Michael Graham - Wind Energy Handbook

(2.37)

where n is frequency, S 12( n ) is the cross‐spectrum of variations at the two points separated by Δ r , and S 11( n ) and S 22( 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 Δ r perpendicular to the wind direction, the coherence C u ( Δ r,n) is:

(2.38)

Here A j (x) = x j K j (x) where K is a fractional order modified Bessel function, and

(2.39)

with c = 1. L uis a local length scale that can be defined as

(2.40)

where Δ y and Δ z are the lateral and vertical components of the separation Δ r , and y L uand z L uare the lateral and vertical length scales for the longitudinal component of turbulence. Normally f u( n ) = 1, but ESDU (1975) suggests a modification at low frequencies where the wind becomes more anisotropic, with f u( n ) = MIN (1.0, 0.04 n −2/3).

The (1999) edition 2 standard allows only an isotropic turbulence model to be used if the von Karman spectrum is used, in which x L u= 2 y L u= 2 z L u, and then L u= x L u, and f u( n ) = 1.

The modified von Karman model described in Eq. (2.26)also uses f u( n ) = 1, but the factor c in Eq. (2.39)is modified instead (ESDU 1985).

For the lateral and vertical components, the corresponding equations are as follows. The analytical derivation for the coherence, based as before on the von Karman spectrum and Taylor's hypothesis, is

(2.41)

for i = u or v , where η iis calculated as in Eq. (2.39)but with L ureplaced by L vor L w, respectively, and with c = 1. Also

(2.42)

and L vand L ware given by expressions analogous to Eq. (2.40).

The expressions for spatial coherence in Eqs. (2.38)and (2.41)are derived theoretically from the von Karman spectrum, although there are empirical factors in some of the expressions for length scales, for example. If a Kaimal rather than a von Karman spectrum is used as the starting point, there are no such relatively straightforward analytical expressions for the coherence functions. In this case a simpler, and purely empirical, exponential model of coherence is often used. The (1999) edition 2 standard, for example, gives the following expression for the coherence of the longitudinal component of turbulence:

(2.43)

where H = 8.8 and L c= L u. This can also be approximated by

(2.44)

with η uas in Eq. (2.39).

The standard also states that this may also be used with the von Karman model, as an approximation to Eq. (2.38). The standard does not specify the coherence of the other two components to be used in conjunction with the Kaimal model, so the following expression is often used:

(2.45)

In the later editions, IEC (2005) and IEC (2019), a slightly modified form is specified, in which H = 12 and L c= L 1u.

The three turbulence components are usually assumed to be independent of one another. This is a reasonable assumption, although it ignores the effect of Reynolds stresses that result in a small correlation between the longitudinal and vertical components near to the ground, an effect that is captured by the Mann model described in Section 2.6.8.

Clearly there are significant discrepancies between the various recommended spectra and coherence functions. Also these wind models are applicable to flat sites, and there is only limited understanding of the way in which turbulence characteristics change over hills and in complex terrain. Given the important effect of turbulence characteristics on wind turbine loading and performance, this is clearly an area in which there is scope for further research.

Alongside the Kaimal model, editions 3 and 4 of the IEC standard (IEC 2005, 2019) give the option to use a rather different form of turbulence model developed by Mann (1994, 1998). The other models described above make use of a one‐dimensional fast Fourier transform (FFT) to generate time histories from spectra, applied to each turbulence component independently. In contrast, the Mann model is based on a three‐dimensional spectrum tensor representation of the turbulence, and one three‐dimensional FFT is then used to generate all three components of turbulence simultaneously. The three‐dimensional spectrum tensor is derived from rapid distortion theory, in which isotropic turbulence described by the von Karman spectrum is distorted by a uniform mean vertical velocity shear. This means that the three turbulence components are no longer independent, as energy is transferred between the longitudinal and vertical components by distortion of the eddies in the flow, resulting in a realistic representation of the correlation between the longitudinal and vertical components described by the Reynolds stress. The spectral density for any three‐dimensional wavenumber vector is derived, and all three components of turbulence are then generated simultaneously by summing a set of such wavenumber vectors, each with the appropriate amplitude and a random phase.

This is in many ways a rather elegant approach, but in practice there are some computational limitations that can make it difficult to use. The summation requires a three‐dimensional FFT to achieve reasonable computation time. The number of points in the longitudinal, lateral, and vertical directions must be a power of two for efficient FFT computation. In the longitudinal direction, the number of points is determined by the length of time history required and the maximum frequency of interest and is therefore typically at least 1024. The maximum wavelength used is the length of the turbulence history to be generated (i.e. the mean wind speed multiplied by duration of the required time series), and the minimum wavelength is twice the longitudinal spacing of points (which is the mean wind speed divided by the maximum frequency of interest). In the lateral and vertical directions, a much smaller number of points must be used, perhaps as low as 32, depending on available computer memory. The maximum wavelength must be significantly greater than the rotor diameter, because the solution is spatially periodic, with period equal to the maximum wavelength in each direction. The number of FFT points then determines the minimum wavelength in these directions. With a realistic number of points, the resulting turbulence spectra are deficient at the high frequency end (Veldkamp 2006). Mann (1998) suggests that this may be realistic, because it represents averaging of the turbulence over finite volumes of space, which is appropriate for practical engineering applications. However, a practical simulation tool will perform all necessary spatial averaging in any case, and so the high frequency variations are really lost. Mann (1998) does suggest a remedy for this, but in practice it is extremely intensive computationally.