Two-dimensional Flame Research Articles

Prediction of the steady flame spread rate by the principle of minimal entropy production

The propagation of a two-dimensional diffusive flame over a combustible material is studied by solving steady-state conservation equations written in the coordinate system fixed on the flame front. The analysis of a two-dimensional problem featuring flame spread rate as an eigenvalue has shown that there is no unique solution relative to the flame spread rate unless some additional condition is involved. A novel approach to the prediction of flame spread rate is proposed using the principles of irreversible thermodynamics. The steady flame propagation is considered as a stationary non-equilibrium thermodynamic state, which can be characterized, according to the formulation of Prigogine, by minimal entropy production. A numerical algorithm for the prediction of flame spread rate has been developed and tested on the investigation of downward flame spread over thin sheets of paper. The adequate physical background of the proposed approach and satisfactory agreement with experimental data have been shown.

Fundamental mechanisms in premixed turbulent flame propagation via flame–vortex interactions: Part I: experiment

A combined experimental and numerical study of the interaction of a two-dimensional vortex with planar laminar premixed flames has been carried out. In such a flow, the flame is subjected to time-varying strain and curvature and, hence, the interaction may be viewed as a model of fundamental processes occurring in premixed turbulent flames. Part I of the paper describes the experimental facility and diagnostics employed and presents results from the experimental investigation of effects of Lewis number, radiative heat losses, and unsteadiness on the interaction. A two-dimensional V-shaped laminar premixed flame is stabilized on a heated wire in a constant-area square duct. A two-dimensional vortex pair, generated from a slot in the duct wall, eventually interacts with one of the flame fronts. Schlieren and smoke flow visualization indicate that the flow field remains two-dimensional over a significant part of the flame–vortex interaction. This feature allows use of line-of-sight diagnostics, and, in particular, a CO ∗ 2 emission imaging technique for determination of quantitative heat release rates. Several fuel and oxidizer mixtures are employed in order to vary the Lewis number from 0.8 to 1.6 and to increase the heat loss parameter by a factor of 2.6. For the conditions investigated, the Lewis and Damköhler numbers are important controlling parameters in the evolution of the heat release rate and radiative losses may be neglected. In Part II of the paper, results from direct numerical simulations of a flame–vortex pair interaction are presented. The simulations were performed using a two-dimensional Navier-Stokes solver in which variable density and temperature-dependent transport coefficients are considered. The simulations utilize the initial conditions of the experiment and were used to investigate the role of multistep chemistry in the flame–vortex interaction.

The nonlinear equation for curved flames applied to the problem of flames in cylindrical tubes

The nonlinear equation for curved stationary flames of realistic expansion coefficients is solved numerically for the problem of flame propagation in cylindrical tubes. Two different configurations of a flame front corresponding to convex and concave flames are obtained. The convex and concave flames propagate with different velocities that depend on the tube radius and on the expansion coefficient of the burning matter. For tubes of a moderate radius the velocity amplification for convex flames exceeds the respective velocity amplification of two-dimensional flames almost twice. For tubes of a large radius unlimited increase of the curved flame velocity with increase of the tube width takes place. The obtained theoretical results are in good quantitative agreement with the results of numerical experiments on flame dynamics in cylindrical tubes.

Steady states of the surface of a nonadiabatic flame near the limits

Within the framework of a weakly linear model, which describes a nonadiabatic flame near the limit of its propagation caused by heat losses, steady states of the combustion-wave from are studied. Three-dimensional structures of the wave front are formed because of diffusion-thermal instability of the planar flame. The limits of propagation of a curved flame front are shown to expand if the diffusion-thermal instability is taken into account: a cellular flame can exist at heat losses higher than the critical value for the two-dimensional flame. The stability of steady solutions, which describe the cellular flame near the limits of its propagation, is studied. For sufficiently high heat losses, steady solutions for a nonadiabatic flame with front discontinities are obtained.

Diffusional-thermal instability in strained diffusion flames with unequal Lewis numbers

Diffusional–thermal instability is analysed for near-extinction counterflow diffusion flames to examine the instability characteristics of strained diffusion flamelets in turbulent flames, with the additional intention of providing a guideline to future experimental investigations. Attention is focused on the linear stability of the instability patterns appearing in the unstrained direction of two-dimensional counterflow diffusion flames, which is treated by the near-equilibrium regime of activation-energy asymptotics with Lewis numbers close to unity. The effects of unequal Lewis numbers for fuel and oxidizer are also taken into account by introducing an effective Lewis number. The resulting formulation describing linear stability of the harmonically decomposed disturbances turns out to be identical to the formulation derived previously for equal fuel and oxidizer Lewis numbers. For effective Lewis numbers less than unity, cellular instability is predicted for the entire range of the equivalence ratio, and the threshold Lewis number maintains a value slightly less than unity. On the other hand, for effective Lewis numbers sufficiently greater than unity, two types of oscillatory instabilities are found. As the effective Lewis number increases from unity, a travelling instability is first encountered for a range of finite wavelengths, and a pulsating instability emerges immediately above the travelling instability. These two types of oscillatory instabilities are predicted only for equivalence ratios sufficiently greater than unity because the threshold Lewis numbers for these instabilities are found to be infinity at unity equivalence ratio. For large values of the equivalence ratio, which is typical of most hydrocarbon flames, oscillatory instabilities are predicted only for flames burning extremely heavy hydrocarbon fuels or for flames heavily diluted by light inert gases.

An adaptive two-dimensional wavelet-vaguelette algorithm for the computation of flame balls

This paper is concerned with the numerical simulation of two-dimensional flame balls. We describe a Galerkin-type discretization in an adaptive basis of orthogonal wavelets. The solution is represented by means of a reduced basis being adapted in each time step. This algorithm is applied to compute the evolution of circular and elliptic thermodiffusive flames. In particular, we study the influence of the Lewis number, the strength of radiative losses and of the initial radius. The results show different scenarios. We find repeated splitting of the flame ball which is studied in some detail, extinction after excessive growth and also obtain quasi-steady flame balls.

Modeling of premixed turbulent flames with second moment methods

Turbulent premixed flames feature complex interactions between turbulent transport of scalars and chemical reaction as well as a strong coupling between velocity and scalar turbulence. The present work concerns the modeling of such flames at the second moment closure level and mainly focuses on the critical role of pressure redistribution/scrambling. It is shown that the common and exclusive practice of past work in this area, which involves a density weighted rewriting of models derived on a constant density basis, is insufficient. To resolve this issue a class of model extensions is proposed to account for variable density effects. The accuracy of the proposed approach is illustrated by consideration of incompressible premixed turbulent flames in the laminar flamelet regime of combustion. However, the proposed approach is general and does not preclude the consideration of other combustion regimes or the use of sophisticated closures for reaction related terms. The full closure is here applied to the simulation of transient one-dimensional flames and fully two-dimensional flames stabilized in counterflow geometries. The obtained level of agreement with experimental data proves most encouraging and constitutes a significant advance over past modeling efforts.

Role of Elementary Reactions in Flame Structure and Unsteady Behavior of Two-Dimensional Fuel Jet Diffusion Flame.

A direct numerical simulation of two-dimensional fuel jet flames developed in a co-flowing air stream was made considering rather complex finite chemistry in order to clarify the role of elementary reactions in flame structure and its unsteady behavior. The governing equations were discretized and numerically integrated using the finite volume method. The temperature dependence of thermodynamical properties was taken into account and the transport properties were calculated according to the simplified transport model proposed by Smooke [Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane-Air Flames, (1991), p.1-28, Springer-Verlag]. Chemical reactions were described by Smooke's skeletal methane-air reaction mechanism. It was found that large-scale fluctuations were produced in the downstream of the jet flame where a decrease in temperature occurred at some instant, leading to local extinction. Upon comparison with the results of a counterflow laminar diffusion flame, it was also found that the hypothesis of the laminar flamelet model could be accepted even for the case of the unsteady jet flame with local extinction. Furthermore, better understanding of the extinction mechanism was obtained : In the turbulent region of the flame, fuel is converted by large vortices into the reaction zone at high supply rates, inducing CH3, CH3O, CH2O and HCO production. At some instant, the increase in mixing rates will result in a decrease in temperature, causing the consumption of available active radicals due to the activity of some exothermic reactions, and the slowdown of the reactions producing new OH, O and H radicals. The radical pool is no longer available for chain reactions to proceed, resulting in an extinction.

The mechanism of two-dimensional pocket formation in lean premixed methane-air flames with implications to turbulent combustion

The mechanism of unburnt pocket formation in an unsteady two-dimensional premixed lean methane-air flame is investigated using direct numerical simulations. Theoretical results for nonlinear diffusion equations combined with analytical examples are used to interpret some of the results. Flame structure and propagation show three distinct stages of pocket formation: (1) flame channel closing involving head-on quenching of flames, (2) cusp recovery, and (3) pocket burnout. The flame channel closing and subsequent pocket burnout are mutual annihilation events that feature curvature, diffusion normal to the flame front, unsteady strain rate effects, and singularities in flame propagation and stretch rate. The results show that during channel closing and pocket burnout thermo-diffusive and chemical interactions result in the acceleration of the flames prior to annihilation; the time scales associated with the final stage of mutual annihilation and the initial stage of cusp recovery are significantly smaller than diffusive and convective time scales. As in earlier one-dimensional studies, the acceleration is attributed to enhanced diffusion and reaction rates, modifications to species profiles leading to shifts in balance between diffusion and reaction, and vanishing species and thermal gradients at the location in the channel where the pocket pinches off. Flame propagation and stretch rate are singular at this location. Enhanced radical production is initiated by a reversal of diffusion of H 2 towards the reaction zone during the early stages of thermo-diffusive interactions. Peak radical concentrations resulting from flame channel closing and pocket burnout exceed peak laminar values by as much as 25%. After the merging of the fuel consumption layers, radical production and flame structure shifts more towards an H 2/CO/O 2 system at the expense of hydrocarbon reactions. Species thermodiffusive interaction times are shorter than the unstrained one-dimensional counterpart due to unsteady strain and convection. Curvature effects on the flame propagation are prominent during pocket burnout and cusp recovery. The recovery stage shows strong dependence on diffusion of radicals left from the channel closing stage. This diffusion is amplified by the strong curvature of the flame cusp.

Modelling of two-dimensional flame spread across a sloping fuel bed

A significant contributing factor to wildland fire development is the slope effect which causes the fire spread rate to increase considerably as compared to horizontal spread. This leads to difficulties in determining the development of the fires hence in coordinating forest fighting efforts. In the present study, a two-dimensional non-stationary model for a fire spreading across a sloping fuel bed made up of Pinus pinaster litter is described. Based on a series of hypotheses, we first defined a medium equivalent to the pine needle litter for which we provided a thermal balance. By coupling this balance to a diffusion flame model we obtained the fire spread model numerically solved by means of the SIMPLEC procedure. The fire spread rates given by the simulations were then compared to experimental results generated by small-scale laboratory fires for a range of slope values. Predicted flow field structure, and temperature field are also discussed.

NO x Mechanisms in Rich Methane-Air Flames

Local production rates of NO have been derived from experiments in vertical flames on a burner commonly used in central heating equipment. The fuel-air equivalence ratio of around 1.9 caused a double-flame structure to develop above the burner, a phenomenon reflected by the temperature and OH concentration fields. The production rate of NO was a maximum in regions of low temperature and low OH concentration, located in the inner premixed flame and prior to the higher temperatures and OH concentrations of the outer diffusion flame. In addition, the peak rate of disappearance of NO was at the core of the premixed flame. These experimental results supported the assumption that the main mechanism of NO formation was the Fenimore Prompt route; also, an important internal “reburn” mechanism was operating. The opposed flow flame code was used here in the largest part of the simulations with the GRI 2.11 mechanism and with two additional mechanisms for a particular condition of flow and fuel-air equivalence ratio. The code was applied for fuel streams containing methane in air and oxidizer streams containing air only; the aim was to draw analogies with the experimental two-dimensional flame. Results from local rates of production and sensitivity analyses and quantitative reaction path diagrams (QRPD) for the nitrogenous species are presented for conditions representative of domestic appliances (cookers, space and water heaters, and central heating units) fired with natural gas. The co-flow experimental flame and the counter-flow theoretical flame at similar strain rates and fuel richness showed significant similarities.

A Thin Front Model Applied to Flame Propagation in Tubes

Velocities of stationary curved flames propagating in tubes with ideally adiabatic and slip walls are studied in the model of a thin flame front. It is found that for a tube width larger than some critical value a flame front acquires a curved shape and propagates with the velocity exceeding the velocity of a planar front. The curved flame velocity depends on the expansion coefficient and on the tube width. However, there is a maximal possible velocity of a curved stationary two-dimensional flame of a fixed expansion coefficient, which cannot be exceeded with the increase of the tube width.

Dimensional effects on the transition from ignition to flame spread in microgravity

Three-dimensional (3-D) and two-dimensional (2-D) simulations of the transition from radiative ignition on a solid fuel to flame spread in an imposed wind were performed in microgravity. Two-dimensional flames were found to quench (due to poor oxygen supply) more easily (i.e., at larger wind speeds) than 3-D flames. Results from the 2- and 3-D simulations were compared during the transition phase at wind speeds that ultimately lead to quenching of the 2-D flame but survival of the 3-D flame. In all locations near and in the flame, oxygen mass flux was larger in the 3-D flames and dominated by diffusion (as opposed to convection). Oxygen was supplied to the core of the 3-D flame due to diffusion from the sides of the flame (in a cross-wind direction). Diffusion in the 2-D flame was limited to directions parallel to the wind. This cross-wind diffusion was most significant at early times during transition when the flame was small and had a relatively large curvature. The 3-D flame, therefore, required less oxygen supply from an external wind to undergo transition to flame spread. Once flame spread was established there was little difference between the 3-D flames (in the centerline plane) and 2-D flames, due to the decreased curvature of the three-dimensional flame relative to the curvature during ignition and transition.

Numerical Study of NOx Emission in High Temperature Air Combustion.

The characteristics of NOx emission in the high-temperature air combustion technology were investigated numerically. Two kinds of methods were used. Detailed chemistry and transport properties were used for a two-dimensional laminar diffusion flame, and a simplified reaction scheme was used for calculating a turbulent flame in a furnace. Results show that the thermal mechanism dominates NOx emission in high-temperature air combustion. If only the preheated air temperature is increased, NOx emission will rise significantly. However, the combination of air preheating and flue gas recirculation not only improves the combustion efficiency, but also suppresses NOx emission in the combustion process.

Role of Elementary Reactions on Flame Structure and Unsteady Behavior of Two-Dimensional Fuel Jet Diffusion Flame.

A direct numerical simulation of two-dimensional fuel jet flames developed in a co-flowing air stream was made considering rather complex finite chemistry in order to clarify the role of elementary reactions on flame structure and its unsteady behavior. The governing equations were discretized and numerically integrated by using a finite volume method. The temperature dependence of thermodynamical properties was taken into account and the transport properties were calculated according to the simplified transport model proposed by Smooke et al. Chemical reactions were described by the Smooke's skeletal methane-air reaction mechanism. It was found that large-scale fluctuations were produced in the downstream where a decrease in temperature occurred at some instants leading to local extinction. Comparing with the results of counterflow laminar diffusion flame, it was also found that the hypothesis of laminar flamelet model can be accepted even for the case of the unsteady jet flame with the local extinction. Furthermore, a better understanding of extinction mechanism was obtained : In the turbulent region of the flame, fuel is converted by the large-scale vortices into the reaction zone at high supply rates inducing CH3, CH3O, CH2O and HCO production. At some instant, the increase of mixing rates will results in a decrease of temperature causing the consumption of available active radicals due to the activity of some exothermic reactions, and the slow-down of the reactions producing new OH, O, H radicals. The radical pool is no longer available for chain reactions to proceed, resulting in an extinction.

The Influence of Hydrodynamic Instability on the Structure of Cellular Flames.

The influence of hydrodynamic instability on the structure of two-dimensional (2D) and three-dimensional (3D) cellular flames is numerically investigated. The equation used is the compressible Navier–Stokes equation including a one-step irreversible chemical reaction. We superimpose an infinitesimal disturbance on the stationary plane flame and calculate the evolution of the disturbed flame front to obtain the relation between the growth rate and the wave number, i.e., the dispersion relation. With an increase in flame temperature, the growth rate increases since hydrodynamic instability becomes stronger. The unstable range normalized by the preheat zone thickness hardly changes, even though the flame temperature increases. The critical wave number, which corresponds to the maximum growth rate, is almost constant. Therefore, the normalized spacing between cells of the cellular flame is independent of the flame temperature. Moreover, we superimpose the disturbance with the critical wave number to investigate the structure of cellular flames. The stationary cellular flame is obtained when the inlet-flow velocity is set to the flame velocity of the cellular flame. The higher the flame temperature, the deeper the cell and the broader the flame surface. In addition, the cell depth and the surface area of the 3D flame are larger than those of the 2D flame. This is caused by the difference in the disposition of cells.

Artificial neural network-based low-dimensional model for spatio-temporally varying cellular flames

In general obtaining a mathematical model from experimental data of a system with spatio-temporal variation is a challenging task. In this article Karhunen-Loéve (KL) decomposition and artificial neural networks (ANN) are used to extract a simple and accurate dynamic model from video data from experiments of two-dimensional flames of a radial extinction mode regime. The KL decomposition is used to identify coherent structures or eigenfunctions of the system. Projections onto these eigenfunctions reduce the data to a small number of time series. The ANN is then used to process these time series. As a result a low-dimensional, nonlinear dynamic model is obtained.

Numerical Simulation of Curved Flames in Cylindrical Tubes

Dynamics of a curved flame front in cylindrical tubes is studied by means of numerical simulation of the complete system of hydrodynamic equations including thermal conduction, fuel diffusion, viscosity and chemical kinetics. The complete three-dimensional system of equations is reduced to a two-dimensional system with the account of cylindrical symmetry of the problem. The dependence of the flame velocity on the tube diameter and the expansion coefficient of the fuel is investigated. It is obtained that the velocity increase due to the curved shape of the flame front is considerably larger for the case of cylindrical tubes compared to the two-dimensional curved flames. The regime of strong initiation of the flame instability is obtained for narrow tubes, when all perturbation modes of small amplitude are stable, but a curved stationary flame is still possible. The simulation results indicate that there is no maximal velocity for curved flames in cylindrical tubes unlike two-dimensional flames. Instead the flame velocity increases with no limit with the increase of the tube diameter. The case of a flame propagating upwards in a gravitational field is also investigated.

Phase-conjugate resonant holographic interferometry applied to NH concentration measurements in a two-dimensional diffusion flame

Resonant holographic interferometry is a diagnostic technique based on the dispersion of light having a frequency close to that of an electronic transition of a molecule. We propose a novel single-laser, two-color setup for the recording of resonant holograms and apply it to two-dimensional (2D) species concentration measurements in a combustion environment. The generation of the second color is achieved by optical phase conjugation from stimulated Brillouin scattering in a cell. The frequency shift of ~8.5 GHz introduced by the phase conjugation matches approximately the linewidth of many molecular transitions at typical flame temperatures and can be implemented to produce holograms of good contrast and diffraction efficiency. Phase-conjugate resonant holographic interferometry is demonstrated in a 2D NH(3) -O(2) flame, yielding interferograms containing information on the NH radical concentration distribution in the flame. Experimental results are quantified by application of a numerical computation of the complex refractive index.

An evaluation of different contributions to flame stretch for stationary premixed flames

The concept of flame stretch is extended to study stationary premixed flames with a finite thickness. It is shown that the analysis results in additional contributions to the stretch rate due to changes in the flame thickness and due to density variations along the flame. Extended expressions are derived that describe the effect of stretch on variations in scalar quantities, such as the enthalpy. These expressions are used to determine local variations in the flame temperature, and it is shown that known results are recovered when a number of approximations are introduced. The extended stretch formalism might be useful to analyze and quantify the different flame stretch contributions and their effects in numerical flame studies. Finally, the different contributions to the total stretch rate and the effects thereof on the flame stabilization are numerically computed for the flame tip of a two-dimensional Bunsen flame as illustration.