## 4 Numerical Results
### Tip Vortex Reconstruction  
Figure 14 shows the iso-contours of flow velocity components at the plane of \(x=-3.15\) m, which is located just behind the propeller. The steepest gradient is observed in the distribution of the \(x\)-directional velocity component. Since the \(x\)-direction is parallel to the freestream velocity and almost normal to the propeller rotational plane, the vortex structure is better captured in terms of the \(x\)-directional velocity component than the others. Therefore, vortex modeling is made by using the velocity component in the \(x\)-direction on the plane normal to the mean flow direction.  
Figure 14: Reynolds-averaged Navier–Stokes (RANS) simulation results on Model-A: (**a**) \(x\)-dir. velocity; (**b**) \(y\)-dir. velocity; (**c**) \(z\)-dir. velocity.  
Figure 13: The starting point of the vortex core trajectory on each propeller.  
The vortex core radius and maximum tangential velocity of the vortex model need to be determined to regenerate tip vortex structure by using the vortex model. The procedure to define these values is schematically presented in Figure 15. The distribution of the axial velocity component, which corresponds to the tangential velocity of the vortex model, is extracted from the black-dashed lines between the red-solid lines shown in Figure 15a. The extracted distributions for seven cores are shown in Figure 15b. Here, the vortex core diameter is defined as the distance between the locations of maximum and minimum velocities, and then naturally, the initial vortex core radius is set to be half of the diameter. The initial tangential velocity is also defined as the difference between the maximum and minimum velocities, as denoted by the blue text and arrow-line in Figure 15b. Because the velocity is zero at the vortex center, the mean value of the maximum and minimum velocities, as denoted by the green dashed line in Figure 15b, is considered the mean flow velocity in the \(x\)-direction. The vortex structure described by Equations (7) and (8) can be determined using the initial vortex core radius and tangential velocity determined in this way.  
After the vortex core trajectory and the properties of the vortex model are determined, the domains where the tip vortex is reconstructed need to be defined. Figure 16 shows the vortex domain constructed for each blade, together with its construction process. The vortex domain is constructed by using the cuboid of the square cross-section, whose center coincides with the vortex core line, as shown in Figure 16a. This cuboid is bent to follow the spiral curve of the vortex core trajectory, as shown in Figure 16c. The cross-section of the bent cuboid at the plane of the constant azimuthal angle is shown in Figure 16b. The tip vortex for each blade is reconstructed by extending the vortex model in the bent cuboid.  
Figure 17 compares the reconstructed tip vortex flow with the RANS result in terms of the axial velocity distribution on the six cross-sections of the vortex domain. It is seen from Figure 17a that the axial velocity distributions on the plane nearest the blade tip are almost the same between the two results. However, it can be found in Figure 17b-f that, as the fluid flows in the downstream direction, the vortex structure in the RANS result disappears quickly, while the vortex structure reconstructed using the vortex model keeps almost its original one. Note that the marginal vortex dissipation occurs in the regenerated tip vortex due to the influence of DVM.  
Figure 15: Schematic diagram to define the maximum tangential velocity and initial core radius of the vortex regenerated using the vortex model. (**a**) Distribution of the velocity component in the \(x\)-direction on a plane normal to mean flow. (**b**) Distribution of \(x\)-directional velocity on the lines of the constant azimuthal angle passing through the point of the maximum radial gradient of \(x\)-direction velocity.