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
subgrid-scale (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 (streamwise filtering using Taylor's
hypothesis). We analyse data taken at the centreline of a cylinder wake at Reynolds
numbers up to Rλ ∼ 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,
v(k, kΔ). 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 kΔ. When
using a spectral cutoff filter in the streamwise direction (with a box-filter in the cross-stream direction) a cusp behaviour near the filter scale is observed. In physical space,
certain features of a wavenumber-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.