# Numerical Study on Tip Vortex Cavitation Inception on a Foil
## 3 Numerical Results and Discussion
### Validation of Non-Cavitating Flows
Moreover, the predicted minimum pressure locations which were very close to the foil tip were found to be almost similar under all numerical conditions, but a change in the pressure downstream of those locations showed different behaviors. The rapid pressure recovery due to the numerical diffusion was seen in the results for the coarser grid resolutions. This numerical trend was more apparent in the pressure variations predicted by the RKE and SKW turbulence modes as previously discussed in Figure 4. This can be attributed to the influence of the increase in numerical diffusion caused by excessive turbulent viscosity. Figure 8 compares the pressures predicted along the horizontal line across the tip vortex core at x/c = 0.2 and x/c = 1.0. In theory, the pressure distribution can be obtained from the integration of the circumferential velocity shown in Figure 6. The pressure profiles obtained using the RSM model with grid resolutions G4 and G5 showed a sharper variation with similar magnitudes. Even in the results computed by the RKE and SKW turbulence models, the pressure gradients were somewhat large near the tip but decreased significantly due to the increase in the numerical diffusion in the downstream of the foil. From the grid dependence test, the finest grid G5 was considered to be suitable for this study, and the following cavitation analysis was done to investigate the effect of the turbulence model on this grid. Figure 6: Angular momentum profiles at the section of x/c = 1.0. A detailed experimental study of the near tip region was performed for the NACA 16-020, NACA 4215M and NACA 66\({}_{2}\)-415 foils by [PERSON] and [PERSON] [9]. As an example of their research results, Figure 9a shows a limiting streamline pattern of developed vortex-boundary layer interaction near the tip region of the NACA 16-020 foil suction side. Note that this result is not available for the NACA 66\({}_{2}\)-415. In Figure 9b, the vortex footprints are provided for the NACA 16-020 and NACA 66\({}_{2}\)-415 foils. This vortex footprint clearly defines the separation and reattachment of the vortex flow [9]. In Figure 10, the predicted vortex footprints on the suction surface of the current NACA 66\({}_{2}\)-415 foil are compared with the experimental results, whereas the numerical results on the pressure side are compared with different turbulence models. The simulations were performed for the finest grid G5 at \(\alpha_{e}=13^{\circ}\) and \(R_{e}=4.8\times 10^{5}\). The current numerical results showed the limiting streamlines overlapped with the magnitude of the wall shear stress. As discussed in the numerical results for the tip vortex flows, the predicted developed vortex-boundary layer interactions were also different depending upon the turbulence model. The RKE result showed somewhat delayed vortex separation and reattachment, while the SKW result showed an improved agreement with the experimental observation. On the other hand, the RSM provided a better agreement with the measured vortex footprint which can be defined as the area contained between the vortex separation and reattachment lines. When examined together with the wall shear stress shown in Figure 10, the vortex footprint is increased to the wall shear stress. Figure 3:0 Pressure coefficient profiles across the tip vortex core: (**a**) x/c = 0.2; (**b**) x/c = 1.0. Figure 3:1 Pressure coefficient profiles along the vortex core: (**a**) global; (**b**) close-up view around the minimum pressure. the RSM result, it is seen that the magnitude of this physical quantity is large inside the vortex footprint. Although the suction side boundary layer characteristics varied greatly according to the turbulence model, the pressure side flow features were similar. Figure 3:2 Limiting streamlines and wall shear stress distribution on the suction and pressure surfaces of the foil: (**a**) suction side, RKE; (**b**) suction side, SKW; (**c**) suction side, RSM; (**d**) pressure side, RKE; (**e**) pressure side, SKW; (**f**) pressure side, RSM.