Non-Integrated Far Field Boundary Conditions
 

I'm implementing low speed preconditioning into a 2D external flow solver, but I am having difficulty with the far field boundary conditions. The standard Riemann invariant treatment does not work because preconditioning changes the characteristics of the system. I have yet to see a closed form solution for the preconditioned compatibility relations.

Turkel in a few of his papers and also a recent paper by Hejranfar use a non-integrated form of the compatibility relations. However, I am having trouble understanding how to first figure out if the flow is entering or leaving the domain since I no longer can explicitly solve for the normal velocity at the boundary. They always have a free stream value, an extrapolated value, and an unlabeled value in their derivations. What value to assign to the unlabeled value confuses me. Does anyone have any experience or advice on this problem?

Note, I found I can get fairly accurate results setting all of the far field boundary to the free stream values (as one of Van Leer's students did), but that limits applications, domain size, etc. down the road.

Thanks.