## 4 Numerical Results Analysis  
### Flow Field Results Analysis  
A comparison of pump performance between the simulation and experimental results is shown in Figure 12 under 6750 r/min. \(Q_{\rm r}\) is equivalent to the flow rate coefficient \(C_{\rm q}=0.0715\). Seen from the figure, the simulated and experimental values of both the head coefficient and efficiency are almost the same at \(Q_{\rm r}\). The maximum error between the test value and the simulated value of the head coefficient is below 5.5% for the full flow rate. From the simulated efficiency curve, the operation point with the highest efficiency is near 0.7\(Q_{\rm r}\). At the small flow rate condition, the test value of efficiency is close to the simulated ones. The maximum error is not higher than 4.3% for the full flow rate. Therefore, the selected calculation model used in the simulation can be considered to accurately predict the performance of the pump and could be used to translate pump internal flow information for further analysis.  
The absolute velocity distribution of the model pump flow field under different flow conditions are presented in Figure 13. As seen from the figure, the fluid velocity appears the highest at impeller exit and shows asymmetry on the circumference due to the rotor-stator interaction caused by the volute tongue. Under all presented flow conditions, swirls appear inside the impeller flow channel, especially at the suction side of blade leading edge and pressure side of the blade trailing edge. At the flow rate exceeding the highest efficiency working point (0.7\(Q_{\rm r}\)), the number of vortices gradually increased, and the size continued to expand, which severely affected the impeller operation and caused noise problems. In addition to 0.7\(Q_{\rm r}\), flow separation near the volute tongue also occurred, which also became a source of flow-induced noise. The special structure of the exit with a 90\({}^{\circ}\) corner also caused a vortex, which resulted in a significant hydraulic problem.  
Figure 11: Contrast chart of frequency spectrum under the full opening of the valve.  
Vorticity is often used to represent the curl characteristic of a flow field. The swirling motion in the fluid medium is the source of flow-induced noise based on the vortex sound theory [20]. The vorticity distributions of the model pump's flow field under different flow conditions are presented in Figure 14. It can be seen that high-energy vortices are mostly concentrated near the velocity vortex areas of the flow field, such as the rotor-stator interaction zones and flow separation zones inside the blade channel. The highest efficiency working conditions present the least distribution of high energy vorticity. The working condition appeared as the least distribution of high energy vorticity at \(0.7Q_{\text{r}}\). Due to insufficient flow, large areas of backflow and secondary flow exist in the impeller flow channel at a low flow rate (\(0.5Q_{\text{r}}\)), showing relatively high vorticity and asymmetric distribution. Under large flow conditions, the vortex diffuses into the impeller flow channel, which is related to the high-speed area at the leading edge of impeller blade. In addition, a small amount of high-energy vortex mass also appears at the exit of the solute. When the vorticity moves in the same direction of the mean flow inside the rotational impeller, the energy dissipation rate on the blade surface will be reduced. On the contrary, it is not conducive to the fluid energy conversion between the impeller and the solute, which explains the head and efficiency dropping at large flow rates. To be fair, the hydraulic design of this model pump is not excellent due to space constraints, resulting in lower hydraulic efficiency and vibration and noise problems.  
Figure 12: Hydraulic characteristics of numerical calculation and experiment.  
Figure 13: Velocity streamline line of model pump: (**a**) \(0.5\,Q_{\text{r}}\), (**b**) \(0.7\,Q_{\text{r}}\), (**c**) \(1.0\,Q_{\text{r}}\), (**d**) \(1.2Q_{\text{r}}\).