Introduction to Modern Wind Turbines - Wind Turbine Technology: Fundamental Concepts of Wind Turbine Engineering - page 98

Spatial Distribution Around the Sampling Circle

R-S turbulence is assumed to be quasi-static, which means that a HAWT rotor makes
several revolutions before any signiicant changes occur in the spatial distribution of the
wind within the rotor's swept area. Dynamic responses of the rotor blades are assumed to
approach a steady state before the wind turbulence changes signiicantly. With these quasi-
static assumptions, the time-varying unsteady wind can be converted to a spatially-varying
steady wind in a harmonic format, as

U RS = U 0 + S ( A n cos n ) y n = 1, 2, . . . (2-33)

where

U RS = rotationally-sampled free-stream horizontal wind speed, quasi-steady in time
(m/s)

A n = amplitude of n th harmonic of wind speed (m/s)

y = azimuthal position in rotor swept area; 0 = down (deg)

Wind turbulence is equal to the standard deviation of the wind speed from its steady value.
For each cosine wave in Equation (6), standard deviation and amplitude are related as fol-
lows:

(2-34)

| A n | = Ö2 n s

While the absolute values of the harmonic amplitudes can be determined from R-S tur-
bulence data, the signs of these amplitudes cannot. Various patterns of positive and negative
signs (equivalent to in-phase and out-of-phase harmonics in Eq. (2-33)) have been used.
For example, Zimmerman et al. [1995] assumed negative amplitudes for odd-numbered har-
monics and positive amplitudes for even-numbered harmonics. Further examination of the
shapes of various vertical wind proiles deined by different combinations of positive and
negative harmonic amplitudes indicates that a reasonable vertical proile is obtained with
negative harmonics except for the third and fourth.

Spatial Distribution Along a HAWT Blade

In order to apply R-S turbulence wind speeds in a structural-dynamics code for calcu-
lating blade fatigue loads it is necessary to deine the wind speed distribution from hub to
tip. Earlier, the assumption was made that the size of each R-S turbulence harmonic varies
linearly with the sampling radius. However, this assumption alone is not suficient to deine
the simultaneous wind speed distribution along a turbine blade. The spanwise distribution of
turbulence also depends strongly on the transverse coherence of the wind. As explained in
Chapter 8, coherence is a dimensionless quantity between zero and unity that represents the
degree to which two unsteady events, separated in space, are alike in their time histories. If
the two time histories are identical their coherence is unity, and if they are completely unre-
lated their coherence is zero.
Coherences of individual harmonics of R-S turbulence were measured by Zimmerman et
al. [1995] using wind speed sensors mounted at two locations 69.2 ft apart on a 150-ft Mod-2
HAWT blade. The coherence between the irst harmonic at the inboard location ( r = 30.8
ft) and the irst harmonic at the outboard station was found to be high, but equivalent coher-
ences of the higher harmonics were all low for this separation distance. To generalize these
observations, it is assumed that the irst harmonic of the R-S turbulence has a coherence of
unity along the entire blade length. Thus, the irst harmonic turbulence acts simultaneously
along the blade from hub to tip. Higher harmonics are assumed to have a coherence of unity