# Numerical Simulation of Flow Field of Submerged Angular Cavitation Nozzle
## 2 Numerical Calculation Method
### Mathematical Model of Angular Cavitation Nozzle
#### 2.2.2. Cavitation Model  
The cavitation model is used to describe the mass transfer mechanism between the two vapor-liquid phases. It treats the bubbly flow as a homogeneous vapor-liquid mixture and assumes that the vapor phase consists of numerous tiny spherical bubbles. The model uses the Rayleigh-Plesset equation for the description of bubble growth and collapse [26]:  
\[R\frac{\mathrm{d}^{2}R}{\mathrm{d}t^{2}}+\frac{3}{2}\bigg{(}\frac{\mathrm{d} R}{\mathrm{d}t}\bigg{)}^{2}=\frac{1}{\rho_{f}}\bigg{(}p_{g}-p_{\infty}+ \frac{2\sigma}{R}\bigg{)} \tag{14}\]  
where \(R\) is the bubble radius, \(p_{g}\) is the pressure in the bubble (assumed to be the saturated vapor pressure at the temperature of the liquid), \(p_{\infty}\) is the pressure in the liquid surrounding the bubble, \(\rho_{f}\) is the density of the liquid, and \(\sigma\) is the surface tension coefficient between the liquid and the gas. Neglecting second-order terms (which is suitable for low vibration frequencies) and surface tension, this equation can be reduced to:  
\[\frac{\mathrm{d}R}{\mathrm{d}t}=\sqrt{\frac{2}{3}\frac{|p_{g}-p_{\infty}|}{ \rho_{f}}} \tag{15}\]  
Figure 2: The sonic velocity of mixture under vapor volume fraction.  
The rate of change in the mass of the bubble is:  
\[\frac{\mathrm{d}m}{\mathrm{d}t}=\rho_{g}\frac{\mathrm{d}V}{\mathrm{d}t}=\rho_{g} \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{4}{3}\pi R^{3}\right)=\rho_{g}\pi R^{2 }\sqrt{\frac{2}{3}\frac{\left|p_{g}-p_{\infty}\right|}{\rho_{f}}} \tag{16}\]  
where \(\rho_{g}\) is the density of the vapor and \(V\) is the volume of the bubble. If \(N_{B}\) is the number of bubbles per unit volume, the volume fraction \(r_{g}\) can be expressed as:  
\[r_{g}=VN_{B}=\frac{4}{3}\pi R^{3}N_{B} \tag{17}\]  
The total rate of interphase to phase mass transfer per unit volume is:  
\[M=N_{B}\frac{\mathrm{d}m}{\mathrm{d}t}=\frac{3r_{g}\rho_{g}}{\mathrm{R}}\sqrt{ \frac{2}{3}\frac{\left|p_{g}-p_{\infty}\right|}{\rho_{f}}}\ \mathrm{sgn}(p_{v}-p_{\infty}) \tag{18}\]  
The cavitation mass transfer process usually includes condensation and evaporation, and their transfer rates are different; thus, Equation (18) could be further optimized as follows:  
\[M_{fg}=F_{vap}\frac{3r_{nuc}\big{(}1-r_{g}\big{)}\rho_{g}}{R_{B}}\sqrt{\frac{2 }{3}\frac{p_{v}-p_{\infty}}{\rho_{f}}} \tag{19}\]  
\[M_{gf}=F_{cond}\frac{3r_{g}\rho_{g}}{R_{B}}\sqrt{\frac{2}{3}\frac{p_{\infty}- p_{v}}{\rho_{f}}} \tag{20}\]  
where \(R_{B}\) is the radius of the cavitation nucleus, \(r_{nuc}\) is the empirical constant, \(F_{vap}\) is the evaporation coefficient, and \(F_{cond}\) is the condensation coefficient. The default parameters of CFX are set as \(R_{B}=1\)\(\upmu\)m, \(r_{nuc}=5\times 10^{-4}\), \(F_{vap}=50\), and \(F_{cond}=0.01\). Hence, the cavitation mass transfer rate between the vapor and fluid is governed by the transport equation:  
\[\frac{\partial}{\partial t}(r_{\theta}\rho_{\theta})+\
abla\cdot(r_{\theta} \rho_{\theta}\mathbf{U}_{\theta})=\left|M_{fg}-M_{gf}\right| \tag{21}\]  
where \(r_{\theta}\) is the single-phase volume fraction, \(\rho_{\theta}\) is the single-phase density, and \(\mathbf{U}_{\theta}\) is the single-phase velocity vector.