Page 148 - Wind Energy Handbook

122 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

d 1 2
F x ¼ rW cC y
dr 2
and the blade torque contribution about the axis of rotation is
d 1 2
Q ¼ rW crC y (3:141)
dr 2
The total torque is found by integrating along each blade and summing over all the
blades, just as for the normal force. Again, the torque on the rotor will vary with
azimuth position so to find the average torque will require a further integration
with respect to azimuth.

3.10.11 Yawing and tilting moments in steady yaw

The asymmetry of the flow through a yawed rotor, caused by the flow expansion,
means that a blade sweeping upwind has a higher angle of attack than when it is
sweeping downwind, as shown in Figure 3.61. The blade lift upwind will therefore
be greater than the lift downwind and a similar differential applies to the forces
normal to the rotor plane. It can be seen, therefore, that there is a net moment about
the yaw (vertical axis) in a direction which will tend to restore the rotor axis to a
position aligned with the wind direction. The yawing moment is obtained from the
normal force of Equation (3.140)

d 1 2
M z ¼ rW cr sin łC x (3:142)
dr 2
which will also vary with the azimuth position of the blade. The total single-blade
yawing moment at each azimuth position is obtained by integrating Equation
(3.142) along the length of the blade. Summing the moments for all blades, suitably
separated in phase, will result in the yawing moment on the rotor.
A similar calculation can be made for the tilting moment, the moment about the
horizontal diametral axis (y-axis) of the rotor:

d 1 2
M z ¼ rW cr cos łC x (3:143)
dr 2
Measured results of rotor yaw moment for the Delft turbine are shown in Figure
3.67 and the corresponding calculated yawing moments are shown in Figure 3.68.
The measured yawing moments were derived from strain gauge readings of the
flat-wise bending strain close to the root of the blade at 129 mm radius. Flat-wise, or
flap-wise bending causes only displacements normal to the rotor plane. The calcu-
lated yawing moments are determined at the same radial position on the blade and
are, therefore, not the true yawing moments about the actual yaw axis.
The comparison between the measured and calculated yaw moments is quite
good taking into account the limitations of the theory. At 308 of yaw the calculated
values underestimate the measurements significantly whereas at the two lower