THESE de DOCTORAT - cerfacs

38 Chapter 2: Computation of noise generated by combustion
1 ∂ 2 p ′
c ∞ ∂τ 2 − ∂2 p ′
∂y 2 i
= q(y, τ) (2.69)
can be obtained by multiplying Eq. (2.68) by p(y, τ) and Eq. (2.69) to G(x, t|y, τ), substracting
them and finally integrating τ between t 0 and t, and y over V. This leads to:
∫ t
t 0
∫V
∫ t
t 0
∫V
[( 1 ∂ 2 ) (
G
c 2 ∞ ∂τ 2 − ∂2 G
1
∂y 2 − δ(t − τ)δ(x − y) p ′ ∂
(y, τ) −
2 p ′
c
i
∞ ∂τ 2 − ∂2 p ′ )
]
∂y 2 − q G(x, t|y, τ) dydτ = 0
i
(2.70)
[
p ′ 1 ∂ 2 G
c 2 ∞ ∂τ 2 − p′ ∂2 G
∂y 2 − G 1 ∂ 2 p ′
c
i
2 ∞ ∂τ 2 + G ∂2 p ′ ]
∂y 2 + qG dydτ − p ′ (x, t) = 0 (2.71)
i
Reorganizing the terms
p ′ (x, t) =
∫ t
t 0
∫V
∫ t
∫ (
qGdydτ − p ′ ∂2 G
to V ∂y 2 i
And finally integrating by parts results in
− G ∂2 p ′ )
∂y 2 − dydτ + 1 ∫ t
(
c
i
2 p
∞ t 0
∫V
′ ∂2 G
∂τ 2 − G ∂2 p ′ )
∂τ 2 dydτ
(2.72)
p ′ (x, t) =
∫ t
t 0
∫V
∫ t
(
qGdydτ − p
t 0
∫S
′ ∂G )
− G ∂p′ n i dσdτ − 1 [∫ (
∂y i ∂y i c 2 p ′ ∂G ) ]
∞ V ∂τ − G ∂p′ dy
∂τ
t 0
(2.73)
The first integral is the convolution of the source q with the pulse response G, the Green’s
function. The second integral represents the effect of differences between the actual physical
boundary conditions on the surface S and the conditions applied to G. When G satisfies the
same locally reacting linear boundary conditions as the actual field, this surface integral vanishes.
In this case, the Green’s function under consideration is called ‘taylored’. On the other
hand, this surface integral can also vanishe when no boundaries conditions are applied (freefield).
Finally, the last integral represents the contribution of the initial conditions at t 0 to the
acoustic field and disappears if the causality condition is applied. Free-field noise computation
can then be performed by the following expression
p ′ (x, t) =
∫ t
t 0
∫V
q(y, τ)G(x, t|y, τ)dydτ (2.74)
If noise is generated by the unsteady heat release rate induced by combustion and if the prop-