Читать книгу Wind Energy Handbook - Michael Barton Graham - Страница 33

The principal effects governing the properties of the boundary layer are the strength of the geostrophic wind, the surface roughness, Coriolis effects due to the earth's rotation, and thermal effects.

The influence of thermal effects can be classified into three categories: stable, unstable, and neutral stratification. Unstable stratification occurs when there is a lot of surface heating, causing warm air near the surface to rise. As it rises, it expands due to reduced pressure and therefore cools adiabatically. If the cooling is not sufficient to bring the air into thermal equilibrium with the surrounding air, then it will continue to rise, giving rise to large convection cells. The result is a thick boundary layer with large‐scale turbulent eddies. There is a lot of vertical mixing and transfer of momentum, resulting in a relatively small change of mean wind speed with height.

If the adiabatic cooling effect causes the rising air to become colder than its surroundings, its vertical motion will be suppressed. This is known as stable stratification. It often occurs on cold nights when the ground surface is cold. In this situation, turbulence is dominated by friction with the ground, and wind shear (the increase of mean wind speed with height) can be large.

In the neutral atmosphere, adiabatic cooling of the air as it rises is such that it remains in thermal equilibrium with its surroundings. This is often the case in strong winds, when turbulence caused by ground roughness causes sufficient mixing of the boundary layer. For wind energy applications, neutral stability is usually the most important situation to consider, particularly when considering the turbulent wind loads on a turbine, because these are largest in strong winds. Nevertheless, unstable conditions can be important because they can result in sudden gusts from a low level, and stable conditions can give rise to significant asymmetric loadings due to high wind shear. There can also be large veer (change in wind direction with height) in this situation.

In the following sections, a series of relationships are presented that describe the properties of the atmospheric boundary layer, such as turbulence intensities, spectra, length scales, and coherence functions. These relationships are partly based on theoretical considerations and partly on empirical fits to a wide range of observations from many researchers taken in various conditions and in various locations.

In the neutral atmosphere, the boundary layer properties depend mainly on the surface roughness and the Coriolis effect. The surface roughness is characterised by the roughness length z_o. Typical values of z_o are shown in Table 2.1.

The Coriolis parameter f is defined as

(2.7)

Table 2.1 Typical surface roughness lengths.

Type of terrain	Roughness length zo (m)
Cities, forests	0.7
Suburbs, wooded countryside	0.3
Villages, countryside with trees and hedges	0.1
Open farmland, few trees and buildings	0.03
Flat grassy plains	0.01
Flat desert, rough sea	0.001

where Ω is the angular velocity of the earth's rotation, and λ is the latitude. In temperate latitudes, the height of the boundary layer is given by

(2.8)

but it is clear from the division that this and the subsequent derivations cannot be valid at the equator, where f = 0, so a pragmatic recommendation is to use a latitude of 22.5° for all tropical regions. Here u* is known as the friction velocity, given by

(2.9)

where κ is the von Karman constant (approximately 0.4), z is the height above ground, and z_o is 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 = 34.5fz/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 m²).

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)