Page 266 - Wind Energy Handbook

240 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

ignored. Because the flow angle, ö, is small at high tip speed ratio, º, the relative air
velocity, W, can be assumed to be changing much more slowly with the wind speed
than C l , so that dW=du can be ignored. As a result,

dL 2 dC l dÆ
1
¼ rW c (5:24)
du 2 dÆ du
where Æ, the angle of attack, is equal to (ö â).
If the blades are not pitching, then the local blade twist, â, is constant, so that
dÆ=du ¼ dö=du. To preserve linearity, it is necessary to assume that the rate of
change of lift coefficient with angle of attack, dC l =dÆ is constant, which is tenable
only if the blade remains unstalled. Assuming, for simplicity, that the wake is
frozen, i.e., that the induced velocity, Ua, remains constant, despite the wind speed
fluctuations, u, we obtain

tan ö ffi (U(1 a) þ u)=Ùr,

so that, for ö small, dj=du ffi 1=Ùr and W ffi Ùr, leading to

dL dC l u dC l
1
1
2
˜L ¼ L L ¼ u ¼ r(Ùr) c ¼ rÙrc u (5:25)
du 2 dÆ Ùr 2 dÆ
Hence

ó L ¼ 1 rÙ dC l rcó u
2 dÆ

Normally dC l =dÆ is equal to 2ð.
If the turbulence integral length scale is large compared to the blade radius, then
the expression for the standard deviation of the blade root bending moment
(assuming a completely rigid blade) approximates to

ð R ð R
2
1
ó M ¼ ó L r dr ¼ rÙ dC l ó u c(r)r dr (5:26)
2 dÆ
0 0
where ó u is the standard deviation of the wind speed incident on the rotor disc
which, by virtue of the ‘frozen wake’ assumption, equates to the standard deviation
of the wind speed in the undisturbed flow. If, as will be the case in practice, the
longitudinal wind fluctuations are not perfectly correlated along the length of the
blade, then
2ð ð
R R
ó 2 ¼ 1 rÙ dC l 2 2 (5:27)
1 2
M 2 dÆ 0 0 k u (r 1 , r 2 ,0)c(r 1 )c(r 2 )r r dr 1 dr 2