Flame Propagation in a Nonuniform Mixture: Analysis of a Propagating Triple- Flame

A situation in which a diffusion flame reaches an end at some position in a medium of non-premixed reactants is studied. The mixing of reactants that takes place ahead of the diffusion flame leads to the formation of a "triple-flame", a structure which consists of a fuel-rich premixed flame, a fuel-lean premixed flame. and a diffusion flame that starts where the two premixed flames meet. An important property of such an end-point is its ability to propagate. The limits oflow heal release. unit Lewis number and large Zel'dovich number are considered. The structure of the triple-flame and the unique relationship between propagation speed and transverse mixture fraction gradient are computed numerically. For the range of values considered here. the end of the diffusion flame is shown to extend itself at a rate that can be substantially reduced. but that remains positive as the gradient of the mixture fraction is increased. INTRODUCTION The burning of initially separate streams of a fuel and an oxidant has been known for a long time to take the form of a diffusion flame located around an interface where reactants mix in stoichiometric proportion (Burke and Schumann, 1928), Mathematically, this relatively passive combustion interface can normally be analysed as a one-dimensional problem, depending only on a co-ordinate normal to the diffusionflame surface, as in the work of Clarke (1971) and Lifian (1974). Even in situations that are not strictly one-dimensional (such as the study of turbulent non-premixed combustion), one can often usefully reduce the problem to the study of one-dimensional diffusion flames. This is done, for example, in "laminar flamelet" modelling of turbulent diffusion flames (Peters, 1984). These one-dimensional analyses typically reveal such fundamental features of diffusion flames as ignition and extinction limits, denoting parameter boundaries beyond which only burning or non-burning steady solutions can exist (Lifian, 1974). However, in more than one dimension a richer picture becomes possible, As reproduced in Figure I, Phillips (1965) observed a flame-structure in an experimental mixing layer between methane and air that contains elements of both premixed and diffusion flames, a structure that has become known as a "triple-flame". It consists of a fuel-rich premixed flame branch that leaves a residue of unburnt fuel behind it, a fuel-lean branch that leaves unburnt oxidant, and a diffusion flame that starts where both premixed branches meet. The diffusion flame branch continues a diffusioncontrolled burning of whatever fuel and oxidant remain behind the two premixed flame branches. Away from the leading end of the triple flame, these premixed branches become progressively weaker as they move further from the diffusion flame into regions of weakening mixture-strength, ultimately becoming practically insignificant. A triple-flame may be thought of as dividing a non-burning part of a mixing layer (between fuel and oxidant) from a part where more-or-Iess one dimensional burning of a diffusion flame predominates. In this sense it describes the typical structure of an 23 D ow nl oa de d By : [ Th e U ni ve rs ity o f M an ch es te r] At : 2 2: 39 3 1 M ay 2 00 7 24 L. J. HARTLEY AND J. W. DOLO FIGURE I A triple flame propagating in a nonuniform medium. BritishCrown Copyright: Reproduced from Phillips (1965). with kind permission of H. Phillips. Health and Safely Executive. Buxton. U.K. open "end" of a diffusion flame (excluding diffusion-flames that end at an impermeably boundary). On the burning side of a triple-flame, fuel and oxidant are unable to mix without being mutually annihilated by chemical processes, while on the other non-burning side, temperatures are typically low enough for reactants to be able to mix without appreciable reaction. Most significantly, the premixed flames impart the ability to propagate to an end of a diffusion flame so that, whenever it advances. the triple flame serves to initiate or to spread a diffusion flame along a mixing layer. Triple-flames may be found in a variety of possible circumstances, generally involving non-uniformly premixed combustion. As in the work of Phillips (1965), for example, it may initiate the burning of a buoyant layer of light gaseous fuel. It may be found in flame-spread over vapourising liquid fuels, such as diesel droplets or volatile combustible pools, or it may occur within a flow of mixing gases, as would be found behind a splitter-plate separating incoming streams of fuel and oxidant. Triple-flames may also arise in strained flows, such as may be formed through a counterflow geometry (Ishizuka, 1986). Interestingly, the propagation speed of a triple-Ilame in strained Ilow can become negative for large enough strain-rates (Dold, 1991), representing the propagation of an extinction front along a diffusive interface rather than a front that initiates combustion. In a turbulent non-premixed system, an end of a diffusion flame would arise whenever a local flow strain-rate becomes large enough to cause a diffusion flame to extinguish at some point. This would cause a "hole" to open in at least some region of the diffusion Ilame sheet (Williams, 1985). This situation leads naturally to a two-dimensional "laminar triple-flumclet" analogue of the one-dimensional "laminar flamclct" model of turbulent combustion (Peters, 1984). Such a model can be used to predict the manner in which any holes in a diffusion flame would respond dynamically to a localilow field, and is discussed by Dold et al. (1991). For example, this response may vary between some degree of incomplete burning and a complete extinction of the combustion. The article by Dold et al. (1991) outlines the background to many of the ideas that are currently being developed concerning the role and behaviour of triple-flames. In particular, this article makes it clear that a great deal of theoretical work remains to D ow nl oa de d By : [ Th e U ni ve rs ity o f M an ch es te r] At : 2 2: 39 3 1 M ay 2 00 7 FLAME PROPAGATION 25 be done in obtaining solutions that describe the structure and propagation of triple flames. In fact, since the pioneering work of Phillips (1965), the first numerical descriptions of triple-flame structures have appeared only comparatively recently (Savas and Gollahalli, 1986; Chen and T'ien, 1986; Ohki and Tsuje, 1986). Asymptotic descriptions, based on large activation energy asyrnptotics, have also now appeared (Dold, 1988, 1989, 1991; Savas and Gollahalli, 1986; Wichman, 1989). These analyses make it clear that the most significant factor that controls the shape and propagation speed of a triple-flame is the gradient of the mixture fraction encountered at the leading end of the flame. In cases where this gradient is very small indeed, the premixed flame branches become only very weakly curved and behave, at any point, as if they were locally planar. Any small part of the premixed flames therefore propagates in a normal direction at the laminar flame speed that is appropriate to the local value of the mixture properties. In the papers by Dold (1988, 1989), increased gradients of the mixture fraction were considered for cases in which flame curvatures playa small but non-vanishing role in determining the properties of a triple flame. Solutions could then be obtained in terms of an asymptotic expansion for "small" mixture-fraction gradients. At still larger gradients, these expansions lose accuracy once the radius of curvature of the premixed flames becomes as small as the thickness of the preheating regions of the premixed flame branches. This article is concerned with obtaining solutions that describe these more rapidly varying triple flames. For this purpose, a simple small heat release model is used, taking Lewis numbers for both fuel and oxidant to be unity. The chemistry is taken to be described by either one of two examples of one-step Arrhenius reaction at large activation energy, although the analysis makes it clear that any more general model that produces a reaction-zone and preheat-zone structure could be used. With this, we find that the propagation speed of the resulting triple flames remains positive for all mixture-fraction gradients that produce preheat-zone thicknesses and flame radii of curvature of similar magnitudes. In a separate paper (Dold, 1991), it is shown that zero or negative propagation speeds become possible only for still more strongly curved triple flames, as may be found in strained flow.