A Ghost Fluid/Level Set Method for boiling flows and liquid evaporation: Application to the Leidenfrost effect

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To link to this article : DOI:10.1016/j.jcp.2016.04.031

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To cite this version : Rueda Villegas, Lucia and Alis, Romain and

Lepilliez, Mathieu and Tanguy, Sébastien A Ghost Fluid/Level Set

Method for boiling flows and liquid evaporation: Application to the

Leidenfrost effect. (2016) Journal of Computational Physics, vol. 316.

pp. 789-813. ISSN 0021-9991

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A

Ghost

Fluid/Level

Set

Method

for

boiling

flows

and

liquid

evaporation:

Application

to

the

Leidenfrost

effect

Lucia Rueda Villegas,

Romain Alis,

Mathieu Lepilliez,

Sébastien Tanguy

∗

IMFTCNRSUMR5502,UniversitédeToulouse,France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Keywords: Evaporation Boiling Level Set Ghost Fluid Leidenfrost

Thedevelopmentofnumericalmethodsforthedirectnumericalsimulationoftwo-phase flows with phase change,in the framework of interfacecapturing or interfacetracking methods,isthe maintopicofthisstudy. Weproposeanovelnumerical method,which allowsdealing with bothevaporationand boiling atthe interfacebetweenaliquidand a gas.Indeed, insome specific situationsinvolving veryheterogeneous thermodynamic conditionsattheinterface,thedistinctionbetweenboilingandevaporationisnotalways possible.Forinstance,itcanoccurforaLeidenfrostdroplet;awaterdroplevitatingabove ahotplatewhose temperatureismuchhigherthantheboilingtemperature.Inthiscase, boilingoccursinthefilmofsaturatedvaporwhichisentrappedbetweenthebottomofthe dropandtheplate,whereasthetopofthewaterdropletevaporatesincontactofambient air. The situation can also be ambiguous for a superheated droplet or at the contact line betweenaliquid and ahot wallwhose temperature is higher thanthe saturation temperatureoftheliquid.Inthesesituations,theinterfacetemperaturecanlocallyreach thesaturationtemperature(boilingpoint),forinstanceneara contactline,andbecoolerin otherplaces.Thus,boilingandevaporationcanoccursimultaneouslyondifferentregions ofthesameliquidinterfaceoroccur successivelyatdifferenttimesofthe historyofan evaporatingdroplet.Standard numerical methodsare not able toperformcomputations inthesetransientregimes,therefore,weproposeinthispaperanovelnumericalmethod to achievethis challengingtask. Finally,wepresent several accuracyvalidations against theoretical solutions and experimental results to strengthen the relevance of thisnew method.

1. Introduction

Boiling isthe phase changeofa pure liquidintoa saturatedvapor. It ariseswhen theliquidis heated(or depressur-izedin thecaseofcavitation)beyonditsboiling point.Tostart,it requirestheactivationofa nucleus, eitherinthebulk (homogeneousnucleation)orona heatedwall(heterogeneous nucleation). Manyprevious worksonthedirect numerical simulationoftwo-phaseflows proposenumericalmethodswhichallowdealing withboilingflows [7,13,21,23,37,40,41,46, 47].

Unlike boiling, evaporationdoes not require any nucleiactivation to start. Indeed, it occursspontaneously, whatever the pressure andthe temperature conditions,at the interface betweena liquidand a gas whosechemical compositions

*

Corresponding author. Tel.: +33 5 34 32 28 08; fax: +33 5 34 32 28 99. E-mailaddress:tanguy@imft.fr(S. Tanguy).

are different, suchastheevaporationofliquidwaterinair,ortheevaporationoffueldroplets inanignitable mixture.A few previous works[6,15,18,26,28,39,45] propose numericalmethods able to compute theevaporationof a liquidin the frameworkofthedirectnumericalsimulationoftwo-phaseflowswithmovinganddeformableinterfaces.

The direct numerical simulation of liquid–vapor phase change is a powerful andpromising tool to perform accurate predictionsandtoimprovetheunderstandingofvariousphysicalphenomenawhichareinvolvedinmanyindustrial appli-cations. Forexample,nucleateboilingisatopicofinterestfortheoptimizationofheatexchangers,sinceitiswell-known that theheattransfercanbe stronglyfavoredbythenucleationofvaporbubblesonthewallofa heatpipe.Ontheother hand,ifthewall heatflux reachesacriticalvalue,the formationofafilm ofsaturatedvaporoccursandleadsto a dras-ticdecreaseintheheattransfercoefficient.Thisphenomenon,knownastheboilingcrisis,isfearedparticularlyinnuclear powerplant.Otherexistingindustrialapplications,e.g.thefluidmanagementofcryogenicliquidsinmicrogravityconditions insidethetanksofspacelaunchers,furthermotivateaccuratestudiesofnucleateboiling.

The evaporationofdroplets isalsoatopicofinterestwhichis involvedinmanypractical situations.Forinstance,itis an importantstepinthedescriptionofthecombustioninautomotiveandaircraftengines. Thepredictionofthiscomplex phenomenon requires an accurate description oftheinteraction betweena cloud ofmoving and vaporizingdroplets and flame fronts.As thedynamicsofdropletscanbe stronglyaffectedby thermal, chemicalandcollectiveeffects,performing directnumericalsimulationsofsuchflowswithawell-resolveddescriptionofthedropletswouldbeastepforwardinthe descriptionandtheunderstandingoftheseflames.Theevaporationofadropletsprayisalsoasignificant phenomenonin thesteelindustry forcoolingsystemsusingliquidjetimpingements.Thislatterinvolvesthedescriptionoftheinteractions ofvaporizingdropletswithaveryhotsteelplate.Thissituationseemsevenmoredifficultthanspraycombustion.Indeed, many complexphenomena arise duringthe impactof adroplet ona hotwall. Forexample,inthe so-called“Leidenfrost regime”, which occurs when the wall superheat is high, the formation of a thinlayer of saturated vapor between the impinging dropletandthehot plateleadsto thedropletlevitation duringits spreading; nocontactlineis formedduring theimpact.Thisregimeisquitesimilartothefilmboilingregime.

The main motivation of this paperis to presenta novel numerical strategy to perform direct numerical simulations of the Leidenfrost droplets. Some previous works propose various strategies to achieve this difficult task. In [16,17,33], Volume-Of-Fluidsimulationsarecarriedoutwithamicroscopicmodelforphasechange.Thismicroscopicmodel,whichis basedontheSchrage’slawfromthekinetictheory[2],allowsdeducingthelocalmassflowrateofphasechangesfromthe discontinuities oflocal thermodynamicsproperties(temperatureandpressure). Whereasitcan provideacceptable results in some situations, manycriticisms can be addressedtothat kind ofapproaches. First, thismicroscopic modelhas been designed to describe the phase change in non-equilibrium thermodynamic conditions,thus the local mass flow rate of phase changewillbedifferentfromzeroiftheinterface temperatureisdifferentintheliquidandinthevapor.However, assuming thelocalthermodynamic equilibrium,thejump conditiononthe entropypointsout thatthe temperaturemust becontinuousattheinterface[19].Letusnotethatthislastassertionisnotstrictlyexactifthepressureisdiscontinuousat theinterface,butthecorrectionsresultingfromtheinfluenceofthepressurejumpconditiononthetemperaturecontinuity areweak,see[21]formoredetailsonthisothertopic.

Moreover, it appearsthat usingthe Schrage’slawdoesnotallow foraccountingonthe dependenceofthe localmass flow rateof phase change withthe local temperature gradient at the interface. Instead, it depends on the temperature discontinuityandonanaccommodationcoefficient,whosevalueisnotalwaysperfectlyknown.Tosummarize,wethinkthat usingthislawinnumericalsimulations,basedonalocalthermodynamicequilibriumassumption,leads toadisagreement with the first lawof thermodynamic (mass flow rateof phase changes doesnot depend directly on the heat flux jump condition),and/orwiththesecondlawofthethermodynamic(thetemperatureisnotconsideredcontinuousattheinterface in contradictionwiththe jumpcondition ontheentropy). Moreover,since thelocalmassflow rateof phasechangescan be deduced fromthe jump condition on the thermalflux if the temperatureis continuous atthe interface, considering an additional law which involves an accommodation coefficient leads to an over-determined problem. Nevertheless, the Schrage’slawisstillfrequentlyusedinvariouscontexts,sinceitallowsdeterminingmuchmoreeasilyamassflowrateof phase changethanbytheclassicalapproach basedon theassumptionoflocalthermodynamic equilibrium.Indeed,inthe lattercasethethermalgradientattheinterfacemustbeaccuratelycomputedwhichisamuchmoredemandingtask.

In[9,10],anotherapproachhasbeendevelopedtoperformnumericalsimulationsofLeidenfrostdroplets.Inthesepapers, a Level SetMethod, used to describe the dropletmotion in an isothermal domain, iscoupled to a lubrication model to accounttothermaltransferinthevicinityofthehotwallthroughanALEauxiliarygrid.Althoughmanyinterestingresults are provided in thisstudy,thistechnique does notaddress the overallproblem, since the thermaleffects andthephase changeareonlysolved inthevicinity ofthehotwall.Moreover, alubricationmodelinthethinvaporlayerisnotableto fullydescribethecomplexandunsteadyfluiddynamicinthismicroscopicregion,whichisstronglycoupledtotheinterface motion.

Therefore,wepresentinthispapersomenewdevelopmentsinordertoperformsimulationsofLeidenfrostdropletswith fully-resolved computationsofthevapor layerdynamic, oftheheat transferandofthephase changes.The key-pointsto achievethesedifficultsimulationsarethedevelopmentofaphasechangemodelwhichcandealbothwithevaporationand withboiling,aswellastheuseofveryrefined gridsclosetothehot wall.Suchrefined gridsmustbe usedwithimplicit algorithms forthe temporaldiscretization ofall thediffusionterms. Moreover,we willsee that asemi-implicittemporal solution oftheextrapolation techniquesbasedon aPDE approach[1,12] can alsobe requiredto alleviatethe numberof temporaliterationsoftheseextrapolationswhentheinterfacecrossesgridcellswithverydifferentresolutions.

Wewillalsopresent,inthenumericalresultssection,severalbenchmarksinordertoassesstheaccuracyofthe numer-icalmethods described inthispaper.Toourknowledge,thesebenchmarkshavenever beenconsideredwithan interface capturingmethodcoupledtoanumericalsolverdedicatedtodropletevaporation.

2. Formalism

In thissection, we first presentthe conditions oflocal thermodynamic equilibrium. Next, the conservation equations fortwo-phaseflows arepresented.As wewillusesharpinterface methodsforthecomputation ofall singularterms,the conservationequationswillbeexpressedwiththe“jumpconditionformulation”[13,38,46].

2.1. Thermodynamiclocalequilibriumconditionsforapureliquidincontactwithagasmixture

Weconsideran interfacebetweenapure liquidandagasmixturewhichiscomposed ofthevapor oftheliquid com-ponent (species 1)andofanother component(species 2),forexamplesteamwaterinnitrogen. Weassume that thetwo gaseouscomponentsbehaveasanidealgasmixture.Therefore,thetotalpressure P0 ofthisgasmixturecanbeexpressed asthesumofthepartialpressuresofthetwocomponents

P0

=

P1

+

P2

,

(1)

where P1 and P2arerespectivelythepartialpressuresofthechemicalspecies1and2,whichcanbedefinedas

P1

V

=

n1R T

,

(2)

P2

V

=

n2R T

,

(3)

where

V

and T are respectively the volume and the temperatureof the gas mixture, R is the ideal gas constant (R

=

8

.

31 J mol−1K−1) andn isthenumberofmolesofthechemicalspecies.Wenowdefine Y1 tobethemassfractionofthe chemicalspecies1

Y1

=

n1M1

n1M1

+

n2M2

,

(4)

withM1 andM2 themolarmassesofspecies1and2,respectively.Fromtheserelations,wecanexpressthemassfraction ofchemicalspecies1withthetotalpressureofthegasmixtureandthepartialpressureofthischemicalspecies:

Y1

=

P1M1

P1M1

+ (

P0

−

P1

)

M2

.

(5)

We consider nowan interface

Γ

betweena liquid anda gas.By writingthe Clausius–Clapeyronlaw, we can determine thesaturatedvaporpressureoftheliquidcomponentattheinterface,whichprovidesthepartialpressureofthischemical speciesattheinterface:

P1|Γ

=

P0e− Lvap M1 R  1 T|Γ−Tsat1 

.

(6)

Inthisequation, Lvap isthe masslatentheat ofvaporization ofchemicalspecies 1,T

|

Γ isthe interface temperatureand

Tsat is the saturation temperature (or boiling temperature) of chemical species 1. Lvap and Tsat depend on the external pressurewhichwillbe assumedconstant inthisstudy(isobaricphenomena).Whenthelocalthermodynamic equilibrium isrespected,theinequality ontheentropyjumpconditionresultingfromthesecondlawofthermodynamicsimpliesthat thetemperatureiscontinuous acrosstheinterface,providedthepressureiscontinuousattheinterface.Letusnoticethat variouseffectscaninvolveweakjumpconditionsintheinterfacetemperaturerelatedtothepressurejumpconditionwhich can be induced by capillary, viscous orphase change effects. In[21],the authors presenta detailedstudy to assessthe correctionsduetothesedifferenteffects.Aswehavecheckedthatthecontributionofthesedifferenteffectswas negligible inthesituationswhicharepresentedinthiswork,wewillnotconsideranyjumpconditionintheinterfacetemperaturein thepresentstudy.Therefore,wewillconsiderintherestofthepaperthat:

[

T

]

Γ

=

0

,

(7)

with

[.]

Γ denotingthejumpconditionoperatordefinedby

[

f

]

Γ

=

fliq

−

fgaz

.

(8)

2.2. Massconservation

Strongthermalgradientsandmixingofdifferentchemicalspeciescaninvolvedensityvariationsinthegasphase. How-ever,intheinterestofsimplification, wewillrestrict ourstudytoincompressibleflows withaconstantdensity.Therefore, wewillnotneedanyequationofstateforthegasphaseorfortheliquidphase.Themassconservationwillbesatisfiedby imposingadivergence-freeconditioninthevelocityfieldineachphase

∇ · 

V

=

0 (9)

andtheappropriatejumpconditioninthevelocityfield attheinterfacewillbeimposedbytakingintoaccountthephase change[32]:

[ 

V

]

Γ

= − ˙

m



1

ρ



Γ



n

,

(10)

withm the

˙

localmassflowrateofphasechange,

ρ

thedensity,and



n thenormalvectorpointingtotheliquidphase.

2.3. Momentumconservation

ThemomentumconservationisexpressedwiththeNavier–Stokesequationsforanincompressibleflow:

ρ

DV



Dt

= −∇

p

+ ∇ · (

2

μ

D

)

+

ρ



g

,

(11)

where p isthepressure,

μ

thedynamicalviscosity, D thetensorofdeformation,and



g theaccelerationduetogravity.An appropriate jump condition inpressuremustalso besatisfied attheinterface toaccount forcapillary,viscous andphase changeeffects[13]:

[

p

]

Γ

=

σ κ

+

2



μ

∂

Vn

∂

n



Γ

− ˙

m2



1

ρ



Γ

,

(12)

where

σ

isthesurfacetensioncoefficient,and

κ

istheinterfacecurvature.

2.4. Energyconservation

Severalformulationscanbeusedfortheenergyconservationequation.Indeed,thefirstlawofthermodynamicscanbe expressed withthe totalenergy,theinternal energyortheenthalpy.Asonlyquasi-isobaricphenomena areconsidered in this paper,we willuse aformulation basedon theenthalpy to expresstheenergyconservation. Thecontribution ofthe viscousdissipationontheenergybalancewillbeneglectedinthispaper,thereforethefollowingsimplifiedrelationcanbe usedtocomputethetemperatureevolutionineachphase:

ρ

Cp

D T

Dt

= ∇ · (

k

∇

T

),

(13)

whereT isthetemperature,Cp isthespecificheatatconstantpressure,andk isthethermalconductivity.Ajumpcondition inthethermalfluxattheinterfacecanbededucedfromthefirstlawofthermodynamics:

[

k

∇

T

· 

n

]

Γ

= ˙

m



Lvap

+ (

Cpliq

−

Cpvap

)(

Tsat

−

T

|

Γ

)



.

(14)

2.5. Chemicalspeciesconservation

Theconservationofchemicalspecies1canbeexpressedwiththefollowingconvection–diffusionequation:

ρ

DY1

D t

= ∇ · (

ρ

Dm

∇

Y1

),

(15)

where Dmisthediffusioncoefficientofthespecies1inthegasphase.Thefollowingjumpconditionmustalsobesatisfied attheinterface:

[

ρ

Dm

∇

Y1

· 

n

]

Γ

= − ˙

m

[

Y1]Γ

.

(16)

Foramono-componentliquid,thisjumpconditioncanbeexpressedasaRobinboundaryconditionattheinterface:

˙

mY1|Γ

+

ρ

gDm

∇

Y1

· 

n

|

Γ

= ˙

m

.

(17)

3. SolvingtheincompressibleNavier–Stokesequationsfortwo-phaseflowswithaprescribedvelocityjumpconditionat theinterface

In thissection,we provide detailson theimplementationof theNavier–Stokessolver forincompressibleflows witha prescribedvelocityjumpconditionatamovinganddeformableinterface.Themethodusedinthispaperhasbeendeveloped by Nguyenetal.[32]todescribe thepropagationofflamefront.Thismethodhasalsobeenusedforthedirectnumerical simulation of boiling flows in [13,46].In [45], the authors point out that it is not perfectlysuited forthe simulation of evaporating droplets. To overcomethis issuethey propose an improvement on thevelocity field extension to insure the liquidvelocity extensionsatisfies thedivergence-freepropertyattheinterface.Since thesimulationswhichare presented inthispaperinvolvedropletevaporation,thisdivergence-freeextrapolationwillbeusedinallthecomputationspresented inthispaper.

3.1. OnthecomputationofviscoustermswithLevelSet/GhostFluidMethod

Thecomputationoftheviscoustermsattheinterfaceisanotherissuetobeaddressedforsimulationswithphasechange intheframeworkoftheGhostFluidMethod.Indeed,inthefirstworkof[22]ontheGhostFluidMethodforincompressible two-phaseflows, theauthorshavedevelopedaspecifictreatmentoftheviscous termsatthe interfacewhichisbasedon the jumpcondition ofthe viscous stresses.However thesejump conditionscannot be appliedto simulations withphase changebecausetheirdeterminationisnotpossibleifthevelocityfieldattheinterfaceisnotcontinuous.

Letusnoticethat inthe case ofthesimulations ofboiling flows in[13],the authorspropose tocompute the viscous termsattheinterfacebyusingthesmootheddeltafunction[43],whereastheoverallalgorithmforphasechangeisbased on asharp interface formulation. Alsointhe framework ofboiling flows, theauthors in[46] usea differentGhost Fluid approachtocomputeviscousterms.Thismethod,whichhasbeenfirstlypresentedin[44],doesnotrequirethatthevelocity fieldiscontinuousattheinterface.

Infact,thisGhostFluidapproachisquitesimilarto theoriginalDeltaFunctionMethodto computetheviscous terms, exceptthatthesharpnessofthemethodispreservedbyusinganinterpolatedviscosityontheborderofthecellswhichare crossedbytheinterface.TheinterpolationoftheviscosityconsistsinaharmonicaveragetypicaloftheGhostFluidMethod forincompressibleflows[22,24,27].

Ina recentstudy[25],an overviewonthe computationof viscousterms withLevel SetMethods ispresented.In this paper, itis demonstrated that the DeltaFunction Method [43], the Ghost FluidMethod of[22] (named GFPM forGhost FluidPrimitive viscous Method), andtheGhost FluidMethod of[44] (named GFCM forGhost FluidConservativeviscous Method)areformally equivalent.Moreover,duetoseveraltest-casesitisshownthat thetwo GhostFluidMethodshavea comparableaccuracyandthat thesetwo methodsoutclasstheDeltaFunctionMethod.Anewmethodisalsoproposedto dealwithviscoustermsintheframeworkoftheGhostFluidMethod.

The main interest ofthis new method (named GFSCM for Ghost Fluid Semi-Conservativeviscous Method) lies in its abilitytodealeasilybothwithphasechanges(astheGFCM)andwithanimplicittreatmentoftheviscous terms.Indeed, thedirectnumericalsimulationofvaporizingdropletsintheLeidenfrostregimerequiresanimplicittemporaldiscretization ofviscoustermsduetotherefinedgridclosetothehotwall.

Wewillpresentinwhat followstheadaptationoftheoriginalmethodof[32] toanimplicittemporaldiscretizationof theGFSCMwithajumpconditiononthevelocityfield.

3.2. ImplicittemporaldiscretizationoftheGhostFluidSemi-ConservativeviscousMethodwithajumpconditioninthevelocityfield

Asitispresentedin[25],thefollowingimplicittemporaldiscretizationoftheviscoustermscanbedevelopedwhenthe GhostFluidSemi-ConservativeviscousMethodisusedwithaprojectionmethodforsolvingincompressibleflows

ρ

n+1V



∗

− 

t

∇ ·



μ

∇ 

V∗



=

ρ

n+1

 

Vn

− 

t



A

 

Vn



− 

g



(18)

withthefollowingdefinitionof



A

( 

Vn

)

toaccountforconvectivetermsifphasechangeoccurs



A

 

Vn



= 

V_ln

· ∇ 

V_ln if

φ >

0 (19)



A

 

Vn



= 

Vn_g

· ∇ 

Vn_g if

φ <

0

.

(20)

Letusnoticethatthistemporaldiscretization,Eq.(18),islikeaHelmholtzequation.Itsmainadvantageliesinitssimplicity, sinceitallowstocomputeseparately eachcomponentofthevelocityfield.Thiswouldnototherwisebepossiblebyusing theGFCM ortheDelta FunctionMethod.Indeed,a fullyimplicittreatment ofviscous termswiththe GFCMortheDelta FunctionMethodleadstoalargeandcomplexlinearsystemsinceallthecomponentsofthevelocityfieldmustbecomputed simultaneously.

FinallytheprojectionofEq.(18)onthex-axisandthey-axisleadsrespectivelytothefollowingrelationonastaggered grid:

ρ

_in₊+₁1_{/2 j}u∗_{i+1/2 j}

− 

t



∇ ·



μ

∇

u∗



_{i+1/2 j}

=

ρ

n_i++₁1_{/2 j}



un_{i+1/2 j}

− 

t



A

 

Vn



· 

ex



_{i+1/2 j}

− 

g



,

(21)

ρ

_{i j}n+₊1_1/2v∗_{i j+1/2}

− 

t



∇ ·



μ

∇

v∗



_{i j+1/2}

=

ρ

_{i j}n+₊1_1/2



vn_{i j+1/2}

− 

t



A

 

Vn



· 

ey



_{i j+1/2}

− 

g



.

(22)

The following scalar equations, Eq. (21) and Eq. (22), must be respectively solved with the two following scalar jump conditionstorespectthemassconservationacrossareactiveinterface:



u∗_{i+1/2 j}



= − ˙

mi+1/2 j



1

ρ



Γ nx_{i+1/2 j}

,

(23)



v∗_{i j+1/2}



= − ˙

mi j+1/2



1

ρ



Γ ny_{i j+1/2}

.

(24)

Thatcanbe done byusing thespatial discretizationproposed in[27] toimpose jumpconditionswhen solvinga Poisson equation.Thenextstepconsistsincomputinganextensionofthevelocityfieldineachsideoftheinterface:

⎧

⎨

⎩



V_l∗

= 

V∗ if

φ >

0



V_l∗ghost

= 

V∗

− ˙

m



1

ρ



Γ



n if

φ <

0 (25)

⎧

⎨

⎩



V∗_g

= 

V∗ if

φ <

0



V∗gghost

= 

V∗

+ ˙

m



1

ρ



Γ



n if

φ >

0 (26)

Next, this extension of the velocity field will be used to compute the right hand side of the Poisson equation for the pressure:

∇ ·



∇

Pn+1

ρ

n+1

=

f

,

(27)

wheretherighthandside f iscomputed asfollows:

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

f

=

∇ · 

V ∗ l



t if

φ >

0 f

=

∇ · 

V ∗ g



t if

φ <

0 (28)

andwiththefollowingpressurejumpconditionwhichisimposedwiththenumericalmethodsproposedin[22,27]:

[

P

]

Γ

=

σ κ

+



μ

∂

Vn

∂

n



Γ

− ˙

m2



1

ρ



Γ

.

(29)

Thefinalcorrectionstepcannextbeapplied:



Vn+1

= 

V∗

−



t

ρ

n+1



∇

Pn+1

−



σ κ

+



μ

∂

Vn

∂

n



Γ

− ˙

m2



1

ρ



Γ



n

δ

Γ

.

(30)

A sharpapproximationof theDiracdistribution

δ

Γ, basedonthe workof[27],mustbe usedto computethiscorrection

step,see[25]formoredetailsonthislastpoint.

3.3. Divergence-freeextrapolationofvelocityfieldforevaporatingdroplets

Finally to restart the next time step,a velocity field extrapolation ofthe corrected velocity field is required,both to evolvetheinterfacewhentheLevelSetequationissolved,andalsotocompute A



( 

Vn

)

.In[45],theauthorshavedeveloped a specificextensionoftheliquidvelocityfieldwhichpreservesthedivergence-freepropertyattheinterface.Itconsistsin solvinganadditionalPoissonequationforaghostpressure Pghost:

∇ ·



∇

Pghost

ρ

n+1

=

∇ · 

Vl∗



t

.

(31)

Thisghostpressureenablesthedefinitionofanextrapolatedvelocityfieldfortheliquidvelocityfield

⎧

⎨

⎩



V_ln+1

= 

Vn+1 if

φ >

0



V_lghost

= 

V_l∗

− 

t

∇

P ghost

ρ

n+1 if

φ <

0 (32)

whereas theextension ofthegas velocitycan bedefinedwithEq.(26).Thisdivergence-freeextrapolationofthevelocity fieldleadstosignificantimprovementsonthemasspredictionofevaporatingdroplets[45]andslightimprovementsonthe masspredictioninthecontextofboilingflows[46].

4. Phasechangemodel

Intheprevioussection ofthispaper,thenumericalmethodsinvolvedtoimposeajumpconditiononthevelocityfield have beendetailedinthe frameworkof theGhost FluidMethod,assuming thatthe massflow rateofphase changewas known. Inthissection, we will presentthedifferentmodels that can be usedto determinethe massflow rateof phase change.

4.1. GhostFluidThermalSolverforBoiling(GFTSB)

Wepresentherethenumericalmodelwhichisproposedin[13] todealwithboilingflows. Thissharpinterfacemodel willbereferredintherestofthispaperastheGhostFluidThermalSolverforBoiling(GFTSB).Consideringthattheinterface temperatureiscontinuousanduniform,theauthorsproposetosolveseparatelythetemperatureineachphasebyimposing aDirichletBoundaryconditionattheinterface[11].Itconsistsinfirstsolvingthefollowingequationintheliquiddomain:

ρ

lCplT_ln+1

− 

t

∇ ·



kl

∇

Tn_l+1



=

ρ

lCpl



T_ln

− 

tV



_ln

· ∇

T_ln



if

φ >

0 (33) T

|

Γ

=

Tsat (34)

andthecorrespondingequationinthegasphase:

ρ

gCpgTng+1

− 

t

∇ · (

kg

∇

Tg

)

=

ρ

gCpg



Tn_g

− 

tV



n_g

· ∇

Tn_g



if

φ <

0 (35)

T

|

Γ

=

Tsat (36)

nextwiththejumpconditionEq.(14),theboilingmassflowratecanbedetermined:

˙

m

=

[

k

∇

T

· 

n

]

Γ Lvap

.

(37)

4.2. GhostFluidThermalSolverMethodforEvaporation(GFTSE)

Wepresentnowthephasechangemodelproposedin[45]todealwithevaporatingdropletsintheframework ofLevel Set/Ghost FluidMethod.It willbe referred to intherest ofthispaper asthe GhostFluidThermalSolverforEvaporation (GFTSE).Letusnoticethatsimilarmodelshavebeendevelopednextin[39]and[28]withVolumeOfFluidmethods.Asthe evaporationofdroplets stronglydependsonthe chemicalcomposition ofthe gasmixture,an additionalequation forthe conservationofchemicalspeciesmustbesolved(forabinarymixtureinthegasphase).Theequilibriumthermodynamics conditionsattheinterfaceEq.(5)andEq.(6)definethemassfractionofevaporatedspeciesattheinterface.

Therefore,theconservationequationofchemicalspeciescanbesolvedinthegasfieldwithanimposedDirichlet bound-aryconditionattheinterface

ρ

gY1n+1

− 

t

∇ ·



ρ

gDm

∇

Y1n+1



=

ρ

g



Y₁n

− 

tV



n_g

· ∇

Y₁n



if

φ <

0

,

(38) Y1|Γ

=

P1|ΓM1 P1|ΓM1

+ (

P0

−

P1|Γ

)

M2

,

(39)

where P1

|

Γ isthepartial pressureofsaturatedvaporofspecies1whichcan bedeterminedwiththeClausius–Clapeyron

law:

P1|Γ

=

P0e− Lvap M1

R (T1|Γ−Tsat1 )

_.

₍₄₀₎

Next,oncethemassfractionfieldhasbeendetermined,themassflowrateofphasechangecanbecomputedthroughthe jumpconditionfortheconservationofchemicalspeciesEq.(16)

˙

m

= −

[

ρ

Dm

∇

Y1

· 

n

]

Γ

[

Y1]Γ

,

(41)

whichcanbesimplifiedforamono-componentliquidwhichevaporatesincontactwithabinarymixtureinthegasphase

˙

m

=

ρ

gDm

∇

Y1

· 

n

|

Γ

1

−

Y1|Γ

.

(42)

Oncem has

˙

beendetermined,theenergyconservationequationcanbesolvedsimultaneouslyinthetwodomainswiththe appropriatejumpconditioninthethermalflux



ρ

CpTn+1

− 

t

∇ ·



k

∇

Tn+1



=

ρ

Cp



Tn

− 

t B

 

Vn

,

Tn



,

[

k

∇

T

· 

n

]

Γ

= ˙

m



Lvap

+ (

Cpliq

−

Cpvap

)(

Tsat

−

T

|

Γ

)



,

(43) with



B

 

Vn

,

Tn



= 

V_ln

· ∇

T_ln if

φ >

0 B

 

Vn

_,

_Tn



_{= }

_Vn g

· ∇

Tng if

φ <

0

.

(44)

4.3. GhostFluidThermalSolverforBoilingandEvaporation(GFTSBE)

We describehereanovelnumericalmethodwhichallowsthe treatmentofbothevaporationandboiling.Indeed,very heterogeneous conditions of vaporization occur during the interaction of vaporizing droplets with hot plates (above the boiling temperature) whetherforlevitating droplets inthe Leidenfrostregime or forsessiledroplets witha contactline. This newmethodcaptures implicitlythespatial transitionbetweenregions where evaporationoccursandregions where boilingoccurs.ItwillbereferredtointherestofthispaperastheGhostFluidThermalSolverforBoilingandEvaporation (GFTSBE). Themain ideaconsistsofsolvingthe thermalfieldina similarwayasintheGFTSBmethod.However, instead ofauniformDirichletboundaryconditionontheinterface temperature,anon-homogeneous Dirichletboundarycondition is imposed. This boundary condition is determined by reversing Eq.(5) and Eq. (6), to define an interface temperature dependingonthelocalmassfractioninthegasphase:

T

|

Γ

=

LvapM1Tsat LvapM1

−

R Tsatln



_P1|Γ P0

,

(45)

with P1

|

Γ determinedfromEq.(5):

P1|Γ

=

−

Y1|ΓP0M2

(

M1

−

M2

)

Y1|Γ

−

M1

.

(46)

Letusremarkthatthetemperaturevariesalongtheinterfacebutremains continuousacrosstheinterfaceasstatedby the second lawofthermodynamics.Next,oncethetemperaturefieldhasbeendeterminedseparatelyineachdomainbyusing thisnon-homogeneousDirichletboundarycondition,themassflowrateofphasechangecanbecomputedinthesameway asfortheGFTSBwiththeEq.(37).Itis noteworthythatifthemassfractionis equalto1inthe gasphase,thisthermal solverbecomesstrictlyidenticalastheoneintheGFTSBsinceT

|

Γ

=

Tsat ifY1

|

Γ

=

1.

Asthemassfractionisrequiredtodeterminetheinterfacetemperature,thenextstepconsistsofsolvingtheequationof themassfraction, Eq.(15),withthesuitablejumpcondition,Eq.(16).Themassflowrateofphase changeisknownfrom Eq.(14)andtheequationsofthermodynamicsequilibriumattheinterfacehavealreadybeenusedwiththethermalsolver. Then themassfractionfieldcan becomputedbysolving thisequation withan imposedRobinboundarycondition atthe interface:

ρ

gY₁n+1

− 

t

∇ ·



ρ

gDm

∇

Y₁n+1



=

ρ

g



Yn₁

− 

tV



n_g

· ∇

Y₁n



if

φ <

0

,

(47)

˙

mY1|Γ

+

ρ

gDm

∇

Y1

· 

n

|

Γ

= ˙

m

.

(48)

Withthisnewformulation,wesolveexactlythesamesetofequationsasfortheGFTSE,buttheoverallalgorithmhasbeen reversed.ItavoidsusingEq.(42)whichcanleadtoindeterminacywhenthegasphaseisvaporsaturated(Y1

|

Γ

=

1).Indeed,

inthiscriticalcaseEq.(42)divergesduetothedivisionby 0.UnliketheGFTSE,thisnewmethodisstableevenifthegas phase issaturatedofvaporsince inthissituation Y1

|

Γ

=

1 and

∇

Y1

|

Γ

=

0. Inthiscase, theRobinboundarycondition is

automaticallysatisfiedsincebothsidesoftheequationareequaltom.

˙

Thismethoddoesnotpresentanysingularitiesifthe interfacetemperatureisequaltothesaturationtemperature,andwewillseeintheresultssectionthatitisperfectlysuited forsimulationsinvolvingheterogeneousregimesofvaporizationwithtransientregionsbetweenevaporationandboiling.

5. Spatialdiscretizationanddetailsonthenumericalimplementationoftheoverallmodel

UsualspatialnumericalschemesareusedtosolvethePDEfarfromtheinterface(fifthorderWENOscheme[20]forthe convective derivativesandsecondorderfinitevolumeforthediffusivederivatives).Thetemporalintegrationiscarriedout withasecondorderRunge–Kuttascheme.Classicaltimestepconstraints,inrelationwithconvectiveterms,surfacetension terms andviscous terms(except ifan implicittemporal discretizationis usedforthe latter),must be satisfiedto ensure thetemporalstability.TheBlack-BoxMultiGrid[4]methodhasbeenimplementedandusedtospeedupmostsimulations (comparedto aICCGsolver) duetoitsgreat efficiencytosolve linearsysteminvolvingheterogeneousphysicalproperties

[29].However, thissolveris notefficientinthecaseoftheLeidenfrostdropletwhereavery refinedgridis usedcloseto thehotwall.

5.1. Secondorderspatialdiscretizationofaboundaryconditiononanimmersedinterface

In [11], the authors propose a rather simple discretization to impose Dirichlet boundary conditions on an immersed interfacewhenaPoissonoraHelmholtzequationisconsidered.Thismethodhasmanyattractivefeaturessinceitissecond order in space and the matrix discretization remains symmetric positive-definite and simple (5 diagonals in 2D and7 diagonals in3D).ThissecondorderdiscretizationisalsousedintheGFTSB[13] tocompute thetemperaturefieldandin the GFTSE [45] to compute the massfraction field.In thiswork, the methodis used ina similar wayasforthe GFTSB, exceptthattheinterfacetemperatureisnothomogeneousanddependsonthelocalmassfraction.

Moreover, asithasbeenpreviously explained,we needtosolvetheconservationequationofchemicalspecieswithan imposedRobinboundaryconditionattheinterface.In[35],theauthorshavepresentedanefficientspatialdiscretizationto

solveaPoissonoraHelmholtzequationwithaNeumannoraRobinboundaryconditionattheinterface.Thisnewmethod, whichisbased onafinitevolume formulation, alsopreservesthematrixsymmetry anda simplestencilofdiscretization with5 points in 2D and7 points in3D. A sub-cell interface reconstruction [30,31] is requiredto compute the interface geometricalpropertiesforthecomputational cellswhich arecrossed bytheinterface. Forthepurposesofthisstudy,this methodhasalsobeenimplementedtosolveEq.(47)withtheimmersedRobinboundarycondition,Eq.(48).

5.2. TemperatureandmassfractionextensionbasedonPDEresolution

Some discontinuous variables must be extrapolatedacross each side ofthe interface in orderto populateghost cells. Thisextrapolationcan be done bysolving iteratively some partialdifferential equationswhichpropagate theinformation that must beextrapolated alongthe normaldirection attheinterface. More details onthe extrapolationmethod andits implementationareprovidedin[1].Letusnoticethattheinfluenceofthisextrapolationontheglobalaccuracyoftheoretical problems has beenextensively andsuccessfullyassessed in [12,14]for Stefanproblems andin[46] inthe framework of boilingflows.

In thisstudy, we willuse linear extrapolationsand quadraticextensions, whichextend the extrapolatedvariable and respectivelyits firstordernormalderivative anditssecond ordernormalderivative,forthetemperaturefield andforthe massfractionfield.

ForthepurposesofthisstudyonLeidenfrostdroplets,a semi-implicittemporalintegrationhasbeenusedforthetime advancement of these PDE in the longitudinal direction. Indeed, by using very refined grids close to a hot wall, if the interfacecrossesgridcellswithverydifferentsizes,themeshpointsclosetotheinterfaceintheveryrefinedzoneswillbe updatedmuchmorequicklythanthemeshpointlocatedinacoarseregion.Thisissuehasbeenfixedbyusinganimplicit temporaldiscretizationinthelongitudinal regiontospeedup theextrapolationsinthewhole domain.Thatsemi-implicit temporalintegrationrequirestosolveatri-diagonallinearsystemateachtemporaliterationofthePDEresolution.Itallowed ustoreduce thenumberoftemporal iterationsfromup to2000toabout50to achievea sufficientlysteadystate inthe interfaceneighborhoodinthecaseofthesimulationoftheLeidenfrostdroplet.

5.3. Interfaceadvectionandre-initializationwithaLevelSetMethod

TheLevelSetMethod[8,34,43]isusedtocomputetheinterfacemotion.Itconsistsofsolvingaconvectionequation for aLevelSetFunction

φ

:

∂φ

∂

t

+ 

Vint

· ∇φ =

0

.

(49)

Positivevaluesof

φ

representtheliquidfieldandthenegativevaluesthegasfield.V



int istheinterfacevelocity:



Vint

= 

Vliq

+

˙

m

ρ

liq



n

= 

Vgas

+

˙

m

ρ

gas



n

.

(50)

Next,are-initializationstep[43]canbeperformedtoensurethatthe

φ

functionremainsasigneddistanceinthe compu-tationaldomain.ItcanbedonebysolvingiterativelythespecificPDE

∂

d

∂

τ

=

sign

(φ)



1

− |∇

d

|



(51)

withd thereinitializeddistancefunction,

τ

afictitiousreinitializationtimeandsign

(φ)

asmoothedsignedfunctiondefined in[43].Twotemporaliterationsoftheredistancingequation,Eq.(51),arerequiredateverytimestep.

Throughthissigneddistancefunction,agoodaccuracyispreservedwhencomputingthenormalvectorandthecurvature attheinterface,withthefollowingsimplerelations:



n

=

∇φ

|∇φ|

,

(52)

κ

(φ)

= −∇ · 

n

.

(53)

5.4. Timestepconstraints

Aclassicaltime-stepconstraint[22,43]accountingforthestabilityconditionsforconvection,viscosity(inthecaseofan explicittemporaldiscretization)andsurfacetensionisimposed.



tconv

=



x max

 

V



,

(54)



tvisc

=

1 4



x2

max

(

υ

liq

,

υ

gaz

)

,

(55)



tsurf_tens

=

1 2



ρ

liq



x3

σ

.

(56)

Finally,theglobaltimesteprestrictioncanbecomputedwiththefollowingrelation: 1



t

=

1



tconv

+

1



tvisc

+

1



tsurf_tens

.

(57) 6. Numericalresults

We presentinthissection some numericalbenchmarksinorder toassessthe relevance andtheaccuracyof thenew method whichis proposedin thispaper.Firstly, inthe nextsection we willdescribe an analytical theoryof astaticand isolateddropletinaninfinitemedium.Thistheorywillprovideuswithanaccuratereferencesolutionthatcanbeusedto assessthedifferentnumericalsolversdescribedpreviously.

6.1. Theoryoftheevaporationofastaticdroplet

The evaporationof a static droplet in an infinite medium can be modeled, see [42] for more details, by solving the following setofequationsforthemassconservation,theenergyconservationandthe chemicalspeciesconservationina 1Dsphericalcoordinatesystem:

1 r2 d dr



r2

ρ

gur



=

0

,

(58) 1 r2 d dr



r2

ρ

gCpgurT



=

1 r2 d dr



kgr2 dT dr

,

(59) 1 r2 d dr



r2

ρ

gurY



=

1 r2 d dr



ρ

gDmr2 dY dr

.

(60)

As theevaporationspeed regressionisverylowfordroplets(unlikeboiling)incomparisontotheStefanflowwhichruns out inthenormaldirectionattheinterface, theunsteadytermscan beneglected besidetheconvective andthe diffusive derivatives.

Thefollowingboundaryconditions,whichcanbededucedfromthejumpconditionsEqs.(10),(14),(16)mustbeapplied onthedropletradiuslocalizedwithr

=

RΓ

(

t

)

:

Vr



r

=

RΓ

(

t

)



=

M

˙

(

t

)

4

π

R2 Γ

(

t

)

ρ

g

,

(61) 4

π

R2_Γ

(

t

)

kg dT dr





r=RΓ(t)

= ˙

M

(

t

)

Lvap

,

(62) 4

π

R2_Γ

(

t

)

ρ

gDm dY dr





r=RΓ(t)

= ˙

M

(

t

)(

YΓ

−

1

),

(63)

whereY

|

Γ isthemassfractionattheinterface, T

|

Γ isthetemperatureattheinterfaceandM

˙

(

t

)

isthetotalmassflowrate

ofevaporation.Afirstintegrationoftheseequationsleadstothefollowingrelations:

ur

(

r

,

t

)

=

˙

M

(

t

)

4

π

r2

_ρ

g

,

(64)

˙

M

(

t

)



Cpg



T

(

r

,

t

)

−

T

|

Γ



−

Lvap



=

4

π

r2kg dT dr

,

(65)

˙

M

(

t

)



Y

(

r

,

t

)

−

1



=

4

π

r2

ρ

gDm dY dr

.

(66)

Asecondintegrationgivesthetemperaturefieldandthemassfractionfield:



_T

₍

_r

_,

_t

₎

₋

_T

_|

Γ

−

L_Cvap_pg T_∞

−

T

|

Γ

−

LCvappg

=

e− M˙(t)C pg 4πkg ₁ r

_,

₍₆₇₎



Y

(

r

,

t

)

−

1 Y_∞

−

1

=

e−  M˙(t) 4πρg Dm 1 r

_,

₍₆₈₎

withthefollowingfar-fieldboundaryconditions:

T

(

r

→ ∞,

t

)

=

T_∞

,

Table 1

Physical properties (SI) of fluids considered in the simulation presented in section6.2and section6.3.

ρ M Cp K σ Tsat Lvap Dm M

Liquid acetone 700 3.26E−6 2000 100 0.005 329 5.18E5 5.2E−5 0.058

Gas mixing 1 1E−7 1000 0.052 – – – – 0.029

Nextbypointingoutthat

˙

M

(

t

)

= −

4

π

rkg Cpg ln



_T

₍

_r

_,

_t

₎

₋

_T

_|

Γ

−

L_Cvap_pg T_∞

−

T

|

Γ

−

LCvappg

= −

4

π

r

ρ

gDmln



Y

(

r

,

t

)

−

1 Y_∞

−

1

,

(69)

weobtainthefollowingequationforTΓ:

T

|

Γ

=

T∞

−

Lvap Cpg



1

−



Y_∞

−

1 Y

|

Γ

−

1

Le

,

(70)

withLe theLewisnumberdefinedby:

Le

=

Dm

ρ

gCpg

k

.

(71)

By using Eq. (39) and Eq. (40), we obtain two non-linear equations for the two unknowns T

|

Γ and Y

|

Γ, for which a

numericalsolutioncanbecomputedeasily.

Next,themassflowrateM

˙

(

t

)

ofthedropletcanbededucedfromEq.(69)andweobtainthefollowingrelationforthe temporalevolutionofthedropletdiameter:

DΓ

(

t

)

=



D2₀

−

4

ρ

gDm

ρ

l ln



Y_∞

−

1 Y

|

Γ

−

1

t

,

(72)

whichismostcommonlyusedasfollows:

DΓ

(

t

)

=



D2₀

−

4

ρ

gDm

ρ

l

ln

(

1

+

BM

)

t

,

(73)

withBM themasstransfercoefficientdefinedas:

BM

=

Y

|

Γ

−

Y∞ 1

−

Y

|

Γ

.

(74)

Letusremarkthattheglobalmassflowrateofevaporationcanbeexpressedasfollows:

˙

M

(

t

)

=

2

π

RΓ

(

t

)

ρ

gDmShBM

,

(75) whereSh istheSherwoodnumber,whichisanon-dimensionalmasstransferflux,definedbythefollowingexpressionin thecaseofastaticevaporatingdroplet:

Sh

=

2ln

(

1

+

BM

)

BM

.

(76)

As thistheoretical solutionassumesthat thetemperature isuniformin theliquidphase, theaverage temperatureofthe droplet isequalto theinterface temperature TΓ.Letusremarkthat thistheory provides a quasi-steadysolutionforthe

dropletevaporationsinceall unsteadytermshavebeenneglectedinthe initialsetofequations,howeverifthedropletis initialized withatemperature whichisdifferent from thesteadytemperature, the droplettemperaturewill initiallyvary (warmingorcooling)untilreachingthesteady-statetemperature.Thislastpointwillbeillustratedinsection6.3.

6.2. Numericalsimulationsoftheevaporationofastaticdroplet

Anumericaltestis proposedinthissection inordertocomparetheaccuracyandtherelevance ofthetwonumerical solvers forevaporationwhich havebeendescribed in thispaper.The theoretical solutionwhich hasbeen derived inthe previoussectionwillprovideareferencesolution.Asthecharacteristictimeofdropletsevaporationisveryslowcompared withthecharacteristictimeofcapillarywaves,alargerangeoftemporalscalesmustbesolvedtoperformthatkindof sim-ulation.Asaresult,thenumberoftemporaliterations, whicharerequiredtoconsiderasufficientdurationofevaporation, islarge(upto100 000iterationsevenifcoarsegridsareused).Toreducethesedrasticconstraints,somephysicalproperties ofthefluidhavebeenmodifiedinordertospeedupthenumericalsimulations.Thephysicalpropertiesoftheliquidphase andofthe vapor phase aresummarized inthe Table 1; thesepropertiesare closeto those ofthe acetonefortheliquid

Fig. 1. Snapshots

of the velocity field, of the interface location (black line), of the temperature field (a) or of the mass fraction field (b) on the computational

grid (256 ×512) with the GFTSBE.

Fig. 2. Comparison

between the GFTSE with linear extensions and theoretical results of the temporal evolution of the dimensionless droplet diameter (a)

and on the dimensionless average droplet temperature (b).

phase andto thoseofamixingofacetonevapor andairforthegasphase.Some valueshavebeenincreasedsuch asthe mass diffusioncoefficient,inorderto speedthe evaporationrate, andother propertieshavebeendecreased,e.g. the sur-facetensionandtheviscositycoefficientsinordertoincreasethetimesteprestrictiondependingonthesevariables.Since thenumericaltestproposedhereisbasedonatheoreticalsolutionwhichdoesnotinvolveanycapillaryeffectsorviscous effects,themodificationofthesevaluesdoesnot changetheexpectedreferencesolution.Thethermalconductivityofthe liquid hasbeen stronglyincreased toguarantee that thetemperature field willbe uniform insidethe droplet,consistent withthetheorydescribedintheprevioussection.Thesimulationsareperformedwithanaxisymmetriccoordinatesystem. The initial diameterofthedropletis equalto D0

=

100 μm.The dimensionsof thecomputational fieldare lr

=

2D0 and lz

=

4D0.Theboundaryconditionsforthetemperaturefieldandforthemassfractionfieldareimposedasthevaluesofthe theoreticalsolution.Asaresulttheseboundaryconditionsevolveslowlywithtimewhenthedropletdiameterisdecreasing. The temperatureandthe massfractionin thefar field (beyondthe computationaldomain) are T_∞

=

700 K and Y_∞

=

0 respectively. Theinitial conditionsonthe temperaturefield andonthe massfractionfieldare alsoimposed byusingthe theoretical solution.In particular,thedroplet temperatureisinitialized by usingthetheoretical steady-statetemperature. Forthevelocityfield,outflowboundaryconditionsareusedtolettheevaporatedgasoutofthecomputationaldomain.

In thisconfiguration, the theoretical steady-state temperature is equal to TΓ

=

294

.

94 K. Several computations have

been performedin orderto assessthe accuracyof theGFTSE andof thenewmethod, GFTSBE,with fourdifferent com-putationalgrids (32

×

64,64

×

128,128

×

256,256

×

512),whichcorrespond respectivelyto 8gridnodes/radius,16grid nodes/radius,32gridnodes/radiusand64gridnodes/radius.Moreover,theinfluenceoftheorderoftheextensionswhich are used toextrapolate thetemperaturefield andthe massfractionfield hasalso beentestedonthe overallaccuracy of thecomputation, sinceallthesimulationspresentedinthissection,havebeenperformedwitheitherlinearextensionsor withquadraticextensions.Thecomputationsareperformeduntilthedropletdiameterreaches0

.

8D0,whichcorrespondsto amassevaporationofabout50%oftheinitialdropletmass.

In Figs. 1(a) and 1(b), snapshots of the simulations have been displayed to visualize the velocity field, the interface positionandrespectivelythetemperaturefieldandthemassfractionfield.InFigs. 2(a)and 2(b)thetemporalevolutionof

Fig. 3. Comparison

between the GFTSE with quadratic extensions and theoretical results of the temporal evolution of the dimensionless droplet diameter (a)

and on the dimensionless average droplet temperature (b).

Fig. 4. Comparison

between the GFTSBE with linear extensions and theoretical results of the temporal evolution of the dimensionless droplet diameter (a)

and on the dimensionless average droplet temperature (b).

thenon-dimensionaldropletdiameterandofthenon-dimensionaldroplettemperatureisplottedfordifferentcomputational gridsifGFTSEisusedinconjunctionwithlinearextensionsonthetemperaturefield andonthemassfractionfield.These figures clearly indicate that the simulations converge to the theoretical solution. A first order accuracy can be roughly extrapolatedfromFig. 2(b),whereas Fig. 2(a)doesnot allowevaluating anaccuracyordersince theerrorisloweronthe coarsestgrid.Thisbehavioristypicalfor simulationswhereoppositeerrorscanbecompensatedandcanleadtomisleading conclusions if sufficient attentionis not paid to the analysisof the simulationresults. Indeed,that can be explained by notingthatthetemperaturedropletisstronglyovervaluedonthecoarsestgridwhichincreasesspuriouslytheevaporation massflowrate,whereasanundervaluationofthedropletdiameterspeedregressioncanresultfromother errors.However, thisnon-monotonicconvergencedisappearsifthinnestgridsareused.

InFigs.3(a)and3(b),thesamevariableshavebeenplottedwiththesamesolverbutwithquadraticextensionsinstead of linearextensions. Theseresults are very similar to the resultsobtained withlinearextensions, butwe can observe a slightgaininaccuracysince,onthethinnestgrid,theerrorontheaveragedroplettemperatureisabout0.1%(about0.3 K) insteadof0.2%(about0.6 K).Suchtrendshavealreadybeenreportedin[14]fortheStefanproblemorin[46] forboiling flows,whereasthegaininaccuracywasmuchmoreimportantinthelatterstudies.

Next,we haveplottedinFigs.4(a) and4(b),the numericalresultsobtainedwithGFTSBEandlinearextensions.These numericalresultsalsodemonstratethattheGFTSBEconvergestotheexpectedsolution.Theorderofaccuracyseemstobe approximatelythesameaswithGFTSE.GFTSBE providesresultswhichare slightlybetter ontheaveragetemperaturebut theerroronthetemporalevolutionofthediameterismoreimportant.Wecanremarkthatwiththissolverthetemperature isslightlyundervalued,unlikeGFTSEwhichoverestimatestheaveragedroplettemperature.

At last, inFigs. 5(a)and 5(b), thenumerical resultsobtainedwith GFTSBE andwithquadratic extensionsare plotted. Again inaccuracyontheaveragetemperatureisobservedincomparisonwithother solvers.Forexample,ifthethinnest gridisused,theerrorontemperatureislessthan0.1%,andtheerroronthetemporalevolutionofthedropletdiameteris

Fig. 5. Comparison

between the GFTSBE with quadratic extensions and theoretical results of the temporal evolution of the dimensionless droplet diameter (a)

and on the dimensionless average droplet temperature (b).

Fig. 6. Comparisons

between theory and numerical simulations of the radial evolution of the temperature and of the mass fraction in the gas phase with

the GFTSBE and with the GFTSE at a given time with the grid 128 ×256.

approximatelyinthesamerangeaswithGFTSE(lessthan5% atthe endofthesimulation). Inconclusionofthissection, wecanassertthatthiscombinationofnumericalmethods(GFTSBE

+

quadraticextension)isthemostaccurateofthefour whichhavebeentestedinthispaper.

InFig. 6,acomparison,betweenthetheoryandthenumericalsimulations,oftheradialevolutionofthetemperatureand ofthemassfraction,hasbeenplottedbothforGFTSBEandGFTSE.Thiscomparisonisconvincingsincethetheoreticalcurves andallthenumericalcurvesaresuperimposedintheneighborhoodoftheinterface.Letusnoticethataslightdiscrepancy appearsclosetotheboundarycondition.Thisslightdiscrepancycanbeexplainedbyconsideringthattheoutflowboundary conditions,whichareusedinthesimulationsdonotperfectlymaintain thesphericalsymmetryofthetheoretical velocity field.

We can concludefromthis firstbenchmark that both GFTSE from[45] andGFTSBE, which ispresented inthis work, convergetoasolutionwhichisveryclosetothetheoreticalsolutiondescribedintheprevioussection.A gaininaccuracy isobservedifGFTSBEandquadraticextensionsareused.However,wemustalsoremarkthatamoredetailedquantification oftheaccuracyseemsnottobepossibleinthesesimulations.Indeedthenumericalsimulationsconvergetowardasolution whichseemstobeslightlydifferentfromthetheoreticalone,ifthetemporalevolutionofthedropletdiameterisconsidered. Variousreasonscouldexplainthisslightdiscrepancybetweenthenumericalsimulationsandthetheory.Forinstance,unlike the numericalsimulations, theunsteadytermsareneglected inthe theory.Anothersourceoferrorscouldcomefromthe outflowboundaryconditionswhichdonot maintain perfectlythe sphericalsymmetryoftheproblem. Finally,the droplet temperatureisassumedtobeuniforminthetheory,whereasthesimulationsdonotperfectlymatchthisassumption.

6.3. Non-steadystaticdropletevaporation

In this section we presentsome numerical results which havebeen obtained withthe sameconfiguration as in the previoussection,exceptforthedropletinitialtemperaturewhichhasbeenchangedinordertocheckifafteratransientstep

Fig. 7. Temporal

evolution of the average dimensionless droplet temperature with the GFTSE (a) for two different initial temperatures and with the GFTSBE

(b) for three different initial temperatures.

thedroplettemperaturewilltendtowards thetheoreticalsteady-statetemperatureofevaporation.GFTSE hasbeentested withtwodifferentinitial temperatures,one whichishigher(T

=

304

.

94 K)than thetheoretical steady-statetemperature andanotherwhichislower(T

=

284

.

94 K).Thetwocomputationshavebeenperformedbyusingquadraticextrapolations, since they provide more accurate results, andwith two computational grids (64

×

128 and 128

×

256). The results are plotted inFig. 7(a) and show clearly that all the computations tend to the correct theoretical steady-state temperature. Moreover,we canobserveinthisfigurethegoodagreementbetweenthesimulations performedonthetwo grids,which furtherdemonstratesthespatialconvergenceoftheoverallsolver.

GFTSBEhasbeentestedinthesameconditionsasGFTSE,butwiththreedifferentinitialtemperatures,thesametwoas forGFTSEandathirdtemperaturewhichishigher(T

=

339 K)thanthesaturationtemperature(T

=

329 K).Theobjective ofthislastconfigurationistoshowthestabilityofGFTSBEeveninthecasewheresuperheatedliquidsareinvolved,unlike GFTSE which is unstable in thissituation due to Eq.(42). The numerical results, summarizedin Fig. 7(b), highlightthe correctbehavior of thisnumericalsolver, even ifa superheated liquidis considered.This second benchmark proves that GFTSBEisabletodealwithbothliquidevaporationandboilingwhereasGFTSEfailstoaddressthisspecificsituation.

Moredetailedinvestigationsonthespatialconvergenceareproposedinanappendixwiththreedifferentcomputational grids32

×

64,64

×

128,128

×

256 foralloftheseconfigurations.

6.4. Movingdropletevaporation

In this section, we are now interested by the evaporation of a moving spherical droplet. The simulations are still performedwithan axisymmetriccoordinatesystem. The initialdiameterofthe dropletisequalto D0

=

100 μm.The di-mensionsofthecomputationalfieldarelr

=

4D0 andlz

=

8D0inordertominimizecontainmenteffects.Weuseauniform gridinside thedropandinthegas phaseup toadistance D0

/

2 fromthe interface.Beyondthisdistance,anon-uniform gridisusedtominimizethenumberofgridcells.Inthisway,themeshissufficientlyrefinedwithrespectively16and32 gridcellstodiscretizeadropletradiusforthetwocomputationalgridswhichcontain respectively64

×

128 and128

×

256 points.As thedropletwilltravel animportantdistancebeforea significantevaporationoccurs,thecomputationsare per-formed witha moving frame inorder toensure that the dropletis maintained atthe centerofthe computational field. Aconstantrelativevelocity Vz

=

2 m s−1 isimposedbetweenthedropletandthegasphasethroughoutthecomputation. The gravity is neglected, andthe computation is initialized withthe static theoretical solution for the temperatureand themassfractionfields.WhensolvingtheNavier–Stokesequations,weuseaninflowboundarycondition atthebottomof the domainto blowthe droplet withthecorrectvelocity, slip boundaryconditionsare imposed on therightside ofthe computationalfieldandoutflowboundaryconditionsareusedatthetopofthedomain.Wesolvetheconservationofthe energy andofthe species equationswith a Dirichletboundary condition at thebottom of the domain T_∞

=

700 K and

Y_∞

=

0,andfortheseequationsaNeumannboundarycondition canbe imposedonthe rightside andonthetopofthe domainsincethecomputationalfieldislargeenoughtoavoidcontainmenteffectsinthefar-field.Inthisconfiguration,the value of theReynolds number isaround 20, andthe Weber numberis sufficiently smallto ensure that thedroplet will remainspherical.Therefore, insucha configurationsomecorrelationscan beused tobuild areferencesolutiontoassess ournumericalmethods withthismorecomplex benchmark.Forexample,thefollowingcorrelation hasbeenproposed in

[36]toincludetheReynoldsnumber,Re,tocorrecttheSherwoodnumbercomparedtothestatictheory

Sh

(

1

+

BM

)

0.7

=

Table 2

Physical properties (SI) of fluids considered in the simulation presented in section6.4.

ρ M Cp K σ Tsat Lvap Dm M

Liquid acetone 700 3.26E−4 2000 0.161 0.0237 329 5.18E5 5.2E−5 0.058

Gas mixing 1 1E−5 1000 0.052 – – – – 0.029

Fig. 8. Temperature

field, streamlines and interface position (thick black line) around a moving droplet which evaporates in a hot gas. Comparison between

the GFTSE (a) and the GFTSBE (b).

wheretheSherwoodnumberSh takesintoaccount themodificationofthemasstransferfluxduetotheconvectionandits couplingwithevaporationandSc istheSchmidtnumberdefinedby

Sc

=

μ

g

ρ

gDm

.

(78)

ThismodifiedexpressionoftheSherwoodnumbercannextbeusedtocomputeamodifiedmassflowrate,M

˙

(

t

)

,depending on Reynolds numberand Schmidt numberin Eq. (75). The correlation, Eq. (77), is based on boundary fitted numerical simulations which guarantee ahigh accuracybutwhich are limitedto a single sphericalor steadyshape droplet,unlike the numericalmethods presentedinthispaperwhich canbe usedwithan interface ofmoving andarbitraryshape.The numericalsimulations performedin[36]involveavolatileliquid(n-heptane)inahot gas(800 K) fora rangeofReynolds numbervaryingbetween100and13.4.

OurnumericalsimulationshavebeencarriedoutwiththephysicalpropertieswhicharegivenintheTable 2,whichare nearlyidenticaltotheTable 1.Amorerealisticvaluethanintheprevioussectionisusedhereforthethermalconductivity, sincewedonotcompareournumericalsimulationagainstatheoreticalstudywhichassumesthatthedroplettemperature is perfectlyuniform. The surface tensionandthe viscosity havebeen increasedinorder to obtainrealistic valuesofthe Reynolds numberandoftheWeber numbersince thisbenchmarkallows thecomputation oftheinfluence ofconvection on the overall process of evaporation. As a result, the number oftemporal iterations required to compute a significant evaporationofthedropletwillbemoreimportantthaninthestaticcase.Therefore,thesecomputationsaremoreexpensive thanthosepresentedintheprevioussection.

Comparisonsbetweenthecorrelation,Eq.(77),anddirectnumericalsimulationswithGFTSEandGFTSBE(withquadratic extensionsinbothcases)arenowpresented.

The temperature field, the streamlines and the interface location have been plotted at a given time in Fig. 8(a) for GFTSE andinFig. 8(b) forGFTSBEto demonstratethequalitative agreementbetweenthesetwo methods.Further results are presented inFig. 9(a)and inFig. 9(b)to show theagreement betweenthe two methods forthemass fractionfield. Thesedepictthestreamlines,insidethedropletandoutsideofthedroplet,inordertobringoutthecouplingbetweenthe externalflowandtheradialvelocityfieldduetoevaporation(Stefanflow).Finally,inFig. 10,theinternaltemperaturefield ofthe dropletisplottedtoillustratethat itisslightlymodified bytheinternal recirculatingzonesince acolder regionis observedinthecenteroftheaxisymmetricvortex.

In Fig. 11(a)the temporalevolutionof thenormalizeddiameterisplotteduntil thediameterdecreases byabout15%. Thisfigureshowsthatthetwonumericalmethodsconvergetowardsasolutionwhichisveryclosetotheresultsobtained byusingthecorrelationinEq.(77).Theagreementbetweenthenumericalsimulationandthecorrelationisgood,especially ifGFTSBEisused.ThetemporalevolutionsoftheaveragedroplettemperatureareplottedinFig. 11(b).Thisfigurefurther demonstratestheaccuratebehaviorandtheconvergenceofallnumericalmethodssincethedeviationsfromthetheoretical steadytemperatureareverysmall(lessthan0.5%)regardlessof thesolverandthecomputationalgridused.

Fig. 9. Mass

fraction field, streamlines and interface position (thick black line) around a moving droplet which evaporates in a hot gas. Comparison between

the GFTSE (a) and the GFTSBE (b).

Fig. 10. Zoom

on the temperature field in the liquid phase with the streamlines and the interface position (thick black line) around a moving droplet which

evaporates in a hot gas with the GFTSE.

Fig. 11. Comparison

between the GFTSBE, the GFTSE (both with quadratic extensions) and a semi-empirical correlation of the temporal evolution of the

dimensionless droplet diameter (a) and of the dimensionless average droplet temperature (b).

Table 3

Physical properties (SI) of fluids considered in the simulation presented in section6.5.

ρ M Cp k σ Tsat Lvap Dm M

Liquid water 1000 0.00113 4180 0.6 0.07 373 2.3E6 2E−5 0.018

Gas mixing 1.226 3E−5 1000 0.046 – – – – 0.029

Fig. 12. Strongly refined computational grid used in the simulation of the Leidenfrost droplet.

Duetothisthirdbenchmark,therelevanceofthetwonumericalsolverstocomputetheevaporationofamovingdroplet has beendemonstrated.As it isthe caseofthe firstbenchmark, a gain inaccuracy hasalsobeen observedifGFTSBE is used.

6.5. Levitationandbouncingofanimpactingdropletonahotplate(Leidenfrosteffect)

Wepresentherethenumericalsimulationofadropletimpactingahotplate.IntheLeidenfrostregime,adropletnever touchestheplatebecausethehighmassflowrateofvaporizationleadstotheformationofaverythinlayerofsaturated vapor. The pressureforce resultingfrom theflow in thisthinlayer is sufficiently highto ensure thesustentation of the droplet above the hotplate. Several difficultiesmake the numericalsimulationof thissingular phenomenon challenging. A majorissuetobeovercomeconsistsindesigningaspecificphasechangemodel,whichissuitedbothforevaporationand boiling,sincethetwophenomenaoccursimultaneously.Anotherdifficulty,thatmustbeaddressed,resultsfromtheuseof a locallyrefinedgridto capturetheformation ofthe thinvapor layerbetweentheplateandthedropduring theimpact. Therefore, animplicittemporal discretizationmustbe appliedto computeall thediffusionterms (viscousterms,thermal conductionandmassfractiondiffusion).Inthesameway,theresolutionofthehyperbolicpartialdifferentialequations[1,14, 46],whichresultsfromextrapolationtechniques,mustalsobeendiscretizedwithanimplicitscheme.Fortheseequations, this implicit temporal discretization is useful only in the refined direction of the mesh. Thus, as it has been explained previously,that canbe done bysolving atri-diagonallinearsystemforeverytime stepoftheextrapolationprocess. This implicitdiscretizationallowstheuseoflargetime stepsduringtheextrapolation.Itensures thattheextrapolationwillbe advanced onanumberofgridpointssufficientlyimportantboth intheregion wherethegridiscoarseandintheregion wherethegridisveryrefined.

Inthesimulationwhichispresentedhere,theinitialdropletdiameterisD0

=

130 μm,theinitialvelocityisV0

=

2 m s−1 whichcorrespondstoaWebernumberWe

=

ρlV

2 0D0

σ

=

7

.

4.TheinitialtemperatureofthedropletisT0

=

290 K andthehot plateisassumedtobe isothermalTwall

=

800 K.ThefollowingNeumannboundarycondition,

∇

Y1

|

wall

=

0,isimposedfor the massfraction on the hot wall in order to ensure that the chemical speciesdo not diffuseinside the solid domain. The physical propertiesofthe liquidphase andofthegas phase aresummarized inTable 3.Letusnotice that,forsake ofsimplicity,the physicalvariablesofthemodelare constantandthereforedonotdepend onthelocaltemperature, nor