The Variational Principle of a Rotor Inner-Passage Shock in the Circumferential Average Through-Flow Inverse Problem of Axial Compressors and Applications

This paper presents an application and validation case for the recently obtained variational principle of a shock stationed in a duct. The streamline curvature method for the circumferentially averaged through-flow and blading design inverse problem remains fundamentally used in current axial compressor design systems and is indispensable as the generator of multi-stage blade coordinates. However, this method inherently smoothens flow discontinuities and thus, to date, cannot provide the stage stall margin, the key performance indicator most critical in the adjustment of high-loading stages, requiring instead a time-consuming CFD validation afterward. Leveraging the variational principle for shock stationarity, this paper acquires a method to show efficiently the stage stall margin by visualizing rotor passage shock rapidly. In the general coaxial rotating relative motion, by modeling the transonic streamlines as a set of layered quasi-one-dimensional duct flows, a variational principle of flow impulse potential energy for the stationary normal shock is derived. It is found that the factors governing the stationarity and location of the normal shock in relative motion include the variable cross-sectional area, the frictional and other on-way losses, and the variable rotational radius of the duct flow. In the applications to transonic rotor cascades, the frictional and other on-way losses are prescribed. First, the discontinuous entropy generation distributions along the cascades of each transonic layer are set to consider the boundary layer, oblique shock, normal passage shock, shock–boundary layer interference, and trailing edge losses. Second, with the total streamline loss fixed by the through-flow design, all shock locations possessing positional stability are determined via the variational principle for each streamline. Third, by comparing with CFD direct problem resu lts, a dimensionless rule governing the actual entropy generation distribution along the layer cascades is established. In three kinds of design cases of axial compressor stage, this method yields consistently 3D curved-surface structures of passage shock that agree well with CFD direct problem solutions, demonstrating its effectiveness and a certain applicability.