Evolving Turbulence

If a LIDAR sensor is used to provide wind preview information to the external controller, the upstream wind velocities measured by the LIDAR cannot be expected to convect towards the turbine unchanged as implied by Taylor’s frozen turbulence hypothesis, in which turbulent velocity variations are simply assumed to be transported downwind at the mean wind speed. Therefore a model of evolving turbulence is provided, such that the upstream turbulence measured by a LIDAR changes before it reaches the turbine. A detailed description of the evolving turbulence model is presented in (Bossanyi, 2012).

The method for modelling evolving turbulence makes use of the linear superposition of the individual waves that can be used to describe a turbulent time history. A turbulent time history of the wind speed in the longitudinal wind direction can be defining using the following parameters; the mean speed \(U\), wind speed standard deviation \(\sigma\), characteristic length scale \(L\), the auto spectral density \(S_{i} = \ S_{uu}(y,z,n_{i})\) and a series of phases \(\phi_{i}\), where \(n_{i}\) is the frequency and \(i = 1,\ldots,\ N\). The variables \(y\) and \(z\) denote the cross wind lateral and vertical directions respectively.

The models for evolving turbulence assumes that high frequency turbulent variations will evolve and change faster than low frequency variations over a given time period \(\Delta t = \Delta x/U\). A coherence \(C_{u}(n_{i})\) model developed by (Kristensen, 1979) is used for this in Bladed. This evolution is modelled using a decay fraction: $f_ = \sqrt{C_u ( n_i )} $ which in the case of Kristensen is defined as

\[ \begin{equation} f_{i}(n,\Delta x) = \exp( - \alpha G(\xi))(1 - \exp( - 1/(2\alpha\min{(\alpha,1)\xi^{2})))} \end{equation} \]

\[ \begin{equation} \alpha(\Delta x) = \frac{\sigma\Delta x}{UL} \end{equation} \]

\[ \begin{equation} \xi(n) = \frac{nL}{U} \end{equation} \]

\[ \begin{equation} G(\xi) = \frac{\xi^{2}\sqrt{\xi + \frac{1}{121}}}{\left( \xi + \frac{1}{33} \right)^{\frac{11}{6}}} \end{equation} \]

The decay fraction is principally a function of the distance (or time \(\Delta t = \Delta x/U\)) and the frequency.

An alternative and more simplified exponential function is also provided. In this case the decay fraction \(f_{i} = \sqrt{C_u ( n_i )}\) takes the following form

\[ \begin{equation} f(n, \Delta x) = \exp( \beta \Delta x k(n) ) \end{equation} \]

where

\[ \begin{equation} k(n) = \frac{2 \pi n}{U}. \end{equation} \]

The decay fraction is principally a function of the distance (or time \(\Delta t = \Delta x/U\)) and the frequency. The user can modify the exponential factor \(\beta\) which must be a positive number.

To represent the evolution of the time history two turbulent wind fields are generated, using different random number seeds or phases. The two sets of random phases are defined as \(\phi_{i,1}\) and \(\phi_{i,2}\). These can be considered to represent two independent realisations of turbulence satisfying the same statistical properties (spectrum and coherence). Likewise, two sets of amplitudes are defined by

\[ \begin{equation} A_{u,1}\left( n_{i} \right) = {0.5\ S}_{i}/df_{i} \end{equation} \]

\[ \begin{equation} A_{u,2}\left( n_{i} \right) = {0.5\ S}_{i}/df_{i} \end{equation} \]

where,
\(df_{i} = f_{i} - f_{i - 1}\).

The first field can be considered to evolve into the second one after a very long time. After a shorter time interval it will have partly mutated, with the low frequencies still similar to those in the first field while the higher frequencies will have mutated further towards those in the second field. By describing the turbulence in terms of frequency components (amplitude and phase) at each grid point, the partially evolved wind field can be reconstructed at each grid point by interpolating the phase of each frequency component between the two wind fields as a function of the along-wind coherence of the turbulence, a quantity which decreases both with the along-wind distance over which the turbulence evolves, and also with increasing frequency of turbulent variations.

When a line of sight measurement is made using the LIDAR module, the evolving turbulence model is used (provided this option has been enabled by the user). The LIDAR module will specify a location \((\Delta x,y,z)\) where the wind module should be interrogated. It is important to note that the wind file is “fed” through the rotor plane such that \(x\) represents the distance travelled by the wind file past the rotor plane and \(t\) is the current simulation time. Therefore, the alongwind location that is interrogated is given by \(x + \Delta x = U(t + \Delta t)\). The distance \(\Delta x = U\Delta t\) is therefore an upwind position measured from the rotor plane which is determined by the LIDAR system.

To compute the evolved wind speed at a point \(\Delta t\) seconds upwind of the turbine at a particular grid point \((y,z)\) the following formula is used

\[ \begin{equation} u(y,z,t + \Delta t) = \ \sum_{i = 1}^{N}{f_{i}{\left( n_{i},\ U\Delta t\ \right)a}_{i}(y,z,n_{i},t + \Delta t) + \sum_{i = 1}^{N}{\sqrt{1 - f_{i}^{2}(n_{i},U\Delta t)}b_{i}(y,z,n_{i},t + \Delta t)\ \ }\ } \end{equation} \]

where,

\[ \begin{equation} a_{i}\left( {y,z,n}_{i},t \right) = A_{u,1}\left( n_{i} \right)\cos\left( 2\pi n_{i}t + \phi_{i,1} \right) \end{equation} \]

\[ \begin{equation} b_{i}\left( y,z,n_{i},t \right) = A_{u,2}\left( n_{i} \right)\cos\left( 2\pi n_{i}t + \phi_{i,2} \right) \end{equation} \]

Last updated 30-08-2024