THESE de DOCTORAT - cerfacs

6.4 Transmitted and reflected acoustic waves in isentropic nozzles 103
with a 1 = ¯M 1 + 1 2 ¯M 2 1 + 1
γ−1 , a 2 = ¯M 1 − 1 2 ¯M 2 1 − 1
γ−1 and a 3 = ¯M 2 + 1 2 ¯M 2 2 + 1
γ−1 . These
equations (6.31, 6.32 and 6.33) are solved for three unknowns out of six: A 1 , A 2 , B 1 , B 2 , E 1 and
E 2 . As a consequence, three of these six coefficients must be imposed on the problem.
6.4 Transmitted and reflected acoustic waves in isentropic nozzles
As already stated in the introductory part of this section, determining appropriate acoustic
boundary conditions for combustion chambers is crucial in order to correctly predict combustion
instabilities. These acoustic boundary conditions depend on the mean flow either upstream
(Compressors) or downstream (Turbines). As a first step, it is useful to consider the
mean flow in these regions as isentropic (Ŵ,
ˆ˙Q = 0) and isenthalpic ( ˆF) = 0. This means that
the combustion chamber is only surrounded by perfect diffusers and nozzles.
The unchocked Nozzle
A subsonic system needs to be acoustically constrained both at the inlet and at the outlet. In order
to acoustically characterize the nozzle independently from boundary conditions, the outlet
is fixed as totally non-reflecting (w − 2
= 0 → E 2 = 0). Eqs. (6.31) and (6.33) are re-arranged to
obtain the following general expressions [55]
R AA = B 1 /A 1 =
R SA = B 1 /E 1 =
( ) ( ) ¯M 1 − ¯M 2 ¯M 1 + ( 1 1 − 1 2 (γ − 1) )
¯M 1 ¯M 2
¯M 1 + ¯M 2
¯M 1 − 1 1 + 1 2 (γ − 1) ¯M 1 ¯M 2
( ) ¯M 2 − ¯M ( )
1
¯M 1
1 − ¯M 1 1 + 1 2 (γ − 1) ¯M 1 ¯M 2
( ) ( ) 2 ¯M 2 1 + ¯M ( T AA = 1 1 + 1 2
A 1 /A (γ − 1) )
¯M 2 2
2
1 + ¯M 2
¯M 1 + ¯M 2 1 + 1 2 (γ − 1) ¯M 1 ¯M 2
( ) ¯M 2 − ¯M ( 1
T SA = ¯M
)
1
2 2
E 1 /A 2
1 + ¯M 2 1 + 1 2 (γ − 1) ¯M 1 ¯M 2
(6.34)
(6.35)
(6.36)
(6.37)
The chocked Nozzle
The mass flow through a chocked nozzle is constrained by