THESE de DOCTORAT - cerfacs

40 Chapter 2: Computation of noise generated by combustion
G(x, t|y, τ) = − c ∞H [(t − τ) − r/c ∞ ]
2π √ c 2 ∞(t − τ) 2 − r 2 (2.79)
where H is the Heavyside function, H(u) = 1 if u > 0 and H(u) = 0 otherwise. Another useful
approach in combustion noise is the study of noise radiation under spectral analysis. Green’s
functions G are therefore expressed in the frequency domain by applying a Fourier transform
defined as follows
Ĝ(x|y, ω) =
∫ ∞
−∞
G(x, t|y, τ)e −iωt dt (2.80)
By doing so, both 3D and 2D Green’s functions in terms of frequency are obtained:
3D spectral Green’s function
Ĝ(x|y, ω) = e−ikr
4πr
(2.81)
2D spectral Green’s function
Ĝ(x|y, ω) = i 4 H(2) 0
(kr) (2.82)
where H (2)
0
(kr) is the Hankel function of second kind and order 0 and r = |x − y|. These
Green’s functions satisfy the wave equation written in the frequency domain
( ∂
2
∂x 2 i
)
+ ω2
c 2 Ĝ(x|y, ω) = δ(x − y) (2.83)
∞