Simulation of sedimentation of two spheres with different densities in a square tube

Abstract

We simulated the sedimentations of two unequal spheres with different densities in a square tube for Galileo numbers (Ga) from 5 to 25, resulting in a Reynolds number range of based on the terminal settling velocity. The sedimentation of spheres with different densities is dynamically more complex than that of identical spheres. At high Ga the spheres oscillate in the centreline plane of the tube, where they are initially released from rest. By contrast, the spheres move to the diagonal or reverse–diagonal plane of the tube at low Ga, reaching a steady or periodic state depending on the density difference between them. A phase diagram illustrates the transitions between different sedimentation behaviours depending on Ga and the density difference. A possible mechanism for these behaviours is also presented. Furthermore, we compare two-dimensional (2-D) and three-dimensional computations for our system to attain a better understanding of the hydrodynamic interactions between two unequal spheres at low but finite Reynolds number. Comparing relative trajectories, periods of oscillation and flow features shows that 2-D circular cylinders oscillate much more strongly and frequently than spheres under the same flow conditions. In particular, spheres do not have the discontinuity in period that arises in the 2-D case from the change in rotation sign of a heavy particle.