# Performance Improvement in a Wavy Twisted Rudder by Alignment of the Wave Peak
## 2 Materials and Methods
### Computation Setup
#### 2.3.3 Boundary Conditions and Grid System  
Figure 4 displays the domain, coordinate system, and boundary conditions used for the analysis. The origin of the domain is the intersection of the after-perpendicular and the waterline, and the symmetry condition is 1.5 y/LPP from the origin. The symmetry condition, which is the lateral boundary of the domain, was applied with zero-gradient extrapolation for both velocity and pressure. The dimensions (\(-2\leq\mathrm{x}/\mathrm{LPP}\leq 2\), \(-1.5\leq\mathrm{y}/\mathrm{LPP}\leq 1.5\), \(-2\leq\mathrm{z}/\mathrm{LPP}\leq 1\)) concerning the origin were chosen in the streamwise (x), starboard (y), and upward (z) direction, respectively. Table 5 lists the boundary physics.  
The numerical grid around the hull included a trimmer grid; the numerical grid around the propeller included a polyhedral grid. Figure 5 displays the grid configuration around the hull. The same grid system as in the previous study [10] was used because the reliability of the experimental results was verified [10]. Table 6 lists the detailed minimum mesh size and grid number. The first grid on the wall was composed of y+ = 60; Figure 6 displays the y+ value of the hull surface.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Resistance** & **Self-Propulsion** & **Rudder Force** \\ \hline Governing equation & & RANS \\ Velocity–pressure & & SIMPLE method \\ coupling & & \\ Turbulence model & & K-\(\epsilon\) turbulence model \\ Flow & & Unsteady \\  & DFBI (Dynamic & \\ Body motion & Fluid–Body & n/a \\  & Interaction motion) & \\ Rotating motion & n/a & RBM method (sliding mesh) \\ Time step & 0.002 s & 5\({}^{\circ}\) rotation of propeller \\ Total physical time & 30 s & 200 revolutions of the propeller \\  & & (approximately 19 s) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Computation analysis conditions.  
Figure 4: Computational domain.  
\begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{Inlet/Bottom/Top}} & \multicolumn{1}{c}{Velocity inlet with defined volume fraction, flow} \\  & speed = 1.964 m/s, turbulence intensity 0.01, viscosity ratio 10 \\ Outlet & Pressure outlet with relative pressure = hydrostatic pressure \\ Side & Wall with slip condition \\ Hull & Wall with no slip condition \\ \hline \hline \end{tabular}
\end{table}
Table 5: Boundary physics.  
To ensure simulation stability, at the free surface the value of CFL must be limited to 1, and superior results with a sharp resolution of the two phases are obtained with a CFL around 0.5. In the present study, the courant number level range is reasonable, as shown in Figure 7.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Resistance** & **Self-Propulsion** & **Rudder Force** \\ \hline Fluid region minimum mesh size & & 0.005625 m \\ Propeller region minimum & n/a & 0.0014062 5 m \\ \begin{tabular}{c} Fluid region no. of grids \\ \end{tabular} & 2.50 \(\times\) 10\({}^{6}\) & 2.85 \(\times\) 10\({}^{6}\) \\ \
\begin{tabular}{c} Propeller region no. of grids \\ \end{tabular} & n/a & 3.20 \(\times\) 10\({}^{5}\) \\
\begin{tabular}{c} Total no. of grids \\ \end{tabular} & 2.50 \(\times\) 10\({}^{6}\) & 3.17 \(\times\) 10\({}^{6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Number of grids and grid size used in CFD analysis.  
Figure 5: Grid configuration for CFD analysis: (a) hull; (b) bow; (c) stern.  
Figure 6: Y+ of the hull surface.  
However, such a limitation is overly restrictive, as other physics calculations with implicit solvers can run at a much higher CFL number. This reduces the computational efficiency of VOF free surface simulations [24]. In implicit unsteady simulations, the time step is determined by the flow properties, rather than the courant number. In this study, the implicit unsteady condition and a very small time step were used.