THESE de DOCTORAT - cerfacs

104 Chapter 6: Boundary conditions for low Mach number acoustic codes
ṁ = p t,1
√
γtTt,1
A cr γ
( γ + 1
2
) −
γ+1
2(γ−1)
(6.38)
where A cr is the critical (throat) section. Derivating the logarithm of Eq. (6.38) results in a
simple expression [41]
û 1
− γ (
¯c 1 2 M ˆp1
1 + 1 )
γ ¯p 1 2 ¯M ˆρ 1
1 = 0 (6.39)
¯ρ 1
which if combined with the expression of waves (section 6.3.1) results in
R AA = 1 − 1 2 (γ − 1) ¯M 1
1 + 1 2 (γ − 1) ¯M 1
(6.40)
R SA =
1
2 ¯M 1
1 + 1 2 (γ − 1) ¯M 1
(6.41)
T AA = 1 + 1 2 (γ − 1) ¯M 2
1 + 1 2 (γ − 1) ¯M 1
(6.42)
T SA =
1
2 ( ¯M 2 − ¯M 1 )
1 + 1 2 (γ − 1) ¯M 1
(6.43)
It is important to emphasize here that since the nozzle is chocked, the flow after the throat
is supersonic. As a consequence acoustic waves should not be imposed at the outlet of the
computational domain. Doing so would result in an ill-posed problem.
6.4.1 Acoustic Response of Chocked and Unchocked Nozzles
As stated before, the compact assumption is very useful to study the acoustic response of a nozzle
when the acoustic wave length λ is much larger than the nozzle length. When acoustic and
entropy disturbances fluctuate at large frequencies, the jump conditions showed previously become
unsuited since they do not account for phase changes across the nozzle. In order to study
the influence of ‘non-compactness’, different methods are found in the literature [12, 59, 55]. In
[55] it is stated that the main influence of the nozzle finiteness is to alter the phase between the
two wave trains. This can be modeled by considering that the mean flow velocity changes linearly
through the nozzle. Nevertheless, this assumption is too restrictive for real systems. For
reliable computations, it is then mandatory to resolve the mean flow in the nozzle. If non-linear
acoustics and complex geometries are considered, the only possibility would be to resolve the
complete set of Euler equations in the time domain. Nevertheless, three dimensional Euler
computations are computationally expensive and the consideration of non-linearities is not al-