Curvature instability of a curved Batchelor vortex

Abstract

In this paper, we analyse the curvature instability of a curved Batchelor vortex. We consider this short-wavelength instability when the radius of curvature of the vortex centerline is large compared to the vortex core size. In this limit, the curvature instability can be interpreted as a resonant phenomenon. It results from the resonant coupling of two Kelvin modes of the underlying Batchelor vortex with the dipolar correction induced by curvature. The condition of resonance of the two modes is analysed in detail as a function of the axial jet strength of the Batchelor vortex. Contrarily to the Rankine vortex, only a few configurations involving m = 0 and m = 1 modes are found to become the most unstable. The growth rate of the resonant configurations is systematically computed and used to determine the characteristics of the most unstable mode as a function of the curvature ratio, the Reynolds number, and the axial flow parameter. The competition of the curvature instability with another short-wavelength instability, which was considered in a companion paper [Blanco-Rodríguez & Le Dizès, Elliptic instability of a curved Batchelor vortex, J. Fluid Mech. 804, 224-247 (2016)], is analysed for a vortex ring. A numerical error found in this paper which affects the relative strength of the elliptic instability is also corrected. We show that the curvature instability becomes the dominant instability in large rings as soon as axial flow is present (vortex ring with swirl).

Numerical stability analysis of a vortex ring with swirl

“…, where the numerical factor 0.638 is peculiar to the Gaussian distribution of vorticity. Here, a 0 is the initial core radius and u ,max and ⌦ (0) max = /(2⇡a 2 0 ) are the maximum circumferential velocity and the maximum angular velocity around the vortex core, respectively; it is pointed out that ⌦ 0max is defined for the initial Gaussian vorticity distribution as in Blanco-Rodríguez & Le Dizès (2017).…”

“…= (1.14, 0.11) in the present work corresponds to (m A , m B ) = (0, 1) and (l A , l B ) = (3, 1) with (k, !) = ( 1.14, 0.11) in Blanco-Rodríguez & Le Dizès (2017). Theoretical values for some typical modes taken from Blanco-Rodríguez & Le Dizès (2016) and Blanco-Rodríguez & Le Dizès (2017) are also included by lines in the figures 8-11.…”

“…However, the vortex rings observed in the experiments usually have smooth vorticity distributions which are often approximated by Gaussian distributions. Recently, Blanco-Rodríguez & Le Dizès (2016) and Blanco-Rodríguez & Le Dizès (2017) have shown theoretically that vortex rings which have Gaussian distributions of azimuthal vorticity and velocity at the leading order are also subjected to elliptic and curvature instabilities. They found that some important features of the instabilities are quite di↵erent from those of Kelvin's vortex ring.…”

Numerical realization of helical vortices: application to vortex instability

“…The short-wavelength instabilities observed in the stability analysis correspond to the elliptic instability, which has been a topic of intense research over the past decades (see [34] for a recent review, and [47] for details regarding the curvature instability, which is another short-wavelength instability in curved Batchelor vortices).…”

“…One also has that ε = Ra/(R 2 + L 2 ), and as before, = L/R is the dimensionless reduced pitch. Approximate relations for the other coefficients appearing in (31) are given in Appendix C of [51] (note also the corrigendum to some of these relations appended to [47]). The Reynolds number used in (31) is defined as Re = Re/(2π).…”