Oxford University Press Inc. , New York, 1991. 572 p.
Lecturer, Department of Physics University of Edinburgh
NOTATION
1 The Semi-Empirical Picture of Turbulent Shear Flow
The equations of fluid motion
A brief statement of the problem
The statistical formulation
Equations for the mean velocity
The energy balance equation for fluctuations
Two-dimensional mean flow as a special case
The boundary-layer equations of motion
The turbulent boundary layer: length and velocity scales
Universal mean velocity distribution near a solid surface
A simple way of calculating the friction drag due to a boundary layer
Empirical relationships for resistance to flow through ducts
Semi-empirical theoretical methods
The eddy-viscosity hypothesis
The Prandtl mixing-length model for flow near a solid boundary
The mixing-length model applied to a free jet
Some experimental results for shear flows
Mean and root mean square velocity distributions in duct flow
The turbulent energy balance
Mean and root mean square velocity distributions in a plane jet
Further reading
Note
References
2 The Fundamental Approach
The Navier-Stokes equation in solenoidal form
The general statistical formulation
Statistical equations and the closure problem
Two-point correlations
Reduction of the statistical equations to the form for channel flow as an example
Homogeneous isotropic turbulence
Isotropic form of the two-point correlation tensor
Length scales for isotropic turbulence
Stationary turbulence
Fourier analysis of the turbulent velocity field
The solenoidal Navier-Stokes equations
Homogeneity and the velocity-field moments
Limit of infinite system volume
The isotropic spectrum tensor
The Taylor hypothesis and the one-dimensional spectrum
Statistical equations in wavenumber space: many-time moments
Statistical equations in wavenumber space: single-time moments
The energy cascade in isotropic turbulence
The energy balance equation
Spectral picture: the Kolmogorov hypotheses
Interpretation in terms of vortex stretching
Closure approximations
The Heisenberg effective viscosity theory
The quasi-normality hypothesis
Some representative experimental results for spectra
One-dimensional energy spectra
The Kolmogorov constant
Energy transfer in wavenumber space
Further reading
References
3 Some Recent Developments in the Study of Turbulence
Measurement techniques and data analysis
Anemometry
The laser anemometer
Time series analysis and computer data processing
Intermittency and the turbulent bursting process
Fine-structure intermittency
Intermittency and the energy cascade
Intermittent generation of turbulence: the bursting process
Numerical computation of turbulent flows
Direct numerical simulation (DNS)
Large-eddy simulation (LES)
The use of the Reynolds-averaged Navier—Stokes equations for practical calculations
An example of a two-equation turbulence model
Turbulent drag reduction by additives
Historical background
Polymer properties
The threshold effect
Maximum drag reduction
Drag reduction in fibre suspensions
Renormalization methods and the closure problem
Renormalization methods in statistical physics
Renormalized perturbation theory
Renormalization group (RG) methods
References
4 Statistical Formulation of the General Problem
Turbulence in the context of classical statistical mechanics
Statistical mechanics of the classical JV-particle system
Kinetic equations in statistical mechanics
The difficulties involved in generalizing statistical mechanics to include turbulence
Functional formalisms for the turbulence problem
The space-time functional formalism
The k-space-time functional formalism
The Hopf equation for the characteristic functional
General remarks on functional formalisms
Test problems in isotropic turbulence
Free decay of turbulence
Stationary isotropic turbulence
Further reading
Note
References
5 Renormalized Perturbation Theory and the Turbulence Closure Problem
Time evolution and propagators
Perturbation methods using Feynman-type diagrams
Equilibrium system with weak interactions: an introduction to renormalized perturbation theory
Interactions in dilute systems and the connection with macroscopic thermodynamics
Primitive perturbation expansion for the configuration integral
Renormalized expansion for the free energy
The electron gas: an example with long-range forces
Phenomenological theory: the screened potential
Perturbation calculation of the free energy
Perturbation expansion of the Navier-Stokes equation
The zero-order isotropic propagators
The primitive perturbation expansion
Graphical representation of the perturbation series
Class A diagrams: the renormalized propagator
Class B diagrams: renormalized perturbation series
Second-order closures
Notes
References
6 Renormalized Perturbation Theories of the First Kind
The direct-interaction approximation (DIA)
The infinitesimal response tensor
Perturbation expansion of the velocity field
Perturbation series for the mean-response and correlation tensors
Second-order equations for the isotropic response and correlation functions
Spectral transport of energy: the inertial range
The DIA energy spectrum in the inertial range
Alteative derivation of DIA by the method of reversion of power series
Concluding remarks
The Edwards-Fokker-Planck theory
The derivation of the Liouville equation
The Edwards-Fokker-Planck equation
Evaluation of the coefficients in the expansion for the probability distribution of velocities
The energy-balance equation
The response equation
Comparison with the DIA
The limit of infinite Reynolds number
Self-consistent field theory
Time-dependent SCF
Other self-consistent methods
Note
References
7 Renormalized Perturbation Theories of the Second Kind
The low-wavenumber catastrophe
The infra-red divergence
Spurious convection effects
Postulate of random Galilean invariance
Response integrals with an arbitrary cut-off in wavenumber
Lagrangian-history direct-interaction theories
The Lagrangian-history formulation
The statistical formulation
DIA adapted to Lagrangian-history coordinates
Abridged LHDI theory
Other Lagrangian theories
Modified EFP theories.
Maximal entropy principle
The response function determined by a local energy balance
Local energy-transfer equations
Local energy-transfer theory of non-stationary turbulence (LET)
The velocity-field propagator
The generalized covariance equation
Equations for the correlation and propagator functions
Comparison with DIA
Near-Markovian model closures
Quasi-normal Markovian approximations
The test-field model
Note
References
8 An Assessment of Renormalized Perturbation Theories
Free decay of isotropic turbulence as a test problem
Calculations of decaying turbulence at low Reynolds numbers
The direct-interaction approximation (DIA)
Comparison of various theories: Herring and Kraichnan
The LET theory
Calculations of decaying turbulence at high Reynolds numbers
The Kolmogorov spectrum as a test problem
Do intermittency corrections have any bearing on our assessment of RPTs?
Is the Kolmogorov 5/3 law correct after all?
Application to non-isotropic turbulence
Application of DIA to inhomogeneous turbulence
The computational problems
Other applications
Appraisal of the theories
Critique of DIA: the wider justification of RPT approaches
Some comments on random Galilean invariance
General remarks
Postscript: some current work
Notes
References
9 Renormalization Group Theories
Background: RG applied to critical phenomena
Ferromagnetism and the Ising model
Block spins and RG
Space dimension and the epsilon expansion
Application of RG to turbulence
Determination of scaling laws
Subgrid-scale modelling
The Forster-Nelson-Stephen (FNS) theory
Formulation of the problem
The perturbation series
The effective viscosity
Recursion relations
Behaviour near the fixed point
Some later conjectures about FNS theory
Application of RG by iterative averaging
General formulation
Partial averaging of the small scales
The statistical equations of motion
Moment hierarchy from partial averaging
A mean field approximation
The RG equations
Second-order calculation of the effective viscosity
The effect of higher-order moments
Concluding remarks
Notes
References
10 Numerical Simulation of Turbulence
Full simulations
Isotropic turbulence
Shear flows
Large-eddy simulations
Assessment of subgrid models
Application of renormalization methods to the subgrid modelling problem
Formulation of spectral LES
Renormalized perturbation theory
Renormalization group
Miscellaneous simulation methods
Methods based on the Navier-Stokes equation
Alteatives to the Navier-Stokes equation
Note
References
11 Coherent Structures
Coherent structures in free turbulent flows
Plane mixing layers
Other free shear flows
Conditional sampling, intermittency, and the turbulent-non-turbulent interface
Transitional structures in boundary layers and pipes
Anatomy of the turbulent spot
Developed structures in boundary layers and pipes
Turbulent bursts
Frequency of turbulent bursts
Streaky structure and streamwise vortices
Relationship between bursts and other types of Intermittency
Theoretical approaches
Numerical simulation of bounded turbulence
Numerical simulation of free shear layers
Deterministic chaos
Wave theories 4
Implications for other turbulence concepts
Further reading
Note
References
12 Turbulent Diffusion: The Lagrangian Picture
Diffusion by continuous movements
The problem of expressing the Lagrangian analysis in Eulerian coordinates
Statement of the problem
Approximations based on the conjecture of Hay and Pasquill
Approximations based on Corrsin's independence hypothesis
Experimental measurements of Lagrangian quantities
Relative diffusion
Richardson's law
Three-dimensional diffusion
The motion of discrete particles in a turbulent fluid
Some asymptotic results
Tchen's analysis
Applications of Taylor's analysis to shear flows
References
13 Turbulent Diffusion: The Eulerian Picture
Heat and mass transfer
Statistical formulation
Single-point equations
Two-point equations
Some experimental measurements in pipes and jets
Scalar transport in homogeneous turbulence
The inertial-convective range of wavenumbers
Universal forms of the scalar spectrum
Renormalized perturbation theory
The motion of discrete particles
Goveing equations for particles
Interpretation in terms of a diffusion coefficient
Random walk models of particle diffusion
Measurements of particle motion in turbulent flows
Turbulent mixing
Note
References
14 Non- Newtonian Fluid Turbulence
Non-Newtonian fluid flow
Rheological aspects
Composite systems: Newtonian fluid with modified boundary conditions
Flow in pipes
Structural turbulence
Isotropic turbulence
Turbulent structure in drag-reducing polymer solutions
Mean velocity distributions
Turbulent intensities
Spectra and correlations
The importance of the region near the wall
Free turbulence
Turbulent structure in drag-reducing fibre suspensions
Mean velocity distributions
Intensities, correlations, and spectra
Mixed fibre-polymer suspensions
The effect of drag-reducing additives on turbulent transport
Heat and mass transfer
Turbulent diffusion
Comparison of polymers and macroscopic fibres as drag-reducing additives
Further reading
References
APPENDICES
A Creation and dissipation of kinetic energy in a viscous fluid
B Probability and statistics
C Symmetry and invariance
D Application of Fourier methods and Green's functions to the Navier-Stokes equation
E Evaluation of the coefficients L(k,j) and L(k,k—j)
F Optical background to laser-Doppler anemometry
G Second-order term in the perturbation series as an example of the diagram calculus
H The Novikov functional formalism
Author Index
Subject Index
Lecturer, Department of Physics University of Edinburgh
NOTATION
1 The Semi-Empirical Picture of Turbulent Shear Flow
The equations of fluid motion
A brief statement of the problem
The statistical formulation
Equations for the mean velocity
The energy balance equation for fluctuations
Two-dimensional mean flow as a special case
The boundary-layer equations of motion
The turbulent boundary layer: length and velocity scales
Universal mean velocity distribution near a solid surface
A simple way of calculating the friction drag due to a boundary layer
Empirical relationships for resistance to flow through ducts
Semi-empirical theoretical methods
The eddy-viscosity hypothesis
The Prandtl mixing-length model for flow near a solid boundary
The mixing-length model applied to a free jet
Some experimental results for shear flows
Mean and root mean square velocity distributions in duct flow
The turbulent energy balance
Mean and root mean square velocity distributions in a plane jet
Further reading
Note
References
2 The Fundamental Approach
The Navier-Stokes equation in solenoidal form
The general statistical formulation
Statistical equations and the closure problem
Two-point correlations
Reduction of the statistical equations to the form for channel flow as an example
Homogeneous isotropic turbulence
Isotropic form of the two-point correlation tensor
Length scales for isotropic turbulence
Stationary turbulence
Fourier analysis of the turbulent velocity field
The solenoidal Navier-Stokes equations
Homogeneity and the velocity-field moments
Limit of infinite system volume
The isotropic spectrum tensor
The Taylor hypothesis and the one-dimensional spectrum
Statistical equations in wavenumber space: many-time moments
Statistical equations in wavenumber space: single-time moments
The energy cascade in isotropic turbulence
The energy balance equation
Spectral picture: the Kolmogorov hypotheses
Interpretation in terms of vortex stretching
Closure approximations
The Heisenberg effective viscosity theory
The quasi-normality hypothesis
Some representative experimental results for spectra
One-dimensional energy spectra
The Kolmogorov constant
Energy transfer in wavenumber space
Further reading
References
3 Some Recent Developments in the Study of Turbulence
Measurement techniques and data analysis
Anemometry
The laser anemometer
Time series analysis and computer data processing
Intermittency and the turbulent bursting process
Fine-structure intermittency
Intermittency and the energy cascade
Intermittent generation of turbulence: the bursting process
Numerical computation of turbulent flows
Direct numerical simulation (DNS)
Large-eddy simulation (LES)
The use of the Reynolds-averaged Navier—Stokes equations for practical calculations
An example of a two-equation turbulence model
Turbulent drag reduction by additives
Historical background
Polymer properties
The threshold effect
Maximum drag reduction
Drag reduction in fibre suspensions
Renormalization methods and the closure problem
Renormalization methods in statistical physics
Renormalized perturbation theory
Renormalization group (RG) methods
References
4 Statistical Formulation of the General Problem
Turbulence in the context of classical statistical mechanics
Statistical mechanics of the classical JV-particle system
Kinetic equations in statistical mechanics
The difficulties involved in generalizing statistical mechanics to include turbulence
Functional formalisms for the turbulence problem
The space-time functional formalism
The k-space-time functional formalism
The Hopf equation for the characteristic functional
General remarks on functional formalisms
Test problems in isotropic turbulence
Free decay of turbulence
Stationary isotropic turbulence
Further reading
Note
References
5 Renormalized Perturbation Theory and the Turbulence Closure Problem
Time evolution and propagators
Perturbation methods using Feynman-type diagrams
Equilibrium system with weak interactions: an introduction to renormalized perturbation theory
Interactions in dilute systems and the connection with macroscopic thermodynamics
Primitive perturbation expansion for the configuration integral
Renormalized expansion for the free energy
The electron gas: an example with long-range forces
Phenomenological theory: the screened potential
Perturbation calculation of the free energy
Perturbation expansion of the Navier-Stokes equation
The zero-order isotropic propagators
The primitive perturbation expansion
Graphical representation of the perturbation series
Class A diagrams: the renormalized propagator
Class B diagrams: renormalized perturbation series
Second-order closures
Notes
References
6 Renormalized Perturbation Theories of the First Kind
The direct-interaction approximation (DIA)
The infinitesimal response tensor
Perturbation expansion of the velocity field
Perturbation series for the mean-response and correlation tensors
Second-order equations for the isotropic response and correlation functions
Spectral transport of energy: the inertial range
The DIA energy spectrum in the inertial range
Alteative derivation of DIA by the method of reversion of power series
Concluding remarks
The Edwards-Fokker-Planck theory
The derivation of the Liouville equation
The Edwards-Fokker-Planck equation
Evaluation of the coefficients in the expansion for the probability distribution of velocities
The energy-balance equation
The response equation
Comparison with the DIA
The limit of infinite Reynolds number
Self-consistent field theory
Time-dependent SCF
Other self-consistent methods
Note
References
7 Renormalized Perturbation Theories of the Second Kind
The low-wavenumber catastrophe
The infra-red divergence
Spurious convection effects
Postulate of random Galilean invariance
Response integrals with an arbitrary cut-off in wavenumber
Lagrangian-history direct-interaction theories
The Lagrangian-history formulation
The statistical formulation
DIA adapted to Lagrangian-history coordinates
Abridged LHDI theory
Other Lagrangian theories
Modified EFP theories.
Maximal entropy principle
The response function determined by a local energy balance
Local energy-transfer equations
Local energy-transfer theory of non-stationary turbulence (LET)
The velocity-field propagator
The generalized covariance equation
Equations for the correlation and propagator functions
Comparison with DIA
Near-Markovian model closures
Quasi-normal Markovian approximations
The test-field model
Note
References
8 An Assessment of Renormalized Perturbation Theories
Free decay of isotropic turbulence as a test problem
Calculations of decaying turbulence at low Reynolds numbers
The direct-interaction approximation (DIA)
Comparison of various theories: Herring and Kraichnan
The LET theory
Calculations of decaying turbulence at high Reynolds numbers
The Kolmogorov spectrum as a test problem
Do intermittency corrections have any bearing on our assessment of RPTs?
Is the Kolmogorov 5/3 law correct after all?
Application to non-isotropic turbulence
Application of DIA to inhomogeneous turbulence
The computational problems
Other applications
Appraisal of the theories
Critique of DIA: the wider justification of RPT approaches
Some comments on random Galilean invariance
General remarks
Postscript: some current work
Notes
References
9 Renormalization Group Theories
Background: RG applied to critical phenomena
Ferromagnetism and the Ising model
Block spins and RG
Space dimension and the epsilon expansion
Application of RG to turbulence
Determination of scaling laws
Subgrid-scale modelling
The Forster-Nelson-Stephen (FNS) theory
Formulation of the problem
The perturbation series
The effective viscosity
Recursion relations
Behaviour near the fixed point
Some later conjectures about FNS theory
Application of RG by iterative averaging
General formulation
Partial averaging of the small scales
The statistical equations of motion
Moment hierarchy from partial averaging
A mean field approximation
The RG equations
Second-order calculation of the effective viscosity
The effect of higher-order moments
Concluding remarks
Notes
References
10 Numerical Simulation of Turbulence
Full simulations
Isotropic turbulence
Shear flows
Large-eddy simulations
Assessment of subgrid models
Application of renormalization methods to the subgrid modelling problem
Formulation of spectral LES
Renormalized perturbation theory
Renormalization group
Miscellaneous simulation methods
Methods based on the Navier-Stokes equation
Alteatives to the Navier-Stokes equation
Note
References
11 Coherent Structures
Coherent structures in free turbulent flows
Plane mixing layers
Other free shear flows
Conditional sampling, intermittency, and the turbulent-non-turbulent interface
Transitional structures in boundary layers and pipes
Anatomy of the turbulent spot
Developed structures in boundary layers and pipes
Turbulent bursts
Frequency of turbulent bursts
Streaky structure and streamwise vortices
Relationship between bursts and other types of Intermittency
Theoretical approaches
Numerical simulation of bounded turbulence
Numerical simulation of free shear layers
Deterministic chaos
Wave theories 4
Implications for other turbulence concepts
Further reading
Note
References
12 Turbulent Diffusion: The Lagrangian Picture
Diffusion by continuous movements
The problem of expressing the Lagrangian analysis in Eulerian coordinates
Statement of the problem
Approximations based on the conjecture of Hay and Pasquill
Approximations based on Corrsin's independence hypothesis
Experimental measurements of Lagrangian quantities
Relative diffusion
Richardson's law
Three-dimensional diffusion
The motion of discrete particles in a turbulent fluid
Some asymptotic results
Tchen's analysis
Applications of Taylor's analysis to shear flows
References
13 Turbulent Diffusion: The Eulerian Picture
Heat and mass transfer
Statistical formulation
Single-point equations
Two-point equations
Some experimental measurements in pipes and jets
Scalar transport in homogeneous turbulence
The inertial-convective range of wavenumbers
Universal forms of the scalar spectrum
Renormalized perturbation theory
The motion of discrete particles
Goveing equations for particles
Interpretation in terms of a diffusion coefficient
Random walk models of particle diffusion
Measurements of particle motion in turbulent flows
Turbulent mixing
Note
References
14 Non- Newtonian Fluid Turbulence
Non-Newtonian fluid flow
Rheological aspects
Composite systems: Newtonian fluid with modified boundary conditions
Flow in pipes
Structural turbulence
Isotropic turbulence
Turbulent structure in drag-reducing polymer solutions
Mean velocity distributions
Turbulent intensities
Spectra and correlations
The importance of the region near the wall
Free turbulence
Turbulent structure in drag-reducing fibre suspensions
Mean velocity distributions
Intensities, correlations, and spectra
Mixed fibre-polymer suspensions
The effect of drag-reducing additives on turbulent transport
Heat and mass transfer
Turbulent diffusion
Comparison of polymers and macroscopic fibres as drag-reducing additives
Further reading
References
APPENDICES
A Creation and dissipation of kinetic energy in a viscous fluid
B Probability and statistics
C Symmetry and invariance
D Application of Fourier methods and Green's functions to the Navier-Stokes equation
E Evaluation of the coefficients L(k,j) and L(k,k—j)
F Optical background to laser-Doppler anemometry
G Second-order term in the perturbation series as an example of the diagram calculus
H The Novikov functional formalism
Author Index
Subject Index