## 4 Results and Discussion
### Hydraulic Radial Force on Each Blade
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Direction**} & \multirow{2}{*}{**Blade No.**} & \(a\) & \(\omega\) & \(\Phi\) & \(b\) & \(R^{2}\) \\  & & **[N]** & **[rad/s]** & **[rad]** & **[N]** & **[N]** & **[N]** \\ \hline \multirow{4}{*}{X} & 1 & 28.722 & 303.687 & 1.172 & \(-\)0.185 & 0.986 \\  & 2 & 28.722 & 303.687 & 0.125 & \(-\)0.185 & 0.986 \\  & 3 & 28.722 & 303.687 & \(-\)0.922 & \(-\)0.185 & 0.986 \\  & 4 & 28.722 & 303.687 & \(-\)1.969 & \(-\)0.185 & 0.986 \\  & 5 & 28.722 & 303.687 & \(-\)3.017 & \(-\)0.185 & 0.986 \\  & 6 & 28.722 & 303.687 & \(-\)4.064 & \(-\)0.185 & 0.986 \\ \hline \multirow{4}{*}{Y} & 1 & 27.187 & 303.687 & \(-\)0.381 & 0.769 & 0.987 \\  & 2 & 27.187 & 303.687 & \(-\)1.428 & 0.769 & 0.987 \\ \cline{1-1}  & 3 & 27.187 & 303.687 & \(-\)2.475 & 0.769 & 0.987 \\ \cline{1-1}  & 4 & 27.187 & 303.687 & \(-\)3.523 & 0.769 & 0.987 \\ \cline{1-1}  & 5 & 27.187 & 303.687 & \(-\)4.570 & 0.769 & 0.987 \\ \cline{1-1}  & 6 & 27.187 & 303.687 & \(-\)5.617 & 0.769 & 0.987 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fitted parameters of hydraulic radial force components on each blade under 0.8\(Q_{d}\). Figure 21: Residual curves of radial hydraulic force components on each blade under 0.8\(Q_{d}\): (**a**) X direction; (**b**) Y direction. \(a\times\sin\left(\omega\times t+\Phi-\pi\right)\) of that from Blade 4. Similarly, the main parts of the fitting functions of Blade 2 and Blade 5, as well as those from Blade 3 and Blade 6, can also cancel each other out. That is, with the impeller center as the symmetric point, the main parts of the fitting functions of two blades that are centrally symmetric to each other can completely cancel each other out. Therefore, the force curve of each blade in the impeller can be regarded as the superposition of the fitting curve and the residual curve. The main parts of the fitting curves can cancel each other out, leaving only the same constant term \(b\). Finally, the force curve from the superposition of all blades is actually the sum of the residual curves of each blade and 6 times the constant term \(b\). With the above findings, the time varying characteristics of the hydraulic radial force in Figure 11 can be well explained. Taking the hydraulic radial force of the impeller in the X-direction under the \(0.8Q_{d}\) flow condition as an example, as shown in Figure 11a, its time varying curve has 6 fluctuation waves within a single cycle, with the peak and trough values approximately \(-3.5\) N and \(2.6\) N, respectively. Thus, its peak-to-peak value is about \(6.1\) N. That force curve can be approximately regarded as the superposition of the force components of the 6 blades of the impeller. The original curves and fitting curves of these 6 blade force components are shown by the solid and dashed lines in Figure 20a, respectively, and the fitting parameter results are listed in Table 4. Despite the peak-to-peak value of the fitting function curve, i.e., \(2a\), is as high as \(57.444\) N, over 9 times that of the impeller's hydraulic radial force, the main parts of the fitting functions for two centrally symmetric blades cancel each other out. Therefore, the impeller's hydraulic radial force curve in the X direction is primarily determined by the fitting residual curves of each blade shown in Figure 21a and the constant term's fitting value. It can be found in Figure 21a that the fitting residual curves of each blade show a prominent fluctuation within one impeller rotation cycle, and in the direction of impeller rotation, the fluctuation interval between adjacent blades is exactly equal to \(T/6\). By superimposing the fitting residual curves of each blade in Figure 21a and then adding 6 times the fitting value of the constant term \(b\), the original hydraulic radial force curve of the impeller in the X direction under the \(0.8Q_{d}\) in Figure 11a can be basically restored.