Wind Energy

p(σ⏐u)
1
0.8
0.6
0.4
0.2
22 Stochastic Small-Scale Modelling of Turbulent Wind Time Series 125
σ(u(i+1)−u(i)) = +1
σ(u(i+1)−u(i)) = −1
0
−5 0
u
5
p(σ⏐u)
1
0.8
0.6
0.4
0.2
σ(u(i+1)−u(i)) = +1
σ(u(i+1)−u(i)) = −1
0
−5 0
u
5
Fig. 22.1. PDF of σ conditioned on the current velocity value for a real turbulent
flow (left) and the process (22.2) (right)
p(σ=+1⏐⏐du⏐)
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
|du|
p(σ=+1⏐⏐du⏐)
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0 0.2 0.4 0.6 0.8 1
du
Fig. 22.2. Probability that an increment of size du is going to the positive direction
for a real turbulent flow (left) and the process (22.2) (right)
from an investigation of turbulent data recorded in a wind tunnel [4] and leads
to focus on sign statistics.
Skewness unravels in the non-zero odd-order structure functions and the
asymmetry of the probability density functions (PDFs) p(u) of the velocity
field and p(du) of velocity increments. This asymmetry can be depicted nicely
in conditional sign-probabilities. Figure 22.1a shows the probability p(σ|u) of
the sign σ of the next increment conditioned on the present value of the
velocity. Note that the turning point utp, where it is equally likely to make
a step to the positive or to the negative direction is not the mean velocity,
which by normalisation has been set to 〈u〉 = 0. The sign probability p(σ|du)
conditioned on the increment size is shown in Fig. 22.2a. This PDF demonstrates
that a big increment is more likely to occur in the positive direction
than in the negative direction. These two asymmetries play together such that
the mean 〈du〉 = 0 is zero.