Orthogonal Shape Functions and Progressive Optimization for the Design of 2D Transonic Cascades

Abstract

This paper describes the main ingredients of a recently-developed very efficient formulation for the inverse design of turbomachinery blades in transonic flow conditions. The flow solution is obtained by means of a highly accurate, basic discretization. A discrete adjoint formulation, using a dissipative auxiliary discretization, is employed for a robust and efficient evaluation of the sensitivities, even in presence of shocks. The progressive optimization relies on the simultaneous convergence of the flow computation, of the adjoint problem and of the optimization process itself, as well as on the use of progressively finer grids; the adjoint equations are always solved at the coarsest level, with a further work reduction. The novel contribution of this paper consists of the development of a method for the generation of a set of orthogonal shape functions for a wide and efficient definition of the blade profile. The proposed approach is tested by generating two sets of ten orthogonal curves for the inverse design of a two-dimensional turbine blade in inviscid transonic flow conditions, with fixed stagger angle and with fixed exit angle. The numerical results demonstrate that the methodology is robust and highly efficient, with a converged design optimization produced in no more than the amount of computational work needed to perform from 3 to 5 converged flow analyses on the finest grid.