*3.4. Pressure Fluctuations in Time–Frequency Domain at the Runner Inlets*

A time–frequency analysis of the transient pressure pulsations at the monitoring points was performed by using the Short Time Fourier Transform (STFT) method [30–32]. From Figures 9–12, at the beginning of the runaway process, the characteristics of pressure pulsations are mainly influenced by the runner. The dominant frequency in the spectrogram is the blade passing frequency (BPF) (7*f* <sup>0</sup> for PT-1; 9*f* <sup>0</sup> for PT-2, PT-3, and PT-4, where *f* <sup>0</sup> is the rotating frequency of the runner rotation), and the rest high frequencies are the integer multiples of the BPF.

**Figure 9.** Frequency spectrums for pressures at the monitoring points of PT-1: (**a**) at hub side, (**b**) at mid span, (**c**) at shroud side.

**Figure 10.** Frequency spectrums for pressures at the monitoring points of PT-2: (**a**) at hub side, (**b**) at mid span, (**c**) at shroud side.

**Figure 11.** Frequency spectrums for pressures at the monitoring points of PT-3: (**a**) at hub side, (**b**) at mid span, (**c**) at shroud side.

**Figure 12.** Frequency spectrums for pressures at the monitoring points of PT-4: (**a**) at hub side, (**b**) at mid span, (**c**) at shroud side.

In the runaway process, each outstanding frequency varies with the change of rotational speed. As a whole, the amplitude of each frequency increases obviously once the working point enters the S-shaped region, which is due to the enhancement of impact at the runner inlet and rotor-stator interaction. In addition, the high-amplitude low-frequency signals occur obviously, and their occurrence time is consistent with the reduction of inlet radial velocity. Once the backflows generate, the amplitude increases rapidly and reaches at the maximum near the runaway point. Previous studies shown that the high-amplitude low-frequency signals are mainly caused by rotating stalls [1].

In contrast, in PT-1 and PT-2, the durations of the maximum amplitude are mainly after the runaway point, while those in PT-3 and PT-4 are before the runaway point, indicating that the evolutions of unstable flow patterns are affected quite differently by the S-shaped characteristics. Because the working points of PT-3 and PT-4 have gone through the RP mode, the amplitudes suddenly decrease obviously at *t* = 10 s (PT-3) and *t* = 5 s (PT-4), and increase at *t* = 16 s (PT-3) and *t* = 10 s (PT-4), respectively. All of these phenomena are caused by the backflow transitions, consistent with the changes of pressure fluctuations in the time domain spectrum in Figure 8.

For each runner, the amplitudes of pressure pulsations in different locations at the runner inlet are also different. In PT-1 and PT-2, the differences of pressure pulsation characteristics at the three monitoring points Figure 8 are quite large, while those in PT-3 and PT-4 are smaller. Taking PT-1 as an example, with the runaway beginning, the radial velocity at the inlet decreases obviously, and the low-frequency signals gradually generate at each monitoring point. Once the backflows occur on the shroud side, the amplitudes increase rapidly. Compared with pressure fluctuations at the three locations, the duration of the low-frequency signals is the longest on the shroud side, and they exist in the whole S-shaped region, because the backflows keep staying at this location all of the time. However, the highest amplitudes of low-frequency signals are at the mid span, while the lowest ones are on the hub side, and there are only low-frequency signals at the runaway point. In PT-2, the same phenomenon as in PT-1 is that the location with the highest amplitudes is also at the mid span, though the backflows occur on the hub side. In PT-3 and PT-4, there are no significant differences in the frequency of pulsations in different locations.

From the analysis mentioned above, we know that the high-amplitude low-frequency signals will generate at the location where backflows occur, which is the most obvious in PT-1 because its inlet height is the largest. These phenomena also have the same laws in PT-3 and PT-4, but the difference is not obvious because their inlet heights are smaller. However, the pressure characteristics in PT-2 is an exception Figure 10, which will be discussed in the later chapter.
