Turbulent Flows. Stephen B. Pope CAMBRIDGE UNIVERSITY PRESS. Cornell University

Transcription

1 Turbulent Flows Stephen B. Pope Cornell University CAMBRIDGE UNIVERSITY PRESS

2 Contents List of tables Preface Nomenclature PART ONE: FUNDAMENTALS 1 Introduction 1.1 The nature of turbulent flows 1.2 The study of turbulent flows 2 The equations of fluid motion 2.1 Continuum fluid properties 2.2 Eulerian and Lagrangian fields 2.3 The continuity equation 2.4 The momentum equation 2.5 The role of pressure 2.6 Conserved passive scalars 2.7 The vorticity equation 2.8 Rates of strain and rotation 2.9 Transformation properties 3 The statistical description of turbulent flows 3.1 The random nature of turbulence 3.2 Characterization of random variables 3.3 Examples of probability distributions 3.4 Joint random variables 3.5 Normal and joint-normal distributions 3.6 Random processes 3.7 Random fields 3.8 Probability and averaging page xv xvii xxi VII

3 viii Contents 4 Mean-flow equations Reynolds equations Reynolds stresses The mean scalar equation Gradient-diffusion and turbulent-viscosity hypotheses 92 5 Free shear flows The round jet: experimental observations A description of the flow The mean velocity field Reynolds stresses The round jet: mean momentum Boundary-layer equations Flow rates of mass, momentum, and energy Self-similarity Uniform turbulent viscosity The round jet: kinetic energy Other self-similar flows The plane jet The plane mixing layer The plane wake The axisymmetric wake Homogeneous shear flow Grid turbulence Further observations A conserved scalar Intermittency PDFs and higher moments Large-scale turbulent motion The scales of turbulent motion The energy cascade and Kolmogorov hypotheses The energy cascade The Kolmogorov hypotheses The energy spectrum Restatement of the Kolmogorov hypotheses Structure functions Two-point correlation Fourier modes Fourier-series representation The evolution of Fourier modes 211

4 Contents ix Wall The kinetic energy of Fourier modes Velocity spectra Definitions and properties Kolmogorov spectra A model spectrum Dissipation spectra The inertial subrange The energy-containing range Effects of the Reynolds number The shear-stress spectrum The : spectral view of the energy cascade Limitations, shortcomings, and refinements The Reynolds number Higher-order statistics Internal intermittency Refined similarity hypotheses Closing remarks flows Channel flow A description of the flow The balance of mean forces The near-wall shear stress Mean velocity profiles The friction law and the Reynolds number Reynolds stresses Lengthscales and the mixing length Pipe flow The friction law for smooth pipes Wall roughness Boundary layers A description of the flow Mean-momentum equations Mean velocity profiles The overlap region reconsidered Reynolds-stress balances Additional effects Turbulent structures

5 x Contents PART TWO: MODELLING AND SIMULATION An introduction to modelling and simulation The challenge An overview of approaches Criteria for appraising models Direct numerical simulation Homogeneous turbulence Pseudo-spectral methods The computational cost Artificial modifications and incomplete resolution Inhomogeneous flows Channel flow Free shear flows Flow over a backward-facing step Discussion Turbulent-viscosity models The turbulent-viscosity hypothesis The intrinsic assumption The specific assumption Algebraic models Uniform turbulent viscosity The mixing-length model Turbulent-kinetic-energy models The k-e model An overview The model equation for e Discussion Further turbulent-viscosity models The k-co model The Spalart-Allmaras model Reynolds-stress and related models Introduction The pressure-rate-of-strain tensor Return-to-isotropy models Rotta's model The characterization of Reynolds-stress anisotropy Nonlinear return-to-isotropy models Rapid-distortion theory Rapid-distortion equations 405

6 Contents xi The evolution of a Fourier mode The evolution of the spectrum Rapid distortion of initially isotropic turbulence Final remarks Pressure-rate-of-strain models The basic model (LRR-IP) Other pressure-rate-of-strain models Extension to inhomogeneous flows Redistribution Reynolds-stress transport The dissipation equation Near-wall treatments Near-wall effects Turbulent viscosity Model equations for k and s The dissipation tensor Fluctuating pressure Wall functions Elliptic relaxation models Algebraic stress and nonlinear viscosity models Algebraic stress models Nonlinear turbulent viscosity Discussion PDF methods The Eulerian PDF of velocity Definitions and properties The PDF transport equation The PDF of the fluctuating velocity The model velocity PDF equation The generalized Langevin model The evolution of the PDF Corresponding Reynolds-stress models Eulerian and Lagrangian modelling approaches Relationships between Lagrangian and Eulerian PDFs Langevin equations Stationary isotropic turbulence The generalized Langevin model Turbulent dispersion 494

7 xii Contents 12.5 The velocity-frequency joint PDF Complete PDF closure The log-normal model for the turbulence frequency The gamma-distribution model The model joint PDF equation The Lagrangian particle method ,6.1 Fluid and particle systems :6.2 Corresponding equations Estimation of means Summary Extensions Wall functions The near-wall elliptic-relaxation model The wavevector model Mixing and reaction Discussion Large-eddy simulation Introduction Filtering The general definition Filtering in one dimension Spectral representation The filtered energy spectrum The resolution of filtered fields Filtering in three dimensions The filtered rate of strain Filtered conservation equations Conservation of momentum Decomposition of the residual stress Conservation of energy The Smagorinsky model The definition of the model Behavior in the inertial subrange The Smagorinsky filter Limiting behaviors Near-wall resolution Tests of model performance LES in wavenumber space Filtered equations 604

8 Contents xiii Triad interactions The spectral energy balance The spectral eddy viscosity Backscatter A statistical view of LES Resolution and modelling Further residual-stress models The dynamic model Mixed models and variants Transport-equation models Implicit numerical filters Near-wall treatments Discussion An appraisal of LES Final perspectives 638 PART THREE: APPENDICES 641 Appendix A Cartesian tensors 643 A.I 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.I The definition of 3(x) 670 C.2 Properties of 6{x) 672 C.3 Derivatives of 5(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.I 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

9 xiv Contents 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