  Combustion instabilities are a major concern in the design of Liquid Rocket
Engines (LREs) and gas turbines. During this PhD work, several directions were
explored to understand and mitigate their effects. First, more efficient and
robust numerical methods for their prediction in complex combustors were
designed. In this matter, a novel type of modal expansion, named a frame
expansion and comparable to the classical Galerkin expansion, was introduced to
build more accurate acoustic Low-Order Models (LOMs), able to account for the
full geometrical complexity of industrial combustors. In particular, the frame
expansion is able to accurately represent the acoustic velocity field near
non-rigid-wall boundaries, a crucial ability that the Galerkin method lacks. An
entire class of novel numerical methods, based on the frame expansion, were
then designed and combined with the state-space formalism to build acoustic
networks of complex systems. The second ingredient in the prediction of
thermoacoustic instabilities is the flame dynamics modeling. This work dealt
with this problem, in the specific case of a cryogenic coaxial jet-flame
characteristic of a LRE. Flame dynamics driving phenomena were identified
thanks to three-dimensional Large Eddy Simulations (LES) of the Mascotte
experimental test rig where both reactants (CH4 and O2) are injected in
transcritical conditions. Several LES with harmonic modulation of the fuel
inflow at various frequencies and amplitudes were performed in order to
evaluate the flame response to acoustic oscillations and compute a Flame
Transfer Function (FTF). The stabilization of this flame in the near-injector
region, which is of primary importance on the overall flame dynamics, was also
investigated thanks to multi-physics two-dimensional Direct Numerical
Simulations (DNS), where a conjugate heat transfer problem is resolved at the
injector lip.
