Stability of the Rankine vortex in a multipolar strain field

In this paper, the linear stability of a Rankine vortex in an n-fold multip olar strain field is addressed. The flow geometry is characterized by two p arameters: the degree of azimuthal symmetry n which is an integer and the s train strength epsilon which is assumed to be small. For n=2, 3 and 4 (dipo lar, tripolar and quadrupolar strain fields, respectively), it is shown tha t the flow is subject to a three-dimensional instability which can be descr ibed 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), wi th the azimuthal wave numbers separated by n, resonate and interact with th e multipolar strain field when their axial wave numbers and frequencies are identical. The inviscid growth rate of each resonant Kelvin mode combinati on is computed and compared to the asymptotic values obtained in the large wave numbers limits. The instability is also interpreted as a vorticity str etching mechanism. It is shown that the inviscid growth rate is maximum whe n the perturbation vorticity is preferentially aligned with the direction o f stretching. Viscous effects are also considered for the distinguished sca lings: nu =O(epsilon) for n=2 and 3, nu =O(epsilon (2)) for n=4, where nu i s the dimensionless viscosity. The instability diagram showing the most uns table mode combination and its growth rate as a function of viscosity is ob tained 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 th an the classical helical modes m=+/-1. For n=3 and 4, the azimuthal structu re of the most unstable Kelvin mode combination is shown to be strongly dep endent on viscous effects. The results are discussed in the context of turb ulence and compared to recent observations of vortex filaments. (C) 2001 Am erican Institute of Physics.