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

TL;DR

This work introduces a mapped Legendre spectral collocation method to perform linear stability analysis of wake vortices in an unbounded domain, formulating the problem in toroidal and poloidal variables. By using algebraically mapped Legendre functions, the authors enforce analyticity at the origin and decay at infinity without domain truncation, producing a standard matrix eigenproblem with high computational efficiency. They distinguish discrete, inviscid critical-layer, and non-normal-spectrum regions, and reveal, for the first time, two distinct viscous critical-layer spectra (σ_c^ν) that arise from regularisation and exhibit Re^{-1/3} scaling of the layer thickness; a secondary continuous “potential” spectrum (σ_p^ν) is also characterized. The study confirms pairing behavior in inviscid critical-layer eigenvalues and demonstrates how optimal map parameters (L_opt) and high resolution (M) consolidate convergence, while pseudospectral analysis supports the continuity of the viscous spectra. These findings deepen understanding of wake vortex stability and provide a robust numerical framework for future non-linear and triadic interaction studies in aerospace applications.

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 (2011). 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.

Figures (21)

• Figure 1: Vortex with circulation $\Gamma$ of length scale $R_0$ and coordinate systems.
• Figure 2: Changes in distribution of the collocation points with respect to $L$ given $N = 52$. Some collocation points at large radii are omitted. The high-resolution region is $0\le r < L$, where half of the collocation points are clustered around the origin. As $L$ increases, the high-resolution region is expanded. However, the mean spacing $\Delta$ grows simultaneously. $L$ should be chosen carefully to balance these anti-complementary effects.
• Figure 3: A comparison of our numerical calculation with that of Mayer1992. Shown is the radial velocity component of the most unstable eigenmode for the validation cases $(a)$$(m, \kappa, q, \Rey) = (1, 0.5, -0.5, \infty)$ and $(b)$$(m, \kappa, q, \Rey) = (0, 0.5, 1, 10^4)$, where the maximum of $\Real(\Tilde{u}_r)$ is normalised to unity. Numerical parameters are $M=80$ and $L=2$. Note that Mayer1992 only plotted the real parts of the eigenmodes.
• Figure 4: Schematic diagrams of the spectra of the eigenvalues of a $q$-vortex of $(a)$$\mathcal{L}_{m\kappa}^{0}$ for inviscid problems where $\nu \equiv 0$Mayer1992Heaton2007Gallay2020 and $(b)$$\mathcal{L}_{m\kappa}^{\nu}$ for viscous problems with finite $\Rey$, including $\nu \rightarrow 0^+$Fabre2006Mao2011. Each schematic exhibits a set of eigenvalues where $m$ and $\kappa$ are fixed. The cases illustrated here assume $m > 0$. These spectra are shown here because they are representative, but they do not embrace all of the different families of spectra. The labels attached here are used throughout the main body of the text. Note that figures of the true numerical spectra computed by us, rather than schematics, follow in § \ref{['inviscidlinearanalysis']} and § \ref{['viscouslinearanalysis']}, and that the viscous critical-layer spectrum, consisting of two distinct curves in $(b)$, were discovered via the present numerical analysis and were not previously identified.
• Figure 5: Critical-layer singularity radial location $r_c$ versus critical layer eigenvalue $\sigma_c$ with fixed $m$, $\kappa$ and $q$. See \ref{['sigma_r_c_qvort']} and \ref{['sigma_r_c_lamb']}. The two illustrated cases where $(m, \kappa, q) = (1, 1.0, \infty)$ and $(m, \kappa, q) = (2, 3.0, 4.0)$ are investigated in later analyses.

Theorems & Definitions (2)

• Definition 1
• Definition 2