# Numerical Analysis of the Take-Off Performance of a Seaplane in Calm Water
## 3 Numerical Model  
### Governing Equations  
The incompressible viscous two-phase flow field surrounding the seaplane was described using the Reynold-Averaged Navier-Stokes (RANS) equations with turbulence model. The Reynolds averaged forms of mass conservation equation and the momentum conservation equation can be written as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial\overline{u}_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho\overline{u_{i}})+\frac{\partial}{\partial x_ {i}}\big{(}\rho\overline{u_{i}u_{i}}\big{)}=-\frac{\partial\overline{p}}{ \partial x_{j}}+\frac{\partial}{\partial x_{j}}\Bigg{(}\mu\Bigg{(}\frac{ \partial\overline{u_{i}}}{\partial x_{j}}+\frac{\partial\overline{u_{j}}}{ \partial x_{j}}\Bigg{)}-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\Bigg{)}+g_ {i} \tag{2}\]  
where, \(x_{i}\) and \(x_{j}\) are Cartesian coordinates, \(\overline{u_{i}}\) and \(\overline{u_{j}}\) represent the averaged components of velocity vector in Cartesian coordinates. \(\overline{p}\) is the averaged pressure, \(\rho\) is the density, and \(g_{i}\) the component of gravity acceleration along the coordinate axes of inertia. In addition, \(\mu\) is the effective viscosity coefficient of dynamic viscosity, and \(\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\) represents the Reynolds stresses. The \(k\)-\(\omega\) shear stress transport (SST) turbulence model was introduced to close the equations. Concretely, the transport equations for \(k\) and \(\omega\) are written as:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_{ i})=\frac{\partial}{\partial x_{i}}\bigg{(}\Gamma_{k}\frac{\partial k}{ \partial x_{i}}\bigg{)}+G_{k}-\mathrm{Y}_{k} \tag{3}\]  
\[\frac{\partial}{\partial t}(\rho\omega)+\frac{\partial}{\partial x_{i}}(\rho \omega u_{i})=\frac{\partial}{\partial x_{i}}\bigg{(}\Gamma_{\omega}\frac{ \partial\omega}{\partial x_{i}}\bigg{)}+G_{\omega}-\mathrm{Y}_{\omega} \tag{4}\]  
where \(G_{k}\) and \(G_{\omega}\) represent the turbulence kinetic energy generation items, and \(\mathrm{Y}_{k}\) and \(\mathrm{Y}_{\omega}\) represent the turbulence dissipation rate of \(k\) and \(\omega\), respectively. A segregated flow solver with Semi Implicit Method for Pressure Linked Equations (SIMPLE) was used to conjugate pressure field and velocity field, and an Algebraic MultiGrid solver was applied to accelerate the convergence of the solution. The convection terms were discretized by 2 nd order upwind schemes. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Value** \\ \hline Wingspan & m & 3.6 \\ \hline Wing area & m\({}^{2}\) & 1.296 \\ \hline Length of fuselage & m & 2.9 \\ \hline Beam width & m & 0.26 \\ \hline Longitudinal position of step(measured from the nose) & m & 1.35 \\ \hline Step height & m & 0.02 \\ \hline Deadrise angle of forebody & deg & 20 \\ \hline Deadrise angle of afterbody & deg & 22 \\ \hline Afterbody keel angle & deg & 8 \\ \hline Weight & kg & 20 \\ \hline Pitching moment of inertia & kg\(\cdot\)m\({}^{2}\) & 7.26 \\ \hline Longitudinal CG(measured from the nose) & m & 1.24 \\ \hline Vertical CG(measured from the forebody keel) & m & 0.28 \\ \hline Distance from thrust line to CG & m & 0.4 \\ \hline Thrust-weight ratio at rest & & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details of the seaplane model. The VOF method was used to capture the water-air interface by introducing a volume fraction \(\alpha\) for each cell, defined as the ratio between the volume occupied by the phase and the volume of the control volume. When a cell is full of water, \(\alpha=1\); while when the cell is full of air, \(\alpha=0\); and when there is a fluid interface in the cell, \(0<\alpha<1\). The transport equation for \(\alpha\) is as follows:  
\[\frac{\partial\alpha}{\partial t}+\
abla\cdot(\alpha u)=0 \tag{5}\]  
In each control volume, the two phases share the same velocity, pressure and temperature.