An estimate of the Kelvin impulse of a transient cavity

Abstract

The Lagally theorem is used to obtain an expression for the Bjerknes force acting on a bubble in terms of the singularities of the fluid velocity potential, defined within the bubble by analytic continuation. This expression is applied to transient cavity collapse in the neighbourhood of boundaries, allowing analytical estimates to be made of the Kelvin impulse of the cavity. The known result for collapse near a horizontal rigid boundary is recovered, and the Kelvin impulse of a cavity collapsing in the neighbourhood of a submerged and partially submerged sphere is estimated. A numerical method is developed to deal with more general body shapes and in particular, bodies of revolution. Noting that the direction of the impulse at the end of the collapse phase generally indicates the direction of the liquid jet that may form, the behaviour of transient cavities in these geometries is predicted. In these examples the concept of a zone of attraction is introduced. This is a region around the body, within which the Kelvin impulse at the time of collapse, and consequent jet formation, is expected to be directed towards the body. Outside this zone the converse is true.

Pulsating, buoyant bubbles close to a rigid boundary and near the null final Kelvin impulse state

“…The Kelvin impulse can be interpreted as a linear impulse of the bubble if one attributes a virtual mass to the bubble which corresponds to the liquid mass moving around the cavity. Since axial jets are associated with bubble migration in the direction of the jet, it was suggested that their occurrence and direction can be predicted by analysing the Kelvin impulse at the end of the bubble collapse (Blake et al, 1986(Blake et al, , 1987(Blake et al, , 1997Best and Kucera, 1992;Best and Blake, 1994).…”

Numerical simulation of violent bubble motion

“…The Kelvin impulse of a bubble was introduced by Benjamin and Ellis, 26 and was developed by Blake et al 2,10,27 to derive a criterion governing the directions of the migration and re-entrant jet for bubbles. Our simulations will be compared with the Kelvin impulse theory.…”

“…Best and Blake 27 obtained an analytical estimation of the Kelvin impulse of a bubble in the neighborhood of a rigid surface using the Lagally theorem and assuming the bubble to remain spheri-cal throughout its lifetime. In particular the Kelvin impulse at the end of the collapse phase is given as follows:…”

“…The buoyancy and Bjerknes attraction of the wall are close to each other in magnitude for this case as ␥␦ϭ0.442; consequently, the jet is roughly in their bisection as predicted by the Kelvin impulse theory. 2,27 To investigate the buoyancy effects, the above case is recalculated at ␦ϭ0.074 (␥␦ϭ0.442/2) and ␦ϭ0.221 (␥␦ ϭ0.442ϫ1.5), a weak buoyancy case and a strong one, respectively. The effects of the wall inclination are investigated by recalculating the case at ␤ϭ/4 and 3/4, respectively, where ␤ is the inclined angle of the wall to the x-axis.…”

Cloud Cavitation Dynamics

“…Their results indicated that such a model may be generally at most 3% in error, for such quantities as the bubble volume, Kelvin impulse, and bubble trajectory toward the boundary. Higher-order approximations were used by Best and Blake [1], again to obtain criteria for the migration of the bubble.…”