THESE de DOCTORAT - cerfacs

96 Chapter 6: Boundary conditions for low Mach number acoustic codes
6.1 Introduction
The prediction of combustion instabilities in combustion chambers embedded in complex systems,
such as aeronautical engines, is extremely difficult due to all the different phenomena
that must be taken into account: the two way interaction between the flame and the most
representative turbulent length scales of the reacting and burnt gases, the two way interaction
between the flame and the radiated acoustic waves produced by this one, the possible coupling
between entropy waves (hot spots generated by the flame) and the acoustic waves produced
due to non-homogeneities in the mean flow and evidently the role played by the boundary
conditions (inlets, outlets and walls) as surfaces that directly interact with the acoustic, entropy
and vortical waves present in the system.
Different strategies have been developed over the years [78, 71, 48, 22, 77] in order to understand
the physics of all these interactions and to create methodologies to control or avoid them.
In the recent years, reacting and compressible Large Eddy-Simulation (LES) has shown its capability
to study the dynamics of turbulent flames [17, 69, 67, 90, 82, 68, 84]. Due to its intrinsic
nature (resolution of the unsteady 3D Navier-Stokes equations) LES is able to predict the interactions
between the flame, the turbulent flow and the acoustic modes of the chamber, and
for some particular configurations [56] to distinguish between stable and unstable combustion
systems. In despite of this, LES remains today very CPU demanding and its use for parametric
studies on aeronautical engines design is excluded. In order to overcome this constraint,
several methods have been proposed:
1) The Navier-Stokes equations are simplified to quasi-1D Linearized Euler equations LEE
[64, 5]. This strategy leads to consider only the main fluctuations on the flow (still remaining
small in comparison to mean flow values) and the flame, which is modeled as a source term.
This source term can be coupled to the acoustic field and be responsible for thermo-acoustic
instabilities or it can be decoupled and be responsible only for sound generation. Another important
characteristic of these methods is that they can also take into account possible couplings
of entropy or vorticity with acoustic modes.
2) The three dimensional wave equation is considered and resolved in the frequency domain.
The underlying numerical tools, commonly known as Helmholtz solvers, are very useful in
order to find the acoustic modes of a real combustor with its geometrical complexity. The
influence of the flame dynamics on the acoustic system can also be accounted for. Several
studies show their ability to predict combustion instabilities [56, 89, 62] in real combustion
chambers. Nevertheless, these methods still present some drawbacks. The wave equation
solved does not contain terms of convection, and as a consequence, the mean flow is neglected,
which can lead to wrong predictions of the propagation speed and wave length which in turn
leads to mis-estimations on the resonant frequencies and the grow rates of the acoustic modes.
Moreover, entropy and hydrodynamic modes are totally left out of the physics under study,
although they may play an important role.