Assume two linear systems coupled by their boundary conditions. Given a first guess solution and corresponding to a first guess of the boundary condition and , respectively, the converged solution can be obtained in one iteration.
This process is still not satisfactory, because the sensitivities are unknown. They can be computed by a previous sweep for both viscous and inviscid solutions.
The cases we are interested in are strongly non linear in their BC, that is: to a small perturbation in the boundary conditions corresponds a (generally) large perturbation of the solution. Furthermore, if a solution does exist, it might not be unique. Here two different cases might occur: there is a multiple solution through the same point; there are multiple solutions through different points. Non-unique solutions have been found at laminar separation and reattachment points, and more generally in decelerating flows, i.e. flows with adverse pressure gradients (Drela, 1985).
The problem schematically formulated is unlikely to be solved in the general case. Except for the study at separation and reattachment points, the literature is unclear as to how the viscous and inviscid solutions behave. Therefore, the problem requires specific algorithms, whose nature is the object of the following analysis. The interaction procedure has to take into account the different nature of the equations: boundary layer equations essentially parabolic, and inviscid flow equations essentially elliptic (for subcritical flows).
At the matching surface the velocity gradient of the inviscid flow is related to the non-dimensional source at the wall. For the viscous flow Eq. 1 has the following shape:
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