## 3 Results and Analysis
### Influence of Pressure Fluctuations and Flow Changes on Dynamic Trajectories  
To investigate the influence of flow evolutions on the runaway stability, the variations of total pressure coefficient \(P_{\mathrm{GN}}\) at the guide vane inlet, the total pressure coefficient \(P_{\mathrm{DN}}\) at the draft tube inlet, and the normalized radial velocity \(Vr_{N}\) in vaneless space for GVOs at 15\({}^{\circ}\), 9\({}^{\circ}\), and 6\({}^{\circ}\) are shown in Figures 4-6, respectively. The layout of monitoring points is illustrated in Figure 7. Here, the variations of radial velocity are the low frequency filtered signals of velocity fluctuations. The total pressure \(P_{\mathrm{T}}\) is composed of static pressure \(P\)s and kinetic pressure \(P_{\mathrm{k}}=\int\rho v^{2}ds/2s\), where \(\rho\) is the fluid density, \(v\) is the fluid velocity, and \(s\) is the area of flow section. The pressure was normalized by the reference pressure \(0.5\rho(\pi n_{0}D_{1}/60)^{2}\), and the radial velocity was normalized by the reference velocity \(\pi n_{0}D_{1}/60\), as expressed in Equations (6) and (7). \[P_{N}=\frac{P_{\mathrm{T}}}{0.5\rho(\pi n_{0}D_{1}/60)^{2}} \tag{6}\]  
\[V_{rN}=\frac{V_{r}}{\pi n_{0}D_{1}/30}. \tag{7}\]  
During the runaway processes, the operating points will move from the turbine mode to the reverse pump mode in the forward direction and then back from the reverse pump mode to the turbine mode in the backward direction. As shown in Figures 4-6, the transient variations of pressure and velocity in the two directions are different. For the GVOs at 15\({}^{\circ}\) and 9\({}^{\circ}\), the total pressure at the section of draft tube inlet first decreases with decreased discharge in the forward direction and then increases with increased discharge in the backward direction (Figures 4 and 5). Meanwhile, the pressure \(P_{\mathrm{DN}}\) at no-load point OP1 is smaller than the pressure at initial time, whereas \(P_{\mathrm{DN}}\) at no-load point OP2 is larger than the pressure at initial time. As a result, the slopes \(dT_{\mathrm{ED}}/dn_{\mathrm{ED}}\) of the dynamic trajectories at the no-load points (OP1 and OP2) both become positive, so the runaway processes are self-excited. On the contrary, for the GVO at 6\({}^{\circ}\), the pressure \(P_{\mathrm{DN}}\) first increases with decreased discharge in the forward direction and then decreases with increased discharge in the backward direction (Figure 6a). Simultaneously, the values of total pressure \(P_{\mathrm{DN}}\) at the no-load points (OP1 and OP2) are both larger than the value at initial time. Therefore, the slopes \(dT_{\mathrm{ED}}/dn_{\mathrm{ED}}\) of dynamic trajectories at the no-load points (OP1 and OP2) both become negative (Figure 6b), which dampens the runaway process. Figure 4: Dynamic trajectories and time histories of operating parameters at GVO 15\({}^{\circ}\). (**a**) Variation of pressure and radial velocity; (**b**) dynamic trajectories. The variations of radial velocity in the vaneless space show that the BFVSs with negative radial velocity emerge at the runner inlet and have regular transition near the no-load point. For the GVOs at 15\({}^{\circ}\) and 9\({}^{\circ}\), the location of BFVSs changing back and forth between the hub side and mid-span (Figures 4a and 5a). For GVO at 6\({}^{\circ}\), the BFVSs only occur at the mid-span (Figure 6a). In general, the BFVSs will block the through flow, leading to increased pressure in the guide vane domain. Figure 5: Dynamic trajectories and time histories of operating parameters at GVO 9\({}^{\circ}\). (**a**) Variation of pressure and radial velocity; and (**b**) dynamic trajectories. Figure 6: Dynamic trajectories and time histories of operating parameters at GVO 6\({}^{\circ}\). (**a**) Variation of pressure and radial velocity; (**b**) dynamic trajectories. Figure 7: Schematic diagram of monitoring points and sections.