A quantitative three-dimensional inverse design rule based on cross-flow control of profiled end wall

Profiled end wall provides a novel and effective solution for end wall flow control in turbines and compressors. However, the application of profiled end walls in compressors still lacks quantitative design rules. This paper presents a quantitative design rule for end wall profiling based on a three-dimensional inverse method and numerically validates it on a highly loaded compressor cascade. The current inverse method can solve the corresponding end wall shape from a given end wall pressure distribution, but the determination of the end wall pressure distribution heavily relies on empirical knowledge. This study establishes a model between streamline curvature and cross-passage pressure gradient (CPG) through the circumferential equilibrium equation in the S1 stream surface, thereby providing a quantitative basis for determining the end wall pressure distribution. The quantitative design rule proposed in this paper is expressed as follows: at the axial position where the separation begins on the suction surface (SS), within the range of 0.1–0.2 pitch away from the SS, the end wall boundary layer fluid with a higher velocity than the corner region average velocity should possess the same streamline curvature as the fluid within the viscous sublayer. The inverse-designed profiled end wall using the quantitative design rule enhances the local cross-flow near the SS by imposing a stronger CPG, thus encouraging the end wall boundary layer fluid with relatively higher momentum to arrive at the SS earlier and enhancing the radial migration on the SS. Consequently, the intensified cross-flow entrains relatively higher momentum into the corner region, while the enhanced radial migration drives the low-momentum fluid away from the corner region towards the midspan. Finally, the inverse-designed profiled end wall reduces the half-span mass-flow-weighted average loss by 4.3%.