## 3 Results and Discussion
### Scenario 2 (Cylinder)  
Then, the shape of the high-pressure bubble was modified to be cylindrical, while the total volume of the bubble was kept the same (see Figure 4a). Moreover, the distance from the lateral side of the cylindrical bubble to the boundary of the domain was also set to 0.4. The height of the cylinder is then calculated as 0.133. Other properties such as the initial pressure and the density of the explosive gas were kept the same as the ones applied in Scenario 1. With respect to the initial distribution of the signed distance function, the expression of \(\phi\) is rather complex and thus it is detailed in Appendix A of this manuscript. Figure 4b displays a slice plot of the initial status of the level-set function. Figure 4: The initial setup of the UNDEX simulation with a high-pressure gas bubble of a cylindrical shape. The configurations of the figure are similar to Figure 1.  
Figure 3: The pressure history recorded at the boundaries of the computational domain for an initially spherical bubble (Scenario 1). Point 1 is located at the position (0.5, 0.0, 0.0), and the position of Point 2 is (0.0, 0.0, 0.5). From the temporal evolution of the pressure field shown in Figure 5, it is seen that the dynamics of the UNDEX under these conditions are similar to the one of Scenario 1. In Figure 5a, an initial outward shock wave is formed, leading to a pressure increase of the water in the vicinity of the interface. Meanwhile, a rarefaction wave is generated inwards, causing a pressure drop inside the cylindrical bubble. Later, at time \(t=0.03\) (Figure 5b), the shock wave along the horizontal direction propagates to the location \(x=0.35\), with a peak value of 1700. By comparing with Scenario 1, we found that the horizontal shock wave originating from this cylindrical bubble propagates faster, but with a lower peak value of the pressure. This is possibly caused by the difference in the cross-sectional area of the initial bubble between these two simulation scenarios. For Scenario 1, the area of the cross section along each direction is \(\pi r^{2}\), where \(r\) is the radius of the sphere. In contrast, in the simulation of the UNDEX with a cylindrical bubble, as the height of the cylinder is \(4r/3\), the cross section along the \(x\)- or \(y\)-axis has an area of \(4r/3\times 2r=8r^{2}/3\), which is smaller than that in Scenario 1. Therefore, the peak value of the shock wave along the horizontal direction is lower for the cylindrical bubble. On the contrary, along the vertical direction, a relatively higher pressure exists in this simulation scenario (see Figure 5b). The peak pressure of the vertical shock wave reaches approximately 2084. The deviation of the pressure value between the horizontal direction and the vertical direction in this simulation is also due to the difference in the area of the cross sections for the initially cylindrical bubble. In this simulation scenario, the cross-sectional area along the \(z\)-axis is \(\pi r^{2}\), which is the same as the one in Scenario 1. Thus, the peak value of the vertical shock wave is roughly similar to Scenario 1 at the same time. However, as the vertical cross section of the initial bubble (\(\pi r^{2}\)) is larger than that along the horizontal direction (\(8r^{2}/3\)), it can be expected that the vertical shock wave is stronger  
Figure 5: The temporal evolution of the pressure field for an initially cylindrical bubble in slice plots (\(y=0.0\)). Note that the range of the legend in each figure is different for a better display. than the horizontal one. Afterwards, the shock wave impacts the boundaries of the computational domain (see Figure 5c), causing a reflection of the wave and also an increase of the pressure. Figure 5c also shows that the pressure exerted on the top and the bottom of the domain by the shock wave is relatively larger than that by the collision of the shock wave with the lateral sides of the domain, which is also due to the difference in the area of the cross sections discussed above.