Filtered Wrinkled Flamelets model for Large-Eddy Simulation of turbulent premixed combustion

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Filtered Wrinkled Flamelets model for Large-Eddy Simulation of turbulent premixed combustion

Renaud Mercier, Cédric Mehl, Benoit Fiorina, Vincent Moureau

To cite this version:

Renaud Mercier, Cédric Mehl, Benoit Fiorina, Vincent Moureau. Filtered Wrinkled Flamelets model

for Large-Eddy Simulation of turbulent premixed combustion. Combustion and Flame, Elsevier, 2019,

205, pp.93-108. �10.1016/j.combustflame.2019.03.025�. �hal-02129890�

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Combustion and Flame 205 (2019) 93–108

ContentslistsavailableatScienceDirect

Combustion and Flame

journalhomepage:www.elsevier.com/locate/combustflame

Filtere d Wrinkle d Flamelets model for Large-Eddy Simulation of turbulent premixed combustion

Renaud Mercier

^a^,^∗

, Cédric Mehl

^b

, Benoît Fiorina

^b

, Vincent Moureau

^c

aSafran Tech, Modelling and Simulation, Rue des Jeunes Bois, Châteaufort, Magny-Les-Hameaux 78114, France

bLaboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay, 3 rue Joliot Curie, Gif-sur-Yvette 91190, France

cCORIA- CNRS UMR 6614-Normandie Université - Université et INSA de Rouen, Campus Universitaire du Madrillet, Saint Etienne du Rouvray, Rouen 76800, France

a rt i c l e i n f o

Article history:

Received 3 October 2017 Revised 12 December 2017 Accepted 18 March 2019 Available online 5 April 2019 Keywords:

Turbulent combustion Premixed ﬂames Flame wrinkling Filtered Wrinkled Flame Large Eddy Simulation

a b s t r a c t

Modelsforcombustion LESbasedonageometricaldescriptionofthereactivelayer arewell suitedto capture the turbulentflame front displacementspeed,butdo not predict the filteredchemical flame structure.Thisarticleaimstodiscussandmodeltheimpactoftheflamesub-filterwrinklinglevelonthe speciesproduction,withafocusoncarbonmonoxideemission.Forthatpurpose,2-D wrinkledflames with asinusoidal pattern, whichincludedetailed chemistryeffects, aremanufactured. Three control- lingparameters areidentified: theflamefilter size,thesub-filterflame wrinklingand the numberof flamepatternscontainedwithinthesub-filtervolume.Thisnewflamearchetype,namedFilteredWrin- kled Flamelets(FWF),may be embeddedinvariouscombustion modeling frameworks.Inthe present paper,itisusedtobuild-upafilteredchemicallook-uptableinordertomodeltheunclosedtermsof thefilteredprogress variable equation.Aprioritests areconductedby analyzinganexistingturbulent premixedflamedatabase.Aposterioritestsconsistinmodelingtheswirlingbluff-bodystabilizedCam- bridgeflame.Resultsanalysisshowsthataccountingforsub-filterflamewrinklingonthechemicalflame structureismandatorytopredictintermediatespeciessuchasCO.

1. Introduction

Introductionofdetailedchemistryeffectsinnumericalsimula- tionofturbulentflamesisanultimateobjectiveforthecombustion modelingcommunity[1].Despitethegreatchallengetoovercome, this effort is mandatory to predict complex phenomena such as flame ignitionor extinctionandalso pollutantformation. Several methodshavebeenconductedtoaccountfordetailedchemistryin aLarge-EddySimulationcontext[2]foraCPUcost,whichremains reasonable, so that the simulation of practical industrial reactive systems is possible. Among them, geometrical formulations have beendesignedtocapturetheflamefrontpropagationandthusto representaccuratelythe flamedynamicsinpremixedorstratified combustionregimesinLES[3].

Examplesofexistingapproachesbasedonageometricalformu- lation are the G-equation [4–7], filtered one-dimensional flames [8–11] andthickened flame model[12–14].These approachesare designed to capture accurately the turbulent flame consumption speed inthe flamelet regimewhether ornot theflame thickness

∗ Corresponding author.

E-mail address: renaud-c.mercier@safrangroup.com (R. Mercier).

and wrinkling are resolved at the LES sub-filter scale. The sub- filterscale flame wrinkling , generally modeled eitherby al- gebraic[15–17]ordynamical[18–20]procedures,iscoupledtothe reactiveflowequationssothatboththesourceandturbulentdif- fusion terms of chemical species balance equations (or progress variable) are enhanced by a factor . Consequently, the turbulent flame front propagates at a speed S=S⁰_L, where S⁰_L is the unstrained laminar flameconsumption speed. Arecent com- parative study, conducted on a turbulent stratified flame config- urationexperimentedatT.U.Darmstadt[21],indeedconfirmsthat geometrical-basednumericalapproachesperformwellwhenfocus- ingontheturbulentflamefrontposition[22].

The impact of the sub-filter scale flame wrinkling on the filteredflamechemical structurehashoweverneverbeenconsid- ered. Indeed, geometrical approaches forflamelet regime usually assume that the filtered flame thickness and chemical profiles arenot affected bysub-filter scaleflame wrinkling.This assumption has been questioned by Moureau et al. [11] who observed, throughthe analysisof theDNSof a turbulentpremixed swirled flame, that filtered quantities are also influenced by the surface of the flame contained at the sub-filter scale. Several families of modeling approaches implicitly account for the impact of sub-filterstate on the resolved quantitiessuch as(i) Linear-Eddy

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94 R. Mercier, C. Mehl and B. Fiorina et al. / Combustion and Flame 205 (2019) 93–108

Fig. 1. Differences between thickened and filtered flame structures. Both filtered and thickened flames have identical thermal layer thicknesses. Legend : — Laminar detailed chemistry solution, — Detailed chemistry solution, filtered at a size = 2 mm, ••• F-TACLES solution..

model[23]whereempiricalmeanofperturbedstatesareused;(ii) PDF-based models were the sub-ﬁlter state is describedthrough a presumed [24] or transported PDF [25]. These approaches are howeverdiﬃculttoexpressinageometricalformalism.

The objective ofthe presentarticleis to modelthe impact of thesub-filter scale flamewrinkling on thechemical flame struc- tureandthereforeon the minorspeciesproductionand destruc- tion,whichis importantforpredictingpollutantemissions. A fo- cus here is made on the carbon monoxide (CO) species. The unclosedtermsoftheprogressvariableequationaremodeledfol- lowinganapproachsimilartotheFilteredTabulatedChemistryfor LES(F-TACLES)model[10],whereachemicallook-uptableisgen- eratedfromFilteredPlanarFlamelet(FPF)elementsthat arecom- puteda priori usinga 1-D flame solver includingdetailedchem- istryand complex transport. The major issueof this approach is to incorporate the flame wrinkling in the look-up table genera- tion. For that purpose, the standard planar flamelet assumption of F-TACLES is revisited by manufacturing a series of 2-D wrinkledflames withanalytical wrinklingpatterns.The paperisorga- nizedasfollows.Section 2 presentsthea priori studyof theim- pact of sub-filter scale flame wrinkling on the filtered chemical flamestructure.Section3describestheproceduretogeneratease- riesof2-Dwrinkledflameswithanalyticalwrinklingpatterns.The explicit filtering of these manufactured solutions serves to com- puteadatabasethat relatesfilteredthermochemicalquantitiesto the flame wrinklingpatterns. A turbulent combustion model us- ingtheFilteredWrinkledFlamelets(FWF) isderivedinSection 5. Itsa priorivalidationagainst afilteredDNSdatabaseisthen con- ductedinSection6.Finally,LESoftheCambridgepremixedflame burner [26] is computed and compared to experimental data in Section7.

2. Aprioristudyoftheimpactofsub-filterscaleflame wrinklingonthefilteredchemicalflamestructure

Two scales are often unresolved in Large Eddy Simulation of turbulent premixed orstratified flames: the flame thickness and thesmall-scaleflamewrinkling.

Thefirstone,thelaminarflamethickness,isingeneralsmaller thanthegridsizeinmostpracticalmeshes[3].Toavoidtheunder- resolution of the reactive layer and therefore a mis-prediction of the flame propagation speed, Thickened Flame model forLES (TFLES)artificiallythickenstheflamefrontbyafactorF[12].TFLES doesnotreproducehoweversomespecificsub-gridscalephenom- ena for intermediate species. An alternative to thickened flame modelareFilteredPlanar Flamelet(FPF)based modelssuch asF- TACLES[10],wherereferenceflamelets are filtered,tabulated and storedinalook-uptable.

FilteredPlanarFlameletapproachisillustratedinFig.1bypost- processing the detailed chemistry solution of a laminar freely- propagating 1-D methane-air flame computed with the REGATH solver[27]usingthemechanismproposed byLindstedt[28]atan equivalenceratioφ=0.83andatmosphericpressure.Thereactive flame thickness, definedhereasthe full widthathalf maximum (FWHM) of theheat release,is δr=0.17mm. The thermalflame thickness,definedfromunburntgastemperatureTû andburntgas temperatureT^b asδl⁰=(^T^b−Tû)/max(∇^T), is δ⁰l =0.53mm. To mimic a referenceideal laminar LES flame solution, the detailed chemistry temperature andCO mass fractionprofiles are filtered byaGaussianfunctionofwidth^.^ProfilesôftemperatureT,nor- malizedprogressvariablereactionrateω˙Y_C/max(ω^˙Y_C)ând^CO^mass fractionY_COareplottedinFig.1for=2mm.

Theeffectiveratio/δ^r=11.2iswellrepresentativeofpractical reactive LES configurations, where the flame thicknessis smaller than the filterassociated to the flame [29].An advantage of the filtering formalism is the conservation of the mass of chemical species contained within the flame layer as discussed in [10]. Figure 1 presents an a priori reconstruction of the filtered temperature andCOmass fractionsextracted froma FPFlook-up table.Fora1-Dunstrainedlaminarflame(i.e.withoutanysub-filter wrinkling) F-TACLES recovers the filteredlaminar flame structure asexpected.

The second unresolved scales are flame wrinkling patterns smaller than the filter size. The impact of sub-filter scale flame wrinkling on the filtered flame propagation speed is well accounted for in F-TACLES models [10]. However this formulation assumes that the filtered chemical flame structure is not altered by sub-filter scale eddies (i.e. the flame is planar at the sub-filter scale). This assumption is challenged by the post- processing of turbulent premixed flame DNS data obtained by Moureau etal.[11]ofthePRECCINSTA combustionchamber [30]. The configuration, which features a plenum, a swirl-injector and a combustion chamber, is representative ofaeronautical combus- tiondevices.DNSflamedataarefilteredatasize =11.2δr.The Favre-filtered temperature and CO mass fraction fields are then conditionally-averaged on iso-contours ata givendistance to the filtered flame front position. This filtered flame front position is defined as the iso-contour c˜=0.8, where c˜is the Favre-filtered normalizedprogressvariable.Theconditionalaveragesareplotted inFig.2. Diamondandplus symbol linesshow referencesprofiles obtainedattwoaxial distancestotheburner head,z=1 cmand z=4 cm,respectively. While the firstaxial position is locatedat the anchoring ofthe flame,the second position is atits tip. The wrinklingoftheflameincreasesfurther downstreaminthecom- bustor, and the two axial positions correspond to two different sub-filterscaleflamewrinklingconditions:=1.2atz=1cm,

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R. Mercier, C. Mehl and B. Fiorina et al. / Combustion and Flame 205 (2019) 93–108 95

Fig. 2. Filtered temperature and CO mass fraction proﬁles obtained by post-processing a DNS solution of the PRECCINSTA conﬁguration. Legend : Filtered DNS solution ( = 1 . 2 ), +—+ Filtered DNS solution ( = 2 . 0 ), F-TACLES solution.

whereas =2 atz=4cm.Itcan benotedthat forthesecond position,wheretheflameishighlycorrugated,multiplefront-front interactions occur, which leadstoan increase ofthe temperature andtheCOmassfractiononthefreshgassideforX<−2mm.The filteredchemical flamestructureapriori predictedby F-TACLESis reconstructed by post-processing the DNS progress variable field with a filtered look-up table built fromPlanar Flamelets filtered at =11.2δ^r ^(FPF) âs ⁱⁿ ^the post-processed DNS. As discussed in [11],the sub-filter scale flamewrinklingtends to increase the filteredflamethickness.Itleadstoaslightunderestimationofthe thermallayerby F-TACLESasobservedinFig.2,whichisempha- sizedwhenincreases.Thediscrepanciesareconsiderablymore significantfortheCOmassfraction.Indeed,neglectingtheimpact ofsub-filterscaleflamewrinklingonthecarbonmonoxideproduc- tioncausesan50%underestimationofCOpeakinthiscase.

While the impact of sub-filter scale flame wrinkling on the flame thermal layer remains marginal, its effect on intermediate species prediction is pronounced. In the following section, a methodology named Filtered Wrinkled Flamelets (FWF) is pro- posedtoincludesub-filterscalewrinklinginthefilteredflamelets formalism.

3. Modelingtheimpactofthesub-filterscalewrinklingonthe filteredflameproperties

The influence of sub-filter scale wrinkling on the chemical flame structure is in practice difficult to anticipate and describe analytically.Inthissection,manufacturedflamesolutionsareused tostudytherelationbetweenthefilteredflameproperties(suchas consumption speed,thickness, andspeciesprofiles)withthegeo- metricalpropertiesofthewrinkledflameatthesub-filterscale.

3.1. Problemmodel

The impact of the geometrical properties of a 2-D wrinkled flame on the filtered flame structure is analyzed. Assuming that the sub-filter flame is in the flamelet regime, the flame front is trackedbyaprogressvariablecequalto0and1infreshandburnt gases,respectively.A2-Dwrinkledflamepattern,showninFig.3, is manufactured by assuming that the coordinates (^xc=c₀,yc=c₀) ofthe iso-linec=c₀ are relatedbya sinusoidalfunction: x_c=_c₀= Asin(²π^y^c=c₀/Ly),withAthe amplitudeofthesinusoidaliso-line and Ly its wavelength. This flame patternis embedded in a 2-D square ofsize ^,^the ^flame^filter^size,^to ^mimicâ ^sub-filter^scale domain.Forsimplicity,theflameis notwrinkledinthespanwise direction. Themodelcouldbeeasily extendedwith3-D manufactured solutions. A 2-D approach is however preferred hereas it simplifiesthemodelformalismaswellasitsanalysisinafirstap- proach.

Fig. 3. Schematic view of sub-filter sinusoidal flame pattern with n = /L y= 2 . The mapping process of a thermochemical quantity onto a 1-D premixed laminar flame computed with detailed chemistry is also presented.

Thenumberofwavelengths containedwithinthe boxisgiven by n=/Ly.As discussed inthe BML analysis[31], 2n corre- spondstotheaveragenumberofflamecrossingsofaspecifiedline perunitdistance^.^We^suppose^here^thatⁿ isaninteger.Fora givenflamefiltersize^,^the^sub-filter^flame^patternîs^then^peri- odicandparametrizedbytwo parameters:n controlsthewave- length while A controlsthe amplitudeof the c=c₀ iso-line. The sub-filter scale wrinkling is determined by the pair (A, n). For n=0orA=0theflameremainsplanarwhileforA>0,theflame iswrinkled assoonasn≥1.The possible rangeofvariation of bothAandn isdiscussedinthefollowing.

3.2.PhysicalrangeofvariationofmodelparametersAandn

Assumingaflameletregime,thesizeofthesmallesteddyhav- ingaturnovervelocitysufficienttowrinkletheflamefrontisthe GibsonlengthL_G[32]:

LG=(^S⁰l)³

=

_S₀

l

u

3

⁽¹⁾

whereS⁰_l is thelaminar flamespeed, u quantifiesthe sub-filter velocityfluctuationsand isthe kinetic energytransferrate, as- sumedherewithin theinertialrange.Theflame wrinklinglength scale Ly/2, illustrated in Fig. 3, should therefore be larger than L_G. As discussed by [3,32] in the flamelet regime (i.e.thin flame regime),L_GvariesbetweentheintegrallengthscaleLt andthesize ofthesmallesteddyη^whichîs^larger^than^the^flame^thicknessδl⁰. Theminimalvalueoftheflamefrontwavelengthreads:

L^min_y =2L_G>2

δ

⁰l (2)

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Fig. 4. Representation of 2-D manufactured thermochemical variables for /δl⁰= 5 , = 3 and n = 2 . Top left: progress variable c . Top right: CO mass fraction Y CO. Bottom left: progress variable reaction rate ω˙ c. Bottom right: CH radical mass fraction Y CH.

Asn isassumedtobean integer,themaximalnumberofﬂame wrinklingpatternsn^max isthereforegivenby:

n^max =E

L^min_y

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whereE[r]istheintegerpartofr.

In the ﬂamelet regime, the local ﬂame front curvature κ ^has

also to remain smaller that a threshold value so that the ﬂame structure is unaltered. Thus, the radius of curvature R=1/κ ^has

tobe largerthan thecut-off lengthscale δ ^whichîsôf^the ôrder

ofthelaminarﬂamethickness[3].Astheﬂamefrontgeometryis modeledbyasinusoidalpatternofnormaln,theanalyticalexpres- sionforκ=∇·nreads:

κ

=A

2

π

Ly

2

|

^sin(²

π

^yc=c0/Ly)

|

[1+A²(²

π

/Ly)²^cos²(²

π

^yc=c0/Ly)^]³^/² ⁽⁴⁾

Themaximal valueofκ ôbservedâlong^the^sinusoidal^flame^front

isobservedfory_c=c₀=Ly/4:

max

κ

(^yc=c0)=A

2

π

Ly

2

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TheﬂameletconditionR>δ^combined^with^Eq.⁽⁵⁾^gives^the^max-

imumpossibleamplitudeA^maxofthesinusoidalpattern:

A^max= L²_y

4

π

²

δ

⁽⁶⁾

Equations (3)and(6)give the maximalphysicalvalue ofboth Aandn parameters sothat thewrinkledflamepatternremains within the flamelet regime. Based on this assumption, the wrin- kledflame structure is reconstructed from a 1-D flamelet inthe followingsection.

3.3. Estimatingfilteredquantitiesfromapresumedsub-filterflame pattern

Assuming that the flame is in the flamelet regime, the 2-D flame structure is manufactured from a 1-D laminar premixed flame computed with detailed chemistry. The shortest distance D(^x,y) ^betweenêach ^point ⁽^x, ^y⁾ ^contained ⁱⁿ^the ^2-D ^sub-filter domainandtheiso-linec=c₀ isintroducedasillustratedinFig.3. The thin reaction zone position c=c₀ is located at the maximal heat releasepoint ofthe flame.The D field is thenused to map anythermochemicalquantityûsing^the^1-D^flamelet^data:

(^x,^y)⁼1D(^D(^x,^y))^, ⁽⁷⁾

where 1D denotes the 1-D ﬂamelet thermochemical quantities.

An exampleof 2-D manufactured ﬁeldsis givenin Fig.4 forthe progressvariablec,itssourcetermω˙c, COmass fractionYCO and CHmassfractionY_CH.

Filteredquantities^are^deﬁned^by:

(x,y)= ₊_∞

−∞

₊_∞

−∞ (^x,y)^F^2D(^x−x,y−y)^d^x^d^y, (8)

where F^2D is a 2-D box filterof size ^.^By introducing F^1D, the 1-D boxfilterofsize ^,^F^2D(^x,y)îs^decomposedâs^F^1D(^x)^F^1D(^y) andâlso^reads:

(x,y)= ₊_∞

−∞ F^1D(^x−x)

₊_∞

−∞ (^x,y)^F^1D(^y−y)^dy

dx. (9)

AstheflamepatternisnLy-periodicatthesub-filterscaleinthe y-direction,thefilteringalongyreducestoanaveragingoperation overasinglewavelength:

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Fig. 5. Sub-filter scale wrinkling as a function of the normalized amplitude A/δl⁰for different n and normalized filter sizes /δl⁰. Legend: —Manufactured wrinkled flamelet integration ( Eq. (12 )); − −Infinitely thin flame assumption ( Eq. (13) ).

(^x)= ₊_∞

−∞ F^1D(^x−x)_L¹

y

₊_L

y/2

−Ly/2 (^x,y)^dy

^y(x)

dx. (10)

Therefore,anyﬁlteredquantityassociatedtotheperiodicwrinkled ﬂamepatternvariesonlyalongxcoordinateandreads:

(^x)= ₊_∞

−∞ F^1D(^x−x)^y(^x)^dx. (11) 3.4. Linkbetweensub-ﬁlterscalepatternandwrinkling

Thesub-filterflamewrinklingisdefinedas:

=S S⁰_l = 1

ρ

0S⁰_l ₊_∞

−∞

ω

˙c(^x)^dx, (12) wherethe filteredprogressvariablereaction rateω˙^c îs^computed from Eq.(11).For a given LES filtersize ^, ^the ^presumed ^flame pattern and its wrinkling are parametrized by both n= /Ly and the amplitude A. An analytical relation between andξ=nA/îsôbtainedûnderînfinitely^thin^flameâssumption (/δ⁰_l →∞):

=

(²

πξ

)²+1

2

π

^EI

2

π

^,¹⁻₍₂

πξ

¹)²⁺¹

, (13)

where EI is the incomplete elliptic integral of the second kind.

However,inthegeneralcasewheretheﬂamethicknessisconsid- ered,anumericalintegrationisneededtocomputeEq.(12). 4. AnalysisofFilteredWrinkledFlameletconsistency

To illustrate the methodology, 2-D wrinkled flames are manufactured for a given pair (A, n) using ^1D ^variables ^from â 1-D CH₄-air flame computed at φ=0.83 using detailed chemistry [28]. The progress variable is defined as c=Yc/Y_cêq, where Yc=Y_CO+Y_CO₂+Y_H₂_O andeqsuperscriptdenotesequilibriumcon- ditions. Manufactured 2-D variables âre ^then numerically fil- teredaccordingtoEq.(11)foragivenfilterwidth^.

4.1. Evolutionof withA

Figure5shows asafunctionofA/δ⁰_l ^for^different^values^of

n and/δ⁰l.Solidlinesareobtainedfromthenumericalintegra- tionofEq.(12),whereasdashedlinesrepresenttheanalyticalsolu- tionsgivenbyEq.(13)underinfinitelythinflameassumption.For a planarflame(A=0),theflamewrinkling equals1.WhenA increases, increasesasexpected.Thedependencyoftothe pair(A,n)isclearlyevidencedinFig.5as,foragivenamplitude Aandfiltersize^,increaseswiththenumberofwavelengths

n.TheanalyticalmodelgivenbyEq.(13)isretrievedaslongasA andnremainsmall,whichisinlinewiththeboundsproposedin Eq.(6). Flamethermalandreactive layers areindeedoverlapping atthesinewavepeakswhenAreachesthecriticalvalueA^maxand thenumericallycomputedwrinklingratiothusdecreasescompare toinfinitelythinflamecase.Itisimportanttonotethatthemodel degeneratestowards DNSwhen→0: asthefiltersizevanishes theboundimposedonnduetotheflamecut-off impliesthatno analyticalwrinklingispossibleandhencetheflamestaysplanar.

Figure 5 also shows that, for a given ând ⁿ, there is a uniquecorrespondencebetweenA and. Filteredvariablescan therefore be parametrized by (,n,) înstead ôf (Â,n,)^. Thisformulationispreferredforpractical reasons:manyaccurate LESmodelforthesubgridscaleflamewrinklingexistsintheliter- ature(unliketothesubgridflamewrinklingamplitude)

4.2.Effectofwrinklingonﬂamethickness

Filtered data are also post-processed to compute, for the set of parameters (,n,)^, ^the ^filtered ^flame ^thickness δc˜= 1/max(|∇^c^˜|)^. δc˜ is plottedin Fig. 6as a function of fordif- ferentvalues of n.A strong dependency ofthe flame thickness

δc˜ to is observedforany ofthe threeﬁltersizes considered.

When/δ⁰_l >2,thenumberofwavelengthsninfluences signif- icantlythefilteredflamethickness.Indeed,foragiven,δc˜de- creaseswhenn increases.Inthislast case,δc˜ tends tothefilter size ^because,^forâ ^given,the amplitudeofthe flamepat- ternAgetssmallercomparedto^.^For^small^valuesôfⁿ,Aneeds tobelarger toreachagivenlevelofwrinkling(seeFig.5) which leadsto anincreasing filteredflamethicknessδc˜.Thisbehavior is alsoillustrated by Fig.7 whichshowsfiltered progressvariablec andfilteredchemicalreactionrateω˙^c^for^different^valuesôf/δl⁰, n and.Fora givenn,theflamethicknessincreasesdueto wrinkling(left plot). Larger wrinklingvalues also modify the filtered reaction rates profiles (right plot) to larger maximum values.Foragivenwrinklingvalue,theflamethicknessdecreases whennincreasessincethesineamplitudeAreducesforconstant .

Thisanalysis showsthat fora given filtersize ^, ^the ^filtered flame structure dependson both andn. It means that isa priorinot sufficientto determinethefilteredflame structure, whichisalsoinfluenced bythegeometricalshapeoftheflameat thesub-filterscale.

4.3.Effectofwrinklingonsub-ﬁlterspeciesmass

Figure8showstheevolutionofthemassofCOm_COcontained in a sub-filter box of size âs â ^function ôf ^the ^wrinkling oftheflame pattern.Toease theinterpretation,m_CO()îs ^nor- malizedbythemassofCOofthe planarflamem_CO(=1)^.^For

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Fig. 6. Comparison of the ratio δc˜/δ_l⁰as a function of the sub-ﬁlter scale wrinkling for different values of n and different normalized ﬁlter sizes /δ⁰_l.

Fig. 7. Favre-filtered progress variable c as a function of dimensionless position x/δ⁰l (left) and filtered reaction rate ω˙ cas a function of c (right) for different values of wrinkling ratio and filter width /δl⁰. Legend: Solid lines: FWF n = 1 ; Dashed lines: FWF n = 2 .

Fig. 8. Normalized mass of CO in the sub-ﬁlter box as a function of the wrinkling ratio for different values of /δl⁰and n . Dashed line shows unitary line f()= .

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Fig. 9. Normalized mass of CH in the sub-ﬁlter box as a function of the wrinkling ratio for different values of /δ⁰l and n . Dashed line shows unitary line f()= .

a givenﬂame patternofwrinkling,m_CO()^is^computed^by integratingρ^YCOoverthesurface:

mCO()= _/2

x=−/2

y=0

ρ

(x,y)^YCO(x,y)^d^y^d^x ⁽¹⁴⁾

Severalcurvesarepresentedfordifferentvaluesof/δl⁰ andn. As expected,themainobservationisthatm_COincreaseswith duetotheincreasedﬂamesurfacewithinthebox.m_COisrelatively independent from the value of n and is however much lower than the unitary line m_CO()=m_CO(=1) ^which ^corresponds tothe theoreticalvalue ofthe integral that wouldbe obtained if the COmass wasconcentrated on the sine pattern(i.e.

for an infinitelythinfront). Thisdeparture of m_CO is mainly due to the non-zeroY_CO value in the burntgases whichimplies that some fractionof COmass is located outsideof the filtervolume whenthemanufacturedwrinkledflameisconstructed.Thisisver- ifiedbycarryingoutthesameanalysisforCHradicalwhichhasa muchlower thickness(asseeninFig.4)andazerovalueinboth freshandburntgases.TheresultingCHmassm_CH showninFig.9 isaccuratelyapproximatedbytheunitaryline.Theremaininggap increases with andisdueto profilethickness effectsin high curvature regions. In this case only, the following approximation thereforeapplies:

m_CH()^≈m_CH(=1) ⁽¹⁵⁾

In the more generalcaseof thickerspecies profiles compared to ôr^toâ ^species^with^non-zero^mass ^fractionsⁱⁿ^freshôr ^burnt gases, thefiltered speciesmassfractions cannotbe approximated bysimplerelationssuchasEq.(15).

4.4. ComparisonofFilteredWrinkledFlamestructureagainstDNS data

The approximations made to manufacture the Filtered Wrin- kledFlameletsneglectseveralphysicalphenomenasuch ascurva- ture or stretch effects. In order to estimate the impact of these assumptions, the FWF database is compared against DNS solution [11,30] of a turbulent premixed flamepreviously introduced inSection2.ToenableadirectcomparisonbetweenDNSandthe FWFdatabase,thefilteredflamethicknessisnowdefinedas:

δ

⁰c˜ =1/

|∇

^c^˜

|

c˜=c˜0 (16) where c˜₀ is a progressvariable iso-line value chosen tobe close to the region of maximal |∇^c^˜|^. ^The ^sub-grid ^scale ^ﬂame ^wrin-

klingiscomputedintheDNSas=|∇^c|/|∇^c|^.^For^a^given^ﬁlter

size,boththefilteredflamethicknessandthesub-gridscaleflame wrinklingarethenestimatedintheDNSsolutionsalongtheflame surfacedefined asc˜₀ iso-line.The filtered flamethickness condi- tioned by the sub-grid scale flamewrinkling isshown in Fig.10 for three values offilter sizes: /δ_l⁰ = 2, 5 and10. The filtered flamethicknessdefinedatc˜₀=0.5iscomputedfromthereference

DNSdata(solidplainline)andtheapproximateFilteredWrinkled Flameletdatabase(solidlinewithsymbols)fordifferentvaluesof n. The gray area represents the variability ofδc⁰˜/δ⁰l for a given wrinklingvalue:theupperandlowerthinlinescorrespondto plusandminustheRMS,respectively.As thefilteredflamethick- nessisstronglydependentontheflamewrinklingintheDNSand onthenparameterintheFWFmodel,thisfiguregivesanindica- tiononthenparameterthatshouldbeusedintheFWFmodelto recovertheDNSfilteredflamethickness.Then valuethatshould be chosen dependson the non-dimensional filter size /δ⁰l: n should increase withthe filter size. Moreover, a singlevalue per filtersizewouldalreadygiveanacceptablemodelsince1≤n≤2 for/δ⁰l ≤10fortheDNSconsideredhere.

The prediction of COﬁltered reaction rate given by the FWF databaseis also assessed by comparisonwith the DNSdatabase.

Forthatpurpose,Fig.11illustratestheimpactofbothn and onthefilteredCOmassfractionprofileY_COforagivenfilterwidth /δ⁰_l ^(left ^plot).^YCO profile thickness andpeak value are clearly impactedbyboth parameters.The solidblue curve(=1) also corresponds to the original F-TACLES. Significant differenceswith the=1.5and=2curvesillustrates thegain ofthemodel improvement. CO filtered reaction rate ω˙CO extracted from FWF database is then compared to the post-processed DNS for both /δ⁰l =5 and10 (right plot). DNSmean andRMS valuesof ω˙CO

conditionedtoc are shownby blacklineandgrayfilledarea, respectively. The trends of the DNS mean profiles are well repro- ducedbytheFWFdatabase.TheFWFprofilesvariationsdueton andarealsoinlinewiththeRMSvaluesofthepost-processed DNS.Theslightdifferencesobservedarecertainlyduetomodeling hypothesis, which includethe sinusoidal shape ofthe flame sur- faceandthe canonicalunstrained 1-D laminarflameprofile used tomanufacturethewrinkledflamelets.ItconfirmsthattheFiltered WrinkledFlamesarchetypesare abletotacklethe impacts ofac- tualsub-filterscalewrinklingpatternsontheresolvedintermedi- atespeciesprofilesthroughnandparameters.

5. Closureoftheﬁlteredprogressvariablebalanceequation usingFilteredWrinkledFlamelets(FWF)

5.1. FilteredWrinkledFlameletdatabase

Aseriesoffilteredwrinkledflameletsismanufacturedbyvary- ing A and ^for â ^given ^wavelength^parameter ⁿ. For each set ofgeometricalparameters,filteredthermochemicalvariablesâre mappedinthefilteredprogressvariablec˜spaceasin[10].Asdis- cussedinSection 4.1,a setofparameters (A,⁾ corresponds,for agivenn,toauniquevalueof.Therefore^can^be^mapped andtabulatedas:

=^[^c,,]_n. (17)

Anillustrationof theFWFtable is showninFig. 7fora CH₄- Air premixed ﬂame at φ=0.83 ﬁltered at =5δ⁰l (top right)

(9)

Fig. 10. Comparison of δ⁰˜c/δl⁰computed for ˜ c 0= 0 . 5 as a function of the sub-ﬁlter scale wrinkling for different values of n and different normalized ﬁlter sizes /δl⁰. Legend: Solid lines with symbols: FWF database. Solid line: DNS Mean value. Filled area: plus and minus the standard deviation.

Fig. 11. Favre-filtered CO mass fraction Y COas a function of dimensionless position x/δ_l⁰(left) and filtered chemical reaction rate ω˙ COas a function of the filtered progress variable ˜ c (right) for different values of wrinkling ratio and filter width /δl⁰. Legend: Solid colored lines: FWF n = 1 ; Dashed colored lines: FWF n = 2 ; Solid black line: DNS Mean value. Filled area: plus and minus the standard deviation.

and =10δ_l⁰ ^(bottom ^right). ^The ^evolution ^along ^the ^c^˜ ^coordi-

nate of the filtered progress variable reaction rate ω˙^c îs ^shown for different values of and n. In both figures, the profiles of ω˙^c ^for =1 match as expected regardless of the value of n.Theyalsocorrespondtothefilteredlaminarreactionratethat wouldbeextractedfromthe1-Dfilteredpremixedflame.Itcorre- spondstothecasewheretheflameisplanaratthesub-filterscale.

However,when =1.4 and=2.5,ω˙^c ^differs ^depending^on n as discussed in Section 3.4. Thiseffect is exacerbated as increases.

5.2. Closureoftheﬁlteredprogressvariableequation

InLES, the ﬁlteredprogressvariable c˜is governedby the fol- lowingequation:

∂ ρ

^c^˜

∂

^t ⁺

∇

^·

ρ

^u^˜^c^˜⁼

∇

^·(−

ρ

^V^c)+

∇

^·

ρ

(^c^˜^u^˜−cu)+

ω

˙^c^, ⁽¹⁸⁾ where ρ îs ^the ^filtered ^density, û^˜ îs ^the ^filtered ^velocity, ^Vc is the molecular diffusion velocity of the progress variable and ω˙^c is the filtered progress variable reaction rate. The RHS terms of

(10)

Eq.(18)arecomputedfromFilteredWrinkledFlamelets(FWF)in- troducedinSection3.Theﬁlteredchemicalsourcetermisdirectly estimatedfromEq.(11)as:

ω

˙^c⁼⁺^∞

−∞ F¹^D(^x⁻^x)

ω

^˙^∗c^y(^x)^dx^, ⁽¹⁹⁾

where thesuperscript ^∗ representsthevalues extractedfromthe manufactured wrinkled flamelet. The filtered molecular diffusion termis closedasρ^V^c=−Dm∇^c^˜^where^D^m îsêstimated^from^the

manufacturedwrinkledﬂameletas:

Dm=−

ρ

^∗^Vc∗/

∇

^c^˜^∗. (20)

Thesub-filterscaleconvectionterm∇·ρ(^c^˜û^˜−cu)îs^modeledâs:

T =

∇

^·

ρ

(^c^˜^u^˜−cu)=

ρ

⁰^S

|∇

^c^˜^∗

|

⁻

|∇

^c^∗

|

, (21)

whereρ0 andS arerespectively thefreshgas densityandcon- sumptionspeed of themanufactured flamelet filtered at ^computed from Eq. (12). This formulation ensures that the filtered progressvariableiso-surfacespropagateatSasshownin[10].For afixedvalueofn,thetermsDm,T andω˙^cârepre-computedand tabulatedasfunctionsofthefilteredprogressvariablec,theflame filtersizeând^the^wrinkling^factoraccordingtoEq.(17).The flamefiltersizeischosentoensurethatthefilteredflamefrontis well resolvedontheLESgrid[29,33].Thewrinklingfactorcanbe estimatedfromtheLESusingeitheratransport equation[34],al- gebraicmodels[12,15–17] ordynamical formulations[18–20].The filteredprogressvariabletransportequation closedwithtabulated FilteredWrinkledFlameletsthereforereads:

∂ ρ

^c^˜

∂

^t ⁺

∇

·

ρ

^c^˜^u^˜ =

∇

·

Dm[c,,]_n

∇

^c^˜

+T[c,,]_n+

ω

˙^c^[^c,,]_n. (22) ThismodelingstrategyissimilartotheF-TACLESmodel[10]ex- cept that Filtered Wrinkled Flamelets (FWF) are used instead of FilteredPlanarFlamelets(FPF).Therefore,thewrinklingfactor becomesacoordinateoftheFWFdatabase-whichintegratesthe impactofwrinklingontheﬁlteredﬂamefeatures-insteadofmul- tiplyingtheRHStermsofEq.(22).

5.3. Estimationofthewavelengthparametern

The FWF framework assumes that representing the sub-filter scale flame wrinkling with a unique wavelength Ly is sufficient tocapturethecorrectfilteredchemical speciesdistribution atthe sub-filterscale.Thishypothesiswillbeassessedusingaprioriand aposteriorianalysisinSections6and7,respectively.Inthecontext ofrealisticmulti-scaleflamewrinkling,thewavelengthparameter n=/Ly representsthe averagenumber offlame crossingsper filterlength ^[35]^.Âssumingân înfinitely^thin^flame^front, ^Bray etal.[35]showthatn reads:

n=

| σ

^y

|

2 (23)

where isthesub-filterflamesurfacedensitywhichexpresses =|∇^c|=|∇^c|^.σ^yîs^the^mean^cosineângleôf^theînstanta-

neousﬂamefrontwiththec=c₀iso-surfaceandisassumedtobe anuniversalconstant[35,36]sothat|σ^y|≈0.5.Equation(23)may alsoberecastas:

n=C

|∇

(

ρ

^c)

|

⁽²⁴⁾

where C=0.25/ρb with ρb is the burnt gases density. In this modelformulationgivenbyEq.(24),nisnotafreeparameteras itisrelatedto∇(ρ^c)^computed^from^the^LESâs^wellâsând, whicharealsotwoFWFtablecoordinates.Inthefollowing,ρ^c^will

becomputeddirectlyfromtheDNSdatabaseintheapriorianaly- sis(Section6)whereasitwillbetransportedfollowingEq.(18)in theaposteriorianalysis(Section7).

5.4.SynopticofFWFmodelworkﬂow

GivenanLESgridandburneroperatingconditions,thestep-by- stepprocedure togeneratea FWFtableanduseit ina turbulent premixedﬂameLESissummarizedhere:

1.Compute the1-D premixedﬂameatselectedoperatingcondi- tionsusingdetailedchemistry.

2.Based on the computed 1-D flame, generate a series of 2-D wrinkled flamelets varying sine amplitude A and wavelength n as detailed in Section 3.1. Knowing the grid cell size ^x^, thesuggestedflamefiltersizeis=5^x^.

3.FilterthedifferentmappedvariablestocomputerT,T,Y_k,ω˙Y_k, Dm,T,ω˙^c^.

4.Tabulatethesevariablesas=^[^c,,]_n

5.PerformtheLESusingrT toclosetheequationofstate.Theﬁl- teredprogressvariableequationisclosedusingDmandT+ω˙^c as diffusion term and reaction rate, respectively. Other op- tional quantities such asY_k andω˙Y_k can be accessed aspost- processingvariables.

Theimplementationofthe FWFmodelanditstabulationpro- cedurearevalidatedby computingaseriesof1-DfilteredCH₄-air flames atφ=0.83for different filtersizes^.În ^these ^1-D ^sim- ulations, Eq. (22) is solved by assuming constant values of the sub-filter scale wrinkling . They are performedto verify that thefilteredflamefronteffectivelypropagatesatS=S⁰_l what- ever the choice of the flame patternparameter n. The simula- tionresults,notdetailedhere,showedthattheresolutioncriterion ^/x≥5 was neededto ensure an errorof less than 3% on S. Thisconclusionwasalsodrawnforotherfilteredflamemodelsim- plementedintheYALES2code[10,29].

6. AprioricomparisonofFWFdistributionsagainstaﬁltered DNSdatabase

Anapriorivalidationofthemethodologyisperformedbychal- lenging the Filtered Density Function (FDF) reconstructed using FWF against the DNS data [11,30] introduced in Section 2. Ref- erenceprogress variable FDFare extractedfrom the PRECCINSTA database by using the procedure detailed in [11] and plotted in Fig.12fordifferentpairs(^c,S_c),whereS_c,theunmixednessfactor, readsSc=c²/(^c(¹−c))^and^two^ﬁlter^size/δ⁰l =2and/δ⁰l = 4.

TheFWFsub-ﬁlterFDFiscomputedbyﬁrstrecastingEq.(10)as follows:

ρ

⁼ ¹

0

₊_∞

−∞

F¹^D(^x−x) Ly

₊_L

y/2

−Ly/2

ρ

δ

(^c⁻^C)^dy

dxdC (25)

where ρ=ρ(^x,y), c=c(^x,y) ^and δ ^is ^the ^dirac distribution.

Then,astatisticalexpressionof^is^obtained^byintroducingP^FWF, theFDFassociatedtoFWF:

= ₁

0 ^P^FWF(C)^dC (26)

where P^FWF(C)= 1

ρ

₊_∞

−∞

F¹^D(^x⁻^x) Ly

₊_L

y/2

−Ly/2

ρδ

(^c−C)^dy

dx (27)

Firstandsecond moments ofthe progressvariableare computed fromtheFWF-FDFasfollow:

c= 1

0

cP^FWF(C)dC (28)

c²= ₁

0 (^c−c)²^P^FWF(C)^dC (29)