## 4 Results and Discussions
### Pressure Fluctuation Analysis of Bladeless Region  
The bladeless region between the runner and the stay guide vane is the area with the strongest rotor-stator interaction. In order to obtain the unsteady fluctuation characteristic of its flow field, 48 monitoring points were uniformly arranged in the circumferential direction. Figure 13 shows the time and frequency domain of pressure fluctuation of monitoring point GR001 under different flow rates. The time domain diagram on the left of the figure shows the variation of pressure fluctuation in a rotation period. The time domain distribution of pressure fluctuation under a rated flow rate and large flow rate had obvious periodicity, while under the small flow rate it was irregular due to flow instability. Figure 14 shows that there were many nonlinear frequencies under the small flow rate at  
Figure 11: Frequency domain of pressure fluctuation near the pressure side of the short blade. Figure 12: Frequency domain of pressure fluctuation near the suction side of the short blade. 0.75 Q\({}_{d}\), and the frequencies with high amplitudes included 5 f\({}_{n}\), 10 f\({}_{n}\), and 20f\({}_{n}\), caused by rotor-stator interaction, as well as the main frequency of 7.8 f\({}_{n}\). The frequencies under a rated flow rate at 1.0 Q\({}_{d}\) and large flow rate at 1.25 Q\({}_{d}\) were mainly the axial frequency multiplier caused by rotor-stator interaction. The first, second, and third main frequencies at 1.0 Q\({}_{d}\) were 10 f\({}_{n}\), 20 f\({}_{n}\), and 5 f\({}_{n}\), and the first, second, and third main frequencies at 1.25 Q\({}_{d}\) were 20 f\({}_{n}\), 10 f\({}_{n}\), and 5f\({}_{n}\), respectively. At the monitoring point, when the flow rate was at 1.0 Q\({}_{d}\), the rotor-stator interaction was strong when the node-diameter number k\({}_{1}\) = \(-\)6. When the flow rate was at 1.25 Q\({}_{d}\), the rotor-stator interaction was strong when the node-diameter number k\({}_{1}\) = 4. This indicates that the flow rate will affect the energy distribution of the rotor-stator interaction with different node-diameter numbers. Figure 13: Time and frequency domain of pressure fluctuation of GR001. In order to reflect the influence of the relative positions of the blades and stay guide vanes on pressure distribution in the bladeless region, Figure 14 shows the pressure distribution of GR001 at different times at 1.0 Q\({}_{\rm d}\), and a time domain diagram is also marked out, with the range from peak to trough of the pressure fluctuation. At t = 0.01758 s, GR001 was close to the pressure surface of the short blade, and the pressure in the passage was relatively large, so the pressure fluctuation was at the peak at this moment. At t = 0.01909 s, GR001 was close to the small pressure area near the head of the short blade, so the pressure fluctuation was negative at this moment. At t = 0.01970 s, a short blade skimmed over GR001. The pressure rose, so the pressure fluctuation became positive at this moment. At t = 0.02152 s, GR001 was far away from the short blade, and the influence of rotor-stator interference was weakened. Under the influence of the low pressure on the suction surface of the short blade, the pressure dropped and the pressure fluctuation became negative. At t = 0.02303 s, GR001 was close to the pressure surface of the long blade, so the pressure fluctuation was at the secondary crest at this moment. At t = 0.02455 s, GR001 was close to the small pressure area near the head of the long blade, so the pressure fluctuation was at a valley value at this moment. This indicates that rotation of the long and short blades will cause different pressure distributions; short blades have a more significant effect on the rise of pressure than long blades, while long blades have a more significant effect on the fall of pressure. Figure 15 shows the circumferential distribution of rotor-stator interaction amplitudes at 1.0 Q\({}_{\rm d}\).