Wind Energy

46
Stability of the Tip Vortices in the Far Wake
behind a Wind Turbine
Valery L. Okulov and Jens N. Sørensen
Summary. The work is a further development of a model developed by one of the
authors Okulov (J. Fluid Mech. 521:319–342 2004) in which linear stability of N
equally azimuthally spaced infinity long helical vortices with constant pitch were
considered. A multiplicity of helical vortices approximates the tip vortices of the far
wake behind a wind turbine. The present analysis is extended to include an assigned
vorticity field due to root vortices and the hub of the wake. Thus the tip vortices are
assumed to be embedded in an axisymmetric helical vortex field formed from the
circulation of the inner part of the rotor blades and the hub. As examples of inner
vortex fields we consider three generic vorticity distributions (Rankine, Gaussian
and Scully vortices) at radial extents ranging from the core radius of a tip vortex to
several rotor radii.
46.1 Theory: Analysis of the Stability
The influence of an assigned flow on the stability of multiple vortices has
so far only been studied for circular arrays of point vortices or straight vortex
filaments (the limiting case of a helical vortex with infinite pitch) [2, 3].
A non-perturbed multiple of helical vortices (Fig. 46.1a rotates with constant
angular velocity Ω = Ω0 + ΩInd + ΩSind and moves uniformly along
its axis with velocity V = V0 + VInd + VSind. Note that in contrast to the
plane case, where the total vortex motion consists of an assigned flow field
(Ω0,V0) and the mutual induction of the vortices (ΩInd,VInd) with VInd =0,
we also include self-induction of each of helical vortex (ΩSind,VSind) [1].
From linear stability, introducing infinitesimal space displacements rk =
α (t)exp(2πiks/N), χk = β (t)exp(2πiks/N), of the k-vortex from the its
equilibrium position(a, 2πk/N + Ωt,V t) a correlation equation B =0was
derived for dimensionless pitch τ = l/a and vortex radius ɛ = rcore/a