Dynamics of the three helical vortex system and instability

International audienceMany systems develop helical vortices in their wake (propellers, wind turbines, helicopters). Such flows can be assumed, at least locally, to be helically symmetric, i.e invariant through combined axial translation of distance ∆z and rotation of angle θ = ∆z/L around the same z-axis, where 2π L stands for the helix pitch. In this context, analytical and numerical works describing stationary vortices are mostly restricted to inviscid filaments and patches. Here, we present results obtained using a new DNS code with built-in helical symmetry able to simulate the viscous dynamics of distributed vorticity profiles. This approach contains the effects of 3D vortex curvature and torsion in a simple way and allows one to reach higher Reynolds numbers when compared to a full 3D DNS. In this framework, the long-time (or equivalently far-wake) dynamics of regularly spaced helical vortices is investigated. We focus here on the case of three identical vortices and simulate their dynamics as their pitch L and Reynolds number is varied. At large L, a “classical” three-vortex merging takes place, which resembles the 2D two-vortex merging however with slight differences. When L is reduced, it takes more and more time for the vortices to merge, as their rotation speed around the system axis is slowed down by self-induced vorticity effects. This phenomenon is explained by following the interplay between vorticity and streamfunction in the co-rotating frame of reference, and tracking the locus of hyperbolic points of the streamfunction. At low L-values, typically less than 1, the exponential instability described by Okulov and Sørensen is obtained, resulting in various grouping and merging scenarii at the nonlinear stage of evolution. At intermediate L-values of the order of 1, only viscous diffusion acts, resulting in a, slow, viscous type of merging. Note that other types of instability which are purely 3D are not described within this purely helical framework. Instead, the helical code run on a short period of time allows one to generate a quasi-steady flow state which may then be used to investigate such instabilities. A full 3D Navier–Stokes code is then used, linearized in the vicinity of the basic state, to extract the dominant instability properties