## 4 Results and Discussion
### Hydraulic Force on the Whole Impeller
Figure 12 shows the variation of the angle of the hydraulic radial force on the impeller with time under different flow conditions. In this paper, the angle of the hydraulic radial force \(\theta\) is defined as follows:  
\[\theta=\arctan 2(F_{y},F_{x}) \tag{6}\]  
In the formula, \(F_{x}\) and \(F_{y}\) represent the X direction and Y direction components of the hydraulic radial force, respectively. The \"\(\arctan 2\)\" function is an extension of the arctangent function in mainstream programming languages such as Matlab. Upon receiving the abscissa and ordinate of a point in the Cartesian coordinate system, it determines the directed angle (with a value range of \((-\pi,\pi]\)) between the line connecting the point to the origin and the positive X axis. This calculation is based on the positive or negative values and magnitudes of \(F_{x}\) and \(F_{y}\), using the rules associated with the arctangent function. Thus, it can more accurately determine the direction of the angle. For ease of observation, the unit of radians in the figure is converted to degrees. Figure 12 indicates that, apart from the significant variations in the amplitude and fluctuation level of the hydraulic radial force, there are also substantial differences in its angle under different flow conditions. Overall, its angle varies less under the rated and high flow conditions, indicating more stable flow. In contrast, under low flow conditions such as \(0.8Q_{d}\) and \(0.9Q_{d}\), the angle fluctuates much more significantly. The above phenomena can be explained by the unsteady flow inside the centrifugal pump under different flow conditions. At the rated flow rate, the incoming flow at the impeller inlet matches well with the blade inlet angle. The flow adheres to the blade surface without significant flow separation or backflow, and the flow directions at the volute and impeller outlet are also well matched. As a result, both the time average values of hydraulic radial force and the pulsation amplitude are small. When the flow rate is low, the velocity at the impeller inlet decreases, leading to significant flow separation on the blade suction surface and the formation of a local low pressure area. This causes backflow and exacerbates flow instability, which is prominently manifested as sudden changes in the angle of the hydraulic radial force. This results in rotating stall phenomena in the blade passages, triggering low frequency and large amplitude pulsations. Then, the vortices generated in the separated flow region shed periodically, superimposing on the low frequency pulsations to form high frequency and small amplitude fluctuations, presenting a complex waveform. In contrast, when the flow rate is too high, the velocity at the impeller inlet is also excessively high. The angle of incidence at the blade leading edge  
Figure 12: Time-varying curves of angle of hydraulic radial force on the impeller. deviates from the design value, causing inlet shock, which leads to local high-pressure areas and high turbulent kinetic energy. This results in a higher circumferential pressure gradient on the impeller, thereby increasing the hydraulic radial force. Meanwhile, the alternating action of the high-speed jet at the blade outlet and the low-speed flow in the wake region forms periodic pressure fluctuations. Since these fluctuations are relatively consistent with the flow direction, a relatively stable waveform of the hydraulic radial force can be formed.