Turbulence effects on the collision kernel. I: formation of velocity deviations of drops falling within a turbulent three-dimensional flow

The fall of drops in three-dimensional stationary homogeneous turbulent flow is investigated. Equations for drop motion in turbulent flow and for drop velocity relative to the air are deduced, in vector and tensor forms. Homogeneous and isotropic turbulence is described using Batchelor's approximation formula which is valid both for the inertial and viscous turbulence ranges. The mean-square values of relative drop-velocity are calculated as a function of the drop size and turbulence-dissipation rate. It is shown that, for small droplets (within radii less than 50 μm) and a dissipation rate as low as 50 cm²s^-3, turbulence-induced relative velocity is of the same order as the still-air terminal fall-speed. For greater dissipation rates, turbulence-induced drop-velocity relative to the air can be several times greater than terminal velocity. The mechanisms of the formation of the relative velocity are analysed. The contribution of the inertial acceleration of the flow is found to be dominant for small drops.