# Numerical Simulation of Flow Field of Submerged Angular Cavitation Nozzle
## 2 Numerical Calculation Method
### Mathematical Model of Angular Cavitation Nozzle
#### 2.2.3 Turbulence Model  
At present, the Reynolds-averaged Navier-Stokes equations (RANS) method, large eddy simulation (LES) method, and direct numerical simulation (DNS) method are usually used to deal with turbulence problems. The RANS method is the most widely used turbulence numerical simulation method for fluid simulation because of its high computational efficiency and solution accuracy [28]. The \(k\)-\(\varepsilon\) model is often used in the simulation of cavitation jets [29]. Three \(k\)-\(\varepsilon\) models are provided in CFX software: the standard \(k\)-\(\varepsilon\) model and the RNG \(k\)-\(\varepsilon\) model. The standard \(k\)-\(\varepsilon\) model has the advantages of wide application scope, economy, and better accuracy. The model assumes that the flow is completely turbulent, and the influence of molecular viscosity can be ignored, so the standard \(k\)-\(\varepsilon\) model is only suitable for the simulation of a completely turbulent flow process [29]. The RNG \(k\)-\(\varepsilon\) model is added to a condition in \(\varepsilon\), taking into account the rotation and flow conditions in the flow, which can better deal with flow with a high strain rate and greater streamline curvature. Combined with the same model and numerical simulation scheme, two turbulence models are used to characterize the cavitation jet flow field, and the results are shown in Figure 3. From the analysis of the boundary layer, the RNG \(k\)-\(\varepsilon\) model has a good analysis of the expansion part. The water jet velocity of the nozzle can be simply calculated by the following Equation [30]:  
\[v_{t}=44.77\sqrt{p_{1}} \tag{22}\]  
where \(v_{t}\) is the jet velocity and \(p_{1}\) is the jet incidence pressure. The axial velocity of the RNG model is closer to the theoretical calculation value of 200 m/s (Figure 3). Hence, the RNG \(k\)-\(\varepsilon\) model is a relatively suitable model with high accuracy and fast convergence rate. The used RNG \(k\)-\(\varepsilon\) model is a RANS-based turbulence model which is in the light of Navier-Stokes equations. For the standard \(k\)-\(\varepsilon\) model cannot accurately predict the impact of transient flow and bending streamline, the RNG \(k\)-\(\varepsilon\) model can reduce the calculation error to a certain extent and expand its field of application by modifying the turbulent dynamic viscosity coefficient [24]. In homogeneous multiphase flow, bulk turbulence equations are solved, which are the same as the single-phase equations except that the mixture density and the mixture viscosity are used. In addition to the dynamic viscosity \(\mu_{0}\) of the fluid itself, turbulence flow usually has a turbulent viscosity \(\mu_{t}\) due to its own flow properties:  
\[\mu_{t}=C_{\mu}\rho_{k}\frac{k^{2}}{\varepsilon} \tag{23}\]  
where \(\rho_{k}\) is the fluid density, \(k\) is the turbulent kinetic energy (the variance of velocity fluctuations), \(\varepsilon\) is the turbulent vortex dissipation rate (the rate of dissipation of velocity fluctuations), and \(C_{\mu}\) is the correction factor. Then, the turbulent effective viscosity \(\mu_{\rm eff}\) can be obtained by:  
\[\mu_{\rm eff}=\mu_{0}+\mu_{t} \tag{24}\]  
Denoting the shear production of turbulence by \(P_{k}\) for convenience:  
\[P_{k}=\mu_{t}\Bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{ j}}{\partial x_{i}}\Bigg{)}\frac{\partial u_{j}}{\partial x_{i}} \tag{25}\]  
Hence, the governing equation of the turbulence model is shown in the following equations:  
\[\frac{\partial(\rho_{k}k)}{\partial t}+\frac{\partial(\rho_{k}ku_{i})}{ \partial x_{i}}=\frac{\partial}{\partial x_{i}}\bigg{(}\alpha_{k}\mu_{\rm eff }\frac{\partial k}{\partial x_{i}}\bigg{)}+P_{k}-\rho_{k}\varepsilon(i,j=1,2) \tag{26}\]  
Figure 3: Velocity vector contour on expansion part. (**a**) The standard model; (**b**) the RNG model.