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An improved embedded-manifold/Navier–Stokes numerical methodology is developed to simulate the propagation of premixed flames within the context of the hydrodynamic theory. The method is computatio...

The objective of this work is to provide physical insight into the mechanisms governing flame–turbulence interactions and explore the impact of the ubiquitous Darrieus–Landau instability on the propagation. It is based on the hydrodynamic theory of premixed flames that considers the flame thickness much smaller than all other fluidynamical length scales. In this asymptotic limit, the flame is thus confined to a surface whilst the diffusion and reaction processes occurring inside the flame zone are accounted for by two parameters: the unburned-to-burned density ratio and the Markstein length. The robust model, which is free of phenomenology and turbulence modelling assumptions, makes transparent the mutual interactions between the flame and the fluid flow, and permits examining trends in flame and flow characteristics while varying the turbulence intensity and mixture properties. It is used in this study to examine the morphological changes of the flame surface that result from the intertwined effects of the turbulence and instability, as demonstrated by the local displacement and curvature of the flame front, the extent of wrinkling and folding of the flame surface, and the overall flame brush thickness. It also provides a direct evaluation of the turbulent flame speed and its dependence on the mean flame curvature and on the hydrodynamic strain that it experiences. Also discussed are the effects of the flame on the flow by examining the various mechanisms of enstrophy and scalar gradient production/destruction, the degree of anisotropy created in the burned gas, and the restructuring of the vortical motion beyond the flame.

A numerical study is carried out to investigate the stabilization and dynamic properties of the edge flame formed in the wake of two merging streams, one containing fuel and the other oxidizer, separated by a splitter plate. Several plates are considered to illustrate the effects of their thermo-physical properties on the edge flame for low- and high-Lewis-number mixtures. The objective is to provide a comprehensive understanding of the effects of the heat recirculation cycle, from the edge flame through the splitter plate and back to the fresh reactants, on the edge flame. A diffusive-thermal model is adopted, with the flow field determined by solving the incompressible Navier–Stokes equations in the vicinity of the plate trailing edge, and the combustion field determined by solving the transport equations with a constant density. Two distinct modes of flame stabilization are identified and examined: a stationary mode, where the edge flame is held stationary at a well-defined distance, whether attached to or lifted from the tip of the plate, and an oscillatory mode where the edge flame undergoes sustained oscillations relative to an (unstable) equilibrium position. To characterize the heat recirculation effect under varying flow/mixture conditions, we introduce the thermal sensing distance, as a heuristic parameter that determines whether substantial thermal interaction between the edge flame and the plate occurs, and the average output heat flux, as a universal measure characterizing the efficiency of the heat recirculation cycle.

We study the propagation of premixed flames, in the absence of external turbulence, under the effect of both hydrodynamic (Darrieus–Landau) and thermodiffusive instabilities. The Sivashinsky equation in a suitable parameter space is initially utilized to parametrically investigate the flame propagation speed under the potential action of both kinds of instability. An adequate variable transformation shows that the propagation speed can collapse on a universal scaling law as a function of a parameter related to the number of unstable wavelengths within the domain nc. To assess whether this picture can persist in realistic flames, a DNS database of large scale, two-dimensional flames is presented, embracing a range of nc values and subject to either purely hydrodynamic instability (DL) or both kinds of instability (TD). With the aid of similar DNS databases from the literature we observe that when adequately rescaled, propagation speeds follow two distinct scaling laws, depending on the presence of thermodiffusive instability or lack thereof. We verify the presence of secondary cutoff values for nc identifying (a) the insurgence of secondary wrinkling in purely hydrodynamically unstable flames and (b) the attainment of domain independence in thermodiffusively unstable flames. A possible flame surface density based model for the subgrid wrinkling is also proposed.

Edge flames are of fundamental importance to the attachment and stabilization of diffusion flames in many practical devices, and serve as transient structures for establishing combustion in locally...

The flame displacement speed (FDS) and flame consumption speed (FCS) are commonly used in numerical studies to characterize flame dynamics. Although for a planar configuration they are both well defined and accurately represent the propagation speed of the flame into the combustible mixture, their definition in more general circumstances is ambiguous. The FDS and FCS are local quantities: the FDS is associated with the displacement of an arbitrarily selected iso-surface, and the FCS is an integrated quantity throughout a region that needs to be approximated, in a direction that is not always uniquely defined. The only unambiguously defined quantity is the global (volumetric) consumption rate obtained by integrating the rate of reactant consumption over the entire combustion volume. However, using it to determine the FCS requires a proper identification of the flame surface area which introduces uncertainty in the results. Indeed, numerical simulations show that combustion properties depend significantly on the choices made in the determination of the FDS and FCS. In order to utilize these quantities in a meaningful way, their limitations are explored by providing a detailed comparison between predictions of numerical simulations and theoretical expressions obtained for weakly-stretched flames. The theory is based on the assumption that the flame is thin relative to the representative hydrodynamic length scale and in this asymptotic limit both, the FDS (commonly referred to as the flame speed) and the FCS, are uniquely and unambiguously defined. Two configurations are examined in this paper: (i) spherical flames, unsteady expanding as well as stationary, where the flow is unidirectional and (ii) steadily propagating cusp-like flames (resulting from the Darrieus-Landau instability) whose structures are spatially varying and where the flow through the flame is nonuniform. The presented comparison validates the accuracy of the asymptotic expressions for the dependence of the FDS and FCS on stretch for one-step chemistry, and demonstrates that the theoretical predictions remain qualitatively, and to a large extent quantitatively valid for detailed chemistry for both, lean and rich flames.

The morphological development of wrinkles along the surface of a Bunsen flame in a weakly-turbulent flow is investigated, with turbulence added solely to disturb the flame front. The resulting flame-flow interactions are examined using a hybrid Navier–Stokes/front-tracking methodology within the context of the hydrodynamic theory. Topological markers based on the skewness of curvature are introduced to distinguish between sub- and super-critical conditions, or the absence/presence of the Darrieus–Landau instability, respectively. We show that for sub-critical conditions disturbances created along the flame surface are dampened when convected downstream along the flame front, and the flame surface is only weakly perturbed. For super-critical conditions, on the other hand, disturbances of the flame front are amplified when advected downstream leading to a highly corrugated surface and a flame brush of increasing thickness. A measure of these dramatic changes is included in the mean local stretch rate which, when properly modulated by a Markstein length, directly affects the turbulent flame speed.

A fundamental study aimed at investigating the stabilization characteristics of edge flames established in the near-wake of two merging streams, one containing fuel and the other oxidizer, is presented, with the main focus placed on the effects of the thermal interaction between the flame and the splitter plate. To this end, a diffusive-thermal model characterized by constant gas density and transport coefficients is used for conditions at which flame liftoff is likely to occur. It is assumed that the incoming streams are of equal strain rates, that the fuel and oxidizer are supplied in stoichiometric proportion, and that the mass diffusivities of the reactants are equal, such that the resulting combustion field is symmetric with respect to the centerline extending from the splitter plate. The results indicate that the plate has a negligible effect on the edge flame unless the tip of the plate intrudes into the preheat zone of the curved premixed flame segment forming the edge flame. In an overall adiabatic system, the heat conducted from the flame to the plate is completely recirculated back to the reactants via the lateral surfaces of the plate, thus supporting an excess enthalpy flame in the near-wake. The average output heat flux, defined as the total heat output through the lateral surfaces of the plate divided by the characteristic length associated with the temperature variation along the plate, is identified as an appropriate measure to characterize the heat recirculation efficiency.

The propagation of outwardly expanding premixed flames in turbulent media is examined within the context of the hydrodynamic theory wherein the flame, treated as a surface of density discontinuity separating fresh combustible mixture from burned products, is propagating at a speed dependent upon local geometric and mixture/flow characteristics. An embedded manifold approach, one adept at handling multi-valued and disjointed surfaces which are frequently observed in real flames, is used to couple the flow and flame evolution. A sensitivity analysis, based on mixtures with different Markstein numbers, is performed to investigate early flame kernel development in addition to its long-term evolution. The focus is to understand the effect of turbulent flow characteristics, distinguished by the intensity of velocity fluctuations and its integral length scale, in addition to intrinsic flame instabilities (predominantly the Darrieus-Landau instability) on flame propagation. The overarching objective is to quantify their influence on the flame morphology and burning rate and to construct scaling laws for the turbulent flame speed through appropriate modifications of Damkohler’s first hypothesis. Flame-turbulence interactions are inferred from statistical quantities based on its developing flame topology, including local flame curvature and hydrodynamic strain, and their combined effects integrated into the flame stretch rate experienced by the flame and the local flame speed deviation from the laminar flame speed.

This study focuses on the dynamics of compressibility-driven flames that emerge in narrow tubes, closed at their ignition end, when conductive heat losses through the walls are appreciable. A narrow gap approximation is used to reduce the governing equations to an effectively one-dimensional problem. In long channels this problem admits traveling-wave solutions which we have investigated numerically for finite values of the Zel’dovich number, and asymptotically in the limit of large Zel’dovich numbers. In particular, we describe the flame structure and the dependence of the propagation speed on the physico-chemical parameters, including the heat loss and compressibility parameters, and examine the transition from compressibility-driven to isobaric flames when systematically reducing the representative Mach number.

The hydrodynamic instability, which results from the large density variations between the fresh mixture and the hot combustion products, was discovered by Darrieus and Landau over seventy years ago, and has been named after its inventors. The instability, which prevents flames from being too flat, was thought to lead immediately to turbulent flames. Recent studies, initiated by weakly nonlinear analyses and extended by two-dimensional simulations suggest that this is not the case. It was established that the flame beyond the onset of instability, develops into a cusp-like structure pointing towards the burned gas region that propagates at a speed substantially larger than the laminar flame speed. In this work, we present for the first time a systematic study of the bifurcation phenomena in the more realistic three-dimensional flow. The computations are carried out within the context of the hydrodynamic theory where the flame is treated as a surface of density discontinuity separating burned gas from the fresh mixture, and propagates at a speed that depends on the local curvature and hydrodynamic strain rate. A low Mach-number Navier–Stokes solver modified by an appropriate source term is used to determine the flow field that results from the gas expansion and the flame is tracked using a level-set methodology with a surface parameterization method employed to accurately capture the local velocity and stretch rate. The numerical scheme is shown to recover the known exact solutions predicted in the weak gas expansion limit and corroborates the bifurcation results from linear stability analysis. The new conformations that evolve beyond the instability threshold have sharp crest pointing towards the burned gas with ridges along the troughs, and propagate nearly 40% faster than planar flames. Indeed, the appearance of sharp folds and creases, which are some manifestations of the Darrieus–Landau instability, have been observed on the surface of premixed flames in various laminar and turbulent settings.

In this work we re-examine the counterflow diffusion flame problem focusing in particular on the flame–flow interactions due to thermal expansion and its influence on various flame properties such ...

Experiments of non-premixed combustion in micro-channels have exhibited different modes of burning. Typically, a diffusion flame is established along or near the axis of a channel spanning the entire mixing layer. It separates a region of fuel and no oxidiser from a region with only oxidiser. Often, however, a periodic sequence of extinction and reignition events, termed collectively as “diffusion flame-streets”, are observed. They constitute a series of separate diffusion flames, each with a tribrachial edge flame structure that is stabilised along the channel. The current work focuses on understanding the underlying mechanism responsible for these unique observations. Numerical simulations were conducted in a thermo-diffusive limit to examine the effects of confinement and heat loss on flames in three dimensional micro-channels with low aspect ratios. An asymptotic analysis was used to reduce the mathematical equations into a two-dimensional problem which effectively captured the three dimensional nature of the combustion process. Two key burning regimes were identified: (i) stable continuous diffusion flames and (ii) stable diffusion flame-streets. The transition between regimes is demarcated primarily by the Damkohler number, defined as the ratio of a diffusion time to a chemical reaction time, but is also influenced by the extent of heat loss. Occasionally within the diffusion flame-street regime, the residual mixture would reignite but would fail to evolve into stationary auxiliary flames. This was generally observed at low flow-rates for Reynolds numbers below a critical value. The behaviour appeared to be periodic in time with a frequency that depended on the removal from criticality.

The propagation of premixed flames in long adiabatic channels is investigated when a combustible mixture is ignited at their close end while the other end remains open to atmospheric pressure. This constraint conditions the flow produced by gas expansion near the flame. The burned gas trapped between the flame and the closed end comes eventually to rest, while the flow sets in the fresh mixture escapes freely at the far end of the channel. Seeking for traveling wave solutions, we find that two possible solutions, corresponding to slow and fast steadily propagating flames, exist under appropriate conditions. The critical conditions are determined when the two solutions merge, and depend on the channel width, the heat release and the Lewis number. Beyond criticality, steadily propagating flames in channels closed at their ignition end are not possible. Numerical simulations of the time-dependent equations in sufficiently long channels confirm the existence of a steady propagation mode, always corresponding to the slow flame solution. Beyond criticality, the flame always accelerate as it travels down the channel.

Diffusion flames established in inert porous media have been reported to present temperatures lower than a comparable gaseous mixture. Therefore, the study of the flame structure, temperature and extinction limits of confined diffusion flames is of importance. In the present work we discuss extinction conditions for such flames. Using an asymptotic model that accounts for the excess/deficient enthalpy at the reaction region, we study the multiscale problem and analyze the effects of the heat exchange between gas and solid phases on the flame structure. When the heat removed from the flame by the solid matrix is large, the flame can extinguish because the lowering in the flame temperature leads to increasingly large leakage of reactants through the flame sheet. We show that this occurs when the porosity or the mass injection rate is small enough. The extinction limit associated with a small value of the mass injection rate adds to the kinetic extinction limit (which is associated with a large value of the mass injection rate) to characterize a dual-extinction-point behavior for this problem. When the porosity of the medium reaches a minimum critical value, these two distinct extinction points collapse, such that for porosities lower than the critical porosity no flame can be established inside the porous chamber. Then, it is possible to construct a flammability map for the confined diffusion flame, where the critical porosity defines an absolute flammability limit.

Here, outwardly propagating spherical hydrogen/air flames are examined theoretically and experimentally with respect to flame propagation speed and the onset of instabilities which develop due to thermal expansion and non-equal diffusivities. Instabilities increase the surface area of the spherical flame, and hence the flame propagation speed. The theory applied here accounts for both hydrodynamic and diffusive-thermal effects, incorporating temperature dependent transport coefficients. Experiments are performed in a spherical combustion chamber over a wide range of equivalence ratios (0.6–2.0), initial temperatures (298–423 K), and initial pressures (1 atm to 15 bar). The evolution of the flame propagation speed as a function of flame radius is compared to predictions from theory showing excellent agreement. Also the wrinkling of hydrogen/air flames is examined under increased pressure and temperature for various equivalence ratios. Critical flame radii, defined as the point of transition to cellular flames, are extracted from high-speed Schlieren flame imaging. Overall, the critical radius is found to decrease with increasing pressure. The predictions yield the growth rate of small disturbances and the critical flame radius. Experimental flame radii, as expected, are underpredicted by the theoretical findings. Experimental data are provided in the form of an approximation formula.

This paper, dedicated to Norbert Peters, follows his lead in developing theoretical understanding of the complex flow-turbulence interactions occurring in the propagation of premixed flames. The work is based on the asymptotic hydrodynamic model of premixed flames, where the flame is modeled by a surface that separates unburned and burned gases and propagates relative to the incoming flow at a speed that depends locally on the flame stretch rate. As such the work may be categorized as corresponding to the “flamelet regime”, based on the turbulent combustion regimes diagram. The results at the present are limited to mixtures corresponding to positive Markstein lengths, and to “two-dimensional turbulent flows”. In this parametric study, the different factors affecting the turbulent flame speed have been examined and scaling laws for the turbulent flame speed are proposed for low-to-moderate turbulence intensities that highlight the dependence on physically measurable quantities. Comparison to various empirical correlations suggested in the literature is presented. The results, devoid of turbulence-modeling assumptions and/or ad-hoc coefficients, can help explaining the influence of varying the system parameters individually and collectively, and formulating physically-based small-scale models for large-scale numerical simulations of turbulent flames.

We examine the dynamics of premixed flames in long, narrow, adiabatic channels focusing, in particular, on the effects of gas compressibility on the propagation. Recognising the importance of the boundary conditions, we examine and compare three cases: flame propagation in channels open at both ends, where the pressure must adjust to the ambient pressure at both ends and the expanding gas is allowed to leave the channel freely, and flame propagation in channels that remain closed at one of the two ends, where the burned/unburned gas remains trapped between the flame and one of the two walls. Earlier studies have shown that a flame accelerates when travelling down a narrow channel as a result of the combined effects of wall friction and thermal expansion. In the present work we show that compressibility effects enhance the transition to fast accelerating flames in channels open at both ends and in channels closed at the ignition end. In both situations, the accelerating flames could reach values that, depe...

The effects of an externally applied electric field on the burning characteristics of a spherically symmetric fuel drop, including the flame structure, flame standoff distance, mass burning rate and flame extinction characteristics of the diffusion flame are studied. A reduced three-step chemical kinetic mechanism that reflects the chemi-ionization process for general hydrocarbon fuels has been proposed to capture the production and destruction of ions inside the flame zone. Due to the imposed symmetry, the effect of the ionic wind is simply to modify the pressure field. Our study thus focuses exclusively on the effects of Ohmic heating and kinetic effects on the burning process. Two distinguished limits of weak and strong field are identified, highlighting the relative strength of the internal charge barrier compared to the externally applied field. For both limits, significantly different charged species distributions are observed. An increase in the mass burning rate is noticed with increasing the strength of the electric field in both limits, with a small change in flame temperature. Increasing external voltages pushes the flame away from the droplet and causes a strengthening of the flame with a reduction in the extinction Damkholer number.

We examine in this study the structure and dynamic properties of an edge flame formed in the near-wake of two initially separated shear flows, one containing fuel and the other oxidiser. A comprehensive study is carried out within the diffusive-thermal framework where the flow field, computed a-priori, is used for the determination of the combustion field. Our focus is on the effects of three controlling parameters: the Damkohler number controlling the overall flow rate, the oxidiser-to-fuel strain rate ratio of the supply streams that determines the extent of oxidiser entrainment towards the mixing zone, and the Lewis number, assumed equal for the fuel and oxidiser, that depends on the mixture composition. Response curves, representing the edge flame standoff distance as a function of Damkohler number, exhibit two distinct shapes: C-shaped and U-shaped curves characterising the response of low and high Lewis number flames, respectively. Stability considerations show that the upper solution branch of the ...

The characteristics of a steady diffusion flame that arises at the interfaces of two condensed phase reactant streams that form an opposed counterflow are discussed. We assume that the flow is due to deformation from compaction or local heating and thermal expansion processes in the microscale environment of composite energetic materials. As a representative example of high temperature combustion of metal/intermetallic reactants, the overall reaction of titanium and boron to create titanium diboride products is considered under near isobaric conditions. The multi-component diffusion description uses a generalized Fick formulation with coefficients related to the binary diffusivities defined in the Maxwell–Stefan relations. A fairly simple depletion form with Arrhenius temperature dependent coefficients is used to describe the reaction rate. Several types of analyses are carried out at increasing levels of complexities: an asymptotic analysis valid in the limit of low strain rates (high residence time in the reaction zone), a constant mixture density assumption that simplifies the flow description, diffusion models with equal and unequal molecular weights for the various species, and a full numerical study for finite rate chemistry, composition-dependent density and strain rates extending from low to moderate values. All are found to agree remarkably well in describing the flame structure, the flame temperature and the degree of incomplete combustion. Of particular importance is the determination of a critical strain rate beyond which steady burning may no longer be observed. The analysis has a general character and can be applied to other condensed phase energetic material systems, where reaction and diffusion occur in the presence of flow and material deformation.

Propagation of premixed turbulent flames is examined using a hybrid Navier–Stokes/front tracking methodology, within the context of a hydrodynamic model. The flame, treated as a surface of density discontinuity separating the burned and unburned gases, propagates relative to the fresh mixture at a speed that depends on the local mixture (through a Markstein length) and flow conditions (through the stretch rate), and the flow field is modified in turn by gas expansion; only positive Markstein length are considered, where thermo-diffusive instabilities are absent. Depending on the Markstein length, we have identified in a previous publication two modes of propagation – sub-critical and super-critical, based on whether the effects of the Darrieus–Landau instability are absent or dominant, respectively. The results were limited to low turbulence intensities where the mathematical representation of the flame front was based on an explicit single-valued function. In the present paper we utilize a generalized representation of the flame surface that allows for multivalued and disjointed interfaces, thus extending the results to higher turbulence intensities. We show that when increasing the turbulence intensity the influence of the Darrieus–Landau instability on the super-critical mode of propagation progressively decreases and in the newly identified highly-turbulent regime the flame is dominated completely by the turbulence for all values of Markstein numbers; i.e., with no distinction between sub- and super-critical conditions. Primary importance is given to the determination of the turbulent flame speed and its dependence on turbulence intensity which, when increasing the turbulence level, transitions from a quadratic to a sub-linear scaling. Moreover, the exponent of the sub-linear scaling for the turbulent flame speed is generally lower than the corresponding exponent for the scaling of the flame surface area ratio, which is often used for experimentally determining the turbulent flame speed. We show that the leveling in the rate of increase of the turbulent flame speed with turbulence intensity, is due to frequent flame folding and detachment of pockets of unburned gas that cause a reduction in the average main surface area of the flame, while the lower exponents in the scaling law for the turbulent flame speed compared to that of the flame surface area ratio is due to flame stretching. Disregarding the effect of flame stretch for mixtures of positive Markstein length results in overestimating the turbulent flame speed. Finally, we characterize the flame turbulence interaction via quantities such as the mean vorticity and mean strain, illustrating the effects of incoming turbulence on the flame and the modification of the flow by the flame on the unburned and burned sides.

The definition of the flame displacement speed (FDS), often used to characterize the dynamical properties of premixed flames, is generally ambiguous because, except for a steadily propagating planar flame, the mass flow rate through the combustion region varies with distance through the flame and one is therefore faced with the difficulty of choosing a proper iso-surface to represent the flame surface. A directly related issue is the determination of the proportionality coefficient in the linear flame speed-flame stretch relation of weakly-stretched flames, known as the Markstein length, which depends strongly on the location inside the flame zone where it is measured or calculated. The objective of the present study is to identify an iso-surface and thereby a definition of the FDS that is well conditioned and less prone to uncertainties, and a consistent and unambiguous expression for the Markstein length. With a selected isotherm to represent the flame surface, the two most common definitions of the FDS are based either on the energy equation with the temperature as the progress variable, or on the kinematic characteristics of the surface (the propagation speed relative to the flow). In this study we examine the spherical flame geometry, a setup that provides an independent determination of the FDS that is not contingent upon an arbitrary selection of the flame surface and thus permits a proper evaluation of the two FDS definitions. A large number of simulations of premixed spherical propane/air flames with equivalence ratio ranging from 0.8 to 1.4 were carried out at various temperatures and pressures using both global single-step and detailed reaction schemes. Outwardly propagating spherical (or cylindrical) flames and inwardly propagating stationary spherical flames were examined. The dependence of the flame speed and flame temperature on stretch, and the corresponding Markstein length were identified for different isotherms selected to represent the flame surface, and the results were carefully compared to the asymptotic theory of weakly-stretched flames. The excellent agreement between theory and simulations provides a clear explanation and quantification of the differences found between the trends in the flame speed-flame stretch relation and the corresponding Markstein lengths, exhibited when the FDS was calculated based on an isotherm in the burned or unburned sides of the flame. We show that the proper isotherm for the evaluation of the FDS which is well-conditioned and properly accounts for the physics must be sufficiently close to the burned side of the flame. This choice is less prone to uncertainty, as the slope of the flame speed-flame stretch relation when reaching the burned side of the flame becomes less dependent on the selected iso-value. On the other hand, the choice of the fresh combustible mixture temperature as a reference location for the calculation of the FDS, and the corresponding unburned Markstein length, is ill conditioned and should be avoided.

Extensive numerical simulations with detailed chemistry and transport are performed to identify the range of dominance (in terms of equivalence ratio and domain size) of the hydrodynamic instability, the shape of the structures that evolve at long times, and their propagation speed. The calculations were performed in two-dimensional domains of lateral extent 3–100 flame thicknesses. Hydrogen/air mixtures ranging from rich ( ϕ = 2 ) to lean conditions ( ϕ = 0.5 ) were considered, expecting that thermo-diffusive effects will start becoming important only at the lean end. The initial growth of a perturbed planar flame front is found to agree qualitatively, and to a large extent even quantitatively, with the asymptotic theoretical predictions. Beyond linearity it is shown that the dynamics depend strongly on the equivalence ratio (or on the effective Lewis number of the mixture) and the domain lateral size. For stoichiometric and rich mixtures, the flame shape is generally characterized by a single-cusp structure that propagates at a constant speed. The propagation speed increases with increasing lateral domain size and asymptotes to a value nearly 24% larger than the laminar flame speed. For the lean mixtures, the flame does not assume a well-defined structure even after a long time. It is regularly contaminated by small cells that result from thermo-diffusive effects and cause a significant increase in the propagation speed (nearly 60% above the laminar flame speed) that varies continuously in time. Except for the lean cases, the simulation results compare well with the asymptotic hydrodynamic theory both in the flame shape and propagation speed.

In this work we extend our earlier asymptotic one-dimensional analysis of flame propagation in long narrow channels open at both ends to two-dimensional flames. The analysis follows two tracks; a multi-scale asymptotic study and a full numerical study of the unsteady propagation. We show that during the early stages of propagation the flame accelerates at a nearly constant rate, independent of the channel height. In sufficiently narrow channels, the flame retains a constant acceleration until it reaches the end of the channel, consistent with our earlier work. In wider channels, however, the flame beyond a certain distance begins to accelerate at a nearly-exponential rate, reaching exceedingly large speeds at the end of the channel. The flame self-acceleration arises from the combined effects of gas expansion and lateral confinement. The gas expansion that results from the heat released by the chemical reactions produces a continuous flow of burned gas directed towards the ignition end of the channel. Due to the frictional forces at the walls and, since the pressure at both ends is maintained constant, the gas motion that develops in the burned gas sets a pressure gradient that drives the fresh unburned gas towards the other end of the channel. Stretching out to reach additional fuel, the flame extends towards the fresh mixture propagating faster. And because of lateral confinement, the gas expansion induces large straining on the elongated flame surface that further increases its propagation speed. The asymptotic approximation properly predicts the initial propagation stage, the location within the channel where the sudden acceleration begins and the early stages of the self-accelerating process. The full numerical study confirms and extends the asymptotic results, showing that in long but finite channels premixed flames could self-accelerate reaching velocities that are ten-to-twenty times larger than the laminar flame speed.

The turbulent propagation speed of a premixed flame can be significantly enhanced by the onset of Darrieus-Landau (DL) instability within the wrinkled and corrugated flamelet regimes of turbulent combustion. Previous studies have revealed the existence of clearly distinct regimes of turbulent propagation, depending on the presence of DL instabilities or lack thereof, named here as super- and subcritical respectively, characterized by different scaling laws for the turbulent flame speed. In this study we present experimental turbulent flame speed measurements for propane/air mixtures at atmospheric pressure, variable equivalence ratio at Lewis numbers greater than one obtained within a Bunsen geometry with particle image velocimetry diagnostics. By varying the equivalence ratio we act on the cut-off wavelength and can thus control DL instability. A classification of observed flames into sub/supercritical regimes is achieved through the characterization of their morphology in terms of flame curvature statistics. Numerical low-Mach number simulations of weakly turbulent two-dimensional methane/air slot burner flames are also performed both in the presence or absence of DL instability and are observed to exhibit similar morphological properties. We show that experimental normalized turbulent propane flame speeds ST/SL are subject to two distinct scaling laws, as a function of the normalized turbulence intensity Urms/SL, depending on the sub/supercritical nature of the propagation regime. We also conjecture, based on the experimental results, that at higher values of turbulence intensity a transition occurs whereby the effects of DL instability become shadowed by the dominant effect of turbulence. © 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

The role of thermal expansion in diffusion edge flames is investigated numerically in the strained mixing layer configuration. We are able to access both the positive as well as the negative edge flame speed regime when density variation, and therefore full hydrodynamic coupling, is present. The computational approach employs a homotopy method to gradually map the solutions from the computationally simpler constant-density flow to the more challenging variable-density case. Particular attention is paid to the role of boundary conditions and how they, in turn, can induce an undesirable streamwise pressure gradient in the trailing diffusion flame that affects the edge flame speed. A new approach is designed to eliminate this adverse pressure gradient. Previous studies observe that the ratio of the edge flame speed to the premixed stoichiometric laminar flame velocity scales approximately as the square root of the ratio of the cold stream density to the stoichiometric density (which is lower). At least for the small set of parameters investigated here, it is found that the speedup of the normalized edge flame velocity might be superlinear on the density ratio, a fact we attribute to the lack of pressure gradient behind the edge flame. This result is new and complements previous results, for different boundary conditions, which strongly suggest that the edge flame speed is a strong function of the particular hydrodynamic boundary conditions employed in the simulations.

Premixed syngas/air flames expanding in turbulent flow fields are investigated using large scale direct numerical simulations. A parametric analysis is performed in circular and spherical geometries for the detailed investigation of the interaction between the flame front and the turbulent flow field. A stoichiometric syngas-air mixture with molar ratio \({ CO}/H_2=3\) is considered at conditions relevant to internal combustion engines. The dependence of the integral heat release rate on the characteristics of the flow field (integral length scale and turbulent intensity) is discussed. The long-term evolution of important global flame quantities is analyzed, and the mechanisms that dominate the growth of the flame kernel are identified. An expression for the speed of turbulent premixed spherical flames is formulated, based on the rate of change of the surface area of the flame.

We present a Gibbs free energy approach to modeling reaction fronts in condensed phase reactive materials. The current interest is in chemical reactions of condensed phase reactants that are initially separated. In energetic materials such reactions are observed to occur extremely fast and at relatively sharp fronts. The condensed phase combustion process differs in several aspects from classical gaseous combustion due to the disparity between the characteristic thermal conductivity length and the mass diffusion lengths and a volume, temperature, stress, mass fraction equation of state that principally depends only on the component reference volumes and the current mixture composition. To retain a simple planar configuration, we consider the two reactants, in solid phase, are in motion towards each other characterized by counter-flow geometry. We apply the model to a simplified Titanium-Boron system and present the analysis of reaction zone length for various strain rates. The numerical results are valida...

We study the propagation of premixed flames in long but finite channels, when the mixture is ignited at one end and both ends remain open and exposed to atmospheric pressure. Thermal expansion produces a continuous flow of burned gas directed away from the flame and towards the end of the channel where ignition took place. Owing to viscous drag, the flow is retarded at the walls and accelerated in the center, producing a pressure gradient that pushes the unburned gas ahead of the flame towards the other end of the channel. As a result the flame accelerates when it travels from end to end of the channel. The total travel time depends on the length of the channel and is proportional to γ −1 ln(1 + γ ), where γ is the heat release parameter.

In this paper we discuss the propagation of premixed flames in long narrow channels that are either open at both ends, or close at one of the two ends. An asymptotic method that exploits the different length scales in the problem is used to reduce the complex multi-dimensional problem to a one-dimensional one, which is then solved numerically. The results describe the influences of thermal expansion, frictional forces and compressibility on flame propagation in adiabatic channels.

Numerical simulations with single-step chemistry and detailed transport are used to study premixed hydrogen/air flames in two-dimensional channel-like domains with periodic boundary conditions along the horizontal boundaries as a function of the domain height. Both unity Lewis number, where only hydrodynamic instability appears, and subunity Lewis number, where the flame propagation is strongly affected by the combined effect of hydrodynamic and thermodiffusive instabilities are considered. The simulations aim at studying the initial linear growth of perturbations superimposed on the planar flame front as well as the long-term nonlinear evolution. The dispersion relation between the growth rate and the wavelength of the perturbation characterizing the linear regime is extracted from the simulations and compared with linear stability theory. The dynamics observed during the nonlinear evolution depend strongly on the domain size and on the Lewis number. As predicted by the theory, unity Lewis number flames are found to form a single cusp structure which propagates unchanged with constant speed. The long-term dynamics of the subunity Lewis number flames include steady cell propagation, lateral flame movement, oscillations and regular as well as chaotic cell splitting and merging.

The dynamics of an edge flame confined in a non-premixed microcombustor model is studied numerically within the context of a diffusive-thermal model. Fuel and oxidizer, separated upstream by a thin plate, flow through a channel with a prescribed velocity. At the tip of the plate, the fuel and oxidizer mix and, when ignited, an edge flame is sustained at some distance from the plate. The objective in this work is to consider the effects of confinement, differential diffusion, and heat loss on the dynamics of an edge flame in a narrow channel. We consider a wide range of channel widths and allow for changing Lewis numbers, and both adiabatic conditions and heat losses along the channel walls. The results illustrate how the flame shape and standoff distance are affected by the channel width, by mixture composition through variations in Lewis numbers and by heat losses. Conditions for flame stabilization, flame oscillations and flame extinction or blowoff are predicted.

We study the propagation of premixed flames in two-dimensional homogeneous isotropic turbulence using a Navier–Stokes/front-capturing methodology within the context of hydrodynamic theory. The flame is treated as a thin layer separating burnt and unburnt gases, of vanishingly small thickness, smaller than the smallest fluid scales. The method is thus suitable to investigate the flame propagation in the wrinkled flamelet regime of turbulent combustion. A flow-control system regulates the mean position of the flame and the incident turbulence intensity. In this context we study the individual effects of turbulence intensity, turbulence scale, thermal expansion, hydrodynamic strain and hydrodynamic instability on the propagation characteristics of the flame. Results are obtained assuming positive Markstein length, corresponding to lean hydrocarbon–air or rich hydrogen–air mixtures. For stable planar flames we find a quadratic dependence of turbulent speed on turbulence intensity. Upon onset of hydrodynamic instability, corrugated structures replace the planar conformation and we observe a greater resilience to turbulence, the quadratic scaling being replaced by scaling exponents less than one. Such resilience is also confirmed by the observation of a threshold turbulence intensity below which the propagation speed of corrugated flames is indistinguishable from the laminar speed. Turbulent speed is found to increase and later plateau with increasing thermal expansion, this affecting the average flame displacement but not the mean flame curvature. In addition, turbulence integral scale is also observed to affect the propagation of the flame with the existence of an intermediate scale maximizing the turbulent speed. This maximizing scale is smaller for corrugated flames than it is for planar flames, implying that small eddies that will be unable to significantly perturb a planar front could be rather effective in perturbing a corrugated flame. Turbulent planar flames, and more so corrugated flames, were observed to experience a positive mean hydrodynamic strain, which was explained in terms of the overwhelming mean contribution of the normal component of strain. The positive straining causes a decrease in the mean laminar propagation speed which in turn can decrease the turbulent speed. The effect of the flame on the incident turbulent field was examined in terms of loss of isotropy and vorticity destruction by thermal expansion. The latter can be mitigated by a baroclinic vorticity generation which is enhanced for corrugated flames.

We investigate the role played by hydrodynamic instability in the wrinkled flamelet regime of turbulent combustion, where the intensity of turbulence is small compared to the laminar flame speed and the scale large compared to the flame thickness. To this end the Michelson–Sivashinsky (MS) equation for flame front propagation in one and two spatial dimensions is studied in the presence of uncorrelated and correlated noise representing a turbulent flow field. The combined effect of turbulence intensity, integral scale, and an instability parameter related to the Markstein length are examined and turbulent propagation speed monitored for both stable planar flames and corrugated flames for which the planar conformation is unstable. For planar flames a particularly simple scaling law emerges, involving quadratic dependence on intensity and a linear dependence on the degree of instability. For corrugated flames we find the dependence on intensity to be substantially weaker than quadratic, revealing that corrug...

The structure and stability of two-dimensional premixed flames in an idealized microcombustor are investigated numerically within the context of a diffusive-thermal model with an imposed flow field satisfying the Navier–Stokes equations. The combustible mixture flows in a straight channel with a bend at its end that forces the flow to turn back and reverts its direction. Heat exchange occurs near the bend along a segment of the wall separating the two opposing streams; the remaining walls are assumed adiabatic. Response curves identifying the dependence of the combustion characteristics on the mass flow rate illustrate the existence of multiple steady states for a certain range of the parameters with hysteresis and bi-stability phenomena. Stable solutions correspond to flames attached to the dividing wall, where intense heat exchange occurs, or stabilized by the flow near the front wall. Depending on the conditions, one or both solutions are physically possible. At high flow rates the flame is quenched by the flow.

In this paper we numerically examine the stabilization of an edge flame in a confined mixing layer. Unlike most previous theoretical studies which have assumed, for convenience, that the density is constant and independent of temperature, the present work realistically accounts for density variations and their effect on the flow field. The focus is on the effect of lateral confinement on the flame structure and standoff distance, for both adiabatic conditions and in the presence of conductive losses to the channel walls. The inability of the reactants to diffuse outwards, as they would in an unlimited mixing layer, promotes mixing and as a result the premixed flame segment extends further in the transverse direction, stands farther away from the tip of the splitter plate, and the trailing diffusion flame is much shorter. In very narrow channels the resulting flame is a planar premixed flame that consumes all the supplied reactants. Heat losses cause a drop in temperature and as a result, the premixed flame segment is limited near the channel axis, with a shorter diffusion flame trailing behind.

The topological derivative provides the variation of a response functional when an infinitesimal hole of a particular shape is introduced into the domain. In this work, we compute higher order topological derivatives for elasticity problems, so that we are able to obtain better estimates of the response when holes of finite sizes are introduced in the domain. A critical element of our algorithm involves the asymptotic approximation for the stress on the hole boundary when the hole size approaches zero; it consists of a composite expansion that is based on the responses of elasticity problems on the domain without the hole and on a domain consisting of a hole in an infinite space. We present a simple example in which the higher order topological derivatives of the total potential energy are obtained analytically and by using the proposed asymptotic expansion. We also use the finite element method to verify the topological asymptotic expansion when the analytical solution is unknown.

The most prominent intrinsic flame instability is the hydrodynamic, or Darrieus–Landau (DL) instability, that results from the gas expansion caused by the heat released during combustion, which induces hydrodynamic disturbances that enhance perturbations of the flame front. The DL instability has many ramifications in premixed combustion; it promotes the creation of corrugated flames with relatively sharp edges pointing towards the burned gas. In this presentation, we first review the developments that led to a better understanding of the roles of viscosity, heat conduction and species diffusion on the flame stability. This includes the work of Markstein that attempted to phenomenologically improve the Darrieus and Landau analyses, and the asymptotic studies that provided an explicit dependence on the physical parameters. We then discuss the nonlinear flame development starting with the weakly-nonlinear analytical studies and proceeding with the more recent fully-nonlinear numerical results. We show that, unlike the implication that may be inferred from the original publications of Darrieus and Landau that premixed flames as a result of the instability will always appear as turbulent flames, the instability leads to the formation of cusp-like conformations with elongated intrusions pointing toward the burned gas region. These structures are stable and, because of their larger surface area, propagate at a speed that is substantially faster than the laminar flame speed. Finally, we show that the DL instability remains relevant in turbulent flames, but their influence appears limited to weak-to-moderate turbulence intensity flows.

Very rich premixed outward propagating spherical flames of n -heptane ( ϕ = 3.5) and iso -octane ( ϕ = 3.9) spherical flames were experimentally investigated under microgravity conditions at 420 K and initial pressures of up to 30 bar. Inert gases with significantly different molecular weights were used as diluents leading to mixtures with varying effective Lewis numbers. The experimental setup consisted of a spherical closed pressurized vessel that enabled optical access. The experiments were performed in a drop tower under microgravity conditions to prevent the influence of buoyancy on the expanding flames. A Schlieren measurement technique combined with a high-speed CCD camera was used to track the expanding flames. During the entire experiment, the flames were laminar and smooth with no wrinkles due to combustion instabilities observed. This allowed for accurate measurements of the propagation speed and its dependence on stretch, which was then compared to theoretical predictions based a hydrodynamic model for relatively thin flames. The experimental and theoretical results show good agreement with each other. Investigation of such rich mixtures that are close to the upper flammability limit has not been reported before.

This paper examines the influence of thermal expansion on the stabilization of an edge-flame in a mixing layer. We find, similar to the earlier predictions based on a constant-density model, two modes of flame stabilization: a steady mode at low injection velocities and an oscillatory mode at higher velocities. The gas expansion has an effect on the flame standoff distance: as a result of the reduced density in the preheat zone the flow accelerates when crossing the flame which consequently forces its edge to relocate at an upstream position where its propagation speed balances the gas velocity. The onset of oscillations at relatively high flow rates is predicted with or without invoking the constant-density approximation; the critical conditions of the onset, however, are affected by density variations.

Flame propagation involves physico-chemical processes that occur over a range of temporal and spatial scales. By use of a multi-scale analysis it is shown that diffusion processes occurring on relatively small scales can be resolved analytically when the overall activation energy of the chemical reactions is large, thus providing, by asymptotic matching, explicit conditions for the state of the gas and for the flow field across the flame zone. The mathematical formulation on the larger hydrodynamic scale reduces to a free-boundary problem, with the free surface being the flame front. The front propagates into the fresh unburned gas at a rate that depends on both the local strain that it experiences and the local curvature, with coefficients that depend on the diffusion rates of heat and mass, the equivalence ratio of the mixture and the chemical kinetic parameters. The simplified model, properly termed a hydrodynamic model, involves the solution of the Navier Stokes equations with different densities and viscosities for the burned and unburned gas. The present work extends earlier studies by including volumetric heat loss, such as radiative loss, which affects the dynamics and may lead to flame extinction.

The focus of this article is on intrinsic combustion instabilities in both premixed and nonpremixed systems, identifying, in particular, the roles of differential and preferential diffusion, thermal expansion, and heat losses. For premixed flames, the hydrodynamic instability resulting from thermal expansion plays a central role and is particularly dominant in large-scale flames. It is responsible for the formation of sharp folds and creases in the flame front and for the wrinkling observed over the surface of expanding flames. In contrast, instabilities in diffusion flames, which give rise to cellular and oscillating flames, are mainly driven by diffusive-thermal effects, with thermal expansion playing a secondary role. The discussion also includes instabilities of edge-flames in unmixed reactants, which possess stability characteristics of both premixed and diffusion flames, but with a distinct mode of instability.

The stabilization and dynamics of an edge-flame in the corner region of two mutually perpendicular streams, one of fuel and the other of oxidizer, is studied within the context of a diffusive-thermal model, with an imposed flow satisfying the Navier–Stokes equations. The formulation allows for non-unity Lewis numbers and finite rate chemistry with an Arrhenius dependence on temperature. Two flow configurations, corresponding to inlet velocity profiles of uniform speed and of constant strain, have been examined. The results identify the dependence of the flame standoff distance on the flow as well as on the properties of the mixture, including the Damkohler D and Lewis numbers. For high flow rates, or small enough D, sufficient pre-mixing occurs in front of the edge-flame, which consequently takes on a tribrachial structure consisting of two premixed branches, one lean and one rich, with a trailing diffusion flame sheet. For large D, however, there is no enough premixing and the chemical reaction occurs in a small kernel very close to the corner, much like a local thermal explosion; further downstream the reaction occurs along a diffusion flame sheet that extends along the symmetry axis. The present results also predict the onset of spontaneous oscillations when the Lewis numbers are sufficiently large provided the flow rate is sufficiently high, or D reduced below a critical value. Oscillations are first sustained when D is reduced below criticality, but depending on the flow conditions, they are either damped leading to flame re-stabilization, or amplified leading to blow-off.

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 instab...

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 flow...