Eddy Viscosity in Two and Three Dimensions

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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 k0, where k0 is a characteristic wavenumber at the bottom of the enstrophy range. If kmk0, the shape of the transfer function does not have a universal form but instead depends on the spectrum shape near k0. Representation of this transfer by an eddy viscosity seems highly unjustified.

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 k0, where k0 is a characteristic wavenumber at the bottom of the enstrophy range. If kmk0, the shape of the transfer function does not have a universal form but instead depends on the spectrum shape near k0. Representation of this transfer by an eddy viscosity seems highly unjustified.

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