Abstract
A comprehensive investigation, using experimental, computational and analytic methods, is reported on the motion of, and the forces on, spheres of different density ratios rolling freely down an incline in a fluid under gravity. The Reynolds number, based on sphere diameter and terminal velocity, ranged up to 1000 for the experiments, and up to 250 for the computer simulations. A modified Reynolds number, incorporating the density ratio, gravitational acceleration and angle of incline, was found to govern the saturated state of the flow. Transition from steady to unsteady flow was sensitive to mass ratio, with lighter spheres undergoing earlier transition. Indeed, positively buoyant spheres develop cross-slope oscillations prior to the onset of shedding. Also of interest, the transition to chaotic wake flow occurs at Reynolds numbers lower than for a sphere forced to roll at a constant speed. In addition to the average sphere motion, flow-induced vibrations were predicted and measured, with quasi-periodic lateral oscillations found to increase as the flow became more unstable, and to decrease with increased density ratio. The study confirms the time-averaged results of a previous experimental study, although closer inspection shows sensitivity to the relative surface roughness of the sphere and plane in experiments; this sensitivity is masked in typical log–log plots of drag against Reynolds number. Physical surface roughness appears to play a role analogous to the necessary imposed gap between the sphere and plane in computations, removing the singularity in drag that would prevent rolling for an incompressible fluid and perfectly smooth surfaces.