Page 345 - Wind Energy Handbook

EFFECT OF ACROSS-WIND TURBULENCE DISTRIBUTION 319

A5.4.1 Formula for normalized co-spectrum
Ð Ð R
R
2
N
It remains to evaluate S Q1 (n 1 ) ¼ (rUC f ) S u (n 1 ) 0 0 ł (r, r9, n) c(r)c(r9)ì 1 (r)ì 1
uu
N
(r9)dr dr9. The normalized co-spectrum, ł (r, r9, n), must decrease as the spacing
uu
[r r9] between the two points considered increases, and intuitively it is to be
expected that the decrease would be more rapid for the higher frequency compo-
nents of wind fluctuation. On an empirical basis, Davenport (1962) has proposed an
exponential expression for the normalized co-spectrum as follows:
N
ł (r, r9, n) ¼ exp[ Cjr r9jn=U] (A5:21)
uu
where C is a non-dimensional decay constant. Davenport noted that measurements
by Cramer (1958) indicated values of C ranging from 7 in unstable conditions to 50
in stable conditions, but recommended the use of the lower figure as being the more
conservative despite the likelihood of stable conditions in high winds. Dyrbye and
Hansen (1997) quote Riso measurements reported by Mann (1994) which indicate a
value of C of 9.4, and they recommend a value of 10 for use in design. A value of 9.2
is implicitly assumed in Eurocode 1 (1997).
There is an obvious inconsistency in the exponential expression for the normal-
ized co-spectrum – when it is integrated up over the plane perpendicular to the
wind direction, the result is positive instead of zero as it should be. This has led to
the development of more complex expressions by Harris (1971) and Krenk (1995).
However, the Davenport formulation will be used here, giving
2
ó 2 ¼ S Q1 (n 1 ) ð n 1
x1 2
2ä k
1
ð ð " 2 #
R R
2
¼ (rUC f ) S u (n 1 ) exp[ Cjr r9jn 1 =U]c(r)c(r9)ì 1 (r)ì 1 (r9)dr dr9 ð n 1 2
0 0 2ä k 1
(A5:22)
The resonant response can be expressed in terms of the first mode component, x 1 ,of
the steady response,
ð R
1
1 2
2 rU C f ì 1 (r)c(r)dr
k 1 0
from Equation (A5.11) giving
ð ð
R R
exp [ Cjr r9jn 1 =U]c(r)c(r9)ì 1 (r)ì 1 (r9)dr dr9
ó 2 ó 2 ð 2
x1 ¼ 4 u R u (n 1 ) 0 0 ! (A5:23)
2
x 2 U 2ä ð R 2
1
c(r)ì 1 (r)dr
0
Hence