THESE de DOCTORAT - cerfacs

6.2 The quasi 1D Linearized Euler Equations - SNozzle 97
The reliability of all these strategies is subjected to the precision at which boundary conditions
are modeled. Useful methods have been developed to model the acoustic impedances of inlets/outlets
and they have shown to be successful for some particular cases [41]. The general
idea is to solve the perturbation equations in a quasi-1D domain, which represents the regions
either upstream (for the diffuser) or downstream (for the nozzle) of the combustion chamber.
Feeding this domain with an incoming acoustic wave, the equivalent impedance can be assessed
as soon as the outgoing wave is computed from the perturbation equations. Such a
strategy was detailed in [41] in the case of perturbations propagating in an isentropic mean
flow. In this chapter, it is generalized to the case where neither the enthalpy nor the entropy or
the mean flow are constant.
This chapter is organized as follows. First, the quasi 1D LEE system is presented. Compared
to [64], these equations contain additional terms that account for the compression or expansion
stages in the system. Second, analytical solutions are derived to validate quasi 1D LEE numerical
tool in three specific cases: isentropic compact nozzles [55], one dimensional flames and
one dimensional compressors.
6.2 The quasi 1D Linearized Euler Equations - SNozzle
After applying the quasi 1D approximation, i.e., neglecting the in-plane correlation terms, the
quasi-1D LEE for mass, momentum and entropy are written in non-conservative form [64]
∂ρ
∂t + u ∂ρ
∂x + ρ ∂u
∂x + ρu ∂S
S ∂x = 0 (6.1)
ρ ∂u
∂t
+ ρu
∂u
∂x + ∂p
∂x − Ϝ = 0 (6.2)
ρT ∂s ∂s
+ ρTu
∂t ∂x − ˙Q − W k = 0 (6.3)
where S, ˙Q, Ϝ and W k stand for the cross section area, an external energy source, the force
added by the compressor, and the work done by the compressor respectively. Except for the
mass balance equation, which now contains an additional term accounting for the transversal
section S, it is observed from Eqs. (6.1) to (6.3) that the quasi 1D LEE system is essentially the
1D restriction of the 3D Euler equations. Considering harmonic perturbations (φ ′ = ˆφe −iωt ),
where φ(x, t) = ¯φ(x) + φ ′ (x, t), the quasi 1D LEE system in the frequency domain becomes
( ∂ū
∂x + ū ∂
∂x + ū
S
) (
∂S ∂ ¯ρ
ˆρ +
∂x ∂x + ¯ρ ∂
∂x + ¯ρ S
)
∂S
û − jω ˆρ = 0 (6.4)
∂x