Asymptotic behaviour of velocity profiles in the Prandtl boundary layer theory

Abstract

Consider the Prandtl boundary layer equation for the steady two-dimensional laminar flow of an incompressible viscous fluid past a rigid wall. On the basis of an arbitrary velocity profile at some initial position on the wall, the analysis presented shows that, for a power law streaming speed U ( x ) ═ C ( x + d ) m ( m ≽ 0), the velocity profile which develops downstream is asymptotically given by the well known Falkner-Skan similarity solution. Moreover, for a streaming speed satisfying (6), the velocity profile which develops downstream is asymptotically unique, though of course the particular form of the resulting profile depends on the precise nature of the exterior stream. The rate of convergence for this asymptotic behaviour is estimated, as well as corresponding rates for the convergence of the skin friction coefficient. This result verifies the tacit assumption of a number of writers that the downstream velocity profile is essentially independent of the initial profile, and also supplies a theoretical justification for the role of similar solutions in boundary layer theory. We also prove the existence of concave velocity profiles whenever the pressure gradient is favourable. It follows that, for streaming speeds which correspond to a favourable pressure gradient, concave velocity profiles play somewhat the same role as similarity profiles do for a power law streaming speed.