## 5 Conclusions  
An improved method to reduce the boundary layer separation loss induced by the first-passage shock foot is investigated for a supersonic compressor cascade with a design inlet Mach number of 1.75. It employs a negative-curvature profile with a constant adverse-pressure gradient ahead of the position of the impingement of the first-passage shock on the suction surface. The numerical results show that the aerodynamic performance of the modified cascade is superior to that of the baseline cascade under design and off-design conditions. In addition, the comparison of flow field characteristics between the baseline model and the modified model provides a detailed analysis of the flow physics of negative-curvature ramps. Some valuable conclusions from this investigation are as follows:  
1. Under design conditions, the negative-curvature ramp splits the first-passage shock foot into a series of compression waves and a weaker-passage shock foot, which reduces the strong adverse-pressure gradient generated by the shock. The shock-induced flow separation is significantly suppressed, resulting in less loss from the interaction between the shockwave and the boundary layer;
2. The heavy adverse-pressure gradient induced by the first-passage shock of the improved cascade is divided into three sections with a moderate pressure rise. There is an increasing tendency with a constant pressure gradient on the negative-curvature profile, which demonstrates the feasibility of the initial design method;
3. Under off-design conditions, the overall aerodynamic performance of the improved supersonic compressor cascade is superior to that of the baseline cascade at various inlet Mach numbers and exit pressure conditions, and the negative-curvature smooth ramp is compatible to some extent.  
It should be noted that the design method of the negative-curvature profile with a constant adverse pressure may not be the best scheme. An optimized design method and experimental verification of the local negative-curvature section will be investigated in subsequent research. The control effects of different local negative-curvature profiles and multiple individual three-dimensional negative-curvature structures on the shock-induced separation will be compared and analyzed. Finally, the design method will be improved and modified according to the experimental data.  
Conceptualization, Q.Z.; Methodology, Y.L. and J.X.; Software, Z.F.; Formal analysis, Q.Z.; Investigation, Y.L.; Data curation, Z.F.; Writing--original draft, Y.L.; Writing--review & editing, W.C.; Supervision, J.X.; Project administration, Q.Z.; Funding acquisition, W.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the national natural science foundation of China grant number [52076124].  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
This work has been carried out with the support of the National Natural Science Foundation of China (No. 52076124). The authors would like to thank them for funding this work and for their permission to publish this paper.  
The authors declare no conflicts of interest.