Two-Time-Variable Perturbation Theory for Liquid-Rocket Combustion Instability

Abstract

Nonlinear, transverse-mode, liquid-propellant-rocket-motor combustion instability is examined for the first time via a two-time-variable perturbation expansion in an amplitude parameter. Both triggered and spontaneous instability domains are studied. A specific coaxial multi-injector example demonstrates the matching process between wave dynamics and injection/combustion mechanisms. The combustion has a characteristic time for mixing, producing a time lag in the energy release rate relative to pressure. The coupled combustion process and wave dynamics are calculated for the first tangential mode. Two first-order ordinary differential equations are developed and solved for the amplitude and phase angle in the slow time. Limit cycles and transient behaviors are resolved. Nonlinear triggering is predicted in certain operational domains; above a critical initial amplitude, the amplitude grows; otherwise, it decays with time. A linear representation of the combustion process suffices to balance nonlinear nozzle damping. This perturbation approach provides better physical understanding than a computational fluid dynamics approach and allows lower-cost computation to determine trends over the key parameter domains. Moving a higher fraction of the propellant flow away from the chamber center has a destabilizing effect on the tangential mode. A most stable Mach-number value is deduced. The reduction to two governing ordinary differential equations benefits future optimization and control analyses.