Michael Graham - Wind Energy Handbook

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, σ u/ u *, and the Richardson number u */( fz 0).

Note that some of the standards used for wind turbine loading calculations prescribe that certain turbulence spectra and/or length scales are to be used. These are often simplified compared to the expressions given above. Thus the Danish standard (DS 472 1992) specifies a Kaimal spectrum with

(2.30)

while the IEC edition 2 standard (IEC 61400‐1 1999) gives a choice between a Kaimal model with

(2.31)

and an isotropic von Karman model with

(2.32)

Editions 3 (IEC 61400‐1 2005) and 4 (IEC 61400‐1 2019) of the IEC standard give a choice of between a slightly different Kaimal model and the Mann model. The Kaimal model has the same form [ Eq. (2.24)] but with

(2.33)

The Mann model has a rather different form and is described in Section 2.6.8.

The Eurocode (EN 1991‐1‐4:2005) standard for wind loading specifies a longitudinal spectrum of Kaimal form with L 1u= 1.7 L i, where

(2.34)

for z < 200 m, with α = 0.67 + 0.05 ln( z 0). This standard is used for buildings but not usually for wind turbines.

With so many variables, it is difficult to present a concise comparison of the different spectra, so a few examples are presented in Figures 2.5and 2.6. These are plots of the normalised longitudinal spectrum nS u(n)/σ u 2against frequency, which means that the area under the curve is representative of the fraction of total variance in any given frequency range. A typical hub height of 80 m has been used, with 50° latitude assumed for the modified von Karman model.

Figure 2.5shows spectra for a typical rated wind speed of 12 m/s. The IEC edition 2 Kaimal spectrum is clearly very similar to DS 472, while the IEC editions 3 and 4 spectrum has clearly moved to lower frequencies, being now more consistent with Eurocode (in fact identical for 80 m height and z 0= 0.01 m). Note the characteristic difference between the Kaimal and von Karman spectra, the latter being rather more sharply peaked. The modified von Karman spectrum is intermediate in shape; with a very small roughness length the peak is at a similar frequency to the IEC edition 2 spectra, but with higher roughness length it comes closer to edition 3.

Figure 2.6shows a similar figure for a typical cut‐out wind speed of 25 m/s. All peaks have moved to higher frequency as expected, but the modified von Karman now matches IEC editions 3 and 4 better with the very small roughness length.

Other spectra may also be used, but to comply with the IEC standard the high frequency asymptotic behaviour must tend to the following relationship:

(2.35)

with Λ 1defined as above: it is a function only of height above ground but differs between edition 2 and edition 3 for heights above 30 m. Expressing this as

(2.36)

the asymptotic parameter A can then be compared for different spectra, as in Figure 2.7. This shows how the DS 472 asymptote is similar to IEC edition 2 (both Kaimal and von Karman spectra in that standard have the same asymptote), but the asymptote becomes much lower for IEC editions 3 and 4 above 30 m height and is now more comparable with Eurocode. The ESDU modified von Karman spectrum is more difficult to characterise because the asymptote now varies also with wind speed, surface roughness, and geographical latitude. Figure 2.7shows the results for 20 m/s wind speed and 50° latitude for two different roughness lengths. The asymptote can be made to match the IEC editions 3 and 4 specification, but only by choosing a very small roughness length – even smaller to match edition 2. However, if such a low roughness length is selected, the turbulence intensity implied by the ESDU model will be much smaller than that required by the standards. It is common practice when using the ESDU model to adjust the surface roughness at each wind speed until the turbulence intensity matches the standard, although this clearly makes little physical sense. Obviously it is not possible to adjust the roughness to match both the required asymptote and the required turbulence intensity at the same time. Compared to the physical model, therefore, the standards are probably conservative, which is surely to be expected. It can also be argued that the physical model is valid for flat terrain, while many wind farms are built in complex terrain where the turbulence intensities will indeed be higher, and the length scales shorter.

Figure 2.7 Some asymptotic limits

Note also that the IEC edition 3 and 4 standards further specify that in the high frequency limit.

The turbulence spectra presented in the preceding sections describe the temporal variation of each component of turbulence at any given point. However, as the wind turbine blade sweeps out its trajectory, the wind speed variations it experiences are not well represented by these single‐point spectra. The spatial variation of turbulence in the lateral and vertical directions is clearly important, because this spatial variation is ‘sampled’ by the moving blade and thus contributes to the temporal variations experienced by it.

To model these effects, the spectral description of turbulence must be extended to include information about the cross‐correlations between turbulent fluctuations at points separated laterally and vertically. Clearly these correlations decrease as the distance separating two points increases. The correlations are also smaller for high frequency than for low frequency variations. They can therefore be described by ‘coherence’ functions, which describe the correlation as a function of frequency and separation. The coherence C ( Δ r,n) is defined by