**4. Discussions of Influences of Runner Shapes**

It can be seen from the above analysis that the starting and staying locations of backflows at the runner inlets are different during the runaway processes in the pump-turbines with different specific speeds. Xia [33] pointed out that the backflow structures are mainly affected by the shape of the blade inlet and centrifugal force, which can change the pressure gradient. Similarly, the initial position of backflows is related to this factor. Figure 17 shows the different blade lean angles of the four pump-turbines, which mean the inclination angles of blade leading edges at the runner inlets. The blade lean angle of PT-1 is negative, and its backflows generate from the shroud side. The blade lean angle PT-2 is positive, and its backflows generate from the hub side. Interestingly, the inlets of PT-3 and PT-4 have no blade lean, but the backflows also generate from the hub side.

**Figure 17.** Lean angles of the blade leading edges of the four pump-turbines: (**a**) PT-1, negative lean angle, (**b**) PT-2, positive lean angle, (**c**) PT-3, no blade lean, (**d**) PT-4, No blade lean.

As shown in Figure 8, it can be seen from the filtered-data that in the early stage of runaway, the pressures at the monitoring points are approximately the same and there is no backflow. With the increase in rotating speed, the centrifugal force increases but the discharge decreases, then the pressure gradient between the hub and shroud sides becomes larger, resulting in water flows from the higher-pressure side to the lower one. Here, the blade lean angle affects the distribution of pressure gradient and leads to the different initial position of backflows. The negative lean angle of PT-1 forces the pressure to increase on the hub side, which makes the water turn from the hub side to middle and shroud ones, leading to backflows on the shroud side Figure 18a. On the contrary, the backflows in PT-2 generate from the hub side due to the existence of a positive lean angle Figure 18b. Although there is no lean angle in PT-3 and PT-4, the pressure gradient distribution in them is consistent with that in PT-2, therefore the backflows all generate from the hub side Figure 18c.

**Figure 18.** Diagram explaining the reason of backflows at runner inlet in four pump-turbines: (**a**) PT-1, (**b**) PT-2, (**c**) PT-3 and PT-4.

Secondly, the different heights of runner inlets affect the development of backflows. The smaller the height of runner inlet, the easier the backflows change location. The inlet height of PT-1 is the largest, therefore the backflows can only exist on the shroud side all the time, and the influence range of backflows is relatively small. Therefore, due to the lowest height of PT-4, in the RP mode, the relative backflow region can be larger than that in PT-3 Figure 3. Because of these differences in backflow transitions, pressure pulsation evolutions get large differences. With the decrease of the inlet height, the differences of the pressure fluctuations between three locations decrease. Hence, the difference of the pressure fluctuations at each location in PT-1 is the largest, while that in PT-4 is the smallest. In the above four runners, except for PT-2, the location where the backflows occur, the pressure amplitudes are the largest. As a special case, the blade inlet design of PT-2 is the main reason that the blade leading edge diameters at the three locations are quite different Figure 19b. Due to the difference of blade leading edge diameters at the three locations, the pressure characteristics in PT-2 is an exception. Therefore, besides the backflows, the size of the vaneless space and distance to the blade should be considered.

**Figure 19.** Differences of blade leading edge diameters of the four pump-turbines: (**a**) PT-1, (**b**) PT-2, (**c**) PT-3, (**d**) PT-4.

In order to further verify the analysis mentioned above, the runaway process of a conventional turbine was also simulated, and the detailed information including the lean angle of blade leading edge and inlet diameter was shown in Figure 20. Though the starting working condition of runaway is not the rated one, it is a large guide-vane opening case, which is near the rated working point and can reflect the main characteristics of backflows and pressure pulsations.

The results show that the macro parameters nearly maintain constant values after *t* = 4 s due to the absence of the S-shaped characteristics, and the period during this time is defined as the no-load mode (Figure 21). The radial velocity and flow patterns are selected (Figures 22 and 23), and it can be seen that the backflows only generate on the hub side, which is similar to those in PT-2 because these two turbines have the same blade lean angle (Figures 17b and 20a) and the same pressure gradient (Figure 18b). Also, the backflows keep staying on the hub side because the runner inlet height is relatively large, which is similar to those in PT-1.

**Figure 20.** Lean angle of blade leading edge and inlet diameter of CT. (**a**) Inlet lean angle and (**b**) inlet diameter.

**Figure 21.** Histories of the macro parameters of CT during runaway processes.

**Figure 22.** The variations of normalized radial velocity *v*r of three monitor points in CT.

**Figure 23.** Flow patterns at the runner inlet of CT.
