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%.

中文翻译：

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基于异形端墙错流控制的定量三维逆设计规则


异形端壁为涡轮机和压缩机的端壁流量控制提供了一种新颖有效的解决方案。然而，异形端壁在压缩机中的应用仍然缺乏定量的设计规则。本文提出了一种基于三维逆法的端壁轮廓定量设计规则，并在高负载压缩机级联上对其进行了数值验证。目前的逆法可以从给定的端壁压力分布中求解相应的端壁形状，但端壁压力分布的确定在很大程度上依赖于经验知识。本研究通过 S1 溪流表面的周向平衡方程建立了流线曲率和交叉通道压力梯度 （CPG） 之间的模型，从而为确定端壁压力分布提供了定量依据。本文提出的定量设计规则表示如下：在分离开始于吸入表面 （SS） 的轴向位置，在距离 SS 0.1–0.2 节距的范围内，速度高于拐角区域平均速度的端壁边界层流体应具有与粘性子层内的流体相同的流线曲率。采用定量设计规则的逆向设计异形端壁通过施加更强的 CPG 增强了 SS 附近的局部错流，从而促进了动量相对较高的端壁边界层流体更早到达 SS，并增强了 SS 上的径向迁移。因此，增强的错流将相对较高的动量带入拐角区域，而增强的径向迁移将低动量流体从拐角区域推向中跨。 最后，逆向设计的异形端墙将半跨质量流加权平均损失降低了 4.3%。