Environmental Engineering Reference
In-Depth Information
R
xx
(z) = [k
L
(z/
Z
)(
x
2
/z
2
) + k
T
(z/
Z
)(1 -
x
2
/z
2
)] s
0,
x
(8-31a)
(8-31b)
x
= y
U
/W
4
r
2
sin
2
y/2 +
x
2
(8-31c)
z =
where
R
xx
=
two-point autocorrelation function
for longitudinal winds at
A
and
B
(m
2
/s
2
)
k
L
( ), k
T
( ) = longitudinal and transverse correlation coefficients evaluated at ( ), refer-
enced to
A-B'
(Eqs. (8-28) to (8-30))
y = azimuthal coordinate from
A
to
B
(rad)
z = distance from
A
to
B'
(m)
Inspection of Equations (8-32) reveals that the Eulerian autocorrelation function is obtained
by setting
r
equal to zero, and the
frozen turbulence
case is obtained with
U
equal to zero.
For non-isotropic turbulence, we can define an equivalent isotropic integral scale length as
Z
=
Z
x
(
x
/z) +
Z
y
(1 -
x
/z)
(8-31d)
where
Z
x
,
Z
y
=
longitudinal and lateral integral scale lengths, respectively (m)
Powell
et al.
[1985] found that the best correlation between calculated and measured
Lagrangian spectra from the Clayton, New Mexico, vertical-plane array was obtained when
the integral length scales
Z
x
and Z
y
were both set equal to about 1.7 times the mid-elevation
of the array. Therefore, a reasonable approach would be to assume
isotropic turbulence
and
treat Z as an empirical parameter, to be evaluated on the basis of available test data. Powell
and Connell [1986b] provide a computer model for simulating rotational data for both
HAWTs and VAWTs and compare calculated spectra to experimental data.
To illustrate the use of Equations 8-31, we will calculate horizontal autocorrelation
functions for a Mod-OA HAWT, using the Dryden correlation coefficients for simplicity.
The configuration parameters for this sample case are as follows:
R =
rotor radius = 19.05 m
W
= rotor speed = 4.19 rad/s
h
=
hub elevation = 30.5 m
Z = 51 m
U =
steady wind speed = 8.21 m/s
s = 0.9 m/s
Figure 8-31 shows typical behavior of the autocorrelation function (normalized by the
square of the turbulence or the
variance
)
with increasing azimuthal angle, at four radial
locations. The valleys are regions of lower correlation which produce the characteristic
peaks in the Lagrangian power spectrum shown in Figure 8-32. This spectrum is obtained
from the autocorrelation function by a standard
fast Fourier transform (FFT)
analysis.
VAWT Rotational Sampling
The two-point spatial correlation for a VAWT is developed in the same manner as that
for the HAWT [Powell and Connell 1986a]. Comparisons of simulated Lagrangian spectra
for VAWTs with equivalent spectra for a HAWT [George 1984] indicate much lower turbu-
lent energy for VAWTs in flow fields with strong vertical gradients, as would be expected.
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