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

A spectral method using associated Legendre functions with algebraic mapping is developed for a linear stability analysis of wake vortices. The functions serve as Galerkin basis functions because they capture the correct boundary conditions of vortex motions in an unbounded domain. Using a poloidal-toroidal decomposition, the incompressible Euler or Navier-Stokes equations linearised on the Batchelor $q$-vortex reduce to standard matrix eigenvalue problems to compute linear perturbation velocity eigenmodes and their eigenvalues (i.e., complex growth rates). The number of basis function elements, the number of collocation points and the map parameter are considered adjustable numerical parameters. Based on this numerical method, eigenmodes and eigenvalues of the strong swirling $q$-vortices are examined in perturbation wavenumbers of order unity via spectral theory. Without viscosity, neutrally stable eigenmodes associated with the continuous eigenvalue spectrum having critical-layer singularities are successfully resolved. These eigenvalues generally appear in pairs, indicating the singular degenerate nature in their exact form. Considering viscosity, two distinct, continuous curves in the eigenvalue spectra are discovered, which have not been identified previously. Their associated eigenmodes exhibit a wave packet structure with modest wiggles at their critical layers whose radial thickness varies in the order of $Re^{-1/3}$, suggesting that they are physically feasible viscous remnants of the inviscid critical-layer eigenmodes. A superset of the viscous critical-layer eigenmodes, so-called the potential family, is also recognised. However, their relatively excessive wiggles and radially slow-decay behaviour imply their physical irrelevancy. The onset of the two curves is believed to be caused by viscosity breaking the singular degeneracies.