An asymptotic approach to the separation of two-dimensional laminar boundary layers

The problem of laminar boundary layer separation is examined using asymptotic methods to identify some overall features of an unsteady boundary layer near a separation point. An asymptotic solution for large distances from the wall is constructed for both steady and unsteady flows. These solutions show that unmatchability can only occur if the boundary layer solution develops a singularity with respect to x for the steady case and in the x-t plane in the unsteady case. The asymptotic solution is then used in conjunction with approximate analyses for the flow near the wall. For steady flows, the unmatchability condition is shown to coincide with the customary criterion of vanishing wall shear. But the real criterion is that the shear should have a singularity, which is of the Goldstein type and happens to occur at zero shear. The same methods of analysis applied to unsteady flow suggest that singular behavior may develop in different ways.