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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.

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.

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.

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.

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.

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 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 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.

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...

We have developed a general theory of non-adiabatic premixed flames that is valid for flames of arbitrary shape that fully accounts for the hydrodynamic and diffusive-thermal processes, and incorporates the effects of volumetric heat losses. The model is used to describe aspects of experimentally observed phenomena of self-extinguishing (SEFs) and self-wrinkling flames (SWFs), in which radiative heat losses play an important role. SEFs are spherical flames that propagate considerable distances in sub-limit conditions before suddenly extinguishing. Our results capture many aspects of this phenomenon including an explicit determination of flame size and propagation speed at quenching. SWFs are hydrodynamically unstable flames in which cells spontaneously appear on the flame surface once the flame reaches a critical size. Our results yield expressions of the critical flame size at the onset of wrinkling and expected cell size beyond the stability threshold. The various possible burning regimes are mapped out in parameter space.

Based on a hydrodynamic length, which is typically larger than the nominal flame thickness, a premixed flame can be viewed as a surface of density discontinuity, advected and distorted by the flow. The velocities and the pressure suffer abrupt changes across the flame front that consist of Rankine–Hugoniot jump conditions, to leading order, with corrections of the order of the flame thickness that account for transverse fluxes and accumulation. To complete the formulation, expressions for the flame temperature and propagation speed, which vary along the flame as a result of local non-uniformities in the flow field and of flame front curvature, are derived. Unlike previous studies that assumed a mixture consisting of a single deficient reactant, the present study uses a two-reactant scheme and thus considers mixtures whose compositions vary from lean to rich conditions. Furthermore, non-unity and general reaction orders are considered in an attempt to mimic a wider range of reaction mechanisms and, to better represent actual experimental conditions, all transport coefficients are allowed to depend arbitrarily on temperature. The present model, expressed in a coordinate-free form, is valid for flames of arbitrary shape propagating in general fluid flows, either laminar or turbulent.

We study the propagation of premixed flames in two-dimensional channels accounting for heat-losses by conduction to the channel’s walls and a prescribed Poiseuille flow. A diffusive-thermal model is used and the calculations reported are based on Arrhenius-type chemistry. Attention is focused on the influence of the magnitude of heat losses, the channel width, and the mean flow velocity. Special attention is devoted to the determination of the global burning rate and to extinction conditions. Depending on the channel width we discuss two possible modes of extinction: total flame extinction brought about in narrow channels by excessive losses, and partial flame extinction near the walls of wider channels. Our predictions of the quenching distance, namely the smallest channel’s width that permits flame propagation, and the dead space in the case of partial extinction are in agreement with experimentally reported values. The sensitivity of the flame to an imposed flow, being directed either towards the fresh mixture or towards the burned gas, is examined with some details.

We derive analytical expressions for the burning rate of a flame propagating in a prescribed steady parallel flow whose scale is much smaller than the laminar flame thickness.In this specific context, the asymptotic results can be viewed as an analytical test of Damkohler's hypothesis relating to the influence of the small scales in the flow on the flame; the increase in the effective diffusion processes is described. The results are not restricted to the adiabaticequidiffusional case, which is treated first, but address also the influence of non-unit Lewis numbers and volumetric heat losses. In particular, it is shown that non-unit Lewis numbereffects become insignificant in the asymptotic limit considered. It is also shown that the dependence of the effective propagation speed on the flow is the same as in the adiabatic equidiffusional case, provided it is scaled with the speed of the planar non-adiabatic flame.

In this paper, we describe the behavior of an edge flame in the mixing layer of two coflowing streams, one of fuel and the other of oxidizer, established at the mouth of a cylindrical injector. The edge flame is stabilized by conductive losses to the rim of the injector and a diffusion flame is trailing behind, either converging to the centerline of the cylinder or extending outward in the oxidizer region. The objective is to determine whether radiative losses alone can drive the edge of the flame to oscillate back and forth, a behavior that has been observed in various experimental configurations. The assumption-of-unity Lewis numbers is adopted to avoid differential-diffusion effects that are known to promote oscillations. Steady and unsteady calculations are reported within a diffusive-thermal approximation, but with finite-rate chemistry. We show that, in the presence of volumetric heat losses, the edge of the flame is stabilized at a larger distance from the rim than in their absence. Furthermore, when the intensity of radiation losses is sufficiently low, the edge remains stationary. When radiative losses become excessive, the edge undergoes sustained oscillations, moving back and forth with a well-defined frequency and amplitude.

The onset of instability and subsequent development of cells on spherically expanding flames is examined theoretically. The model used accounts for both hydronamic and diffusive-thermal effects and, in contrast to earlier theories, is valid for variable transport properties over a wide range of equivalence ratios. The analysis yields predictions for a number of flame properties, including growth rate of small disturbances, critical flame size for the instability onset, cell size beyond the threshold, and an estimate of the speed of the developing turbulent flame. It is shown that results using the more realistic temperature-dependent transport coefficients are more commensurate with experimental data concerning the critical conditions, that is, flame size or Peclet number, at the transition from one burning regime to another.

Both theory and experiment predict a linear relationship between flame speed and stretch, and the sensitivity of this dependence is given in terms of the Markstein number. In this note the dependence of Markstein number on mixture strength is explicitly determined. Results are presented for hydrogen-air, hydrocarbon-air, and alcohol-air mixtures over a range of equivalence ratio.

We present a theoretical study aimed at understanding the basic mechanisms responsible for the onset of oscillations in diffusion flames. A simple one-dimensional configuration is considered with one reactant supplied in a uniform stream and the other diffusing against the stream. The analysis allows for unequal non-unity Lewis numbers as well as for incomplete combustion. It is found that oscillations are possible when the Damkohler number is sufficiently small, namely at near-extinction conditions. They occur when the reactant diffusing against the stream is more completely consumed and the corresponding Lewis number is sufficiently large (typically larger than one). The conditions also require the Lewis number of the reactant supplied in the stream to be within a certain range (typically also larger than one). In accord with experimental results the onset of oscillations is found to be sensitive to stoichiometric conditions (or mixture strength) and to the temperature differential between the supply co...

The propagation of premixed flames in weak two-dimensional homogeneous turbulent flows is studied numerically via a hybrid Navier–Stokes/front capturing methodology within the context of a hydrodynamic model, which treats the flame as a surface of density discontinuity separating the burnt and unburnt gases. The focus is the influence of the Darrieus-Landau instability on the turbulent flame, which has been recognized recently to have a dramatic effect on its structure and the turbulent flame speed. Such instability, controlled by a parameter inversely proportional to the Markstein length, can be triggered in a laboratory setting by variations in system pressure or in fuel type and composition. Particular attention in this study is devoted to the influence of the Darrieus-Landau instability on a turbulent, statistically planar flame. Results are therefore limited to positive Markstein length corresponding to lean hydrocarbon–air or rich hydrogen–air mixtures. We show that, although the planar flame under similar but laminar conditions is stable, it is nonetheless affected by the instability in the presence of a turbulent incident flowfield. The turbulent flame speed is observed to exhibit, in addition to the effect of thermal expansion, a nontrivial dependence on the instability parameter and on the turbulence integral scale both effects modulating, in the weak turbulence regime, the well established quadratic dependence of turbulent flame speed on turbulence intensity.

The determination of the turbulent flame speed is a central problem in combustion theory. Early studies by Damkohler and Shelkin resorted to geometrical and scaling arguments to deduce expressions for the turbulent flame speed and its dependence on turbulence intensity. A more rigorous approach was undertaken by Clavin and Williams who, based on a multi-scale asymptotic approach valid for weakly wrinkled flames, derived an expression that apart from a numerical factor recaptures the early result by Damkohler and Shelkin. The common denominator of the phenomenological and the more rigorous propositions is an increase in turbulent flame speed due solely to an increase in flame surface area. Various suggestions based on physical and/or experimental arguments have been also proposed, incorporating other functional parameters into the flame speed relation. The objective of this work is to extend the asymptotic results to a fully nonlinear regime that permits to systematically extract scaling laws for the turbulent flame speed that depend on turbulence intensity and scale, mixture composition and thermal expansion, flow conditions including effects of curvature and strain, and flame instabilities. To this end, we use a hybrid Navier–Stokes/front-capturing methodology, which consistently with the asymptotic model, treats the flame as a surface of density discontinuity separating burned and unburned gases. The present results are limited to positive Markstein length, corresponding to lean hydrocarbon–air or rich hydrogen–air mixtures, and to wrinkled flames of vanishingly small thickness, smaller that the smallest fluid scales. For simplicity we have considered here two-dimensional turbulence, which although lacks some features of real three-dimensional turbulence, is not detrimental when using the hydrodynamic model under consideration, because the turbulent flame retains its laminar structure and its interaction with turbulence is primarily advective/kinematic in nature.

The initial development of a “tulip flame,” often observed during flame propagation in closed tubes, isattributed to a combustion instability. The roles of hydrodynamic and of the diffusional-thermal processes on the onset of instability are investigated through a linear stability analysis in which the growth or decay of small disturbances, superimposed on an otherwise smooth and planar flame front, are followed. A range of the Markstein parameter, related to the mixture composition through an appropriately defined Lewis number, has been identified where a tulip flame could be observed. For a given value of the Markstein parameter within this range, a critical wavelength is identified as the most unstable mode. This wavelength is directly related to the minimal aspect ratio of the tube where a tulip flame could be observed. The time of onset of instability is identified as the time when the most unstable disturbance, associated with the critical wavelength, grows at a faster rate than the flame front itself and exceeds a certain threshold. This occurs after the flame has propagated a certain distance down the tube: a value which has been explicitly determined in terms of the relevant parameters. Experimental records on the tulip flame phenomenon support the findings of the analysis. That is, the tulip flame forms after the flame has traveled half the tube's length, it does not form in short tubes, and its formation depends on the mixture composition and on the initial pressure in the tube.

The hydrodynamic model of flame propagation is based on an asymptotic theory that exploits the multi-scale structure of a premixed flame, characterized essentially by a small diffusion thickness compared to the hydrodynamic length associated with the spatial domain of the vessel where the combustion process takes place. In this paper we show that its numerical implementation using a hybrid Navier-Stokes/front-capturing technique yield novel results on the dynamics of premixed flames in laminar as well as turbulent environments. In laminar flow, we characterize the structure and speed of the corrugated flame that evolves as a result of the hydrodynamic or Darrieus-Landau instability, and the subsequent induced flow that is ultimately responsible to the flame’s straining. The nature of the hydrodynamic model, where the flame is of vanishing thickness, smaller than the smallest turbulence scales, makes it an ideal tool to also study the propagation in the flamelet regime of turbulent combustion. In this context we obtain a functional relationship for the turbulent flame speed in terms of turbulence characteristics, such as scale and intensity, and flame characteristics, such as thermal expansion and mixture composition.

In this study we numerically implement the hydrodynamic model for a premixed flame as a nonlinear free boundary problem where the flame is tracked via a level set equation and the flow is described by a solution of the variable density Navier–Stokes equations. Unlike an earlier similar study, the present model is enriched by fully accounting for hydrodynamic strain in the flame stretch relation which, in turn, affects the local flame speed. The objective is to comprehensively analyze the effect of strain on the onset of the hydrodynamic instability and on the nonlinear development that takes place beyond its inception. The initial evolution is corroborated with the results of a linear stability analysis for which strain rate effects are fully included. We show that while strain provides an additional stabilizing effect on the short wavelength disturbances, thereby delaying the onset of the hydrodynamic instability, it acts to sharpen the cusps near the troughs of the corrugated flame that develops beyond the stability threshold resulting in a larger flame surface area and a higher propagation speed.

Previously, the stability of a premixed flame in a stagnation flow was discussed for a restricted class of disturbances that are self-similar to the basic undisturbed flow; thus, flame fronts with corrugations only in the cross stream direction were considered. Here, we consider a more general class of three-dimensional flame front perturbations which also permits corrugations in the streamwise direction. It is shown that, because of the stretch experienced by the flame, the hydrodynamic instability is limited only to disturbances of short wavelength. If in addition diffusion effects have a stabilizing influence, as would be the case of mixtures with Lewis number greater than one, a stretched flame could be absolutely stable. Instabilities occur when the Lewis number is below some critical value less than one. Neutral stability boundaries are presented in terms of the Lewis number, the strain rate, and the appropriate wavenumbers. Beyond the stability threshold, the two-dimensional self-similar modes always grow first. However, if disturbances of long wavelength are excluded, it is possible for the three-dimensional modes to be the least stable one. Accordingly, the pattern that will be observed on the flame front, at the onset of instability, will consist of either ridges in the direction of stretch or the more common three-dimensional cellular structure.

The evolution of a premixed flame under conditions of confinement is studied theoretically. The analysis is based on a hydrodynamic model in which the flame is treated as a surface of discontinuity. The flame structure is assumed to be quasi-steady with a high activation energy and a large heat release. Its resolution in the postignition period yields a coupled system of equations for the determination of the pressure and the burning rate. The analysis also resolves the thermal and flow fields on either side of the flame and determines the instantaneous location of the flame front together with the overall time required for the flame to reach the end of the tube. The results indicate that qualitatively distinct behaviors are possible for mixtures depending on whether their Lewis numbers Le are less or greater than one.

In this study, the stability of plane stretched flames, more specifically plane flames in straining fields, has been examined. It is shown that flame stretch stabilizes long wavelength disturbances and so can suppress, in this regime the hydrodynamic instability. If in addition, the mixture is deficient in the reactant which is also the weakly diffusing component, and hence the Lewis number is greater than unity, thermal effects will stabilize the short wavelength disturbances. Thus sufficiently strong stretch can render a flame absolutely stable. The instability, which first appears by reducing the strain rate from the critical value, is in the form of longitudinal cells with ridges in the direction of stretch. By reducing the strain rate further a cellular structure will probably emerge.

We describe flame propagation between two opposed reactive streams which may differ in their composition and temperature. A two-dimensional counterflow configuration and an irreversible Arrhenius reaction are adopted, along with the constant density approximation. Attention is focused on the influence of two nondimensional parameters. The first one, denoted by γ, represents the difference in the enthalpy of the feed streams. The second one, ϵ, quantifies the ratio between the characteristic chemical time and the strain time. After a general formulation of the problem, we begin by an analysis of the one-dimensional case consisting of two parallel planar flames of unequal strength. The flames behavior is described analytically and numerically. In particular, two extinction regimes are identified: for values of γ smaller than a critical value γ∗, the flames extinguish by quenching against each other at the stagnation plane; for γ > γ∗ they extinguish while at a finite distance from each other which increases with γ. These behaviors are similar to those, known in the literature, associated with the influence of Lewis numbers on the extinction of twin-flames. We then describe the propagation of two-dimensional flame fronts along the stagnation line, in a direction perpendicular to the plane of strain. The flame front is thus curved under the combined effects of the flow field and the transverse enthalpy gradient in the frozen mixture ahead of it; far behind the state of the gas is that of the pair of flat flames introduced above. The problem is studied numerically and complemented by an analytical description of the fast-chemistry situations corresponding to small values of ϵ. In particular we describe, for different fixed values of γ, the evolution of ignition fronts, characterized by a positive propagation speed, to extinction fronts, characterized by negative speeds, as ϵ is increased. In addition to the marked change in the flame shape, the most noticeable effect of an increase in γ is the decrease in the propagation speed of the flame front. These effects are associated with the increased front curvature for higher values of γ, along with a shift of the front leading edge towards the stream with higher enthalpy.

The structure of a simple one-dimensional spray diffusion flame is examined by means of asymptotics and numerics. The results emphasize the importance of the heat and mass transfer rates, the supply conditions and the droplet loading factor on the burning process. Particular attention is given to the conditions for flame extinguishment and to the onset of spontaneous flame oscillation.

We examine the dynamic characteristics of the diffusion flame surrounding a droplet of liquid fuel in a reduced oxidant environment. The study is motivated by the experimental observations of candle flames in microgravity, which bear some similarity to the spherical diffusion flame that surrounds a burning liquid fuel droplet. The microgravity experiments reveal that, when the oxidant concentration in the chamber is sufficiently low, oscillations develop before extinction. A similar phenomenon is predicted here. An oscillatory state results for a sufficiently low oxidant concentration when radiative losses from the flame are appreciable. The frequencies of oscillations we predict are in good agreement with the experimental observations.

The strong coupling of a flame to the flow through which it propagates is caused primarily by the thermal expansion of the gas undergoing chemical reaction. This coupling has a prominent effect on the response of a flame to hydrodynamic disturbances. In the first stability analysis carried out independently by Darrieus (1938) and Landau (1945), the linear stability of a plane unbounded flame was studied. Their results reveal that the effect of thermal expansion is always to destabilize the flame. This instability is commonly referred as to the hydrodynamic instability; see the discussion on Flame Stability by J. Buckmaster in these proceedings. Since all combustion processes are accompanied by thermal expansion, the hydrodynamic instability is always present. However, other effects such as curvature, flame stretch, gravity and diffusion, may have stabilizing influences. These effects are briefly examined in the following.

This paper is based on an invited presentation at the Conference on Turbulent Mixing and Beyond held in the Abdus Salam International Center for Theoretical Physics, Trieste, Italy (August 2007). It consists of a summary of recent investigations aimed at understanding the nature and consequences of the Darrieus–Landau instability that is prominent in premixed combustion. It describes rigorous asymptotic methodologies used to simplify the propagation problem of multi-dimensional and time-dependent premixed flames in order to understand the nonlinear evolution of hydrodynamically unstable flames. In particular, it addresses the effect of background turbulent noise on the structure and propagation of large-scale flames.