## 3 Results and Analysis
### Influence of Flow Transitions on Discharge Change Rate  
The above investigations demonstrate that the sharp change in discharge around the no-load points is detrimental to the runaway oscillation stability. Meanwhile, a remarkable relevance can be observed between the backflow transitions at the runner inlet and the discharge change rate. During the runaway process for large GVOs (9\({}^{\circ}\), 15\({}^{\circ}\), and 24\({}^{\circ}\)), the location of BFVSs on the hub side will turn to the mid-span as the operating point enters into the turbine braking region in the forward direction, and the falling rate of discharge will increase. Afterward, the location of BFVSs will change in a reverse process as the operating point moves out from the turbine braking region in the backward direction, and the rising rate of discharge will increase dramatically. For 6\({}^{\circ}\) GVO, the BFVSs only occur at the mid-span of the runner inlet, and the radial velocity only changes in magnitude but without direction. Although the discharge changes in falling and rising stages also are different, their amplitudes are quite small. Figures 12 and 13 show the distribution changes in the entropy production rate \(Ep\) along the streamwise direction of flow passages around no-load points OP1 and OP2 for 15\({}^{\circ}\) and 6\({}^{\circ}\) GVOs, respectively. As shown in Figure 12, the entropy production rate has a sudden change around the no-load points for 15\({}^{\circ}\) GVO, which means that the transition of BFVSs at the runner inlet leads to a dramatic change in energy dissipation and causes a sharp change in the flow resistance, which results in a rapid change in discharge. Moreover, the change range of the entropy production rate within the guide-vane domain in the backward direction is much larger than that in the forward direction (Figure 12). Therefore, the change in the energy dissipation inside the flow passages in the stage with increased discharge is higher than that in the stage with decreased discharge, which leads to significant differences in the discharge change rate in both directions (Figure 8a). Conversely, the entropy production rate changes very slightly around no-load points for 6\({}^{\circ}\) GVO (Figure 13). Therefore, the change in discharge at no-load points for 6\({}^{\circ}\) GVO is slighter than that for 15\({}^{\circ}\) GVO. Figure 11: Changes of dynamic trajectory by neglecting the influence of kinetic pressure for 6\({}^{\circ}\) GVO. Figure 14 shows the vortex structures in the guide-vane and runner passages at no-load points OP1 and OP2 for 15\({}^{\circ}\) and 6\({}^{\circ}\) GVOs. The \(Q\)-criterion is employed to identify the vortex structures [34]. Here, the iso-surface of the \(Q\)-criterion was set to 500,000 s\({}^{-2}\) and colored by the magnitude of the entropy production rate. For 15\({}^{\circ}\) GVO, the runner inlet was severely blocked by the unstable vortices, but the region with separation vortices at no-load point OP1 is larger than at no-load point OP2, especially nearby the runner outlet (Figure 14a). This explains why there is a higher entropy production rate near the runner outlet at OP1 than at OP2 (see Figure 12). For 6\({}^{\circ}\) GVO, the runner passages are also blocked severely by separation vortices (Figure 14b). However, the vortices in the guide-vane domain have a smaller value of the entropy production rate. In other words, smaller energy is dissipated there. Although the energy dissipation of separation vortices at no-load points for 15\({}^{\circ}\) GVO is higher than that for 6\({}^{\circ}\) GVO, the change rate in energy dissipation for 15\({}^{\circ}\) GVO has much more influence on the change in discharge, which is unfavorable to the oscillation stability of a pump turbine in a runaway process. Figure 12: Changes of the distribution of entropy production rate \(Ep\) along the streamwise direction of flow passages at no-load points OP1 and OP2 for 15\({}^{\circ}\) GVO; (**a**) in the forward direction; (**b**) in the backward direction.