Widnall instabilities in vortex pairs

Abstract

In this article we analyze the cooperative three-dimensional short-wave instabilities developing on concentrated vortex dipoles that have been obtained by means of two-dimensional direct numerical simulations. These dipoles are characterized by their aspect ratio a/b where a is the radius of the vortices based on the polar moments of vorticity and b is the separation between the vortex centroids. In the inviscid case, we show that the selection of the antisymmetric eigenmode smoothly increases with a/b: for a/b=0.208, the amplification rate of the antisymmetric eigenmode is only 1.4% larger than the amplification rate of the symmetric eigenmode. When a/b=0.288, this difference increases up to 7%. The results of the normal mode analysis may be compared to those of an asymptotic stability analysis of a Lamb–Oseen vortex subjected to a weak straining field, following Moore and Saffman [Proc. R. Soc. London, Ser. A 346, 413 (1975)]. This theory shows that the instability may occur whenever two Kelvin waves exist with the same frequency ω, the same axial wavenumber k and with azimuthal wavenumbers m and m+2. Contrary to the case of a Rankine vortex [Tsai and Widnall, J. Fluid Mech. 73, 721 (1976)], the presence of critical layers in a Lamb–Oseen vortex prevents a large number of possible resonances. For example, resonances between m=−2 and m=0 modes lead to damped modes. The only resonances that occur are related to the stationary (ω=0) bending waves (m=±1) obtained for specific values of the axial wavenumber. All these predictions are found to be in good agreement with the results obtained by the stability analysis of the considered vortex pairs. At last, we present a nonautonomous amplitude equation which takes into account all effects of viscosity, i.e., the viscous damping of the amplification rate of the perturbation but also the increase of the dipole aspect ratio a/b due to the viscous diffusion of the basic flowfield. The low-Reynolds number experiment of Leweke and Williamson [J. Fluid Mech. 360, 85 (1998)] is revisited under the light of these theoretical results. We show that these theoretical results yield predictions for the amplification rate and for the wavenumber that agree with the experimental observations.