Statistical Theory and Modeling for Turbulent Flows	3
	Contents	9
	Preface	13
	Preface to second edition	13
	Preface to first edition	13
	Motivation	14
	Epitome	15
	Acknowledgements	15
	Part I FUNDAMENTALS OF TURBULENCE	17
	1 Introduction	19
	1.1 The turbulence problem	20
	1.2 Closure modeling	25
	1.3 Categories of turbulent flow	26
	Exercises	30
	2 Mathematical and statistical background	31
	2.1 Dimensional analysis	31
	2.1.1 Scales of turbulence	34
	2.2 Statistical tools	35
	2.2.1 Averages and probability density functions	35
	2.2.2 Correlations	41
	2.3 Cartesian tensors	50
	2.3.1 Isotropic tensors	52
	2.3.2 Tensor functions of tensors	53
	Exercises	58
	3 Reynolds averaged Navier–Stokes equations	61
	3.1 Background to the equations	62
	3.2 Reynolds averaged equations	64
	3.3 Terms of kinetic energy and Reynolds stress budgets	65
	3.4 Passive contaminant transport	70
	Exercises	72
	4 Parallel and self-similar shear flows	73
	4.1 Plane channel flow	74
	4.1.1 Logarithmic layer	77
	4.1.2 Roughness	79
	4.2 Boundary layer	81
	4.2.1 Entrainment	85
	4.3 Free-shear layers	86
	4.3.1 Spreading rates	92
	4.3.2 Remarks on self-similar boundary layers	92
	4.4 Heat and mass transfer	93
	4.4.1 Parallel flow and boundary layers	94
	4.4.2 Dispersion from elevated sources	98
	Exercises	102
	5 Vorticity and vortical structures	107
	5.1 Structures	109
	5.1.1 Free-shear layers	109
	5.1.2 Boundary layers	113
	5.1.3 Non-random vortices	118
	5.2 Vorticity and dissipation	118
	5.2.1 Vortex stretching and relative dispersion	120
	5.2.2 Mean-squared vorticity equation	122
	Exercises	124
	Part II SINGLE-POINT CLOSURE MODELING	125
	6 Models with scalar variables	127
	6.1 Boundary-layer methods	128
	6.1.1 Integral boundary-layer methods	129
	6.1.2 Mixing length model	131
	6.2 The k –å model	137
	6.2.1 Analytical solutions to the k –å model	139
	6.2.2 Boundary conditions and near-wall modifications	144
	6.2.3 Weak solution at edges of free-shear flow	151
	6.3 The k –ù model	152
	6.4 Stagnation-point anomaly	155
	6.5 The question of transition	157
	6.5.1 Reliance on the turbulence model	160
	6.5.2 Intermittency equation	161
	6.5.3 Laminar fluctuations	163
	6.6 Eddy viscosity transport models	164
	Exercises	168
	7 Models with tensor variables	171
	7.1 Second-moment transport	171
	7.1.1 A simple illustration	172
	7.1.2 Closing the Reynolds stress transport equation	173
	7.1.3 Models for the slow part	175
	7.1.4 Models for the rapid part	178
	7.2 Analytic solutions to SMC models	185
	7.2.1 Homogeneous shear flow	185
	7.2.2 Curved shear flow	188
	7.2.3 Algebraic stress approximation and nonlinear eddy viscosity	192
	7.3 Non-homogeneity	195
	7.3.1 Turbulent transport	196
	7.3.2 Near-wall modeling	197
	7.3.3 No-slip condition	198
	7.3.4 Nonlocal wall effects	200
	7.4 Reynolds averaged computation	210
	7.4.1 Numerical issues	211
	7.4.2 Examples of Reynolds averaged computation	214
	Exercises	229
	8 Advanced topics	233
	8.1 Further modeling principles	233
	8.1.1 Galilean invariance and frame rotation	235
	8.1.2 Realizability	237
	8.2 Second-moment closure and Langevin equations	240
	8.3 Moving equilibrium solutions of SMC	242
	8.3.1 Criterion for steady mean flow	243
	8.3.2 Solution in two-dimensional mean flow	244
	8.3.3 Bifurcations	247
	8.4 Passive scalar flux modeling	251
	8.4.1 Scalar diffusivity models	251
	8.4.2 Tensor diffusivity models	252
	8.4.3 Scalar flux transport	254
	8.4.4 Scalar variance	257
	8.5 Active scalar flux modeling: effects of buoyancy	258
	8.5.1 Second-moment transport models	261
	8.5.2 Stratified shear flow	262
	Exercises	263
	Part III THEORY OF HOMOGENEOUS TURBULENCE	265
	9 Mathematical representations	267
	9.1 Fourier transforms	268
	9.2 Three-dimensional energy spectrum of homogeneous turbulence	270
	9.2.1 Spectrum tensor and velocity covariances	271
	9.2.2 Modeling the energy spectrum	273
	Exercises	282
	10 Navier–Stokes equations in spectral space	285
	10.1 Convolution integrals as triad interaction	285
	10.2 Evolution of spectra	287
	10.2.1 Small-k behavior and energy decay	287
	10.2.2 Energy cascade	289
	10.2.3 Final period of decay	292
	Exercises	293
	11 Rapid distortion theory	297
	11.1 Irrotational mean flow	298
	11.1.1 Cauchy form of vorticity equation	298
	11.1.2 Distortion of a Fourier mode	301
	11.1.3 Calculation of covariances	303
	11.2 General homogeneous distortions	307
	11.2.1 Homogeneous shear	309
	11.2.2 Turbulence near a wall	312
	Exercises	316
	Part IV TURBULENCE SIMULATION	319
	12 Eddy-resolving simulation	321
	12.1 Direct numerical simulation	322
	12.1.1 Grid requirements	322
	12.1.2 Numerical dissipation	324
	12.1.3 Energy-conserving schemes	326
	12.2 Illustrations	329
	12.3 Pseudo-spectral method	334
	Exercises	338
	13 Simulation of large eddies	341
	13.1 Large eddy simulation	341
	13.1.1 Filtering	342
	13.1.2 Subgrid models	346
	13.2 Detached eddy simulation	355
	Exercises	359
	References	361
	Index	369