Dynamics of a high-Reynolds-number bubble rising within a thin gap

Abstract

AbstractWe report an experimental analysis of path and shape oscillations of an air bubble of diameter $d$ rising in water at high Reynolds number $\mathit{Re}$ in a vertical Hele-Shaw cell of width $h$. Liquid velocity perturbations induced by the relative movement have also been investigated to analyse the coupling between the bubble motion and the wake dynamics. The confinement ratio $h/ d$ is less than unity so that the bubble is flattened between the walls of the cell. As the bubble diameter is increased, the Archimedes and the Bond numbers increase within the ranges $10\leq \mathit{Ar}\leq 1{0}^{4} $ and $6\ensuremath{\times} 1{0}^{\ensuremath{-} 3} \leq Bo\leq 140$. Mean shapes become more and more elongated. They first evolve from in-plane circles to ellipses, then to complicated shapes without fore–aft symmetry and finally to semi-circular-capped bubbles. The scaling law $\mathit{Re}= 0. 5\mathit{Ar}$ is valid for a large range of $\mathit{Ar}$, however, indicating that the liquid films between the bubble and the walls do not contribute significantly to the drag force exerted on the bubble. The coupling between wake dynamics, bubble path and shape oscillations evolves and a succession of different regimes of oscillations is observed. The rectilinear bubble motion becomes unstable from a critical value ${\mathit{Ar}}_{1} $ through an Hopf bifurcation while the bubble shape is still circular. The amplitude of path oscillations first grows as $\mathit{Ar}$ increases above ${\mathit{Ar}}_{1} $ but then surprisingly decreases beyond a second Archimedes number ${\mathit{Ar}}_{2} $. This phenomenon, observed for steady ellipsoidal shape with moderate eccentricity, can be explained by the rapid attenuation of bubble wakes caused by the confinement. Shape oscillations around a significantly elongated mean shape start for $\mathit{Ar}\geq {\mathit{Ar}}_{3} $. The wake structure progressively evolves due to changes in the bubble shape. After the break-up of the fore–aft symmetry, a fourth regime involving complicated shape oscillations is then observed for $\mathit{Ar}\geq {\mathit{Ar}}_{4} $. Vortex shedding disappears and unsteady attached vortices coupled to shape oscillations trigger path oscillations of moderate amplitude. Path and shape oscillations finally decrease when $\mathit{Ar}$ is further increased. For $\mathit{Ar}\geq {\mathit{Ar}}_{5} $, capped bubbles followed by a steady wake rise on a straight path.