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.

Coaxial Jet Flame Subject to Long-Wavelength Acoustic Oscillations

“…The wave dynamics equation [16,22] includes a term with the first time derivative of E, the rate of energy conversion per unit volume. This quantity E will oscillate when acoustic oscillations occur.…”

“…Under oscillation, it will create kinematic entropy waves as well as directly modifying the longer-wavelength acoustic oscillations. Those shorterlength kinematic waves have been filtered in the wave-dynamics equation [16,22] under the assumption that turbulent mixing eliminates quickly those short waves with higher gradients. We follow that assumption here and neglect entropy waves.…”

“…In a more recent work, Sirignano and Krieg [22] use the Sirignano and Popov [16] model as the basis for study of nonlinear, transverse-mode, LPRE combustion instability via a two-time-variable perturbation expansion in an amplitude parameter. Both triggered and spontaneous instability domains are studied.…”

“…Here, in this paper, the goal is to develop via closed-form analysis a model of the oscillatory coaxial jet that can be used to provide the energy release rate for the coupling of the combustion process with the perturbation model of the chamber wave dynamics. A linear treatment of combustion has been shown [22,23] to be sufficient for matching a nonlinear transverse wave oscillation in a circular LPRE chamber to third order in an expansion in the amplitude parameter. Therefore, both linear and nonlinear (triggered) combustion instabilities may be studied using the model developed here.…”

“…Although this work was originally motivated by the need for a simplified coaxial, oscillatory, diffusion-flame model for the aforementioned combustion-instability perturbation analysis [22], the model is capable of wider use.…”