Ordinary differential equations for the adjoint Euler equations

TL;DR

This paper derives and demonstrates ordinary differential equations for the adjoint Euler equations in 2D and 3D, providing a foundation for analytical and numerical sensitivity analysis in fluid flow models.

Contribution

It introduces a method to derive ODEs for the adjoint Euler equations using characteristics, extending known results from the direct system to adjoint fields in 2D and 3D.

Findings

Derived ODEs along streamtraces for adjoint Euler equations in 2D.

Extended the ODE framework to 3D adjoint equations.

Numerical validation performed on flow around an airfoil.

Abstract

Ordinary Differential Equations are derived for the adjoint Euler equations firstly 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 C+ and C- 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. Beside 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.