Diffusional-thermal instability of diffusion flames

Abstract

The diffusional–thermal instability, which gives rise to striped quenching patterns that have been observed for diffusion flames, is analysed by studying the model of a one-dimensional convective diffusion flame in the diffusion-flame regime of activation-energy asymptotics. Attention is focused principally on near-extinction conditions with Lewis numbers less than unity, in which the reactants with high diffusivity diffuse into the strong segments of the reaction sheet, so that the regions between the strong segments become deficient in reactant and subject to the local quenching that leads to the striped patterns. This analysis differs from other flame stability analyses in that the complete description of the dispersion relation is obtained from a composite expansion of the results of an analysis with the conventional convective-diffusive scaling and one with reaction-zone scaling. The results predict that striped patterns will occur, for flames sufficiently close to quasi-steady extinction, with a finite wavenumber that in convective–diffusive scaling is proportional to the cube root of the Zel'dovich number. The convective–diffusive response contributes to the stabilization of long-wavelength disturbances by through positive excess enthalpies by which the flame becomes more resistant to instability, while the reaction-zone response provides stabilization of short-wavelength disturbances by transverse diffusion, within the reactive inner layer, which relaxes the perturbed scalar fields towards their unperturbed states. As quasi-steady extinction is approached, marginal stability arises first at an intermediate range between these two scalings. Parametric results for this bifurcation point are obtained through numerical solutions of the associated generalized eigenvalue problems. Comparisons with measured pattern dimensions for different sets of reactants and diluents reveal excellent qualitative agreement.

Diffusive-thermal instability of counterflow flames at low Lewis number

“…9 At sufficiently low Da, reactant leakage through the flame front occurs 30 which causes partial premixing of reactants and thus the local heat release rate becomes dependent on chemical reaction rates in addition to mixing rates. This in turn enables a diffusive-thermal instability to occur for both the socalled diffusion flame regime of nonpremixed flames 9,31 in which an O(1/β) amount of both reactants leak through the flame front, as well as the premixed flame regime of nonpremixed flames 32 in which an O(1) amount of one reactant leaks though the flame front. Similar results are predicted for both the theoretically-attractive convective diffusion flame 9,31 and the experimentally-convenient strained counterflow configuration.…”

“…It is well known theoretically and experimentally that both premixed flames 1,2,3,4,5,6 and nonpremixed flames 7,8,9 exhibit diffusive-thermal instability (DTI) that leads to cellular flame structures at low Lewis number (Le) of the stoichiometrically limiting reactant (fuel or oxidant). Here Le is defined as the ratio of the thermal diffusivity of the bulk mixture to the mass diffusivity of the reactant.…”

Period doubling cascade in diffusion flames

“…Early studies by Matalon, Kim and coworkers used high activation energy asymptotic techniques to study the corresponding linear stability problem, see [2,9,10,11,14]. Vance et al [17] studied numerically the eigenvalues of the complete linearized system.…”

Near Quenching Limit Instabilities of Concurrent Flame Spread over Thin Solid Fuel

“…It is well-known that adiabatic diffusion flames are described by the S-shaped response curve (Liñán 1974), whose upper ignition branch approaches the Burke-Schumann limit at large Damköhler numbers and terminates at the blow-off limit at the other extreme of the Damköhler number. The BurkeSchumann limit turns out to be unconditionally stable (Cheatham and Matalon 2000;Kim et al 1996), whereas near the blow-off limit the flame may suffer different kinds of instabilities, as has been discussed in the Introduction.…”

“…Chen et al (1992) experimentally studied the diffusive-thermal instability of diffusion flames, and identified cellular instability near the blow- A c c e p t e d M a n u s c r i p t 8 off limit. Subsequently, cellular and pulsating instabilities of diffusion flames near the blow-off limit have been extensively studied (Cheatham andMatalon 1996, 2000;Kim 1997;Kim and Lee 1999;Kim et al 1996;Kukuck and Matalon 2001;Metzener and Matalon 2006;Sohn et al 1999). In addition to the low Damköhler number blow-off limit, when heat loss effect is incorporated, the diffusion flame may develop a quenching limit at large Damköhler numbers.…”