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f
h
;
Dz
I
u
þ
/
j
2
c
s
;
h
R
t
¼
ð
6
:
26
Þ
f
h
;
Dz
I
u
þ
/
j
2
c
teff
Formulation (
6.26
) permits easily to add the turbulence intensity produced by
the turbines during operation to the upstream turbulence intensity
(I
u,ef
2
= I
u
2
? I
u,
2
). Following Barthelmie et al. (
2003
) the additional turbulence,
I
u,t
can be parameterized as a function of the thrust coefficient (
6.13
) using a mean
turbine distance normalized by the turbine diameter s:
r
1
;
2C
T
s
2
I
u
;
t
¼
ð
6
:
27
Þ
The upper frame in Fig.
6.3
—in displaying R
t
from (
6.24
)—shows how much
the wind speed at hub height will be reduced as a function of the atmospheric
instability and the surface roughness. The presented results have been found for
turbines with a hub height of 92 m, a rotor diameter of 90 m and a mean distance
between two turbines in the park of 10 rotor diameters. It becomes obvious that the
reduction is smallest (a few percent) for unstable thermal stratification of the
atmospheric boundary layer and high surface roughness. I.e., the reduction is
smallest over a rough land surface with trees and other obstacles for cold air
flowing over a warm surface (usually during daytime with strong solar insolation).
The largest reduction (up to 45 %) occurs for very smooth sea surfaces when warm
air flows over cold waters. This may happen most preferably in springtime. The
lower frame of Fig.
6.3
translates this wind speed reduction into a reduction of the
available wind power by plotting the third power of R
t
from (
6.26
). The strong
stability dependence of the reduction of the available power can be confirmed from
measurements at the Nysted wind park in Denmark (Barthelmie et al.
2007
).
The dependence of wind and available power reduction as function of surface
roughness has consequences for offshore wind parks which will become the major
facilities for wind power generation in the near future. The lower turbulence
production due to the relative smoothness of the sea surface compared to land
surfaces hampers the momentum re-supply from the undisturbed flow above. In
order to limit the wind speed reduction at hub height in the interior of the wind
park to values known from onshore parks, the turbines within an offshore wind
park must have a larger spacing than within an onshore park. Roughly speaking,
the number of turbines per unit area in an offshore park with roughness
z
0
= 0.001 m must be approximately 40 % lower than in an onshore park with
z
0
= 0.1 m in order to have the same power yield for a given wind speed and
atmospheric stability.
Inversely, Eq. (
6.26
) may be used to determine the optimal areal density of
turbines in a large wind park for given surface roughness and atmospheric stability
conditions.
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