Wind Energy

250 V.L. Okulov and J.N. Sørensen
Γ 0
Ω 0
x
Z
r hub
a
V 0
h = 2πl
Γ Γ
2r core
Fig. 46.1. Sketch of N tip helical vortices embedded in assigned flow (Ω0,V0) and
relative velocity plots about vortex axis for (a) unstable and (b) stable far wakes
4πa2 �
2 1+τ
B = s (N − s)
Γ �3/2 τ 3
N − 2
− 2N +2
τ 2
1
+
τ (1 + τ 2 ) 3/2
��
τ 2 − 1
�� �
E + ψ −
4
s
�
−
N
N
�
s
+ 3
4 − 2τ 2 � �
2 1+τ
− ln Nε
� �� 3/2
+
τ
1+2τ2 τ (1 + τ 2 ) 1/2
τ
+
3 �
2τ 4 − 6τ 2 + 3
�
1.20206
4 N 2
4πa2
+
Γ BA
(1 + τ 2 ) 9/2
y
a
b
(46.1)
where s = [0; N − 1] is the subharmonic wave number; E = 0.5772. . . is
the Euler constant and the psi-function ψ(·) takes values such as ψ(−1/2) =
0.0395..., ψ(−1/3) = 1.6818..., ψ(−2/3) = −1.6320... (see, e.g. [1]). The
last term of BA in (46.1) describes the influence of the assigned velocity
field due to root vorticity and the hub of the rotor. Three important examples
of the central axisymmetrical helical vortices have been considered with
R- (step-shape), L- (Gaussian) and S- (hat-type) axial vorticity distribution
corresponding to Rankine, Lamb and Scully vortices at τ →∞[4]. For those
types of assigned flow field the BA-term takes form:
�
−1, r < δ,
BA =4Nγ
1 � τ 2δ2 (2-R)
,r≥ δ,
1
BA =4Nγ
1+δ2 � 2 1+τ
τ 2
δ2 �
− 1 , (2-L)
1+δ2