Table Of ContentTURBULENCE AND TRANSITION MODELLING
ERCOFTAC SERIES
VOLUME 2
Series Editors
P. Hutchinson, Chairman ERCOFTAC,
Cranfield University, Bedford. UK
W. Rodi, Chairman ERCOFTAC Scientific Programme Committee,
Universitdt Karlsruhe , Karlsruhe. Germany
Aims and Scope of the Series
ERCOFfAC (European Research Community on Flow, Turbulence and Combus-
tion) was founded as an international association with scientific objectives in 1988.
ERCOFfAC 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, ERCOFfAC organizes several meetings in
the form of workshops, conferences and summerschools, where ERCOFfAC
members and other researchers meet and exchange information.
The ERCOFfAC Series will publish the proceedings of ERCOFfAC 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,
multi-phase flows, non-Newtonian flows, astrophysical flows, laminar, turbulent
and transitional flows.
The titles published in this series are listed at the end of this volume.
Turbulence and
Transition Modelling
Lecture notes from the
ERCOFTACI/UTAM Summerschool
held in Stockholm, 12-20 June, 1995
Edited by
M.HALLBACK
Department ofMechanics.
Royal Institute ofTechnology,
Stockholm, Sweden
D. S. HENNINGSON
Department ofMechanics,
Royal Institute ofTechnology,
Stockholm, Sweden
and
Aeronautical Research Institute ofSweden,
Bromma, Sweden
A. V. JOHANSSON
Department ofMechanics ,
Royal Institute ofTechnology,
Stockholm, Sweden
and
P. H. ALFREDSSON
Department ofMechanics,
Royal Institute ofTechnology,
Stockholm, Sweden
Springer-Science-Business Media, B.Y.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4707-6 ISBN 978-94-015-8666-5 (eBook)
DOI 10.1007/978-94-015-8666-5
Printed on acid-free paper
All Rights Reserved
© 1996 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1996.
Softcover reprint of the hardcover 1st edition 1996
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
CONTENTS
Preface xi
1 Introduction 1
1\1. Hallb ack , D.S. Henningson, A.V. Johansson and P.H. Alfredsson
1.1 Early developments . . . . . . . . 1
1.2 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Kinetic energy equation . . . . . . . . . . . . . . . . . 6
1.2.2 The equation for a disturbance on a laminar base flow . 6
1.2.3 The Reynolds decomposition . . . . . 7
1.2.4 Filtering the Navier-Stokes equations 9
References . . . . . . . . . . . . . . . . . . . . . . . 10
2 Stability and transition 13
D.S. Henningson and P.H. Alfredsson
2.0 Nomenclature . . . . .. . . . . 13
2.1 Introduction . ... ... . . . . 16
2.1.1 Definitions of stability . 17
2.1.2 The need for linear growth mechanisms 18
2.1.3 Linear stability equations. 19
2.2 Inviscid linear stability theory . . . . . . . . 21
2.2 .1 The Rayleigh equation . . . . . . . 21
2.2.2 Dispersive effects and wave-packets 23
2.2.3 The lift-up effect and the algebraic instability 25
2.3 Viscous instability analysis 26
2.3.1 The Orr-Sommerfeld and Squire equations . 27
2.3.2 Numerical solutions to the stability problem 29
2.3.3 Squires transformation . . . . . . . . . . . . 30
2.3.4 Eigenfunct ion expansion and transient growt h 31
2.3.5 Optimal disturbances . . . . . . . 33
2.4 Stability of complex boundary layer flows 35
2.4.1 Two-dimensional boundary layers 35
2.4 .2 3D boundary layers . . . . . . . . 40
2.4.3 Effects of cur vat ure and rotation . 44
2.5 Transition scenarios 50
2.5.1 Receptivity .. . . . . . . . . .. . 51
2.5.2 Transition emanating from exponential inst abilities 53
2.5 .3 Byp ass t ransit ion 60
2.6 Transitionmodeling 71
References . . . . . . . . . . . . 74
v
vi
3 The basics of turbulence modelling 81
M. Hallback , A.V. Johansson and A.D. Burden
3.1 Introduction . .. . . . .. 81
3.2 Nomenclature . . . .. . .. . . . ... . 82
3.3 The physics of turbulence . . . . . . . . 84
3.3.1 The energy cascade in isotropic turbulence 87
3.3.2 Basic properties of near-wall turbulence . 88
3.4 Single-point transport equations 92
3.4.1 The dissipation rate equation . . . . . . . . 95
3.5 The hierarchy and history of single-point closures . 96
3.5.1 The eddy viscosity hypothesis 96
3.5.2 One-equation models . . . . . . . 98
3.5.3 Two-equation models . . . . . . . 98
3.5.4 Reynolds stress transport models 99
3.5.5 Algebraic Reynolds stress models 99
3.6 What should a closure fulfill? . . . 100
3.6.1 Coordinate invariance . . . 100
3.6.2 Material frame indifference 100
3.6.3 Invariant modelling . . 101
3.6.4 Realizability ....... . 103
3.6.5 Near-wall asymptotics . . . 105
3.7 The K:e and other two-equation models 107
3.7.1 The K-e: model 108
3.7.2 The K-w model . . . . . . . . . 109
3.7.3 The K :r model . . . . . . . . . 110
3.7.4 A comparison between the different models . 110
3.8 Differential Reynolds stress models . . 110
3.8.1 The dissipation rate tensor . . . . . . . . . . 112
3.8.2 The pressure-strain rate term 114
3.8.3 Rotating channel flow - an illustrative example 123
3.9 Modelling the s-equation . . . . . . . . . . 124
3.9.1 Influence of mean flow strain rate . . 126
3.10 Models for turbulent transport . . . . . . . . 127
3.10.1 Passive scalars and gradient diffusion 128
3.10.2 Mean kinetic energy of the turbulence, K . 131
3.10.3 Auxiliary quantities such as e . . . . 132
3.10.4 Turbulent transport in DRST models . . 133
3.11 Algebraic Reynolds stress models . . . . . . . . . 137
3.11.1 Explicit algebraic Reynolds stress models 139
3.12 Near-wall treatment ' " . . . . . . . . . . . . . 143
3.12.1 Boundary conditions in the log-layer. . . 143
3.12.2 Low Reynolds number model formulations 144
3.13 Model development and validation tools . . . . . . 147
3.13.1 DNS as a tool for model development and validation 147
3.13.2 Rapid Distortion Theory 148
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
vii
4 Constitutive relations and realizability of single-point
turbulence closures 155
T.-H. Shih
4.1 Turbulence constitutive relationships . . . . . . . . . . . . 155
4.1.1 Method of constructing consti t ut ive relationships 156
4.1.2 Constitutive relationships for U iUj and U i8 . . . . 157
4.1.3 Constitutive relationships for second ord er closures 160
4.2 Realizability in turbulence modeling . . . . . . . . . . . . . 170
4.2.1 Realizability . .. . .. . . . . .. . . . . . .. . . . 171
4.2 .2 Application of realizability in Reynolds st ress algebra ic equa-
tion models . . . . . . . . . . . . . . . . . . . . . . . 173
4.2 .3 Application of realizability in second-order closures 176
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5 Advanced turbulence models for industrial application 193
B.E. Launder
5.1 Introduction.... .... ... .. 193
5.2 Non-Linear Eddy Viscosity Models 195
5.2.1 Introductory Remarks. . . 195
5.2.2 The Connection between Stress and Mean Velocity 196
5.2.3 Transport Equations for the 2-Equation Version . . 198
5.2.4 Some Applications to Inhomogeneous Flows with Complex
Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 198
5.2.5 Further Developments with the Non-Linear Eddy Viscosity
Model. . . . . . . . . . . . . . . . . . . . 200
5.3 New Developments in Stress-Transport Closures. . . . . . . . . 210
5.3.1 Introductory Remarks. . . . . . . . . . . . . . . . . . . 210
5.3.2 The Stress Transport Equation and the Initial Steps to
Closure . . . . . . . . . . . . . 210
5.3.3 Extension to Near-Wall Flows 214
5.3.4 Current Research 218
5.4 Concluding Remarks . . . . . . . . . . 225
5.A Appendix . . . . . . . . . . . . . . . . 226
5.A.1 Coefficients for the 3-equation N-LEVM 226
5.A.2 Closure Form Adopted by Suga (1995) for ih Transport
Equation 226
References . . . . . . . . . . . . . . . . . . . . 228
6 One-point closures applied to transition 233
A.:\I. Savill
6.1 Introduction and historical background . . . . . . . . . . 233
6.2 Basic concepts : eddy viscosity and other simple approaches 235
6.2.1 Low-Reynolds number transport models 236
6.2.2 Alternative correlation and intermittency models 237
6.2.3 Initial results for simple test cases . . . . . . . . 239
6.3 Eddy-viscosity model refinement for predicting transition . 241
6.3.1 Integral methods . . . . . . . . . . . . . . . . . . 242
viii
6.3.2 One-equation models . 242
6.3.3 Two-equation models . 243
6.3.4 Other possible approaches 246
6.4 Results from by-pass transition simulations 246
6.4.1 Analysis of the transition simulations 246
6.4.2 Results of some simulation 'experiments' 251
6.4.3 Implications for models . . . . . . . . . . 252
6.5 Using and refining RST models to predict transition 253
6.5.1 Low Reynolds number approaches . . . . . . 253
6.5.2 The SLY low-Re model for predicting transition 253
6.5.3 Comparison of transition test case predictions . 257
6.5.4 Non-local pressure-strain modelling refinements 260
6.6 Towards practical computations for engineering flows 262
6.7 Conclusions and summary of best choice current models 263
References . . . . . . . . . . . . . . . . . . . . . . . . 265
7 Large-Eddy Simulations: theory and applications 269
U. Piomelli and J.R. Chasnov
7.1 Introduction . 269
7.2 Governing equations and filters 271
7.2.1 The filtering operation 271
7.2.2 Filtered Navier-Stokes equations 272
7.2.3 Energyequations . 274
7.3 Principles of small scale modeling . 276
7.3.1 Universality of small scales . 276
7.3.2 Dissipation set by the large scales 278
7.3.3 Basic requirements of subgrid models 279
7.3.4 Eddy viscosity and eddy noise 280
7.4 Subgrid-scale modeling . 284
7.4.1 Eddy viscosity models . 284
7.4.2 Modeling in Fourier space 284
7.4.3 Modeling in real space 289
7.5 Numerical methods . . . . . . 292
7.5.1 Spatial discretization 292
7.5.2 Time advancement . 293
7.5.3 Boundary conditions 294
7.5.4 Initial conditions . . . 295
7.5.5 Implementation on parallel computers . 296
7.6 Applications . 297
7.6.1 Homogeneous turbulence . 297
7.6.2 Wall-bounded flows . 309
7.6.3 Transitional and relaminarizing flows 318
ix
7.6.4 Separated or highly three-dimensional flows 323
7.7 Conclusions 328
References . . . . 331
8 Transition modeling based on the PSE 337
F.P. Bertolotti
8.1 Introduction . 337
8.2 Preliminary . 338
8.3 The PSE formulation: Basics 340
8.4 The linearized PSE . . . . . . 343
8.4.1 The auxiliary condition 345
8.4.2 Discretization of the PSE equations 345
8.4.3 The initial condition: Local solutions 346
8.4.4 Measures of growth . . . . . . . . . . 348
8.4.5 Effect of the auxiliary condition on PSE results 349
8.5 The nonlinear PSE . . . . . . . . 352
8.5.1 The boundary conditions . 355
8.5.2 Adding new modes . . . . . . . . . . . . . . 356
8.5.3 Forced transition and wave-triad resonances 357
8.5.4 Forced transition in the Blasius boundary layer 359
8.6 Receptivity . . . . . . . 361
8.6.1 Steady modes from surface undulation 361
8.6.2 Receptivity to sound . 362
8.6.3 Receptivity to vortical disturbances 364
References . . . . . . . . . . . . . . . . . . . . . . 366
More than one hundred participants attended the summerschool at KTH in June 1995. In the front (from left)
lecturers Burden, Alfredsson, Shih, Chasnov, Bertolot ti , Launder , Piomelli, Johansson and Hallback.
