*3.3. Pressure Fluctuations in the Time Domain at the Runner Inlets*

The dimensionless pressure fluctuations at each monitoring point in the vaneless space are analyzed by comparing with the pressures at the initial time. The normalized pressure was calculated by equation:

$$\mathcal{C}\_{\text{P}} = \frac{p - p\_{\text{initial}}}{0.5 \rho u\_1^2} \tag{2}$$

where p is the instantaneous pressure signals, *p*initial is mean initial pressure values at the initial time, ρ is the water density, and *u*<sup>1</sup> is the tip velocity of the runner blade leading edge. In addition, *C*<sup>p</sup> (O) and *C*<sup>p</sup> (L) in Figure 8 are the original and low-pass filtered data, respectively, and the upper frequency limit of the low-pass filtered data is 2 Hz.

**Figure 8.** Variations of normalized pressure *C*p at the three monitoring points of runner inlet: (**a**) PT-1, (**b**) PT-2, (**c**) PT-3, (**d**) PT-4.

Previous research has shown that after runaway, backflows will enhance the rotor–stator interactions and greatly increase the amplitudes of pressure pulsations [1]. Figure 8 shows the pressure pulsations in the time domain at the runner inlets of the four pump-turbines. On the whole, under the same total rotational inertia of runner and generator, the amplitudes in PT-1 and PT-3 are relatively large, while those in PT-2 and PT-4 are relatively small. It is found that the longer fluctuation periods of PT-1 and PT-3 mean the longer residence time in the S-shaped region and larger pressure pulsations. In addition, with the variations of rotational speed and discharge, the pressure pulsations present regular changes, with the amplitudes reach the maximum near the runaway point. Due to the different working conditions, there are obvious different characteristics of pressure pulsations.

1. PT-1 and PT-2: The working points only go through the T and TB modes, and the filtered data only slightly vary with the changes of rotational speed and discharge, while the amplitudes of high-frequency signals have no obvious change.

2. PT-3 and PT-4: The trends of pressure pulsations in these two pump-turbines are basically the same, and before the RP mode, they are all similar to those in PT-1 and PT-2 because the low-pass filtered pressure has a shut down when the backflow occurs. However, with the conversion from the TB mode to the RP mode, the low-frequency signals have a significant increase. And when the reverse discharge increases to the maximum value, the low-frequency signals also reach at the maximum. This is because the rotating energy of the runner and rotor is converted to the water head of the pump-turbine. In addition, the pressure variations in PT-3 and PT-4 in the RP mode are still quite different. In particular, the low-frequency signals decrease slowly in PT-3, while those in PT-4 decrease rapidly. The reason is that the backflows at the runner inlet are quite different during this period. In PT-3, backflows are mainly at the mid span, contributing to poor flow capacity to get water out of the blade passages, forming flow blockage at the inlet and increasing pressure [1]. However, in PT-4, backflows occur at the mid span and on the shroud side at the same time, with strong flow capacity and rapid pressure reduction. For the high-frequency signals, the amplitudes of those in the T mode gradually increase, while those in the TB and RP modes decrease.

Compared with the velocity pulsations in Figure 3, it is found that when the working points of PT-3 and PT-4 enter the RP mode, the velocity pulsation always keeps high amplitude characteristics, while the amplitude of pressure pulsations decreases rapidly, which means the unsteady development of the flow patterns cannot accurately reflect the true values of the pressure pulsations. Figures 4–7 not only show the flow pattern development at the runner inlets, but also show the magnitude of turbulent kinetic energy. It can be seen that the turbulent kinetic energy at the runner inlet is relatively low after entering the RP mode, indicating that pressure pulsations will decrease rapidly when the turbulent kinetic energy becomes small.
