PART ONE:FUNDAMENTALS 1
1 Introduction 3
1.1 The nature of turbulent flows 3
1.2 The study of turbulent flows 7
2 The equations of fluid motion 10
2.1 Continuum fluid properties 10
2.2 Eulerian and Lagrangian fields 12
2.3 The continuity equation 14
2.4 The momentum equation 16
2.5 The role of pressure 18
2.6 Conserved passive scalars 21
2.7 The vorticity equation 22
2.8 Rates of strain and rotation 23
2.9 Transformation properties 24
3 The statistical description of turbulent flows 34
3.1 The random nature of turbulence 34
3.2 Characterization of random variables 37
3.3 Examples of probability distributions 43
3.4 Joint random variables 54
3.5 Normal and joint-normal distributions 61
3.6 Random processes 65
3.7 Random fields 74
3.8 Probability and averaging 79
4 Mean-flow equations 83
4.1 Reynolds equations 83
4.2 Reynolds stresses 86
4.3 The mean scalar equation 91
4.4 Gradient-diffusion and turbulent-viscosity hypotheses 92
5 Free shear flows 96
5.1 The round jet:experimental observations 96
5.1.1 A description of the flow 96
5.1.2 The mean velocity field 97
5.1.3 Reynolds stresses 105
5.2 The round jet:mean momentum 111
5.2.1 Boundary-layer equations 111
5.2.2 Flow rates of mass,momentum,and energy 115
5.2.3 Self-similarity 116
5.2.4 Uniform turbulent viscosity 118
5.3 The round jet:kinetic energy 122
5.4 Other self-similar flows 134
5.4.1 The plane jet 134
5.4.2 The plane mixing layer 139
5.4.3 The plane wake 147
5.4.4 The axisymmetric wake 151
5.4.5 Homogeneous shear flow 154
5.4.6 Grid turbulence 158
5.5 Further observations 161
5.5.1 A conserved scalar 161
5.5.2 Intermittency 167
5.5.3 PDFs and higher moments 173
5.5.4 Large-scale turbulent motion 178
6 The scales of turbulent motion 182
6.1 The energy cascade and Kolmogorov hypotheses 182
6.1.1 The energy cascade 183
6.1.2 The Kolmogorov hypotheses 184
6.1.3 The energy spectrum 188
6.1.4 Restatement of the Kolmogorov hypotheses 189
6.2 Structure functions 191
6.3 Two-point correlation 195
6.4 Fourier modes 207
6.4.1 Fourier-series representation 207
6.4.2 The evolution of Fourier modes 211
6.4.3 The kinetic energy of Fourier modes 215
6.5 Velocity spectra 219
6.5.1 Definitions and properties 220
6.5.2 Kolmogorov spectra 229
6.5.3 A model spectrum 232
6.5.4 Dissipation spectra 234
6.5.5 The inertial subrange 238
6.5.6 The energy-containing range 240
6.5.7 Effects of the Reynolds number 242
6.5.8 The shear-stress spectrum 246
6.6 The spectral view of the energy cascade 249
6.7 Limitations,shortcomings,and refinements 254
6.7.1 The Reynolds number 254
6.7.2 Higher-order statistics 255
6.7.3 Internal intermittency 258
6.7.4 Refined similarity hypotheses 260
6.7.5 Closing remarks 263
7 Wall flows 264
7.1 Channel flow 264
7.1.1 A description of the flow 264
7.1.2 The balance of mean forces 266
7.1.3 The near-wall shear stress 268
7.1.4 Mean velocity profiles 271
7.1.5 The friction law and the Reynolds number 278
7.1.6 Reynolds stresses 281
7.1.7 Lengthscales and the mixing length 288
7.2 Pipe flow 290
7.2.1 The friction law for smooth pipes 290
7.2.2 Wall roughness 295
7.3 Boundary layers 298
7.3.1 A description of the flow 299
7.3.2 Mean-momentum equations 300
7.3.3 Mean velocity profiles 302
7.3.4 The overlap region reconsidered 308
7.3.5 Reynolds-stress balances 313
7.3.6 Additional effects 320
7.4 Turbulent structures 322
PART TWO:MODELLING AND SIMULATION 333
8 An introduction to modelling and simulation 335
8.1 The challenge 335
8.2 An overview of approaches 336
8.3 Criteria for appraising models 336
9 Direct numerical simulation 344
9.1 Homogeneous turbulence 344
9.1.1 Pseudo-spectral methods 344
9.1.2 The computational cost 346
9.1.3 Artificial modifications and incomplete resolution 352
9.2 Inhomogeneous flows 353
9.2.1 Channel flow 353
9.2.2 Free shear flows 354
9.2.3 Flow over a backward-facing step 355
9.3 Discussion 356
10 Turbulent-viscosity models 358
10.1 The turbulent-viscosity hypothesis 359
10.1.1 The intrinsic assumption 359
10.1.2 The specific assumption 364
10.2 Algebraic models 365
10.2.1 Uniform turbulent viscosity 365
10.2.2 The mixing-length model 366
10.3 Turbulent-kinetic-energy models 369
10.4 The κ-εmodel 373
10.4.1 An overview 373
10.4.2 The model equation for ε 375
10.4.3 Discussion 382
10.5 Further turbulent-viscosity models 383
10.5.1 The κ-ω model 383
10.5.2 The Spalart-Allmaras model 385
11 Reynolds-stress and related models 387
11.1 Introduction 387
11.2 The pressure-rate-of-strain tensor 388
11.3 Return-to-isotropy models 392
11.3.1 Rotta's model 392
11.3.2 The characterization of Reynolds-stress anisotropy 393
11.3.3 Nonlinear return-to-isotropy models 398
11.4 Rapid-distortion theory 404
11.4.1 Rapid-distortion equations 405
11.4.2 The evolution of a Fourier mode 406
11.4.3 The evolution of the spectrum 411
11.4.4 Rapid distortion of initially isotropic turbulence 415
11.4.5 Final remarks 421
11.5 Pressure-rate-of-strain models 422
11.5.1 The basic model(LRR-IP) 423
11.5.2 Other pressure-rate-of-strain models 425
11.6 Extension to inhomogeneous flows 428
11.6.1 Redistribution 428
11.6.2 Reynolds-stress transport 429
11.6.3 The dissipation equation 432
11.7 Near-wall treatments 433
11.7.1 Near-wall effects 433
11.7.2 Turbulent viscosity 434
11.7.3 Model equations for κ and ε 435
11.7.4 The dissipation tensor 436
11.7.5 Fluctuating pressure 439
11.7.6 Wall functions 442
11.8 Elliptic relaxation models 445
11.9 Algebraic stress and nonlinear viscosity models 448
11.9.1 Algebraic stress models 448
11.9.2 Nonlinear turbulent viscosity 452
11.10 Discussion 457
12 PDF methods 463
12.1 The Eulerian PDF of velocity 464
12.1.1 Definitions and properties 464
12.1.2 The PDF transport equation 465
12.1.3 The PDF of the fluctuating velocity 467
12.2 The model velocity PDF equation 468
12.2.1 The generalized Langevin model 469
12.2.2 The evolution of the PDF 470
12.2.3 Corresponding Reynolds-stress models 475
12.2.4 Eulerian and Lagrangian modelling approaches 479
12.2.5 Relationships between Lagrangian and Eulerian PDFs 480
12.3 Langevin equations 483
12.3.1 Stationary isotropic turbulence 484
12.3.2 The generalized Langevin model 489
12.4 Turbulent dispersion 494
12.5 The velocity-frequency joint PDF 506
12.5.1 Complete PDF closure 506
12.5.2 The log-normal model for the turbulence frequency 507
12.5.3 The gamma-distribution model 511
12.5.4 The model joint PDF equation 514
12.6 The Lagrangian particle method 516
12.6.1 Fluid and particle systems 516
12.6.2 Corresponding equations 519
12.6.3 Estimation of means 523
12.6.4 Summary 526
12.7 Extensions 529
12.7.1 Wall functions 529
12.7.2 The near-wall elliptic-relaxation model 534
12.7.3 The wavevector model 540
12.7.4 Mixing and reaction 545
12.8 Discussion 555
13 Large-eddy simulation 558
13.1 Introduction 558
13.2 Filtering 561
13.2.1 The general definition 561
13.2.2 Filtering in one dimension 562
13.2.3 Spectral representation 565
13.2.4 The filtered energy spectrum 568
13.2.5 The resolution of filtered fields 571
13.2.6 Filtering in three dimensions 575
13.2.7 The filtered rate of strain 578
13.3 Filtered conservation equations 581
13.3.1 Conservation of momentum 581
13.3.2 Decomposition of the residual stress 582
13.3.3 Conservation of energy 585
13.4 The Smagorinsky model 587
13.4.1 The definition of the model 587
13.4.2 Behavior in the inertial subrange 587
13.4.3 The Smagorinsky filter 590
13.4.4 Limiting behaviors 594
13.4.5 Near-wall resolution 598
13.4.6 Tests of model performance 601
13.5 LES in wavenumber space 604
13.5.1 Filtered equations 604
13.5.2 Triad interactions 606
13.5.3 The spectral energy balance 609
13.5.4 The spectral eddy viscosity 610
13.5.5 Backscatter 611
13.5.6 A statistical view of LES 612
13.5.7 Resolution and modelling 615
13.6 Further residual-stress models 619
13.6.1 The dynamic model 619
13.6.2 Mixed models and variants 627
13.6.3 Transport-equation models 629
13.6.4 Implicit numerical filters 631
13.6.5 Near-wall treatments 634
13.7 Discussion 635
13.7.1 An appraisal of LES 635
13.7.2 Final perspectives 638
PART THREE:APPENDICES 641
Appendix A Cartesian tensors 643
A.1 Cartesian coordinates and vectors 643
A.2 The definition of Cartesian tensors 647
A.3 Tensor operations 649
A.4 The vector cross product 654
A.5 A summary of Cartesian-tensor suffix notation 659
Appendix B Properties of second-order tensors 661
Appendix C Dirac delta functions 670
C.1 The definition of δ（x） 670
C.2 Properties of δ（x） 672
C.3 Derivatives of δ（x） 673
C.4 Taylor series 675
C.5 The Heaviside function 675
C.6 Multiple dimensions 677
Appendix D Fourier transforms 678
Appendix E Spectral representation of stationary random processes 683
E.1 Fourier series 683
E.2 Periodic random processes 686
E.3 Non-periodic random processes 689
E.4 Derivatives of the process 690
Appendix F The discrete Fourier transform 692
Appendix G Power-law spectra 696
Appendix H Derivation of Eulerian PDF equations 702
Appendix I Characteristic functions 707
Appendix J Diffusion processes 713
Bibliography 727
Author index 749
Subject index 754