Statistical Approach to Wall Turbulence

Wall turbulence is encountered in many technological applications as well as in the atmosphere, and a detailed understanding leading to its management would have considerable consequences. Many inspired works on the topic have been brought forth by experimenters, theoreticians, engineers, and mathematicians in previous decades. This book has the intention of giving an updated and synthetic view of the progress that has been made in recent years. It includes an introduction to the topic, as well as chapters that detail basic notions in homogeneous isotropic and axisymmetric turbulence, phenomenology and closure in wall turbulence, scaling and spectra, Reynolds number effects on the mean velocity profile, turbulent intensities and fine structure, vorticity and vorticity transport near the wall, self-sustaining near-wall coherent structures and their identification, detection, and effects on turbulence production, genesis and regeneration of wall coherence, turbulent transport phenomena near the wall, dissipation and fine scale vorticity, very large structures, and the dynamic system approach to wall turbulence.

Sedat Tardu is Associate Professor at Grenoble University in France where he teaches turbulent shear flows, heat transfer, fluid dynamics, and chaos. His research area includes canonical and non-canonical wall turbulence, active and passive control of near wall turbulence, unsteady wall layers, mixing and micro-mixing, and microfluidics. He has been a visiting researcher at several universities (Ecole Polytechnique Federale de Lausanne, Imperial College London, Beihang University Beijing) and he is author of more than 200 papers in peer-reviewed journals, and international conferences and of several book chapters.

Foreword	p. ix
Introduction	p. xi
Basic Concepts	p. 1
Introduction	p. 1
Fundamental equations	p. 1
Euler equations	p. 2
Notation	p. 4
Reynolds averaged Navier-Stokes equations	p. 4
Basic concepts of turbulent transport mechanisms	p. 6
Turbulent energy transport	p. 6
Inter-component transport	p. 8
Correlation tensor dynamics	p. 11
Homogeneous turbulence	p. 15
Isotropic homogeneous turbulence	p. 20
Axisymmetric homogeneous turbulence	p. 33
Turbulence scales	p. 35
Taylor hypothesis	p. 39
Approaches to modeling wall turbulence	p. 40
Direct numerical simulations	p. 40
Measurements	p. 41
Preliminary Concepts: Phenomenology, Closures and Fine Structure	p. 45
Introduction	p. 45
Hydrodynamic stability and origins of wall turbulence	p. 46
Linear stability	p. 46
Secondary stability, non-linearity and bypass transition	p. 48
Reynolds equations in internal turbulent flows	p. 55
Scales in turbulent wall flow	p. 55
Eddy viscosity closures	p. 56
Exact equations for fully developed channel flow	p. 61
Shear stress field	p. 61
Friction coefficient	p. 63
"Laminar/turbulent" decomposition	p. 64
Algebraic closures for the mixing length in internal flows	p. 65
Some illustrations using direct numerical simulations at low Reynolds numbers	p. 69
Turbulent intensities	p. 69
Fine structure	p. 70
Transport of turbulent kinetic energy and reformulation of the logarithmic sublayer	p. 72
Transport of the Reynolds shear stress -uv	p. 74
Transition to turbulence in a boundary layer on a flat plate	p. 76
Equations for the turbulent boundary layer	p. 77
Mean vorticity	p. 81
Integral equations	p. 83
Scales in a turbulent boundary layer	p. 85
Power law distributions and simplified integral approach	p. 85
Outer layer	p. 88
Izakson-Millikan-von Mises overlap	p. 89
Integral quantities	p. 91
Wake region	p. 94
Drag coefficient in external turbulent flows	p. 96
Asymptotic behavior close to the wall	p. 98
Coherent wall structures-a brief introduction	p. 101
Inner and Outer Scales: Spectral Behavior	p. 105
Introduction	p. 105
Townsend-Perry analysis in the fully-developed turbulent sublayer	p. 107
Spectral densities	p. 110
Longitudinal fluctuating velocity	p. 110
Spanwise fluctuating velocity	p. 118
Fluctuating wall-normal velocity	p. 119
Reynolds shear stress	p. 121
Summary: active and passive structures	p. 123
Clues to the k_x^l behavior, and discussion	p. 124
Spectral density E^vv and cospectral density E^uv	p. 129
Two-dimensional spectral densities	p. 131
Reynolds Number-Based Effects	p. 137
Introduction	p. 137
The von Karman constant and the renormalization group	p. 140
Renormalization group (RNG)	p. 140
The von Karman constant derived from the RNG	p. 141
Complete and incomplete similarity	p. 146
General considerations. Power law distributions	p. 146
Implications for mixing length	p. 154
Symmetries and their consequences	p. 155
Lie symmetries	p. 155
Application to wall turbulence	p. 157
Principle of asymptotic invariance. Approach of W.K. George	p. 163
Internal flows	p. 163
External (boundary-layer) flows	p. 178
Mean velocity distribution. Summary	p. 185
Townsend's attached eddies	p. 185
Concept of attached eddies	p. 185
Cross-correlations	p. 185
Mean shear	p. 192
Generalization. Correspondence between constants and the Coles parameter	p. 193
Spectral compatibility. Generalization	p. 196
Generalization for the intensity of longitudinal velocity fluctuations	p. 197
Turbulent intensity of spanwise velocity fluctuations	p. 203
Turbulent intensity of wall-normal fluctuations	p. 210
Reynolds shear stress -uv	p. 215
Overlap region in internal flows	p. 228
Two-point correlations	p. 230
Symmetries of two-point correlations	p. 230
Comparison with experimental results	p. 234
Active and passive Townsend eddies	p. 239
Decomposition and scales	p. 240
Effects of wall-layer (active) structures and outer-layer (passive) structures on wall dynamics	p. 245
Fine structure	p. 249
Vorticity	p. 259
Introduction	p. 259
General characteristics of vorticity	p. 259
Reynolds shear stress and vorticity transport	p. 261
Characteristics of the vorticity field close to a wall	p. 264
Statistics and fine structure	p. 270
Vorticity transport	p. 277
Mean vorticity	p. 277
Transport of vorticity and enstrophy correlation terms	p. 279
Estimating the importance of non-linearity close to the wall	p. 284
Measurements	p. 287
Notations Used	p. 291
Subscripts and superscripts	p. 293
Greek letters	p. 294
Abbreviations	p. 295
Bibliography	p. 297
Index	p. 309
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Statistical Approach to Wall Turbulence

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