### Turbulence

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Turbulence in the wind results from the continuous creation and disappearance of small eddies (or gusts), which convert the kinetic energy dissipation of the wind into thermal energy. Turbulent winds may have relatively stable averages over time spans of an hour or longer, but over shorter periods of time (minutes or less) it may vary frequently. The change in wind appears to be random on the surface, but in fact it has several different characteristics. These characteristics are reflected by several statistical properties: turbulence intensity; probability density function autocorrelation of wind speed; integration time scale/length scale; power spectral density function.

Turbulent wind consists of longitudinal, lateral and vertical components. The longitudinal component (in the prevailing wind direction) is denoted by u(z,t). The lateral component (perpendicular to U) is v(z, t) and the vertical component is w(z, t), each of which is often assumed to be determined by the short-term average wind speed, for example U, plus a fluctuating wind speed with a mean value of zero constitutes:

where u is the instantaneous longitudinal wind speed, z is the height above the ground, and t is the time. The lateral and vertical components can also be similarly decomposed into mean and fluctuation components. For the sake of clarity, the correlation between height and time will not be explained in the following formulas.

Note that the short-term average wind speed U here refers to the average wind speed over a certain time period Δr (shorter), and this time period is greater than the characteristic time of turbulent fluctuations. This time period is usually taken as 10 minutes, but can also be as long as one hour, which is calculated as:

The instantaneous turbulent wind is actually observed discontinuously; it is actually sampled at relatively high velocities. Assuming that the sampling interval is δt, such that Δt=N_{s}δt where N_{s} is the number of samples in each short-term interval, then the turbulent wind can be represented by a sequence ui. The short-term average wind speed can be expressed as:

The short-term mean longitudinal wind speed U is often used in time series observations.

**1. Turbulence intensity**

The most basic measure of turbulence is turbulence intensity. is defined as the ratio of the standard deviation of the wind speed to the mean. In the calculation of mean wind speed and standard deviation, the time span is longer than the turbulent fluctuation time, and is shorter than the time length of other wind speed variation types (such as diurnal variation). This length of time is generally not longer than one hour, and it is customary to take 10 minutes in wind energy projects. The sampling rate is generally at least once per second (1Hz), and the turbulence intensity, TI is defined as:

TI=σ_{u}/U (1.4)

where σ_{u} is the standard deviation, which is given by the following formula according to the sampling method:

Turbulence intensity is usually in the range of 0.1 to 0.4. The highest values of turbulence intensity generally occur when wind speeds are lowest, and the value of the low limit for a given area depends on the specific topographical characteristics and ground conditions of the site. The mean wind speed of the data is 10.4m/s and the standard deviation is 1.63m/s, so the flow intensity for the 10-minute period is 0.16.

**2. Wind speed probability density function**

The probability that the wind speed has a certain value is described by a probability density function (pdf). Experience has shown that wind speeds tend to be closer to the average than farther away, and that below-average and above-average speeds are roughly equally likely. The best probability density function to describe this property of turbulence is the Gaussian or normal distribution. The normally distributed probability density function for continuous data represented by variables is given by:

**3. Autocorrelation**

The probability density function of wind speed provides a measure of the likelihood of wind speed at various specific values. As described in Appendix C, the regularized autocorrelation function for the turbulent wind speed sample data is given by:

Where r is the delay number.

The autocorrelation function can be used to determine the turbulent integration time scale as described below.

**4. Integration time scale/length scale**

If any trend is removed before the sampling process begins, the autocorrelation function will decay from a value of 1.0 at the initial sampling (zero delay) to a zero value, and then tend to small positive or negative values as the sampling delay time increases. A measure of the average time period in which the wind speed fluctuations are correlated with each other can be obtained by integrating the autocorrelation function from the delay of zero to the time of occurrence of the first zero value. This integral value is called the turbulence integral time scale. Typically this value is less than 10 seconds, and the integration time scale is a function of site, atmospheric stability, and other factors, and can be much greater than 10 seconds. The wind speed of a gust rises and falls relatively coherently (closely related), and it has a characteristic time of the same order of magnitude as the integration time scale.

The integral length scale is obtained by multiplying the mean wind speed by the integral time scale. The integration length scale is more constant than the integration time scale in a certain wind speed range, so it can better reflect the characteristics of the site.

According to the autocorrelation function shown above, the integration time scale is 50.6s and the average wind speed is 10.4m/s. Thus, the size of the turbulent eddies in the mean airflow, or the magnitude of the integrated length scale, is 526 m.

**5. Power spectral density function**

Wind fluctuations can be seen as the result of superimposing a set of sinusoidally varying winds on the basis of the average steady wind. These sinusoidally varying winds have various frequencies, amplitudes and phases. The term “spectrum” is used to describe a frequency function. The frequency function used to characterize turbulent flow is called the “spectral density” function. Since the mean value of any sine function is zero, its magnitude is represented by its mean square value. This form of analysis originates from power applications, where the mean square value of voltage or current is proportional to power. Therefore, the full name of the function describing the relationship between the frequency and amplitude of the sinusoidally varying waves that make up the fluctuating wind speed is “power spectral density”.

Two points about the properties of power spectral densities (psds) are particularly important. First, the average power of turbulent flow in a certain frequency range can be obtained by integrating the power spectral density between two frequencies. Second, the integral of the power spectral density over all frequencies equals the total change.

Power spectral density is often used in kinetic analysis. If a representative turbulent power spectral density does not exist for a given site, a variety of power spectral density functions can be used as models for wind energy engineering. One such suitable model is similar to the wind tunnel turbulence model developed by von Karman (Freris, 1990), given by equation (1.8), called the von Karman power spectral density.

where f is the frequency (Hz), L is the integral length scale, and U is the average wind speed at the height of interest. Other forms of power spectral density are also used in wind engineering applications.