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

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.

[1]  Christopher M. Danforth,et al.  Complex dynamic behavior during transition in a solid combustion model , 2009 .

[2]  A. Merzhanov,et al.  Three-dimensional modes of unsteady solid-flame combustion , 2003 .

[3]  Hans G. Kaper,et al.  Bifurcation of Pulsating and Spinning Reaction Fronts in Condensed Two-Phase Combustion , 1984 .

[4]  S. Margolis.  An Asymptotic Theory of Condensed Two-Phase Flame Propagation , 1983 .

[5]  Yiyong Huang,et al.  Boron-based pronucleophiles in catalytic (asymmetric) C(sp3)–allyl cross-couplings , 2012 .

[6]  B. Matkowsky,et al.  Flame Propagation in Channels: Secondary Bifurcation to Quasi-Periodic Pulsations , 1985 .

[7]  Stephen B. Margolis,et al.  New routes to quasi-periodic combustion of solids and high-density fluids near resonant Hopf bifurcation points , 1991 .

[8]  Stephen B. Margolis,et al.  Chaotic combustion of solids and high-density fluids near points of strong resonance , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[9]  J. Yu,et al.  Weakly Nonlinear and Numerical Analyses of Dynamics in a Solid Combustion Model , 2005, SIAM J. Appl. Math..

[10]  Marc Garbey,et al.  Quasi-periodic waves and the transfer of stability in condensed-phase surface combustion , 1992 .

[11]  B. Novozhilov.  Non-linear SHS phenomena: Experiment, theory, numerical modeling , 1992 .

[12]  Robert C. Armstrong,et al.  Two Asymptotic Models for Solid Propellant Combustion , 1986 .

[13]  L. K. Gross,et al.  The Onset of Linear Instabilities in a Solid Combustion Model , 2001 .

[14]  B. Novozhilov.  Nonlinear oscillations of combustion velocity of powder , 1966 .

[15]  Gregory I. Sivashinsky,et al.  Chaotic dynamics in solid fuel combustion , 1993 .

[16]  Bernard J. Matkowsky,et al.  Dynamics of hot spots in solid fuel combustion , 2002 .

[17]  Michael L. Frankel,et al.  Thermo-kinetically controlled pattern selection , 2000 .

[18]  Bernard J. Matkowsky,et al.  Interaction of pulsating and spinning waves in condensed phase combustion , 1986 .

[19]  Bernard J. Matkowsky,et al.  Fronts, relaxation oscillations, and period doubling in solid fuel combustion , 1987 .

[20]  Gregory I. Sivashinsky,et al.  A Sequence of Period Doublings and Chaotic Pulsations in a Free Boundary Problem Modeling Thermal Instabilities , 1994, SIAM J. Appl. Math..

[21]  A. Merzhanov,et al.  Propagation of a pulsating exothermic reaction front in the condensed phase , 1971 .

[22]  A. Bayliss,et al.  Period doubling gained, period doubling lost , 1989 .

[23]  A. Bayliss,et al.  Two routes to chaos in condensed phase combustion , 1990 .