Eddy viscosity and the statistical theory of turbulence

doi:10.1016/S0377-0265(97)00012-2

Dynamics of Atmospheres and Oceans

Volume 27, Issues 1–4, January 1998, Pages 233-241

https://doi.org/10.1016/S0377-0265(97)00012-2 Get rights and content

Abstract

We examine, for the case of stationary turbulence, the correlation between fully resolved turbulence (in the sense of Fourier modes) and a system composed of the same equations on a reduced wavenumber span. This ‘reduced system’ is subjected to a wavenumber-dependent eddy viscosity contrived so that the full and reduced systems have the same energy spectrum, on the reduced span. The particular numerical problem studied is isotropic turbulence, and the model of turbulence is a Langevin representation of the test-field model (Kraichnan (1971, J. Fluid Mech., 47: 513–524). The correlation between the full and reduced systems is surprisingly small, a fact we attribute to be related to the unpredictability of the underlying Navier-Stokes equations. The (qualitative) form of the eddy viscosity introduced by such modeling is also discussed, and it is noted that at large Reynolds numbers, there exists an optimal cut-off wavenumber which permits an elimination of explicit empiricism from this form of large eddy simulation (LES). Our computations permit a spectral estimate of the magnitude of the backscatter term in LES, along the lines recently proposed by Schumann (1995, Proc. R. Soc. London, Ser. A, 451: 293–318).

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Citation Excerpt :
The pioneering study in this area was the parameter-free Eulerian direct interaction approximation (DIA) of Kraichnan [29] developed for isotropic homogeneous turbulence, followed by the self-consistent field theory of Herring [30], the eddy-damped quasi-normal Markovian (EDQNM) model implemented by Leith [31], and the local energy transfer theory of McComb [32]. Statistical dynamical subgrid closure models pertaining mainly to three-dimensional homogeneous turbulence include [5,22,28,33–40]. Further references, including using renormalisation group methods, are given in the review of Zhou [41].
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Eddy viscosity and the statistical theory of turbulence

Abstract

NonGaussian statistics in isotropic turbulence

Phys. Fluids A

Parameterization of small scales of three-dimensional isotropic turbulence utilizing spectral closure

J. Atmos. Sci.

The predictability of quasi- geostrophic flows

Comparison of some approximations for isotropic turbulence

Energy transfer and constrained simulations in isotropic turbulence

An almost markovian Galilean invariant turbulence model

J. Fluid Mech.