Synthetic Inflow Condition for Large Eddy Simulation (Synthetic - KTH

3.2.3 Higher order statistics
CHAPTER 3. THE SYNTHETIC EDDY METHOD
Moreover in order to understand better the behavior of the signal, higher order
moments as the skewness and the flatness, will be computed which will allow us to
check afterwards the validity of the SEM implementation.
3.2.3.1 Skewness
To express the final equations of these statistics we have to start with another
formulation of the SEM velocity fluctuations by rearranging Eq. (3.3) as
u ′
i(x, t) = 1
√ N
with, X (k)
i
N�
k=1
X (k)
i
= aijε k j fσ(x − x k )
(3.20)
Xk i being independent random variables following the same distribution, the
central limit theorem can be applied. This states that when N tends towards infinity,
the probability density function of u ′
i (x, t) tends towards a Normal distribution
N(µi, σ 2 i ) of mean µi = 〈X k i 〉 and of variance σ2 i = 〈(Xk i )2 〉. In this method it
would mean, when the number of eddies N tends towards infinity, the signature of
each eddy in the final synthetic signal becomes more faint and the final signal tends
toward a universal Gaussian state.
From there we can thus express the skewness of the velocity signal using Eq.(3.20).
Sui = 〈u′ 3
i 〉
〈u ′ = 2
i 〉 3/2
which is shown to be zero
1
(NRii) 3/2
��
N�
X
k=1
(k)
�3�
i
Sui
Actually using the multinomial theorem,
(x1 + x2 + ... + xm) n =
�
k1,k2,...,km
(3.21)
= 0 (3.22)
n!
k1!k2!...km! xk1 1 xk2 2 ...xkm m
(3.23)
with � ki = n for any positive integer m and any non-negative integer n, we obtain,
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