Environmental Engineering Reference
In-Depth Information
Spatial Turbulence Models
Spatial turbulence models are mathematical descriptions of the fluctuating wind field
in two- and three-dimensions. These models are important to the design of wind turbine
control systems, to the experimental verification of power output, to the prediction of
asymmetric forces on rotors resulting from nonuniform gusts over the swept area, and many
other responses of the turbine to unsteady, non-uniform wind conditions.
Information about spatial wind fields is normally based on a time-history of wind speed
measured by an isolated anemometer measurement and the application of
Taylor's hypo-
thesis
[Hinze 1975]. Taylor's hypothesis assumes the fluctuating velocity of the wind at
a fixed point,
g
(
x
0
,t
),
can be converted using the relationship
x = x
0
+
Ut
,
producing a
spatial relationship at a fixed time,
g
(
x, t
0
)
.
This is also referred to as the
frozen turbulence
concept, since the spatial variation is assumed to remain unchanged during the averaging
time of
U.
To illustrate the relation between time and distance inherent in Taylor's
hypothesis, consider two points in space,
A
and
B
,
in which
B
is a distance D
x
downwind
of
A.
According to the frozen turbulence concept
g
a
(
B
,
t
) =
g
a
(
A
,
t
-
D
x
/
U
)
where, as before, the subscript a
represents each of the three components of the wind.
Spatial variation at two or more points in space can be modeled without recourse to
Taylor's hypothesis using wind speed data from an array of anemometers. Spatial vari-
ations in the vertical direction can be measured with a single tower instrumented at different
levels, and in both the vertical and lateral directions with multiple towers supporting
anemometers in a two-dimensional pattern called a
vertical plane array
(Figure 2-24).
Statistical Parameters
Quantitative estimates of spatial variations in wind fields are provided by the following
three statistical parameters:
--
correlation coefficient
--
two-point spectrum (or two-point correlation)
--
coherence function
Full mathematical descriptions of these parameters are given in a number of references [
e.g.
,
Papoulis 1965, Bendat and Piersol 1971, Panofsky and Dutton 1984]. Brief physical
descriptions of some models will be given here.
Correlation Coefficient
A correlation coefficient is a measure of how well fluctuations in a wind speed compo-
nent measured at one position in space correspond or correlate with fluctuations in a wind
speed component at a second point. The full three-dimensional correlation coefficient,
correlating each of three components at the first point with each of the three at the second,
is a
nine-component tensor.
However, for the conditions of isotropic, homogeneous turbu-
lence normally assumed in wind turbine analysis, the correlation coefficient reduces to two
components: a
longitudinal correlation coefficient
for wind fluctuations parallel to a line
joining the two points in space, and a
transverse correlation coefficient
for wind fluctuation
components perpendicular to this line. Assuming a separation distance of z, some
examples of these two types of correlation are as follows:
Search WWH ::
Custom Search