## 4 Results and Discussions  
### Pressure Fluctuation Analysis of Runner  
Figure 8 shows pressure fluctuation distributions at the mid-span of the runner under different flow rates within five rotation cycles. It can be seen from the figure that the pressure fluctuation of the runner from the inlet to the outlet gradually increased. The fluctuation near the trailing edge pressure surface of the splitter blades was the most intense, due to the influence of rotor-stator interaction. The pressure fluctuation intensity at a small flow rate at 0.75 Q\({}_{d}\) was large, and the pressure fluctuation on both sides of the trailing edge of the splitter blades was particularly significant. With the increase of flow, the pressure fluctuation inside the runner decreased gradually, and the strong fluctuation region was mainly concentrated in the pressure surface at the trailing edge of the splitter blades. This indicates that, at the same opening and under the condition of a small flow, the flow instability inside the runner leads to a greater intensity of internal pressure fluctuation and intensifies the rotor-stator interaction at the trailing edge of the blades. Moreover, the fluctuation near the pressure surface at the trailing edge of the splitter blades and the suction surface is particularly violent under a small flow. Additionally, the flow pattern inside the runner will be improved when the flow rate increases, and the rotor-stator interaction at mid-span in the runner and the stay guide vanes will also be improved to some extent, thus reducing the pressure fluctuation intensity inside the runner, especially at the suction surface of the trailing edge of the blades. Figures 8-11 show the frequency domain of pressure fluctuation near both sides of the long and short blades through FFT transform processing. The x coordinate is the frequency (f is the actual frequency in the figure and f\({}_{n}\) is the axial frequency), and the y coordinate is the names of the monitoring points near the blades. According to the pressure fluctuation coefficient C\({}_{p}\), the color reflects the frequency domain of pressure fluctuation and its amplitude variation trend. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Frequency**} & **Experimental Value** & **Calculated Value** & \multirow{2}{*}{**Relative Error (\%)**} \\  & \(dH/H\) (\%) & \(dH/H\) (\%) & \\ \hline
5 f\({}_{n}\) & 0.989 & 0.887 & \(-\)10.31 \\
10 f\({}_{n}\) & 2.068 & 2.322 & \(+\)12.28 \\
20 f\({}_{n}\) & 1.425 & 1.594 & \(+\)11.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison between calculation and test results of pressure fluctuation in the bladeless region. Figure 8: Pressure fluctuation distributions at the mid-span of the runner. Figures 9 and 10 show the frequency domain of pressure fluctuation near the pressure surface and suction surface of the long blade under different flow rates at monitoring points RN01-RN05 and RN16-RN20, respectively. Figure 8 shows that the frequency domain of pressure fluctuation near the pressure side of the long blade decreased gradually from the blade outlet to the inlet and changed with different flow rates. Under a small flow rate at 0.75 Q\({}_{d}\), there was also a nonlinear frequency with a high amplitude between 9 f\({}_{n}\) and 10 f\({}_{n}\), whose main frequency was about 9.2 f\({}_{n}\), in addition to the guide vane passage frequency of 16 f\({}_{n}\) caused by rotor-stator interaction. The main reason for this phenomenon is that there was a strong flow instability under the working conditions. Under the rated flow rate at 1.0 Q\({}_{d}\), the amplitude of pressure fluctuation decreased, especially for the fluctuation corresponding to the nonlinear frequency. The main frequency was the guide vane passage frequency of 16 f\({}_{n}\), and the low-axis frequency doubling was f\({}_{n}\)-4f\({}_{n}\).