Viscous-Inviscid Interaction

• Semi-Inverse Methods

The boundary layer equations are solved in the inverse mode, that is: a guess in the boundary layer development yields an estimate of velocity gradient distribution at the matching surface. The mismatch between viscous and inviscid solutions is used in an updating formula to improve the results, Fig. 1. By using the linearizations Eqs 8 and 9 (given in direct method), the interaction equation is

The most rigorous approach to the semi-inverse method is due to Le Balleur. He used a local Fourier analysis of the small perturbation equations and the integral boundary layer equations to study the stability of his semi-inverse method (Le Balleur, 1978). His correction formula is a particular case of Eq. 12

where is the Prandtl-Glauert factor ( ), and is the largest Fourier wave number. If is the length scale of the grid, then . Eq. 13, that was originally developed at maximal wave number, has been found convergent at all wave numbers.

The method of Carter is a prime integral of Eq.12. The difficulty of implementing Eq. 13 with the differential boundary layer equations lies in the fact that the coefficients of the linearized viscous operator are unknown. This is the reason why a rational semi-inverse method with this type of equations has never been developed. Thus, most of the people working with differential equations have resorted to a quasi-simultaneous method.