Helical vortex systems: linear analysis and nonlinear dynamics

The near wake behind helicopter rotors, wind turbines and more generally behind rotating devices are dominated by helical vortices. Investigating their stability properties is a necessary step to predict their dynamics. Instabilities in such vortex systems have mainly been studied theoretically (Widnall 1972 [1], Okulov [2] and Sørensen 2007 [3]) in an inviscid framework for small core size vortices. The aim of the present study is to generalize these works to the viscous framework for arbitrary core sizes and vorticity profiles.The base flows considered here are helically symmetric: fields are invariant through combined axial translation of distance z and rotation of angle θ= Δz/L around the z-axis, where 2π L denotes the helix pitch.We first perform a linear temporal stability analysis of these base flows, using an Arnoldi [4] procedure coupled to two different codes : (i) a linearised version of the helical DNS code HELIX [5], (ii) another linear code called HELIKZ, which computes the dynamics of arbitrary perturbations in the vicinity of a helically symmetric base flow. These two codes permit the investigation of different types of instability modes: (i) modes having the same helical symmetry as the base flow which generalize the Okulov modes ; (ii) modes depending on z as exp ikz which generalize the Widnall modes. In the first case (i), instabilities are found to be dominated by displacement modes of the type presented in figure 1 for the case of two vortices. In the second case (ii), modes will be compared to those observed in recent experimental work (Leweke et al. 2014 [6]). We then compute the nonlinear dynamics of a basic flow perturbed with a linear mode of type (i) set at a small initial amplitude. In the helical framework, the displacement mode is shown to be responsible for leap-frog dynamics (cf. figure 2) and/or vortex merging (cf. figure 3) with characteristics depending on the various parameters