Synthetic Inflow Condition for Large Eddy Simulation (Synthetic - KTH

The random variables x k j and ek j
independent thus
CHAPTER 3. THE SYNTHETIC EDDY METHOD
involved in the mean
�
aijε k j fσ(x)(x − x k � �
) = aijε k � �
j fσ(x)(x − x k �
)
�
aijεk j fσ(x)(x − xk �
) are
(3.8)
�
The term aijεk �
�
j simplifies further to aijεk � �
j = aij εk �
j = 0 since the intensities
of the eddies is either 1 or -1 with equal probability. Then we substitute these
relations into Eq. (3.7), the mean of the velocity signal ui is simply the input mean
velocity Ui,
< ui >= Ui
and the fluctuations u ′
i around the mean velocity are
u ′
i = 1
√ N
(3.9)
N�
aijε k j fσ(x)(x − x k ) (3.10)
k=1
The Reynolds stresses < u ′
iu′ j > of the synthesized signal is then calculated.
Using the above expression and the linearity of the statistical mean, we obtain
< u ′
iu ′
j >= 1
N
N�
k=1 l=1
N�
aimajn < ε k mε l nfσ(x − x k )fσ(x − x l ) > (3.11)
Using again the independence between the positions x k j and the intensities εk j of
the eddies, Eq. (3.11) becomes
< u ′
iu ′
j >= 1
N
N�
k=1 l=1
N�
aimajn < ε k mε l n >< fσ(x − x k )fσ(x − x l ) > (3.12)
If k �= l or m �= n the random variables ε k m and ε l n are independent and hence
< ε k mε l n >=< ε k m >< ε l n >= 0. If k = l and m = n, then < ε k mε l n >=< (ε k m) 2 >= 1
by definition of the intensities of the eddies. Hence we can write,
< ε k mε l n >= δklδmn
26
(3.13)