Stability of the Rankine vortex in a multipolar strain field

In this paper, the linear stability of a Rankine vortex in an n-fold multipolar strain field is addressed. The flow geometry is characterized by two parameters: the degree of azimuthal symmetry n which is an integer and the strain strength epsilon which is assumed to be small. For n=2, 3 and 4 (dipolar, tripolar and quadrupolar strain fields, respectively), it is shown that the flow is subject to a three-dimensional instability which can be described by the resonance mechanism of Moore and Saffman [Proc. R. Soc. London, Ser. A 346, 413 (1975)]. In each case, two normal modes (Kelvin modes), with the azimuthal wave numbers separated by n, resonate and interact with the multipolar strain field when their axial wave numbers and frequencies are identical. The inviscid growth rate of each resonant Kelvin mode combination is computed and compared to the asymptotic values obtained in the large wave numbers limits. The instability is also interpreted as a vorticity stretching mechanism. It is shown that the inviscid growth rate is maximum when the perturbation vorticity is preferentially aligned with the direction of stretching. Viscous effects are also considered for the distinguished scalings: nu =O(epsilon) for n=2 and 3, nu =O(epsilon (2)) for n=4, where nu is the dimensionless viscosity. The instability diagram showing the most unstable mode combination and its growth rate as a function of viscosity is obtained and used to discuss the role of viscosity in the selection process. Interestingly, for n=2 in a high viscosity regime, a combination of Kelvin modes of azimuthal wave numbers m=0 and m=2 is found to be more unstable than the classical helical modes m=+/-1. For n=3 and 4, the azimuthal structure of the most unstable Kelvin mode combination is shown to be strongly dependent on viscous effects. The results are discussed in the context of turbulence and compared to recent observations of vortex filaments. (C) 2001 American Institute of Physics.