## 1 Introduction  
Aero-propulsion systems emphasize a higher thrust-to-weight ratio along with durability. The key parts are the compressor and fan. The higher thrust emphasizes the change in flow from subsonic to transonic and then to supersonic. The increment in relative speed archives the aerodynamic penalties along with the generation of shockwaves. The main separation is induced by the strong shockwave/boundary layer interaction (SWBLI) inside the vane passages [1,2]. It has been confirmed that about 30% of the total pressure loss comes directly from SWBLId [3; 4] since the aerodynamic performance of high-payload compressor vanes is dependent on the flow around the vanes and the overall configuration of the shock system. Thus, the supersonic cascade passage has a significant influence on aerodynamic performance. This influence is regarded as the viscous loss of the SWBLL. Therefore, it is necessary to suppress the boundary layer separation induced by the shock foot along with the pressurization of the strong shock system in the supersonic compressor cascade [5; 6]. The loss of a shock-induced flow separation depends on the Mach number of the inflow and the local curvature of the vane surface [7; 8; 9]. The concave-convex curvature is closely related to the boundary layer structure [10; 11]. The change in wall curvature can alter the growth and rupture of flow vortices within the boundary layer, affecting the stability of the boundary layer [12; 13; 14]. Suppressing shock-induced boundary layer separation losses through local blade curvature changes is the most suitable method. The curvature variation of a body surface is important in the interaction of a shockwave and a boundary layer along with the alteration in the streamwise pressure profile. It has been observed that variation in a local surface curvature close to the shock impingement position provides potential revenues for the reduction in shock-induced flow separations. A type of structure called a shock control bump (SCB) has been found to obtain global aerodynamic gains from local vane surface curvature modifications, which first emerged in the late 1970s as a local humped profile of a transonic wing aerofoil to mitigate high-shock strength and wing drag [15]. In general, there exist three groups of SCB in its applications. The first one mainly focuses on transonic and supercritical wings with a beneficial smearing effect on the near-normal shockwave close to the wing surface. A suitable SCB structure would be employed to transform an original single stronger shock on the baseline suction surface to a set of weaker (oblique) shocks or compression wave systems on the modified suction surface. This would subsequently lead to a lower loss of stagnation pressure gained for a greater decrease in wave drag [16; 17]. In previous studies, a series of optimization design techniques have been adopted to provide 2D and 3D SCB profiles to improve their performances [18; 19; 20]. Furthermore, researchers have balanced the comprehensive performance of design points and variable operating conditions through adjoint optimization and numerical optimization methods [21; 22; 23]. The second group of SCB applications, first reported in 2005, are concentrated on a supersonic/hypersonic mixed intake. The last single normal shock can be transitioned to a \(\lambda\)-shock structure with local flow deflection by a 3D SCB structure. The concept of a deformable 2D bump was introduced in the design of a hypersonic inlet with the formation of an adaptive curvature of a local passage wall to suppress the area of shock-induced flow separations and widen the range of the Mach number. A comparison of experiments and computations showed that shock-induced flow separation was suppressed with a precompression effect on the windward section of a bump with a height 0.33 times that of the local boundary layer thickness [24; 25]. The third group of SCB applications, first reported in 2015, concerns transonic compressors in turbomachinery.