Wind Energy

h(u)
0.5
0.3
0.1
20 Superposition Model for Atmospheric Turbulence 117
0 5 10 15
u [m/s]
On 1
On 2
On 3
Off
Fig. 20.2. Symbols represent measured mean velocity distributions (averaged over
10 min) of four different atmospheric data sets. Solid lines are fits according to
(20.3)
� ∞
p(uτ )= dūh(ū) · p(uτ |ū). (20.2)
We assume h(ū) to be a Weibull distribution
h(ū) = k
A
0
� �k−1 ū
exp
A
�
−
� �k ū
A
�
(20.3)
which is well established in meteorology [3]. In Fig. 20.2 it is shown that a
Weibull distribution is a good representation of h(ū).
Inserting (20.3) and (20.1) into (20.2) the following expression for atmospheric
PDFs is obtained
p(uτ )= k
2πA k
� ∞ � ∞
dū
0
× 1
exp
λσ2 0
�
− u2 τ
2σ 2
dσ ū k−1 � �
ū
exp −
�
exp
A
�k �
�
− ln2 (σ/σ0)
2λ2 �
. (20.4)
Parameters A and k play a similar role as σ0 and λ 2 in the Castaing
distribution. With this approach intermittent atmospheric PDFs can be
approximated for any location as long as the mean velocity distribution is
known.
In Fig. 20.3 probability density functions of velocity increments are shown
for different scales τ (increasing from top to bottom). The left figure corresponds
to a measurement with an ultrasonic anemometer in a non-stationary
atmospheric flow. The right figure corresponds to a measurement with a