Читать книгу Wind Energy Handbook - Michael Barton Graham - Страница 38

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 C_u (Δr,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_u is a local length scale that can be defined as

(2.40)

where Δy and Δz are the lateral and vertical components of the separation Δr, and ^yL_u and ^zL_u are 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, 0.04n^−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 ^xL_u = 2 ^yL_u = 2 ^zL_u, and then L_u = ^xL_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_i is calculated as in Eq. (2.39) but with L_u replaced by L_v or L_w, respectively, and with c = 1. Also

(2.42)

and L_v and L_w are 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_u as 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.