High Reynolds number Navier-Stokes solutions and boundary layer separation induced by a rectilinear vortex

Abstract

We compute the solutions of Prandtl's and Navier-Stokes equations for the two dimensional flow induced by a rectilinear vortex interacting with a boundary in the half plane. For this initial datum Prandtl's equation develops, in a finite time, a separation singularity. We investigate the different stages of unsteady separation for Navier-Stokes solution at different Reynolds numbers $Re=10^3-10^5$, and we show the presence of a large-scale interaction between the viscous boundary layer and the inviscid outer flow. We also see a subsequent stage, characterized by the presence of a small-scale interaction, which is visible only for moderate-high Re numbers $Re=10^4-10^5$. We also investigate the asymptotic validity of boundary layer theory by comparing Prandtl's solution to Navier-Stokes solutions during the various stages of unsteady separation.