On the collision rate of small particles in turbulent flows

A theoretical framework is developed to predict the rate of geometric collision and the collision velocity of small size inertialess particles in general turbulent flows. The present approach evaluates the collision rate for small size, inertialess particles in a given instantaneous flow field based on the local eigenvalues of the rate-of-strain tensor. An ensemble average is then applied to the instantaneous collision rate to obtain the average collision rate. The collision rates predicted by Smoluchowski (1917) for laminar shear flow and by Saffman & Turner (1956) for isotropic turbulence are recovered. The collision velocities presently predicted in both laminar shear flow and isotropic turbulence agree well with the results from numerical simulations for particle collision in both flows. The present theory for evaluating the collision rate and the collision velocity is also applied to a rapidly sheared homogeneous turbulence to assess the effect of strong anisotropy on the collision rate. Using (ε/v)1/2, in which ε is the average turbulence energy dissipation rate and v is the fluid kinematic viscosity, as the characteristic turbulence shear rate to normalize the collision rate, the effect of the turbulence structure on the collision rate and collision velocity can be reliably described. The combined effects of the mean flow shear and the turbulence shear on the collision rate and collision velocity are elucidated.