Migration of pulsating buoyant bubbles beneath a free surface

This paper presents a systematic investigation of bubble migration behavior beneath a free surface through a combination of experimental, numerical, and theoretical approaches. In the experiment, it is observed that the bubble subject to a large buoyancy effect transitions into a stable upward-migrating vortex ring after jet penetration, which subsequently interacts with the free surface and produces distinct jetting phenomena. Through numerical simulations based on the boundary integral method, we quantitatively determined the Kelvin impulse I and migration displacement d of the bubble at the end of its first cycle across a broad parameter space with stand-off distance (γ) ranging from 0.5 to 8 and buoyancy parameters (δ) from 0.02 to 0.4. It is observed that for bubbles with sufficiently weak buoyancy, the displacement during the first cycle tends to follow d∼γ−1.77 when γ is relatively large. In contrast, for bubbles situated far from the free surface, where buoyancy dominates the migration, the displacement follows a scaling relationship of d∼δ1.1. Special emphasis is placed on the reversal of migration direction observed in buoyant bubbles at relatively small stand-off distances. These initially downward-migrating bubbles eventually undergo directional reversal and ascend due to the dominant effect of buoyancy in the long run. By employing a theoretical framework based on Kelvin impulse theory, we demonstrate that this directional reversal is not solely attributable to the interplay between Bjerknes and buoyancy forces but is also modulated by the temporal variation in the bubble's added mass resulting from its volumetric oscillations.