## 4 Discussion and Analysis of Results
### A Discrimination Criterion for Propeller Cavitation Inception Based on the Bubble Dynamics Model
As shown in Figure 8, at stage I, with the decrease in the cavitation number, the number of incipient cavitation bubbles does not increase significantly, but presents a slow upward trend on the whole with a small absolute value of the slope. The reasons are as  
Figure 8: The curve of the number of incipient cavitation bubbles with different cavitation numbers (\(J=0.9983\)). Figure 7: Curves of the number of incipient cavitation bubbles with different diameters. follows: the cavitation number is larger at this stage, although part of nuclei are sectioned into the tip vortex core, and there are not many nuclei growing explosively to an incipient cavitation bubble due to the higher pressure at the vortex core. However, at stage II, with the further decrease in the cavitation number, the number of incipient cavitation bubbles increases significantly and the absolute value of the slope increases dramatically, which contrasts sharply with stage I. This is because the vortex core pressure decreases with a further decrease in the cavitation number and more nuclei captured by the tip vortex grow explosively to incipient cavitation bubbles, resulting in cavitation on propellers at this moment. Once combined with the variation regulations of the number of incipient cavitation bubbles and the physical mechanism of bubble generation in stages I and II, it indicates that with the decrease in the cavitation number, the 'inflection point' of the slope in the transition from stage I to stage II can be considered as a cavitation inception point. It is worth noting that the experimental incipient cavitation number \(\sigma\) = 6.389 is exactly around the 'inflection point' of the slope. This means that the incipient cavitation number under this advance coefficient can be obtained by counting up incipient cavitation bubbles at different cavitation numbers and finding the 'inflection point' of the slope representing the transition of quantity trends; thus, we can implement an effective numerical prediction for the incipient cavitation of propellers, constraining the error within a reasonable range. In order to further verify the reliability of this method and referring to the incipient cavitation operating conditions in experimental reports, we conduct numerical calculations for the two operating conditions with advance coefficients of \(J\) = 0.8964 and \(J\) = 0.9725, respectively, count up the incipient cavitation bubbles at different cavitation numbers and obtain the numerical prediction results for incipient cavitation numbers through fitting, which are compared with experimental results. We select 13 operating conditions with different cavitation numbers under each advance coefficient, as shown in Table 2, in which, the experimental incipient cavitation numbers of the two operating conditions with the two advance coefficients mentioned above are \(\sigma\) = 7.859 and \(\sigma\) = 6.2859, respectively. As shown in Figure 9, it can be observed that the number of incipient cavitation bubbles first increases slowly with the decrease in the cavitation number at stage I likewise. However, compared to stage I, it increases sharply with the decrease in the cavitation number at stage II and the absolute value of the slope increases meanwhile. This is in accord with a cavitation vortex strip phenomenon occurring suddenly, as observed during experiments, and further demonstrates the accuracy of this method. It is worth noting that the red dotted line box represents the relative error band of the incipient cavitation number (\(-\)10-10%), and it can be observed that the slope 'inflection point' of the fitting curve of the number of incipient cavitation bubbles is within this relative error band. This indicates that the relative errors between the experimental results and the prediction results of cavitation inception obtained with this discrimination method for cavitation inception can be constrained within a range of 10%.