TOC-Pope-Turbulent Flow

* pope- turbulent flow
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PART ONE: FUNDAMENTALS
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

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

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

6 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
6.4.3 The kinetic energy of Fourier modes

6.5 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
6.6 The spectral view of the energy cascade

6.7 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

7 wall flows
7.1 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
7.2 Pipe flow
7.2.1 The friction law for smooth pipes
7.2.2 Wall roughness
7.3 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
7.4 Turbulent structures

PART TWO: MODELLING AND SIMULATION
8 An introduction to modelling and simulation
8.1 The challenge
8.2 An overview of approaches
8.3 Criteria for appraising models

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

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–ε model
10.4.1 An overview
10.4.2 The model equation for ε
10.4.3 Discussion
10.5 Further turbulent-viscosity models
10.5.1 The k–ω 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
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
11.5 Pressure–rate-of-strain models
11.5.1 The basic model (LRR-IP)
11.5.2 Other pressure–rate-of-strain models
11.6 Extension to inhomogeneous flows
11.6.1 Redistribution
11.6.2 Reynolds-stress transport
11.6.3 The dissipation equation
11.7 Near-wall treatments
11.7.1 Near-wall effects
11.7.2 Turbulent viscosity
11.7.3 Model equations for k and ε
11.7.4 The dissipation tensor
11.7.5 Fluctuating pressure
11.7.6 Wall functions
11.8 Elliptic relaxation models
11.9 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

12.5The 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
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 fields
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
13.613.713.5.2 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
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 filters
13.6.5 Near-wall treatments
Discussion
13.7.1 An appraisal of LES
13.7.2 Final perspectives
PART THREE: APPENDICES

Appendix A Cartesian tensors
A.1  Cartesian coordinates and vectors
A.2  The definition of Cartesian tensors
A.3 Tensor operations
A.4  The vector cross product
A.5 A summary of Cartesian-tensor suffix notation

Appendix B Properties of second-order tensors

Appendix  C Dirac delta functions
C.1  The definition of δ(x)
C.2  Properties of δ(x)
C.3  Derivatives of δ(x)
C.4  Taylor series
C.5  The Heaviside function
C.6  Multiple dimensions
Appendix D Fourier transforms
Appendix  E Spectral representation of stationary random processes
E.1  Fourier series
E.2  Periodic random processes
E.3  Non-periodic random processes
E.4  Derivatives of the process
Appendix F The discrete Fourier transform
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