Model for the radial distribution function of polydisperse inertial spheres settling in homogeneous, isotropic turbulence

While particle inertia is widely known to cause substantial clustering of monodisperse particles in turbulent flows, differential sedimentation of polydisperse particles can rapidly decorrelate their relative positions and attenuate clustering. This paper presents a simple analytical model for the radial distribution function (RDF) of inertial particles settling in homogeneous, isotropic turbulence over a broad range of particle Stokes numbers, settling parameters, size ratios, and interparticle separations. We first draw on previous theories and direct-numerical simulations (DNS) to develop a simple comprehensive fit for the RDF of monodisperse particles without sedimentation. Even in the absence of gravity, the relative positions of polydisperse particles decorrelate as a result of turbulent accelerations, which have been treated as an acceleration-driven relative diffusivity of unequal size particles balancing the radial diffusion and drift resulting from turbulent shearing motions. We develop a similar model to describe the orientational averaged pair distribution function or RDF in the presence of differential sedimentation. The model is validated by comparison with DNS results for polydisperse settling particles. Juxtaposition of the model predictions with a variety of experimental measurements provides a perspective on current empirical knowledge of the RDF of sedimenting particles in turbulence. A sample calculation is then performed to illustrate the effect of preferential concentration in the presence of differential sedimentation on the coalescence rate of cloud droplets.