Wind Energy

246 J.N. Sørensen and V.L. Okulov
wz(r, χ) ∼ = ΓN
� �
1
+
2πl 0
Γ
4√
l2 + a2 2πl 4√
l2 + r2 Re
N�
� i(χ−2πn/N)
{±}e
e
n=1
{∓}ξ − ei(χ−2πn/N) + l
� 2 2 3r − 2l
24 (l2 + r2 ) 3/2 + 9a2 +2l2 (l2 + a2 ) 3/2
� �
ln 1 − e ξ+i(χ−2πn/N)��
wθ(r, χ) =(ΓN/2πr) − (lwz(r, χ)/).
(45.2)
Here the terms in the braces are defined such that the upper one corresponds
to ra. The helical vortex parameters a, l, Γare
introduced in Fig. 45.1, where ΓT refers to the circulation of a tip vortex and
Γ0 refers to the circulation of the inner vortex, and
e ξ � � √
r l + l2 + a2 =
� exp �√ l2 + r2�� � � √
a l + l2 + r2 � exp �√ l2 + a2��. From (45.2) it is seen that the azimuthally averaged axial velocity is constant
in the wake. This is a direct implication of one of the basic assumptions of
Joukowski, that the circulation along each rotor blade is constant (≡ Γ) and
that the total circulation is zero. If we look at experimental data, however, a
constant axial velocity profile is rarely seen [1, 5].
In order to develop a model that is capable of reproducing measured velocity
distributions, we extend the model of Joukowski by assuming the tip
vortices to be embedded in an axisymmetric helical vortex field formed from
the circulation Γ0 of the rotor blades and the hub (see Fig. 45.1b). The resulting
axial and azimuthal velocity components are described by the following
formulas:
uz = V0 − wz + Γ0f
2πl , uθ = wθ − Γ0f 1
, f =
2πr δ2 � �
2 2 δ − r ,r