Viscous critical-layer analysis of vortex normal modes

The linear stability properties of an incompressible axisymmetrical vortex of axial velocity W-0(r) and angular velocity Omega(0)(r) are considered in the limit of large Reynolds number. Inviscid approximations and viscous WKBJ approximations for three-dimensional linear normal modes are first constructed. They are then shown to be singular at the critical points r(c) satisfying omega=mOmega(0)(r(c))+kW(0)(r(c)), where omega is the frequency, k and m the axial and azimuthal wavenumbers. The goal of this paper is to resolve these singularities. We show that a viscous critical-layer analysis is analytically tractable. It leads to a single sixth-order equation for the perturbation pressure. This equation is identical to the one obtained in stratified shear flows for a Prandtl number equal to 1. Integral expressions for typical solutions of this equation are provided and matched to the outer inviscid and viscous approximations in the complex plane around r(c). As for planar flows, it is proved that the critical layer solution with a balanced behavior matches a non-viscous approximation in a 4pi/3 sector of the complex-plane. As a consequence, the Frobenius expansions of a non-viscous mode on each side of a critical point r(c) differ by a pi phase jump.