This paper develops a discrete adjoint formulation for the constrained aerodynamic shape optimization in a multistage turbomachinery environment. The adjoint approach for viscous internal flow problems and the corresponding adjoint boundary conditions are discussed. To allow for a concurrent rotor/stator optimization, a nonreflective adjoint mixing-plane formulation is proposed. A sequential-quadratic programming algorithm is utilized to determine an improved airfoil shape based on the objective function gradient provided by the adjoint solution. The functionality of the proposed optimization method is demonstrated by the redesign of a midspan section of a single-stage transonic compressor. The objective is to maximize the isentropic efficiency while constraining the mass flow rate and the total pressure ratio.

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