Statistics of volumes, swept by spherical particles in a turbulent flow

Collisions between non-spherical particles (ice crystals) give rise to formation of aggregates. Collisions of non-spherical crystals with cloud droplets is the main mechanism of graupel production. The rate of riming and that of ice-ice collisions is not well known even in a pure gravity case. At the same time these collisions take place as a rule in the regions of enhanced turbulence in cumulus clouds. In spite of its high importance, the problem of collisions of such particles in a turbulent flow is not yet solved. In this work we present a novel method of collision kernels calculation between small (effective radius is less than 30 microns) prolate and oblate spheroid particles of different aspect ratios. The collision kernel between two spheroids is defined in terms of fluxes of particles of one sub-population (consisting of identical particles of a certain mass and aspect ratio) to the particles of another sub-population. In this study hydrodynamic interaction between particles is not taken into account, so that the collision kernel is equal to the swept volume of colliding particles. Scale analysis indicates that spatial and time characteristic scales of Lagrangian acceleration and turbulent shears are significantly larger then the corresponding scales characterizing particles collisions. The results of this analysis allow one to calculate collisions between particle pairs within small volumes of turbulent flow, in which Lagrangian accelerations and shears can be considered to be constant. Statistics of Lagrangian accelerations and turbulent shears as well as geometrical characteristics of particles fully determine statistics of swept volumes. A statistical model of turbulent flow has been used for generation of a long series of acceleration/shear pairs with probability distribution functions (PDF) at high Reynolds numbers and dissipation rates, as they were obtained in recent laboratory and theoretical studies. Thus, the characteristic properties of atmospheric turbulence such as very high Reynolds numbers and turbulent intermittency are taken into account in the swept volume calculations. An approximate analytical solution of spheroid motion has been derived. This solution allowed us to find approximate probability distribution functions (PDF) of spheroid velocities (translation and angular) and orientations for any given realization of a turbulent field. Using these PDFs, we calculated PDF, as well as mean values of swept volumes. These results were obtained for a wide range of turbulent flow intensity (different Reynolds numbers and energy dissipations) typical of actual clouds from stratiform to deep cumulus clouds. The estimations were performed also for different values of aspect ratios: from a plate-like spheroid (aspect ratio 0.05) up to a needle-like one (aspect ratio 20) and different particles sizes. The results manifest that: PDF of swept volumes differs significantly from the Gaussian one and the difference increases with turbulent flow intensity and the rate of the particle non-sphericity. It was found that turbulence significantly increases swept volumes and collision rate. It was also found that the increase of swept volumes of non-spherical particles significantly stronger than that for spherical droplets of similar mass. The effect enlarges with turbulence intensity. The swept volume enhancement is especially significant for small particles and particles of similar mass.