## 2 Numerical Methods
### Computational Domain and Boundary Conditions
#### 2.4.2 Propeller behind Hull Condition  
The propeller KP505 behind the KCS hull with a rudder in calm water is also tested in the presented work. A propeller disk (cylinder grid block) is required for applying both the average and rotating blade body force propeller models. The same CFD grid system including the propeller disk, and the setup is shared between the two propeller models. Figure 5 explains the computational domain and boundary conditions detailed in Table 2 accordingly. Figure 6 shows the propeller disk is attached on the ship stern bulb with the propeller boss and in front of the rudder. The disk has a central tunnel, which is also  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Grid Block Name** & \(N_{i}\) & \(N_{j}\) & \(N_{k}\) & **Grid Number** \\ \hline Background & 149 & 121 & 159 & 2,866,611 \\ Propeller cube & 125 & 73 & 64 & 584,000 \\ Total: 3,450,611 \\ \hline \end{tabular}
\end{table}
Table 1: Grid size (\(N_{i}\), \(N_{j}\), \(N_{k}\) are the grid number in the \(x\), \(y\), \(z\) direction, respectively).  
Figure 4: Overset grid system for the open water propeller (the grids are plotted on propeller plane at \(x\)/\(L=0\)): (a) far view; (b) near view.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Boundary** & **Condition** & \(u\) & \(v\) & \(w\) & \(P_{S}\) & \(k\) & \(\omega\) & \(\upsilon_{t}\) & \(\phi\) \\ \hline Inlet & Uniform inflow & \(U\) & 0 & 0 & 0 & 0 & 0 & 0 & \(z=0\) as free surface \\ Outlet & Exit & \(\
abla^{2}u=0\) & \(\
abla^{2}v=0\) & \(\
abla^{2}w=0\) & \(\
abla P_{S}=0\) & \(\
abla k=0\) & \(\
abla\omega=0\) & \(\
abla\psi_{t}=0\) & \(\
abla\phi=0\) \\ Sides & Zero-gradient & \(\
abla u=0\) & \(\
abla v=0\) & \(\
abla w=0\) & \(\
abla P_{S}=0\) & \(\
abla k=0\) & \(\
abla\omega=0\) & \(\
abla\psi_{t}=0\) & \(\
abla\phi=0\) \\ Top & Far Field \#2 & \(U\) & 0 & 0 & \(\
abla P_{S}=0\) & \(\
abla k=0\) & \(\
abla\omega=0\) & \(\
abla\upsilon_{t}=0\) & \(\
abla\phi=1\) \\ Bottom & Far Field \#1 & \(U\) & \(\
abla v=0\) & \(\
abla w=0\) & 0 & \(\
abla k=0\) & \(\
abla\omega=0\) & \(\
abla\upsilon_{t}=0\) & \(\
abla\phi=-1\) \\ Solid surface & No-slip & 0 & 0 & 0 & \(\
abla P_{S}=0\) & 0 & \(\frac{\phi 0}{\text{Re}\phi\times 7}\) & \(\
abla\upsilon_{t}=0\) & \(\
abla\phi=0\) \\ \hline \end{tabular}
\end{table}
Table 2: Boundary conditions.  
the propeller boss surface. The rotating hub velocity, i.e., propeller rotational velocity, is specified on the inner surface of the disk central tunnel. In this section, the ship vertical motions including heave and pitch are considered to predict the sinkage and trim in CFD simulation using dynamic overset grid performed by [PERSON]. An identical grid system was used in the previous study [41] as well.  
Figure 5: Computational domain and boundary conditions for propeller behind hull condition [41].  
Figure 6: Propeller disk for behind hull condition [41].  
The entire grid system is composed of nine blocks to describe the complicated geometry: starboard and portside for hull, cover (hull top), stern (including shaft tube and dummy boss), and rudder (attached under transom) and one background. The total grid number is around 13 M (million). To capture turbulence, the minimum size of the grid approximately is \(1\times 10^{-6}\) normal to the solid surface corresponding to \(y+<1\). Fine grids are generated in the \(x\)-direction near the ship and the \(z\)-direction near the free surface to capture and maintain the propagating waves.