Effect of thermal expansion on the linear stability of planar premixed flames for a simple chain-branching model: The high activation energy asymptotic limit

The linear stability of freely propagating, adiabatic, planar premixed ames is investigated in the context of a simple chain-branching chemistry model consisting of a chain-branching reaction step and a completion reaction step. The role of chain-branching is governed by a crossover temperature. Hydrodynamic effects, induced by thermal expansion, are taken into account and the results compared and contrasted with those from a previous purely thermal-di�usive constant density linear stability study. It is shown that when thermal expansion is properly accounted for, a region of stable ames predicted by the constant density model disappears, and instead the ame is unstable to a long-wavelength cellular instability. For a pulsating mode, however, thermal expansion is shown to have only a weak e�ect on the critical fuel Lewis number required for instability. These e�ects of thermal expansion on the two-step chain-branching ame are shown to be qualitatively similar to those on the standard one-step reaction model. Indeed, as found by constant density studies, in the limit that the chain-branching crossover temperature tends to the adiabatic ame temperature, the two-step model can be described to leading order by the one-step model with a suitably de�ned e�ective activation energy

Linear stability of planar premixed flames: reactive Navier-Stokes equations with finite activation energy and arbitrary Lewis number

A numerical shooting method for performing linear stability analyses of travelling waves is described and applied to the problem of freely propagating planar premixed flames. Previous linear stability analyses of premixed flames either employ high activation temperature asymptotics or have been performed numerically with finite activation temperature, but either for unit Lewis numbers (which ignores thermal-diffusive effects) or in the limit of small heat release (which ignores hydrodynamic effects). In this paper the full reactive Navier-Stokes equations are used with arbitrary values of the parameters (activation temperature, Lewis number, heat of reaction, Prandtl number), for which both thermal-diffusive and hydrodynamic effects on the instability, and their interactions, are taken into account. Comparisons are made with previous asymptotic and numerical results. For Lewis numbers very close to or above unity, for which hydrodynamic effects caused by thermal expansion are the dominant destablizing mechanism, it is shown that slowly varying flame analyses give qualitatively good but quantitatively poor predictions, and also that the stability is insensitive to the activation temperature. However, for Lewis numbers sufficiently below unity for which thermal-diffusive effects play a major role, the stability of the flame becomes very sensitive to the activation temperature. Indeed, unphysically high activation temperatures are required for the high activation temperature analysis to give quantitatively good predictions at such low Lewis numbers. It is also shown that state-insensitive viscosity has a small destabilizing effect on the cellular instability at low Lewis numbers

A stability index for detonation waves in Majda's model for reacting flow

Using Evans function techniques, we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in [GZ,BSZ], yielding useful necessary conditions for stability. Here, we carry out the analysis in the context of the Majda model, a simplified model for reacting flow; the method is extended to the full Navier-Stokes equations of reacting flow in [Ly,LyZ]. The resulting stability condition is satisfied for all nondegenerate, i.e., spatially exponentially decaying, weak and strong detonations of the Majda model in agreement with numerical experiments of [CMR] and analytical results of [Sz,LY] for a related model of Majda and Rosales. We discuss also the role in the ZND limit of degenerate, subalgebraically decaying weak detonation and (for a modified, ``bump-type'' ignition function) deflagration profiles, as discussed in [GS.1-2] for the full equations.Comment: 36 pages, 3 figure

Pointwise Green function bounds and stability of combustion waves

Generalizing similar results for viscous shock and relaxation waves, we establish sharp pointwise Green function bounds and linearized and nonlinear stability for traveling wave solutions of an abstract viscous combustion model including both Majda's model and the full reacting compressible Navier--Stokes equations with artificial viscosity with general multi-species reaction and reaction-dependent equation of state, % under the necessary conditions of strong spectral stability, i.e., stable point spectrum of the linearized operator about the wave, transversality of the profile as a connection in the traveling-wave ODE, and hyperbolic stability of the associated Chapman--Jouguet (square-wave) approximation. Notably, our results apply to combustion waves of any type: weak or strong, detonations or deflagrations, reducing the study of stability to verification of a readily numerically checkable Evans function condition. Together with spectral results of Lyng and Zumbrun, this gives immediately stability of small-amplitude strong detonations in the small heat-release (i.e., fluid-dynamical) limit, simplifying and greatly extending previous results obtained by energy methods by Liu--Ying and Tesei--Tan for Majda's model and the reactive Navier--Stokes equations, respectively

On the One-dimensional Stability of Viscous Strong Detonation Waves

Building on Evans function techniques developed to study the stability of viscous shocks, we examine the stability of viscous strong detonation wave solutions of the reacting Navier-Stokes equations. The primary result, following the work of Alexander, Gardner & Jones and Gardner & Zumbrun, is the calculation of a stability index whose sign determines a necessary condition for spectral stability. We show that for an ideal gas this index can be evaluated in the ZND limit of vanishing dissipative effects. Moreover, when the heat of reaction is sufficiently small, we prove that strong detonations are spectrally stable provided the underlying shock is stable. Finally, for completeness, the stability index calculations for the nonreacting Navier-Stokes equations are includedComment: 66 pages, 7 figure