Bubble velocities induced by trailing vortices behind neighbours

Abstract

Although potential flow, including viscous dissipation, explains quite well the flow around individual bubbles of about 1mm radius rising in water, and e.g. predicts their drag quite accurately, this model cannot explain the homogeneous rise of a bubbly suspension. From numerical and analytical work it follows that eventually all bubbles cluster together. On the other hand it has been shown that velocity fluctuations of the bubbles of sufficient intensity, expressed in terms of a critical (pseudo) temperature, prevents clustering.

Bubbles with radius above 0.8 mm rising in water perform zigzag or spiralling motions. Recently experimental and numerical work has made it clear that such bubbles have a wake behind them consisting of twin vortical threads carrying vorticity of opposite sign in the direction of motion. It is the purpose of this contribution to make an estimate of the velocity fluctuations induced by these trailing vortices in neighbouring bubbles. To this end the two-threaded wake is represented as a horseshoe vortex similar to the wake behind an airfoil. A pair of bubbles is considered and first the velocity induced by the horseshoe vortex behind one of the pair at the centre of the other is calculated. After this the force exerted on the latter based on the induced velocity and on the relative velocity of the bubbles, due to hydrodynamic interaction is calculated. Then the motion of one bubble in the pair is analysed under the influence both of this force and the hydrodynamic forces already there in the absence of the horseshoe vortex. Using these results and appropriate averaging, an estimate is made of the intensity of the velocity fluctuations of bubbles, and the corresponding temperature.