Long-wavelength asymptotics of unstable crossflow modes, including the effect of surface curvature

Abstract

Stationary vortex instabilities with wavelengths significantly larger than the thickness of the underlying three-dimensional boundary layer are studied with asymptotic methods. The long-wavelength Rayleigh modes are locally neutral and aligned with the direction of the local inviscid streamline. For a spanwise wave numberβ≪ 1, the spatial growth rate of these vortices isO(β3/2). WhenβbecomesO(R-1/7), the viscous correction associated with a thin sublayer near the surface modifies the inviscid growth rate to the leading order. Asβis further decreased through this regime, viscous effects assume greater significance and dominate the growth-rate behaviour. The spatial growth rate becomes comparable to the real part of the wave number whenβ=O(R-¼). At this stage, the disturbance structure becomes fully viscous-inviscid interactive and is described by the triple-deck theory. For even smaller values ofβ, the vortex modes become nearly neutral again and align themselves with the direction of the wall-shear stress. Thus the study explains the progression of the crossflow-vortex structure from the inflectional upper branch mode to nearly neutral long-wavelength modes that are aligned with the wall-shear direction.