On stability of time marching in numerical solutions of rayleigh-plesset equation for ultrasonic cavitation

Abstract

Ultrasonic irradiation approach has become one of the most popular methods applied in chemical processing including lignocellulosic biomass pretreatment and industrial cleansing. The phenomenon of ultrasonic cavitation can be indeed delineated via the Rayleigh-Plesset equation (RPE), which governs the transient radius of the bubble. Nonetheless, the time marching in the numerical solutions for RPE is highly unstable, which cannot be assured using von Neumann analysis. High sensitivity of RPE to time step may lead to extremely long computational time. The lack of numerical investigation into the time stepping issue of RPE has hindered in-depth simulation of ultrasonic cavitation. Therefore, the purpose of this paper is to investigate the stability criterion of time stepping for RPE in different time progression schemes, namely Euler explicit, 2nd order Taylor’s method, 4th order Runge-Kutta, Runge-Kutta Fehlberg and Cash-Karp Runge-Kutta method. A simple modified adaptive time step method and α independence study has been introduced in this paper for fast, stable and accurate computation of RPE. Compared with the traditional constant time marching method, the new model is able to improve the computational cost significantly without affecting the time marching stability and resolution of the results. Among the investigated method, Runge-Kutta family solvers have higher computational accuracy, with the cost of higher critical α value. The model is also applied to compute the pressure and temperature hike during bubble collapse due to different sonication power. The simulation results show that the ultrasonic irradiation with higher sonication power could produce a higher energy to break the lignocellulose wall.