4.1. 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.
Figure 8,
Figure 9,
Figure 10 and
Figure 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.
Figure 9 and
Figure 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. As the monitoring points became far away from the guide vane, the amplitude of 16 f
n attenuated rapidly, and its main frequency gradually changed from 16 f
n to f
n from the blade outlet to the inlet. By comparing
Figure 9 with
Figure 10, it can be seen that the frequency domain distribution of pressure fluctuation near both sides of the long blade was similar, but the amplitudes were different. Compared with RN01, the amplitude of the guide vane passage frequency 16 f
n at RN16 decreased by about 42.6%, 44.2%, and 51.9%, respectively from 0.75 Q
d to 1.25 Q
d, while the amplitude of f
n only decreased by −1.5%, 5.3%, and 9.7%, respectively. Therefore, the low-axis frequency doubling near the suction side of the long blade was more prominent. Above, the flow condition inside the runner can be improved by increasing the flow rate. The amplitude of nonlinear frequency was greatly reduced, and the main frequency was the axial frequency doubling caused by rotor–stator interaction.
Figure 11 and
Figure 12 show the frequency domain of pressure fluctuation near the pressure surface and suction surface of the short blade under different flow rates at the monitoring points RN11~RN15 and RN06~RN10, respectively. Compared with the figures above, it can be seen that the frequency domain distribution of pressure fluctuation of the long and short blades was similar, but the fluctuation amplitude of the pressure side was reduced, while that of the suction side increased at the monitoring points of the short blade. In addition, under the influence of potential flow interference at the outlet of the adjacent long blades, the amplitude of the guide vanes passing through frequency 16 f
n near the pressure side of the short blade attenuated from the outlet of the blades, but there was an obvious surge at RN13, which was at the middle position of the pressure surface. Compared with RN12, the amplitude of 16 f
n at RN13 increased by about 89.8%, 63.6%, and 61.5%, respectively from 0.75 Q
d to 1.25 Q
d, which indicates that the influence of the outlet potential flow of the long blade on the pressure surface of the short blade was more significant under a small flow rate.
4.2. 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 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.
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
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
d. It can be seen that the variation period of 5 f
n was about 90°, which means that if the circumference area is divided into four quadrants, the distribution of this component in each quadrant is roughly the same. The variation period of 10 f
n and 20 f
n varied with a period of 22.5°, which was the same as the angle between each blade. It shows that the blade frequency and frequency doubling amplitude values were mainly affected by the different positions of the monitoring points relative to the adjacent guide vanes. In addition, the amplitude variation of the main frequency 10 f
n at the monitoring points GR037~GR039 was different from that at other points.
Figure 15 shows the velocity distribution around GR001~GR003 and GR037~GR039 at 1.0 Q
d. It can be seen in
Figure 16 that the high-speed flow field at the outlet of the blade impinged the guide vane, resulting in delamination on the trailing edge of the guide vane. The velocity direction was toward monitoring point GR002, so this monitoring point was greatly affected by the blade frequency. At monitoring points GR001 and GR003, at which the occurrence and attenuation of stay guide vane rotor–stator interaction occurred, respectively, the blade frequency amplitude was small. The variation trend of blade frequency amplitude at monitoring points GR037~GR039 was the same as that at monitoring points GR001~GR003. However, the upstream channel was near the tongue, so the insufficient overcurrent capacity led to a small velocity in the upstream, the runner’s outflow was squeezed, and the velocity of the channel at the monitoring point increased. At monitoring point GR038, the velocity along the guide vane passage direction was relatively high while the upstream velocity was relatively low. The rotor–stator interaction effect generated after hitting the trailing edge of the guide vane was relatively small, resulting in almost no increase in the blade frequency amplitude at GR037 and GR038. Therefore, the blade frequency amplitude of GR038 and GR039 was obviously lower than that of the other channels at the same position.
As shown in
Figure 17,
Figure 18 and
Figure 19, the pressure fluctuation distributions at different vane heights under different flow rates were significantly different. The section near the roof was span0.1, span0.5 was the middle section, and span0.9 was the section near the bottom ring. At a low flow rate at 0.75 Q
d, the pressure fluctuation intensity distribution at different guide vane heights had the most significant difference. Pressure fluctuation in span0.1 was small, and the strong fluctuation region was mainly concentrated below the trailing edge of the guide vane. The pressure fluctuation under the trailing edge of the guide vane at the middle height was significantly intensified, which affected the pressure fluctuation distribution in the bladeless region. The pressure fluctuation under the trailing edge of the active guide vane in span0.9 weakened, but the pressure fluctuation intensity in the bladeless region near the outlet of the runner was greater. In addition, due to the strong flow separation near the trailing edge of the guide vane at a low flow rate at 0.75 Q
d, the pressure fluctuation in the channel above the trailing edge of the guide vane was also obvious, especially near the bottom ring. With the increase of flow, the overcurrent capacity in the cascade channel increases, and the pressure fluctuation greatly reduces. The strong fluctuation region was mainly concentrated in the bladeless region near the runner’s outlet, and the pressure fluctuation intensity near the bottom ring was relatively large. This indicates that the pressure fluctuation intensity at 0.75 Q
d is relatively large, and the flow separation at the trailing edge of the guide vane has a great influence on the pressure fluctuation in the bladeless region and the cascade channel. The pressure fluctuation intensity at 1.0 Q
d and 1.25 Q
d was relatively small, and the pressure fluctuation distributions in the bladeless region were mainly affected by the rotor–stator interaction at the outlet of the runner.
It can be seen from the above figures that the pressure fluctuation in the bladeless region at different vane heights has significant differences, and the high-amplitude fluctuation frequency caused by rotor–stator interaction is mainly 10 f
n and 20 f
n.
Figure 20 shows the circumferential distribution of 10 f
n and 20 f
n amplitudes at different vane heights and under different flow rates. As can be seen from the figure, the amplitude difference of 10 f
n at different vane heights was large, while the amplitude difference of 20 f
n was relatively insignificant. For 10 f
n, at a small flow rate at 0.75 Q
d, the amplitudes near the middle height and the bottom ring were relatively large, while the amplitude near the roof was significantly reduced, indicating that the rotor–stator interaction near the bottom ring was relatively strong. With the increase of the flow rate, the amplitude of the middle height gradually decreased, close to that found near the roof, and the amplitude near the bottom ring was relatively large, especially at 1.25 Q
d. Except for the monitoring points near the trailing edge of the guide vanes, the amplitudes of 10 f
n at the other monitoring points were higher than 20 f
n, indicating that the node-diameter number k
1 = −6 dominated the rotor–stator interaction in the bladeless region.
4.3. Pressure Fluctuation Analysis of Volute
Figure 21 shows the pressure fluctuation distributions at the mid-span of the volute in five rotation cycles under different flow rates. It can be seen that the flow rate had a significant influence on the intensity and distribution characteristics of pressure fluctuation in the volute. At a low flow rate at 0.75 Q
d, the pressure fluctuation at the tongue was the most violent, and the fluctuation intensity of the pressure surface decreased gradually with the increase of the section area, forming the pressure fluctuation gradient, which reached the minimum value at the farthest distance from the tongue and then maintained stability at the outlet direction. The pressure fluctuation intensity at a rated flow rate at 1.0 Q
d was small and uniformly distributed, and there was no gradient change. As the flow rate increased, compared with other positions, the pressure fluctuation of the flow passage with a small section near the tongue increased significantly, but it was still about 10
−1 times that of the same position with a small flow rate. This indicates that deviated conditions, especially low flow conditions, will lead to a significant increase in pressure fluctuation of the flow passage at a small cross-section near the tongue, which may seriously induce strong vibrations.
In the pump working condition, as a downstream component, the unsteady pressure distribution of the volute is susceptible to the influence of rotor–stator interaction in the upstream, especially in the bladeless region.
Figure 22 shows the distribution of 5 f
n and 10 f
n amplitudes for each monitoring point in the volute. As can be seen from the figure, the pressure fluctuation amplitude at the mid-span of the volute at a low flow rate at 0.75 Q
d was strong at 7.8 f
n, and the spectral feature was nonlinear, which is consistent with the fluctuation characteristic in the bladeless region. Monitoring point SC01, which was near the tongue, had the maximum amplitude, while monitoring point SC08 far from the tongue had the minimum amplitude. With the increase of flow, the pressure fluctuation at the mid-span of the volute was mainly concentrated in the rotor–stator interaction frequencies of 5 f
n, 10 f
n, 15 f
n, and 20 f
n, especially in the blade frequency of 10 f
n and the single-type blade passing frequency of 5 f
n. In order to further analyze the distribution rules of these two frequencies at the mid-span of the volute,
Figure 23 shows the distribution of 5 f
n and 10 f
n amplitudes in the volute at rated flow rates at 1.0 Q
d and 1.25 Q
d. It can be seen that, for the blade frequency of 10 f
n, the amplitude at 1.25 Q
d was higher than that at 1.0 Q
d, and the distribution of the two in the volute were similar. From SC01 to SC05, 10f
n had a high attenuation rate, reached a minimum value at SC08–SC09 (far from the tongue), and was stable near the outlet of the volute. For a single-blade passing frequency of 5 f
n, the amplitude of the blade from the small section to the large section gradually decreased at the rated flow rate at 1.0 Q
d. While at the large flow rate at 1.25 Q
d, the amplitude of SC01–SC06 at the small section near the tongue was less than the rated flow rate, but the amplitude of SC07–SC12 at the large section near the exit was significantly increased and higher than the rated flow rate. This indicates that the change of flow rate changes the propagation characteristic of 5 f
n, thus changing the amplitude distribution rule corresponding to 5 f
n. In general, the main frequency of pressure fluctuation in the small section of the volute flow passage was 10 f
n, while in the big section it was 5 f
n, indicating that the flow difference at the outlet of the splitter blades had a greater influence on the pressure fluctuation distribution than the blade passing interaction did.
4.4. Pressure Fluctuation Analysis of Draft Tube
As an inflow component, the rotor–stator interaction and flow instability of the draft tube were mainly near the outlet of the draft tube.
Figure 24 shows the distributions of the standard deviation coefficient of pressure fluctuation at the outlet of the draft tube under different flow rates within five rotation cycles. It can be seen that pressure fluctuation distributions under different flow rates at the outlet were similar. The pressure fluctuation intensity near the center was small and increased continuously along the radial direction. The pressure fluctuation intensity at the tube wall was about 10
2 times that at the center. In addition, the fluctuation intensity was the highest at 0.75 Q
d and the lowest at 1.0 Q
d, indicating that deviated working conditions, especially at a small flow rate, will aggravate the pressure fluctuation at the outlet of the draft tube, which damages the stable operation of the unit.
In order to further analyze the time and frequency domain characteristics of the pressure fluctuation of the draft tube under different flow rates,
Figure 25 shows the time and frequency domain of pressure fluctuation at the outlet of the draft tube at monitoring points DT01, DT02, and DT03 within a rotational cycle. The results show that the pressure fluctuation distributions of DT01 and DT03 at the tube wall were similar. There are five peaks and valleys in a rotation period, which are affected by the rotor–stator interaction between the long blade and the outlet of the draft tube. In the time domain figures, the main frequency is 5 f
n, and the secondary main frequency is 10 f
n. The fluctuation amplitude under a rated flow rate at 1.0 Q
d was relatively small, while the negative amplitude under a small flow rate at 0.75 Q
d was relatively large, resulting in large pressure fluctuations under this working condition. In addition, by comparing the time domain figure of each flow rate, it can be found that the change of flow rate also has an obvious influence on the phase of pressure fluctuation. For the central monitoring point DT02, the time domain distribution showed an irregular periodic fluctuation, and the fluctuation amplitude under 1.25 Q
d was large, but the fluctuation amplitude was about two orders of magnitude lower than that of the monitoring point at the tube wall. In the frequency domain figures, it is shown that the main frequencies are axis-doubling frequencies, the main frequency is 5 f
n, and the amplitude at 1.25 Q
d is relatively large, indicating that the influence of rotor–stator interaction between the runner and the draft tube on the middle position of the interface is relatively significant at a large flow rate.
Figure 26 shows the frequency domain of pressure fluctuation of DT04–DT12 under different flow rates. The results show that the main frequencies of pressure fluctuation at these positions were axis-doubling frequencies, in which 10 f
n, 16 f
n, and 20 f
n were caused by the upstream transmission of the rotor–stator interaction between the bladeless region and the runner. Overall, with the increase of the flow rate, the pressure fluctuation amplitude increased gradually from the inlet of the draft tube to the straight taper tube, and the amplitude at monitoring points DT04–DT09, which were near the elbow position and outer wall of the draft tube, was larger. Moreover, the amplitude of the outer wall of the tube was slightly larger than that of the inner wall, which was mainly due to the rotor–stator interaction between the runner and the draft tube, as well as the strong impact of the water flowing through the elbow tube on the outer wall.