Rate of formation of caustics in heavy particles advected by turbulence

TL;DR

This paper investigates how often caustics form in heavy particles within turbulent flows, revealing an exponential relationship with the Stokes number, which impacts collision rates in natural phenomena.

Contribution

The study provides the first extensive numerical analysis of caustic formation rates across different turbulent flow regimes, establishing an exponential model for the rate as a function of the Stokes number.

Findings

Caustic formation rate follows an exponential dependence on inverse Stokes number.

Numerical simulations cover both 2D and 3D turbulence.

The model applies in the limit of small Stokes numbers.

Abstract

The rate of collision and the relative velocities of the colliding particles in turbulent flows is a crucial part of several natural phenomena, e.g., rain formation in warm clouds and planetesimal formation in a protoplanetary disks. The particles are often modeled as passive, but heavy and inertial. Within this model, large relative velocities emerge due to formation of singularities (caustics) of in the gradient matrix of the velocities of the particles. Using extensive direct numerical simulations of heavy particles in both two (direct and inverse cascade) and three dimensional turbulent flows we calculate the rate of formation of caustics, $J$ as a function of the Stokes number (${\rm St}$).The best approximation to our data is $J \sim \exp(-C/{\rm St})$, in the limit ${\rm St} \to 0 $ where $C$ is a non-universal constant.