Aerodynamic design of turbine blades using an adjoint equation method

Abstract

Aerodynamic design of turbine blades using an adjoint equation method is studied. Two design cases are tested. The first one is an inviscid design case for a VKI turbine stator, and the design objective is to minimize the entropy generation rate of the blade subject to a prescribed blade loading. The second case is a viscous design case for a standard configuration 4 turbine stator. The design objective is to minimize the entropy generation rate subject to a prescribed mass-averaged exit flow angle. The penalty function method is applied to deal with the constrained optimization problems. A resultant cost function is defined as a weighted sum of the original cost function and the deviation from the constraint. The formulations of the adjoint systems are derived for both cases based on the flow governing equations and the design objectives. Numerical programs are implemented to perform the optimization design. For the inviscid design case, the method is able to effectively reduce the entropy generation rate while the constraint is precisely satisfied. Reduction of shock wave strength is observed. For the viscous design case, results using the Baldwin-Lomax turbulence model and results using laminar flow solutions are presented. The program is effective for both transonic and subsonic conditions, which means the method is able to deal with frictional effects in addition to reducing shock wave strength.