Page 140 - Wind Energy Handbook

114 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

actual rotor disc. The distance upstream or downstream of a point on the actual
rotor disc from the plane of the disc normal to the wake axis determines the value
of the root vortex influence function h(ł). The value of h(ł) will lie between 0.0 and
2.0, being equal to 1.0 at points on the vertical diameter.
The velocity induced by a semi-infinite line vortex of strength ˆ lying along the x-
axis from zero to infinity at a point with cylindrical co-ordinates (x-, ł-, r-) is,
using the Biot–Savart law,
2 3
0 2 0 3
6 7
! ˆ 6 x- 7 6 7
V- ¼ 6 1 þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 ¼ 4 v- 5 (3:126)
4ðr- 4 x- þ r- 2 5
2
0
0
The induced velocity when x- ¼1 is twice that when x- ¼ 0 and is zero when
x- ¼ 1.
For a point on the rotor disc (0, ł, r) the corresponding co-ordinates (x-, ł-, r-)
in normal disc axes are
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
x- ¼ y- sin ÷ ¼ r sin ł sin ÷, r- ¼ r cos ł þ cos ÷ sin ł 2
and

r r
cos ł- ¼ cos ł, sin ł- ¼ sin ł cos ÷ (3:127)
r- r-
the induced velocity at the same point is

x-
v- ¼ Ùr-a9 1 þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (3:128)
2
x- þ r- 2
So, transforming the velocity of (3.126) to the rotating axes in the plane of the rotor
disc

2 3 2 cos ÷ sin ÷ 3 2 1 0 0 32 0 3
1 0 0 0 6 76 7
! 6 7 6 7 6 76 7
7
6
6
V0 ¼ 4 0 cos ł sin ł 5 sin ÷ cos ÷ 0 6 0 cos ł- sin ł- 7 v- 7
5
4
4
54
5
0 sin ł cos ł 0 0 1 0 sin ł- cos ł- 0
(3:129)
Substituting (3.127) and (3.128) into (3.129) gives
2 3
cos ł sin ÷
! 6 7
V0 ¼ 4 cos ÷ 5Ùra9(1 þ sin ł sin ÷) (3:130)
0