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

Abstract

A spectral method using associated Legendre functions with algebraic mapping is developed for a linear stability analysis of wake vortices. These functions serve as Galerkin basis functions, capturing correct analyticity and boundary conditions for vortices in an unbounded domain. The incompressible Euler or Navier–Stokes equations linearised on a swirling flow are transformed into a standard matrix eigenvalue problem of toroidal and poloidal streamfunctions, solving perturbation velocity eigenmodes with their complex growth rate as eigenvalues. This reduces the problem size for computation and distributes collocation points adjustably clustered around the vortex core. Based on this method, strong swirling$q$vortices with linear perturbation wavenumbers of order unity are examined. Without viscosity, neutrally stable eigenmodes associated with the continuous eigenvalue spectrum having critical-layer singularities are successfully resolved. The inviscid critical-layer eigenmodes numerically tend to appear in pairs, implying their singular degeneracy. With viscosity, the spectra pertaining to physical regularisation of critical layers stretch out toward an area, referring to potential eigenmodes with wavepackets found by Mao & Sherwin (J. Fluid Mech., vol. 681, 2011, pp. 1–23). However, the potential eigenmodes exhibit no spatial similarity to the inviscid critical-layer eigenmodes, doubting that they truly represent the viscous remnants of the inviscid critical-layer eigenmodes. Instead, two distinct continuous curves in the numerical spectra are identified for the first time, named the viscous critical-layer spectrum, where the similarity is noticeable. Moreover, the viscous critical-layer eigenmodes are resolved in conformity with the$Re^{-1/3}$scaling law. The onset of the two curves is believed to be caused by viscosity breaking the singular degeneracy.