Global Interactive Boundary Layer (GIBL) for a channel

We consider a laminar, steady, two-dimensional flow of an incompressible Newtonian fluid in a channel at high Reynolds number. When the walls are slightly deformed, adverse pressure gradients are generated and separation can occur. The analysis of the flow structure has been done essentially by Smith [7]. Later, a systematic asymptotic analysis has been performed by Saintlos and Mauss [6]. More recently, the modelling of channel flow has been examined by Lagrée et al. [5]. These analyses show that there is no external flow region and the asymptotic models for the flow perturbations are mainly based on an inviscid rotational core flow region together with boundary layers near the walls; a comprehensive discussion of this structure can be found in Sobey [8]. Here, we use the Successive Complementary Expansion Method, SCEM, in which we assume a uniformly valid approximation (UVA) based on generalized expansions. This method, developed by Cousteix and Mauss [1, 2], has been used by Dechaume et al. [4]. The first step consists of an inviscid approximation which applies far from the walls. This approximation must be improved near the walls by adding a correction which takes into account the effects of viscosity. Thanks to generalized expansions, a strong coupling occurs between the viscous and inviscid regions. This notion is called “interactive boundary layer” (IBL). This means that the effect of the boundary layer on the inviscid flow and the reciprocal effect are considered simultaneously. The construction of the UVA does not require any matching principle, only the boundary conditions of the problem are applied.