Viscous Perturbation to Inviscid Wake Vortices - Perturbation Theory in Vortex Stability

We investigate how viscosity affects the linear stability of a q-vortex, the typical model of aircraft wake vortices, by means of the degenerate and non-degenerate perturbation theories that were originally developed for quantum mechanics for studying how energy levels of atoms and molecules changed subject to perturbations. For the regular, inviscid eigenmodes of the q-vortex, we verify that the non-degenerate perturbation theory yields the same viscous eigenmodes and eigenvalues that can be found numerically without the use of perturbation theory by solving for the eigenmodes of the full, viscous Navier-Stokes equations. For the singular, inviscid eigenmodes with singularities at their critical layers, it was found in our numerical analysis that the corresponding viscous eigenmodes appeared in two bifurcated sets of eigenmodes with slightly different complex growth rates. We quantitatively describe the reason for the bifurcation of the viscous eigenmodes using the degenerate perturbation theory and relate it to the two-fold degeneracies in the inviscid critical-layer eigenmodes. This approach, however, lacks the regular information inside the critical layers as it treats them as singular points. As a complement, asymptotic expansion of the critical layers should follow.