Caustics of finitely dense inertial particles

Abstract

Estimating collision rates is of immense importance in particle-laden flows. An economical way of doing this is to directly identify incidences of caustics, or extreme clustering, by tracking particle velocity gradients in the neighborhoods of individual particles. The objective of this work is two-fold. (i) We find conditions under which caustics form, in point-vortex flow and in two-dimensional turbulence. While caustics are known to form in regions of strain, we show that the velocity alignment with strain directions is key. Particles must remain in compressional strain throughout the process to form caustics, whereas survivor particles: which visit high strain but do not form caustics, briefly go through extensional strain during the early part of the process. This enables survivor particles to attain significantly straighter paths, and to move faster, whereas caustics particles follow paths of high curvature and move slower. As a result, caustics particles stay longer in high-strain regions than survivors. (ii) We ask about the effect of finite particle density, where the particle is denser than the background fluid. We show that finite-density particles need to sample stronger background strain than infinite-density ones to trigger caustics, but our other findings are universal across particle density.