Characteristics of the Wind - Wind Turbine Technology: Fundamental Concepts of Wind Turbine Engineering - page 518

Spectral Analysis of Rotationally-Sampled Wind
Spectral analysis of the three time-series illustrated in Figure 8-28 gives us further
information on the distinctions between the Eulerian (fixed coordinate) and Lagrangian
(rotating coordinate) 1 representations of the same wind field, and these spectra are shown
in Figure 8-29. Here the power amplitude, f( n ), has been multiplied by the circular
frequency, n , to accentuate the higher-frequency portion of the spectrum. The spectrum
of the center anemometer provides a reference in the Eulerian system. The rotationally-
sampled spectrum contains several large “spikes”, the largest of which occurs at the
simulated rotor speed of 7.8 rad/s. The average magnitude of the Lagrangian spectrum in
the frequency range of the spikes is much greater than that in the Eulerian spectrum.
Conversely, there is a considerable loss of energy in the decade of frequency below the
simulated rotor speed, compared to the Eulerian spectrum. Connell [1981] points out that
rotational sampling transfers energy from intermediate frequencies to energy at higher
frequencies and collects a portion of this energy into narrow frequency bands at harmonics
of the rotational speed. The total turbulent energy, of course, cannot be changed by a
change of coordinate system. As expected, the spectrum of the array-average wind speed
contains less energy than that of the center anemometer, because of the low-pass filtering
mentioned earlier.
Figure 8-29. Power spectra of the time-series wind speeds illustrated in Figure 8-28.
(a) Synthesized wind speed in the Lagrangian (rotating) coordinate system (b) Center ane-
mometer in the Eulerian (fixed) coordinate system (c) Array-averaged wind speed [Verholek
1978]
1 Note that these are not the same definitions as used in classical fluid mechanics.