Distribution of relative velocities in turbulent aerosols

TL;DR

This paper analyzes the distribution of relative velocities of heavy particles in turbulent flows using a one-dimensional model, revealing universal behaviors and validating findings with numerical simulations.

Contribution

It introduces a universal form for the relative velocity distribution in turbulent aerosols and extends the principles to multiple spatial dimensions.

Findings

Distribution follows a universal form at small separations

Results agree with numerical simulations in 2D flows

Quantitative match with published turbulence data

Abstract

We compute the distribution of relative velocities for a one-dimensional model of heavy particles suspended in a turbulent flow, quantifying the caustic contribution to the moments of relative velocities. The same principles determine the corresponding caustic contribution in $d$ spatial dimensions. The distribution of relative velocities \Delta v at small separations R acquires the universal form \rho(\Delta v,R) ~ R^{d-1}|\Delta v|^{D_2-2d} for large (but not too large) values of |\Delta v|. Here D_2 is the phase-space correlation dimension. Our conclusions are in excellent agreement with numerical simulations of particles suspended in a randomly mixing flow in two dimensions, and in quantitative agreement with published data on direct numerical simulations of particles in turbulent flows.