THESE de DOCTORAT - cerfacs

30 Chapter 2: Computation of noise generated by combustion
yields
1
c 2 ∞
∂ 2 p
∂t 2 − ∂2 p
∂xi
2 = ∂2 ( ) ∂ ρui u j − 2 ρ
τ ij −
∂x i ∂x j ∂t 2 + 1 ∂ 2 p
c 2 ∞ ∂t 2 (2.32)
A density-pressure relation for the region far from noise sources (subindex ∞) is now added
∂ 2 p ∞
∂t 2
to the RHS of Eq. (2.32). This term ( 1
c 2 ∞
considered at rest and isentropic. Eq. (2.32) becomes
− ∂2 ρ ∞
) is in fact zero since at infinity the flow is
∂t 2
1
c 2 ∞
∂ 2 p
∂t 2 − ∂2 p
∂xi
2 = ∂2 ( ) ∂ ρui u j − 2 [
τ ij −
∂x i ∂x j ∂t 2 (ρ − ρ ∞ ) − 1 ]
c 2 (p − p ∞ )
∞
(2.33)
The last term of Eq. (2.33), discussed by Morfey [61] in 1973, is known as the excess of density
ρ e = (ρ − ρ ∞ ) − 1
c 2 ∞
(p − p ∞ ) and can be seen as an estimator of the deviation from the adiabatic
relation dρ = dp/c 2 ∞. Eventually, the Lightill analogy reads
1
c 2 ∞
∂ 2 p
∂t 2 − ∂2 p
∂xi
2 = ∂2 ( ) ∂ ρui u j − 2 ρ e
τ ij −
∂x i ∂x j ∂t 2 (2.34)
For an isentropic flow the excess of density ρ e is clearly zero. In such a case, unsteady turbulence
is responsible for noise generation. The term
∂2
∂x i ∂x ρui
( )
j
u j − τ ij is known as the aerodynamic
source of noise. Moreover, at high Reynolds number the contribution of the viscous
tensor τ ij is negligible in comparison to the Reynolds tensor ρu i u j . Reactive flows are, on the
contrary, of course non-isentropic. It will be seen in the following that in such flows the term
ρ e is responsible for most of the noise production. The purpose now is to express the material
derivative of the excess of density Dρ e
Dt
as function of the material derivative of density Dρ
Dt .
Dρ e
Dt
= ∂ρ e
∂t + u ∂ρ e
j = ∂ρ e
∂x j ∂t + ∂
∂u j
(ρ e u j ) − ρ e (2.35)
∂x j ∂x j
= ∂ρ e
∂t + ∂ (ρ e u j ) + ρ e Dρ
∂x j ρ Dt
(2.36)
Replacing by the definition of ρ e
[
D
(ρ − ρ ∞ ) − 1 ]
Dt
c 2 (p − p ∞ ) = ∂ρ e
∞
∂t + ∂ [
(ρ e u j ) + (ρ − ρ ∞ ) − 1 ] 1 Dρ
∂x j c 2 (p − p ∞ )
∞ ρ Dt
Dρ
Dt − 1 Dp
c 2 ∞ Dt = ∂ρ e
∂t + ∂ (ρ e u j ) + Dρ
∂x j Dt − ρ ∞ Dρ
ρ Dt − 1
c 2 ∞ρ (p − p ∞) Dρ
Dt
(2.37)
(2.38)