Effect of pressure on Hydrogen/Oxygen coupled flame-wall interaction

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Effect of pressure on Hydrogen/Oxygen coupled

flame-wall interaction

Raphaël Mari, Bénédicte Cuenot, Jean-Philippe Rocchi, Laurent Selle, Florent

Duchaine

To cite this version:

Raphaël Mari, Bénédicte Cuenot, Jean-Philippe Rocchi, Laurent Selle, Florent Duchaine. Effect of

pressure on Hydrogen/Oxygen coupled flame-wall interaction. Combustion and Flame, Elsevier, 2016,

168, pp.409-419. �10.1016/j.combustflame.2016.01.004�. �hal-01320335�

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Eprints ID : 15676

To link to this article :

DOI:10.1016/j.combustflame.2016.01.004

URL :

http://dx.doi.org/10.1016/j.combustflame.2016.01.004

To cite this version :

Mari, Raphaël and Cuenot, Bénédicte and Rocchi, Jean-Philippe and

Selle, Laurent and Duchaine, Florent Effect of pressure on

Hydrogen/Oxygen coupled flame-wall interaction. (2016)

Combustion and Flame, vol. 168. pp. 409-419. ISSN 0010-2180

Any correspondence concerning this service should be sent to the repository

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Effect

of

pressure

on

hydrogen/oxygen

coupled

flame–wall

interaction

Raphael

Mari

a ,_∗

_,

_Benedicte

_Cuenot

a

_,

_{Jean-Philippe}

_Rocchi

a

_,

_Laurent

_Selle

b

_,

Florent

Duchaine

a

a CERFACS, 42 av G. Coriolis, 31057 Toulouse, France b CNRS, IMFT, F-31400 Toulouse, France

Keywords:

Real-gas thermodynamics Flame–wall interaction Conjugate heat transfer

a

b

s

t

r

a

c

t

Thedesign and optimizationofliquid-fuelrocketenginesis amajorscientific and technological chal-lenge.Oneparticularly criticalissueisthe heatingofsolidpartsthat aresubjectedtoextremely high heatfluxes whenexposed tothe flame. Thisinturn changesthe injector liptemperature, leading to possiblydifferentflamebehaviorsandafullycoupledsystem.Asthechamberpressureisusuallymuch largerthanthecriticalpressureofthemixture,supercriticalflowbehaviorsaddevenmorecomplexity tothethermalproblem.Whensimulatingsuchphenomena,thesethermodynamicconditionsraiseboth modelingandnumericalspecificissues.Inthispaper,bothsubcriticalandsupercriticalhydrogen/oxygen one-dimensional,laminarflamesinteractingwithsolidwallsarestudiedbyuseofconjugateheattransfer simulations,allowingtoevaluatethewallheatfluxand temperature,theirimpactontheflameaswell astheirsensitivitytohighpressureandrealgasthermodynamicsupto100barwhererealgaseffects areimportant.Atlow pressure,resultsare foundingoodagreementwithpreviousstudiesintermsof wallheatfluxand quenchingdistance,and thewallstaysclosetoisothermal.Onthecontrary,dueto importantchangesofthefluidtransportpropertiesandthe flamecharacteristics,the wallexperiences significantheatingathighpressureconditionandtheflamebehaviorismodified.

1. Introduction

Mostofhighperformancepropulsion devicessuch asturbines,

rocket engines or scramjets operate in wall-bounded flows. The

interaction betweenflame andwalls hasa direct andstrong

im-pact on combustion, pollutant emissions and combustion

cham-berlifetime. Understandingthemechanismsatplayinflame–wall

interaction (FWI) is therefore necessary to further gain in

per-formance, safety,fuel consumption andunburnt gasemission. As

shown in [1,2] , local FWI may be described in simple laminar

flowswheregenericflameconfigurationsmaybeintroduced.

Dur-ingtheflame–wallinteractionprocess,theflamespeedand

thick-ness decrease,beforefull quenchingata few microns awayfrom

the wall. When the flame approaches the wall, the temperature

decreases fromburnt gases (approximately 3000 Kfor hydrogen

(H2)/oxygen(O2)flamesat1bar)towalllevelsthataremaintained

in the 300–800 K range to avoid damaging. This high

tempera-turevariation occursina very thinlayer,lessthan 1mm , leading

toverystrongtemperaturegradientsandmakingexperimental

ob-servationofFWIquitedifficult.

∗ _{Corresponding author.}

E-mail address: [email protected] (R. Mari).

Ezekoye et al. [3] experimentally studied the impact of wall

temperature and the equivalence ratio on the wall heat flux for

propane and methane flames. It was shown that the maximum

wall heatflux decreaseswhen thewalltemperatureincreases.Lu

et al. [4] investigated FWI in the side wall quenching

configura-tion where the flame propagates along the wall and found that

the ratioofthewall heat fluxto theheat releaseintheflame is

roughlyconstantandequalto0.3–0.4.Basedonexperimental

cor-relations, Boustetal.[5] proposed a theoreticalrelation between

the normalized wall heat flux andthe quenching Pecletnumber,

defined astheflame positionnormalized by theflame thickness,

formethane-airflameswheretheyobservethatthewallheatflux

isinverselyproportionaltotheflamequenchingdistance.

Many numerical studies have been conducted on laminar

flame–wall interactions [6–10] . It has been shown by Popp

et al. [9] that in the low wall temperature regime (300 K <

T w < 400 K) the wall can be assumed chemically inert. Kim

et al.[11] experimentallyconfirmed thisresultusingseveral

sur-face materials andwall temperatures. Dabireauet al.[7] , Gruber

et al.[12] andOwstonetal.[10] ,demonstrateda strongly

differ-entbehaviorforhydrogenflamescomparedtohydrocarbonflames.

Hydrogen flame quenching occurs much closer to the wall

rela-tively to the flame thickness. Normalized wall heat flux is also

largelydifferentfromhydrocarbonflamesandequalto_∼0.12.

Nomenclature

Symbol Units Description

Cp J K −1 kg −1 Heat capacity

Dth m 2 s −1 Thermal diffusivity

e J m −2 K −1 s −1/2 _Effusivity Pc Pa Critical pressure

Pe – Peclet number based on heat release rate

PeF _{– Peclet number based on fuel consumption rate}

Q W m −3 _{Heat release rate}

Q∗ _{– Heat release rate non-dimensionalized with Q} 0

l /δ

Q0

l W m −2 Laminar flame power

S0

l m s −1 Flame speed

Sc – Schmidt number

Tc K Critical temperature

TS K Solid temperature

Tw K Fluid-solid interface temperature

Z – Compressiblity factor

Greek symbols

δl m Thermal flame thickness

δ m Diffusive flame thickness

1H J kg −1 _{Heat per kg of fuel}

κ – Ratio of wall and fluid effusivities

λ W m −1 K −1 _{Thermal conductivity}

8w W m −2 Heat flux 8∗

w – Heat flux non-dimensionalized with Q l0

τ s Flame characteristic time

Superscripts

IF F _{Infinitely Fast Flame} C _Coupled U _Uncoupled b _burnt u _unburnt Subscripts Q quenching w wall

Inallthesestudies,resultshavebeenprovidedforwall

bound-ary conditions either adiabatic or isothermal. However in reality

heattransferoccurringbetweenthesolidwallandthefluidresults

inapossibleincreaseofwalltemperatureandanon-zeroheatflux,

i.e. neitherisothermalnoradiabaticwallbehavior.Inaddition,the

wall temperatureisusually unknownandintroduces asignificant

uncertainty onthe predictedheatflux. Finally,FWI beinga

tran-sientphenomenon,eventuallyleading toflamequenching,the

so-lution cannotbe searchedforasa steadystatesolution and

sim-ulations describing the unsteady coupling of heat conduction in

the wall withfluid dynamicsandheat transfer arerequired [13] .

Such approachavoids toimposethe walltemperatureatan

arbi-traryvalue,andallowsittoadapttothevaryingfluidtemperature,

consequentlysignificantlymodifyingthewallheatflux.

Toaddressthisissue,thepresentstudyconsiderstheunsteady

behavior of a stoichiometric laminar one-dimensional premixed

hydrogen–oxygenflameimpingingonacoldwallincluding

conju-gateheattransfer.Thecontextisliquid-fuelrocketengines(LREs),

which operateat verylow temperature andhighpressurewhere

thethermodynamicpropertiesdepartfromidealgaslaws.Indeed,

beyondthe critical point ,definedby(P c , T c )valuesspecifictoeach

species, surface tension disappears and the distinction between

gaseous andliquid phasesvanishes.This stateof matteris called

supercritical, wherephasechangeisreplaced byasteep but

con-tinuous variation of the density and thermodynamic properties.

Therefore the objective of the study istwofold: first, the role of

conjugate heat transfer inFWI is studied; second, theimpact on

FWIofhighpressure,uptosupercriticalconditions,isevaluated.

As shown in Fig. 1 , the chosen configuration corresponds to

head-on quenching (HOQ), where the flame propagates towards

the wall with the characteristics of a free flame before

interact-ingwiththewall.Inthissimplifiedconfigurationin-depthanalysis

canbemadeandagoodunderstandingofbasicphenomenacanbe

achieved. TheHOQconfigurationappearsasanecessaryfirststep

Fig. 1. Flame–wall interaction (FWI): head-on quenching (HOQ) configuration. Ini- tial wall temperature T i

w is set equal to fresh gas temperature T u .

tostudyboth effectsofhighpressureandconjugateheat transfer

priortothethermalstudyofrealisticconfigurations.

2. Numericalsetupandmethodology

Simulationswereperformedbyrunningsimultaneouslyafluid

(AVBP) and a thermal (AVTP) solver in a coupled framework.

Forcomparisonpurposes,uncoupled simulationswithan

isother-mal wall were also computed. In AVBP, the compressible

re-active Navier–Stokes equations are solved with a third order

in space and fourth order in time, two-steps Taylor–Galerkin

scheme [14,15] along with a second order Galerkin scheme for

diffusion terms.The parallel conductionsolver AVTP is based on

thesame datastructure thanAVBP andalsouses a second order

Galerkindiffusion scheme. Timeintegration is done with an

im-plicitfirst orderforward Eulerscheme.The resolutionofthe

im-plicitsystemisdonewithaparallelmatrixfreeconjugategradient

method.

Thecouplingmethodologyconsistsinan exchangeofvariables

at the wall surface between both codes: the fluid solver sends

a heat flux and the heat conduction code sends back a

temper-ature. Data are exchanged through a supervisor using OpenPalm

libraries [16] . Between two coupling events, the flow and wall

thermalconductionare advancedintime by aquantity

α

f

τ

f and

α

w

τ

w respectively, where

τ

f and

τ

w are the flow andheat

con-duction characteristictimes.To respect simultaneity,the physical

timecomputedbythecodesmustbethesamebetweentwodata

exchanges:

α

_f

τ

_f =

α

w

τ

w .Thisensures continuityoftheheatflux

andtemperatureatthewallsurface.Moredetailsandvalidationof

thecouplingmethodologycanbefoundin[17] .

Computationswererun onuniformgrids,ofsimilargrid

spac-ing(seeTable 4 ),inboththefluidandthesolid.Theywere

initial-ized witha free stationarypremixed flame previously calculated

underthesamethermodynamic conditions(pressure and

temper-ature),andlocatedfarenoughfromthewalltoassumeno

interac-tionatthestartofthesimulation.Forthesamereasontheinitial

walltemperature T _w i wastakenequaltothefreshgastemperature

T u . The fluid boundary condition at the open endis a

pressure-imposedoutlet,usingthecharacteristicformulationfor

compress-ibleflow[18] .Thetemperatureisimposedattheleftsolid bound-arytotheinitialwalltemperature T _w i .Thesolidissufficientlylong

toensurethatthisboundaryconditiondoesnotinfluenceFWI.

3. Chemicalkinetics

Computationswerecarriedoutwitha purehydrogen(H2)and

pureoxygen(O2)mixtureatstoichiometry.Thecombustionof

hy-drogen and oxygen is modeled using a skeletal mechanism

ac-countingfor8speciesand12reactionsfromBoivinetal.[19] ,

re-portedinTable 1 .Itisderivedfromthe21-stepSanDiegodetailed

mechanism[20] ,usedinmanyhydrogencombustionapplications.

Fig. 2. 1D flame profiles of (a) temperature T , (b) heat release rate Q , (c) HO 2 and (d) H mass fractions for (–) San Diego [20] and (- -) Boivin [19] mechanisms. Case 2a:

pressure is 1 bar and fresh gas temperature 300 K.

Table 1

Rate coefficients in Arrhenius form k = AT n exp ₍

−E/R 0 T ) as in [19] . Reaction Aa _{n E}a R1 H + O 2 ⇌ OH + O kf 3.52 10 16 −0 .7 71 .42 kb 7.04 10 13 −0 .26 0 .60 R2 H 2 + O ⇌ OH + H kf 5.06 10 4 2 .67 26 .32 kb 3.03 10 4 2 .63 20 .23 R3 H 2 + OH ⇌ H 2 O + H kf 1.17 10 9 1 .3 15 .21 kb 1.28 10 10 1 .19 78 .25 R4 H + O 2 + M → HO 2 + M b k0 5.75 10 19 −1 .4 0 .0 k∞ 4.65 10 12 0 .44 0 .0 R5 HO 2 + H → 2 OH 7.08 10 13 0 .0 1 .23 R6 HO 2 + OH ⇌ H 2 + O 2 kf 1.66 10 13 0 .0 3 .44 kb 2.69 10 12 0 .36 231 .86 R7 HO 2 + OH → H 2 O + O 2 2.89 10 13 0 .0 −2 .08 R8 H + OH + M ⇌ H 2 O + M c kf 4.00 10 22 −2 .0 0 .0 kb 1.03 10 23 −1 .75 496 .14 R9 2 H + M ⇌ H 2 + M c kf 1.30 10 18 −1 .0 0 .0 kb 3.04 10 17 −0 .65 433 .09 R10 2 HO 2 → H 2 O 2 + O 2 3.02 10 12 0 .0 5 .8 R11 HO 2 + H 2 → H 2 O 2 + H 1.62 10 11 0 .61 100 .14 R12 H 2 O 2 + M → 2 OH + M d k0 8.15 10 23 −1 .9 207 .62 k∞ 2.62 10 19 −1 .39 214 .74 a Units are mol, s, cm 3 , kJ and K.

b Chaperon efficiencies H

2 : 2.5, H 2 O: 16.0, 1.0 for all other species. Troe falloff

with F c = 0 . 5 .

c Chaperon efficiencies H

2 : 2.5, H 2 O: 12.0, 1.0 for all other species. d Chaperon efficiencies H

2 : 2.5, H 2 O: 6.0, 1.0 for all other species. Troe falloff

with F c = 0 . 265 exp (−T / 94) + 0 . 735 exp (−T / 1756) + exp (−T / 5182) .

predictpremixedflamespeed,autoignitiondelay,burntgases

tem-peratureandextinctionlimitsundermanyconditionsofpressure,

temperatureandcomposition[21] andisconsideredasareference.

InordertovalidateBoivin’sschemeinthethermodynamic

condi-tionsofinterest,i.e.,freshgasat150K,300Kand750Kand

pres-sure up to 100 bar, premixed flames have been computedusing

CANTERA[22] andcomparedwiththedetailedmechanism.Results

are shown here forflames corresponding to Cases 2a and 2c of

Table 3 .Thestoichiometriclaminarflamespeedsobtainedwiththe Boivin scheme(10.76ms−1 _and9.03ms−1_{, respectively)are very}

close to the values computed with the reference scheme of San

Diego (10.61ms−1 _and9.46ms−1_{, respectively). Theflame}

struc-tures shownin Figs. 2 and 3 demonstratethat both mechanisms

are inverygoodagreementintermsoftemperature,heat release

rate, andspecies(includingradicals)massfractionprofiles,atlow

andhighpressure.Inparticular,thegoodpredictionofspecieslike

HO2 is criticalforFWI,aswillbe seenlater. Similar resultswere

obtainedfortheothercasesconditions.

4. Real-gasequations

For high pressure computations, real-gas thermodynamics

are accounted for through the Peng–Robinson equation of

state [23] (PR-EOS).Thegeneralformofacubicequation ofstate

isgivenby:

P

(

v

_,T

₎

₌ RT

v

_{− b}−

a

(

T

)

(

v

₊

δ

₁b

₎₍

v

₊

δ

₂b

₎

(1)

where P is the pressure, T the temperature, v the molar

vol-ume and R theperfect-gasconstant. The coefficients a and b

ac-count respectivelyforlong-rangeandshort-rangeinteractions

be-tween molecules.In the Peng–Robinson equation the parameters

(

δ

1,

δ

2)are

(

1+√2,1−√2

)

.Allthermodynamiccoefficientsmust

bemodifiedtotakeintoaccountrealgaseffects.Atlowpressure,a

standard technique consistsintabulatingorusingpolynomial fits

Fig. 3. 1D flame profiles of (a) temperature T , (b) heat release rate Q , (c) HO 2 and (d) H mass fractions for (–) San Diego [20] and (- -) Boivin [19] mechanisms. Case 2c:

pressure is 100 bar and fresh gas temperature 300 K.

Table 2

Species critical-point temperature T c and pressure P c , and Schmidt numbers.

Parameters H 2 O 2 H 2 O O H OH H 2 O 2 HO 2 Tc [K] 33 154 .6 647 .1 105 .3 190 .8 105 .3 141 .3 141 .3

Pc [bar] 12 .6 49 .7 217 .7 70 .0 306 .0 70 .0 47 .3 47 .3

Sc 0 .28 0 .99 0 .77 0 .64 0 .17 0 .65 0 .65 0 .65

extended toaccountforpressuredependencebykeepingthe

tab-ulation forlow pressurereferencevaluesandusedeparture

func-tions basedontheEOStocomputetheinfluenceofpressure[24] .

For example to calculate the constant-pressure heat capacity C p ,

onestartstowritetheGibbsfunctionGas:

G

(

P,T

)

=G0+P

v

_{− RT+}

Z v₀ v

P

(

v

¯,T

)

d

v

¯ (2)

where v 0 and G 0arerespectivelythemolarvolumeandtheGibbs

energyatareference low pressure.Theenthalpy h isthen

classi-callydefinedas:

h=G− T

µ ∂

_∂

G_T

¶

P

(3)

aswellastheconstant-pressureheatcapacity:

Cp =

µ ∂

_∂

_Th

¶

P

(4)

Thispointsoutthatlow-pressuredata,combinedwiththePR-EOS,

allowtocomputeallthermodynamicpropertiesofthefluidathigh

pressure.

The viscosity and thermal conductivity are modeled

follow-ing the method of Chung [25] , based on the theory of

corre-sponding states, linking low- and high-pressure values through

semi-empirical functions expressedin reduced variables T /T c and

P /P c .The low-pressure (ideal gas) reference values are computed

from the Chapman–Enskog equation. Species diffusion velocities

are expressed as functions of the species gradients using the

Hirschfelder–Curtisapproximation andconstant Schmidtnumbers

S _c .Itwasindeedverifiedwithdetailedcalculationsusingthe

soft-ware CANTERA [22] , that in the considered cases the Schmidt

numbersofmostspeciesdonot stronglyvarythrough theflame,

andtake thevaluesreportedinTable 2 .Soret andDufoureffects

arenotincluded.

The criticalpoint coordinates of theintermediate speciesOH,

O, H, H2O2, HO2 (for which no experimental values are

avail-able)are estimatedasin[26] ,usingtheLennard–Jones

potential-welldepth,andthemolecular diameter,takenfromthetransport

databaseoftheSanDiegomechanism[20] .

The ability of the AVBP solver to accurately reproduce

super-criticalandtranscriticalflows andflames hasbeen demonstrated

invarious configurationscorresponding toLRE conditions[27,28] .

Notethatreal-gasthermodynamicsalsohaveanimpactonthe

for-mulationofboundaryconditionsandJacobianmatricesofthe

nu-mericalschemes.

5. Flamewallinteraction(FWI)

Flame–wallinteractionisfirstcharacterizedwiththewallheat

8

w =

λ

w

∂

_∂

T_x

¯

¯

¯

¯

_w (5)

where

λ

w isthethermalconductivityofthefluidevaluatedatthe

wall.Thewallheatfluxisstronlgylinkedtotheflame

characteris-tics:thethermalflamethickness

δ

l iscalculatedfromthe

temper-aturegradient:

δ

_l = T

b _{− T}u

(

_∇

T

)

max (6)

where(

_∇

T )max isthemaximumofthetemperaturegradient.This

flamethicknessmaybealsoestimatedfromtheflameparameters

usingthediffusiveflamethickness

δ

[7] givenby:

δ

=

λ

u

ρ

u _Cu p Sl 0

(7) where S _l 0isthelaminarflamespeed.Thelaminarflamepower Q _l 0

isdefinedas:

Q_l 0=

ρ

u Y_F u S0_l

1

H (8)

where Y _F u is the fuel mass fraction in unburnt gases and

1

H[Jkg−1_{]theheatproducedperkilogramoffuelconsumed.}

The wallheatflux isnon-dimensionalizedby theflamepower

as

8

∗

w =

8

w /Q 0

l ,whereasthenon-dimensionalflameheatrelease

rateis Q ∗_=Q

_δ

_/Q 0

l .Inaddition,the flamecharacteristictime

τ

=

δ

/S 0

l is used to non-dimensionalize the time as t ∗=t/

τ

, while

spacedimensionsarenon-dimensionalizedby theflamethickness

as x ∗_{= x/}

_δ

_.

Becauseofcomplexchemistry,thedefinitionoftheflame

posi-tionisnotunique.Itcanbeeitherlocatedatthemaximumofheat

releaserate Q max (x Q max) oratthemaximumoffuel consumption

rate

ω

˙F,max (x ω˙F,max).Bothlocationsaredifferentandmaybeused

todefinePecletnumberswhichcharacterizetheratiobetween

dif-fusionandconvectivecharacteristictimes:

• theheatreleasePecletnumberis

Pe=xQ max

δ

(9)

• thefuelPecletnumberis

PeF =xω˙F,max

δ

(10)

Assumingthat noreactionoccursatthewall,thetemperature

difference T b − Tu divided by the flamequenching distancegives

anestimateofthewalltemperaturegradient.Asshownin[5] ,this

leads to a simplerelationship betweenthe non-dimensional wall

heat flux andthePecletnumber(either fromthe heatreleaseor

fuelconsumption)

8

∗

w ∼ 1/Pe or,takingintoaccountthewallheat loss:

8

∗

w ∼ 1/

(

1+Pe

)

(11)

Theoretical model: the infinitely fast flame model

Theroleandimportanceofthecouplingbetweenthesolidand

thefluidthermalproblemsmaybeunderstoodfromthelimitcase

ofinfinitelyfastflame[17] (IFF),inwhichthecharacteristicflame

time scaleisnegligiblecomparedtothesolidconductiontime.In

thiscasetheconfigurationreducestothesimplerproblemoftwo

semi-infinite domains having different temperatures and a

com-mon contact surface. Solving this classical heat transfer problem

leadstothefollowingexpressionfortheinterfacetemperature:

T_w IF F =ew _eTw +ef Tf

w +ef (12)

where T w (T _f )isthesolid(resp.fluid)temperature,and e w (e _f )the solid(resp.fluid)effusivitydefinedby

e=

p

λρ

Cp (13)

Table 3

Summary of test cases: fresh gases properties at stoichiometry and compressibility factor calculated using NIST software REFPROP [30] . Case Tu _{Pressure ρ}u _{Compressibility factor}

[K] [bar] [kg m −3 ] [Dimensionless] 1 750 1 0.1931 1.0 0 0 2a 1 0.4824 1.0 0 0 2b 300 10 4.8476 0.995 2c 100 48.342 0.998 3 150 100 108.75 0.887

where

λ

isthe heat conductivity,

ρ

the density and C p the heat

capacityofthesolid(w )orthefluid(f ).

Introducing the effusivity ratio parameter

κ

=e w /e _f ,

Eq. (12) canbewritten T_w IF F =

κ

T

_κ

w +Tf

+1 (14)

Eq. (14) showsthat theinterfacetemperaturedependsonthe

pa-rameter

κ

: for large values of this ratio, the temperature atthe

solid/fluid interface stays close to the wall temperature and the

wall may be considered isothermal; on the contrary, low values

of

κ

allow significant heating of the wall which is then neither

isothermal nor adiabatic. In this last case the resolution of the

unsteady coupledproblemisnecessarytoobtain thecorrectwall

heatflux.

6. Casesdescription

Several FWI cases for laminarstoichiometric premixed flames

were performedandaresummarizedinTable 3 .Forallcases,the

initial wall temperature T _w i andthe freshgas temperature T u are

takenthesameandnon-coupled,isothermalsimulations(denoted

U _{) are compared to fluid-thermal solid coupled simulations}

(de-noted C ).Case 1is presentedforvalidation purposesandwill be

compared toprevious studies [7,10,12] .Cases 2a,2band2callow

to evaluate the influenceof thepressure on FWI andextendthe

results tovery highpressure.FinallyCase 3corresponds to

cryo-genic flames typical of LREs operatingconditions, with very low

freshgastemperature.

Thefirsteffectofpressureincreaseisthereductionoftheflame

thickness,whichmaybeapproximatedbyapowerlaw:

δ

l

(

P

)

=

δ

l

(

P0

)

³

_P

P0

´

α

(15)

where P 0 is a reference pressure and

α

dependson the

temper-ature andthefuel.Inthe caseofstoichiometrichydrogen/oxygen

mixtureat300K

α

∼ −1.21[29] wasfound,whichmeansthatthe

thermalflamethicknessdecreaseswithpressure.Thisaprioriwill

have a strong impact on FWI, withan expectedincrease of wall

heatfluxwithpressure.

As shown inTable 3 ,the freshgasdensity

ρ

u increases

dras-ticallywithincreasingpressureanddecreasingtemperature,upto

200 times(Case 3) higherthanthe reference Case2a atambient

conditions.Lookingatthecompressibilityfactor,givenby:

Z=

_ρ

P_rT (16)

where r isthe specific gasconstant,the deviationfromthe ideal

gas lawstays closeto1aslongasthetemperatureremains

rela-tivelyhigh.Forthesecasesnostrongrealgaseffectsareexpected.

Withthedecreaseofthefreshgastemperature, Case3leadstoa

compressibility factor of0.887, i.e. presentingsignificant realgas

Case Tu _{P T}b_{− T} u _S 0

l δl δ Ql0 Mesh cell size [K] [bar] [K] [m s −1 ] [m] _{[m] [W m} −2 ] [m]

1 750 1 2380 34.27 2.59e −4 1.07e −5 8.66e7 2.0e −6 2a 1 2770 10.76 2.23e −4 6.96e −6 6.87e7 2.0e −6 2b 300 10 3090 12.49 1.21e −5 5.85e −7 8.22e8 2.0e −7 2c 100 3430 9.03 1.18e −6 9.46e −8 6.25e9 1.0e −8 3 150 100 3544 3.96 1.23e −6 5.93e −8 5.47e9 1.0e −8

Table 5

Fluid and wall thermal effusivity, effusivity ratio κand interface temperature predicted by the IFF model T IF F

w . Thermal effusivity unit is [W m −2 K −1 s −1/2 ]. Case Ti

w Pressure Fluid effusivity e f Wall effusivity e w κ T IF Fw [K] [bar] [SI] [SI] [Dimensionless] [K]

1 750 1 6 .09 9280 1524 751.6

2a 1 4 .47 6186 1383 302

2b 300 10 13 .92 6186 4 4 4 307

2c 100 96 .61 6186 64 352.8

3 150 100 93 .62 4914 52.5 216.2

Fig. 4. Comparison of H 2 O 2 (left) and HO 2 (right) profiles in free propagating flames between Case 2c (solid line) and Case 3 (dashed line).

Flameproperties,computedwiththeBoivinscheme,areshown

inTable 4 forthevariouscases,togetherwiththemeshresolution.

The temperature difference T b _{− T}u and flame thickness change

largely whenthepressure increases,from2770Kand223

µ

m for

Case 2a to 3430K and1.18

µ

m for Case2c. The flame thickness

hasadirectconsequenceonthemeshcellsizethatischanged

ac-cordinglytoresolvetheflamefront.

The flame speed first increases with pressure, until _∼15bar,

where it reaches _∼12.5m s−1_{, before decreasing for higher}

pres-sure, to reach _∼9.0m s−1_{. This non-monotonic behavior was}

al-ready shownin[31] andisduetothechangeofchain-branching

tostraight-chainkinetics.Theflamespeedalsoincreaseswiththe

freshgas temperature T u ,whichhasa directeffecton the

chem-istrybutalsomodifiesthethermaldiffusivity D u _th =

λ

u /

ρ

u C p u .

In-creasingthefreshgastemperatureatambientpressureleadstoa

strongincreaseoftheflamevelocityandmoderatechangeofburnt

gastemperatureandflamethickness(Case1).Finallythecryogenic

condition (Case 3)givesa hotbutslowflame.Its structureis

de-tailedbelow.

6.1. Cryogenic premixed flame

The cryogenic, supercritical flame (Case 3) exhibits a

particu-lar structure.When compared to Case 2c, thefirst impact ofthe

lowertemperature,amplifiedbytherealgasthermodynamics,isto

significantly increase thedensityinthe freshgas,from48kgm−3

(Case2c)to108kgm−3 _{(Case3),whiletheburntgastemperature}

isonlyslightlylower.Theimportantdecreaseofthelaminarflame

speedis mostlyrelatedtothe decreaseofthe thermaldiffusivity

D u _th ,from9.2610−7_m2_s−1_{inCase2cto2.2310}−7_m2_s−1_inCase3,

associatedto supercriticaltransport properties.The most

remark-ablefeatureofCase3isthechangeofthechemicalstructureinthe

inductionzoneaheadoftheflame.Figure 4 showsHO2 andH2O2

massfraction profilesfor Cases2c and3.As alreadyobserved in

manystudies[6,7,12] ,premixedflamesarecharacterizedby

chem-ical reactions occurring in the induction zone between reactants

and radical species that diffuse from the main reaction zone. In

thecaseofH2/O2flames,thesereactionsleadtotheformationof

HO2 andH2O2intheinductionzone.InCase3,realgastransport

strongly limits radical diffusion, so that even zero-activation,

re-combinationreactionssuchasR4,R8orR9ofTable 1 cannotoccur.

As aconsequence,radical speciesdo notappear inthe induction

zoneinCase3,asclearlyvisibleinFig. 4 .Cryogenic,supercritical

flamesthereforehavenoreactiveinductionzoneandallreactions

start simultaneously when thetemperature reachesa sufficiently

highvalue. Thiswillhavedirect consequenceson theflame–wall

interactionfortheseflames.

Table 5 summarizesthe fluid andwall effusivities for all test

cases.Bothquantitiesincrease withtemperature, but e _f increases

evenmorestronglywithpressure.Theresultinginterface

temper-atures predicted by the IFF model, where the wall temperature

Fig. 5. Profiles of temperature (left) and dimensionless heat release rate (right) at various instants of FWI. Maximum non-dimensional heat release rate is 0.352. Case 1, coupled.

Fig. 6. Profiles of HO 2 (left) and H 2 O 2 (right) mass fractions at various instants of FWI. Case 1, coupled.

fluid temperature has been taken to the burnt gas temperature

T b , stays close to the initial interface temperature for high

val-ues of

κ

, in Cases 1 and 2a. As the fluid thermal effusivity

in-creases from Cases 2a to 2c, the ratio

κ

decreases and the final

walltemperaturemovesawayfromtheinitialtemperature.Finally

Cases 2cand 3,withlow

κ

, show a significantwall temperature

increase.

7. Resultsanddiscussion

7.1. Validation case 1

Case1isfirstpresentedforvalidationpurposes,asitiscloseto

theisothermalwallcasestudiedinpreviouspublications[7,10,12] .

Indeed in this case the high effusivity ratio

κ

=1524 leads to a

theoretical wall temperature T _w IF F =751.6K,very closetothe

ini-tial wall temperature, so that the isothermal wall assumption is

fullyvalid,andnostrongdifferenceswiththecoupledsolutionare

expected. Figure 5 (left)showsthetemperatureprofilesatseveral

instants,illustratingthetime-dependencyofFWIandthe

quench-ingprocess.Toallowcomparisonbetweencases,timeissetto0at

thestartofFWI,i.e.,whenthewallheatfluxstartstoincrease.As

a consequencetheflame firstpropagates freely towardsthe wall,

keepingafree flamestructure until t ∗ _{∼ 0.Then theflame starts}

to interactwith the wall, and becomes thinner while

approach-ing thewall until t ∗ _{∼ 20. Atthis time,there isno sufficient}

re-mainingfuelinthecoldgasandtheflamequenches.Inthesame

time, a transient process occurs fromthe start of FWI, where a

verylargeincreaseoftheheatreleaserateatthewallisobserved

(Fig. 5 (right)).Thisislinkedtoachangeofthechemicalbehavior

oftheinductionzonewhenapproachingthewall.Infreely

propa-gating flames,preliminarydecompositionofthefueloccursinthe

inductionzonethroughhigh-energy-activationreactionswith

rad-icals suchas R 2and R 3(Table 1 ).DuringFWI,thetemperaturein

the induction zone decreases down to the wall temperature and

these reactions get frozen, leading to a longer persistence of O2

than H2 near thewall. At the same time, andfor the same

rea-son, zero-activation-energy, exothermic,radical recombination

re-actions suchas R 4and R 8becomedominant, andlead tothe

ob-servedpeakofheatreleaserateandproductionrateofHO2(Fig. 6

(left)).Hence,throughthelow-activation-energy,propagation

reac-tion R 10hydrogenperoxide(H2O2)isalsoproduced(Fig. 6 (right)).

All thesechemicalmechanisms werealreadyobservedin

isother-malFWI[7,10,12] .Byincreasingthewalltemperaturegradient,this

strongpeakofheatreleaseatthewallhasadirectimpactonthe

wall heat flux. Inaddition, it leads to a zero quenching distance

which can thereforenot be used to evaluate the heat flux as in

Eq. (11) .

Figure 7 (left) showsthe time evolution ofthe wall heat flux

and walltemperature duringFWI.The wall temperature

progres-sively increasesto avalueof755.5K,i.e.,slightlyhigherthan the

IFF model value of 751.6K. The maximum wall heat flux is

ob-tained when theflame quenchesat t ∗ _{∼ 18,andreaches}

₈

_{w,Q =}

18.9MW m−2 ₍

₈

∗

w,Q =0.218).Afterflamequenching,thewallheat

fluxexperiencesfirstafastdecrease,thenamuchslowerdecrease

(_∝1/√t ∗) corresponding to theheat diffusion inthe fluid andin

the solid.DuringFWI,theflamepropagates towardthewalluntil

the remaining fuel is toolow to sustain the flame and

compen-sate for thewall heat loss. Thefuel quenchingdistance is

there-foremainlycontrolled bytheflamepowerandthewall

tempera-ture. Inthe present case, thefuel Peclet numberatquenchingis

Fig. 7. Temporal evolution of (left) wall heat flux and wall temperature and (right) fuel Peclet number. Case 1, coupled.

Fig. 8. Time evolution of temperature profiles in the solid wall T S . Case 1, coupled.

evolution of the fuel Peclet number during FWI. Both the

non-dimensionalfluxandthequenchingfuelPecletnumberaresmaller

than usualvaluesobtainedinFWI(_{∼ 0.3}and_{∼ 3.0}respectively)

andmaybeexplainedbythehighwalltemperature.Thedecrease

of themaximumwall heat fluxwithincreasing wall temperature

was alsodescribed in[3] .Thismaybe enhancedby thehigh

dif-fusivityofH2 andthehighheatreleaseatthewallduetoradical

recombinationasalreadymentioned.Thistrendandthevaluesof

the wallheat fluxandquenchingdistanceobtainedinCase1are

ingoodagreementwiththeresultsof[7,12] or[10] wherea

max-imumwallheatflux_∼18MWm−2 _{wasfoundforthesamecase.}

Finally,Fig. 8 showsthetemporalevolutionofthetemperature

inthesolid wall.Onecanobservethat thecouplingmethodology

is able to transfer the heat flux to the wall, whichthen diffuses

inthesolid.Notethattheheatpenetrationismuchslowerinthe

solid than inthe fluid, which isconsistent with thehigher solid

effusivity.

7.2. Effect of pressure

InthissectiontheeffectofpressureonFWIisinvestigatedwith

Cases2a(1bar)to2c(100bar).AlthoughCase2cisathigh

pres-sure,therelativelyhightemperatureleadstoacompressibility

fac-torcloseto1andnorealgaseffectsareexpectedhere.Fromthe

above IFFanalysis, resultsareexpectedtobecomparabletothose

obtainedinFWI withan isothermalwall forCases 2aand2b.

In-deed,theIFFinterfacetemperaturedoesnotexceedtheinitialwall

temperature by more than 2K and 7K, respectively. In Case 2c

however, theburnt gaseffusivity e _{f =}96.6Wm−2_K−1_s−1/2 _being

much higher,the predictedinterface temperatureincreasesup to

Fig. 9. Temporal evolution of the non-dimensional maximum heat release at the wall for Cases 2a, 2b and 2c, coupled. For all cases, time is set to 0 at the start of FWI.

T _w IF F =352Kandthecoupledsimulationisexpectedtogive

signif-icantlydifferentresultsfromthecorrespondingisothermalFWI.

Overall, similar trends as in the validation caseare observed,

withaheatreleasepeakandproductionofH2O2 andHO2radicals

occurringatthe wall duringthe FWI.However, asthe wall

tem-peratureissmaller,theeffectissignificantlyamplifiedin

compari-sontoCase1.Indeedthenon-dimensionalmaximumheatrelease,

showninFig. 9 isabout2ordersofmagnitudelargerduringFWI

thaninthe freeflame inCase2a,whereas itwas onlyone order

ofmagnitude larger inCase 1 (Fig. 5 (right)).The effectis

how-everdecreasingwithpressure,comingbackinCase2ctothesame

orderofmagnitudethaninCase1.

Figure 10 showsthetemporalevolutionofthenon-dimensional

heat flux and the temperature at the wall for the three cases.

Duetofasterchemistryandsmallerflamethickness, FWIisfaster

athighpressure. The maximum wall heat flux isobtained when

flame quenches at t ∗ _{∼ 11, t} ∗ _{∼ 8 and t} ∗ _{∼ 5 for Cases 2a, 2b}

and 2c respectively, and slightly decreases with pressure, from

8

∗

w,Q =0.388forCase2ato

8

∗w,Q =0.333forCase2c,consistently

withthe lower wall heat release effectat highpressure. Overall,

themaximumwallheatfluxislittlesensitivetopressureandstays

inthe range0.3–0.4,i.e., similar tohydrocarbonflames withlow

wall temperatures [4,32,33] . Note however that the dimensional

wallheatflux increaseswithpressure,from

8

w,Q =26.4MWm−2

forCase 2ato

8

_{w,Q =}2.09GWm−2 _{for Case2c, i.e., reaching}

ex-tremelyhighvalues.

AsexpectedfromtheIFFmodel,theinterface temperature

in-creasesonlyslightlyatlowpressure(Cases2aand2b),butreaches

Fig. 10. Temporal evolution of non-dimensional wall heat flux (left) and wall temperature (right). Cases 2a, 2b, 2c, coupled.

Fig. 11. Time evolution of wall heat flux difference 18w = 8Uw −8Cw between isothermal wall condition and coupled computation. Case 2c. 8C

w = 2 . 09 e 9 W m −2 .

that the increase is always stronger than predicted by the IFF

model. This difference is dueto the strong heat release, both in

theflameandatthewall,duringFWIinthe coupledsimulations

andwhichisnottakenintoaccountintheIFFmodel.Thismakes

theheatfluxstrongerandincreasestheinterfacetemperature.This

justifies a posteriori the use of fully coupled simulations for the

predictionofheattransfer.

The interface temperatureincreasealso explainsthewall heat

flux decreasewithpressure.Figure 11 showstheevolutionofthe

difference betweenthe wall heat flux obtainedin the uncoupled

(calculated withan isothermal wall condition at T w =300K)

8

U _w

andthecoupledcomputation

8

C _w ofCase2c.Themaximum

differ-enceisobservedjustbeforequenching,wheretheisothermalwall

assumption leads to an overestimationof the maximal wall heat

fluxby 200MWm−2_{,i.e., approximately10% ofthewall heatflux}

inthecoupledcase,whichissignificantforthethermalfatigueof

solid materials. Thiscorresponds toa non-dimensional wall heat

fluxof

8

U∗

w,Q =0.352,i.e.,closertothelowpressurecasesthanthe

coupledcase.

Figure 12 (left) shows the fuel Peclet number obtained at quenchingforthethreecases.Thequenchingdistanceof Pe F _{Q =}4.1

forCase 2a islarger than forCase 1dueto the lower wall

tem-perature. It is slightly larger than the value of _{∼ 3} typically

ob-served inpreviousnumericalandexperimentalstudiesfor

hydro-carbonsfuels[4,32,33] ,whichmaybeduetothehighdiffusivityof

H2. When pressureincreases,the quenchingdistance slightly

de-creases, down to Pe F _{Q =}3.2 forCase 2c, still stayingin the range 3− 4.The slightdecrease of Pe F _Q withpressure maybe again

at-tributed to the increase of the interface temperature which

al-lows fuel oxidation reactions to occur closer to the wall. As

al-ready mentioned, the non-dimensional maximum wall heat flux,

also reportedin Fig. 12 (right), decreases withpressure. This

be-haviorwasalreadyobservedinotherstudies[5,34] forlower

pres-sureranges(0.5–3.5bar)andisconfirmedhereforhigherpressure

levelsandconjugateheattransfer.Thisalsodemonstratesthat,

al-though thesimpleexpressionEq. (11) stillholdsintermsoforder

ofmagnitude,itisnotabletodescribeacomplexbehaviorsuchas

the simultaneous decreaseofboth

8

∗

w,Q and Pe F Q withincreasing

pressure. Thisindeedistheresultofchemical phenomena

occur-ingatthewallandcannotbepredictedfromfreeflameparameters

suchastheflamethickness

δ

_l .

7.3. Supercritical case

This section presentsthe resultsobtainedforthe supercritical

case (Case 3)where thefreshgas temperaturehasbeen lowered

downto T u ₌150K.Thecompressibilityfactorinthatcaseis0.887

meaning that real gas effects have to be taken into account. As

shown inTable 5 , theeffusivity ofthe burntgas islarge insuch

thermodynamic conditions,thus requiring the fluid/solidthermal

coupling tosimulatethe transientFWI andpredictthe final wall

temperature. FWIwithanisothermalwallat150Kleads tostrong

watercondensationwhenthecombustionproductsreachthewall,

so that direct comparison of coupled or uncoupled cases is not

possibleinthiscase.

Figure 13 (left)reportsthetemperatureprofilesduringFWI.The

overall process is similar toall previous casesandis comparable

to Case2c,alsoathighpressure.AsinCase2c, theinteractionis

quite fast,with quenching occurring at t ∗ _{∼ 8, andheat release}

peakonthewallisstillobserved(Fig. 13 (right)).Howeveraswas

observedinthecryogenicfreeflame,theinductionzoneisfrozen

due tothe low temperatureanddoesnot interactwith thewall.

Neither H2O2 orHO2 arepresentoutsidetheflamezoneandthey

start to build on thewall only whenthe flamereaches thewall.

Compared to Case 2c, the increase of heat release atthe wall is

delayedandstartsshortlybefore quenching.Asa result,although

theincreaseiscomparabletoCase2c,its impactonthewallheat

fluxisreduced.

In supercritical conditions, the fluid properties differ largely

fromtheperfectgas,withathermaldiffusivitydividedby4when

compared toCase2c. (Case2c:

λ

u /

ρ

u C p u ₌9.2610−7_m2_s−1 _and

Case 3 :

λ

u /

ρ

u C u _{p =}2.2310−7_m2_s−1_{). This, combined with the}

Fig. 12. Effect of pressure on the quenching fuel Peclet number Pe F

Q ( ◦) (left) and on the dimensionless maximum wall heat flux 8∗w,Q ( ◦) (right) for Cases 2a,b,c, coupled simulations.

Fig. 13. Profiles of temperature at various instants of FWI (left) and time evolution of the maximum heat release at the wall (right). Case 3, coupled.

Fig. 14. Temporal evolution of wall heat flux and wall temperature. Case 3, coupled.

corresponding to Pe F _Q =6.0.As a consequence,the wall

tempera-tureincreasesslowly,remaininglowduringthequenchingprocess

and still increasing after the flamehas extinguished (Fig. 14 ). As

theheatreleaseatthewallstayszeroforalongtimeandstartsto

increasejustbeforequenching,itdoesnotcontributemuchtothe

wall temperatureincrease which stays closeto the predictedIFF

temperature(T _w IF F =216.2K).Thenon-dimensionalmaximumwall heat fluxreachesavalue of0.36(

8

w,Q =1.97GWm−2_{), i.e.,stays}

in therange0.3–0.4, mainlythanks to thelarge temperature

dif-ference T b − Tu .Inthiscase,Eq. (11) doesnotholdanymore. This

againdemonstratesthatthequenchingdistanceandthemaximum

wall heatflux arenot directlylinked butstrongly dependon the

interfacetemperature,requiringtheuseofcoupledsimulations.

8. Conclusions

The interaction between premixed flames and non-adiabatic

wallshasbeeninvestigatedinaconjugateheattransferapproach,

where the fluid and the solid wall are thermally coupled. To

be representative of liquid rocket engines, stoichiometric H2–O2

mixtures in ambient andcryogenic (low temperature, high

pres-sure)conditionshavebeenconsidered.Aunique framework,

cou-pling both fluid and heat transfer solvers, was used in order to

take into account the wall heating transient phenomena. It was

demonstratedthatiftheeffusivityoftheburntgasbecomes

non-negligible comparedto that of the solid, theisothermal

assump-tiondoesnotholdanymore.Itwasfoundthatthissituationmainly

occurs at high pressure, requiring the use of fluid–solid

ther-mal coupling. When pressure increases, the more powerful and

much thinner flame leads to important quenching distance

de-crease and maximum wall heat flux increase by two orders of

magnitude compared to atmospheric conditions. However, when

non-dimensionalized with the flame thickness and flame power,

both quantities become almost insensitive to pressure and take

typicalvaluesalreadyobservedinhydrocarbonflames.Still,the

in-creaseofwalltemperatureduetoconjugateheattransfer,andthe

heatreleaseatthewallduetoradicalrecombination,are

responsi-bleforaslightdecreaseofthequenchingdistanceandmaximum

wall heat flux when pressure increases.Finally, low-temperature,

high-pressurecryogenicconditionswhichleadtosupercriticalfluid

largequenchingdistance.Howeverthenon-dimensionalmaximum

wallheatfluxstays comparabletothepreviouscases.Inthiscase

also,significant impactofthe conjugateheattransfer isobserved

and requires fluid–solid thermal coupling to describe accurately

thewall temperatureandtheflamebehavior. Thesefindingsmay

haveimportantimplicationsforflamestabilizationandthermal

fa-tigue in practical systems such as liquid rocket engine injectors.

The demonstrated feasibility and relevance of thermally coupled

fluid–solidsimulationsallowstoremovetheuncertaintyaboutthe

wallthermalconditionsandimprovethepredictionanddesignof

optimumburnergeometries.

Supplementarymaterial

Supplementary material associated with this article can be

found, in the online version, at 10.1016/j.combustflame.2016.01.

004 .

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