, 59655 Villeneuve d’Ascq, France e-mail: [email protected] M. Marquillie e-mail: [email protected] W. Elsner · L. Kuban Technical University of Czestochowa, Institute of Thermal Machinery, Czestochowa, Poland e-mail: [email protected] L. Kuban e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_37, © Springer Science+Business Media B.V. 2011

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increasingly applied for modeling of more complex geometries and higher Reynolds number flows. The accuracy of LES solutions depends strongly on two main sources of errors, introduced on one hand by discretization method and on the other hand by given subgrid-scale model. The more so, as it was presented by Meyers et al. [4, 5], this two errors interacts strongly one with the other. Subgrid-scale models, applied for wall-bounded flows, should have suitable properties and should produce an eddy viscosity with correct near-wall behavior. The discretization error, which influences the level of artificial dissipation, depends not only on chosen discretization method but also on its implementation. The difficulties arise for such a group of wall bounded flows in which turbulent boundary layer separates from curved wall due to geometry induced adverse pressure gradient. In such cases the structure of the flow influences separation and the more so reattachment point. To have a proper representation of such a flow it is necessary to resolve both large-scale structures present in a recirculation zone as well as details of the thin separated shear layer. The aim of the paper was an assessment of discretization methods and various subgrid models implemented in two different codes, one, MFLOPS3D, developed at the Lille Mechanics Laboratory (LML) and the other, SAILOR, developed at TU of Czestochowa. As a test benchmark, a turbulent channel flow with inlet conditions corresponded to the Reynolds number Reτ = 395 and varying pressure gradient calculated with DNS by Marquillie et al. [3] was chosen. The adverse pressure gradient in the second part of the channel together with a wall curvature results in small separation of the flow at that side. The models considered in the paper are the classical Smagorinsky with dumping function, Dynamic Smagorinsky (DSM) and WALE model of Nicoud & Ducros [6].

2 Numerical Code Both SAILOR and MFLOPS3D have some similarities, however the scope of these two codes was different. SAILOR was initially designed to solve free flows problems like iso- and nonisothermal (variable density) free flows (round and plane jets, temporal and mixing layers, diffusion flames). For the purpose of WALLTURB project it was redesigned to treat the wall bounded flows as well [8]. The code is based on the projection method [3] for the pressure–velocity coupling. The time integration is performed by the corrector–predictor scheme Adams– Bashforth/Adams–Multon. The spatial derivatives are approximated by Fourier approximation in the spanwise direction and a high-order compact scheme [1] in the wall normal and streamwise direction. The compact scheme for the first derivative discretization is of 6th order for the inner nodes i.e. 3, 4, . . . , N − 3, N − 2, where N is the number of nodes, 4th order for the near boundary nodes i.e. 2 and N − 2 and the 3th order for the boundary nodes. The high-order discretization allows for coarsening of the grid and in this way to decrease the computational cost. Because of instability of the simulation and in order to avoid backflow at the domain outlet the filtration of the velocity field has been applied for the last ten nodes of the domain.

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MFLOPS3D also relies on the projection method for the pressure–velocity coupling. The spectral spanwise and normal discretization combines the advantage of a good accuracy with a fast integration procedure compared to standard numerical procedure for complex geometries. MFLOPS3D use spectral discretization in both normal and spanwise directions and a second-order backward Euler differencing is used for time integration. Both codes are structured grid solvers, where instead of writing the Navier–Stokes system in curvilinear coordinates, the wall curvature was obtained by a mathematical mapping of the partial differential operators from physical coordinates to Cartesian ones. More details about MFLOPS3D and the mapping of coordinates can be found in [3].

3 Subgrid-Scale Models It is well known that, in wall bounded flows, the critical point is to properly describe the behavior of small turbulent structures created near the wall. It means that subgrid-scale models have not only to model properly the energy transfer from the resolved scales to the subgrid ones, but also to present correctly eddy viscosity distribution in wall normal direction. In the paper sample results obtained with classical Smagorinsky, dynamic Smagorinsky and WALE model are presented, where classical Smagorinsky and WALE were applied in SAILOR code and dynamic Smagorinsky and WALE in MFLOPS3D code. Because the eddy-viscosity in classical Smagorinsky model is not properly scaled near the wall the damping function D similar to the van Driest damping formula was used. The function was defined as: + n y (1) D = 1 − exp 26 In MFLOPS3D, Chebyshev-collocation used in the wall-normal y-direction can lead aliasing errors. This aliasing is usually not critical for DNS because the level of kinetic energy is very low for the highest modes since the discretization must be sufficient to resolve all turbulent scales. In the case of LES, and especially coarse LES, the kinetic energy at the largest modes is significant and leads to spurious aliasing errors. In order to overcome this effect, an explicit filtering was used in the normal direction only. The explicit filter must be smooth enough to remove the aliasing effect but sharp enough to keep a sufficient spectral resolution in this direction. We finalized our choice on to the “raised cosine filter”. This filter was used by Pasquetti et al. in order to weaken the Gibbs phenomenon induced by a pseudopenalization method [7]. In 1D, the raised cosine filter is defined in the spectral space by: 1 g(k) ˆ = (1 − cos(2πk/N )); |k| < N/2 (2) 2 where k is the Fourier mode. No explicit filtering were used in the spanwise and streamwise direction. The Lilly formulation [2] of the dynamic Smagorinsky model

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Table 1 Parameters of the DNS and LES. The space discretization in wall units (x + , y + , z+ ) are the maximum values in the streamwise direction but computed with the friction velocity at the submit of the bump for each simulation. C is the constant of the subgrid-scale model. Nz is the number of grid point in the spanwise size of π even if the simulation with SAILOR code were performed on a domain twice as large Nx

Ny

Nz

x +

z+

C

MFLOPS3D

1536

257

384

5.1

5.1

–

WALE

MFLOPS3D

256

129

48

30.9

40.2

0.4

WALE

MFLOPS3D

256

65

48

29.3

38.2

0.4

Model

Code

DNS

WALE

MFLOPS3D

256

65

48

29.3

38.2

0.2

DSM

MFLOPS3D

256

65

48

29.3

38.2

–

NONE

MFLOPS3D

256

65

48

31.2

40.5

–

WALE

MFLOPS3D

256

65

24

29.3

74.8

0.4

WALE

MFLOPS3D

192

129

48

41.3

40.3

0.4

WALE

MFLOPS3D

192

65

48

39.4

38.4

0.4

WALE

SAILOR

192

96

32

25.3

37.8

0.4

SM

SAILOR

192

96

32

25.3

37.8

0.1

SM

SAILOR

96

96

32

50.9

37.8

0.1

NONE

SAILOR

96

96

32

50.9

37.8

–

WALE

SAILOR

96

96

32

50.9

37.8

0.4

WALE

SAILOR

96

96

32

50.9

37.8

0.5

SM

SAILOR

96

96

16

50.9

75.6

0.1

with a sharp spectral cut-off test filter in spanwise direction (keeping half Fourier modes) was used. An explicit five points filter with three vanishing moments is used in physical space for the streamwise direction: 1 1 5 1 1 ui = − ui−2 + ui−1 + ui + ui+1 − ui+2 (3) 16 4 8 4 16 √ The test filter width was estimated as 3 4 where = 3 x y z . The results of the estimated constant by the dynamic procedure was averaged in the homogeneous spanwise direction. The WALE model is defined in [6] and the effect of the constant will be discussed.

4 Test Case Description The statistics of LES with 3 subgrid-scale models are compared with several grid refinements. The simulation parameters are summarized in Table 1. The spatial resolution of each LES is given in the table. For the simulations with MFLOPS3D, the normal resolution corresponds to the number of Chebyshev modes (before the filtering process of (3)). The three normal resolutions used for the LES are compared to

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Fig. 1 Comparison of the normal spatial resolution at the top of the bump of the three LES grids and the DNS grid. The distance are normalized with the friction velocity at the top of the bump

Fig. 2 Comparison of the skin friction coefficient at the bottom curved wall (lower curves) and at the top flat wall (upper curves, shifted up by +0.01) for LES with WALE model on different grids (MFLOPS3D)

the DNS in Fig. 1. The normal resolution used by the SAILOR code with Ny = 96 is comparable to the normal resolution Ny = 129 used by MFLOPS3D in most of the viscous sublayer and in the buffer layer (2 < y + < 60). The time step was chosen constant (t = 10−3 ) for all the LES with MFLOPS3D in order to avoid possible effect on the results and to focus only on the model effect. In case of both codes the inlet were generated for each time step from a precursor LES of flat channel flow at the same Reynolds number.

5 Results The effect of the grid on the friction velocity is presented in Fig. 2 for the LES using the WALE model and performed with MFLOPS3D. The main conclusion from this comparison is that 256 grid points are necessary in the streamwise direction to be able to capture the recirculation region at the lower wall. All the case does lead to a recirculation region at the lower wall, but the LES with only 192 points in streamwise direction clearly under-predict this region. Surprisingly, the case with the higher spanwise and normal resolution does not lead to the best results at the

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Fig. 3 Comparison of the skin friction coefficient at the bottom curved wall (lower curves) and at the top flat wall (upper curves, shifted up by +0.01) for LES with WALE and Smagorinsky model on two different grids (SAILOR)

Fig. 4 Comparison of the skin friction coefficient at the bottom curved wall (lower curves) and at the top flat wall (upper curves, shifted up by +0.01) for LES with WALE and DSM model on two different grids (MFLOPS3D)

lower wall, but better reproduce the intensity and position of the minimum friction velocity at the upper wall. The case with a very coarse spanwise resolution (24) leads to acceptable results at the lower wall but does predict a slight separation at the upper wall. The above observations were also confirmed by the mesh dependence study performed with SAILOR code. The results, not shown here, exhibit that coarsening of the mesh in spanwise direction (from 32 to 16) causes the small separation on the upper wall and the shift of minimum Cf position downstream the flow by approximately 0.5 units on the upper and 0.3 on the lower wall. The slight delay of reattachment point on the lower wall was also confirmed. The more significant influence of streamwise grid spacing was demonstrated in Fig. 3. One can observe that coarsening of the mesh in this direction (from 192 to 96) leads to increased discrepancy from DNS especially downstream the reattachment point. The WALE and DSM models with MFLOPS3D were compared on a sufficiently fine grids which leads to correct results. The comparison of the skin friction coefficient Cf is shown in Fig. 4. The figure shows that the effect of the models is clearly much weaker than the effect of the grid. However, the WALE model (Cw = 0.4) behave slightly better than the DSM but the difference between the two models is small as compared to the effect of the spatial resolution. The conclusions on the subgridscale model influence are identical for simulations at larger resolution (384, 129, 96)

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Fig. 5 Comparison of the skin friction coefficient at the bottom curved wall (lower curves) and at the top flat wall (upper curves, shifted up by +0.01) for LES with WALE model with different model constants (SAILOR)

and lower resolution (96, 65, 24) (not shown here). The effect of the subgrid-scale model is slightly stronger for the results obtained with SAILOR (see Fig. 3). One can observe this influence in particular for a coarse grid (96, 96, 32). For WALE and grid (192, 96, 32), the skin friction coefficient distribution is reproduced almost perfectly, while the other model predict longer recirculation zone. The higher differences between models than observed for MFLOPS3D is due to the fact that WALE model was compared with classical Smagorinsky model which is more dissipative especially near the wall. The effect of the constant of the WALE model was also investigated. The results with MFLOPS3D on the same grid as before but with a lower value of the constant (Cw = 0.2) is shown in Fig. 4. On this grid, and from a general point of view, the WALE model with the lower constant better predict the averaged velocity and averaged pressure except in the recirculation region at the lower wall. Therefore, the optimum constant of the model, even for a single grid, is difficult to define as it depends of the statistics which are compared. In order to test the influence of the model, a simulation without any model, but with the same explicit filtering, was conducted on the same grid. The skin friction is compared with the two other subgrid-scale models (see Fig. 4). The recirculation region at the lower wall is under-predicted, however, the results is comparable to the case with 192 grid points in the streamwise direction (see Fig. 2). This shows that the effect of the explicit filtering, even on the single normal direction, is not negligible in the global turbulence modeling. Supplementary conclusions could be formulated based on the results obtained with SAILOR code and WALE model. Figure 5 presents the skin friction coefficient Cf for two constant values Cw = 0.4 and 0.5. To show the differences the coarse mesh was chosen. For comparison, an additional simulation without any model was also performed. As it is clear from the plots, the optimum constant is difficult to suggest. The best prediction for the lower wall is with Cw = 0.5, while for the upper wall, calculation with Cw = 0.4 gives more consistent results with DNS. The effect of the absence of the model is similar to what has been observed for MFLOPS3D results. The impact of LES solver was analyzed based on results obtained with WALE model on similar grid and related to a filtered DNS using the same explicit filtering as with MFLOPS3D. For this purpose Reynolds stresses have been presented in Fig. 6. It has been shown already that a streamwise discretization with 192 grid points with MFLOPS3D was not enough

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+

+

Fig. 6 Comparison of u (upper graph) and v (lower graph) for LES with WALE model using the two different codes. FDNS is the results of the DNS filtered with the explicit filter used for MFLOPS3D LES simulations +

to correctly predict the skin friction. At this resolution, u is comparable with the + filtered DNS, but the v clearly differs from the reference at the lower wall. The appearance and the intensity of the wall peak is different in the decelerating part of the domain, but the intensity returns to a correct value after the bump. The statistics of + LES with SAILOR code are comparable to MFLOPS3D, but the wall peak of u is + more intense and the profile of v is closer to the reference even on the downstream part of the bump. This effect could results from the higher order approximation in streamwise direction (6th order) as compared with MFLOPS3D.

6 Conclusions Three subgrid-scale models have been tested on an adverse pressure gradient flow with two different codes. The grid effect as well as the constant parameters were investigated and the result were compared to a DNS performed in the exact same configuration. The main conclusion is that the results are sensitive to the numerical

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code (derivation schemes) and highly dependent to the number of grid points in the configuration. A sufficient streamwise resolution is required in order to capture the small separation region at the lower wall. This resolution is different for the SAILOR code using 6th order compact scheme and MFLOPS3D using only 4th order explicit finite differences. The optimal constant for the WALE model has been shown to be of the order of 0.4 in most configurations, but this value may vary with the grid. However, the results are not strongly dependent of the constant. The DSM and WALE model give very similar results for a large range of grid size. Smagorinsky with dumping function gives slightly worse results particularly for the coarse grid. Acknowledgements This work was supported by WALLTURB (A European synergy for the assessment of wall turbulence) which is funded by the EC under the 6th framework program (CONTRACT: AST4-CT-2005-516008). The DNS was performed through DEISA Extreme Computing Initiatives (DEISA is a Distributed European Infrastructure for Supercomputing Applications) and the LES with MFLOPS3D were performed at IDRIS (French CNRS Computing Facilities).

References 1. Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1991) 2. Lilly, D.K.: A proposed modification of the Germano subgrid-scale closure method. Phys. Fluids A 4, 633–635 (1992) 3. Marquillie, M., Laval, J.P., Dolganov, R.: Direct numerical simulation of separated channel flows with a smooth profile. J. Turbul. 9(1), 1–23 (2008) 4. Meyers, J., Geurts, B.J., Baelmans, M.: Database analysis of errors in large-eddy simulation. Phys. Fluids 15(9), 2740–2755 (2003) 5. Meyers, J., Sagaut, P., Geurts, B.: A computational error assessment of central finite-volume discretizations in large-eddy simulation using a Smagorinsky model. J. Comput. Phys. 227, 156–173 (2007) 6. Nicoud, F., Ducros, F.: Subgrid-scale stress modelling based on the square of the velocity gradient tensor. Flow Turbul. Combust. 62, 183–200 (1999) 7. Pasquetti, R., Bwemba, R., Cousin, L.: A pseudo-penalization method for high Reynolds number unsteady flows. Appl. Numer. Math. 58, 946–954 (2008) 8. Tyliszczak, A.: An efficient implementation of compact-pseudospectral method for turbulence modeling. TASK Q. 10(2), 125–138 (2006)

Large-Scale Organized Motion in Turbulent Pipe Flow Sebastian Große, Dirk Jan Kuik, and Jerry Westerweel

Abstract Turbulent pipe flow is supposed to be highly confined by the pipe geometry and an interaction of the wall layers developing on ‘opposite’ sides of the center line is expected. The negative two-point correlation of streamwise fluctuations across the centerline is an obvious evidence of this interaction. To investigate this interaction and to also study the large-scale and very large-scale motion in the logarithmic and wake region of the pipe flow shear layers, measurements using stereoscopic high-speed PIV were performed in a plane perpendicular to the mean flow at bulk Reynolds numbers Reb = 10 000÷44 000. Individual recording sequences cover more than 150 bulk scales based on Ub and R such that even the largest expected flow structures are captured. The results will be compared to existing DNS and hot-wire data available in the literature. The uniqueness of the present data is that the entire azimuthal plane is covered such that further valuable information about the large-scale organized structure of turbulent pipe flow can be assessed.

1 Introduction Turbulent pipe flow is supposed to be highly dominated by the confinement of the flow geometry and the wall layers developing on ‘opposite’ sides of the center line are expected to interact with each other even at high Reynolds numbers. The negative two-point correlation of streamwise fluctuations across the centerline Ruz uz (r), which has been detected in experimental data, is an obvious evidence of this interaction. However, it is not clear, how a possible interaction takes place and how far a ‘sloshing’, i.e., a meandering of the pipe flow core region, is related to the large-scale (LSM) and very large-scale motion (VLSM) in the logarithmic and wake region [1, 4, 7, 10]. S. Große () Laboratory for Aero- and Hydrodynamics, Delft University of Technology, Leeghwaterstraat 21, 2628 CA Delft, The Netherlands e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_38, © Springer Science+Business Media B.V. 2011

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To study fully developed turbulent pipe flow in more detail, measurements using stereoscopic high-speed PIV are performed in the flow facility of the Laboratory of Aero- and Hydrodynamics at TU Delft at Reynolds numbers Reb = 10 000÷44 000. From the data all three components of the velocity field can be reconstructed. Only a single measurement plane is observed. To reconstruct the quasi-instantaneous streamwise extension of the flow field, Taylor’s hypothesis is applied. Due to a sufficiently high recording frequency we are able to ‘track’ the development of coherent motion from instantaneous velocity fields along the streamwise direction and with approximately 150 bulk scales, which are continuously being recorded, even the largest expected scales reaching up to seven times the pipe radius [7] can be captured. The simultaneous assessment of the flow field at different heights in the wall layer will allow to gain valuable information on the growth and development of coherent motion. Furthermore, the recorded data spans the entire azimuthal plane allowing to investigate whether or not the meandering of large-scale low-momentum zones in the logarithmic region spans over the entire circumference of the pipe, which would corroborate the idea of a strong interaction of ‘opposite’ wall layers in pipe flow. In this paper, first results on the azimuthal scaling of coherent motion in the logarithmic and wake region will be presented.

2 Flow Facility, Experimental Setup, and PIV Processing The pipe flow facility of the Laboratory of Aero- and Hydrodynamics at TU Delft is build from acrylic glass with a diameter of D = 40 mm and a total length of 28 m. The measurement section is made from glass at the same diameter. The entire pipe is thermally isolated. The fluid used in the measurements is filtered tap water at T = 20°C. The temperature during the measurements is constant within less than 0.1°C. The Reynolds number based on the bulk velocity Reb = Ub D/ν is determined from the volume flux V , which can be measured to within 0.4% accuracy. The flow enters the pipe section through a settling chamber with a flow straightener of approximately 5 mm core size and several meshes and a contraction with a 5 : 1 contraction ratio (based on the diameter). This design allows to maintain the flow in the pipe laminar up to Reb = 30 000 such that the flow needs to be tripped artificially downstream of the pipe inlet. All measurements are carried out at about 25 m downstream of the tripping device, i.e., at 600Le /D, to ensure fully developed flow conditions at the measurement section [2, 15]. A further description of the setup can be found in [13, 14]. A schematic is shown in Fig. 1. Stereoscopic high-speed PIV recordings at 500÷1 000 Hz and at Reynolds numbers of Reb = 10 000÷44 000 were performed. The recording time of individual sequences spans approximately 150 integral scales based on the bulk velocity Ub and the radius of the pipe R. In total, 3 000 bulk scales (overturning times defined as D/Ub ) of low-speed (50 Hz) and high-speed data (500÷1 000 Hz) were recorded such that the data can be evaluated statistically. The SPIV images are recorded with two high-speed cameras (Photron Fastcam APX) operated at the aforementioned frame rate. The two cameras observe the flow

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Fig. 1 Schematic of the pipe flow facility

Fig. 2 (a) Schematic of measurement setup for stereoscopic PIV recordings. (b) Coordinate frame

at angles of +45° and −45° to the light sheet in a Scheimpflug configuration [12]. A schematic of the experimental setup is given in Fig. 2(a). The frame of coordinates used in the following discussion is illustrated in Fig. 2(b). The flow is illuminated by a 1.0–1.5 mm thick light sheet formed by a dual cavity frequency-doubled pulsed Nd:YLF high-speed laser with a maximum energy of 10 mJ at 2 000 Hz (New-Wave Pegasus). Nearly neutrally buoyant hollow glass spheres with a mean diameter of 10 µm are used as seeding material. The evaluation of vector fields from the PIV-images is performed with commercial PIV-software (DaVis 7.2, LaVision). The stereoscopic view of the measurement plane allows the determination of all three velocity components. A multi-pass correlation is applied with a final 16 × 16 px interrogation windows size and 50% overlap leading to a vector spacing of approximately 0.32 mm, corresponding to 8 · 10−3 D. The size of the PIV interrogation windows corresponds to approximately 8, 18, and 36l + at the Reynolds numbers in the experiments, respectively. Since the focus of the present study is on the large-scale motion of the flow, this averaging at the small scales should have negligible influence.

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Fig. 3 (a) Instantaneous velocity field at Reb = 20 000. Contours indicate the streamwise velocity uz . Vectors show the cross-plane velocity field ur , uΘ . Only every second vector is shown. (b) Close-up of the region highlighted in (a) at full vector resolution

Fig. 4 Spatial correlation of streamwise fluctuations as function of radius Ruz uz (r/R) across the centerline (a) at Reb = 10 000 and (b) at Reb = 44 000, respectively

3 Discussion of First Results In this section the results from the stereo-PIV investigation of turbulent pipe flow at Reb = 10 100÷44 000 will be discussed and compared to data in the literature. An exemplary instantaneous velocity field at Reb = 20 000 is given in Fig. 3(a). The coexistence of low-speed and high-speed regions in the logarithmic layer and wake region, which are related to the existence of nearly streamwise vortices, is evident. A close-up of the highlighted region in Fig. 3(a) is given in Fig. 3(b) evidencing the high resolution of the available data. The two-point correlation of streamwise velocity fluctuations across the centerline Ruz uz (r) at Reb = 10 000 and 44 000 is shown in Fig. 4. The correlation function reaches negative values at distances from the centerline r/R = 0.0÷0.5 for positions

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Fig. 5 Distribution of the spatial correlation Ruz uz (Θ) at Reb = 10 000 (a, b) and Reb = 44 000 (c, d), respectively, in a stretched Cartesian scale and the original polar form. Note, these graphs can only be read along lines of constant radius. Dashed lines in (a, c) indicate Θ = 100l + /(2pi)

of the reference point from the centerline being less than −0.4r/R. The reference point being closer to the wall, the correlation across the centerline decays toward zero. A similar finding holds for square duct flow. Data of 2D-channel flow will still need to be further investigated to study the effects of the 2D flow confinement in channels on the interaction of opposite wall layers. Whether or not the negative trend of the two-point correlations across the centerline can also be observed in DNS data of turbulent pipe flow will need further investigation, especially concerning the influence of the length of the computational domain along the streamwise direction. There is evidence from instantaneous streamwise velocity fields indicating a strong growth of near-wall coherent motion into the near-center wake region causing a strong influence on the entire flow field, but further investigation is necessary. The characteristics of the interaction and a possible existence of a ‘sloshing’, i.e., a meandering of the core region, which might be related to or induced by the large-scale (LSM) and very large-scale motion (VLSM), are still subject of current research. In Fig. 5 the distribution of Ruz uz (Θ) exhibits a negative correlation at Θ ≈ π close to the centerline. At smaller wall-distances, hence, smaller y/R, lobe-like regions of negative correlation indicate the co-existence of low- and high-momentum zones aligned in the azimuthal plane. The distance between the extrema decreases with decreasing y/R. In Figs. 5(a) and 5(b) two distinct lobes very close to the wall

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Fig. 6 Azimuthal distribution of spatial correlation of streamwise velocity fluctuations Ruz uz at different radial positions r/R at (a) Reb = 10 000 and (b) Reb = 44 000. Azimuthal length-scale lz in different scalings (c, d). lz is the azimuthal dimension of the two-sided correlation peak using a threshold of Ruz uz ≥ 0.05

at Reb = 10 000 are obviously separated by a region of less negative correlation values at approximately y/R = 0.1÷0.3 from the large-scale lobes at higher values of y/R. This could be an indication of a larger order of different scales present in the flow, whereas the strong negative correlation in the regions at y/R ≤ 0.1 and y/R ≥ 0.3 indicate a rather uniform scale of structures. From Fig. 5(a) it can be observed that the distance between the negative lobes in the vicinity of the wall reaches approximately 100l + , i.e., 100 viscous scales, which is in good agreement with the spacing of the near-wall cycle reported in the literature (e.g. [3, 8, 9, 11]). At higher Reynolds numbers this near-wall region is not sufficiently resolved. A similar trend can be observed in the distributions of Ruz uz (Θ) at distinct heights in the wall layer,

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Fig. 7 Contours of the streamwise velocity fluctuations at different height y/R in the logarithmic and wake region at Reb = 44 000

which are given in Figs. 6(a) and 6(b), where only the data in Fig. 6(a) shows the existence of a near-wall cycle at reduced azimuthal dimension. To further assess the scaling of coherent motion, the azimuthal dimension of the correlation function has been estimated following the procedure used by [1, 6, 10]. The distance from the reference point, at which the correlation value drops below a level of Ruz uz ≤ 0.05, is defined by lz /2. Figures 6(c) and 6(d) show the azimuthal dimension lz as a function of y/R in comparison to values in the literature. The azimuthal dimension in these graphs are shown scaled on the one hand, with the local value of 2πr and on the other hand, with R. The results at the highest Reynolds

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number evidence excellent agreement with the data of [1, 10]. At lower Reynolds number the azimuthal scaling exhibits a decrease close to the pipe wall and the distributions of lz drop to values rather similar to those reported for turbulent boundary layer flow [5, 6]. This might indicate the near-wall cycle in the vicinity of the wall to be only weakly influenced by the geometric confinement that the flow experiences at higher regions. On the other hand, it is remarkable that the fields of streamwise velocity fluctuations shown in Fig. 7 indicate similar structures of meandering bands of low-momentum and high-momentum fluid almost throughout an region ranging from 0.05 ≤ y/R ≤ 0.50, which would rather suggest a strong interaction of the flow field at different heights in the wall layer.

4 Outlook First findings from the measurements show the high quality of the recorded PIV data and their agreement with data in the literature. The sufficiently long integral time, over which individual recordings could be performed, the high recording frequency, the coverage of the entire azimuthal velocity plane, and the simultaneous assessment of all three velocity components possess the potential for a valuable and in-depth investigation of further features of turbulent pipe flow including characteristics of the recently found super-structures, an ‘inter-scale’ interaction and a possible meandering (‘sloshing’) of the core region. First statistics of turbulent pipe flow presented in this work indicate very good agreement with data obtained by the hot-wire arrays covering a limited azimuthal region at a constant radius. Further study of the data will be necessary to bring more light to the influence of the confinement on the flow in the log-layer and the wake region of turbulent pipe flow and towards a possible large-scale interaction of the wall layers on opposite sites of the centerline.

References 1. Bailey, S.C.C., Hultmark, M., Smits, A.J., Schultz, M.P.: Azimuthal structure of turbulence in high Reynolds number pipe flow. J. Fluid Mech. 366, 121–138 (2008) 2. Durst, F., Fischer, M., Jovanovi´c, J., Kikura, H.: Methods to set up and investigate low Reynolds number. Fully developed turbulent plane channel flows. J. Fluids Eng. — Trans. ASME 120, 496–503 (1998) 3. Große, S., Schröder, W.: Two-dimensional visualization of turbulent wall-shear stress using micro-pillars. AIAA J. 47(2), 314–321 (2009). doi:10.2514/1.36892 4. Guala, M., Hommema, S.E., Adrian, R.J.: Large-scale and very-large-scale motions in turbulent pipe flow. J. Fluid Mech. 554, 521–542 (2006) 5. Hutchins, N., Marusic, I.: Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28 (2007) 6. Hutchins, N., Hambleton, W.T., Marusic, I.: Inclined cross-stream stereo particle image velocimetry measurements in turbulent boundary layers. J. Fluid Mech. 541, 21–54 (2005) 7. Kim, K.C., Adrian, R.J.: Very large-scale motion in the outer layer. Phys. Fluids 11(2), 417– 422 (1999)

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8. Kreplin, H.P., Eckelmann, H.: Propagation of perturbations in the viscous sublayer and adjacent wall region. J. Fluid Mech. 95(2), 305–322 (1979) 9. Lee, M.K., Eckelman, L.D., Hanratty, T.J.: Identification of turbulent wall eddies through the phase relation of the components of the fluctuating velocity gradient. J. Fluid Mech. 66(1), 17–33 (1974) 10. Monty, J.P., Stewart, J.A., Williams, R.C., Chong, M.S.: Large-scale features in turbulent pipe and channel flows. J. Fluid Mech. 589, 147–156 (2007) 11. Moser, R.D., Kim, J., Mansour, N.N.: Direct numerical simulation of turbulent channel flow up to Reτ = 590. Phys. Fluids 11, 943–945 (1999) 12. Prasad, A.K.: Stereoscopic particle image velocimetry. Exp. Fluids 29(2), 103–116 (2000) 13. van Doorne, C.W.H.: Stereoscopic PIV on transition in pipe flow. Ph.D. thesis, TU Delft University (2004) 14. van Doorne, C.W.H., Westerweel, J.: Measurement of laminar, transitional and turbulent pipe flow using stereoscopic-PIV. Exp. Fluids 42, 259–279 (2007) 15. Zagarola, M.V., Smits, A.J.: Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 33–79 (1998)

Session 10: Skin Friction

• Near-Wall Measurements and Wall Shear Stress T.G. Johansson • Oil Film Wall Shear Stress Measurements Through Separation P. Nathan and P.E. Hancock (no paper) • Measurements of Near Wall Velocity and Wall Stress in a Wall-Bounded Turbulent Flow Using Digital Holographic Microscopic PIV and Shear Stress Sensitive Film O. Amili and J. Soria • Friction Measurement in Zero and Adverse Pressure Gradient Boundary Layer Using Oil Droplet Interferometric Method P. Barricau, G. Pailhas, Y. Touvet, and L. Perret

Near-Wall Measurements and Wall Shear Stress T. Gunnar Johansson

Abstract The near wall region in turbulent boundary layer flows is important for many reasons and is at the focus of many investigations. It is however very difficult to study this region experimentally, partly because of strong demands on spatial resolution, but also because the presence of the wall influences the performance of the measurements. In this contribution we try to detect and quantify problems with very near wall measurements using laser Doppler anemometry, and, in particular, with the determination of the wall shear stress. Several experiments are considered: a wall jet, and smooth and rough boundary layers, both with zero and favorable pressure gradients. Three methods were used to obtain the wall shear stress. Two of them are based only on velocity measurements and are referred to as “the momentum integral method” and “the wall gradient method”. The third method, oil film interferometry was used only in the case of the wall jet. In all experiments, problems with spatial resolution were encountered, with the exception of the smooth flat plate at very low speed. This was the only case in which the wall gradient of the mean velocity could be determined with reasonable accuracy. The momentum integral method worked for the boundary layers, but not for the wall jet, where a small secondary motion was present in spite of purely two-dimensional boundary conditions. The momentum integral method was the only one that could be applied to the rough boundary layers. The oil film method worked well in the wall jet, but could naturally not be applied on the rough surfaces.

1 Introduction The velocity field very near walls in turbulent boundary layers is of considerable interest. This is for example the region where most of the production of turbulence T.G. Johansson () Chalmers University of Technology, 41296 Gothenburg, Sweden e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_39, © Springer Science+Business Media B.V. 2011

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takes place, and, even closer to the wall, where the wall shear stress, τw , manifests itself through the mean velocity gradient right at the wall. It would be of considerable value if the wall shear stress could be measured as a bi-product of the measurement of the velocity field. However, only in rare circumstances can this be accomplished. Pitot tubes are simply too big, and hot-wire signals start to become contaminated by wall-interference at about 0.2 mm from the wall. Laser-Doppler anemometry (LDA) appears to offer a solution to this problem, since the diameter of the probe volume of an LDA system can be made quite small, say 30 µm or so. This should be a good enough spatial resolution at least in low Reynolds number flows, where the velocity gradient at the wall isn’t too steep. However, as have been witnessed in several investigations, also this measurement method is faced with problems in the very near wall region. Two examples of near wall measurements using LDA are shown in Fig. 1. In both cases an LDA system with a probe volume diameter of 58 µm was used, which in case 1(a), a smooth flat plate, was less than one viscous length unit, and thus, the spatial resolution was expected to be sufficient to measure the mean velocity gradient at the wall. In this figure it is immediately observed that this was not the case. In the second case 1(b) the spatial resolution was less favorable, but still expected to be good enough to give a reasonable estimate of the velocity gradient. In Fig. 1(a) we notice that the mean velocity profile does not cut the horizontal axis at the origin. This might be interpreted as being due to an error in the measured distance from the wall. However, as this deviation corresponds to almost two full diameters of the probe volume, it is too large to make sense. In this case the method used to determine the distance from the wall was to slowly traverse the probe volume from a position “inside” the wall and outwards, and simultaneously observing the signal on an oscilloscope screen. Since the surface in this case was not perfectly smooth (an aluminum plate) the small irregularities on the surface acted as scatterers, and the intensity of the scattered light should increase in proportion to the intensity of the incident light right at the surface. This implies that the maximum scattered intensity should be observed when the center of the probe volume coincided with the surface. This method is quite sensitive, and a precision better than 10% of the probe volume diameter is expected based on observations of the amplitude of the signal versus traversed distance. Obviously some other factor affected the near wall measurements. The measurements shown in Fig. 1(b) were obtained in a wall jet experiment on a very smooth glass plate. In this case the method used to obtain the data in Fig. 1(a) didn’t work since the surface was too smooth to scatter any light. Instead the laser beams were reflected, and actually formed a probe volume also after reflection. It was hoped that this effect could have been used to detect the precise position of the surface by taking advantage of the expected symmetry of the profile before and after reflection. This didn’t work out as expected as can be seen in Fig. 1(b), and, again, we have to conclude that some other factor is at play. An alternative method to obtain the wall shear stress is based on the momentum integral, which in its simplest form is τw = ρU02 ∂θ/∂x. However, even in its full form, it is very sensitive to secondary flows, and, as will be demonstrated, also

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Fig. 1 Measured mean velocity profiles very near a wall. (a) Flat aluminum plate, (b) wall jet with glass plate

in cases where secondary flows are unlikely, minute deviations from perfectly 2D flow can completely destroy the measurement of the wall shear stress. However, the method does not require measurements very close to the wall, and is therefore useful for example for measurements on rough surfaces. In this paper we will try to explain why the LDA system used for very near wall measurements and for the determination of the wall shear stress sometimes didn’t perform as expected.

2 Very Near Wall Measurements Using LDA In this chapter we propose an explanation for the malfunctioning of LDA systems when measuring very close to walls. A simple model is put forward, and used to quantify approximately the error involved, and the conditions under which the problem can be minimized. The basic idea is that when measuring very close to a wall the difference in velocity at the top and the bottom of the probe volume is large; the mean velocity at the outer rim of the probe volume may be many times larger than the speed at the lower rim. In the extreme case of a probe volume touching the wall this ratio would be infinite. This means that when a scattering particle passes through the lower part of the probe volume the passage time is large, and, due to the higher speed in the outer part, a large volume of fluid will pass through the outer part. On the other hand when a scattering particle passes through the outer part of the probe volume a comparatively smaller fluid volume passes through the lower part. Given a certain concentration of scattering particles in the fluid there is thus a larger probability that a second particle will be present within the probe volume during the passage time of the first particle if the first particle is close to the wall. LDA processors are designed to reject signals from two (or more) scattering particles simultaneously present in the probe volume, and, thus, measurements on near wall particles, i.e., low speed samples, will be systematically removed from the measurement record, while samples from higher speed particles will survive. In this way the measured average of very near wall measurements will always be too high.

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2.1 Analysis of Bias in Near Wall Measurements The mean velocity profile very close to the wall can be approximated by u2τ y (1) ν We also assume that the probe volume can be approximated by a circular cylinder with diameter d and length L, and its center at a distance δ from the wall. The time it takes for a scattering particle to pass through the probe volume at a distance ys from the wall is then 2 2 d4 − (ys − δ)2 (2) ts = u2τ ν ys U=

During the passage time a certain amount of fluid flows through the probe volume, and since this fluid carries scattering particles there is a certain likelihood that one will have more than one scattering particle inside the probe volume at the same time. The total volume of fluid either present within the probe volume from the start or entering into it during the time interval ts is δ+ d2

Vs = ts

U (y)L dy +

π 2 d L 4

(3)

δ− d2

Denoting the number concentration of scattering particles per unit volume of the fluid by n the total number of scattering particles present during some time interval during the passage time of the original particle is (not counting the original particle) δ 2 π 2 4δ 1 ys Ns = nVs = n d L 1 + (4) − 1−4 4 π d ys /d d d We see that the probability to find a second scattering particle within the probe volume increases with • concentration of scattering particles (by number) • the volume of the probe volume • a certain function involving the distance from the wall of the primary particle and the position and diameter of the probe volume The function involving the distance from the wall of the primary particle and the position and diameter of the probe volume is shown in Fig. 2. It is immediately observed that the average number of particles passing through the probe volume during the passage time of a primary particle is larger when the primary particle passes through closer to the wall. This confirms the idea that there is a higher probability that low speed samples are systematically rejected. We also notice that this effect is stronger for probe volumes closer to the wall, but also that the effect fades out quite fast. This means that it is only in the extreme proximity of the wall that this biasing

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Fig. 2 Normalized number of scattering particles present within the probe volume during the passage time of a near wall particle. The curves show (from left to right) δ/d = 0.5, 0.7, 1, 2

Fig. 3 Maximum permitted concentration of scattering particles in the fluid for a 1% probability to destroy a near wall sample

effect is active, and farther away from the wall, although still quite close, we can trust the measurements. A different view of the effect is obtained by computing the probability that the maximum average number of particles passing through the probe volume during the passage time of a primary particle is less than, say, one percent. This is found directly from Eq. 4, and is shown in Fig. 3. We notice that, if the mid-point of the probe volume is at a distance from the wall larger than one probe volume diameter the concentration of particles in the fluid should not exceed 1 per 250 probe volumes (1/0.004). This is an inconveniently low concentration, since it will render a low data rate. We note in particular that this result is independent of the wall shear stress. The explanation for this behavior is that when the wall shear stress increases, the fluid volume passing through the probe volume per unit time increases in proportion. This is however compensated for by a proportionate decrease in the passage time of the primary particle.

3 Momentum Integral Method The momentum integral method, see [1] or [2], takes as its starting point the x-component of the momentum equation integrated from the wall and outwards,

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Fig. 4 Contributions to the wall shear stress in the momentum integral equation. Open circles: viscous shear stress, filled circles: (minus) Reynolds shear stress, filled rectangles: sum of momentum integrals, solid line: pressure, open rectangles: sum of all terms. Data were obtained on a smooth flat plate, 1.4 m downstream of the leading edge at a free stream speed of 20 m/s

∂U τw =ν − uv − ρ ∂y y +

y

2

∂U dy + U ∂x

0

∂v 2 ∂U0 dy + U0 y ∂x ∂x

y 0

∂U dy − ∂x

y

∂u2 dy ∂x

0

(5)

0

The key point is that the left hand side of the equation depends only on x, and, thus, so must the sum of all terms on the right hand side. By the measurement of the mean velocity profile U , the normal Reynolds stresses, u2 and v 2 , and the Reynolds shear stress uv, the right hand side of the equation can be computed and the wall shear stress determined. An example is shown in Fig. 4. The contributions from the normal Reynolds stresses are not shown since they are small. The dominating terms are the first four terms and, somewhat surprisingly, the last term, the pressure gradient, in Eq. 5. Note that only the sum of the third and fourth terms is shown in Fig. 4. The viscous term is, of course, of importance only in the very near wall region. Outside the viscous sub-layer the Reynolds shear stress dominates over a considerable fraction of the boundary layer, and further out the momentum integral terms take over. The sum of all terms is approximately constant across the whole boundary layer, and provides a measure of the wall shear stress. It is of some importance to study the contributions of the various terms in more detail. It can be argued that the degree to which the sum of all terms is constant is a measure of the accuracy of the method. We notice however that close to the wall the sum deviates considerably from the value attained further out. A close-up of the near-wall region reveals that the deviation is caused by the combined action of the viscous stress and the Reynolds shear stress. We have already established that the determination of the viscous stress is inaccurate very close to the wall. It also turns out that the statistical error in the measurement of the Reynolds shear stress is comparatively large. The number of samples in each point was about 5000, and using the known value of the ratio of the rms of the shear stress to its average of about 3 in the near wall region, we conclude that the statistical uncertainty in the Reynolds shear stress is about 7%. It should be noticed however, that this does not

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Fig. 5 Momentum equation balance for a wall jet showing (in order of increasing gray level) the viscous term (very light gray), the Reynolds shear stress term, the sum of the two momentum integral terms, and the sum of all terms (black)

seriously affect the determined value of the wall shear stress, since further out these terms become small compared to the integral terms. It should be noted that the pressure gradient in the experiment was quite low, corresponding to a relative change in the free stream speed over a distance of 1 meter of 0.16%, yet, as can be seen in Fig. 4, it cannot be neglected. There is however a more severe problem associated with the integral terms; an error close to the wall will affect also the values obtained further out. It is unlikely however, that this could happen without also showing a non-constant sum of the shear stress contributions also in the inner part of the boundary layer, where the Reynolds shear stress contributes significantly to the sum. The problem will arise if there is a secondary flow present, and this is the case in the wall jet experiment shown in Fig. 5, for details see [3]. The experiment was carefully set up, and the boundary conditions were symmetric, yet a small secondary flow was created. The secondary flow showed up in the momentum balance. As can be seen in Fig. 5 the sum of all terms is not constant across the boundary layer as it should be. The value of the wall shear stress computed from the sum of all terms in the outermost part of the boundary layer differs from the value obtained using the oil film interferometry method by a factor larger than 3. It is worth noting that the asymptotic value in the outer part of the boundary layer is equal to U02 ∂θ/∂x. This should serve as a warning against uncritical use of this simple formula. In spite of the occasional failure of the momentum integral method in obtaining an accurate value of the wall shear stress it may still pay off to compute the momentum integral balance, since it is a sensitive test on an unwelcome presence of small secondary flows. There is yet another case where the momentum integral method is useful for the measurement of the wall shear stress: rough boundary layers. In these cases the wall gradient method cannot be used, [2].

4 Conclusions The wall shear stress is an essential parameter in theoretical descriptions of boundary layers, and therefore of importance to measure accurately. In this paper two

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methods are considered, one based on the measurement of the mean velocity gradient at the wall, the other on the stream-wise momentum integral. Both methods are theoretically exact, but both are also prone to errors that are not always easily foreseen. To measure the mean velocity gradient at the wall requires very accurate measurements of the mean velocity in several points in the extreme vicinity of the wall, say within y + < 3. This is possible only when the wall shear stress is small, which is the case when the Reynolds number is small or in adverse pressure gradients. In other cases the spatial resolution is never adequate. However, also in many cases when the spatial resolution is expected to be good enough, it turns out that this is not the case. An explanation for this unexpected behavior of LDA systems is given, and the conditions when the wall gradient method can be used is presented. The basic mechanism for the poor functioning of LDA systems measuring very near walls is that during the passage time of a scattering particle in the lower part of the probe volume, a large volume of fluid is advected through its outer part, and thereby there is a severe risk that a second particle is swept through the probe volume during the same time interval. When this happens the LDA processors reject the sample. Low speed samples are for this reason systematically rejected at a higher rate than high speed samples, and, very close to walls, one always measures a too high mean velocity. The system works best at low particle concentrations, and for very small probe diameters. The momentum integral method has the advantages that it can be used also for high Reynolds numbers, and for rough surfaces where the wall shear stress is not easily measured using other methods. This study has also pointed out a few peculiarities of the method: it is very sensitive even to very small secondary flows, and it has been found that the pressure gradient can in practice never be neglected. These features of the method are not necessarily drawbacks. The method can be used to detect if a secondary flow is present, and to investigate to what extent a small variation in the free stream speed affects the force balance in the boundary layer. Acknowledgements The experiments on which this paper is based have been carried out in collaboration with Prof. Luciano Castillo, Mr. Faraz Medhi, Dr. Raul Bayoan Cal, Dr. Brian Brzek, and Prof. Jonathan Naughton. Their extensive contributions are gratefully acknowledged.

References 1. Johansson, T.G., Castillo, L.: Near-wall measurements in turbulent boundary layers using laser Doppler anemometry. In: FEDSM2002. 2002 Joint US ASME–European Fluids Engineering Summer Conference, July 14–18, 2002, Montreal, Quebec, Canada 2. Brzek, B., Cal, R.B., Johansson, G., Castillo, L.: Inner and outer scalings in rough surface zero pressure gradient turbulent boundary layers. Phys. Fluids 19(6), 065101 (2007) 3. Johansson, T.G., Mehdi, F., Naughton, J.W.: Some problems with near-wall measurements and the determination of wall shear stress. In: AIAA-2006-3833, 25th AIAA Aerodynamic Measurement Technology and Ground Testing Conference, San Francisco, June 5–8, 2006

Measurements of Near Wall Velocity and Wall Stress in a Wall-Bounded Turbulent Flow Using Digital Holographic Microscopic PIV and Shear Stress Sensitive Film Omid Amili and Julio Soria

Abstract In wall-bounded turbulent flows, a large portion of total turbulence production happens in the near wall region. Although the viscous and buffer layers are relatively easy for numerical simulations due to local low Reynolds numbers of the important structures, they are very difficult to study experimentally because of their physically small geometry. The aim of this paper is to measure the wall shear stress in a fully developed turbulent channel flow at moderately high Reynolds numbers using a novel stress meter which is capable of measuring surface forces over an extended region of the model. In addition, near wall region in a wall-bounded flow by means of digital holographic microscopic particle image velocimetry will be investigated. DHMPIV provides a solution to overcome the poor axial accuracy and the low spatial resolution. The paper addresses the experimental techniques, articulate calibration procedure, data acquisition, and analysis procedure.

1 Introduction In most applications of fluid mechanics, a knowledge of measuring shear stress over a solid surface is essential for fundamental understanding of the flow and the performance of the system. The time-averaged wall shear stress can be used to determine body-averaged properties like skin friction drag. The instantaneous surface stress can be used for flow control purposes like drag reduction of separation delay. In addition, it is theoretically possible to have a better knowledge of near wall flow by knowing the wall shear stress distribution. The small group of direct shear stress measuring methods are ideal as they directly measure the applied force to the wall by the fluid and do not directly depend O. Amili () · J. Soria LTRAC, Department of Mechanical and Aerospace Engineering, Monash University, VIC 3800, Melbourne, Australia e-mail: [email protected] J. Soria e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_40, © Springer Science+Business Media B.V. 2011

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on the flow field or fluid properties [7]. Direct skin friction measuring techniques mostly involve implementation of floating elements which measures the force applied by the wall shear stress on the element of the wall. The shear stress is directly measured with no assumption for the flow domain, but it is accompanied with different problems mainly associated with measuring very small forces, pressure gradients, misalignment of floating element, necessary gap around the floating element, temperature change, heat transfer, boundary layer injection or suction, leaks, gravity or acceleration, and transient normal forces [11]. In the following section, film shear stress sensor, a novel approach to measure wall shear stress in an accurate way is addressed.

2 Wall Shear Stress Sensor The film shear stress sensor consists of mounting a thin film made of a elastic material on the surface of the solid model. The geometry and mechanical properties of the elastomer are known, particles acting as markers are applied to the top surface of the film, and an optical technique is used to measure the film deformation caused by the flow. The film is a linear elastic solid and the shear stress distribution over the film is determined by implementing the shear strain-shear stress relationship which will be elaborated in the following subsection. The elastomer behaves in the same manner as an incompressible fluid whereas unlike standard fluids, it tries to recover its original shape after removal of the force. The film deforms under the normal or shear load but will not compress or yield. The local thickness is a function of the applied force, the film initial thickness, and the film shear modulus. This technique was first introduced by Tarasov and Orlov in early 1990s as a direct method for measuring wall shear stress [3]. The hardware was very similar to the pressure sensitive paint (PSP) technique included a surface or volumedistributed transducer, a specialized light source, an image acquisition system, a date acquisition system. The main drawback of the mentioned work is that the film response estimation is based on the finite element modeling of the film under unit normal and tangential applied loads. Governing equations for determining the shear stress in terms of the displacements are stated in the following. A schematic of shear stress film in a cavity of a model is shown in Fig. 1. The equations are based on the linear continuum mechanics concepts [4]. By considering the displacement vector and the displacement gradient of a typical point under loading and its neighbouring point, changes of lengths undergoing small deformations can be calculated known as the strain tensor. It is assumed that the film is a linear elastic solid which satisfies the following conditions: the deformations are very small; the relationship between applied loads and deformations is linear; upon removal of applied loading, deformations are completely removed; the loading rate has no effect on deformations. As a result, the relationship between stress tensor, T, and strain tensor, E, upon isotropic assumption is stated as the following equation: T = λeI + 2μE

(1)

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Fig. 1 The schematic of a film shear stress sensor under loading

where I is the unit matrix, e is dilatation (first invariant of the strain tensor), and λ and μ are the Lame’s coefficients which are determined by the film’s mechanical properties as: λ=

νEy , (1 + ν)(1 − 2ν)

μ=G=

Ey 2(1 + ν)

(2)

in which Ey , G, and ν are Young Modulus, shear Modulus and Poisson’s ratio of film respectively. Under assumption of very low compressibility, dilatation would be zero and consequently each component of stress tensor can be stated as Tij = 2GEij . To find the stress tensor on the film surface, it is enough to calculate the stress vector, t where is the multiplication of stress tensor by normal unit vector to the surface, n: t = Tn.

2.1 Sensor Calibration and Application The film is created by forming the material into a flat cavity with a smooth surface to form the elastic layer. The 11 µm Potters hollow spherical particles are applied just to the top surface of the film and the shear modulus is statically and dynamically measured. The static calibration involves of applying a known weight on the top surface of the film and measuring film deformation under different tilted positions. The schematic of the static calibration setup is shown in Fig. 2a and a typical example of measuring the shear modulus curve is given in Fig. 2b. All the errors associated with the calibration step including the tilted angle, film thickness, specified weight, and load contact area were considered and the overall uncertainty of the shear modulus based on the 95% of confidence was estimated. It is worthwhile to mention that to avoid any possible errors caused by changes of the film thickness and mechanical properties, it was tested before and after each shear stress measuring experiment. The non-zero intercept arises from the pre-loading during the measurement which can be reduced by increasing the resolution of reading the tilted angle. The reason that the fitted loading and unloading lines do not align is the that the material has

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Fig. 2 (a) The schematic of the static calibration setup. (b) An example of the shear stress-shear strain curve in a static calibration

a visco-elasticity term. This hysteresis effect which adds an error to the estimated wall shear stress is different from one sensor to the other. In order to find the film frequency response and its first natural frequency of tangential oscillations, there is a need for dynamic calibration. The setup consists of a accurate vibration exciter capable of making vibrations in the range of 10 Hz to 20 kHz, a signal generator, a power amplifier, and a high-speed camera to record the film deflection. The calibration involves measuring the amplitude of the film oscillation respect to the amplitude of the applied force at different frequencies. The natural film frequency is expected to be a function of the film shear modulus, thickness, and density. However, all shown results in the present work is based on the static calibrations.

2.2 Experimental Setup Wall shear stress measurements were performed in a open-circuit wind tunnel facility of LTRAC laboratory at Monash University. The flow is driven by a three-phase 5.5 KW electrical motor coupled with an in-line centrifugal blower. The working section has the aspect ratio of 9.75:1 and the sensor was flush mounted at the lower wall of the wind tunnel at the position of 4.09 m behind the ending of the tunnel contraction. The film was formed into a cavity with the dimensions 100 mm × 70 mm using a perspex plate with the dimensions 300 mm × 300 mm and it was confirmed that mounting the plate at the tunnel wall does not cause any edges which may disturb the local flow field. The experiments were conducted at Reynolds numbers ranging from 93,000 to 130,000 based on the bulk flow velocity and the full channel height. PIV measurements confirm the existence of a fully developed turbulent flow in the working section. Film deformation was imaged at 12.7 fps by using a PCO Pixelfly camera in combination with a 55 mm Nikkon Micro-Nikkor lens with the F-stop of 2.8.

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Table 1 Mean wall shear stress and skin friction coefficient measured by the film sensor (h = 1 mm & G = 83 Pa) Rem

U (m/s)

τw (Pa)

uτ (m/s)

Cf

9.38 × 104

13.40

0.466

0.618

4.25×10−3

1.01 × 105

14.45

0.535

0.662

4.20×10−3

1.08 × 105

15.45

0.604

0.704

4.15×10−3

1.15 × 105

16.45

0.672

0.742

4.07×10−3

1.23 × 105

17.54

0.749

0.784

3.99×10−3

1.30 × 105

18.55

0.827

0.823

3.94×10−3

2.3 Results and Discussion To show the capability of the shear stress measuring technique, different film sensors which had the thickness between 1 mm and 2 mm and the shear modulus between 50 Pa and 450 Pa were used at different Reynolds numbers. The average wall shear stress was examined based on the static calibration and the film deformation was calculated by a planar multi-grid cross correlation PIV algorithm [9] of flow-off and flow-on images. In order to remove any possible rigid-body motion of the sensor caused by motor vibrations, a reference pattern stuck on the perspex was simultaneously recorded with a similar imaging system and that movements were subtracted from the film deformation images. Table 1 shows mean wall shear stress, friction velocity, and skin friction coefficient measured at different flow conditions using film sensor (h = 1 mm & G = 83 Pa). Mean wall shear stress calculation is based on the time-average and also the spatial-average of the stress over the film located in the entire FOV. Mean wall skin friction coefficient measured at different Reynolds numbers is shown in Fig. 3 and compared with experimental data for different channel flows. The measured skin friction which has the overall uncertainty of 3.5% compares favourably with measurements by [2, 6, 12] and also the logarithmic skin friction relation by Zanoun et al. (2003, 2005) cited in [12]. In addition to an uncertainty for measuring ρ and U , there are several error sources mostly arise from the calibration stage. An ideal calibration results in a linear curve with zero intercept and exactly the same slope of loading and unloading paths which is not the case for any sensors. The intercept of the calibration curve for used sensors is between 0.006 Pa and 0.05 Pa indicating the pre-loading, hysteresis or any bias error in the measurement system. Owing to the low imaging rate (≈12 Hz) with current camera, measured shear stress is not suitable to obtain spectra.

3 Velocity Profile Measurement In the present section, application of digital holographic PIV to measure near wall velocity profile is demonstrated. While digital holographic PIV offers a solution for

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Fig. 3 Mean skin friction coefficient, Cf measured by the film sensor (shear modulus = 84 Pa & thickness = 1 mm) in the channel flow with the aspect ratio of 9.75:1 compared with the experimental data in the literature

volumetric measurements of the three-dimensional velocity fields in complex flows, it is limited by the inherent depth of focus problem which leads to a low accuracy in the axial positioning especially in case of imaging a volume of particles [5]. One solution is recording magnified holograms on a CCD camera in a similar optical microscope setup in which the light source is replaced with a collimated, coherent laser beam. It has been demonstrated that the method has the ability to examine the spatial distribution and velocity of a high density of particles in an extended volume depth [8, 10]. It has been confirmed that the typical elongation of the reconstructed particles in the axial direction (depth of focus problem) is reduced at least one order of magnitude in comparison to lens-less in-line holography [1, 8]. However, due to high magnification, the field of view is relatively small. The schematic diagram of the experimental setup for digital holographic microscopic PIV is shown in Fig. 4. Appropriate spherical converging lenses were used to expand the laser beam generated by two Nd:YAG pulsed lasers (532 nm and 350 mJ per pulse) and suitable neutral density filters were implemented to reduce the laser energy. The collimated laser beam illuminates the object volume; diffracted light beams from the particles form the object waves while un-diffracted ones are reference waves. A 10× SemiApochromat microscope objective located in front the a PCO Pixelfly camera was used to magnify holograms. The channel has the aspect ratio of 7:1 and the flow was generated by a centrifugal pump with the nominal flow rate of 600 L/hr at 1 m head, and it was seeded by 11 µm Potters hollow spherical particles. In order to measure the imaging resolution, a USAF test target was located in the working plane of the microscope objective where the object was perfectly infocus. More detailed information for hologram recording, numerical particle field reconstruction, and imaging resolution is available in the work by [1, 10]. The flow field was reconstructed at the middle plane of the channel; an instantaneous velocity profile is shown in Fig. 5 compared with theoretical Poiseuille flow. The elongated dimension of 11 µm particle in depth direction was calculated as 130 and 70 µm based on the 75% of the peak intensity with the imaging resolution of 1.16 and 0.508 µm/px respectively. This shows the ability of DHM to improve the depth of focus problem where in a non-magnified reconstruction this elongated

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Fig. 4 The schematic setup for digital holographic microscopic PIV Fig. 5 Normalised velocity profile obtained from the reconstruction at the middle plane of the channel with the aspect ratio of 7:1; imaging resolution is 2.48 µm/px

length is more than two orders of magnitude [5]. As a result, the technique is applicable to measure near wall velocity profile over the turbulent boundary layer in the region of y + < 5 and consequently to measure wall shear stress. This technique will be used over the turbulent channel flow to compare with the wall shear stress obtained from the film sensor.

4 Conclusion In the present work, a novel shear stress sensor which is capable of measuring surface forces over an extended area of the model was introduced. The sensor application and its calibration was addressed and it was successfully applied to a fully developed turbulent channel flow. The technique can be used in air or water and its sensitivity can be tuned for different flow conditions. In addition, digital in-line

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holographic microscopic PIV was successfully used to examine the flow field in a micro channel flow. By means of increasing the imaging resolution of recorded holograms, the technique is able to improve axial resolution as well as the lateral accuracy in a very high dense cloud of particles. Acknowledgements The financial support to conduct this research by the Australian Research Council is appreciatively acknowledged. In addition, O. Amili was supported by the scholarships provided by Monash Research Graduate School while undertaking this research.

References 1. Amili, O., Soria, J.: Application of digital holographic microscopic PIV to a water jet. In: Proceedings of the 5th Australian Conference on Laser Diagnostics in Fluid Mechanics and Combustion, pp. 51–54 (2008) 2. Christensen, K.T.: Experimental Investigation of Acceleration and Velocity Fields in Turbulent Channel Flow. Ph.D. Thesis, University of Illinois at Urbana-Champaign, USA (2001) 3. Fonov, S., Jones, G., Crafton, J., Fonov, V., Goss, L.: The development of optical techniques for the measurement of pressure and skin friction. Meas. Sci. Technol. 17(6), 1261–1268 (2006) 4. Lai, W.M., Rubin, D., Krempl, E.: Introduction to Continuum Mechanics, 3rd edn. Butterworth–Heinemann, Stoneham–London (1993) 5. Meng, H., Pan, G., Pu, Y., Woodward, S.H.: Holographic particle image velocimetry: from film to digital recording. Meas. Sci. Technol. 15(4), 673–685 (2004) 6. Monty, J.P.: Developments in smooth wall turbulent duct flows. Ph.D. Thesis, University of Melbourne, Australia (2005) 7. Naughton, J.W., Sheplak, M.: Modern developments in shear-stress measurement. Prog. Aerosp. Sci. 38(6–7), 515–570 (2002) 8. Sheng, J., Malkiel, E., Katz, J.: Digital holographic microscope for measuring threedimensional particle distributions and motions. Appl. Opt. 45(16), 3893–3901 (2006) 9. Soria, J.: Multigrid approach to cross-correlation digital PIV and HPIV analysis. In: Proceedings of the 1998 Thirteenth Australasian Fluid Mechanics Conference, Melbourne, Australia, 13–18 December 1998, pp. 381–384 (1998) 10. Soria, J., Amili, O., Atkinson, C.: Measuring dynamic phenomena at the sub-micron scale. In: International Conference on Nanoscience and Nanotechnology (ICONN 2008), pp. 129–132 (2008) 11. Winter, K.G.: Outline of the techniques available for the measurement of skin friction in turbulent boundary layers. Prog. Aerosp. Sci. 18(1), 1–57 (1977) 12. Zanoun, E.S., Nagib, H.M., Durst, F.: Refined Cf relation for turbulent channels and consequences for high Re experiments. Fluid Dyn. Res. 41, 1–12 (2009)

Friction Measurement in Zero and Adverse Pressure Gradient Boundary Layer Using Oil Droplet Interferometric Method Philippe Barricau, Guy Pailhas, Yann Touvet, and Laurent Perret

Abstract In the framework of the WALLTURB project, the oil droplet interferometric technique has been used to investigate the skin friction distribution along a zero and adverse pressure gradient boundary layer developing in the Laboratoire de Mécanique de Lille wind tunnel. Skin friction values close to 0.01 Pa have been measured with this optical method.

1 Introduction In opposition to the majority of skin friction measurement techniques based upon analogical laws or assuming the universality of the logarithmic law, the oil droplet technique is able to provide directly the absolute skin friction measurement without any assumption about the form of the velocity profile.

2 Oil Film Interferometric Method The oil film technique is based upon the analysis of the law governing the spreading of a very thin film of viscous fluid under the action of the wall friction [7, 8]. If the P. Barricau () · G. Pailhas · Y. Touvet Aerodynamics and Energetics Department, ONERA, BP 74025, 2 Avenue E. Belin, 31055 Toulouse Cedex 4, France e-mail: [email protected] G. Pailhas e-mail: [email protected] Y. Touvet e-mail: [email protected] L. Perret Laboratoire de Mécanique de Lille, Villeneuve d’Ascq, France e-mail: [email protected] Present address: L. Perret École Centrale de Nantes, Nantes, France M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_41, © Springer Science+Business Media B.V. 2011

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Fig. 1 Generation of the interferometric pattern close to the oil film

wall friction only depends upon x, it can be demonstrated [2] that the film height h is given by relation (1): x dx μoil (1) h(x, t) = √ √ τw (x)t x0 τw (x) where μoil is the oil viscosity, τw the wall friction, x the abscissa and t the time. The oil film thickness variation can be measured using interferometric technique [3]. One part of the illuminating light Ψi is partially reflected by the air/oil interface (Fig. 1) whereas the remaining part passes through the film and is reflected by the solid surface and then travels back again through the film. These two parts of light (Ψ1 and Ψ2 ) interfere together and produce a Fizeau fringe pattern. Using basic interference optic formulas, it can be demonstrated that the height of the oil film is related to the phase difference Φt of the light intensity signal obtained from the fringe pattern between the leading edge of the film and a local X station by the relation (2): h=

1 λΦt 4π n2oil − n2air sin θi

(2)

where λ is the light wavelength, noil and nair respectively represent the refractive index of oil and air and θi the incidence angle of the light. Thus, if we are able to calculate the phase difference Φt at any X stations in the fringe pattern by using the Hilbert transformation (HT), we can determine the oil film thickness at these considered X locations. Drawing inspiration from the work of Naughton and Brown [4], ONERA/DMAE developed a data reduction algorithm based on the Hilbert transformation of the recorded light signal. The HT of the signal I (x) is the convolution of the original 1 . In the Fourier domain, the convolution can be written: signal with πx

1 ∗ F If (x) I˜f (x) = H If (x) I˜f (x) = F −1 F (3) πx where H and F respectively represent the Hilbert and Fourier transformations. The local phase at each pixel in the interferogram can be extracted from relation (4): ˜ If (x) (4) Φ = tan−1 If (x) Then, the total phase Φt can be calculated. Once the total phase Φt is known, the height of the oil film at each corresponding pixel can be calculated using relation (2). Though this technique was validated in the case of a standard ZPG two-dimensional turbulent boundary layer, it presents some drawbacks in the case of more

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Fig. 2 Starting skin friction distribution

Fig. 3 Oil film profiles derived from skin friction distribution

complex flows. The phase determination using the Hilbert transformation can be questionable. In separated or recirculated flow region, the friction vector may exhibit a sign reversal. In fact, we make the assumption that the oil film thickness increases in a monotonous way from the leading edge of the film; this hypothesis is not at all realistic when the flow is submitted to a strong wall friction gradient. Let us consider a given friction distribution (Fig. 2). This friction distribution gives from relation (4) two oil film height evolutions for two different times (Fig. 3). From these oil film thickness evolutions, we can simulate a “theoretical” I (x) intensity profile as indicated in Fig. 4. By applying the Hilbert transformation on the previous simulated intensity profile, we obtain the monotonous h profile (Fig. 5) in disagreement with the real h evolution; the sign reversal occurring on the slope of the oil film height curve cannot be detected with the Hilbert transformation. From this point, a veering of the computed τw distribution with respect to the real skin friction distribution can be noticed (Fig. 6). The sudden skin friction variations occurring in the region of a boundary layer submitted to a strong pressure gradient cannot be reproduced. So, a new implementation of the interferometric method was required.

396 Fig. 4 Intensity profile derived from h1 profile

Fig. 5 Oil film profiles

Fig. 6 τw distribution

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Fig. 7 Fringe spacing versus time

3 Oil Droplet Interferometric Method An alternative approach for measuring skin friction in high wall shear gradient region consists in applying small oil droplets [6], in order to cut out this region in small area elements over which the wall shear stress can be considered as constant, providing interferometric patterns with a constant interfringe, given by the relation (5): Int =

λx 2noil h

(5)

Then, the film height h is given by the relation (6): h(x, t) =

μoil x τw t

(6)

The replacement of h (5) by its expression in the above relation leads to the following expression for the wall shear stress: τw =

2noil μoil ca λ

(7)

where c is the calibration coefficient (mm/pixel) and a the slope of the interfringe versus time curve (pixels/s). Once the interfringe is calculated for each interferometric pattern acquired during the run, the curve describing the fringe spacing versus time can be plotted (Fig. 7). Then the slope of the best fitted straight line, obtained with a least square method, is calculated to extract a and hence the skin friction value τw according to formula (7), provided that the oil characteristics and the optical calibration coefficient are known. If we refer to relation (7), the accuracy of the wall shear stress measurement depends upon the potential uncertainties in the valuation of the oil viscosity, the calibration coefficient, the fringe spacing through the fitting of the interfringe versus time curve, the optical characteristics of the oil (noil ) and the wavelength of the emitted light (λ). Finally, the shear stress has been measured with an uncertainty of ±3%, which was confirmed by repeated measurements.

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Fig. 8 Example of chevron-patterned interferogram produced by dusts

4 Test Surface The interferometric method imposes constraints on the surface on which the oil droplet is deposited. First of all, the surface must be perfectly flattened, without any roughness in order to prevent the apparition of chevron-patterned interferograms (Fig. 8). Furthermore, it is mostly necessary to add a specific material on the test surface in order to enhance the fringe contrasts, defined by relation (8): √ Imax − Imin 2 I1 I2 = (8) C= Imax + Imin I1 + I2 where I1 and I2 represent the intensity of the two interfering waves (Fig. 1). By introducing the coefficient of reflection for the two concerning interfaces (Rair/oil and Roil/wall ), we can write: I1 = Rair/oil Ii with

Rair/oil =

nair − noil nair + noil

I2 = (1 − Rair/oil )2 Roil/wall Ii 2

Roil/wall =

nwall − noil nwall + noil

(9) 2 (10)

Finally, we obtain the relation (11):

2(1 − Rair/oil ) Rair/oil Roil/wall C= Rair/oil + Roil/wall (1 − Rair/oil )2

(11)

The characteristics of various test surfaces are given in Table 1, assuming nair = 1 and noil = 1.4. The Mylar (nwall = 1.67) is a good material for the test surface, able to increase twofold the fringe contrast, in comparison with a glass wall for example.

5 Experimental Tests: ZPG and APG Cases The experiments were carried out in the Laboratoire de Mécanique de Lille boundary-layer wind tunnel [5]. The boundary layer developing on the floor of the

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Fig. 9 Optical set-up

Table 1 Test surface reflective properties Test surface (wall)

nwall

Rair/oil

Roil/wall

C

Polished stainless steel

∞

0.03

1

0.33

Mylar

1.67

0.03

0.01

0.81

Glass and Plexiglas

1.5

0.03

0.001

0.39

15 m long channel, enters a 5 m long test section (2 × 1 m2 , wxh) equipped with glass panels. The maximum external velocity (10 ms−1 ) allows Reynolds numbers Reθ based upon the momentum thickness between 7.5 × 103 and 2 × 104 to be obtained. The wind tunnel velocity and temperature are kept constant with accuracy better than 1% and 0.2°C respectively during time periods of 8 hours. To measure the local wall shear stress, the oil droplet interferometric technique was applied. For this purpose, the optical setup shown in Fig. 9 was mounted on a traversable gantry around the test section of the wind tunnel. The oil droplet deposited on the surface was illuminated by a halogen lamp. The interference pattern was captured by a digital CCD camera (Lavision sensicam 1280 × 1024 pixels) initially dedicated to PIV measurements. An optical line filter (λ = 590 nm – FWHM = 3 nm) was mounted in front of the camera sensor to filter the white illuminating light. The camera was fitted with a long distance microscope objective (Questar QM1) featured by a working range of 560 to 1520 mm. The distance between the microscope and the oil droplet was about 1200 mm. At this working distance, the camera provides a field of view of 7.7 × 6.2 mm2 . Every sequence of interferometric images was recorded at a fixed acquisition period (8 seconds). An example of interferometric images obtained in the present experiment is given in Fig. 10. To enhance the fringe pattern produced, glass panel and Mylar stamps were used as test surfaces. APG measurements were performed on a black paint-backed Mylar stamp of 1.5 × 1.5 cm2 glued onto the test surface. The glass insert was also used as support surface in the rear constant slope region of the bump (Fig. 11). For the zero pressure gradient boundary layer configuration (ZPG), friction measurements were carried out at two streamwise locations X = 18 m and X = 19.6 m from the entrance of the wind tunnel, where the characteristics of the turbulent

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Fig. 10 Example of fringe patterns captured during the tests

Fig. 11 Mylar stamp and glass surface mounted on the test bump

Fig. 12 Mean skin friction measurements (ZPG case)

boundary layer were known from hot wire experiments. ZPG experiments were conducted at four external velocities U0 : 3, 5, 7, 10 ms−1 . The results are presented in Fig. 12 and are in fair agreement with other wall shear stress determinations (µPIV, Clauser plot). The very low levels, down to 0.01 Pa, must be noticed. To analyze the boundary layer submitted to a significant adverse pressure gradient (APG) a bump was mounted onto the test section floor. This bump has a maximum height of 0.33 m and a length of 3.40 m [1]. For the APG case, measurements

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Fig. 13 Mean skin friction measurements (APG case)

were carried out at a fixed reference flow velocity of 10 ms−1 , at eleven streamwise locations from X = 17 m, corresponding to the top of the bump to X = 18.24 m in the rear part of the bump sloping region (Fig. 13). The low deviation between the various repeated measurements at the end of the sloping region of the bump, where the magnitude of the shear stress is close to zero, is a significant result.

6 Conclusion In this experiment, the oil droplet method has been used to measure the skin friction in a flow submitted to zero and strong adverse pressure gradients. The present experimental study demonstrated that this technique can be used for the measurement of very weak skin friction values, even in a strongly destabilized boundary layer. The overall measurement uncertainty is estimated at about 3% with a 95% confidence interval. This is quite good compared to other measurement techniques and quite interesting for the bump test case.

References 1. Bernard, A., Foucaut, J.M., Dupont, P., Stanislas, M.: Decelerating boundary layer: a new scaling and mixing length model. AIAA J. 41(2), 248–255 (February 2003) 2. Brown, J.L., Naughton, J.W.: The thin oil film equation. NASA/TM-208767 (1999) 3. Monson, D.J., Mateer, G.: Boundary layer transition and global skin friction measurement with an oil fringe imaging technique. SAE Paper 932550 (1993) 4. Naughton, J.W., Brown, J.L.: Surface interferometric skin-friction measurement technique. In: 19th AIAA Advanced Measurement and Ground Testing Technology Conference, June 17–20, 1996 5. Pailhas, G., Barricau, P., Touvet, Y., Perret, L.: Friction measurement in zero and adverse pressure gradient boundary layer using oil film interferometric method. Exp. Fluids 47(2), 195–207 (2009) 6. Ruedi, J.D., Nagib, H., Osterlund, J., Monkewitz, P.A.: Evaluation of three techniques for wallshear measurements in three-dimensional flows. Exp. Fluids 35, 389–396 (2003)

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7. Squire, L.C.: The motion of a thin oil sheet under the steady boundary layer on a body. J. Fluids Mech. 11, 161–179 (1961) 8. Tanner, L.H., Blows, L.G.: A study of the motion of oil films on surfaces in air flow, with application to the measurement of skin-friction. J. Phys. E 9(3), 194–202 (1976)

Session 11: Modified Wall Flow

• Scaling of Turbulence Structures in Very-Rough-Wall Channel Flow D.M. Birch and J.F. Morrison • Characterizing a Boundary Layer Flow for Bubble Drag Reduction M. Harleman, R. Delfos, J. Westerweel, and T.J.C. van Terwisga • Direct and Large Eddy Numerical Simulations of Turbulent Viscoelastic Drag Reduction L. Thais, A.E. Tejada-Martínez, T.B. Gatski, G. Mompean, and H. Naji • DNS of Supercritical Carbon Dioxide Turbulent Channel Flow M. Tanahashi, Y. Tominaga, M. Shimura, K. Hashimoto, and T. Miyauchi

Scaling of Turbulence Structures in Very-Rough-Wall Channel Flow David M. Birch and Jonathan F. Morrison

Abstract The streamwise velocity statistics in fully-developed rough-wall channel flow have been investigated for the cases of two very different surface topologies with similar Reynolds numbers. The first surface consisted of a sparse and isotropic grit having a highly non-Gaussian distribution, while the second surface was covered with a uniform, anisotropic mesh comprised of twisted, rectangular elements. The flow was demonstrated to be fully-developed and two-dimensional up to the fourth moment of velocity. Though the flow over both surface types appear to exhibit a limited logarithmic region, the regions of inner and outer scaling over the mesh surface fail to overlap. The influence of the spanwise periodicity of the mesh upon the outer scaling is discussed.

1 Introduction The scaling of flows over rough walls has been of considerable recent interest. The model originally proposed by Hama [9], Clauser [5], Rotta [12] and Townsend [13, 14] suggests that the near-wall inhomogeneities in the flow arising from the local effect of the individual roughness elements are limited to a thin ‘roughness sublayer’, analogous to the viscous sublayer in smooth-wall flows. Any influence of the specific roughness topology upon the flow is therefore assumed to be contained within this layer (typically taken to be about 5 roughness heights in thickness), so any influence of the roughness upon the flow outside of this layer relative to the smooth-wall case must be attributable to the global increase in the wall friction velocity uτ alone. For roughness which is large relative to the viscous length scale D.M. Birch () University of Surrey, Guildford, UK GU2 7XH e-mail: [email protected] J.F. Morrison Imperial College London, London, UK SW7 2AZ M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_42, © Springer Science+Business Media B.V. 2011

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ν/uτ (but still small relative to the integral length scale δ), the roughness size replaces viscosity in the near-wall scaling while sufficient scale separation remains that an overlap region may exist. Then, the usual log-law may be expressed in the form 1 u y −d (1) = ln uτ κ y0 where y0 is a geometry-dependent roughness length which is typically on the order of 0.1k (where k is the roughness height), and d is a zero-plane displacement representing the height at which momentum is extracted, usually on the order of 0.1k [4, 10]. Despite its comparative simplicity, considerable support exists for this model. After a review of available literature, Jimenez [11] proposed the condition k/δ 2.5% (where k is the roughness height) for these scaling conditions to be met. Flack et al. [7] demonstrated collapse of the outer-scaled velocity moments up to the third order for k/δ < 2.2%, while appropriately-scaled mean velocities have been shown to scale well for values of k/δ as high as 20% in rough-wall boundary layers [4, 6, 8]. Internal flows, on the other hand, are constrained by the additional boundary condition and are therefore more likely to exhibit evidence of inner-outer interaction for larger values of k/δ. In the present study, the streamwise velocity statistics in very rough-wall channel flow (with k/ h ranging up to ∼8%, where h is the channel halfheight) have been studied in order to investigate the extent of the similarity. Special attention is given to the higher-order velocity moments, as these quantities are more sensitive to any changes in the flow structure.

2 Experimental Technique Experiments were carried out in a purpose-built rectangular channel flow facility with half-height h = 50.8 mm, width W = 15h and total streamwise fetch L = 134h. The driving pressure was provided by a 4.7 m3 /s squirrel-cage blower with a series of screens and honeycombs providing flow conditioning upstream of an 8:1, two-dimensional contraction. Roughness could be applied to the upper and lower wetted surfaces only, owing to interference with the channel instrumentation. The smooth side-wall conditions were found to have no observable effect on the velocity statistics up to the fourth order, and the underprediction of the wall shear was found to be within the overall experimental error. Two surface topologies were tested; the first was a grit-type roughness, which was a commercial 16-grit open-type abrasive having a sparse, isotropic and highly non-Gaussian distribution. The second was a mesh-type roughness, which was an expanded aluminium sheet consisting of twisted, 2.36 × 1.6 mm rectangular elements forming a diamond pattern with a spanwise-to-lengthwise aspect ratio Lz /Lx = 2.6. Both surface topologies were digitised using a scanning laser profilometer (with a resolution of 1 µm); the results and roughness distributions are shown in Fig. 1. The key roughness parameters are also listed in Table 1.

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Fig. 1 Laser profilometer results and roughness height distributions for the (a) grit-type and (b) mesh-type roughness surfaces Table 1 Key roughness parameters Surface

kmax (mm)

k (mm)

krms (mm)

k/ h (%)

kmax / h (%)

Grit

2.0

0.74

0.41

1.46

3.94

Mesh

4.0

1.27

1.31

2.50

7.87

Time-domain u-component velocity measurements were made using 5-µm diameter platinum hot-wire probes with sensing lengths of approximately 1 mm. The sensors were driven by custom-built constant-temperature anemometers (described in detail by Birch & Morrison [2]) having a bandwidth of approximately 12 kHz. The sensors were periodically re-calibrated in situ throughout the experiments, and the linearisation error arising from the use of an unsteady reference was estimated according to the method of Breuer [3] and was found to be typically less than 0.6%. Data were sampled at 20 to 60 kHz using a Data Translation DT9836 16-bit data acquisition system, and between 222 and 225 samples were collected. The wall-shear velocities were obtained by measuring the streamwise pressure gradient by means of a series of pressure tappings in the channel side walls spaced at intervals of 2h. The fully-rough condition was verified by ensuring that the skin

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Table 2 Experimental parameters for grit and mesh roughness. k + based on kmax Ucl (m s−1 )

U (m s−1 )

Reτ

Reh × 10−4

d y0 k+ (mm) (mm)

Grit (G1)

24.5

21.0

4780

7.27

1.35

0.13

186

359

Grit (G2)

26.6

22.8

5190

7.87

1.30

0.13

200

389

–··–·

Grit (G3)

28.5

24.4

5530

8.43

1.45

0.13

216

415

-----

Mesh (M1)

21.0

17.3

5290

6.23

1.90

0.43

410

1402

–––

Mesh (M2)

23.1

18.8

5890

6.84

2.60

0.43

458

1561

——

Mesh (M3)

25.0

20.5

6280

7.38

2.00

0.43

493

1664

Line

Case

–·–·· —–

ks+

friction coefficient was independent of Reynolds number, and velocity scans were compared at several spanwise stations to ensure two-dimensionality. The estimated error bounds on the experimental results are as follows: uτ , 6%; mean velocities, 0.5%; second-order moments, 1.5%; velocity skewness and kurtosis, 6% and 10%, respectively.

3 Results and Discussion In the present study, the values of the roughness length y0 and zero-plane offset d were determined by fitting the time-mean velocity profiles to (1), assuming that κ = 0.41 and requiring that the logarithmic region began at the wall-normal distance where the flow became spanwise homogeneous. The critical flow parameters for the various cases tested are included in Table 2. Figure 2(a) shows the inner-scaled mean velocity profiles for all cases tested. The excellent collapse observed was, to some extent, artificially imposed by the manner in which the scaling parameters y0 and d were determined. A limited logarithmic region seems to be apparent over both the grit-type and mesh-type roughness for 15 (y − d)/y0 30, followed by an extensive wake region. Figure 2(b) shows the outer-scaled velocity deficit profiles, together with selected rough-wall channel results from Bakken et al. [1], at similar Reτ . The profiles over the grit surface collapse with those of Bakken et al., and are well-represented by the expression y −d 1 Ucl − Uc ln =− + 0.8 (2) uτ 0.41 h for (y − d)/ h 0.06. The profiles over the mesh-type surface do not collapse below (y − d)/ h ∼ 0.30; significantly, in inner scaling, this corresponds to a wall-normal distance of (y − d)/y0 ∼ 35. This result indicates that the regions of inner- and outer-scaling over the mesh-type surface did not overlap, and consequently a logarithmic region could not have existed. The apparent agreement of the data with (1) observed in Fig. 2(a) may have been spurious, artificially imposed by the presumption of overlap in the best-fit determination of the scaling parameters. In addition,

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Fig. 2 Mean velocity profiles. (a) Inner scaling, -·-: (1); (b) outer scaling, symbols: additional data included from Bakken et al. [1], -·-: (2)

velocity statistics over the mesh-type surface were obtained at a single spanwise location only, at the centre of the depression. Figure 3(a) shows the inner-scaled second velocity moment u2 profiles over the grit and mesh surfaces. The profiles collapse remarkably well, and are indistinguishable above (y − d)/y0 50 and 30, respectively. The outer-scaled profiles are included in Fig. 3(b), together with selected results from Bakken et al. The data of Bakken et al. collapse into two families of curves for (y − d)/ h < 0.5, and the authors suggest that this apparent bifurcation is attributable to a Reynolds number effect. However, all of the present data lie on the low-Reτ curve of Bakken et al. despite having Reynolds numbers above their proposed high-Reτ threshold. The present data collapse together regardless of surface type for (y − d)/ h 0.2, though Reτ -independent collapse with the data of Bakken et al. is only observed for (y − d)/ h 0.4. Figure 4 shows the inner-scaled third and fourth velocity moments, together with selected data from Bakken et al. [1] and boundary layer data from Flack et al. [7] where available. Very-long-time measurements (with a coarse spatial resolu-

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Fig. 3 Second velocity moment profiles. (a) Inner scaling; (b) outer scaling, symbols: additional data included from Bakken et al. [1]

tion) were also made of the fourth-order moment to ensure sufficient statistical convergence. The inner-scaled third-order moments (Fig. 4a) exhibit some Reynoldsnumber dependence, and collapse only over a half-decade of (y − d)/y0 . The outerscaled third-order moments (Fig. 4b) collapse over the range (y − d)/ h > 0.4, and agree reasonably well with the results of Bakken et al. despite the considerable experimental scatter. The results of Flack et al. also fall into the broad range of experimental uncertainty, though it should be noted that some decrease in the velocity moment may be due to the free-stream boundary conditions. The inner-scaled fourth-order moments (Fig. 4(c)) exhibit remarkably good collapse over nearly a decade, though the range of similarity and the degree of statistical convergence is slightly greater for the grit than for the mesh. Unlike the thirdorder moment, there is no apparent Reynolds number dependence. The outer-scaled fourth-order moments (Fig. 4(d)) also collapse surprisingly well for (y −d)/ h > 0.2 given the high order.

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Fig. 4 Higher-order velocity moments. (a) Inner-scaled third moment; (b) outer-scaled third moment, symbols: additional data included from Bakken et al. [1] and Flack et al. [7]; (c) inner-scaled fourth moment, symbols are data taken from very long samples to ensure convergence (t + = tu2τ /ν > 3.0 × 108 ) and (d) outer-scaled fourth moment

4 Conclusions Experimental measurements of the streamwise velocity statistics in very-rough-wall channel flow over mesh- and a grit-like roughness surfaces have been carried out. The inner-scaled mean velocity near the wall demonstrated good agreement with previous studies, and the flow over both the mesh and the grit surfaces appeared to exhibit a limited logarithmic region; however, the lack of simultaneous overlap of the regions of inner and outer scaling over the mesh indicated that a logarithmic region could not exist. The lack of an overlap region may have been the result of the spanwise variation in the flow over the mesh, and the scaling of spanwise spatial averages of the time-mean velocity profiles is presently being examined. Statisticallyconverged measurements of the velocity moments up to the fourth order were made,

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and results agreed well with previous studies. The even-order moments exhibited a far better collapse than the odd-order moments.

References 1. Bakken, O.M., Krogstad, P.A., Ashrafian, A., Andersson, H.I.: Reynolds number effects in the outer layer of the turbulent flow in a channel with rough walls. Phys. Fluids 17, 065101 (2005) 2. Birch, D., Morrison, J.F.: An off-the-shelf monolithic constant temperature anemometer. Technical Report 06-004, Imperial College Department of Aeronautics (2007) 3. Breuer, K.S.: Stochastic calibration of sensors in turbulent flow fields. Exp. Fluids 19(2), 138– 141 (1995) 4. Castro, I.P.: Rough-wall boundary layers: mean flow universality. J. Fluid Mech. 585, 469–485 (2007) 5. Clauser, F.H.: The turbulent boundary layer. Adv. Appl. Mech. 4, 1–51 (1956) 6. Connelly, J.S., Schultz, M.P., Flack, K.A.: Velocity defect scaling for turbulent boundary layers with a range of relative roughness. Exp. Fluids 40, 188–195 (2006) 7. Flack, K.A., Schultz, M.P., Shapiro, T.A.: Experimental support for Townsend’s Reynolds number similarity hypothesis on rough walls. Phys. Fluids 17, 035102 (2005) 8. Flack, K.A., Schultz, M.P., Connelly, J.S.: Examination of a critical roughness height for outer layer similarity. Phys. Fluids 19, 095104 (2007) 9. Hama, F.: Boundary-layer characteristics for smooth and rough surfaces. Trans. Soc. Naval Arch. Marine Eng. 62, 333–358 (1954) 10. Jackson, P.S.: On the displacement height in the logarithmic velocity profile. J. Fluid Mech. 111, 15–25 (1981) 11. Jiménez, J.: Turbulent flows over rough walls. Annu. Rev. Fluid Mech. 36, 173–196 (2004) 12. Rotta, J.C.: Turbulent boundary layers in incompressible flow. Prog. Aerosp. Sci. 2, 1–219 (1962) 13. Townsend, A.A.: The Structure of Turbulent Shear Flow, 1st edn. Cambridge University Press, Cambridge (1956) 14. Townsend, A.A.: The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press, Cambridge (1976)

Characterizing a Boundary Layer Flow for Bubble Drag Reduction Marc Harleman, René Delfos, Jerry Westerweel, and Thomas J.C. van Terwisga

Abstract Laboratory experiments of several research groups show that injection of bubbles in a developing turbulent boundary layer can significantly reduce the drag caused by skin friction (Madavan et al. in Phys. Fluids 27(2):356–363, 1984). With the increasing fuel price it is appealing to apply this phenomenon to reduce the friction drag of ships. Despite the progress of experiments and numerical simulations (Ferrante and Elghobashi in J. Fluid Mech. 543:93–106, 2005) the mechanism responsible for this phenomenon and its scale effects are still not well understood. This paper describes the development in drag reduction research and the characterization of a zero pressure gradient boundary layer with an LDA system. The project aim is to investigate low Reynolds number flows with bubbles with a diameter of only a few viscous lengthscales and measure drag reductions with a direct shear stress sensor. Simultaneous optical bubble concentration, stereo PIV and shear stress measurements are scheduled to validate proposed mechanisms from numerical simulations.

1 Review of Work on Drag Reduction by Air Bubbles Laboratory experiments of several research groups show that injection of bubbles in a developing turbulent boundary layer can significantly reduce the drag caused by skin friction [8, 12, 15]. With increasing fuel prices it is appealing to apply this phenomenon to reduce the friction drag of ships. Although drag reductions up to 80% are reported it should be kept in mind that these results are typically obtained by the injection of large gas volumes. In order to reduce the required propulsion M. Harleman () Delft University of Technology, Laboratory of Aero and Hydrodynamics, Leeghwaterstraat 21, 2628 CA Delft, Netherlands e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_43, © Springer Science+Business Media B.V. 2011

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power of a ship, the gain from a reduction in frictional drag should be larger than the energy required to pump the gas underneath the ship. Experiments with model ships led to the estimation that net power reductions of 5% to 10% should be possible. Experiments with a 116 m long Japanese training ship SEIUN-MARU in 2001 (partly described in [5]) surprisingly resulted in just 2% power reduction for one of the tested operating conditions, while a power increase was found for the other conditions. Although this was partly explained by entrainment of bubbles in the propeller inflow and by separation of the bubble cloud from the ship wall, it made it clear that more knowledge is required about the mechanisms and scaling of drag reduction before successful applications to ships are feasible. Many researchers have attempted to find a universal relation between the amount of drag reduction, water flow velocity and gas volume fraction. In general it is found that the observed drag reduction is largest for high gas concentrations and for low water velocities. In addition drag reduction is strongest shortly after injection and decreases quickly downstream, but it can persist up to 50 m after injection as shown by towing tank experiments [5] in Japan. Several scaling methods are proposed to describe the trends found above, many of them based on Cf Qa = Cf 0 Qa + Qw

(1)

where Cf 0 is the skin friction coefficient without gas injection, Qa is the gas flux and Qw is a water flux. In channel flow the water flux is typically chosen to be the total flux of water, while in boundary layer flow Qw is commonly related to boundary layer parameters such as the boundary layer thickness, δ, displacement thickness, δ ∗ or momentum thickness θ . This makes Qw position dependent so that the downstream development can be described as well. Although the proposed scaling methods usually describe the results of a single set of experiments very well, no universal scaling is found that describes the results of experiments from different researchers [12]. A possible explanation for the inability to find a universal scaling is the intuitive idea that the bubble size should be of some influence on the drag reduction mechanism and efficiency. Most bubble injectors used in experiments, like porous plates, have the problem that the bubble size is determined not only by the injector geometry, but also by the gas flux and the local shear and therefore by the flow velocity. It is therefore very difficult, if not impossible, to vary both the flow velocity, gas flux and bubble size independent of each other. Secondly, bubbles in a turbulent flow can split or coalesce depending on the local shear, so it can be argued that far downstream of injection the bubble size distribution will reach an equilibrium based on the local turbulence level. Despite these difficulties Moriguchi and Kato [9] managed to temporarily vary the bubble size between 0.5 and 2.5 mm by placing the injector in either a converging or diverging inlet of their channel, but found no effect on the drag reduction. Shen et al. [13] varied the bubble size from 44 to 476 µm by adding surfactants and salts to the water and also found no significant effect on drag reduction. Kawamura et al. [4] on the other hand found a higher drag reduction for 1.3 mm bubbles than for 0.3 mm bubbles, but since the smaller bubbles where

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distributed over a larger wall normal distance than the larger bubbles this is more likely to indicate the importance of the local bubble concentration than the direct effect of bubble size. While no significant change in drag reduction with varying bubble size was observed in dedicated experiments, comparison of work by different researchers show a large difference in drag reduction efficiency if this is defined as ηDR =

1 Cf α Cf0

(2)

with α the volume averaged bubble void fraction. The majority of the experiments with bubbles have an efficiency, ηDR of about 1, meaning that for 10% drag reduction about 10% volume percent of gas should be added to the channel or boundary layer. Ferrante and Elghobashi [2] performed a numerical simulation where rigid, spherical point bubbles are described in a Lagrangian frame and are force-coupled to a turbulent boundary layer flow calculated in an Eulerian frame. They obtained 3.5% and 20% of drag reduction with 0.1% and 2% of bubbles (ηDR = 35 and 10), so at least an order of magnitude larger than what is found in most experiments. The difference between the simulations and typical experiments are not only the small 60 µm bubbles, but also a very low Reynolds number of Reθ = 1430, so the bubbles have a diameter of 2.4 viscous lengthscales. Only a limited number of experiments are reported with comparable low Reynolds numbers, but Ortiz-Villafuerte [11] found 42% drag reduction with 5.1% void fraction in their low velocity channel flow with 15 µm bubbles. Olivieri et al. [10] injected 2.1% of 100 µm bubbles (d + = 3) in a boundary layer with Reθ = 3060 and obtained 20% drag reduction. Both the experiments by Ortiz-Villafuerte and Olivieri approximated the wall shear stress from the velocity profile obtained by Particle Tracking Velocimetry (PTV). In a later study Ferrante and Elghobashi [3] reduced the bubble size to 40 µm (d + = 1.5) and got a twice as efficient drag reduction at the same Reynolds number; 22% drag reduction with only 1% volume of bubbles. Secondly they increased the Reynolds number to Reθ = 2900 and for the same bubble size (40 µm, but d + = 3.3) they now found 19% drag reduction. The authors conclude that drag reduction efficiency decreases with increasing Reynolds number, which seems consistant with previous observations that lower drag reduction rates are found at higher flow velocities. Van den Berg et al. [15] injected bubbles with a diameter of about 0.5 mm in a Taylor Couette system and found the highest drag reduction for rotation rates corresponding to Weber numbers larger than one. They argued that bubble deformability is essential to obtain drag reduction at higher Reynolds numbers. Injection of rigid hollow glass spheres did indeed not give a measurable drag reduction at higher Reynolds numbers, while bubbles were still effective. Lo et al. [7] showed analytically that compressible bubbles can lead to higher drag reductions than incompressible bubbles. Since the shear in a turbulent boundary layer, given by τ =μ

∂U (y) − ρu v , ∂y

(3)

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consists of a viscous part and a turbulent part, which includes the fluid density, many proposed mechanisms for drag reduction are based on a fluid mixture with modified density and viscosity. Madavan et al. [8] compared bubble drag reduction to polymer drag reduction and suggested that the viscosity increase and density reduction of the two-phase flow in the buffer layer leads to an increased dissipation and a reduced shear. Using similar arguments, Legner [6] constructed a simple stress model based on density reduction and turbulence modification to describe drag reduction observations. A modification of turbulence is clearly observed in numerical simulations where the velocity gradient near the wall decreases, the peak in Reynolds stress moves away from the wall, and as a consequence turbulence production decreases [2]. From studies on particle and bubble laden flows it is known that clustering is most pronounced if the particle/bubble timescale is of the same magnitude as the smallest fluid timescale; the Kolmogorov timescale. This typically means that bubbles should have a diameter of not more than a few viscous lengthscales. Since highest drag reduction efficiencies are observed for such small bubbles it is worth investigating if bubble clustering is indeed contributing to the turbulence modification that leads to drag reduction. The current project aims to investigate by direct shear measurements whether bubbles with a size of a few viscous length scales can indeed reduce the drag with the high efficiencies as obtained in numerical simulations. In addition the local bubble concentration will be measured by shadowgraphy and fluid statistics will be obtained with stereo Particle Image Velocimetry (PIV). The direct shear stress measurements and bubble concentration profiles are expected to clarify whether bubble clustering is contributing to efficient drag reduction or whether turbulence modification and drag reduction can be explained by a local density and viscosity modification alone.

2 Preparation of a Zero Pressure Gradient Developing Boundary Layer The water channel of the Laboratory of Aero and Hydrodynamics in Delft consists of a 5 m long PMMA test section with a cross section of 0.6 × 0.6 m. The water is lead into the test section via an inlet chamber with honeycombs and wire meshes followed by a 6:1 area ratio contraction. As a consequence a very homogeneous flow is created with a low turbulence level (σu < 0.01U0 ). The growing boundary layers on the four walls of the closed water channel would create a gradual acceleration of the main flow. This effect is most pronounced at low velocities when thick boundary layers are formed. In the case of a 0.2 m/s bulk flow, the boundary layer is about 100 mm thick at the end of the 5 m long test section, reducing the effective cross section and thereby accelerating the bulk flow by about 8%. Since it is convenient to compare velocity profiles and statistics of drag reduced flow with well documented, analytically described, zero pressure gradient boundary layers, an adjustable bottom is installed to compensate for the reduction in cross section by the growing boundary layers. With this bottom the bulk velocity over the

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Table 1 Characteristic boundary layer scales at six positions downstream of a 6 mm trip wire placed at x = 0 m x [m]

U0 [m/s]

δ [mm]

δ ∗ [mm]

θ [mm]

H [-]

Reθ [-]

uτ [mm/s]

0.55 0.85 1.32 1.67 2.17 2.55

0.195 0.195 0.200 0.199 0.201 0.202

33.4 40.8 50.8 58.1 68.7 75.0

5.2 6.2 7.8 8.8 10.5 11.5

3.6 4.4 5.4 6.2 7.4 8.0

1.44 1.42 1.44 1.42 1.43 1.43

694 846 1082 1232 1471 1617

9.8 9.5 9.4 9.1 8.9 8.9

Fig. 1 Boundary layer development; Fits to the log-law region (κ = 0.41, B = 5) are shown for the 6 positions in Table 1. Each profile is vertically shifted in steps of 5u+ . The figure shows the growth of the log-law region and the development of the wake region with increasing Reθ

whole measurement section is maintained constant within 0.5% of the mean velocity of 0.2 m/s, which is within the measurement uncertainty of the LDA system used (Dantec, BSA enhanced, with a fiber coupled probe with a f = 310 mm lens). Velocity profiles at several places after a 6 mm trip wire are measured with a two component LDA system. The characteristic boundary layer scales are shown in Table 1 and the velocity profiles normalized with inner variables are shown in Fig. 1. The wall shear velocity is obtained by fitting a logarithmic profile in the region 30δν < y < 0.3δ, with the classical coefficients κ = 0.41 and B = 5. Although this method depends on the choice of κ and B, for this low Reynolds number flow it is not possible to obtain uτ from an extrapolation of the shear towards the wall. This can be seen in the lower graph in Fig. 2, where the total shear is not a smooth

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Fig. 2 Reynolds stress and shear stress distribution at 2.17 m behind the trip wire; Reθ = 694. The solid lines in the top figures represent DNS data from Spalart [14] at Reθ = 670

function of y + near the wall and extrapolation would lead to an overestimation of τw . The same is found by DeGraaff and Eaton [1]. Comparison of the measured Reynolds stress components with the results from simulations by Spalart [14] show good qualitative agreement. Again, the underestimation of u v and the u u in the outer boundary layer is also observed by the LDA measurements by DeGraaff and Eaton (not shown here). Although measurements of the velocity at several positions in the water channel within a few hours showed that the false bottom correctly compensated for any flow acceleration, the bulk velocity, U0 , reported in Table 1 is not constant. This might be caused by fluctuations in temperature, pump frequency stability or pollution of wire meshes with seeding particles during the measurement period of two weeks. Since the water flow rate is not automatically controlled, monitoring should be improved in order to manually compensate for possible fluctuations. With that adjustment made, the measurement facility will be highly suitable to study the changes to a classical boundary layer created by bubble injection.

3 Outlook Electrolysis will be used to generate circa 100 µm bubbles in a 0.2 m/s flow past a plate so that d + is about 1. Mixtures with such small bubbles have a limited optical access if the volume fraction is around 1% or higher. Therefore the wall shear stress can no longer be accurately determined with LDA or PIV and a direct measurement

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is preferable. At 0.2 m/s shear forces are very small and no commercial, load cell based, sensor is available that is able to measure 1% variation in such a small force. Therefore a sensor is under development which consists of a 150 mm diameter measurement plate suspended on 4 long, weak springs so that the displacement of the plate is a measure for the applied force. In order to minimize the gap between the plate and its housing the spring suspension can be moved by an electrical traverse to counteract the displacement of the measurement plate. The system is designed to measure shear forces in flows between 0.1 and 1 m/s with 1% accuracy. The boundary layer and turbulence statistics will be measured by a stereo PIV system with a Spectra-Physics Quanta Ray 400 mJ double pulsed Nd:YAG laser with two PCO 2000 Sensicam cameras with 14 bit resolution. With this dynamical range it is expected that tracer particles can be imaged without bubbles saturating the cameras. Simultaneously a third camera will be used to record the bubble shadows when they move in front of a strong light source. Bubble size and distribution can than be obtained with developed image processing software. Acknowledgements tion STW (07781).

This research was financially supported by the Dutch Technology Founda-

References 1. DeGraaff, D., Eaton, J.: Reynolds-number scaling of the flat-plate turbulent boundary layer. J. Fluid Mech. 422, 319–346 (2000) 2. Ferrante, A., Elghobashi, S.: On the physical mechanisms of drag reduction in a spatially developing turbulent boundary layer laden with microbubbles. J. Fluid Mech. 503, 345–355 (2004) 3. Ferrante, A., Elghobashi, S.: Reynolds number effect on drag reduction in a microbubble-laden spatially developing turbulent boundary layer. J. Fluid Mech. 543, 93–106 (2005) 4. Kawamura, T., Fujiwara, A., Takahashi, T., Kato, H., Matsumoto, Y., Kodama, Y.: The effects of the bubble size on the bubble dispersion and skin friction reduction. In: 5th Symposium on Smart Control of Turbulence (2004) 5. Kodama, Y., Kakugawa, A., Takahashi, T.: Microbubbles: drag reduction mechanism and applicability to ships. In: Twenty-Fourth Symposium on Naval Hydrodynamics (2002) 6. Legner, H.H.: A simple-model for gas bubble drag reduction. Phys. Fluids 27(12), 2788–2790 (1984) 7. Lo, T.S., L’vov, V.S., Procaccia, I.: Drag reduction by compressible bubbles. Phys. Rev. E 73(3), 036308 (2006) 8. Madavan, N.K., Deutsch, S., Merkle, C.L.: Reduction of turbulent skin friction by microbubbles. Phys. Fluids 27(2), 356–363 (1984) 9. Moriguchi, Y., Kato, H.: Influence of microbubble diameter and distribution on frictional resistance reduction. J. Marine Sci. Technol. 7(2), 79–85 (2002) 10. Olivieri, A., Jacob, B., Cancello, A., Van Oostrum, P., Campana, E., Piva, R.: The effect of microbubbles on a flat plate turbulent boundary layer. In: 2nd International Symposium on Seawater Drag Reduction (2005) 11. Ortiz-Villafuerte, J., Hassan, Y.A.: Investigation of microbubble boundary layer using particle tracking velocimetry. J. Fluids Eng. — Trans. ASME 128(3), 507–519 (2006) 12. Sanders, W.C., Winkel, E.S., Dowling, D.R., Perlin, M., Ceccio, S.L.: Bubble friction drag reduction in a high-Reynolds-number flat-plate turbulent boundary layer. J. Fluid Mech. 552, 353–380 (2006)

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13. Shen, X.C., Ceccio, S.L., Perlin, M.: Influence of bubble size on micro-bubble drag reduction. Exp. Fluids 41(3), 415–424 (2006) 14. Spalart, P.: Direct simulation of a turbulent boundary-layer up to R-theta = 1410. J. Fluid Mech. 187, 61–98 (1988) 15. van den Berg, T.H., Luther, S., Lathrop, D.P., Lohse, D.: Drag reduction in bubbly Taylor– Couette turbulence. Phys. Rev. Lett. 94(4), 044501 (2005)

Direct and Large Eddy Numerical Simulations of Turbulent Viscoelastic Drag Reduction Laurent Thais, Andres E. Tejada-Martínez, Thomas B. Gatski, Gilmar Mompean, and Hassan Naji

Abstract This work deals with direct numerical simulations (DNS) and temporal large eddy simulations (TLES) of turbulent drag reduction induced by injection of heavy-weight long-chain polymers in a Newtonian solvent. The phenomenon is modelled for the three-dimensional wall-bounded channel flow of a FENE-P dilute polymer solution. The DNS are undertaken with an optimized hybrid high-order finite difference spectral code running in parallel using domain decomposition (mpi) and threading (openmp) on each mpi process. The flows computed have friction Reynolds numbers ranging from Reτ = 180 to 590. Various Weissenberg numbers and polymer molecular lengths are considered to obtain percent drag reductions from 28 to 59%. Results of DNS show that viscoelastic drag reduction is characterized by a marked anisotropy of the Reynolds L. Thais () · G. Mompean · H. Naji Laboratoire de Mécanique de Lille, CNRS-UMR 8107, Université Lille I-Sciences et Technologies, Polytech’Lille, Villeneuve d’Ascq, France e-mail: [email protected] G. Mompean e-mail: [email protected] H. Naji e-mail: [email protected] A.E. Tejada-Martínez Department of Civil and Environmental Engineering, University of South Florida, Tampa, FL, USA e-mail: [email protected] T.B. Gatski Laboratoire d’Etudes Aérodynamiques, CNRS-UMR 6609, Université de Poitiers, ENSMA, Poitiers, France e-mail: [email protected] T.B. Gatski Center for Coastal Physical Oceanography and Ocean, Earth and Atmospheric Sciences, Old Dominion University, Norfolk, VA, USA M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_44, © Springer Science+Business Media B.V. 2011

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stress and extra-stress tensors, thus confirming the recent findings of Frohnapfel et al. (J. Fluid Mech. 577:457–466, 2007) who postulate that strong anisotropy is a common feature to all drag reduction situations. The large eddy simulations (TLES) are based upon temporal causal filters (Pruett et al. in Humphrey et al. (eds.) Proc. of the 4th International Symposium on Turbulence and Shear Flow Phenomena, Williamsburg, 2005, vol. 2, pp. 705–710). A key point of the TLES model lies in subfilter modeling of the stretching terms and of the non-linear spring force in the conformation tensor equation, which proves necessary to obtain correct predictions for the percent drag reduction with respect to DNS.

1 Direct Numerical Simulations (DNS) We consider in this study the turbulent channel flow of a solution of a FENE-P fluid characterized by an elastic time scale λ, a zero-shear polymeric viscosity ηp0 , and a maximum molecular extent L, diluted in a Newtonian solvent with viscosity ηs and density ρ.

1.1 DNS Model Equations The transport equations are scaled with the bulk velocity Ub , and the channel halfgap h. The flow has two dimensionless groups, the bulk Reynolds number Reb = ρUb h/η0 based on the total zero shear rate viscosity η0 = ηs + ηp0 , and the bulk Weissenberg number Web = λUb / h. The scaled evolution equations for momentum and for the conformation tensor cij (which represents the ensemble average squared norm of the end-to-end vector of polymer molecules) are ∂ui ∂ui ∂p β0 ∂ 2 ui (1 − β0 ) ∂τij + uj =− + + + ei δi1 , 2 ∂t ∂xj ∂xi Reb ∂xj Reb ∂xj 2 ∂uj ∂cij ∂cij ∂ cij ∂ui 1 + uk − ckj − cki + τij = . ∂t ∂xk ∂xk ∂xk Prc Reb ∂xk2

(1) (2)

The Cartesian coordinates are (x, y, z) = (xi )i=1,3 in the channel streamwise, wallnormal and spanwise directions, the instantaneous velocity field is (ui )i=1,3 , and t denotes time. The formalism of (1) and (2) assumes a uniform polymer concentration governed by the retardation ratio β0 = ηs /η0 . The quantity ei δi1 is the driving pressure gradient in the axis of the channel. The extra-stress τij is related to the conformation tensor through τij =

f ({c})cij − δij , Web

(3)

assuming the Peterlin approximation f ({c}) = (L2 − 3)/(L2 − {c}), ({.} is the trace operator).

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Table 1 Parameters used for viscoelastic FENE-P direct numerical simulations. Reτ 0 is the friction Reynolds number; Web the bulk Weissenberg number; Weτ 0 the friction Weissenberg number; L the dimensionless maximum polymeric chain extension; DR the percent drag reduction Mesh size

Reτ 0

Web

Weτ 0

L

DR

256 × 129 × 128

180

5

55

30

28%

256 × 129 × 128

180

10

115

60

51%

256 × 129 × 128

180

10

115

100

59%

512 × 257 × 128

395

5

115

100

59%

1024 × 257 × 192

590

3.6

116

100

58%

2048 × 513 × 384

1000

2.3

115

100

58%

Equation (2) includes a stress Prandtl number Prc = η0 /ρκc , defined as the ratio of the total kinematic zero shear rate viscosity (η0 /ρ) to a numerical stress diffusivity κc . This term is introduced to stabilize the algorithm.

1.2 Numerical Method Equations (1) and (2) are solved with a hybrid high-order spectral code running in parallel with domain decomposition (mpi) and threading (openmp) on each mpi process. The spatial discretization is Fourier spectral in the two homogeneous directions (x and z) whereas a compact 6th order finite difference scheme [6] is used in the wall-normal direction. Time marching is 4th order Adams–Bashforth for explicit terms and 3rd order Adams–Moulton for implicit viscous terms. The domain decomposition consists of a two-dimensional mpi Cartesian grid, sometimes referred to as ‘pencils’ decomposition. This decomposition allows to compute locally to each mpi process the 1d-FFTs along x and z, as well as the finite difference solutions in the wall-normal direction y. To permit these local computations, the three-index arrays are successively transposed with the help of the transfer routines of the highly scalable package p3dfft1 developed by D. Pekurovsky at the San Diego Supercomputing Center. The resulting DNS code has been shown to scale properly up to 16384 cores on the Blue Gene/P at IDRIS-CNRS, France.

1.3 DNS Results Table 1 summarizes the DNS database which has been established. These simulations have been conducted at constant pressure gradient in a channel of dimensions Lx = 8πh × Lz = 1.5πh. Two different strategies were used to collect the 1 http://www.sdsc.edu/us/resources/p3dfft/.

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Fig. 1 Streak snapshots at y+ = 15 for friction Reynolds number Reτ 0 = 1000; Newtonian flow (left); FENE-P flow with L = 100 and 58% drag reduction (right); only the first half of the channel is shown (x < 4πh)

viscoelastic turbulent flow statistics. At small Reynolds number (Reτ 0 ≤ 395), the simulations were initialized with a Newtonian flow field, leaving the mass flow rate of FENE-P flows to increase and reach a new steady state before collecting statistics. At higher Reynolds numbers, the simulations were initialized with the FENE-P flow field at the immediately smaller Reynolds number previously computed, and statistics were again collected after the transient. Figure 1 displays a snapshot of streaklines at y+ = 15 for the Newtonian and FENE-P flows at the highest Reynolds number reached for now (Reτ 0 = 1000). These pictures show the energy transfer from smaller towards larger turbulent scales when high drag reduction occurs. This result is in line with recent findings by Samanta et al. [9] who recently demonstrated with the help of a Karhunen–Loeve analysis that viscoelastic turbulence is characterized by a systematic increase in the time scales of dynamic turbulent events. Figure 2 shows converged FENE-P flow statistics at Reτ 0 = 395 and L = 100 compared with the Newtonian flow statistics at the same Reynolds number. The turbulence enhancing property of polymers is markedly anisotropic, the addition of polymer serving to enhance Txx while suppressing Tyy and Tzz , Tij denoting the ensembleaveraged Reynolds stress components (averaged in time and in the two homogeneous spatial directions). This confirms recent findings by Frohnapfel et al. [3] who postulate that strong anisotropy is a common feature to all drag reduction situations, also observed for instance in Newtonian supersonic flows. We also note that the slope of the log region increases at this high level of percent drag reduction which is in accordance with experimental observations [10] and with other direct numerical simulations at smaller Reynolds numbers [1, 2, 4].

2 Temporal Large Eddy Simulations (TLES) We present in this section the first attempt to simulate turbulent drag reduction with large eddy simulation. Our approach is based upon Temporal Large Eddy Simulation (TLES) which uses filtered equations in the time domain [7]. A temporal filtering approach utilizing a temporal approximate deconvolution method (ADM) to account for the subfilter scale motions is well suited to viscoelastic fluid flows. Such temporal ADM methods can be written in differential relaxation equations in a manner analogous to stress relaxation effects being incorporated into viscoelastic constitutive equations.

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Fig. 2 DNS statistics of FENE-P flow (symbols) compared with Newtonian statistics (full lines) at the same friction Reynolds number Reτ 0 = 395. Mean flow (upper left); streamwise rms-velocity (upper right); wall-normal rms-velocity (bottom left); spanwise rms-velocity (bottom right)

2.1 TLES Model Equations The TLES equations of the FENE-P flow are obtained by temporally filtering the momentum and the conformation tensor equations: ∂Mij ∂ui ∂ui ∂p β0 ∂ 2 ui + uj =− + − + χu (w i − ui ) 2 ∂t ∂xj ∂xi Reb ∂xj ∂xj +

(1 − β0 ) ∂τ ij (1 − β0 ) ∂Sij + + ei δi1 , Reb ∂xj Reb ∂xj

∂uj ∂cij ∂cij ∂ui + uk − ckj − cki + τ ij ∂t ∂xk ∂xk ∂xk 2 ∂ cij 1 = + Pij + Qij − Sij + χc (γ ij − cij ). Prc Reb ∂xk2

(4)

(5)

The filtered equations are closed by deconvolution of the resolved-scale variables. The quantity Mij is the modeled residual Newtonian stress (resolved subfilter-stress) also present in Newtonian TLES, whereas Sij is a subfilter term related to the nonlinear restoring force and (Pij , Qij ) are subfilter terms due to polymer stretching. These subfilter terms are obtained by solution of auxiliary ordinary differential equations (ODEs) in time domain, equivalent to the application of causal exponential

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time filter kernels of respective bandwidths u (for momentum) and c (for the conformation tensor), i.e. Mij = vi vj − v i v j , vi =

p

m=0 (m+1) ∂ui

(6a)

Cm ui(m+1) , (m)

(6b) (m+1)

− ui , 1 ≤ m ≤ max(p, q), ∂t

u ∂vi vj vi vj − vi vj ∂v i vi − v i , , = = ∂t

u ∂t

u f ({α})α ij − δij f ({α})αij − δij Sij = T ij − , Tij = , Web Web ∂v j ∂vj ∂vi ∂v i αkj − α kj , Qij = αki − α ki , Pij = ∂xk ∂xk ∂xk ∂xk p (m+1) Cm cij , αij = =

m=0 (m+1) ∂cij

∂t

=

ui

(m)

(6c) (6d) (6e) (6f) (6g)

(m+1)

cij − cij

1 ≤ m ≤ max(p, q),

,

c

(6h)

αij − α ij Tij − T ij ∂T ij ∂α ij = = , , ∂t

c ∂t

c ∂ ∂vi 1 ∂vi ∂vi αkj = αkj − αkj , ∂t ∂xk

c ∂xk ∂xk ∂vj 1 ∂vj ∂ ∂vj αki = αki − αki . ∂t ∂xk

c ∂xk ∂xk

(6i) (6j) (6k)

The quantities χu (w i − ui ) and χc (γ ij − cij ) in (4) and (5) are dissipative regularization terms which represent unresolved subfilter-contributions (for this study χu = χc = 1). These last terms are obtained through the resolution of wi = γij =

q m=0 q m=0

(m)

(m+1)

,

(m+1)

,

Dm ui

Dm cij

wi − w i ∂w i = , ∂t

u ∂γ ij ∂t

=

γij − γ ij

c

(7a)

.

(7b)

denotes any m-times temporally filtered quantity Q, with In these ODEs, Q m ≤ max(p + 1, q + 1). Here, we used primary deconvolution of degree p = 3, and secondary deconvolution of degree q = 2. For these degrees of deconvolution, theoptimal primary deconvolution coefficients [8] are [C0 , C1 , C2 , C3 ] = √ √ √ √ √ [0, 6, 4 + 2 6 − 2 6, 1 − 4 + 2 6 + 2 6], and the optimal secondary deconvolution coefficients are [D0 , D1 , D2 ] = [15/8, −9/8, 1/4].

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Fig. 3 Left: ensemble-averaged streamwise velocity U + for FENE-P flow (L = 100) at Reτ 0 = 180. Right: ensemble-averaged first normal conformation tensor component. DNS (solid line); TLES (); DNS on TLES mesh (×)

Fig. 4 Reynolds stress components for FENE-P flow (L = 100) at Reτ 0 = 180. DNS (solid line); TLES subfilter-stress (broken line); TLES total stress = resolved + subfilter ()

2.2 TLES Results at Reτ 0 = 180 The TLES model was applied to predict the three FENE-P viscoelastic flows at Reτ 0 = 180 which exhibit percent drag reductions ranging from 28% to 59% (see Table 1). For these TLES simulations a coarse grid with 48 × 49 × 24 mesh points was used. The time step was t = 0.004h/Ub , i.e. 4 times the time step of DNS. Figures 3 and 4 display the mean flow and selected conformation tensor and Reynolds stress components for high drag reduction (L = 100). From the left picture of Fig. 4, it is clear that TLES is able to predict the high level of anisotropy of turbulence (Txx is even overestimated by TLES). A marginal over prediction by TLES of the shear component is observed in the near wall region, which coincides with a slightly flatter mean flow in the buffer layer. Yet, the TLES percent drag reduction comes out at 60% in line with 59% for DNS. For successful percent drag reduction prediction the momentum (ru = u / t) and conformation tensor (rc =

c / t) filter bandwidth ratios had to be tuned. For the high drag reduction case discussed above, L = 100, we used ru = 256 and rc = 176. From this preliminary study at low Reynolds number, a rule of thumb seems to be that both filter width

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ratios need to increase with the percent drag reduction. For Newtonian flow at the same Reynolds number, ru = 128 is the optimal value. As can be seen from (6a)–(6k) and (7a), (7b), the TLES model requires solution at every grid point of many ODEs, which represents a non-negligible computational and memory overhead. Although the overall reduction in the TLES workload upon DNS is a factor 50 at Reτ 0 = 180, the complexity of the proposed model might become penalizing at higher Reynolds number. A possible way of improvement could be to use the deconvolved velocity field for the conformation tensor filtered equation, in the same way as proposed by Kuerten [5] for spatial LES of particleladen channel flow. Acknowledgements The “Institut du Développement et des Ressources en Informatique Scientifique” (IDRIS-CNRS) provided the extensive computational resources and assistance needed for this work under project N° 092277. Additional support is acknowledged from the “Centre de Ressources en Informatique de Haute Normandie” (CRIHAN) under project No 2007008.

References 1. Dimitropoulos, C.D., Sureshkumar, R., Beris, A.N.: Direct numerical simulation of viscoelastic turbulent channel flow exhibiting drag reduction: effect of the variation of rheological parameters. J. Non-Newton. Fluid Mech. 79, 433–468 (1998) 2. Dubief, Y., White, C.M., Terrapon, V.E., Shakfeh, E.S.G., Moin, P., Lele, S.K.: On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows. J. Fluid Mech. 514, 271–280 (2004) 3. Frohnapfel, P., Lammers, P., Jovanovic, J. Durst, F.: Interpretation of the mechanism associated with turbulent drag reduction in terms of anisotropy invariants. J. Fluid Mech. 577, 457–466 (2007) 4. Housiadas, K.D., Beris, A.N.: Polymer-induced drag reduction: effects of the variations in elasticity and inertia in turbulent viscoelastic channel flow. Phys. Fluids 15, 2369–2384 (2003) 5. Kuerten, J.G.M.: Subgrid modelling in particle-laden channel flow. Phys. Fluids 18, 025108 (2006) 6. Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103(1), 16–42 (1992) 7. Pruett, C.D., Gatski, T.B., Grosch, C.E., Thacker, W.D.: The temporally filtered Navier–Stokes equations: properties of the residual stress. Phys. Fluids 15, 2127–2140 (2003) 8. Pruett, C.D., Thomas B.C., Grosch, C.E., Gatski, T.B.: A temporal approximate deconvolution model for LES. In: Humphrey, J.A.C., Gatski, T.B., Eaton, J.K., Friedrich, R., Kasagi, N., Leschzimer, M.A. (eds.) Proc. of the 4th International Symposium on Turbulence and Shear Flow Phenomena, Williamsburg, 2005, vol. 2, pp. 705–710 9. Samanta, G., Oxberry, G.M., Beris, A.N., Handler, R.A., Housiadas, K.D.: Time-evolution K-L analysis of coherent structures based on DNS of turbulent Newtonian and viscoelastic flows. J. Turbul. 9, 345 (2008). doi:10.1080/14685240802491 10. Warholic, M.D., Massah, J., Hanratty, T.J.: Influence of drag reducing polymers on turbulence: effects of Reynolds number, concentration and mixing. Exp. Fluids 27, 461–472 (1999)

DNS of Supercritical Carbon Dioxide Turbulent Channel Flow Mamoru Tanahashi, Yasuhiro Tominaga, Masayasu Shimura, Katsumi Hashimoto, and Toshio Miyauchi

Abstract Direct numerical simulation (DNS) of supercritical CO2 turbulent channel flow has been performed to investigate the heat transfer mechanism of supercritical fluid. Due to effects of the mean density variation in the wall normal direction, mean velocity in the cooling region becomes high compared with that in the heating region. The mean width between high- and low-speed streaks near the wall decreases in the cooling region. From the turbulent kinetic energy budget, it is found that compressibility effects related with pressure fluctuation and dilatation of velocity fluctuation can be ignored even for supercritical condition. However, the effect of density fluctuation on turbulent kinetic energy cannot be ignored. In the cooling region, low kinematic viscosity and high thermal conductivity in the low speed streaks modify fine scale structure and turbulent transport of temperature, which results in high Nusselt number in the cooling condition of the supercritical CO2 .

1 Introduction The supercritical fluid is frequently used in many industrial applications such as chemical process and food production. Since the supercritical fluid shows very strong heat transport capability near its pseudo-critical point, it is considered to be a good working fluid in heat transfer. For example, in supercritical water-cooled reactor, the supercritical water is used as the medium of heat transfer instead of liquid water [5]. Supercritical CO2 is also one kind of promising heat transfer medium. Since the critical temperature of CO2 (about 304 K) is near normal temperature, it M. Tanahashi () · Y. Tominaga · M. Shimura · T. Miyauchi Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan e-mail: [email protected] K. Hashimoto Central Research Institute of Electric Power Industry, 2-6-1 Nagasaka, Yokosuka-shi, Kanagawa, 240-0196, Japan M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_45, © Springer Science+Business Media B.V. 2011

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is a good refrigerant in hot water heating. A CO2 heat pump for domestic hot tap water supplier with high heat transfer efficiency has been developed and commercialized in Japan. To develop higher performance and further compact heat pump, high-pressure side heat exchanger of the CO2 heat pump should have better performance and be smaller than conventional one. Therefore, it is very important to understand the heat transfer mechanism of supercritical flow. In many cases, the supercritical flows are in turbulent or transitional state and the fluid flow is rather complicated. However, the research works on turbulent (or transitional) supercritical flow are very rare until now [1]. In this study, DNS of supercritical CO2 turbulent channel flow is performed to investigate the turbulent heat transfer mechanism of supercritical flow by considering temperature dependence of physical properties of CO2 at 8MPa. Since the background of this study is development of the supercritical CO2 domestic heat pump, the temperature range in this DNS is chosen within normal temperature, and forced convective heat transfer is focused by neglecting buoyancy effects.

2 Numerical Method Full compressible Navier–Stokes equations and energy conservation equation are solved to take into account temperature dependence of thermal and transport properties of the supercritical fluid. Since the pressure fluctuation is very low due to low Mach number, the enthalpy can be assumed only the function of temperature. The following Peng–Robinson equation [6] is used for equation of state. p=

aα RT − , Vm − b Vm2 + 2bVm2 − b2

(1)

where Vm represents the molar volume and Vm = (0.044 kg/mol)/ρ for CO2 and R is the universal gas constant. Other parameters are given as follows: α = [1 + (0.37464 + 1.54226ω − 0.26992ω2 )(1 − Tr0.5 )]2 , Tr = T /Tc ,

a = 0.45724R 2 Tc2 /pc ,

b = 0.07780RTc /pc ,

where Tc = 304.1282 K is the critical temperature and pc = 7.3773 MPa is the critical pressure. The acentric factor ω is tuned according to experimental data [7], and set to ω = 0.3. Since the pressure fluctuation is very small relative to mean pressure, the transport and thermal properties such as viscosity, specific heat and thermal conductivity are assumed to be only the function of temperature. Partition parabolic interpolation equations are introduced to calculate these properties. The interpolated values agree well with the results of Span et al. [7]. To approximate the convective terms in momentum and energy equations, 7thorder upwind finite difference scheme is used. For other terms, 8th-order central finite difference scheme is used. Time advancement is implemented by a 3rdorder TVD type Runge–Kutta scheme [2]. The periodic conditions are adopted in streamwise and spanwise directions by applying periodic wall-temperature in

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Fig. 1 Profiles of mean velocity and Van Dirst mean velocity for case 1

the streamwise direction. The wall temperature is set to be the maximum value in 4π < x < 20π and the minimum value in 28π < x < 44π . The maximum wall temperature is 321.15 K for case 1 and 319.15 K for case 2, and the minimum wall temperature is 311.15 K for case 1 and 309.15 K for case 2. DNS is conducted for the temperature range higher than the pseudo-critical temperature. Uniform body force which corresponds to Reτ = 180 in incompressible channel is applied to drive the flow. Computational domain is set to Lx × Ly × Lz = 48πδ × 2δ × 2πδ. In the streamwise direction, length of the computational domain is chosen enough to investigate effects of variable density and physical properties on the turbulence structure and heat transfer. DNS was conducted by 2304 × 193 × 160 grid points for case 1 and 2880 × 256 × 192 grid points for case 2.

3 Turbulence Statistics In the present DNS, friction Reynolds number (Reτ ) is not constant in the streamwise direction. Since the added body force corresponds to Reτ = 180 for the incompressible case, Reτ fluctuates near 180. From the characteristics of CO2 near the pseudo-critical point, kinematic viscosity is low in the cooling region (28π < x < 44π ). Therefore, Reτ in the cooling region is higher than that in the heating region. + Figure 1 shows mean streamwise velocity and Van Direst mean velocity (UVD = u+ √ + ρ/ ¯ ρ¯w du ) at x = 15.9π (in the heating region) and x = 41.4π (in the cool0 ing region) for case 1. The mean velocity in the cooling region is much higher than that in the heating region, whereas the Van Direst velocities in two regions agree with each other. These facts suggest that the mean velocities are affected by the density variation, but not by acoustic effects or intrinsic compressibility effects. Figure 2 shows the r.m.s. of velocity fluctuations normalized by local uτ at x = 15.9π and x = 41.4π for case 1. The r.m.s. of velocity fluctuations in the cooling region is higher than that in the heating region. As turbulent intensities are normalized by local uτ , the difference between heating and cooling region cannot be explained only from physical properties of supercritical CO2 .

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Fig. 2 Profiles of r.m.s. of velocity fluctuations for case 1

Fig. 3 Distribution of turbulent kinetic energy production term (P ), viscous dissipation term (ε), viscous diffusion term (D) and turbulent diffusion term (T )

The mean width of the streaks which was estimated from autocorrelation of the streamwise velocity fluctuation decreases in the cooling region. It has been shown that near-wall streamwise eddies, which is one of coherent fine scale eddies [9], tend to exist near boundary of high- and low-speed streaks. Therefore, the increase of streaks suggests that lots of the coherent fine scale eddies appear. Since the nearwall coherent fine scale eddy has an important role in turbulent heat transfer, the increase of the fine scale eddy enhances the heat transfer ability of the supercritical flow.

4 Turbulent Kinetic Energy Budget Turbulent kinetic energy budget at x = 15.9π and 41.4π are evaluated. The production term (P ), viscous dissipation term (−ε), viscous diffusion term (D) and turbulent diffusion term (T ) are dominant, and other terms are small relative to these four terms. In Fig. 3, the dominant four terms are compared in the cooling and heating regions for the two cases. This balance of the energy budget is similar to those of incompressible turbulence [3, 4, 9]. It should be noted that the convection terms are not exactly zero for the present case because of the periodic cooling and heating in the streamwise direction. However, the results of DNS show that the convection terms are very small compared with the major terms. The production term and viscous dissipation terms show large magnitude in the cooling region. In

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Fig. 4 Distribution of pressure dilatation term (Πd ), dilatation dissipation term (εd ) and the term associated with density fluctuation (M)

the buffer layer, the increase of the production term balances with that of the viscous dissipation term, which implies that turbulence structure in the buffer layer is significantly different in the cooling and heating region. Near the wall, the viscous diffusion term in the cooling region is larger than that in the heating region due to viscosity variation. Note that oscillations in Fig. 3 is caused by short average time due to the huge DNS, and will disappear if the average was taken in longer time. In variable density turbulent flows, the pressure-dilatation term (Πd ), dilatation dissipation term (εd ), which is included in the viscous dissipation term (ε), and the term associated with density fluctuation (M) have possibilities to change the energy budget. In Fig. 4, these three terms are shown in the cooling and heating region. Both of pressure dilatation and dilatation dissipation terms have weak contribution for energy transport, whereas the terms associated with density fluctuation show effective values. These terms show a peak at y + ≈ 10, and the peak value is greater than 5% of the production term. It is well known that compressibility effects always depress the development of turbulence. However, density fluctuations effects do not always work for depression in the present study. These terms show positive values in the heating region and negative values in the cooling region. In the cooling region, since the wall temperature is close to the pseudo-critical point, the density fluctuation effect is relatively strong. Therefore, this effect will be more significant if the temperature range was more close to the pseudo-critical point.

5 Heat Transfer Characteristics Nusselt number is very important property to characterize heat transfer efficiency. Nusselt number in the cooling region is higher than that in the heating region. According to physical properties of supercritical CO2 , the Prandtl number increases dramatically if temperature is close to the pseudo-critical point, which leads to high Nusselt number in the cooling region. Figure 5 shows the turbulent heat flux v T at x = 15.9π and −v T at x = 41.4π for case 1. Compared with the heating region, the enhanced turbulence in the cooling region leads to higher turbulent heat flux. The difference of the peak magnitude of the heat flux is about 30%. The location of the peak is shifted into large y + in the cooling region. The high local Reτ in the

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Fig. 5 Distribution of turbulent heat flux for case 1

Fig. 6 Contour of temperature on a x–z plane (y + ≈ 10) with iso-surface of Q = 1.0 for case 1

cooling region leads to the high turbulent heat flux. This enhancement of heat flux results in high Nusselt number in the cooling region. Figure 6 shows distribution of temperature on a x–z plane (y + ≈ 10) with isosurfaces of the second invariant of velocity gradient tensor (Q = 1.0) in case 1. The positive Q regions can represent fine scale vortical structures [8, 9]. Near the wall, the positive Q regions in the normalization based on the outer quantities represent near-wall eddy structures such as streamwise eddies and hairpin-vortices. Temperature is denoted by colors (high temperature is denoted by red). Note that high temperature streaks correspond to low speed streaks in the heating region and those do high speed streaks in the cooling region. The number of the fine scale eddy in the heating region is obviously less than that in the cooling region. In the heating region, the coherent fine scale eddies tend to exist in high temperature and low speed streaks. Since high temperature regions possesses high kinematic viscosity, fine scale eddies might disappear with the heating of the fluid in the downstream. On the other hand, in the cooling region, the fine scale eddies tend to exist in low temperature and high speed streaks. Contrary to the heating region, low temperature regions possesses low kinematic viscosity. Therefore, fine scale eddies might be created continuously in the downstream. These relations between fine scale eddy and temperature have great importance in the heat transfer near the wall. In the cooling region, high Prandtl number due to low temperature enhances temperature fluctuation in small scales, which results in effective mixing by the fine scale eddy. These

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Fig. 7 Iso-surfaces of Q = 1.0 (green) with iso-surfaces of v T = −1.0 × 10−4 (blue), 36π < x < 40π in case 1

combined effects of turbulence structure and turbulent mixing have significant contributions and high Nusselt number in the cooling region. The modifications of turbulence structure and mixing mechanism also affect the turbulent heat flux near the wall. In Fig. 7, instantaneous turbulent heat flux (v T ) is shown in cooling region with iso-surfaces of Q for case 1. The threshold of the turbulent heat flux is selected to v T = ±1.0 × 10−4 which is about three times larger (or smaller) than that of mean values shown in Fig. 5. In the heating region (not shown here), high heat flux regions exist along the fine scale eddies and are localized near the wall. However, in the cooling region, high heat flux region are extended into the channel center. The high heat flux region far from the wall possesses hairpin-like feature, and hairpin-like fine scale eddies, which are enhanced by local high Reynolds number effect, also exist in these regions. These results suggest that the modified turbulence structure causes high turbulent heat flux in the cooling region.

6 Summary DNS of supercritical CO2 turbulent channel flow is performed by considering temperature dependence of thermal and transport properties. From DNS results, difference between the heating and cooling process of supercritical CO2 is discussed in the temperature range higher than the pseudo-critical point. The mean velocity is affected by density variation, but not by acoustic effects or intrinsic compressibility effects. The effects of density fluctuation on turbulent kinetic energy cannot be ignored. The terms related to density fluctuations become significant near the pseudo-critical temperature. The mean width between high- and low-speed streaks near the wall decreases in the cooling region. The low speed and low temperature regions near the wall enhances fine scale motion of the fluid in the cooling side. Since low temperature regions in the cooling side correspond to high thermal conductivity regions, temperature mixing by the fine scale eddy is enhanced. The turbulent heat flux is also intensified by the modification of the turbulence struc-

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tures due to the local high Reynolds number effects. These combined effects result in high Nusselt number in the cooling condition of the supercritical CO2 .

References 1. Bae, J.H., Yoo, J.Y.: Direct numerical simulation of turbulent supercritical flows with heat transfer. Phys. Fluids 17(10), 105104 (2005) 2. Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996) 3. Kim, J., Moin, P., Moser, R.D.: Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166 (1987) 4. Moser, R.D., Kim, J., Mansour, N.N.: Direct numerical simulation of turbulent channel flow up to Reτ = 590. Phys. Fluids 11(4), 943–945 (1999) 5. Oka, Y., Koshizuka, S.: Design concept of once-through cycle super-critical pressure light water cooled reactors. In: Proc. 1st Int. Symp. Supercritical Water-Cooled Reactor Design (SCR2000), Paper No. 101. The University of Tokyo, Tokyo (2000) 6. Peng, D.Y., Robinson, D.B.: New 2-constant equation of state. Ind. Eng. Chem. Fundam. 15(1), 59–64 (1976) 7. Span, R., Wagner, W.J.: A new equation of state for carbon dioxide covering temperature to 110 K at pressure up to 800 MPa. J. Phys. Chem. Ref. Data 25(6), 1509–1596 (1996) 8. Tanahashi, M., Miyauchi, T., Ikeda, J.: In: Simulation and Identification of Organized Structures in Flows, pp. 131–140. Kluwer Academic, Dordrecht (1999) 9. Tanahashi, M., Kang, S.J., Miyamoto, T., Shiokawa, S., Miyauchi, T.: Scaling law of fine scale eddies in turbulent channel flows up to Reτ = 800. Int. J. Heat Fluid Flow 25, 331–340 (2004)

Session 12: Industrial Modeling

• Evaluation of v 2 –f and ASBM Turbulence Models for Transonic Aerofoil RAE2822 J.J. Benton (Airbus UK) • Turbulence Modelling Applied to Aerodynamic Design V. Levasseur, S. Joly, and J.-C. Courty (Dassault Aviation)

Evaluation of v 2 –f and ASBM Turbulence Models for Transonic Aerofoil RAE2822 Jeremy J. Benton

Abstract Linear and nonlinear v 2 –f turbulence models and an algebraic structurebased model (ASBM) are evaluated and compared to k–ω SST for transonic flow over aerofoil RAE2822. Tests on channel and flat plate flow are used to select the baseline linear model as the “code-friendly” k–ε–φ–α model (Billard et al. in 7th Int. ERCOFTAC Symp. on Eng. Turb. Modelling and Measurements, ETMM7, Cyprus, 2008) and later developments, which exploits elliptic blending to handle wall effects. This is extended with the nonlinear stress-strain relationship (Pettersson-Reif in Flow Turbul. Combust. 76:241–256, 2006), and also provides the linear scale equations and elliptic blending parameter for the ASBM (Langer and Reynolds in Tech. Rep. TF-85, Mech. Engng. Dept., Stanford Univ., 2003, and later developments). For the aerofoil with separation the linear model placed the shock aft of the measured location, the nonlinear model gave a useful forward shift, and both SST and ASBM gave good location. All models overpredict the pressure over the rear part of the aerofoil, though trailing edge pressure is noticeably improved for the ASBM. All models underpredict boundary layer recovery downstream of the shock.

1 Introduction In CFD for the aerodynamic design of aircraft, two linear turbulence models dominate: the 1-equation Spalart–Allmaras model [25] and the k–ω SST model by Menter [19]. These have proved to be very effective in many applications involving small separation with limited 3-dimensionality. In particular for high speed wing design these models are effective in placing the shock in the right location. However many of these flow computations would benefit from improved physics modelling. A nonlinear stress-strain relationship or other form of stress modelling is J.J. Benton () Aerodynamics Design and Data, Building 09B, Airbus, Filton, Bristol, BS99 7AR, UK e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_46, © Springer Science+Business Media B.V. 2011

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necessary if the anisotropy of the normal stresses is to be resolved. This is especially important near walls and if complex three-dimensional effects are to be captured such as may occur in strongly curved flows or secondary flows in junction regions. Also, elliptic blending techniques can be adopted to handle the non-local blocking effect of the wall and avoids the use of more ad-hoc wall damping. Traditional stress models derive the Reynolds stresses only from information about the magnitude of the fluctuating velocity components — the componentality. Structure-based models (SBM) on the other hand introduce additional dependence on one-point turbulence structure statistics. Using the exact transport equations as the basis, these structure tensors and their transport equations have been evolved [11, 12, 23] to follow the acceptable performance of simpler models for weakly strained flow, to embody a rational treatment of rapid distortions and anisotropy, and to handle the effects of strong flow curvature or frame rotation. A practical SBM has been developed in algebraic (ASBM) form [14, 23]. This also uses elliptic relaxation to implement wall-blocking effects, here being applied on the turbulent structures based on eddy orientation constraints near a wall. The required turbulent time and length scales for the algebraic model can be obtained from an eddy viscosity model (EVM). Conventionally this would implement a length scale equation in the form of the small scale dissipation, which is recognised as a weakness. In [23] an improvement is sought wherein the EVM length scale equation is sensitised to the structural information from the ASBM. This idea is pursued in the development of a novel large scale enstrophy equation (LSE) [24]. In [14] modelling of the LSE is further refined for use with the ASBM and this forms one of the main additional innovations along with development of the ASBM itself. This combination of ASBM + LSE is further explored in [15]. In contrast the conventional EVMs in the form of the k–ω model [26] and a “code-friendly” v 2 –f model [17] provide the scale equations for testing the ASBM in [9] and [13]. All the SBM studies so far referenced involve application to channel flow, including effects of rotation. In [10] this is extended, using the v 2 –f scale equations, to low Reynolds number 2D boundary layers with zero pressure gradient (ZPG) and adverse pressure gradient (APG). In [22] the ASBM + v 2 –f approach is evaluated for a 2D backstep and asymmetric diffuser. In the WALLTURB project the ASBM has been further developed, and in conjunction with v 2 –f and LSE equations has been tested for a 2D higher Reynolds number ZPG boundary layer. Following on from the above, the objectives of the present study are to evaluate two anisotropy resolving models in the form of the quadratic stress-strain relationship due to Pettersson-Reif [21] for v 2 –f models, and the ASBM here using EVM scale equations. The evaluation is to be performed by implementing the models in a class of compressible flow solver common in the industry, and testing of the ASBM is to be extended to a two-dimensional transonic aerofoil. The RANSMB solver is used as a development platform for the WALLTURB project. This is a three-dimensional, Jameson-type [8], cell-centred, central differenced, compressible method for structured meshes. It was developed from an earlier two-dimensional code [1]. Third order stabilising dissipation is minimised by scaling with local viscosity, and with Mach number near the wall, making it negligible

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for turbulence transport except near the boundary layer edge. The impact of 1st order dissipation outside of shocks is minimised by use of a shock sensor. Transport equations are hyperbolic in time with local time stepping to the steady state solution. Elliptic blending equations are solved by treating the imbalance between left and right hand sides as a time derivative. Free stream turbulence is maintained by minima on the production terms.

2 Turbulence Model Selection and Test on Channel and Flat Plate Results are presented for channel flow for Reτ = 395 including comparison with DNS [20], and a flat plate boundary layer up to Rex = 10.106 for which Reθ 15000, with transition fixed at Rex = 0.1.106 . Centre-line or free-stream Mach is set at 0.2. The channel half-height has 56 cells and the flat plate has around 80 cells in the boundary layer. For both cases the first three near-wall cells have the same height y + = 1. For the original Durbin k–ε–v 2 –f model [5] numerical difficulties are expected in explicit segregated solvers of the type used here. These stem from the near wall asymptotic dependence v 2 = O(y 4 ) and f ∝ v 2 /y 4 , leading to high order sensitivities in the coupling of the equations. The model was implemented and converged results were obtained for the channel and boundary layer, but no consistent convergence could be achieved for the transonic aerofoil. To avoid such problems Lien and Durbin [17] introduced a “code-friendly” k–ε–v 2 –f model with f changed so that f = 0 at the wall, decoupling it from v 2 . In Laurence et al. [16] v 2 is changed to φ = v 2 /k. As k = O(y 2 ) this leads to a lower order wall asymptote φ = O(y 2 ). These models were not used, however, in favour of the following. The recent k–ε–φ–α model described by Billard et al. (2008) [3] retains φ = v 2 /k as the 3rd variable but replaces the elliptic equation for f by a much simpler one for α, following the form adopted by Manceau [18] in the context of stress models: αwall = 0 α → 1 in far-field α − L2α ∇ 2 α = 1 3 1/4 3/2 ν k with CL = 0.161, Cη = 90. , Cη Lα = CL max ε ε

(1) (2)

Results for the channel flow, Fig. 1, show good near wall k and ε. For the boundary layer, Fig. 2, U + is somewhat low compared to the expected log-law behaviour, this being associated with high wall shear, Fig. 3. The k–ε–φ–α model has undergone significant further development to improve its Reynolds number dependence and the low U + . A version of this was supplied to the author [2] and will be referred to as the “2009 version”. The completed model is to be published as a Ph.D. thesis by Billard. For the channel Fig. 1 shows a reduced near wall k peak (ε is not shown due to the difficulty of extracting it). For the

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Fig. 1 Results for channel flow Reτ = 395. Symbols = DNS; k–ε–φ–α (2009) and nonlinear k–ε–φ–α (2009) (same results); ASBM + k–ε–φ–α (2008)

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k–ω SST; linear linear k–ε–φ–α (2008);

Fig. 2 Mean velocity U + for flat plate boundary layer at Rex = 9.6.106 , Reθ = 13800–15500. k–ω SST; linear k–ε–φ–α (2009) and nonlinear k–ε–φ–α (2009) linear (same results); k–ε–φ–α (2008); ASBM + k–ε–φ–α log-law, (2008); 2 lines: U + = 1/.41 ln(y + ) + 5.0 U + = 1/.38 ln(y + ) + 4.08

boundary layer, Fig. 2, the improved U + is now slightly high and wall shear slightly low, Fig. 3. The nonlinear stress-strain relationship [21] is used with the k–ε–φ–α (2009) model due to the latter’s reasonable flat plate results. As expected for channel and flat plate equilibrium boundary layer flow, the normal stress anisotropy introduced by the nonlinear form makes no difference to the plotted results, Figs. 1, 2, and 3. The ASBM routines require as input the local rate of strain and rotation tensors normalised by the time scale i.e. T Sij and T ij , and a wall blockage tensor Bij . Normalised Reynolds stresses τij /2k are output at the same location. The time scale T is simply k/ε with a Kolmogorov lower limit. A good estimate for T and k and hence for ε is required from the scale equations, extending through the wall region.

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Fig. 3 Surface friction Cf for flat plate boundary layer. k–ω SST; linear k–ε–φ–α (2009) and nonlinear k–ε–φ–α (2009) linear (same results); k–ε–φ–α (2008); ASBM + k–ε–φ–α correlations: (2008); Fernholz (lower Cf ) Karman–Schoenherr (higher Cf ). Symbols = experiment, Wieghardt

The blockage tensor Bij is computed from a blockage parameter Φ and its gradients to give the necessary wall orientation. Φ is defined by an elliptic equation L2Φ ∇ 2 Φ = Φ with Φwall = 1 and Φ → 0 in far-field, 3 1/4 3/2 ν k with CL = 0.17, Cη = 80. , Cη LΦ = CL max ε ε

(3) (4)

In this study, scale equations for k and ε are provided for the ASBM by the k–ε–φ–α (2008) model [3]. This choice was prompted by the direct availability of ε from this model, its good near wall k and ε for the channel flow, and its robust numerical behaviour and ease of solving the elliptic equation (1) for α. Any deficiencies in this model’s prediction of stress from eddy viscosity when used as an EVM alone are less relevant here. Another factor was the obvious similarity between the elliptic terms in the k–ε–φ–α and ASBM models, comparing (1) for α and (3) for Φ, and (2) and (4) for the respective length scales Lα and LΦ . If Lα = LΦ then Φ = 1 − α so only 1 elliptic equation need be solved. CL and Cη have very similar values in (2) and (4), so Lα = LΦ was achieved by setting CL and Cη to the values required for the ASBM in (4) as the wall blocking would be more critical for the ASBM than for the k–ε–φ–α model. For the channel, Fig. 1 shows improved near wall k and similar ε compared to k–ε–φ–α (2008) alone. For the flat plate U + , Fig. 2, is now very close to the expected log-law behaviour but accompanied by low wall shear, Fig. 3.

3 Results for RAE2822 Aerofoil Details of the experiment are given by Cook et al. [4]. In the present study measured ordinates are used and results are presented for transonic cases 9 and 10. Case 9 is attached, and case 10, on the basis of CFD results, has as shock induced separation

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Fig. 4 RAE 2822 aerofoil. a: Fine mesh; b: Mach contours for case 10 using ASBM + k–ε–φ–α (2008), M = 0.01: light blue M < .3; dark blue M < .754; green M < 1; red M > 1

followed by reattachment. Tunnel corrections were reviewed by RAE/DRA for the EUROVAL project [7] resulting in: for case 9 M = .734, α = 2.54°, Rec = 6.5.106 ; for case 10 M = .754, α = 2.57°, Rec = 6.2.106 ; and for both a camber correction z/c = .006x/c(1 − x/c). These are used here except that for case 10α is increased to 2.77° to obtain an improved match for forward Cp . The two-dimensionality of the experiment has been questioned [6]. However used together and staying close to the above flow conditions, the two cases still seem useful for comparing results with current practise for the onset of shock-induced separation. All results were produced on medium and fine meshes. Small differences appear only near the shock, so results are shown for the fine mesh only. The number of points on the upper surface is 104 for medium and 144 for fine, which also adds bunching around and aft of the shock. The number of points through the boundary layer at 20%, 50%, and 100% chord are 36, 41, and 59 for medium and 60, 70, and 120 for fine. In both meshes the first three near-wall cells were adjusted as far as possible to have height y + = 1. In contrast to the channel and flat plate, the ASBM now incurred isolated numerical failures but all outside of the boundary layer itself. These were handled by restricting the ASBM to the boundary layer, and reverting locally to the EVM stress from the scale equations if ASBM failures occurred near the boundary layer edge. Figure 4 shows the fine mesh and typical flow field. Surface pressure and friction coefficients (Cp , Cf ) are shown in Fig. 5 for both cases, and in more detail for case 9 in Figs. 6(a) and 6(b) and for case 10 in Figs. 7(a) and 7(b). Deviations in Cp near the leading edge may be associated with effects of the transition trip at x/c = .03. The main problems expected for case 10 are the aft shock location returned by many models, overprediction of pressure aft of the shock, and underprediction of boundary layer recovery. The linear k–ε–φ–α (2009) model exhibits the aft shock problem, similar to Menter’s k–ω baseline (BSL) model (not shown), and all models presented here show the overprediction of pressure. The nonlinear relationship [21] applied to the k–ε–φ–α (2009) model gives a forward movement of the shock, though not as much as the shift in moving from k–ω BSL to the k–ω SST model. Good shock location is achieved by the k–ω SST model, with further small improvement for ASBM + k–ε–φ–α (2008). This is a good result for the ASBM indicating

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Fig. 5 RAE 2822 aerofoil, cases 9 and 10. Surface pressure Cp and friction Cf . k–ω SST; linear k–ε–φ–α (2009); nonlinear k–ε–φ–α Symbols = experiment; ASBM + k–ε–φ–α (2008) (2009);

its potential to be used in aerodynamic design CFD with no degradation in current flow prediction capability up to separation onset, yet offering the potential of improved modelling in more complex three-dimensional flow regions. It is also encouraging to note that the ASBM trailing edge Cp for case 10 in Fig. 7(a) is close to the measured value. For case 9 Cp results are generally satisfactory, with similar trends to those seen in case 10 but to a much lower degree. The friction (Cf ) results in Figs. 5, 6 and 7 show a tendency for k–ω SST to go closer to separation for case 9 and return a larger separated length for case 10 compared to other models. While variations between models are expected aft of the shock, the Cf from SST over the forward part is larger than for the ASBM and k–ε–φ–α models, this broadly following the trend seen in the flat plate results. The measured values are not conclusive as to what should be the correct result, and the difficulty of friction measurements in this flow should be born in mind.

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Fig. 6 Results for RAE 2822 aerofoil, case 9. a, b: pressure Cp and friction Cf , over rear part; c–h: mean velocity U/Ue on upper surface at and aft of shock, where Ue = edge velocity. k–ω SST; linear k–ε–φ–α (2009); nonlinear k–ε–φ–α Symbols = experiment; ASBM + k–ε–φ–α (2008) (2009);

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Fig. 7 Results for RAE 2822 aerofoil, case 10. a, b: pressure Cp and friction Cf , over rear part; c–h: mean velocity U/Ue on upper surface at and aft of shock, where Ue = edge velocity. k–ω SST; linear k–ε–φ–α (2009); nonlinear k–ε–φ–α Symbols = experiment; ASBM + k–ε–φ–α (2008) (2009);

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Figures 6 and 7 show mean flow velocity profiles on the upper surface at and aft of the shock and into the near wake. For both cases 9 and 10 the profiles for k–ω SST and ASBM + k–ε–φ–α (2008) are very close and generally show more flow deceleration near the shock and less recovery downstream, compared to the linear and nonlinear k–ε–φ–α (2009) results. This is more marked for case 10, where the lack of recovery for all models is clearly seen for x/c ≥ 0.9. The excessive momentum deficit in the boundary layer, combined with flow curvature over the aft part of the aerofoil, may be a contributor to the overprediction of pressure here. The profile for x/c = .75 for case 10, Fig. 7(e), does not follow the trend seen at x/c = .65 and .9, the measurements appearing to be displaced from the surface in an anomalous, possibly erroneous manner. For case 9 Fig. 6(e) shows a similar inconsistency but to a much lower extent.

4 Conclusions The code-friendly k–ε–φ–α model [2, 3] was adopted as a baseline linear v 2 –f model and used as scale equations for the ASBM, in which role a good synergy was possible and satisfactory results obtained for channel and flat plate flows, though with rather low wall friction for the flat plate. For the two-dimensional transonic aerofoil RAE2822 results were compared with k–ω SST. For case 10, a shock-induced separation is expected and the linear k–ε–φ–α model placed the shock significantly too far aft, the nonlinear form [21] gave a useful forward shift, and the SST and ASBM models showed good shock placement. The ASBM also gave the best trailing edge Cp . The ASBM thus has potential to be used in aerodynamic design CFD without compromising the good results expected up to separation onset using current models such as SST or Spalart–Allmaras, while offering potential improvements for more complex threedimensional flow regions. All models showed overprediction of pressure over much of the surface aft of the shock, and underprediction of boundary layer recovery. For the attached flow case 9, generally satisfactory results were obtained with similar trends to case 10 but to a much lower degree. Future work will focus on testing the ASBM on a wider range of adverse pressure gradient flows, three-dimensional cases, and adoption of the LSE scale equations. Acknowledgements This work has been performed under the WALLTURB project. WALLTURB (A European synergy for the assessment of wall turbulence) is funded by the CEC under the sixth framework program (CONTRACT No. AST4-CT-2005-516008). Thanks also to Flavien Billard (k–ε–φ–α) and Carlos Langer (ASBM) for providing and assisting with these turbulence models.

References 1. Benton, J.J.: Chap. 4.2 in: Haase, W., Brandsma, F., Elsholz, E., Leschziner, M., Schwamborn, D. (eds.) EUROVAL — A European Initiative on Validation of CFD Codes, eds: Vieweg, Wiesbaden (1993)

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2. Billard, F.: Improvement and tuning of a Reynolds averaged Navier Stokes model. Univ. of Manchester, UK (Jan-2009). Private communications 3. Billard, F., Uribe, J.C., Laurence, D.: A new formulation of the v 2 –f model using elliptic blending and its application to heat transfer prediction. In: 7th Int. ERCOFTAC Symp. on Eng. Turb. Modelling and Measurements, ETMM7, Cyprus, 2008 4. Cook, P.H., McDonald, M.A., Firmin, M.C.P.: Aerofoil RAE 2822 — pressure distributions, and boundary layer and wake measurements. In: Experimental Data Base for Computer Program Assessment, AGARD AR-138, Chap. A.6 (1979) 5. Durbin, P.A.: Separated flow computations with the k–e–v2 model. AIAA J. 33(4), 659–664 (1995) 6. Garbaruk, A., Shur, M., Strelets, M.: Numerical study of wind-tunnel walls effects on transonic airfoil flow. AIAA J. 41(6), 1046–1054 (2003) 7. Haase, W., Brandsma, F., Elsholz, E., Leschziner, M., Schwamborn, D. (eds.): EUROVAL — A European Initiative on Validation of CFD Codes. Vieweg, Wiesbaden (1993) 8. Jameson, A., Schmidt, W., Turkel, E.: Numerical solutions of the Euler equations by finite volume methods using Runge–Kutta time-stepping. Tech. Rep. AIAA paper 81-1259 (1981) 9. Kalitzin, G., Iaccarino, G., Langer, C.A., Kassinos, S.C.: Combining eddy-viscosity models and the algebraic structure-based Reynolds stress closure. In: Proceedings of the Summer Program, Center for Turbulence Research, NASA Ames/Stanford Univ. (2004) 10. Kassinos, S., Langer, C.: A new algebraic structure-based model with proper handling of strong rotation. In: Proceedings of the 5th GRACM Int. Congress on Computational Mechanics, Limassol, Cyprus, 2005 11. Kassinos, S.C., Reynolds, W.C.: A structure-based model for the rapid distortion of homogeneous turbulence. Tech. Rep. TF-61, Mech. Engng. Dept., Stanford Univ. (1994) 12. Kassinos, S.C., Langer, C.A., Haire, S., Reynolds, W.C.: Structure-based turbulence modeling for wall-bounded flows. Int. J. Heat Fluid Flow 21, 599–605 (2000) 13. Kassinos, S.C., Langer, C.A., Kalitzin, G., Iaccarino, G.: A simplified structure-based model using standard turbulence scale equations: computation of rotating wall-bounded flows. Int. J. Heat Fluid Flow 27, 653–660 (2006). Also in 4th Int. Symp. on Turb. Shear Flow Phenomena 14. Langer, C.A., Reynolds, W.C.: A new algebraic structure-based turbulence model for rotating wall-bounded flows. Tech. Rep. TF-85, Mech. Engng. Dept., Stanford Univ. (2003) 15. Langer, C.A., Kassinos, S.C., Haire, S.: Application of a new algebraic structure-based model to rotating turbulent flows. In: 6th Int. ERCOFTAC Symp. on Eng. Turb. Modelling and Measurements, ETMM6, Italy, 2005 16. Laurence, D.R., Uribe, J.C., Utyuzhnikov, S.V.: A robust formulation of the v2–f model. Flow Turbul. Combust. 73, 169–185 (2004) 17. Lien, F.S., Durbin, P.A.: Non-linear k–e–v2 modeling with application to high-lift. Annual Research Briefs, Center for Turbulence Research, NASA Ames/Stanford Univ., pp. 5–22 (1996) 18. Manceau, R.: An improved version of the elliptic blending model. Application to nonrotating and rotating channel flows. In: Proc. 4th Int. Symp. Turb. Shear Flow Phenomena, Williamsburg, VA, USA, 2005 19. Menter, F.R.: Zonal two equation k–w turbulence models for aerodynamic flows. In: 24th Fluid Dynamics Conference (1993) 20. Moser, R.D., Kim, J., Mansour, N.N.: Direct numerical simulation of turbulent channel flow up to Re = 590. Phys. Fluids 11(4), 943–945 (1999) 21. Pettersson-Reif, B.A.: Towards a nonlinear eddy-viscosity model based on elliptic relaxation. Flow Turbul. Combust. 76, 241–256 (2006) 22. Radharkrishnan, H., Pecnik, R., Iaccarino, G., Kassinos, S.C.: Computation of separated turbulent flows using the ASBM model. In: Proceedings of the Summer Program, Center for Turbulence Research, NASA Ames/Stanford Univ. (2008) 23. Reynolds, W.C., Kassinos, S.C., Langer, C.A., Haire, S.L.: New directions in turbulence modeling. In: Third Int. Symp. on Turbulence, Heat, and Mass Transfer, Nagoya, Japan, 2000

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24. Reynolds, W.C., Langer, C.A., Kassinos, S.C.: Structure and scales in turbulence modeling. Phys. Fluids 14(7), 2485–2492 (2002) 25. Spalart, P.R., Allmaras, S.R.: A one-equation turbulence model for aerodynamic flows. In: 30th Aerospace Sciences Meeting, Reno. AIAA 92-0439 (1992) 26. Wilcox, D.C.: Turbulence Modelling for CFD, 1st edn. DCW Industries, Inc., La Cañada (1993)

Turbulence Modelling Applied to Aerodynamic Design Vincent Levasseur, Sylvain Joly, and Jean-Claude Courty

Abstract This paper presents the state of the art of the turbulence modelling at Dassault Aviation and the ability to compute complex flows of industrial interest. It describes developments performed to achieve an accurate and efficient simulation capacity, used as an engineering tool for aerodynamic design. The development is performed within an in-house code used at Dassault for the aerodynamic design of both military aircrafts and business jets. Non-exhaustive challenging industrial applications are presented, for which turbulence modelling improvements lead to a major impact on key design issues. RANS as well as Reynolds Stresses Models and LES/DES approaches will be assessed.

1 Introduction Reynolds Averaged Navier–Stokes (RANS) simulations have reached a mature and validated capacity [4, 6, 11]. Their range of application is well identified and they are routinely used for aerodynamic design. A detailed review of the state of the art in CFD for industrial aerodynamics can be found in [3], and the description of the numerics can be found in [5]. However, RANS models lack accuracy for flow with strong non equilibrium and high anisotropy. Such flow features are observed for example in the presence of large recirculating flows. Related aerodynamic problems include the study of off-design points for civil aircraft like post-stall or strong buffet, the analysis of flow associated with unconventional shapes dictated by stealth, or flow with fluidic control devices. V. Levasseur () · S. Joly · J.-C. Courty Dassault Aviation — Advanced Aerodynamics and Aeroacoustics, 78 quai Marcel Dassault, 92552 Saint-Cloud CEDEX, France e-mail: [email protected] S. Joly e-mail: [email protected] J.-C. Courty e-mail: [email protected] M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling, ERCOFTAC Series 14, DOI 10.1007/978-90-481-9603-6_47, © Springer Science+Business Media B.V. 2011

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The paper goes through non-exhaustive academic to industrial applications with connection to the turbulence modelling, but without aiming at a fine analysis of the flow phenomena. We intend to show how improvements of the turbulence modelling can increase the efficiency of the CFD-based design. For each presented model, an application will be presented underlining the improvements as well as the remaining progress. The following organization will be followed: new technologies dealing with RANS models will first be addressed. Efforts are focused first on two-equation models, Sect. 2, and then on second-order Reynolds stresses models, Sect. 3. Both internal and external flows will be addressed, and will be seen to profit from these improvements. On the other hand, many challenging applications such as curved air inlet ducts, weapon bays or airbrake design require unsteady computations to estimate unsteady distorsion or structural loads, and finally control their effects using passive or active devices. The latest developments and use of unsteady turbulence modelling will be described in Sect. 4. In particular, DES will be seen to have brought a huge improvement for industrial design [6]. Finally, conclusions and perspectives will be addressed in Sect. 5.

2 Reynolds Averaged Navier–Stokes Modelling Many different RANS models have been developed and as most of CFD tools, many of them are implemented in our in-house Navier–Stokes solver Aether. Nevertheless, the standard for aerodynamic design is a two-layer1 k–ε model. Full aircraft N.-S. simulations are now used at all stages of design with very good validations at cruise conditions. For the latest F7X business jet, CFD has been intensively used and aerodynamic design for cruise condition has been based on computations, so that wind tunnel experiments were limited to intermediate and final check-out, given sufficient validation is obtained at flight Reynolds number. An example of the computed pressure coefficient distribution compared with the experiment is displayed in Fig. 1 for two Mach numbers (M = 0.8 and M = 0.85). The challenges now lay upon the computation of off-design configuration, involving separating or recirculating flows. For such flows, a k–ε model fails recovering the right recirculating zone, the wall pressure or the shear stress. Catris and Aupoix [2] proposed a constraint approach of any k–φ model, with φ a length-scale to be determined, accounting for compressibility effects and adverse pressure gradient. This leads to the formulation of the k–kl model which is depicted by: with φ = kl k 5/2 2νk νt −−→ Dk = Pk − − 2 + div ν + grad(k) (1) Dt φ σk y the two-layer formulation, the ε variable “close” to the wall is computed as ε = k 3/2 / lε and √ −R −R the turbulent viscosity is νt = Cμ klμ , with lε = Cl y(1 − exp 2Cly ), lμ = Cl y(1 − exp 70y ) and

1 In

−3/4

Cl = 0.41Cμ

.

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Fig. 1 Pressure coefficient distribution on a generic F7X Falcon wing. M = 0.8 (left), M = 0.85 (right) Table 1 k–kl model constants k–kl model

Cφ1

Cφ2

σk

σφ1

σφ2

Cφφ

Cφk

Ckk

Inner zone

1

0.58

1.8

1.03

1.03

−1.72

0.96

0

Outer zone

1.1

0.58

0.55

0.686

0.4

−2.384

1.665

0.05

νt −−→ Dφ φ φ 5/2 3/2 = Cφ1 Pk − Cφ2 k − Cφp fp 2 √ + div ν + grad(φ) Dt k σφ ν y νt −−→ νt −−→ −−→ −−→ + Cφφ grad(φ) · grad(φ) + Cφk grad(φ) · grad(k) φ k νt φ −−→ −−→ + Ckk 2 grad(k) · grad(k) k φ νt = fμ Cμ √ k

(2) (3)

Furthermore, two sets of constants have been numerically optimized for wallbounded flows and for free-shear flows, Table 1. Combined with switch functions, this leads to the so-called zonal k–kl model. Eventually, in order to account for high anisotropic flows, EARSM extension of the k–kl model have been implemented, based on Bézard and Daris formulation [1]. As already mentioned we are particularly interested in enlarging the range of application to off-design, especially to landing and take-off configurations. The current high-lift devices include mobile flaps and slats, and, combined with a high angleof-attack, major separation zones appear. For this kind of flow, k–kl models w/o EARSM extension seem fully adapted. An interesting test-case is then the adverse pressure gradient boundary layer as obtained downstream a bump, leading to a steady separation. The Surrey bump configuration, experimented for EC project WALLTURB, has been computed. A view of the mesh is seen in Fig. 2. Figure 3 shows the streamwise velocity and stream-

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Fig. 2 View of the Surrey bump

Fig. 3 Iso-contours of streamwise velocity and streamlines downstream the bump. k–ε (left), k–kl (right)

lines obtained by the k–ε and k–kl models. It clearly highlights that the k–kl model provides a boundary layer more robust to an adverse pressure gradient and shows no separation compared with the k–ε model. This can also be seen looking at the pressure distribution along the bump, Fig. 4. It turns out that the k–kl actually underprescribed the separation, while adding an anisotropic contribution thanks to the EARSM extension enables to recover the right separation, and then the pressure distribution perfectly matches the experiment. The industrial application is then the design of high-lift devices for landing and take-off. The Eurolift configuration has been computed. It is a generic Airbus shape with slats 20° and flaps 22°, i.e. in take-off configuration. Figure 5 shows the wall stresses and the β coefficient, which is correlated to the separation, for the k–ε, k–kl, zonal k–kl and k–kl EARSM models. As expected the k–ε model seems to provide the largest separation on the main body, close to the fuselage. Figure 6 shows the pressure distribution along the DV1 section which is close to the fuselage. Actually, it seems there is no real separation. Besides, the under-pressure zone at the leadingedge of the three bodies is always better recovered with the k–ε compared to the different k–kl models. Therefore, in take-off configuration, the lift prediction is still better predicted by the k–ε model. For landing configuration (with flaps around 40°) this is no more true. The flow behind the flaps is separated and has a major impact on the prediction of the lift coefficient. That’s why, k–kl modelizations lead to better assessment of drag and lift, see Fig. 7.

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Fig. 4 Pressure distribution along the bump

Fig. 5 Wall stresses and iso-contours of β coefficient on a generic Airbus wing in take-off configuration. (a) k–ε, (b) k–kl EARSM, (c) k–kl, (d) zonal k–kl

Eventually, the k–kl model, and the EARSM extension show some very interesting abilities to recover a flow separation but still lack efficiency in free-shear flows. This is still on-going work.

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Fig. 6 Pressure coefficient distribution around DV1 section

Fig. 7 Generic Falcon F7X shape in landing configuration

3 Reynolds Stress Modelling As the EARSM approach also shows some limits (see the following), the solution of the Reynolds stresses equations turns out to be a real improvement in the efficiency of the computation of high anisotropic flows. As the present paper doesn’t pretend to deal with theoretical modelling, a generic shape of equation is simply recalled. Assuming incompressibility, and denoting the filtering operation of a scalar f = f¯ + f , the transport equation for the Reynolds stresses reads: ∂(ρui uj ) ∂t

+

∂(ρui uj u¯ k ) ∂xk

=

∂Dij k + ρ(Pij + φij − εij ) ∂xk

(4)

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Fig. 8 Geometry of the NASA S-duct

with ∂ u¯

∂ u¯ i • Pij = −ρui uk ∂xkj − ρuj uk ∂x : the production term k • εij : the dissipation, which is assumed to be isotropic, following Launder and Shima, i.e. εij = 23 δij ε

• Dij k : the diffusion term, with a turbulent part modelled as CS kε ρul uk ∂u u μ(T ) ∂xi k j

∂ui uj ∂xl

and a

viscous part modelled as • φij = φij 1 + φij 2 : the redistribution term decomposed into 2 terms: φij 1 a nonlinear term associated to the velocity fluctuations and responsible for the redistribution of the energy on the normal Reynolds stresses, modelled as proposed by Rotta (1951) [12]: φij 1 = C1 ρεaij , with aij the anisotropy tensor, and φij 2 a linear term associated to the gradient of the mean velocity, modelled as proposed by Launder et al. [8]: φij 2 = C2 (Pij − 23 δij P ). The RSM approach is expected to bring major improvements for non-equilibrium flows such as the flows in a curved air-intake. Such inlets are needed to shield the Compressor Entry Plane (CEP) of stealth aircraft, in particular strike UAVs. An essential design objective is to reduce the total length of the aircraft, this can be achieved by a reduction of the length of the diffuser. It thus requires highly curvature and can lead to very distorted flows. The design must keep pressure recovery and flow distortion at an acceptable level. An S-duct configuration is computed at Mach number M = 0.6, see Fig. 8. In this kind of flows, standard RANS models yield poor predictions of the recirculation areas inside the inlet, so that only anisotropic models will be presented, i.e. k–kl EARSM and the RSM model. Figure 9 shows a difference of the length of the recirculation given by the each model. Looking at the Mach number in the CEP, Fig. 10, RSM results perfectly match the experiment, especially regarding the recirculation bubble.

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Fig. 9 Iso-contours of Mach number in y = 0 plane. EARSM (up), RSM (down)

Fig. 10 Iso-contours of Mach number in the CEP. EARSM (left), RSM (right)

Nevertheless, for highly-curved U-shape duct, RSM approach also fails giving the correct total pressure recovery. This is due to the intrinsic unsteadiness of the flow. Besides, a key-feature for the engine manufacturers is the dynamic flow distortion in the CEP, which can only be achieved by a fully unsteady approach.

4 LES/DES Modelling Large-Eddy Simulation (LES) has already proven to be a valuable technique for simulating turbulent flows. The key idea is to truncate the solution of the Navier–Stokes equations and compute a filtered velocity field. This leads to a scale separation between the energy containing ones, i.e. the largest eddies, and the mainly dissipative small scales, which are, in practice, the scales that are not represented by the mesh. The Variational Multiscale approach (VMS) proposed by Hughes et al. [7] has been successfully implemented in Dassault-Aviation’s CFD code, in a filtering formulation based on the Smagorinsky model, see [9] for further details. For an aircraft manufacturer such as Dassault Aviation, one particular field which requires unsteady computations is the aerodynamic design of stealth aircraft. One of the most relevant and challenging problem is the design of weapon bays, in which large amplitude aerodynamic loads develop leading to structural vibrations that

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Fig. 11 Pressure spectra at the bottom of the cavity. Left: without control; Middle: with rod; Right: with spoiler

Fig. 12 Iso-surfaces of Q-criterion (Q ≈ 230

2 U∞ ): L2

rod-in-crossflow (left), flat-top spoiler (right)

could endanger the integrity of the aircraft. Since Rossiter’s work (1964), the aeroacoustic coupling is known to induce very strong periodic pressure fluctuations. The sum of the so-called Rossiter modes and of the broadband noise associated to the shear layer create extremely high vibration loads. Accurate aerodynamic predictions will avoid structural over-design and enable the assessment of efficient palliatives. LES perfectly recovers the pressure fluctuations at the bottom of the cavity, Fig. 11, in both controlled and uncontrolled cases. Here, the two studied passive devices are the flat-top spoiler and the rod-in-crossflow. Beside its design interest, LES also enables to have a valuable insight into the physics of control. Figure 12 displays the coherent structures downstream both control devices and highlights very different flow features, and therefore different control mechanisms. See [10] for further details. Nevertheless, the use of LES for flows of industrial interest and for aerodynamic design is still very limited because of the excessive computational cost especially in near-walls areas. That’s why DES approach, [13], is very appealing and has been

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Fig. 13 Compressor Entry Plane of a generic fighter air intake: Up: RMS pressure; Down: Total pressure recovery

widely used and improved throughout the last decade. In its original formulation, DES appears to be an hybrid RANS/LES method which relies on the existence of a gray area between RANS and LES regions whose location depends only on the grid. Based on the Spalart–Allmaras model, which formally reads, in incompressible form: Cb2 −−→ −−→ ν˜ 2 1 D ν˜ −−→ = Cb1 S˜ ν˜ + div (ν + ν˜ ) grad ν˜ + grad ν˜ · grad ν˜ − Cω1 fω 2 (5) Dt σ σ d where dω is the distance to the closest wall. The main idea of the DES approach relies on a new length scale d˜ replacing dω and defined following: d˜ = min(dω , CDES )

(6)

where is the grid spacing. This original formulation suffers from some identified drawbacks. In particular, in case of “ambiguous” density grid, a premature switch to the LES region can occur in a boundary layer resulting in a depletion of the Reynolds stresses. One of the derived method developed to avoid this phenomenon is the Delayed DES (DDES), [14], in which the length scale d˜ is replaced following: d˜ = dω − fa max(0, dω − CDES ) 3 fa = 1 − tanh (8rd ) νt + ν rd = Ui,j Ui,j κ 2 dω 2

(7) (8) (9)

As already mentioned in Sect. 3 statistical approaches fail predicting the flow in a highly-curved duct because of the large unsteadiness of the motion. Figure 13 shows the interesting abilities of the DES to recover the flow in a generic fighter jet inlet. The precise level of turbulence seems still incorrect but the flow topology is very well described.

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Fig. 14 View of the coherent vortices around the Falcon 7X with the airbrakes deployed

Finally, DES has proven very efficient for the design of airbrakes on business jets. It is a typical design problem that extends beyond the prediction of the flow at cruise conditions. Up to now, RANS calculations have been able to assess airbrake efficiency, viz. the relative influence of the airbrake deflection on aircraft lift and drag is correctly predicted. Furthermore, DES enables now to compute the pressure fluctuations and then structural loads on the airbrakes, so that design could be optimized. Eventually, the interaction between the airbrakes wake and the horizontal stabilizer must be carefully analyzed, Fig. 14, in particular at high angle of attack.

5 Conclusions and Perspectives The maturity of turbulence modelling, along with the computational resources, have made CFD an unavoidable tools for aerodynamic design. Full aircraft N.-S. simulations with RANS models are daily used with very good validations and have a major impact on the design of aircraft at cruise conditions. This leads to a time- and cost-gain as well as a better and optimized design. The perpetual development of turbulence modelling, and the account for an increasing complexity in the turbulence features, from two-equation models, to EARSM and second-order Reynolds stress models, enable to compute always more intricate flows with an improved efficiency. The key aerodynamic problems include on the one hand off-design points, like take-off and landing configurations, strong buffet, airbrakes design, reverse, . . . . On the other hand, unconventional shapes, dictated by stealth or green development, also require these improvements. Besides, the achievement of unsteady turbulent simulations, now available as an engineering tool, has increased the precision of the prediction but also gives access to a number of new data, like pressure fluctuations or dynamic distorsion, very valuable in the design process. Nevertheless a number of shortcomings of turbulence models are identified. One can cite, in particular, the difficulty to compute external flows with RSM models because of some numerical relaminarization problems, or else the problem of the gray zone in DES and the generation of turbulence. In the future, efforts will be put

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on these two main directions: RSM modelling, and hybrid RANS/LES formulation, in order to assess very different flows, many of them were not presented here. Finally, it is interesting to note that each design problems can be split into some basic flow features, like compressibility effects, adverse pressure gradient, transition, shock/boundary layer interaction, aeroacoustic coupling, . . . , and turbulence research is concentrated to address these key issues. Eventually, let’s mention that, still in order to improve the CFD-based design, future trends include, beyond the turbulence modelling, the use of innovative high-order numerical methods, differentiation oriented toward optimization, aeroelasticity, or aeroacoustics — opening new areas to simulation, adaptive mesh, etc. Acknowledgements Part of the work was funded under a grant by the European Commision through the DESIDER and WALLTURB projects. The authors want to thank the French Ministry of Defense (DGA) which supported this work through research grants.

References 1. Bézard, H., Daris, T.: Calibrating the length scale equation with an explicit algebraic Reynolds stress constitutive relation. In: Rodi, W., Mulas, M. (eds.) Sixth International Symposium on Engineering Turbulence Modelling and Measurements, Sardinia, May 23–25, 2005, pp. 77– 86. Elsevier, Amsterdam (2005) 2. Catris, S., Aupoix, B.: Towards a calibration of the length-scale equation. Int. J. Heat Fluid Flow 21(5), 606–613 (2000) 3. Chalot, F.: Industrial Aerodynamics. Encyclopediae of Computational Mechanics, vol. 3. Wiley, New York (2004) 4. Chalot, F., Mallet, M.: The stabilized Finite Element Method for compressible Navier–Stokes simulations: review and application to aircraft design. In: Franca, L.P. (ed.) Finite Element Methods: 1970’s and Beyond. CIMNE, Barcelona (2004) 5. Chalot, F., Mallet, M., Ravachol, M.: A comprehensive finite-element Navier–Stokes solver for low and hight speed aircraft design. AIAA Paper 94-0814, Reno, January 1994 6. Chalot, F., Levasseur, V., Mallet, M., Petit, G., Réau, N.: LES and DES simulations for aircraft design. AIAA paper 2007-0723 7. Hughes, T.J.R., Mazzei, L., Jansen, K.E.: Large eddy simulation and the variational multiscale method. Comput. Vis. Sci. 3, 47–59 (2000) 8. Launder, B.E., Reece, G.J., Rodi, W.: Progress in the development of a Reynolds-stress turbulence closure. J. Fluid Mech. 68, 537–566 (1975) 9. Levasseur, V., Sagaut, P., Chalot, F., Davroux, A.: An entropy-variable-based VMS/GLS method for the simulation of compressible flows on unstructured grids. Comput. Methods Appl. Mech. Eng. 195, 1154–1179 (2006) 10. Levasseur, V., Sagaut, P., Mallet, M., Chalot, F.: Unstructured large-eddy simulation of the flow in a three-dimensional open cavity with passive control. J. Fluids Struct. 24(8), 1204– 1215 (2008) 11. Rostand, P.: New generation aerodynamic and multidisciplinary design methods for FALCON business jets. Application to the F7X. In: 23rd AIAA Applied Aerodynamics Conference, Toronto, 2005. AIAA Paper 2005-5083 12. Rotta, J.: Statistische Theorie nichthomogener Turbulenz, 1. Mitteilung. Z. Phys. 129, 547– 572 (1951) 13. Spalart, P., Jou, W.H., Strelets, M., Allmaras, S.R.: Comments on the feasibility of LES for wings and on a hybrid RANS/LES approach. In: 1st AFSOR Int. Conf. on DNS/LES, Ruston, 1997, Proceedings, pp. 137–147 14. Spalart, P.R., Deck, S., Shur, M.L., Squires, K.D., Strelets, M.K., Travin, A.: A new version of detached eddy simulation, resistant to ambiguous grid densities. Theor. Comput. Fluid Dyn. 20, 181–195 (2006). doi:10.1007/s00162-006-0015-0

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