Inverse Problem in Aerodynamic Shape Design of Turbomachinery Blades

Abstract

There are various methods for aerodynamic shape design in turbomachinery blades, but at the state of the art the shape design has still been a formidable problem. Optimal shape design based on adjoint method has been developed rapidly in the last decades in aeronautic field with the start of Jameson’s work. As a gradient-based optimization, the adjoint method introduces an adjoint system and the sensitivity derivative is computed by solving a linear adjoint equation, which makes the computational cost almost independent of the number of design variables. Because the adjoint method realizes the quick and exact sensitivity analysis and saves large computational resources, it has been the highlight in aerodynamic shape design of CFD field. Combining the continuous adjoint method with quasi-Newton method, we developed an optimization algorithm for turbomachinery aerodynamic design governed by two-dimensional Euler equations in this paper. The blade shape to be optimized is parameterized by non-uniform B-spline and the computational domain is discreted with H-grids. Then the adjoint equations and their boundary conditions are deduced in detail, both in computational and physical spaces, and are solved numerically by using time-marching finite difference method based on Jameson’s diffusion scheme. With the solved adjoint variables, the final expression of the cost function gradient with respect to the design variables is formulated. Finally, several numerical cases of turbomachinery blade aerodynamic design are presented and analyzed to validate the present optimization algorithm.