Nonpremixed and premixed flamelets LES of partially premixed spray flames using a two-phase transport equation of progress variable

Title

Nonpremixed and premixed flamelets LES of partially

premixed spray flames using a two-phase transport equation of

progress variable

Author(s)

Hu, Yong; Kurose, Ryoichi

Citation

Combustion and Flame (2018), 188: 227-242

Issue Date

2018-02

URL

http://hdl.handle.net/2433/230559

Right

© 2017 The Author(s). Published by Elsevier Inc. on behalf of

The Combustion Institute. This is an open access article under

the CC BY-NC-ND license.

(http://creativecommons.org/licenses/by-nc-nd/4.0/)

Type

Journal Article

Textversion

publisher

ContentslistsavailableatScienceDirect

Combustion

and

Flame

journalhomepage:www.elsevier.com/locate/combustflame

Nonpremixed

and

premixed

flamelets

LES

of

partially

premixed

spray

flames

using

a

two-phase

transport

equation

of

progress

variable

Yong

Hu

∗

,

Ryoichi

Kurose

DepartmentofMechanicalEngineeringandScience,KyotoUniversity,Kyotodaigaku-Katsura,Nishikyo-ku,Kyoto615–8540,Japan

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received1June2017 Revised24July2017 Accepted3October2017 Availableonline5November2017

Keywords:

Mixedreactionregime Largeeddysimulation Flamelet

Sprayeffect

Reactionprogressvariable

a

b

s

t

r

a

c

t

Partiallypremixedsprayflamesaresimulatedwithflamelet-basedtabulatedchemistryparameterizedby themixturefractionandprogressvariable.ThetransportequationofthereactionprogressvariableCis reconsidered,anditsformulationforthereactingtwo-phaseflowsisderivedandemployed,whichallows theinclusionofsprayimpactsthroughanewspraysourcetermthatisabsentinitsgaseousform.Both thenonpremixedandpremixedflameletsassumingsinglereactionregimeareimplementedinLES,and theirvaliditiesinsprayflamesand dependenceontheevaporationeffectwhenconsideringtwo-phase

Cequationareexamined.Theeffectofspray,reactionandturbulenceinteractionistheninvestigatedin comparisonwith experimentsofSydneyreacting acetone sprays,covering the rich,leanand stoichio-metriccases.Thecomputedresultsgenerallyfollowtheexperimentaldata,butadisagreementbetween two flameletsimulationsis observedespecially inrich and leanflames. The premixed flamelets tend tocapture thedownstreamjetspreadingwhileoverestimating thepeaktemperaturecomparedtothe nonpremixedchemistry.Flameindexanalysisindicatesthatinthepresentsprayflamesan evaporation-dominatedregimeexists insidethe upstream core jetand it promotesthe coexistence ofsubsequent interactingpremixedandnonpremixedreactionzones,whichimpedesaccurateflamepredictionbythe singleregimeflamelets.Furthermore,thespraysourcetermappearinginthederivedCequationis iden-tifiedtoactasscalarfluxesdrivenbysprays inflameletstructures. Includingthisnew sourcetermis foundtobeimportanttoaccountforthedissipationeffectinducedbyevaporationonthereactionzone intheflameletsimulationofturbulentsprayflames.

© 2017 The Author(s). Published by Elsevier Inc. on behalf of The Combustion Institute. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

The use of liquid fuel in turbulent combustion is prevalent in many industrial devices. The process in those systems in-volves complex multi-physics and features interactions among spray evaporation, turbulent transport and vapor fuel/air mix-ing, aswell aschemicalreactions that determinethe behaviorof such combustion devices in relation to both stability and pollu-tant emissions.Because oftheprevaporizationeffects and disper-sionoflocalfueldroplets,sprayflamesareoftencharacterizedby a partially premixed reaction mode [1,2] , exhibiting the proper-ties of both premixed andnonpremixed flames, and show addi-tional evaporation-dependent flame structures compared to pure gascombustion [1,3,4] .Consequently,theincreasedcomplexitiesin

∗ _{Correspondingauthor.}

E-mailaddress: [email protected] (Y. Hu).

reactingspraysmakeitchallengingforsimulation,andselectingor developingpropernumericaltoolsforspraycombustionmodeling isan importantissuewhenthedesignofmoreefficientandclean combustionsystemsisdesired.

Large-eddysimulation(LES)hasgainedincreasing attentionin recentyears andproved its abilityto yield reliable computations ofcomplexreactingsprayflows [5–10] .Unsteady turbulent struc-turesandmixingareexplicitlyresolvedinLES,butthemodelingof subgridscale(SGS)chemicalreactionsremainsamajorissuesince thecombustionprocessoccurspredominantlyinasmallscalewell belowthe LES filterwidth [11] .Severaldifferent LES combustion models havebeen successfully applied in previous spray studies, whicharebasedoneitherassumedPDF approaches,such as con-ditionalmomentumclosure [5] andtheflameletapproach [6,7] ,or PDF-like models [12] , such asthe linear-eddy model [8] and the transportedPDF method [9,10] .Among thesecombustion models, theflamelet-based tabulatedchemistry approach showsthemost attractivein thatwith adramaticallyreduced computationalcost

https://doi.org/10.1016/j.combustflame.2017.10.004

0010-2180/© 2017TheAuthor(s).PublishedbyElsevierInc.onbehalfofTheCombustionInstitute.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. (http://creativecommons.org/licenses/by-nc-nd/4.0/ )

thedetailedchemistrycanbeincorporatedinthemodeling,which is important to accurately describe the emission formation and transientphenomenahighlyinfluencedbythefinite-ratechemistry andsprayevaporation [13] .

For the proper calculations of two-phase flows when using flameletmodels,aspecialcareshouldbepaidtothemodelingand inclusionof inter-phasecouplings [24] .In both nonpremixedand premixedgaseous flameletsmodeling,thegeneratedflameletsare usually characterized using two control parameters, the mixture fractionandthe reactionprogressvariable,where theformer de-scribesthemixednessoffuelandair,andthelatterrepresentsthe progressoflocalreaction.Thisparameterizationofflamelet struc-turesshowedits most robust capabilityin predictingthevarious flamefeatures [25,26] . In theimplementation offlamelet models for turbulent flame simulation, the transport equations of these twocontrolvariablesaresolvedtogetherwithotherflowvariables andturbulent effects on the chemistry are accountedfor in this manner.The effects, such astheevaporation insprayflames can usually be included in additionalsource terms of their transport equations.Manystudieshavebeendevotedtotheexaminationof the evaporationinfluence on the mixture fraction field [6,27,28] , but few studies have focused on the reaction progress and the spray effects on reaction progress of local mixtures in reacting sprayflows.

On the other hand, when applied to the spray flames with mixedinteracting combustionregimes, the predictive capabilities ofclassical flameletmodels need tobe reconsidered. Aturbulent flamein flamelet concept is considered asan ensemble of lami-narflames(i.e.,theso-calledflamelets) [14] andtwomain strate-giesto generatethe flamelet structures can be found in the lit-erature, which rely on either nonpremixed or premixed flames

[15–17] . These two types of flamelet formulations are, therefore, arguably apropos to the description of flames only in the sin-gle nonpremixed orpremixed burning regime. Recently Knudsen andPitsch [18] attemptedtocombinenonpremixedandpremixed flamelets in one simulation of an LES spray combustor, where a combustionregime indexwasusedto differentiate thelocal pre-mixedanddiffusionmodesofburning. However,thisindexinits ownformpresentsmorecomplexityandneglectsthesubfilter con-tributions where evaporationis significant. Thus, because of the lackofa morereliable flameletmodel forsprayflames,the clas-sicalsingleflameletmodelisstillusedbroadlyinstudieson react-ing spraysimulations [6,7,19,20] . EI-Asraget al. [19] useda non-premixedflamelet(i.e.,theflamelet-progressvariableapproach)in the large eddy simulation of a lean direct injector combustor to studythe emission characteristics.The sameflamelet model was adopted by Tachibana et al. [20] in an investigation of combus-tion instability of a model aircraftcombustor. Among recentLES studies, the premixed flamelets were reported in the simulation oftheSydneypilotedsprayflames [6] .Thesestudies showedthat thesingleflameletmodelscansomehowreproducesomefeatures ofsprayflames,butquestionsstill remainregardingtowhich ex-tent thedifferent singleflamelet models candescribe the flames atthepartiallypremixedoperatingconditionsandhowthiswould affectthecouplingswithspraydynamics.Thoughtheperformance ofdifferentflamelet-basedtabulationapproacheshas been inves-tigatedinafewstudiesofgaseousflames [21,22] ,ithasnotbeen completelyidentified in the context ofspray combustion. Partic-ularly, spray flames feature more complex local flame structures with more distributed and coupled multi-reaction regimes com-paredtothegaseouscounterpart [23] .

Themainobjectiveofthepresentworkistoidentifythespray effects on the reaction progress in the flamelet-based LES mod-eling and thus, the transport equation of reaction progress vari-able is reconsidered, and its formulation for two-phase flames is derived. To understand the capabilities and limitations of

single-regime flamelets with respect to the prediction of spray flamesandtheirperformancewithsprayimpactswhenintegrating this two-phase transport equation of progress variable, both the nonpremixedandpremixedflamelets LESsimulations are applied toexperimentalSydneypartiallypremixedsprayflameswithcases featuringvariousinletequivalenceratio [29] .Theremainderofthis paperisorganizedasfollows.Thegoverningequationsforgasand liquidphasesare introducedinthe nextsection. The emphasisis onthediscussionofthederivedequationfortheprogressvariable and the closure method for the unclosed terms. Section 3 gives the experimental setup and computational details. The main re-sultsanddiscussionare presentedin Section 4 ,whichisfollowed bytheconclusionsection.

2. Methodology

2.1. LESgoverningequations

IntheLESofreactivetwo-phaseflows,basedonthedilute ap-proximationthefilteredconservationequationsofgas-phasemass, momentum, and energy, neglecting the volume displacement of dispersedphase,aresolvedandtheyaregivenas

∂

ρ

¯

∂

t +

∂

(

ρ

¯u˜i

)

∂

xi =S˙v, (1)

∂(

ρ

¯u˜j

)

∂

t +

∂

(

ρ

¯u˜iu˜j

)

∂

xi =−

∂

p¯

∂

xj +

∂

∂

xi

2

μ

˜

˜ Si j− 1 3

δ

i jS˜kk

−

∂

τ

¯ sgs i j

∂

xi +S˙m,j, (2)

∂(

ρ

¯h˜

)

∂

t +

∂

(

ρ

¯u˜ih˜

)

∂

xi =

∂

∂

xi

¯

ρ

α

˜h

∂

˜ h

∂

xi

−

∂

J sgs h

∂

xi + ˙ Se. (3)

Intheaboveequations,theresolveddensity-weightedfiltered vari-ableis f¯₌

ρ

f_/

ρ

¯_,andthe overbarrepresentsthe spatialfiltering.

¯

ρ

isthegasdensity,u˜ithevelocity, p¯thepressure,S˜i j the

rate-of-straintensorgivenby

˜ Si j= 1 2

∂

u˜i

∂

xj+

∂

u˜j

∂

xi

, (4)

andthesubgrid,unsolvedstress

τ

¯_{i j}sgs₌

ρ

¯u_iu_j−

ρ

¯u˜_iu˜_j;basedonan eddyviscosityassumption, theSmagorinskymodelisusedto ap-proximatethedeviatoricpartofthistermatthescaleofcellwidth

as

τ

¯_{i j}sgs=−2

μ

tS˜i jwith

μ

t=

ρ

¯

(

Cs

)

2

(

2S˜i jS˜i j

)

1/2.Here,themodel

coefficientCsisobtainedusingadynamicprocedure [30] .S˙v,S˙m,j

andS˙e arethesourcetermsformass,momentumandenergy,

re-spectively,accountingfortheexchangesbetweenthegasand liq-uidphases. h˜ isthe totalenthalpy, andit isincluded herein or-der to evaluate the gas-phase temperature that accounts for the spray effectusing the correction

T=

(

_h˜_−hc

₎

_{/Cp[1] .As} empha-sized by BabaandKurose [1] ,the heat lossdueto droplet evap-oration is relevant in spray flames. The enthalpy hc taken from

flamelet libraries does not include the evaporation effect and a temperature modification should be applied when the enthalpy solved in flowfield outstrips the lower limits of that in flamelet table. Thegaseous temperature obtainedinthiswayis alsoused inthedropletevolutionequations.hcandheatcapacityCparethe

tabulatedvaluesintheflameletdatabaseintroducedbelow. In reacting flows, the evolution of reactants and products in-volvedin thechemicalreactions arealsorelevant,anditinvolves theformationandtransportation ofthousandsofspecies depend-ingon thefuelconsidered [12] .Adirect solutionofthetransport

equationsforallspeciesisusuallynotviableinpractical3D com-bustion simulations. Alternatively, a reduced set of control vari-ablesareadoptedinaflameletmodeltoparameterizeandtabulate thedetailedchemistrydatabase.UsuallythemixturefractionZand progressvariable Careselected andtheir transport equationsare solved inconjunctionwiththeflowfield.In two-phaseflows,the Favre-filtered equation for the mixturefraction can be expressed as,

∂

(

ρ

¯Z˜

)

∂

t +

∂(

ρ

¯u˜iZ˜

)

∂

xi =

∂

∂

xi

¯

ρ

α

˜Z

∂

˜ Z

∂

xi

−

∂

J sgs Z

∂

xi + ˙ Sv, (5)

where the effects of spray evaporation are added in the source termS˙v.

The reactionprogress variableincludes theinformation ofthe extentofreactionprogressofthereactant mixture.Often,the re-actingmixtureisdescribedintermsofthemassfractionofmajor species,thelinearcombinationofwhichisusually usedtodefine theprogressvariable [15,16,25] .Inthiswork,accordingto [31,32] , the following formulation of progress variable in a normalized formischosen

C= Yc Yceq

, (6)

whereYc isthesumofCO2,CO,H2OandH2massfractions

Yc=YCO2+YCO+YH2O+YH2, (7)

which is consistent with previous studies on hydrocarbon flames [15] ,andYceq isthechemicalequilibriumvalueofYc inthe

reactant mixture, which depends on the mixture fraction Z. The progress variable C defined by Eq. (6) is in the range of [0, 1], where C=0 corresponds to the unburnt mixture andC=1 the burnt mixture, andit serves as a useful markerfor the descrip-tion of reaction zone transition in partially premixed combus-tion [32,33] .Itsbalanceequationforsprayflows,asderivedin Ap-pendixA,isgivenas

∂

(

ρ

¯C˜

)

∂

t +

∂(

ρ

¯u˜iC˜

)

∂

xi =

∂

∂

xi

¯

ρ

α

˜C

∂

˜ C

∂

xi

−

∂

J sgs C

∂

xi +

ω

¯¯˙c+S˙c, (8) ¯¯˙

ω

c=

ω

¯˙c+C 1 Yceq d2_Yeq c dZ2

ρχ

Z

ω(I) +2 1 Yceq dYceq dZ

ρχ

Z,C

ω(II) , (9) ˙ Sc=−C 1 Yceq dYceq dZ

(

S˙v−ZS˙v

)

, (10)

where atthe righthand side of Eq. (8 ), inaddition to the diffu-siontermandchemicalreactionterm

ω

¯¯˙cthatwillbeencountered

in the C equation forgas flames, the last term S˙c is new, andit

representsthesourcetermthatstemsfromsprayevaporation.The closed formsof thesetwo sources arediscussed inthe next sec-tion.Here,Jsgs₌

ρ

¯u_i

₋

ρ

¯u˜_i

˜ (

₌[h_,Z_,C])istheresidualsubgrid scalarflux,anditismodeledas

Jsgs =−

ρα

¯ t,

∂

˜

∂

xi,

(11)

wheretheturbulenteddydiffusivity

α

t, isdeterminedby

α

t,=

μ

t/

(

ρ

Sct

)

with Sct=0.4 [34] .It isimportant to note that in

ad-ditiontothenewspraysourcetermappearingin Eq. (8) ,the un-closed reaction rateincludes two more terms

ω

_(I) and

ω

_(II) that are associated with the scalar dissipation terms of

χ

Z and

χ

Z,C,

whichareabsent inthefullypremixedcombustionandrepresent the contributions fromthe diffusion mode ofburning. Their im-portancein aliquid-fueled partiallypremixedcaseremains to be discussed.

2.2.Closureofreactionrate

Thefilteredchemical reactionrateinthetransport equationof progressvariable isobtained by theconvolution of thetabulated chemistrydatabasewiththe probabilitydensityfunction(PDF) to accountforthesub-gridturbulentfluctuationeffects,whichis ex-pressedas ¯˙

ω

c= 1 0 1 0 ˙

ω

c

(

Z,C

)

P˜

(

Z,C

)

dZdC, (12) and

ω

(I)=C˜

ρ

¯

χ

˜Z 1 0 1 Yceq

(

Z

)

d2_Yeq c

(

Z

)

dZ2 P˜

(

Z

)

dZ, (13)

ω

(II)=2

ρ

¯

χ

˜Z,C 1 0 1 Yceq

(

Z

)

dYceq

(

Z

)

dZ P˜

(

Z

)

dZ. (14)

Here,

ω

˙c

(

Z,C

)

is the chemical reaction ratewhich is read in the

flameletdatabase. P˜

(

Z,C

)

is thejointfiltereddensityfunction de-scribingthesubfilterdistributionofthecontrol variablesZandC. Theequilibriumvalue ofYceq

(

Z

)

is afunction ofmixturefraction,

anditisevaluatedintheflamelettabulationprocedure.

Thescalardissipation

ρ

¯

χ

˜Z=

ρα

|∇

Z

|

2,decomposedintothe

re-solvedandsubfilterpart

χ

_Zsgs,isdeterminedinthefollowing man-ner [35] ¯

ρ

χ

˜Z=

ρα

¯

|∇

Z˜

|

2+

χ

Zsgs, (15) with

χ

sgs Z =

β

Z

ρ α

¯ t

2Z2, (16)

whereZ2_{isthevarianceofmixturefractionand}

β

Zamodel

con-stant.

ρ

χ

˜Z,C=

ρα

∇

Z·

∇

C in Eq. (14) corresponds to the cross-scalar

dissipationrateofthemixture fractionZ andprogressvariableC, andit describesthe transport ofthe reactant mixture across the iso-surface ofZ. Thisterm is modeled by the square rootof the productofscalardissipationratesforthemixturefraction

χ

˜Z and

progressvariable

χ

˜C,as [32,36] ˜

χ

Z,C=

˜

χ

Z×

χ

˜C, (17)

where

χ

˜C isapproximatedbyacommonlyusedmodelakintothe

formulationof

χ

˜Zas [37]

¯

ρ

χ

˜C=

ρα

¯

|∇

C˜

|

2+

χ

_Csgs and

χ

_Csgs=

β

C

ρ α

¯ t

2C2. (18)

In Eqs. (16) and (18) ,themodelcoefficients

β

Band

β

Carethetime

scale ratios andassigned with the value 1.0 [37] in the present study.

2.3.Flameletmodeling

Bothnonpremixedandpremixedtabulationtechniquesareused in this work to generate flamelet databases for the prescription ofreactionratedescribedin theabove section.The nonpremixed flameletmodelassumesthe1Ddiffusionflameasthebasic chem-icalstructurecomposingtheturbulentflames,wherethechemical sourcetermisdeemedtobemainlybalancedbythediffusion pro-cesses [14]

−

ρχ

Z

∂

2_Y

i

∂

Z2 =

ω

˙Yi. (19)

Ontheotherhand,thepremixedflameletstructures areobtained bythesolutionofalaminarsteadypremixedflame [17]

ρ

uSl,u

∂

Yi

∂

x =

∂(ρ

Vi,xYi

)

Here, Yi is the species mass fraction and

ω

˙Y_i the corresponding

chemicalreactionrate.

ρ

uandSl,uaretheunburntmixturedensity

andthe laminarflame speed, respectively. Vi,x denotes the mass

diffusionvelocityofspeciesi.

Intermsof Eq. (19) ,thediffusionflameletsarecalculatedwith thescalardissipation rate

χ

Z ranging froma verysmallvalue to

the extinction value, the solution of which would correspond to an S-curve [15] .The steadyflamelet structures together withthe unstableflameletsolutions alongthiscurve arethen tabulated in atablelookupparameterizedbythemixturefractionandprogress variable.Ontheother hand, thepremixedflamelet table consists ofpremixedflamestructuresgeneratedbymeansof Eq. (20) with variousinitialfuel/airmixingstates.

For application to the LES simulation, the generated flamelet databases need to be formulated to the Favre-filtered quantities byintegratingthejointPDF P˜

(

Z,C

)

.Inthewidelyusedpresumed PDF modeling [6,7,12] , under the assumption of statistical inde-pendence,the jointPDF is oftenexpressed astheproduct of the marginalPDFofthedependentvariables,P˜

(

Z,C

)

=_P˜

₍

_Z

₎

_P˜

₍

_C

₎

_.

Inthisstudy,thepresumedPDFmethodisadoptedwitha beta-PDFdistribution forZ anda deltafunction forCin nonpremixed flameletmodelingas ˜

ψ

= 1 0

ψ

˜ P

Z;Z˜,Z2_,C˜

dZ, (21)

and with a beta PDF describing the distribution of C and delta functionfortheZinpremixedflameletmodeling

˜

ψ

= 1 0

ψ

˜ P

C;C˜,C2_,Z˜

dC, (22)

where

ψ

denotesthereactionrateandspeciesmassfractionthat are obtained from the flamelet tables. Z2 _{and C}2 _{are the}

fil-tered variance of mixture fraction and progress variable, respec-tively,andusedintheevaluationofthebetaPDFdistribution.They aredeterminedintheLEScalculationbytheiralgebraicmodel

2₌

β

v

2

∂

˜

∂

xi

2 (23)

withthemodelconstant

β

vsetto0.15accordingto [6,39,40]

It isworth mentioning thatthe combinationofflamelet mod-elswiththetransportedPDFmethodisanotherreliablealternative approachinspraycombustionsimulations [26,41] thatavoidsthe statisticalindependenceassumptionforthejointPDFanddirectly solvesthetransport equation ofjointPDFofthe mixturefraction andother considered control variables, although additional com-putationalcost mayarise duetothesolutionofhigh-dimensional PDFtransportequation.Thisapproachisoutsidethescopeofthis study,andthereadercanrefertoRef. [42] formoreinformation.

2.4.Sub-modelsforliquidphase

The liquid phase is assumed to be dilute spray consisting of spherical single-component droplets. The droplet coalescence or breakup is not considered. The dilute spray evolves accordingto a set of Lagrangian equations describing the dynamics of fuel dropletsincludingtheirtemperature,Td,mass,md,velocity,vd,and

trajectory,xd,inthecontinuousgasphase.Withtheassumptionof

heavyparticles,theforcesconsideredtohaveasignificant contri-butiontothedropletsmotionincludethedragforce,gravitational forceandarandomforceduetothesubgridfluctuationsinLES

dxd=vddt, (24) dvd=

˜ u−vd

τ

d

dt+gdt+

C0ksgs

τ

c

1/2 dW, (25) dTd= Nu 3Pr

cp,g cp,l

_f 2

_˜ T−Td

τ

St d dt+ LV cp,l

dmd md

, (26) dmd=− md

τ

St d

Sh 3Sc

ln

(

1+BM

)

dt. (27)

Here,u˜,andT˜arethelocalgasproperties,namelygasvelocityand temperature, respectively, atthedroplet position.LV isthe latent

heat ofevaporation, gthegravitationalacceleration,cp,g andcp,l

thespecificheatcapacitiesofthegasandliquidphase,NuandSh

the NusseltnumberandSherwoodnumber.

τ

St d =2

ρ

lr

2

d/

(

9

μ

)

the

particlerelaxationtime inStokes regime.f2 isthe correction

fac-torfortheinterphasethermaltransferofevaporatingdroplets [43] .

ksgs_=C−83

s

(

ρμ¯t

)

2isthesubgridkineticenergyandC0=1 [44] .dW

denotes the incrementof a stochastic Wiener process.

τ

_d is the particlekineticresponsetimeandcanbeexpressedas

τ

d= 8 3

ρ

l

ρ

g rd CD

|

u˜−vd

|

, (28)

whererd istheparticle radius,

ρ

g thegasdensity,

ρ

l thedensity

ofdroplets andCD thedragcoefficient givenby an empirical

ex-pression [1] CD= 24 Red

_{1+0.0545Red+0.1Re}0.5 d

(

1−0.03Red

)

1+b

|

Reb

|

c

, (29) b=0.06+0.077e(−0.4Red) c=0.4+0.77e(−0.04Red) ₍₃₀₎

in which Red=2

ρ

grdusl/

μ

and Reb denote the droplet Reynolds

numbersbasedontheslipvelocity usl=

|

u˜−vd

|

andtheblowing

velocityu_b= dm_d

dt

(

4

π

rd2

ρ

g

)

−1,respectively.

Accordingto [44] ,

τ

cdenotesatypicaltimescaleforthe

interac-tionsbetweentheparticleandturbulence,anditisevaluatedwith

τ

c=

τ

d2a

√ ksgs

1−2a , and a=0.8. (31) The mass transfer number BM is the normalized fuel flux

around thedropletsurface, which involvesthefuel massfraction insurroundinggasY˜F andthatatthedropletsurfaceYF,s

BM= YF,s−Y˜F

1−YF,s

(32)

withYF,sdeterminedbytheClausius–Clapeyronrelation [1,45] .

Additionally,studies have shownthe important effectsof SGS scalar fluctuations in the modeling of spray properties, includ-ing auto-ignition [46] .In thisstudy,we adopta techniqueinthe framework offlameletmodeling thatuses thesubfilter presumed PDF toprescriberandomgas quantitiesofY˜F andT˜forthe

evap-orating droplets [28] . In this algorithm, the pairing procedure of thedroplets withastochastic value ofY˜F orT˜ isreinitiatedafter

anintermittentcouplingprocess,whichwassetbasedontheSGS turbulencetimescale,

τ

t=

2

max(α,αt),asappliedinthestudyofDe

andKim [28] .In thiswork, a time ofmin

(

τ

t,

τ

c

)

isused instead

toensurethedroplet/gassubfiltercorrelationisrenewedwhenthe dropletbreaksawayfromaneddyortheeddyisdissipated,andto numericallyavoidtheappearanceofspurious long-duration corre-lation.

2.5. Spraysourceterms

The spray source terms in the gas-phase governing equations accountforthetwo-phasecouplingthroughheatandmass trans-fers.Byusingtheparticle-source-in-cell(PSI-Cell)method [1] ,the

mass,momentum, andenergyexchangeterms,which areS˙v,S˙m,

andS˙e,respectively,in Eqs. (1) –(3) areexpressedas

˙ Sv=− 1

V Nd k=1 nd,k d dt

md,k

(33) ˙ Sm=− 1

V Nd k=1 nd,k d dt

md,kvd,k

(34) ˙ Se=− 1

V Nd k=1 nd,k

₁ 2 d dt

md,kv2d,k

+_dtd

md,kcp,lTd,k

(35)

wherethespraysourcetermsineachcellvolume

Vareobtained fromthesummationofallthe dropletparcelslocatedwithin this cell. Droplet groupingsare used,andn_d,k representsthenumber ofrealfueldropletsinonecomputationalparcelk.

Thesource termS˙c appearing inthenewlyderived Cequation

(8) canbeobtainedwith

˙ Sc=−C˜

˙ Sv−Z˜S˙v

1 0 1 Yceq

(

Z

)

dYceq

(

Z

)

dZ P˜

(

Z

)

dZ, (36)

neglecting thehigher-ordercorrelationsinthe spraysource, mix-turefractionandprogressvariablefield.

3. Flameconfigurationandcomputationdetails 3.1. Testcases

TheflamesconsideredinthepresentLEScomputationsarethe piloted turbulent spray flames, which were experimentally stud-ied at the University of Sydney [29] . This Sydney piloted spray burnerbearsanopenannularconfigurationandiswelldesignedto berepresentativeofreactivesprayflows stabilizedby hotgaseous mixtures that arewidely encounteredin realengineapplications. The burner geometry comprises a central spray jet along with a mixtureofprevaporizedfuelandair,surroundingwhichisan an-nulusofouterdiameter25mmsupplyingthehot-pilotstreamfor thestabilizationofthemainjet.Astoichiometricmixtureof acety-lene,hydrogenandairismaintainedforthepilotflow.Thesprayis generatedby an ultrasonicnebulizerplaced 215mm upstreamof the jet exitplane, andthe centralnozzle diameterD=10.5mm. Moreover,thereisanairco-flowwithadiameterof104mmand bulk velocity of 4.5 m/s. A series of cases involving nonreacting and reacting sprayswith acetone andethanol fuel droplets have been investigated with this burner, ofwhich three withacetone fuelaresimulatedinthiswork.TheyarereferredtoasAcF3,AcF4 andAcF6,whichfeaturetherich,leanandstoichiometricoperating condition,respectively.

3.2. Computationalmethodsandboundaryconditions

A sketch of the computational domain is presented in Fig. 1 . The simulations are performed using an in-house LES code FK3 _{[1,47,48] with the finite difference formulation in a}

Carte-sian coordinate system. The spatial gradients in the momentum equation are approximated witha fourth-order central difference scheme,andaWENOschemeisusedforthediscretizationof non-lineartermsinthescalars’governingequations.Thetime integra-tion is based on a third-order explicitRunge–Kutta method. The computationaldomainextendsto48D_×11D_×11Dinthree direc-tions andconsists of around 5 M grid points with finer meshes neartheinletandshearlayer.Thediffusionandpremixedflamelet librariesforacetone/aircombustionaregeneratedusingthe config-urationsofa1Dcounterflowdiffusionflameandapremixedfreely propagating flame,respectively, withthe FlameMaster code [38] .

Fig.1. Sketchofthecomputationaldomain..

Inflameletequations,theboundaryconditionsatZ₌1andZ₌0 are set as pure fuel acetone andair at temperatureof 300 K. A detailedreactionmechanismwith83speciesand419element re-actions,developedbyPichonetal. [49] isusedtomodelthe ace-tone oxidation. The stoichiometric mixture fraction Zst is 0.095.

The diffusion flamelet solutions comprise the solutions with the scalar dissipation rate varying from 0.1_×10−2_{/sto the extinction}

valueof102/s,resultinginatotalof126differentsteadyand un-steadyflamelet solutions. The flammableregion in the premixed acetoneflamecalculationisintheequivalenceratiorangeof(0.39, 2.55), anda linearly interpolated mixingstate is applied outside thisflammabilitylimit.Itisnoteworthythatincaseofoccurrence of envelope flame, where chemical reaction can happen around each droplets, thismixing assumption might be violated for the mixtureattheleanside [50] duetotheincreasedtemperatureand speciesgradientinthegasareaaroundthisenvelopeflame.Butin boththeexperimentalandcomputationalstudiesofpresentdilute sprayflames [28,29] ,there isno clearindicationoftheexistence ofenvelopeflame,whichisthusassumedtobenegligibleandnot consideredinthepresentstudy.

Theboundarydataofthegasphaseandliquidphaseatthefirst experimental cross-section are used to determine the inlet com-putationalprofiles.Adigitalfiltertechniqueisemployed to gener-atethe pseudo-turbulenceforthe jet velocitiesatinlet based on themethodproposed byKleinetal. [51] .The progressvariablein the pilot-stream is set to unity and zero for other inlet bound-aries. The inlet liquid particles are randomly distributed around each gridpoint, andthedroplet sizeis assignedwiththe Rosin– Rammlerdistribution,matchingthemeasured Sautermean diam-eter.Thedroplet groupingis useddependingonthesize,andfor each sizegroup, particle velocity isassignedbased on the veloc-ity distribution of specific size class given by experiments. The number of droplets within each parcel is determined such that themeasured liquidfuelmassflowrateispreserved.Thenumber of droplet parcels in the computational domain remains around 6.5_×105_{,andafewcaseswithhigherandlowerparticlenumbers}

havebeenstudiedtoensurethesuitabilityoftheparticlenumber usedin the presentcomputations.The statistics are collectedfor each case over eight flow-through times,and all simulations are performedusingCRAY: XE6atthe ACCMS, KyotoUniversity with 576coresandapproximately100hofwallclocktime.

Table1

Experimental inletconditionsofacetone fuelsprayflames, AcF3,AcF4andAcF6,experimentalsetB[29] .

TestCase AcF3 AcF4 AcF6

Bulkjetvelocity(m/s) 24 24 36

Hot-pilotstreamvelocity(m/s) 11.4 11.4 11.4 Bulkco-flowvelocity(m/s) 4.5 4.5 4.5 Aircarrierflow-rate(g/min) 150 150 225 Liquidfuelflow-rate(g/min) 19.4 10.4 21.6 Vaporfuelflow-rate(g/min) 25.7 13.0 23.4

Equivalenceratio 1.6 0.8 1.0

The inlet boundary conditions for the three cases of reacting acetonespray(AcF3,AcF4andAcF6)arelistedin Table 1 .

4. Results

4.1.Sprayflamescalculatedwithdifferentflameletdatabases

Partially premixed acetone spray flames have been simulated andtheacetone/aircombustionisdescribedwiththetabulated de-tailed chemistry by counterflowdiffusion flamelet and premixed flamelet, respectively. The Sydney spray flames characterized by different rates of pre-vaporization, covering cases of rich (AcF3), lean(AcF4),andstoichiometric(AcF6)premixingmixturesatinlet, areconsideredwithan attempttothoroughlyexaminethe perfor-manceofdifferenttabulated chemistriesandtheeffectson spray dynamics.

4.1.1. Gastemperature

Figure 2 shows the snapshots of the instantaneous gas-phase temperaturefor spray flames of AcF3, AcF4 and AcF6, where for each casethe figures at the left and righthand side correspond to the resultsfrom the nonpremixed andpremixed flamelet cal-culations,respectively. Bycomparing three flamecases, itcan be seen that the computed temperature in the premixed and non-premixedcasesdiffer fromeach other,especially nearthe center partofthejet.Thediffusionflamelet(left-sidefigureforeachcase) computesanearliercombustionwiththecenterflamefront estab-lishedclosertothenozzleexit,andcomparatively,thisinnerflame brush is anchored further downstream in the simulations with

premixedflamelets (right-sidefigure ineach case).Thisis partic-ularlyevidentinthe richcaseofsprayflameAcF3. Meanwhile,it can be observed that thiscenter high-temperature reaction zone initiatesearlierinthe lean sprayflameAcF4 than inAcF3 orthe stoichiometriccaseAcF6, whichhas themostreactive mixtureat theinlet. Thereasonforthiswillbe discussedin thesubsequent section. Since the injected fuel droplets move towards this inner flame front,the predicteddistinct temperatureis expectedto af-fectthesprayevaporation,thestatisticsofwhichwillbediscussed in Section 4.1.2 .

With the general idea obtained from the above comparison,

Fig. 3 compares the experimental measurements with the radial profilesofmeangastemperatureatdifferentaxiallocationsx/D=

10, 20, and 30, computed using nonpremixed and premixed flameletsforthreeflamecases.Overall,thetrendsofchangesthat the experimental values suggest towards the downstream of the jetarecapturedbytwoflameletmodelcomputations.Inthethree cases, at the jet exit spray flames are characterized by the par-tially premixed vapor fuel/air mixtures at the ambient tempera-ture.Wheninjectedintothecombustionfield,thiscenterjet mix-turetogetherwithfueldropletsissheathedbythepilotflame,and duetothehighinitial momentumofthejet carrier,the immedi-ateinward propagation ofthe hot-pilotstream isretarded. How-ever, when moving downstream away from the nozzle exit, be-causeof the turbulent mixinganddroplet evaporation, the main jet is slowlyheatedup, asevidenced by the increasing valuesof gastemperaturenearthe centerlineindicatedby theexperiments andbothnonpremixedandpremixedcomputations.Incomparison, amarkeddifferenceisfoundinthepredictionsbythetwoflamelet databases. The nonpremixed flamelet predicts a higher tempera-turecomparedtothepremixedflamelet,indicatingtheimportant influenceof turbulence/chemistryinteractions on theinner flame propagating.In general,the premixedflameletshowsa better re-sult in capturing the flame spreading in the radial direction to-wardsthedownstreamlocations.

Additionally,itisobservedthat,neartheflameedgeofthe ra-dial position r/D=1, where the stoichiometric mixtures are lo-cated,thepremixedflamelet slightlyoverestimatesthepeak tem-perature. As pointed out by Ramaekers et al. [22] in the study ofdifferentflamelet modelsforthe simulations ofSandiaflames, the difference between flamelets in the species mass fraction

Fig.2. RepresentativeinstantaneousfilteredgastemperatureforsprayflameAcF3,AcF4andAcF6,wherethepredictedresultsfromnonpremixedandpremixedflamelet modelsarepresentedatleftandrighthandside,respectively,foreachcase.

Fig.3. ComparisonofradialdistributionofmeangastemperatureprofilepredictedbynonpremixedandpremixedflameletdatabaseforsprayflamesAcF3(left),AcF4 (mid-dle)andAcF6(right)atthreedownstreamlocationsofx/D=10,20,and30.Solidline:nonpremixedflameletcalculations;Shortdash:premixedflameletcalculations;Filled dots:experimentaldata [29] .

predictions canbe attributedto thefact thatinthe nonpremixed flamelet,speciesaretransportedbetweentheiso-mixturefraction linepassing through the Zst plane whereintense reaction occurs,

while they only diffuse inthe C-direction inpremixed flamelets. Thiscanbeusedtoexplaintheoverestimatedpeakofflame tem-peratureinpremixedflameletmodeling.Thediffusionofheatfrom thereactionzone(Z₌Zst)tothesurroundingmixtures is

guaran-teedindiffusionflamelets.

Furthermore, note that, in AcF6, although reasonable agree-ment withthe experimental data is observed forthe predictions at the downstream locations, a considerable underprediction of gas temperature near the centerline by both the nonpremixed andpremixed simulations isfound atthe upstream cross-section

x/D=10.Similardisagreementshavebeenreportedinother stud-ies [5,39,40] .Itwasarguedthatthisdeviationfromthe experimen-taldatacanarisefromtheuncertaintyinthemeasurements.Inthe experiments,thegas-phasetemperaturewasmeasuredusing ther-mocouples, whichcan lead to significant errorsin thehigh tem-peraturezoneoftwo-phaseflows.

4.1.2. Spraystatistics

In this section, the influences of diffusion and premixed flameletcalculationsonthedropletproperties,namelyevaporation anddispersion,arestudiedincomparisonwiththeavailable exper-imentaldata [29] .

In Fig. 4 , thecomputed radial profiles ofdroplet Sautermean diameter (SMD) at four different cross-sections x/D = 5, 10, 20, and30arecomparedwithexperimentaldata.Inthetwodifferent flameletmodelings,thesimulatedvaluesgenerallyfollowthe mea-suredprofilesofdropletSMDinallthreecases,althoughan over-estimatedSMDisobtainedbybothflamelets inthesimulationsof sprayAcF4forthecross-sectionsx/D_> 10.Thesehighervaluescan be attributedtothe overpredictedgas-phase temperatureseenin

Fig. 3 forAcF4atx/D=10.Similarapparentdiscrepanciesarealso observed in AcF3 at x_/D₌30_, where the nonpremixed flamelet yields a much larger SMD than both the premixed flamelet and measurements, andthis isconsistent with thenoted disparityin temperature profiles shown at the same location in Fig. 3 . This observed concordance betweenthe predicted SMD and tempera-ture can be due to the poly-dispersity of present spray flames.

A wide range of droplets differing in size and dynamic history dictatethe injected sprays,within which thesmall droplets tend to evaporate fasterand are more likelyaffected by the gas tem-perature.Comparatively,thelargerdroplets cansurvivefar down-stream,andmoretimeisneededforheatingupthelargedroplets becauseofthesizedependenceofrelaxationtime [26] .Also, itis notedthat,whenapproachingtheflameregion0.4_<r/D_<0.8(the shearlayerbetweenthemainjetandpilotflame),thepredictions yieldaslightlyhigherSMDcomparedtothemeasureddata,which suggeststhedropletsonthejetedgeevaporatemorerapidlythan thoseintheinnerregionofthespray.

Figure 5 shows the radial profiles of the axial mean, Ud and

fluctuating,U_d velocitiesofdropletsatfourcrosssections. Gener-ally,thecomputationsshowgoodagreement withmeasurements. The calculated mean droplet axial velocities from flamelet mod-els well capture the trend of droplet dispersion when traveling farawayfromthejetexit,butadistinguishabledisagreement be-tween the two flamelet predictions isfound. This can be related to the thermal expansion effect induced by heat release in the mainjet.Asdiscussedin Figs. 2 and 3 ,differentpredictionsonthe progress of combustion in the corezone of jet have been made bythenonpremixedandpremixedflameletsimulations. Concern-ing thefluctuatingvelocity U_d,farfrom thecenterline, the com-putedvaluestend toexceed theexperimental data.However, the present LES calculation shows a better agreement than that ob-tainedin theRANS simulation [52] , inwhichthe velocity fluctu-ationsare underpredicted owing to the inadequate estimation of turbulent intensity.Similar observationson theoverestimation of dropletfluctuatingvelocityweremadeinotherLESstudies [6,39] . The predicted highertemperature and the enhanced evaporation associated with it could have caused this discrepancy. The con-sideration of an adequate dispersion model is also expected to improve the results since the overprediction can be a result of insufficient droplets near the flame edge forobtaining the spray statistics.

4.1.3. Reactionzone

Asdiscussedabove,thecombustionchemistrydescribedby dif-fusionand premixedflamelets leadsto differentflame structures in terms of the gas temperature and spray evaporation. In this

Fig.4. ComparisonofSautermeandiameter(SMD)predictedbynonpremixedandpremixedflameletsimulations withexperimentaldatafor sprayflamesAcF3(left), AcF4(middle)andAcF6(right)atfouraxiallocationsofx/D=5,10,20,and30.Solidline:nonpremixedflameletcalculations;Shortdash:premixedflameletcalculations; Dots:experimentaldata [29] .

Fig.5. ComparisonofdropletmeanandfluctuatingaxialvelocityUd,Ud predictedbynonpremixedandpremixedflameletsimulationswithexperimentaldataforspray flamesAcF3(left),AcF4(middle)andAcF6(right)atfouraxiallocationsofx/D=5,10,20,and30.Solidanddash-dotlines:nonpremixedflameletcalculations;Shortdash anddotlines:premixedflameletcalculations;Square:measureddataofUd;Circle:measureddataofUd[29] .

section,the reactionmode isexplored tofurther analyzethekey mechanism.

Figure 6 illustratestheinstantaneousfieldofreactionrate over-lapped with the isoline of gas temperature T=738 K in AcF6 predictedbythe two flamelet calculations.Firstly, asexpected, it isnoted that in the two figures the isoline of T=738 K, which is the ignition temperature of the acetone/air mixture, gener-allyembracesthe region inwhich the mainreaction occurs. The high-temperature reaction zone initiated from the reactive pilot

developedwithincreasingdistancefromtheexitplanethrough ei-ther the coflowentrainment or ignition ofcentral premixed fuel mixture, which could be delayed or prompted by the evapora-tion.Ontheotherhand,itisseenthatthecombustionreactionin thesetwoflameletcomputationsshowssignificantlydifferent pat-terns.Thepremixedflameletcomputestwoevident areasof reac-tiononeitherside ofthepilot-stream whicharemainlyattached totheflow interfacebetweenthepilot withmainjetandcoflow. Incomparison, thenonpremixedflamelet leadsto amore widely

Fig.6. Snapshotsofthereactionratepredictedbynonpremixed(left)andpremixed (right)flameletsimulationsofsprayflameAcF6.Blacksolidline:isolineofgas tem-peraturewithvalueof738K.

distributedreaction.Comparisonsarealsoconductedfortwoother test flames (not shownhere), wherea similar observation is ob-tained.

Figure 7 showsscatterplotsofgas-phasetemperature,Tagainst theequivalence ratio,

φ

forthethreetest flamesobtainedby us-ing the nonpremixed (top) and premixed (bottom) flamelet. The samples in the plots are collected fromall the filtercells atthe across-section x/D =10. Ingeneral, aclosedcircleformed bythe scattersisobserved inthe T−

φ

map, whichsubstantially differs fromtheonethatwouldbeobservedinthegaseous flames oper-atedusingasimilarpilotedburner [53] .Comparatively,aclusterof scatterdataexistsontherich-sideoftheflame(

φ

>1)thatvaries with temperatureand equivalenceratio, which iscreated mainly bysprayevaporationinthecentraljet.Dependingonthedistance fromthepilotedreactingfront,fueldropletsshow adifferent

de-greeofevaporation. Thefuelpocketsleftby theevaporationthen mix withtheoxidiserandform acoredomain ofstratified com-bustiblegasescharacterisedbyanincreasingtemperaturewiththe increaseofequivalence ratio.Fora givenvalue ofequivalence ra-tio,anumberofscatterswithaslowlyrisingreactionrateare ob-served,correspondingtoapremixedburningmode.Thispremixed propagating layer heading towards the inner unburnt core zone persiststillthebaseofflamenearthecenterline isestablishedat thedownstream.InthecaseofAcF4withanonpremixedflamelet model,sincethe innerflame frontis formedfurtherupstream as shownin Fig. 2 , its scatter plot showsfewer scatter data at the fuel-richside(seethefigureattop-middle).

Inaccordance withthe observationin Fig. 6 , inthe premixed flamelet,twoapparent reactionzonesarefound,andthe stoichio-metric mixtures (

φ

=1) in the pilot stream stay in the equilib-riumstatewitha negligiblereactionrate, incontrasttothe non-premixed flamelet case, which shows intense reaction. It is also worth notingthat, onthe lean side, thepremixedflamelet starts thereactionatapproximatelyT=1100Kcompared toT=550K in caseof diffusion flamelet, which mayexplain the higher pre-dicted temperature in this lean region (r/D>1) by the diffusion flamelet,asshownin Fig. 3 .

In flamelet simulations, the chemistry properties are deter-minedfromtheflameletlookuptablethroughthesolved mixture fraction,Z,andprogressvariable,C,whichincludetheeffectof tur-bulenttransportorchemicalreactionsandsprayevaporation. Illus-tratedin Fig. 8 isthegeneraldistributionofthecomputedmixture fractionandprogressvariableintheflow fieldofthe threespray flames,whichisdisplayedintermsoftheiso-contourofjoint nor-malized histogram of Z and C. The difference between the solid anddash-dot-dot linesin the figures isthat the lattertakes into accountthesamplesonlyinthecentralfuelstream.

Thecontourshowstwomainregionswithsignificantvariations ofZ andC. The first region, with a low progress variable in the Apart, corresponds to theevolution of premixedjet mixtures at the centerflow. The second, referred to as theB zone,featuring ahighCandabroaderrangeofmixturefractionaroundthe stoi-chiometricvalue Zst, associateswith themixinglayer around the

Fig.7. Scatterplotsofthegastemperaturevs.equivalenceratioobtainedinnonpremixed(top)andpremixed(bottom)flameletsimulationsofsprayflamesAcF3,AcF4and AcF6atcross-sectionx/D=10.Dotsarecoloredwiththereactionrate.

Fig.8. Iso-contours(solidanddash-dot-dotlines)ofjointnormalizedhistogram ofmixturefraction,Zagainst progressvariable,Cobtainedinnonpremixed(top)and premixed(bottom)flameletcalculationsofthreetestflames,whichareoverlaidwiththecontoursofreactionratedeterminedfromthecorrespondingflameletlibraries. Thickdashlineindicatestheregionwithsignificantreactionrate.Verticaldashline:locationofstoichiometricmixturefractionZst.

Fig.9. InstantaneousfieldofsprayevaporationrateforsprayflameAcF6fromthe simulationswithnonpremixed(left)andpremixed(right)flamelet.Theoverlapped solid-linesdenotelocationsoftheintensereactionregionmarkedbyYOH×YCH2O.

pilot stream. In the three flames investigated here, the mixture fractionsattheinlet determinedbased onthe pre-vaporizedfuel andair flow rate listed in Table 1 are 0.146 (AcF3), 0.08 (AcF4) and0.094(AcF6),whicharereflectedonthedifferentinitialpoints withC=0intheAregionforthethreeflames.Starting fromthis

initialpremixedmixture,thespreadingofpossiblereactantstates towardstheupperregioninZ₋Cspaceconfirmsthemixing pro-cessesbetweenthestreams ofmainjetandhotpilot when mov-ing farther downstream. Owing to the evaporative fuel addition inthe spray jet,a curvedspreading domain, indicatingthis two-stream interaction, is observed instead ofa straight domain that connects upper and lower regions directly which would be ex-pected in gaseous flames with only the dominant effect of tur-bulentmixing. In theB zone, a two-wing structure appears. The leanbranchdelineatestheentrainmentofaircoflowtothe react-ingmixtureofthepilot,whileattherichside,thepremixed mix-turesendupfullyburntthroughacombustion trajectoryaffected bytheinitialcondition,dropletevaporationandturbulence.

InAcF3, asillustrated inthe temperatureprofileof Fig. 3 ,the premixedflameletshowsan underpredictedtemperaturenearthe axis region compared to the values obtained from the diffusion flamelet and the experiments. The central flow temperature is linkedtotheevolutiontrajectoryofpartA,asshownhereforAcF3. In the premixed case, the contour of reaction ratehas a narrow inverted-triangledistribution, and dueto the lack ofdiffusion in the Z direction for the area with low C, the evaporated fuel ex-periences theinadequately predictedreaction progress rate. Near the nozzleexit,the burningratein thecentral flow is morelike diffusioncontrolled.Thus,theheattransferbetweenthe neighbor-ingmixedfuel/airpocketsinthisregionisnottakenintoaccount properly by the premixed flamelet database. Nevertheless, when one moves downstream,even the nonpremixedflamelet leads to a higher prediction of gas temperature, which indicates that no

Fig.10. Contoursofsourcetermsinprogressvariable Eq. (8) fromnonpremixedflameletcalculationsofAcF6.(a):reactionrateω¯˙c(left)andspraysourcetermS¯˙c(right); (b):sourcetermsofω(I)(left)andω(II)(right),respectively.Whitesolidline:isolineofstoichiometricmixturefraction.

single-regimeflameletthatisderivedfromthescenarioof asymp-toticpremixedornonpremixedflamescan appropriately simulate the spray combustion. Spray flames under consideration, as dis-cussed in the subsequent section, are characterized by a struc-tureofevaporation-dominantreactionzones,presentingboth non-premixedandpremixedbehavior.

It shouldbe alsonotedthatinAcF4 underthelean condition, the reaction path of spray jet mixtures initialized in the A zone tends to reach upwards of the stoichiometric condition in the B sidethroughastraightlinesinceAcF4hasthesmallestliquidmass loadingatinlet amongthethreecases.Hence,the effectofspray evaporationisnotapparentinAcF4withasmallincrementof mix-turefractionforunburntcentral gases.Comparatively,AcF6 hasa similar amountofinletpremixedreactantbutamuchhigher liq-uid injectionrate, resultinginthe rich combustionofthe central freshmixtureinBzonewithacurvedreactionpath.Thisexplains whytheinnerflamefrontstartsearlierinAcF4thaninAcF6ashas beenrevealedin Fig. 2 .IfthesprayeffectisneglectedforAcF6,its combustion trajectorywouldbe similar tothat observed inAcF4, wherethecentraljet mixesquicklywiththestoichiometric burn-ingmixtureinthepilot.Thisimpliesthatevaporatingdropletscan changethechemicalstructuresoftheflamesignificantlyandleave distinctfootprintsintheZ−Cspace.

4.2. Spray-reactioninteraction

As demonstrated inabove discussions, theaccurate prediction of spray flames is affected by the representative flamelet struc-turesandtheclosecouplingwithspraysneedstobeaccountedfor withcare.Inthiswork,aformoftheCgoverningequationnewly derived for two-phase flows is employed, accountingfor the in-fluenceofspray/reactioninteraction,afurtheranalysisofwhichis givenbelowbasedonthepredictionsresultingfromthetwo differ-entflameletsimulationsofcaseAcF6withthehighestliquidmass loading,andasimilarobservationcanbemadefortwootherspray cases.

Figure 9 shows the pattern of spray evaporation interacting with intense reaction regions (indicated by the isoline of YOH×

YCH₂O) in the nonpremixed (left) and premixed (right) flamelet

simulations of AcF6. As can be seen in thecase of nonpremixed flamelet simulation, the inner reaction zone occurs in the shear

layerwhichisdirectedinwardsthecentraljet,wherethehotpilot transfers the heat tothe spray mixture,promoting droplet evap-orationcloseto thelayer interface.In turn,theevaporationfuels the reaction zone for further expansion. With increasing down-streamdistance,thecentraljetbreaksdownaroundx_/D₌15_, fol-lowing which combustion reaction establishes atthe axisregion andsubsequentlyenhancestheinteractionofthereactionwith up-comingsprays.Forthepremixedflameletcase,thereactionatthe inner side tends to be broader in space interacting withcentral sprays.Thiscanberelatedtothefactthatpremixedflamelet struc-turescontaininformationregardingthespeciesandheatfluxesin progressvariablespace,promotingthepropagationofthereaction zoneto thelower C area,which canbe observed in Fig. 8 . Over-all,inbothnonpremixedandpremixedflameletcomputations,the evaporating sprays show a pronounced interaction with reaction, eventhoughdifferentpatternsareobserved.

To investigate the influence of spray effects on the reaction progress,thecontourofsourcetermsinthetwo-phaseC Eq. (8) is shownin Figs. 10 and 11 . When a nonpremixedflamelet is con-sidered,asdisplayedin Fig. 10 ,itcan be seenthat thedominant areaofsprayevaporationofS¯˙c overlaps withtheintense reaction

zone of

ω

¯˙c in space, and inmost areasespecially within the

in-ner shearlayer, theevaporationtends todecreasethe local com-bustion intensity witha negative S¯˙c and to slowthe progress of

reactantstoachieveequilibrium.Asimilarobservationismadefor thepremixedflameletcalculationin Fig 11 exceptforthebroader interaction zone for the spray and reaction. As forthe terms of

ω

(I) and

ω

(II),they arenot sprayrelated, showingtheirmain

dis-tribution out of the area where the interaction of spray/reaction dominates. The term

ω

(I) associated with d2Yceq/dZ2 presents a

local maximum with a negative value along the stoichiometric line,andit contributesmainly to flamelet reactionin a diffusion mode [32] . On the other hand,

ω

(II) remains small with a

pos-itive value compared to

ω

_(I). Note that the model used in this work for the cross-dissipation rate

χ

˜Z,C in

ω

(II) may lead to the

overestimationof

χ

˜Z,C [54] .Thus, it can be deduced that the

in-fluenceof

ω

(I) and

ω

(II) isnegligible inthe presentflames.

How-ever, as noted by Bray et al. [32] , this may need reconsidera-tion in a partially premixed flame, where flame propagation is highlyaffectedbythecloselycoupledZandCfieldswithasteeper gradient.

Fig.11. Contoursofsourcetermsinprogressvariable Eq. (8) frompremixedflameletcalculationsofAcF6.(a):reactionrateω¯˙c(left)andspraysourcetermS¯˙c(right);(b): sourcetermsofω(I)(left)andω(II)(right),respectively.Whitesolidline:isolineofstoichiometricmixturefraction.

Fig.12. Comparisonofradialprofilesofmeangastemperature(left),mixturefraction(middle)andprogressvariable(right)computedforcaseAcF6.Solid anddash-dot linesdenote,respectively,thecomputationsfromnonpremixedflameletwithandwithouttheconsiderationofsourcetermS¯˙c.Dash anddot linesdenote, respectively,thecomputationsfrompremixedflameletwithandwithouttheconsiderationofsourcetermS¯˙c.Dots:experimentaldataonlyavailablefortemperature [29] .

Plotted in Fig. 12 are the radial profiles ofgas-phase temper-ature,mixturefractionandprogressvariable computedbyuse of twodifferentflameletchemistriesforAcF6.Thecomputationsthat neglect the spraysource termS¯˙c in the C equation are included

for comparison. It can be seen that the profile of the progress variable is shifted slightly outwards in the calculations withthe spray source term, which can be linked to the observed sup-pression effect of evaporating sprays on the reaction zone (see e.g., Figs. 10 and 11 ). The inward flame propagation is delayed whenconsideringthespraysourceterm,andasubsequentsmaller temperature profile is found at the downstream position x/D=

30. Consistent with the observations made on the mean values, the results of RMS gas temperature shown in Fig. 13 also indi-cate that the cases considering spray source term tend to pre-dictadecreasedtemperaturefluctuations. Thereactionintense in

combustion region is affected by the source term in C equation, andbecauseofthedissipationeffectofdropletsthegasphaseRMS temperature declines accordingly. Since the spray source term is onlyincluded inthefilteredtransport equation ofmeanprogress variable,thepredictedRMSofmixturefractionandprogress vari-ableshowsasmallsensitivitytothesprayeffects.Asindicatedin theworkofDeetal. [28] byconsidering spraytermsinequation ofSGSvarianceofZ,thesprayevaporationcanbeimportantinthe leancasesofSydneysprayflames.Thesamestudycanbecarried out forthe progress variable variance by deriving a correspond-ingtwo-phaseequation ofC,whichwillnotbe discussedinthis study.IngeneralasmalleffectofS¯˙contheflameletpredictionsis

observedinthedilutesprayconfigurationconsideredhere. ItisworthnoticingthatthenewspraysourcetermS¯˙cservesto

Fig.13. ComparisonofradialprofilesofRMSgastemperature(left),mixturefraction(middle)andprogressvariable(right)computedforcaseAcF6.Solid anddash-dot linesdenote,respectively,thecomputationsfromnonpremixedflameletwithandwithouttheconsiderationofsourcetermS¯˙c.Dash anddot linesdenote, respectively,thecomputationsfrompremixedflameletwithandwithouttheconsiderationofsourcetermS¯˙c.

similarformasthatfoundinthesprayflameletequationofspecies mass fraction derived by Olguin andGutheil [4] as Sv

(

Z−1

)

∂_∂Y_Zi.

There,inaconfigurationofcounterflowingsprayflame,itwas ob-served that thissource term arising fromevaporation can domi-natethetransportequationofproductswithaconsiderable contri-butiontothedissipation.Thisisconsistentwiththefindingsofthe presentwork,andgiventhatS¯˙c isoneorderofmagnitudesmaller

than

ω

¯˙cinthisflame,amoreapparenteffectonchemicalreactions

isexpectedfortheimplementationindensesprays.

4.3. Combustionregime

In the above sections, it was shown that the chemical struc-tures of spray flames can not be well captured by accessing a single flamelet database from either the nonpremixed or pre-mixedflameletmethod.Below, thecombustionregime inthis pi-lotedspray flameisanalyzed,andthecalculationsfromthe non-premixedflameletmodelingofthethreetestflamesarediscussed. For theinvestigation of flamestructures, the flame indexis a usefultool,whichisusuallydeterminedasthenormalizedproduct ofmassfractiongradientoffuelandoxidizer [55]

=

∇

YFuel·

∇

YO2/

|∇

YFuel

||∇

YO2

|

Incorrespondencewiththepremixed-likeanddiffusion-like reac-tion regime,

takes a positive and negativevalue, respectively, anditsvaluelocateswithintheregionof(−1,1).

Thecenterlinedistributionofmeantemperatureandmass frac-tionoffuelandoxygen,aswellastheevaporationrateandflame index shown in Fig. 14 reveals that there exist three combus-tion domainsmarked bya,b andc, wherethediffusionand par-tiallypremixedflamestructuresarecoupledwiththe evaporation-dominated regime.In thea region,the jet flowis more evapora-tiondictated,featuringasmalltemperaturebutaslowincreaseof fuel mass fraction.The evaporationof injected droplets is driven by the initial momentum difference between the spray and gas carrierandpartlybythe heatingeffectofthe pilotstream,while the heat diffused from the pilot is offset in a large part by the heat loss becauseof dropletevaporation. When moving far away fromthenozzleexit,once theflowtemperaturestartstoescalate withadramaticdecreaseofYO2,thetransitiontothebregionofa

Fig.14. Axialdistributionofmeanmass fractionoffuel(solidline)andoxygen (dashline),meantemperature(dotline),sprayevaporationtermS˙v/1000(dash-dot line)aswellasflameindex(dash-dot-dotline)alongthecenterlineforspray flamesAcF3,AcF4andAcF6.

premixed-type flame is observed and the flame index increases fromnegativetothepositivevalue.Thedropletsinthecoreregion ofthejet begintoevaporatefasterandoutstriptheconsumption offuelinthepreheat zone,leadingtothe continuousincrease of

Fig.15. Contourofflameindexneartheaxisregion.Isoline indicatesthespray regionwithsignificantevaporation.Filleddots:representativedropletsintheflow fieldcoloredbytheirindividualevaporaterate.

stage,Y_Fuel experiences a bigslump resulting fromthe establish-mentofthemainpremixedburningzonenearthecenterline. Fur-ther downstream, the diffusion flame in the c domain develops alongwithadecliningnegativeflameindexbecauseoftheleakage offuel fromthe bregion andthe oxidizerentrained fromcoflow. Interms of temperatureprofiles, a dual-reaction structure is ob-served,particularlyin leanflame AcF4,withtwo peaksat down-streamlocationscorrespondingtothepartiallypremixedflameand asubsequentdiffusionflame.Itisalsoobservedthatthe evapora-tionrateincreasespromptlywhenspraysreachthevicinityof pre-mixedflamefrontinbdomain,althougharelativelyhigh evapora-tionisalsofoundclosetotheinletduetotheslipvelocitybetween theinjecteddropletsandtheturbulentflow.

Forfurtherunderstandingthedistributionofdifferent combus-tionregimes,thecontourofflameindexoverlaidwiththeisoline of a region with significant evaporation is illustrated in Fig. 15 . It is clear that mixed burning regimes exist in this spray flame, andthey are placed closelyin space, forming multilayer coupled flame structures. Apart from the middle part of the flame zone ina premixed-type indicated by the positive

, an evaporation-controlled regime and diffusion flame burning are found at the near- and far-field areas. Close to the inlet, owing to the accu-mulatedvapor fuel from droplets evaporating in a relatively low temperaturecondition,anegative

isassignedtotheevaporation regime,which reachesits intenseregion atthebase ofpremixed flame.

Thismulti-regimereactioninsprayflamepresentsasignificant challengeforthepredictive capabilityofmodelingbasedon clas-sical single-regime flamelet methods.Knowing this, one research effortproposed acombinedimplementationofboth nonpremixed andpremixed flamelets in a singlesimulation by identifying the local combustion mode with flame index and reasonable results were obtainedinthe simulations of gasflames [16,23,31] . As in-dicated in this work, however, the simple form of

can fail to differentiatetheevaporation-dominatedregimefromthediffusion burningsince they could hold the samenegative

.Another at-tempt resorts to the establishment of a model flamelet configu-ration that fits the spray case, where the inherent spray effect

on flameletstructures is incorporatedinthe tabulated chemistry. Thisincludesthesprayflameletapproach [56] ,wheretheflamelet structures,obtainedfromacounterflowdiffusionsprayflame,well predictedtheturbulentsprayflamescontrolledbyevaporationand nonpremixedcombustion.Anextensionofthismodelwasattained in [26] whena partiallypremixedsprayflameaccountingforthe pre-vaporization effect was considered for spray flamelet gener-ation, but its performance for various spray and gas conditions needsfurthervalidation.

Insummary,forsprayflames amore comprehensive flamelet-basedapproach shouldrecognize thecoexistenceofdiffusion-like and premixed-like reactions and should be able to include the regime withsignificant impacton chemicalstructures dueto the sprayevaporation.

5. Conclusions

Largeeddysimulationsofpartiallypremixedsprayflamewere performed with detailed chemistry effects included by use of flamelet-based combustion models. A new formulation of the progress variable C equation for two-phase flows was devel-oped and used, and it includes a spray source term in contrast to the gaseous counterpart. The single-regime nonpremixed and premixed flamelets parameterized by the mixture fraction and progressvariablewerebothadopted,andtheirperformanceinLES of partiallypremixed sprayflames and dependency onthe spray effectwhenincludingtheproposedtwo-phaseCequationwere ex-amined.ThedilutesprayphasewasmodeledinaLagrangianway, andtheSGSfluctuationeffectontheevaporationwasincorporated byarefinedstochasticapproach.Thecomputationswerediscussed and compared with experiments of the Sydney reacting acetone sprays, whichrange fromrich and lean to stoichiometric operat-ingconditions.

It was found that two flamelet predictions considering the source termof progressvariable generallyshoweda good agree-mentwiththemeasurementsintermsofgastemperature,droplet size, andvelocity.Yet amarked difference betweennonpremixed and premixed flamelet calculations was observed. Generally, the premixedflamelet captured the downstreamjet spreading in the radial direction whileoverestimating the peak temperature com-pared to the non-premixed chemistry. The diffusion flamelet, on theotherhand,yieldedanearliercombustionforcentralpremixed spray mixtures. The disagreement between two flamelet simula-tions tendstobecomesmallinthestoichiometric flame.The rea-son forthisdifference isduetothe differenceof reaction propa-gationin themixture fractionandprogressvariable subspace in-dicated by thesetwo flamelet models. The analysis of Z−C his-togramsshowedthatapartfromtheeffectofdiffusionand chem-icalreaction,evaporating dropletsadd anotherfreedimension af-fectingthecombustiontrajectoryofpremixedreactantsinthecore jet,whichinturninfluencesthestabilizationofinnerflamefront. Thus, these Sydney spray flames characterize a hybrid combus-tion regime, where combustions occurring in diffusion-like and premixed-like regimes are closely located in space and strongly coupled by the turbulent transport and spray evaporation. The present study indicated that the single-regime flamelet can not fully describe the spray flames, and a more reliable modeling would rely on the flamelet structures that should recognize the multi-regimereactionsinteractedwithevaporatingsprays.

It wasalsofound that thenew spraysource termS˙c in theC

equationisidentifiedtoactasanimportantscalarfluxinducedby sprayevaporationinflameletstructuresandthustheincorporation ofS˙c canpotentiallyplayanimportantroleinthepredictive

capa-bilitiesofflameletmodels.Inthepresentcalculationsofturbulent spray flame,it contributedmainly to the retardationof chemical