Michael Graham - Wind Energy Handbook

The earlier GL rules (GL 1993) simply specified 20% turbulence intensity, but the later edition (GL 2003) follows IEC edition 2.

Figure 2.4shows example longitudinal turbulence intensities for the GL, IEC, and Danish standards. The low value for the Danish standard is for 90 m height with roughness length 0.01 m; the high value is for 30 m height with roughness length 0.3 m. The high values for IEC editions 2, 3, and 4 are the same, but the low value is significantly lower in editions 3 and 4 than in edition 2.

Figure 2.4 Turbulence intensities according to various standards

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 −5/3at 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:

Kaimal:

(2.24)

von Karman:

(2.25)

where S u( n ) is the autospectral density function for the longitudinal component and L 1uand L 2uare length scales. 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= 2.329 L 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 2uis identified as the integral length scale of the longitudinal component in the longitudinal direction, denoted x L uand defined as where κ(r x) is the cross‐correlation function between the turbulence component u at two points separated longitudinally by a distance r xand measured simultaneously (similar definitions apply to the integral length scales of the longitudinal component of turbulence in the lateral and vertical directions, y L uand z L u, which are used in the definitions of cross‐spectra below, and also to the integral length scales of the lateral and vertical components in the three directions). It is important to recognise that the power spectra and accompanying length scales are theoretical constructs, and attempts to fit them to real atmospheric data result in semi‐empirical models in which the length scales may not be fully consistent with theory.

Figure 2.5 Comparison of spectra at 12 m/s

The Kaimal spectrum has a lower, broader peak than the von Karman spectrum: see Figures 2.5and 2.6. More 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):

(2.26)

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 1vand L 1w, respectively. The von Karman spectrum for the i component ( i = v or w) is

(2.27)

Figure 2.6 Comparison of spectra at 25 m/s

where L 2v= x L vand L 2w= x L w. For the modified von Karman spectrum of Eq. (2.26), it is

(2.28)

To use the spectra defined above, it is necessary to define the appropriate length scales. Additional parameters β 1, β 2, F 1, and F 2are also required for the modified von Karman model.

The length scales are dependent on the surface roughness z 0as 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 z ', the height above ground should be corrected for the effect of these by assuming that the effective ground surface is at a height z ' − 2.5 z 0(ESDU 1975). Far enough above the ground, i.e. for z greater than some height z i, the turbulence is no longer constrained by the proximity of the surface and becomes isotropic. According to ESDU (1975), z i= 1000 z 0 0.18, and above this height x L u= 280 m, and y L u= z L u= x L v= z L v= 140 m. Even for very small roughness lengths z 0, the isotropic region is well above the height of a wind turbine, and the following corrections for z < z ishould be applied:

(2.29)

together with x L w= y L w= 0.35 z (for z < 400 m). Expressions for y L vand z L ware not given. The length scales x L u, x L v, and x L wcan be used directly in the von Karman spectra. For the Kaimal spectra we already have L 1u= 2.329 x L u, and to achieve the same high frequency asymptotes for the other components we also have L 1v= 3.2054 x L v, L 1w= 3.2054 x L w.