Transition scenario of a sphere freely falling in a vertical tube

Abstract

The 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.

Sedimentation of an ellipsoidal particle in narrow tubes

“…ρ f , and the Galileo number or "Archimedes number." 17 The Galileo number is defined as [17][18][19][20]…”

Path instability on a sphere towed at constant speed

“…Wake instabilities of the later kind have already been reported by Newton [1] when studying the drag of spheres in falling liquids. Over the last 15 years, with the advent of direct numerical simulation (DNS) at higher Reynolds number and high speed imaging techniques, this problem has received renewed attention [2,3,4,5,6,7]. A big variety of motions have been reported for such a system, a review on this subject can be found in [8] and a detailed historical review in [4].…”

Quasi-Steady Convergence of Multistep Navier–Stokes Icing Simulations

“…Available turbulence models include the widely used Spalart-Allmaras and A'-«j-Shear Stress Transport (SST) models. The chimera approach is also implemented [23,24]. Validations have been presented at the AIAA validation workshop series, namely drag predictions [25], high lift [26] and aeroelastic [27], and in a number of European framework program reports [28].…”