Michael Graham - Wind Energy Handbook

Turbulence is clearly a complex process and one that cannot be represented simply in terms of deterministic equations. Clearly it does obey certain physical laws, such as those describing the conservation of mass, momentum, and energy. However, to describe turbulence using these laws it is necessary to take account of temperature, pressure, density, and humidity as well as the motion of the air itself in three dimensions. It is then possible to formulate a set of differential equations describing the process, and in principle the progress of the turbulence can be predicted by integrating these equations forward in time starting from certain initial conditions and subject to certain boundary conditions. In practice of course, the process can be described as ‘chaotic’ in that small differences in initial conditions or boundary conditions may result in large differences in the predictions after a relatively short time. For this reason it is generally more useful to develop descriptions of turbulence in terms of its statistical properties.

There are many statistical descriptors of turbulence that may be useful, depending on the application. These range from simple turbulence intensities and gust factors to detailed descriptions of the way in which the three components of turbulence vary in space and time as a function of frequency.

The turbulence intensity is a measure of the overall level of turbulence. It is defined as

(2.6)

where σ is the standard deviation of wind speed variations about the mean wind speed , usually defined over 10 minutes or an hour. Turbulent wind speed variations can be considered to be roughly Gaussian, meaning that the speed variations are normally distributed, with standard deviation σ , about the mean wind speed . However, the tails of the distribution may be significantly non‐Gaussian, so this approximation is not reliable for estimating, say, the probability of a large gust within a certain period.

The turbulence intensity clearly depends on the roughness of the ground surface and the height above the surface. However, it also depends on topographical features, such as hills or mountains, especially when they lie upwind, as well as more local features, such as trees or buildings. It also depends on the thermal behaviour of the atmosphere: for example, if the air near to the ground warms up on a sunny day, it may become buoyant enough to rise up through the atmosphere, causing a pattern of convection cells that are experienced as large‐scale turbulent eddies.

Clearly as the height above ground increases, the effects of all these processes that are driven by interactions at the earth's surface become weaker. Above a certain height, the air flow can be considered largely free of surface influences. Here it can be considered to be driven by large‐scale synoptic pressure differences and the rotation of the earth. This air flow is known as the geostrophic wind . At lower altitudes, the effect of the earth's surface can be felt. This part of the atmosphere is known as the boundary layer . The properties of the boundary layer are important in understanding the turbulence experienced by wind turbines.

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 oare shown in Table 2.1.

The Coriolis parameter f is defined as

(2.7)

Table 2.1 Typical surface roughness lengths.

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)