## 3 Results and Discussion
### Scenario 3 (Cuboidal)  
In this simulation scenario, the shape of the high-pressure bubble was varied to be cuboidal, whereas all other parameters were kept the same (Figure 7a). The distance from the lateral side of the cuboid-shaped bubble to the side boundary of the domain was still set to 0.4 m. The height of the cuboid was thus calculated as \((4\pi r^{3}/3)/(4r^{2})=\pi r/3=0.105\), where \(r=0.1\). The initial distribution of the level-set function is given in Appendix A. The principle of setting up this initial level-set function is to divide the computational domain into parts, and then define the level-set function in each part separately. Figure 7b gives a slice plot of the initial distribution of the signed distance function. Figure 6: The pressure history recorded at the boundaries of the computational domain for an initially cylindrical bubble (Scenario 2). The temporal behavior of the pressure field under this situation is visualized in Figure 8. Similarly, at the beginning of the explosion, an outward compressive shock wave and an inward rarefaction wave are formed (see Figure 8a). Similar to Scenario 2, we found that in the application of the initially cuboid shape of the bubble, the shock wave propagates faster along the horizontal direction than the vertical direction. The pressure peak of the horizontal shock wave is 4815, while a higher pressure exists in the vertically propagating shock wave. As time passes by, at \(t=0.03\) (Figure 8b) the pressure of the shock wave along the vertical direction has a peak value of 2460, while the maximum pressure of the horizontal shock wave is only 1688. This is because the area of the vertical cross section of the initial cuboid-shaped bubble is \(4r^{2}\), where \(r=0.1\), while the horizontal cross section possesses an area of only \(2\pi r^{2}/3\). The difference in the area of the cross sections dominates the peak values of the pressure along these two directions. By comparing this simulation with Scenario 2, we found that under the condition of the cuboidal bubble, the shock wave along the horizontal direction is weaker than that in the cylindrical bubble simulation. In contrast, the vertical shock wave in this scenario is relatively stronger than that obtained in Scenario 3. It is possibly because the area of the vertical cross section of the cuboidal bubble (\(4r^{2}\)) is larger than that of the cylindrical bubble (\(\pi r^{2}\)), while the horizontal cross section of the cuboid-shaped bubble (\(2\pi r^{2}/3\)) is smaller than that in the cylindrical situation (\(8r^{2}/3\)). From these comparisons, we infer that if the cross section is larger, the energy of the initial explosives is liberated and transmitted through a wider channel towards the boundaries of the domain. As a result, during the traveling of the shock wave in the water, the attenuation from the sides is weaker, thus maintaining a relatively higher pressure region. The stronger vertical shock wave and weaker horizontal shock wave are also indicated at later time steps (see Figure 8c,d). Figure 7: The initial setup of the UNDEX simulation with a high-pressure gas bubble in a cuboidal shape. The configurations of the figure are similar to Figure 1.  
In Figure 9, the time record of the pressure exerted at the boundary Points 1 and 2 by the expansion of the initial cuboidal high-pressure bubble is shown. It is seen that, after the arrival of the shock wave on the boundaries of the domain, the pressure attains a maximum value, and then drops rapidly. Again, it is found that the vertical shock wave is stronger than the horizontal shock wave. The maximum of the pressure recorded at the top of the boundary reaches 1714, whereas the peak value obtained at the lateral boundary is 1340. It is also noticed that the duration of the high pressure at the top boundary is shorter than that at the side boundary.