Transient growth of perturbations in a vortex column

Abstract

Linear transient growth in a normal-mode-stable vortex column is studied by extracting ‘optimal modes’ of perturbations. Amplifications occur over a wide range of azimuthal wavenumbers $m$ and axial wavenumbers $k$, and can be more than three orders of magnitude even at moderate vortex Reynolds numbers $\hbox{\it Re}$ (checked up to $10^4$). Transient growth is unbounded in the inviscid limit. For given $\hbox{\it Re}$ and $k$, axisymmetric ($m\,{=}\,0$) modes undergo the largest volume-integrated energy growth, whereas maximum core energy growth occurs for bending waves ($|m|\,{=}\,1$). At fixed $m$ and $\hbox{\it Re}$, growth decreases with increasing $k$, due to the damping effect of viscosity. At fixed $m$ and $k$, growth increases rapidly with $\hbox{\it Re}$ – pointing to the significance of transient growth in high-$\hbox{\it Re}$ practical flows, such as the trailing vortex. Inviscid effects not only cause transient growth, but also its subsequent arrest. There are two distinct mechanisms for growth. First, two-dimensional perturbations amplify because the streamlines have ‘positive tilt’, contributing $uv\,{>}\,0$ stress necessary for growth; here $u$ and $v$ are the radial and azimuthal velocity perturbations, respectively. Second, three-dimensional perturbations grow through azimuthal stretching of spiral vortex filaments containing radial vorticity. Decay in both cases is due to the differential advection – of axial vorticity by the mean swirl – transforming the perturbation streamlines to predominantly ‘negative tilt’, producing $uv\,{<}\,0$ stress. The transient growth mechanism is explained in terms of the distinct effects of the strain and vorticity components of the mean flow, which play counteractive roles. While strain amplifies energy, vorticity limits transient growth by inducing wave motions, in which radial vorticity is depleted by vortex line coiling, i.e. by the tilting of radial vorticity into axial and azimuthal components. Since the strain-to-vorticity ratio varies with radius in the vortex, the competition between strain and vorticity selects a preferred radius of localization of an ‘optimal perturbation’. With increasing growth, axisymmetric optimal modes are localized at progressively larger radii and their growth rates progressively diminished – both limiting the physical significance of such modes in high-$\hbox{\it Re}$ practical flows. An optimal bending wave, on the other hand, is localized closer to the vortex column, where a vorticity perturbation external to the core can resonantly excite vortex core waves. This leads to substantial growth of core fluctuation energy and, probably, to core transition to turbulence. Such resonant growth may be the mechanism for the appearance of bending waves in a vortex in a turbulent field.