Rate of formation of caustics in heavy particles advected by turbulence

The rate of collision and the relative velocities of the colliding particles in turbulent flows are a crucial part of several natural phenomena, e.g. rain formation in warm clouds and planetesimal formation in protoplanetary discs. The particles are often modelled as passive, but heavy and inertial. Within this model, large relative velocities emerge due to formation of singularities (caustics) of 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, [Formula: see text] as a function of the Stokes number ([Formula: see text]). The best approximation to our data is [Formula: see text], in the limit [Formula: see text] where [Formula: see text] is a non-universal constant. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.