## 4 Discussion and Analysis of Results
### A Discrimination Criterion for Propeller Cavitation Inception Based on the Bubble Dynamics Model  
The comparative verification between the aforementioned calculation results and experimental results merely illustrates the reliability of this numerical method and is not yet sufficient to implement a numerical prediction for propeller cavitation inception. If we have the intention of using this method to conduct a cavitation inception prediction instead of verification, it is necessary to establish a set of discrimination criteria applicable to propeller TVC inception. However, most of existing studies based on microscopic bubbles only mention the specific size to which a nucleus needs to grow to be determined as incipient cavitation and do not specify a quantity threshold for incipient cavitation bubbles used to determine the cavitation inception. Therefore, if we have the intention of implementing a numerical prediction for propeller TVC inception, it is necessary to establish a set of discrimination criteria for the propeller TVC inception based on bubble dynamics through associating incipient cavitation bubbles with cavitation inception. That is, it is necessary to specify a diameter threshold that is used to determine whether a nucleus is an incipient cavitation bubble to allow for an accurate prediction of propeller TVC inception. The definition of an incipient cavitation bubble has been given in many experimental studies [59], that is, a nucleus grows to a size threshold. For example, as early as the last century, [PERSON] et al. [39] argued that if a nucleus grows to 1 mm, which can be observed with the naked eye, it can be defined as an incipient cavitation bubble. In this paper, we respectively count up the nuclei growing to 1 mm, 2 mm, 3 mm, and 4 mm under different cavitation numbers in the TVC inception operating condition (advance coefficient \(J=0.9983\)), as shown in Figure 7.  
It can be observed that, although under the circumstance of the same cavitation number, the larger the diameter used for defining an incipient cavitation bubble, the fewer the incipient cavitation bubbles, the variation trends of the number of incipient cavitation bubbles obtained under different definition criteria with the cavitation number are identical, which indicates that although the number of incipient cavitation bubbles obtained by adopting different definition criteria under the circumstance of the same cavitation number are different, the variation trends of the number of incipient cavitation bubbles with the cavitation number are identical. In order to be consistent with the optics criterion adopted in previous studies, we select the size of 1 mm as the discrimination criterion for an incipient cavitation bubble, that is, as long as the diameter of a nucleus increases to 1 mm, the nucleus will be determined as an incipient cavitation bubble. Figure 6: Comparison of the cavitation distribution with experimental results [55]. (**a**) Experimental result; (**b**) Numerical result obtained by proposed numerical prediction method based on the Eulerian-Lagrangian framework; (**c**) Numerical result based on the traditional ZGB cavitation model under the Eulerian framework. In order to better reflect the variation trend of the number of incipient cavitation bubbles with the cavitation number, we count up all incipient cavitation bubbles under the circumstance of different cavitation numbers, respectively, and fit them with an exponential function curve, as shown in Figure 8. The scatter points in Figure 8 are the results calculated with numerical simulations, and the curve is the fitting result. The R-Square value of the curve fitting is 0.957, which indicates that the fitting has a higher confidence level and can fully reflect the variation trend of scatter points. It should be noted that although the number of incipient cavitation bubbles presents an upward trend with the decrease in the cavitation number on the whole, the numbers within different cavitation number ranges are not the same.