The rate of collisions of small cloud droplets in turbulent flows

Coalescence growth of droplets is a fundamental process for liquid cloud evolution. The initiation of collisions and coalescence occurs when a few droplets become large enough to fall. Gravitational collisions represent the most efficient mechanism for multi-disperse solutions, when droplets span a large variety of sizes. However, turbulence provides another mechanism for droplets coalescence, taking place also in the case of uniform condensational growth leading to narrow droplet-size spectra. We consider the problem of estimating the rate of collisions of small droplets dispersed in a highly turbulent medium. The problem is investigated by means of high-resolution direct numerical simulations of a three-dimensional turbulent flow, seeded with inertial particles, up to resolutions of 2048^3 grid points. Rate of collision is estimated in terms of the probability to find particles at close positions, and of the statistics of particles velocity. In particular, we show that the statistics of velocity differences between inertial particles suspended in an incompressible turbulent flow is extremely intermittent. When particles are separated by distances of the order of their diameter, the competition between quiet regular regions and multivalued caustics leads to a quasi bi-fractal behavior of the particle velocity statistics, with high-order moments bringing the signature of caustics. This results in large probabilities that close particles have important velocity differences. Together with preferential concentration of particles in low-vorticity regions, caustics contribute to speed-up collisions between inertial particles. Implications for the early stage of rain droplets formation are discussed.