ABSTRACT
When an explosive explodes in water, a shockwave will generate at the outset and then a pulsating bubble. The shockwave has strong discontinuity and the bubble motion is a two phase problem. The period of shockwave for a charge exploding underwater is about O(1 ms) while that of the bubble is about O(1 s). The difference between these two processes brings difficulties to the relative simulations. The Discontinuous Galerkin (DG) method is popular as well as the Boundary Element Method (BEM) which is used to solve the shockwave propagation and the bubble motion separately and highly good results are obtained. But they do not have sufficient applications in the simulation of both shockwave and bubble motion up till now. In this work, we focus on the features of shockwave propagation and bubble pulsating features under different boundary conditions. Based on the Eulerian Finite Element Method (EFEM), the Eulerian equation is discretized. In addition, the charge denotation is simulated by JWL equation. Using this method, the whole process of the shockwave emission and bubble pulsation is studied. The numerical models are validated through both theoretical and numerical methods. Then the bubble pulsations near disparate boundary conditions, such as free surface and solid wall are simulated. It is found that there are two pressure peaks in one pressure curve in these conditions, the first one comes from the shockwave and the second one is from the reflection of the shockwave at the boundary. When the bubble is near the free surface, the bubble is repelled by the free surface and a downward jet is formed, which is contrary to the case when a bubble is near the solid wall.
There has been a steady growth of interest in studying the bubble dynamics for the past few years owing to its wide applications in numerous aspects, such as underwater explosion (Meng et al., 2019; Ming et al., 2016; Chen and Yao, 2016), the medical and imaging enhancement (Constantin and Ronald, 2008; Lindner, 2004; Chen and Hwang, 2013), the ship and ocean engineering field (Graaf et al., 2014; Chen et al., 2008; Thompson, 2003) and so on. When a charge explodes underwater, both a shockwave and a bubble will be generated. According to the different time scales, the whole process of underwater explosion is segmented into the shockwave stage and bubble pulsation stage, which are usually investigated respectively. In the shockwave stage, the duration of the shockwave generated by the explosion is only at the millisecond order and it exhibits strong discontinuity and nonlinearity. In the vicinity of the free surface or structure, a cavitation region may occur under the combined effect of an incident shockwave and sparse wave. The DG method has been adopted to simulate the shockwave in recent years, because it can capture the discontinuities of flow field with high resolution. Cockburn and Shu (1989; 1998; 1989) solved the conservation equations using the DG method for exploring the algorithm efficiency and calculation accuracy, and also investigated the characteristics of the shockwave propagation in different media. However, there are still many technical problems using this method to study the bubble motions.
