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STABILITY ANALYSIS OF A RANKINE VORTEX WITH RADIAL DENSITY STRATIFICATION IN THREE DIMENSIONS

V.R.K. Reddy
Indian Institute of Technology Hyderabad

Harish N Dixit
Indian Institute of Technology Hyderabad, Telangana

DOI: 10.1615/IHMTC-2017.120
pages 73-77

Abstract

In this work, we present a 3D linear stability analysis of a radially stratified Rankine vortex of radius r = α. The flow is assumed to be inviscid and the density distribution axisymmetric with a single density jump at radial location r = rj. Recently a 2D stability analysis of the same base state was studied in [1] and [2] where it was shown that a light-cored vortex, i.e. (ρ1 < ρ2), may become unstable while heavy-cored vortex, i.e. (ρ1 > ρ2), may be stabilized. Further it is shown in [2] that shear can cause the destabilization of a light-cored vortex or stabilization of a heavy-cored vortex. We extend the 2D stability results to include three-dimensional perturbations. It is known that a 3D Rankine vortex with constant density supports an infinite number of discrete Kelvin modes which tremendously increases the possibilities of wave interactions. For step density jumps at arbitrary radial locations outside the vortex core, we derive the complete dispersion relation analytically. The transcendental nature of the dispersion relation makes obtaining the growth rates a very complex task. We therefore carry out linear stability analysis on a smooth tanh vortex using Chebyshev spectral collocation method. The unstable eigenvalues obtained numerically for the smooth profile are used for determining the roots of the complex dispersion relation. The problem is fully governed by four non-dimensional parameters: azimuthal wavenumber, m, axial wavenumber, , location of density jump, rj, and density stratification parameter, At = (ρ1 − ρ2)/(ρ1 + ρ2). Results are presented in the form of contours of growth rates in the At−rj/α, At−kα and rj/α−kα plane for fixed m and vortex radius, α, both for light-cored and heavy cored vortices. In agreement with the surprising 2D result, light-cored vortex is again found to be unstable to 3D perturbations.

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