Instabilities in helical vortex systems: linear analysis and nonlinear dynamics
Résumé
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 improve understanding and prediction of their dynamics. This is of importance since coherent structures in the wake are responsible for additional stresses and losses of performance in wind turbine farms for instance.
Instabilities can also trigger transition to the highly non-stationary Vortex Ring State which is responsible for some helicopter crashes. The near wake flow can be considered, at least locally, helically symmetric: fields are invariant through combined axial translation of distance $\Delta z$ and rotation of angle $\theta = \Delta z/L$ around the rotor $z$-axis, where $2\pi L$ denotes the helix pitch.
In order to simulate the evolution of such flows, we use a DNS\footnote{Delbende, Rossi and Daube, Theor. Comput Fluid Dyn. 26, 141-160 (2012).} code based on an original formulation in which the helical symmetry is enforced into the equations. We obtain basic states which consist of multiple diffusing helical vortices with or without a central hub vortex, for various values of the helical pitch and vortex core sizes.
Such vortex systems are subject to multiple types of instabilities which have mainly been studied analytically\footnote{Okulov and Sørensen, J. Fluid Mech. 576, 1-25 (2007).} and experimentally\footnote{Bolnot et al, (2012) \textit{Instabilités de tourbillons hélicoïdaux$:$ application au sillage des rotors} Ph.D. Thesis.
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In the present study we first perform a linear temporal stability analysis of these base flows. In order to extract the dominant unstable modes, we use a linearised version of the helical DNS code coupled to an Arnoldi procedure. In the helical framework, instabilities are dominated by the displacement mode presented in figure~\ref{fig1} for the case of two vortices. The influence of the presence of a hub vortex will be also presented. Nonlinear evolutions have then been computed$:$ the above displacement mode is shown to be responsible for a leap-frog dynamics which will be compared to the one observed in vortex rings (see figure~\ref{fig2}). Instability properties as well as nonlinear dynamics will be characterized for helical pitch values ranging from large ones (quasi-2D behaviour) to small ones more pertinent for helicopter and wind turbine applications.