### Steady Wind: Variation of wind speed with altitude

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As shown in Figure 1 (van der Tempel, 2006), the actual wind speed varies with space and time. Due to turbulence effects, the actual wind speed at any location will always vary around its average value with time and direction. Most importantly, the graph clearly shows that the average wind speed increases with height, a phenomenon known as wind shear.

Wind shear affects both wind resource assessment and wind turbine design.

First, an assessment of wind resources over a broad geographic area may require the correction of wind measurements from various sources to a uniform altitude. Second, in terms of design, the fatigue life of rotor blades is affected by cyclic loads, which are generated by the rotation of the rotor through a vertically varying wind field. Therefore, models of wind speed as a function of height are required in wind energy applications. The following describes several popular models currently used to predict wind speed as a function of height above ground.

In wind energy research, two mathematical models or “laws” are generally used to simulate vertical profiles of wind speeds over homogeneous, flat terrain (fields, deserts, grasslands). The first method is the logarithmic law, which originated in the study of fluid mechanics and atmospheric boundary layer flow, and was established from the results of theoretical and experimental studies. The second method is the power law, used by many wind energy researchers. Both methods have certain uncertainties due to the complexity and variability of turbulent flow (Hiester and Pennell, 1981). Each law and its general application are described below.

**1. Logarithmic wind profile (logarithmic law)**

There are several methods for predicting logarithmic wind profiles (eg, mixing length theory, eddy viscosity theory, and similar theories), only the mixing length analysis method given by Wortman (1982) is described here.

The momentum equation for regions near the Earth’s surface simplifies to:

where x and z are the horizontal and vertical coordinates, p is the pressure, τ_{zx} is the shear stress in the x direction, and its normal direction is consistent with z.

In this region the pressure is independent of x and the integration gives:

τ_{zx}=τ_{0}+z(∂p/∂x) (1.2)

where τ_{0} is the shear stress on the surface. The pressure gradient is smaller near the ground, so the second term on the right can be ignored. Using Prandtl’s mixing length theory, the shear stress can be expressed as:

τ_{zx}=pl^{2}(∂U/∂z)^{2} (1.3)

where ρ is the air density, U is the horizontal component of velocity, and l is the mixing length. Note that using U here means that turbulence effects have been averaged out.

Combining formulas (1.2) and (1.3), we can get

In the formula, U* is defined as the friction velocity.

For a smooth surface, l=kz, where k=0.4 (von Karman constant), then equation (1.4) can be integrated directly from z_{0} to z, where z_{0} is the surface roughness length, which represents the roughness characteristics of the surface. so:

This formula is called the logarithmic wind profile.

The reason why the integral starts from the lower limit z_{0} instead of 0 is that the natural surface cannot be flat and smooth.

Equation (1.5) can also be written as:

ln(z)=(k/U^{*})U(z)=ln(z_{0}) (1.6)

This formula can be drawn as a straight line on semi-logarithmic graph paper. Its slope is k/U^{*}, and U^{*} and z_{0} can be calculated using plots of experimental data. The logarithmic law is often used to extrapolate the wind speed at another height from a reference height z_{r}, using the following relationship:

U(z)/U(z_{r})=ln(z/z_{0})/ln(z_{r}/z_{0}) (1.7)

Sometimes the mixing of the air flow on the Earth’s surface is considered, and the mixing length is expressed as l=k(z+z_{0}) to correct the logarithmic law. The logarithmic wind profile then becomes

U(z)=(U^{*}/k)ln(z+z_{0}/z_{0}) (1.8)

2. Power-law wind profile

The power law is a simple model representing the vertical wind profile, and its basic form is:

U(z)/U(z_{r})=(z/z_{r})^{α} (1.9)

where U(z) is the wind speed at height z, U(z_{r}) is the reference wind speed at height z_{r}, and a is the power exponent. Early work in this area showed that under certain conditions, α is equal to 1/7, where the wind profile corresponds to the flow over a flat plate (see Schlichting, 1968). In fact, the exponent α is a highly variable quantity.

The following example emphasizes the importance of α variation: If U_{0} = 5 m/s at 10 m high, what are U and P/A at 30 m? Note that at 10m, P/A=75.6 W/m², for three different α, the calculated wind speed at 30m is listed in Table 1. When calculating P/A, assume ρ=1.225 kg/m³.

Alpha has been found to vary with altitude, time of day, season, topographic features, wind speed, temperature, and various thermodynamic, mechanical mixing parameters. Some researchers have developed methods to calculate α from the parameters of the logarithmic law. But many researchers believe that these complex approximations reduce the simplicity and applicability of general power laws, and wind energy experts should accept the empirical nature of power laws and choose the α that best matches existing wind data. Several commonly used empirical methods for determining representative power-law exponents are reviewed below.

Relationships for power exponents as a function of wind speed and altitude

A method for dealing with this variation is proposed by Jutus (1978). Its expression is as follows:

In the formula, the unit of U is m/s, and the unit of zref is m.

Relation depending on surface roughness

The relationship proposed by Counihan (1975) is as follows:

α=0.096log_{10}z_{0}+0.016(log_{10}z_{0})^{2}+0.24 (1.11)

In the formula, 0.001m<z0<10m, where z0 represents the surface roughness in m (see the example values in Table 1).

Relationship based on surface roughness (z_{0}) and velocity

The formula for α suggested by wind researchers at NASA is based on surface roughness and wind speed U_{ref} at a reference altitude (see Spera, 1994).

3. Comparison of predicted wind speed profiles with actual data

The importance of wind speed versus height characteristics, or wind shear, for power generation wind turbines at a given site cannot be overemphasized. That is, such features are necessary to accurately predict wind turbine output power. For wind power developers, it is most important to accurately know the wind speed variation characteristics at the height of the wind turbine rotor (usually between 60 meters and 100 meters) and at different heights of the wind turbine. Recent research in this area, including the use of high-tower wind data series, shows that using log and exponential laws to predict wind shear is not very different, and in some cases, no matter which method is adopted, it cannot give an accurate prediction of the mean wind speed at the height of the hub. This conclusion is based on topographical experimental data: (1) flat terrain, no trees; (2) hillside terrain, no trees; (3) forest terrain. For all three terrains, the study found errors between predicted and actual measured wind speeds at wheel hub heights ranged from 1% to a maximum of 13%.

In practice, it should be recognized that wind shear is a function of several variables:

atmospheric stability;

ground roughness;

changes in ground conditions;

Terrain shape.

Because wind shear models do not always reflect actual characteristics, wind shear extrapolation can introduce large uncertainties.