THESE de DOCTORAT - cerfacs

114 Chapter 6: Boundary conditions for low Mach number acoustic codes
(1 + ¯M 1 )/ ¯c 1 A 1 − (1 − ¯M 1 )/ ¯c 1 B 1 = (1 + ¯M 2 )/ ¯c 2 A 2 − (1 − ¯M 2 )/ ¯c 2 B 2 (6.55)
¯π c b 1 A 1 + ¯π c b 2 B 1 = b 3 A 2 + b 4 B 2 (6.56)
with b 1 = ( 1 + 0.5 ¯M 2 1 + ¯M 1
)
, b2 = ( 1 + 0.5 ¯M 2 1 − ¯M 1
)
, b3 = ( 1 + 0.5 ¯M 2 2 + ¯M 2
)
and b4 =
(
1 + 0.5 ¯M 2 2 − ¯M 2
)
In addition the acoustic condition at the inlet ( ˆp = 0) must be taken into account. This is done
simple by doing A 1 = B 1 e iπ = −B 1 . The coefficient B 2 which is related to the forced wave at
the outlet must be given by the user. The system is resolved then to find the values of A 1 , B 1
and A 2 .
6.6.2 The mean flow in SNozzle
The main effect of a compressor is to add energy to the flow by increasing its total pressure ¯p t .
A typical profile of ¯p t corresponding to a compressor of thickness δ c is shown in Fig. 6.11(a).
This jump of total pressure is built from the following hyperbolic tangent function
[ ( )]
k(x − xm )
¯p t = ¯p t1 + 0.5 ¯p t1 ( ¯π c − 1) 1 + tanh
x m − x ups
(6.57)
which satisfies the isentropic expression:
¯π c = ¯p ( ) γ/(γ−1)
t2 ¯T t2
=
(6.58)
¯p t1
¯T t1
The indices [1] [2] stand for upstream and downstream of the compressor respectively. The
Mach number profile ¯M for the case in which ¯M 2 = 0.025 is shown in Fig. 6.11(b).
All mean quantities satisfy the Euler equations for steady and isentropic flows. Mean profiles
of pressure, velocity, density and temperature are shown in Figs. 6.12 and 6.13 for the case in
which ¯π c = 2 and ¯M 2 = 0.025.
6.6.3 Introducing π c in the momentum equation
Let us recall the momentum equation of the Quasi-1D LEE system. It reads