Statistical Theory and Modeling for Turbulent Flows

Statistical Theory and Modeling for Turbulent Flows

Contents

Preface

Preface to second edition

Preface to first edition

Motivation

Epitome

Acknowledgements

Part I FUNDAMENTALS OF TURBULENCE

1 Introduction

1.1 The turbulence problem

1.2 Closure modeling

1.3 Categories of turbulent flow

Exercises

2 Mathematical and statistical background

2.1 Dimensional analysis

2.1.1 Scales of turbulence

2.2 Statistical tools

2.2.1 Averages and probability density functions

2.2.2 Correlations

2.3 Cartesian tensors

2.3.1 Isotropic tensors

2.3.2 Tensor functions of tensors; Cayley–Hamilton theorem

Exercises

3 Reynolds averaged Navier–Stokes equations

3.1 Background to the equations

3.2 Reynolds averaged equations

3.3 Terms of kinetic energy and Reynolds stress budgets

3.4 Passive contaminant transport

Exercises

4 Parallel and self-similar shear flows

4.1 Plane channel flow

4.1.1 Logarithmic layer

4.1.2 Roughness

4.2 Boundary layer

4.2.1 Entrainment

4.3 Free-shear layers

4.3.1 Spreading rates

4.3.2 Remarks on self-similar boundary layers

4.4 Heat and mass transfer

4.4.1 Parallel flow and boundary layers

4.4.2 Dispersion from elevated sources

Exercises

5 Vorticity and vortical structures

5.1 Structures

5.1.1 Free-shear layers

5.1.2 Boundary layers

5.1.3 Non-random vortices

5.2 Vorticity and dissipation

5.2.1 Vortex stretching and relative dispersion

5.2.2 Mean-squared vorticity equation

Exercises

Part II SINGLE-POINT CLOSURE MODELING

6 Models with scalar variables

6.1 Boundary-layer methods

6.1.1 Integral boundary-layer methods

6.1.2 Mixing length model

6.2 The k –å model

6.2.1 Analytical solutions to the k –å model

6.2.2 Boundary conditions and near-wall modifications

6.2.3 Weak solution at edges of free-shear flow; free-stream sensitivity

6.3 The k –ù model

6.4 Stagnation-point anomaly

6.5 The question of transition

6.5.1 Reliance on the turbulence model

6.5.2 Intermittency equation

6.5.3 Laminar fluctuations

6.6 Eddy viscosity transport models

Exercises

7 Models with tensor variables

7.1 Second-moment transport

7.1.1 A simple illustration

7.1.2 Closing the Reynolds stress transport equation

7.1.3 Models for the slow part

7.1.4 Models for the rapid part

7.2 Analytic solutions to SMC models

7.2.1 Homogeneous shear flow

7.2.2 Curved shear flow

7.2.3 Algebraic stress approximation and nonlinear eddy viscosity

7.3 Non-homogeneity

7.3.1 Turbulent transport

7.3.2 Near-wall modeling

7.3.3 No-slip condition

7.3.4 Nonlocal wall effects

7.4 Reynolds averaged computation

7.4.1 Numerical issues

7.4.2 Examples of Reynolds averaged computation

Exercises

8 Advanced topics

8.1 Further modeling principles

8.1.1 Galilean invariance and frame rotation

8.1.2 Realizability

8.2 Second-moment closure and Langevin equations

8.3 Moving equilibrium solutions of SMC

8.3.1 Criterion for steady mean flow

8.3.2 Solution in two-dimensional mean flow

8.3.3 Bifurcations

8.4 Passive scalar flux modeling

8.4.1 Scalar diffusivity models

8.4.2 Tensor diffusivity models

8.4.3 Scalar flux transport

8.4.4 Scalar variance

8.5 Active scalar flux modeling: effects of buoyancy

8.5.1 Second-moment transport models

8.5.2 Stratified shear flow

Exercises

Part III THEORY OF HOMOGENEOUS TURBULENCE

9 Mathematical representations

9.1 Fourier transforms

9.2 Three-dimensional energy spectrum of homogeneous turbulence

9.2.1 Spectrum tensor and velocity covariances

9.2.2 Modeling the energy spectrum

Exercises

10 Navier–Stokes equations in spectral space

10.1 Convolution integrals as triad interaction

10.2 Evolution of spectra

10.2.1 Small-k behavior and energy decay

10.2.2 Energy cascade

10.2.3 Final period of decay

Exercises

11 Rapid distortion theory

11.1 Irrotational mean flow

11.1.1 Cauchy form of vorticity equation

11.1.2 Distortion of a Fourier mode

11.1.3 Calculation of covariances

11.2 General homogeneous distortions

11.2.1 Homogeneous shear

11.2.2 Turbulence near a wall

Exercises

Part IV TURBULENCE SIMULATION

12 Eddy-resolving simulation

12.1 Direct numerical simulation

12.1.1 Grid requirements

12.1.2 Numerical dissipation

12.1.3 Energy-conserving schemes

12.2 Illustrations

12.3 Pseudo-spectral method

Exercises

13 Simulation of large eddies

13.1 Large eddy simulation

13.1.1 Filtering

13.1.2 Subgrid models

13.2 Detached eddy simulation

Exercises

References

Index