THESE de DOCTORAT - cerfacs

2.3 Hybrid computation of noise: Acoustic Analogies 33
˙ω ′ T : p ′ (x, t) = R( ˆp(x)e −iωt ) (2.44)
˙ω ′ T(x, t) = R( ˆ˙ω T (x)e −iωt ) (2.45)
where ω = 2π f and f is the oscillation frequency in Hertz. Equation (2.43) then becomes
∂ 2 ˆp
∂x 2 i
+ ω2 (γ − 1)
c 2 ˆp = −iω ˆ˙ω
∞
c 2 T (2.46)
∞
Equation (2.46) is commonly known as the Helmholtz equation. Expressions (2.43) and (2.46)
can be easily solved by means of the Green’s functions as shown in section 2.3.4.
2.3.3 Phillips’ Analogy
Lighthill’s theory considers the radiation of sound throughout a uniform medium caused by
stationary or moving sources. In spite of the fact that directivity is accounted for in Lighthill’s
formulation, this directivity is only caused by the convective pattern of sources (if any), which
is not completely true. In real flows the propagation medium is non-homogeneous at least in
the vicinity of the sources region (within about a wavelength or so) and most of the directivity
encountered in an acoustic field is originated by the acoustic-mean flow interactions that take
place in this non-homogeneous near field [31].
Phillips’ rearrangement of the Navier-Stokes equations is considered as the first significant
work that takes into account some of all possible acoustic-flow interactions. In this movingmedia
wave equation, acoustic-mean flow interactions are accounted for in the respective
acoustic operator. Combining the first and second law of thermodynamics results in
1 Ds
c p Dt = 1 Dp
γp Dt − 1 Dρ
ρ Dt
(2.47)
A new variable is introduced by Phillips [66] and is defined as π = 1 γ ln(p/p ∞). It can be
straightforward included in Eq. (2.47)
1 Ds
c p Dt = Dπ
Dt − 1 Dρ
ρ Dt
(2.48)
and by using the continuity equation