Finite-difference solution for three- dimensional boundary layers with large positive and negative crossflows

Abstract

This paper shows how finite-difference methods can he used to analyze three-dimensional boundary layers on axisymmetric bodies with large positive and negative crossflows. The particular problem considered is the flow around a sphere in a single layer of spheres. The external flow is given by the potential solution and is obtained by doing experiments in an electrolytic tank. The computational scheme was found to be stable provided that the components of the wall velocity gradient along the meridians were positive. The solution covered regions of both positive and negative crossflows so that the angle between the limiting steamlines and the meridians varied between -4~33° and —90°. These computational results are consistent with an analysis of the stability of the numerical scheme. INITE-DIFFERENCE methods are now widely used to solve the equations for two-dimensional boundary layers. This paper shows how such techniques can be applied to an axisymmetric body having a three-dimensio nal boundary layer with large positive and negative crossflows. We consider the laminar boundary layer on a sphere in a single layer of spheres for which the external flow is given by potential theory. Because of the very large crossflows, particular attention had to be paid to the stability of the calculations. The computational scheme developed is also of interest in that the external flow was determined from experiments in an electrolytic tank and therefore is given in tabular form. Boundary-layer separation can cause the flow external to the boundary layer to be different from that predicted by potential theory. Therefore the calculated results can be expected to disagree with measurements, particularly near separation. The two methods that have been most widely used for three-dimensional boundary layers involve series expansions or integral forms of the momentum equations. (Series solutions have been reviewed by Crab tree, Kuchemann, and Sowerby.1) They are limited in that they are conveniently applied only in a region close to the birthplace of the boundary layer. Momentum integral methods have usually been formulated in an system of coordinates constructed from the projections of the external streamlines and their orthogonals. They have been most useful when a asmall crossflow can be made whereby the velocity components in the boundary layer perpendicular to the direction of the external streamlines can be assumed to be small. A review of this category of solutions is given by Cooke and Hall.2 Integral methods are not conveniently applied to the problem considered in this paper because equations for an intrinsic set of coordinates are not easily derived and because a small crossflow approximation would not be applicable. Very recently Der and Raetz,3 Hall,4 and Dwyer5 applied finite-difference methods to three-dimensional boundary layers. The difference approximations used are implicit in the direction normal to the solid surface and explicit in the tangential directions. The results of Hall and Dwyer are