Viscous and inviscid centre modes in the linear stability of vortices: the vicinity of the neutral curves

Abstract

In a previous paper, We have recently that if the Reynolds number is sufficiently large, all trailing vortices with non-zero rotation rate and non-constant axial velocity become linearly unstable with respect to a class of viscous centre modes. We provided an asymptotic description of these modes which applies away from the neutral curves in the (q, k)-plane, where q is the swirl number which compares the azimuthal and axial velocities, and k is the axial wavenumber. In this paper, we complete the asymptotic description of these modes for general vortex flows by considering the vicinity of the neutral curves. Five different regions of the neutral curves are successively considered. In each region, the stability equations are reduced to a generic form which is solved numerically. The study permits us to predict the location of all branches of the neutral curve (except for a portion of the upper neutral curve where it is shown that near-neutral modes are not centre modes). We also show that four other families of centre modes exist in the vicinity of the neutral curves. Two of them are viscous damped modes and were also previously described. The third family corresponds to stable modes of an inviscid nature which exist outside of the unstable region. The modes of the fourth family are also of an inviscid nature, but their structure is singular owing to the presence of a critical point. These modes are unstable, but much less amplified than unstable viscous centre modes. It is observed that in all the regions of the neutral curve, the five families of centre modes exchange their identity in a very intricate way. For the q vortex model, the asymptotic results are compared to numerical results, and a good agreement is demonstrated for all the regions of the neutral curve. Finally, the case of ‘pure vortices’ without axial flow is also considered in a similar way. In this case, centre modes exist only in the long-wave limit, and are always stable. A comparison with numerical results is performed for the Lamb–Oseen vortex.