# Study on the Influence of a Rigid Wall on Cavitation in Underwater Explosions Near the Free Surface
## 2 Numerical Method  
The compressible nature of the multiphase flow involving explosive gas, air, liquid, and vapor water in underwater explosions near a free surface is well captured by considering these as compressible fluids. The utilization of a 4-equation model effectively portrays the dynamic behavior of these diverse, multiphase fluids throughout their motion process [29] as follows:  
\[\frac{\partial Q}{\partial t}+\frac{\partial F}{\partial x}+\frac{\partial G}{ \partial y}=0 \tag{1}\]  
where \(Q=\left[\rho,\rho u,\rho v,\rho E,\rho Y_{k}\right]^{T}\), \(F=\left[\rho u,\rho u^{2}+p,\rho uv,(\rho E+p)u,\rho Y_{k}u\right]^{T}\), \(G=\left[\rho v,\rho uv,\rho v^{2}+p,(\rho E+p)v,\rho Y_{k}v\right]^{T}\), \(\rho\) is the fluid density, \(u\) and \(v\) are velocity components of the \(x,y\)-axis, respectively, \(p\) is the pressure, \(E\) is the total energy per unit mass, and \(Y_{k}\) is the mass fraction for the \(k\)-th species in the mixture fluid. The equation of state for each phase fluid is described by the Noble-Abel stiffened-gas (NASG) EOS [31].  
The 4-equation system (1) does not take into account the mass transfer between the liquid water and its vapor phase. In other words, the liquid water and its vapor phase in the mixture solved by Equation (1) are out of thermodynamic equilibrium. During the phase transition process, the mixture specific volume \(v=1/\rho\) and internal energy \(e\) do not vary under the assumptions of constant pressure and velocity. The mass fractions for the liquid and its vapor phase remain constant, although the pressure and temperature vary and reach their equilibrium values (\(p^{\star}\), \(T^{\star}\)) so that the phase transition model will change to compute the equilibrium state (\(p^{\star}\), \(T^{\star}\), \(Y_{\text{k=1,2}}\)) from the state (\(v\), \(e\), \(p\), \(T\), \(Y_{\text{k}}\)). Therefore, under the assumption of mechanical and thermal equilibriums, the mixture fluid satisfies [29; 30] the following:  
\[\left\{\begin{array}{l}T=T_{k},\quad p=p_{k},\quad\forall k\\ g_{1}=g_{2}\\ v=\sum\limits_{k=1}^{N}Y_{k}v_{k};\quad\quad e=\sum\limits_{k=1}^{N}Y_{k}e_{k }\end{array}\right. \tag{2}\]  
This system contains four unknown variables with four equations, which can be solved using the Newton iterative method. Therefore, the phase transition model ensures that the final equilibrium state (\(p^{\star}\), \(T^{\star}\)) is saturated at each step. The numerical method of the multiphase compressible fluid model in this study has been previously validated in various cases involving phase transition in underwater explosion applications [13; 14].  
The control system (1) and (2) can be solved using a simple fractional step method that consists of the homogeneous hyperbolic step (step 1) and is followed by a relaxation step (step 2). In step 1, Equation (1) is solved using the MUSCL-Hancock method, using [PERSON]'s slope limiter and HLLC approximate Riemann solver. After the preliminary solution of the flow is provided, Equation (2) is solved using the Newton iterative method to obtain the final equilibrium solution after phase transition in step 2.