THESE de DOCTORAT - cerfacs

102 Chapter 6: Boundary conditions for low Mach number acoustic codes
R AA → ŵ − 1 /ŵ+ 1
→ with ŵ S 1 = 0
R SA → ŵ − 1 /ŵS 1
→ with ŵ + 1 = 0
T AA → ŵ + 2 /ŵ+ 1
→ with ŵ S 1 = 0
T SA → ŵ + 2 /ŵS 1
→ with ŵ + 1 = 0
T AS → ŵ S 2 /ŵ+ 1
→ with ŵ S 1 = 0
Table 6.1: Wave coefficients definition
w 1
+
Subsonic Region Subsonic Region
w
+
2
w 1
+
Subsonic Region Supersonic Region
w
+
2
M 1
w
-
w
-
1 2
M 2
M 1
w
-
w
-
1 2
M 2
w 1
S
w 2
S
w 1
S
w 2
S
Compact Domain
Compact Domain
(a)
(b)
Figure 6.2: Some possible configurations under the compact assumption
as function principally of the Mach number
¯M = ū/ ¯c.
A 1 = γ ˆp
2 |ŵ+ 1 |, B 1 = γ ˆp
2 |ŵ− 1 |, A 2 = γ ˆp
2 |ŵ+ 2 |, B 2 = γ ˆp
2 |ŵ− 2 |, E 1 = c p |ŵ S 1 |, E 2 = c p |ŵ S 2|
(6.30)
Here A and B have the dimension
(
of a
)
pressure signal (Pa) while E has (kJ/kgK) units. Considering
also a perfect gas ˆT¯T = ˆp¯p − ˆρ¯ρ , Eqs. (6.21) to (6.23) become
A 1 (1 + ¯M 1 )/ ¯c 1 − B 1 (1 − ¯M 1 )/ ¯c 1 − ρ 1 ¯c 1 ¯M 1 E 1 /c p =
A 2 (1 + ¯M 2 )/ ¯c 2 − B 2 (1 − ¯M 2 )/ ¯c 2 − ρ 2 ¯c 2 ¯M 2 E 2 /c p
(6.31)
(1 + ¯M 1 ) 2 A 1 + (1 − ¯M 1 ) 2 B 1 − ρ 1 ¯c 2 1 ¯M 2 1 E 1/c p + ˆ F =
(1 + ¯M 1 ) 2 A 2 + (1 − ¯M 2 ) 2 B 2 − ρ 2 ¯c 2 2 ¯M 2 2 E 2/c p
(6.32)
(1 + ¯M 1 )a 1 ¯c 1 A 1 + (1 − ¯M 1 )a 2 ¯c 1 B 1 − ρ 1 ¯c 3 1 ¯M 3 1 E 1/2c p + Ŵ/ ¯c 1 + ˆ˙q/ ¯c 1 =
(1 + ¯M 2 )a 3 ¯c 2 A 2 + (1 − ¯M 2 )a 4 ¯c 1 B 2 − ρ 2 ¯c 3 2 ¯M 3 2 E 2/2c p
(6.33)