Calculation of Small Deformations of a Radially Convergent Shock Wave Inside a Cavitation Bubble

Abstract

The possibility of increasing the efficiency of calculation of small axisymmetric non-sphericity of a radially convergent shock wave in a collapsing cavitation bubble by the Godunov method of increased order of accuracy is shown in the case the surfaces of the bubble and the shock wave are presented as a combination of the spherical component and its small perturbation in the form of a spherical harmonic of some degree $n$. The dynamics of the vapor in the bubble and the surrounding liquid in the final high-speed stage of collapse is governed by the equations of gas dynamics closed by wide-range equations of state. Non-uniform moving radially-divergent grids are applied, condensing to the bubble surface. An increase in the calculation efficiency is achieved by decreasing the apex angle of the computational domain from (normally accepted) $\pi/2$ to a value that is the minimum among nonzero angles $\theta$ corresponding to the local extrema of the Legendre polynomial of degree $n$ in $\cos{\theta}$. This way of increasing the calculation efficiency was used to study the growth of small axisymmetric non-sphericity of a radially convergent shock wave in a collapsing cavitation bubble in acetone with a temperature of 273.15 K and a pressure of 15 bar in the case of the initial non-sphericity of the bubble in the form of even harmonics of degree $n=6-18$. It was found that in the initial stage of convergence of the shock wave, where it turns into a strong one, its non-sphericity increases more slowly than during the subsequent convergence. In the initial stage, the growth rate of non-sphericity decreases with increasing $n$. During the subsequent convergence, the non-sphericity of the shock wave grows, independently of $n$, proportionally to its radius to the power of $-1.12$.