Hopf Bifurcation Analysis of the Combustion Instability in a Liquid Rocket Engine

The bifurcation process of self-sustained combustion instability pressure perturbations in a liquid rocket combustor is investigated based on the Helmholtz equations and a pressure dependent flame describing function. The modal frequency and growth rates are numerically resolved by the commercial software COMSOL multiphysics. Validation of the numerical approach is firstly conducted on a Rijke tube combustor, and a supercritical bifurcation for the first longitudinal mode is observed. The bifurcation diagrams for the first transverse mode for different time delays and gain index of the flame describing function are analyzed. Only the supercritical bifurcation presents for this configuration. The trajectory of Hopf points and the bifurcation diagram feature period motions with increasing the time delay. The effect of flame length distributions on the bifurcation diagrams is analyzed by considering a non-uniform flame length distribution model. Results show that the distribution has a large impact on the bifurcation process, e.g., the first transverse mode is more unstable for the non-uniform distribution. Finally, a subcritical bifurcation is found when a more complicated flame describing function is considered; the bistable region presents and the condition for this is discussed.