## 4 Results and Discussion  
### Hydraulic Force on the Whole Impeller  
Firstly, following the common practice reported in existing literature, the impeller is regarded as a whole. The hydraulic radial force and hydraulic axial force acting on the impeller within two impeller rotation cycles under different flow conditions are plotted, as shown in Figure 8a,b, respectively. For all the plots of time-series data in this paper, including Figure 8, the time is uniformly represented by the multiple of the impeller rotation cycle. One impeller rotation cycle, that is, 1.0\(T\), corresponds to approximately 0.0207 s.  
As shown in Figure 8, as the flow rate increases, both the hydraulic radial force and hydraulic axial force acting on the impeller generally show an increasing trend, which is consistent with the findings of [PERSON] et al. [39] and [PERSON] et al. [40]. Meanwhile, the hydraulic axial force on the impeller under each flow condition remains basically stable, with a very limited fluctuation range, and its magnitude is also much lower than that of the hydraulic radial force. A possible explanation for the trend of the hydraulic axial force is that the liquid in the impeller changes its flow direction from the axial direction to the radial direction, and the momentum of the liquid also changes accordingly. An increase in the flow rate means that the mass of the liquid passing through the impeller per unit time increases, so the rate of change of the liquid momentum also increases. Correspondingly, the reaction force exerted by the liquid on the impeller, that is, the axial force, also increases. However, compared with the hydraulic axial force, the variation behaviors of the hydraulic radial force are more complex. The statistics of the average value and peak-to-peak value of the time-varying curve of the hydraulic radial force under each flow condition are determined; the results are plotted in Figure 9. It can be observed that near the rated flow rate, both the average value and peak-to-peak value of the hydraulic radial force are relatively small, and the minimum values of both occur under the flow condition of 0.9\(\Omega_{d}\). The further away from the rated flow rate, the larger the average value and peak-to-peak value will be, and the level of the hydraulic radial force under the high flow condition is higher than that under the low flow condition. Furthermore, it can also be seen from Figure 8a that under relatively low flow conditions (0.8\(\Omega_{d}\)-1.0 \(Q_{d}\)), the curves of the hydraulic radial force show a pattern of obvious waveforms superimposed on small waveforms, while when the flow rate is relatively high (1.1\(Q_{d}\) and 1.2\(Q_{d}\)), there are basically only large waveforms. To obtain the dominant frequency of the fluctuations, a FFT analysis is performed for the hydraulic radial force curves in Figure 8a; the results are shown in Figure 10. It is evident that under all flow conditions, the blade passing frequency (BPF) is the dominant frequency of the fluctuations, superimposed with the shaft frequency and other multiples of the BPF, which is consistent with other research reports [41; 42]. Moreover, the hydraulic radial force on the impeller under each flow condition is decomposed into two components of the X-axis and Y-axis directions, as shown in Figure 11a,b, respectively. The time varying curves of these two components exhibit similar fluctuation characteristics: the fluctuation is minimal at the rated flow rate, and it increases as the flow rate deviates from the rated value. Additionally, the waveforms under larger flow conditions are smoother than those under smaller flow conditions. Figure 8: Time-varying curves of hydraulic force on the impeller: (**a**) Radial force; (**b**) Axial force. Figure 11: Time-varying curves of hydraulic radial force components on the impeller: (**a**) X direction; (**b**) Y direction. Figure 10: FFT analysis curves of hydraulic radial force on the impeller. Figure 9: Mean and peak to peak values of the hydraulic radial forces.