## 4 Numerical Results
### Vortex Core Trajectory  
Before the tip vortex is regenerated by using the vortex model, the vortex core center needs to be identified for the swirling flow from propeller blades. Coherent vortex structure is identified using the \(\lambda_{2}\)-criterion of the RANS simulation results. The vortex core center is defined as the location of the minimum value of \(\lambda_{2}\). Figure 9 presents the distribution of the \(\lambda_{2}\)-criterion on the circular plane just behind the propeller. Figure 9a shows the overall distribution of the \(\lambda_{2}\)-criterion in a range between \(-2400\) and \(2400\). It is seen that the seven similar regions are formed in the flow field, as can be expected from the fact that the propeller consists of seven blades. Under the understanding that the large negative value of the \(\lambda_{2}\) indicates the existence of a strong coherent vortex structure, the regions of high negative values are detailed by adjusting the contour levels, together with the projected view of the propeller blades, as shown in Figure 9b. In addition to the contours of the \(\lambda_{2}\)-criterion, the circular band contours in the center region denoting the circular angle are added to specify the vortex core location. The vortex core centers are identified by using the minimum value of the \(\lambda_{2}\) for each of the blades, and their exact locations are presented with the azimuthal angles, as shown in Figure 9b. Figure 10 presents the distribution of the \(\lambda_{2}\) on the black-colored circle shown in Figure 9b. The seven sharp peaks can be identified in the curve, which implies that the vortex center can be identified with high resolution. The trajectories of the tip vortex core are formed by tracing the locations of the minimum \(\lambda_{2}\) values identified in this way along with the downstream direction. Figure 11 shows the formed trajectory curves of the tip vortex for two types of propellers.  
Figure 10: A variation of \(\lambda_{2}\) along the circular line in Figure 9b.  
Figure 9: Distribution of \(\lambda_{2}\)-contours (**a**) in a full wide range of \(\lambda_{2}\) and (**b**) in a range of \(\lambda_{2}\) less than 0.  
The tip vortex core trajectory is known to be of a spiral curve along the downstream direction. It is seen in Figure 11 that the vortex core trajectory predicted using the minimum \(\lambda_{2}\)-criterion of the RANS results well follows this curve. Moreover, the predicted trajectories of the tip vortex core seem to be similar to the experimental ones shown in Figure 7. However, it noted that the vortex core trajectory predicted for the Model-A is attached to the blade tip while detached from the Model-B blade tip. This difference is one of the points to characterize the flow driven by the propellers with different sweep angles.  
For further analysis, Figure 12 presents the distribution of the minimum values of \(\lambda_{2}\) according to the blade location for two propellers. It is seen that the minimum values of \(\lambda_{2}\) for all blades of the Model-B propeller are lower than those of the Model-A propeller. This result reveals that the tip vortex of Model-B with a higher sweep angle is weaker than that of Model-A with a lower sweep angle, and the tip vortex cavitation is also possibly delayed for Model-B in comparison with Model-A. Figure 13 shows the zoomed plot of the starting point of the vortex core trajectory for each propeller shown in Figure 11. It is confirmed that the vortex core trajectory of the Model-B propeller is formed more apart from the blade tip. These results reveal that the underwater propeller of a higher sweep angle induces a weaker tip vortex than a lower sweep angle.  
Figure 11: Vortex core trajectory along with the downstream: (**a**) Model-A; (**b**) Model-B.  
Figure 12: Comparison of minimum \(\lambda_{2}\) of each blade between two propellers of Model-A and Model-B.