## 4 Results and Discussion
### Hydraulic Radial Force on Blade 1  
Based on the findings in Section 4.3, under a certain flow condition, the curves of the hydraulic radial forces acting on each blade have identical waveforms in terms of size and shape, but there are certain time differences in the fluctuation sequence. The time difference between the force waveforms of adjacent blades is \(T/6\). Therefore, if the time varying curve of the hydraulic radial force of any one blade is obtained, then curves of all other blades can be derived through time translation. Taking Blade 1 as the research object, the curves of the hydraulic radial force components in the X and Y directions under various flow conditions are plotted as solid lines in Figure 22a,b, respectively. The fitting function curves are also plotted as dashed lines in Figure 22, and the fitting residual curves are shown in Figure 23. The corresponding fitting parameters are listed in Table 5.  
As can be observed from Figure 22, generally, the time varying curves of the two force components in the X and Y directions under each flow condition are very close to sine curves. The time varying curves of the same hydraulic radial force component have high similarities in terms of waveform shape and phase, and the amplitude increases continuously as the flow rate increases. This finding can be quantitatively verified by the fitting parameter results in Table 5. Taking the curve of the hydraulic radial force component in the X direction as an example, the amplitude \(a\) obtained through sine function fitting has a fitted value of \(28.722\) N under the \(0.8Q_{d}\). As the flow rate gradually increases to \(1.2Q_{d}\)the fitted value also increases to 37.809 N. Meanwhile, the fitting results of the initial phase under each flow condition are relatively close, with a radian value of approximately 1.16. In addition, although the fitted value of the constant term \(b\) changes with the flow rate, its magnitude and fluctuation level are basically negligible compared with the amplitude \(a\) and the fitting residual value. Figure 23: Residual curves of radial hydraulic force components on blade 1 under different flow conditions: (**a**) X direction; (**b**) Y direction. Figure 22: Fitted curves of radial hydraulic force components on blade 1 under different flow conditions: (**a**) X direction; (**b**) Y direction. The above phenomena can be explained from the perspective of unsteady flow inside the centrifugal pump. When the time is near an integer multiple of the impeller rotation period \(T\), it corresponds to the moment when Blade 1 sweeps past the volute tongue. During the operation of pump, there is a strong interaction between the impeller and the volute tongue, which triggers a series of prominent unsteady flow phenomena, leading to significant changes in the hydraulic radial force on the impeller. According to the research of [PERSON] et al. ([PERSON] et al. (2018), when the impeller blade sweeps past the volute tongue, the jet-wake flow pattern at the impeller outlet interacts with the volute tongue. A high-speed jet region is formed on the pressure side of the blade, and a low-speed wake region appears on the suction side. This makes the pressure distribution in the flow field near the volute tongue extremely non-uniform, generating a significant radial force. Meanwhile, there are complex vortex structures inside the centrifugal pump. When the impeller blade approaches the volute tongue, a high vortex region is formed at the blade trailing edge. The interaction between the vortices in this region and the volute tongue changes the local flow field pressure distribution. The generation, development, and shedding of these vortices cause pressure fluctuations, resulting in significant fluctuations in the radial force on the impeller. In addition, when the impeller blade sweeps past the volute tongue, flow passage blockage may occur.