Probability distributions of angle of approach and relative velocity for colliding droplets in a turbulent flow

Prediction of the effect of air turbulence on statistics relevant to a collision-coalescence process represents a key challenge in the modeling of cloud microphysics. In this paper, collision-related statistics for gravity-driven motion of droplets are considered and various probability distributions associated with geometric configuration and relative motion of colliding droplets are theoretically derived. The theoretical results agree well with numerical results obtained from direct numerical simulations (DNSs). In the absence of air turbulence, the probability distributions, calculated at the beginning of the time steps used for collision detection, nontrivially depend on the time step size. Next, a novel theory is developed to quantify the effect of turbulence on the angle-of-approach θ and radial relative velocity w_r,c for colliding pairs. A logical decomposition is used to construct extended collision volumes for a specific level of radial motion caused by air turbulence. It is shown that the inward relative motion due to turbulent fluctuations dominates the effect of turbulence in modifying the probability distributions of θ and w_r,c . Two key dimensionless parameters are identified in the theory: one measures the effect of finite time step size in numerical collision detection and the second measures the relative magnitude of air turbulence. The theory is compared with 11 numerical experiments from DNS. It is shown that the theory captures the essential physics of the effect of air turbulence and provides a quantitatively good representation of the statistics for θ. For most numerical experiments, the theory predicts (θ) to within 5%. The probability distribution of w_r,c is more sensitive to the influence of air turbulence and shows larger intermittency at large w_r,c than what is assumed in the theory. The theoretical framework developed here may be of value to other problems involving gravitational settling and weak turbulence, such as parameterization of collision kernel and hydrodynamic interactions of droplets in warm rain processes.