Statistical geometry of subgrid-scale stresses determined from holographic particle image velocimetry measurements

Abstract

Three-dimensional velocity distributions of a turbulent flow in the core region of a square duct at Re_H = 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.