# Dynamic Analysis of Cavitation Tip Vortex of Pump-Jet Propeller Based on DES
## 2 Numerical Method and Computational Setup
### Cavitation Model  
The cavitation model is composed of a set of mathematical equations which describe the mass transfer rate between liquid phase and vapor phase. In previous studies, [PERSON] et al. [43] used the SST \(k\)-\(\omega\) turbulence model and Z-G-B cavitation model to research the cavitating turbulent flow near the rotor suction surface. Further, combining with the experiment method, the cavitation performance curve showed good agreement with test data. [PERSON] et al. [44] can predict tip vortex cavitation and sheet cavitation near hub of propeller well based on commercial solver Fluent and STAR-CCM+, which is validated by experimental observation. [PERSON] et al. [45] conducted a cavitating numerical simulation with Z-G-B cavitation for a water-jet pump, through comparing with experimental results, the predicted cavitation performance curves exhibits reasonable agreement with them. In addition, the time-dependent vapor iso-surfaces are in accordance with high-speed video at different times in a typical cycle. Therefore, given that pump-jet propeller has similar geometry with propeller and water-jet pump, this study will employ Z-G-B cavitation model to calculate the mass transfer rate between liquid phase and vapor phase.  
This study employs the Zwart-Gerber-Belamri cavitation model in CFX 17.2 software, which assumes that all bubbles in a system have the same initial diameter. The mass change rate for a single bubble is expressed by:  
\[\frac{dm_{B}}{dt}=\ \rho_{v}\frac{dV_{B}}{dt}=\ 4\pi{R_{B}}^{2}\rho_{v}\sqrt{ \frac{2}{3}\frac{p_{v}\text{-}p}{\rho_{l}}} \tag{11}\]The vapor volume fraction is calculated by:  
\[\alpha_{v}=\ N_{B}V_{B}=\frac{4}{3}\pi R_{B}{}^{2}N_{B} \tag{12}\]  
where \(m_{B}\) is the bubble mass, \(V_{B}\) is the bubble volume, \(\rho_{v}\) is the vapor phase density, \(\rho_{l}\) is the liquid density, \(p_{v}\) is the vaporization pressure of liquid at local temperature, \(p\) is the liquid pressure around the bubble, \(\alpha_{v}\) is the vapor volume fraction, \(N_{B}\) is the bubble number per unit volume, and \(R_{B}\) is the bubble radius.  
The total interphase mass transfer rate is calculated as:  
\[m_{t}=\ N_{B}\frac{dm_{B}}{dt}=\frac{3\alpha_{v}\rho_{v}}{R_{B}}\sqrt{\frac{2} {3}\frac{p_{v}-p}{\rho_{l}}} \tag{13}\]  
The cavitation collapse process can be described by the following equation:  
\[R_{e}=\ F\frac{3\alpha_{v}\rho_{v}}{R_{B}}\sqrt{\frac{2}{3}\frac{\left|p_{v}-p \right|}{\rho_{l}}}sign(p_{v}-p) \tag{14}\]  
where \(R_{e}\) is vapor generation rate. The concrete expression of Zwart-Gerber-Belamri cavitation model can be obtained by replacing the \(\alpha_{v}\) term with \(\alpha_{nuc}\)(1-\(\alpha_{v}\)) as:  
\[m=\ F_{vup}\frac{3\alpha_{nuc}(1-\alpha_{v})\rho_{v}}{R_{B}}\sqrt{\frac{2}{3 }\frac{p_{v}-p}{\rho_{l}}},p\ \leq\ p_{v} \tag{15}\]  
\[m=\ F_{cond}\frac{3\alpha_{v}\rho_{v}}{R_{B}}\sqrt{\frac{2}{3}\frac{p-p_{v}}{ \rho_{l}}},p>\ p_{v} \tag{16}\]  
in which the vapor volume fraction \(\alpha_{nuc}=5\times 10^{-4}\), the empirical correction factor for vaporization \(F_{vup}=50\), and the condensation empirical correction factor \(F_{cond}=0.001\).