## 3 Result and Discussion
### Flow within the Leakage Vortex
#### 3.2.2 Effect of the Shock Wave on the Leakage Vortex  
Figure 11 shows the breakdown area and process of the leakage vortex change in five regions of the outlet Mach number.  
Before \(\text{Ma}_{2}\) equals 1.15 (Figure 11e-f), the shape of the leakage vortex is symmetrical in the initial area of breakdown, like a \"cone\" expansion shape. When the \(\text{Ma}_{2}\) is higher than 1.15, the low-velocity region in the vortex core becomes an asymmetrical shape after passing through the shock.  
Therefore, it indicates that the effect of the shock wave interference of the latter is stronger than that of the former and the leakage vortex core still breaks up by centrifugal instability after the shock wave. Meanwhile, Figure 11f shows that the expansion of the vortex core at this time causes an obvious strong shock wave in the mainstream.  
As the shock wave interference increases, the centrifugal force caused by the rotating motion of the vortex core cannot hold by the pressure gradient of the main flow to the vortex core. Figure 11f,h shows that the breakdown caused by the centrifugal force occurs instantaneously and the influence of the interaction extends to the shear layer flow on the surface of the leakage vortex.  
Combined with the research of [PERSON] [33], the intensity of the interaction between the shock wave and the stream-wise vortex directly determine the breakdown pattern of the vortex and the flow pattern after breakdown, accordingly the loss behind the shock wave. This can be observed from the change of the leakage vortex pattern in Figure 10. However, the unsteady nature of this process needs further investigation and will not be discussed in the present study.  
Figure 11a-d shows that when there is no shock wave in the passage (before Figure 11c), the change of the leakage vortex breakdown position is mainly affected by the leakage vortex intensity and the main flow. Moreover, the vortex instability breakage is the main reason for the loss of the leakage vortex.  
When a shock wave appears in the passage, the interaction between the shock wave and the leakage vortex is the main reason for the leakage vortex breakdown. This can be concluded from the change of the breakdown position with the shock wave interference position.  
As Figure 11f-h (region 5) shows, as the outlet Mach number increases, the position of the trailing edge incident shock hit on the suction side moves backward, the length of the low-velocity area decreases and the low-velocity area after the vortex breakdown disappears quickly. The reduction of the length indicates the increment of the velocity gradient. Therefore, stronger mixing occurs between the low-velocity flow in the leakage vortex and the supersonic flow of the mainstream.  
Moreover, it is found that the flow in the surface shear layer of the leakage vortex with strong interference is unstable and strong mixing loss between the vortex core flow and the main flow occurs after the breakdown.  
Therefore, the strong interference caused by the enhancement of the shock wave and the strong mixing loss with the mainstream after a breakdown are the main reasons for the increase of the leakage vortex loss. The intensity and position of the shock wave have a direct impact on the breakdown of the tip-leakage vortex after the shock wave appears. The interaction changes the breakdown position of the leakage vortex and the mixing strength of the leakage vortex with the mainflow.