## 4 Numerical Results
### Numerical RANS Solutions  
Figure 7 compares the RANS simulation results with the experimental results in terms of cavitating flow shapes. The snapshots of cavitating flow shape recorded by using the high-speed camera in the test section of the LCT are compared with the iso-surface of vapor volume fractions obtained from the numerical simulation results. Although the TVC and sheet cavitation on the suction sides of propeller blades are well captured, the cavitating tip vortex trajectory observed clearly in the experiments cannot be found in the corresponding numerical results. It is challenging to capture the cavitating tip vortex trajectory by using the RANS solver without enormous numerical cost. This difficulty will be overcome effectively by applying the vortex model and the bubble dynamics and trajectory equations.  
Figure 6: Comparison of numerical bubble radius with experiment.  
Further analysis is carried out to confirm the validity of numerical results in terms of hydrodynamic performances. Table 5 compares the predicted thrust with the measured ones. The thrust is compared in terms of non-dimensional values: the thrust coefficient of \(\text{K}_{\text{T}}\) (\(=\frac{T}{\rho n^{2}D^{3}}\)). It can be seen that the maximum relative percentage error is less than 2%.  
Figure 8 shows the distribution of axial inflow velocity in the region 0.14D upstream from the propeller tip, where D denotes the propeller diameter. The axial velocity is averaged during one rotation on the circle of the propeller tip radius, and its distribution is presented according to the azimuthal angle defined on the left side of Figure 8. The axial velocity distribution shows the deep narrow valleys at the circular positions of 0\({}^{\circ}\), 90\({}^{\circ}\), 180\({}^{\circ}\), and 270\({}^{\circ}\), which coincide with the locations of the rudders. Moreover, the deepest valley is located at 90\({}^{\circ}\), which manifests the additional effects of the sail. Because the reduced axial velocity at these locations locally increases the angle of attack of the propeller blade, the onset of cavitation can occur more likely on the surface of the propeller blade passing through these locations, mostly at 90\({}^{\circ}\). This phenomenon affects the so-called sheet cavitation inception, as shown in Figure 7. The sheet cavitation inception was not expected because the targeted propellers were originally designed to investigate the tip vortex cavitation that is known to occur firstly among the other types of cavitation. However, it is confirmed that the hydrodynamic flow field obtained from the RANS simulation includes the effects of wake flow originating from the boundary layers of the submarined body, especially, the rudders and the sail.  
Figure 8: Inflow axial velocity averaged during one rotation on the circular plane upstream by 0.14D from the propeller tip: (**a**) the region for extracted inflow axial velocity; (**b**) averaged inflow axial velocity.  
Figure 7: Comparison of predicted cavitating flow of underwater propellers of a submarine with the experimental ones for (**a**) Model-A and (**b**) Model-B.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Numerical Result** & **Experimental Result** & **Error (\%)** \\ \hline
**Model-A** & 0.235 & 0.239 & 1.86 \\
**Model-B** & 0.233 & 0.236 & 1.30 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of predicted thrust coefficients with measured ones.