Direct numerical simulations of anisotropic diffusion of spherical particles in sedimentation

We investigated the scaling of the hydrodynamic velocity fluctuations and self-diffusion in the sedimentation of monodispersed spherical particles via direct numerical simulations using the smoothed profile method over a moderate range of volume fractions (0.01≤ϕ≤0.12). Hydrodynamic velocity fluctuations are visible at large Peclet numbers (Pe), and they scale as (ϕL/a)1/2 at low volume fractions (ϕ≤0.04). Their characteristics become independent of volume fraction at moderate volume fractions (0.06≤ϕ≤0.12). Both vertical and horizontal self-diffusion coefficients scale as (L/a)3/2ϕ1/2 at low volume fractions. At moderate volume fractions, the vertical diffusion scales as (L/a)3/2ϕ−1/2; in contrast, the horizontal diffusion is saturated with respect to volume fraction. The diffusion anisotropy increases with increasing Pe and saturates at high Pe values. The saturated value remains unchanged at low volume fractions, whereas further increase in the volume fraction decreases this anisotropy. The reduction of this anisotropy is attributed to the ϕ−1/2 scaling of the vertical relaxation time at moderate volume fractions; however, the horizontal relaxation time is independent of the volume fraction at this regime.