Statistical geometry of subgrid-scale stresses determined from holographic particle image velocimetry measurements
Authors: TAO BO , KATZ JOSEPH , MENEVEAU CHARLES 
Published: July 2002

Three-dimensional velocity distributions of a turbulent flow in the core region of
a square duct at ReH = 1.2 × 105 are measured using holographic particle image
velocimetry (HPIV). Spatial filtering of the 136 × 130 × 128 velocity vector maps
enables calculation of subgrid-scale (SGS) stresses and parameters based on the filtered
velocity gradients, such as the filtered strain-rate tensor and vorticity vector.
Probability density functions (p.d.f.) of scalar parameters characterizing eigenvalue
structures confirm that the most probable strain-rate topology is axisymmetric extension,
and show that the most probable SGS stress state is axisymmetric contraction.
Conditional sampling shows that high positive SGS dissipation occurs preferentially
in regions with these preferred strain-rate and stress topologies. High negative SGS
dissipation (backscatter) occurs preferentially in regions of axisymmetric contracting
SGS stress topology, but is not associated with any particular strain-rate topology.
The nonlinear model produces the same trends but tends to overpredict the likelihood
of the preferred stress state.

Joint p.d.f.s of relative angles are used to investigate the alignments of the SGS
stress eigenvectors relative to the vorticity and eigenvectors associated with eddy viscosity
and similarity/nonlinear models. The results show that the most extensive SGS
stress eigenvector is preferentially aligned at 32° to the most contracting strain-rate
eigenvector. This alignment trend persists, with some variations in angle and peak
probability, during conditional samplings based on the SGS dissipation rate, vorticity
and strain-rate magnitudes. The relative alignment of the other two stress and strain-rate
eigenvectors has a bimodal behaviour with the most contracting and intermediate
stress eigenvectors ‘switching places’: from being aligned at 32° to the most extensive
strain-rate eigenvector to being parallel to the intermediate strain-rate eigenvector.
Conditional sampling shows that one of the alignment configurations occurs preferentially
in regions of high vorticity magnitude, whereas the other one dominates in
regions where the filtered strain-rate tensor has axisymmetric contracting topology.
Analysis of DNS data for isotropic turbulence at lower Re shows similar trends.

Conversely, the measured stress eigenvectors are preferentially aligned with those
of the nonlinear model. This alignment persists in various regions of the flow (high
vorticity, specific flow topologies, etc). Furthermore, the alignment between the strain-rate
and nonlinear model tensors also exhibits a bimodal behaviour, but the alignment
angle of both configurations is 42°. Implications of alignment trends on SGS dissipation
are explored and conditions for high backscatter are identified based on the
orientation of the stress eigenvectors. Several dynamical and kinematical arguments
are presented that may explain some of the observed preferred alignments among
tensors. These arguments motivate further analysis of the mixed model, which shows
good alignment properties owing to the dominance of the Leonard stress on the
alignments. Nevertheless, the data also show that the mixed model produces some
unrealistic features in probability distributions of SGS dissipation, and unphysical
eigenvector alignments in selected subregions of the flow.