Caustic formation in a non-Gaussian model for turbulent aerosols

TL;DR

This paper investigates how non-Gaussian statistical models explain the formation of caustics in turbulent aerosols, revealing the importance of velocity-gradient tails and aligning with numerical simulations.

Contribution

It introduces a non-Gaussian model for caustic formation at small Stokes numbers, explaining discrepancies with Gaussian models and numerical results.

Findings

Caustic formation rate depends on velocity-gradient distribution tails.

Optimal fluctuations for caustic formation are similar in Gaussian and non-Gaussian models.

Numerical simulations show close but not identical optimal fluctuations compared to the model.

Abstract

Caustics in the dynamics of heavy particles in turbulence accelerate particle collisions. The rate $\mathscr{J}$ at which these singularities form depends sensitively on the Stokes number St, the non-dimensional inertia parameter. Exact results for this sensitive dependence have been obtained using Gaussian statistical models for turbulent aerosols. However, direct numerical simulations of heavy particles in turbulence yield much larger caustic-formation rates than predicted by the Gaussian theory. In order to understand possible mechanisms explaining this difference, we analyse a non-Gaussian statistical model for caustic formation in the limit of small St. We show that at small St, $\mathscr{J}$ depends sensitively on the tails of the distribution of Lagrangian fluid-velocity gradients. This explains why different authors obtained different St-dependencies of $\mathscr{J}$ in numerical-simulation studies. The most-likely gradient fluctuation that induces caustics at small St, by contrast, is the same in the non-Gaussian and Gaussian models. Direct-numerical simulation results for particles in turbulence show that the optimal fluctuation is similar, but not identical, to that obtained by the model calculations.