Planar Flame Front Research Articles

Existence of pulsating waves of advection–reaction–diffusion equations of ignition type by a new method

The goal of this paper is to study the advection–reaction–diffusion equation of ignition type: u t − Δ u + q ( x , y ) ⋅ ∇ u = f ( u ) . Xin [J.X. Xin, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Ration. Mech. Anal. 121 (1992) 205–233], Berestycki and Hamel [H. Berestycki, F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math. 55 (2002) 949–1032] proved the existence of pulsating waves, using the theory of degenerate elliptic equations. Our goal is to give an alternative and rather natural proof of the same result: first we transform the problem so that the existence of pulsating waves is equivalent to the existence of 1-periodic in time solutions of a nonlinear parabolic equation with 1-periodic in time coefficients; next we prove the existence of such periodic solutions, using a continuation method, based on the implicit function theorem.

The role of duct thickness on the quenching process of premixed flame propagation

The propagation of premixed laminar flame in ducts of circular cross-section considering a thermal-diffusive model is investigated numerically. Heat losses by conduction to the channels walls are taken into account using the thermally thin wall regime. The effects and the relationship between thickness and diameter of the tube with the flame speed propagation are studied and the quenching condition is obtained as a function of the heat-loss parameter. The mathematical model employs the axisymmetric energy and species equations. The calculations are based on a two-step chemistry, with an Arrhenius, energetically neutral, radical production reaction followed by an exothermic radical recombination reaction. For large values of the heat-loss parameter, the wall temperature is close to the free stream temperature and all the heat losses through the wall are convected away. No heat feedback occurs. On the other hand, for small values of the heat-loss parameter, a feedback mechanism occurs by transferring heat from the burned gas to the fresh mixture along the tube wall. For values of the heat-loss parameter of order unity, the heat feedback mechanism is able to sustain the flame propagation and the quenching condition disappears, producing an almost planar flame front as the propagation velocity reduces. For this two-step reaction mechanism, the radical species behaviour at the duct walls seems to have negligible effect on the quenching process.

An Experimental Investigation on Combustion Synthesis of Transition Metal Silicides V5Si3, Nb5Si3, and Ta5Si3.

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Mathematical modelling of the heating of tree elements by a plane flame front

Mathematical modelling of the heating of tree elements by a plane flame front

An experimental investigation on combustion synthesis of transition metal silicides V 5Si 3, Nb 5Si 3, and Ta 5Si 3

Preparation of transition metal silicides V 5Si 3, Nb 5Si 3, and Ta 5Si 3 was conducted by self-propagating high-temperature synthesis (SHS) from elemental powder compacts at their stoichiometries. The propagation mode of self-sustaining reactions, flame-front velocity, combustion temperature, and phase composition of end products were extensively studied. For the reactant compacts from the mixtures of V:Si = 5:3 and Ta:Si = 5:3, a planar flame front propagating in a steady manner was observed, with or without sample preheating. On the other hand, a preheating temperature of 200 °C or above was required by the Nb–Si powder compact to initiate the self-sustaining combustion. Moreover, it was found that the planar reaction font was not stable in the samples of Nb:Si = 5:3 and was transformed into a localized reaction zone propagating in a spinning mode. For all three types of the samples, as the sample density and preheating temperature increased, the combustion temperature was found to increase and the flame-front velocity was correspondingly enhanced. According to the XRD analysis, formation of single-phase silicides V 5Si 3 and Nb 5Si 3 from the reactant compacts of 5V + 3Si and 5Nb + 3Si, respectively, was confirmed. The samples of 5Ta + 3Si yielded predominantly the silicide Ta 5Si 3, along with an intermediate phase Ta 2Si in a small amount. Based upon the temperature dependence of combustion wave velocity, the activation energies associated with combustion synthesis of V 5Si 3, Nb 5Si 3, and Ta 5Si 3 were determined to be 111.5, 259.2, and 105.2 kJ/mol, respectively.

Numerical simulation of flames as gas-dynamic discontinuities

The dynamics of thin premixed flames is computationally studied within the context of a hydrodynamic theory. A level-set method is used to track down the flame, which is treated as a free-boundary interface. The flow field is described by the incompressible Navier–Stokes equations, with different densities for the burnt and unburnt gases, supplemented by singular source terms that properly account for thermal expansion effects. The numerical scheme has been tested on several benchmark problems and was shown to be stable and accurate. In particular, the propagation of a planar flame front and the dynamics of hydrodynamically unstable flames were successfully simulated. This includes recovering the planar front in narrow domains, the Darrieus–Landau linear growth rate for long waves of small amplitude, and the nonlinear development of cusp-like structures predicted by the Michelson–Sivashinsky equation for a small density change. The stationary flame of a Bunsen burner with uniform and parabolic outlet flows were also simulated, showing in particular a careful mapping of the flow field. Finally, the evolution of a hydrodynamically unstable flame was studied for finite amplitude disturbances and realistic values of thermal expansion. These results, which constitute one of the main objectives of this study, elucidate the effect of thermal expansion on flame dynamics.

COMPARISON OF FLAME-FRONT INSTABILITIES WITH INSTABILITIES OF ABLATION FRONTS IN INERTIAL-CONFINEMENT FUSION

ABSTRACT There are both similarities and differences between instabilities of planar laminar flame fronts and ablation fronts in inertial-confinement fusion (ICF). The mathematical formulation of the problem in ICF is the same as for flames propagating upward; the differences stem from the different order of magnitude of the Froude number and thermal conductivity variations. When the thermal conductivity varies strongly, as in ICF, a wide range of characteristic diffusive lengths occurs across the wave structure. For disturbances with intermediate wavelengths, there is then a universal diffusive relaxation rate of thermal waves that is independent of the thermal conductivity. A kinematic relation for the ablation front is discussed here that includes this thermal relaxation. The resulting formulation facilitates comparison of the two types of fronts.

Propagation of a planar flame front studied by molecular dynamics

In this paper we study the propagation of a planar traveling wave in a nonisothermal autocatalytic reaction diffusion system. This is done by performing a series of molecular dynamics simulations for different densities and heat of reactions. In the special case where the reaction is isothermal, it is shown that the molecular dynamics simulation qualitatively resembles the approximative macroscopic description that is based on the reaction diffusion equation. Simulations for different densities and heat releases further show that the speed of propagation is not simply the minimum speed of the corresponding macroscopic equation and that the speed is always constant even though the Lewis number is greater than unity.

A similarity solution describing the collision of twoplanar premixed flames

The problem of the head-on collision of two planar premixed flame fronts is considered using the idealized model of a single step reaction controlled by the deficient species and an Arrhenius reaction law. Density changes are neglected. It is shown that a similarity solution exists if the deficient species and temperature are equidiffusive (Lewis number unity). The similarity solution, derived using activation energy asymptotics, is valid in the intermediate region when the flames are close enough that their pre-heat zones overlap but their reaction zones may be considered to be well separated. A one–dimensional numerical simulation shows good agreement with the analytical solution.

Traveling and spiral waves for sequential flames with translation symmetry: coupled CGL–Burgers equations

The dynamics of traveling and spiral waves resulting from the oscillatory instability of a uniformly propagating planar flame front governed by a sequential reaction is investigated. The nonlinear dynamics of this system, which enjoys translation symmetry, is described by two coupled equations: a complex Ginzburg–Landau equation for the oscillation amplitude, and a Burgers equation for the slow Goldstone mode due to the symmetry, describing frontal deformation. The pure CGL equation in 2D exhibits spiral wave solutions as well as uniformly propagating wave solutions. We present a stability analysis and numerical computations of the coupled system to show how the coupling modifies traveling and spiral wave solutions of the pure CGL equation. In particular, we show that the coupled system exhibits new types of instabilities as well as new dynamical behavior, including bound states of two or four spirals, “liquid spiral” states, superspiral structures, oscillating cellular structures separated by chaotically merging and splitting domain walls, and others.

Stability of Pole Solutions for Planar Propagating Flames: II. Properties of Eigenvalues/Eigenfunctions and Implications to Stability

In a previous paper (Part I) we focused our attention on pole solutions that arise in the context of flame propagation. The nonlinear development that follows after a planar flame front becomes unstable is described by a single nonlinear PDE which admits pole solutions as equilibrium states. Specifically, we were concerned with coalescent steady states, which correspond to steadily propagating single-peak structures extended periodically over the infinite domain. This pattern is one that is commonly observed in experiments. In order to examine the linear stability of these equilibrium solutions, we formulated in Part I the corresponding eigenvalue problem and derived exact analytical expressions for the spectrum and the corresponding eigenfunctions. In this paper, we examine their properties as they relate to the stability issue. Being based on analytical expressions, our results resolve earlier controversies that resulted from numerical investigations of the stability problem. We show that, for any period 2L, there always exists one and only one stable steady coalescent pole solution. We also examine the dependence of the eigenvalues and eigenfunctions on L which provides insight into the behavior of the nonlinear PDE and, consequently, on the nonlinear dynamics of the flame front.

Stability of Pole Solutions for Planar Propagating Flames: I. Exact Eigenvalues and Eigenfunctions

It is well known that the nonlinear PDE describing the dynamics of a hydrodynamically unstable planar flame front admits exact pole solutions as equilibrium states. Such a solution corresponds to a steadily propagating cusp-like structure commonly observed in experiments. In this work we investigate the linear stability of these equilibrium states---the steady coalescent pole solutions. In previous similar studies, either a truncated linear system was numerically solved for the eigenvalues or the initial value problem for the linearized PDE was numerically integrated in order to examine the evolution of initially small disturbances in time. In contrast, our results are based on the exact analytical expressions for the eigenvalues and corresponding eigenfunctions. In this paper we derive the expressions for the eigenvalues and eigenfunctions. Their properties and the implication on the stability of pole solutions is discussed in a paper which will appear later.

Bifurcations in a planar propagating flame as the size of the domain increases

The partial differential equation (PDE) describing the dynamics of hydrodynamically unstable planar flame front has exact pole solutions which satisfy a set of ordinary differential equations (ODEs). This set of ODEs prohibits the creation of new poles in the complex plane, or the appearance of cusps in the physical space, as observed experimentally. The contribution of this paper is to show that most exact pole solutions are unstable solutions for the PDE. Even the one-peak, coalescent solutions (whose number of poles is maximal) is unstable as soon as the number of poles exceeds a certain (rather small) threshold. As the size of the domain increases, the front undergoes bifurcations which can be described as follows: the one-pole, one-peak coalescent solution is neutrally stable for small intervals. As the length of the interval increases, it becomes unstable and the two-pole one-peak coalescent solution is neutrally stable. For larger intervals, the two-pole solution is unstable, the three-pole solution becomes stable. As the interval length increases further, the steady one-peak, coalescent solutions are no longer stable and bifurcations to unsteady states occur.

Stability of a planar flame front in a compressible flow

The effect of flow compressibility on the hydrodynamic stability of a planar flame front of finite thickness is investigated. It is shown that a flame front becomes more unstable with the increase of the Mach number of the flame generated flow. Particularly, even for a flow with relatively small Mach numbers ∼0.1–0.2 the maximal growth rate of the flame instability becomes 2–3 times larger than the growth rates for a flame in the incompressible flow.

Stability of pole solutions for planar propagating flames.

It is well known that the partial differential equation (PDE) describing the dynamics of a hydrodynamically unstable planar flame front admits exact pole solutions. For such solutions, the original PDE can be reduced to a set of ordinary differential equations (ODE's). The situation, however, is paradoxical since the steady solutions obtained by numerically integrating the PDE differ, in general, from the exact solutions governed by the ODE's. For example, if the initial condition is a one-pole steady solution, provided that the size of the domain considered is larger than a (small) critical length, the number of poles increases with time in the PDE while it remains constant in the ODE's. In previous studies, this generation of poles was thus believed to be an artifact or product of external noise, rather than a dynamical process intrinsic to the PDE. In this paper, we show that the phenomenon is due to the fact that most exact steady pole solutions are unstable for the PDE. In certain cases, such solutions are unstable for the ODE's, in other cases, they are neutrally stable for the ODE's but unstable for the PDE. The only steady pole solutions which are neutrally stable for both the ODE's and the PDE correspond to small interval lengths; both their number of poles and propagation speed are maximal (among all possible steady solutions corresponding to the interval considered) and all their poles are aligned on the same vertical axis in the complex plane (i.e., such solutions are coalescent). For a given interval of small length, there is only one such solution (up to translation symmetry). \textcopyright{} 1996 The American Physical Society.

Scaling, propagation, and kinetic roughening of flame fronts in random media

We introduce a model of two coupled reaction-diffusion equations to describe the dynamics and propagation of flame fronts in random media. The model incorporates heat diffusion, its dissipation, and its production through coupling to the background reactant density. We first show analytically and numerically that there is a finite critical value of the background density, below which the front associated with the temperature field stops propagating. The critical exponents associated with this transition are shown to be consistent with mean field theory of percolation. Second, we study the kinetic roughening associated with a moving planar flame front above the critical density. By numerically calculating the time dependent width and equal time height correlation function of the front, we demonstrate that the roughening process belongs to the universality class of the Kardar-Parisi-Zhang interface equation. Finally, we show how this interface equation can be analytically derived from our model in the limit of almost uniform background density.

Stability of a deflagration front in compressible media

A two-dimensional linear analysis of the planar flame front stability for a compressible fluid is presented. The analysis shows that there are two types of perturbations. The first type, corresponding to waves in incompressible media, has already been studied by Landau. It predicts absolute instability of the flame front. The second type of perturbations is due to fluid compressibility and the dependence on upstream flow parameters of the flame front velocity. Three different regimes for these perturbations are possible: stable, acoustically unstable, and absolutely unstable. The instability results in a pronounced pressure wave generation.

Thermal stability of distorted gas flame in porous media

The stability of a plane flame front with respect to spatial perturbations is investigated for chemically inert media. The influence of the gas-filtration rate on the stability is analyzed. In the long-wave approximation, the dependence of the flame front velocity on the surface curvature is obtained.

The initial development of a tulip flame

The initial development of a “tulip flame,” often observed during flame propagation in closed tubes, isattributed to a combustion instability. The roles of hydrodynamic and of the diffusional-thermal processes on the onset of instability are investigated through a linear stability analysis in which the growth or decay of small disturbances, superimposed on an otherwise smooth and planar flame front, are followed. A range of the Markstein parameter, related to the mixture composition through an appropriately defined Lewis number, has been identified where a tulip flame could be observed. For a given value of the Markstein parameter within this range, a critical wavelength is identified as the most unstable mode. This wavelength is directly related to the minimal aspect ratio of the tube where a tulip flame could be observed. The time of onset of instability is identified as the time when the most unstable disturbance, associated with the critical wavelength, grows at a faster rate than the flame front itself and exceeds a certain threshold. This occurs after the flame has propagated a certain distance down the tube: a value which has been explicitly determined in terms of the relevant parameters. Experimental records on the tulip flame phenomenon support the findings of the analysis. That is, the tulip flame forms after the flame has traveled half the tube's length, it does not form in short tubes, and its formation depends on the mixture composition and on the initial pressure in the tube.

Stability of a planar flame front in the slow-combustion regime.

The hydrodynamic stability of a planar flame (deflagration) is determined by solving the complete system of equations, including thermal conduction and energy release due to chemical reactions, for the case in which the Lewis number is equal to unity. In the asymptotic limit of large-wavelength perturbations, the developed theory provides a rigorous justification of the Darrieus-Landau assumption that the flame-front velocity is constant, which is the necessary supplementary condition in the model of discontinuous flame front. The analytical solution for the suppression of the flame-front instability is obtained for an arbitrary activation energy. It is shown that the obtained solution does not depend on the specific form of the energy release. The perturbation growth rate is also found numerically by solving the eigenvalue problem.