## 4 Numerical Method and Verification
### Reliability Verification of Numerical Calculation  
Through the energy performance repeatability test of the vertical axial flow pump unit, the hydraulic efficiency of the model pump unit was calculated by taking the test data after the measurement results were stable. The calculation formula is shown in Equation (14):  
\[\eta=\frac{30\rho gQ_{i}H}{\pi n(M-M\prime)} \tag{14}\]  
where \(Q_{i}\) is the flow rate, \(\mathrm{L/s}\); \(H\) is the head, \(\mathrm{m}\); \(n\) is the speed, \(\mathrm{r/min}\); \(M\) is the input torque, \(\mathrm{N\cdot m}\); \(M\) is the mechanical loss torque, \(\mathrm{N\cdot m}\).  
Based on the results of the numerical calculations and three-dimensional constant value calculations, the head \(\mathrm{H}\) and efficiency of the pump unit were predicted with static total pressure at each grid node of the inlet and outlet sections of the pump unit according to Equations (15) and (16):  
\[H=\left[\sum_{i=1}^{N_{\mathrm{out}}}\frac{\left(\frac{P_{\mathrm{out}}}{\rho g }\right)_{i}}{N_{\mathrm{out}}}+\sum_{i=1}^{N_{\mathrm{out}}}\frac{\left(\frac{ P_{\mathrm{out}}^{2}}{\rho g}\right)_{i}}{N_{\mathrm{out}}}\right]_{\mathrm{out}}- \left[\sum_{i=1}^{N_{\mathrm{in}}}\frac{\left(\frac{P_{\mathrm{in}}}{\rho g} \right)_{i}}{N_{\mathrm{in}}}+\sum_{i=1}^{N_{\mathrm{in}}}\frac{\left(\frac{ P_{\mathrm{in}}^{2}}{2g}\right)_{i}}{N_{\mathrm{is}}}\right]_{\mathrm{in}}+Z_{\mathrm{out}}-Z_{ \mathrm{in}} \tag{15}\]  
\[\eta=\frac{30\rho gQ_{i}H}{\pi nT_{p}} \tag{16}\]  
where \(P_{\mathrm{in}}\) and \(P_{\mathrm{out}}\) are the static pressure values of each node of the inlet and outlet sections, \(Z_{\mathrm{in}}\) and \(Z_{\mathrm{out}}\) are the geometric center potential energies of the inlet and outlet sections, \(V_{\mathrm{in}}\) and \(V_{\mathrm{out}}\) are the absolute speed of each grid node of the inlet and outlet sections, \(N_{\mathrm{in}}\) and  
Figure 11: Analysis of the grid independence.  
\(N_{\text{out}}\) are the total number of grid nodes of the inlet and outlet sections, \(Q_{i}\) is the flow rate of the pump unit, \(n\) is the speed of the impeller and \(T_{p}\) is the torque of the impeller.  
At a 2200-r/min speed, the comparison between the model test and numerical calculation results of the vertical axial flow pump device with the siphon outlet conduit is shown in Figure 12. The predicted performance curve of the vertical axial flow pump unit with the siphon outlet conduit is basically in accordance with the change trend of the test curve, and the curve is in good agreement. Near the optimum operating point, the \(Q_{i}\)-\(H\) curve calculated by the numerical method basically coincides with the \(Q_{i}\)-\(H\) curve of the physical model test. Under the calculation conditions, the whole \(Q_{i}\)-\(\eta\) curve calculated by the numerical method was higher than that of the physical model test. At the optimum operating point, the efficiency error of the numerical calculation and model test was 2.0%, and the error of each point was within 3%. The predicted performance curve of the numerical calculation agreed well with the test results, which indicates the accuracy and reliability of the calculation results. The numerical calculations can reflect the internal flow characteristics of the pump unit more accurately.