Clustering of Rapidly Settling, Low-Inertia Particle Pairs in Isotropic Turbulence. II. Comparison of Theory and DNS

We developed a stochastic theory for the relative positions of monodisperse, low-inertia
particle pairs that are settling rapidly in isotropic turbulence. The theory involved the development of closures for the drift and diffusion fluxes in the PDF equation for pair relative positions. Two closure forms are derived specifically for the drift flux. The first is based on the assumption that the fluid velocity gradient along primary particle trajectories has a Gaussian distribution. In the second closure form for the drift, instead of the fluid velocity gradient being Gaussian, we assume that the ``seen" strain-rate and rotation-rate tensors
scaled by the dissipation rate and enstrophy, respectively, are normally distributed. The second closure accounts for the two-time autocorrelations and cross-correlations of dissipation rate and enstrophy. The theory predicts that particle clustering has a power-law dependence on pair separation. The power-law exponent obtained from the theory is in reasonable agreement with the DNS data for $Fr = 0.006$. In conformity with the DNS, the theory shows that the clustering of $St_\eta \ll 1$ particles is only weakly anisotropic.