Spectral and hyper eddy-viscosity in high Reynolds number turbulence

S. Cerutti1, C. Meneveau1,2, and O.M. Knio1,2
1Department of Mechanical Engineering
The Johns Hopkins University, Baltimore MD 21218
2Center for Environmental and Applied Fluid Mechanics
The Johns Hopkins University, Baltimore MD 21218

ABSTRACT: For the purpose of studying the spectral properties of energy transfer between large and small scales in high Reynolds-number turbulence, we measure the longitudinal SGS dissipation spectrum, defined as the co-spectrum of the SGS stress and filtered strain-rate tensors. An array of four closely spaced X-wire probes enables us to approximate a two-dimensional box filter by averaging over different probe locations (cross-stream filtering) and in time (stream-wise filtering using Taylor's hypothesis). We analyze data taken at the centerline of a cylinder wake at Reynolds numbers up to Rl~450. Using the assumption of local isotropy, the longitudinal SGS stress and filtered strain-rate co-spectrum is transformed into a radial co-spectrum, which allows us to evaluate the spectral eddy-viscosity, n(k,kc). In agreement with classical two-point closure predictions, for graded filters, the spectral eddy-viscosity deduced from the box-filtered data decreases near the filter wavenumber k. When using a spectral cutoff filter in the stream-wise direction (with a box-filter in the cross-stream direction) a cusp behavior near the filter scale is observed. In physical space, certain features of a wave-number dependent eddy-viscosity can be approximated by a combination of a regular and a hyper-viscosity term. A hyper-viscous term is also suggested from considering equilibrium between production and SGS dissipation of resolved enstrophy. Assuming local isotropy, the dimensionless coefficient of the hyper-viscous term can be related to the skewness coefficient of filtered velocity gradients. The skewness is measured from the X-wire array and from direct numerical simulation of isotropic turbulence. The results show that the hyper-viscosity coefficient is negative for graded filters and positive for spectral filters. These trends are in agreement with the spectral eddy-viscosity measured directly from the SGS stress-strain rate co-spectrum. The results provide significant support, now at high Reynolds numbers, for the ability of classical two-point closures to predict general trends of mean energy transfer in locally isotropic turbulence.

J. Fluid Mech. 421, p. 307. (2000)

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