THESE de DOCTORAT - cerfacs

2.2 Direct Computation of noise through Large Eddy Simulation 25
From this definition it is possible to write the system of equations that govern LES for nonreacting
flows. It yields [74]
Mass
∂ ¯ρ
∂t + ∂ ( ) ¯ρũj = 0 (2.10)
∂x j
Momentum
∂ ¯ρũ i
∂t
+ ∂
∂x j
( ¯ρũi ũ j
) +
∂ ¯p
∂x i
=
∂ [
¯τij − ¯ρ ( )]
ũ i u j − ũ i ũ j
∂x j
(2.11)
Chemical species
∂ ¯ρỸ k
∂t
+ ∂ ( )
¯ρũ j Ỹ
∂x k = ∂ [
)]
V
j ∂x k,j Y k − ¯ρ
(ũj Y k − ũ j Ỹ k + ¯˙ω k (2.12)
j
Enthalpy
∂ ¯ρ˜h s
∂t
[
∂
( )
¯ρũ j˜hs = Dp
∂x j Dt + ∂ λ ∂T
) ] ∂u j
− ¯ρ
(ũj h s − ũ j˜hs + τ ij
∂x j ∂x j ∂x i
)
ρ V k,j Y k h s,k
− ∂
∂x j
(
N
∑
k=1
+ ¯˙ω T
(2.13)
where h s stands for the sensible enthalpy of the mixture. The quantities (ũ i u j − ũ i ũ j ), (ũ j Y k −
ũ j Ỹ k ) , (ũjh s − ũ j˜hs ), V k,i Y k and ( ¯˙ω k ) are known as the unresolved Reynolds stresses, unresolved
species flux, unresolved enthalpy flux, filtered laminar diffusion fluxes and filtered chemical
reaction rate respectively. These terms, which need to be modeled, are the quantities that account
for the influence of the small structures on the entire physical system. LES system of
equations for reactive flows may vary somehow with respect to Eq. (2.10) - Eq. (2.13) due to
the combustion model applied. As an example in the Thickened Flame combustion model TF
[11, 17, 74], viscous terms (as for instance the filtered diffusive species flux ¯J j,k and the filtered
heat flux ¯q i ) and notably the filtered chemical reaction rate ¯˙ω k are weighted by several factors
in order to account for the ‘new’ thickness of the flame.
Large Eddy Simulation has become an important tool for the simulation and post-processing
analysis of turbulent flows. It offers the best promise in the foreseeable future for the estimation
of noise from flows at Reynolds numbers of interest in both open and closed systems. In
aeroacoustics, LES plays an important role in the study of aerodynamical generated noise of
numerous practical cases that range from air jets, high-lift devices or landing gears in an aircraft
to the rear-view mirror of a car or the blades of a wind turbine [65, 8]. In reactive flows,
LES has been successfully applied to partially premixed and non-premixed open flames [70, 36]