In 1950, Wu  suggested that the three-dimensional inverse (design) problem of turbomachine flow may be solved approximately by a Taylor series expansion in the circumferential direction based on the known flow variation over an S2 stream surface in the midchannel of the blade passage. This idea has been realized recently. A new coordinate transformation has been developed. The coordinate surfaces are coincident with the S2 stream surfaces. This transformation leads to a new method to calculate the aerodynamic equations of three-dimensional flow. By the use of this transformation, a high-order expansion is realized to determine the shape of the blade surfaces from the fluid state on the S2m stream surface directly. Computation in this manner soon leads to the discovery that theoretically the distribution of flow parameter (usually Vθr) on S2m prescribed by the designer should satisfy a constraint condition, which guarantees that the S1 stream surfaces along the hub and shroud obtained from circumferential extension of the S2m surface are surfaces of revolution. An approximate method is suggested to meet this condition.
A Simple Method for Solving Three-Dimensional Inverse Problems of Turbomachine Flow and the Annular Constraint Condition
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Zhao, X., Sun, C., and Wu, C. (April 1, 1985). "A Simple Method for Solving Three-Dimensional Inverse Problems of Turbomachine Flow and the Annular Constraint Condition." ASME. J. Eng. Gas Turbines Power. April 1985; 107(2): 293–300. https://doi.org/10.1115/1.3239714
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