Statistics of particle pair relative velocity in the homogeneous shear flow

Small scale clustering of inertial particles and relative velocity of particle pairs have been fully characterized for statistically steady homogeneous isotropic flows. Depending on the particle Stokes relaxation time, the spatial distribution of the disperse phase results in a multi-scale manifold characterized by local particle concentration and voids and, because of finite inertia, the two nearby particles have high probability to exhibit large relative velocities. Both effects might explain the speed-up of particle collision rate in turbulent flows. Recently it has been shown that the large scale geometry of the flow plays a crucial role in organizing small scale particle clusters. For instance, a mean shear preferentially orients particle patterns. In this case, depending on the Stokes time, anisotropic clustering may occur even in the inertial range of scales where the turbulent fluctuations which drive the particles have already recovered isotropy. Here we consider the statistics of particle pair relative velocity in the homogeneous shear flow, the prototypical flow which manifests anisotropic clustering at small scales. We show that the mean shear, by imprinting anisotropy on the large scale velocity fluctuations, dramatically affects the particle relative velocity distribution even in the range of small scales where the anisotropic mechanisms of turbulent kinetic energy production are sub-dominant with respect to the inertial energy transfer which drives the carrier fluid velocity towards isotropy. We find that the particles’ populations which manifest strong anisotropy in their relative velocities are the same which exhibit small scale clustering. In contrast to any Kolmogorov-like picture of turbulent transport these phenomena may persist even below the smallest dissipative scales where the residual level of anisotropy may eventually blow-up. The observed anisotropy of particle relative velocity and spatial configuration is suggested to influence the directionality of the collision probability, as inferred on the basis of the so-called “ghost collision” model.