THESE de DOCTORAT - cerfacs
THESE de DOCTORAT - cerfacs THESE de DOCTORAT - cerfacs
34 Chapter 2: Computation of noise generated by combustion 1 Ds c p Dt = Dπ Dt + ∂u j (2.49) ∂x j The variable π can also be introduced in the momentum equation. It yields Du i Dt + 1 ∂p − 1 ∂τ ij = 0 → Du i ∂π + c2 − 1 ∂τ ij = 0 (2.50) ρ ∂x i ρ ∂x j Dt ∂x i ρ ∂x j The material derivative D Dt momentum equation. is now applied to the Eq. (2.49) and the spatial derivative ∂ ∂x i to the D Dt ( ) 1 Ds − D c p Dt Dt ( ) Dπ − D ( ) ∂uj = 0 (2.51) Dt Dt ∂x j ∂ Du i ∂x i Dt + ∂ ( c 2 ∂π ) − ∂ ( ) 1 ∂τ ij = 0 (2.52) ∂x i ∂x i ∂x i ρ ∂x j The operation D Dt ( ∂ ∂x j ) is not commutative. As a consequence D Dt ∂ ∂x i = ∂ D ∂x i Dt = ∂ ∂x i D Dt = D Dt ∂2 ∂ + 2 u j (2.53) ∂t∂x i ∂x i ∂x j ∂2 ∂x i ∂t + ∂ ( ) ∂ u j = ∂2 ∂x i ∂x j ∂x i ∂t + ∂u j ∂ ∂ + 2 u j (2.54) ∂x i ∂x j ∂x i ∂x j ∂ ∂x i + ∂u j ∂x i ∂ ∂x j (2.55) Replacing the index j to i in Eq. (2.49) and accounting for the previous relation, Eq. (2.51) becomes D Dt ( ) 1 Ds − D2 π c p Dt Dt 2 − ∂ Du j ∂x i Dt + ∂u j ∂u i = 0 (2.56) ∂x i ∂x j Finally, Eqs. (2.52) and (2.56) are summed leading to the final expression for Phillip’s analogy. D 2 π Dt 2 − ∂ ( c 2 ∂π ) = ∂u j ∂u i − ∂ ( ) 1 ∂τ ij + D ∂x i ∂x i ∂x i ∂x j ∂x i ρ ∂x j Dt ( ) 1 Ds c p Dt (2.57) In order to have an explicit expression for Phillip’s analogy with all the possible sources in a
2.3 Hybrid computation of noise: Acoustic Analogies 35 reacting flow, the energy equation Eq. (2.28) is combined with Eq. (2.47) leading to [ 1 Ds (γ − 1) = c p Dt ρc 2 Phillip’s equation for reacting flows reads D 2 π Dt 2 − ∂ ( c 2 ∂π ) = ∂u j ∂u i − ∂ ∂x i ∂x i ∂x i ∂x j + D Dt ] ∂J ˙ω T + ∑ h k k − ∂q i + τ ij : ∂u i + ˙Q + D (ln r) (2.58) ∂x k i ∂x i ∂x j Dt ∂x i ( 1 ρ [ ( (γ − 1) ρc 2 ) ∂τ ij ∂x j ∂J ˙ω T + ∑ h k k − ∂q i + τ ij : ∂u i + ˙Q (2.59) ∂x k i ∂x i ∂x j Dt 2 (ln r) + D2 Equation (2.59) was introduced by Chiu & Summerfield [14] and Kotake [39]. This expression accounts for some acoustic-flow interactions, since gradients of the sound velocity c are included in the acoustic operator. It has several advantages compared to Lighthill’s analogy, since mean temperature and density of the propagation medium are not assumed homogeneous in space, which is clearly the case for reacting flows. The term responsible for noise generation due to hydrodynamic fluctuations is now defined as ∂u j ∂x i ∂u i ∂x j . The density contribution in this term has been removed, contrary to its form in Lighthill’s analogy ∂2 ∂x i ∂x j (ρu i u j ). It is clear then that for low Mach numbers there is not anymore ambiguity in this source definition since acoustic-flow interactions are not anymore included in this term [4]. Nevertheless, Lilley [52] demonstrated that when the Mach number is not negligible, acoustic-flow interactions are still contained in the compressible part of u i . Another interesting remark done by Bailly [4] is that in this equation the source responsible for indirect noise (the acceleration of density inhomogeneities) no longer appears. It is recommended therefore that if indirect noise is not negligible and no strong changes in the mean flow are present, Lighthill’s analogy should be the formulation under consideration. )] Simplifying Phillips’ Analogy Phillips’ equation (Eq. 2.59), as Lighthill’s formulation, is an exact rearrangement of the fluid dynamics equations. Similar assumptions as those made in section 2.3.2 are considered here. It is assumed then that the diffusion of species J k , the dissipation function τ ∂u i ∂x j , external energy sources ˙Q, heat fluxes ∂q i ∂x i , the viscosity stresses τ ij , changes in the molecular weight of the mixture D Dt (ln r) and the Reynolds stress tensor ρu iu j are neglected in comparison to the monopolar source of noise ∂ ˙ω T ∂t . The simplified Phillips’ equation then reads:
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34 Chapter 2: Computation of noise generated by combustion<br />
1 Ds<br />
c p Dt = Dπ<br />
Dt + ∂u j<br />
(2.49)<br />
∂x j<br />
The variable π can also be introduced in the momentum equation. It yields<br />
Du i<br />
Dt + 1 ∂p<br />
− 1 ∂τ ij<br />
= 0 → Du i ∂π<br />
+ c2 − 1 ∂τ ij<br />
= 0 (2.50)<br />
ρ ∂x i ρ ∂x j Dt ∂x i ρ ∂x j<br />
The material <strong>de</strong>rivative D Dt<br />
momentum equation.<br />
is now applied to the Eq. (2.49) and the spatial <strong>de</strong>rivative<br />
∂<br />
∂x i<br />
to the<br />
D<br />
Dt<br />
( ) 1 Ds<br />
− D c p Dt Dt<br />
( ) Dπ<br />
− D ( ) ∂uj<br />
= 0 (2.51)<br />
Dt Dt ∂x j<br />
∂ Du i<br />
∂x i Dt + ∂ (<br />
c 2 ∂π )<br />
− ∂ ( ) 1 ∂τ ij<br />
= 0 (2.52)<br />
∂x i ∂x i ∂x i ρ ∂x j<br />
The operation D Dt<br />
(<br />
∂<br />
∂x j<br />
)<br />
is not commutative. As a consequence<br />
D<br />
Dt<br />
∂<br />
∂x i<br />
=<br />
∂ D<br />
∂x i Dt =<br />
∂<br />
∂x i<br />
D<br />
Dt = D Dt<br />
∂2 ∂<br />
+ 2<br />
u j (2.53)<br />
∂t∂x i ∂x i ∂x j ∂2<br />
∂x i ∂t + ∂ ( )<br />
∂<br />
u j = ∂2<br />
∂x i ∂x j ∂x i ∂t + ∂u j ∂ ∂<br />
+ 2<br />
u j (2.54)<br />
∂x i ∂x j ∂x i ∂x j<br />
∂<br />
∂x i<br />
+ ∂u j<br />
∂x i<br />
∂<br />
∂x j<br />
(2.55)<br />
Replacing the in<strong>de</strong>x j to i in Eq. (2.49) and accounting for the previous relation, Eq. (2.51) becomes<br />
D<br />
Dt<br />
( ) 1 Ds<br />
− D2 π<br />
c p Dt Dt 2 − ∂ Du j<br />
∂x i Dt + ∂u j ∂u i<br />
= 0 (2.56)<br />
∂x i ∂x j<br />
Finally, Eqs. (2.52) and (2.56) are summed leading to the final expression for Phillip’s analogy.<br />
D 2 π<br />
Dt 2 − ∂ (<br />
c 2 ∂π )<br />
= ∂u j ∂u i<br />
− ∂ ( ) 1 ∂τ ij<br />
+ D ∂x i ∂x i ∂x i ∂x j ∂x i ρ ∂x j Dt<br />
( ) 1 Ds<br />
c p Dt<br />
(2.57)<br />
In or<strong>de</strong>r to have an explicit expression for Phillip’s analogy with all the possible sources in a
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