Читать книгу Wind Energy Handbook - Michael Barton Graham - Страница 36

To use the spectra defined above, it is necessary to define the appropriate length scales. Additional parameters β₁, β₂, F₁, and F₂ are also required for the modified von Karman model.

The length scales are dependent on the surface roughness z₀ as well as on the height above ground (z): proximity to the ground constrains the size of turbulent eddies and thus reduces the length scales. If there are many small obstacles on the ground of typical height z', the height above ground should be corrected for the effect of these by assuming that the effective ground surface is at a height z' − 2.5z₀ (ESDU 1975). Far enough above the ground, i.e. for z greater than some height z_i, the turbulence is no longer constrained by the proximity of the surface and becomes isotropic. According to ESDU (1975), z_i = 1000z₀^0.18, and above this height ^xL_u = 280 m, and ^yL_u = ^zL_u = ^xL_v = ^zL_v = 140 m. Even for very small roughness lengths z₀, the isotropic region is well above the height of a wind turbine, and the following corrections for z < z_i should be applied:

(2.29)

together with ^xL_w = ^yL_w = 0.35z (for z < 400 m). Expressions for ^yL_v and ^zL_w are not given. The length scales ^xL_u, ^xL_v, and ^xL_w can be used directly in the von Karman spectra. For the Kaimal spectra we already have L_1u = 2.329 ^xL_u, and to achieve the same high frequency asymptotes for the other components we also have L_1v = 3.2054 ^xL_v, L_1w = 3.2054 ^xL_w.

Later work based on measurements for a greater range of heights (Harris 1990; ESDU 1985) takes into account an increase in length scales with the thickness of the boundary layer, h, which also implies a variation of length scales with mean wind speed. This yields more complicated expressions for the nine length scales in terms of z/h, σ_u/u*, and the Richardson number u*/(fz₀).

Note that some of the standards used for wind turbine loading calculations prescribe that certain turbulence spectra and/or length scales are to be used. These are often simplified compared to the expressions given above. Thus the Danish standard (DS 472 1992) specifies a Kaimal spectrum with

(2.30)

while the IEC edition 2 standard (IEC 61400‐1 1999) gives a choice between a Kaimal model with

(2.31)

and an isotropic von Karman model with

(2.32)

Editions 3 (IEC 61400‐1 2005) and 4 (IEC 61400‐1 2019) of the IEC standard give a choice of between a slightly different Kaimal model and the Mann model. The Kaimal model has the same form [Eq. (2.24)] but with

(2.33)

The Mann model has a rather different form and is described in Section 2.6.8.

The Eurocode (EN 1991‐1‐4:2005) standard for wind loading specifies a longitudinal spectrum of Kaimal form with L_1u = 1.7L_i, where

(2.34)

for z < 200 m, with α = 0.67 + 0.05 ln(z₀). This standard is used for buildings but not usually for wind turbines.

With so many variables, it is difficult to present a concise comparison of the different spectra, so a few examples are presented in Figures 2.5 and 2.6. These are plots of the normalised longitudinal spectrum nS_u(n)/σ_u² against frequency, which means that the area under the curve is representative of the fraction of total variance in any given frequency range. A typical hub height of 80 m has been used, with 50° latitude assumed for the modified von Karman model.

Figure 2.5 shows spectra for a typical rated wind speed of 12 m/s. The IEC edition 2 Kaimal spectrum is clearly very similar to DS 472, while the IEC editions 3 and 4 spectrum has clearly moved to lower frequencies, being now more consistent with Eurocode (in fact identical for 80 m height and z₀ = 0.01 m). Note the characteristic difference between the Kaimal and von Karman spectra, the latter being rather more sharply peaked. The modified von Karman spectrum is intermediate in shape; with a very small roughness length the peak is at a similar frequency to the IEC edition 2 spectra, but with higher roughness length it comes closer to edition 3.

Figure 2.6 shows a similar figure for a typical cut‐out wind speed of 25 m/s. All peaks have moved to higher frequency as expected, but the modified von Karman now matches IEC editions 3 and 4 better with the very small roughness length.