# Comprehensive Investigations on Fluid Flow and Cavitation Characteristics in Rotating Disk System
## 3 Simulation Analysis
### CFX Model Analysis
#### 3.1.2 Cavitation Models  
There is a convective exchange of mass and volume fractions between the gas and liquid phases in cavitation flow [35], where the transport equation for the volume content of the gas phase is:  
\[\frac{\partial(\lambda_{\varphi}\rho_{\varphi})}{\partial t}+\
abla\cdot( \lambda_{\varphi}\rho_{\varphi}u)=\dot{m} \tag{26}\]  
where \(\lambda_{\varphi}\) is volume fraction of gas phase; \(\rho_{\varphi}\) is gas density, kg/m\({}^{3}\); \(u\) is gas vector velocity, m/s; \(\dot{m}\) is the source term for evaporation and condensation.  
The Rayleigh-Plesset equation describes bubble growth in liquids:  
\[R_{B}\frac{d^{2}R_{B}}{dt^{2}}+\frac{3}{2}\bigg{(}\frac{dR_{B}}{dt}\bigg{)}^{2 }+\frac{2\tau}{\rho_{f}R_{B}}=\frac{p_{v}-p}{\rho_{f}} \tag{27}\]  
where \(R_{B}\) is bubble radius, m; \(p_{v}\) is the pressure in bubbles, Pa; \(p\) is the pressure surrounding the bubble, Pa; \(\rho_{f}\) is liquid density, kg/m\({}^{3}\); \(\rho_{v}\) is gas density, kg/m\({}^{3}\); \(\tau\) is the coefficient of surface tension between liquid and gas, N\(\cdot\)m.  
Equation (27) can be simplified by ignoring the second-order derivative term and the surface tension term:  
\[\frac{dR_{B}}{dt}=\sqrt{\frac{2}{3}\frac{p_{v}-p}{\rho_{f}}} \tag{28}\]  
The rate of change in bubble volume and mass can thus be calculated as follows:  
\[\frac{dV_{B}}{dt}=\frac{d}{dt}\bigg{(}\frac{4\pi}{3}R_{B}^{3}\bigg{)}=4\pi R_ {B}^{2}\sqrt{\frac{2}{3}\frac{p_{v}-p}{\rho_{f}}} \tag{29}\]  
Figure 13: Schematic diagram of homogeneous model (R is radius of bubble; yellow area is liquid; bubbles are gas).  
\[\frac{dm_{B}}{dt}=\rho_{v}\frac{dV_{B}}{dt}=4\pi\rho_{v}R_{B}^{2}\sqrt{\frac{2}{3} \frac{p_{v}-p}{\rho_{f}}} \tag{30}\]  
If the quantity of bubbles in fluid is \(N_{B}\), the fluid's vapor volume fraction can be demonstrated as:  
\[\lambda_{v}=V_{B}N_{B}=\frac{4}{3}\pi R_{B}^{3}N_{B} \tag{31}\]  
Then, the total interphase mass transport rate between phases is:  
\[\dot{m}_{fv}=F\frac{3\lambda_{v}\rho_{v}}{R_{B}}\sqrt{\frac{2}{3}\frac{p_{v}-p }{\rho_{f}}}\text{sgn}(p_{v}-p) \tag{32}\]  
where \(F\) is an empirical correction factor for the difference in evaporation or condensation rates, with evaporation being faster than condensation in general [40]. Because evaporation occurs primarily at the nucleation and causes a decrease in density there, the mass transport rate equation for the evaporation process must be rectified as:  
\[\dot{m}_{fv}=F\frac{3\lambda_{nuc}(1-\lambda_{v})\rho_{v}}{R_{B}}\sqrt{\frac{ 2}{3}\frac{p_{v}-p}{\rho_{f}}}\text{sgn}(p_{v}-p) \tag{33}\]  
where \(\lambda_{nuc}\) is the volume fraction of nucleation centers.  
When \(p\leq p_{v}\), evaporation occurs as described in Equation (28); when \(p\geq p_{v}\), condensation occurs as described in Equation (27). The Rayleigh-Plesset cavitation model parameters in the CFX simulation are \(R_{B}\) = 2 \(\upmu\)m, \(\lambda_{nuc}\) = \(5\times 10^{-4}\), evaporation coefficient \(F_{vap}\) = 50, and condensation coefficient \(F_{coud}\) = 0.01.