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

Abstract

Abstract. 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 NavierStokes 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.

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

“…These equations have been non-dimensionalized using the standard scales employed in previous linear stability analysis of the one-step model [9,12], with which we seek to compare. Thus…”

“…Hence note that X 0 = 0 in the region x > 0 by equation (12). Since, apart from for the pressure, the steady, flame solution is independent of the Prandtl number, and P r is known to have only a very weak effect on flame stability [6,12], throughout this paper we set P r = 0.75.…”

“…Since, apart from for the pressure, the steady, flame solution is independent of the Prandtl number, and P r is known to have only a very weak effect on flame stability [6,12], throughout this paper we set P r = 0.75. Here will consider the case n = 2 in order to retain analytical simplicity of the steady flame structure (the equation for Y 0 is then linear and decoupled from the T 0 and F 0 equations), and such that it has precisely the same structure as in the CDM considered previously [14], with which we seek to compare.…”

“…We hence now seek asymptotic solutions to equation (24) valid near the fresh or burnt states, as x → ±∞, which can then be used as initial conditions for numerical integration of the equation. In order to obtain higher order terms in these asymptotic solutions, which are necessary for implementation of straightforward shooting methods [12,19], it is beneficial to use one of the steady state solution variables as the independent variable [12,19]. Given the structure of the steady solution in equations (14)- (16), and given that Y 0 → 0 as x → ±∞, here we choose Y 0 as the independent variable for both the fresh and burnt state analyses.…”

“…These asymptotic one-step reaction studies also entail a near-equidiffusional flames (NEF) approximation, i.e they are valid for Lewis numbers asymptotically close to unity. Finite activation energy numerical linear stability studies, which do not invoke the asymptotic limit, have also been performed for the one-step reaction model, both in terms of the CDM [10] and the Reactive Navier-Stokes equations [11,12]. These studies show that, when the Lewis number is sufficiently far from unity, very high activation energies may be required for the asymptotic results to be quantitatively predic-tive.…”