Coalescence rate of polydisperse sub-Kolmogorov droplets settling in a turbulent gas*

Cloud droplets coalesce due to the coupled effects of differential sedimentation and turbulent shear.  Droplets of diameter 10 to 50 mm represent the bottleneck to precipitation and exhibit little deformation, interfacial mobility, or colloidal force so that the non-continuum nature of the gas is critical to allow coalescence. We evaluate the collision rate of inertia-less droplets settling in a local linear flow field and interacting by a uniformly valid hydrodynamic mobility function that captures non-continuum lubrication at small separations and full continuum hydrodynamic interactions at larger separations. Calculations are first performed for steady biaxial extensional flows with different orientations with respect to gravity providing a “persistent” model of turbulence similar to the classic work of Saffman and Turner. Next, the local linear flow seen by the drops is evolved using Girimaji and Pope’s Lagrangian stochastic model. The collision efficiency is determined for a broad range of droplet size ratios, Knudsen numbers (ratio of gas mean-free path to mean drop radius), and ratios of settling velocity to turbulent shear induced interparticle velocities.