Weakly Nonlinear and Numerical Analysis of Auto-Oscillatory Dynamics in a Solid Propellant Combustion Model

Michael Reardon,Jun Yu

doi:10.1155/2013/580561

Abstract

In solid combustion, a planar traveling flame wave may lose stability to oscillatory modes in certain parameter regimes. The loss of stability is commonly achieved through a Hopf bifurcation in which linearly unstable modes grow to fixed amplitudes. This situation has been well studied in models of strictly condensed combustion and has been observed experimentally. In this paper, we describe the onset and evolution of oscillatory modes arising in a free-interface model of solid propellant combustion. After a brief introduction, the structure and stability of small perturbations to a planar flame wave are then investigated. Multiple-scale expansions are utilized to describe changes in the amplitude and the phase of a single, weakly unstable mode, and the results are compared to numerical predictions. It is found the over much of the parameter space that the analytical and numerical solutions predict a supercritical Hopf bifurcation similar to those found in models of strictly condensed combustion.

Highlights

Introduction and Problem FormulationThe departure from steadyplanar combustion results from the amplification of small disturbances to the basic traveling wave structure due to instability
An increase in the temperature and propagation speed of the reaction zone is eventually counteracted by a decrease in the thermal energy content of the reagent immediately ahead of the front as well as increased abstraction of thermal energy from the reaction zone
A decrease in temperature and propagation speed will be followed by a relative buildup of thermal energy ahead of the front which is eventually reabsorbed into the reaction zone

Summary

Introduction and Problem Formulation

The departure from steadyplanar combustion results from the amplification of small disturbances to the basic traveling wave structure due to instability. The authors of [20] assume a small Mach number and that the pressure is large and approximately constant with the majority of the heat rise occurring in the solid phase preheat zone These assumptions effectively suppress acoustically driven forcing and instability found in other models of propellant combustion. The resulting instability is intrinsic and similar to that found in strictly condensed combustion With these assumptions the gas velocity in the x3 direction can be related to the regression rate of the sublimation surface via mass continuity yielding ũg = (1 − (ρs/ρg))(∂Φs/∂̃t) which is used to remove ũg from the model equations.

Weakly Nonlinear Analysis

A Comparison of Numerical and Analytical Results

Concluding Remarks