On bubbles rising in line

Abstract

The dynamic behaviour of equal-sized spherical gas bubbles rising in vertical line was studied numerically at Reynolds number ( Re) 50–200. A force-law model was suggested for hydrodynamic interactions within the bubble chain. Three forces acted on each bubble: buoyancy, viscous drag, and inviscid inertia forces. Both local (nearest-neighbour approximation) and non-local (distant effects) interactions between bubbles were considered. The viscous non-local interactions consisted in creation of velocity disturbances by the passage of bubbles and resulted in progressive drag reduction down the chain. Due to the balance between creation and decay of the disturbances, the distant interactions affected only a certain number of the anterior bubbles until the limit chain drag was reached. This drag then applied to all remaining bubbles independent of their positions and the distant effects were thus eliminated. The inviscid non-local interactions consisting in inertial coupling between distant bubbles were found weaker than the viscous non-local interactions and were, therefore, neglected. Two kinds of bubble chain with different boundary conditions were distinguished: the free-end chain and the fixed-end chain. The free-end chain tended to split into smaller bubble groups that subsequently interacted and produced a highly fragmented long-time chain structure. The typical phenomena in bubble interactions were the following: merger, separation, pairing-off, re-pairing, and oscillation. The fixed-end chain allowed for uniform spacing, rose faster than an isolated bubble, and supported propagation of concentration disturbances. Uniform spacing became unstable at low Re and bubbles displayed chaotic behaviour. The results produced by the model were compared with data published in the literature. The model was able to predict and explain the key phenomena observed in experiments. This correspondence was obtained in the nearest-neighbour approximation (the distant effects may not be very strong in real systems). Intensive non-local interactions affected the chain behaviour substantially. The analogy between hydrodynamic and mechanical ‘masses-on-spring’ systems was pointed out.