Wind Energy Handbook. Michael Barton Graham

Чтение книги онлайн.

СКАЧАТЬ

      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.

2.6.8 The Mann model of turbulence

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 СКАЧАТЬ