## 2 Geometric Model
### Modified Supersonic Compressor Cascade  
The modified profile of supersonic compressor cascades was obtained by small revisions of the vane suction surface on the baseline profile. The position and area of the geometric throat remained the same during modifications to the vane suction surface. A schematic diagram of the interference between shockwave and boundary layer is shown in Figure 2. Due to the strong shockwave in the first channel, the separation bubble and Mach stem structure are generated at the incident position. Part \(a\) is the windward side of the bulge of the boundary layer, which induces the generation of compression waves. Part \(b\) is the convex part, generating expansion waves. The aerodynamic profile of Part \(c\) is concave, which produces recompression waves. According to the change in boundary layer curve, the wall curve in this region is locally improved, as shown in Figure 3. The starting position of the negative-curvature slope of the improved blade cascade is the position where the shockwave induces an increase in wall pressure, namely, point \(O\). The length is the upstream influence range of shockwave and boundary layer interference, as shown in _OO_\({}_{1}\). _O_\({}_{1}\)_Q_ is the plateau zone, and _QQ_\({}^{\prime}\) is the leeward zone. The positions of these four points, \(O\), _O_\({}_{1}\), \(Q\), and _Q_\({}^{\prime}\), are marked in purple in Figure 3. Moreover, the joint connections with the regions _OO_\({}_{1}\), _O_\({}_{1}\)_Q_, and _QQ_\({}^{\prime}\) should be smoother. In supersonic operation, it was observed that a series of compression waves extended from the negative-curvature concave surface in the cascade channels, which considerably weakened the shock foot through a decrement in the pre-shock Mach number. It was observed that the air velocity at the plateau region _O_\({}_{1}\)_Q_ accelerated through expansion waves and then recompressed at the leeward region _QQ_\({}^{\prime}\). The boundary layer separation can be ascribed to the fact that the flow momentum inside the boundary layer does not suffer from the adverse-pressure gradient induced by the shockwaves [31]. Thus, a negative-curvature profile with a moderate adverse-pressure gradient was designed to mitigate flow separation at the designed inlet condition. The details of the design can be seen in Figure 4.  
Figure 3: Improvement in local wall curvature. Figure 2: Structure of interaction between shockwave and boundary layer. The length of the negative-curvature profile is \(a\) and acts as the upstream influence length (the distance between the starting point of pressure rise and the shock foot in the inviscid flow model); the height at point \(O^{\prime}\) is 35% of the boundary layer displacement thickness at point \(O\). Without considering the viscous effect, the pressure distribution of the negative-curvature profile is assumed to have a constant adverse-pressure gradient. Equation (1) is the constant adverse-pressure gradient distribution equation:  
\[p=k(x-x_{0})+p_{x_{0}} \tag{1}\]  
where parameter \(k\) stands for the constant adverse-pressure gradient. Based on the pressure at the initial position, the pressure distribution of the wall curve is obtained using the linear equation of the constant adverse-pressure gradient.