Stability of the Rankine vortex in a multipolar strain field
Résumé
In this paper, the linear stability of Rankine vortex in a 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 e which is assumed to be small. For n=2, 3 and 4 (dipolar, tripolar and quadripolar strain field respectively), it is shown that the flow is subject to a three-dimensional instability which can be described by the resonance mechanism of Moore & Saffman [Proc. Roy. Soc. Lond. A 346, 413 (1975)]. In each case, two normal modes (Kelvin modes), with azimuthal wavenumbers separated by n, resonate and interact with the multipolar strain field when their axial wavenumbers 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 wavenumbers 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(e) for n=2 and 3, [nu]= O(e^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 wavenumbers m=0 and m=2 is found to be more unstable than the classical helical modes m= 1 or -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 finally discussed in the context of turbulence and compared to recent observations of vortex filaments.