# Numerical Study on Tip Vortex Cavitation Inception on a Foil
## 4 Conclusions  
In the current research, the tip vortex cavitation inception on a hydrofoil was numerically estimated under weak water condition and new scaling laws with variable scaling exponent were further suggested for cavitation inception index. Numerical simulations of the wetted and cavitating flows around an elliptic planform foil were carried out using the RANS approach with a Eulerian cavitation model. The influence of the turbulence models on the steep velocity field inside the vortex was larger than the grid dependence. The Reynolds stress model for the finest grid showed a good agreement with the velocity field measured across the tip vortex. In addition, this model also showed a good correlation with the experimental results of boundary layer characteristics in the tip region of the foil surface where the tip vortex and cavitation developed. A comparison between the wetted and cavitating flows revealed that the presence of the cavitation in the tip vortex core, which is a phase change accompanying the mass transfer phenomenon, decreased the pressure and velocity gradients resulting in a decrease in the vortex rotation. The minimum pressure near the foil tip decreased as the Reynolds number increased, and its location slightly moved downstream from the tip of the foil. This was the trend because the tip vortex gained more strength as the vortex sheet rolled up into the tip vortex at higher Reynolds numbers. Since the present cavitation simulation was included in a weak water problem with a sufficiently large supply of nuclei, the tip vortex cavitation started at a small distance off the tip of the hydrofoil. The starting point of the cavitation moved closer to the tip of the foil as the Reynolds number increases. In the comparison of the cavitation inception prediction, the wetted and cavitating flow simulation approaches showed a reasonable agreement between the simulation and experimental cavitation index values. Finally, the present study has suggested new empirical formulas as a function of the Reynolds number substitutable for the two classic constant scaling exponents. These  
Figure 21: Suggested scaling exponents in weak water as a function of the Reynolds number for different scaling laws. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(R_{e}\) & \(5.0\times 10^{5}\) & \(1.0\times 10^{6}\) & \(1.0\times 10^{7}\) & \(1.0\times 10^{8}\) & \(1.0\times 10^{9}\) \\ \hline \(m\) & 0.32 & 0.31 & 0.29 & 0.26 & 0.23 \\ \(n\) & 0.39 & 0.38 & 0.34 & 0.30 & 0.26 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Suggested scaling exponents for five main Reynolds numbers. formulas were derived by extrapolation based on the current numerical cavitation index and show an asymptotic converge as the Reynolds number increases. It was found that the new [PERSON]' exponent \(m\) was in the range 0.32-0.23 and the [PERSON]'s exponent \(n\) in the range 0.39-0.26 for the Reynolds number in the range 5.0 \(\times\) 10\({}^{5}\)-1.0 \(\times\) 10\({}^{9}\). To conclude, the present study showed how the scaling law changes with the Reynolds number and provided quite useful formulas to properly determine the scaling exponent for a given Reynolds number. In the future, this suggested model is expected to be useful in full scale applications. Conceptualization, I.P., B.P. and H.S. ; methodology, I.P. and J.K.; software, J.K.; validation, J.K. and I.P. ; investigation, I.P. and J.K.; writing--original draft preparation, I.P. ; writing--review and editing, I.P., B.P. and H.S. ; supervision, B.P. and H.S. ; project administration, B.P. and H.S. All authors have read and agreed to the published version of the manuscript. This research was funded by Ministry of Trade, Industry and Energy and Defense Acquisition Program Administration and Agency for Defense Development.