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Transition scenario of a sphere freely falling in a vertical tube. (English) Zbl 1275.76077

Summary: 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 [M. Jenny et al., J. Fluid Mech. 508, 201–239 (2004; Zbl 1065.76068)]. 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.

MSC:

76D25 Wakes and jets
76D05 Navier-Stokes equations for incompressible viscous fluids
76M22 Spectral methods applied to problems in fluid mechanics

Citations:

Zbl 1065.76068
Full Text: DOI

References:

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