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.