## 4 Analyses of Numerical Simulation Results
### Flow Physics on Aerodynamic Performance Under Design Conditions
#### 4.1.2 Flow Field Analysis
Figure 16 depicts the flow angle and Mach number distributions of the baseline cascade and the improved cascade along the three cuts across the first-passage shock foot. Compared to the baseline cascade, the pre-shock Mach numbers of the improved cascade are reduced by the compression waves. The Mach numbers of lines 1 and 2 are reduced from 1.55 and 1.63 to 1.32 and 1.45, respectively. For line 3, the reduction in Mach number caused by the first-passage shock of both cascades is consistent, mainly because the aerodynamic parameters ahead of the shock are not affected by the compression wave. In addition, the position of reflected shock moves at 2.62% of the axial chord length upstream along line 3, indicating that the angle of reflected shock increases and its intensity decreases. The flow angle distributions along line 1 and line 2 near the impingement position of the first passage shock indicate that earlier and more moderate flow deflection takes place on the improved cascade suction surface. The angle distribution curves of the SWBLI region in both cascades tend to rise at first and then decrease. The difference is that the variation in flow angle for the baseline cascade is due to the separation and reattachment of the boundary layer, while the change in flow angle of the improved cascade can be attributed to the variation in vane profile curvature on the suction surface. As for the flow angle distribution along line 3, the flow direction of both vanes after the first-passage shockwave is the same. As the boundary layer separation is diminished, the flow angle behind the shock of the improved cascade is lower than that of the baseline cascade, as shown with a blue dotted box in Figure 16. That is, the flow deflection angle of the improved cascade exceeds that of the baseline cascade across the SWBLI region. This shows that a larger vane loading level is obtained. Figure 15: Shock function contours of both cascades. The static pressure coefficient can quantify the aerodynamic load distribution of the vane surface, which is defined as  
\[C_{p}=\frac{p-p_{\text{in}}}{P_{\text{in}}^{*}-p_{\text{in}}} \tag{11}\]  
where \(P^{*}_{\text{in}}\) and \(p_{\text{in}}\) are the total pressure and the static pressure at the cascade inlet, respectively, and \(p\) is the local static pressure on the vane surface. The surface static pressure coefficient distribution curves across the SWBLI region on the vane suction surface of both cascades are shown in Figure 12:0 A large adverse-pressure gradient emerges, which is generated by the first-passage shock for the baseline. However, for the improved cascade, the corresponding adverse-pressure gradient vanishes and is replaced by a three-stage moderate pressure rise. In the first stage, the vane surface static pressure coefficient curve indicates a continuously increasing tendency with a constant pressure gradient, which is induced by compression waves on the negative-curvature vane profile. The result is essentially in accordance with the initial desired design, demonstrating the feasibility of the design method with a constant adverse-pressure gradient. Then, the second stage  
Figure 12:1 Variation in flow angle and Mach number along lines 1, 2, and 3 of both cascades. of pressure gain can be attributed to the weaker first-passage shock. Finally, the third stage involves recompression shockwaves at the end of the negative-curvature vane profile, which drives the surface pressure to rise again. Compared with the static pressure rise induced by the first passage shock of the baseline cascade, the adverse pressure gradient generated by the three-stage moderate pressure rise in the improved cascade is reduced by about 30%. Thus, with the precompression effect ahead of the incident shock foot, the adverse-pressure gradient induced by the shock can be significantly reduced. Analyses of the amount of loss due to the first-passage shock foot and the shock-induced separation of the two cascades are performed by integrating the entropy production rate.