THESE de DOCTORAT - cerfacs

36 Chapter 2: Computation of noise generated by combustion
D 2 π
Dt 2 − ∂ (
c 2 ∂π )
= ∂ ( ) γ − 1
∂x i ∂x i ∂t ρc 2 ˙ω T
(2.60)
All quantities are decomposed into their mean and fluctuating parts. Small acoustic perturbations
with respect to the mean pressure are considered, so that π = p ′ /(γ ¯p). Another assumption
usually made is to consider the Mach number characterizing the flow ¯M as small. In
doing so, the convective part of the material derivative vanishes and D Dt ≈ ∂ ∂t
. This version of
Phillips’ equation would read
1 ∂ 2 p ′
γ ¯p ∂t 2 − ∂ ( c
2
∂p ′ )
= ∂ ( ) γ − 1
∂x i γ ¯p ∂x i ∂t ρc 2 ˙ω T
(2.61)
It should be noted that no assumptions about uniformity of the propagation medium until
now have been done. Nevertheless this is necessary if solvable a Phillips equation is sought.
It is then assumed that the fluctuations of speed of sound can be neglected. Note anyway that
changes in the mean flow are still considered ( ∂ ¯c
∂x i
̸= 0)
∂ 2 p ′
∂t 2
− ∂ ( )
¯c 2 ∂p′ = (γ − 1) ∂ ˙ω′ T
∂x i ∂x i ∂t
(2.62)
Equation (2.62), as done for Lighthill’s case, can also be expressed in the frequency domain.
Usually, a spectral evaluation of acoustics presents several advantages. In linear acoustics, for
instance, it is possible to study the contribution of each frequency separately since they do not
interact which each other. In doing so, acoustic boundary conditions can be well characterized
by an acoustic property called impedance, which most of the time is a function of the frequency
of oscillation. Dealing with boundary conditions is more challenging when considering
the time domain formalism. Another advantage of a frequential definition is that in a spectral
evaluation there is no necessity of any transient computation to reach a stationary state. Applying
harmonic perturbations (Eqs. 2.44 and 2.45) on p ′ and ˙ω
T ′ results in a Helmholtz equation
written as
(
∂
¯c 2 ∂ ˆp )
+ ω 2 ˆp = −iω(γ − 1) ˆ˙ω T (2.63)
∂x i ∂x i
There is not known analytical Green’s functions associated with this wave operator [4], and so
no integral formulation giving the far field pressure. Most of the time this equation is solved
numerically as in the present study.