Analytic Adjoint Solutions for the 2D Incompressible Euler Equations Using the Green's Function Approach

TL;DR

This paper develops analytic adjoint solutions for 2D incompressible Euler equations using Green's functions, revealing simple forms for drag and singularities in lift solutions related to flow features.

Contribution

It introduces a Green's function-based method to derive explicit adjoint solutions for lift and drag in 2D Euler flows, highlighting singularities and sensitivities.

Findings

Drag adjoint solution is simple and smooth.

Lift adjoint solution exhibits singularities at rear stagnation points.

Singularities influence upstream flow sensitivities.

Abstract

The Green's function approach of Giles and Pierce is used to build the lift and drag based analytic adjoint solutions for the two-dimensional incompressible Euler equations around irrotational base flows. The drag-based adjoint solution turns out to have a very simple closed form in terms of the flow variables and is smooth throughout the flow domain, while the lift-based solution is singular at rear stagnation points and sharp trailing edges owing to the Kutta condition. This singularity is propagated to the whole dividing streamline (which includes the incoming stagnation streamline and the wall) upstream of the rear singularity (trailing edge or rear stagnation point) by the sensitivity of the Kutta condition to changes in the stagnation pressure.