New Approach to Inviscid Flow/ Boundary-Layer Matching

Abstract

M problems in fluid dynamics require a of flow solutions that are calculated in adjacent regions governed by different types of equations. A classical problem of this kind is the calculation of unseparated flow about an airfoil, requiring the of inviscid flow and boundarylayer solutions. In the latter case, the governing equations are generally nonlinear and their solution is further complicated by the fact that surface velocity and boundary-layer displacement thickness must simultaneously satisfy the inviscid flow and boundary-layer boundary value problems. The process that determines the particular combination of surface velocity and boundary-layer displacement thickness satisfying simultaneously the inviscid flow and boundarylayer problems is termed matching in this Note. In other words, is defined here as the simultaneous solution of linear or nonlinear boundary value problems coupled through their boundary conditions. In that sense, is distinguished from the applied in the context of perturbation theory. > The latter provides the pertinent flow equations and conditions, but does not, in general, give a solution of the resulting coupled boundary value problems. The classical approach to the calculation of the flow about an airfoil, including both inviscid and boundary-layer domains, is an iterative process that usually begins by calculating the inviscid flow without any representation of viscous flow effects. This provides an initial estimate of the surface velocity, which is then used as the basis for a boundary-layer calculation. The latter provides a first estimate of the boundary-layer displacement thickness. It is well known that the effect of the boundary-layer on the inviscid flow can be approximately taken into account by adding that displacement thickness to the airfoil surface geometry and repeating the inviscid flow calculation. The iteration is frequently terminated at this point, and the results so obtained do provide a useful first-order correction for flows that are only weakly interactive. However, in many flow problems the inviscid flow/boundary-layer interaction is not weak, and it is frequently desired to continue the iteration to seek a fully converged solution. The conventional approach of recalculating the inviscid and boundary-layer flows in successive fashion rarely converges without some assistance, and when it does the convergence is relatively slow. The present Note shows why the conventional approach so frequently fails and demonstrates a new method of that seems to be clearly superior.