Clustering of Rapidly Settling, Low-Inertia Particle Pairs in Isotropic Turbulence. I. Drift and Diffusion Flux Closures*

We present the development and analysis of a stochastic theory for characterizing the relative positions of monodisperse, low-inertia particle pairs that are settling rapidly in homogeneous isotropic turbulence.  The theory is applicable in the limits of particle Stokes number $St_\eta \ll 1$ and Froude number $Fr \ll St_\eta \ll 1$.    Here $St_\eta$~is the ratio of the particle viscous relaxation time to the Kolmogorov time scale, and $Fr$ is the ratio of the Kolmogorov scale of acceleration and the magnitude of gravitational acceleration.  In these parametric limits, closures are developed for the drift and diffusion fluxes in the probability density function (PDF) equation for the pair relative positions. The theory focuses on the relative motion of particle pairs in the dissipation regime of turbulence, i.e., for pair separations smaller than the Kolmogorov length scale.  In this regime, the theory approximates the fluid velocity field in a reference frame following the primary particle as locally linear.  We present the derivation of closure approximation  for the drift and diffusion fluxes in the PDF equation for the 
pair relative positions $\mathbf{r}$.