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This article was downloaded by: [University of Southern California]On: 18 March 2011Access details: Access Details: [subscription number 911085157]Publisher Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 3741 Mortimer Street, London W1T 3JH, UKCombustion Science and TechnologyPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title content t713456315Edge-Flames and Their StabilityJ. BuckmasteraaDepartment of Aeronautical and Astronautical Engineering, University of Illinois, Urbana, ILTo cite this Article Buckmaster, J.(1996) 'Edge-Flames and Their Stability', Combustion Science and Technology, 115: 1, 41— 68To link to this Article: DOI: 10.1080/00102209608935522URL: http://dx.doi.org/10.1080/00102209608935522PLEASE SCROLL DOWN FOR ARTICLEFull terms and conditions of use: f-access.pdfThis article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.

1996 OPA (Overseas Publishers Association)Amsterdam B.V. Published in The Netherlands underlicense by Gordon and Breach Science Publishers SAPrinted in IndiaCombust. Sci. Tech . \996, Vol. 11:5, pp. 41-68Reprints available: directly from the: publisherPhotocopying permitted by license onlyDownloaded By: [University of Southern California] At: 17:44 18 March 2011Edge-Flames and Their StabilityJ. BUCKMASTERDepartment of Aeronautical and Astronautical Engineering,University of Illinois, Urbana, IL 61801(Received 2 August 1995; in final form 20 February 1996)Flame sheets that arise in non-premixed combustion often have edges. The leading edge of a flamespreading over a fuel-bed, solid or liquid, is one example. The edge of a hemi-spherical candleflame in microgravity is another. We construct a one-dimensional model which contains some ofthe essential physics of these edge-flames, and use this model to describe stationary solutions andtheir stability. The model corresponds to a new class of combustion waves which resembledeflagrations in some respects yet exhibit important differences. Thus, in a uniform flow. wave-likesolutions are possible with positive, negative or vanishing wave-speeds. depending on anassignable Damkohler number. At large activation energy, reaction is concentrated primarily in athin region (the edge) but it persists, in diminished form, behind the edge. This residuaJ reactionplays a key role in defining the flame - or edge-temperature which, in turn, controls the dynamicsof the structure. The familiar Lewis-number stability boundaries of deflagrations are present inmodified form and provide tentative explanations of pulsations and cellular structures that havebeen observed experimentally.LIST OF SYMBOLSB.BCi iC3,C. 1,2,3Pre-exponential factorcoefficients of side-termsG coefficient in expansion of C 3specific heatDarnkohler numberDD,GDDxDrEhoxygen diffusion coefficientfuel diffusion coefficientactivation energyflame-edge locationHhl041

42JkILLeyMQDownloaded By: [University of Southern California] At: 17:44 18 March 2011RstTT;,UVxXX.XiyYY;.Y,zZZiQ(,P" P2Yx, yYBee,A paTJ. BUCKMASTERrc, T;,IQ) iJ Zwave-number(Le-I )IBtransverse length scalefuel Lewis numbertransverse mass fluxheat of reactiongas constantzfl.timetemperature'Burke-Schumann flame temperatureedge speednon-dimensional edge speedtransverse distanceoxygen mass fractionequilibrium X when T T;,coefficient functions in X-expansion in flame edgedistance parallel to edgefuel mass fractionequilibrium Ywhen T T;,coefficient functions in Y-expansion in flame edgedistance along flamefuel mass fraction outside of the flame edgecoefficient functions in Z-expansionEq. (36)Eqs. (61 b), (67b)stoichiometric coefficientsRT;,IETIT;,coefficient functions in O-expansion in flame edgeheat conduction coefficientdistance in flame-edgedensitys-hnon-dimensional timeEq.(15)o outside of the flame-edgecoefficient functions in iJ-expansion

EDGE-FLAMES'I-'n,n( ),(),()w()'Downloaded By: [University of Southern California] At: 17:44 18 March 20110-43(CpTjQ)(Ii-liw) (Y- Yw)reaction rateexpansion coefficientssteady statesupply valuesunsteady perturbationsvalues to the left of the flame-edgeINTRODUCTIONEdge-flames are a common characteristic of non-premixed combustion. Forexample, a flame propagating over a fuel-bed will have a leading edge standingsome distance above the bed (Fig. 1). A candle burning in microgravity willhave a hemispherical shape with a well-defined circular edge (Fig. 2), Dietrich,OxygenIIedget 1FIGURE IedgeFIGURE 2Flame spreading over a fuel bed.edgeHemi-spherical candle flame under microgravity conditions.

Downloaded By: [University of Southern California] At: 17:44 18 March 201144J. BUCKMASTERRoss, and T'ien (1994). And a turbulent diffusion flame will have a hole rippedin it where the local scalar dissipation rate is large enough, and this hole willhave an edge.Edge-flame dynamics is an important subject. Thus in the turbulent contextwe would like to know what conditions will cause the hole to expand, whatconditions will cause it to shrink, and whether or not there are meaningfulconditions under which the hole will do neither, Dold et al. (1991), §4.Systematic experimental studies of edge-flames have not been carried out,but there have been observations that are relevant to our study, and we shallrefer to three.Dietrich et al. (1994) have observed candle flames in microgravity whichextinguish after some period due to asphyxiation. Large scale oscillationsoccur prior to extinction, with the leading edge moving over the hemispherical surface defined by the steadily-burning flame so that the solid anglesubtended by the flame at the sphere center oscillates between a value close to2n and a much smaller value. Note that the heavy hydrocarbons that fuel acandle flame will define a Lewis number that is significantly greater than 1.Chan and T'ien (1978) have examined flames in a Kirkby-Schmitz apparatus, but with the fuel (ethanol and other choices) supplied as vapor from anevaporating pool. The flame is a circular disk confined within a circular tubewith a gap between the disk edge and the tube surface. Prior to extinction,oscillations are observed in which the disk expands and shrinks in a symmetricfashion, so that the distance between the flame edge and the tube walloscillates. An ethanol flame will have an effective Lewis number greater than 1.Chen, Bradley, and Ronney (1992) have examined burner flames in a varietyof atmospheres. When the atmosphere is chosen so that the effective Lewisnumber is small, cellular flame structures are observed.The oscillations observed in the first two studies for Lewis numbers greaterthan I, and the cellular structures observed in the third for Lewis numbers lessthan I, bring to mind the familiar stability boundaries of deflagrations (e.g.Buckmaster and Ludford (1983)). But although a significant degree of mixingoccurs at the edges of edge-flames, they are not deflagrations. And, of course,they are not diffusion flames. They are something in between,a hybrid, and ourpurpose here is to construct a simple model which contains some of theessential characteristics of this hybrid structure. This model admits wave-likesolutions in which the edge advances, retreats, or stands still, and we shalldiscuss the stability of these solutions for the special case when the Lewisnumber of the oxidizer is I, but that of the fuel differs from 1. The asymptoticstructures valid in the limit of infinite activation energy have features familiar

EDGE-FLAMES45from deflagration studies, but, as we shall see, there are crucial, if subtle,differences. One consequence is that the familiar closure problem for stabilitystudies is resolved in a novel fashion that only resembles a NEF strategy(Near-Equidiffusional Flames, Buckmaster and Ludford (1983), p. 52) fordeflagrations,Downloaded By: [University of Southern California] At: 17:44 18 March 2011One-Dimensional Model of an Edge-FlameA cartoon which identifies key features of our model is shown in Figure 3. Thislooks like a flame in a channel, but it should not be interpreted in such a literalfashion. It is simply a device to display some important characteristics.Amongst other things, it does not show any flame-structures at the edge thatcan arise from mixing to the left of the edge (see, for example, Kioni et al.(1993)).The flame is affected by two boundaries, as is always the case in nonpremixed combustion. One boundary is the oxygen-supply boundary at whichthe characteristic value of the oxygen concentration is X w- This would belocated at infinity for the candle flame, at the upper end of the tube for theKirkby-Schmitz apparatus.The second boundary, on the other side of the flame, is the fuel-supplyboundary, where the characteristic value of the fuel concentration is Y",. Itcorresponds to the wick for the candle flame, the pool for the Kirby-Schmitzapparatus.TOxygen supply boundaryxwwtransversefluxesflameedgeLztransverse fluxesFuel supply boundaryTwywFIGURE 3 Cartoon of the edge-flame model.

Downloaded By: [University of Southern California] At: 17:44 18 March 201146r BUCKMASTERThe combustion field is assumed to have a charactreistic length scale Linthe direction perpendicular to the flame. An appropriate choice for L for thecandle flame would be the flame radius, for example.Each boundary will have a characteristic temperature and, for simplicity, weshall adopt the single value T w ' T'; will usually be much smaller than the flametemperature, so that this not an important point.Finally, we assume that there is no applied flow over the edge (in thez-direction) in the adopted frame, but transverse flow is not ruled out. The edgeitself can move in the z-direction.With this scenario in mind, let us examine typical model equations governing the combustion field in a Cartesian coordinate system. These are:(I a)(1b)(Ie)Note that the terms which correspond to transverse fluxes-the 'sideterms'-have been enclosed in brackets. n is the reaction term, but we are notready to define it at this moment.We now simplify the side-terms by replacing them in the following fashion:(2a)(2b)(2c)The essential ideal is that the key physical contribution of the side terms is heatloss to the boundaries (T T,vl, oxygen gain from the oxygen boundary(X w X), and fuel gain from the fuel boundary (Yw Y).

Downloaded By: [University of Southern California] At: 17:44 18 March 2011EDGE-FLAMES47This is a bold move (or crude, depending on ones point of view) but is notnovel. An example which is closer to the spirit of our investigation than anyother is afforded by the work of Weber, Mercer, Gray, and Watt (1995) on atwo-dimensional reactive thermal problem which is reduced to a one-dimensional problem. An earlier example is the one-dimensional heat loss analysis ofSpalding (1957) for a flame in a tube.There is no a priori assumption that the side terms are small. Although itmight seem that a large heat loss (via the term in C I ) would quench the flame,this effect is negated by large fluxes of reactants via the C z, C 3 terms.C"C z , and C 3 are constants, but since C, can be scaled out, viz.(3)we may, without loss of generality (wlog), take it to be I.EquilibriumEquilibrium is defined by the constant solutions of Eqs. (1), (2)0). It corresponds to a balance between the side terms and thereaction. It follows, because of the linear nature of Eqs. (2), that, in equilibrium, X and Yare linear functions of T (C,).("f,,-Tw)/Q CzpD x(X w - Xe)fyx C 3 P Dy(Yw - Ye)fyy)· We define T. as the Burke-Schumannflame temperature for the underlying diffusion-flame system. In other words T.is the flame temperature of the equilibrium state when the Damkohler numberequals infinity. We then define X a and Ya by(a/at a/az (4)where X; and Ye are the linear equilibrium functions.With these definitions, the reactions rate is chosen to be(5)the second defining ingredient of our model.There are several points to be made in connection with the formula (5). Notethat, following the substitutions (2), X and Yare representative or averagevalues of the concentrations so that (5) is a resolution or the closure problem

J. BUCKMASTER48Downloaded By: [University of Southern California] At: 17:44 18 March 2011for the reaction rate. Also, for large activation energy, reaction will only besignificant for values of T close to 7;" and the true concentrations in theneighborhood of the reaction zone will be small, vanishing in the limit ofinfinite Darnkohler number, so that it is natural to associate (X - X a ) and(Y- y,,) with these local concentrations. Clearly related to this is the fact that,with the choice (5), we guarantee that equilibrium for infinite Darnkohlernumber (essentially 8-(0) is characterized by T - 7;,.Non-Dimensional EquationsConsider a stationary structure that propagates with speed U to the left(Fig. 3) so that in a frame attached to the edge, a/at- u a/az. We introducenon-dimensional variables by the formulaszTpC L- C LZs - 0 - V - P-U Q P-QL'7;,'-.Ie'C LZ.Ie'RTB P e-l/'B 8 aA'E '(6a,b,c,d)(6e,l)so that the governing equations becomedOdZOQ -V - - - z -(0-0 ) - Qds dsWC P 7;, ,(7a)dX dZXV- --dZ Cz(Xw-X)-[Yx]Q,dss(7b)(