Turbulent Flows

by Stephen B. Pope

• ISBN13: 9780521591256
• ISBN10: 0521591252
• Format: Hardcover
• Copyright: 2000-09-25
• Publisher: Cambridge University Press

Summary

Turbulent Flows is an up-to-date and comprehensive graduate text on this important topic in fluid dynamics. The book consists of two parts: Part I provides a general introduction to turbulent flows, how they behave, how they can be described quantitatively, and their fundamental physical processes. Part II is concerned with different approaches for modeling, or simulating, turbulent flows. Key appendices present the necessary mathematical techniques. While primarily intended for engineering graduate students, this book will also be valuable to students in applied mathematics, physics, oceanography and atmospheric sciences, as well as to researchers and practicing engineers.

Table of Contents

List of tables	xv
Preface	xvii
Nomenclature	xxi
PART ONE: FUNDAMENTALS	1	(332)
Introduction	3	(7)
The nature of turbulent flows	3	(4)
The study of turbulent flows	7	(3)
The equations of fluid motion	10	(24)
Continuum fluid properties	10	(2)
Eulerian and Lagrangian fields	12	(2)
The continuity equation	14	(2)
The momentum equation	16	(2)
The role of pressure	18	(3)
Conserved passive scalars	21	(1)
The vorticity equation	22	(1)
Rates of strain and rotation	23	(1)
Transformation properties	24	(10)
The statistical description of turbulent flows	34	(49)
The random nature of turbulence	34	(3)
Characterization of random variables	37	(6)
Examples of probability distributions	43	(11)
Joint random variables	54	(7)
Normal and joint-normal distributions	61	(4)
Random processes	65	(9)
Random fields	74	(5)
Probability and averaging	79	(4)
Mean-flow equations	83	(13)
Reynolds equations	83	(3)
Reynolds stresses	86	(5)
The mean scalar equation	91	(1)
Gradient-diffusion and turbulent-viscosity hypotheses	92	(4)
Free shear flows	96	(86)
The round jet: experimental observations	96	(15)
A description of the flow	96	(1)
The mean velocity field	97	(8)
Reynolds stresses	105	(6)
The round jet: mean momentum	111	(11)
Boundary-layer equations	111	(4)
Flow rates of mass, momentum, and energy	115	(1)
Self-similarity	116	(2)
Uniform turbulent viscosity	118	(4)
The round jet: kinetic energy	122	(12)
Other self-similar flows	134	(27)
The plane jet	134	(5)
The plane mixing layer	139	(8)
The plane wake	147	(4)
The axisymmetric wake	151	(3)
Homogeneous shear flow	154	(4)
Grid turbulence	158	(3)
Further observations	161	(21)
A conserved scalar	161	(6)
Intermittency	167	(6)
PDFs and higher moments	173	(5)
Large-scale turbulent motion	178	(4)
The scales of turbulent motion	182	(82)
The energy cascade and Kolmogorov hypotheses	182	(9)
The energy cascade	183	(1)
The Kolmogorov hypotheses	184	(4)
The energy spectrum	188	(1)
Restatement of the Kolmogorov hypotheses	189	(2)
Structure functions	191	(4)
Two-point correlation	195	(12)
Fourier modes	207	(12)
Fourier-series representation	207	(4)
The evolution of Fourier modes	211	(4)
The kinetic energy of Fourier modes	215	(4)
Velocity spectra	219	(30)
Definitions and properties	220	(9)
Kolmogorov spectra	229	(3)
A model spectrum	232	(2)
Dissipation spectra	234	(4)
The inertial subrange	238	(2)
The energy-containing range	240	(2)
Effects of the Reynolds number	242	(4)
The shear-stress spectrum	246	(3)
The spectral view of the energy cascade	249	(5)
Limitations, shortcomings, and refinements	254	(10)
The Reynolds number	254	(1)
Higher-order statistics	255	(3)
Internal intermittency	258	(2)
Refined similarity hypotheses	260	(3)
Closing remarks	263	(1)
Wall flows	264	(69)
Channel flow	264	(26)
A description of the flow	264	(2)
The balance of mean forces	266	(2)
The near-wall shear stress	268	(3)
Mean velocity profiles	271	(7)
The friction law and the Reynolds number	278	(3)
Reynolds stresses	281	(7)
Lengthscales and the mixing length	288	(2)
Pipe flow	290	(8)
The friction law for smooth pipes	290	(5)
Wall roughness	295	(3)
Boundary layers	298	(24)
A description of the flow	299	(1)
Mean-momentum equations	300	(2)
Mean velocity profiles	302	(6)
The overlap region reconsidered	308	(5)
Reynolds-stress balances	313	(7)
Additional effects	320	(2)
Turbulent structures	322	(11)
PART TWO: MODELLING AND SIMULATION	333	(308)
An introduction to modelling and simulation	335	(9)
The challenge	335	(1)
An overview of approaches	336	(1)
Criteria for appraising models	336	(8)
Direct numerical simulation	344	(14)
Homogeneous turbulence	344	(9)
Pseudo-spectral methods	344	(2)
The computational cost	346	(6)
Artificial modifications and incomplete resolution	352	(1)
Inhomogeneous flows	353	(3)
Channel flow	353	(1)
Free shear flows	354	(1)
Flow over a backward-facing step	355	(1)
Discussion	356	(2)
Turbulent-viscosity models	358	(29)
The turbulent-viscosity hypothesis	359	(6)
The intrinsic assumption	359	(5)
The specific assumption	364	(1)
Algebraic models	365	(4)
Uniform turbulent viscosity	365	(1)
The mixing-length model	366	(3)
Turbulent-kinetic-energy models	369	(4)
The k-ε model	373	(10)
An overview	373	(2)
The model equation for ε	375	(7)
Discussion	382	(1)
Further turbulent-viscosity models	383	(4)
The k-ω model	383	(2)
The Spalart-Allmaras model	385	(2)
Reynolds-stress and related models	387	(76)
Introduction	387	(1)
The pressure-rate-of-strain tensor	388	(4)
Return-to-isotropy models	392	(12)
Rotta's model	392	(1)
The characterization of Reynolds-stress anisotropy	393	(5)
Nonlinear return-to-isotropy models	398	(6)
Rapid-distortion theory	404	(18)
Rapid-distortion equations	405	(1)
The evolution of a Fourier mode	406	(5)
The evolution of the spectrum	411	(4)
Rapid distortion of initially isotropic turbulence	415	(6)
Final remarks	421	(1)
Pressure-rate-of-strain models	422	(6)
The basic model (LRR-IP)	423	(2)
Other pressure-rate-of-strain-models	425	(3)
Extension to inhomogeneous flows	428	(5)
Redistribution	428	(1)
Reynolds-stress transport	429	(3)
The dissipation equation	432	(1)
Near-wall treatments	433	(12)
Near-wall effects	433	(1)
Turbulent viscosity	434	(1)
Model equations for k and ε	435	(1)
The dissipation tensor	436	(3)
Fluctuating pressure	439	(3)
Wall functions	442	(3)
Elliptic relaxation models	445	(3)
Algebraic stress and nonlinear viscosity models	448	(9)
Algebraic stress models	448	(4)
Nonlinear turbulent viscosity	452	(5)
Discussion	457	(6)
PDF methods	463	(95)
The Eulerian PDF of velocity	464	(4)
Definitions and properties	464	(1)
The PDF transport equation	465	(2)
The PDF of the fluctuating velocity	467	(1)
The model velocity PDF equation	468	(15)
The generalized Langevin model	469	(1)
The evolution of the PDF	470	(5)
Corresponding Reynolds-stress models	475	(4)
Eulerian and Lagrangian modelling approaches	479	(1)
Relationships between Lagrangian and Eulerian PDFs	480	(3)
Langevin equations	483	(11)
Stationary isotropic turbulence	484	(5)
The generalized Langevin model	489	(5)
Turbulent dispersion	494	(12)
The velocity-frequency joint PDF	506	(10)
Complete PDF closure	506	(1)
The log-normal model for the turbulence frequency	507	(4)
The gamma-distribution model	511	(3)
The model joint PDF equation	514	(2)
The Lagrangian particle method	516	(13)
Fluid and particle systems	516	(3)
Corresponding equations	519	(4)
Estimation of means	523	(3)
Summary	526	(3)
Extensions	529	(26)
Wall functions	529	(5)
The near-wall elliptic-relaxation model	534	(6)
The wavevector model	540	(5)
Mixing and reaction	545	(10)
Discussion	555	(3)
Large-eddy simulation	558	(83)
Introduction	558	(3)
Filtering	561	(20)
The general definition	561	(1)
Filtering in one dimension	562	(3)
Spectral representation	565	(3)
The filtered energy spectrum	568	(3)
The resolution of filtered fields	571	(4)
Filtering in three dimensions	575	(3)
The filtered rate of strain	578	(3)
Filtered conservation equations	581	(6)
Conservation of momentum	581	(1)
Decomposition of the residual stress	582	(3)
Conservation of energy	585	(2)
The Smagorinsky model	587	(17)
The definition of the model	587	(1)
Behavior in the inertial subrange	587	(3)
The Smagorinsky filter	590	(4)
Limiting behaviors	594	(4)
Near-wall resolution	598	(3)
Tests of model performance	601	(3)
LES in wavenumber space	604	(15)
Filtered equations	604	(2)
Triad interactions	606	(3)
The spectral energy balance	609	(1)
The spectral eddy viscosity	610	(1)
Backscatter	611	(1)
A statistical view of LES	612	(3)
Resolution and modelling	615	(4)
Further residual-stress models	619	(16)
The dynamic model	619	(8)
Mixed models and variants	627	(2)
Transport-equation models	629	(2)
Implicit numerical filters	631	(3)
Near-wall treatments	634	(1)
Discussion	635	(6)
An appraisal of LES	635	(3)
Final perspectives	638	(3)
PART THREE: APPENDICES	641	(86)
Appendix A Cartesian tensors	643	(18)
A.1 Cartesian coordinates and vectors	643	(4)
A.2 The definition of Cartesian tensors	647	(2)
A.3 Tensor operations	649	(5)
A.4 The vector cross product	654	(5)
A.5 A summary of Cartesian-tensor suffix notation	659	(2)
Appendix B Properties of second-order tensors	661	(9)
Appendix C Dirac delta functions	670	(8)
C.1 The definition of δ(x)	670	(2)
C.2 Properties of δ(x)	672	(1)
C.3 Derivatives of δ(x)	673	(2)
C.4 Taylor series	675	(1)
C.5 The Heaviside function	675	(2)
C.6 Multiple dimensions	677	(1)
Appendix D Fourier transforms	678	(5)
Appendix E Spectral representation of stationary random processes	683	(9)
E.1 Fourier series	683	(3)
E.2 Periodic random processes	686	(3)
E.3 Non-periodic random processes	689	(1)
E.4 Derivatives of the process	690	(2)
Appendix F The discrete Fourier transform	692	(4)
Appendix G Power-law spectra	696	(6)
Appendix H Derivation of Eulerian PDF equations	702	(5)
Appendix I Characteristic functions	707	(6)
Appendix J Diffusion processes	713	(14)
Bibliography	727	(22)
Author index	749	(5)
Subject index	754