Simulation of bubble motion in a compressible liquid based on the three dimensional wave equation

A simulation model for bubble motion in a compressible liquid is developed based on the linear wave equation. At the initial stage, the bubble is assumed to be spherical and the wave equation is simplified as a one dimensional ordinary equation in the radial direction through the prophase approximation. When time becomes much larger than that required for a disturbance to travel across the bubble at the speed of the sound, the obtained integral equation is approximated by keeping the term of the first order in terms of the characteristic Mach number, through the anaphase approximation. An equation is introduced to unify the approximations in these two phases, which is then used over the entire simulation period. The problem at each time step is solved by a three dimensional boundary element method. The convergence study has been first taken with meshes and time steps. Comparison is made with the analytical solution for spherical bubble in compressible liquid and good agreement is found. Further comparison is made for a bubble in an incoming acoustic wave. Extensive simulations are then made for a bubble in various conditions, including the cases with solid boundary effect, free surface effect, buoyancy effect, as well as for interactions between bubbles.