An approximate solution of two flat plate boundary layer problems

Abstract

The method presented here for obtaining an approximate solution of the laminar boundary-layer equations is based on the iteration process of Piercy and Preston. It leads to a simple analytical approximation of good accuracy for Blasius' solution of the boundary-layer flow past a flat plate. The main purpose of this paper is, however, the application of the method to a generalisation of Blasius' problem, namely the case of a flat plate in a uniform stream when there is a suction velocity normal to the plate proportional to xpower-1/2 where x is the distance along the plate from its leading edge. This generalisation was first given by Schlichting and Bussmann, and has also been considered by Thwaites and Watson. For the simpler problem of the flat plate in a uniform stream it is well known that by means of Blasius' transformation the solution is obtained from that of a third-order non-linear differential equation. The iteration method of Piercy and Preston for the solution of this consists in replacing the velocity where it occurs.in the equation by an inferior approximation and solving the resultant linear equation to obtain a superior approximation. To start the process the velocity was assumed to be that of the stream, giving Oseen's solution as the next approximation. Here the start is made in a different manner. We take as the initial approximation to the velocity one of two choices - (i) a constant value or (ii) a linear function-and in either case have a parameter at our disposal. The iteration is performed, giving a second approximation containing this parameter, which we then determine by substituting the second approximation in the momentum equation. The necessary integrations can be performed analytically, and the quantities which characterise the boundary layer are readily determined.