A comparison of Navier-Stokes solutions with the theoretical description of unsteady separation

Abstract

Numerical solutions of the unsteady two–dimensional boundary–layer and Navier–Stokes equations are considered for the flow induced by a thick–core vortex above an infinite plane wall in an incompressible flow. Vortex–induced flows of this type generally involve unsteady separation, which results in an eruption of high–vorticity fluid within a narrow streamwise region. At high Reynolds numbers, the unsteady separation process is believed to pass through a series of asymptotic stages. The first stage is governed by the classical non–interactive boundary–layer equations, for which solutions are given, and terminate in the Van Dommelen singularity. As an eruption develops, the boundary layer thickens and provokes a viscous–inviscid interaction leading to the second stage of unsteady separation. The third stage occurs when the normal pressure gradient becomes important locally within the boundary layer. In order to identify these asymptotic stages at large, but finite, Reynolds numbers, solutions of the full Navier–Stokes equations are obtained for the flow induced by a thick–core vortex. These results generally support this sequence of events; however, a large–scale viscous–inviscid interaction is found to begin at a time much earlier than allowed for by the asymptotic theory; that is it begins to occur prior to the formation of a spike within the boundary layer. Some consequences of these results on our understanding of unsteady separation are discussed.