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

By B. Tao1, J. Katz1,2, & C. Meneveau 1,2

1 Department of Mechanical Engineering,

2 Center for Environmental and Applied Fluid Mechanics,

The Johns Hopkins University Baltimore, MD 21218, USA

**ABSTRACT:** Three-dimensional velocity distributions of
a turbulent flow in the core region of a square duct at ReH=120,000
are measuredusing holographic particle image velocimetry. Spatial
filtering of the 136x130x128 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. Pdfs 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 non-linear model produces the same trends but tends to
over predict the likelihood of the preferred stress state.
Joint probability density functions (pdf) 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/non-linear models. The results
show that the most extensive SGS stress eigenvector is preferentially
aligned at 32º 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 bi-modal behavior with the most contracting
and intermediate stress eigenvectors “switching places”:
from being aligned at 32º 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 non-linear
model. This alignment persists in various regions of the flow
(high vorticity, specific flow topologies, etc.). Furthermore,
the alignment between the strain-rate and non-linear model
tensors also exhibits a bi-modal behavior, but the alignment
angle of both configurations is 42º. 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
due 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
sub-regions of the flow.

*J.
Fluid Mech*, (2002) **467**, p. 35-78,

full
pdf article

(© CUP,
see http://uk.cambridge.org/journals/flm).