In strong wind conditions, the mean wind speed is primarily influenced by two factors: height above the ground and surface roughness. The latter, characterized by the aerodynamic roughness parameter z₀, represents the texture of the terrain's surface. Under the assumption of a sufficiently large area with homogeneous roughness and flat terrain, an equilibrium boundary layer forms. This boundary layer establishes itself when the airflow can proceed undisturbed over a sufficient distance, allowing a balance between the acting forces. The vertical distribution of the mean wind speed can then be described using a power law approach of the form
| ū(z) | mean wind speed at height z |
| ūref |
mean wind speed at reference height zref |
|
z
|
height above ground |
| Zref | reference height |
| d0 | displacement height |
| α | profile exponent |
In the Prandtl layer, the velocity profile can also be described using a logarithmic approach. In meteorology, the power law approach is no longer used. In this context, the terms mean:
|
u* |
friction velocity |
|
κ |
von Kármán constant, κ ≈ 0,41 |
| z0 | roughness length |
The determination of the displacement height d0 can be carried out using various methods, as described by Counihan, Cook, and Karimpour. In neutral atmospheric stratification, turbulence superimposes on the mean wind speed. These turbulences are typically characterized by turbulence intensities, spectra, and integral length scales. For terrain types that are dependent on wind direction, the most representative terrain type for each inflow direction should be simulated. DIN EN 1991-1-4 provides reference values for terrain roughness, roughness lengths, and profile exponents, in addition to the values in Table 1.
Table 1: Constants A to E of the different approaches and the definition of the dimensionless frequency according to [18] to [23] and [26] based on [2]
| Class | Terrain Description | z0 [m] | α [-] | d0 [m] |
|---|---|---|---|---|
| 1 | Open sea or lakes, mudflats, snow-covered flat plains, desert without special features, asphalt and concrete, with an open take-off distance of several kilometers | 0,0002 to 0,005 | 0,08 to 0,09 | 0 |
| 2 | Land without special features and without noticeable obstacles as well as with negligible vegetation; e.g., beaches, ice without large-scale elevations, swamps, snow-covered or fallow open land | 0,005 to 0,01 | 0,09 to 0,13 | 0 |
| 3 | Flatland with low vegetation (e.g., grass) and with individual obstacles at distances of at least 50 times the obstacle height; e.g., grassland without windbreaks, heath, moorland, tundra, runways at airports | 0,03 to 0,1 | 0,1 to 0,16 | 0 |
| 4 | Farmland with uniformly low vegetation or moderately open land with occasional obstacles (e.g., low hedges, individual trees, individual farmhouses) at relative horizontal distances of at least 20 times the obstacle height | 0,1 to 0,25 | 0,14 to 0,22 | 0 |
| 5 | Newly developed "young" landscape with high vegetation or vegetation of variable height, and scattered obstacles (e.g., dense windbreaks, vineyards) at relative distances of around 15 obstacle heights | 0,25 to 0,5 | 0,16 to 0,27 | 2/3 h |
| 6 |
|
0,5 to 1 | 0,18 to 0,33 | 2/3 h |
| 7 | Completely and fairly evenly covered land with large-scale obstacles of similar size and with open areas almost as high as obstacles; e.g., mature uniform forests, homogeneous cities or towns | 0,23 to 0,43 | 0,23 to 0,43 | 2/3 h |
| 8 | Centers of large cities with a mix of low and high buildings; similarly, irregular large forests with many clearings | ≥2 | 0,27 to 0,62 | 0 |
The profile exponent α is calculated from z₀ according to Wieringa [8] using the approximate equation:
where a height range of 10-80 m is chosen for z1 and z2.
The fluctuating components of the wind speed are captured using the standard deviation σi (z). Thus, the turbulence intensity Ii (z) is defined as the ratio of the fluctuating component to the mean speed:
From the proportionality of the standard deviations of the individual speed fluctuations and the friction velocity, the turbulence intensities for a logarithmic wind profile are given by:
The strong anisotropy of the turbulence is described by Ai, which is derived as follows:
Analogous to the mean velocity profile, the vertical distribution of turbulence intensity can also be described by a power law:
The boundary layer model according to Deaves and Harris [30],[31] is valid throughout the entire atmospheric boundary layer and includes equations for both the velocity profile and longitudinal turbulence intensity, as provided in Chapter 8.2.
The frequency distribution of the longitudinal component of turbulence is represented by the dimensionless spectral density function, or turbulence spectrum, in the following form:
| f | Frequency of velocity fluctuations |
| Suu | Spectral density function |
| fdim | Dimensionless frequency |
| A, B, C, D, E | Selectable constants, according to the specific model parameters |
Table 2: Constants A to E of Various Approaches and Definition of Dimensionless Frequency According to [18] to [23] and [26] as per [2]
| Reference | A | B | C | D | E | fdim |
|---|---|---|---|---|---|---|
| Kaimal et al. (1972) | 16.8 | 33.0 | 1 | 5/3 | 1 | f.z/ū(z) |
| Simiu und Scanlan (1996) | 32.0 | 33.0 | 1 | 5.3 | 1 | f.z/ū(z) |
| Olesen et al. (1984) | 40.42 | 60.62 | 1 | 5.3 | 1 | f.z/ū(z) |
| Tieleman (1995) | 20.53 | 475.1 | 5/3 | 1 | 1 | f.z/ū(z) |
| Fichtl und McVehill (1970) | 54.375 | 36.532 | 0.845 | 5/(3*C) | 1 | f.z/ū(z) |
| von Kármán (1948) | 4 | 70.78 | 2 | 5/(3*C) | 1 | f.Lux(z)/ū(z) |
| CEN (2005) | 6.8 | 10.2 | 1 | 5/3 | 1 | f.Lux(z)/ū(z) |
The height dependence of the dimensionless turbulence spectrum is considered in various approaches, including EN 1991-1-4, where the dimensionless frequency can be formed either with the height z or with the height dependence of the integral length scale of the u-component in the flow direction (Lux). Table 2 provides an overview of these approaches, whose validity is mainly limited to the Prandtl layer.
Integral length scales, such as the integral length scale of the u-component in the flow direction X Lux serve as indicators of the spatial correlation of turbulence and can be interpreted as average eddy sizes. They vary depending on terrain type and height above ground, with Lux decreasing as roughness length z₀ increases; however, this behavior reverses at about 200 meters in height. EN 1991-1-4 describes the increase in the integral length scale with height using a power law approach, limited to a maximum height of 200 meters. There are three common approaches for calculating Lux:
- Definition of the correlation coefficient of velocity fluctuations:
- In the Prandtl layer:
- From the spectral density function:
In addition to Lux, there are eight other integral length scales for the lateral and vertical components, which can be approximately given as fractions of Lux, according to Counihan [1], for the lateral and vertical components as follows:
The Taylor hypothesis of "frozen turbulence" plays an important role in these considerations, as it assumes that velocity fluctuations remain unchanged during the movement of the vortex center between two spatial points.