## 3 Result and Discussion
### Breakdown of the Leakage Vortex Core  
#### 3.3.1 Leakage Vortex Breakdown  
In this section, the results with back pressure of 510 kPa and 300 kPa are compared to investigate the instability of the tip-leakage vortex. The average outlet Mach Numbers of the two cases are 0.79 and 1.20, indicating that the flows are subsonic and supersonic, respectively. Figure 12 shows the streamlines within the core of the tip-leakage vortices. It is observed that near the trailing edge of the blade, the streamline representing the tip-leakage vortex core expands and the vortex core region is taken by the chaotic flow. Figure 12a,b indicates that for a subsonic flow, the core region expands gradually as it flows downstream, while it expands almost suddenly in a sudden in the supersonic flow. Moreover, it is found that a bubble region and the reserved flow form in the vortex core. Figure 13 shows the Mach number distribution on the streamlines near the tip-leakage vortex core. The distortion in Figure 13 is mainly because the selected streamline is not accurately at the vortex core. It is observed that for different back pressures, the flow in the tip-leakage vortex core accelerates to different levels and then decelerates to the stagnation position of the vortex core breakdown. Then, the flow velocity remains at a low level and strong oscillations occur. Instability of the rotating jet flow [34] indicates that such flow with a stagnation point and a recirculation bubble can be considered as the vortex breakdown. Accordingly, it is concluded that the reversed flow is induced by the breakdown of the tip-leakage vortex and the vortex is unstable near the trailing edge. In the case with a backpressure of 510 kPa, deceleration of the vortex core begins at x = 0.59 \(C_{\rm x}\) for the subsonic case and the velocity drops to zero at 0.89 \(C_{\rm x}\). The latter point is the stagnation point, which is defined as the breakdown position of the leakage vortex. Accordingly, the deceleration distance is 0.3 \(C_{\rm x}\). On the other hand, in the case with a backpressure of 300 kPa, the flow of the vortex core begins deceleration and its velocity drops to zero rapidly between 0.87 \(C_{\rm x}\) and 0.95 \(C_{\rm x}\). In this case, the axial distance is 0.08 \(C_{\rm x}\), which is only 27% that of the subsonic case. The flow velocity in Figure 10(c) indicates that for the backpressure of 510 kPa, a supersonic region forms on the suction surface. Figure 12: Streamline line in leakage vortex core and contour of wall shear stress on blade. Figure 13: Velocity distributions on the streamline near the core of the tip-leakage vortex. It is found that in region 3 with the backpressure range from 450-400 kPa, a supersonic flow appears in the leakage vortex and the vortex breaks up under large pressure gradients similar to the normal shock wave. This phenomenon can be used to explain the forward movement of the leakage vortex breakdown in region 3. It also shows that the leakage loss is related to the location of the leakage vortex breakdown. Taking the stagnation position of the leakage vortex core, distribution of the stagnation position of the leakage vortex breakdown can be obtained along the outlet Mach number, as shown in Figure 14. Accordingly, five regions corresponding to the leakage vortex loss are detected, indicating that the leakage vortex loss is a function of the location of the leakage vortex breakdown. The breakdown position in different regions has different behaviors, moving forward or backward. Moreover, the variation amplitude of the axial position is up to 0.13 \(C_{\mathrm{x}}\). It is inferred that when the backpressure fluctuates unsteadily, the breakdown position may fluctuate greatly. It should be indicated that these fluctuations are quite probable because of the influence of the downstream blade in turbine stage. Since pressure fluctuations originate from the leakage vortex breakdown on the rotor blade tip area, these fluctuations may result in the blade vibration.