Newton Solution of Coupled Euler and Boundary-Layer Equations

At present most methods for the numerical calculation of steady viscous-inviscid interactions iterate between an inviscid solver, which calculates the outer flow using either potential or time marching Euler methods, and a viscous solver, which calculates the boundary-layer flow using integral or finite-difference methods. In direct-coupling schemes the inviscid solver calculates the edge pressure gradient dp/dξ, which is passed to the viscous solver, which then calculates the surface displacement thickness δ*, which is passed back to the inviscid solver. With adverse pressure gradients present, this iterative procedure either is very slow or fails outright. A faster and more robust method is the semiinverse coupling scheme, in which δ* is specified for both the inviscid and the viscous solvers as developed by Carter [13] and Wigton and Holt [343]. The resultant mismatch in the pressure gradients dp/dξ, from the two solutions is used to update the specified values of δ*. The quasi-simultaneous technique of Veldman [358] is a more refined method where the viscous equations and a small-perturbation representation of the inviscid flow are solved simultaneously at each iteration. The convergence rate in all these schemes is limited by the accuracy of the δ* updating formula or the small-perturbation assumption, the latter becoming invalid in shocked flows.