Numerical Simulation of Three-Dimensional Bubble Oscillations by a Generalized Vortex Method

Abstract

A numerical method is implemented for simulating the simultaneous three-dimensional volume and shape oscillations of a compressible vapor or gas bubble suspended in an inviscid ambient fluid in the presence of interfacial tension. The flow generated by the bubble expansion, contraction, and deformation is represented by an interfacial distribution of potential dipoles supplemented by a point source situated inside the bubble, accounting for changes in the bubble volume. The mathematical formulation is completed by setting the strength of the point source proportional to the integral of the density of the double-layer potential over the interface. The motion of marker points distributed over the interface is computed using a boundary-element implementation of Baker's generalized vortex method in which the normal component of the interfacial velocity is computed in terms of tangential derivatives of the vector potential associated with the dipoles, whereas the tangential component of the interfacial velocity is computed in terms of the surface gradient of the scalar harmonic potential. The density of the double-layer distribution is computed by solving an integral equation of the second kind using an iterative method, while the evolution of the interfacial distribution of the harmonic potential is computed using Bernoulli's equation for irrotational flow. The onset of interfacial irregularities due to numerical instabilities is prevented by truncating the Fourier–Legendre spectrum of the interfacial distribution of the harmonic potential. With smoothing implemented, the numerical method is capable of describing simultaneous volume and shape oscillations for an indefinite period of time.