Ordinary differential equations for the adjoint Euler equations

Abstract

Ordinary differential equations are derived for the adjoint Euler equations first using the method of characteristics in 2D. For this system of partial-differential equations, the characteristic curves appear to be the streamtraces and the well-known [Formula: see text] and [Formula: see text] curves of the theory applied to the flow. The differential equations satisfied along the streamtraces in 2D are then extended and demonstrated in 3D by linear combinations of the original adjoint equations. These findings extend their well-known counterparts for the direct system and should serve analytical and possibly numerical studies of the perfect-flow model with respect to adjoint fields or sensitivity questions. In addition to the analytical theory, the results are demonstrated by the numerical integration of the compatibility relationships for discrete 2D flow fields and dual-consistent adjoint fields over a very fine grid about an airfoil.