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This paper develops a continuous adjoint formulation for the aerodynamic shape design of a turbine in a multi-stage environment based on S 2  surface governed by the Euler equations with source terms. First, given the general expression of the objective function, the adjoint equations and their boundary conditions are derived by introducing the adjoint variable vectors. Then, the final expression of the objective function gradient only includes the terms pertinent to the physical shape variations. The adjoint system is solved numerically by a finite-difference method with the Jameson spatial scheme employing first and third order dissipative flux and the time-marching is conducted by Runge-Kutta time method. Integrating the blade stagger angles, stacking lines and passage perturbation parameterization with the Quasi-Newton method of BFGS, a gradient-based aerodynamic optimization design system is constructed. Finally, the application of the adjoint method is validated through the blade and passage optimization of a 2-stage turbine with an objective function of entropy generation. The efficiency increased by 0.37% with the deviations of the mass flow rate and the pressure ratio within 1% via the optimization, which demonstrates the capability of the gradient-based system for turbine aerodynamic design.

Aerodynamic Optimization Design of Multi-stage Turbine Using the Continuous Adjoint Method