A numerical method for studying nonlinear bubble oscillations in acoustic cavitation

Abstract

A numerical method is developed for the study of the behaviour of a gas bubble in ultrasonically induced cavitation. This method is based on the solution of the full Navier-Stokes equation for the two-fluid system consisting of the gas inside the bubble and the liquid surrounding it, following ideas originally introduced for the analysis of multi-component fluid flows. Analysis of acoustic cavitation must be done taking into account the compressibility of the gas bubble and for this purpose the Navier-Stokes equation is coupled with an equation of state for the gas; our model also considers the presence of viscosity and surface tension, thus allowing surface oscillations of the bubbles. To avoid numerical problems in the solution of the Navier-Stokes equation two different grids are introduced, an Eulerian one for the ‘background’, where the Navier-Stokes equation is solved, and another moving one for the interface; this second grid is explicitly tracked and properly modified during motion and is responsible for the behaviour of the bubble. The transfer of information between the Eulerian grid and the interface grid is obtained with the aid of a lattice modified distribution function. The method is tested analyzing forced oscillations of cavitation bubbles excited by ultrasonic standing waves at different frequencies and pressure amplitudes, showing characteristic behaviour of nonlinear dynamical systems; frequency spectra are obtained, stability analysis is performed and strong dependence from initial conditions is showed; comparisons with previous different approaches are also performed.