Linear dynamics of the Lamb-Chaplygin dipole in the two-dimensional limit

Abstract

The dynamics of the Lamb-Chaplygin dipole in the large-wavelength limit is investigated by means of linear analysis. Taking the three-dimensional spectrum of the Lamb-Chaplygin dipole calculated by Billant [“Three-dimensional stability of a vortex pair,” Phys. Fluids 11, 2069 (1999)] as a reference, we first show that additional families of unstable modes are present in the spectrum. Among them, a family of symmetric modes and one of antisymmetric modes, both present in the large-wavelength limit, are the main topic of this investigation. The most unstable modes in these two families being purely two-dimensional, the two-dimensional dynamics is more particularly investigated. Our calculations show that the amplification rate of the antisymmetric instability is significantly larger than that of the symmetric instability. Also, while the two-dimensional symmetric mode is stationary and induces a shift of the dipole toward the upstream or downstream direction relatively to the dipole self-propagation velocity vector, the antisymmetric mode is unsteady and displaces the dipole into wavy oscillations about its initial straight trajectory. The leading physical mechanism of these two-dimensional instabilities is the vortex shedding that occurs downstream of the dipole. This shedding is the reaction of the external flow to the dipole motion and basically enables the conservation of the flow impulse. The destabilizing action of the wake is amplified as the displaced dipole generates more shedding. The dipole motion and the wake thus reinforce each other, leading to the instability. The strain mutually exerted by the vortices of the dipole, known as an essential mechanism of three-dimensional vortex pair instabilities, is also shown to participate to this destabilization.