turbulence in fluids, 4th edition

TURBULENCE IN FLUIDS
FLUID MECHANICS AND ITS APPLICATIONS
Volume 84
Series Editor: R. MOREAU
MADYLAM
Ecole Nationale Supérieure d'Hydraulique de Grenoble
Boîte Postale 95
38402 Saint Martin d'Hères Cedex, France
Aims and Scope of the Series
The purpose of this series is to focus on subjects in which fluid mechanics plays a
fundamental role.
As well as the more traditional applications of aeronautics, hydraulics, heat and mass
transfer etc., books will be published dealing with topics which are currently in a state
of rapid development, such as turbulence, suspensions and multiphase fluids, super and
hypersonic flows and numerical modeling techniques.
It is a widely held view that it is the interdisciplinary subjects that will receive intense
scientific attention, bringing them to the forefront of technological advancement. Fluids
have the ability to transport matter and its properties as well as to transmit force,
therefore fluid mechanics is a subject that is particularly open to cross fertilization with
other sciences and disciplines of engineering. The subject of fluid mechanics will be
highly relevant in domains such as chemical, metallurgical, biological and ecological
engineering. This series is particularly open to such new multidisciplinary domains.
The median level of presentation is the first year graduate student. Some texts are
monographs defining the current state of a field; others are accessible to final year
undergraduates; but essentially the emphasis is on readability and clarity.
For a list of related mechanics titles, see final pages.
Turbulence in Fluids
Fourth Revised and Enlarged Edition
By
MARCEL LESIEUR

Fluid Mechanics Professor,
Grenoble Institute of Technology,
Member of the French Academy of Sciences
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4020-6434-0 (HB)
ISBN 978-1-4020-6435-7 (e-book)
Published by Springer,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
www.springer.com
Cover figure: Numerical simulation of positive-Q isosurfaces and passive-scalar cross sections
in coaxial jets of same uniform density (courtesy G. Balarac).
Printed on acid-free paper
All Rights Reserved
© 2008 Springer
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming, recording
or otherwise, without written permission from the Publisher, with the exception
of any material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
Library of Congress Control Number: 2007941158
This fourth edition would have never been possible without the love, enthusiasm
and support of my children Alexandre, Guillaume, Juliette and St´ephanie.
Contents
Preface xvii
1 Introduction to Turbulence in Fluid Mechanics 1
1.1 Isitpossibletodefine turbulence? 1
1.2 Examplesofturbulentflows 4
1.3 Fully-developedturbulence 13
1.4 Fluidturbulence and“chaos” 14
1.5 “Deterministic”andstatistical approaches 16

1.5.1 Mathematical and philosophical considerations . . . . . . . . 17
1.5.2 Numericalsimulations 18
1.5.3 Stochastictools 19
1.6 Whystudyisotropicturbulence? 20
1.7 One-point closure modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.8 Outline ofthefollowingchapters 21
2 Basic Fluid Dynamics 25
2.1 Eulerian notation and Lagrangian derivatives . . . . . . . . . . . . . . . 25
2.2 Thecontinuityequation 27
2.3 Theconservationofmomentum 27
2.3.1 Variabledynamicviscosity 29
2.3.2 Navier–Stokesand Eulerequations 30
2.3.3 Geopotentialform 31
2.3.4 First Bernoulli’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Thethermodynamicequation 33
2.4.1 Second Bernoulli’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.2 Liquid 35
2.4.3 Idealgas 36
2.5 Compressible Navier–Stokes equations in flux form . . . . . . . . . . 38
2.6 The incompressibility assumption . . . . . . . . . . . . . . . . . . . . . . . . . . 38
viii Contents
2.6.1 Liquid 39
2.6.2 Idealgas 39
2.7 Thedynamicsofvorticity 41
2.7.1 Helmholtz–Kelvintheorems 42
2.8 PotentialvorticityandRossbynumber 44
2.8.1 Absolute vortexelements 44
2.8.2 Ertel’stheorem 45
2.8.3 Moleculardiffusion ofpotentialvorticity 47
2.8.4 TheRossbynumber 48

2.8.5 Proudman–Taylortheorem 50
2.8.6 Taylorcolumn 51
2.9 Boussinesqapproximation 52
2.9.1 Liquid 53
2.9.2 Idealgas 53
2.9.3 Vorticitydynamicswithin Boussinesq 54
2.10 Internal-inertialgravitywaves 55
2.10.1 Internalgravitywaves 57
2.10.2 Roleofrotation 60
2.11 Barr´ede Saint-Venantequations 62
2.11.1 Derivationoftheequations 62
2.11.2 Thepotentialvorticity 64
2.11.3 Surfaceinertial-gravitywaves 65
2.12 Gravitywavesinafluidofarbitrarydepth 69
2.12.1 Supersonic shocks and wakes of floating bodies . . . . . . . . 71
3 Transition to Turbulence 73
3.1 Introduction 73
3.2 TheReynoldsnumber 74
3.3 Linear-instability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.3.1 Two-dimensionaltemporalanalysis 77
3.3.2 The two-dimensional Orr–Sommerfeld equation . . . . . . . 79
3.3.3 TheRayleighequation 80
3.3.4 Three-dimensional temporal normal-mode analysis . . . . . 83
3.3.5 Non-normalanalysis 87
3.4 Transitioninfree-shearflows 91
3.4.1 Mixinglayers 91
3.4.2 Roundjets 104
3.4.3 Planejets andwakes 104
3.4.4 Convective and absolute instabilities . . . . . . . . . . . . . . . . . 107
3.5 Wall flows 108

3.5.1 The boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.5.2 Poiseuille flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.6 Thermalconvection 114
Turbulence in Fluids ix
3.6.1 Rayleigh–B´enardconvection 114
3.6.2 Other types ofthermal convection 118
3.7 Transition, coherent structures and Kolmogorov spectra . . . . . . 118
4 Shear Flow Turbulence 121
4.1 Introduction 121
4.1.1 Useofrandomfunctions 121
4.2 Reynoldsequations 121
4.2.1 Themixing-lengththeory 122
4.2.2 Application of mixing length to turbulent-shear flows . . 123
4.3 Characterizationof coherentvortices 135
4.3.1 TheQcriterion 136
4.4 Coherentvorticesinfree-shearlayers 136
4.4.1 Spatialmixinglayer 136
4.4.2 Planespatialwake 138
4.4.3 Roundjets 140
4.4.4 Coaxialjets 145
4.5 Coherentvorticesinwallflows 145
4.5.1 Vortexcontrol 151
4.6 Turbulence,orderandchaos 151
5 Fourier Analysis of Homogeneous Turbulence 155
5.1 Introduction 155
5.2 Fourierrepresentationofaflow 155
5.2.1 Flow“withinabox” 155
5.2.2 IntegralFourierrepresentation 157
5.3 Navier–StokesequationsinFourierspace 158
5.4 BoussinesqequationsinFourierspace 160

5.5 Crayadecomposition 161
5.6 Complexhelical-wavesdecomposition 163
5.7 Utilization of random functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.8 Moments of the velocity field, homogeneity and stationarity . . . 167
5.9 Isotropy 169
5.9.1 Definition 169
5.9.2 Longitudinalvelocitycorrelation 169
5.9.3 Transversevelocitycorrelation 170
5.9.4 Crossvelocitycorrelation 170
5.9.5 Helicity 170
5.9.6 Velocity correlation tensor in physical space . . . . . . . . . . 171
5.9.7 Scalar-velocitycorrelation 173
5.9.8 Velocity spectral tensor of isotropic turbulence . . . . . . . . 174
5.10 Kinetic-energy, helicity, enstrophy and scalar spectra . . . . . . . . . 176
5.10.1 Kineticenergyspectrum 176
x Contents
5.10.2 Helicityspectrum 177
5.10.3 Enstrophy 177
5.10.4 Scalarspectrum 178
5.11 Alternativeexpressionsofthespectraltensor 179
5.12 Axisymmetricturbulence 182
5.13 Rapid-distorsiontheory 184
6 Isotropic Turbulence: Phenomenology and Simulations 187
6.1 Introduction 187
6.2 Triadinteractionsand detailedconservation 187
6.2.1 Quadratic invariants in physical space . . . . . . . . . . . . . . . . 190
6.3 Transferandflux 193
6.4 Kolmogorov’s 1941 theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.4.1 Kolmogorov 1941 in spectral space . . . . . . . . . . . . . . . . . . 197
6.4.2 Kolmogorov wave number . . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.4.3 Integralscale 199
6.4.4 Oboukhov’stheory 200
6.4.5 Kolmogorov 1941 in physical space . . . . . . . . . . . . . . . . . . 201
6.5 Richardsonlaw 202
6.6 Characteristicscalesofturbulence 205
6.6.1 Degreesoffreedomofturbulence 205
6.6.2 Taylormicroscale 207
6.6.3 Self-similarspectra 208
6.7 Skewnessfactorandenstrophydivergence 209
6.7.1 Skewnessfactor 209
6.7.2 Does enstrophy blow up at a finite time in a perfect
fluid? 211
6.7.3 Theviscouscase 215
6.8 Coherentvorticesin3Disotropicturbulence 216
6.9 Pressurespectrum 220
6.9.1 Noiseinturbulence 220
6.9.2 Ultravioletpressure 220
6.10 Phenomenology of passive scalar diffusion . . . . . . . . . . . . . . . . . . 221
6.10.1 Inertial-convectiverange 223
6.10.2 Inertial-conductive range . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
6.10.3 Viscous-convectiverange 227
6.11 Internalintermittency 227
6.11.1 Kolmogorov–Oboukhov–Yaglom theory . . . . . . . . . . . . . . 229
6.11.2 Novikov–Stewart (1964) model . . . . . . . . . . . . . . . . . . . . . . 230
6.11.3 Experimentalandnumericalresults 231
Turbulence in Fluids xi
7 Analytical Theories and Stochastic Models 237
7.1 Introduction 237
7.2 Quasi-Normalapproximation 239
7.2.1 Gaussianrandomfunctions 239

7.2.2 FormalismoftheQ.N.approximation 240
7.2.3 SolutionoftheQ.N.approximation 242
7.3 Eddy-DampedQuasi-Normaltypetheories 243
7.3.1 Eddydamping 243
7.3.2 Markovianization 244
7.4 Stochasticmodels 245
7.5 Closuresphenomenology 250
7.6 Decayingisotropicnon-helicalturbulence 253
7.6.1 Non-localinteractions 255
7.6.2 Energyspectrumand skewness 257
7.6.3 Enstrophy divergence and energy catastrophe . . . . . . . . . 261
7.7 Burgers-M.R.C.M.model 264
7.8 Decayingisotropichelicalturbulence 265
7.9 Decayofkinetic energyandbackscatter 270
7.9.1 Eddyviscosityandspectralbackscatter 270
7.9.2 Decaylaws 273
7.9.3 Infraredpressure 277
7.10 Renormalization-Grouptechniques 277
7.10.1 R.N.G.algebra 278
7.10.2 Two-point closure and R.N.G. techniques . . . . . . . . . . . . . 282
7.11 E.D.Q.N.M.isotropicpassivescalar 284
7.11.1 Asimplified E.D.Q.N.M. model 287
7.11.2 E.D.Q.N.M.scalar-enstrophyblowup 289
7.11.3 Inertial-convective and viscous-convective ranges . . . . . . 291
7.12 Decayoftemperaturefluctuations 292
7.12.1 Phenomenology 293
7.12.2 Experimentaltemperaturedecaydata 299
7.12.3 DiscussionofLESresults 301
7.12.4 Diffusionin stationaryturbulence 302
7.13 Lagrangianparticlepairdispersion 303

7.14 Single-particlediffusion 305
7.14.1 Taylor’sdiffusionlaw 305
7.14.2 E.D.Q.N.M. approach to single-particle diffusion . . . . . . 306
8 Two-Dimensional Turbulence 311
8.1 Introduction 311
8.2 Spectral tools for two-dimensional isotropic turbulence . . . . . . . 314
8.3 Fjortoft’stheorem 316
8.4 Enstrophycascade 317
xii Contents
8.4.1 Forcedcase 317
8.4.2 Decayingcase 318
8.4.3 Enstrophydissipationwavenumber 319
8.4.4 Discussionon the enstrophycascade 320
8.5 Coherentvortices 322
8.6 Inverseenergytransfers 325
8.6.1 Inverseenergycascade 325
8.6.2 Decayingcase 328
8.7 Thetwo-dimensionalE.D.Q.N.M. model 330
8.7.1 Forcedturbulence 334
8.7.2 Freely-decayingturbulence 334
8.8 Diffusionofapassivescalar 339
8.8.1 E.D.Q.N.M. two-dimensionalscalaranalysis 341
8.8.2 Particles-pairdispersionin2D 342
8.9 Pressurespectrumintwodimensions 343
8.9.1 “Ultraviolet”case 343
8.9.2 Infraredcase 344
8.10 Two-dimensional turbulence in a temporal mixing layer . . . . . . 346
9 Beyond Two-Dimensional Turbulence in GFD 349
9.1 Introduction 349
9.2 Geostrophicapproximation 350

9.2.1 Hydrostaticbalance 351
9.2.2 Geostrophicbalance 352
9.2.3 GeneralizedProudman-Taylortheorem 353
9.2.4 Atmosphereversusoceans 354
9.2.5 Thermalwindequation 354
9.3 Quasi geostrophic potential vorticity equation . . . . . . . . . . . . . . . 354
9.4 Baroclinic instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
9.4.1 Eadymodel 357
9.4.2 Displacedfluidparticle 358
9.4.3 Hyperbolic-tangentfront 359
9.4.4 Dynamic evolutionofthebaroclinicjets 360
9.4.5 Baroclinic instability in the ocean . . . . . . . . . . . . . . . . . . . 364
9.5 TheN-layerquasigeostrophicmodel 366
9.5.1 Onelayer 368
9.5.2 Twolayers 369
9.5.3 Spectralverticalexpansion 371
9.6 Ekmanlayer 372
9.6.1 GeostrophicflowaboveanEkmanlayer 373
9.6.2 Theupper Ekmanlayer 376
9.6.3 Oceanic upwellings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
9.7 Tornadoes 378
Turbulence in Fluids xiii
9.7.1 Lilly’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
9.7.2 Ahairpin-vortexbased model 379
9.8 Barotropicandbaroclinicwaves 379
9.8.1 PlanetaryRossbywaves 379
9.8.2 ReflectionofRossbywaves 381
9.8.3 TopographicRossbywaves 382
9.8.4 BaroclinicRossbywaves 383
9.8.5 Otherquasigeostrophicwaves 384

9.9 Quasigeostrophicturbulence 386
9.9.1 Turbulenceandtopography 386
9.9.2 TurbulenceandRossbywaves 387
9.9.3 Charney’stheory 389
10 Statistical Thermodynamics of Turbulence 393
10.1 TruncatedEulerequations 393
10.1.1 Application to three-dimensional turbulence . . . . . . . . . . 393
10.1.2 Application to two-dimensional turbulence . . . . . . . . . . . . 397
10.2 Two-dimensional turbulence over topography . . . . . . . . . . . . . . . 399
10.3 Inviscid statistical mechanics of two-dimensional point vortices 402
11 Statistical Predictability Theory 403
11.1 Introduction 403
11.2 E.D.Q.N.M. predictability equations . . . . . . . . . . . . . . . . . . . . . . . 407
11.3 Predictability of three-dimensional turbulence . . . . . . . . . . . . . . . 408
11.4 Predictability of two-dimensional turbulence . . . . . . . . . . . . . . . . 412
11.4.1 Predictability time in the atmosphere . . . . . . . . . . . . . . . . 413
11.4.2 Predictability time in the ocean . . . . . . . . . . . . . . . . . . . . . 414
11.4.3 Unpredictability and cohence . . . . . . . . . . . . . . . . . . . . . . . 414
11.5 Two-dimensional mixing-layer unpredictability . . . . . . . . . . . . . . 415
11.5.1 Two-dimensional unpredictability and three-
dimensionalgrowth 416
12 Large-Eddy Simulations 419
12.1 DNSofturbulence 419
12.2 LESformalisminphysicalspace 420
12.2.1 Largeandsubgridscales 420
12.2.2 LESofatransportedscalar 422
12.2.3 LES and the predictability problem . . . . . . . . . . . . . . . . . . 423
12.2.4 Eddy-viscosityassumption 424
12.2.5 Eddy-diffusivityassumption 425
12.2.6 LESofBoussinesqequations 425

12.2.7 Compressibleturbulence 426
12.2.8 Smagorinsky model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
xiv Contents
12.3 LESinspectralspace 428
12.3.1 Sharpfilter inFourierspace 428
12.3.2 Spectraleddy viscosityanddiffusivity 428
12.3.3 LESofisotropicturbulence 431
12.3.4 The anomalous spectral eddy diffusivity . . . . . . . . . . . . . . 435
12.3.5 Alternativeapproaches 437
12.3.6 SpectralLESfor inhomogeneousflows 438
12.4 Newphysical-spacemodels 440
12.4.1 Structure-functionmodel 440
12.4.2 Selectivestructure-functionmodel 443
12.4.3 Filtered structure-functionmodel 444
12.4.4 Scale-similarityandmixedmodels 446
12.4.5 Dynamicmodel 448
12.4.6 Otherapproaches 451
12.5 LESoftwo-dimensionalturbulence 452
13 Towards “Real World Turbulence” 455
13.1 Introduction 455
13.2 Stably-stratifiedturbulence 456
13.2.1 Theso-called“collapse”problem 456
13.2.2 Numericalapproachtothe collapse 460
13.2.3 Otherconfigurations 466
13.3 Rotatingturbulence 467
13.3.1 FromlowtohighRossbynumber 467
13.3.2 Linear instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
13.3.3 Mixinglayersandwakes 468
13.3.4 Channels 475
13.3.5 Sometheoreticalconsiderations 479

13.3.6 Initiallythree-dimensionalturbulence 482
13.4 Separated flows 483
13.4.1 Meanreattachmentlength 484
13.4.2 Coherentvortices 485
13.4.3 Instantaneousreattachmentlength 487
13.4.4 Rotatingbackstep 488
13.5 Compressibleflows 489
13.5.1 Compressiblemixinglayer 489
13.5.2 Barocliniceffectsinfree-shearflows 494
13.5.3 Compressiblewake 496
13.5.4 Boundary layer upon a heated plate . . . . . . . . . . . . . . . . . 496
13.5.5 Compressionramp 497
13.5.6 Compressible boundary layer . . . . . . . . . . . . . . . . . . . . . . . 499
13.6 Book’sconclusions 502
Turbulence in Fluids xv
References 509
Index 545
Preface
Turbulence is a dangerous topic which is often at the origin of serious fights
in the scientific meetings devoted to it since it represents extremely different
points of view, all of which have in common their complexity, as well as an
inability to solve the problem. It is even difficult to agree on what exactly is
the problem to be solved.
Extremely schematically, two opposing points of view had been advoc-
ated during these last thirty years: the first one was “statistical”, and tried
to model the evolution of averaged quantities of the flow. This community,
which had followed the glorious trail of Taylor and Kolmogorov, believed in
the phenomenology of cascades, and strongly disputed the possibility of any
coherence or order associated to turbulence.
On the other bank of the river standed the “coherence among chaos”

community, which considered turbulence from a purely deterministic point of
view, by studying either the behaviour of dynamical systems, or the stability
of flows in various situations. To this community were also associated the
experimentalists and computer simulators who sought to identify coherent
vortices in flows.
Situation is more complex now, and the existence of these two camps is
less clear. In fact a third point of view pushed by people from the physics
community has emerged, with the concepts of renormalization group theory,
multifractality, mixing, and Lagrangian approaches.
My personal experience in turbulence was acquired in the first group since
I spent several years studying the stochastic models (or two-point closures)
applied to various situations such as helical turbulence, turbulent diffusion, or
two-dimensional turbulence. These techniques were certainly not the ultimate
solution to the problem, but they allowed me to get acquainted with various
disciplines such as aeronautics, astrophysics, hydraulics, meteorology, ocean-
ography, which were all, for different reasons, interested in turbulence. It is
xviii Preface
certainly true that I discovered the fascination of fluid dynamics through the
somewhat abstract studies of turbulence.
This monograph is in fact an attempt to reconcile the statistical point of
view and the basic concepts of fluid mechanics which determine the evolution
of flows arising in the various fields envisaged above. These basic principles,
accompanied by the instability-theory predictions and the results of numer-
ical simulations, give valuable information on the behaviour of turbulence and
of the structures which compose it. But a statistical analysis of these struc-
tures can, at the same time, supply information about strong nonlinear energy
transfers within the flow.
I have tried to present here a synthesis between two graduate courses
given in Grenoble during these last years, namely a “Turbulence” course and
a “Geophysical Fluid Dynamics” course. I would like to thank my colleagues

of the Ecole Nationale d’Hydraulique et M´ecanique and Universit´eJoseph
Fourier, who offered me the opportunity of giving these two courses. The stu-
dents who attended these classes were, through their questions and remarks,
of great help. I took advantage of a sabbatical year spent at the Aerospace-
Engineering Department of University of Southern California to write the first
draft of this monograph: this was rendered possible by the generous hospitality
of John Laufer and his collaborators. The second edition benefitted also from
a graduate course taught at Stanford University during a visit to the Center
for Turbulence Research. The support and extra time offered through my ap-
pointment to the “Institut Universitaire de France” made the third edition
possible. The fourth edition was written thanks to a CNRS delegation and a
sabbatical semester offered by Grenoble Institute of Technology (INPG).
The organization into 13 chapters of the third edition has been kept:
1. Introduction to turbulence in fluid mechanics; 2. Basic fluid dynamics;
3. Transition to turbulence; 4. Shear-flow turbulence; 5. Fourier analysis for
homogeneous turbulence; 6. Isotropic turbulence: phenomenology and sim-
ulations; 7. Analytical theories and stochastic models; 8. Two-dimensional
turbulence; 9. Beyond two-dimensional turbulence in geophysical fluid dy-
namics; 10. Statistical thermodynamics of turbulence; 11. Statistical predict-
ability theory; 12. Large-eddy simulations; 13. Towards real-world turbulence.
In Chapter 1, the book introduces clear definitions of turbulence in fluids
and of coherent vortices. It provides several industrial and environmental ex-
amples, with numerous illustrations. Chapter 2 develops at lenght equations
of fluid dynamics (velocity and energy) for flows of arbitrary density (incom-
pressible and compressible), including Boussinesq equations (with a study of
internal-gravity waves). It reviews the main theorems of vorticity dynamics
and scalar transport for non-rotating or rotating flows. It looks also in de-
tails at Barr´e de Saint-Venant equations for shallow layers. Chapter 3 studies
linear-instability theory of parallel shear flows (free and wall-bounded) in two
Turbulence in Fluids xix

and three dimensions (with effects of rotation), as well as thermal convection.
It provides an experimental and numerical review of transition in shear flows.
Chapter 4 is devoted to free or wall-bounded turbulent shear flows. They are
studied both statistically (we derive for instance the logarithmic boundary
layer profile) and deterministically, with emphasis put on coherent vortices and
coherent structures. Recent results illuminating the structure of round jets and
turbulent boundary layer without pressure gradient are given. Chapter 5 gives
mathematical details on Fourier analysis of turbulence, with informations on
rapid-distorsion theory. Chapter 6 is devoted to three-dimensional isotropic
turbulence, looked at phenomenologically and from a coherent-vortex point of
view. Passive-scalar diffusion, important for combustion studies, is included
in the chapter. It contains also new results concerning noise in turbulence,
associated with pressure spectrum. Chapter 7 contains the two-point clos-
ure approaches of three-dimensional isotropic turbulence, with applications to
passive scalars. The closure derivation of an helicity cascade superposed to the
Kolmogorov kinetic-energy cascade, and verified by numerical large-eddy sim-
ulations, is certainly an important result of the book. Helicity is important in
the generation of atmospheric tornadoes and of Earth magnetic field (dynamo
effect). The chapter underlines also the important role of spectral backscatter,
which is confirmed by numerical simulations. Chapter 8 is devoted to strictly
two-dimensional turbulence from a phenomenological, closure and numerical
viewpoint. It gives a clear theoretical exposition of the double enstrophy and
inverse-energy cascades, with experimental validations. It gives new numerical
results on energy and pressure spectra. Chapter 9 deals essentially with quasi
two-dimensional turbulence from an external-geophysical point of view. It con-
tains a very detailed presentation of difficult questions: quasi-geostrophic the-
ory, baroclinic instability, atmospheric storms, N-layer models, Rossby waves
(including topographic ones), Ekman layers. It discusses also of tornado gen-
eration, and finishes with Charney’s theory of quasi-geostrophic turbulence.
Chapter 10 presents the statistical thermodynanics of truncated Euler equa-

tions. In fact it turns out that such an approach is far from the reality of
turbulence. Chapter 11, on statistical unpredictability in three and two di-
mensions, is mostly unchanged with respect to former editions. The role of
spectral backscatter in the inverse error cascade is very important. Results
of this chapter show that a deterministic numerical simulation of a turbulent
flow is subject to important errors beyond the predictability time. Chapter 12
is an up to date review of large-eddy simulation techniques, which are be-
coming extremely powerful. It contains a detailed presentation of “classical
models” such as Smagorinsky’s or Kraichan’s, as well as new “dynamic” or
“selective” models allowing the eddy coefficients to adjust automatically to
the local turbulence. Finally, Chapter 13 presents turbulence in more prac-
tical situations. We consider successively the effects of stratification, rotation
xx Preface
(universality of free- and wall-bounded shear flows in anticyclonic regions is
astonishing), separation and compressibility. Here again, our concern is both
statistical and structural.
This book is of great actuality on a topic of upmost importance for engin-
eering and environmental applications, and proposes a very detailed present-
ation of the field. The fourth edition incorporates new results coming from
research works which have been done since 1997, and revisits the older points
of view in the light of these results. Many come from direct and large-eddy
simulations methods, which have provided significant advances in most chal-
lenging problems of turbulence (isotropy, free-shear flows, boundary layers,
compressibility, rotation). The book proposes many aerodynamic, thermal-
hydraulics and environmental applications.
It is obvious that problems are evolving, and so do the applications: de-
veloping faster planes may be less crucial (except for defense problems) than
clean, economic and silent engines. Energy issues such as fusion will push the
numerical modellers towards much more complicated problems involving very
hot plasma. Alarming questions posed by climate evolution about a global

warming will oblige to develop full three-dimensional atmospheric and oceanic
codes based at least on Boussinesq equations. This will be eased by the con-
tinuous spectacular development of computers.
Particular thanks go to the staff and graduate students of the team MOST
(“Mod´elisation et Simulation de la Turbulence”) at the Laboratory for Geo-
physical and Industrial Flows (LEGI, sponsored by CNRS, INPG and UJF),
for their important contribution (visual in particular) to the book. Pierre
Comte and Olivier M´etais provided their great expertise in the domains of
transition, coherent vortices, compressible, stratified or rotating turbulence,
and numerical methods. I am also indebted to all the sponsoring agencies and
companies who showed a continuous interest during all these years in the de-
velopment of fundamental and numerical research on Turbulence in Grenoble.
Rosanne Alessandrini, Patrick B´egou, Eric Lamballais and Akila Rachedi
were very helpful for handling figures, and Yves Gagne, Jack Herring, Sherwin
Maslowe and Jim Riley for editing part of the material (first three editions). I
am greatly indebted to Frances M´etais who corrected the English style of the
first edition. I also hope that this monograph will help the diffusion of some
French contributions to turbulence research.
I am grateful to numerous friends around the world who encouraged me
to undertake this work.
The first three editions were written using the TEX system. This would
not have been possible without the help of Claude Goutorbe and Evelyne
Tournier, of Grenoble Applied Mathematics Institute.
Turbulence in Fluids xxi
Finally I thank Ren´e Moreau and Springer for offering me the possibility
of presenting these ideas.
Grenoble, May 2007
Plates
Plate 1: Two-dimensional numerical simulation of Brown and Roshko’s experiment
shown in Fig. 1.4. Top: passive dye contours. Bottom: vorticity contours (courtesy

X. Normand).
Plate 2: Vorticity contours in the two-dimensional numerical simulations of the mix-
ing layer reported in Lesieur et al. [420] and Comte [134].
xxiv Plates
Plate 3: Evolution with time of the vorticity field in a two-dimensional direct-
numerical simulation of the flow above a backward-facing step (courtesy A. Silveira,
C.E.N.G. and I.M.G.).
Turbulence in Fluids xxv
Plate 4: Visualization of a horizontal section of turbulence in a tank rotating fastly
about a vertical axis: the eddies shown are quasi-two-dimensional, due to the effect
of rotation (courtesy E.J. Hopfinger).
Plate 5: Satellite picture of the temperature field on the surface of the Atlantic ocean
close to the Gulf Stream (courtesy NASA and EDP-Springer [424]).
xxvi Plates
Plate 6: Circulation on Jupiter (courtesy Jet Propulsion Laboratory, Pasadena, and
EDP-Springer [424]).
Turbulence in Fluids xxvii
Plate 7: Direct-numerical simulation of a two-dimensional temporal mixing layer:
left, vorticity field; right, passive scalar field; one can see the formation of the primary
vortices, and the subsequent pairings; (from Comte [134]).