Читать книгу Wind Energy Handbook - Michael Barton Graham - Страница 39

Alongside the Kaimal model, editions 3 and 4 of the IEC standard (IEC 2005, 2019) give the option to use a rather different form of turbulence model developed by Mann (1994, 1998). The other models described above make use of a one‐dimensional fast Fourier transform (FFT) to generate time histories from spectra, applied to each turbulence component independently. In contrast, the Mann model is based on a three‐dimensional spectrum tensor representation of the turbulence, and one three‐dimensional FFT is then used to generate all three components of turbulence simultaneously. The three‐dimensional spectrum tensor is derived from rapid distortion theory, in which isotropic turbulence described by the von Karman spectrum is distorted by a uniform mean vertical velocity shear. This means that the three turbulence components are no longer independent, as energy is transferred between the longitudinal and vertical components by distortion of the eddies in the flow, resulting in a realistic representation of the correlation between the longitudinal and vertical components described by the Reynolds stress. The spectral density for any three‐dimensional wavenumber vector is derived, and all three components of turbulence are then generated simultaneously by summing a set of such wavenumber vectors, each with the appropriate amplitude and a random phase.

This is in many ways a rather elegant approach, but in practice there are some computational limitations that can make it difficult to use. The summation requires a three‐dimensional FFT to achieve reasonable computation time. The number of points in the longitudinal, lateral, and vertical directions must be a power of two for efficient FFT computation. In the longitudinal direction, the number of points is determined by the length of time history required and the maximum frequency of interest and is therefore typically at least 1024. The maximum wavelength used is the length of the turbulence history to be generated (i.e. the mean wind speed multiplied by duration of the required time series), and the minimum wavelength is twice the longitudinal spacing of points (which is the mean wind speed divided by the maximum frequency of interest). In the lateral and vertical directions, a much smaller number of points must be used, perhaps as low as 32, depending on available computer memory. The maximum wavelength must be significantly greater than the rotor diameter, because the solution is spatially periodic, with period equal to the maximum wavelength in each direction. The number of FFT points then determines the minimum wavelength in these directions. With a realistic number of points, the resulting turbulence spectra are deficient at the high frequency end (Veldkamp 2006). Mann (1998) suggests that this may be realistic, because it represents averaging of the turbulence over finite volumes of space, which is appropriate for practical engineering applications. However, a practical simulation tool will perform all necessary spatial averaging in any case, and so the high frequency variations are really lost. Mann (1998) does suggest a remedy for this, but in practice it is extremely intensive computationally.