# Numerical Study on Tip Vortex Cavitation Inception on a Foil
## 3 Numerical Results and Discussion  
### Flow Condition and Grid Resolution  
An elliptic planform foil with the NACA 66\({}_{2}\)-415 section was tested in the present numerical analysis of tip vortex flows. The foil as shown in Figure 1 had a chord length of 0.081 m and a semi-span of 0.095 m as those used in the experiment [7]. The current simulations were overall separated into two parts. The first part considered the validation of the present numerical solutions for tip vortex flows. Non-cavitating flows were computed for the foil with the effective angle of attack \(\alpha_{e}(=\alpha-\alpha_{o})=12^{\circ}\) at the Reynolds number of \(R_{e}=5.2\times 10^{5}\), where the corresponding flow speed was \(U_{o}=6.5\) m/s. It should be noted that the practical angle of attack \(\alpha\) was 9.5\({}^{\circ}\), and the zero-lift angle of attack \(\alpha_{o}\) was \(-2.5^{\circ}\) as mentioned in [PERSON] et al. [7]. The boundary layer characteristics near the foil tip predicted at \(R_{e}=4.8\times 10^{5}\) and \(\alpha_{e}=13^{\circ}\) was validated through the experimental result of [PERSON] and [PERSON] [9]. The simulations of cavitating flows around the foil were carried out at the cavitation numbers (\(\sigma\)), 0.58 and 1.15, and the Reynolds number of \(R_{e}=5.3\times 10^{5}\), in which the foil had the effective angle of attack, \(\alpha_{e}=9.5^{\circ}\). In the second part, the cavitation inception of the foil with the effective angle of attack \(\alpha_{e}=9.5^{\circ}\) was investigated with the variation of the Reynolds number and cavitation number, where the practical angle of attack corresponded to \(\alpha=7.0^{\circ}\). The Reynolds number had the range \(5.2\times 10^{4}\)-\(1.2\times 10^{6}\). In these computations, the cavitation number was increased from a lower value at which the tip vortex cavitation was clearly shown with a large amount until the cavitation was difficult to find around the tip of the foil at each Reynolds number. The present cavitation inception analyses were compared with the cavitation index measured by [PERSON] [8]. In addition, the minimum pressure coefficients of wetted flows were also used to estimate the cavitation inception number as follows:  
\[\sigma_{i}=-C_{p,\ min} \tag{12}\]  
The computational domain of the foil as shown in Figure 2 had the same test section of 0.19 m square cross-section with a length of 1.25 m as the experimental facility. The velocity inlet boundary was located approximately six chord lengths upstream of the foil, and the pressure outlet boundary was placed twelve chord lengths downstream of the foil. The wall boundary condition was applied on the foil and other boundaries. As briefly mentioned in the previous section, one of the key tasks of numerical analysis of tip vortex flows is to find a proper grid resolution around the tip vortex core. A grid refinement along the vortex core, as shown in Figure 3, was performed after identifying the tip vortex trajectory from the numerical result for an initial coarse grid. A circular cylindrical refinement region of 14 mm diameter was placed along the tip vortex which extended 2.5 times downstream of the foil. The grid distributions on the surface of the foil and in the streamwise and transverse directions are presented in Figure 3. Five different grid resolutions with the dimensionless wall distance, \(y^{+}\approx 1\) were generated as shown in Table 1. All grids had 20 prism grid layers to have a proper resolution of the boundary layer around the foil. It can be found from the experiment [7] that the tip vortex core at x/c = 1.0 had an approximate radius of 1.1 mm for the foil with the effective angle of attack of 12\({}^{\circ}\) at a Reynolds number of 5.2 \(\times\) 10\({}^{5}\). The number of grid cells and their sizes across the tip vortex diameter in the grid refinement region are provided in Table 1 for each grid resolution. In the current simulations, 8, 11, 16 and 25 grid cells across the tip vortex core were used for G1, G2, G3 and G4, respectively.