# A Performance Test and Internal Flow Field Simulation of a Vortex Pump
## 4 Numerical Calculation of Pump's Internal Flow Field  
### Governing Equation  
The flow field was calculated based on selecting the best condition of pump performance test. The internal flow field of the vortex pump is very complex, which is three-dimensional turbulent flow dominated by strong swirling flow. It is affected by many factors, such as curvature, rotation, and so on. Choosing the turbulence model for the calculation of the internal flow field is very important. The previous calculation used the standard \(k\)-\(\epsilon\) two-equation model [12; 13; 14], which is a local equilibrium model that cannot accurately predict turbulence with drastic changes in mean flow, such as separation flow and swirling flow. Later, the researchers applied the modified \(k\)-\(\epsilon\) and the Reynolds stress equation model--which was derived from the anisotropic conditions--to the numerical calculation of the strong cyclonic flow field. The renormalization group \(k\)-\(\epsilon\) turbulence model is called the RNG \(k\)-\(\epsilon\) model. The coefficient in the RNG \(k\)-\(\epsilon\) turbulence model is derived by using the RNG model theory. There is an additional item R in the \(k\)-\(\epsilon\) equation, which represents the effect of the average strain rate on the turbulent dissipation rate \(\varepsilon\). The RNG \(k\)-\(\varepsilon\) turbulence model can simulate the flow separation and vortex well. Moreover, because of the complexity of the Reynolds stress model and the large amount of computational resources, we chose the RNG \(k\)-\(\epsilon\) model to numerically calculate the three-dimensional flow in the vortex pump [15].  
There are three expressions of the governing equation:  
(1) The continuity equation  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho\overline{u_{i}})}{ \partial x_{i}}=0 \tag{1}\]  
(2) The momentum equations  
\[\frac{\partial(\rho\overline{u_{i}})}{\partial t}+\frac{\partial \big{(}\rho\overline{u_{i}}\overline{u_{j}}\big{)}}{\partial x_{j}}=-\frac{ \partial\overline{p}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left[ \mu\frac{\partial\overline{u_{i}}}{\partial x_{j}}-\rho\overline{u_{i}^{ \prime}u_{j}^{\prime}}\right] \tag{2}\]  
\[-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}=\mu_{t}\left[\frac{\partial \overline{u_{i}}}{\partial x_{j}}+\frac{\delta\overline{u_{j}}}{\delta x_{i}} \right]-\frac{2}{3}\left[\rho k+\mu_{t}\frac{\delta\overline{u_{j}}}{\delta x _{i}}\right]\delta_{ij} \tag{3}\]  
Figure 3: Experimental pump performance curves.  
(3) The \(k\) and \(\varepsilon\) equation  
\[\rho\frac{Dk}{Dt}=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{ \sigma_{k}}\right)\frac{\partial k}{\partial x_{j}}\right]+2\mu_{t}\overline{S_ {ij}}\frac{\partial\overline{u_{i}}}{\partial x_{j}}-\rho\varepsilon \tag{4}\]  
\[\rho\frac{D\varepsilon}{Dt}=\frac{\partial}{\partial x_{j}}\left[\left(\mu+ \frac{\mu_{t}}{\sigma_{\varepsilon}}\right)\frac{\partial\varepsilon}{ \partial x_{j}}\right]+2C_{1k}\frac{\varepsilon}{k}\mu_{t}\overline{S_{ij}} \frac{\partial\overline{u_{i}}}{\partial x_{j}}-C_{2\varepsilon}^{*}\rho\frac {\varepsilon^{2}}{k} \tag{5}\]  
where  
\[\mu_{t}=C_{\mu}\frac{k^{2}}{\varepsilon} \tag{7}\]  
\[S_{ij}=\frac{1}{2}\left(\frac{\partial\overline{u_{i}}}{\partial x_{j}}+\frac {\delta\overline{u_{j}}}{\delta x_{i}}\right) \tag{8}\]  
\[C_{2\varepsilon}^{*}=C_{2\varepsilon}+\frac{C_{\mu}\eta^{3}\left(\frac{1- \eta}{\eta_{0}}\right)}{1+\beta\eta^{3}} \tag{9}\]  
\[\eta=\sqrt{2\overline{S_{ij}}S_{ij}}\frac{k}{\varepsilon} \tag{10}\]  
where \(\delta_{ij}\) is unit tensor, \(\eta_{0}=4.38\), \(C_{\mu}=0.0845\), \(\beta=0.012\), \(C_{1\varepsilon}=1.42\), \(C_{2\varepsilon}=1.68\), \(\alpha_{k}=0.7194\), and \(\alpha_{\varepsilon}=0.7194\).  
The environment temperature is 25 \({}^{\circ}\)C. The medium is water, and the density \(\rho\) is 997 kg/m\({}^{3}\). The dynamic viscosity \(\mu\) is 0.89 mPa-s. The quantities with over-bar (such as \(\overline{u_{i}}\)) are mean values.