THESE de DOCTORAT - cerfacs
THESE de DOCTORAT - cerfacs
THESE de DOCTORAT - cerfacs
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116 Chapter 6: Boundary conditions for low Mach number acoustic co<strong>de</strong>s<br />
δ C<br />
!!"# !!"$ ! !"$ !"#<br />
$"*<br />
$")<br />
$"(<br />
&$!<br />
&!!<br />
%+!<br />
δ C<br />
¯ρ (kg/m 3 )<br />
$"'<br />
$"&<br />
$"%<br />
$"#<br />
$"$<br />
¯T (K )<br />
%*!<br />
%)!<br />
%(!<br />
%'!<br />
%&!<br />
$<br />
!!"# !!"$ ! !"$ !"#<br />
X (m)<br />
(a) Mean <strong>de</strong>nsity<br />
%%!<br />
X (m)<br />
(b) Mean temperature<br />
Figure 6.13: Typical Profiles<br />
( 1¯ρ<br />
∂ ¯c 2<br />
∂x + ū ∂ū<br />
¯ρ ∂x + ¯c2¯ρ<br />
) (<br />
∂ ∂ū<br />
ˆρ +<br />
∂x ∂x + ū ∂ )<br />
(<br />
∂ ¯p<br />
û + (γ − 1) ¯T<br />
∂x<br />
1¯p ∂x + ∂ )<br />
ŝ − jωû − ̂Ϝ = 0 (6.59)<br />
∂x<br />
From Eq. (6.13), it is known that F(x) = ∫ Ϝdx, or in other words Ϝ = ∂F<br />
∂x<br />
, where F is given<br />
by Eq. (6.51). Since in the Quasi-1D LEE F is function of x, Eq. (6.50) can be re-stated as<br />
∫ ∂<br />
F(x) = ∆p t + ∆e k + (ρ(x)u(x)) dx (6.60)<br />
∂t<br />
∫ ∂<br />
F(x) = p t (x) − p t,1 + e k (x) − e k,1 + (ρ(x)u(x)) dx (6.61)<br />
∂t<br />
Ϝ(x) = ∂<br />
∂x [p t(x) + e k (x)] + ∂ [ρ(x)u(x)] (6.62)<br />
∂t<br />
In seek of readibility, the argument (x) is dropped. Linearizing and consi<strong>de</strong>ring harmonic oscillations:<br />
ˆϜ = ∂ ˆp t<br />
∂x + ∂ (<br />
¯ρūû + 1 )<br />
∂x 2 ˆρū2 − iω ( ¯ρû + ˆρū) (6.63)<br />
In or<strong>de</strong>r to allow a fair comparison with the analytical solution (section 6.6.1), it is necessary<br />
that the computation is ma<strong>de</strong> in such a way that π ′ c = 0. This means that
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