Simulation of the final stage of a cavitation bubble collapsing near a rigid wall

During the collapse of an initially spherical cavitation bubble near a rigid wall, a reentrant jet forms from the side of the bubble furthest from the wall. In the final stage of the collapse, this jet impacts and penetrates the bubble surface closest to the wall. During penetration a continuous liquid-liquid impact occurs, creating a shear layer along the penetration interface. This process is modeled with potential flow theory. Before penetration the bubble is treated as a regular surface and the conventional boundary integral equation is used in the numerical scheme. During penetration the bubble surface is transformed to a ring bubble attached to a vortex sheet (non-regular surface) which models the shear layer. To apply the boundary integral technique to this topology the conventional integral equation is modified and the hypersingular integral equation is introduced. The corresponding numerical scheme with an appropriate time stepping technique is developed to carry out the simulation of the final stage of bubble collapse continuously from before penetration into the penetration process. The characteristics of the penetration phenomenon revealed numerically are as follows: (1) a high pressure region around the penetration interface is generated by the impact causing a sudden change in pressure gradient in the flow field; (2) the normal velocity of the shear layer at the bubble axis of symmetry is always directed toward the wall at a speed less than the velocity of the reentrant jet before impact; (3) circulation is induced by the impact and remains constant during penetration for a path that pierces the vortex sheet once at the axis of symmetry and extends around the ring bubble; (4) the source of the circulation is divided between the vortex sheet and the ring bubble; (5) the energy loss due to the liquid-liquid impact can be as much as 13% of the total mechanical energy under certain conditions; (6) the details of the features of the penetration process strongly depend on {dollar}Zsb0{dollar}, the initial distance of the bubble centroid to the wall.