Some Remarks on Integral Moment Methods for Laminar Boundary Layers with Application to Separation and Reattachment

Abstract

It is well known that the Kármán-Pohlhausen integral method is a rather poor approximation for the analysis of laminar boundary layers in regions of adverse pressure gradient, particularly when separation occurs. Perhaps not so well known, however, is the fact that the Kármán-Pohlhausen method may be completely inadequate downstream of separation, between the separation and reattachment points. When a flow disturbance, such as a forward facing step or incident shock wave, is of sufficient strength to cause extensive separation, the static pressure variation along the surface takes the general appearance shown in Figure 1. The region where the static pressure is virtually constant (plateau) gives rise to much of the difficulty, since the Kármán-Pohlhausen method must produce an attached, Blasius type velocity profile whenever the pressure gradient vanishes. Hence the Kármán-Pohlhausen method must predict reattachment upstream of the plateau, whereas in reality it occurs downstream of the plateau. Apparently what is needed is an integral method which exhibits velocity profiles containing reverse-flow for vanishingly small adverse pressure gradients analogous to the "lower branch" solutions of the Falkner-Skan equation, which were found by Stewartson. The purpose of the present report is to demonstrate that the method first proposed by Walz and modified by Tani does indeed produce velocity profiles with reverse flow, even in the limit of constant pressure, and would, therefore, appear to be a very promising method for predicting the behavior of separated flows. Furthermore this method eliminates the need for a certain amount of empiricism inherent in other existing methods, such as is required at the present time with the Crocco-Lees theory.