THESE de DOCTORAT - cerfacs

24 Chapter 2: Computation of noise generated by combustion
ρ De t
Dt = − ∂q i
∂x i
− ∂
∂x i
(
puj
) +
∂
∂x j
(
τij u j
) + ˙Q + ρ f i u i (2.7)
Here q i represents the energy flux and and ˙Q is the heat source term. ˙Q can be, for instance, the
energy released by an electrical spark, a laser or a radiative flux. It should not be confused with
the heat released by combustion. The balance equation of enthalpy is the one used to account
for the energy balance in the present study:
ρ Dh
Dt = Dp
Dt − ∂q i
∂x i
+ τ ij
∂u i
∂x j
+ ˙Q + ρ f i u i (2.8)
This equation is derived by combining the balance equation for the internal energy e with the
∂u
definition of enthalpy dh = de + d(p/ρ). The term τ i ij ∂x j
is known as the dissipation function
and represents the work done by the viscous stresses due to the deformation of a fluid particle.
2.2.2 Large Eddy Simulation
The dynamics of reactive flows is exactly described by the Eqs. (2.4)-(2.8). Nevertheless, no analytical
solution exists for such a non-linear and coupled differential system of equations. There
is only one way to follow so that Navier-Stokes equations become solvable: discretize them.
The numerical schemes used for such discretizations must be of high order so that numerical
stability and precision is assured, and the grid must be extremely refined so that ‘physical’ solutions
are obtained: reactive flows contain fluid structures that range from the smallest scales
of turbulence (the Kolmogorov scale η ∼ Re −3/4 m) or the flame thickness (∼ 10 −4 m) to the
scales that can be of the size of the entire physical domain as the acoustic waves length (∼ 1 m).
It is then understandable that with today computer’s performance it is impossible to resolve
such equations for system sizes that exceed some centimeters. A system as big as a combustion
chamber is therefore unresolvable. Fortunately, a solution approach has been proposed [79] in
which only the bigger scales of turbulence are explicitly computed and the smallest are modeled.
This technique is known as the Large Eddy Simulation LES. In order to solve only the
large structures of turbulence, the Navier Stokes equations must be filtered:
∫
¯f (x) =
f (x ′ )L(x − x ′ )dx ′ (2.9)
where L stands for the LES filter. The Favre filter is the one usually applied and is defined
as 2 ˜f = ρ f / ¯ρ. The filtered quantity ( ¯f or ˜f ) is explicitly resolved in the numerical simulation
whereas f ′ = f − ¯f corresponds to the unresolved part.
2 The meaning of the symbols ¯ () and ˜() given in this section stands only for this section.