Intermittency and collisions of fast sedimenting droplets in turbulence

We study theoretically and numerically spatial distribution and collision rate of droplets that sediment in homogeneous isotropic Navier-Stokes turbulence. It is assumed that, as it often happens in clouds, typical turbulent accelerations of fluid particles are much smaller than gravity. This was shown to imply that the particles interact weakly with individual vortices and, as a result, form a smooth flow in most of the space. In weakly intermittent turbulence with moderate Reynolds number Reλ, rare regions where the flow breaks down can be neglected in the calculation of space averaged rate of droplet collisions. However, increase of Reλ increases probability of rare, large quiescent vortices whose long coherent interaction with the particles destroys the flow. Thus, at higher Reλ, that apparently include those in the clouds, the space averaged collision rate forms in rare regions where the assumption of smooth flow breaks down. This intermittency of collisions implies that rain initiation could be a strongly nonuniform process. We describe the transition between the regimes and provide collision kernel in the case of moderate Reλ describable by the flow. The distribution of pairwise distances (radial distribution function or RDF) is shown to obey a separable dependence on the magnitude and the polar angle of the separation vector. Magnitude dependence obeys a power law with a negative exponent, manifesting multifractality of the droplets' attractor in space. We provide the so far missing numerical confirmation of a relation between this exponent and the Lyapunov exponents and demonstrate that it holds beyond the theoretical range. The angular dependence of the RDF exhibits a maximum at small angles quantifying particles' formation of spatial columns. We provide typical dimensions of the columns, which belong in the inertial range. We derive the droplets' collision kernel using that in the considered limit the gradients of droplets' flow are Gaussian. We demonstrate that as Reλ increases the columns' aspect ratio decreases, eventually becoming one when the isotropy is restored. We propose how the theory could be constructed at higher Reλ of clouds by using the example of the RDF.