THESE de DOCTORAT - cerfacs
THESE de DOCTORAT - cerfacs 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
6.6 Transmitted and Reflected Waves through an ideal Compressor: the enthalpy jump case 115 δ C 0 0.1 0.2 0.3 0.4 ¯pt (Pa) x 10 5 2 1.5 1 ¯M 0.05 0.04 0.03 0.02 δ C 0.5 0.01 0 0 0.1 0.2 0.3 0.4 X (m) 0 X (m) (a) Total Pressure ¯p t . ¯π c = 2 (b) Mach number ¯M Figure 6.11: Mean Flow. Typical Profiles . δ C !!"# !!"$ ! !"$ !"# $* δ C ¯p (Pa) # $"% $"' $"& $"# $ !"% !!"# !!"$ ! X (m) !"$ !"# #"# ()$!* (a) Mean pressure ū (m/s) $) $( $' $& $# $$ $! % X (m) (b) Mean velocity Figure 6.12: Typical Profiles
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6.6 Transmitted and Reflected Waves through an i<strong>de</strong>al Compressor: the enthalpy jump case 115<br />
δ C<br />
0 0.1 0.2 0.3 0.4<br />
¯pt (Pa)<br />
x 10 5<br />
2<br />
1.5<br />
1<br />
¯M<br />
0.05<br />
0.04<br />
0.03<br />
0.02<br />
δ C<br />
0.5<br />
0.01<br />
0<br />
0 0.1 0.2 0.3 0.4<br />
X (m)<br />
0<br />
X (m)<br />
(a) Total Pressure ¯p t . ¯π c = 2 (b) Mach number ¯M<br />
Figure 6.11: Mean Flow. Typical Profiles<br />
.<br />
δ C<br />
!!"# !!"$ ! !"$ !"#<br />
$*<br />
δ C<br />
¯p (Pa)<br />
#<br />
$"%<br />
$"'<br />
$"&<br />
$"#<br />
$<br />
!"%<br />
!!"# !!"$ !<br />
X (m)<br />
!"$ !"#<br />
#"# ()$!*<br />
(a) Mean pressure<br />
ū (m/s)<br />
$)<br />
$(<br />
$'<br />
$&<br />
$#<br />
$$<br />
$!<br />
%<br />
X (m)<br />
(b) Mean velocity<br />
Figure 6.12: Typical Profiles
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