## Fixed-scale statistics and the geometry of turbulent dispersion at high reynolds number via numerical simulation

##### Abstract

The relative dispersion of one fluid particle with respect to another is
fundamentally related to the transport and mixing of contaminant species in
turbulent flows. The most basic consequence of Kolmogorov's 1941 similarity
hypotheses for relative dispersion, the Richardson-Obukhov law that mean-square
pair separation distance grows with the cube of time
at intermediate times in the inertial subrange, is notoriously difficult to
observe in the environment, laboratory, and direct numerical simulations (DNS).
Inertial subrange scaling in size parameters like the mean-square pair separation requires
careful adjustment for the initial conditions of the dispersion process as well
as a very wide range of scales (high Reynolds number) in the flow being studied.
However, the statistical evolution of the shapes of clusters of more than two
particles has already exhibited statistical invariance at intermediate times in
existing DNS. This invariance is identified with inertial-subrange scaling and
is more readily observed than inertial-subrange scaling for seemingly simpler quantities such as the mean-square pair separation
Results from dispersion of clusters of four particles (called tetrads) in
large-scale DNS at grid resolutions up to 4096 points in each of three directions and Taylor-scale Reynolds
numbers from 140 to 1000 are used to explore the question of
statistical universality in measures of the size and shape of tetrahedra in
homogeneous isotropic turbulence in distinct scaling regimes at very small times
(ballistic), intermediate times (inertial) and very late times (diffusive).
Derivatives of fractional powers of the mean-square pair separation with respect to time normalized by the
characteristic time scale at the initial tetrad size constitute a powerful
technique in isolating cubic time scaling in the mean-square pair separation. This technique
is applied to the eigenvalues of a moment-of-inertia-like tensor formed from the
separation vectors between particles in the tetrad. Estimates of the
proportionality constant "g" in the Richardson-Obukhov law from DNS at a
Taylor-scale Reynolds number of 1000 converge towards the value g=0.56 reported in
previous studies. The exit time taken by a particle pair to first reach
successively larger thresholds of fixed separation distance is also briefly
discussed and found to have unexplained dependence on initial separation
distance for negative moments, but good inertial range scaling for positive
moments. The use of diffusion models of relative dispersion in the inertial
subrange to connect mean exit time to "g" is also tested and briefly discussed
in these simulations.
Mean values and probability density functions of shape
parameters including the triangle aspect ratio "w," tetrahedron
volume-to-gyration radius ratio, and normalized moment-of-inertia
eigenvalues are all found to approach invariant forms in the inertial subrange
for a wider range of initial separations than size parameters such as
mean-square gyration radius. These results constitute the
clearest evidence to date that turbulence has a
tendency to distort and elongate multiparticle configurations more severely in
the inertial subrange than it does in the diffusive regime at asymptotically
late time. Triangle statistics are found to be independent of
initial shape for all time beyond the ballistic regime.
The development and testing of different schemes for parallelizing the cubic
spline interpolation procedure for particle velocities needed to track particles in DNS is also covered. A "pipeline" method of moving batches of particles
from processor to processor is adopted due to its low memory overhead, but there are challenges in achieving good performance scaling.