Flames as gasdynamic discontinuities
TL;DR: In this article, an equation for the propagation of the discontinuity surface for arbitrary flame shapes in general fluid flows is derived, where the structure of the flame is considered to consist of a boundary layer in which the chemical reactions occur, located inside another boundary layer, in which transport processes dominate.
Abstract: Early treatments of flames as gasdynamic discontinuities in a fluid flow are based on several hypotheses and/or on phenomenological assumptions. The simplest and earliest of such analyses, by Landau and by Darrieus prescribed the flame speed to be constant. Thus, in their analysis they ignored the structure of the flame, i.e. the details of chemical reactions, and transport processes. Employing this model to study the stability of a plane flame, they concluded that plane flames are unconditionally unstable. Yet plane flames are observed in the laboratory. To overcome this difficulty, others have attempted to improve on this model, generally through phenomenological assumptions to replace the assumption of constant velocity. In the present work we take flame structure into account and derive an equation for the propagation of the discontinuity surface for arbitrary flame shapes in general fluid flows. The structure of the flame is considered to consist of a boundary layer in which the chemical reactions occur, located inside another boundary layer in which transport processes dominate. We employ the method of matched asymptotic expansions to obtain an equation for the evolution of the shape and location of the flame front. Matching the boundary-layer solutions to the outer gasdynamic flow, we derive the appropriate jump conditions across the front. We also derive an equation for the vorticity produced in the flame, and briefly discuss the stability of a plane flame, obtaining corrections to the formula of Landau and Darrieus.
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01 Jan 1988
TL;DR: In this article, it is shown that the inner structure of the flamelets is one-dimensional and time dependent, and a new coordinate transformation using the mixture fraction Z as independent variable leads to a universal description.
Abstract: The laminar flamelet concept covers a regime in turbulent combustion where chemistry (as compared to transport processes) is fast such that it occurs in asymptotically thin layers—called flamelets—embedded within the turbulent flow field. This situation occurs in most practical combustion systems including reciprocating engines and gas turbine combustors. The inner structure of the flamelets is one-dimensional and time dependent. This is shown by an asymptotic expansion for the Damkohler number of the rate determining reaction which is assumed to be large. Other non-dimensional chemical parameters such as the nondimensional activation energy or Zeldovich number may also be large and may be related to the Damkohler number by a distinguished asymptoiic limit. Examples of the flamelet structure are presented using onestep model kinetics or a reduced four-step quasi-global mechanism for methane flames. For non-premixed combustion a formal coordinate transformation using the mixture fraction Z as independent variable leads to a universal description. The instantaneous scalar dissipation rate χ of the conserved scalar Z is identified to represent the diffusion time scale that is compared with the chemical time scale in the definition of the Damkohler number. Flame stretch increases the scalar dissipation rate in a turbulent flow field. If it exceeds a critical value χ q the diffusion flamelet will extinguish. Considering the probability density distribution of χ , it is shown how local extinction reduces the number of burnable flamelets and thereby the mean reaction rate. Furthermore, local extinction events may interrupt the connection to burnable flamelets which are not yet reached by an ignition source and will therefore not be ignited. This phenomenon, described by percolation theory, is used to derive criteria for the stability of lifted flames. It is shown how values of ∋ q obtained from laminar experiments scale with turbulent residence times to describe lift-off of turbulent jet diffusion flames. For non-premixed combustion it is concluded that the outer mixing field—by imposing the scalar dissipation rate—dominates the flamelet behaviour because the flamelet is attached to the surface of stoichiometric mixture. The flamelet response may be two-fold: burning or non-burning quasi-stationary states. This is the reason why classical turbulence models readily can be used in the flamelet regime of non-premixed combustion. The extent to which burnable yet non-burning flamelets and unsteady transition events contribute to the overall statistics in turbulent non-premixed flames needs still to be explored further. For premixed combustion the interaction between flamelets and the outer flow is much stronger because the flame front can propagate normal to itself. The chemical time scale and the thermal diffusivity determine the flame thickness and the flame velocity. The flamelet concept is valid if the flame thickness is smaller than the smallest length scale in the turbulent flow, the Kolmogorov scale. Also, if the turbulence intensity v′ is larger than the laminar flame velocity, there is a local interaction between the flame front and the turbulent flow which corrugates the front. A new length scale L G =v F 3 /∈ , the Gibson scale, is introduced which describes the smaller size of the burnt gas pockets of the front. Here v F is the laminar flame velocity and ∈ the dissipation of turbulent kinetic energy in the oncoming flow. Eddies smaller than L G cannot corrugate the flame front due to their smaller circumferential velocity while larger eddies up to the macro length scale will only convect the front within the flow field. Flame stretch effects are the most efficient at the smallest scale L G . If stretch combined with differential diffusion of temperature and the deficient reactant, represented by a Lewis number different from unity, is imposed on the flamelet, its inner structure will respond leading to a change in flame velocity and in some cases to extinction. Transient effects of this response are much more important than for diffusion flamelets. A new mechanism of premixed flamelet extinction, based on the diffusion of radicals out of the reaction zone, is described by Rogg. Recent progress in the Bray-Moss-Libby formulation and the pdf-transport equation approach by Pope are presented. Finally, different approaches to predict the turbulent flame velocity including an argument based on the fractal dimension of the flame front are discussed.
1,268 citations
TL;DR: In this article, a review of recent developments in flame theory is provided, in sufficient detail to give the reader a comprehensive introduction to the field, including the stability and flammability limits of planar fronts, cellular flames, flame stretch, turbulent and self-turbulizing flames, hydrodynamic interactions between weakly turbulent gas flows and wrinkled flame fronts, molecular diffusion effects of intermediate species involved in chain reactions.
912 citations
TL;DR: In this paper, a detailed kinetic mechanism for the pyrolysis and combustion of a large variety of fuels at high temperature conditions is presented, and the authors identify aspects of the mechanism that require further revision.
817 citations
TL;DR: In this paper, a unified view of the concept of flame stretch is provided on the basis of a novel derivation of stretch in terms of strain rate and curvature effects, which is used to describe the structure and extinction mechanisms of turbulent flames.
Abstract: When a flame propagates in a nonuniform flow it experiences strain and curvature effects. The fractional rate of change of the flame area constitutes the flame stretch. This quantity is often used to describe the structure and extinction mechanisms of turbulent flames. It also occurs in many recent studies of premixed laminar flames. This article provides a unified view of this concept on the basis of a novel derivation of stretch in terms of strain rate and curvature. The flame stretch, the rate of change of the normal to the flame front and the rate of change of the curvature are deduced from a general transport theorem. As an illustration, the components of flame stretch are evaluated in the case of a direct numerical simulation of the interaction between a pair of vortices and a laminar flame. Another application of flame stretch concerns the determination of the available flame surface density. A balance equation is derived for this quantity and cast in various useful forms thus providing a ...
559 citations
TL;DR: In this paper, the effects of stretch on the flame structure, and by allowing for mixture nonequidiffusion, the flame responses, especially the flame speed, can be quantitatively as well as qualitatively modified.
518 citations
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TL;DR: In this article, an analytical theory for the stability properties of planar fronts of premixed laminar flames freely propagating downwards in a uniform reacting mixture is developed for an arbitrary expansion of the gas across the flame.
Abstract: An analytical theory is developed for the stability properties of planar fronts of premixed laminar flames freely propagating downwards in a uniform reacting mixture. The coupling between the hydrodynamics and the diffusion process is described for an arbitrary expansion of the gas across the flame. Viscous effects are included with an arbitrary Prandtl number. The flame structure is described for a large value of the reduced activation energy and for a Lewis number close to unity. The flame thickness is assumed to be small compared with the wavelength of the wrinkles of the front, this wavelength being also the characteristic lengthscale of the perturbations of the flow field outside the flame. A two-scale method is then used to solve the problem. The results show that the acceleration of gravity associated with the diffusion mechanisms inside the front can counterbalance the hydrodynamical instability when the laminar-flame velocity is low enough. The theory provides predictions concerning the instability threshold. In particular, the dimensions of the cells are predicted to be large compared with the flame thickness, and thus the basic assumption of the theory is verified. Furthermore, the quantitative predictions are in good agreement with the existing experimental data.The bifurcation is shown to be of a different nature than predicted by the purely diffusive–thermal model.The viscous diffusivities are supposed to be independent of the temperature, and then the viscosity is proved to have no effect at all on the dynamical properties of the flame front.
467 citations
TL;DR: In this paper, the effects of flow inhomogeneities on the dynamics of laminar flamelets in turbulent flames, with account taken of influences of the gas expansion produced by heat release, were investigated.
Abstract: To study effects of flow inhomogeneities on the dynamics of laminar flamelets in turbulent flames, with account taken of influences of the gas expansion produced by heat release, a previously developed theory of premixed flames in turbulent flows, that was based on a diffusive-thermal model in which thermal expansion was neglected, and that applied to turbulence having scales large compared with the laminar flame-thickness, is extended by eliminating the hypothesis of negligible expansion and by adding the postulate of weak-intensity turbulence. The consideration of thermal expansion motivates the formal introduction of multiple-scale methods, which should be useful in subsequent investigations. Although the hydrodynamic-instability mechanism of Landau is not considered, no restriction is imposed on the density change across the flame front, and the additional transverse convection correspondingly induced by the tilted front is described. By allowing the heat-to-reactant diffusivity ratio to differ slightly from unity, clarification is achieved of effects of phenomena such as flame stretch and the flame-relaxation mechanism traceable to transverse diffusive processes associated with flame-front curvature. By carrying the analysis to second order in the ratio of the laminar flame thickness to the turbulence scale, an equation for evolution of the flame front is derived, containing influences of transverse convection, flame relaxation and stretch. This equation explains anomalies recently observed at low frequencies in experimental data on power spectra of velocity fluctuations in turbulent flames. It also shows that, concerning the diffusive-stability properties of the laminar flame, the density change across the flame thickness produces a shift of the stability limits from those obtained in the purely diffusive-thermal model. At this second order, the turbulent correction to the flame speed involves only the mean area increase produced by wrinkling. The analysis is carried to the fourth order to demonstrate the mean-stretch and mean-curvature effects on the flame speed that occur if the diffusivity ratio differs from unity.
452 citations
TL;DR: In this paper, the effects of type of fuel, mixture composition, and pressure on flame-front stability have been studied, and a first-order perturbation treatment of flamefront stability gave results in qualitative agreement with experiments when the dependence of burning velocity on flamefront curvature was taken into account.
Abstract: Rich hydrocarbon flames burning in tubes were found to assume a cellular structure in the absence of turbulence in the approach stream. The effects of type of fuel, mixture composition, and pressure on the phenomenon have been studied. A first-order perturbation treatment of flame-front stability gave results in qualitative agreement with experiments when the dependence of burning velocity on flame-front curvature was taken into account. NOTATION a = acceleration A1…A4 = constants cp = specific heat at constant pressure d = average cell size of cellular flames Di = diffusion coefficient of ith species F = aL/(Su0)2 f(x). g(y. t) = functions h = 2π/λ = wave number i = − 1 k = thermal conductivity L = characteristic length of order of flame front thickness m = ρu0Su = mass flow M = molecular weight of fuel Mi = molecular weight of ith species ni = concentration of ith species Nij = Mi(νij/ρDi) ρ = pressure Qi = heat released in jth reaction r = radial coordinate R = radius of curvature of flame front Su = burning velocity t = time T = absolute temperature u, v = velocity components wi = rate of jth reaction x, y = coordinates X = x/L Y0 = T Yi = ni, i = 1, …, n α = δ/ub0h βi = k/cpρDi γ = a/h(Su0)z δ = stability parameter ∈ = ρ u 0 / ρ b 0 φ = angle between tangent of flame front and y- axis χ = Lh = 2πL/λ νij = number of molecules of ith species generated in the jth reaction (νij is negative for those consumed) μ = parameter that determines curvature dependence of burning velocity λ = wave length λmax. = wave length for maximum instability ρ = density σ = μLh τ = parameter that determines curvature dependence of temperature of burned gas ξ , η = coordinates of an element of flame front ∝ = proportional to Subscripts u = unburned b = burned Superscripts 0 = zero order 1 = first order Presented at the 1950 Heat Transfer and Fluid Mechanics Institute, Los Angeles, June 28–30, 1950. Received August 24, 1950.
366 citations
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