Linearized oscillations of a vortex column: the singular eigenfunctions

Abstract

Lord Kelvin analysed the linearized inviscid oscillations of a Rankine vortex as part of a theory of vortex atoms. These eponymously named neutrally stable modes are, however, exceptional regular oscillations that make up the discrete spectrum of the Rankine vortex. In this paper, we examine the singular oscillations that make up the continuous spectrum (CS) and span the entire base state range of frequencies. In two dimensions, the CS eigenfunctions have a twin-vortex-sheet structure similar to that known from earlier investigations of parallel flows with piecewise linear velocity profiles. The vortex sheets are cylindrical, being threaded by axial lines, with one sheet at the edge of the core and the other at the critical radius in the irrotational exterior; the latter refers to the radial location at which the fluid co-rotates with the eigenmode. In three dimensions, the CS eigenfunctions have core vorticity and may be classified into two families based on the singularity at the critical radius. For the first family, the singularity is a cylindrical vortex sheet threaded by helical vortex lines, while for the second family it has a localized dipole structure with radial vorticity. The presence of perturbation vorticity in the otherwise irrotational exterior implies that the CS modes, unlike the Kelvin modes, offer a modal interpretation for the (linearized) interaction of the Rankine vortex with an external vortical disturbance. It is shown that an arbitrary initial distribution of perturbation vorticity, both in two and three dimensions, may be evolved as a superposition over the discrete and CS modes; this modal representation being equivalent to a solution of the corresponding initial value problem. For the restricted case of an initial axial vorticity distribution in two dimensions, the modal representation may be generalized to a smooth vortex. Finally, for the three-dimensional case, the analogy between rotational flows and stratified shear flows, and the known analytical solution for stratified Couette flow, are used to clarify the singular manner in which the modal superposition for a smooth vortex approaches the Rankine limit.

Linear stability of a rotating liquid column revisited

“…The spectrum is seen to borrow its traits from two constituent casesthe Rayleigh-Plateau configuration involving only surface tension (figure 1a) and the Rankine vortex involving only rotation (figure 1b). Much like the Rankine vortex, the rotating liquid column supports an infinite sequence of primarily Coriolis-force-driven modes (henceforth referred to as the Coriolis modes), with the inner dispersion curves corresponding to perturbations with an increasingly fine-scaled radial structure; note that the Rankine vortex has recently been shown to also possess a continuous spectrum on account of the irrotational shear in the column exterior (Roy & Subramanian 2014;Roy et al 2021). The Coriolis modes in both these problems have frequencies σ r ∈ (−2, 2).…”

On the linear receptivity of trailing vortices

“…To the best of our knowledge no analytic expression of the resolvent for three-dimensional inhomogeneous perturbations about smooth viscous vortices exists today (e.g. Ash & Khorrami 1995, p. 321; Roy & Subramanian 2014, p. 439). However, it can be shown that normality of a linear operator implies normality of its resolvent (Kato 1980, pp.…”

“…(Different from Antkowiak 2005, stating that combination of discrete and (unbounded) continuous spectrum.) Roy & Subramanian (2014, p. 405) demonstrate how the inclusion of singular modes pertaining to the inviscid continuous spectrum enables interaction between vortex and free stream, suggesting that a linear model of receptivity to ambient turbulence is intimately related to the inviscid continuous spectrum.…”

“…Considering an inviscid fluid, an additional inviscid continuous spectrum exists as a consequence of a critical-layer singularity of the homogeneous problem (Le Dizès 2004, p. 319; Roy & Subramanian 2014, § 3.2). For non-vanishing viscosity it degenerates to a discrete spectrum of a large number of stable discrete modes (Heaton & Peake 2007, p. 282) which are algebraically localised in the core vicinity, referred to as potential modes by Mao & Sherwin (2011, p. 2).…”

On the Linear Stability of Vortex Columns in the Energy Space

“…In a celebrated paper [18], Lord Kelvin considered the particular case of Rankine's vortex and proved that the linearized operator has a countable family of eigenvalues on the imaginary axis. The corresponding eigenfunctions, which are now referred to as Kelvin's vibration modes, have been extensively studied in the literature, also for more general vortex profiles [8,13,17]. An important contribution was made by Lord Rayleigh in [15], who gave a simple condition for spectral stability with respect to axisymmetric perturbations.…”