Elliptic instability in a strained Batchelor vortex

Abstract

The elliptic instability of a Batchelor vortex subject to a stationary strain field is considered by theoretical and numerical means in the regime of large Reynolds number and small axial flow. In the theory, the elliptic instability is described as a resonant coupling of two quasi-neutral normal modes (Kelvin modes) of the Batchelor vortex of azimuthal wavenumbers m and m + 2 with the underlying strain field. The growth rate associated with these resonances is computed for different values of the azimuthal wavenumbers as the axial flow parameter is varied. We demonstrate that the resonant Kelvin modes m = 1 and m = −1 which are the most unstable in the absence of axial flow become damped as the axial flow is increased. This is shown to be due to the appearance of a critical layer which damps one of the resonant Kelvin modes. However, the elliptic instability does not disappear. Other combinations of Kelvin modes m = −2 and m = 0, then m = −3 and m = −1 are shown to become progressively unstable for increasing axial flow. A complete instability diagram is obtained as a function of the axial flow parameter for several values of the strain rate and Reynolds number.The numerical study considers a system of two counter-rotating Batchelor vortices in which the strain field felt by each vortex is due to the other vortex. The characteristics of the most unstable linear modes developing on the frozen base flow are computed by direct numerical simulations for two axial flow parameters and compared to the theory. In both cases, a very good agreement is obtained for the most unstable modes. Less unstable modes are also identified in the numerics and shown to correspond to peculiar resonances involving Kelvin modes from branches of different labels.

Long- and short-wave instabilities in helical vortices

“…The conditions of resonance are such that, when there is no axial flow within the vortex, the most unstable perturbation involves two helical stationary waves, of azimuthal wavenumbers m = -1 and m = 1, and deforms the vortex core in a sinusoidal way (figure 5a). In the presence of an axial core flow, two different waves, with m = 0 and m = 2, may become the most unstable resonant configuration [17], producing a double-helix perturbation of the core (figure 5b).…”

“…0.15 (!/2h 2 ) and a wavelength % " 5.7 a. The elliptic instability properties of the current flow can be estimated using the results obtained by Lacaze et al [17] for straight vortices. In the absence of axial core flow, the most unstable mode involves a sinusoidal displacement of the vortex centre line (figure 5a), with a perturbation vorticity similar to the one shown in figure 6(b).…”

Curvature instability of a curved Batchelor vortex

“…For the elliptic instability, the resonance occurs with a quadripolar correction generated by the background strain field (Moore & Saffman 1975), while for the curvature instability, it is associated with a dipolar correction created by the vortex curvature (Fukumoto & Hattori 2005). Numerous works have concerned the elliptic instability in the context of straight vortices (Tsai & Widnall 1976;Eloy & Le Dizès 1999;Fabre & Jacquin 2004a;Lacaze et al 2007). The specific case of the curved Batchelor vortex has been analysed in BRLD16.…”

“…In smooth vortices, Kelvin modes are affected by the presence of critical layers (Le Dizès 2004) which introduce singularities and damping (Sipp & Jacquin 2003;Fabre et al 2006). These singularities have to be monitored and avoided in the complex plane to be able to obtain the properties of the Kelvin modes from the inviscid equations as shown in Lacaze et al (2007). In the present work, we shall also use the asymptotic theory of Le Dizès & Lacaze (2005) to obtain an approximation of the Kelvin mode dispersion relation and analyse the condition of resonance.…”

Linear stability analysis of a counter rotating vortex pair of unequal strength

“…All computations in the stability investigation were carried out at Re = 20000 . Lacaze et al [5] employed the frozen baseflow approach to investigate the elliptic instability of a counter rotating equal strength (cres) vortex pair with axial flows.…”