a NATO Advanced Study Institute

LES HOUCHES

SESSION LXXIV

3 1 July - 1 September 2000

New trends in turbulence

Turbulence : nouveaux aspects

Edited by

M. LESIEUR, A. YAGLOM and F. DAVID

Springer

Les Ulis, Paris, Cambridge Berlin. Heidelberg, New York, Barcelona, Hong Kong, London

Milan, Paris, Tokyo

Published in cooperation with the NATO Scientific Affair Division

Preface

The phenomenon of turbulence in fluid mechanics has been known for many centuries. Indeed, it was for instance discussed by the Latin poet Lucretius who described in "de natura rerum" how a small perturbation ("clinamen") could be at the origin of the development of a turbulent order in an initially laminar river made of randomly agitated atoms. More recently, Leonardo da Vinci drew vortices. Analogous vortices were sketched by the Japonese school of artists called Utagawa in the 19th century, which certainly influenced van Gogh in "The Starry Night". However, and notwithstanding decisive contributions made by Benard, Reynolds, Prandtl, von Karman, Richardson and Kolmogorov, the problem is still wide open: there is no exact derivation of the famous so-called Kolmogorov k-5'3 cascade towards small scales, nor of the value of the transitional Reynolds number for turbulence in a pipe. Besides these fundamental aspects, turbulence is associated with essential practical questions in hydraulics, aerodynamics (drag reduction for cars, trains and planes), combustion (improvement of engine efficiency and pollution reduction), acoustics (the reduction of turbulence-induced noise is an essential issue for plane reactors), environmental and climate studies (remember the huge damage caused by severe storms in Europe at the end of 1999), and astrophysics (Jupiter's Great Red Spot and solar granulation are manifestations of turbulence). Therefore, there is an urgent need to develop models that allow us to predict and control turbulence effects.

During the last 10 years, spectacular advances have been made towards a better physical understanding of turbulence, concerning in particular coherent- vortex self-organization resulting from strong nonlinear interactions. This is due both to huge progress in numerical simulations (with the development of large- eddy simulations in particular) and the appearance of new experimental techniques such as digitized particle-image velocimetry methods. New theoretical tools for the study of fluid turbulence in three and two dimensions have appeared, sometimes borrowed fiom statistical thermodynamics or condensed-matter physics: multifractal analysis, Lagrangian dynamics and mixing, maximum-entropy states, wavelet techniques, nonlinear amplitude equations ... These advances motivated the organization of the Les Houches 2000 School on "New Trends in Turbulence", and this book. Theoretical aspects have been blended with more practical viewpoints, where the influence on turbulence of boundaries, compressibility, curvature and rotation, helicity, and magnetic fields is looked at. Various applications of turbulence modelling and control to certain industrial or environmental issues are also considered.

The first introductory course by Akiva Yaglom deals with a "century of turbulence theory". Here the problem of turbulence is reviewed from both the linear and nonlinear points of view. Then the two main achievements of 20th

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century turbulence theory are considered, namely the logarithmic velocity profiles (found by T. von Karman and L. Prandtl) for the intermediate layers of flows in circular pipes, plane channels and boundary layers without pressure gradients, and the Kolmogorov-Obukhov theory of the universal statistical regime of small-scale turbulence in any flow with a large enough Reynolds number (revised in 1962 by the authors to consider intermittency, which led to predictions of the scaling for structure functions of various orders). Since Yaglom was a student of Kolmogorov and participated in the developments of the second of the above-mentioned subjects, his presentation is based on first- hand knowledge of the history of these classic discoveries and of their present status.

The basis of turbulence theory is complemented by Katepalli Sreenivasan and S. Kurien (Course 2), who provide also a very complete account of fully developed turbulence experimental data (concerning structure functions and spectra of velocity and passive scalars) both in the laboratory and the atmosphere at very high Reynolds number (up to Rh = 20 000). Sreenivasan insists on departures from Kolmogorov scaling in the light of anisotropy effects, which are studied with the aid of a SO(3) decomposition. This allows us to extract an isotropic component from inhomogeneous turbulence.

Olivier Metais presents in detail "Large-eddy simulations of turbulence" (LES) in Course 3. These new techniques are powerful1 tools allowing us to simulate deterministically the coherent vortices formation and evolution. He reviews also methods for vortex identification based upon vorticity, pressure and the second invariant of the velocity-gradient tensor. He applies direct-numerical simulations and LES to free-shear flows and boundary layers. Metais is also involved in particular aspects related to thermal stratification and rotation, with application to deep-water formation in the ocean and atmospheric severe storms. He concludes with applications of LES to compressible turbulence in gases, with reentry of hypersonic bodies into the atmosphere, and cooling of rocket engines.

In Course 4, Michael Leschziner describes methods used for predicting statistical industrial flows, where the geometry is right now too complex to allow the use of LES. He shows how these methods can be, in simple geometric cases, assessed by comparison with DNS and LES methods. He presents very encouraging results obtained with nonlinear eddy-viscosity models and second- moment closures. Interesting applications to turbomachinery flows are provided. Still on the industrial-application side, Reda Mankbadi (NASA) provides in Course 5 a very informative review of computational aeroacoustics, with many applications to aircraft noise (in particular jet noise in plane engines). He shows linear, nonlinear and full LES methods.

The remaining chapters are more fundamental. Keith Moffatt (Course 6) presents the basis of topological fluid dynamics and stresses the importance of helicity in neutral and magnetohydrodynamic (MHD) flows. He shows in the latter case how helicity at small scales can generate a large-scale magnetic field (a-effect), a mechanism which might explain the magnetic-field generation in

planets and stars. Moffatt discusses also the possibility of a finite-time singularity within Euler and even Navier-Stokes equations. During his oral presentation, he made an analogy with Euler's disk. His practical demonstration of the latter was one of the School's highlights, especially when people realized that the experiment could be done as well with the restaurant plates, which was less appreciated by the cook. Uriel Frisch and J. Bec (Course 7) speak also of finite-time singularities, but mostly on the basis of Burgers equations in one or several dimensions, with the formation of multiple shocks. They describe methods that allow us to solve these equations analytically and numerically, and the kinetic-energy decay problem. He shows how Burgers equations (in three dimensions) can, in cosmology, apply approximately to the formation of large scales in the universe.

Course 8 is a very complete account of two-dimensional turbulence provided by Joel Sommeria (Grenoble). He first presents numerous examples of 2D turbulence in the laboratory (rotating or MHD flows, plasmas), in the ocean and in planetary atmospheres (Jupiter), insisting in particular on the absence of kinetic-energy dissipation in these flows. He reviews the double cascade of enstrophy (direct) and energy (inverse) proposed by Kraichnan and Batchelor. He presents also very clearly the point-vortex statistical-thermodynamics analysis of Onsager, and the generalization of this model to fmite blobs of vorticity (maximum-entropy principle). Sommeria shows applications of this model to mixing layers and to 2D oceanic flows with differential rotation. Course 9 (Marie Farge and K. Schneider) is a useful presentation of wavelet techniques, a further interesting application of which (not detailed in the book) concerns data compression. In Course 10 (Gregory Falkovich, K. Gawedzki and M. Vergassola), the Lagrangian mixing of passive scalars and the relative dispersion of several particles is discussed. For two particles in particular, Richardson's pioneering law (predicting a dispersion rate proportional to the separation raised to the 413 power) is revisited in the light of Kraichnan's renormalized analysis.

Let us finish this preface with some information regarding the Les Houches 2000 School "New Trends in Turbulence". A computing centre was set up thanks to Patrick Begou during the duration of the programme. The machines (Compaq ds2012proc, Sun ultra80/4proc, IBM 44p/2proc, HP j560012proc, HP kayak linux.2proc) were kindly lent by the respective companies. With this important computing power, we could organize for students five working groups under the direction of a professor or senior researcher, and using databases generated using several computational programs. These groups were: (1) Jets and wakes (Olivier Mdtais); (2) Isotropic turbulence (Marcel Lesieur); (3) Industrial applications (Franco Magagnato); (4) Transition (Laurette Tuckerman); and (5) Environment (Elisabeth Wingate). Among the topics treated in these groups were: analysis of helicity in a wake (group l), vortices in LES of 3D isotropic turbulence (group 2), simulation of flows in complex geometries (around cars) using Magagnato's SPARC code (group 3), transitional plane Couette flow

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(group 4), and dispersion of tracers in the stratosphere (group 5). The computing centre was always crowded in the non-course periods. It also gave students a very good formation in scientific computing and image processing.

The level of the students (who came fiom a wide range of countries) was extremely good, and their sharp questions embarrassed sometimes even the best professors. Relations between the students and the professors were excellent, and discussions continued between the courses and during the meals, in the evenings, and even on the mountain trails above Le Prarion. We had a wide participation of students coming fiom countries in eastern Europe, and it gave the School a distinct flavour. Contacts between the students were numerous, fliendly and extremely tolerant.

Acknowledgments

We thank very much the lecturers for their efforts in preparing and delivering their courses, and then writing the above lecture notes. We thank also Compaq, Sun, IBM and Hewlett-Packard, who lent and maintained the (extremely efficient) computers. Particular thanks go to Patrick Begou (LEGI-Grenoble) for the huge energy he spent contacting the computer suppliers, setting up the computing centre, installing the machines, making them work perfectly during the programme, and (last but not least) de-installing everything at the end.

The practical organization provided by the Les Houches Physics School was very good (especially thanks to the efforts and the patience of Ghyslaine &Henry, Isabel Lelievre and Brigitte Rousset), with excellent housing conditions. The food was great, and the restaurant offered a very fliendly atmosphere, thanks to Claude Cauneau and his staff, who prepared unforgettable fondues, raclettes and tartiflettes. The latter dish, made of Reblochon, an excellent and not very well known cheese, is excellent.

The School was sponsored by Grenoble University, the European Commission as a ,High-Level Scientific Conference, NATO as an Advanced Study Institute, the CEA, CNRS (fixmation permanente), and the French Ministry for Defense. We thank them very much for their support, without which "New Trends in Turbulence" could never have been organized. We hope that the reader will enjoy this book which provides an up-to-date account on what Feynman called the last unsolved problem of classical physics.

Marcel Lesieur Akiva Yaglom Frangois David

CONTENTS

Lecturers

Participants

Prkface

Preface

Contents

xi . . .

Xll l

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Course 1. The Century of Turbulence Theory: The Main Achievements and Unsolved Problems

by A. Yaglom 1

1 Introduction 3

2 Flow instability and transition to turbulence 6

3 Development of the theory of turbulence in the 20th century: Exemplary achievements 11 3.1 Similarity laws of near-wall turbulent flows . . . . . . . . . . . . . 11 3.2 Kolmogorov's theory of locally isotropic turbulence . . . . . . . . . 30

4 Concluding remarks; possible role of Navier-Stokes equations 39

Course 2. Measures of Anisotropy and the Universal Properties of Turbulence

by S. Kurien and K. R. Sreenivasan 53

1 Introduction 56

2 Theoretical tools 58 2.1 The method of SO(3) decomposition . . . . . . . . . . . . . . . . . 58 2.2 Foliation of the structure function into j-sectors . . . . . . . . . . . 62 2.3 The velocity structure functions . . . . . . . . . . . . . . . . . . . . 63

2.3.1 The second-order structure function . . . . . . . . . . . . . 64

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2.4 Dimensional estimates for the lowest-order anisotropic scaling exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3 Some experimental _considerations 69 3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2 Relevance of the anisotropic contributions . . . . . . . . . . . . . . 69 3.3 The measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Anisotropic contribution in the case of homogeneity 74 4.1 General remarks on the data . . . . . . . . . . . . . . . . . . . . . 74 4.2 The tensor form for the second-order structure function . . . . . . 76

4.2.1 The anisotropic tensor component derived under the assumption of axisymmetry . . . . . . . . . . . . . . . . . . 76

4.2.2 The complete j = 2 anisotropic contribution . . . . . . . . . 81 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 Anisotropic contribution in the case of inhomogeneity 85 5.1 Extracting the j = 1 component . . . . . . . . . . . . . . . . . . . 85

6 The higher-order structure functions 88 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.2 Method and results . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2.1 The second-order structure function . . . . . . . . . . . . . 89 6.2.2 Higher-order structure functions . . . . . . . . . . . . . . . 92

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Summary 93

7 Conclusions 98

Appendix 99

A Full form for the j = 2 contribution for the homogeneous case 99

B The j = 1 component in the inhomogeneous case 105 B.l Antisymmetric contribution . . . . . . . . . . . . . . . . . . . . . . 105 B.2 Symmetric contribution . . . . . . . . . . . . . . . . . . . . . . . . 107

C Tests of the robustness of the interpolation formula 109

Course 3 . Large-Eddy Simulations of Turbulence

by 0 . Me'tais 113

1 Introduction 117 1.1 LES and determinism: Unpredictability growth . . . . . . . . . . . 118

2 Vortex dynamics 119 2.1 Coherent vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 2.1.2 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 2.1.3 The Q-criterion . . . . . . . . . . . . . . . . . . . . . . . . . 120

2.2 Vortex identification . . . . . . . . . . . . . . . . . . . . . . . . . . 121 2.2.1 Isotropic turbulence . . . . . . . . . . . . . . . . . . . . . . 122

. . . . . . . . . . . . . . . . . . . . . 2.2.2 Backward-facing step 123

3 LES formalism in physical space 125 3.1 LES equations for a flow of constant density . . . . . . . . . . . . . 125 3.2 LES Boussinesq equations in a rotating frame . . . . . . . . . . . . 128 3.3 Eddy-viscosity and diffusivity assumption . . . . . . . . . . . . . . 128 3.4 Smagorinsky's model . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4 LES in Fourier space 131 . . . . . . . . . . . . . . . . 4.1 Spectral eddy viscosity and diffusivity 131

4.2 EDQNM plateau-peak model . . . . . . . . . . . . . . . . . . . . . 132 4.2.1 The spectral-dynamic model . . . . . . . . . . . . . . . . . 134 4.2.2 Existence of the plateau-peak . . . . . . . . . . . . . . . . . 135

4.3 Incompressible plane channel . . . . . . . . . . . . . . . . . . . . . 137 4.3.1 Wall units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.3.2 Streaks and hairpins . . . . . . . . . . . . . . . . . . . . . . 138 4.3.3 Spectral DNS and LES . . . . . . . . . . . . . . . . . . . . 139

5 Improved models for LES 143 5.1 Structure-function model . . . . . . . . . . . . . . . . . . . . . . . 143

5.1.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.1.2 Non-uniform grids . . . . . . . . . . . . . . . . . . . . . . . 145 5.1.3 Structure-function versus Smagorinsky models . . . . . . . 145 5.1.4 Isotropic turbulence . . . . . . . . . . . . . . . . . . . . . . 146 5.1.5 SF model, transition and wall flows . . . . . . . . . . . . . . 146

5.2 Selective structure-function model . . . . . . . . . . . . . . . . . . 146 . 5.3 Filtered structure-function model . . . . . . . . . . . . . . . . . . . 147

5.3.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.4 A test case for the models: The temporal mixing layer . . . . . . . 147 5.5 Spatially growing mixing layer . . . . . . . . . . . . . . . . . . . . 149 5.6 Vortex control in a round jet . . . . . . . . . . . . . . . . . . . . . 151 5.7 LES of spatially developing boundary layers . . . . . . . . . . . . . 153

6 Dynamic approach in physical space 158 6.1 Dynamic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7 Alternative models 161 7.1 Generalized hyperviscosities . . . . . . . . . . . . . . . . . . . . . 161 7.2 Hyperviscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7.3 Scale-similarity and mixed models . . . . . . . . . . . . . . . . . . 162 7.4 Anisotropic subgrid-scale models . . . . . . . . . . . . . . . . . . . 163

8 LES of rotating flows 163 8.1 Rotating shear flows . . . . . . . . . . . . . . . . . . . . . . . . . . 164

8.1.1 Free-shear flows . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.1.2 Wall flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8.1.3 Homogeneous turbulence . . . . . . . . . . . . . . . . . . . 169

9 LES of flows of geophysical interest 169 9.1 Baroclinic eddies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

9.1.1 Synoptic-scale instability . . . . . . . . . . . . . . . . . . . 171 9.1.2 Secondary cyclogenesis . . . . . . . . . . . . . . . . . . . . . 172

10 LES of compressible turbulence 173 10.1 Compressible LES equations . . . . . . . . . . . . . . . . . . . . . . 174 10.2 Heated flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

10.2.1 The heated duct . . . . . . . . . . . . . . . . . . . . . . . . 176 10.2.2 Towards complex flow geometries . . . . . . . . . . . . . . . 177

11 Conclusion 181

Course 4 . Statistical Turbulence Modelling for the Computation of Physically Complex Flows

by M.A. Leschziner

1 Approaches to characterising turbulence 189

2 Some basic statistical properties of turbulence and associated implications 196

3 Review of "simple" modelling approaches 203 3.1 The eddy-viscosity concept . . . . . . . . . . . . . . . . . . . . . . 203 3.2 Model categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 3.3 Model applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

4 Second-moment equations and implied stress-strain interactions 212 4.1 Near-wall shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 4.2 Streamline curvature . . . . . . . . . . . . . . . . . . . . . . . . . . 217 4.3 Separation and recirculating flow . . . . . . . . . . . . . . . . . . . 218 4.4 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Irrotational strain 220 4.6 Heat transfer and stratification . . . . . . . . . . . . . . . . . . . . 221

5 Second moment closure 222

6 Non-linear eddy-viscosity models 228

7 Application examples 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Overview 233

. . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Asymmetric diffuser 235

. . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Aerospatiale aerofoil 235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Cascade blade 238

7.5 Axisymmetric impinging jet . . . . . . . . . . . . . . . . . . . . . . 239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Prolate spheroid 240

7.7 Round-terectangular transition duct . . . . . . . . . . . . . . . . . 242 7.8 Winglflat-plate junction . . . . . . . . . . . . . . . . . . . . . . . . 245

. . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Fin-plate junction 246 7.10 Jet-afterbody combination . . . . . . . . . . . . . . . . . . . . . . 251

8 Concluding remarks 251

Course 5 . Computational Aeroacoustics

by R . Mankbadi

1 Fundamentals of sound transmission 261 1.1 One-dimensional wave analysis . . . . . . . . . . . . . . . . . . . . 262

1.1.1 General solution of the wave equation . . . . . . . . . . . . 263 1.1.2 The particle velocity . . . . . . . . . . . . . . . . . . . . . . 263

1.2 Three-dimensional sound waves . . . . . . . . . . . . . . . . . . . . 264 1.3 Sound spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

1.3.1 Spectral composition of a square pulse . . . . . . . . . . . . 266 1.3.2 Spectral composition of a harmonic signal . . . . . . . . . . 266

1.4 Logarithmic scales for rating noise . . . . . . . . . . . . . . . . . . 266 1.4.1 The Sound Power Level (PWL) . . . . . . . . . . . . . . . . 266 1.4.2 Sound Pressure Levels (SPL) . . . . . . . . . . . . . . . . . 267 1.4.3 Pressure Band Level (PBL) . . . . . . . . . . . . . . . . . . 267 1.4.4 Pressure Spectral Level (PSL) per unit frequency . . . . . . 267 1.4.5 Acoustic intensity . . . . . . . . . . . . . . . . . . . . . . . 269 1.4.6 Overall pressure levels . . . . . . . . . . . . . . . . . . . . . 269 1.4.7 Subjective noise measures . . . . . . . . . . . . . . . . . . . 269 1.4.8 Perceived Noise Level (PNL) . . . . . . . . . . . . . . . . . 270 1.4.9 Tone Corrected Perceived Noise Level (PNLT) . . . . . . . 270 1.4.10 Effective Perceived Noise Level (EPNL) . . . . . . . . . . . 270

2 Aircraft noise sources 271 2.1 Noise regulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 2.2 Contribution from various components . . . . . . . . . . . . . . . . 271 2.3 Engine noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

2.3.1 Elements of the generation process . . . . . . . . . . . . . . 272 2.3.2 Noise measurements . . . . . . . . . . . . . . . . . . . . . . 272

3 Methodology for jet noise 278 3.1 Jet noise physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 3.2 CAA for jet noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 3.3 Wave-like sound source . . . . . . . . . . . . . . . . . . . . . . . . . 281 3.4 Lighthill's theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

3.4.1 Application of Lighthill's theory . . . . . . . . . . . . . . . 287 3.5 Kirckhoff's solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

4 Algorithms and boundary treatment 290 4.1 Algorithms for CAA . . . . . . . . . . . . . . . . . . . . . . . . . . 290

4.1.1 The 2-4 scheme . . . . . . . . . . . . . . . . . . . . . . . . . 292 4.1.2 The compact scheme . . . . . . . . . . . . . . . . . . . . . . 292 4.1.3 The Dispersion-Relation-Preserving scheme . . . . . . . . . 293

4.2 Boundary treatment for CAA . . . . . . . . . . . . . . . . . . . . . 293 4.2.1 Wall boundary conditions . . . . . . . . . . . . . . . . . . . 295 4.2.2 Outflow boundary treatment . . . . . . . . . . . . . . . . . 295 4.2.3 Radiation boundary condition . . . . . . . . . . . . . . . . . 295 4.2.4 Inflow treatments . . . . . . . . . . . . . . . . . . . . . . . . 296

5 Large-eddy simulations and linearized Euler 298 5.1 Large-eddy simulation . . . . . . . . . . . . . . . . . . . . . . . . . 298

5.1.1 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 5.1.2 Filtered equations . . . . . . . . . . . . . . . . . . . . . . . 301 5.1.3 Modelling subgrid-scale turbulence . . . . . . . . . . . . . . 304

5.2 Linearized Euler equations . . . . . . . . . . . . . . . . . . . . . . . 308

Course 6 . The Topology of Turbulence

by H.K. Moffatt

1 Introduction 321

2 The family of helicity invariants 322 2.1 Chaotic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 2.2 Simply degenerate fields . . . . . . . . . . . . . . . . . . . . . . . . 324 2.3 Doubly degenerate fields . . . . . . . . . . . . . . . . . . . . . . . . 324

3 The special case of Euler dynamics 325

4 Scalar field structure in 2D flows

5 Scalar field structure in 3D flows

6 Vector field structure in 3D flows

7 Helicity and the turbulent dynamo 7.1 The kinematic phase . . . . . . . . . . . . . . . . . . 7.2 The dynamic phase . . . . . . . . . . . . . . . . . . .

8 Magnetic relaxation 8.1 The analogy with Euler flows . . . . . . . . . . . . .

9 The blow-up problem 9.1 Interaction of skewed vortices . . . . . . . . . . . . .

Course 7. "Burgulence"

by U. Frisch and J . Bec

Introduction 1.1 The Burgers equation in cosmology . . . . . . . . . . . 1.2 The Burgers equation in condensed matter and statistical physics . 347 1.3 The Burgers equation as testing ground for Navier-Stokes . . . . . 347

Basic tools 348 2.1 The Hopf-Cole transformation and the maximum representation . 348 2.2 Shocks in one dimension . . . . . . . . . . . . . . . . . . . . . . . . 350 2.3 Convex hull construction in more than one dimension . . . . . . . 354 2.4 Remarks on numerical methods . . . . . . . . . . . . . . . . . . . . 355

The Fourier-Lagrange representation and artefacts 356

The law of energy decay 358

One-dimensional case with Brownian initial velocity 363

Preshocks and the pdf of velocity gradients in one dimension 367

The pdf of density 370

Kicked burgulence 373 8.1 Forced Burgers equation and variational formulation . . . . . . . . 373

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8.2 Periodic kicks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 8.3 Connections with Aubry-Mather theory . . . . . . . . . . . . . . . 380

Course 8 . Two-Dimensional Turbulence

by J . Sonimeria

1 Introduction 387

2 Equations and conservation laws 391 2.1 Euler vs . Navier-Stokes equations . . . . . . . . . . . . . . . . . . 391 2.2 Vorticity representation . . . . . . . . . . . . . . . . . . . . . . . . 392

. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Conservation laws 393 2.4 Steady solutions of the Euler equations . . . . . . . . . . . . . . . . 396

3 Vortex dynamics 397 3.1 Systems of discrete vortices . . . . . . . . . . . . . . . . . . . . . . 398 3.2 Vortex pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

. . . . . . . . . . . . . 3.3 Instability of shear flows and vortex lattices 403 . . . . . . . . . . . . . . . . 3.4 Statistical mechanics of point vortices 404

4 Spectral properties. energy and enstrophy cascade 411 . . . . . . . . . . . . . . . . 4.1 Spectrally truncated equilibrium states 412

4.2 The enstrophy and inverse energy cascades of forced turbulence . . 415 . . . . . . . . 4.3 The enstrophy cascade of freely evolving turbulence 422

. . . . . . . . . . 4.4 The emergence and evolution of isolated vortices 423

5 Equilibrium statistical mechanics and self-organization 425 . . . . . . . . . 5.1 Statistical mechanics of non-singular vorticity fields 425

5.2 The Gibbs states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 5.3 Tests and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 432

6 Eddy diffusivity and sub-grid scale modeling 435 6.1 Thermodynamic approach . . . . . . . . . . . . . . . . . . . . . . . 435 6.2 Kinetic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

7 Conclusions 442

Course 9 . Analysing and Computing Turbulent Flows Using Wavelets

by M . Farge and K . Schneider

1 Introduction

I Wavelet Transforms

2 History 456

3 The continuous wavelet transform 457 3.1 One dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

3.1.1 Analyzing wavelet . . . . . . . . . . . . . . . . . . . . . . . 457 3.1.2 Wavelet analysis . . . . . . . . . . . . . . . . . . . . . . . . 458 3.1.3 Wavelet synthesis . . . . . . . . . . . . . . . . . . . . . . . . 459 3.1.4 Energy conservation . . . . . . . . . . . . . . . . . . . . . . 459

3.2 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 3.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460

4 The orthogonal wavelet transform 4.1 One dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.1 1D Multi-Resolution Analysis . . . . . . . . . . . . . . . . . 4.1.2 Regularity and local decay of wavelet coefficients . . . . . .

4.2 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Tensor product construction . . . . . . . . . . . . . . . . . . 4.2.2 2D Multi-Resolution Analysis . . . . . . . . . . . . . . . . . 4.2.3 Periodic 2D Multi-Resolution Analysis . . . . . . . . . . . .

4.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I1 Statistical Analysis

5 Classical tools 468 5.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

5.1.1 Laboratory experiments . . . . . . . . . . . . . . . . . . . . 468 5.1.2 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 468

5.2 Averaging procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 469 5.3 Statistical diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . 470

5.3.1 Probability Distribution Function (PDF) . . . . . . . . . . 470 5.3.2 Radon-Nikodyn's theorem . . . . . . . . . . . . . . . . . . . 471 5.3.3 Definition of the joint probability . . . . . . . . . . . . . . . 471 5.3.4 Statistical moments . . . . . . . . . . . . . . . . . . . . . . 471 5.3.5 Structure functions . . . . . . . . . . . . . . . . . . . . . . . 472 5.3.6 Autocorrelation function . . . . . . . . . . . . . . . . . . . . 472 5.3.7 Fourier spectrum . . . . . . . . . . . . . . . . . . . . . . . . 472 5.3.8 Wiener-Khinchin's theorem . . . . . . . . . . . . . . . . . . 473

6 Statistical tools based on the continuous wavelet transform 473 6.1 Local and global wavelet spectra . . . . . . . . . . . . . . . . . . . 473 6.2 Relation with Fourier spectrum . . . . . . . . . . . . . . . . . . . . 474 6.3 Application to turbulence . . . . . . . . . . . . . . . . . . . . . . . 475

7 Statistical tools based on the orthogonal wavelet transform 476 7.1 Local and global wavelet spectra . . . . . . . . . . . . . . . . . . . 476 7.2 Relation with Fourier spectrum . . . . . . . . . . . . . . . . . . . . 477 7.3 Intermittency measures . . . . . . . . . . . . . . . . . . . . . . . . 478

I11 Computation

8 Coherent vortex extraction 478 8.1 CVS filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

8.1.1 Vorticity decomposition . . . . . . . . . . . . . . . . . . . . 479 8.1.2 Nonlinear thresholding . . . . . . . . . . . . . . . . . . . . . 479 8.1.3 Vorticity and velocity reconstruction . . . . . . . . . . . . . 480

8.2 Application to a 3D turbulent mixing layer . . . . . . . . . . . . . 480 . . . . . . . . . . . . . 8.3 Comparison between CVS and LES filtering 481

9 Computation of turbulent flows 484 9.1 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . 484

9.1.1 Velocity-pressure formulation . . . . . . . . . . . . . . . . . 484 9.1.2 Vorticity-velocity formulation . . . . . . . . . . . . . . . . . 484

9.2 Classical numerical methods . . . . . . . . . . . . . . . . . . . . . . 485 9.2.1 Direct Numerical Simulation (DNS) . . . . . . . . . . . . . 485 9.2.2 Modelled Numerical Simulation (MNS) . . . . . . . . . . . 486

9.3 Coherent Vortex Simulation (CVS) . . . . . . . . . . . . . . . . . . 487 9.3.1 Principle of CVS . . . . . . . . . . . . . . . . . . . . . . . . 487 9.3.2 CVS without turbulence model . . . . . . . . . . . . . . . . 488 9.3.3 CVS with turbulence model . . . . . . . . . . . . . . . . . . 488

10 Adaptive wavelet computation 489 10.1 Adaptive wavelet scheme for nonlinear PDE's . . . . . . . . . . . . 489

10.1.1 Time discretization . . . . . . . . . . . . . . . . . . . . . . . 490 10.1.2 Wavelet decomposition . . . . . . . . . . . . . . . . . . . . . 490 10.1.3 Evaluation of the nonlinear term . . . . . . . . . . . . . . . 492 10.1.4 Substraction strategy . . . . . . . . . . . . . . . . . . . . . 493 10.1.5 Summary of the algorithm . . . . . . . . . . . . . . . . . . . 494

. . . 10.2 Adaptive wavelet scheme for the 2D Navier-Stokes equations 494 10.3 Application to a 2D turbulent mixing layer . . . . . . . . . . . . . 497

10.3.1 Adaptive wavelet computation . . . . . . . . . . . . . . . . 497 10.3.2 Comparison between CVS and Fourier pseudo-spectral DNS 497

IV Conclusion

xxxvii

Course 10 . Lagrangian Description of Turbulence

by G . Falkovich, K . Gawedzki and M . Vergassola 505

1 Particles in fluid turbulence 507 1.1 Single-particle diffusion . . . . . . . . . . . . . . . . . . . . . . . . 508

. . . . . . . 1.2 Two-particle dispersion in a spatially smooth velocity 510 1.2.1 General consideration . . . . . . . . . . . . . . . . . . . . . 510 1.2.2 Solvable cases . . . . . . . . . . . . . . . . . . . . . . . . . . 515

1.3 Two-particle dispersion in a nonsmooth incompressible flow . . . . 518 1.4 Multiparticle configurations and breakdown of scale-invariance . . 525

1.4.1 Absolute and relative evolution of particles . . . . . . . . . 526 . . . . . . . . . 1.4.2 Multiparticle motion in Kraichnan velocities 527

1.4.3 Zero modes and slow modes . . . . . . . . . . . . . . . . . . 529 1.4.4 Perturbative schemes . . . . . . . . . . . . . . . . . . . . . . 532

2 Passive fields in fluid turbulence 536 2.1 Unforced evolution of passive fields . . . . . . . . . . . . . . . . . . 536 2.2 Cascades of a passive scalar . . . . . . . . . . . . . . . . . . . . . . 538

2.2.1 Passive scalar in a spatially smooth velocity . . . . . . . . . 539 2.3 Passive scalar in a spatially nonsmooth velocity . . . . . . . . . . . 543

3 Navier-Stokes equation from a Lagrangian viewpoint 546 3.1 Enstrophy cascade in two dimensions . . . . . . . . . . . . . . . . . 546 3.2 On the energy cascades in incompressible turbulence . . . . . . . . 549

4 Conclusion 551