# Study on the Influence of a Rigid Wall on Cavitation in Underwater Explosions Near the Free Surface
## 3 Results and Discussions
### Effect of Rigid Wall on Small-Charge Underwater Explosion near Free Surface  
In order to analyze the role of phase transition in the formation of cavitation and atomization during underwater explosions near the free surface, a two-dimensional axisymmetric model was employed to simulate an explosion generated by a 5.2 g equivalent of trinitro-toluene (TNT) at depths of 0.13 m and 80 m. To investigate the influence of a rigid wall on cavitation evolution, two simulation cases were designed, as depicted in Figures 2 and 3. It should be noted that the rigid boundary conditions of two-dimensional axisymmetric models are equivalent to the wall conditions of rigid cylinders in a three-dimensional space. Figure 1: Numerical results for shock tube test with a two-phase mixture made of liquid water, vapor water, and air. The dashed lines represent the initial conditions. The short-dashed lines represent the solutions without phase transition (no _P-T_), and the thick lines represent the phase relaxation algorithm (_P-T_). Figure 2: Schematic of underwater explosion near a free surface without a rigid wall (**a**) and mesh strategy of the fluid domain (**b**). Every 1 in 10 lines is plotted in the right figure. Figure 2 illustrates the simulation setup without a rigid wall and showcases the mesh strategy employed. A non-uniform mesh scheme was utilized to enhance the computational efficiency, with the computational domain extending from (0, 7) m to (\(-\)7, 2) m. The initial radius of the charge, denoted as R0, was set to 0.009 m. The computational domain included a uniform zone (with a mesh size of \(\Delta\)x = \(\Delta\)y = R\({}_{0}\)/10) measuring 40R\({}_{0}\)\(\times\) 80R\({}_{0}\), which facilitated the modeling of the motion of the explosion bubble. The symmetric axis (y-axis) was assigned an impenetrable boundary condition, while non-reflecting boundary conditions were applied to the surrounding boundaries. Figure 3 presents the simulation setup with a rigid wall and outlines the corresponding mesh strategy. In this case, a rigid wall was positioned at the right boundary. The initial conditions for both simulation cases were considered as follows:  
\[ \begin{split}&(\rho,u,p,T,Y_{1},Y_{2},Y_{3},Y_{4})=\\ &\quad\left\{\begin{array}{ll}\left(1.18,\ 0,\ 10^{5},\ 295,\ 0,\ 10^{-9},\ 0.999,\ 0\right)& \text{for \emph{air}}\\ \left(1054,\ 0,\ 10^{5},\ 295,\ 0.9999,\ 10^{-8},\ 4.73\times 10^{-7},\ 0\right)& \text{for \emph{water}}\\ \left(1606,\ 0,\ 10^{9},\ 1080,\ 0,\ 10^{-8},\ 0,\ 0.999\right)&\text{for \emph{gas}}\end{array}\right.\end{split}  \tag{4}\]  
By employing these simulation setups, the study aimed to investigate the effects of phase transition and the presence of a rigid wall on the formation and evolution of cavitation and atomization during underwater explosions. The chosen depth variations allowed for the examination of different scenarios and the assessment of the impact of water depth on the observed phenomena. These simulations enable a comprehensive understanding of the complex dynamics involved in underwater explosion-induced cavitation and contribute to the development of strategies for blast mitigation and the design of protective measures in marine and underwater engineering. Figures 4 and 5 illustrate the density, pressure, and cavitation domain characteristics during an underwater explosion near the free surface, with and without a rigid wall, at different time intervals. It is evident that the cavitation domain shapes in both cases are identical before 0.5 ms. This occurs because the compression wave reflected on the wall has not yet reached the cavitation zone, as depicted by the pressure contours in Figure 5. At \(t\) = 0.5 ms, the reflected compression wave reaches the surface of the cavitation domain, resulting in minimal differences between the cavitation domains in the two cases. As the reflected compression wave propagates from the rigid wall, the cavitation domain becomes smaller compared to the case without a rigid wall.