Wind Energy

124 J. Cleve and M. Greiner
accomplished very successfully by means of random multiplicative cascade
processes (RMCP) [5–7]. The basic idea of our model is to extend fBM with
multifractal properties of RMCPs.
fBM BH (t) is an extension of ordinary Brownian motion where the
variance of increments 〈(BH(t) − BH(t ′ )) 2 〉 ∼ (t − t ′ ) 2H have a Hurst
exponent 0 ≤ H ≤ 1 other than H = 1
2 ; see e.g. [2] for details. Regarding
RMCPs we � pick the most successful variant [6], where the dissipation
� t
ε(x, t) = exp t−T dt′ � x+g(t−t ′ )
x−g(t−t ′ ) dx′ γ(x ′ ,t ′ �
) is defined as a causal integral
over an independently, identically distributed Lévy-stable white-noise field
γ(x, t) ∼ Sα((dxdt) α−1−1σ, −1,µ) with index 0 ≤ α ≤ 2. The function
g(t) = 1 LT η
2 L(T −t) is defined such that a desired universal multifractal model
correlation structure of the dissipation is established. L and T are a correlation
length and time, respectively.
Because energy dissipation is proportional to the square of velocity derivatives
our process is defined as the product
∂u
∂t
= Mε∆BH
(22.1)
of a multifractal field Mε ∝ √ ε and an incremental fBm ∆BH with Hurst
exponent H =1/3. The process (22.1) is a direct generalisation of the multifractal
random walk proposed in [1]. Note that the ansatz (22.1) multiplies two
fields defined at the dissipation scale. Although beyond the scope of this contribution,
it would be very interesting to compare this approach with an implementation
where velocity increments associated with an energy flux evolve
from the integral scale down to the dissipation scale, which appears closer to
the intuitive physical picture of a turbulent energy cascade, see e.g. [3].
With this ansatz it is guaranteed that the correlation structure of the
dissipation ε ∝ (∂u/∂t) 2 ∝ (Mε∆BH ) 2 is determined solely by the RMCP.
Because Mε and ∆BH are uncorrelated, the two-point correlation function
〈ε(x1)ε(x2)〉 ∝ 〈(Mε(x1)) 2 (Mε(x2)) 2 〉〈(∆BH (x1)) 2 (∆BH (x2)) 2 〉 factorises.
The second factor does not depend on the two-point distance x2 − x1.
Similarly, the properties of the velocity field are given predominantly by
the properties of fBm. The standard way to characterise the statistics of the
velocity field are structure functions Sn(r) =〈(u(x+r)−u(x)) n 〉∝r ζn . Plain
fBm would result in the linear spectrum of K41 theory. The Mε-extension
leads to the non-linear spectrum of the scaling exponents.
22.3 Refined Modelling: Stationarity and Skewness
The simple combination (22.1) of the two processes would fall short of
modelling turbulence for at least two reasons. fBm is a non-stationary and
symmetric process whereas turbulence is stationary and skewed. A generalisation
of multifractal fractional Brownian motion is needed. Guidance comes