Large-Reynolds-number asymptotic analysis of viscous centre modes in vortices

Abstract

This paper presents a large-Reynolds-number asymptotic analysis of viscous centre modes on an arbitrary axisymmetrical vortex with an axial jet. For any azimuthal wavenumber m and axial wavenumber k, the frequency of these modes is given at leading order by ω0 = mΩ0 + kW0 where Ω0 and W0 are the angular and axial velocities of the vortex at its centre. These modes possess a multi-layer structure localized in an O(Re−1/6) neighbourhood of the vortex. By a multiple-scale matching analysis, we demonstrate the existence of three different families of viscous centre modes whose frequency expands as ω(n) ∼ ω0 + Re−1/3ω1 + Re−1/2ω(n)2. One of these families is shown to have unstable eigenmodes when H0 = 2Ω0k(2kΩ0 − mW2) < 0 where W2 is the second radial derivative of the axial flow in the centre. The growth rate of these modes is given at leading order by σ ∼ (3/2)(H0/4)1/3Re−1/3. Our results prove that any vortex with a jet (or jet with swirl) such that Ω0W2 ≠ 0 is unstable if the Reynolds number is sufficiently large. The spatial structure of the viscous centre modes is obtained and simple approximations which capture the main feature of the eigenmodes are also provided.The theoretical predictions are compared with numerical results for the q-vortex model (or Batchelor vortex) for Re ≥ 105. For all modes, a good agreement is demonstrated for both the frequency and the spatial structure.