Wind Energy

74 François G. Schmitt
13.2 Scaling and Intermittency of Velocity Fluctuations
The scaling and intermittent properties of wind velocity have been studied in
the turbulence community for several decades (see reviews in [1,3,4]). We recall
here some basic properties of this framework. The Richardson–Kolmogorov
energy cascade develops on the inertial range, between the large injection
scale (of the order of several tens or hundreds of meters) towards dissipative
scales (of the order of millimeters). In the inertial range, wind velocity
fluctuations are scaling: they possess a power-law spectrum
E(k) � k −β
(13.1)
with β =5/3 for K41 turbulence [5]; intermittency however leads to a β value
slightly larger than 5/3. Here we may define intermittency as the property
of having large fluctuations at all scales, having a correlated structure:
large fluctuations are much more frequent than what would be obtained
for Gaussian processes [1, 3, 4]. Intermittency is then classically characterized
considering the probability density function (pdf) or the moments of
the velocity fluctuations ∆Vℓ = |V (x + ℓ) − V (x)|. For large mean velocities,
Taylor’s hypothesis may be invoked to relate time variations ∆Vτ =
|V (t+τ)−V (t)| to spatial fluctuations; one may then study time intermittency
of wind velocity considering the scaling property of moments of order q>0
of the fluctuations ∆Vτ
〈(∆Vτ ) q 〉�τ ζ(q) , (13.2)
where ζ(q) is the scale invariant moment function, which seems rather universal
for fully developed turbulent flows in the laboratory or the atmosphere,
for moments small than 7. This function is nonlinear and concave; ζ(3) = 1
is a fixed point; ζ(2) = β − 1 relates the second order moment to the
power spectrum scaling exponent; the knowledge of the full (q, ζ(q)) curve for
integer and noninteger moments provides a full characterization of velocity
fluctuations at all scales and all intensities.
13.3 Gusts for Fixed Time Increments
and Their Recurrent Times
Wind gusts may be generally defined choosing a time increment τ and a
threshold velocity δ: a gust corresponds to a situation when |V (t1)−V (t2)| >δ
with |t1 − t2| < τ [6]. In practice, for such event to occur, the condition
|V (t1) − V (t2)| > δ should also be realized for some times verifying
|t1 − t2| = τ. We then choose here to fix a small time increment τ and to consider
large fluctuations at this scale. In the following we choose τ = 3 since
3 − s extreme gust correspond to a standard of the American Society of Civil