THESE de DOCTORAT - cerfacs

2.2 Direct Computation of noise through Large Eddy Simulation 23
The extensions to reactive flows of Lighthill’s, Phillips’ and Lilley’s analogies in addition to
LEE and APE formulations have been proved successful for the study of noise generated by
open flames [40, 99, 16, 10, 36]. Studying noise radiated by such flames is important in order to
characterize the position, directivity and strength of each of the sources present [95]. Anyway,
in order to go further in the study of combustion noise, the inclusion of reflecting surfaces is
mandatory. This is an issue where aeroacoustics is a head on top of combustion acoustics. In
aeroacoustics theory, Curle extended Lighthill’s analogy in order to account for turbulencebody
interactions and introduced a corresponding additional source mechanism that involves
the reaction force exerted by the body on the surrounding field, equivalent to an acoustic dipole
[20, 85]. This acoustic analogy has been deeply studied in cavity noise. When these bodies are
moving at not negligible velocities, Curle’s analogy must be re-formulated. This was done
by Ffowcs-Williams and D. L. Hawkings at the end of the sixties [106]. Ffowcs-Wiliams and
Hawkings theory has been broadly implemented in the study of noise radiation by wing airfoils
[60, 15] and aerodynamic rotors [27].
2.2 Direct Computation of noise through Large Eddy Simulation
2.2.1 Governing equations of turbulent reacting flows
(
In terms of the material derivative D
Dt = ∂ ∂t + u j
momentum read
)
∂
∂x j
the Navier-Stokes equations for mass and
Dρ
Dt = −ρ ∂u j
∂x j
(2.4)
ρ Du i
Dt = − ∂p
∂x i
+ ∂τ ij
∂x j
+ ρ f i (2.5)
where ρ, u, p, τ ij and f i stand for the density, the velocity, the pressure, the viscous tensor and
the volume force respectively. The indices i = 1 to 3 and j = 1 to 3 denote the component of
the specific vector/tensor. The equation that characterizes reacting flows is the one denoting
the balance of species k.
ρ DY k
Dt
= ˙ω k − ∂
∂x i
(ρV k,i Y k ) = ˙ω k − ∂J k
∂x i
(2.6)
where Y k is the mass fraction of species k, ˙ω k is the reaction rate of the species k, V k,i is the
diffusion velocity of species k in the direction i and J k is the diffusion flux of species k. The
energy equation can be defined in multiple forms [74]. The equation for the total energy e t =
e + u 2 i
/2 is