Homogeneous, isotropic turbulence : phenomenology, renormalization and statistical closures / W. David McComb

Author:: McComb, W. D.
Published:: Oxford : Oxford University Press, 2014.
Physical Description:: 1 online resource : illustrations (black and white).

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International series of monographs on physics ; 162

Contents:

Machine generated contents note: 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, μ) 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, μ) 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) -- and Contents note continued: 11.4.2.Iterative averaging in wavenumber space -- 11.4.3.Relationship of iterative averaging to Rose's (1977) method -- 11.4.4.Improved iterative averaging -- 11.5.Problems with filtered averages -- 11.6.The two-field theory -- 11.6.1.The hypothesis of local chaos -- 11.6.2.The recursion relations of two-field theory -- 11.7.Improved two-field theory -- 11.7.1.Non-Gaussian perturbation theory -- 11.8.Applications and developments of iterative averaging -- 11.9.Is field-theoretic RG a theory of turbulence? -- 11.9.1.Differential recursion relations -- References -- 12.Work in progress and future directions -- 12.1.Turbulence response -- 12.1.1.Fluctuation-response relations (FRRs) -- 12.1.2.Numerical assessment -- 12.2.Renormalized perturbation theories -- 12.2.1.Extension of Edwards' (1964) theory to the two-time covariance C(k; t, t') -- 12.2.2.Recovering the LET theory -- 12.3.Renormalization group -- 12.3.1.Power-law forcing and the renormalization group -- 12.3.2.Application of the Edwards (1964) pdf to RG mode elimination -- 12.4.Towards shear flows -- 12.4.1.Application of the two-field theory to LES of shear flows -- 12.5.Postscript: The nature of the problem -- References -- pt. IV APPENDICES -- Appendix A Implications of isotropy and continuity for correlation tensors -- References -- Appendix B Properties of Gaussian distributions -- B.1.Discrete systems: real scalar variables -- B.1.1.Two-point correlations -- B.2.Discrete systems: complex scalar variables -- B.3.Scalar fields -- B.3.1.Extension to wavenumber and time -- B.3.2.The generating functional -- B.4.Vector fields -- B.5.Isotropic fields -- B.6.Inhomogeneous vector fields -- References -- Appendix C Evaluation of the L(k, j) coefficient -- C.1.Derivation of the closed covariance equation -- C.2.Evaluation of L(k, j) -- C.2.1.A note on numerical evaluation in closures -- References.

Summary:

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.

Subject(s):

ISBN:

9780191768255 (ebook)

Bibliography Note:

Includes bibliographical references and index.

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