Analytic adjoint solutions for the 2-D incompressible Euler equations using the Green's function approach

The Green's function approach of Giles and Pierce (J. Fluid Mech., vol. 426, 2001, pp. 327-345) 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.

Automated summary

The Green's function approach of Giles and Pierce is used to construct lift and drag based analytic adjoint solutions for the incompressible Euler equations in two dimensions. The drag-based solution is smooth throughout the flow domain, while the lift-based solution exhibits singularity at rear stagnation points and sharp trailing edges due to the Kutta condition. This singularity propagates to the entire dividing streamline upstream of the rear singularity through the sensitivity of the Kutta condition to changes in stagnation pressure.