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

Abstract

Abstract This paper advances the study of the shock phenomena in axial flow compressors to that of the behavior and effects of the rotor passage shock. Based on the design results of the S2m streamline curvature through flow inverse problem and blading of the axial flow compressors, taking the relative supersonic streamlines of the rotor as equivalent to a group of layers of the quasi-one-dimensional duct flow, this paper deduced a variational principle of the normal passage shock, interrupting the flow actually, stationed inside each layer of the rotor passage. It is found that the factors affecting the stationarity of the rotor passage shock include the variable cross-sectional area, the frictional and other on way losses, and the variable rotational radius of the duct flow. According to the variational principle, the stationary locations of the shock in these equivalent duct flows affected by three factors are obtained by the momentum relaxation method, and the location stability of these shocks is analyzed. In the applications to various types of transonic axial compressor rotors, first, the discontinuous entropy generation loss distributions along the cascades of each supersonic layer are set, to consider the boundary layer, oblique shock, normal passage shock, shock boundary layer interference, and trail edge losses. Second, applying the variational principle to the duct flows affected by these three factors for each streamline, all the shock locations that possess location stability are detected. Third, by comparing with the flow field results of the direct problem of Computational Fluid Dynamics, the dimensionless distribution law of the real entropy generation loss along the layer cascades is decided. Finally, by combining the shock lines from each equivalent duct flow corresponding to each streamline, a curved surface structure of the normal passage shock in a rotor passage is established. In the given design examples of three kinds of axial compressor stages, the three-dimensional structures of the normal passage shock obtained by this method are consistently in good agreement with the results of the direct problem of Computational Fluid Dynamics. These afford a first-term verification to this method for its effectiveness and wide applicability. This method provides a theory and a technology, in the through flow and blading inverse problem design phase of an axial compressor, to quickly predict the location and the curved surface shape of the normal passage shock, and to characterize approximately and evaluate relatively whether the design surge margin of a transonic stage is sufficient.