## 4 Analyses of Numerical Simulation Results
### Flow Physics on Aerodynamic Performance Under Design Conditions
#### 4.1.2 Flow Field Analysis  
In order to reveal the effects of the flow field characteristics of the improved cascade on SWBLIs, Mach number contours under design conditions are adopted to sketch the shock configuration for both cascades, as shown as Figure 14. For the baseline cascade, \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Baseline Cascade** & **Improved Cascade** & **Differences (\%)** \\ \hline \(M_{2}\) & 0.86 & 0.87 & 1.16\% \\ \(\beta_{2}\) & 73.75 & 73.69 & \(-\)0.03\% \\ \(\omega\) & 0.142 & 0.1354 & \(-\)4.6\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Aerodynamic performance parameters of both cascades under design conditions. Figure 12: Calculated unique incidence curve. Figure 13: Mass-averaged aerodynamic parameters at the exit plane. due to the increase in the boundary layer thickness near the first-passage shockwave, the deflection angle of the flow near the shock impingement point on the suction side increases. Mach reflection occurs when the passage flow is over-deflected, generating a quasi-normal shockwave (Mach stem) near the shock foot. There exists a separation bubble within the boundary layer, and its bifurcation point E is located above the suction surface. Compared to the baseline cascade, a series of compression waves are generated from the negative-curvature profile ahead of the first-passage shock of the improved cascade, which weakens the strength of the shock foot. The boundary layer separation bubble evidently disappears, and the impingement position of the shockwave moves downstream (about 1.23% of the axial chord length). Furthermore, the compressive wave and reflected shockwave converge at point E\({}^{\prime}\), and this point, corresponding to the bifurcation point E, is far away from the suction side of the improved vane profile as it has moved up 6.2% of the local vane channel width. This phenomenon can be attributed to the interaction between the compression waves and the first passage shock foot, which changes the reflected shock position. In addition, recompression waves are generated on the leeward region behind the plateau section on the vane suction surface, and a new shockwave system structure is displayed in the cascade passage. The structure of the shockwave system can be reflected by the shock function, defined as  
\[\text{shock}=\frac{\overset{\rightarrow}{V}\cdot\
abla p}{a|\
abla p|} \tag{10}\]  
where \(\overset{\rightarrow}{V}\) is the velocity vector, \(\
abla p\) is the pressure gradient, and \(a\) is the local speed of sound. The shockwave function contours of both cascades under design conditions are shown in Figure 15. The reflected shockwave system formed by the first-passage shock is split into two parts. The two reflected shockwave systems cross-propagate in the cascade passage and end up with the last-passage normal shock. According to flow physics, the strong shock system is decomposed into multiple weak shock systems, which can reduce the flow separation loss. In this way, the potential of shockwave pressurization can be maximized. Figure 14: Mach number contours of both cascades. In order to reveal the flow physics of the negative-curvature profile ahead of the shock foot, the aerodynamic parameters across the SWBLI region of the two cascades are analyzed in detail. Three cuts through the first-passage shock are marked by the parallel dashed lines, as plotted in Figure 14. They correspond to several characteristic heights near the first passage shock foot from the vane suction surface. Line 1 and line 2 intersect with the Mach stem of the baseline cascade and the compression waves of the modified cascade. Line 3 intersects with the first-passage shock of both cascades and their reflected shock. In order to more clearly compare the changes in flow field between the baseline and improved cascades, the positions of lines 1, 2, and 3 presented for the baseline remain the same in the improved case.