# Numerical Study on Tip Vortex Cavitation Inception on a Foil
## 3 Numerical Results and Discussion
### Validation of Non-Cavitating Flows  
To validate the quality of the present numerical results, flow velocities across the tip vortex core were evaluated. The different grid resolutions were first tested with the RSM turbulence model for numerical evaluations, followed by dependency examination of the turbulence model on the finest grid. Figure 4 compares the vertical velocity profiles extracted along the horizontal line across the tip vortex core measured and calculated at the section of x/c = 1.0 at \(R_{e}\) = 5.2 \(\times\) 10\({}^{5}\) and \(\alpha_{e}\) = 12\({}^{\circ}\). According to the measured results of [PERSON] et al. [7], the steep gradient of the velocity and its magnitude persisted far downstream of the foil. In these numerical results, it is seen that the influence of grid resolution on the sharpness of the vertical velocity distribution is as large as those of the turbulence model. The SKW and RKE turbulence models underestimated the vertical velocity, resulting in a large difference from the experimental value. On the other hand, the RSM turbulence model shows a better agreement with the experimental data even for \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Grid Name**} & \multirow{2}{*}{\begin{tabular}{c} **Total No. of** \\ **Grids (M)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **No. of Grids in** \\ **Vortex Core** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Grid Size (mm)** \\ **Streamwise** \\ **Direction** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Grid Size (mm)** \\ **Transverse** \\ **Direction** \\ \end{tabular} } \\ \hline G1 & 2.5 & 8 & 0.560 & 0.280 \\ G2 & 5.1 & 11 & 0.400 & 0.200 \\ G3 & 10.4 & 16 & 0.280 & 0.140 \\ G4 & 15.7 & 25 & 0.174 & 0.087 \\ G5 & 30.6 & 28 & 0.156 & 0.078 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computational grid resolutions. Figure 3: Grid distribution on the foil surface and in transverse and streamwise directions: (a) surface distribution and transverse distribution; (b) close-up view of vortex core region; (c) streamwise distribution. the coarsest grid resolution than the two-equation turbulence models. In the current study, the vertical velocity predicted by the RSM model provided the best agreement with the measured velocity profile for the finest grid resolution G5, followed closely by the next higher grid resolution G4. The accuracy of the grid resolution G5 in the streamwise direction on the vertical velocity can be found in Figure 5. The streamwise grid size in the tip vortex region is given in Table 1. The magnitude and gradient of the vertical velocity was slightly underpredicted at the further downstream of the foil, x/c = 2.0. Figure 4: Vertical velocity profiles at the section of x/c = 1.0. Figure 5: Vertical velocity profiles at the sections of x/c = 1.0 and 2.0. The circumferential velocity profiles calculated under the same flow conditions as in Figure 4 for different grid resolutions and turbulence models were compared with the experimental data in Figure 6. This flow property is a plot of the product of circumferential velocity and radius. The numerical results showed a good prediction of the asymmetry of the tip vortex between the suction and pressure sides. The circumferential velocities predicted by the two-equation turbulence models near the center of the tip vortex core were slightly lower than the measured result. Figure 7 presents the pressures along the tip vortex core for different grid resolutions and turbulence models. The predicted pressures decreased to minimum values with large changes in slope near the tip of the foil where the initial vortex rollup develops. A grid convergence appeared in the estimation of the minimum pressure, but the influence of the turbulence model was somewhat large in the magnitude and behavior of the pressure coefficient.