Coaxial Jet Flame Subject to Long-Wavelength Acoustic Oscillations

Abstract

A linear model is developed to relate the response of energy release rate per unit volume to velocity and pressure oscillations in a liquid-propellant rocket engine. Coaxial injection of fuel (outer flow) and oxygen (inner flow) is considered with chemical reaction in a thin flame and turbulent mixing as the controlling factor in an axisymmetric diffusion flame. The combustion process has a characteristic time for mixing, producing a time lag in the energy release rate relative to pressure. The model applies to an individual injector but can be used to couple the multipoint combustion processes and wave dynamics for a multi-injector chamber. In particular, the impacts of long-wavelength oscillations of pressure and velocity in the surrounding chamber gas on the mixing and burning rates are determined. The results are developed in a way that feedback to the chamber oscillations can be determined for either a computational analysis or a perturbation analysis of the chamber wave dynamics. Both the steady-state behavior and the unsteady linear perturbation for the coaxial jet are studied using an axisymmetric Green's function. Flame temperature, flame shape, and burning rates are found as a function of mixture ratio and injector size (mixing time). The combustion response factor for oscillatory behavior is also determined.

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

“…Calculations have been made to analyze the solutions for the amplitude and phase as functions of time using the perturbation theory. The results are obtained using the oscillating coaxial jet turbulent diffusion flame model developed by Sirignano and Krieg [27]. Results are presented here considering methane and gaseous-oxygen propellants with a fuel-to-oxygen mass stoichiometric coefficient of 1∕4.…”

“…2 show that the in-phase component of the combustion response factor decreases as R o increases. Basically, this the Strouhal number effect [27]. Namely, the mixing time increases as Fig.…”

“…The model for an oscillating coaxial jet turbulent diffusion flame developed by Sirignano and Krieg [27] will be used to describe the mass burning rate and, accordingly, the rate of energy release per unit volume. Only the salient results of that theory are given here; the reader can pursue details in the reference.…”

“…The downstream distance for the coaxial jet flame is x. The functions V 1 x and V 2 x, the flame radial position R f x, and the flame temperature T f x are reported by Sirignano and Krieg [27].…”

“…= tangential velocity component, m∕s V 1 x = burning rate parameter defined by Sirignano and Krieg [27], m −1 V 2 x = integral quantity defined by Sirignano and Krieg [27] L IQUID-PROPELLANT-ROCKET-ENGINE (LPRE) combustion instability has been a longstanding natural phenomenon, which causes problems and creates the need for control. In a LPRE, the combustion process produces a very high-energy release rate per unit volume that, in many circumstances, has characteristic times that result in reinforcement of the acoustical oscillations and produces very high-pressure and -velocity amplitudes.…”