A Rayleigh instability in a vortex-induced unsteady boundary layer

Abstract

Numerical solutions of the unsteady, two-dimensional Navier–Stokes equations are considered for the flow induced by a thick-core vortex with a finite difference and spectral element code. The adverse pressure gradient imposed on the surface by the vortex leads to the development of inflectional velocity profiles and the formation of a recirculation region within the boundary layer, and the presence and nature of an instability are considered that may dominate the flow development at high Reynolds numbers. The instability ensues in the form of high-frequency oscillations in vorticity and streamwise pressure gradient along the wall, and the dominant wavenumber of the instability is O(Re1/2), consistent with a Rayleigh instability. The existence of a Rayleigh instability is further confirmed through evaluation of the Rayleigh equation for velocity profiles obtained from a boundary-layer calculation, and the dominant wavenumber of the instability in the Navier–Stokes solutions agrees very closely with that predicted by the Rayleigh solutions.