Читать книгу Wind Energy Handbook - Michael Barton Graham - Страница 35

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^−5/3 at 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_1u and L_2u are 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_2u is identified as the integral length scale of the longitudinal component in the longitudinal direction, denoted ^xL_u and 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_x and measured simultaneously (similar definitions apply to the integral length scales of the longitudinal component of turbulence in the lateral and vertical directions, ^yL_u and ^zL_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 2.5 and 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_1v and 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 = ^xL_v and L_2w = ^xL_w. For the modified von Karman spectrum of Eq. (2.26), it is

(2.28)