Transition from spherical cap to toroidal bubbles

Large gas bubbles rising under the effect of buoyancy are known to adopt either a spherical cap shape or to undergo a topological transition after which they become toroidal. We carry out an axisymmetric numerical investigation of the evolution of such large bubbles in the presence of both capillary and viscous effects. The numerical approach is of the volume of fluid type (it solves the Navier-Stokes equations on a fixed grid and transports the local volume fraction of one of the fluids), but does not involve any explicit reconstruction of the interface. The transition from spherical cap to toroidal bubbles is studied in the parameter space built on the Bond (Bo) and Archimedes (Ar) numbers, which compare the strength of inertial effects to that of capillary and viscous effects, respectively. Preliminary tests show that the position of this transition is very sensitive to the grid resolution; these tests are used to select grid characteristics that yield grid-independent results. Two markedly different transition scenarios, corresponding to the limit of large Ar and large Bo, respectively, are then identified. In the first case, the front of the bubble is pierced by an upward jet coming from the rear of the bubble. In contrast, in the limit of large Bo, a downward jet develops at the front part and pierces the rear of the bubble, unless viscous effects are sufficient to stabilize the front. We also determine the position of the transition for intermediate values of Bo and Ar and discuss the connection between present axisymmetric results and experimental situations in which the bubble is followed by a turbulent wake. We finally examine a puzzling feature of these large bubbles which is that, given an initial gas volume, the final bubble topology appears to depend dramatically on the initial conditions. Indeed, we find that initially oblate bubbles may result in stable spherical cap bubbles for values of Bo and Ar well beyond those for which initially spherical bubbles of similar volume undergo the topological transition. This remarkable influence of the initial shape is shown to be due to the influence of the oblateness on both the bubble acceleration and the hydrostatic pressure difference between the two bubble poles.