Nonparallel spatial stability of the boundary layer induced by Long’s vortex on a solid plane perpendicular to its axis

Abstract

We consider the linear, viscous stability of the boundary layer induced by an unbounded vortex whose outer inviscid structure coincides near the axis with Long's vortex. The viscous boundary layer induced by the interaction of such a vortex with a solid plane perpendicular to the axis has a known self-similar structure. The spatial stability of this self-similar solution is analyzed here for axisymmetric and nonaxisymmetric perturbations propagating towards the axis of rotation. Viscous and nonparallel effects on the stability of the perturbations are retained up to their first order in the inverse of the local Reynolds number (nondimensional radius). The resulting parabolic stability equations are solved numerically using a spectral collocation method varying both the nondimensional frequency and radius. It is found that the flow is unstable to axisymmetric perturbations far away from the axis (inviscid instability). The growth rate of this inertial instability mode first increases and then decreases as the Reynolds number decreases (as the axis is approached). However, before this inviscid mode becomes stabilized, new viscous instabilities for both axisymmetric and nonaxisymmetric perturbations show up, which finally become stabilized at moderate Reynolds numbers. We characterize the critical Reynolds numbers and frequencies for the stability of these unstable perturbations as functions of their azimuthal wave number. It is found that the last perturbations that become stable as the axis is approached are nonaxisymmetric, corotating, perturbations with an azimuthal wave number n=4 .