Discrete Adjoint of the Navier-Stokes Equations for Aerodynamic Shape Optimization

Kurzfassung

A discrete adjoint of the Navier-Stokes equations has been developed in the unstructured finite-volume solver the DLR-TAU-code. The method consists of the explicit construction of the exact Jacobian of the spatial discretization with respect to the unknown variables allowing the adjoint equations to be formulated and solved. A wide range of the spatial discretizations available in TAU have been differentiated, including the Spallart-Almaras-Edwards one-equation, and the Wilcox k-omega two-equation turbulence models. The aim of this paper is to give an overview of the capabilities of the discrete adjoint to perform aerodynamic shape optimization in viscous flow. The strategy developed is extensively validated on 2D cases. The adjoint based design method is first validated by comparing the gradients of the drag, lift and pitching moment it produces with the approximate gradients obtained by finite-differences. Then the accuracy and efficiency of the approach are demonstrated for transonic airfoil design by considering geometric as well aerodynamic constraints, single- as well as multi-point design. Finally, the flap design of a multi-element airfoil in take off configuration confirms the capability of the discrete adjoint to solve wide range of aerodynamic problems.