This paper presents an adjoint optimization technique and its application to the design of a transonic turbine cascade. Capable of a quick and exact sensitivity analysis and using little computational resources, the adjoint method has been a focus of research in aerodynamic shape design optimization. The goal of this work is to extend the adjoint method into turbomachinery design applications for viscous and compressible flow, and to further improve the aerodynamic performance. In the work, the minimization of the entropy generation rate with the mass flow rate constraint was considered as the cost function of the optimization, and was applied in the direct design process. The adjoint boundary conditions of the corresponding cost function were derived in detail, using the nonslip boundary condition on the blade wall, while the flow viscous effect on the cascade inlet and outlet was neglected. Numerical techniques used in Computational Fluid Dynamics (CFD) were employed to solve the adjoint linear partial difference equations. With the solved adjoint variables, the final expression of the cost function gradient with respect to the design variables was formulated. Combined with quasi-Newton algorithm, an aerodynamic design approach based on the adjoint method for turbine blades was presented, which was independent of the Navier–Stokes solver being used. Finally, to validate the present optimization algorithm, the aerodynamic design cases of a transonic turbine blade with and without mass flow rate restriction were performed and analyzed.

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