Analytic Full Potential Adjoint Solution for Two-dimensional Subcritical Flows

TL;DR

This paper derives and analyzes the analytic adjoint solutions for 2D subcritical potential flows, linking them to compressible adjoint variables and exploring the influence of the Kutta condition.

Contribution

It provides an explicit analytic formulation of the adjoint solution for 2D potential flows and connects it to compressible adjoint variables, including the effect of the Kutta condition.

Findings

Derived explicit adjoint solutions using Green's function approach.

Linked potential flow adjoints to compressible adjoint variables.

Analyzed the influence of the Kutta condition on adjoint solutions.

Abstract

The analytic properties of adjoint solutions are investigated for the two-dimensional (2D) full potential equation. For subcritical flows, the Green's function approach is used to derive the analytic adjoint solution for a cost function measuring aerodynamic force. The connection of the adjoint problems for the potential flow equation and the compressible adjoint Euler equations reveals that the adjoint potential and stream function correspond to linear combinations of the compressible adjoint variables measuring the influence of point mass and vorticity sources. The solutions for the adjoint potential and stream function corresponding to aerodynamic lift contain two unknown functions encoding the effect of perturbations to the Kutta condition. The properties of these functions are analyzed from an analytic viewpoint and also by examining numerical adjoint solutions. Based on this analysis, a possible formulation of the Kutta condition within the adjoint framework is also discussed.