Spatially localized radiating diffusion flames

O

pen

A

rchive

T

OULOUSE

A

rchive

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uverte (

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Eprints ID : 18561

To link to this article : DOI:10.1016/j.combustflame.2016.10.002

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https://doi.org/10.1016/j.combustflame.2016.10.002

To cite this version : Lo Jacono, David and Bergeon, Alain and

Knobloch, Edgar Spatially localized radiating diffusion flames. (2017)

Combustion and Flame, vol. 176. pp. 117-124. ISSN 0010-2180

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Spatially

localized

radiating

diffusion

flames

David

Lo

Jacono

a,∗

_,

_Alain

_Bergeon

a

_,

_Edgar

_Knobloch

b

a Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS-INPT-UPS, Toulouse, France b Department of Physics, University of California, Berkeley, CA 94720, USA

Keywords: Diffusion flames Localized states Instabilities

a

b

s

t

r

a

c

t

AsimplemodelofradiatingdiffusionflamesconsideredbyKavousanakisetal.(2013)[1]isextendedto twospatialdimensions.Alargevarietyofstationaryspatiallylocalizedstatesrepresentingthebreakup oftheflamefrontnearextinctioniscomputedusingnumericalcontinuation.Thesestatesareorganized byaglobalbifurcationinspacethattakesplaceataparticularvalueoftheDamköhlernumberandtheir existenceisconsistentwithcurrentunderstandingofspatiallocalizationindrivendissipativesystems.

1. Introduction

Inpremixedflames,strongcouplingbetweenaerodynamicsand reaction/diffusionprocessesarisingfromstrongthermalexpansion givesrise toDarrieus–Landauinstabilities [2].In diffusionflames, incontrast,thermalexpansionplays aminorroleinthe thermal-diffusive processesthatarebehindthepresenceofinstabilitiesin non-premixed flames [3]. The origin ofthese instabilities, which canleadtospatiallyuniformpulsationortostationaryor oscillat-ing cellularstructures, hasbeenreviewedbyMatalon [4,5]. Mod-elsofnon-premixedflamesthat neglectaerodynamiceffectshave proved particularly useful in studies of turbulent partially pre-mixedcombustion.Awell-knownexampleisprovidedby the tur-bulentflameletmodelofPeters [6]whosedynamicsaredominated by diffusive processes, and not advection. The now classical flat unstrained flameintroduced byKirkbeyandSchmitz [7]hasbeen extensively studied, and numerous stability analyses [8–11] have shownthattheLewisnumbersofbothreactantsplayanessential roleinselectingthenatureofthepossiblethermal-diffusive insta-bilitiesthat appearwhenapproaching extinction(herenecessarily vialeanmixtures).Examplesoftheseinstabilitiescanbefoundin axisymmetric jet configurations, both experimentally [12–14] and numerically [15],as well as in tubular flames [16,17] andin un-strainedflames [18,19].

The presentpaper is devoted to shedding additional light on the properties and dynamics of diffusion flames near flame

ex-tinction. Of particular interest in this connection is the dynami-cal behavior preceding extinction,including intermittency, break-up,hopping,andothertypesoftimedependence.Diffusionflames typicallyexhibit cellularstructures,oftenaccompanied by tempo-ral oscillations. Existing studies range from detailed simulations ofmodels that retain as manyof thebasic processesas possible to highly simplified model systems, designed to exhibit an un-derstanding of the qualitative properties of such flames, usually throughtheuseoflinearstabilityanalysisintime,e.g., [20].Such modelstudiesareusefulindevelopingbothphysicalintuitionand amathematicalunderstandingoftheobservedtransitions.The lat-ter oftenreliesonbifurcation theoryandrelevantdynamical sys-temstheory [1].

Thispaperfocusesonthespatialstructureofdiffusionflamesin theextinctionregime,butgoesmuchbeyondlinearstability anal-ysis.Specifically, we showthat a well-knownmechanism respon-sibleforthepresence ofspatially localizedstructures in continu-ous systemsdescribed by partialdifferential equations applies to simplemodelsofradiatingdiffusionflames,andexploreits conse-quencesforthepredictionsofthemodel.Thismechanismis math-ematicalinnature,andemploys an understandingofthistype of model developed using ideasbased on the notion of spatial dy-namics:treating the spatialprofile ofastationarysolution ofthe equationsasaconsequenceofevolutioninthespatialvariable,in other words, as ifspace were like time [21]. This is a powerful ideathatmakesmostsenseinsystemswithoneunbounded direc-tion.Ofcourse,realsystemsaredefinedbyboundary-valued prob-lemswhereastime-likeproblemsaresolvedwithinitialconditions. It turnsthat this difference isnot crucial, andthat much canbe deducedbased onthisapproach evenwhen thedomain isfinite, provideditissufficientlylarge.Thisapproachisextendedhereto

Fig. 1. The S-shaped branch of one-dimensional equilibria in terms of the average temperature  T  ≡(1 / 2) 1

−1T¯(x) dx as a function of the Damköhler number Da for

Leo = Le f = 1 , T b = 0 . 1 , T a = 1 and R = 0 . 2 , showing the folds A and B (solid dots)

and a Hopf bifurcation at Da ≈ 1800 ( H , open square symbol) on the upper branch S+ . The labels S 0 and S −_{indicate the middle and lower branches. Stable (unstable)}

branches are shown in solid (broken) lines.

twospatialdimensionsandusedtocomputealargevarietyof sta-tionaryspatially localized states representingthe breakup ofthe flamefrontnearextinction.Thesestatesareorganizedbyaglobal bifurcation in spacethat takes place ata particular value of the Damköhlernumberandtheir existenceisconsistent withcurrent understandingofspatiallocalizationindrivendissipativesystems.

Weconsiderasimplemodelofaradiatingdiffusionflame stud-ied in [1]. Specifically, we consider a one-dimensional flame be-tween a pair of porous walls that allow fuel (mass fraction Yf)

to diffuse in from the left (x=−1) and oxidizer (mass fraction

Yo) to diffuse in from the right (x=1), both assumed to have

the same temperature Tb. The burning process is described by

a binary one step process of Arrhenius type with reaction term

w₌DaYoYfexp

(

−Ta/T

)

,whereTistheinstantaneoustemperature

and the constant Ta represents the activation temperature. This

temperature-activated process releases heat that is redistributed viaradiation.Convectionisignored.Thesystemisdescribedbythe nondimensionalequations

∂

T

∂

t =

∂

2_T

∂

x2 +w− RDa



T4_{− T}4 b



, (1) Leo

∂

Yo

∂

t =

∂

2_Y o

∂

x2 − w, (2) Lef

∂

Yf

∂

t =

∂

2_Y f

∂

x2 − w (3)

withtheboundaryconditions

T=Tb, Yf=1, Yo=0 at x=−1, (4)

T=Tb, Yf=0, Yo=1 at x=+1. (5)

HereLeo andLef arethe Lewis numbersofthe oxidizer andfuel,

respectively,Risaparameter,andDaistheDamköhlernumber. Steady solutions

(

T¯

(

x

)

,Y¯o

(

x

)

,Y¯f

(

x

))

of this boundary value

problem are independent of Leo, Lef and reveal the presence of

theclassic S-shapedresponsecurve asafunction ofthe Damköh-lernumberfirstidentified byLiñán [22]andPeters [23]. Figure 1

showsa typicalresultintermsofthespatialaverageofthe tem-perature,



T



≡

(

1/2

)

₋₁1 T¯

(

x

)

dx,plottedasafunctionofDa.Inthe followingwe refertothestatesontheupperbranchofthecurve asS+ while thoseon thelower branchare labeledS−;the states

inbetweenarelabeledS0_{(Fig.1).Thestabilityofthissolutionwas}

examinedin [1]forLeo=Lef=1.ThemiddlesegmentS0ofthe

S-shapedresponsewasfoundtobeunstable,witharealeigenvalue passingthroughzeroatboththeleftandrightfolds,labeledAand

Bin Fig.1andrepresentedassolidcircles,asexpectedonthebasis ofstandardbifurcationtheory.Inaddition,theauthorsidentifieda sequenceofHopfbifurcationsontheS+ branch,thefirstofwhich destabilizesS+ as Daincreases(square symbol in Fig. 1), leading to temporal oscillations.Other Hopf bifurcationsrestabilize S+ at largerDa(notshown),sothatinall casesthelargeDapartofS+

wasfoundtobestable.Thisisofcoursetheignitedstate.Thusthe resultsof [1]canbeinterpretedasshowingthat,withinthismodel atleastandforappropriateparametervalues,theflameundergoes oscillationspriortoextinctionasDadecreases.

Inthispaper,weareinterestedinthesteadysolutionsthat bi-furcate fromthe foldsAandBonthe S-shapedbranchwhenthe problem is extended to the (x, y) plane, _−∞<y_<∞. We refer tothe y directionasthe transversedirection. Inother words, we studysteadysolutionsoftheproblem

∂

T

∂

t =

∂

2_T

∂

x2 +

∂

2_T

∂

y2 +w− RDa



T4_{− T}4 b



, (6) Leo

∂

Yo

∂

t =

∂

2_Y o

∂

x2 +

∂

2_Y o

∂

y2 − w, (7) Lef

∂

Yf

∂

t =

∂

2_Y f

∂

x2 +

∂

2_Y f

∂

y2 − w (8)

with the y-independent boundary conditions (4)–(5) on x=±1 andperiodicboundaryconditionsiny withalargespatialperiod

Ly 1.Theresultingsystemisspatiallyreversible,i.e.,itis

invari-antunder thetransformationy→−y.This isan important prop-ertyofthemodelthathasimportantconsequencesforthespatial eigenvaluesofthebase(y-invariant)state [21].

As in theone-dimensional case steadysolutions of this prob-lem are necessarily independent of the Lewis numbers Leo, Lef.

Moreover,oneclassofsolutionsconsistsofthosefoundin [1], ex-tended uniformlyiny.However, whenthestabilityofthese solu-tions isexamined withrespectto y-dependent perturbations one findsthatother,y-dependent,solutionsmaybepresent.Thisisso despite the fact that there is no bifurcation to periodic states in the y direction, theso-called stripedflames. This fact canbe es-tablishedbyexaminingthelinearproblemforaninfinitesimal per-turbation (a(x,y, t), b(x,y, t), c(x, y, t)) ofthe one-dimensional time-independent solution

(

T¯

(

x

)

,_Y¯_o

₍

_x

₎

_,Y¯_f

₍

_x

₎₎

_{. This linear} prob-lemcan be separatedby writing

(

a

(

x,y,t

)

,b

(

x,y,t

)

,c

(

x,y,t

))

=

(

a

(

x

)

,b

(

x

)

,c

(

x

))

exp

(

σ

t+iky

)

. The resulting k-dependent linear problem has no solutions with zero growth rate

σ

when k = 0, indicating the absence of a pattern-forming Turing instability. However, whenk=0thelinearproblemdoeshavetwolocations where

σ

=0.ThesearepreciselythefoldsAandBontheS-shaped branch,where– asalreadymentioned– thestabilityproblemfor theone-dimensionalsolution

(

T¯

(

x

)

,Y¯o

(

x

)

,Y¯f

(

x

))

necessarilyhasa

zeroeigenvalue.

The paper is organized as follows. In Section 2, we provide a brief summary of the essential input from dynamical systems theory that guides our study. Detailed results are presented in

Section3andtheseconfirmthebifurcationstructureanticipatedin

Section2.Thepaperconcludeswithabriefdiscussionand conclu-sions. The numerical procedure usedto compute localized struc-turesinthepresentsystemisdescribedinthe Appendix.

2. Abriefreviewofthetheory

The theory summarized belowapplies to systemswith a sin-gle unbounded direction such as the y direction in the present problem.WesupposethatthesystemexhibitsanS-shapedbranch

of y-independent states. In this caseit is known [24–26] that a branchofy-dependentsolutionsbifurcatesfromeachfold,andthat in both cases it doesso inthe inward direction. An explicit ex-ample of a weakly nonlinear calculation demonstrating this fact can be found in Appendix C of Ref. [24] or in the Appendix of Ref. [26]. The solutions that bifurcate fromthe folds are not pe-riodic, but take the form of a gentle dip in S+ (fold A) or gen-tle bumpontopofS− (foldB),bothwithy-dependentprofiles of theformasech2by.Herea∝

|

Da− Dasn

|

1/2 andb∝

|

Da− Dasn

|

1/4,

where Dasn is theDamköhler numbercorresponding to fold A or

B (and hence the location of the saddle-node bifurcation). Both statesare reflection-symmetricwithrespecttothe reflectiony→ −y.Withincreasingdistancefromthefold Athedipdeepensand the solution localizes in they directionforming a state that has been calleda hole: thesolution connectsthe upperbranch state

S+ tothelowerbranchstate S− andback again.Theshape ofthe fronts connecting these states is determined by the value of Da

andishighlynonlinear.As onefollowsthisbranch,using numer-ical continuation, for example, one finds that the hole broadens as the S− state invades the domain and displaces the fronts to either side away from the center. Following the solution branch further one observes that the peak amplitudebeginsto decrease while the solutioncontinues to broaden,thereby forming a peak on topof theS− state. Thisprocess continuesasone followsthe branch yet furtheruntil the solution branchterminates atfold B

onthebranchofy-independentstates.ThusthefoldsAandBare infact connectedby a branchofy-dependent steadysolutionsof problem. We refer to the resulting y-dependent solutions as lo-calized states, andlabel the associated branch with the letter L. Whenperiodicboundaryconditionswithalargespatialperiodare imposed in the y direction, as typically done in numerical com-putations, the above situation persists, although the bifurcations tothey-dependentstatesshiftfromthefoldstonearbyvaluesof theDamköhlernumber.Allofthesetransitionscanbeclearlyseen inrecentwork ontheLugiato–Lefeverequation arising in nonlin-ear optics [26]; thisequation also exhibitsan S-shapedresponse curvebutthecalculationsaresimplersincetheequationasposed in [26]isone-dimensional.

The detailed behavior of theL branch dependson the nature ofthespatialeigenvalues

λ

ofS±.Theseeigenvaluesare obtained by linearizingthe time-independent Eqs.(6)–(8)aboutthesteady state

(

T¯

(

x

)

,_Y¯_o

₍

_x

₎

_,Y¯_f

₍

_x

₎₎

_{and looking for steady solutions of the} form

(

a

(

x,y

)

,b

(

x,y

)

,c

(

x,y

))

=

(

a

(

x

)

,b

(

x

)

,c

(

x

))

exp(

λ

y). Because ofthesymmetryy→−yif

λ

isaneigenvaluesois−

λ

;if

λ

is com-plex

λ

∗,−

λ

and−

λ

∗_{arealsoeigenvalues.Thusgenericallyspatial}

eigenvaluescomeinquartets.Inparticular,whentheleading spa-tialeigenvalues(theeigenvalueswhoserealpartisclosesttozero) of S± are both realthe front connectingS+ toS− will be mono-tonic,leavingS+alongthe

λ

+_>0directioninphasespaceand ap-proachingS−alongthecorresponding−

λ

−_{<0direction;thefront}

at the other end of the hole will also be monotonic, leaving S−

alongthe

λ

−_>0directionandapproachingS+alongthe₋

λ

+ _< 0 direction.InthiscasetheLbranchisalsomonotonic:itextendsto therightfromtheleftfoldatAandtotheleftfromtherightfold atB,becomingverticalatanintermediateDavalue,hereafterDaM,

correspondingtotheformationofafrontbetweenS+andS−; ow-ing to the symmetry y→−ythis solutionin fact corresponds to the formationof aheteroclinic cycle, whenthe systemisviewed asevolvinginyfrom_−∞to_+∞.ThiscycleconnectsS+toS−and backtoS+,orequivalentlyS− toS+ andbacktoS−.Unfortunately all thesesolutions are unstable withrespect toinfinitesimal per-turbationsgrowingintime.

If, however, the spatialeigenvalues

λ

+ of S+ are complex the frontswillnolongerbemonotonic,andwillexhibitgrowing oscil-lations asthey depart from S+ before approaching S− along the −

λ

− _{direction. The presence of these oscillations is reflected in}

growingoscillations inthebranchLastheholedeepensandfills withS−.These oscillationsare importantbecausetheLsegments withapositiveslope(i.e.,betweenafoldontheleftandthenext foldontheright)inthebifurcationdiagramwillbestablewith re-specttoinfinitesimalperturbations intime [26];therestoftheL

branch,andinparticularthesegmentsnearAandBremain unsta-ble.Thusinthiscaseonefindstemporallystablespatiallylocalized solutionslyingonportionsofthebranchL– anewtype ofstable diffusionflame.SincetheintervalsofstablesolutionsonLlie be-lowS+ (see below)thesespatially localizedflames willbe cooler thanthespatiallyhomogeneousflameS+.Note,inparticular,that thesestablelocalizedsolutionsarepresentdespitetheabsenceof spatiallyperiodicy-dependentstates.Intheliteraturethistypeof behaviorisknownascollapsedsnaking [25,27].

Theabovestatementsareconsequencesofthetheoryofspatial dynamicsapplied to problemsofthis type [21,28]andhenceare notspecific tothe problemunderdiscussion.Inthe nextsection, wedemonstratethattheseconclusionsindeedapplytoEqs.(6)–(8)

withthe boundary conditions (4)–(5), anddiscuss the additional featuresthatcomeaboutwhentheproblemisposedonaperiodic domainintheydirectionwithperiodLy<∞.

Infactthesituationisyetmorecomplexasweshowinexplicit computations,alsointhenextsection.

3. Results

Our numerical results employ the same parameter values as usedin [1],viz.Leo=Lef=1,Tb=0.1,Ta=1andR=0.2.All

cal-culationsareperformedinadomainofarea S≡ 2× Ly with

pe-riodicboundaryconditionsaty=0,Ly andLy=32.Werepresent

ourresultsinbifurcation diagramsshowing



T



_{≡ S}−1_ST

(

x_,y

)

dS_,

i.e.,theaveragetemperatureinthedomain.Thisrepresentationof the solutionis preferred since it is capable ofdistinguishing be-tween solutions withtemperaturepeaksof differentheights, un-like diagrams showing, for example, the maximum temperature along the center line x=0. Stability properties of the solutions are determinedusing theArnoldi method whichis usedto track thedominanttemporaleigenvalues ofthesolutions asafunction oftheDamköhler number.Thefact thatthe domaininy isfinite modifiesincertain respects thepredictions ofthegeneraltheory summarizedintheprecedingsection,asalsoexplainedbelow.

3.1. Primarybranches

The finite domain size Ly < ∞ has two main effects: (i) it

forces the wavenumber k to take on integer values, and (ii) it limits, for example, the width of the S− hole that can develop within the S+ state. The former implies that the bifurcations at theleft andrightfoldsbreak upintoa seriesofsuccessive bifur-cations generating solutions with k=1,2,3,..., with k=1 rep-resenting one wavelength within the y domain, k=2 represent-ing two wavelengths etc. Since modulation of

(

T¯

(

x

)

,Y¯o

(

x

)

,Y¯f

(

x

))

with wavenumber 0 < k 1 is no longer possible these bi-furcations will be displaced from the folds, with the bifurcation generating k₌1 closest to the fold, k₌2 further from the fold, etc. Figure 2(a)shows theS-shaped branchof y-independent so-lutionscomputedin [1](Fig.1) together withfive branchesofy -dependentstatesconnectingtheregionnearfoldAwiththeregion nearfoldB.Allofthesebranches,hereafterLk,behaveinasimilar

way,with the k=1 closest to the theoretically predicted behav-ior.Near theprimary bifurcation pointnearB thesolution corre-spondstotheslowlyvaryingpeak state;by thetime onereaches the firstfold on the left thispeak state hasgrown inamplitude totheamplitudeoftheS+ stateandhasself-focusedintoasingle spatially localizedpulsewhose shape isdetermined by thevalue oftheDamköhlernumberatthisfold(Fig.3,lowestpanel).Atthis

Fig. 2. (a) Bifurcation diagram showing the quantity  T  ≡ S −1 

ST(x, y ) dS as a

function of the Damköhler number Da for Le o = Le f = 1 , T b = 0 . 1 , T a = 1 , R = 0 . 2

and L y = 32 . The branches of k -pulse localized states are denoted L k . Stable (unsta-

ble) branches are indicated in solid (broken) lines. (b) A close-up of the left folds on each of the L k branches shown in (a) showing the presence of tertiary branches

of multipulse states of unequal height (stability not indicated).

Fig. 3. Solutions T (0, y ) at the left folds on the L k branches for the parameter values

of Fig. 2 and appropriate values of the Damköhler number Da . The horizontal lines indicate the amplitude of the plateau corresponding to Da = Da M ; the vertical lines

indicate the axes of symmetry of the different solutions.

foldthebranchacquiresstabilityandthepulseremainsstableuntil thenextfoldontherightwhereitlosesstabilityagain.Sincethere isinfactaninfinitenumberofoscillationsinthebranchas



T



in-creases(impossibletoseeon thescaleofthefigure)thisprocess repeats,generatingan exponentiallydecreasingsequenceof inter-valsinDawithstablepulses.Thesesuccessivepulsesdifferinthe widthoftheplateauatamplitudeS+ thatisenclosedbythe oscil-latingfrontsoneitherside,whoseprofileisfixedbythe Damköh-lernumberDa≈ DaM=1318.Theotherbranchesbehavesimilarly,

exceptthatL2 self-focusesintotwoidenticalequidistantpulses,L3

self-focusesintothreeidenticalequidistantpulses,etc.The pulses comprisingtheL1,L2,L3,... solutionsatthefirstfoldsontheleft

differ only very slightly,owing tothe slightly different valuesof

Daatthesefolds, andthisremains soprovided they remain suf-ficientlyseparated.Ofcourse,sincethepulsewidthisdetermined by Daonlya finitenumberofpulses canfitinto theavailable y -domain.Forthis reasonthesolution brancheswithk >3 depart fromthetheoreticallypredictedbehavior:thecorresponding solu-tionscannolongerbeconsideredtobespatiallylocalizedand sig-nificantinteraction betweenpulsestakesplace.Forthesereasons theLkbrancheswithk> 3nolongeroscillate,andindeeddonot

undergo theabrupt growth inthe pulse widthpredicted to take placenearDaM.

Thevalue Da=DaM corresponds totheformationofa

hetero-clinic cycleconnectingthehomogeneous statesS+ andS− atthis value ofDa, andplays the role ofa Maxwell point familiar from systemsexhibitinggradientdynamics [21]. Figure4showsseveral profilesatDa≈ DaMwithincreasingvaluesof



T



.Atthefirstright

foldatthelowerendofthenear-verticalsegmentoftheL1branch

the one-pulse state shown in Fig. 3 (bottompanel) starts to de-velop a dimple in the center (Fig. 4, profile 1). This is a conse-quence ofthecomplexspatialeigenvalues ofS+.As



T



increases the pulse broadens (Fig. 4, profiles 2 and 3). In an infinite sys-temthisprocesscontinuesindefinitely, resultingintheformation oftheheterocliniccycleconnectingS+ andS−.Ina domainof fi-niteperiodthisprocessinevitablyterminatesandthebranchturns atthetoptowards lowerDamköhler numbers. Figure4,profile4, shows that in this regime the minima at either end start to lift up,creatinga hole-likemodulationofthe hightemperaturestate

S+.WithdecreasingDathisholebecomesshallowerandshallower andattheterminationpointnearfold A theholedisappearsand thestate S+ isrecovered.Ofcourse,becauseoftranslation invari-ance,theholecanbemovedtothecenteroftheperiodicdomain. One sees that such a state therefore represents an “anti-pulse”. Thus on a finite domain the branchof pulses bifurcating fromB

connectstothebranchofholesoranti-pulsesthatbifurcatesfrom

A [26].Analogous evolution takesplacealong branchesLk,k >1,

withthesamevalueofDaM.

We emphasize that the oscillatory approach of the solution branchesin Fig.2(a) toDa=DaM andhence thepresence ofthe

foldson the Lk branchesis adirect consequence ofthe fact that

the leading spatial eigenvalues of the state S+ at Da=DaM are

complex.TheseoscillationsinturnstabilizetheportionsoftheLk

branches with positive slope and hence lead to multistability of differentlocalizedstatesinthevicinityofDaM.

3.2. Secondarybranches

For k > 1 the primary states produced by the above process resemble periodic arrays of pulses. It is therefore not surprising that the foldson thesebranches generate a secondary family of hole-likestates,i.e.,spatiallymodulatedarraysofpulses.The mech-anism is essentially the same,except that this time the stability problemisofFloquettype andthecomputationofthemodulated statesnearthe foldscannot be accomplishedanalytically evenin one spatialdimension. Figure 2(b)showsthe secondarybranches neartheleft foldsontheprimary branchesL_koflocalizedstates. Themodulationalinstabilitygeneratessolutionsconsistingofa cer-tainnumberofhigherandlower peakstakenfromtheupperand lower brancheson either side ofthe L_k fold. We usethe integer

,  < k, to denote the number of higher peaks in the solution (k−  is then the numberof lower peaks), and labelthe result-ing modulatedsolutions using thenotationM_k.In each casewe observe the presence of k− 1 secondary branches. In particular, when k=1 there is nopossible modulation, andindeed no sec-ondarybifurcationisobserved.Whenk=2amodulationcentered ony=0 caneitherincreaseordecreasetheheight ofthecentral peak(Fig.5).However,translationinvarianceofthesysteminthey

Fig. 4. Detail of Fig. 2 in the vicinity of Da = Da M showing the abrupt evolution of the solution profile that takes place in this regime. The profiles T (0, y ) shown on the right

correspond to the numbered locations: 1 : Da = 1321 . 6974 , 2 : Da = 1318 . 1069 , 3 : Da = 1318 . 0387 , 4 : Da = 1221 . 3835 . The horizontal lines in the right panels indicate the plateau corresponding to Da = Da M ; the vertical lines indicate the axis of symmetry of the solutions.

Fig. 5. Comparison of the L 2 solution (top panel) with the M 21 solution (lower pan-

els) at x = 0 . The solutions in the lower panels are related by translation through Ly /2. The thin horizontal line connects the peaks with the smaller amplitude and is

drawn to emphasize the height differences arising from spatial modulation of the L 2

state in the top panel. The Damköhler numbers are ( L 2 ) 1229.7331, ( M 21 ) 1245.8670,

and ( M 21 ) 1244.0886.

directionshowsthattheresultingtwostatesareinfactequivalent; thus only one branch of modulated pulses, hereafter M21,

bifur-catesfromthevicinityoftheleftfoldoftheL2branch(Fig.2(b)).

When k=3a modulationcentered ony=0 can eitherdecrease theamplitudeofthetwo middlepeaksorincreaseit.The result-ing statesare nowdistinct andhencetwo branchesofsecondary modulatedsolutionsbifurcatefromthevicinity ofthe lowestfold ontheLkbranch(Fig.6).Whenk=4therearethreedistinct

sec-ondary brancheslabeled M43,M42 andM41 (Fig. 7),andsimilarly

fork=5(Fig.8).Aseachofthesebranchesisfollowedfromtheir

Fig. 6. Comparison of the L 3 solution (top panel) with the M 31 and M 32 solu-

tions (lower panels) at x = 0 . The Damköhler numbers are ( L 3 ) 1229.6859, ( M 32 )

1249.4207 and ( M 31 ) 1250.0697.

birthinthedirectionofincreasingDa,all replicatethesametype ofsnakinginthe vicinityofDaM astheprimary branches.

More-over,similar tertiary bifurcationstake place atallthe other folds ontheLk branches. Forasimilar construction inadifferent

two-dimensionalsystem,see [29].

TheaboveanalysisisimportantforthestabilityoftheLkstates.

WehavealreadynotedthattheL1branchisinitiallyunstablewith

asingleunstableeigenvalue.This isbecauseitbifurcates fromS0

above the right fold, where S0 _{is once unstable. This bifurcation}

issupercritical, inthesense that it isinthe directionof increas-inginstabilityofS0 _{(decreasingDamköhlernumber).Consequently} L1 isonceunstable.IfnoadditionalbifurcationstakeplaceL1 will

Fig. 7. Comparison of the L 4 solution (top panel) with the M 43 solution (second

panel), the M 42 solutions (panels three and four) and the M 41 solution (last panel),

all at x = 0 . The solutions in panels three and four are related by translation by Ly /4. The Damköhler numbers are ( L 4 ) 1229.1817, ( M 43 ) 1249.7594, ( M 42 ) 1249.6024,

( M 42 ) 1249.6946 and ( M 41 ) 1249.7307.

S0 _{which alsorestabilizesataleft fold(foldA).Thus oneexpects}

thesinglepulsestatetobestableabovethefirstleftfoldofL1;this

stateisdestabilizedbythesameeigenvalueatthenextfoldonthe right,andsoonallthewayuptheL1branch.Similarly,thebranch L2istwiceunstablenearthebifurcationthatcreatesit(itis

unsta-blewithrespecttoaspatiallyuniformeigenfunctionandalsowith respecttosingle-pulse perturbations).Thefirst ofthese eigenval-uesrestabilizes at the left fold of L2 whilethe other is involved

inthebifurcationtothemodulatedstateM21thattakesplacenear

thisfold.Thusafterthesetwobifurcationsatorneartheleftfold thetwo-pulsestateacquiresstability.Atthesametimeweseethat thesecondarybifurcationtotheM21 stateissubcritical,andhence

thisstateisunstablenearthebifurcationthatcreatesit(Fig.2(b)). Bythesame argumentthethree-pulse state L3 alsoacquires

sta-bilityaftertheleft fold onL3 andthetwo successivebifurcations

creatingM31andM32,etc.Theseresultsarecompletelyconsistent

withournumericalcontinuationresults.

4. Discussionandconclusions

In thispaper, we have examined a simple model of diffusive flames with radiative temperature redistribution, focusing on its propertiesneartheextinctionregime (foldlabeledA).Despiteits simplicitythemodelexhibitsinteresting properties.Using numer-icalbranch-following methods, motivatedby a theoretical under-standing of the origin of spatially localized structures [21], we identifiedasequenceofbranchesconsistingofkequidistant tem-perature peaks along the flame front. These solutions represent statesgenerated by the breakup of the front, and correspond to cooler structures centered on the middle of the channel (x=0) andlocalizedinthetransversedirection.Wehaveseenthatthese statesarelocatednearthe“Maxwell” pointDa=DaM

correspond-Fig. 8. Comparison of the L 5 solution (top panel) with the M 54 solution (second

panel), two distinct M 53 solutions (panels three and four), two distinct M 52 solu-

tions (panels five and six), and one M 51 solution (panel seven). The branches of

the M 52 and of the M 53 solutions are distinct but coincide in the representation

used to plot them, to within numerical error. The Damköhler numbers are ( L 5 )

1222.9785, ( M 54 ) 1239.7467, ( M 53 ) 1239.7574, ( M 53 ) 1239.3069, ( M 52 ) 1238.0365,

( M 52 ) 1239.8294 and ( M 51 ) 1239.7314.

ing to the appearance of a heteroclinic connection between the cold homogeneous state and the homogeneous flame, and that theycanbe stableprovidedthespatialeigenvaluesofthehot ho-mogeneousstate are complex.We haveconfirmedthatthisis in-deedthecasefortheparameter valuesconsidered in [1].1 Apply-ing thesame ideasto thelocalized statesLk, nowviewed as

pe-riodicstateswithperiodLy,wediscoveredthepresenceof

multi-ple tertiary branchesoriginating fromthe vicinity ofall folds on theLk branches,andconsistingofkpeaksofdifferentheights.

Al-thoughwedidnotdeterminethestabilitypropertiesofthese ter-tiarystates we haveargued thattheir presence is infact keyfor the stabilityofthe multipulse states Lk,k > 1,since bifurcations

tothesestatesare requiredtostabilizethemultipulse statesnear

1 These eigenvalues become pure imaginary at and beyond the Hopf bifurcation

Da=DaM.Theseresultssuggestasimpleprocedureforidentifying

thelocationinparameterspacewherelocalizedflamesarepresent – itsufficestocompute thedependenceofDaM onthemodel

pa-rametersandusetheresulttotrackDaMthroughparameterspace.

WeremarkthateachofourLksolutionscorrespondstoa

peri-odicstateontherealline,−∞<y<∞,eachwithadifferentbasic wavelength.Eachoftheseperiodicstatesundergoesa foldonthe leftmuchlikethemuch-studiedSwift–Hohenbergequationwitha quadratic-cubicnonlinearity [30].Itfollowsthatapairofbranches ofspatiallylocalizedpulsetrainsnecessarilybifurcatesfromtheleft foldofsuchreplicatedLkstatesandthattheseundergohomoclinic

snakingasdescribedin [21,28],incontrasttothecollapsedsnaking exhibitedbyLk.Thusonlarge domains,admittingmodulation

ex-tending overmanywavelengths,a yetgreater varietyoflocalized structures, consistingofcoexistinggroupsofpulseswithdifferent numbersofpulses,canbefound.However,thecorresponding cal-culations appearprohibitive andwe havenot pursued this direc-tionfurther.

Themodelstudiedhere,despiteitssimplicity,capturesthe es-sentialqualitativepropertiesof“real” diffusive flamesseen in ex-periments.Todemonstratethatthisissoa movieispresentedas supplemental material. This movie records an experiment in the first version of a novel one-dimensional burner [18]. The exper-iment consists of a closed square combustion chamber with the reactants and productspositioned in a counterflowconfiguration. This is done using 625 injection needles (1 mm O.D.) through which the reactant is fed in while the hot air/products exit the chamberinbetweentheneedles.Theoxidizermixture(O2–CO2)for

thismoviewasfixedat70%O2byvolume(62.92%mass),withthe

reactant mixture(H2–CO2)slowlydecreasingfrom24%to12%H2

byvolume(1.41%to0.73%mass).Theflowratewasmaintainedat

Qo=3.33× 10−5m3/s(33.3ml/s),Qf=3.00× 10−5m3/s(30ml/s)

with a cross-sectional area of 46 mm2_{. The movie wascaptured}

by a digital camcorder (Sony, DSR 300P), at an oblique angleof about 45° with respect to the flat and horizontal flame. Frames fromthismovieare shownasstill picturesin [18].The movie il-lustratesthetransitionprocessfromaflatflametoafullycellular flameallthewaytospatiallylocalizedflamelets.Themoviebegins witha flatflame,andasthereactantmixture becomespoorerin H2 cellformationstartsattheflameedgesadjacenttothe

cham-ber walls. We note the cellsare not spatially locked tothe tube grid.AstheH2concentrationisreducedfurther,thecellularregion

grows towards the center ofthe flame. After some time (around 1min),theentireflameconsistsofcellsandthetransitiontothe fullycellularregimeisconsidered complete.Atthispoint,thecell structures (flame“patches”)are notstationary.AstheH2

concen-tration isreduced evenfurther, thenumberofcellsprogressively decreasesuntiltheextinctionlimitisreached.Webelievethatthe flameundergoes breakupinto spatiallylocalizedstructures inthe time interval1.0–1.5min.It isinthisregionthatourinsightinto the presenceofa large multiplicityoflocalizedstructures witha specificcharacteristicsizebecomesparticularlyrelevant.

The key new observation for near-extinction flame dynamics is the observationthat the fold A representing extinction of the uniform (y-invariant) flame is responsible for the presence of a large number of simultaneously stable spatially localized flames at nearbyvalues ofthe Damköhler number. Webelieve that this propertycarriesovertomoresophisticatedmodelsandhencehas broad relevance to diffusion flames in general. This newclass of states becomes particularly relevant when the uniform flame S+

becomes unstable to oscillations already nearA, forexample, by movingtheHopfbifurcationHin Fig.1tolowervalues(by chang-ingthevalueoftheparameterRortheLewisnumbersLeo,Lef)so

thatthelocalizedflamesarealsotime-dependent.

Giventhesuccessofourapproachitisnowofconsiderable in-teresttoexploreextensionsofthemodeltomorerealistic

configu-rations.Infutureworkweshallexploretheeffectsofdifferentfuel andoxidizerLewisnumbersonthedynamicalpropertiesofthe lo-calizedstateswehavefound(whichareindependentofLeo,Lef),as

well asincludingtheeffects ofhydrodynamicflow (cf. [20]),and extending the calculationsto three dimensions in orderto iden-tifyspot-likeflamesintheextinctionregion.Morerealisticmodels shouldalsoincludedensitydependenceofthemodelparameters.

Acknowledgment

TheworkofEKwassupportedinpartbytheNationalScience FoundationundergrantDMS-1211953.

Appendix

In order to compute steady solutions of Eqs. (6)–(8) with the boundary conditions (4)–(5) as a function of the Damköhler numberDa, we use a standard numerical arclength continuation method [31]based on aNewton solver forthetime-independent version ofthe equations [32].This methodsallows one to follow branchesofstableorunstablesteadystates,compute their linear stability,locatesecondarybifurcationsandfollowtheresulting sec-ondarysolutionbranches.

Todo so,the equationsare spatially discretized usinga spec-tralelement method inwhich the domain[0, 2] × [0, Ly] is

de-composed intoNe spectral elementsof size[0,2]× [kly,

(

k+1

)

ly]

withNely=Lyand0≤ k≤ Ne− 1.Ineachelement,thefields(T,Yo,

Yf)are approximatedbyahigh-orderinterpolantthroughnx × ny

Gauss–Lobatto–Legendrepoints.Inthecontinuationprocedure,the solutionsarecalculatedasafunctionofthebrancharclength.Each pointofthisbranchcorrespondstoasteadystatethatincludesthe valuesY¯≡

(

_T¯_j,Y¯_oj,Y¯_{f j}

₎

_{1≤ j≤N of (T, Yo, Yf) at the N=Nenxny} dis-cretization pointsand the corresponding Damköhler number Da. TheunknownsX¯≡

(

Y¯,Da

)

areobtainedby solvingthediscretized steadystateequationsbyaNewtonmethod.Thearclengthisthen incrementedandtheprocedurerepeated.

TheNewtonsolverthat weusederivesfromafirst-ordertime integration scheme for the time-dependent problem [33,34] and we usea scheme in whichthe diffusive linear partof the equa-tions is treatedimplicitlywhile the nonlinear partis treated ex-plicitly. Eachtime step requiresthe inversion ofthree Helmholtz problems.ThediscreteversionoftheweakformofeachHelmholtz problemisfirsttransformedintoasetofnxone-dimensional

prob-lemsafterapartialdiagonalizationoftheoperator, eachofwhich issolvedusingaSchurdecompositionmethod.

Once a steady state is obtained, its linear stability properties arestudiedbymeansofArnoldi’smethod.Thismethodallowsone tocompute themostunstableeigenvalues andthecorresponding eigenvectors, andthesecan be used to locatesecondary bifurca-tions.Oncea steadybifurcation islocated,the corresponding un-stableeigenvectorandassociatedsteadystate areused tobuilda predictorforasolutionalongtheemergingbranch;thecodethen usescontinuationtofollowthenewbranch.Again,the implemen-tationofthe Arnoldiprocedureusesthe firstordertime-stepping code following the procedure originally proposed by Tuckerman

[33].

The technique has been used with success to compute time-independentlocalizedstructuresinseveralsystemsarisinginfluid mechanics [35–38].

Supplementarymaterial

Supplementary material associated with this article can be found, in the online version, at 10.1016/j.combustflame.2016.10. 002.

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