This paper investigates the variety of dynamics displayed by two identical interwoven helical vortices for different values of their helical pitch. We present simplified approaches aimed at understanding their nonlinear time evolution: in order of increasing complexity, we use models of nonlinear interaction of two point-vortex alleys, of two vortex-ring filament alleys, and of two inviscid helical filaments. Each of these inviscid models leads to a dynamical system of few degrees of freedom that can be analyzed in terms of orbits in a phase space structured by elliptical and hyperbolic points. At low pitch, the basic state with the two helical vortices in symmetric configuration is an unstable equilibrium, and it corresponds to hyperbolic points in the phase space. An initial perturbation induces different types of dynamics depending on its direction and amplitude. Features observed for increasing helical pitches in direct numerical simulations or in experiments are gradually understood as the model complexity is increased. At small pitch, leapfrog cycles are triggered by axial or azimuthal displacements of the vortices, while radial perturbations trigger overtaking; for both regimes, a periodic change in the interwaving order of the vortices occurs. Above a critical pitch value, small-amplitude perturbations trigger a new regime called fluttering, mainly characterized by a periodic evolution of the vortex radii without a change of the interwaving order; large-amplitude displacements then still allow for leapfrog dynamics far from equilibrium. The amplitudes of the radial excursions of the vortices induced by small perturbations are characterized over the whole range of helical pitches, up to the linear stabilization threshold. Mimicking viscous effects in the point-vortex alley model sheds some light on how dissipation causes overtaking to turn into leapfrog (the reverse is not observed).

• Received 19 October 2020
• Accepted 6 July 2021

DOI:https://doi.org/10.1103/PhysRevFluids.6.084701