Abstract
The effect of air turbulence on the geometric collision kernel of cloud droplets can be predicted if the effects of air turbulence on two kinematic pair statistics can be modeled. The first is the average radial relative velocity and the second is the radial distribution function (RDF). A survey of the literature shows that no theory is available for predicting the radial relative velocity of finite-inertia sedimenting droplets in a turbulent flow. In this paper, a theory for the radial relative velocity is developed, using a statistical approach assuming that gravitational sedimentation dominates the relative motion of droplets before collision. In the weak-inertia limit, the theory reveals a new term making a positive contribution to the radial relative velocity resulting from a coupling between sedimentation and air turbulence on the motion of finite-inertia droplets. The theory is compared to the direct numerical simulations (DNS) results in part 1, showing a reasonable agreement with the DNS data for bidisperse cloud droplets. For droplets larger than 30 μm in radius, a nonlinear drag (NLD) can also be included in the theory in terms of an effective inertial response time and an effective terminal velocity. In addition, an empirical model is developed to quantify the RDF. This, together with the theory for radial relative velocity, provides a parameterization for the turbulent geometric collision kernel. Using this integrated model, we find that turbulence could triple the geometric collision kernel, relative to the stagnant air case, for a droplet pair of 10 and 20 μm sedimenting through a cumulus cloud at Rλ=2×104 and ϵ=600 cm2 s−3. For the self-collisions of 20 μm droplets, the collision kernel depends sensitively on the flow dissipation rate.
Highlights
The effect of air turbulence on the geometric collision kernel of cloud droplets can be predicted if the effects of air turbulence on two kinematic pair statistics can be modeled
A new theory has been developed to describe the radial relative velocity between a pair of droplets sedimenting under gravity in a turbulent flow
This is a reasonable assumption for cloud droplets as the sedimentation dominates the droplet relative motion, with the air turbulence introducing a moderate modification to the relative motion
Summary
As in part 1, we focus here on geometric collision of sedimenting droplets without considering the local droplet–droplet aerodynamic interactions. In part 1, it was confirmed by DNS that the collision kernel for sedimenting droplets in a turbulent air can be expressed in terms of the kinematic pair statistics as. The other factor g12(r ) in equation (2) is known as the RDF measuring the effect of preferential concentration on the pair number density at separation distance r. Where |wr | / |wr | g represents the enhancement by air turbulence on the relative pair motion and will be referred to as the turbulent transport effect, g12 represents the effect of local nonuniform pair concentration due to interaction of droplets with local flow structures, and this contribution will be referred to as the accumulation effect. The task of predicting the turbulent collision kernel can be divided into two parts: a theory for the turbulent transport effect and a formulation for the accumulation effect Each of these will be considered in this paper. Most of the theoretical studies discussed below were published outside the atmospheric sciences literature
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