Intricate relations among particle collision, relative motion and clustering in turbulent clouds: computational observation and theory

Considering turbulent clouds containing small inertial particles, we investigate the effect of particle collision, in particular collision-coagulation, on particle clustering and particle relative motion. We perform direct numerical simulation (DNS) of coagulating particles in isotropic turbulent flow in the regime of small Stokes number (St = 0.001-0.54) and find that, due to collision-coagulation, the radial distribution functions (RDFs) fall off dramatically at scales r similar to d (where d is the particle diameter) to small but finite values, while the mean radial component of the particle relative velocity (MRV) increases sharply in magnitude. Based on a previously proposed Fokker-Planck (drift-diffusion) framework, we derive a theoretical account of the relationship among particle collision-coagulation rate, RDF and MRV. The theory includes contributions from turbulent fluctuations absent in earlier mean-field theories. We show numerically that the theory accurately accounts for the DNS results (i.e., given an accurate RDF, the theory could produce an accurate MRV). Separately, we also propose a phenomenological model that could directly predict MRV and find that it is accurate when calibrated using fourth moments of the fluid velocities. We use the model to derive a general solution of RDF. We uncover a paradox: the past empirical success of the differential version of the theory is theoretically unjustified. We see a further shape-preserving reduction of the RDF (and MRV) when the gravitational settling parameter (S-g) is of order O(1). Our results demonstrate strong coupling between RDF and MRV and imply that earlier isolated studies on either RDF or MRV have limited relevance for predicting particle collision rate.