Philippe Metzener

The propagation of premixed flames in closed tubes

A nonlinear evolution equation that describes the propagation of a premixed flame in a closed tube has been derived from the general conservation equations. What distinguishes it from other similar equations is a memory term whose origin is in the vorticity production at the flame front. The two important parameters in this equation are the tubes aspect ratio and the Markstein parameter. A linear stability analysis indicates that when the Markstein parameter α is above a critical value α c the planar flame is the stable equilibrium solution. For α below α c the planar flame is no longer stable and there is a band of growing modes. Numerical solutions of the full nonlinear equation confirm this conclusion. Starting with random initial conditions the results indicate that, after a short transient, a at flame develops when α>α c and it remains flat until it reaches the end of the tube. When α c , on the other hand, stable curved flames may develop down the tube. Depending on the initial conditions the flame assumes either a cellular structure, characterized by a finite number of cells convex towards the unburned gas, or a tulip shape characterized by a sharp indentation at the centre of the tube pointing toward the burned gases. In particular, if the initial conditions are chosen so as to simulate the elongated finger-like flame that evolves from an ignition source, a tulip flame evolves downstream. In accord with experimental observations the tulip shape forms only after the flame has travelled a certain distance down the tube, it does not form in short tubes and its formation depends on the mixture composition. While the initial deformation of the flame front is a direct result of the hydrodynamic instability, the actual formation of the tulip flame results from the vortical motion created in the burned gas which is a consequence of the vorticity produced at the flame front.

Premixed flames in closed cylindrical tubes

We consider the propagation of a premixed flame, as a two-dimensionalsheet separating unburned gas from burned products, in a closedcylindrical tube. A nonlinear evolutionequation, that describes the motion of theflame front as a function of its meanposition, is derived. The equation containsa destabilizing term that results from thegas motion induced by thermal expansionand has a memory term associated withvorticity generation. Numerical solutionsof this equation indicate that, whendiffusion is stabilizing, the flameevolves into a non-planar form whose shape, and its associated symmetry properties, aredetermined by the Markstein parameter, andby the initial data. In particular, weobserve the development of convexaxisymmetric or non-axisymmetric flames, tulip flames and cellular flames.

Diffusive-thermal instabilities of diffusion flames: Onset of cells and oscillations

A comprehensive stability analysis of planar diffusion flames is presented within the context of a constant-density model. The analysis provides a complete characterization of the possible patterns that are likely to be observed as a result of differential and preferential diffusion when a planar flame becomes unstable. A whole range of physical parameters is considered, including the Lewis numbers associated with the fuel and the oxidizer, the initial mixture fraction, and the flow conditions. The two main forms of instability are cellular flames, obtained primarily in fuel-lean systems when the Lewis numbers are generally less than one, and planar pulsations, obtained in fuel-rich systems when the Lewis numbers are generally larger than one. The cellular instability is predominantly characterized by stationary cells of characteristic dimension comparable to the diffusion length, but smaller cells that scale on the reaction zone thickness are also possible near extinction conditions. The pulsating instability is characterized by planar oscillations normal to the flame sheet with a well-defined frequency comparable to the reciprocal of the diffusion time; high-frequency modes are also possible just prior to extinction. The analysis also alludes to other possible patterns, such as oscillating cellular structures, which result from competing modes of instability of comparable and/or disparate scales. The expected pattern depends of course on the underlying physical parameters. Consequently, stability boundaries have been identified for the onset of one or another form of the instability. The conditions for the onset of cellular and pulsating flames, as well as the predicted cell size and the frequency of oscillations, compare well with the experimental record.

Strong resonance in two‐dimensional non‐Boussinesq convection

Two‐dimensional convection is described and analyzed in the case of strong 2:1 resonance. The density variation due to the temperature to linear order and the viscous heating as perturbations to the usual Boussinesq approximation are introduced. In addition the temperature dependence of the viscosity is taken into account. In this special setting one is able to quantify the influence of these effects on the dynamics. They break the symmetry with respect to reflections at the horizontal midplane that is artificially induced when adopting the Boussinesq approximation, and give rise to an O(2) symmetric bifurcation problem. Time‐dependent solutions appear and become stable when the Prandtl number lies between two critical values, between zero and one. In this case there also exists a structurally stable heteroclinic orbit very close to the onset of motion in a small amplitude limit. A similar orbit exists outside this limit if Pr is sufficiently small and the viscosity ratio is relatively large. This behavior is in contrast to previous results obtained by adopting the Boussinesq approximation where only stationary convection patterns persist. Finally a brief survey over real fluids is given and the resulting dynamics is described.

Onset and two-dimensional patterns of convection with strongly temperature-dependent viscosity

The onset of two‐dimensional convection with strongly temperature‐dependent viscosity has been considered for a fluid obeying an Arrhenius law. The critical Rayleigh number Rc and the basic features of the flow field at criticality have been identified based on a linear stability analysis. Convective flow patterns near and beyond criticality have been determined based on a direct numerical simulation. It is shown that, as the Rayleigh number R increases beyond Rc, steady rolls first emerge supercritically and that at sufficiently large values of R there is a secondary Hopf bifurcation corresponding to pulsating cells; the peculiar structure of the flow field in each case has been described.

The effect of thermal expansion on diffusion flame instabilities

In this paper we examine the effect of thermal expansion on the stability of a planar unstrained diffusion flame and provide a comprehensive characterization of diffusive thermal instabilities while realistically accounting for density variations. The possible patterns that are likely to be observed as a result of differential and preferential diffusion are identified for a whole range of parameters including the distinct Lewis numbers associated with the fuel and oxidizer, the initial mixture strength and the flow conditions. Although we find that thermal expansion has a marked influence on flame instability, it does not play a crucial role as it does in premixed combustion. It primarily affects the parameter regime associated with the onset of the instabilities and the growth rate of the unstable modes. Perhaps the most surprising result is that its has a different influence on the various modes of instability a destabilizing influence on the formation of cellular structures and a stabilizing influence on the onset of oscillations.

Long Wave and Short Wave Oscillatory Patterns in Rapid Directional Solidification

During directional solidification of dilute binary alloys the planar solid/liquid interface might be destabilized via mechanisms that involve corrugations of long or short wavelengths. For some operating conditions both instabilities are present at criticality; situations that lead to competition between different scales. Here we focus on the instabilities of Hopf type; combining nonequilibrium thermodynamics, thermal effects and weak flows in distinguished limits it is possible to retain most of the physics at criticality and to derive an integro-differential Ginzburg-Landau equation whose marginal stability curve does not have a parabolic profile. Firstly we analyze the stability characteristics of uniform wavetrains near Hopf bifurcations with non-standard marginal curves; a generalized Eckhaus instability criterion for long wave disturbances results. Secondly numerical simulations have been performed in cases where multiple minima are present along the marginal curve. They show that when the periodic patterns are unstable, the realized solution can either be the superposition of travelling and standing waves of long and short wavelengths respectively or have domain structures (localization) with fronts separating regions of different amplitudes and length scales. This example shows that a single generalized Ginzburg-Landau equation might describe qualitatively different microstructures which are observed in actual experiments.