Prediction of Drag Coefficient and Secondary Motion of Free-Falling Rigid Cylindrical Particles with and without Curvature at Moderate Reynolds Number

Laboratory experiments have been conducted to understand the behavior of negatively buoyant cylindrical particles of density $\rho$, length $L$, diameter $d$, with and without curvature, freely falling in a fluid of density ${\rho }_w$ at Reynolds numbers based on $d$ of 200–6,000. The paper proposes a parameter based on the cylinder density ratio $S=\rho /{\rho }_w$ and aspect ratio $E=L/d$ that is able to predict the onset of different modes of secondary motion ranging from oscillations to tumbling. The same parameter can also predict the maximum amplitude of the oscillations of the cylinder, on the basis of comparing the magnitudes of the oscillation velocity with the fall velocity. Contrary to previous work that has treated the oscillations and drag coefficient dependence as independent phenomena, this paper argues that the secondary motion reduces the time-averaged projected surface area of the cylinder during its descent, leading to a lower observed drag coefficient $C_D$ computed using the nominal projected area of length times diameter, $Ld$. Curved cylinders adopt an average inclination to the horizontal and an oscillation pattern that depends on the curved particle’s arc angle and its specific gravity. The inclined particle has a smaller projected area than $Ld$, which, similarly to a straight cylindrical particle, leads to a reduced drag coefficient.