Synthetic Inflow Condition for Large Eddy Simulation (Synthetic - KTH

ΘT = ˜ T − 1
The state equation is written [10]:
2ρCυ
ΘP ≈ ρRΘT
τ t kk
CHAPTER 2. BACKGROUND
(2.10)
(2.11)
The subgrid viscosity µsm with a scale of length l0 and a time scale t0 characteristics
of the subgrid quantities. The models will have to determine these characteristic
scales and here we define two of them, the Smagorinsky and the WALE model which
is used for the computations of this project.
2.2.2 Smagorinsky model
This is an old model, at the origin of other important models. It is based in the
resolved scales and uses a similar approach to the model of Prandtl mixture length.
If we assume that the cutoff scale △c imposed by the filter is representative of the
subgrid modes, then we obtain the length scale l0 :
l0 = Cs△c
(2.12)
A time scale is defined by assumption of local equilibrium between the kinetic
energy production rate, the dissipation energy rate by viscosity in internal energy
and the kinetic energy flux through the cutoff imposed by the filter:
1
t0
= (2SijSij) 1/2
(2.13)
The eddy viscosity of the subgrid structures ({νsm}) is proportional to the product
of a length by a velocity, thus it is written
νsm = C 2 s ∆ 2�
2SijSij
with Cs the constant of the model to determine and (here)
Sij = 1
�
∂Ui
+
2 ∂xj
∂Uj
�
∂xi
(2.14)
(2.15)
If we assume the energetic cascade of Kolmogorov, the constant Cs can be
evaluated (Ref.[23]) in a way such that the subgrid dissipation is equivalent to
the unresolved scales dissipation:
6