## 3 Result and Discussion  
### Analysis of the Leakage Vortex Loss  
#### 3.1.1 Analysis Method  
The loss caused by the leakage vortex is one of the most concerning issues in the current study and it shows the effect of the shock wave on the leakage vortex loss and instability. The local total-pressure loss is expressed as follows:  
\[Y=\ \left(\overline{p}_{\mathrm{t,in}}-p_{\mathrm{t}}\right)/\left(\overline{p} _{\mathrm{t,in}}\right) \tag{2}\]  
where \(\overline{p}_{\mathrm{t,in}}\) is the averaged relative total pressure at the rotor inlet and \(p_{\mathrm{t}}\) is the local relative total pressure at the study section, 0.2 \(C_{\mathrm{x}}\) downstream of the blade trailing edge.  
Figure 8 shows the typical span-wise distribution of the total pressure loss at the study section, whose outlet Mach number is 1.10. The distributions for other cases with different \(\mathrm{Ma}_{2}\) are similar. It should be indicated that the data on the outlet section is averaged integrally in the circumferential direction.  
Figure 8 shows that different flow loss components can be distinguished from the total-pressure loss according to the loss analysis method of [PERSON] et al. [29] in the compressor cascade. In addition, the tip-leakage vortex loss can be studied directly. Near the mid-span, the loss is mainly 2D lightly influenced by the secondary flow and the tip-leakage flow. Therefore, the loss near the mid-span is approximated as the blade profile loss \(Y_{\mathrm{p}}\). Although the rotor blade is 3D, the loss \(Y_{\mathrm{p}}\) can be considered as the average value along the entire blade. The profile loss is mainly due to the viscous effect in the blade boundary layer, main flow and wakes, including shock wave, expansion wave and their interactions with the boundary layer.  
Figure 7: Static pressure distribution on the blade surface.  
Therefore, the span-wise distribution of the local-pressure loss is presented and divided into several parts. The total-pressure loss \(Y_{\text{a}}\) of the entire blade is calculated by the integration of Y along the span-wise direction, which is expressed as:  
\[Y_{a}=\int\text{Y}\text{d}h \tag{3}\]  
Figure 8 shows that the tip-leakage vortex loss \(Y_{\text{t}}\) can be approximated as the integration of the green area which results from subtracting the blade profile loss \(Y_{\text{p}}\) from the local total-pressure loss \(Y\) within the tip-leakage vortex region (\(h_{1}\leq h\leq h_{2}\)). Therefore, \(Y_{\text{t}}\) can be expressed as follows:  
\[Y_{\text{t}}=\int_{h_{1}}^{h_{2}}{(Y-Y_{\text{p}})\text{d}h} \tag{4}\]  
Here, \(h_{2}\) is the spanwise position of the lowest loss point near the casing. The loss above \(h_{2}\) can be regarded as the direct loss related to the boundary layer of the casing wall. Although the tip-leakage flow has a direct effect on the vortex in the casing passage [30], the boundary influenced by the leakage vortex can be set as the lowest loss position \(h_{1}\) between the casing passage vortex and the tip-leakage vortex. The peak loss of the leakage vortex can be remarked as \(Y_{\text{pk}}\) and the spanwise scale \(\Delta\) of the leakage vortex can be recognized from Figure 8 and expressed as \(\Delta=h_{2}-h_{1}\). Therefore, the peak loss \(Y_{\text{pk}}\) and spanwise scale \(\Delta\) can be used to characterize the shape of the tip leakage vortex.