Instability Of An Elliptical Flow:
Weakly Nonlinear Analysis
Author(s)
Y. Hattori, Y. Fukumoto & K. Fujimura
Abstract
Elliptical flows are of interest since they appear in various vortical flows in
which circular vortices are deformed by straining flow. We study the nonlinear
evolution of an elliptical flow by weakly nonlinear analysis. Two sets of amplitude
equations are derived for different situations. First, the evolution of bending waves
is considered. Nonlinear interaction of the two base Kelvin waves results in cubic
nonlinear terms, which leads to saturation of the elliptical instability. Next, the
secondary instability is considered. Three Kelvin waves, one of which is a bending
wave, form a resonant triad thanks to freedom of wavenumber shift. As a result
three-wave equations augmented with linear terms are obtained as amplitude
equations. They explain the previous numerical results on the secondary instability
obtained by Kerswell J. Fluid Mech., 382, pp. 283–386, 1999.
Keywords: elliptical flow, weakly nonlinear analysis, triad resonance, secondary
instability.
1 Introduction
Large-scale coherent vortical structures are ubiquitous in turbulent flows. They
are often seen to destabilize and collapse to small-scale disorder. There are
several important mechanisms of the destabilization among which are the
elliptical instability [1–5] and the curvature instability [6, 7]. Both instabilities
are the consequences of parametric resonance of two Kelvin waves mediated by
Keywords
elliptical flow, weakly nonlinear analysis, triad resonance, secondary
instability.
Related Book
Other papers in this volume
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