Page 160 - Wind Energy Handbook

134 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

Because of the nature of the Legendre polynomials only one term in the series of
Equation (3.159) will produce a net thrust and only one term will produce a yawing
moment, which is a first moment. Similarly only one term will produce a second
moment, and so on.
The unique term in Equation (3.159) which yields a yawing moment is that for
1
which m ¼ 1, n ¼ 2 and C 6¼ 0, therefore
n
p ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
P (v) ¼ 3v 1 v ¼ 3ì 1 ì 2 (3:174)
2
2
and
p ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffi i
1
2
2
Q (iç) ¼ 3iç 1 þ ç tan 1 3i 1 þ ç þ p ffiffiffiffiffiffiffiffiffiffiffiffiffi , (3:175)
2 ç 1 þ ç 2
so
1
Q (i0) ¼ 2i (3:175a)
2
A zero pressure gradient at the rotor axis is not appropriate in this case because the
pressure distribution is anti-symmetric about the yaw axis, therefore,
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
1
1
1
2
p( ì, ł) ¼ P ( ì)Q (i0)D sin ł ¼ 6iD ì 1 ì sin ł (3:176)
2 2 2 2
The pressure distribution is shown in Figure 3.72.
The yawing moment coefficient is defined by
M z
C mz ¼ (3:177)
1 2 3
rU ðR
1
2
As before, if the pressure in Equation (3.176) is non-dimensionalized by the free-
2
stream dynamic pressure (1=2)rU , then
1
ψ

Figure 3.72 The Form of the Pressure Distribution which Yields a Yawing Moment