## 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.**} & \(\mathbf{a}\) & \(\mathbf{\omega}\) & \(\mathbf{\Phi}\) & \(\mathbf{b}\) & \multirow{2}{*}{\(\mathbf{R^{2}}\)} \\  & & **[N]** & **[rad/s]** & **[rad]** & **[N]** & \\ \hline \multirow{6}{*}{X} & 1 & 28.799 & 301.705 & 1.215 & \(-\)0.009 & 0.987 \\  & 2 & 28.956 & 305.445 & 0.090 & \(-\)0.196 & 0.987 \\  & 3 & 28.577 & 304.349 & \(-\)0.934 & \(-\)0.132 & 0.986 \\  & 4 & 28.754 & 304.127 & \(-\)1.976 & \(-\)0.133 & 0.986 \\  & 5 & 28.952 & 303.642 & \(-\)3.017 & \(-\)0.167 & 0.986 \\  & 6 & 28.594 & 301.777 & \(-\)4.024 & \(-\)0.055 & 0.987 \\ \hline \multirow{6}{*}{Y} & 1 & 27.373 & 307.990 & \(-\)0.466 & 0.896 & 0.990 \\  & 2 & 27.311 & 303.867 & \(-\)1.432 & 0.782 & 0.987 \\ \cline{1-1}  & 3 & 27.022 & 302.914 & \(-\)2.458 & 0.688 & 0.987 \\ \cline{1-1}  & 4 & 27.168 & 301.636 & \(-\)3.481 & 0.830 & 0.988 \\ \cline{1-1}  & 5 & 27.420 & 303.051 & \(-\)4.560 & 0.838 & 0.987 \\ \cline{1-1}  & 6 & 27.016 & 303.994 & \(-\)5.621 & 0.702 & 0.987 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Preliminary fitted parameters of hydraulic radial force components on each blade under 0.80\({}_{d}\). Figure 18: Preliminary fitted curves of hydraulic radial force components on each blade under 0.80\({}_{d}\): (**a**) X direction; (**b**) Y direction. Table 4 shows the improved fitting method can also ensure that the goodness of fit value \(R^{2}\) of each curve is not less than 0.986. By controlling the values of the same amplitude \(a\), constant term \(b\), and the successively decreasing initial phase \(\Phi\) for the force curves of each blade corresponding to the same hydraulic radial force component under the same flow condition, it better helps researchers interpret the physical laws of the hydraulic radial force data. The parameter fitting values for the flow conditions from 0.9\(Q_{d}\) to 1.2\(Q_{d}\) are listed in Tables A1-A4 in Appendix A. According to the fitting results above, it is evident that, within the research scope, regardless of the flow condition, the force curves of each blade for the same hydraulic radial force component can be grouped into a cluster of sine curves with identical amplitude, constant term, and angular frequency, achieving a goodness of fit of at least 0.96. The angular frequency in the fitting function is equal to the angular frequency of the impeller rotation. Moreover, there is a 60\({}^{\circ}\) phase difference in the initial phase of the sine functions in the same curve cluster in the order of blade rotation. According to the induction formula of trigonometric functions, for any trigonometric function sin (\(\alpha\)), sin (\(\alpha+2\pi\))= sin (\(\alpha\)) and sin (\(\alpha+\pi\)) = \(-\) sin (\(\alpha\)). Since there is a 60\({}^{\circ}\) initial phase difference between each blade for the same hydraulic radial force component under the same flow condition, the main part \(a\times\) sin (\(\omega\times t+\Phi\)) (after removing the constant term \(b\)) of the trigonometric force curve from Blade 1 can exactly cancel out the main part