Effects of surface roughness on the drag coefficient of spheres freely rolling on an inclined plane

Abstract

An experimental investigation identifying the effects of surface roughness on the drag coefficient ( $C_{D}$ ) of freely rolling spheres is reported. Although lubrication theory predicts an infinite drag force for an ideally smooth sphere in contact with a smooth wall, finite drag coefficients are obtained in experiments. It is proposed that surface roughness provides a finite effective gap ( $G$ ) between the sphere and panel, resulting in a finite drag force while also allowing physical contact between the sphere and plane. The measured surface roughnesses of both the sphere and panel are combined to give a total relative roughness ( $\xi$ ). The measured $C_{D}$ increases with decreasing $\xi$ , in agreement with analytical predictions. Furthermore, the measured $C_{D}$ is also in good agreement with the combined analytical and numerical predictions for a smooth sphere and wall, with a gap approximately equal to the root-mean-square roughness ( $R_q$ ). The accuracy of these predictions decreases for low mean Reynolds numbers ( $\overline {Re}$ ), due to the existence of multiple scales of surface roughness that are not effectively captured by $R_{q}$ . Experimental flow visualisations have been used to identify critical flow transitions that have been previously predicted numerically. Path tracking of spheres rolling on two panels with different surface roughnesses indicates that surface roughness does not significantly affect the sphere path or oscillations. Analysis of sphere Strouhal number ( $St$ ) highlights that wake shedding and sphere oscillations are coupled at low $\overline {Re}$ but with increasing $\overline {Re}$ , the influence of wake shedding on the sphere path diminishes.