Synthetic Inflow Condition for Large Eddy Simulation (Synthetic - KTH

CHAPTER 2. BACKGROUND
turbulence models used to close the equations are provided. Although the software
elsA solves the compressible Navier-Stokes equation, we will restrict ourselves to
the simulation of incompressible flows where the mass density ρ and the dynamic
viscosity µ are constant. In this case, the Navier-Stokes equations governing the
evolution of the velocity u and pressure p of the fluid read
ρ ∂u
∂t
+ ∇.(ρu ⊗ u) = −∇p + ∇.σ (2.1)
∇.u = 0 (2.2)
The viscous stress tensor is σ = 2µS for a Newtonian fluid where
is the strain rate tensor.
S = 1
2 (∇u + (∇u)T ) (2.3)
However since the direct numerical simulation of the above Navier-Stokes equations
requires an extreme computational cost due to the several scales involved, a
smoothing operator has to be applied to the exact solution u of the Navier-Stokes
equations. The complexity of the system to be solved will thus be reduced.
Assuming that the LES spatial filter and the derivative operators commute,
governing equations for the filtered/averaged quantities u and p can then be derived,
where u corresponds to the LES averaging operators
ρ ∂u
∂t
+ ∇.(ρu ⊗ u) = −∇p + ∇.(σ + τ) (2.4)
∇.u = 0 (2.5)
And the subgrid-scale stress tensor is τ = −ρ(u ⊗ u − u ⊗ u)
Hence the LES representations of the flow field are seen as two levels of description
of the exact solution u of the Navier-Stokes equations.
Labourasse and Sagaut [20] defined the LES spatial filtering operator (see Eq.
(2.6)) in the general framework of multilevel methods relying on different scale
separation operators.
In fact the separation of large scales and small scales is done via a low-pass
filtering operation, which is defined as a convolution product
� +∞
u = G ⋆ u = u(y)G△ (x − y)dy (2.6)
−∞
4