# Numerical Investigation of Methodologies for Cavitation Suppression Inside Globe Valves
## 2 Numerical Methods  
The structure of the investigated globe valve is shown in Figure 2; here the globe valve with a different number of cages and the geometry of the cage are displayed. The two-stage globe valve is self-designed to be used in the pipe system with a high-pressure drop, and the model is created using commercial software SOLIDWORKS. The nominal pipe diameter of the globe valve, \(D\), is 50 mm, and the entrance pipe length is 3 \(D\) and the downstream pipe length is 8 \(D\). The throttling orifices on each cage are the same and are evenly distributed along the circumference. The throttling orifices on two cages can either be installed aligned or be installed with an installation angle, \(\alpha\), as shown in Figure 2.  
Fluid flow through the two-cage globe valve with a fully opened state is numerically solved. Liquid water and water vapor under different temperatures are considered as the incompressible working fluid, and the properties of the working fluid are shown in Table 2. The continuity and the momentum equations are solved using commercial software ANSYS Fluent, and the governing equations and the equations of the Realizable \(k\)-\(\varepsilon\) turbulence model can be found in the previous work [26]. The cavitation model proposed by [PERSON] et al. [38] is used and the equations describing cavitation are shown below.  
\[\frac{\partial}{\partial t}(\alpha\rho_{v})+\
abla(\alpha\rho_{v}u_{v})=R_{e} -R_{c} \tag{1}\]  
Figure 2: Structure of the investigated globe valve.  
\[R_{e}=F_{\mathit{vap}}\frac{3\alpha_{\mathit{nuc}}(1-\alpha_{v})\rho_{v}}{R_{B}} \sqrt{\frac{2}{3}\frac{p_{v}-p}{\rho_{l}}}\ \ \ (p\leq p_{v}) \tag{2}\]  
\[R_{c}=F_{\mathit{cond}}\frac{3\alpha_{v}\rho_{v}}{R_{B}}\sqrt{\frac{2}{3}\frac{p -p_{v}}{\rho_{l}}}\ \ \ (p\geq p_{v}) \tag{3}\]  
here \(\alpha\) is vapor volume fraction, subscript \(v\) stands for vapor phase, \(\rho\) stands for density, \(u\) stands for velocity, \(R_{e}\) and \(R_{c}\) are related to the growth and collapse of the vapor bubbles, \(R_{B}\), \(F_{\mathit{vap}}\), and \(F_{\mathit{cond}}\) are model constant. During the simulation, the pressure-based SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm is utilized, and the second-order upwind discretization scheme is applied. The residuals are set as 0.001 and the area-weighted inlet velocity is monitored, and the solution is considered as converged if the residual is below 0.001 and the inlet velocity is constant.  
Pressure inlet and pressure outlet are applied as the main boundary conditions. The inlet pressure is 1.17 MPa, and the outlet pressure varies from 0.1 to 0.25 MPa. The no-slip wall with the scalable wall function is used. The temperature variation is neglected during the simulation.  
A polyhedral mesh with boundary layer is utilized in this study, and the zone near the throttling window of the cage is refined. Figure 3 shows the mesh of the investigated two-cage globe valve. Mesh independence is done with a different number of grid cells when the outlet pressure is 0.25 MPa and the temperature is 80 \({}^{\circ}\)C. The inlet velocity and pressure drop are compared and shown in Table 3, and it is clear that the results are close when the cell number is 824,206 and 1,550,930, thus 824,206 cells are adopted eventually.