A Simple Method for Solving Three-Dimensional Inverse Problems of Turbomachine Flow and the Annular Constraint Condition

In 1950, Wu [1] 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 S_2m stream surface directly. Computation in this manner soon leads to the discovery that theoretically the distribution of flow parameter (usually V_θr) on S_2m 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 S_2m surface are surfaces of revolution. An approximate method is suggested to meet this condition.