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To find the coefficients Cij(k) we calculate the covariance tensor of Eq. (84) obtaining
C ∗ ik(k)Cjk(k)
1
=
V 2
〈ui(x)uj(x
(B) B B
′ )〉e ik·x e −ik·x′
dxdx ′
1
=
V 2
Rij(x −x
(B)
′ )1B(x)1B(x ′ )e ik·(x−x′ ) ′
dxdx ,
where 1B(x) = 1 if x ∈ B and 0 otherwise. Using the change of variables r = x −x ′ and
s = x +x ′ having the Jacobian |∂(r,s)/∂(x,x ′ )| = 8 we get
1
Cik(k)Ckj(k) =
8V 2
Rij(r)e
(B)
−ik·r
s +r s −r
1B 1B dsdr (86)
2 2
The inner integration can be carried out according to
⎧
⎪⎨
3
s +r s −r 2(Ll −|rl|) for |rl| < Ll for all l
1B 1B ds =
2 2 ⎪⎩ l=1
0 otherwise
so, usingthe convolutiontheoremand notingthat the Fouriertransform ofL−|r| (for |r| < L
and else 0) is L2sinc 2 (kL/2), we get
C ∗ ik (k)Cjk(k)
= Φij(k ′ 3
) sinc 2
(kl −k ′ l )Ll
dk
2
′ , (88)
l=1
where sincx ≡ (sinx)/x. For Ll ≫ L, the sinc 2 -function is ‘delta-function-like’, in the sense
that it vanishes away from kl much faster than any change in Φij, and the area beneath the
sinc 2 -curve is 2π/Ll. Therefore, we get
The solution to Eq. (89) is
(85)
(87)
C ∗ ik(k)Cjk(k) = (2π)3
V(B) Φij(k). (89)
Cij(k) = (2π)3/2
V(B) 1/2Aij(k) = (∆k1∆k2∆k3) 1/2 Aij(k) (90)
with A ∗ ik Ajk = Φij and ∆kl = 2π/Ll. This result should be expected when comparing Eq.
(26) to (83).
Twoproblems occurby simulatingafieldby the Fourierseries Eq. (83) with the coefficients
Eq. (90). The first is that for many applications the dimensions of the simulated box of
turbulence need not to be much larger than the length scale of the turbulence model L.
Therefore Eq. (89) may not be a good approximation to Eq. (88). The second problem is
that the simulated velocity field Eq. (83) is periodic in all three directions. Both problems
have been addressed in Mann (1998).
The algorithms above simulate a three-dimensional vector field on a three-dimensional
domain, but it can easily be modified to simulate one- or two-dimensional vectors in a 2- or
3-D domain (Mann, 1998). The algorithms are not needed for a one-dimensional domain, i.e.
simulation of wind fluctuations in one point as a function of time.
The implementation of the model includes three steps:
1. Evaluate the coefficients Cij(k), either by Eq. (90) or a modification of this (Mann,
1998).
2. Simulate the Gaussian variable nj(k) and multiply.
3. Calculate ui(x) from Eq. (83) by FFT.
The time consumption in the first step is proportional to the total number of points N =
N1N2N3 in the simulation. The required time to perform the FFT is O(N log 2N) (Press
et al., 1992). In practice, simulating a three-dimensional field, used for load calculations on
wind turbines, with millions of velocity vectors takes of the order of a few minutes on a
modern pc.
68 DTU Wind Energy-E-Report-0029(EN)