THESE de DOCTORAT - cerfacs

6.3 The 1D linearized Euler equations for compact systems 101
where ŵ + and ŵ − represent the acoustic waves propagating respectively in the direction of the
flow at speed (ū + ¯c), and against the flow at speed ( ¯c − ū). ŵ S stands for the entropy wave
that is convected at the mean flow velocity (ū).
The compact assumption is realistic when the length of the system L is small if compared to
the respective wavelength λ. When considering acoustics in nozzles, it is shown in [41] that
this assumption is valid in nozzles for frequencies up to 700 Hz, which is already an important
bandwidth. When a wave passes through a compact domain, it is assumed then that only its
amplitude changes. The phase, on the contrary remains constant. This can be easier understood
from Fig. (6.1).
Wave
Compact Domain
Figure 6.1: A compact nozzle acting on a wave
As a result, the compact assumption consists simply in expressing all fluctuating quantities as
quasi-stationary (ω ≈ 0). Pressure, velocity and entropy fluctuations can be defined then as
function only of their amplitudes
ˆp
γ ¯p = 2ŵ+ 1 + 1 2ŵ− ≈ 1 2 |ŵ+ | + 1 2 |ŵ− | (6.27)
û
¯c = 1 2ŵ+ − 1 2ŵ− ≈ 1 2 |ŵ+ | − 1 2 |ŵ− | (6.28)
ŝ
= ŵ S ≈ |ŵ S |
c p
(6.29)
The reflection R and transmission T coefficients are now defined. The coefficient R AA is the
reflection coefficient that measures the amplitude of the outgoing acoustic wave w − 1
with respect
to an incoming wave w + 1
when it is assured that no incoming entropy waves are present
ŵ1
S = 0. R SA is also a reflection coefficient, but instead of R AA , relates the amplitude of an
outgoing acoustic wave w − 1
with respect to an incoming entropy wave ŵS 1
. In this case, no incoming
acoustic waves are present ŵ + 1
= 0. The transmitted waves T are defined in a similar
way. This definition is shown in table 6.1. All the possible waves ŵ that can appear in a 1D
system are shown in Fig. (6.2)
Six other coefficients are now defined, which are helpful to express Eqs. (6.21), (6.22) and (6.23)