Modern wind-power generators (wind turbines)often are grouped together in order to ensure anincreased operating efficiency at the expense of usinga large number of turbines in the same terrain area.However, in this case, one or several turbines can turnout to be in the wake beyond other turbines. As isshown in Fig. 1, the airflow beyond a wind turbine isrepresented by a system of intense rotational helicalvortices determining the dynamics of a far wake. Thewake produces a significant periodic load for the struc-ture of a wind turbine present in the wake. This fact,naturally, results in a reduction of the turbine in-ser-vice time. However, as is well known, for certain oper-ating regimes of wind turbines, the vortex wakebecomes unstable and breaks down, as is shown inFig. 1b. If a turbine is located in a stable vortex wake(Fig. 1a), then the interaction with it may have signifi-cant consequences than in the case of wake breakdown(Fig. 1b). Therefore, analysis of stability conditions fora vortex system simulating the vortex wake beyondwind turbines, screws, and aerodynamic propellers isof great practical importance.At a sufficiently large distance behind a wind tur-bine or an aerodynamic propeller, the vorticity is con-centrated in N blade-tip vortices of helical shape. Thevortices are located on a cylindrical surface of radius R and have the same azimuth shift by the angle . Leteach of the vortices have circulation Γ and a radius ofthe vortex core equal to e . In addition, let a rectilinearvortex with the opposite total circulation — N Γ existalong the system axis. Such an ( N + 1)-vortex system2πN------ with the constant helical pitch 2 π l (or, in the dimen-sionless form, τ = l / R ) is the simplest model of a wakebehind aerodynamic propellers and wind turbines [1].In this approximation, the problem under consider-ation is reduced to the analysis of stability of the( N + 1)-vortex system that moves in uniform airflowat a constant wind speed V (Fig. 2). Since the flowbehind the wind turbine has a wake-like profile, weanalyze a system of left-handed helical vortices [2]. Itis worth mentioning that the system of ( N + 1) vorticeswas studied previously only in the case of point vorti-ces or rectilinear vortex filaments (the ultimate case ofhelical vortices with an infinitely large pitch) forwhich their instability was determined (see, e.g., [3]).It is clear that this result obtained for the simplest par-ticular case ( τ = ∞ ) is inconsistent with the visualiza-tions of wakes behind aerodynamic propellers andwind turbines (see, e.g., [4]). In addition, the linearanalysis of stability for a simpler equilibrium con-figuration of N helical vortices without a centralvortex, which model a pair, triplet, etc., of helical vor-tices arising in the tornado core after the vortex break-down [5, 6] cannot be applied for solving the problemposed. Therefore, in the present paper, the stabilityanalysis for wakes behind aerodynamic propellers,screws, and wind turbines is generalized to the case ofa system consisting of N + 1 left-handed vortices.In accordance with the ideas of [7], outside the vor-tex cores, the components of the velocity induced bythe system of N + 1 left-handed vortices, which movesat constant wind speed V and additionally rotates in thefield generated by the central circulation vortex — N Γ ,can be written out in the formu
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Apr 1, 2009
Manikandan Ramasamy + 3
Open Access