Weakly Nonlinear and Numerical Analyses of Dynamics in a Solid Combustion Model

Abstract

Abstract. This paper contains qualitative and quantitative comparisons between a weakly nonlinear analysis and direct numerical simulations of a free-boundary problem. The former involves modulating the most linearly unstable mode, taking a small perturbation of the neutrally stable value νc of a parameter ν related to the activation energy. Analogously, we perform the direct numerical computations near the marginally unstable value, namely, ν = νc − 2 , where is rather small.We delineate the role of a different parameter σ (related to the Arrhenius kinetics) in the combustion dynamics when ν = νc − 2 . In particular, the numerics show that varying σ produces a period-doubling scenario when lies approximately between 0.08 and 0.12. We describe the σ intervals within which complex dynamics occur for various values of and for ν fixed at νc − 2 . When drops to approximately 0.06, the asymptotic and numerical solutions agree well for all physical values of σ.

Complex dynamic behavior during transition in a solid combustion model

“…as the boundary conditions are now evaluated at η = 0 [6]. The free-boundary problem admits a traveling-wave solution.…”

“…where t 0 = τ , and t 1 = τ and t 2 = 2 τ are slow times, A(t 1 , t 2 ) is complex, and "CC" stands for complex-conjugate terms [6]. In the velocity expansion, we have…”

“…the Crank-Nicholson method to solve the problem in a front-attached coordinate frame, reformulating the boundary condition (3) for robustness (see [6] for details). As for the asymptotic solution, the previous section describes the order-perturbation to the travelingwave solution (6). Figure 1 reveals that from t = 0 to approximately t = 30, the small front velocity perturbation is linearly unstable, and its amplitude grows exponentially in time.…”

“…Figure 8 shows the σ -interval as a function of (with the values of the line obtained by polynomial interpolation). Varying quantifies the domain of applicability of the weakly nonlinear analysis and delineates the role of σ in the dynamics [6].…”

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

“…In this case, the basic solution loses stability via a supercritical Hopf bifurcation which cannot be predicted by the linear analysis alone and requires more advanced analytical methods or numerical simulation. The most common analytical technique is the method of multiple scales which has been employed in [8,22,23] among others. In this study, we apply similar methods to a mathematical model of solid propellant combustion which was developed by Margolis and Armstrong in [20].…”