THESE de DOCTORAT - cerfacs

100 Chapter 6: Boundary conditions for low Mach number acoustic codes
Considering now harmonic oscillations (φ ′ = ˆφe −iωt ) leads to:
[
cp ¯T t ( ¯ρû + ρ ′ ū) + ¯ρū(c p ˆT + ūû) ] ∣ ∣ ∣∣
x 2
x 2 ∫ x2
( ¯ρû + ˆρū)
∣ = iω ˆρdx (6.18)
x x 1 1
( ˆp + ˆρū 2 + 2 ¯ρūû ) ∣ x 2
∫ x2
∣∣ − F ˆ = iω ( ¯ρû + ˆρū) dx (6.19)
x x 1 1
∫ x2
[
− ˆ˙q − Ŵ = iω ˆρcp ¯T t + ¯ρ ( c p ˆT + ūû )] dx (6.20)
x x 1 1
The contribution of the RHS terms on the above equations goes to zero for compact regions
(x 1 → x 2 ). Equations (6.18), (6.19) and (6.20) are then simplified to
¯ρ 1 û 1 + ˆρ 1 ū 1 = ¯ρ 2 û 2 + ˆρ 2 ū 2 (6.21)
ˆp 1 + ˆρ 1 ū 2 1 + 2 ¯ρ 1ū 1 û 1 + ˆ F = ˆp 2 + ˆρ 2 ū 2 2 + 2 ¯ρ 2 ū 2 û 2 (6.22)
c p ¯T t1 ( ¯ρ 1 û 1 + ˆρ 1 ū 1 ) + ¯ρ 1 ū 1 (c p ˆT 1 + ū 1 û 1 ) + ˆ˙q + Ŵ = c p ¯T t2 ( ¯ρ 2 û 2 + ˆρ 2 ū 2 ) + ¯ρ 2 ū 2 (c p ˆT 2 + ū 2 û 2 )
(6.23)
where the indices 1 and 2 denote the regions upstream or downstream of any infinitely thin
compact element; ˆ˙q, Fˆ
and Ŵ being the total heat, force and work associated to this element.
Equations (6.21) - (6.23) generalize the results of Dowling [22] to the case where the compact
interface between states 1 and 2 generates a force Fˆ
and the associated work Ŵ.
6.3.1 The transmitted and reflected waves
In the frequency domain, acoustic and entropy waves ŵ are defined by their amplitude |ŵ| and
phase φ, so that ŵ = |ŵ|e iφ . They read [55]
ŵ + =
ŵ − =
ŵ S =
ˆp
γ ¯p + û
¯c = |ŵ+ |e iωx/ ¯c(1+ ¯M)
(6.24)
ˆp
γ ¯p − û
¯c = |ŵ− |e −iωx/ ¯c(1− ¯M)
(6.25)
ˆp
γ ¯p − ˆρ¯ρ = |ŵS |e iωx/ū (6.26)