Turbulent Flows Stephen B. Pope Cornell University
CAMBRIDGE UNIVERSITY PRESS
Contents
List of tables Preface Nomenclature
page xv xvii xxi
PART ONE: FUNDAMENTALS
1
1
Introduction 1.1 The nature of turbulent flows 1.2 The study of turbulent flows
3 3 7
2
The 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
equations of fluid motion Continuum fluid properties Eulerian and Lagrangian fields The continuity equation The momentum equation The role of pressure Conserved passive scalars The vorticity equation Rates of strain and rotation Transformation properties
10 10 12 14 16 18 21 22 23 24
3
The 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8
statistical description of turbulent flows The random nature of turbulence Characterization of random variables Examples of probability distributions Joint random variables Normal and joint-normal distributions Random processes Random fields Probability and averaging
34 34 37 43 54 61 65 74 79
VII
viii
Contents
4
Mean-flow equations 4.1 Reynolds equations 4.2 Reynolds stresses 4.3 The mean scalar equation 4.4 Gradient-diffusion and turbulent-viscosity hypotheses
83 83 86 91 92
5
Free shear flows 5.1 The round jet: experimental observations 5.1.1 A description of the flow 5.1.2 The mean velocity field 5.1.3 Reynolds stresses 5.2 The round jet: mean momentum 5.2.1 Boundary-layer equations 5.2.2 Flow rates of mass, momentum, and energy 5.2.3 Self-similarity 5.2.4 Uniform turbulent viscosity 5.3 The round jet: kinetic energy 5.4 Other self-similar flows 5.4.1 The plane jet 5.4.2 The plane mixing layer 5.4.3 The plane wake 5.4.4 The axisymmetric wake 5.4.5 Homogeneous shear flow 5.4.6 Grid turbulence 5.5 Further observations 5.5.1 A conserved scalar 5.5.2 Intermittency 5.5.3 PDFs and higher moments 5.5.4 Large-scale turbulent motion
The scales of turbulent motion 6.1 The energy cascade and Kolmogorov hypotheses 6.1.1 The energy cascade 6.1.2 The Kolmogorov hypotheses 6.1.3 The energy spectrum 6.1.4 Restatement of the Kolmogorov hypotheses 6.2 Structure functions 6.3 Two-point correlation 6.4 Fourier modes 6.4.1 Fourier-series representation 6.4.2 The evolution of Fourier modes
182 182 183 184 188 189 191 195 207 207 211
Contents
6.5
6.6 6.7
7
6.4.3 The kinetic energy of Fourier modes Velocity spectra 6.5.1 Definitions and properties 6.5.2 Kolmogorov spectra 6.5.3 A model spectrum 6.5.4 Dissipation spectra 6.5.5 The inertial subrange 6.5.6 The energy-containing range 6.5.7 Effects of the Reynolds number 6.5.8 The shear-stress spectrum The :spectral view of the energy cascade Limitations, shortcomings, and refinements 6.7.1 The Reynolds number 6.7.2 Higher-order statistics 6.7.3 Internal intermittency 6.7.4 Refined similarity hypotheses 6.7.5 Closing remarks
Wall flows
7.1
7.2
7.3
7.4
Channel flow 7.1.1 A description of the flow 7.1.2 The balance of mean forces 7.1.3 The near-wall shear stress 7.1.4 Mean velocity profiles 7.1.5 The friction law and the Reynolds number 7.1.6 Reynolds stresses 7.1.7 Lengthscales and the mixing length Pipe flow 7.2.1 The friction law for smooth pipes 7.2.2 Wall roughness Boundary layers 7.3.1 A description of the flow 7.3.2 Mean-momentum equations 7.3.3 Mean velocity profiles 7.3.4 The overlap region reconsidered 7.3.5 Reynolds-stress balances 7.3.6 Additional effects Turbulent structures
An introduction to modelling and simulation 8.1 The challenge 8.2 An overview of approaches 8.3 Criteria for appraising models
335 335 336 336
9
Direct numerical simulation 9.1 Homogeneous turbulence 9.1.1 Pseudo-spectral methods 9.1.2 The computational cost 9.1.3 Artificial modifications and incomplete resolution 9.2 Inhomogeneous flows 9.2.1 Channel flow 9.2.2 Free shear flows 9.2.3 Flow over a backward-facing step 9.3 Discussion
344 344 344 346 352 353 353 354 355 356
10 Turbulent-viscosity models 10.1 The turbulent-viscosity hypothesis 10.1.1 The intrinsic assumption 10.1.2 The specific assumption 10.2 Algebraic models 10.2.1 Uniform turbulent viscosity 10.2.2 The mixing-length model 10.3 Turbulent-kinetic-energy models 10.4 The k-e model 10.4.1 An overview 10.4.2 The model equation for e 10.4.3 Discussion 10.5 Further turbulent-viscosity models 10.5.1 The k-co model 10.5.2 The Spalart-Allmaras model
11 Reynolds-stress and related models 11.1 Introduction 11.2 The pressure-rate-of-strain tensor 11.3 Return-to-isotropy models 11.3.1 Rotta's model 11.3.2 The characterization of Reynolds-stress anisotropy 11.3.3 Nonlinear return-to-isotropy models 11.4 Rapid-distortion theory 11.4.1 Rapid-distortion equations
387 387 388 392 392 393 398 404 405
Contents
11.5
11.6
11.7
11.8 11.9
11.10
11.4.2 The evolution of a Fourier mode 11.4.3 The evolution of the spectrum 11.4.4 Rapid distortion of initially isotropic turbulence 11.4.5 Final remarks Pressure-rate-of-strain models 11.5.1 The basic model (LRR-IP) 11.5.2 Other pressure-rate-of-strain models Extension to inhomogeneous flows 11.6.1 Redistribution 11.6.2 Reynolds-stress transport 11.6.3 The dissipation equation Near-wall treatments 11.7.1 Near-wall effects 11.7.2 Turbulent viscosity 11.7.3 Model equations for k and s 11.7.4 The dissipation tensor 11.7.5 Fluctuating pressure 11.7.6 Wall functions Elliptic relaxation models Algebraic stress and nonlinear viscosity models 11.9.1 Algebraic stress models 11.9.2 Nonlinear turbulent viscosity Discussion
12 PDF methods 12.1 The Eulerian PDF of velocity 12.1.1 Definitions and properties 12.1.2 The PDF transport equation 12.1.3 The PDF of the fluctuating velocity 12.2 The model velocity PDF equation 12.2.1 The generalized Langevin model 12.2.2 The evolution of the PDF 12.2.3 Corresponding Reynolds-stress models 12.2.4 Eulerian and Lagrangian modelling approaches 12.2.5 Relationships between Lagrangian and Eulerian PDFs 12.3 Langevin equations 12.3.1 Stationary isotropic turbulence 12.3.2 The generalized Langevin model 12.4 Turbulent dispersion
12.5 The velocity-frequency joint PDF 12.5.1 Complete PDF closure 12.5.2 The log-normal model for the turbulence frequency 12.5.3 The gamma-distribution model 12.5.4 The model joint PDF equation 12.6 The Lagrangian particle method 12,6.1 Fluid and particle systems 12:6.2 Corresponding equations 12.6.3 Estimation of means 12.6.4 Summary 12.7 Extensions 12.7.1 Wall functions 12.7.2 The near-wall elliptic-relaxation model 12.7.3 The wavevector model 12.7.4 Mixing and reaction 12.8 Discussion 13 Large-eddy simulation 13.1 Introduction 13.2 Filtering 13.2.1 The general definition 13.2.2 Filtering in one dimension 13.2.3 Spectral representation 13.2.4 The filtered energy spectrum 13.2.5 The resolution of filtered 13.2.6 Filtering in three dimensions 13.2.7 The filtered rate of strain 13.3 Filtered conservation equations 13.3.1 Conservation of momentum 13.3.2 Decomposition of the residual stress 13.3.3 Conservation of energy 13.4 The Smagorinsky model 13.4.1 The definition of the model 13.4.2 Behavior in the inertial subrange 13.4.3 The Smagorinsky filter 13.4.4 Limiting behaviors 13.4.5 Near-wall resolution 13.4.6 Tests of model performance 13.5 LES in wavenumber space 13.5.1 Filtered equations
Contents 13.5.2 Triad interactions 13.5.3 The spectral energy balance 13.5.4 The spectral eddy viscosity 13.5.5 Backscatter 13.5.6 A statistical view of LES 13.5.7 Resolution and modelling 13.6 Further residual-stress models 13.6.1 The dynamic model 13.6.2 Mixed models and variants 13.6.3 Transport-equation models 13.6.4 Implicit numerical 13.6.5 Near-wall treatments 13.7 Discussion 13.7.1 An appraisal of LES 13.7.2 Final perspectives
xiii
filters
PART THREE: APPENDICES Appendix A.I A.2 A.3 A.4 A.5
A Cartesian tensors Cartesian coordinates and vectors The definition of Cartesian tensors Tensor operations The vector cross product A summary of Cartesian-tensor suffix notation
C Dirac delta functions The definition of 3(x) Properties of 6{x) Derivatives of 5(x) Taylor series The Heaviside function Multiple dimensions
Appendix D Fourier transforms
678
Appendix E.I E.2 E.3 E.4
683 683 686 689 690
E Spectral representation of stationary random processes Fourier series Periodic random processes Non-periodic random processes Derivatives of the process
