Weakly nonlinear saturation of short-wave instabilities in a strained Lamb–Oseen vortex

Abstract

A Lamb–Oseen vortex in a planar straining field is known to be subject to 3D (three-dimensional) short-wave instabilities which are due to the resonance of the straining field and two stationary Kelvin waves characterized by the same axial wave number and by azimuthal wave numbers equal to −1 and +1. The linear regime has been described by Moore and Saffman (1975). In this article, we extend this analysis to the weakly nonlinear regime. The emerging eigenmode is characterized by a complex amplitude A=|A|eiφ, whose behavior is governed by an amplitude equation. It is shown that the unstable perturbation corresponds to an oscillation of the vortex in a plane inclined at an angle φ, while the amplitude of these oscillations is proportional to |A|. The vortex centers are defined as the points where the velocity of the vortex is zero, which also corresponds to the points where the pressure is minimum. We show that these instabilities saturate. The saturation amplitudes are evaluated numerically and expressed in terms of oscillation amplitudes of the vortex centers. If a denotes the internal radius of the vortex and if the straining field is due to a counter-rotating vortex of same strength, located at a distance b, then the maximum amplitude Δ of the vortex oscillations is Δ/b=6.1a2/b2. This result is in agreement with those of the experiments of Leweke and Williamson (1998) for which a/b=0.2. It also shows that in aeronautical situations, for which a/b is smaller, i.e., a/b<0.1, the considered short-wave instability will saturate at very low amplitude.