A NUMERICAL INVESTIGATION OF THE FORMATION OF SECONDARY VORTICES IN LABORATORY-SIMULATED TORNADOES.

Abstract

Two numerical models, described in detail herein, have been constructed and used to investigate the formation of secondary vortices in axisymmetrically-forced rotating flows. The particular type of 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 vortex flows based on a range of swirl ratios. The second model tested those flows for instability by simulating the behavior of small amplitude, axially asymmetric, linear perturbations superimposed on the flows: amplification of the perturbations indicated instability whereas damping indicated stability. For those flows found to be unstable, the linear perturbations of various azimuthal wavenumbers were analyzed in detail, and from the perturbation growth rates, structures, phase speeds, and energetics, the nature of the instability could be studied. 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 around 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. Structurally, 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, but at a somewhat lower angular velocity at the surface. These linear results are all consistent with observed laboratory behavior. From this, it was concluded that linear theory is capable of explaining many important aspects of secondary vortices. An analysis of the perturbation energy equation revealed that 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.