## 2 Numerical Scheme
### Governing Equations  
The RANS CFD method with the VOF model includes pressure boundary conditions on the free surface of fluid flow [24]. Assuming a homogeneous multi-phase flow model using the VOF method allows one to ignore the relative motion between phases.  
With the homogeneous multiphase flow assumed in most implementations of the VOF method, it can ignore the relative motion between the phases. The interface momentum transfer and velocity fields are also resolved using only one set of momentum conservation equations in both phases. The governing equation for homogeneous flows uses the following mass and momentum conservation equations:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_{i})=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x_{j}}\big{(} \rho u_{i}u_{j}\big{)}=-\frac{\partial p}{\partial x_{j}}+\mu\frac{\partial^ {2}u_{i}}{\partial x_{j}\partial x_{j}}+b_{i} \tag{2}\]where \(b_{i}\) is the external body force, \(p\) is pressure, and \(u_{i}\) and \(x_{i}\) are the Cartesian velocity and coordinate tensors. Only one momentum conservation equation is used for both fluids. The homogenous approach uses a dynamic viscosity (\(\mu\)) and density (\(\rho\)) of the fluid mixture calculated using the volume fraction (\(r_{k}\)) with the constraint \(\sum r_{k}=1\), as expressed in Equations (3) and (4).  
\[\rho=\sum\rho_{k}r_{k} \tag{3}\]  
\[\mu=\sum\mu_{k}r_{k} \tag{4}\]  
For inhomogeneous fluid flow, the interfacial momentum and mass transfer is directly dependent on the interface surface area between the two phases. It is characterized by the interaction area per unit volume between two phases. Therefore, the interface momentum transfer and the velocity field are solved for both phases. An inhomogeneous viscous compressible multiphase flow with two phases (\(\alpha\) and \(\beta\)) can be described by Equations (5) and (6).  
\[\frac{\partial}{\partial t}(r_{k}\rho_{k})+\frac{\partial}{\partial x_{i}}(r _{k}\rho_{k}u_{i})=m+\Gamma_{\alpha\beta} \tag{5}\]  
\[\frac{\partial}{\partial t}(r_{k}\rho_{k}u_{i})+\frac{\partial}{\partial x_{ j}}(r_{k}\rho_{k}u_{i}u_{j})=-r_{k}\frac{\partial p}{\partial x_{i}}+\frac{ \partial}{\partial x_{j}}\Bigg{(}r_{k}\mu_{k}\frac{\partial u_{i}}{\partial x _{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Bigg{)}+M^{r_{k}}+b_{i} \tag{6}\]  
where \(\Gamma_{\alpha\beta}\) is the mass transfer between phases, \(m\) is the mass source expressed in Equation (7), \(M^{r_{k}}\) is the force on the interface caused by the presence of phase \(\beta\) as expressed in Equation (8), and \(M^{r_{k}}\) is interphase momentum transfer caused by mass transfer. The interface momentum transfer (\(M^{r_{k}}\)) needs to be taken into account as it ties the fluid velocity fields. This term may be modelled by drag force (\(M^{D}\)) as the dominant force acting on the fluid interface in the sloshing problem, as described in Equation (8).  
\[\Gamma_{\alpha\beta}=\dot{m}_{\alpha\beta}A_{\alpha\beta}=\Gamma_{\alpha\beta }^{+}-\Gamma_{\beta\alpha}^{-} \tag{7}\]  
\[M^{r_{k}}\approx M^{D} \tag{8}\]  
where, the interphase drag force (\(M^{D}\)) is expressed using the drag coefficient, as follows:  
\[C_{D}=\frac{D}{0.5\rho\big{|}u_{\alpha}-u_{\beta}\big{|}^{2}A} \tag{9}\]  
where \(A\) is interfacial area, \(D\) is drag, \(\rho\) is density, and \(\big{|}u_{\alpha}-u_{\beta}\big{|}\) is the velocity between the phases \(\alpha\) and \(\beta\). For the current Newtonian flow regime, a drag coefficient of 0.45 is used [28].