Wind Energy Handbook. Michael Barton Graham
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Название: Wind Energy Handbook
Автор: Michael Barton Graham
Издательство: John Wiley & Sons Limited
Жанр: Физика
isbn: 9781119451167
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2.6.6 Asymptotic limits
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 Λ1 defined 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 2.7 shows 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
2.6.7 Cross‐spectra and coherence functions
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 (Δr,n) is defined by
(2.37)
where n is frequency, S12(n) is the cross‐spectrum of variations at the two points separated by Δr, and S11(n) and S22(n) are the spectra of variations at each of the points (usually these can be taken as equal).
Starting from von Karman spectral equations, and assuming Taylor's frozen turbulence hypothesis, an analytical expression for the coherence of wind speed fluctuations can be derived. Accordingly for the longitudinal component at points separated by a distance Δr perpendicular to the wind direction, the coherence Cu (Δr,n) is:
Here Aj(x) = xj Kj(x) where K is a fractional order modified Bessel function, and
with c = 1. Lu is a local length scale that can be defined as