# Numerical Investigation on the Water Entry of Several Different Bow-Flared Sections
## 2 Mathematical Formulation and Numerical Methods  
Referring to previous studies on the water entry of ship sections [11,12] and asymmetric wedges [24,25], two fluids (water and air) were assumed to be immiscible and incompressible. Then, the RANS model consisting of the time-averaged instantaneous continuity and momentum equations can be written as follows:  
\[\frac{\partial u_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial u_{i}}{\partial t}+u_{j}\frac{\partial u_{i}}{\partial x_{j}}= f_{i}-\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+v\frac{\partial^{2}u_{i}}{ \partial x_{j}\partial x_{j}}-\overline{u_{i}^{\prime}u_{j}^{\prime}} \tag{2}\]  
where \(u_{i,j}\), \(i,j\in[1,\ 2,\ 3]\) denotes the time-averaged velocity components in the \(x_{1},x_{2},\ \text{and}\ x_{3}\) directions, respectively; \((x_{1}\,x_{2}\,x_{3})=(X,Y,Z)\); \(f_{i}\) is the time-averaged body force; \(u_{i}^{\prime}\) and \(u_{j}^{\prime}\) are the fluctuating velocity components; \(\overline{u_{i}^{\prime}u_{j}^{\prime}}\) is the Reynolds stress tensor, which is sometimes expressed as \(\tau_{ij}\); \(v\) is the kinematic viscosity of the fluids; \(\rho\) is the density of the fluids; and \(p\) is the time-averaged pressure.  
The interface between air and water was captured using the volume of fluid (VOF) technique [24]. An effective fluid representing the two-phase flow of water and air was introduced to solve the RANS model. Effective density \(\rho_{eff}\) and effective viscosity \(\
u_{eff}\) can be calculated using the following equations:  
\[\rho_{eff}=\gamma\rho_{1}+(1-\gamma)\rho_{2} \tag{3}\]  
\[\
u_{eff}=\gamma\
u_{1}+(1-\gamma)\
u_{2} \tag{4}\]  
where \(\rho_{1}\) and \(\
u_{1}\) are the density and viscosity of air, while \(\rho_{2}\) and \(\
u_{2}\) are the density and viscosity of water. \(\gamma\) is the volume fraction of air; \(\gamma\) = 0 denotes that the fluid in the cell is water, whereas \(\gamma\) = 1 is air. If \(0<\gamma<1\), it means that a combination of air and water exists in the cell. By means of Equations (3) and (4), the property of a cell can be predicted. The governing equation of \(\gamma\) is as follows:  
\[\frac{D\rho}{Dt}=0\rightarrow\frac{D(\gamma\cdot\rho_{1}+(1-\gamma)\rho_{2}}{ Dt}=0 \tag{5}\]  
\[\frac{D\gamma}{Dt}=\frac{\partial\gamma}{\partial t}+u\cdot\
abla\gamma=0 \tag{6}\]  
The commercial CFD solver StarCCM+ was used to solve the RANS model. The chosen turbulence model was realizable k-epsilon two-layer turbulence model by referring to previous studies [24,26]. Temporal integration was carried out using the Euler implicit scheme. The governing equations were discretized using the central difference scheme in space except the convection term, which used a second-order upwind scheme. The segregated flow solver based on the semi-implicit method for pressure-linked equation (SIMPLE) algorithm was used to solve the pressure and velocity coupling problem during the water entry. The motion of the grid around the ship section was numerically realized using the overset mesh method, and the transfer of physical quantities between the overset zone and the background zone was realized using the linear interpolation method.