## 3 Results and Discussion  
In this section, we present the simulations of UNDEXs originating from various shapes of high-pressure gas bubbles immersed in a water environment. In this study, the governing equations were solved in terms of dimensionless variables. In order to perform the non-dimensionalization, we introduced the reference parameters, \(l_{0}=1\,\mathrm{m}\), \(\rho_{0}=1.0\,\mathrm{kg\,m^{-3}}\) and \(p_{0}=1.013\times 10^{5}\,\mathrm{Pa}\). As a result, the reference sound speed \(a_{0}\) is calculated as \(a_{0}=\sqrt{p_{0}/\rho_{0}}\), and the reference time is computed as \(t_{0}=l_{0}/a_{0}\). After the implementation of these dimensionless parameters, the forms of the governing equations such as the Euler Equation (1) are kept the same as before, which enables the application of the original numerical schemes. As a result, the value of the dimensional velocity can be obtained by multiplying the simulated dimensionless velocity with the reference sound speed \(a_{0}\). In addition, the dimensional time is achieved by multiplying the dimensionless time with the reference time \(t_{0}\). In the simulations of this study, a computational domain with a size of \([-0.5,0.5]\times[-0.5,0.5]\times[-0.5,0.5]\) was taken, and the number of grid nodes was set to \(61\times 61\times 61\). We had also increased the number of the grid nodes along each direction to 81 (see Section 2 of the Supplementary Materials). It was found that in the finer-mesh computations, the peak value of the underwater shock is generally larger than that in the original \(61\times 61\times 61\) simulations, which is attributed to the weaker numerical diffusion. However, no significant change was found in the main features of the computational results such as the location of the shock. As the three-dimensional computations of the UNDEX are extremely costly, the choice of the present mesh size (\(61\times 61\times 61\)) is a result from the balance between the computational resources and the numerical stability. The CFL (Courant-Friedrichs-Lewy) number 0.1 is used in this 3-D computation to estimate the length of each time step according to the CFL criterion [45]. At first, a spherical high-pressure gas bubble with a radius of 0.1 is located in the center of the computational domain, surrounded by water (Scenario 1, see Figure 1a). The high-pressure gas inside the bubble is assumed to occupy an initial density of \(1270\,\mathrm{kg\,m^{-3}}\) and a pressure of \(8.183\times 10^{3}\) (corresponding to \(8.29\times 10^{8}\) Pa) according to [46], which represents a typical explosive gas after the detonation of the warhead. Indeed, there exists a better choice for the EOS of the explosive gas, namely the Jones-Wilkins-Lee (JWL) equation [47]. However, the JWL equation contains many parameters that need to be fitted based on the experimental results. In addition, different values are given to these parameters for different types of the detonation product gases. In contrast to that, the ideal-gas EOS for the explosive gas is less accurate, but relatively simpler and more convenient to be implemented in the model. As the focus of the present study is laid on investigating the impact of the shape change of the warhead on the dynamics of the UNDEX, rather than precisely capturing the pressure exerted on the submerged structure, we used the ideal-gas EOS for the explosive gas instead, and the update of the EOS was left for future work. A heat capacity ratio of \(\gamma=2.0\) was used for the high-pressure gas [46]. The initial pressure ratio between the gas and water is 8183:1. Later, different shapes of the high-pressure gas bubble such as cylindrical (Scenario 2) and cuboidal (Scenario 3) were applied. We kept the distance from the boundary of the gas bubble to the lateral sides of the computational domain the same, \(0.4\,\mathrm{m}\). Thus, the pressure history recorded at the lateral boundaries of the domain can be compared between different simulation scenarios.