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Homogeneous, isotropic turbulence : phenomenology, renormalization and statistical closures /

This book addresses the idealised problem posed by homogeneous, isotropic turbulence. It is written from the perspective of a theoretical physicist, but is designed to be accessible to all researchers in turbulence, both theoretical and experimental, and from all disciplines.

Bibliographic Details
Main Author: McComb, W. D. (Author)
Corporate Author: Oxford University Press
Format: eBook
Language:English
Published: Oxford : Oxford University Press, 2014.
Series:International series of monographs on physics ; 162.
Subjects:
Turbulence.
Turbulence > Statistical methods.
Turbulence > Méthodes statistiques.
TECHNOLOGY & ENGINEERING > Hydraulics.
Turbulence
Turbulence > Statistical methods
Electronic books.
Online Access:Connect to the full text of this electronic book
Table of Contents:
  • Pt. I THE FUNDAMENTAL PROBLEM, THE BASIC STATISTICAL FORMULATION, AND THE PHENOMENOLOGY OF ENERGY TRANSFER
  • 1.Overview of the statistical problem
  • 1.1.What is turbulence?
  • 1.1.1.Definition and characteristic features
  • 1.1.2.The development of turbulence
  • 1.1.3.Homogeneous, isotropic turbulence (HIT)
  • 1.2.The turbulence problem
  • 1.2.1.The turbulence problem in real flows
  • 1.2.2.Formulation of the turbulence problem in HIT
  • 1.3.The characteristics of HIT
  • 1.4.Turbulence as a problem in quantum field theory
  • 1.5.Renormalized perturbation theory (RPT): the general idea
  • 1.5.1.Primitive perturbation series of the Navier
  • Stokes equations
  • 1.5.2.Application to the closure problem: the response function
  • 1.5.3.Renormalization
  • 1.5.4.Vertex renormalization
  • 1.5.5.Physical interpretation of renormalized perturbation theory
  • 1.6.Renormalization group (RG) and mode elimination
  • 1.6.1.RG as stirred hydrodynamics at low wavenumbers
  • 1.6.2.RG as iterative conditional averaging at high wavenumbers
  • 1.6.3.Discussion
  • 1.7.Background reading
  • References
  • 2.Basic equations and definitions in x-space and k-space
  • 2.1.The Navier
  • Stokes equations in real space
  • 2.2.Correlations in x-space
  • 2.2.1.The two-point, two-time covariance of velocities
  • 2.2.2.Correlation functions and coefficients in isotropic turbulence
  • 2.2.3.Structure functions
  • 2.3.Basic equations in k-space: finite system
  • 2.3.1.The Navier
  • Stokes equations
  • 2.3.2.The symmetrized Navier
  • Stokes equation
  • 2.3.3.Moments: finite homogeneous system
  • 2.4.Basic equations in k-space: infinite system
  • 2.4.1.The Navier
  • Stokes equations
  • 2.4.2.Moments: infinite homogeneous system
  • 2.4.3.Isotropic system
  • 2.4.4.Stationary and time-dependent systems
  • 2.5.The viscous dissipation
  • 2.6.Stirring forces and negative damping
  • 2.7.Fourier transforms of isotropic correlations, structure functions, and spectra
  • References
  • 3.Formulation of the statistical problem
  • 3.1.The covariance equations
  • 3.1.1.Off the time-diagonal: C(k; t, t!)
  • 3.1.2.On the time diagonal: C(k; t, t) = C(k, t)
  • 3.2.Conservation of energy in wavenumber space
  • 3.2.1.Equation for the energy spectrum: the Lin equation
  • 3.2.2.The effect of stirring forces
  • 3.3.Conservation properties of the transfer spectrum T(k, t)
  • 3.4.Symmetrized conservation identities
  • 3.5.Alternative formulations of the triangle condition
  • 3.5.1.The Edwards (k, j, [MARC+6F]) formulation
  • 3.5.2.The Kraichnan (k, j, l) formulation
  • 3.5.3.Conservation identities in the two formulations
  • 3.6.The L coefficients of turbulence theory in the (k, j, [MARC+6F]) formulation
  • 3.7.Dimensions of relevant spectral quantities
  • 3.7.1.Finite system
  • 3.7.2.Infinite system
  • 3.8.Some useful relationships involving the energy spectrum
  • 3.9.Conservation of energy in real space
  • 3.9.1.Viscous dissipation
  • 3.10.Derivation of the Karman
  • Howarth equation
  • 3.10.1.Various forms of the KHE
  • 3.10.2.The KHE for forced turbulence
  • 3.10.3.KHE specialized to the freely decaying and stationary cases
  • References
  • 4.Turbulence energy: its inertial transfer and dissipation
  • 4.1.The test problems
  • 4.1.1.Test Problem 1: free decay of turbulence
  • 4.1.2.Test problem 2: stationary turbulence
  • 4.2.The Lin equation for the spectral energy balance
  • 4.2.1.The stationary case
  • 4.2.2.The global energy balances
  • 4.3.The local spectral energy balance
  • 4.3.1.The energy flux
  • 4.3.2.Local spectral energy balances: stationary case
  • 4.3.3.The limit of infinite Reynolds number
  • 4.3.4.The peak value of the energy flux
  • 4.4.Summary of expressions for rates of dissipation, decay, energy injection, and inertial transfer
  • 4.5.The Karman
  • Howarth equation as an energy balance in real space
  • 4.6.The Kolmogorov (1941) theory: K41
  • 4.6.1.The `2/3' law: K41A
  • 4.6.2.The `4/5' law
  • 4.6.3.The `2/3' law again: K41B
  • 4.7.The Kolmogorov (1962) theory: K62
  • 4.8.Some aspects of the experimental picture
  • 4.8.1.Spectra
  • 4.8.2.Structure functions
  • 4.9.Is Kolmogorov's theory K41 or K62?
  • References
  • pt. II PHENOMENOLOGY: SOME CURRENT RESEARCH AND UNRESOLVED ISSUES
  • 5.Galilean invariance
  • 5.1.Historical background
  • 5.2.Some relativistic preliminaries
  • 5.3.Galilean relativistic treatment of the Navier
  • Stokes equation
  • 5.3.1.Galilean transformations and invariance of the NSE
  • 5.4.The Reynolds decomposition
  • 5.4.1.Galilean transformation of the mean and fluctuating velocities
  • 5.4.2.Transformation of the mean-velocity equation to S
  • 5.4.3.Transformation of the equation for the fluctuating velocity to S
  • 5.5.Constant mean velocity
  • 5.6.Is vertex renormalization suppressed by GI?
  • 5.7.Extension to wavenumber space
  • 5.7.1.Invariance of the NSE in k-space
  • 5.7.2.The Reynolds decomposition
  • 5.8.Moments of the fluctuating velocity field
  • 5.9.The covariance equations
  • 5.9.1.Covariance equation for t ≠ t'
  • 5.9.2.The covariance equation for t = t'
  • 5.10.Two-time closures
  • 5.11.Filtered equations of motion: LES and RG
  • 5.12.Concluding remarks
  • References
  • 6.Kolmogorov's (1941) theory revisited
  • 6.1.Standard criticisms of Kolmogorov's (1941) theory
  • 6.1.1.The effect of intermittency
  • 6.1.2.Local cascade or `nonlocal' vortex stretching?
  • 6.1.3.Problems with averages
  • 6.1.4.Anomalous exponents
  • 6.2.The scale-invariance paradox
  • 6.2.1.Scale invariance
  • 6.2.2.The paradox
  • 6.2.3.Resolution of the paradox
  • 6.3.Scale invariance and the `-5/3' inertial-range spectrum
  • 6.3.1.The scale-invariant inertial subrange
  • 6.3.2.The inertial-range energy spectrum
  • 6.3.3.Calculation of the Kolmogorov prefactor
  • 6.3.4.The limit of infinite Reynolds number
  • 6.4.Finite-Reynolds-number effects on K41: theoretical studies
  • 6.4.1.Batchelor's interpolation function for the second-order structure function
  • 6.4.2.Effinger and Grossmann (1987)
  • 6.4.3.Barenblatt and Chorin (1998)
  • 6.4.4.Qian (2000)
  • 6.4.5.Gamard and George (2000)
  • 6.4.6.Lundgren (2002)
  • 6.5.Finite-Reynolds-number effects on K41: experimental and numerical studies
  • 6.6.Discussion
  • References
  • 7.Turbulence dissipation and decay
  • 7.1.The mean dissipation rate
  • 7.2.Dependence on the Taylor
  • Reynolds number
  • 7.3.The behaviour of the dissipation rate according to the Karman
  • Howarth equation
  • 7.3.1.The dependence of the dimensionless dissipation rate on Reynolds number
  • 7.4.A reinterpretation of the Taylor dissipation surrogate
  • 7.4.1.Reinterpretation of Taylor's expression based on results from DNS
  • 7.5.Freely decaying turbulence: the background
  • 7.5.1.Variation of the Taylor microscale during decay
  • 7.5.2.The energy spectrum at small wavenumbers
  • 7.5.3.The final period of the decay
  • 7.5.4.The Loitsiansky and Saffman integrals
  • 7.6.Free decay: the classical era
  • 7.6.1.Taylor (1935)
  • 7.6.2.Von Karman and Howarth (1938)
  • 7.6.3.Kolmogorov's prediction of the decay exponents
  • 7.6.4.Batchelor (1948)
  • 7.6.5.The non-invariance of the Loitsiansky integral
  • 7.7.Theories of the decay based on spectral models
  • 7.7.1.Two-range spectral models
  • 7.7.2.Three-range spectral models
  • 7.8.Free decay: towards universality?
  • 7.8.1.The effect of initial conditions
  • 7.8.2.Fractal-generated turbulence
  • References
  • 8.Theoretical constraints on mode reduction and the turbulence response
  • 8.1.Spectral large-eddy simulation
  • 8.1.1.Statement of the problem
  • 8.1.2.Spectral filtering to reduce the number of degrees of freedom
  • 8.2.Intermode spectral energy fluxes
  • 8.2.1.Low-k partitioned energy fluxes
  • 8.2.2.High-k partitioned energy fluxes
  • 8.2.3.Energy conservation revisited
  • 8.3.Semi-analytical studies of subgrid modelling using statistical closures
  • 8.4.Studies of subgrid models using direct numerical simulation
  • 8.5.Stochastic backscatter
  • 8.6.Conditional averaging
  • 8.7.A statistical test of the eddy-viscosity hypothesis
  • 8.8.Constrained numerical simulations
  • 8.8.1.Operational LES
  • 8.9.Discussion
  • References
  • pt.
  • III STATISTICAL THEORY AND FUTURE DIRECTIONS
  • 9.The Kraichnan
  • Wyld
  • Edwards covariance equations
  • 9.1.Preliminary remarks
  • 9.1.1.RPTs as statistical closures
  • 9.1.2.Perceptions of RPTs
  • 9.1.3.Some general characteristics of RPTs
  • 9.2.The problem restated: the exact covariance equations
  • 9.2.1.The general inhomogeneous covariance equation
  • 9.2.2.Centroid and difference coordinates
  • 9.2.3.The exact covariance equations for HIT
  • 9.3.A short history of closure approximations
  • 9.4.The KWE covariance equations: the problem reformulated
  • 9.4.1.Comparison of quasi-normality with perturbation theory
  • 9.4.2.The KWE covariance equations
  • 9.5.Renormalized response functions as closure approximations
  • 9.5.1.Failure of the EFP and DIA closures
  • 9.5.2.The Local Energy Transfer (LET) theory
  • 9.6.Numerical assessment of closure theories
  • 9.6.1.Some recent calculations of LET and EDQNM
  • 9.7.Conclusions
  • References
  • 10.Two-point closures: some basic issues
  • 10.1.Perturbation theory and renormalization
  • 10.2.Quantum-style formalisms: Wyld
  • Lee and Martin
  • Siggia
  • Rose
  • 10.2.1.The improved Wyld
  • Lee formalism
  • 10.2.2.The Martin
  • Siggia
  • Rose formalism
  • 10.3.How general are the formalisms?
  • 10.4.Galilean invariance and the DIA
  • 10.5.Lagrangian-history theories
  • References
  • 11.The renormalization group applied to turbulence
  • 11.1.Formulation of conditional mode elimination for turbulence
  • 11.2.Renormalization group
  • 11.3.Forster
  • Nelson
  • Stephen theory of stirred fluid motion
  • 11.3.1.Application of the RG to stirred fluid motion with asymptotic freedom as k → 0
  • 11.3.2.Differential RG equations
  • 11.3.3.FNS theory in terms of conditional averaging
  • 11.4.Turbulence RG theories based on filtered averages
  • 11.4.1.Iterative averaging: McComb (1982)