# Numerical Study on Tip Vortex Cavitation Inception on a Foil
## 3 Numerical Results and Discussion
### Scaling of Tip Vortex Cavitation Inception
The cavitation inception numbers predicted by the wetted flow simulations showed slightly higher values than those estimated by the cavitation simulations. If the minimum volume observable at a cavitation inception point is used, the difference between the two methods can be slightly larger. The slope of the cavitation index change decreased with increase in the Reynolds number. Figure 17: Cavitation volume changes with the variation of the Reynolds number and cavitation number. Figure 18: Tip vortex cavitations predicted at different Reynolds numbers and cavitation numbers: (**a**) \(R_{e}\) = 5.0 \(\times\) 10\({}^{4}\), \(\sigma\) = 1.0, 1.1, 1.15 and 1.20; (**b**) \(R_{e}\) = 1.0 \(\times\) 10\({}^{5}\), \(\sigma\) = 1.68, 1.90, 2.0 and 2.1; (**c**) \(R_{e}\) = 3.0 \(\times\) 10\({}^{5}\), \(\sigma\) = 2.5, 2.8, 3.0 and 3.15; (**d**) \(R_{e}\) = 1.0 \(\times\) 10\({}^{6}\), \(\sigma\) = 3.0, 3.4, 3.6 and 3.8. In Figure 20, the predicted cavitation inception numbers are compared over a range of lift coefficient with the cavitation index measured in weak water at a constant velocity, \(R_{e}\) = 1.0 \(\times\) 10\({}^{5}\)[8]. The experimental curve was obtained by adjusting the measured value of cavitation inception number by the amount of measured value of tension [42]. Although the lift coefficient and cavitation index were obtained at different Reynolds number and angles of attack, the current numerical result showed a reasonable trend with the experiment. The cavitation inception numbers obtained from the numerical analysis were further compared with those calculated from the empirical formulas for cavitation inception scaling  
Figure 19: Cavitation inception index with the variation of the Reynolds number. Figure 20: Cavitation inception index in weak water with the variation of the lift coefficient. derived based on the experimental results (see Table 2). The first scaling equation was [PERSON]'s original scaling law to relate inception to the angle of attack as follows:  
\[\sigma_{i}=\alpha^{1.4}R_{e}^{m} \tag{13}\]  
In Equation (13), the Reynolds number exponent \(m\) was suggested by [PERSON] [22] to be approximately 0.35. The next scaling equation was [PERSON]'s law [5,8] that uses the lift coefficient of the foil for the Reynolds scaling as follows:  
\[\sigma_{i}=0.068C_{L}^{2}R_{e}^{n} \tag{14}\]  
The constant of 0.068 in Equation (14) is named as \(K\), and it varies owing to a secondary effect caused by differences in vortex roll-up for different blade sections and can be determined from cavitation experiments [5]. The Reynolds number exponent \(n\) is generally accepted to be approximately 0.4. In the Table 2, both the cavitation scaling equations with the original exponent of the Reynolds number overestimated the cavitation inception index when compared to the present numerical results. This is possibly because of certain numerical errors and a pending problem of imperfection in the scaling of vortex cavitation inception from model tests to full scale conditions. In the current study, new Reynolds number exponents for the two scaling laws were suggested based on the predicted cavitation index. Each Reynolds number's exponent m and n varied with the Reynolds number and tended to decrease lower continuously than the classic constants of 0.35 and 0.4. As shown in Figure 21, when extrapolated over the range of the Reynolds number used in the current simulations, the suggested scaling exponents behaved as a function of the Reynolds number and asymptotically converge as the Reynolds number increased. The extracted empirical formulas of the exponents over the asymptotic region larger than the Reynolds number of \(R_{e}=3.0\times 10^{5}\) are as follows:  
\[m=-0.012\ln(R_{e})+0.48 \tag{15}\]  
Figure 15:0  
Table 3 shows the values of the current two scaling exponents in logarithmic form for the Reynolds numbers in the range 5.0 \(\times\) 10\({}^{5}\)-1.0 \(\times\) 10\({}^{9}\).