Page 131 - Wind Energy Handbook

THE AERODYNAMICS OF A WIND TURBINE IN STEADY YAW 105

analysis of Section 3.4. The vorticity g ł has a direction which remains parallel to the
yawed disc and assuming it to be uniform (not varying with the azimuth angle),
using the Biot–Savart law, induces an average velocity at the disc of aU 1 sec ÷=2ina
direction which bisects the skew angle between the central axis of the skew angle,
as shown in Figure 3.54. The average axial induced velocity, normal to the rotor
plane, is aU 1 , as in the unyawed case. In the fully developed wake the induced
velocity is twice that at the rotor disc.
Because the average induced velocity at the disc is not in the rotor’s axial
direction, as is assumed for the momentum theory of Sections 3.10.1 and 3.10.2, the
force F on the disc, which must be in the axial direction, cannot be solely res-
ponsible for the overall rate of change of momentum of the flow; there is a change
of momentum in a direction normal to the rotor axis.
The velocity components at the rotor disc define the skew angle:

÷ ÷
U 1 sin ª a tan 2 tan
2
tan ÷ ¼ ¼ 2 (3:109)
U 1 (cos ª a) ÷ 2
1 tan
2
From which it can be shown that a close, approximate relationship between ÷, ª
and a is
÷ ¼ (0:6a þ 1)ª (3:110)

Using the velocities shown in Figure 3.54 a fresh analysis can be made of the flow.
The average force on the disc can be determined by applying Bernoulli’s equation
to both the upwind and downwind regions of the flow.

Upwind

1 2 þ 1 2
p 1 þ rU 1 ¼ p þ rU d
d
2 2

γ χ
F 2
χ χ
2 2U a sec

χ 2
U a sec

2
U

Figure 3.54 Average Induced Velocities Caused by a Yawed Actuator Disc