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CHAPTER 1 Introduction1

1.1 Turbulence and symmetries1

1.2 Outline of the book11

CHAPTER 2 Symmetries and conservation laws14

2.1 Periodic boundary conditions14

2.2 Symmetries17

2.3 Conservation laws18

2.4 Energy budget scale-by-scale21

CHAPTER 3 Why a probabilistic description of turbulence？21

3.1 There is something predictable in a turbulent signal27

3.2 A model for deterministic chaos31

3.3 Dynamical systems36

3.4 The Navier-Stokes equation as a dynamical system37

CHAPTER 4 Probabilistic tools：a survey40

4.1 Random variables40

4.2 Random functions45

4.3 Statistical symmetries46

4.4 Ergodic results49

4.5 The spectrum of stationary random functions52

CHAPTER 5 Two experimental laws of fully developed turbulence52

5.1 The two-thirds law57

5.2 The energy dissipation law67

CHAPTER 6 The Kolmogorov 1941 theory72

6.1 Kolmogorov 1941 and symmetries72

6.2 Kolmogorov's four-fifths law76

6.2.1 The Kámán-Howarth-Monin relation for anisotropic turbulence77

6.2.2 The energy flux for homogeneous turbulence79

6.2.3 The energy flux for homogeneous isotropic turbulence81

6.2.4 From the energy flux relation to the four-fifths law84

6.2.5 Remarks on Kolmogorov's four-fifths law86

6.3 Main results of the Kolmogorov 1941 theory89

6.3.1 The Kolmogorov-Obukhov law and the structure func-tions89

6.3.2 Effect of a finite viscosity：the dissipation range91

6.4 Kolmogorov and Landau：the lack of universality93

6.4.1 The original formulation of Landau's objection93

6.4.2 A modern reformulation of Landau's objection94

6.4.3 Kolmogorov and Landau reconciled？97

6.5 Historical remarks on the Kolmogorov 1941 theory98

CHAPTER 7 Phenomenology of turbulence in the sense of Kolmogorov 1941100

7.1 Introduction100

7.2 Basic tools of phenomenology101

7.3 The Richardson cascade and the localness of interactions103

7.4 Reynolds numbers and degrees of freedom106

7.5 Microscopic and macroscopic degrees of freedom109

7.6 The distribution of velocity gradients111

7.7 The law of decay of the energy112

7.8 Beyond phenomenology：finite-time blow-up of ideal flow115

CHAPTER 8 Intermittency120

8.1 Introduction120

8.2 Self-similar and intermittent random functions121

8.3 Experimental results on intermittency127

8.4 Exact results on intermittency133

8.5 Intermittency models based on the velocity135

8.5.1 The β-model135

8.5.2 The bifractal model140

8.5.3 The multifractal model143

8.5.4 A probabilistic reformulation of the multifractal model146

8.5.5 The intermediate dissipation range and multifractal universality149

8.5.6 The skewness and the flatness of velocity derivatives according to the multifractal model155

8.6 Intermittency models based on the dissipation159

8.6.1 Multifractal dissipation159

8.6.2 Bridging multifractality based on the velocity and mul-tifractality based on the dissipation163

8.6.3 Random cascade models165

8.6.4 Large deviations and multifractality168

8.6.5 The lognormal model and its shortcomings171

8.7 Shell models174

8.8 Historical remarks on fractal intermittency models178

8.9 Trends in intermittency research182

8.9.1 Vortex filaments：the sinews of turbulence？185

8.9.2 Statistical signature of vortex filaments：dog or tail？188

8.9.3 The distribution of velocity increments192

CHAPTER 9 Further reading：a guided tour195

9.1 Introduction195

9.2 Books on turbulence and fluid mechanics195

9.3 Mathematical aspects of fully developed turbulence199

9.4 Dynamical systems,fractals and turbulence203

9.5 Closure,functional and diagrammatic methods206

9.5.1 The Hopf equation207

9.5.2 Functional and diagrammatic methods212

9.5.3 The direct interaction approximation217

9.5.4 Closures and their shortcomings219

9.6 Eddy viscosity,multiscale methods and renormalization222

9.6.1 Eddy viscosity：a very old idea222

9.6.2 Multiscale methods226

9.6.3 Applications of multiscale methods in turbulence230

9.6.4 Renormalization group（RG）methods235

9.7 Two-dimensional turbulence240

9.7.1 Cascades and vortices241

9.7.2 Two-dimensional turbulence and statistical mechanics243

9.7.3 Conservative dynamics‘punctuated’by dissipative events249

9.7.4 From Flatland to three-dimensional turbulence251

References255

Author index283

Subject index289