## 3 Results and Discussion
### Scenario 1 (Sphere)  
The initial setup of simulation scenario 1 is shown in Figure 1. A spherical hot-gas bubble is located at the center of the computational domain (see Figure 1a). The initial pressure inside the bubble is \(8.183\times 10^{3}\), corresponding to \(8.29\times 10^{8}\) Pa as mentioned above. The initial pressure of the water surrounding the bubble is \(1.0\) (\(1.013\times 10^{5}\) Pa). Both the high-pressure bubble and the water are assumed initially quiescent. Two recording points, Point 1 with the position (0.5, 0, 0) and Point 2 (0, 0, 0.5), were placed at the lateral boundary and the top of the computational domain, respectively (see Figure 1a). By doing so, we were able to obtain the time history of the pressure exerted at the boundaries of the domain by the UNDEX with different shapes of the bubble. With respect to the initial value of the level-set function, the distance function \(\phi\) is defined as \(\phi=0.1-\sqrt{x^{2}+y^{2}+z^{2}}\), in which \(\phi\geq 0\) denotes the high-pressure gas and \(\phi<0\) represents the water, and a 2-D plot of the initial distribution of the signed function is shown in Figure 1b. The evolutions of the pressure field of the UNDEX are presented in Figure 2. In Figure 2a, it is seen that at an early stage of the explosion, \(t=0.006\) (corresponding to \(19\,\upmu\)s), an initial shock wave is generated and propagates outwards in the water medium. The peak value of the shock wave is approximately \(3600\). As a result, the water medium is compressed, and thus the density increases. In contrast to that, inside the bubble, a rarefaction wave is formed, propagating from the interface to the center of the bubble, which reduces the gas pressure inside. The density of the gas is also lowered (not shown here). The wave patterns after the detonation are found to be consistent with the findings in previous modeling studies [34; 43; 44], which indicates the correctness of the model results. As time passes by, the compressive wave attenuates when propagating in the water. At the time \(t=0.03\) (see Figure 2b), the shock wave has arrived at the location of approximately \(x=0.3\), and the peak of the pressure has dropped to a value of \(2000\). By comparing the pressure distributions along all directions, it is not surprising to find that the propagation of the shock wave is nearly symmetrical due to the initial spherical shape of the bubble. However, it should be noted that a slight asymmetry occurs in the computational results with respect to the planes \(x=0\) and \(z=0\). This is caused due to the existence of rounding errors, and it happens when the interface is initially located in the position  
Figure 1: The initial setup of the underwater explosion (UNDEX) simulation with a high-pressure gas bubble in a spherical shape. (**a**) the initial distribution of the pressure, shown in a slice plot (surfaces \(y=0\) and \(z=0\)). Two recording points, Point 1 with the position (0.5, 0.0, 0.0) and Point 2 (0.0, 0.0, 0.5) were placed at the boundaries of the computational domain; (**b**) a 2-D plot of the initial level-set function \(\phi\) in the \(x\)-\(z\) surface (\(y=0\)). of the grid nodes. However, this asymmetry affects mostly the low-pressure regions, instead of the propagating high-pressure shock (see Section 3 of the supplements). Moreover, the deviation of the pressure exerted on the lateral domain boundaries between opposite directions is only within 1%. Since the focus of the present study is laid on the impacts of the underwater shock generated by various shapes of high-pressure bubbles on the submerged structures nearby, the merits of the present study are not significantly influenced by this asymmetry. Later on, when the shock wave arrives at the lateral boundary of the computational domain at the time \(t=0.08\) (Figure 2c), a reflection of the pressure occurs due to the existence of the solid wall, which results in a superimposition of the pressure. Consequently, a maximum of the pressure is achieved, and the peak value reaches 1530 at this time.