Wind Energy

88 J. Mann
vector between two points is − 4
5 times the dissipation of kinetic energy ε times
the separation distance r. It is approximately true for high Reynolds number
flows when r is in the inertial subrange, and, assuming an infinitely long
inertial subrange, can be derived rigourously. Despite this fact it simplifies the
simulation algorithms immensely to assume the velocity field to be a gaussian
random field. Furthermore, it is not clear how important the departure from
gaussianity is for the dynamic loads, though some attempts to quantify this
has been undertaken. For example, for a turbine in rather complex terrain
some loads increased by 15% comparing a non-gaussian simulation with a
gaussian with the same second order structure [7].
In this paper we from hereon assume gaussianity. We furthermore assume
Taylor’s hypothesis to be valid
ũ(x, y, z, t) =ũ(x − Ut,y,z,0), (15.1)
where x is the coordinate in the mean flow direction, U the mean wind speed.
The velocity vector is denoted by ũ and the fluctuations around the mean is
u. All second order statistics of the fluctuations can be expressed in terms of
the covariance tensor
Rij(r) =〈ui(x)uj(x + r)〉 . (15.2)
It does not depend on x if homogeneity is assumed. This is usually a good
assumption for the horizontal directions x and y, but less optimal for the
vertical.
The Fourier transform of the covariance tensor
Φij(k) = 1
(2π) 3
�
Rij(r)exp(−ik · r)dr, (15.3)
is called the spectral velocity tensor. This has been modelled by Mann [4] for a
homogeneous shear layer, and compared well to atmospheric surface-layer data
over flat terrain. Based on this model a Fourier simulation algorithm, which
essentially is a superposition of sine-waves in 3D space, may be devised [5].
The model by Veers [10], which was improved by Winkelaar [11], is based
on one-point spectra and coherences, i.e. the absolute value of the normalized
cross-spectra. The phase of the cross-spectra is typically ignored and
so is incompressibility. Also, empirical information of all cross-spectra is not
available.
A group at Risø has simulated 3D fields by the model and sampled the
velocities on a helix emulating the passage of a point on a wind turbine blade
through the air [8]. They compared this with measurements made by a pitot
tube mounted on a rotating blade. The comparison is shown in the right plot
of Fig. 15.1. The left plot in this figure shows the helix sampled spectra from
a Veers simulation of only the longitudinal wind component (lower points)
and of the two horizontal component. It seems to be important to simulate
all three components of the velocity field to obtain realistic spectra.