Page 54 - Wind Energy Handbook

28 THE WIND RESOURCE
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2
0:747 2ðn
ç u ¼ ˜r þ c (2:35)
L u U

with c ¼ 1. L u is a local length scale which can be defined as:
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y
z
2
( L u ˜y) þ ( L u ˜z) 2
L u (˜r, n) ¼ 2f u (n) (2:36)
2
˜y þ ˜z 2
where ˜y and ˜z are the lateral and vertical components of the separation ˜r,
z
y
and L u and L u are the lateral and vertical length scales for the longitudinal
component of turbulence. Normally f u (n) ¼ 1, but ESDU (1975) suggests a mod-
ification at low frequencies where the wind becomes more anisotropic, with
f u (n) ¼ MIN(1:0, 0:04n 2=3 ).
The IEC (1999) standard gives an isotropic turbulence model for use with the von
x
z
x
y
Karman spectrum, in which L u ¼ 2 L u ¼ 2 L u , and then L u ¼ L u , and f u (n) ¼ 1.
The modified von Karman model described in Equation (2.25) also uses f u (n) ¼ 1,
but the factor c in Equation (2.35) is modified instead (ESDU, 1985).
For the lateral and vertical components, the corresponding equations are as
follows. The analytical derivation for the coherence, based as before on the von
Karman spectrum and Taylor’s hypothesis, is
0:597
2
C i (˜r, n) ¼ 2 [4:781ª A 5=6 (ç i ) A 11=6 (ç i )] (2:37)
i
2:869ª 1
i
for i ¼ u or v, where ç i is calculated as in Equation (2.35) but with L u replaced by L v
or L w respectively, and with c ¼ 1. Also

ç i L i (˜r, n)
ª i ¼ (2:38)
˜r
and L v and L w are given by expressions analogous to Equation (2.36).
The expressions for spatial coherence in Equations (2.34) and (2.37) above are
derived theoretically from the von Karman spectrum, although there are empirical
factors in some of the expressions for length scales for example. If a Kaimal rather
than a von Karman spectrum is used as the starting point, there are no such
relatively straightforward analytical expressions for the coherence functions. In this
case a simpler and purely empirical exponential model of coherence is often used.
The IEC (1999) standard, for example, gives the following expression for the
coherence of the longitudinal component of turbulence:
0 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

n
@ 0:12 2 2 A
C u (˜r, n) ¼ exp 8:8˜r þ
U
L u
ffi exp( 1:4ç u ) (2:39)