A Continuous Adjoint Method for the Minimization of Losses in Cascade Viscous Flows

Abstract

A continuous adjoint formulation for the minimization of viscous losses in laminar cascade flows is presented. The losses are expressed in terms of entropy generation due to the boundary layer formation and development. The minimization of the entropy difference between the inlet to and outlet from the flow domain results from the minimization of a field integral, expressed in terms of the velocity gradient. For the latter, appropriate field adjoint equations along with boundary conditions are derived, leading to sensitivity derivatives depending only upon wall boundary terms. The Lagrange multiplier penalty method is used to handle geometrical constraints related to the minimum allowed thickness of the designed cascade airfoils. For the sake of comparison, a discrete adjoint method was also programmed and used for the solution of the same problem, in which the total pressure losses, instead of the entropy increase, was used as the objective function.