Kinematic and dynamic collision statistics of cloud droplets from high-resolution simulations

We study the dynamic and kinematic collision statistics of cloud droplets for a range of flow Taylor microscale Reynolds numbers (up to 500), using a highly scalable hybrid direct numerical simulation approach. Accurate results of radial relative velocity (RRV) and radial distribution function (RDF) at contact have been obtained by taking advantage of their power-law scaling at short separation distances. Three specific but inter-related questions have been addressed in a systematic manner for geometric collisions of same-size droplets (of radius from 10 to 60μm) in a typical cloud turbulence (dissipation rate at 400 cm² s^-3). Firstly, both deterministic and stochastic forcing schemes were employed to test the sensitivity of the simulation results on the large-scale driving mechanism.

We found that, in general, the results are quantitatively similar, with the deterministic forcing giving a slightly larger RDF and collision kernel. This difference, however, is negligible for droplets of radius less than 30μm. Secondly, we have shown that the dependence of pair statistics on the flow Reynolds number R_λ or larger scale fluid motion is of secondary importance, with a tendency for this effect to saturate at high enough R_λ leading to R_λ-independent results. Both DNS results and theoretical arguments show that the saturation happens at a smaller R_λ for smaller droplets. Finally, since most previous studies of turbulent collision of inertial particles concerned non-sedimenting particles, we have specifically addressed the role of gravity on collision statistics, by simultaneously simulating collision statistics with and without gravity. It is shown that the collision statistics is not affected by gravity when a < ac, where the critical droplet radius ac is found to be around 30μm for the RRV, and around 20μm for the RDF. For larger droplets, gravity alters the particle–eddy interaction time and significantly reduces the RRV. The effect of gravity on the RDF is rather complex: gravity reduces the RDF for intermediate-sized droplets but enhances the RDF for larger droplets. In addition, we have studied the scaling exponents of both RDF and RRV, and found that gravity modifies the RDF scaling exponents for both intermediate and large particles, in a manner very similar to the effect of gravity on the RDF at contact. Gravity is shown to cause the scaling exponents for RDF and RRV to level off for large droplets, in contrast to diminishing exponents for non-sedimenting particles.