How turbulence enhances coalescence of settling particles with applications to rain in clouds

Abstract

From theoretical, numerical and experimental studies of small inertial particles with density equal to β (>1) times that of the fluid, it is shown that such particles are ‘centrifuged’ out of vortices and eddies in turbulence. Thus, in the presence of gravitational acceleration g , their average sedimentation velocity V T in a size range just below a critical radius a cr is increased significantly by up to about 80%. We show that in fully developed turbulence, a cr is determined by the circulation Γ k of the smallest Kolmogorov micro-scale eddies, but is approximately independent of the rate of turbulent energy dissipation ϵ , because Γ k is about equal to the kinematic viscosity ν . It is shown that a cr varies approximately like and is about 20 μm (±2 μm) for water droplets in most types of cloud. New calculations are presented to show how this phenomena causes higher collision rates between these ‘large’ droplets and those that are smaller than a cr , leading to rapid growth rates of droplets above this critical radius. Calculations of the resulting droplet size spectra in cloud turbulence are in good agreement with experimental data. The analysis, which explains why cloud droplets can grow rapidly from 20 to 80 μm irrespective of the level of cloud turbulence is also applicable where a cr ∼1 mm for typical sand/mud particles. This mechanism, associated with unequal droplet/particle sizes is not dependant on higher particle concentration around vortices and the results differ quantitatively and physically from theories based on this hypothesis.