*3.1. Flow Deflection in the Guide Vane Domain*

Figure 7 shows an example of a typical flow state in the shutdown process (GVO = 6.5◦~1.82◦), including the velocity contour and streamline at the Z = 0 plane (middle plane of the GV domain). It can be observed that the main stream in the GV domain is deflected when it passes through the gap in the GVs.

At GVO = 6.5◦, the water from the runner outlet flows out along the channel between the movable guide vanes, and its direction is consistent with the overflow channel. When the GVO decreases to 3.1◦, it can be seen that the direction of the water flow is no longer towards the right side of GV #4, but is deflected by nearly 90◦ and flows towards the fixed guide vane. The main flow in the deflection process is very unstable. When the GVO is further reduced to 1.82◦, the water flow is completely attached to the right wall of GV #5. According to the streamline distribution, a clockwise circulation is formed between the movable guide vane and the fixed guide vane.

The reverse process is found during the startup, as shown in Figure 8. At GVO = 2◦, the water flow from the runner outlet is completely attached to the right wall of GV #5. When the GVO increases to 3.3◦, it can be seen that the direction of the water flow is no longer attached to GV #5, but is deflected by nearly 90◦, and the water flows towards the fixed guide vane. There are vortices with opposite rotation directions on both sides of the main stream. When the GVO is further decreased to 4.45◦, the main stream flows out along the channel between the movable guide vanes, and its direction is consistent with the overflow channel.

We gave definitions for the two flow types in the pump mode, which are sketched in Figure 9. Type I is defined as the flow type with a direction consistent with the incoming flow direction, while Type II is the flow type with a deflected direction that is attached to the guide vane head. This belongs to the state of the transition between the two.

**Figure 7.** Global velocity contours of the shutdown process (Z = 0 plane).

**Figure 8.** Global velocity contours of the startup process (Z = 0 plane).

**Figure 9.** A schematic diagram of the flow pattern definitions.

The main flow in the gap of GVs #26 and #1 was chosen to demonstrate the formation of these two flow patterns. Figure 10 shows the velocity resolution of the main flow near the tailing edge of GV #26. *Vr* is the radial velocity, and *V<sup>θ</sup>* is the tangential velocity. In the plot, a subscript of 1 stands for a small opening, and 2 stands for a large opening. It can be seen that, near the tailing edge of the GV, the radial velocity is greater than the tangential velocity. At the small opening, the incidence angle is too large, and this large incidence angle can lead to flow separation near the tailing edge of GV #26. Then, this separation forces the main flow to turn toward the leading edge of GV #1. Because the curvature radius of the GV surface is too large, the Coanda effect [24,25] can make the main flow adhere to the surface of GV #1 so that the main flow stays in Type II. However, at the large opening, the incidence angle can be reduced with a change in the vane angle; thus, the flow separation is weakened and the main flow eventually stays in Type I.

**Figure 10.** Analysis of the resultant velocity: (**a**) small opening; (**b**) large opening.

#### *3.2. Flow-Field Characteristics of Startup and Shutdown*

Because the flow pattern of each guide vane has little difference, we selected some guide vanes (GV03, GV04, GV05, GV06) for the flow-field analysis. In order to analyze the characteristics of flow deflection between moving and stationary guide vanes in detail, Figure 11 shows the local velocity contours and pressure contours of the shutdown process. During this process, the GVs gradually closed from a max opening of 19.69% to a max opening of 5.91% (decrease in the GVO from 6.5◦ to 1.95◦).

**Figure 11.** *Cont*.

**Figure 11.** Local velocity contours (**left**) and pressure contours (**right**) of the shutdown process (Z = 0 plane, GVO = 6.5◦~1.95◦).

When GVO = 6.5◦~4.58◦, the main flow stays in Type I. At GVO = 6.5◦, the direction of the main flow is towards the left side of the fixed guide vane directly in front of the flow channel. The velocity coefficient *kv* is about 0.9. The velocity at the throat between the two GVs is the highest where *kv* is about 1.2. Therefore, the pressure here is also the lowest relative to the flow field inside and outside the GV. The velocity on the right side of the main stream is very low with *kv* ≈ 0.1.

As the GVO decreases, the main flow tends to shift away from the left movable guide vane, as shown at GVO = 4.58◦. When the GVO decreases, the velocity at the throat gradually increases and the pressure further decreases, which is caused by the decrease in the cross-sectional area of the overflow.

In the range of GVO = 4.58◦~2.08◦, the main flow is in the transition stage between Type I and Type II. When GVO = 3◦, the direction of the main flow is deflected by nearly 90 degrees. At the same time, there is a high-pressure area at the left GV, where *cp* ≈ 0.8. The trailing-edge of the main stream impinges on the other fixed guide vane (transferred from left to right) and flows along the fixed guide vane wall to both sides. With the decrease in the GVO, the deflection angle of the main flow increases further and changes to flow in the flow channel between the fixed guide vanes, as shown at GVO = 2.08◦. At this time, the pressure coefficient inside the GV increases to *cp* ≈ 1.2.

When the GVO further decreases to 1.95◦, the main flow state changes to Type II. At this opening, the main stream flows around the leading edge of the GV. The throat position becomes the boundary between the high-pressure area and low-pressure area.

Moreover, the main flow stays in Type I when the GVO is more than 6.24◦, and the flow pattern stays in Type II when the GVO is less than 1.95◦. The pressure plot shows that the pressure at the tailing edge of the GV is higher than that at the leading edge of the GV; thus, the water coming from the runner domain pushes the GV to close, and the resultant moment should be positive. Furthermore, with the decrease in the GVO, the HT must theoretically increase.

During startup, the flow goes through the process of opposite deflection, as shown in Figure 12. With the increase in the GVO, the main flow starts deflecting from Type II to Type I. Unlike in the shutdown process, when GVO = 2.8◦, the main flow is still in Type II, while the main flow is converted to Type I.

**Figure 12.** *Cont*.

**Figure 12.** Local velocity contours (**left**) and pressure contours (**right**) of the startup process (Z = 0 plane).

The pressure plot shows that the pressure at the tailing edge of the GV is larger than the pressure at the leading edge of the GV; thus, the resultant moment on the GV helps the GV to close. In this paper, the moment that forces the GV to close is defined as a positive moment (HT > 0).

The behaviors of the deflections during the startup and shutdown processes are not the same. Figure 13 shows a comparison of the GVO ranges of the deflections between the shutdown and startup processes. The specific deflecting range of the shutdown process is larger than that of the startup process. For the shutdown process, the deflection angle ranges from 1.99◦ to 5.32◦ on average. For the startup process, the range is reduced to 2.83◦ to 4.11◦ on average.

**Figure 13.** A comparison of GVO ranges of the deflections between the shutdown and startup processes.

During the shutdown process, the GVO of the starting deflection fluctuates greatly on different guide vanes. However, for the ending deflection, the fluctuation is small.

#### *3.3. Hydraulic Torque on the Guide Vanes*

Several GVs were chosen for analysis in this part because the torque change trends of most GVs are similar.

Figure 14 presents the GVO-*Cm* plot for GV03, GV04, GV05, GV06, and GV10 during the simulation of the startup process. In the beginning, *Cm* gradually decreases with the increase in the GVO, but when the GVO goes up to 2.8◦, *Cm* abruptly increases. This sudden increase in *Cm* is indicated by the red line. When the GVO is about 4.05◦, which is indicated by the blue line, *Cm* of GV10 is larger than that of the others. As can be seen in Figure 15, when GVO = 4.05◦, the deflection speed of the main stream of GV10 is faster than that of GV06, so the torque of GV10 at this time is greater than that of the other GVs. When the GVO is around 4.22◦ (indicated by the green line), the flow deflection is complete, and the trend of *Cm* slows down. In addition, it can be seen that during the deflection process (GVO ranging from 2.8◦ to 4.2◦), the values of *Cm* have phase differences between each other. This difference is caused by the unstable flow behavior of the main flow during the deflection process, as mentioned above.

**Figure 14.** GVO-*Cm* plot of the startup process.

By comparing the *Cm* plot with the velocity contours of the main flow (Figure 12), it can be seen that the main flow deflection and the increase in *Cm* occur at the same opening. When the deflection occurs abruptly, the velocity near the GV surface on the side of the stay vane decreases quickly; thus, the pressure on the surface of the GV increases. In addition, the resultant moment on the GV suddenly increases.

The GVO-*Cm* plot for GV01, GV03, GV04, GV05, and GV06 obtained from the calculation of the shutdown process shows that *Cm* increases with the gradual decrease in the GVO. However, when the GVO ranges from 5.26◦ to 3.15◦ (which is indicated by the red line and blue line in Figure 16), *Cm* has a turbulent trend, and the GV near the nose of the spiral case (GV01) has a large moment value. This phenomenon is due to the instability of the main flow when the GVO is in the specific range (from a max opening of 19.39% (about 6.4◦) to a max opening of 5.91% (about 1.95◦)) that was mentioned before. The velocity near the surface of the GV fluctuates, and this fluctuation leads to the turbulent fluctuation of *Cm*. When the GVO is less than 3.15◦, *Cm* has a stable trend. Moreover, when the GVO

is about 2.2◦, *Cm* descends sharply, as indicated by the green line in the plot. In addition, the main flow stays in Type II when *Cm* flattens (which is indicated by the yellow line in Figure 16).

**Figure 15.** Comparison of deflection speeds between the main flow of GV10 (**right**) and that of GV06 (**left**) at GVO = 4.05◦.

**Figure 16.** GVO-*Cm* plot of the shutdown process.

It can be inferred that the deflection led to a sudden change in the velocity near the GV, and the pressure on the GV's surface abruptly changed. Consequently, the change in pressure resulted in a sudden increase in the *Cm*; if the transmission of the distributer cannot immediately adapt to this change, there will be a relative displacement between the shaft of the GV and the friction device, which will produce violent friction. This violent friction will lead to vibrations and abnormal sounds.

## **4. Conclusions**

Numerical studies were presented for the startup and shutdown processes of an RPT in the pump mode when GVO is between 1.5 and 6.5 degrees. A dynamic meshing technique was used to investigate the flow deflection of the main flow between the guide vanes. The primary findings include:


**Author Contributions:** Methodology, Q.J., G.W.; formal analysis, Q.J., H.F., W.L.; investigation, G.W.; resources, H.F.; data curation, Q.J., G.W.; writing—original draft preparation, Q.J.; writing—review and editing, Q.J., H.F., W.L.; supervision, H.F., W.L.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by [National Natural Science Foundation of China] [Grant No. 51879140, No. 51679196], [State Key Laboratory of Hydroscience and Hydraulic Engineering] [Grant No. 2021-KY-04], [Creative Seed Fund of Shanxi Research Institute for Clean Energy of Tsinghua University], and [Tsinghua-Foshan Innovation Special Fund(TFISF)] [Grant No. 2021THFS0209].

**Acknowledgments:** The authors would like to thank the National Natural Science Foundation of China (No. 51879140, No. 51679196), State Key Laboratory of Hydroscience and Hydraulic Engineering (Grant No. 2021-KY-04), Creative Seed Fund of Shanxi Research Institute for Clean Energy of Tsinghua University and Tsinghua-Foshan Innovation Special Fund(TFISF) 2021THFS0209 for their financial support.

**Conflicts of Interest:** The authors declare no conflict of interest.
