# Effect of Waves on the Behavior of Emergent Buoyantly Rising Submarines Using CFD
## 2 Theory
### Fluid EOM  
The fluids in the simulation include two phases, namely air and water, both of which are assumed to be incompressible. The motion of each fluid phase can be expressed by the RANS equations, and the free surface between the water and air is captured by the volume-of-fluid (VOF) method. The following equations represent the mass conservation and momentum conservation, respectively:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\left(\rho\mathbf{U}\right)=0 \tag{4}\]\[\frac{\partial\rho U_{i}}{\partial t}+\frac{\partial\rho U_{i}U_{j}}{\partial x_{j }}= -\frac{\partial P}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left[\mu \left(\frac{\partial U_{i}}{\partial x_{j}}+\frac{\partial U_{j}}{\partial x_{ i}}\right)\right]-\frac{\partial}{\partial x_{j}}\rho\overline{u_{i}^{\prime}u_{j}^{ \prime}}+F_{B} \tag{5}\]  
The instantaneous fluid velocity, \(U\), is denoted by the mean component, \(U_{i}\), and the fluctuating component, \(u_{i}^{\prime}\); \(\rho\) indicates the fluid density, \(\mu\) is the viscosity of the fluid, and \(P\) is the mean component of the instantaneous pressure. Moreover, \(F_{B}\) on the right-hand side of Equation (5) denotes the body force:  
\[F_{B}=-\rho\left(\frac{d^{2}R_{0}}{dt^{2}}+\frac{d\Omega}{dt}\times r_{1}+ \Omega\times(\Omega\times r_{1})+2\Omega\times U_{i}\right) \tag{6}\]  
where \(R_{0}\) is the position vector in the inertial coordinates, \(\Omega\) is the angular velocity vector in the body coordinates (\(\Omega=pe_{x}+qe_{y}+re_{2}\)), and \(r_{1}\) is a vector representing a fluid particle in the body axes. The body force depends on the results obtained using Equations (2). The linear acceleration in the body-fixed coordinates is as follows:  
\[\frac{d^{2}R_{0}}{dt^{2}}=U^{\prime}+\Omega\times U \tag{7}\]  
where \(U\) is the velocity vector of the submarine (\(U=(u,v,w)\) and \(U^{\prime}=(\dot{u},\dot{v},\dot{w})\)).  
The Reynolds stress, \(\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\), in Equation (5) was obtained using the shear-stress-transport (SST) \(k-\omega\) turbulence model [11] in this simulation. The frequently used \(k-\epsilon\) turbulence model assumes that the turbulence is fully developed and the viscosity between molecules can be neglected; thus, the calculation for flow fields distant from the boundary layers is highly accurate, whereas it is less accurate for flow fields near the boundary layers [12; 13; 14]. The \(k-\omega\) turbulence model performs significantly better when predicting the flow near the boundary layers, especially when predicting flow separation under adverse pressure gradients. The SST model uses the \(k-\epsilon\) model for the distant fields and the \(k-\omega\) model near the boundary layers; therefore, this model is highly accurate and is widely used in ship and ocean engineering simulations [15].  
The water surface was captured using the VOF method [16]. This method can represent the structure and motion of the free surface, and the solution is considerably simpler than that of other methods, such as the MAC method [17] or Level Set method [18]. The following equations express the density and viscosity of fluid on the water surface:  
\[\rho=\alpha_{w}\rho_{w}+(1-\alpha_{w})\rho_{a} \tag{8}\]  
\[\mu=\alpha_{w}\mu_{w}+(1-\alpha_{w})\mu_{a} \tag{9}\]  
In these equations, \(\alpha_{w}\) is the volume fraction of water in the control volume, \(\rho_{w}\) and \(\rho_{a}\) are the densities of water and air, respectively, and \(\mu_{w}\) and \(\mu_{a}\) are the viscosities of water and air, respectively. \(\alpha_{w}\) meets Equation (10).  
\[\frac{\partial\alpha_{w}}{\partial t}+\frac{\partial\alpha_{w}U_{j}}{\partial x _{j}}=0 \tag{10}\]