THESE de DOCTORAT - cerfacs

2.1 Introduction 21
In their experiment they fill soap bubbles with a reactive mixture and recorded the farfield
pressure generated by the burning of each isolated bubble. The noise (far-field pressure signal
p ′ ) was then expressed in terms of the volume acceleration over the reactive region induced by
the non-steady combustion. This expression yields
p ′ (r, t) = ρ ∞ d 2 ∆V
4πr dt 2 (2.1)
where ρ ∞ is the far field air density at the measurement location r and r designates the distance
between the compact flame and the observation point. Two years later, Price [76] and Hurle
[34] proposed a relation that accounts for the rate of consumption of reactants as responsible
for this density dilatation
p ′ (r, t) = ρ ( ) [ ]
∞ ρu dQ
4πr ρ b − 1 dt
t−τ
(2.2)
In this expression ρ u /ρ b designates the volumetric expansion ratio of unburnt to burnt gases,
Q the total volumetric rate of consumption of reactants and τ is the time required for acoustic
propagation from the combustion region to the measurement point r. Anyway, the first to
generalize these studies starting from the Navier-Stokes equations and deriving an analytical
expression was Strahle [98]. Following the procedure developed by Lighthill [50, 51], Strahle
proposed a rearrengement of the Navier-Stokes equations in which the acoustic field is separated
from the generators of combustion noise [99]. In order to obtain a compact formulation,
he considers that the acoustic wave length λ is large compared to any flame length scale L
(λ ≫ L). This expression, which is valid for low Mach numbers, is recognized as the main
result in combustion noise theory [13], and reads:
p ′ (r, t) = γ − 1 ∫ (
∂ ˙ω T
4πrc 2 1 V c ∂t
r 0 , t − |r − r 0|
¯c
)
dV(r 0 ) (2.3)
Here γ is the specific heat ratio, c 1 is the sound speed in the burnt gases and ˙ω T the volumetric
heat release rate in the combustion zone. Equation (2.3) proves the relation (2.2) obtained on a
more intuitive basis. The reactive source ∂ ˙ω T /∂t is an efficient acoustic radiator, a monopole,
that has been proved to be much more important when compared to the classical aerodynamic
source from Lighithill’s analogy, the double divergence of the Reynolds tensor, which is in fact
a quadrupolar term [32]. A dimensional analysis performed on the sound power produced
by these sources in combustion systems shows that the acoustic power from the combustion
process is of the order of 1/M 4 larger than the acoustic power generated by turbulence [102, 38].
Hence, for low Mach-number flows, the quadrupole source term can be neglected. Note that
to decrease pressure losses induced by high speed flows, most combustion systems indeed
operate at low Mach numbers [105].