A Two-Dimensional Linear Stability Analysis of the Multiple Vortex Phenomenon

Abstract

Two numerical models have been constructed and used to investigate the formation of secondary vortices in axisymmetrically forced rotating flows. The vortex flow examined is that developed in a laboratory vortex simulator where secondary vortices have been produced and extensively studied. The first numerical model generated a collection of steady-state, axisymmetric, two-dimensional vortex flows for a range of swirl ratios. The second model tested those flows for instability by simulating the behavior of small-amplitude, linear perturbations superimposed on the flows: amplification of the perturbations indicated instability, whereas damping indicated stability. The results of the instability study show that the vortex is stable for the lowest swirl ratios but that, above a certain value, instability persists indefinitely. The most rapidly growing wavenumber shifts steadily with increasing swirl from 1 to approximately 5 in the swirl range investigated. Growth rates were found to be high enough for secondary vortices to form in the laboratory simulator in just a few seconds. The perturbation fields were found to have a helical tilt and to be centered near the radius of maximum vertical vorticity in the axisymmetric vortex. They propagated in the same azimuthal direction as the rotation of the axisymmetric flow. These linear results we consistent with observed laboratory behavior, as well as with a full three-dimensional numerical multiple-vortex simulation by Rotunno. From this, it was concluded that linear theory is capable of explaining many important aspects of secondary vortices in the simulator. At the higher swirl ratios, the perturbation received most of its energy from the deformation of the axisymmetric flow due to the radial distribution of azimuthal velocity, while for low swirl, the primary source was from the radial distribution of the vertical velocity. No other component of the axisymmetric vortex ever contributed more than about 25% of these terms.