## 3 Results and Discussion
### Scenario 4 (Bullet-Like)  
In this simulation scenario, we adopted a specially designed bullet-like shape for the initial high-pressure bubble. The initial hollow-cone shape of the bubble is shown in Figure 10a. Warhead shells of a similar shape have been widely used in many fields such as oil exploration and warship protection. We kept the volume of the initial bubble and the distance from the center of the bottom circle to the lateral boundary of the domain, which is 0.4 m, unchanged compared to the previously discussed cases. The radius of the outer bottom circle of the cone is 0.17, and the radius of the inner circle is 0.1. The apex angle of the cone is assumed as 90\({}^{\circ}\). The initial setup of the distance function is shown in Figure 10b and presented in Appendix A.  
Figure 9: The pressure history recorded at the boundaries of the computational domain for an initially cuboid-shaped bubble (Scenario 3). The computational results of the pressure evolution under these conditions are shown in Figure 11a-d. It is seen that the expansion of this bullet-like high-pressure bubble leads to a complicated transmission of the pressure and energy. At the time \(t=0.006\) (Figure 11a), the underwater gas bubble with the complex shape begins to expand due to the high pressure, and a compressive shock wave is generated outwards. Thus, in the vicinity of the outer surface of the conical bubble, the pressure of the water increases. In contrast, the pressure inside the bubble drops rapidly due to the formation of the inward rarefaction wave. However, as a result of the particular shape of this air bubble, at the central axis of the cone, the outward shock wave concentrates. Therefore, an extremely high pressure region is formed, and the peak value reaches 10,242. In contrast to that, the pressure of the shock wave in other regions has a value within the range of only 2000-4000. Moreover, due to the symmetry of the shape of the initial air bubble, the ultra-high pressure region has only the horizontal velocity along the positive direction of the \(x\)-axis. Later, when \(t=0.03\) (Figure 11b), due to the horizontal velocity in the high pressure region, this strong shock wave at the central axis is \"pushed out\" towards the positive direction of the \(x\)-axis. Although the shock wave decays after traveling in the water, the peak value of the pressure still reaches 4028, which is far more than the values in the other simulation scenarios at the same time. Then, the collision of the shock wave and the lateral boundary of the domain occurs. It is seen in Figure 11c that the average pressure under this situation is normally higher than those in Scenarios 1, 2 and 3 at the same time. At the end of the simulation (Figure 11d), as the shock wave along the central axis of the cone has already returned back to the location \(x=0.25\), at the upper-right (\(x=0.5\) and \(z=0.5\)) and the lower-right (\(x=0.5\) and \(z=-0.5\)) boundaries, the reflection just occurs, resulting in a high pressure of 1641. In contrast, at the upper-left and lower-left boundaries (\(x=-0.5\) and \(z=\pm 0.5\)), the shock wave has not arrived yet. Figure 10: The initial setup of the UNDEX simulation with a high-pressure gas bubble in a bullet-like shape. The configurations of the figure are similar to Figure 1.  
The time-dependent pressure curves recorded at Points 1 and 2 are also shown in Figure 12. The extremely high pressure value of the shock wave (2313.0) at the lateral side of the domain is obtained. In contrast to that, the pressure peak felt at the top of the domain occurs later, at the time \(t=0.072\). In addition, the peak value is 1142, approximately only half of that obtained at Point 1. However, the duration of the extremely high pressure recorded at Point 1 is rather short. The pressure drops from the peak to the amount less than 230 within 0.03 (corresponding to 91 us). This relatively short high-pressure duration is also possibly caused by the smaller effective thickness of the bubble along the horizontal direction.