Hill R. J. (March 2005): Geometric collision rates and trajectories of cloud droplets falling into a Burgers vortex. Phys. Fluids, 17 (3), 037103. doi:10.1063/1.1858191Full text not available from this repository.
Droplet velocities, concentrations, and geometric collision rates are calculated for droplets falling into Burgers vortices as a step toward understanding the role of turbulence-induced collisions of cloud water droplets. The Burgers vortex is an often used model of vortices in high Reynolds number turbulence. Droplet radii considered are 10, 20, and 40 μm; those radii are relevant to warm rain initiation. A method of calculating the concentrations of droplets along their trajectories by means of differential geometry is derived and implemented. A generalization of the rate of geometrical collisions of inertial particles is derived; the formulation applies for any local vorticity and rate of strain, and the classic collision-rate formula is obtained in the process. The relative velocities of droplets of different radii and their spatial variation of concentration affects spatial variation of collision rate; greater variation exists for a stronger vortex. The physical effects included in the droplet equation of motion are inertia, gravity, viscous drag, pressure and shear stress, added mass, the history integral, and the lift force. The lift force requires calculation of droplet angular velocity, the equation for which contains rotational inertial and viscous drag. An initial condition is found that does not cause an impulse in the history integral. The important terms in the droplets’ equations of motion are found such that simpler approximate equations can be used. It is found that the lift force is negligible, the history integral is not. For smaller droplets in regions of lower vorticity, the time derivative of the difference of slip velocity and gravitationally induced drift velocity may be neglected. The present study suggests that acceleration-induced coalescence is most significant for droplets that are entrained into or formed within an intensifying vortex as distinct from falling toward the vortex.
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