A Theory of the Two-Dimensional Laminar Boundary Layer Over a Curved Surface

Abstract

The purpose of this paper is to present a theory to account for surface curvature effects on the two-dimensional boundary-layer flow which approaches a potential flow at free stream. The problem of two-dimensional viscous flow is first formulated by using the streamlines and their orthogonal trajectories as the generalized coordinates. A boundary-layer approximation is applied to the Navier-Stokes equations and the Gauss equation in the generalized coordinates to yield the boundary-layer equations. The conditions under which similar solutions of the boundary-layer equations exist are determined. By a simple transformation, the governing differential equation can be expressed in a form which reduces to the Falkner-Skan equation for zero surface curvature. Numerical results for a similar solution which corresponds to a flow over a curved surface with zero surface pressure gradient have been obtained. The velocity profiles in the boundary layer and the wall skin-friction distribution for concave and convex surfaces are presented. The wall skin friction for a convex wall is found to be higher than the Blasius value for a flat plate. On the other hand, for a concave wall, the skin friction will drop below the Blasius value as the curvature increases, but it appears to reach a minimum, and beyond this minimum point it will increase again. The same flow problem was treated by Murphy by a different method of analysis. Comparison of Murphy's results with those obtained by the present method reveals some basic differences in the boundary-layer characteristics. In particular, Murphy's results indicate that the wall skin friction for a convex surface is smaller than the Blasius value, while for a concave wall it is higher.