Page 268 - Wind Energy Handbook

242 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

Step 1 – Derivation of the autocorrelation function at a fixed point: The von Karman
spectrum

S u (n) ¼ 4L x u (5:31)
ó 2 x 2 5
u U(1 þ 70:8(nL =U) ) 6
u
is chosen as the input power spectrum of the along-wind wind speed fluctuations
at a fixed point in space. It can be shown that Equation (5.30) yields the following
expression for the corresponding auto correlation function:
1
2ó 2 ô=2 3 ô
u
k u (ô) ¼ K1 (5:32)
ˆ 1 T9 3 T9
3
x
where T9 is related to the integral length scale, L , by the formula
u

ˆ 1 x x
T9 ¼ 3 p ffiffiffi L u ffi 1:34 L u (5:33)
ˆ 5 ð U U
6
ˆ( ) is the Gamma function and K1(x) is a modified Bessel function of the second
3
1
kind and order ı ¼ . The general definition of K ı (x) is:
3

1
ð X (x=2) 2m (x=2) ı (x=2Þ ı
K ı (x) ¼ (5:34)
2 sin ðı m! ˆ(m ı þ 1) ˆ(m þ ı þ 1)
m¼0
Step 2 – Derivation of the autocorrelation function at a point on the rotating blade: This
derivation makes use of Taylor’s ‘frozen turbulence’ hypothesis, by which the
instantaneous wind speed at point C at time t ¼ ô is assumed to be equal to that at a
point B a distance Uô upwind of C at time t ¼ 0, U being the mean wind speed.
o
Thus, referring to Figure 5.17, the autocorrelation function k (r, ô) for the along-
u
wind wind fluctuations seen by a point Q at radius r on the rotating blade is equal
to the cross correlation function k u (~ s, 0) between the simultaneous along-wind wind
s
fluctuations at points A and B. Here A and C are the positions of point Q at the
s
beginning and end of time interval ô respectively, B is Uô upwind of C and ~ s is the
vector BA. (Note that the superscript 8 denotes that the autocorrelation function
relates to a point on a rotating blade rather than a fixed point. The same convention
will be adopted in relation to power spectra.)
Batchelor (1953) has shown that, if the turbulence is assumed to be homogeneous
s
and isotropic, the cross correlation function, k u (~ s, 0) is given by:
2
s 1
s
k u (~ s,0) ¼ (k L (s) k T (s)) þk T (s) (5:35)
s
where k L (s) is the cross correlation function between velocity components at points
B
A
A and B, s apart, in a direction parallel to AB (v and v in Figure 5.17), and k T (s)is
L L