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Robert H. Kraichnan

Abstract

The test-field model for isotropic turbulence is used to examine the effective eddy viscosity acting on wavenumbers <km due to interactions with subgrid-scale wavenumbers, defined as wavenumbers >km. In both two and three dimensions, the effective eddy viscosity for kkm. is independent of k and of local spectrum shape. In two dimensions, this asymptotic eddy viscosity is negative. The physical mechanism responsible for the negative eddy viscosity is the interaction of large-spatial-scale straining fields with the secondary flow associated with small-spatial-scale vorticity fluctuations. This process is examined without appeal to turbulence approximations. For kmkkm, the effective eddy viscosity rises sharply to a cusp at k=km if km lies in a long energy-transferring inertial range in either two or three dimensions or in a long enstrophy-transferring inertial range in two dimensions. The cusp behavior is associated with a diffusion in wavenumber due to random straining, by large spatial scales, of structures with wavenumber close to km. This behavior makes the use of a k-independent eddy viscosity substantially inaccurate for the three-dimensional inertial range. In the two-dimensional enstrophy inertial range, the cusp region contributes most of the enstrophy transfer across km. The transfer function is squeezed into a region about km whose width is of order k 0, where k 0 is a characteristic wavenumber at the bottom of the enstrophy range. If kmk 0, the shape of the transfer function does not have a universal form but instead depends on the spectrum shape near k 0. Representation of this transfer by an eddy viscosity seems highly unjustified.

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Jackson R. Herring, James J. Riley, G. S. Patterson Jr., and Robert H. Kraichnan

Abstract

Computer simulators are made of the growth of the difference-velocity field for pairs of realizations of isotropic, three-dimensional turbulence at Reynolds number R&lambda≈40. The simulations involve full-scale integration of the Navier-Stokes equation in the Fourier representation. It is found that the difference-velocity variance (error energy) grows with time even when the initial difference-velocity is confined to wave numbers strongly damped by viscosity. The numerical integrations are compared with results of the direct-interaction approximation (DIA). It is found that the DIA gives reasonably satisfactory quantitative agreement for the evolution of the error energy and the error. energy spectrum. What discrepancies there are represent an underestimate of error energy growth by the DIA. This is explained by theoretical analysis of the approximation.

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