Recent developments of finite-difference approximations for boundary-layer equations

Abstract

Finite-difference solutions for Prandtl's boundary-layer equations are described for steady, two- and three-dimensional laminar and turbulent flows. For three-dimensional flows only boundary sheets are considered and curvature effects in the direction normal to the wall are being neglected. The governing equations are presented in form of a matrix-vector equation. Its numerical stability is discussed for elementary finite-difference molecules. Non-orthogonal coordinates are shown to affect the stability limits for the convective terms. If the momentum equations and/or the energy equation are decoupled by splitting the main part of the differential equations additional conditions must be observed for stable solutions. Finite-difference approximations with truncation error of fourth order are introduced to enable either increased accuracy or shortened calculation times, in particular, for three-dimensional problems. Studies of the behaviour of the overall error of the solution and several applications to realflow situations supplement the general considerations. Finally, a brief discussion is given for second-order closure problems.