Page 158 - Wind Energy Handbook

132 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

conditions also depend upon the angle of yaw of the disc. The integration continues
until a point on the disc is reached where the induced velocity is to be determined.
The particular induced velocity component that is most important for determin-
ing the angle of attack on a blade element is normal to the rotor disc, i.e., the axial
induced velocity. Mangler and Squire (1950) calculated the axial induced velocity
distribution as a function of yaw angle by expressing the velocity as a Fourier series
of the azimuth angle ł.
!
1
u A 0 ( ì, ª) X
¼ C T þ A k ( ì, ÷)sin kł (3:167)
U 1 2
k¼1
For the pressure distribution of Equation (3.166) the Fourier coefficients are

15 2 2 1=2
A 0 ( ì, ª) ¼ ì (1 ì ) (3:168a)
8
15ð ª
2
A 1 ( ì, ª) ¼ ì(9ì 4)tan (3:168b)
256 2

45ð ª 3
3
A 3 ( ì, ª) ¼ ì tan (3:168c)
256 2
Higher-order odd terms are zero. There are also even terms which have the general
form
k=2
2
2
3 k þ v 9v þ k 6 3v 1 v ª k=2
(k 2)=2
A k ¼ ( 1) þ tan
2
2
2
4 k 1 k 9 k 9 1 þ v 2
2
2
where v ¼ 1 ì and k is an even integer greater than zero.
The average value of the axial induced flow factor is independent of yaw angle
and is given by
1
u 0
a 0 ¼ ¼ C T
U 1 4
where u 0 is the average axial induced velocity.
Thus, the average value of the axial flow induced flow factor is related to the
thrust coefficient by
C T ¼ 4a 0 (3:169)

compared with the momentum theory C T ¼ 4a 0 (1 a 0 ) or compared with any of
the expressions developed for yawed conditions (Equations (3.91), (3.106) and
(3.112)).
Because of the assumption that the induced velocity is small compared with the
flow velocity, a 0 is small compared with 1. Clearly, the acceleration potential
method only applies if the value of C T is much less than 1.