Collision efficiency of rapidly settling particle pairs in a turbulent flow

Abstract

We investigate the collision dynamics of hydrodynamically interacting inertialess spherical particle pairs sedimenting in a homogeneous isotropic turbulent flow. The analysis focuses on the rapid-settling limit, in which the particle settling time across a Kolmogorov eddy is much shorter than the Kolmogorov time scale. We also consider continuum breakdown during lubrication interactions, which is important when the separation the particles is comparable to the $O(100)$ nm mean-free path of a gaseous media. Owing to the sub-Kolmogorov particle sizes considered here, we approximate the local flow field in the vicinity of a particle pair as a stochastic linear flow induced by the background turbulence. In the rapid-settling regime, the cumulative effect of turbulent strain fluctuations is weak, and the relative particle motion may therefore be described as a diffusive process. In addition, hydrodynamic interactions generate a net relative drift between the particle pairs. We obtain the hydrodynamic diffusivity and relative drift velocity from the Lagrangian autocorrelation function of the fluid velocity gradient evaluated along the settling trajectory. The rapid-settling assumption further enables us to relate the autocorrelation function to the turbulence energy spectrum. Using these results, we solve the advection-diffusion equation for the pair probability density function to determine the collision rate. We show that the ideal collision rate increases monotonically with increasing relative strength of gravity to turbulence, whereas the collision efficiency decreases monotonically over the same range.