Clusters of sedimenting high-Reynolds-number particles

Abstract

We report experiments wherein groups of particles were allowed to sediment in an otherwise quiescent fluid contained in a large tank. The Reynolds number of the particles, defined as Re = aU/ν, ranged from 93 to 425; here, a is the radius of the spherical particle, U its settling velocity and ν the kinematic viscosity of the fluid. The characteristic size of a cluster, in a plane transverse to gravity, was measured by a ‘cluster variance’(〈r2t〉); the latter is defined as the mean square of the transverse coordinates of all constituent particles, averaged over a series of runs. The cluster variance, when plotted as a function of time, exhibited two regimes. There was a quadratic growth in the variance at short times(〈r2t〉 ∝ t2), while for long times, the cluster variance exhibited a slower sublinear growth with 〈r2t〉 ∝ t0.67. A theory, based on isotropic repulsive hydrodynamic interactions between particles, predicts the cluster variance to grow as t2/3 in the limit of long times. The theoretical framework was originally proposed to describe the long-time self-similar evolution of dilute clusters in the limit Re ≪ 1 Subramanian & Koch (J. Fluid Mech., vol. 603, 2008, p. 63), when the probability of wake-mediated interactions between particles remains asymptotically small; the latter requirement is satisfied for homogeneous spherical clusters larger than a critical radius, and is evidently satisfied for planar clusters oriented transversely to gravity. The isotropy of the interactions therefore stems from the isotropy, at large distances, of the disturbance velocity field produced by a single sedimenting particle outside its wake(which contains the compensating inflow to satisfy mass conservation). Herein, the theory is extended to large Re using an empirical correlation for the drag on a sedimenting particle. This allows one to predict, as a function of Re, the numerical prefactors in the expressions for the cluster variance of both spherical and planar clusters; the predictions for the growth exponent remain unchanged. The agreement between the theoretical and experimental growth exponents supports the hypothesis of a self-similar expansion at long times. The prefactor determined from the experimental observations is found to lie between the theoretical predictions for planar and spherical clusters.