Triple-flame propagation in a parallel flow: an analytical study

We present an analytical study of triple-flame propagation in a two-dimensional mixing layer against a parallel flow. The problem is formulated within a constant density thermo-diffusive model, and solved analytically in the asymptotic limit of large activation energy of the chemical reaction for flames thin compared with their typical radius of curvature. Explicit expressions are obtained in this limit, describing the influence of the flow on the triple-flame. The results are expected to be applicable when the ratio between the flow-scale and the flame-front radius of curvature (which is mainly dictated by concentration gradients) is of order unity, or larger. When this ratio is large, as in the illustrative case of a Poiseuille flow in a porous channel considered here, the flow is found to negligibly affect the flame structure except for a change in its speed by an amount which depends on the stoichiometric conditions of the mixture. On the other hand, when this ratio is of order unity, the flow is able to significantly wrinkle the flame-front, modify its propagation speed, and shift its leading edge away from the stoichiometric line. The latter situation is investigated in the illustrative case of spatially harmonic flows. The results presented describe, in particular, how the leading-edge of the flame-front can be determined in terms of the flow amplitude A which is critical in determining the flame speed. The latter is found to depend linearly on A in the first approximation with a correction proportional to the flame thickness multiplied by , for |A| sufficiently large. The effect of varying the flow-scale on flame propagation in this context is also described, with explicit formulae provided, and interesting behaviours, such as non-monotonic dependence on the scale, identified.     

Experimental and numerical study of a conical turbulent partially premixed flame

The structure and dynamics of a turbulent partially premixed methane/air flame in a conical burner were investigated using laser diagnostics and large-eddy simulations (LES). The flame structure inside the cone was characterized in detail using LES based on a two-scalar flamelet model, with the mixture fraction for the mixing field and level-set G-function for the partially premixed flame front propagation. In addition, planar laser induced florescence (PLIF) of CH and chemiluminescence imaging with high speed video were performed through a glass cone. CH and CH₂O PLIF were also used to examine the flame structures above the cone. It is shown that in the entire flame the CH layer remains very thin, whereas the CH₂O layer is rather thick. The flame is stabilized inside the cone a short distance above the nozzle. The stabilization of the flame can be simulated by the triple-flame model but not the flamelet-quenching model. The results show that flame stabilization in the cone is a result of premixed flame front propagation and flow reversal near the wall of the cone which is deemed to be dependent on the cone angle. Flamelet based LES is shown to capture the measured CH structures whereas the predicted CH₂O structure is somewhat thinner than the experiments.     

Flame propagation in a small-scale parallel flow

We consider the propagation of laminar premixed flames in the presence of a parallel flow whose scale is smaller than the laminar flame thickness. The study addresses fundamental aspects with relevance to flame propagation in narrow channels, to the emerging micro-combustion technology, and to the understanding of the effect of small scales in a (turbulent) flow on the flame structure. In part, the study extends the results of a previous analytical study carried out in the thick flame asymptotic limit which has in particular addressed the validity of Damköhler's second hypothesis in the context of laminar steady parallel flows. Several new contributions are made here.

Analytical contributions include the derivation of an explicit formula for the effective speed of a premixed flame U _T in the presence of an oscillatory parallel flow whose scale ? (measured with the laminar flame thickness δ _L ) is small and amplitude A (measured with the laminar flame speed U _L ) is (1). The formula shows a quadratic dependence on both the amplitude and the scale of the flow. The validity of the formula is established analytically in two distinguished limits corresponding to (1) frequencies of oscillations (measured with the natural frequency of the flame U _L /δ _L ), and to higher frequencies of (A/?) (the natural frequency of the flow). The analytical study yields partial support of Damköhler's second hypothesis in that it shows that the flame behaves as a planar flame (to leading order) with an increased propagation speed which depends on both the scale and amplitude of the velocity fluctuation. However our formula for U _T contradicts the formula given by Damköhler in his original paper where U _T has a square root dependence on the scale and amplitude.

Numerical contributions include a significant set of two-dimensional calculations which determine the range of validity of the asymptotic findings. In particular, these account for volumetric heat loss and differential diffusion effects. Good agreement between the numerics and asymptotics is found in all cases, both for steady and oscillatory flows, at least in the expected range of validity of the asymptotics. The effect of the frequency of oscillation is also discussed. Additional related aspects such as the difference in the response of thin and thick flames to the combined effect of heat loss and fluid flow are also addressed. It is found for example that the sensitivity of thick flames to volumetric heat loss is negligibly affected by the parallel flow intensity, in marked contrast to the sensitivity of thin flames. Interestingly, and somewhat surprisingly, thin flames are found to be more resistant to heat loss when a flow is present, even for unit Lewis number; this ceases to be the case, however, when the Lewis number is large enough.     

Spatiotemporal measurements of flame stretch and propagation rates for lean and rich CH4/air premixed flames interacting with a turbulent-wake

A 1.5 m long turbulent-wake combustion vessel with a 0.15 m × 0.15 m cross-sectional area is proposed for spatiotemporal measurements of curvature, strain, dilatation and burning rates along a freely downward-propagating premixed flame interacting with a parallel row of staggered vortex pairs having both compression (negative) and extension (positive) strains simultaneously. The wanted wake is generated by rapidly withdrawing an electrically-controlled, horizontally-oriented sliding plate of 5 mm thickness for flame–wake interactions. Both rich and lean CH₄/air flames at the equivalence ratios  = 1.4 and  = 0.7 with nearly the same laminar burning velocity are studied, where flame–wake interactions and their time-dependent velocity fields are obtained by high-speed, high-resolution DPIV and laser-tomography. Correlations among curvature, strain, stretch, and dilatation rates along wrinkled flame fronts at different times are measured and thus their influences on front propagation rates can be analyzed. It is found that strain-related effects have significant influence on front propagation rates of rich CH₄/air (diffusionally stable) flames even when the curvature weights more in the total stretch than the strain rate does. The local propagation rates along the wrinkled flame front are more intense at negative strain rates corresponding to positive peak dilatation rates but the global propagation rate averaged along the rich flame front remains constant during all period of flame–wake interaction. For lean CH₄/air (diffusionally unstable) flames, the curvature becomes a dominant parameter influencing the structure and propagation of the wrinkled flame front, where both local and global propagation rates increase significantly with time, showing unsteady flame propagation. These experimental results suggest that the theory of laminar flame stretch can be applicable to a more complex flame–wake interaction involving unsteadiness and multitudinous interactions between vortices.     

Front Propagation Techniques to Calculate the Largest Lyapunov Exponent of Dilute Hard Disk Gases

A kinetic approach is adopted to describe the exponential growth of a small deviation of the initial phase space point, measured by the largest Lyapunov exponent, for a dilute system of hard disks, both in equilibrium and in a uniform shear flow. We derive a generalized Boltzmann equation for an extended one-particle distribution that includes deviations from the reference phase space point. The equation is valid for very low densities n, and requires an unusual expansion in powers of 1/|ln n|. It reproduces and extends results from the earlier, more heuristic clock model and may be interpreted as describing a front propagating into an unstable state. The asymptotic speed of propagation of the front is proportional to the largest Lyapunov exponent of the system. Its value may be found by applying the standard front speed selection mechanism for pulled fronts to the case at hand. For the equilibrium case, an explicit expression for the largest Lyapunov exponent is given and for sheared systems we give explicit expressions that may be evaluated numerically to obtain the shear rate dependence of the largest Lyapunov exponent.     

Effect of the reversibility of the chemical reaction on triple flames

We examine a new aspect of triple flames, namely the effect of the reversibility of the chemical reaction on flame propagation. The study is carried out in the configuration of the two-dimensional strained mixing layer formed between two opposing streams of fuel and oxidiser. The chemical reaction is modelled as a single reversible reaction following an Arrhenius law in the forward and backward directions. The problem is formulated within the constant-density (thermo-diffusive) approximation, the main non-dimensional parameters relevant to this study being a reversibility parameter R (equal to zero in the irreversible case), a non-dimensional measure of the strain rate ? and a stoichiometric parameter S. We provide analytical (asymptotic) expressions for the local burning speed of the triple flame in terms of ?, S, and R. In particular we describe how the propagation speed of the front is decreased by an increase in R and how the location of its leading edge is affected by reversibility. For example, it is found that the leading edge shifts towards the fuel stream for S > 1 and towards the oxidiser if S < 1, as R is increased. A detailed numerical study is conducted covering all propagation regimes ranging from weakly stretched positively propagating (ignition) fronts to thick negatively propagating (extinction) fronts. In the weakly stretched cases we show that the numerics are in good agreement with the asymptotic findings. Furthermore, the results allow the determination of the domains of the distinct propagation regimes, mainly in the terms of R and ?. In line with our physical intuition, it is found that reversibility reduces the domain of ignition fronts and promotes extinction. The results provide a systematic investigation which can be considered as a first step when considering a more realistic chemistry, or poorly explored aspects (such as the existence of a temperature gradient in the unburnt mixture), when analyzing triple flames.     

Interfacial reaction barriers during thin-film solid-state reactions: The crystallographic origin of kinetic barriers at the NiS2/Si(111) interface

Epitaxial NiSi₂ islands have been grown on Si(111) substrates by the direct reaction of nickel vapour with the silicon substrate in ultra-high vacuum at 400° C. Growth kinetics was shown to depend on the orientation of the islands: A-oriented islands grow about ten times faster than B-oriented ones, with the ratio of the advance rates of the main growth fronts even reaching 30. Applying plan-view transmission electron microscopy and high-resolution electron microscopy of cross sections, a corresponding difference was found in the structure of the NiSi₂/Si(111) growth front: Steps at the B-oriented growth front were of three or six interplanar (111) spacings in height, whereas at the A-oriented growth front step-like defects of less than one interplanar (111) spacing in height were observed. These observations are explained by an atomic-scale model of the solid-state reaction, which involves the diffusion of nickel to the interfaces and the nucleation and subsequent lateral propagation of interfacial steps. The difference in the reaction kinetics originates from the presence of kinetic reaction barriers at the NiSi₂/Si(111) growth fronts, the barrier at the B-front being higher owing to the lower formation rate of steps of triple atomic height than that of steps of lower height at the A-NiSi₂/Si(111) growth front.     

Computational analysis of the steady states of the Sivashinsky model of hydrodynamic flame instability

In this work a numerical analysis of an integro-differential equation modelling the Darrieus–Landau instability of plane flame fronts was undertaken. A relatively new computational method based on saturated asymptotic approximations was used. Within the considered computational times a steady limiting shape of the flame front was not reached in large enough computational domains of size L?>?L_c . Instead, a smooth surface of an almost steadily shaped flame is repeatedly disturbed by small perturbations, resembling small cusps, appearing and disappearing randomly in time. The nature of these small cusps as well as of the steady limiting shape of the flame front was studied with a relatively new computational method.

The correlation between the critical length L_c and parameters of the computational algorithm and the computer precision was investigated. The calculations confirmed that, unlike the round-off errors, there is no significant link between the approximation accuracy of the algorithm and L_c . The obtained dependence of the critical length L_c on the magnitude of the round-off errors, considered as an external noise, was compared with the predictions given by other researchers. The agreement supports the idea of high sensitivity of solutions of the Sivashinsky equation to the external noise. A similarity between the appearance of small cusps on the surface of large enough flames governed by the Sivashinsky equation and streamwise streaks accompanying the loss of stability of the classic Hagen–Poiseuille flow was noted.     

Semi-analytical validation of a dynamic large-eddy simulation procedure for turbulent premixed flames via the G-equation

The performance of a dynamic subgrid model for the turbulent burning speed of a premixed flame is investigated for a series of idealized test cases where the flame front is wrinkled by a multiple-scale shear flow; a rigorous asymptotic subgrid model is also implemented for comparison. Explicit formulae for the flame wrinkled shape and turbulent speed are available to generate a reference database. The role of the subgrid wrinkling models is to achieve the same overall flame shape and propagation speed in a simulation where only the largest scales of the flow are explicitly accounted for. Very good results are obtained when the subgrid burning speed enhancement is estimated using the asymptotic subgrid model. On the other hand, the dynamic model attempts to exploit the scaling observable in the simulation to extrapolate the turbulent burning speed enhancement in the original system. The performance of this strategy is adequate for some regimes but poor for others; the source of the problem is traced back to the existence of a scaling transition that occurs as the flame propagating speed is adjusted during the large-eddy simulation. A modification to the scaling of the enhanced burning is implemented to account for the existence of the two distinct scaling ranges; it improves significantly the predictions of the dynamic model away from the transition, but results in the near-critical range remain predictably very poor compared with the rigorous asymptotic model results. These conclusions based on a priori performance for the reference steady data are confirmed by comparing unsteady large-eddy and direct simulations. Results based on rigorous mathematical tools are possible here because of the separation of length scales in the special class of idealized flow fields used in this study: their relevance to more realistic flows is also discussed.     

Stability of convective patterns in reaction fronts: a comparison of three models

Autocatalytic reaction fronts generate density gradients that may lead to convection. Fronts propagating in vertical tubes can be flat, axisymmetric, or nonaxisymmetric, depending on the diameter of the tube. In this paper, we study the transitions to convection as well as the stability of different types of fronts. We analyze the stability of the convective reaction fronts using three different models for front propagation. We use a model based on a reaction-diffusion-advection equation coupled to the Navier-Stokes equations to account for fluid flow. A second model replaces the reaction-diffusion equation with a thin front approximation where the front speed depends on the front curvature. We also introduce a new low-dimensional model based on a finite mode truncation. This model allows a complete analysis of all stable and unstable fronts.     

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.     

Diesel flame lift-off stabilization in the presence of laser-ignition: a numerical study

Diesel flame lift-off and stabilization in the presence of laser-ignition were numerically investigated with the method of Eulerian stochastic fields. The aim was to scrutinise the interaction between the lifted diesel flame and an ignition kernel upstream of the lifted flame. The numerical simulation was carried out in a constant-volume combustion vessel with n-heptane as fuel. The process was studied previously in an experiment employing Diesel #2 as the fuel in the same combustion vessel. In the experiment a lifted flame was first established at a position downstream of the nozzle. An ignition kernel was then initiated using a high-energy pulse laser at a position upstream of the natural lift-off position of the diesel flame. The laser-ignition kernel was modelled using a high-temperature (～2000 K) hot spot. In both experiment and simulations the upstream front of the ignition kernel was shown to remain around the initial laser ignition site for a substantially long period of time, while the downstream front of the ignition kernel propagates rapidly towards the natural lift-off position downstream of the laser ignition site. The lift-off position oscillated before the final stabilization at the natural lift-off position. The structures and the propagation speed of the reaction fronts in the laser-ignition kernel and the main flame were analysed. Two different stabilization mechanisms, the auto-ignition mechanism and the flame propagation mechanism, were identified for the naturally lifted flame and the laser-induced reaction front, respectively. A mechanism was proposed to explain the oscillation of the lift-off position.     

Study of transients in the propagation of nonlinear waves in some reaction diffusion systems

We study the transient dynamics of single species reaction diffusion systems whose reaction terms f(u) vary nonlinearly near u ≈ 0, specifically as f(u) ≈ u² and f(u) ≈ u³. We consider three cases, calculate their traveling wave fronts and speeds analytically and solve the equations numerically with different initial conditions to study the approach to the asymptotic front shape and speed. Observed time evolution is found to be quite sensitive to initial conditions and to display in some cases nonmonotonic behavior, ascribable to the disparity in time scales between the evolution of the front interior and the front tail.     

Role of large scale flow features on cycle-to-cycle variations of spark-ignited flame-initiation and its transition to turbulent combustion

This study investigates the influence of large-scale flow features, including flow structure and velocity magnitude, on the early-burn period variability in a homogenous-charge spark-ignited engine fueled with premixed propane-air mixture. Particle image velocimetry and in-cylinder pressure measurement data from a previous study - were processed to enable simultaneous flow characterization and flame-front tracking as well as apparent heat-release analysis. By combining probability analysis of flame development with conditional sampling of fast and slow early-burn cycles using 10% fuel mass fraction burned, it is shown that an undesirable flow structure produces an asymmetric flame development at the initial flame growth period. This asymmetric flame structure persists through the whole initial-to-turbulent transition period until the flame becomes fully turbulent. The undesirable flow condition is characterized by large-scale convective flows near spark plug rather than flows that lead to increased flame spread in multiple directions. The simultaneous flow and flame characterization enables the quantifications of flame-front propagation speed, unburned fuel-air mixture velocity ahead of flame front and local burning velocity at flame surface. Here the local burning velocity is referred to as laminar or turbulent flame speed. A simplified approach is introduced to derive integrated values for these quantities per crank-angle-degree, enabling the quantitative comparison of the trend-wise difference in these integrated metrics between fast and slow early-burn cycles. It is revealed that for the transition period, the CCV in the velocity magnitude of unburned fuel-air mixture ahead of the flame front accounts for nearly 50% to the variability of flame propagation speed. The burning velocity provides the remaining source of the flame propagation variability in this period. The flame propagation variations in the initial flame growth and fully turbulent periods are smaller than those in the transition period and are primarily dependent on the variability of large-scale flow features.     

Influence of heat losses on nonlinear fingering dynamics of exothermic autocatalytic fronts

Across traveling exothermic autocatalytic fronts, a density jump can be observed due to changes in composition and temperature. These density changes are prone to induce buoyancy-driven convection around the front when the propagation takes place in absence of gel within the gravity field. Most recent experiments devoted to studying such reaction-diffusion-convection dynamics are performed in Hele-Shaw cells, two glass plates separated by a thin gap width and filled by the chemical solutions. We investigate here the influence of heat losses through the walls of such cells on the nonlinear fingering dynamics of exothermic autocatalytic fronts propagating in vertical Hele-Shaw cells. We show that these heat losses increase tip splittings and modify the properties of the flow field. A comparison of the differences between the dynamics in reactors with respectively insulating and conducting walls is performed as a function of the Lewis number Le, the Newton cooling coefficient α quantifying the amplitude of heat losses and the width of the system. We find that tip splitting is enhanced for intermediate values of α while coarsening towards one single finger dominates for insulated systems or large values of α leading to situations equivalent to isothermal ones.     

Premixed turbulent flame front structure investigation by Rayleigh scattering in the thin reaction zone regime

Premixed turbulent flames of methane–air and propane–air stabilized on a bunsen type burner were studied using planar Rayleigh scattering and particle image velocimetry. The fuel–air equivalence ratio range was from lean 0.6 to stoichiometric for methane flames, and from 0.7 to stoichiometric for propane flames. The non-dimensional turbulence rms velocity, u′/S_L, covered a range from 3 to 24, corresponding to conditions of corrugated flamelets and thin reaction zones regimes. Flame front thickness increased slightly with increasing non-dimensional turbulence rms velocity in both methane and propane flames, although the flame thickening was more prominent in propane flames. The probability density function of curvature showed a Gaussian-like distribution at all turbulence intensities in both methane and propane flames, at all sections of the flame.The value of the term Dκ, the product of molecular diffusivity evaluated at reaction zone conditions and the flame front curvature, has been shown to be smaller than the magnitude of the laminar burning velocity. This finding questions the validity of extending the level set formulation, developed for corrugated flames region, into the thin reaction zone regime by increasing the local flame propagation by adding the term Dκ to laminar burning velocity.     

Surface tension driven flow on a thin reaction front

Surface tension driven convection affects the propagation of chemical reaction fronts in liquids. The changes in surface tension across the front generate this type of convection. The resulting fluid motion increases the speed and changes the shape of fronts as observed in the iodate-arsenous acid reaction. We calculate these effects using a thin front approximation, where the reaction front is modeled by an abrupt discontinuity between reacted and unreacted substances. We analyze the propagation of reaction fronts of small curvature. In this case the front propagation equation becomes the deterministic Kardar-Parisi-Zhang (KPZ) equation with the addition of fluid flow. These results are compared to calculations based on a set of reaction-diffusion-convection equations.     

Propagation and extinction of an unsteady spherical spray flame front

Experimental evidence seems to indicate that the life of a laminar spherical flame front propagating through a fresh mixture of air and liquid fuel droplets can be roughly split into three stages: (1) ignition, (2) radial propagation with a smooth flame front and (3) propagation with flame front cellularization and/or pulsation. In this work, the second stage is analysed using the slowly varying flame approach, for a fuel rich flame. The droplets are presumed to vaporize in a sharp front ahead of the reaction front. Evolution equations for the flame and evaporation fronts are derived. For the former the combined effect of heat loss due to droplet vaporization and radiation plays a dominant explicit role. In addition, the structure of the evaporation front is deduced using asymptotics based on a large parameter associated with spray vaporization. Numerical calculations based on the analysis point to the way in which the spray modifies conditions for flame front extinction. Within the framework of the present simplified model the main relevant parameters turn out to be the initial liquid fuel load in the fresh mixture and/or the latent heat of vaporization of the fuel.