# Numerical Simulation of Flow Field of Submerged Angular Cavitation Nozzle
## 2 Numerical Calculation Method
### Mathematical Model of Angular Cavitation Nozzle
#### 2.2.1 Multiphase Flow Mixture Model  
The cavitation jet belongs to a gas-liquid two-phase flow, and a reference analysis [26] shows that for the current nozzle flow, the relative motion of the two phases can be ignored, so we consider the vapor-liquid flow to be a uniform bubble-liquid mixture. The nozzle flow simulations can be performed using the homogeneous equilibrium model. For the mixture model, the basic equations of continuity and momentum are the same as those in single-phase flow except that the density and dynamic viscosity in the cavitating flow of nozzles use the mixture density \(\rho\) and the mixture dynamic viscosity \(\mu\):  
\[\rho=r_{l}\rho_{l}+r_{v}\rho_{v} \tag{1}\]  
\[\mu=r_{l}\mu_{l}+r_{v}\mu_{v} \tag{2}\]  
where \(r_{l}\) is the volume fraction of the liquid, \(r_{v}\) is the volume fraction of the vapor, \(r_{l}+r_{v}=1\), \(\rho_{l}\) is the density of the liquid, \(\rho_{v}\) is the density of the vapor, \(\mu_{l}\) is the dynamic viscosity of the liquid, and \(\mu_{v}\) is the dynamic viscosity of the vapor. The mixture velocity vector \(\mathbf{U}\) is as follows:  
\[\mathbf{U}=\frac{1}{\rho}(r_{l}\rho_{l}\mathbf{U}_{l}+r_{v}\rho_{v}\mathbf{U}_{v}) \tag{3}\]  
Figure 1: Schematic diagram of physical model. where \(\mathbf{U}_{l}\) is the velocity vector of the liquid and \(\mathbf{U}_{v}\) is the velocity vector of the vapor. Thus, assuming an incompressible liquid, the momentum conservation equation and continuity equation of cavitation flow are as follows:  
\[\rho\frac{\partial\mathbf{U}}{\partial t}+\rho\mathbf{U}\cdot\
abla\mathbf{U}=-\
abla p+\mu \
abla^{2}\mathbf{U} \tag{4}\]  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{U})=0 \tag{5}\]  
where \(p\) is the peripheral pressure. Experimental observations show that cavitation caused in the high-speed submerged water jet appears in the form of a bubble cloud [27]. Corresponding to the variation of surrounding liquid pressure, bubble clouds expand and break up unsteadily. Thus, the density of the two-phase mixture varies sharply with bubble oscillation, and it becomes essential to consider the effect of fluid compressibility in the numerical simulation of intensive cavitating flows. When considering the compressibility of the mixture, from the conservation of mass, Equation (1) can be derived as:  
\[\rho\mathrm{d}V=r_{l}\rho_{l}\mathrm{d}V_{l}+r_{v}\rho_{v}\mathrm{d}V_{v} \tag{6}\]  
where \(\mathrm{d}V\) is assumed as the change in total volume and \(\mathrm{d}V_{l}\) and \(\mathrm{d}V_{v}\) are the volume changes of fluid and gas, respectively. Further, the result can be written as:  
\[\frac{1}{\rho}\mathrm{d}\rho=\frac{r_{l}}{\rho_{l}}\mathrm{d}\rho_{l}+\frac{r _{v}}{\rho_{v}}\mathrm{d}\rho_{v} \tag{7}\]  
In accordance with the definition of sonic velocity, Equation (7) is expressed as follows:  
\[\frac{1}{\rho c^{2}}\mathrm{d}p=\frac{r_{l}}{\rho_{l}c_{l}^{2}}\mathrm{d}p_{l }+\frac{r_{v}}{\rho_{v}c_{v}^{2}}\mathrm{d}p_{v} \tag{8}\]  
where \(c\) is the sonic velocity of a mixture, \(c_{l}\) is the sonic velocity of the fluid, and \(c_{v}\) is the sonic velocity of the vapor. In the absence of surface tension, according to a local equilibrium state, one obtains that \(p=p_{l}=p_{v}\). Hence, the sonic velocity of the mixture is given by:  
\[\frac{1}{\rho c^{2}}=\frac{r_{l}}{\rho_{l}c_{l}^{2}}+\frac{r_{v}}{\rho_{v}c_{ v}^{2}} \tag{9}\]  
In other words, the acoustic impedance \(1/\rho c^{2}\) for the mixture is simply given by the average of the acoustic impedance of the components weighted according to their volume fractions. Supposing the gas is perfect and the process is isentropic,  
\[\mu=r_{l}\mu_{l}+r_{v}\mu_{v} \tag{2}\]0  
where \(k\) is the adiabatic index and \(p_{v}\) is the gas pressure. Therefore, Equation (9) may be written as:  
\[\mu=r_{l}\mu_{l}+r_{v}\mu_{v} \tag{2}\]1  
Note that, in most applications, dramatic effects occur when one of the components is a vapor, which is much more compressible than the other component such as liquid.