## 3 Result and Discussion
### Analysis of the Leakage Vortex Loss
#### 3.1.2 Loss in the Leakage Vortex  
In Figure 9, the red line shows the tip-leakage vortex loss by simulations with different outlet Mach numbers. It is worth noting that the exponential curves fitted to the loss are presented with a black dotted line. Moreover, Figure 9 shows the ratio of the leakage vortex loss \(Y_{\text{t}}\) to the total loss of the blade passage \(Y_{\text{a}}\) (green dotted line).  
It is observed that the overall increment of the leakage vortex loss of the rotor is almost exponential. As the outlet Mach number Ma\({}_{2}\) increases from 0.67 (Pb = 600 kPa) to 1.15 (Pb = 325 kPa), the loss in the leakage vortex increases by nearly 10 times, whose proportion to the total loss increases from 6.5% to 30.2%. Moreover, it continually increases as the increase of outlet Mach number increases. Moreover, it is found that the ratio of the profile loss Y\({}_{\text{p}}\) in the total loss in this rotor is almost between 40% and 50%.  
Figure 8: Span-wise distribution of the total pressure loss coefficient (Pb = 350 kPa, Ma2 = 1.10).  
Compared with the pure exponential growth, the change curve of the leakage vortex loss \(Y_{\text{t}}\) is more complicated. Figure 9 illustrates that the growth is divided into five regions according to the change of the leakage vortex loss. The green dotted line shows that when the outlet Mach number is lower than 0.67, the proportion of the leakage vortex loss \(Y_{\text{t}}/Y_{\text{a}}\) decreases as the Mach number increases.  
According to the definition of the critical Mach number [5], the critical Mach number can be set as 0.67 and marked as region 1, in which the flow within the turbine passage is subsonic. It should be indicated that the leakage vortex grows slowly in region 1. The growth rates of the leakage loss in regions 2 and 3 are greater than that of the exponential curve. The growth rate in region 2 gradually increases and the growth rate in region 3 gradually decreases. The growth rate in region 4 is smaller than that of the exponential curve. Moreover, the growth rate in region 5 becomes equivalent to that of the exponential curve.  
Therefore, the loss within the leakage vortex increases sharply and becomes the main part of the overall loss as the outlet Mach number increases, resulting in a drop of the efficiency by more than 5 percent. Moreover, the leakage loss changes differently in various ranges of the Mach number. Therefore, leakage loss is an important aspect of a transonic turbine.  
Figure 10 shows the span-wise scale \(\Delta\) and loss peak \(Y_{\text{pk}}\) of the leakage vortex along the outlet Mach number. It also shows that there are five regions corresponding to those in Figure 9. In region 2, 3 and 4, the span-wise scale \(\Delta\) of the leakage vortex between Mach number 0.67 and 1.10 increases for the first time, while the span-wise scale increases from 0.16 to 0.195 (about 1.2 times). However, in region 5 (Mach number above 1.10), a rapid increment is observed. From region 2 to region 4, the leakage vortex scale only doubles and remains within a certain range, while the peak leakage loss increases rapidly. In region 5, both the scale and the loss peak of the leakage vortex increase significantly, which corresponds to the approximate exponential increment of the leakage loss in region 5. Therefore, it indicates that the influence on loss is correlated to the change of the tip-leakage vortex.  
Figure 9: Tip-leakage vortex loss along the outlet Mach number.