A fully-simultaneous method is based on the assumption that there is no precise hierarchy between viscous and inviscid solutions. This statement has been proved with the triple-deck boundary layer theory (Stewartson, 1974). The simultaneous solution of the two sets of equations is performed by a Newton-like iteration process after assuming linearizations of the type Eq. 8 and 9 (direct method).
The response to small perturbations of a (generally) non-linear system is linear. Perturbations to the Eqs. 3 and 4 (strongly coupled solutions) yield respectively
The simultaneous solution of Eqs. 16 and 17 at the current point
The solution of Eq. 18 and 19 involves the inversion of the operator , which is a full matrix. The essentially newtonian structure of this coupling scheme allows the simultaneous solution of the inviscid and viscous flow equations. For the boundary layer only integral equations have been used so far, both Euler equations (Drela-Giles, 1987), panel methods (Drela, 1988) and full potential equation.
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