Page 49 - Wind Energy Handbook

TURBULENCE 23

nS u (n) 4nL 1u =U
Kaimal: ¼ (2:23)
ó 2 u (1 þ 6nL 1u =U) 5=3
nS u (n) 4nL 2u =U
von Karman: ¼ (2:24)
ó 2 (1 þ 70:8(nL 2u =U) )
2 5=6
u
where S u (n) is the autospectral density function for the longitudinal component
and L 1u and L 2u are length scales. In order for these two forms to have the same
high-frequency asymptotic limit, these length scales must be related by the ratio
(36=70:8) 5=4 , i.e., L 1u ¼ 2:329L 2u . The appropriate length scales to use are discussed
in the next section.
According to Petersen et al. (1998), the von Karman spectrum gives a good
description for turbulence in wind tunnels, although the Kaimal spectrum may give
a better fit to empirical observations of atmospheric turbulence. Nevertheless the
von Karman spectrum is often used for consistency with analytical expressions for
the correlations. The length scale L 2u is identified as the integral length scale of the
longitudinal component in the longitudinal direction, denoted x L u . The Kaimal
spectrum has a lower, broader peak than the von Karman spectrum (see Figures 2.5
to 2.7).
Recent work suggests that the von Karman spectrum gives a good representation
of atmospheric turbulence above about 150 m, but has some deficiencies at lower
altitudes. Several modifications have been suggested (Harris, 1990) and a modified
von Karman spectrum of the following form is recommended (ESDU, 1985):

nS u (n) 2:987nL 3u =U 1:294nL 3u =U
¼ â 1 þ â 2 F 1 (2:25)
ó 2 (1 þ (2ðnL 3u =U) ) (1 þ (ðnL 3u =U) )
2 5=6
2 5=6
u
All three of these spectra have corresponding expressions for the lateral and vertical
components of turbulence. The Kaimal spectra have the same form as for the
longitudinal component but with different length scales, L 1v and L 1w respectively.
The von Karman spectrum for the i component (i ¼ v or w)is

2
nS i (n) ¼ 4(nL 2i =U)(1 þ 755:2(nL 2i =U) )
ó 2 (1 þ 283:2(nL 2i =U) ) (2:26)
2 11=6
i
x
x
where L 2v ¼ L v and L 2w ¼ L w . For the modified von Karman spectrum it is
2
8
nS i (n) 2:987(nL 3i =U)(1 þ (4ðnL 3i =U) ) 1:294nL 3i =U
3
¼ â 1 þ â 2 F 2i (2:27)
ó 2 (1 þ (4ðnL 3i =U) ) (1 þ (2ðnL 3i =U) )
2 5=6
2 11=6
i
2.6.5 Length scales and other parameters
To use the spectra defined above, it is necessary to define the appropriate length
scales. Additional parameters â 1 , â 2 , F 1 , and F 2 are also required for the modified
von Karman model.