*3.2. Radial Velocity Variations and Backflow Transitions at the Runner Inlets*

The aforementioned fluctuations of dynamic trajectories are closely related to the unstable flow patterns near the runner inlets and outlets [29]. The variations of flow velocity at the runner inlet can reasonably demonstrate the characteristics of flow evolutions during the runaway processes. Figure 3 show the variations of normalized radial velocity *v*<sup>r</sup> at the three monitoring points (HS, MS, and SS shown in Figure 1f, namely hub side, mid span and shroud side, respectively) in the four runners. The normalized velocities were defined by:

$$w\_{\mathbf{r}} = \frac{60U\_{\mathbf{r}}}{\pi n\_1 D\_1} \tag{1}$$

where *U*<sup>r</sup> is the instantaneous radial velocity, *n*<sup>1</sup> is the initial rotational speed, and *D*<sup>1</sup> is the runner inlet diameter. Here, positive values of *v*r are defined as the direction of water flowing into the runner passages, while negative values of *v*<sup>r</sup> mean the backflows from the runner passages to the vaneless space. In addition, *v*r (O) and *v*r (L), in Figure 3, are the original and low-pass filtered data, respectively, and the upper frequency limit of low-pass filtered data is 2 Hz.

**Figure 3.** Variations of the normalized radial velocity *v*r at the three monitor points: (**a**) PT-1, (**b**) PT-2, (**c**) PT-3, (**d**) PT-4.

In general, during the beginning period of the runaway process, the rotational speed increases, the inflow attack angle decreases, and the velocity pulsations increase due to the growing impact at the runner inlet. When the backflows occur at the runner inlet (the reverse direction of *v*r), the velocity pulsations suddenly increase. Also, the velocity pulsations are almost the largest near this critical time. The lower the specific speed, the smaller the differences of velocity pulsations in different monitoring points. Consistent with the features in Figure 2, the velocity pulsations in PT-1 and PT-2 are the largest, and those in PT-4 is the smallest. In addition, though the discharge varies periodically, the variations of radial velocity in PT-1 and PT-2 are not obviously, especially at the location where the backflows occur, which are affected by the absence of flow transitions. But for PT-3 and PT-4, the variation period of radial velocity is corresponding to that of discharge. Overall, with the changes of flow rate, there are significant differences in flow features at the runner inlets.

1. PT-1: The dynamic trajectory of PT-1 only goes through the turbine (T) and turbine braking (TB) modes, and the macro parameters only fluctuate in relatively small amplitudes, therefore, the radial velocity (low-pass filtered data) cannot vary violently. At around *t* = 3.6 s (in the T mode), the radial velocity direction at the shroud side alters, indicating the appearance of backflows. At the same time, the velocity fluctuations increase significantly, namely the flow instability is intensified. However, the radial velocity directions on the hub side and mid span keep unchanged, and the increased

values (high-frequency data) indicate that the water flow can rush into the blade passages more easily. Although the rotational speed and flow rate fluctuate greatly, the radial velocity direction at the runner inlet remains unchanged after *t* = 3.6 s (Figure 3a).

2. PT-2: Though the working modes experienced are the same as those of PT-1, the developments of backflows show different characteristics because the backflows start from the hub side (*t* = 2.1 s) in the turbine mode and have transitions. At the early stage of backflow generations, the radial velocity at the mid span increases briefly and then decreases gradually, while that on the shroud side increases rapidly. At about *t* = 8–10 s (in the TB mode), there are significant transitions of radial velocity directions, namely the backflows occur suddenly at the mid span and shroud side at the same time, while those at the hub side disappear for a short time. After a short stay, backflows return to the hub side again. Similar to the phenomenon in PT-1, although the speed and discharge still fluctuate afterward, backflows keep staying at one location, and there is no transition (Figure 3b).

3. PT-3 and PT-4: Besides the turbine and turbine modes, the dynamic trajectories of these two pump-turbines also go through the reverse pump mode and the backflow transitions are basically similar. All of them generate from the hub side (in the T mode), then turn to the mid span and shroud side (in the TB mode). However, the only difference is that when the working point enters the reverse pump mode, the backflows in PT-3 mainly alternate between the hub side and mid span, while those in PT-4 also spread to the shroud side (Figure 3c,d).

In order to further explore the flow patterns at the runner inlets, Figures 4–7 show backflows at typical times in a single passage. Generally speaking, when the working points leave from the optimal ones, the water will impact on the blades and form backflows, making some water returning to the vaneless space and some water jumping over and impacting the next blade.

**Figure 4.** Flow patterns at the runner inlet in PT-1: (**a**) *t* = 3.6 s (turbine (T)), (**b**) *t* = 10.0 s (turbine braking (TB)), and (**c**) *t* = 15.0 s (T).

**Figure 5.** Flow patterns at the runner inlet in PT-2: (**a**) *t* = 5.0 s (TB), (**b**) *t* = 8.0 s (TB), and (**c**) *t* = 15.0 s (TB).

**Figure 6.** Flow patterns at the runner inlet in PT-3: (**a**) *t* = 8.0 s (T), (**b**) *t* = 10.0 s (TB), (**c**) *t* = 13.0 s (reverse pump (RP)).

**Figure 7.** Flow patterns at the runner inlet in PT-4: (**a**) *t* = 4.0 s (T), (**b**) *t* = 5.8 s (TB), (**c**) *t* = 7.4 s (RP).

1. PT-1: The backflows generate from the shroud side, while the water flows into the blade passage easily on the hub side. Because the inlet height of PT-1 is relatively large, the backflows are mainly maintained near the shroud over the entire runaway process, and just influence the normal inflow at the mid span (Figure 4a).

2. PT-2: Backflows generate from the hub side and gradually evaluate to other locations. Compared with those in PT-1, the inlet height of PT-2 is smaller, and the backflows are easy to expand to the whole inlet. There is an obvious transition in the flow patterns, and the backflows suddenly occur on the shroud side and at the mid span (Figure 5b, *t* = 8.0 s), which is consistent with the transition of *v*<sup>r</sup> in Figure 3b. But with the speed and discharge tending to steady, the backflows keep stay on the hub side.

3. PT-3 and PT-4: The inflow attacks on the blades at the mid span, leading to the upward deviation of the normal inflow on the hub side, then backflows generate and evaluate to other locations. Once entering the reverse pump mode, the backflows at the mid span in PT-3 have less influence to the hub and shroud sides, while those in PT-4 affect the shroud side obviously (Figure 7c).
