For a given first solution of the inviscid flow we provide as boundary conditions for the boundary layer equations (this means that the boundary layer is solved directly). The boundary layer solution provides a new source , and hence a surface transpiration velocity that is used to update the boundary conditions of the inviscid flow. From here the process is repeated according to the scheme of Fig. 1 below, that obeys to the operator of
Eq. 7 implicitly assumes . This is in fact a critical assumption. To illustrate the convergence and the effect of under-relaxation on this process we consider a local linearization of the operators and given in strongly coupled methods (Eqs. 3-4)
where is the inverse operator of : . This linearization is shown in Fig 1. Analytic expressions of the coefficients of Eq. 9 have been effectively found for integral boundary layer equations (Lock-Williams, 1987). It has been proven that the gradient changes sign at separation point, being negative/positive for attached/separated boundary layers, respectively. For differential equations the coefficients have never been found, and it is likely that only numerical experimentation can lead to a rational convergence analysis. However, we assume that this is property inherent to the boundary layer development, rather than to the type of the equations that describe it. A local Fourier analysis of the small perturbation equation (Le Balleur, 1978) has shown that is uniformly increasing with the source . From geometric arguments we find:
Under-relaxation is used for accelerating the convergence of the process when the flow is still attached, and to obtain convergence in situations where the flow, although attached, would not converge, due to the large adverse pressure gradients (as indicated by the slope ). If , the under-relaxation formula is
For linear problems convergence can be obtained in one step. Strictly speaking is a local property of the viscous-inviscid coupling. This can be evidenced by a vector analog of the above equations.
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