Michael Graham - Wind Energy Handbook

where κ is the von Karman constant (approximately 0.4), z is the height above ground, and z ois the surface roughness length. Ψ is a function that depends on stability: it is negative for unstable conditions, giving rise to low wind shear, and positive for stable conditions, giving high wind shear. For neutral conditions, ESDU (1985) gives Ψ = 34.5 f z/ u *, which is small compared to ln(z/z o) for situations of interest here. If Ψ is ignored, the wind shear is then given by a logarithmic wind profile:

(2.10)

A power law approximation,

(2.11)

is often used, where the exponent α is typically about 0.14 onshore and lower offshore but varies with the type of terrain. However, the value of α should also depend on the height interval over which the expression is applied, making this approximation less useful than the logarithmic profile.

The wind turbine design standards typically specify that a given exponent should be used; the International Electrotechnical Commission (IEC) and Germanischer Lloyd (GL) standards, for example, specify an exponent of α = 0.20 for normal wind conditions onshore and α = 0.14 for normal wind conditions offshore. Both standards specify an exponent of α = 0.11 for extreme wind conditions (onshore and offshore). For conservatism, edition 4 of the IEC standard (IEC 61400‐1 2019) allows a higher exponent (0.3) to be used for turbines of ‘medium’ size (swept area from 200 to 1000 m 2).

If there is a change in the surface roughness, the wind shear profile changes gradually downwind of the transition point, from the original to the new profile. Essentially, a new boundary layer starts, and the height of the boundary between the new and old boundary layers increases from zero at the transition point until the new boundary layer is fully established. The calculation of wind shear in the transition zone is covered by, for example, Cook (1985).

By combining Eqs. (2.8)and (2.9), we obtain the wind speed at the top of the boundary layer as

(2.12)

This is similar to the so‐called ‘geostrophic wind speed’, G , which is the notional wind speed driving the boundary layer as calculated from the pressure field. The geostrophic wind speed is given by

(2.13)

where, for neutral conditions, A = ln 6 and B = 4.5. This relationship is often referred to as the geostrophic drag law.

The effect of surface roughness is not only to cause the wind speed to decrease closer to the ground. There is also a change in direction between the ‘free’ pressure‐driven geostrophic wind and the wind close to the ground. Although the geostrophic wind is driven by the pressure gradients in the atmosphere, Coriolis forces act to force the wind to flow at right angles to the pressure gradient, causing a characteristic circulating pattern. Thus in the northern hemisphere, wind flowing from high pressure in the south to low pressure in the north will be forced eastwards by Coriolis effects, in effect to conserve angular momentum on the rotating earth. The result is that the wind circulates anti‐clockwise around low‐pressure areas and clockwise around high‐pressure areas, or the other way round in the southern hemisphere. Close to the ground, these flow directions are modified due to the effect of surface friction. The total direction change, α, from the geostrophic to the surface wind is given by

(2.14)

The turbulence intensity in the neutral atmosphere clearly depends on the surface roughness. For the longitudinal component, the standard deviation σ uis approximately constant with height, so the turbulence intensity decreases with height. More precisely, the relationship σ u≈ 2.5 u * may be used to calculate the standard deviation, with the friction velocity u * calculated as in the previous section. More recent work (ESDU 1985) suggests a variation given by

(2.15)

where

(2.16)

(2.17)

This approximates to σ u= 2.5 u * close to the ground, but gives larger values at greater heights. The longitudinal turbulence intensity is then

(2.18)

The lateral (v) and vertical (w) turbulence intensities are given (ESDU 1985) by

(2.19)

(2.20)

Note that specific values of turbulence intensity for use in design calculations are prescribed in some of the standards used for wind turbine design calculations, and these may not always correspond with the above expressions. For example, the now superseded Danish standard (DS 472 1992) specified

(2.21)

with I v= 0.8 I uand I w= 0.5 I u.

The IEC edition 2 standard (IEC 61400‐1 1999) gives

(2.22)

where I 15= 0.18 for ‘higher turbulence sites’ and 0.16 for ‘lower turbulence sites’, with corresponding values of a of 2 and 3, respectively. For the lateral and vertical components, a choice is allowed: either I v= 0.8 I uand I w= 0.5 I u, or an isotropic model with I u= I v= I w.

Editions 3 (IEC 61400‐1 2005) and 4 (IEC 61400‐1 2019) of the IEC standard specify

(2.23)

where I ref= 0.16, 0.14, or 0.12 depending on the wind class. For lateral and vertical components, I vmust be at least 0.7I u, and I wat least 0.5I u. Standard deviations are assumed constant with height, so the turbulence intensity will change with height as the mean wind speed changes due to wind shear.