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2.6.4 Turbulence spectra
The spectrum of turbulence describes the frequency content of wind-speed variations. According to the Kolmogorov law, the spectrum must approach an asymptotic limit proportional to n~5=3 at high frequency (here n denotes the frequency, in Hz). This relationship is based on the decay of turbulent eddies to higher and higher frequencies as turbulent energy is dissipated as heat.
Two alternative expressions for the spectrum of the longitudinal component of turbulence are commonly used, both tending to this asymptotic limit. These are the Kaimal and the von Karman spectra, which take the following forms:
nSu(n) 4nL2u/U n - ,, von Karman: -r— =-=-— (2.24)
where Su(n) is the autospectral density function for the longitudinal component and L1u and L2u 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., L1u = 2.329L2u. 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 L2u is identified as the integral length scale of the longitudinal component in the longitudinal direction, denoted xLu. 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):
nSJjn) = ft 2987nW_U + 1294 nL3u/U o 2u F1(1 + (2n nL3u/U )2)5/6 F2(1 + (n nL3u /U )2)5/6 1 ( )
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, L1v and L1w respectively. The von Karman spectrum for the i component (i = v or w) is
L2v nSi(n) _ 4(nL2i/U)(1 + 755.2(nLzf/U)2) a2 _ (1 + 283.2( nL2i/U )2)11/6
cLv and L2w = xLw. For the modified von Karman spectrum it is
nSi( n) = 2.987( nL3l/U )(1 + ^ nL3l/U )2) 1.294nL3t/U F
a2 Pl (1 + (4rcnL3i/U)2)11/6 ^2(1 + (2^nL3l/U)2)5/6 2i ( : )
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 Fi, and F2 are also required for the modified von Karman model.
The length scales are dependent on the surface roughness zo, as well as on the height above ground (z); proximity to the ground constrains the size of turbulent eddies and thus reduces the length scales. If there are many small obstacles on the ground of typical height z9, the height above ground should be corrected for the effect of these by assuming that the effective ground surface is at a height z' — 2.5zo (ESDU, 1975). Far enough above the ground, i.e., for z > z/, the turbulence is no longer constrained by the proximity of the surface and becomes isotropic. According to ESDU (1975), z, = 1000z018 and above this height xL„ = 280 m, and yL„ = zL„ = xLv = zLv = 140 m. Even for very small roughness lengths zo, the isotropic region is well above the height of a wind turbine and the following corrections for z < z, should be applied:
xLu = 280(z/z;)0'35 yL„ = 140(z/zi)°'38 zL„ = 140(z/zj )°'45 xLv = 140(z/zi)0'48 zL„ = 140(z/z;)0'55
together with xLw = yLw = 0.35z (for z < 400 m). Expressions for yLv and zLw are not given. The length scales xL„, xLv and xLw can be used directly in the von Karman spectra. For the Kaimal spectra we already have L1u = 2.329xL„, and to achieve the same high frequency asymptotes for the other components we also have L1„ = 3.2054xL„, L2w = 3.2054xLw.
Later work based on measurements for a greater range of heights (Harris, 1990; ESDU, 1985) takes into account an increase in length scales with the thickness of the boundary layer, h, which also implies a variation of length scales with mean wind speed. This yields more complicated expressions for the nine length scales in terms of z/h, a„/m* and the Richardson number u*/(/z0).
Note that some of the standards used for wind turbine loading calculations prescribe that certain turbulence spectra and/or length scales be used. These are often simplified compared to the expressions given above. Thus the Danish standard (DS 472,1992) specifies a Kaimal spectrum with
while the IEC standard (IEC, 1999) recommends either a Kaimal model with
L1u = 170.1 m, or 5.67z for z < 30 m Liv = 0.3333 Li« Liw = 0.08148 Li«
or an isotropic von Karman model with
Eurocode 1 (1997) specifies a longitudinal spectrum of Kaimal form with L1u 1.7Li, where
for z < 300 m, with e varying between 0.13 over open water to 0.46 in urban areas. This standard is used for buildings, but not usually for wind turbines.
Figure 2.5 compares these various longitudinal turbulence spectra at 30 m height, for a mean wind speed of 10 m/s. The surface roughness is 0.001 m, corresponding to very flat land or rough sea, and the latitude is 50°. There is reasonable agreement between the various spectra in this situation, apart from the Eurocode spectrum which is shifted to somewhat lower frequencies. Note the characteristic difference between the Kaimal and von Karman spectra, the latter being rather more sharply peaked. The improved von Karman spectrum (Equation 2.25) is intermediate in shape.
30m height, 10m/s, z0 = 0.001m, 50 latitude ~l I I I I I I I I
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