On the viscous–viscous and the viscous–inviscid interactions in Computational Fluid Dynamics

Abstract

We consider here the exterior boundary value problem for compressible viscous flow around airfoils. In a first approximation, the viscosity effects are neglected at some distance to the airfoil. The unbounded domain is decomposed by an artificial boundary into a bounded computational domain (near field) and an associated far field. The complete system of conservation laws, modelling viscous flow in the near field is coupled with simplified models for inviscid flow in the far field. The use of the heterogeneous domain decomposition method including physically and mathematically justified transmission conditions at the artificial interface provides one with a quite accurate approximate solution, modelling the viscous–inviscid interaction between the two model zones. However, such a solution does not take into account the viscosity in the far field and does not satisfy the natural transmission conditions at the artificial interface (i.e. continuity of the solution and of the normal flux). In order to get some information for the a-posteriori improvement of this solution, we introduce one-dimensional transmission-boundary value problems, obtained by an appropriate dimensional reduction of the coupled problems from CFD. The one-dimensional problems are analyzed in the framework of singular perturbation theory. We consider formal asymptotic expansions to construct appropriate boundary layer corrections of the coupled problem modelling the viscous–inviscid interaction. Our one-dimensional analysis seems to allow an extension to higher dimensions and therefore could be used in the computation of the solution to the compressible Navier–Stokes problem by updating the solution of the approximation by a (degenerate) Navier–Stokes/Euler problem with boundary layer viscosity correction terms.