## 4 Discussion and Analysis of Results
### A Discrimination Criterion for Propeller Cavitation Inception Based on the Bubble Dynamics Model
In contrast, the relative error of the incipient cavitation number obtained with the traditional minimum pressure coefficient method and the same grid settings is obviously larger than 10%, as illustrated in Table 3. This is because the minimum pressure coefficient method requires grids with fairly good quality, whereas the propeller, a complex and distorted unstructured grid for geometric purposes, is bound to have grids with extremely poor quality near the edge, which result in large numerical errors. The bubble dynamics model has low requirements for grid quality and can implement a more accurate prediction under the circumstance of unstructured propeller grids, which further indicates that the numerical calculation method adopted in \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \(J\) & \(N\) (r/s) & & & & & & & **Cavitation Number \(\sigma\)** & & & & & \\ \hline
0.8649 & 25 & 4 & 5 & 6 & 6.619 & 7 & 7.5 & 7.859 & 8 & 8.25 & 8.5 & 10 & 12 & 15 \\
0.9725 & 25 & 3 & 4 & 4.225 & 4.5 & 5 & 5.5 & 6 & 6.285 & 6.5 & 8 & 10 & 12 & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculation cases. this paper and the corresponding discrimination criterion for cavitation inception have a higher accuracy compared to the minimum pressure coefficient method. The variation trend of the number of incipient cavitation bubbles with the cavitation number under the operating condition with the advance coefficient of \(J\) = 0.9725 is identical with that of \(J\) = 0.8649, and adopting the same method to determine the range of the incipient cavitation number can effectively constrain the relative error within 10% likewise, as shown in Figure 10. The two relative errors of the incipient cavitation number under the two operating conditions obtained with the traditional minimum pressure coefficient method are obviously larger than 10%, as illustrated in Table 3.  
Figure 10: The curve of the number of incipient cavitation bubbles at different cavitation numbers (\(J\) = 0.9725). Figure 9: The curve of the number of incipient cavitation bubbles at different cavitation numbers (\(J\) = 0.8649). In addition, it is worth noting that although the variation trends of the number of incipient cavitation bubbles with the cavitation number under the two different advance coefficients are identical, the number of incipient cavitation bubbles under the operating condition with an advance coefficient of \(J=0.8649\) is obviously larger than that of \(J=0.9725\), which indicates that nuclei are more prone to be su  
Figure 11 shows the variation curves of the coefficient of the pressure on the nucleus surface and nucleus radius under the operating condition with an advance coefficient of \(J=0.9725\). In order to analyze in detail the influence of a tip vortex flow on nuclei evolution in the flow field in which the nuclei are located, we select six representative times--\(t_{1}\), \(t_{2}\),..., \(t_{6}\)--in the growth and collapse processes of nuclei under the influence of the tip vortex, presenting the typical process of nuclei being su  
Figure 11: The variation curves of the coefficient of the pressure on the nucleus surface and nucleus radius. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(J\) & **Experimental Result** & \
\begin{tabular}{c} **The Minimum Pressure** \\ **Coefficient Method** \\ \end{tabular} &
\begin{tabular}{c} **The Relative Error** \\ \end{tabular} \\ \hline
0.8649 & 7.859 & 6.712 & \(-14.6\%\) \\
0.9725 & 6.285 & 5.292 & \(-15.8\%\) \\
0.9983 & 6.389 & 5.265 & \(-17.6\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Numerical calculation error of the minimum pressure coefficient method. It can be observed that around the guide edge of a blade tip, nuclei commence to approach the tip vortex for the first time and the diameters of nuclei fluctuate within a narrow range during the period from time \(t_{1}\) to \(t_{2}\).