Inverse Design of Turbomachinery Blading for Arbitrary Blade Thickness in Three-Dimensional Transonic Flow

A three-dimensional inverse design of turbomachinery blading for arbitrary blade thickness was obtained by using two periodic bound vortex sheets representing the pressure side and suction side of a blade row. The mean swirl distribution and blade tangential thickness distribution are specified in the present inverse design method. The prescribed mean swirl distribution is split into two fractions to form the strength of two bound vortex sheets. However, the designed results are uniquely determined by the specification of the mean swirl distribution and blade tangential thickness distribution, while splitting the mean swirl distribution into any two fractions for two bound vortex sheets is irrelevant. The resulting velocity field is composed of three parts: the first is sawtooth integrated from two bound vortex sheets; the second is axisymmetrical to provide an irrotational flow outside the two bound vortex sheets; and the last is potential to ensure mass conservation. The blade shape is determined from either the pressure side or suction side boundary condition, without a difference. Numerical results of a subsonic stator blade row designed by the present inverse design have been compared with three-dimensional Euler solutions and show a good agreement. For transonic calculation, a special form of retarding density was implemented to avoid transformation of the coordinate. However, due to the nonisentropic and rotational nature of shock wave, the present inverse solution does not give a correct answer after shocks. Coupling the entropy change and generation of vorticity after shocks with the present analytical formulation is recommended in the future work.