# Evaluation and Validation of Viscous Oil Cavitation Model Used in Torque Converter
## 2 Materials and Methods
### Multiphase Model  
The key to cavitation simulation is to establish a proper cavitation model [21] which governs the mutual transformation between liquid and vapor. There are mainly two kinds of cavitation models--the state equation model and transport equation model. The state equation model simulates cavitation by assuming the function relationship between density and pressure, thus its accuracy relies on the assumed density-pressure function. The transport equation model determines the multi-phase mass transfer rate by local pressure and it is widely used in cavitation simulation, many multiple transport equation cavitation models were developed and validated by numerous research now, such as Zwart-Gerber-Belamri model [22], Kunz model [23], Schnerr-Sauer model [24], and Singhal model [25]. Compared with other models, Zwart cavitation model not only has higher accuracy and better robustness, but also is easy to converge and obtain obvious cavitation movement [26]. Therefore, the cavitation phenomenon of hydraulic torque converter was studied based on the Zwart cavitation model in this paper. Under most engineering conditions, it is assumed that there are enough gas nuclei to induce cavitation inception. Therefore, the emphasis is put on the reasonable description of cavitation bubble growth and collapse. Based on the assumption that the cavitation bubble is spherical and there is zero slip velocity between fluid and cavitation bubbles, the dynamic equation of cavitation can be deduced from the Rayleigh-Plesset equation  
\[R_{B}\frac{d^{2}R_{B}}{dt^{2}}+\frac{3}{2}\bigg{(}\frac{dR_{B}}{dt}\bigg{)}^{ 2}+\frac{2\sigma}{\rho_{f}R_{B}}=\frac{p_{v}-p}{\rho_{f}} \tag{1}\]  
where \(R_{B}\) is the cavitation vapor bubble radius, \(p\) is the liquid pressure, \(p_{v}\) represents the pressure in the bubble, \(\rho_{f}\) is the liquid density, and \(\sigma\) is the coefficient of surface tension. According to the Rayleigh-Plesset equation, the volume change of a single bubble dependson the difference between vaporization pressure and liquid pressure. The above equation reduces to  
\[\frac{dR_{B}}{dt}=\sqrt{\frac{2}{3}\frac{p_{v}-p}{\rho_{f}}} \tag{2}\]  
Therefore, the volume change rate of bubble can be described by  
\[\frac{dV_{B}}{dt}=\frac{d}{dt}\left(\frac{4}{3}\pi R_{B}^{3}\right)=4\pi R_{B} ^{2}\sqrt{\frac{2}{3}\frac{p_{v}-p}{\rho_{f}}} \tag{3}\]  
The mass change of bubble is as follows,  
\[\frac{dm_{B}}{dt}=\rho_{S}\frac{dV_{B}}{dt}=4\pi R_{B}^{2}\rho_{v}\sqrt{\frac {2}{3}\frac{p_{v}-p}{\rho_{f}}} \tag{4}\]  
where \(\rho_{v}\) represents the density of the vapor. [PERSON] calculated the volume fraction (\(r_{g}\)) by multiplying the number of bubbles per unit volume (\(N_{\rm B}\)) and the mass transfer rate of bubble volume (\(V_{\rm B}\)) [22]. The expression is as follows,  
\[r_{v}=V_{B}N_{B}=\frac{4}{3}\pi R_{B}^{3}N_{B} \tag{5}\]  
And the total interphase mass transfer rate is obtained as  
\[\dot{m}_{fv}=N_{B}\frac{dm_{B}}{dt}=\frac{3r_{v}\rho_{v}}{R_{B}}\sqrt{\frac{ 2}{3}\frac{p_{v}-p}{\rho_{f}}} \tag{6}\]  
This expression depends on the direction of phase transition. For the growth/vaporization of bubbles, it is given by the equation  
\[\dot{m}_{fv}=F\frac{3r_{v}\rho_{v}}{R_{B}}\sqrt{\frac{2}{3}\frac{|p_{v}-p|}{ \rho_{f}}}\text{sgn}(p_{v}-p) \tag{7}\]  
where \(F\) is a factor expressing the mass conversion rate direction, which includes vaporization (\(F_{\rm vap}\)) and condensation (\(F_{\rm cond}\)) processes. \(F_{\rm vap}\) is different from \(F_{\rm cond}\) because the condensation process is much slower than the vaporization process [27]. Although Equation (7) can be applied to the vaporization and condensation processes, the equation needs to be modified during the vaporization process. The vaporization process begins with the nuclei and grows up to bubbles.