Caustic formation in a non-Gaussian model for turbulent aerosols

Caustics in the dynamics of heavy particles in turbulence accelerate particle collisions. The rate $J$ at which these singularities form depends sensitively on the Stokes number $\mathrm{St}$, the nondimensional 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. To understand the possible mechanisms explaining this difference, we analyze a non-Gaussian statistical model for caustic formation in the limit of small $\mathrm{St}$. We show that at small $\mathrm{St}$, $J$ depends sensitively on the tails of the distribution of Lagrangian fluid-velocity gradients. This explains why different authors obtained different $\mathrm{St}$-dependencies of $J$ in numerical-simulation studies. The most likely gradient fluctuation that induces caustics at small $\mathrm{St}$, 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.