Bending waves on inviscid columnar vortices

Abstract

Bending waves, perturbation modes leading to deflections of the vortex centreline, are considered for an infinitely long straight vortex embedded in an irrotational flow of unlimited extent. We first establish the general form of the dispersion relation for long waves on columnar vortices with arbitrary distributions of axial and azimuthal vorticity by a singular perturbation analysis of the Howard-Gupta equation. The asymptotic results are shown to compare favourably with numerical solutions of the Howard-Gupta equation for wavelengths comparable to the vortex core radius and longer. Dispersion relations are then found numerically for specific models of vortex core structures observed experimentally; here no restrictions are placed on wavelength. The linear dispersion relation has an infinite number of branches, falling into two families; one with infinite phase speed at zero wavenumber (which we call ‘fast’ waves), the other with zero phase speed (‘slow’ waves). In the long-wave limit, slow waves have zero group velocity, while the fast waves may have finite non-zero group speeds that depend on the form of the velocity profiles on the axis of rotation. Weakly nonlinear waves are described under most circumstances by the nonlinear Schrödinger equation. Solitons are possible in certain ‘windows’ of wavenumbers of the carrier waves. An example has already been presented by Leibovich & Ma (1983), who compute solitons and soliton windows on a fast-wave branch for a vortex with a particular core structure. Experimental data of Maxworthy, Hopfinger & Redekopp (1985) reveal solitons which appear to be associated with the slow branch, and these are computed for velocity profiles fitting their data. The nonlinear Schrödinger equation is shown to fail for long waves, and to be replaced by a nonlinear integro-differential equation.

Linearized oscillations of a vortex column: the singular eigenfunctions

“…Our focus is on what happens to these modes as a smooth vorticity profile approaches the Rankine limit. In the absence of an axial flow, centre-mode behaviour is observed only in the limit of small axial wavenumbers (Leibovich, Brown & Patel 1986). Each of the structured modes of a smooth vortex becomes a centre mode for k → 0, the corresponding co-grade eigenfunction being characterized by a vanishingly small radial length scale (the focus here is on the co-grade modes; the retrograde modes transform to inviscidly damped singular oscillations best interpreted in terms of a superposition of decaying quasi-modes).…”

“…Thus, the dispersion curve in the nearly convected limit has the asymptotic form ω − mΩ 0 ∼ O(k 2p/(2p−1) ), consistent with a vanishing group velocity for k → 0, while the radial extent of the oscillatory boundary layer around the rotation axis is O(k/mΩ 2p ) 1/(2p−1) which determines the radial scale of the eigenmode. The case p = 1, for which (k) ∼ O(k 2 ) and (4.2) reduces to the hypergeometric equation, was analysed by Leibovich et al (1986). For large p, the reduction in the radial length scale ( ∝ k 1/2p ) of the eigenfunction with decreasing k becomes increasingly gradual, and correspondingly, the transition to a centre-mode behaviour with a vanishing group velocity occurs at an increasingly small value of k. In the limit p → ∞, there is no boundary layer, and consequently no centre-mode behaviour, with ω − mΩ 0 ∼ O(k) and a finite group velocity at k = 0.…”

Experiments on the elliptic instability in vortex pairs with axial core flow

“…The self-induced dynamics of vortices with arbitrary (axisymmetric) velocity profiles was analysed by Widnall et al (1971), and subsequently by Moore & Saffman (1973), Leibovich, Brown & Patel (1986) and Klein & Knio (1995), in the limit of long wavelengths. It can be deduced from this work that a vortex, with azimuthal and axial velocity distributions v φ (r) and v z (r) evolving on a characteristic radial scale a, exhibits the same self-induced dynamics as an equivalent Rankine vortex, having the same circulation Γ and a core radius Crow (1970).…”

Instability and unsteadiness of aircraft wake vortices

“…, correspond to co-rotating waves. The long-wave asymptotic behaviour of these branches has been given by Leibovich et al [29] …”

“…This mode takes the form of a helical displacement of the vortex core as a whole, and it actually corresponds to the self-induced oscillation mode of a filament vortex (see [39]). The frequency of this mode in the limit of long wavelength (ka 1) has been obtained using asymptotic methods by Moore and Saffman [34] and Leibovich et al [29]:…”