Statistical Approach to Wall Turbulence

Sédat Tardu is an associate professor at Grenoble University in France.

Foreword ix
Ivan MARUSIC

Introduction xi

Chapter 1. Basic Concepts 1

1.1. Introduction 1

1.2. Fundamental equations 1

1.3. Notation 4

1.4. Reynolds averaged Navier-Stokes equations 4

1.5. Basic concepts of turbulent transport mechanisms 6

1.6. Correlation tensor dynamics 11

1.7. Homogeneous turbulence 15

1.8. Isotropic homogeneous turbulence 20

1.9. Axisymmetric homogeneous turbulence 33

1.10. Turbulence scales 35

1.11. Taylor hypothesis 39

1.12. Approaches to modeling wall turbulence 40

Chapter 2. Preliminary Concepts: Phenomenology, Closures and Fine Structure 45

2.1. Introduction 45

2.2. Hydrodynamic stability and origins of wall turbulence 46

2.3. Reynolds equations in internal turbulent flows 55

2.4. Scales in turbulent wall flow 55

2.5. Eddy viscosity closures 56

2.6. Exact equations for fully developed channel flow 61

2.7. Algebraic closures for the mixing length in internal flows 65

2.8. Some illustrations using direct numerical simulations at low Reynolds numbers 69

2.9. Transition to turbulence in a boundary layer on a flat plate 76

2.10. Equations for the turbulent boundary layer 77

2.11. Mean vorticity 81

2.12. Integral equations 83

2.13. Scales in a turbulent boundary layer 85

2.14. Power law distributions and simplified integral approach 85

2.15. Outer layer 88

2.16. Izakson-Millikan-von Mises overlap 89

2.17. Integral quantities 91

2.18. Wake region 94

2.19. Drag coefficient in external turbulent flows 96

2.20. Asymptotic behavior close to the wall 98

2.21. Coherent wall structures – a brief introduction 101

Chapter 3. Inner and Outer Scales: Spectral Behavior 105

3.1. Introduction105

3.2. Townsend-Perry analysis in the fully-developed turbulent sublayer 107

3.3. Spectral densities 110

3.4. Clues to the 1x k _ behavior, and discussion 124

3.5. Spectral density vv E and cospectral density uv E 129

3.6. Two-dimensional spectral densities 131

Chapter 4. Reynolds Number-Based Effects 137

4.1. Introduction 137
 
4.2. The von Karman constant and the renormalization group 140

4.3. Complete and incomplete similarity 146

4.4. Symmetries and their consequences 155

4.5. Principle of asymptotic invariance. Approach of W.K. George 163

4.6. Mean velocity distribution. Summary 185

4.7. Townsend’s attached eddies 185

4.8. Overlap region in internal flows 228

4.9. Two-point correlations 230

4.10. Active and passive Townsend eddies 239

4.11. Fine structure 249

Chapter 5. Vorticity 259

5.1. Introduction 259

5.2. General characteristics of vorticity 259

5.3. Reynolds shear stress and vorticity transport 261

5.4. Characteristics of the vorticity field close to a wall 264

5.5. Statistics and fine structure 270

5.6. Vorticity transport 277

5.7. Estimating the importance of non-linearity close to the wall 284

5.8. Measurements 287

Notations Used 291

Subscripts and superscripts 293

Greek letters 294

Abbreviations 295

Bibliography 297

Index 309