Kinematics of a bubble freely rising in a thin-gap cell with additional in-plane confinement

Abstract

We analyze the behavior of air bubbles freely rising at high Reynolds numbers in a planar thin-gap cell filled with distilled water. The gap thickness of the cell is fixed to h≃2.8 mm (or h≃1 mm in additional experiments) and its in-plane width W is varied from 2.4 to 21 cm. This allows us to investigate the evolution from unconfined thin-gap situations (i.e., large W and h≪W) controlled by the bubble characteristic lengths (diameter in the cell plane d>h and thickness close to the gap size h) to doubly confined situations controlled by the channel dimensions. As the bubble size d increases, and beyond a critical value that depends on W, we observe a mean rise velocity of the bubble, Vb, lower than that for larger W, along with a modification of the bubble shape. The departure occurs for oscillating bubbles of approximate elliptical shape, which becomes closer to circular as the lateral confinement increases. We further investigate how the bubble oscillatory motion is impacted by the transverse confinement. Assuming that the wall effect is related to the strength of the downward flow generated by the bubble, we introduce the relative velocity Urel=Vb/ζ, where ζ=1-d/W is the confinement ratio and found Urel≃Vb,∞ for all the cell widths considered, where Vb,∞ is the mean rise velocity in the absence of the transverse confinement (i.e., for W sufficiently large). This provides an estimation, at leading order, of the bubble velocity, Vb≃0.8ζ(h/d)1/6gd, that generalizes the expression proposed by Filella J. Fluid Mech. 778, 60 (2015)JFLSA70022-112010.1017/jfm.2015.355 and accounts for the additional drag experienced by the bubble due to the lateral walls. We then show that, for given d and ζ, the frequency and amplitudes of the oscillatory motion can be predicted using the characteristic length and velocity scales, d and Urel. As the bubble size is increased further, the bubble behavior becomes fully dominated by the channel dimensions. Cylindrical-capped shapes emerge, corresponding to a radius of curvature Rc at the front of the bubble, Rc≃0.31W, independent of the bubble size and of the gap thickness. At the same time, the mean rise velocity of the bubble saturates at a constant value, corresponding to a constant Froude number, Fr=Vb/gW, that depends on the gap thickness h of the cell.