# Numerical Study on Tip Vortex Cavitation Inception on a Foil
## 3 Numerical Results and Discussion
### Scaling of Tip Vortex Cavitation Inception  
This section shows the numerical deduction of a scaling law for the tip vortex cavitation inception for the current elliptic planform foil with the NACA 662-415 section. The tip vortex flow was simulated at different Reynolds numbers: \(5.0\times 10^{4}\), \(1.0\times 10^{5}\), \(3.0\times 10^{5}\), \(5.3\times 10^{5}\), \(7.0\times 10^{5}\), \(1.0\times 10^{6}\) and \(1.2\times 10^{6}\). In these computations, the cavitation number \(\sigma\) is increased until the tip vortex cavitation was difficult to find around the tip of the foil at each Reynolds number. To compare the deduced cavitation index using the minimum pressure coefficients, the wetted flows at the above same Reynolds numbers were simulated. Figure 15 shows the pressure coefficients along the vortex core at different Reynolds numbers. As the Reynolds number increased, the minimum pressure decreased, and its location moved slightly downstream from the tip of the foil. This trend was because the tip vortex gained more strength as the vortex sheet rolled up to the tip vortex at higher Reynolds number. Figure 16 shows the time histories of the cavitation volumes (\(V_{cav}\)) predicted with respect to the variation in cavitation number at the following selected Reynolds numbers,  
Figure 14: Pressure, axial and vertical profiles at x/c = 1.0 for different flow conditions: (**a**) axial velocity; (**b**) vertical velocity; (**c**) pressure coefficient. Figure 15: Pressure coefficient profiles along the vortex core at different Reynolds numbers. \(R_{e}\) = 5.0 \(\times\) 10\({}^{4}\), 1.0 \(\times\) 10\({}^{5}\), 3.0 \(\times\) 10\({}^{5}\) and 1.0 \(\times\) 10\({}^{6}\). The simulation time was 1 s, and it is seen that the volume change of the tip vortex cavitation occurring under all flow conditions converged after 0.5 s.  
In Figure 17, the variations in cavitation amount with respect to the cavitation number at different Reynolds numbers are compared. Although the cavitation numbers at each Reynolds number condition were different, they were selected to roughly compare the generated tip vortex cavitation within a similar volume at different Reynolds numbers. It was difficult to accurately estimate the cavitation volume generated under a flow condition which was judged as the cavitation inception point in the experiment. Since the cavitation volume detectable at an inception number was very small, in this study, the cavitation inception number was assumed to be the point where the cavitation volume was nearly zero. The cavitation inception numbers for each Reynolds number, as shown in the figure, were obtained by the extrapolation of the numerical results. Figure 18 shows the iso-surfaces of the predicted tip vortex cavitation in the order of cavitation number from small to large values at the given Reynolds numbers mentioned in Figure 17. In the figures, the iso-surfaces of the tip vortex cavitation predicted by a roughly similar volume are compared between different Reynolds numbers. This shows that the tip vortex cavitation became slightly thinner and longer as the Reynolds number increased. Since the present cavitation simulation was included in a weak water problem with a sufficiently large supply of nuclei, it can be found that the cavitation inceptions occurred at a small distance off the tip of the hydrofoil as discussed by [PERSON] [5]. Unlike the behavior of the minimum pressure shown in Figure 15, the starting point of the tip vortex cavitation moved closer to the tip of the foil as the Reynolds number increased. Mostly, the tip vortex cavitation started to occur at the region preceding the location of the minimum pressure for all given cavitation numbers. For this reason, the variation of the pressure according to the Reynolds number in this region as shown in Figure 15 deter-mined the starting point of cavitation occurrence. Figure 19 compares the predicted cavitation inception numbers for the variation of the Reynolds number by wetted flow simulations and cavitation simulations.