Transition scenario of a sphere freely falling in a vertical tube

Abstract

AbstractThe paper presents the results of direct numerical simulations of the fall of a single freely moving sphere in a vertical circular tube. Most results are obtained for the solid–fluid density ratio ${\rho }_{s} / \rho = 2$. The parametric investigation is carried out depending on the Galileo number defined in Jenny, Dušek & Bouchet J. Fluid Mech., vol. 508, 2004, pp. 201–239. A qualitatively new scenario is found, as compared to that of an unconfined sphere. The primary bifurcation making the sphere deviate from a vertical fall along the tube axis at a constant velocity is of Hopf type. It sets in at a Galileo number (between 155 and 160) similar to that for an unconfined sphere. We find evidence for two stages of the primary regime: a planar trajectory at $G= 160$ and a helical one (at $G= 165$ and 170). At these Galileo numbers, the regime is perfectly periodic, with a slow period corresponding to a Strouhal number only slightly above 0.01. The dynamics is identified as a periodic wake–wall interaction. The helical regime is found to give way directly to chaos between $G= 170$ and $G= 180$. This transition is associated with the onset of vortex shedding in the wake of the falling sphere and with a complex interaction between the unsteady wake and the wall marked by intermittent wake extinction. The effect of density ratio is partly investigated at $G= 250$ by considering three density ratios: 2, 3 and 5. A significant change of behaviour is found between the ratios 3 and 5.