THESE de DOCTORAT - cerfacs

2.3 Hybrid computation of noise: Acoustic Analogies 39
agation of noise takes place in free-field, the simplified Lighthill equation Eq. (2.43) applies
1
c 2 ∞
The farfield pressure is then given by
∂ 2 p ′
∂t 2 − ∂2 p ′
∂x 2 i
=
(γ − 1)
c 2 ∞
∂ ˙ω ′ T
∂t
(2.75)
p ′ (x, t) =
∫ t
t 0
∫V
(γ − 1)
G(x, t|y, τ)
c 2 ∞
On the other hand, the 3D temporal Green’s function is defined as
∂ ˙ω
T
′ dydτ (2.76)
∂t
G(x, t|y, τ) = − δ(t − r/c ∞)
4πr
(2.77)
where r = |x − y| . As a result, the farfield pressure radiated by a free turbulent flame can be
obtained
p ′ (x, t) =
(γ − 1)
4πrc 2 ∞
∫
∂
∂t V
˙ω ′ T
(
y, t − r
c ∞
)
dy (2.78)
Expression (2.78) can be found in the work of Strahle [98, 99] for instance. It can be noted
from this expression that the farfield pressure radiated from a turbulent flame results from
the contribution over the combustion region of elementary sources formed by the local time
rates of change of the heat release rate. Each source produces an elementary contribution at
the measurement location after a time delay corresponding to propagation between the source
inside the combustion region and the point of observation in the farfield. Scaling rules can also
be derived from Eq. (2.78) for the thermo-acoustic efficiency η ta , defined as the ratio between
the radiated acoustic power and the thermal power released in the flow (typically η ta ∼ 10 −6 −
10 −5 ) for combustion noise in absence of instabilities.
Useful Green’s functions
The 3D free-field Green’s function expressed in the temporal domain has already been given in
Eq. (2.77). This function has been widely used in combustion noise studies since it is associated
with the radiation of sound from a monopolar ‘spherical’ source, which is the basis of combustion
noise theory. In addition to this 3D Green’s function, the 2D temporal Green’s function can
also be useful for the study of noise radiation in academic cases. It reads