An extensive numerical study of the burning dynamics of wildland fuel using proposed configuration space

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An extensive numerical study of the burning dynamics of wildland fuel using proposed configuration space

Kai Zhang, Aymeric Lamorlette

To cite this version:

Kai Zhang, Aymeric Lamorlette. An extensive numerical study of the burning dynamics of wildland

fuel using proposed configuration space. International Journal of Heat and Mass Transfer, Elsevier,

2020, 160, pp.120174. �10.1016/j.ijheatmasstransfer.2020.120174�. �hal-03232086�

An extensive numerical study of the burning dynamics of wildland fuel using proposed configuration space

Kai Zhang

^∗

, Aymeric Lamorlette

M2P2 , Aix-Marseille University , CNRS. 38 rue Joliot-Curie, Centrale Marseille Plot 6 13451 Marseille, France

a r t i c l e i n f o

Article history:

Received 6 February 2020 Revised 26 June 2020 Accepted 6 July 2020 Available online 15 July 2020 Keywords:

Heat transfer Forestfirefoam Flame model CFD Porous Combustion

a b s t r a c t

Physics-basedflamemodelscapableofpredictingsmall-scalefirebehaviorsreducecomputationalpower neededforpredictingfiresoflarge-andgiga-scale.However,classicalmodelcorrelationsareoftende- veloped for‘free fires’withoutconsidering vegetationaround. Thesemodelsmay result in inaccurate firemodelingduetowrong‘prior’flameshapeestimatedfromθ ^{~ windspeed.Toovercome thisde-} fect,three-dimensionalsmall-scalefireswithfirelineintensityof100KW/marenumericallysimulated usinglargeeddy simulation.Firebehaviorssuchasflametilt angleand heattransfermechanisms are extensivelystudiedusinganewlyproposedconfigurationspace{NC,CdLAI}.Theformeronerepresents theratiobetweenfiretowindpower,andthelatteroneconsideringthevegetationeffectisforthefirst timeintroduced inflamemodels.Usingthe configurationspace, twomodel correlationsforflame tilt angleandradiativeheatpowerreachingtheunburntfuelsareproposed.Theflametiltangleθ^isdirectly relatedtoCdLAI (C_dαsσsHF/2),whileinverselyrelatedtoNC (2gI/ρ0C_p,0T0U₀³), incontrasttothe model proposedforradiative heatpower.Comparisons withseveralclassicalmodels evidencedthecapability ofnewflamemodelsinpredictingbothfreeandnon-freefires.Thelimitsofthevalidityofthenewly proposedmodelsarealsodiscussed.

1. Introduction

The burning behavior of wildland fuels represents one of the verycomplexheatandmasstransferproblemsbecauseitisgreatly affectedbymanyphysicalparameterssuchasthecharacteristicsof vegetation,thetopography, andtheenvironmental conditions[1]. Alloftheseparameters havepotentially influencedthemass,mo- mentum,andenergy(heat)exchangesbetweenthesolidfuelpar- ticles andthe surrounding atmosphere. As a result of thesemu- tualexchanges,fireregimetransitionanditsassociatedheattrans- fer mechanismsemergeastwoprimary factorsco-controllingthe burningdynamics ofwildlandfuels. Depending ondimensionless numbers suchasFroude (F_C), squaredFroude(F_C²)or Byramcon- vective number(N_C) [2,3], wildland firesare often classified into tworegimes:aplume-dominatedregime andwind-drivenregime.

Forplume-dominatedfire,radiativeheattransferdominatestothe fire frontpropagationwhereas forwind-driven fire,bothradiative andconvectiveheattransfercontributestotheignitionofunburnt fuels.

∗ Corresponding author.

E-mail addresses: [email protected] (K. Zhang), aymeric.lamorlette@univ- amu.fr (A. Lamorlette).

Nelson [4] analysed previous experimental data of Morandini andSilvani[5]basedonatriangularflamemodelandanalterna- tivewaytodefineflameheightandtiltangle.Theflameheightde- finedF_C²(denotedasF_CH²)bestdescribedthemodeofheattransfer tounburntfuels.Theothertwopotential indicatorsoffireregime transitionand heat transfermechanism are flame height defined F_C (denoted as F_H) and the N_C. For N_C < 2, a wind-driven fire withlarge tiltangleisdominatedbyconvective heattransfer;for N_C > 10,aplume-dominatedfire withtiltanglesmallerthan20° isgovernedbyradiation.Amixedheattransfermechanismoccurs when2<N_C<10.Theseanalysesareinagreementwiththatre- portedbyMorvanandFrangieh[6]inwhoseworkselecteddataof laboratoryfiresatvariousexperimentalconditionsarere-visited.

Although the burning dynamics of wildland fires have been broadly discussed in the literature [7–12] using N_C (2gI/

ρ

0C_p,0T₀U₀³), the role of this dimensionless number is ques- tioned from a physical point of view especially for small-to- medium scale fires. In 2015, Lamorletteet al. [13]discussed the relevance betweenseveral dimensionlessnumbers andthe cylin- dricaltypeparticle(solidfuel)ignitiontimet_s,ig.Amongstseveral dimensionlessnumberssuchasthePrandtlnumberPr,theradia- tiveBiotnumberBi[14],theparticlequasi-staticReynoldsnumber Re_σs andtheNusseltnumberNu,onlytwoparameterswereidenti- fiedmostrelevantforsolidfuelignitionundernaturalconvection:

Nomenclature

Bi Biotnumber

C_d Dragcoefficient

C_p,0,C_p,s Specificheatcapacityforambientairandsolid (Jkg⁻¹K⁻¹)

D spanwisewidthofburner/flame(m) D₀ flamedepth(m)

F_C Froudenumber

F_D Dragforce(N)

g Gravitationalacceleration(ms⁻²)

^HC heatofcombustion(Jkg⁻¹) H_F Vegetationheight(m) H_flame Heightofflame(m)

h,h_T Heattransferandtotalheattransfercoefficient (Wm⁻²K⁻¹)

I Firelineintensity(KW) J Irradiance(Wm⁻²) LAI Leafareaindex,

α

^s

σ

^sHF/2

˙

m mass burning rate per unit area (mass flux) (kgm⁻²s⁻¹)

Nu Nusseltnumber

N_C Byramconvectivenumber,2gI/

ρ

0C_p,0T₀U₀³ Pr,Pr_t PrandtlandturbulentPrandtlnumber R_i Richardsonnumber

Re,Re_d, Re_σs Reynolds, particle Reynolds and quasi-static Reynoldsnumber.

s Stochiometricfuel/oxygenmassratio

T,Ts,T₀,T_f Gas,solid,ambient,andflametemperature(K) T_ig,T_s,ig Gasandsolidignitiontemperature(K) U₀,Ue Windspeed,entrainmentvelocity(ms⁻¹) w₀ Characteristicbuoyancyvelocity(ms⁻¹) w_flame Widthofflame(m)

Greeksymbols

α

^g,

α

^{s Volumefractionofgasandsolid}

σ

Stefan-Boltzmannconstant

σ

^{s Surfacetovolumeratio(SVR)(m−1)}

μ

^,

ν

^,

ν

^{t Dynamic, kinematic, and turbulent kinematic}

viscosity

τ

^react,

τ

^res,

τ

k Chemical reaction, fire residence, and kernel characteristictime(s)

λ

s Solidthermalconductivity(Wm⁻¹K⁻¹)

φ

surf Surfaceheatflux(KWm⁻²)

η

^{fractionofimpingingair}

ρ

^,

ρ

0,

ρ

s Gas,ambientair,andsoliddensity(kgm⁻³)

δ

ij Kroneckerdelta

θ

^{Flametiltangle(degree)}

theCdLAI(C_d

α

^s

σ

^sHF/2)andthe

ψ

⁽

α

²s

υ

²/gH_F).Theformeroneac- countsforinducedmomentumandin-depthradiationeffects,and thelatteronerepresentsaninducedsublayerhydrodynamiceffect.

Apart from these two parameters, the minor effect of Re_σs wasalsohighlighted.Withinthefuellayer,the

ψ

=

α

²s

ν

²/gH_F and T_b^∗=

(

^T−T₀

)

/T togetherdefinedparticleReynoldsnumberRe_σs=

1/

ϕ

^CdLAI

(

^T−T₀

)

/T

)

^{. By roughly} considering that ignition oc- cursat T_b^∗=0.5and fuelignition temperatureT_s,ig equals to the surrounding temperature T_ig, i.e., T_s,ig ~ T_ig, a physical limit of

(

^CdLAI

)

lim=2×10⁻⁶wasobtainedthatawayfromthisvalue,Re_σs effecton the particle ignition time t_s,ig is negligible. This is be- cause away from the limit, sublayer flow is far from the point ofvortex sheddingtransition[15,16] at whichheat transfer from gas to solid phase (and hence t_s,ig) is modified. In the present study,because

ψ

^{isroughly 0.027 and0.01}≤ CdLAI≤ 0.5, giving

ψ

^CdLAI2×10⁻⁶,theparticlequasi-staticReynoldsnumberhas negligibleinfluenceonthefuelparticleignitiontime.

Therefore, the only two dimensionless numbers which affect the fuel particle ignition by heat transfer are indeed the CdLAI and

ψ

^{. Because}

ψ

^{accountsfor induced sublayer flow, its effect}

can also be replaced by a Byram convective number N_C, which measures the competition effect between wind and fire power.

The N_C is known to control fire behaviours such as the rate of spread,mass lossrate, flame residencetime,heat transfer mech- anismetc.incaseswhenanexternalforcedflowexists[17–19].To showtherelevancebetween

ψ

^andNC,onemaystartwithhowN_C (2gI/

ρ

0C_p,0T₀U₀³)isdefined.Thisparametermodifiessublayer flow by imposingan atmosphericboundarylayer (ABL)viaan implicit wind speed dependency on Y₀ and H_F. The former is the burnt fuel roughness layer and the latter is the vegetation height, i.e., NC∼Y0−1∼HF−1.Itcanbeusedtoreplace

ψ

^{becauseithassim-}

ilaritiesto

ψ

=

α

s²

υ

²/gH_F that

ψ

∼H⁻¹_F aswell.Moreover,thein- termediate relationshipofN_C∼Y₀⁻¹,withY₀ approximatelythree timestheH_F,is affectedby Kelvin-Helmholtzinstability [20] that therelevantturbulentlengthscalel_TisalsostronglyrelatedtoH_F [21,22].

As aresultof aboveobservations, itis reasonabletoconclude thatfire globalbehaviorsarenot solelydependentonN_C,butare governedbyaconfigurationspace{N_C,CdLAI}.Thesetwoparame- tersareinternallycoupledtoaffectlocalfuelignitiontimebyheat and mass transfer. However, none of previous studies has tried to correlate fire regime transition and heat transfer mechanism withthisconfigurationspace, i.e.,previous flamemodelwasonly builtbasedonN_Calone.Anaccuratesmall-tomedium-scaleflame modelconsideringtheroleofvegetationcharacteristicsCdLAIises- sentialtoextendthelimitofvalidityoflarge-scalefiremodels.

Furthermore,theimportanceandrelevanceofthisnewconfigu- rationspace{N_C,CdLAI}onsolidfuelburningbehaviorcanalsobe explained by thefact that fire global behavior hasa dependency ontwokindsofflows:an atmosphericboundarylayerflowanda mixinglayerflow. Aheadofthefirepropagatingfront,thebound- arylayer tomixinglayer transitionoccursroughly atCdLAI=0.1 [23],whereas behindthe fire propagating front, ashlayer modi- fiedatmosphericboundarylayereffect[24]isnon-ignorableandis implicitlygovernedbyN_C,.Thelocalfuelignitiontime andglobal fire behaviorsarehencea resultofthe competitionbetweentwo kindsofflows governedbythe configurationspace{N_C,CdLAI}.It wasobservedbyLamorletteetal.[13]thatforfireunderno-wind conditionwithnearzerorateofspread,thesublayersmouldering orpyrolysis process isinfluenced solely by CdLAI,while thecou- plingeffectofN_CandCdLAIremainunclear.

Overall, this study is motivated by the lack of understanding oftheeffectofconfiguration space{N_C,CdLAI}on controllingthe burningdynamicsofthewildlandfuelsviathetwokeyfactors:the fire regime transitionandtheheat transfermechanism. As faras theauthorsareaware,theonlyworkwhichhasdiscussedtheef- fectofLAI,similarasCdLAI,onthefireregimetransitionistheone by Morvan andLamorlette [25]. Theyreported that fora plume- dominated fire under moderatewind speed (N_C ~ 23), LAIhas a notableinfluenceontheamountofradiationreachingtheunburnt solid fuel.With LAIincreasing from0.35 to 4.2, fueltemperature ahead of flame front approaches the gas temperature indicating an extra contributionof radiativeheat transfer. Thismay suggest that the effect ofcanopy is toprevent the fire regime transition fromplume-dominatedtowind-driven.Studiesbasedondifferent N_C andCdLAI needtobe performedto advancetheknowledgeof thewildlandfuelsburningbehaviorsandpromotethebasicunder- standingofheatandmasstransferacrossthevegetation.

This paper is organized as follows. Section 2 describes the 3D numerical configuration of the representative fire and in- troduces the mathematical models employed for the simulation.

Section3discussesthefirebehaviorsunderdifferentN_CandCdLAI valueswhichcomposetheconfigurationspace{N_C,CdLAI}.Finally, conclusionsofthepresentstudyaregiveninSection4.

2. Methods 2.1. Numericaldetails

Inthisstudy,3Dsmall-scalewildfiresstabilizingonvegetation layers of differentcharacteristics (CdLAI) are numericalsimulated usingan inertvegetationversion ofacompressiblesolver Forest- FireFoam(FFF),anextensionofFireFoamsolverdevelopedatM2P2 lab, Aix-Marseille University.Itwaspreviously demonstrated that the multiphase solverFFF is ableto capturewildlandfire behav- iorsinbothno-windandwindconditionswiththesub-modelim- provements[26,27].TheinertversionFFFsolverisprimarilybased onthestandardGaussianfinite-volumeintegrationmethodandas- sumesaninertvegetationlayerwithoutconsideringtheprocesses suchaspyrolysis,charcombustion,etc.tosimplifytheanalysis.To representthekey processesforwildlandfire, a burneris usedto injectCOintothecomputationaldomaintomimicthemainprod- uct ofpyrolysisandcharoxidation. Thegasphase chemicalreac- tionrateisthendrivenbythecompetitionbetweentheCO–airtur- bulentandmoleculardiffusionrates,i.e.,animprovedmodelbased onEddyDissipationConcept[28].

For high-wind speed induced wall-bounded flames, a wall- adaptivelocaleddy-viscosity(WALE) modelisemployedtoreturn thecorrectwallasymptoticbehavior[29]andtocalculatethetur- bulentdiffusionratebysolvinganimprovedsub-gridscalekinetic energyequation[30].Thesolidfueltemperatureisessentiallygov- ernedbyboth convectiveandradiative heattransferfromthegas phase,whileanimposedcoldgaugeofTs=295Konthevegetation layerprohibitsthesolidtemperaturetoincreaseforthepurposeof measuring surface heat flux. The radiative intensity I_rad obtained fromsolvingradiative heattransferequationsusinga discreteor- dinate method (fvDOM)[31] is integratedfor a finite numberof solidanglesinordertocalculatethetotalirradianceJfromwhich the radiativesourcetermQ_rad^s isdefinedby consideringthe radi- ationextinctioncoefficient

α

^s

σ

^{s/4forconvexparticles[32].Detail}

descriptionsoftheemployedmodelsareavailablein[26,27].

The computationaldomain is7.8m longin thestreamwisex- direction, 2.5m high in the vertical y-direction and0.5 m wide in thespanwise z-direction partially given inFig. 1.A vegetation layer of5m long,0.05m highand0.5m wide isimposed from the leading edge ofthe CO burner sitting atx = 0.0 m. Forthe present study, a small-scale fireline intensity of 100KW/m corre- spondingtoaCOinjectionvelocityof0.173636m/sandtheburner size 0.05× 0.5m² is used. Periodic boundary conditions are ap- pliedonthewallsinthespanwisedirectiontoreducetherequired computationalpowerwithoutlosingtoomany3D flamefrontfea-

Fig. 1. Visualization of the computational domain using instantaneous isocontour of Q -criterion at 150 s ⁻², clipped to x max= 2 . 5 m , y max= 1 m .

turesofwildfires[33]suchastheflametowerinducedconvective heating/coolingofsolidparticles[34].Atripwireof0.005mlong, 0.5mwideand0.005mhighisplacedatx=(−0.105)mtoperturb theincomingboundarylayerflowgivenby,

U_∞

(

^Y

)

^=A×Ure f×ln

_Y+Y0

Y0

(1) WherethesurfaceroughnessY₀ isapproximated asone tenth ofthevegetationheightH_F=0.05mtomimictheashesleftfrom burningwildlandfuels[35].TheconstantAiscalculatedusingan openwindspeedoftendefinedat10mreferenceheight[36,37], U10

Y =Y_{re f} =10m

=A×U10×ln

_Y

Y0 +1

;A=0.1315546732

(2) To accurately capture the in-depth radiation and the ABL/canopy turbulence, the grid spacing used along stream- wise x-direction is 0.005mforx<1.0m(mainflameregion)and vertical y-direction is 0.002 m on average fory<0.05m, smaller thanthecellsize(࢞)requirementfollowingthecriterion[10],

<min

_H

F

3,2HF

LAI

=min

(

^{0.0167m,0.02m}

)

⁽³⁾

With0.1≤LAI≤5forthepresentstudy.

Consideringtheoverallexpansionratioofthemeshintheless importantplume regions responsible forlittle radiation,the total numberofcellsis3.5millionwithmore detailsavailable in[12]. Statistical datais collectedusingthe last 15 s out of30 sof the total simulation time. The Courant–Fredrichs–Lewy (CFL) number [38]isrestricted to besmaller than0.5 using automatictime step adjustment. Each simulation requires CPU time of about 2500 h using48processorsonhigh-performancecomputing(HPC)cluster ofAix-MarseilleUniversity.

2.2.Thecoldgauge,inertvegetationassumption

Because of the high computational power required for inves- tigating the 3D propagating wildland fire behaviors, the present studyassumesan inertvegetationonwhicha cold gaugeofTs= 295Kisimposed tomeasure theradiative heattransfer fromthe stationaryfirebodyconfinedbyT=500K.Asmentionedabove,a detaildescriptionofthechosen sub-modelsisavailable in[26,27], while the inert version FFF does not take evaporation, pyrolysis andcharoxidation rates intoaccount leadingto thefollowing Ts

equation:

Cp,s

α

^s

σ

^sd_dt^Ts=Qrad^s +Q_con^s _v (4) Theequation indicatesthatthetimeevolutionofsolidtemper- ature(energybalance)is controlledby heattransfer betweengas andsolid viaradiationQ_rad^s =

α

^s

σ

^s/4×

(

^J−4

σ

^Ts⁴

)

^{andconvection} Q_con^s _v=h

α

^s

σ

^s×

(

^T−Ts

)

^.

Despitethe ease ofretrieving radiative heat transfer fromJ≈ 4Q_rad^s /

α

^s

σ

^{s,thecoldgauge,inertvegetationassumption[39]brings}

aquestionoftowhatextentdoestheTs=T₀∼295Kinfluencethe fire dynamics compared to that of a propagating fire. In a real propagatingcase, theheatflux reachingthevegetationisrespon- sibleforthesolidtemperatureelevationanddecompositionofthe fuelfollowing the threesteps:evaporation, pyrolysis, andsmoul- dering[40–42].Theevaporationprocesscan beneglected consid- eringthechosendeadpineneedlesasfuel,i.e.,zerofuelmoisture content,while thepyrolysis andsmoulderingratesmaybe influ- encedby the coldvegetation assumptionsince solid temperature isnotallowedtorise.Infact,withoutimposingthecoldgauge,the inertvegetationtemperaturewillfinallyrisetothefluidtempera- tureTs=T∼1900Kfora stationaryflame,equivalenttoa quasi- staticpropagationfire,i.e.,atanytimeduringpropagation,theveg- etationisatsteady-state.Neither thecold northehot vegetation

Table 1

Relevant parameters for calculation.

α^s σ^{s( m −1) H} F( m ) C d CdLAI

5 . 33 ×10 ⁻⁴∼2 . 66 ×10 ⁻² 7500 0.05 0.1 0.01–0.5

ρs(kg m ⁻³) C p,s( Jk g ⁻¹K ⁻¹) h T( W m ⁻²K ⁻¹) λs( W m ⁻¹K ⁻¹) T s,ig(K)

831.7 2069.7 10–20 0.12 600

casemayberepresentativeofarealpropagatingfirewithoutcon- sideringthecompetitionbetweenthefire residencetime

τ

^{res and}

thesolidparticlereactiontime

τ

^react.

For

τ

res

τ

react:vegetationhasnotimetoadapttotheprop- agationoftheflamefrontanditisreasonabletoassumeT_S=T₀∼ 295Kuntilignitionoccurs,equivalenttoacoldvegetationcase.

For

τ

res

τ

react:vegetationhasenough timeto reachsteady- state T_S=T∼1900K before the flame front moves leading to a quasi-staticpropagationfire,equivalenttoahotvegetationcase.

FollowingthestudyofBurrows[43],thefireresidencetimefor aroundwoodparticlehaving

σ

^s=7500m⁻¹canbecalculatedas,

τ

^res=208487

σ

s^1.236

≈3.4s (5)

In terms of solid fuel chemical reaction time, the best rep- resentation may be the characteristic time of a thermal ker- nel according to the study of Lamorlette and Candelier [44]. TheBiot numberB_i=h_TL/

λ

^{s andthe} dimensionless number

ϕ

=

φ

sur fL/

λ

^s

(

^Ts,ig−T₀

)

co-determine the validity of either the ther- mallythick orthermallythinapproach.Forthepresentstudy,the totalheattransfercoefficient h_T,whichconsidersboth convection andradiant-emissionrangesfrom10to 20Wm^-2K^-1 accordingto severalstudies[45–47],thecharacteristiclengthL=2/

σ

^s=2.67× 10⁻⁴m, the solid heat conductivity

λ

s=0.12Wm⁻¹K⁻¹ and the solidignitiontemperatureT_s,ig≈600K.Inthesolver,asuitedNus- seltcorrelationisusedtocalculatetheh_T,whileradiationisfully resolvedasexplainedin[26,27]. Thevalue ofhT=10–20Wm^-2K^-1 is then chosen only to evaluate B_i and the solid chemical reac- tiontime. Withthe

φ

sur f≈42KWm⁻² obtainedfromsimulations (ahead of flame front), the B_i ≈ 0.044 and

ϕ

≈ 0.31 correspond toathermallythinbehaviorofpineneedlefuels.Hence,thesolid chemical reactiontime can be represented by thethermally thin kernelcharacteristictime

τ

k[48],giving,

τ

^react=

τ

k=

ρ

^sCp,s

hT

σ

^{s =11.5∼23s (6)}

Where

ρ

s=831.7kgm⁻³andC_p,s=2069.7Jkg⁻¹K⁻¹.

Following the above calculation where

τ

^res <

τ

^{react,it is rea-}

sonabletoretainacoldgauge,inertvegetationassumptionbecause thevegetationhasnotimetoadapttothepropagationoftheflame frontinarealcase.Relevantparametersusedfortheabovecalcu- lationandthebelowsimulationsaresummarizedinTable1.

3. Resultsanddiscussion 3.1.Fireregimetransition

Tounderstand theeffectoftheconfigurationspace{N_C,CdLAI} onthefireregimetransition,oneimportantaspectmightbetoex- plorehowdoestheconfigurationspacechangetheflametiltangle

θ

^{andwhetherthe}controllingmechanismusingtheconfiguration spacecanprovideabetterestimationofthetiltanglecomparedto previousmodelsconsideringtheroleofN_Calone.Fig.2showsthe roleplayedbyCdLAIforalow windspeed casewithN_C=20.In- creasingCdLAIfrom0.01to0.5hasclearlyledtoanincreaseoftilt anglefrom52.7to 70.94° representingthe changeof fire regime fromplume-dominated towind-driven. The tangent of theflame

Fig. 2. The time-averaged velocity vector for N C = 20 with length of arrow indicat- ing velocity magnitude. Flame body (grey) is confined by T = 500 K.

tiltangleisdefinedastheratioofflamewidthw_flametotheflame heightH_flame giveninFig.3.

Complete data points collected fromprediction are shown in Fig. 4 for N_C ranging from2 to 20and CdLAI ranging from 0.01 to0.5.ThechosenvaluesforN_CandCdLAIarebasedontwojudg- ments: first,the fire data should cover two fire regimes [4], the plume-dominated NC > 10 andwind-driven NC < 2; second, the CdLAImustcovertheABLtoMLflametransitionpointatCdLAI= 0.1[23](see Section1).Becauseitisobservedthatflameiscom- pletely wind-driven forN_C=2andthe flameangleis hardly cal- culatedduetoirregular flameshape, nopredictionforN_C < 2is performed.

FromFig.4a,itisobservedthat(a)forsmallN_C,theflametilt angleis almostindependent ofCdLAI; (b)forsmall CdLAI(sparse vegetation), flame tilt angle seems to be linearlyproportional to N_C:thesmallertheN_C,thelargertheflametiltanglecorrespond- ing to flame regime transition from plume-dominated to wind- driven; (c)the effectoflarge CdLAI (densevegetation) onchang- ing flametiltangleisnon-ignorable forlow wind speed,highN_C conditionsthat(d)thelargertheCdLAI,thelargertheflametiltan- gleimplyingatendencyofthefire regimetransitionfromplume- dominatedtowind-driven.

Table 2

Two representative model correlations.

Eq. Model correlations Fuel Ref.

(7) tan (θ)= C ×(²_T^T_f⁰)^1/5×K a ^−1/5 Heptane, Ethanol, and Acetone [50,51]

(8) tan (θ)= {^{C 1×}α^1/2N _C^−1/3(N C< 10)

C 2×η²N _C^−2/3(N C > 10) Long leaf pine, slash pine litter, etc. [3]

Fig. 3. Representation of flame tilt angle definition. Variables U 0, D 0, w flame, H flame, and H Frepresent wind speed, flame depth, flame width, flame height, and vegeta- tion height respectively.

Fig. 4. The predicted flame tilt angle θfor different N Cand CdLAI .

FromFig.4b,itseemsthat otherthanthedatapointsforN_C= 2,flametiltanglechangesexponentiallywithCdLAI,i.e.,

θ

^~eCdLAI.

ForN_C=2,thereislittleeffectofCdLAIonchangingtheflametilt angle.

Lam andWeckman[49]reviewedseveraltiltanglemodelcor- relationsforpoolfireandwildlandfire,theinnerrelationshipsfor 13 models were examined andreason for different behaviors of thesemodels were highlighted.Based onthat study,Table 2lists

Table 3

A summary of parameters used to calculate flame tilt angle.

ρ0(kg m ⁻³) C p,0(Jk g ⁻¹K ⁻¹) T 0(K) T f(K) D (m) I (KW)

1.19 1000 295 1000 0.5 100

U 0(m s ⁻¹) N C R i K a K ⁻¹a^/5 N _C^−1/3

0.65356 20 8.0875 161.75 0.3616 0.368

0.8234 10 5.0952 50.952 0.4556 0.464

0.97629 6 3.6243 21.7458 0.54 0.55

1.408 2 1.7425 3.485 0.779 0.7937

Fig. 5. The predicted flame tilt angle vs . N _C^−1/3for different CdLAI .

tworepresentativemodelcorrelationsproposed recentlywiththe equation(8)notdiscussedin[49].

TheEquation(7)wasoriginallydefinedbyHuetal.[51],given as,

tan

( θ )

^=C×

ρ

^0Cp,0

^Tf

˙

mD²

^HC ×

T0

g

^Tf 2

1/5

(9)

Considering the fireline intensity I=m˙

^HCD, Byram con- vective number N_C=2gI/

( ρ

0U₀³C_p,0T₀

)

^{, Richardson number R}i= g

^TfD/T_fU₀²,andassuming Ka=N_C×R_i,Eq.(9)can be simplified to,

tan

( θ )

=C×

2×

ρ

^0Cp,0T0U₀³

2gI × U₀²T_f g

Tf−T0

D×T0

Tf

1/5

=C×

2T0

T_f

1/5

×[Ka]^−1/5 (10) Table3providesasummaryofthepropertiesusedtocalculate

θ

^{, N}C, R_i,Ka andtheir exponential values. Surprisingly,it is seen that theK_a^−1/5 andN_C^−1/3 sharealmost the samevalue indicating thatEquation (7)mayonlybe suitable forN_C < 10.Indeed,after carefulexaminationof theexperimental datain [51], thepresent authorsnotice that the flametilt angles predictedwith Equation (7)deviatefromexperimentaltiltanglesmainlyforlargeN_Cvalues (theN_Cisabout800forinaccuratefittings).

Nevertheless,Fig.5showsaplotforthepredictedflameangles vs N_C^−1/3. A good agreement between the model correlation and predicteddatacanbe clearlyobservedforCdLAI=0.01,N_C < 10.

Because ofthe similaritybetween N_C^−1/3 andK_a^−1/5, thishas also evidencedthe effectivenessofthe modelcorrelationproposed by Hu et al. [51]. The constant C in Equation (7) for heptane and ethanolwas9.1 [51], andthe value foracetonewas4.16 [50]. In

Fig. 6. The velocity vectors close to the vegetation height H F= 0 . 05 m ; Length of arrow indicates velocity magnitude.

thepresentstudy,thisconstant isroughly 5.70.From the deriva- tionofEquation(8)byNelsonetal.[3],thedifferentvaluesofthat constant might be a resultof different fractionof the impinging airenteringtheflamepartiallyaffectedbythedragforcefromthe vegetationandthepool [49]. Despitethe relativeeffectiveness of theN_C^−1/3 andK_a^−1/5 correlationsforlowCdLAI,theeffectofhigh CdLAIonflametiltanglewasessentiallynotconsideredbyanyof theavailablemodelsinthepast.

Besides, Fig. 5has alsoshownthat when wind speed islarge correspondingtoN_C=2,theeffectofCdLAIonchangingflametilt angleissmallasobservedinFig.4a.ThisisnotbecausetheCdLAI hasnoeffectonchangingthewindprofileinthevegetationlayer consideringthewindisvery strongbutisbecauseofthefullen- trainmentofairintotheflameabovethevegetation(seeFig.6).In otherwords,forN_C=2,theeffectofMLflowinthevegetationon flamebehavior isnegligiblecompared tothat ofthe rapiddevel- opmentofABLflowabovethevegetation(HF=0.05m).

Because ofthedifficulty toconsiderthedatapointsatN_C=2, theEquation(8)isfurtherexaminedforN_C>10.AccordingtoNel- sonetal.[3],themodelcorrelationwasbuiltupontheconsidera- tionofthecompetingeffectbetweenthedragforceF_d∼C_d

ρ

0U_e²HL andthebuoyancyforceF_b∼

ρ

^w20HLwheretheC_disthedragcoef- ficientfortheinclinedflameandthe

η

=Ue/U₀representsthefrac- tionoftheimpingingairenteringtheflame.TheinvolvementofC_d and

η

^{providesagoodstartingpointtoconsidertheroleofCdLAI.}

FromFig.6,itisveryobviousthattheroleofCdLAIistosuppress the sub-layer flow while improvethe air entrainment above the vegetation.Because

η

^{isdefinedfor thefractionof impinging air}

above the vegetation,it is reasonable to write

η

^{~ CdLAI. While,}

consideringthatonlythenaturallogpartoftheentrainedairpar- ticipatesin the combustion dueto the log wind profile given in Eq.(1),thecorrelationmaybe furtherwrittenasln

η

^{~ CdLAI,giv-}

ing,

ln

η

^=C1CdLAIor

η

^{=e(C1CdLAI) (11)}

Theentrainmentvelocityisthenwrittenas,

Ue=e^(C1CdLAI)U0 (12)

The tangentoftheflametiltangleistheratioofdragforce to buoyancyforce,

tan

( θ )

^{= F}_F^d

b

∼

ρ

0U_e²HL

ρ

^w20HL =

ρ

0e^(2C1CdLAI)U₀²HL

ρ

^w20HL =

ρ

0

ρ

^×e2C1CdLAI

×N_C^−2/3=C₂e^2C1CdLAI×N_C^−2/3 (13)

Fig. 7. The best fit model correlation for different N Cand CdLAI .

WhereU₀/w₀=N_C^−1/3 because thecharacteristic buoyancyve- locityw₀[52]canbewrittenas,

w0=

2gI

ρ

^0Cp,0T0 1/3

andNC= 2gI

ρ

^0Cp,0T0U₀³ (14) The Eq.(13) is then used to fitthe predicted flametilt angle data for N_C=6, 10 and 20 though the equation seems to work only forN_C > 10.The comparison betweenthe best fit equation withtheconstantsC₁=0.5366287,C₂=10.76685369andthecol- lapsed data is shownin Fig. 7. A good matching is obtainedfor N_C >10,whilepredicteddataslightlydeviatefromthatpredicted usingthemodelEq.(13)forN_C=6andthemodelfailsforN_C=2. In fact, our previous study [35] showed that the flame regime transition from plume-dominated to wind-driven occurs roughly atN_C=5.6byobservingtheflameshapesforthecaseofCdLAI= 0.01.It istherefore concludedthat the proposed modelEquation (7)worksprimarilyforplume-dominatedfire.

The model correlation tan

( θ )

=10.76685369e^{1.0732574CdLAI}× N_C^−2/3is,insomeextent,agreeswellwiththefindingsfromFig.4b that

θ

^~eCdLAI.

3.2. Radiativeheatpower

Theeffect oftheconfigurationspace{N_C,CdLAI}ontheradia- tive heat power to the unburnt vegetation layer in front of the fire front is demonstrated in Fig. 8. The radiative heat power is defined as the CdLAI weighted total radiation written as P_rad=

∫Q_rad^s d

v

/CdLAI.The chosen of theradiative heat poweris forthe purposeofconsideringthein-depthradiationeffectcausedbythe characteristicofvegetation,i.e.,theCdLAI.

FromFig.8,itisfound that(a)theeffectofCdLAIonradiative heat power is large for both plume-dominated and wind-driven

Fig. 8. The predicted radiative heat power P radfor different N Cand CdLAI .

Fig. 9. The predicted radiative heat power P radvs . tan θfor different N Cand CdLAI .

fires,whileitseffectissmallfortransitionalfireroughlyatN_C=6 (or 5.6 from a previous study [35]); (b) wind (or N_C) has little effect on radiative heat power for large CdLAI, while it plays an important role when vegetation is sparse (CdLAI=0.01); (c) ex- cluding the data points for N_C=2, there seems to exist a rela- tionship between P_rad and the proposed configuration space, i.e.

P_rad~f(N_C,CdLAI).

To correlate the configuration space with the radiative heat power, thefirst stepmight be toconsider theeffectof flametilt angle

θ

^{ontheradiativeheatpowerP}rad.Fig.9ishenceprovided inordertocorrelatethesequantities.Becauseamodelcorrelation hasbeenproposedforthetangentofthetiltangle

θ

^andthecon-

figurationspace{N_C,CdLAI},tan(

θ

^{) ratherthan}

θ

^{isconsidered in}

thefigure.Theproposedmodelisagaingivenas,

tan

( θ )

^{=10.76685369e1.0732574CdLAI×N}C^−2/3 (15)

AgoodfittingofEq.(15)withthepredicteddatawasobtained forN_C > 10, whiledata slightlydeviated from theprediction for NC=6 and the model failed forNC=2. Indeed, from Fig. 9,the black color circled data points for N_C=2 are far from the main trendoftheotherpointsthatabestfitmodelproposedexcluding

Fig. 10. The fitting curve of the radiative heat power P radvs . ta n ⁻²θfor different N C

and CdLAI .

Fig. 11. The best fit model correlation for different N Cand CdLAI .

Fig. 12. Irradiance or radiative heat flux on the vegetation surface.

Fig. 13. Temperature contours for different N Cand CdLAI . Left: CdLAI = 0 . 01 ; Right: CdLAI = 0 . 5 .

thosepointsisgivenas,

Prad=5148.793tan⁻²

( θ )

^{withR2=0.89 (16)}

The fittingcurve isthen givenseparately inFig.10forclarity.

Itis noteworthy that themodel correlation doesnot fit the data mainlyforthe casesof high{N_C,CdLAI}and low{N_C,CdLAI} rep- resentingthetwoextremeconditions:the{lowwindspeed,dense vegetation}and the {high wind speed, sparse vegetation}. These observationsmightbecausedbytheassumptionofaconstant

χ

rad, theratioofthermalradiationto thetheoretical heatreleaserate, whichismoreinfluencedbythetwoextremeconditions.Aproper calibration of

χ

rad ~ f(N_C,CdLAI) from experiments may provide more accurate mathematical correlations since the value of

χ

rad

changesforthecombustionofdifferentfluid/solidfueltypeatdif- ferentconditions[53,54].

Consequently,substitutingEq.(15)intoEq.(16)leadstothere- lationshipbetweenP_radand{N_C,CdLAI}givenas,

Prad=C2e^2C1CdLAIN_C^4/3 (17)

WithC₁=−1.0732574andC₂=44.41481followingtheformat ofthederivedequation(7).

Indeed, this model correlation P_rad∼ f

(

^NC^4/3,CdLAI

)

^{is not far} fromthe observationin Fig.8 that P_rad ~ f(N_C,CdLAI). The fitting curveisthen showninFig.11withcompletelywrongfittingsfor thecasesinextremeconditionsduetoamplifiederrorfromsub- stitution.Formoderate{N_C,CdLAI},goodfittingisobtained.

3.3.Radiativeheatfluxandgeneralbehaviors

Other thandiscussingthebehavioroftheradiativeheatpower totheentirevegetationlayersaheadoftheflamefront,theradia-

tiveheatfluxortheirradianceJonthetopsurfaceofthevegeta- tion ispresented inFig.12.From Fig.12a,it isobserved that N_C hasastronginfluenceontheradiativeheatfluxacrossthesparse vegetationsurface(CdLAI=0.01),whileitseffectrapidlydecreases fordenser vegetation(CdLAI=0.5). The minimal effect ofN_C for thecaseofCdLAI=0.5leadstotheoverlappingofthecurvesim- plyingthatN_Calonecanbeusedtopredictradiativeheatfluxfrom flamebodytothemajoritytypesoftheforestfuellayersasitwas reportedthatthemostforestfuellayerssuchasneedlesorshrubs exhibitCdLAI>0.5[13,55].Moreover,itwaswidelyacceptedthat N_C maybe used independently to classifythe fire regime transi- tion definedby theratio of radiative to convective heat transfer.

ForCdLAI=0.5in Fig. 12a which is a typical case forforest fire study,thelittleeffectofN_Contheradiativeheatfluxindicatesthat theN_C ismainlyresponsibleforchangingconvectiveheattransfer tothevegetation,andthereforechangingthe ratioofradiative to convectiveheattransfer.ForCdLAI<0.5oftenforsmallscalelitter fires,bothradiativeandconvectiveheattransferareinfluencedby N_C.

Onthe other hand, Fig.12bexplicitly demonstratesthe effect ofCdLAI onthe radiativeheat fluxacross unburntvegetation un- der differentwindconditions.ForN_C=20representinglow wind power, increasing CdLAI leadsto the shift of the peak irradiance totheright(downstream);whileasN_Cdecreases,thisobservation is reversed that the peak irradiance shifts more to the left (up- stream) following the decrease of N_C from 10 to 2. It might be interesting to knowwhat do the peaksrepresent and doesit in some extent relate to the validity of themodel correlationspro- posedinSections3.1and3.2.Afirstimplication isthattheposi- tion ofthe peaksisa resultof localheat releaseratewithin the flamebody,andhenceitmayalsoberelatedtothestochiometric

fuel/airratioandfurtherthefractionofimpingingairenteringthe flame.

Fig. 13 shows the temperaturecontour plots ofthe fires. The stochiometric fuel/air ratio and the vegetation are labeled with the white solid line. The former one has an irregularshape and is referred to asa stochiometric linein thefollowing discussion, andthelatteronehasarectangularshapewiththetopsitting at Y=H_F =0.05m.Interestingly,theintersectionpointsbetweenthe stochiometriclineandthevegetationtopreflectalmostexactlythe positionoftheirradiancepeaksobservedinFig.12.

However, it is also noticed that the stochiometric line can- not always go beyond the surface top that for cases such as {NC= 6,CdLAI=0.01}and{NC=2,CdLAI=0.01},nointersection points can be observed. These configuration spaces are the low {N_C, CdLAI} cases discussed in Figs. 10 and11 wheremost inac- curate fittings were found.Practically speaking, whetherthe sto- chiometriclinecangobeyondthevegetationsurfaceornotrelates strictly withthefractionofimpinging airintotheflame.Because thefractionofentrainmentwithinthevegetationlayerisnotcon- sideredintothemodelcorrelationofEq.(17)duetothefactthat theequationinheritsthefeaturesofEq.(15)fortheflametiltangle definedabovethevegetation,theaboveobservationhasessentially explainedwhythetwoproposedmodecorrelationsdoesnotfitall availablefiredata.

4. Conclusion

This study was motivated by the lack of knowledge on how does vegetation characteristic (defined as CdLAI in the present study) may affectthe small-scale fire regime transition andheat transfer mechanism.Previous studieshavehighlightedthe roleof Byram convectivenumber N_C on describing thebehavior of‘free’

fires, i.e., fires without vegetationaround (for pool fire) or with- out vegetationpropertydistinguished(forwildlandfire)[4,6],the responsibility ofvegetationproperty on fire behaviorwere hence remainunclear.Theflamemodelsestablishedbasedonthesefires may result in inaccurate large- and giga-scale fire modeling due towrong‘prior’flameshapeestimatedfrom

θ

^{~ windspeed[56].}

An accurate flamemodelconsidering the role ofvegetationchar- acteristics is essential to extend the limit of validity of large- scale fire models. Therefore, thepresent workfirst discussed the non-ignorableeffectofthevegetationcharacteristicsontheflame tilt angleandtheradiative heat transfer,andthen proposed new modelcorrelationsforpredictingthesetwoquantitiesusingacon- figuration space {N_C,CdLAI}. The main results are summarized as follows:

• TwopreviousexperimentalbasedflametiltanglemodelsbyHu etal.[51]andNelsonetal.[3]arefound tohavesimilar per- formance forN_C < 10.For N_C > 10, the modelcorrelation by Nelsonetal.fitswiththepredictedflametiltanglemainlyfor smallCdLAI.Neitherofthemodelscanbeextensivelyusedfor differenttypesofvegetationcharacteristicsduetolackofcon- sideringCdLAI.

• For N_C > 6, a new flame tilt angle model is proposed as tan

( θ )

=C₂e^2C1CdLAI×N_C^−2/3 with C₁ being positive, i.e., flame tiltangleisdirectlyrelatedtoCdLAIwhile inverselyrelatedto N_C.

• Anewmodelforradiative heatpowerreachingthevegetation ahead of the flame front is proposed as P_rad=C₂e^2C1CdLAIN_C^4/3 with C₁ being negative, i.e., radiative heat power is reversely relatedtoCdLAIwhiledirectlyrelatedtoN_C.

• Theradiationmodelisvalidformoderate{N_C,CdLAI}primarily duetotherole ofCdLAI onchangingthefractionofimpinging airintotheflameabovethevegetationlayer, andmayalsobe aresultoftheconstant

χ

rad usedforprediction.

• Overall, thepresentworkshowedthe importanceofconsider- ing the vegetationcharacteristic inthe flame models that the proposed modelcorrelationsbased onthe configurationspace {N_C,CdLAI}shouldbe usedforlarge- orgiga-scaleflamemod- elinginsteadofthemoreclassicalfreefiremodelcorrelations.

DeclarationofCompetingInterest Nonedeclared.

CRediTauthorshipcontributionstatement

Kai Zhang: Conceptualization, Methodology, Formal analysis, Writing-originaldraft,Writing-review&editing.AymericLam- orlette: Writing -review & editing,Supervision, Project adminis- tration.

Acknowledgments

This work is supported by Labex MEC (ANR-10-LABX-0092) andthe A^∗MIDEX project (ANR-11-IDEX-0001-02), funded by the

“Investissementsd’Avenir”.

This work was granted access to the HPC resources of Aix- MarseilleUniversityfinancedbytheprojectEquip@Meso(ANR-10- EQPX-29-01)oftheprogram“Investissementsd’Avenir” supervised bytheAgenceNationalepourlaRecherche.

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