I am looking to derive a conformal map for the problem illustrated in this image. I've read a bit about how to map a square onto a circle, but I'm struggling to extend the concepts for the domain at hand. I don't have a rigorous mathematical background (mech. engineer in computational fluid dynamics), so I would appreciate if someone here could advise me on the route that I should take in order to derive such a conformal mapping.

The end application is to generate a smooth computational mesh that looks like this. I have generated a mesh like this using other means, but the smoothness of the mesh vertices is not sufficient for extremely fine meshes. This results in spurious oscillations in the numerical problem I am trying to solve.