## 2 Mathematical Models and Methods
### Governing Equations of the Flow Field  
The Euler equations which represent the flow field of a compressible, non-viscous, non-heat-conducting medium are used in this study to describe the gas and water. The specific form of the Euler equations is as follows:  
\[\vec{U}_{t}+\vec{F}_{x}+\vec{G}_{y}+\vec{H}_{z}=0, \tag{1}\]  
in which  
\[\begin{split}&\vec{U}=(\rho\quad\rho u\quad\rho v\quad\rho w \quad E)^{\mathsf{T}},\\ &\vec{F}=(\rho u\quad\rho u^{2}+p\quad\rho uv\quad\rho uw\quad u (E+p))^{\mathsf{T}},\\ &\vec{G}=(\rho v\quad\rho vu\quad\rho v^{2}+p\quad\rho vw\quad v (E+p))^{\mathsf{T}},\\ &\vec{H}=(\rho w\quad\rho wu\quad\rho wv\quad\rho w^{2}+p\quad w (E+p))^{\mathsf{T}},\\ & E=\rho e+\frac{1}{2}\rho(u^{2}+v^{2}+w^{2}).\end{split} \tag{2}\]  
In Equation (1), \(\vec{U}\) is the matrix form of the variables to be estimated. \(t\) denotes time. \(\vec{F}\), \(\vec{G}\) and \(\vec{H}\) are the numerical fluxes in the matrix form along the \(x\)-, \(y\)- and \(z\)-directions, respectively. In Equation (2), \(\rho\) is the density of the fluid, and \(u\), \(v\) and \(w\) are its velocity components along the \(x\)-, \(y\)- and \(z\)-directions, respectively. \(p\) in Equation (2) denotes the pressure. \(E\) represents the total energy of a unit volume of the fluid, and \(e\) is the specific internal energy of the fluid medium.  
For the closure of the system, the Equations Of State (EOS) for the media on both sides of the interface are solved. The EOS of an ideal gas is as follows:  
\[p=\rho RT, \tag{3}\]in which \(R\) is the gas constant. \(T\) denotes the temperature of the gas. The specific internal energy of the gas, \(e\), is then calculated as  
\[e=\frac{p}{(\gamma-1)\rho}, \tag{4}\]  
where \(\gamma\) denotes the ratio of the heat capacities. With respect to the water, the Tait Equation (5) obtained in experiments [36] is used, namely  
\[p=\mathrm{B}\left(\frac{\rho}{\rho_{\mathrm{w}}}\right)^{\gamma_{\mathrm{w}}} -\mathrm{B}+\mathrm{A}. \tag{5}\]  
In this equation, the reference density \(\rho_{\mathrm{w}}\) is given as \(\rho_{\mathrm{w}}=1.0\times 10^{3}\,\mathrm{kg}\) m\({}^{-3}\) and \(\gamma_{\mathrm{w}}=7.15\). The constants \(\mathrm{A}\) and \(\mathrm{B}\) have the values of \(10^{5}\,\mathrm{Pa}\) and \(3.31\times 10^{8}\,\mathrm{Pa}\), respectively. The specific energy of the water medium, \(e\), is thus estimated as:  
\[e=\frac{\mathrm{B}\rho^{\gamma_{\mathrm{w}}-1}}{(\gamma_{\mathrm{w}}-1)\rho_{ 0}^{\gamma_{\mathrm{w}}}}+\frac{\mathrm{B}-\mathrm{A}}{\rho}. \tag{6}\]  
As the pressure and the internal energy in the Tait Equations (5) and (6) are only a function of density, it is unnecessary to solve the energy equation in the simulation of the liquid medium.  
The discretization of the governing equations of the flow field in the present study is performed by using the TVD (Total Variation Diminishing) scheme proposed by [PERSON] [37] and [PERSON] [38]. The forms of the difference equations of this numerical scheme can be found in [37]. It was demonstrated by [PERSON] [37] that this non-oscillatory scheme is robust and able to provide a solution of a second-order accuracy to the Euler equations.