Linear instability analysis of wake vortices by a spectral method using mapped Legendre functions

A spectral-Galerkin method using mapped Legendre functions with a poloidal-toroidal decomposition is presented to find eigenmodes of q-vortices in an unbounded domain with correct boundary conditions. In the inviscid limit, the method can resolve a continuous spectrum of neutral modes despite the presence of critical-layer singularities. Spurious instabilities are removed as numerical parameters, including a map parameter, are adjusted to improve spatial resolution. When viscosity is given, a novel set of numerically converging modes is found, forming two contiguous curves in the spectrum. By fine-tuning the map parameter, their associated eigenvalues densely fill up the curves unlike the discrete ones. Each of the modes commonly exhibits a dominant wave structure localized in the vicinity of the critical layer due to viscous regularization and a minor structure comparable with its inviscid counterpart. The results strongly imply that these curved spectra are true viscous remnants of the inviscid continuous spectrum, which are known to contribute to destabilization via transient growth. The bifurcation of the spectrum is thought to be related to two-fold degeneracies due to the critical-layer singularities in the inviscid singular modes.