Application of the Galerkin Method in the Solution of Non-linear Axial Combustion Instability Problems in Liquid Rockets

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Abstract
A nonlinear analysis of the stability of longitudinal combustion-driven oscillations in liquid propellant rocket motors is presented. A modified version of the Galerkin method is used to obtain solutions valid for moderate amplitude instabilities in rocket combustors having a low Mach number mean flow. Linear stability of a variety of liquid-rocket motors is investigated. Computed nonlinear results show that the resultant instability will exhibit a shock-type behavior with the number of shocks present in the system determined by the characteristics of the unsteady combustion process. These predictions are in qualitative agreement with available experimental data. Contrary to other solution techniques, the approach presented in this paper can predict both the transient and final periodic behavior of the instability. The final limit cycles are independent of the nature of the initial conditions. The relationship between the final limit cycles and the characteristics of the unsteady combustion process are discussed. Finally, the application of the Galerkin Method requires relatively little computation time and it offers considerable physical insight into the behavior of the instability.