Effect of Green's function on aerodynamic lift and drag

Question

I am trying to determine the effect of inserting a Green's function on the lift and drag exerted on a body inmersed in an inviscid, incompressible and irrotational flow in 2D. One concrete example is the vortex $\vec{v}=\Gamma\nabla \theta/2\pi$ where $\theta$ is the polar angle or the point source $\vec{v}=Q\nabla Log(r)/2\pi$. By placing it close to a body immersed in fluid flow, it alters the flow and hence the forces on the body. The question is how to determine the new forces.

Answer 1

It turns out that the procedure I was using was correct, but I had missed the contribution of the Kutta condition (extra lift that has to be added to prevent singularities at sharp trailing edges). With that additional contribution, analytic adjoint solutions can be computed as shown in

https://arxiv.org/abs/2201.08128