THESE de DOCTORAT - cerfacs

6.3 The 1D linearized Euler equations for compact systems 99
∂ρ
∂t + ∂ (ρu) = 0
∂x
(6.9)
∂ρu
+ ∂
∂t ∂x (ρu2 ) = − ∂p
∂x + Ϝ (6.10)
∂ρh t
+ ∂
∂t ∂x (ρuh t) = ˙Q + W k (6.11)
Integrating over the 1D domain between positions x 1 and x 2 results in
∫ x2
x 1
∫
∂ρ
∂t dx + x2
∫ x2
x 1
x 1
∂
∂x (ρu)dx = 0
∂ρ
∂t dx + (ρu) ∣ ∣∣∣
x 2
x 1
= 0
(6.12)
∫ x2
x 1
∂ρu
∂t dx + ∫ x2
x 1
∫ x2
∫
∂
∂p
∂x (ρu2 )dx +
x 1
∂x dx − x2
Ϝdx = 0
x 1
∫ x2
x
∂ρu
x 1
∂t dx + 2
x 2
(ρu2 )
∣ + p
∣ − F = 0
x 1 x 1
(6.13)
∫ x2
x 1
∂ρh t
∂t dx + ∫ x2
x 1
∂
∂x (ρuh t)dx −
∫ x2
x 1
∫ x2
x 1
∂ρh t
∂t dx + (ρuh t)
∫ x2
˙Qdx − W k dx = 0
x 1
x 2
∣ − ˙q − W = 0
x 1
(6.14)
where the following notations have been introduced: ∫ ˙Qdx = ˙q, ∫ Ϝdx = F and ∫ W k dx =
W. After linearizing and recalling that h t = c p T t , Eqs. (6.12), (6.13) and (6.14) become
( ¯ρu ′ + ρ ′ ū ) ∣ ∣ ∣∣
x 2
(
p ′ + ρ ′ ū 2 + 2 ¯ρūu ′) ∣ ∣ ∣∣
x 2
[
cp ¯T t ( ¯ρu ′ + ρ ′ ū) + ¯ρū(c p T ′ + ūu ′ ) ] ∣ ∣ ∣∣
x 2
x 1
= −
x 1
− F ′ = −
x 1
− ˙q ′ − W ′ = −
∫ x2
x 1
∂ρ ′
∂t
∫ x2
x 1
∫ x2
x 1
dx (6.15)
∂ ( ¯ρu ′ + ρ ′ ū ) dx (6.16)
∂t
∂ [
ρ ′ c p ¯T t + ¯ρ ( c p T ′ + ūu ′)] dx
∂t
(6.17)