Synthetic Inflow Condition for Large Eddy Simulation (Synthetic - KTH

u ′
(j) =
N�
k=−N
CHAPTER 2. BACKGROUND
bkr(j + k) (2.36)
with bk, the filter coefficients, N, connected to the support of the filter and
r(j + k), the random number generated at point (j + k) with a normal distribution
ℵ(0, 1). Thus the two-point correlations between points j and (j + m) depend on
the filter choice and read
< u ′
(j)u ′
(j + m) >=
N�
k=−N+m
bkbk−m
(2.37)
An extension of this procedure is realized to the time-dependent computation of
synthetic velocity field on a plane (Oyz) by generating for each velocity component
m, a three-dimensional random field rm(i, j, k). The indices i, j, and k correspond to
the time t, the direction y and the direction z respectively. A three-dimensional filter
bijk is obtained by the convolution of three one-dimensional filters bijk = bi ˙ bj . . . bk.
This is used to filter the random data rm(i, j, k) in the three directions t, y and z,
u ′
m(j,k) =
Nx �
Ny �
Nz �
i ′ =−Nx j ′ =−Ny k ′ =−Nz
b i ′ j ′ k ′ rm(i ′
, j + j ′
, k + k ′
) (2.38)
The random numbers are convected at each time step through the inlet plane
using Taylor’s frozen turbulence hypothesis ri,j,k −→ ri+1,j,k and new random numbers
are generated on the plane i = 1. At the next time step, the new random field
is filtered and so on. If the objective would be only to produce homogeneous turbulence,
the procedure would thus stop at this moment. However this is generally
not the case, for this reason the signal is reconstructed at each time step following
Eq. (2.34) to generate target mean velocity and Reynolds stresses profiles. The
fluctuations generated have to reproduce exactly the target two-point correlations
< u ′
(j)u ′
(j + m) > therefore the filter coefficients bk should be computed by inverting
Eq. (2.37). Klein [25] assumed a Gaussian shape depending on one single
parameter, the lengthscale L, since the two-point autocorrelation tensor is rarely
available. Then without inverting Eq. (2.37), the coefficients can be computed
analytically. Hence Klein [25] could test with this procedure the influence of the
length and time scales (uniform over the inlet plane) on the development of a plane
jet and the primary break-up of a liquid jet.
Additionally to the above presented methods which play in fact an important
role in this project, other techniques are introduced briefly as following.
18