Thermo-diffusive Model Research Articles

Propagation de flammes gazeuses dans la limite d'une diffusion massique nulle

In this Note, we study a thermo-diffusive model for the propagation of 1D gaseous flames for complex chemistry with open graph. We use a new approach in order to prove the existence of a traveling wave and we study the behavior of the solution in the limit of zero mass diffusion coefficients: we show the continuity of the wave velocity and of the temperature and mass fraction profiles. To cite this article: F. Laurent et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 405–410.

The role of unequal diffusivities in ignition and extinction fronts in strained mixing layers

We have studied flame propagation in a strained mixing layer formed between a fuel stream and an oxidizer stream, which can have different initial temperatures. Allowing the Lewis numbers to deviate from unity, the problem is first formulated within the framework of a thermo-diffusive model and a single irreversible reaction. A compact formulation is then derived in the limit of large activation energy, and solved analytically for high values of the Damköhler number. Simple expressions describing the flame shape and its propagation velocity are obtained. In particular, it is found that the Lewis numbers affect the propagation of the triple flame in a way similar to that obtained in the studies of stretched premixed flames. For example, the flame curvature determined by the transverse enthalpy gradients in the frozen mixing layer leads to flame-front velocities which grow with decreasing values of the Lewis numbers.The analytical results are complemented by a numerical study which focuses on preferential-diffusion effects on triple flames. The results cover, for different values of the fuel Lewis number, a wide range of values of the Damköhler number leading to propagation speeds which vary from positive values down to large negative values

Analysis of combustion-driven acoustics

Combustion-driven acoustic oscillations are investigated by performing a one-dimensional stability analysis of a burner-stabilized premixed flame. In contrast to other investigators, no initial acoustic wave is assumed in the analysis; the downstream acoustic field results from flame instability. Two models are considered: the thermodiffusive model (uncoupled model) and the fully coupled thermodiffusive-hydrodynamic model. The fully coupled problem exhibits instability at a much lower critical Lewis number than the uncoupled problem.

Triple flames in mixing layers with nonunity lewis numbers

The present paper is devoted to the study of the effects of nonunity Lewis numbers on triple-flame propagation in nonuniform mixtures. For definiteness, the case of a strained reactive mixing layer is considered. The fuel and oxidizer that are fed to the mixing layer are allowed to have different initial temperatures. Specifically, we examine how the triple flames encountered in this context are influenced by (a) the transverse gradients in the temperature and composition of the fresh reactive mixture and (b) by differential-diffusion effects. The analysis is carried for a single irreversible reaction with a large activation energy and using the thermo-diffusive model. Analytical expressions describing the flame shape, the local burning speed, and the propagation velocity of the triple flame are obtained. In particular, it is found that the Lewis numbers affect the propagation of the triple flame in a way similar to that obtained in the studies of stretched premixed flames. For example, the flame curvature determined by the transverse gradients in the frozen mixing layer leads to flame-front velocities that grow with decreasing values of the Lewis numbers.

Massively parallel computation of stiff propagating combustion fronts

Gas combustion, solid combustion as well as frontal polymerization are characterized by stiff fronts that propagate with nonlinear dynamics. The multiple-scale phenomena under consideration lead to very intense computations that require parallel computing in order to reduce the elapsed time of the computation. We develop a methodology to build on the MIMD architecture a parallel numerical method based on the property of the solution, i.e. a stiff quasi-planar two-dimensional combustion front. We illustrate our methodology using two models of the combustion process. The first is a thermo-diffusive model of a two-step chemical reaction exhibiting two transition layers. The second is a thermo-diffusive model of a one-step chemical reaction coupled with a hydrodynamical model using the stream function - vorticity formulation of the Navier - Stokes equations written in the Boussinesq approximation. This methodology makes use of efficient domain decomposition methods, combined with asymptotic analytical qualitative results to adapt the interface position, to solve the transition layer(s) of the solution accurately and operator splitting to take advantage of the quasi-planar property of the frontal process. Then, it provides three complementary levels of parallelism. A first level of parallelism based on the domain decomposition, thus a priori limited to the number of transition layers in the problem. A second based on an explicit parallelism in the orthogonal direction of the front propagation. A third based on the spread of equations on subnetworks of processors. The parallel implementation using the message passing library concept on the Paragon and iPSC860 MIMD computers are discussed. An efficient parallel algorithm to solve the space-periodic stream-function in the second model, based on Fourier modes decomposition combined with the first and second level of parallelism is provided. The direct numerical simulation provided by our numerical method allows us to explore the physical parameter space of the combustion process in order to understand the mechanism of instabilities. Some examples of hydrodynamical and thermal instabilities are given.

Massively parallel computation of stiff propagating combustion fronts

Gas combustion, solid combustion as well as frontal polymerization are characterized by stiff fronts that propagate with nonlinear dynamics. The multiple-scale phenomena under consideration lead to very intense computations that require parallel computing in order to reduce the elapsed time of the computation. We develop a methodology to build on the MIMD architecture a parallel numerical method based on the property of the solution, i.e. a stiff quasi-planar two-dimensional combustion front. We illustrate our methodology using two models of the combustion process. The first is a thermo-diffusive model of a two-step chemical reaction exhibiting two transition layers. The second is a thermo-diffusive model of a one-step chemical reaction coupled with a hydrodynamical model using the stream function - vorticity formulation of the Navier - Stokes equations written in the Boussinesq approximation. This methodology makes use of efficient domain decomposition methods, combined with asymptotic analytical qualitative results to adapt the interface position, to solve the transition layer(s) of the solution accurately and operator splitting to take advantage of the quasi-planar property of the frontal process. Then, it provides three complementary levels of parallelism. A first level of parallelism based on the domain decomposition, thus a priori limited to the number of transition layers in the problem. A second based on an explicit parallelism in the orthogonal direction of the front propagation. A third based on the spread of equations on subnetworks of processors. The parallel implementation using the message passing library concept on the Paragon and iPSC860 MIMD computers are discussed. An efficient parallel algorithm to solve the space-periodic stream-function in the second model, based on Fourier modes decomposition combined with the first and second level of parallelism is provided. The direct numerical simulation provided by our numerical method allows us to explore the physical parameter space of the combustion process in order to understand the mechanism of instabilities. Some examples of hydrodynamical and thermal instabilities are given.

Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders

The paper is concerned with the long-time behaviour of the solutions of a certain class of semilinear parabolic equations in cylinders, which contains as a particular case the multidimensional thermo-diffusive model in combustion theory. We prove, under minimal conditions on the initial values, that the solutions eventually become monotone in the direction of the axis of the cylinder on every compact subset; this implies convergence to travelling fronts. This result is applied to propagation versus extinction problems: given a compactly supported initial datum, sufficient conditions ensuring that the solution will either converge to 0 or to a pair of travelling fronts are given. Additional information on the corresponding equations in finite cylinders is also obtained.

Bifurcations of travelling waves in the thermo-diffusive model for flame propagation

The main topic of this paper is the study of steady-state bifurcations occurring in the two-dimensional thermo-diffusive model in the framework of large activation energies.

Two-dimensional fully adaptive solutions of reaction-diffusion equations

We present an adaptive Rothe method for two-dimensional problems combining an embedded Runge-Kutta scheme in time and a multi-level finite element discretization in space. The spatial discretization error is controlled by a posteriori error estimates based on interpolation techniques. A computational example for a thermodiffusive flame propagation model illustrates the high accuracy that is possible with the proposed method.

FLAME PROPAGATION AND EXTINCTION IN SPATIALLY-PERIODIC LONGITUDINAL VELOCITY FIELDS

ABSTRACT Propagation of one-dimensional “laminar” flame in combustible mixtures with spatially-periodic longitudinal velocity oscillations is investigated within the thermo-diffusive flame model using the large activation energy asymptotic technique. The ratio of the flame propagation velocity in the periodically disturbed versus undisturbed fields X is examined as a function of the amplitude parameter Γ for the complete range of wavelength parameter y (0, ∞). It is found that for infinitely longwave lengths γ →0 the dimensionless flame velocity X decreases monotonically with Γ from unity towards zero without any extinction. For intermediate wavelengths X first increases to a maximum value above unity and then decreases with Γ and eventually at a critical value Γ > Γe becomes less than unity as it continues to decrease towards zero. Therefore, for a given wavelength a critical amplitude of velocity oscillation is identified that results in a maximum burning velocity for the combustible mixture. In the lim...

Surface tension anisotropy and cell-like structure formation in free solidification

This work addresses the role of a surface tension anisotropy during solidification of a pure material from a supercooled melt. We study the formation of a pre-dendritic cell-like structure using fully time-dependent nonlinear numerical solutions of the thermodiffusion model avoiding simplifications of mathematical nature. The introduction of surface tension anisotropy to the model results in changes of cell tip radii as well as in some variations in the morphology of the solid-liquid interface. Time dependent mechanisms lead to the establishment of near constancy of cell tip radii and the restoration of cell tip radii following induced tip perturbations; both phenomenon are found for zero and nonzero surface tension anisotropy parameter values.

On the pulsating instability of two-dimensional flames

We consider a well-known thermo-diffusive model for the propagation of a premixed, adiabatic flame front in the large-activation-energy limit. That model depends only on one nondimensional parameter β, the reduced Lewis number. Near the pulsating instability limit, as β↓β0= 32/3, we obtain an asymptotic model for the evolution of a quasi-planar flame front, via a multi-scale analysis. The asymptotic model consists of two complex Ginzburg–Landau equations and a real Burgers equation, coupled by non-local terms. The model is used to analyse the nonlinear stability of the flame front.

Stability of travelling fronts in a model for flame propagation part I: Linear analysis

We investigate the stability of travelling wave solutions of the multidimensional thermodiffusive model for flame propagation with unit Lewis number. This model consists in a system of two nonlinear parabolic equations posed in an infinite cylinder, with Neumann boundary conditions. In this paper, we prove that every travelling wave solution of this model is linearly stable. Our tools are exponential decay estimates for solutions of elliptic equations in a cylinder, and the Maximum Principle for parabolic equations.

Stability of travelling fronts in a model for flame propagation Part II: Nonlinear stability

This paper is concerned with the nonlinear stability of the travelling-wave solutions of the multidimensional thermodiffusive model for flame propagation, with unit Lewis number. The model consists in a semilinear parabolic equation in an infinite cylinder, with Neumann boundary conditions. We prove that any solution which is initially close to a travelling wave will converge to a translate of that wave.

Stability and instability in crystal growth. III. Reduction of the thermodiffusion equation for a paraboloidal needle crystal.

An exact treatment is given of the thermodiffusion model for the growth of a paraboloidal needle crystal from a supercooled melt. By means of successive transformations, the stability analysis of this system is reduced to the solution of a set of eigenvalue equations. This analysis is valid for all values of the supercooling and paves the way for a calculation of the evolution of the dendrite-tip shape with time.

Numerical Investigation of the Extinction Limit of Curved Flames

Abstract This paper is devoted to the numerical study of the propagation of a premixed curved flame in a two-dimensional tube with non adiabatic walls and to the extinction limits of this flame. The thermo-diffusive model is solved using a finite-element method combined with an adaptive gridding procedure. The two-dimensional numerical results are compared to the analytical results for volumetric heat losses in a one-dimensional setting.

On some adaptive numerical approaches of thin flame propagation problems

This paper presents a collection of experiments in which we investigate the question of grid adaption for flame propagation problems. Local refinement or mesh deformation procedures are applied to a sample of spatial approximations, including finite elements, finite differences and spectral methods. Comparisons are performed using the simplified thermo-diffusive model and a more realistic model involving compressible aerodynamics and combustion.

Numerical analysis of the two-dimensional thermo-diffusive model for flame propagation

On presente l'analyse numerique du modele thermodiffusif employe en theorie de la combustion pour decrire la propagation d'une flamme plissee dans un tube rectangulaire infini. La methode numerique utilise un maillage mobile adaptatif, ce qui conduit a resoudre un systeme d'equations integrodifferentielles non lineaires de type reaction-diffusion. On etablit l'existence et l'unicite d'une solution de ce probleme et on prouve la convergence de l'approximation numerique

Stability and instability in crystal growth. II. Asymmetric solutions of the thermodiffusion model.

Stability and instability in crystal growth. II. Asymmetric solutions of the thermodiffusion model.