## 2 Numerical Calculation Model
### Discrete Bubble Model  
TVC inception is mainly influenced by the coupling effect of cavitation nuclei, the low pressure action at a vortex core and the action time of the low pressure [51]. In the process of TVC inception, nuclei gather towards vortex core low-pressure regions under the action of the pressure gradient of a flow field. They commence to expand and grow explosively when the pressure reaches a certain level and collapse rapidly when the pressure rises again. Therefore, it is necessary to fully consider the growth and collapse effects of nuclei. However, the minimum pressure coefficient method based on the traditional Eulerian framework cannot solve for nuclei's motion and growth states, thus neglecting their significant influences on cavitation inception. In this paper, we use a discrete spherical bubble model to describe the motion state of the nuclei under the action of forces and the growth-collapse behavior of the nuclei in a tip vortex flow. A discrete spherical bubble model is adopted to describe the motion and dynamics of nuclei. The motion trajectory of a nucleus that is subject to forces is decided by [PERSON]'s second law given by Equations (4) and (5):  
\[\frac{dx}{dt}=u_{b} \tag{4}\]  
\[m_{b}\frac{du_{b}}{dt}=F \tag{5}\]  
where \(x\), \(u_{b}\), and \(m_{b}\) represent the position, velocity, and mass of a bubble, respectively. The force \(F\) consists of all kinds of forces imposing on a bubble. As for the cavitation inception problem, the drag \(F_{D}\), pressure gradient force \(F_{PG}\), volume change force \(F_{V}\) and additional mass force are the main factors affecting the motion trajectory of a nucleus in a flow field. Therefore, we only consider the influences of the four forces mentioned above on a nucleus, and Equation (5) becomes  
\[\frac{du_{b}}{dt}=F_{D}+F_{PG}+F_{V}=\frac{3}{4}\frac{C_{D}}{R_{b}}(u-u_{b})|u-u _{b}|-\frac{3}{\rho_{l}}\
abla p+\frac{3}{R_{b}}(u-u_{b})\frac{dR_{b}}{dt} \tag{6}\]  
where \(F_{D}\), \(F_{PG}\), and \(F_{V}\) are the drag, pressure gradient force, and volume change force, respectively. Additionally, \(\frac{1}{2}\rho_{1}V_{b}\left(\frac{du}{dt}-\frac{du_{b}}{dt}\right)\) is the additional mass force. \(R_{b}\) is the nucleus radius, \(\rho_{1}\) is the Eulerian field density, \(p\) is the Eulerian field, and \(C_{D}\) is the resistance coefficient given by the following empirical formula:  
\[C_{D}=\frac{24}{Re_{b}}\left(1+0.197 Re_{b}^{0.63}+2.6\cdot 10^{-4}Re_{b}^{1.38}\right) \tag{7}\]  
where \(Re_{b}\) is the nucleus Reynolds number:  
\[Re_{b}=2\rho R|u-u_{b}|/\mu \tag{8}\]  
Moreover, nuclei dynamics are another point to note. In order to describe the growth and collapse behaviors of nuclei in a pressure field precisely, the classic Rayleigh-Plesset (R-P) equation is used to solve the nuclei changes in the radial direction. Meanwhile, due to the velocity difference between a moving nucleus and the local fluid, there is an additional pressure term acting on a nucleus surface with the magnitude of \((u-u_{b})^{2}/4\). The R-P equation can be denoted as  
\[R\ddot{R}+\frac{3}{2}\dot{R}^{2}=\frac{1}{\rho_{l}}\left( \begin{array}{c}P _{v}+P_{g0}\left(\frac{R_{0}}{R}\right)^{3\gamma}-\\ P_{encounter}-\frac{2S}{R}-\frac{4\mu_{l}}{R}\dot{R}\end{array} \right)+\frac {(u-u_{b})^{2}}{4} \tag{9}\]  
where \(R\) is the nucleus radius, \(\dot{R}\) is the velocity of a nucleus wall, \(\ddot{R}\) is the acceleration of a nucleus wall, \(P_{v}\) is the saturated vapor pressure, and \(S\) is the surface tension. The initial pressure of nuclei \(P_{g0}\) is given by  
\[P_{g0}=P_{0}+\frac{2S}{R_{0}}-P_{v} \tag{10}\]  
A previous study [52] shows that the effect of liquid compressibility should not be neglected in the growth and collapse processes of nuclei. Therefore, we adopt the piecewise Rayleigh-Plesset governing equation in this paper, that is, adopting the traditional R-P equation to control the growth process and the R-P equation considering the effect of liquid compressibility to control the collapse process.