# Numerical Simulation of the Hydrodynamic Performance and Self-Propulsion of a UUV near the Seabed
## 3 Numerical Methods
### Governing Equations  
The Reynolds average equation includes: the continuum equation, and the momentum equation, as shown in Equations (12) and (13).  
\[\frac{\partial\rho}{\partial t}+\frac{\partial\rho\overline{u}_{i}}{\partial t }=0 \tag{12}\]  
\[\frac{\partial}{\partial t}(\rho\overline{u}_{i})+\frac{\partial}{\partial t} \left(\rho\overline{u}_{i}\overline{u}_{j}\right)=-\frac{\partial P}{ \partial x_{i}}+\frac{\partial}{\partial x_{j}}\left[\mu\left(\frac{\partial \overline{u}_{i}}{\partial x_{j}}+\frac{\partial\overline{u}_{j}}{\partial x _{i}}-\frac{2}{3}\delta_{ij}\frac{\partial\overline{u}_{i}}{\partial x_{i}} \right)\right]-\frac{\partial}{\partial x_{j}}\left(\rho\overline{u^{\prime}_{ i}u^{\prime}_{j}}\right)+F_{i} \tag{13}\]  
\(u_{i}\) is the fluid velocity component. \(P\) is the total pressure. \(F_{i}\) is the volume force term. Additionally, \(\mu\) is the dynamic viscosity coefficient. The \(SST-kw\) turbulence model is used to simulate the turbulence terms in the \(RANS\) equations. This model is able to simulate turbulent shear stress transport and accurately predict the flow separation under the inverse pressure gradient. Turbulence kinetic energy and turbulence dissipation rate are calculated by the following equations.  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_{ i})=\frac{\partial}{\partial x_{j}}\left(\Gamma_{k}\frac{\partial k}{ \partial x_{j}}\right)+G_{k}-Y_{k}+S_{k} \tag{14}\]  
\[\frac{\partial}{\partial t}(\rho w)+\frac{\partial}{\partial x_{i}}(\rho wu_{ i})=\frac{\partial}{\partial x_{j}}\left(\Gamma_{w}\frac{\partial w}{ \partial x_{j}}\right)+G_{w}-Y_{w}+D_{w}+S_{w} \tag{15}\]  
To investigate the effect of \(Hd\) and \(Re_{L}\) on the hydrodynamic performance of the self-propelled UUV and the coupling effect of the hull and the propeller, firstly, it is necessary to obtain the rotational speed of the propeller under different operating conditions, and then solve the hydrodynamic performance and self-propulsion factor of the UUV. The rotational speed of propeller is corrected by the DEFINE_ZONE_MOTION macro once every 1000 iterations. The rotational speed for self-propelled point is considered to be obtained when the rotational speed value of the previous step does not change more than 0.01% with respect to the previous rotational speed correction value and the relative error  
\begin{table}
\begin{tabular}{c c c c} \hline
**Group** & **Description** & \(Hd\) & \(U\) \\ \hline \multirow{2}{*}{1} & Effect of seabed & \multirow{2}{*}{(1.5, 2, 2.5, 3, 5, 7, 9, 10)} & (0.5, 1, 2, 4) \\  & distance on bare UUV & & \\  & Subsea distance effect & & \\
2 & on self-propelled UUV & & \\ \hline \end{tabular}
\end{table}
Table 3: Simulation case classification.  
between the drag of the UUV and the thrust of the propeller does not exceed 0.01%. The propeller rotational speed is calculated according to the following equation.  
\[N=N^{\prime}+\frac{F_{uuv}+F_{balle}}{F_{uuv}}\delta_{n} \tag{16}\]  
\(N^{\prime}\) is the currently rotational speed. \(F_{uuv}\) is the drag of the UUV without propeller. \(F_{balle}\) is the thrust of the propeller. \(\delta_{n}\) is the convergence coefficient.