Cagliari-Italy
Development of the TFC Turbulent Premixed
Combustion Model: the Combustion Rates and the
Counter-gradient Transport Phenomenon.
Vladimir Zimont
Abstra t.
Weanalyze two "bottlene ks" of turbulent premixed ombustionmodeling
at developed turbulen e orresponding to modern lean-premixed large-s ale
industrialgas-turbine ombusterswheninstantaneous ombustion(heatrelease)
takes pla e inthin strongly wrinkled surfa es( amelet sheets). They are as
follows: 1) fundamental failure of statisti al ombustion models in terms of
momentorPDFequations toresolvethe stru ture ameletrea tionzonesand
small-s alesheetwrinkles ontrollinga tual ombustionrates(so alled hallenge
ofturbulent ombustion);2)well-knownthe ounter-gradientturbulenttransport
phenomenonthatrendersdire tappli ationofexistingturbulen emodeltogether
with ombustionmodelsimpossibleatsimulationofpremixed ombustion.
The rstproblems was solvedinanalyzed our TFC ombustion modelby
assumptionofequilibriumsmall-s alestru tureifthese ameletsheetstru ture,i.
e. infa tbyusingimpli itlyKolmogorovmethodologydevelopedforsimulation
non-rea tive turbulent ows. In thisreport we develop Kolmogorov ideas for
ombustion modeling, analyze their possibilities and restri tions, these results
providethene essaryba kgroundtotheTFCmodel.
Forsolvingthese ondproblemwedevelopedinthisreportsomegas-dynami s
modelofpressure-driven ounter-gradient omponentof transport thatgavean
opportunitytogetherwiththe ombustionmodelandtraditional"k-e"turbulen e
modeltosimulatethe ounter-gradienttransport phenomenonand ittransition
tothegradienttransportatvariationsonthe ombustionoperationregime.
Sothereport ontains:
1. Statement and analysis of the method of atta k to the problem of
orre t des ription in ontext of model statisti al equations ombustion rates
at strongturbulen e and fast hemistry when ombustion ratesare ontrolled
byunreservedsmall-s ale ouplingofturbulen eand hemistry,someanswerat
pra ti al ombustionsimulationstothe" hallengeofturbulent ombustion";
2. Analysis of the ounter-gradient transport phenomenon observed in
premixed turbulent ames,des riptionofdevelopedgas-dynami smodelofthe
pressure-driven me hanism of it, results of numeri al simulationsof turbulent
ames withquantitative analysisofthisphenomenon and its transitionto the
gradient turbulent di usion, omparison with re ently published experimental
datawherethistransitionwasthe rstobserved.
We dis ussed this results from the industrial standpoint at ombustion
simulationsofreale ologi ally leanleanpremexed ombustion hambers.
1. Introdu tion
We analyze two "bottlene ks" of turbulent premixed ombustion modeling at
developed turbulen e orresponding to modern lean-premixed large-s ale industrial
gas-turbine ombusterswhen instantaneous ombustion (heatrelease) takespla ein
thinstronglywrinkledsurfa es( ameletsheets). Theyareasfollows: 1)fundamental
failure of statisti al ombustion models in terms of moment or PDF equations to
resolvethestru ture ameletrea tionzonesandsmall-s alesheetwrinkles ontrolling
a tual ombustionrates(so alled hallengeofturbulent ombustion);2)well-known
the ounter-gradientturbulenttransportphenomenonthatrendersdire tappli ation
ofexistingturbulen emodeltogetherwith ombustionmodelsimpossibleatsimulation
ofpremixed ombustion.
I. The term " hallenge" has been used last years in s ienti papers [1℄-[2℄ to
emphasizethat there isthe fundamental obsta leon the way ofdevelopinggeneral-
purposeturbulent ombustionmodelsforsolvingpra ti alproblems,i. e. themodels
areintended for a tualturbulent ombustion simulationsof realburners that would
give not only quantitative agreementwith experimental data (it is in itself is very
diÆ altproblem)butalsotohaveabilitytopredi ttheturbulent ombustionpro ess
trendsat variationsof ompositions, theoperation onditions,burner geometry and
soon. Thereasonofitis onne tedwiththefa tthatatrealturbulen eand hemi al
ombustionrea tionratesatreal onditionsare ontrolledbysu hsmall-s ale oupling
of hemistry,mi roturbulentmixingandmole ulardi usionthat annotberesolvedin
framesofmoment,PDFandLES ombustionmodels[2℄. So"thea uratepredi tionof
meanrea tionrates,whi h anbein uen edstronglybymole ulardi usion ausedby
small-s aleturbulentmixing,representsthe entralproblemand hallengeofturbulent
ombustion"[2℄. Thisproblemisespe iallypressinginthe aseofpremixed ombustion
as for nonpremixed ombustion at suÆ iently fast hemistry the problem of ame
aerodynami s and ombustion rates simulation often an be redu ed to analysis of
mixing,i. e. this ouplingbe omesthese ondaryimportan efa tor.
This hallenge onne ted with the inability to des ribe mentioned small-s ale
oupling is the most distin t at premixed ombustion at real Reynolds R e and
Damkohler Da numbers when takes pla e the amelet ombustion me hanism with
very thin sheet-type rea tion zones. It means in fa t that turbulent ombustion
models annotpredi t,forexample,thelo aldistributionoftheaveraged ombustion
rate a ross the ame W(x) =
u U
f
(x) or the turbulent ombustion speed U
t
=
U
f (S=S
0
) = (1=
u )
R
W(x)dx in relation to the fuel, the air ex ess oeÆ ient,
initial temperature of a mixture, the pressure and so on over more or less range
ofthese parameters. HereU
f
isthe amelet velo ity(at theoreti alestimations U
f
often assumed to be the laminar ombustion velo ity U
l
), is the amelet surfa e
density,(S=S
0
)isthedimensionless ameletsurfa earea,
u
isthedensityofunburned
mixture.
Itshould benotedthat ProfessorKenBraywhoworkedmanyyearsinthe eld
ofturbulent(mainlypremixed) ombustionandhavefundamentalresultsinthis eld
titledhisreview HottelPlanaryle ture"Challengeofturbulent ombustion"instead
of, for example, "A hevements of turbulent ombustion" be ause as he wrote "the
hallange appearing in the title of this paper is dire ted to the whole ombustion
resear h ommunity". Proposedpaperisanattempttogivea onstra tiveanswerto
nonresolved ouplingthat anbeusedat designingofturbulent ombustionmodels.
At rstglan e, thisproblem seemsintra table: we annotresolvein theframes
of ombustionmodel equationsspa eandtime s alesthatarene essaryfora orre t
des ription of hemi al kineti s, mole ular transfer pro esses and their oupling. If
it were so it would mean that to design physi ally reasonable ombustion model
ontaining the key ombustion me hanisms (and between them one of the main
is mentioned oupling) would be impossible. It would mean that every possible
ombustionmodelevenifitreprodu essomelimitedexperiments(bytuningempiri al
onstants) ould notdes ribealargeensemble ofexperimental dataand, in general,
ouldnotgive orre ttrends.
Oneoftheaimsofthispaperistoshowthat fromtheappli ation-orientedpoint
of view this problem doesnot look unresolved, and that there are ideasthat ould
give positiveapplied results. Our approa h is based on an assumption of universal
small-s aleturbulentstru tureofrea tionzonesthatisinequilibriumwithlarge-s ale
turbulentmovement. Wetrytoformulate apra ti ablelineof inquirytoanswerthis
hallengeand analyzeitspra ti alappli ationto simulationofpremixed ombustion
for large R e and Da numbers. The hara ter of this hallenge and the main idea
of our answer in fa t is not new. A similar hallenge was in past that of pra ti al
turbulen e modeling asexisted methods ould not predi t a urately the turbulent
dissipationrates,whi h wasin uen edstronglybysmall-s alevortexes. Kolmogorov
gave a pra ti al answer to this hallenge and it was in fa t the ornerstone of all
turbulen e models for large R e numbers. In both ases, turbulen e and turbulent
ombustion, ontrolling pro esses take pla e in su h small s ales that ould not be
resolvedbymodelequations.
Kolmogorov answer[3℄ was based on the assumption of equilibrium ne-s ale
turbulen e (where a tual dissipation took pla e) due to the Ri hardson as ade
me hanism. This assumptionyields that the turbulentdissipation rate is ontrolled
by large s ale turbulen e and at large Reynolds numbers the averaged dissipation
rate " does not depend on the mole ular vis osity oeÆ ient: " = Cu 03
=L, where
C 1. Next year Kolmogorovdesigned using these ideas the rst two parametri
turbulen emodel("k !")[4℄,wherethedissipationratewasdes ribednotinterms
ofthemole ularvis osity oeÆ ient,butthroughthisphysi alme hanism.
In this paper we extend this Kolmogorov methodology to model turbulent
ombustionatlargeReynoldsandDamkohlernumbers. Weinfa tusedtheseideasin
past[5℄andTFCmodelthathassomeappli ation(in"Fluent5",forexample)isbased
onthem but wedid notformulate is asaprin iple. Here weformulate and analyze
thefeasibilityoftheseideasfor ombustionmodeling,thisideasgiveandopportunity
to understandbettersomelimitation ofexisting ombustionmodelsandprovidethe
ne essaryba kgroundtotheTFCmodel.
II. Counter-gradient transport is a phenomenon whi h ommonly o urs in
turbulent premixed ombustion. It is onne ted with thedi erential a elerationof
hotand old uidvolumesunder the ame pressuredrop generatedby heatrelease.
Ithasbeenexperimentallyobservedforexampleinopen[1℄andimpinging[2℄ ames.
This phenomenon is losely onne ted with the amelets ombustion me hanismin
turbulentpremixed ames,when ombustiontakespla einthinandstronglywrinkled
layers alled amelet sheets. In laboratory ows with relatively low turbulen e
ReynoldsnumbersR e
t
=u 0
L= these amelet sheets are laminar ameswith speed
s andthi knessÆ highly wrinkledbysmalls aleturbulen e.
Anewphenomenon onsistinginthetransitionfrom ounter-gradienttogradient
transport has been re ently experimentally observed [3℄. A ording to these
experiments in open premixed ames, ounter-gradient transport observed at some
xeddistan efromtheburnerinlettransformsintogradienttransportwhentheratio
betweentheturbulentvelo ity u tuationu 0
andthelaminar amespeeds
L
in reases
as onsequen eofthevariationinthefuel/airmixtureequivalen eratio. Inthepresent
paperwefo usthemainattentiononthetheoreti alanalysisofthisphenomenon,its
numeri alsimulationand omparisonwiththeselaboratoryexperimentaldata.
Another issue addressed in the report is the existen e of ounter-gradient
transport and the phenomenon of transition to gradient transport in large-s ale
high velo ity ames in industrial ombustors where dire t measurements are very
diÆ ult. Our point of view in fa t is that in industrial burners hara terized by
mu hlargerReynoldsandDamkohler(Da
t
=
h
ratiobetweentheintegralturbulen e
and hemi altimess ales)numbers,presen eof ounter-gradienttransportis highly
probable. Atlarge turbulent Reynoldsnumbersin fa t,turbulen e has averywide
ontinuousspe trumofeddiesin ludingverysmalldissipativevorti es. AtDamkohler
numbersinindustrial burners (Da'10)thesize ofthe smallestKolmogoroveddies
=LR e 3=4
t
(whereL istheintegrallengths aleofturbulen e) anbe omparable
orlessthan thelaminar amethi kness, i.e. Æ
L
. In this asewe havewrinkled
amelets whi h an have thi kness Æ
f
larger than laminar one Æ
L
be ause of the
intensi ationin thetransportpro essesbysmall-s aleeddieswithsizelessthanÆ
f .
If at the same time the turbulentintegral length s ale L is mu h largerthat Æ
f , it
meansthat ombustion takespla e in a stronglywrinkled sheet that isnot laminar
butappearsbroadenedbysmalls aleturbulen e.
As the ounter-gradient phenomenon is losely onne ted with the amelet
ombustion me hanism, it is important to stress that the su essive falling of more
andmorelargeeddiesinintheexpandingbroadened amelethasalimit. Equilibrium
thi knessisinfa ta hievedwhen onve tion,di usionandheatreleaseintensityhave
thesameorderofmagnitude[4,5℄. Su hexpandedequilibrium amelet,ina ordan e
with the theoreti al estimations at Da >> 1, is always hara terized by Æ
f
> Æ
L .
Existingdire t experimental datademonstratethat Æ
f
'(3 5)Æ
L
[6℄ su hthat the
typi alorderofmagnitudeofÆ
f
is'1mm.
Unfortunately,thereisnodire teviden ethat ombustiontakespla einthinand
stronglywrinkledsheetsalsoinindustrial ombustors,whereu 0
anbequitelarge,as
therearenotdire tmeasurementsofthe ameletparametersinthesekindofburners.
In spite anyway of the in rease in the ratio Æ
f
=L for in reasing u 0
(whi h gives a
de rease in Da), estimations presented in [5℄ show that extin tion due to amelet
stret h takes pla e long before the formation of distributed ombustion zones with
Æ
f
=L1. Wethereforebelievethat ounter-gradienttransportandthephenomenon
of transition to gradient transport exist also in ames in industrial burners. We
must nevertheless mention that a di erentpointof view, whi h we do notshare, is
often found in literature. A ording to this, industrial ombustion o urs with the
distributedvolumeme hanism orrespondingtothemodelofstirredrea tor. Clearly,
inthis aseawell-de ned ounter-gradientphenomenon isnotlikelytobepossible.
Thee e tofsmall-s aleturbulen ewithsizelargerthanÆ
f
istomakethe amelet
sheet wrinkled. This in reases the amelet surfa e/turbulent ame ross se tional
areasratioS=S
0
whi h,togetherwiththeexpanded amelet ombustionvelo ityU
f ,
ontrols the turbulent burning velo ity, i.e. U
t
=U
f S=S
0
. It is verysigni ant to
tothein reaseinS=S
0
isgivenbysmall-s alewrinklesgeneratedbyeddieswithsizes
Æ
f
d<<L, whilelarge-s alewrinkles aused by eddieswithdL mainly ontrol
the amebrushwidthand giveonlyrelativelysmall ontributionto S=S
0 .
The onsequen e of this situation is that immediately after ombustion has
started,wehavenotonlyequilibrium broadened ameletswith U
f
= onst but also
equilibrium small-s ale wrinklesstru ture of the amelet sheet. On the otherhand,
in the aseof u 0
>> U
f
and during arelativelylong period of time, the larges ale
wrinklesstru turewillnotbeinequilibrium(a ordingtoestimationsin [7℄duringa
times aleoft
t (u
0
=U
f )
2
). Thepra ti alresultofthisisthatduringthisrelatively
longperiodof time,anintermediate propagation ombustion regime,pro eedingthe
formationofastationary ombustionfrontstru turewith onstantU st
t andÆ
st
t ,takes
pla e. Premixed ombustion withinthis intermediateregimetakespla ethereforein
ameswith in reasing amebrush width ontrolled byturbulent di usion and with
pra ti ally onstant turbulent ombustion velo ity U
t
onst (more exa tly with
slowlyin reasingU
t
due torelativelysmall ontributiontothe amelet sheetareaof
non-equilibrium large wrinkles). Stri tly speaking this arguments are valid only in
thehypotheti al aseof ombustion at onstantdensity. Leaninguponexperimental
observations,wehavepostulatedthesamephysi alpi tureandthepossibilitytouse
theasso iated ombustionmodelalso inthereal aseof variabledensity.
Prudnikov was probably the rst who established this pe uliarity of premixed
ombustion during the late '50s. He in fa t showed experimentally that turbulent
premixed amesstabilizedinuniformdu t owswithstrongarti ialturbulen ehad
temperature pro les orrespondingto theintegralprobabilitydistribution, i.e. lose
to thenormal amelet probability density fun tion,and amebrushwith in reasing
widthwhi hwas ontrolledmainlybythe old owturbulentdi usion oeÆ ient. This
resultwas rstpublishedin1959inanowadayshard-to- ndpreprintandsu essively
presentedin[8℄. Theanalysisofthelatestexperimentaldatainthepaper[32℄ on rms
alsothatturbulentpremixed ombustiono ursin ameswithin reasingbrushwidth.
This parti ular regime of turbulent premixed ombustion has been named
IntermediateSteadyPropagation(ISP)regime[5,7℄(intermediateasymptoti between
theinitialstageofformationofanequilibriumsmall-s alewrinkledstru tureand nal
stageofaturbulent ame ompletelyinequilibrium withU
t
= onstandÆ
t
= onst)
inorder toemphasizethe on eptof ames hara terizedbyapproximately onstant
turbulent amespeed( onstantinthe aseofhomogeneousturbulen eandequilibrium
inmore omplexsituations)anda amebrushthi knesswhi hgrowsa ordingtothe
turbulent dispersion law. The reason of ISP ames prevailing within an industrial
ombustorisgivenbythethefa tthat ombustioninthisburnersisstillwellfarfrom
the nalstageoffullysteady onditions( onstantU
t andÆ
t
)astheme hanismwhi h
is responsible for ompensating the turbulent growth of the ame brush thi kness
be omese e tiveonatimes alemu hlargerthantheaverageresiden etime.
In the ase of open axisymmetri premixed ame analyzed in this paper, this
on eptmeansthatapremixed ameissimplyaturbulentmixinglayerwithin reasing
width(similarly tothemixing layerof anonrea tive ow),propagating inthe fresh
mixture with equilibrium ombustion velo ity U
t
. This propagating mixing layer
rossesat the ombustor axislong before theformation of the ame with omplete
equilibriumstru ture.
Given that a ame in omplete equilibrium with U
t
= onst and Æ
t
= onst is
unattainableinpra ti aldevi esunderstrongturbulen e onditions,itisreasonableto
anddoesnot ontaintheme hanismsresponsibleforrea hingthe nalfullequilibrium
amestru ture. Thismodelwhi hwasproposedandinvestigatedinthepapers[4,5℄
and alled in [9℄ the Turbulent Flame Closure (TFC) ombustion model, has been
further developedand usedhereforthedes riptionof ounter-gradientandgradient
transport as well astheir transition in orresponden e of variations in the fuel/air
mixtureequivalen eratio.
Thereasonsofexperimentallyobservedturbulentpremixed ameswithin reasing
widthmightnotbe ompletely learat thisstage,deservingthen further omments.
A ontradi tioninfa t apparentlyarisesbetweentheideaofaturbulent amebrush
whi h grows in thi kness a ording to the turbulent dispersion law (i.e. a ording
toapositiveturbulentdi usion oeÆ ient)andtheexperimentallyobserved ounter-
gradientnature of theprogressvariable transport whi h hasbeenso farthe subje t
of many theoreti al and numeri alstudies. Forexample Moss [1℄ has observed this
phenomenon in his early experiments for an open premixed ame; subsequently
ounter-gradienttransporthasbeenobservedintheexperimentsperformedin[2,10℄
and in dire t numeri al simulations of turbulent premixed ame [11℄. It has been
onsequently argued that the assumption of gradient transport for the progress
variable or the rea tive omponents is in general not appropriate for turbulent
premixed ames and sometimes also that, as a onsequen e of " ounter-gradient
di usion", the ame brush thi kness de reases instead than in reasing [11℄. The
possibility to model the turbulent transport of rea tive omponents in turbulent
premixed ombustion using models based on gradienttransport and a onveniently
estimated turbulent di usion oeÆ ient has been therefore abandoned a long time
agoandtheattentionhasbeentotally on entratedonse ondordermoment losure
methods[12,13℄.
The point of view developed in this paper is that the term " ounter-gradient
di usion"oftengivenintheliteraturetodes ribetransportoftheprogressvariablein
owswithheat releasehasitsoriginintheinterpretation ofthistransport asapure
turbulent di usion ux and in the attempts sometimes to onne t it to a turbulent
di usion oeÆ ient, that for agreement with experimental data must be negative
( ounter-gradient, i. e. negative di usion). We believe insteadthat this transport
is ontrolled both by turbulen e (turbulent di usion) and by spe i gas-dynami s
displa ement of hot and old volumes driven by pressurevariation a rossthe ame
brush (in open ames the pressure is de reasingbe ause of heat release). We will
thereforeuse hereaftertheterm transport insteadthan turbulent di usionespe ially
inthe aseofthe ounter-gradientphenomenoninordertoemphasizethepresen eof
anadditionaltransport me hanism,usually prevailingin turbulentpremixed ames,
whi h isnotdire tlyasso iatedwiththeturbulentpulsationsofvelo ities.
Itwillbeshownthattheprogressvariabletransportinopenturbulentpremixed
ameshasgradient hara teratthebeginningofthe ame,whereturbulentdi usion
prevails over the gas-dynami s transport omponent ( ounter-gradient) be ause of
the small ame brush width, and has ounter-gradient hara ter at larger distan e
wherethegasdynanami se e tstartstodominate. Itisworthalsomentioningthat,
if we reate in a ombustor turbulent premixed ow a nonuniform distribution of
verysmallpassive ontaminants,thereare stronggroundstobelievethatthenature
of its transport would be of the gradient type ontrolled by the turbulentdi usion
oeÆ ient. Thisapproa hwasinfa tusedinhisexperimentalworkbyPrudnikov[14℄
where measurements of turbulent intensity and turbulent di usion oeÆ ient were
di usionwakeofluminousparti lesbehindapointsour epla edinpremixed ames.
These experimental resultsdid not show any " ounter-gradientturbulent di usion"
atleastattheba kpartofthe ameswheremeasurementswere made.
In the TFC ombustion model there is no ontradi tion between the
experimentallyobservedin reaseofthe amebrushwidthandthemeasured ounter-
gradient transport of the progress variable. The onve tive nature of the ounter-
gradient omponent does notdire tly a e t in the model the ame brush thi kness
whi h, similarly to a non rea tive mixing layer, is ontrolled by physi al turbulent
di usion. In the present simulations of the ounter-gradient phenomenon and its
transitionintogradienttransportwehaveestimatedtheturbulentdi usion omponent
usingthestandardk modelandthe ounter-gradienttransportusinganovelgas-
dynami smodeldevelopedhere. Thesesimulationsdemonstrategoodagreementwith
thetrendshown bytheexperimentaldatain thetransition from ounter-gradientto
gradienttransport.
Asimilarideawasdevelopedearlierinthepaper[7℄wherehighReynoldsnumber
turbulentpremixed ombustionin a hannelwassimulatedusingtheTFCmodel. In
that ase anyway a very rough and less quantitatively a urate, estimation of the
gasdinami s transport omponent was used. This was based on the assumption of
onstant onditionallyaveragedvelo itiesintheunburntandburntmixturesandgave
thepossibleupperboundaryforthe ounter-gradienttransport omponent. It must
bementionedthat on lusions onsistentwiththisassumptionfollowedalsofromthe
analysis of DNS resultsof turbulent premixed ames for the ase of low turbulen e
levels(u 0
=s
L
<1)performedbyVeynanteetal [11℄whenthetwo onditionalvelo ities
were foundapproximately onstanta rossthe ame brushandrespe tively equalto
s
L
and s
L
, where =
u
=
b
is the ratio between the unburnt and burnt gases
densities.
Inour aseofstrongturbulen e(u 0
=s
L
>>1)su hassumptionisnotvalid. Asit
willbeseenbelowin fa ttheassumptionof onstant onditionalvelo itiesa rossthe
turbulent amebrushresultsin amu h largere e t. Comparingwithexperimental
dataavailablefromMoss[1℄foranopenturbulentpremixed ame,itisshownthatthe
modeldevelopedin[7℄overestimatesthe ounter-gradientpartofthetotalturbulent
s alar ux byapproximatelythreetimes.
In the present work a new improved model for the pressure-driven transport
omponentisdeveloped. Thisisbasedontheassumptionofequal onditionalaverage
pressurep
u and p
b
respe tivelyin theunburntandburnt uidvolumes. Thismodel
givesex ellentagreementwithMossexperimentaldata.
Itisfurthermoredemonstratedthatthe apabilityto orre tlymodelthepressure-
driventransport omponentdire tlya e tsthea ura ywithwhi hwe anestimate
theheatreleasedistribution. Consistentlywiththeexperimentalobservationmadein
[10,15℄, itis shown, in fa t,thattheimprovedmodel for ounter-gradienttransport
developed here yields an average heat releasewhi h is skewed towards value of the
progress variable > 0:5 with respe t to the symmetri theoreti al distribution of
the Bray-Moss-Libby (BML) model (also predi ted by our previous assumption of
onstant onditionalvelo itiesa rossthe amebrush).
Experimental eviden e supporting the present analysis is represented by the
re entexperimental datafrom[3℄. A ordingto theseexperimentsinopenpremixed
ames a transition from ounter-gradient to gradient transport is observed at
a given distan e from the burner inlet when the ratio between the turbulent
velo ity u tuation and the laminar ame speed u 0
=s in reases, suggesting that
physi al turbulentdi usion be omes dominant on the gas-dynami s pressure-driven
me hanism. Theseexperimentshavebeen onsideredhereforassessingthefeasibility
oftheproposedmodelingideaforthe ounter-gradienttransportphenomenon.
2. Kinemati spi ture ofpremixed ombustion ame
In the ase under review ombustion takes pla e in thin highly wrinkled amelet
sheetsthat separatestherea tantsfrom theprodu tsand propagatesrelativeto the
rea tantswith velo ity U
f
. In ourmodel the amelet is notthe laminar ame with
the ombustionvelo ityU
l
,but isthi kenedbysmalls aleturbulen e amelet with
the ombustionvelo ityU
f
>U
l .
As a preliminary we assume that gas density = onst, i. e. ombustion
does not hange hydrodynami owand turbulen e. Let denote P
u , P
b and P
f the
probabilitiesofunburnedmixture(rea tants),burnedmixture(produ ts)and amelet
ompositionsin everypointof theturbulent ame. Forkinemati sdes riptionofthe
ameme hanismweassumethatP
u +P
b
=1asP
f
<<1(the amelet ombustionon
me hanism). Wewillusethe ombustionprodu tprobabilityP
b
(~x ;t)ortheaveraged
progressvariable (~x ;t)(at= onst, =P
b
)fordes riptionofturbulent ames.
At rst we will analyze a 1-D non-stationary ame front in gas moving along
thex-axis at u 0
>> U
f
. Assume that at theinitial time t =0forx <0 = 1and
for x > 0 = 0. For t > 0the plane boundary be omeswrinkled at t >0 and its
dimensional area(S(t)=S
0
)grows. So at the beginning we have a ombustion front
within reasingturbulent ombustionvelo ityU
t
(t)andin reasing amebrushwidth
Æ
t (t).
Let F(k;t) be a spe trum of the amelet sheet thought as being a random
surfa e x = h(y;z;t), where k is the wave-number (k = 2= and is the
wave length). The dispersion of the amelet sheet is determined by the large
waves of the random amelet sheet (small k): 2
(t) = (x x ) 2
= R
1
0
F(k;t)dk,
whereasthe amelet areais determinedbythe smallwaves(largek): (S(t)=S(0))=
onst R
1
0 k
2
F(k;t)dk, where the onst is order unity (in the aseof Gauss random
surfa eit anbe al ulatedexa tly). Without ombustion(S(t)=S(0))in reasesvery
fast(approximatelyexponentially),butinthe aseof ombustionthe ameletprogress
suppressesthisfastin rease. Soaftersometime,thatprobablyistheorderofso alled
Gibsontime
G
=L
G
=U
f
=
t (U
f
=u 0
) 2
,where
t
=L=u 0
[9℄,wewouldhaveU
t
onst
(moreexa tlyveryslowin reasingofU
t ).
For
G
< t <
, when suppression of large wrinkles of the sheet by moving
amelet is negligible, wehave ame with in reasing brush width Æ
t
that is growing
in a ordan ewith theturbulent di usionlawÆ
t
(
2
) 1=2
=(2D
t t)
1=2
, where 2
is
thedispersionofthe amelet sheetand D
t
is theturbulentdi usion oeÆ ient. We
estimatedthetime
2D
t
=U 2
f
2
t (u
0
=U
f )
2
fromthe ondition(
2
(
))
1=2
U
f
,
i. e. when transport dueto turbulent di usion anddue to the amelet progressare
ofthesameorder(obviouslythisisalowerestimation,sin edueto u tuationofthe
ameletangles,theaveraged ameletvelo ityinsomedire tionlessU
f .
So for times
G
< t <
we have a ame with in reasing brush width
and pra ti ally onstant turbulent ombustion velo ity. We all these ombustion
fronts Intermediate Steady Propagation (ISP) ames and believe that they should
be re ognized asaspe ial lassof ames. They orrespondto a ombustion regime
whi h is intermediate between the initial stage at t <
G
, when we have intensive
forming of wrinkles on the amelet sheet and fast in reasing of U
t
(t) and the nal
stage orrespondingtot>>
,whenwehavestationaryturbulent ombustionfronts
withU st
t
= onst. and Æ st
t
= onst.
Simpleestimations showthat for realindustrial ombustors,asarule,the time
is largerthat the residen etime (that ould be 10
t
), i. e. in real ombustors
a amerea hesawalland ombustionis ompletedlongbeforeforminga amewith
onstantbrushwidth. As
G
is mu h lessthanthe residen etime in many aseswe
anassumethat ISPregime takespla e att<
. Wewill seebelowthat properties
ofISPandstationary amesandtheir ontrollingme hanismsarequitedi erent.
The3-DISP amekinemati s equationfor= onst (asthespe ial aseofthe
modelequationforthepartiallypremixed ombustion)wasproposedin[6℄:
(P
b
)=(t)+r(u P
b
)=r(D
t rP
b )+U
t jrP
b
j; (1)
whereD
t
istheturbulentdi usion oeÆ ient.
Thestatementofthisse tion: Premixed ombustionatintensiveturbulen eu 0
>>
U
f and
t (U
f
=u 0
) 2
<t <
t (u
0
=U
f )
2
takes pla e in intermediate steady propagation
(ISP) ames, i. e. in ames with U
t
onst (as approximate equilibrium between
generation anddissipation ofsmall-s aleswrinklesofthe ameletsheet ontrollingits
area is rea hed) and in reasing by turbulent di usion of the ame brush width (as
atthese timesthe regime isfar from the equilibriumbetween formationof large-s ale
sheetwrinkles ontrollingthiswidthandtheir onsumptiondueto ameletmovement.
Combustion models that based dire tly or indire tly on the only time
t
in fa t does
not ontainthis regime.
3. Mean rea tion ratesand TFC ombustionmodel
The main attention in this se tion would be devoted to the physi al equilibrium
me hanismsatdevelopedturbulen ewhi henableustointrodu eintotheTFCmodel
equationthe ouplingbetweenturbulen eand hemistry. Wewillanalyzetheintegral
ombustion rate U
t
dependen e on ontrollingparametersand thelo al ombustion
rateW distributiona rossthepremixed ame. Itwouldbeshownthatin ameswith
knownintegral hara teristi sU
t andÆ
t
thedistribution ofW and ounter-gradient
transportphenomenonare losely onne ted.
The main physi o- hemi alme hanism ontrollingU
t :
The main ontrolling me hanism responsible for mu h weaker U
t
dependen e
on hemistry in omparison with the laminar ame velo ity U
l
(more exa tly with
the a tual amelet velo ity U
f
) is the smoothing of wrinkled amelet sheet due to
moving of their elements with the velo ity U
l
. It means, for example, that amore
fast hemistry in reasesU
l
, but at thesame time impliesa de rease of the amelet
sheetarea(S=S
0
)dueto onsumption bymorefast ameletofadditionalsmall-s ale
wrinklesand makesit moresmooth. We havewhat we allhydrodynami " amelet
ombustionself- ompensationme hanism".
For stationary ames at u 0
>> U
l
in a ordan e with Damkohler[10℄,
Sh helkin[11℄ ideas U st
t
does not depends on hemistry (U st
t
u 0
), i. e. omplete
ompensation: (S=S
0
) 1=U
l
. This situation is similar to turbulentdissipation at
largeR enumbers[3℄,whenthee e tofthekinemati vis osityin reaseis ompletely
ompensatedbyade reaseof instantaneousvelo itygradients.
Wewill see belowthat for ISP ame, that takespla e at t<
t (u
0
=Uf) 2
, takes
pla eapartial ombustionvelo ity ompensation: U dependson hemistrybutmu h
weakerthanU
l orU
f
andthis ametakespla eatt<
t
(u=Uf) . Atthesametime
themostpartof ombustionmodelequationsgiveatt>
t
thestationary ame(see,
forexample,resultofsimulationin [12℄)andthis1-D amedependen eon hemistry
infa t orrespondstolaminar ames. Itisworthnotingthatthemodelthat ontains
theresultasanlimiting aseU st
t
u 0
wasdevelopedre entlybyPeters[13℄.
The parameters ofthi kened amelet and its area, U
t
dependen e:
a. Ina ordan e with [5℄transferpro essinside the thi kened amelet depend
on vorti esfrom equilibrium inertial interval and the value of the relevant transfer
oeÆ ient follows dire tly from dimensional analysis
f
"
1=3
Æ 4=3
f
, whi h is in
fa t the well-known Ri hardson law of the turbulent di usion for s ales inside the
inertialinterval(Æ
f
isthe amelet width). U
f and Æ
f
are fun tion of
f
and of the
hara teristi hemi altime
h
,andbyusingdimensionalanalysis onsiderations(for
laminar amesU
l
( =
h )
1=2
,Æ
l
(
h )
1=2
),weobtain:
U
f
(
f
=
h )
1=2
u 0
(Da) 1=2
; Æ
f
(
f
h )
1=2
L(Da) 3=2
;
f
D
t (Da)
2
: (2)
Thisformulas anbederivedstraightforwardusingtheinertialintervalspe trum
E(k)=Ck 5=3
[5℄. Therelationships(2)areequivalenttothefa tthat,ina oordinate
systemwherethethi kened amelet is xed, theheat uxesin thefrontdueto heat
transferand onve tionareofthesameorderofmagnitudeoftheheatreleasedue to
hemi al rea tions [5℄. Noti ethat hemi al dependen e of U
f
in (2) is identi al to
thatofthelaminar ame.
b. Fortheestimationof(S=S
0
)>>1dimensionalanalysisisnotsuÆ ientandit
isne essarytousealsosomegeneralpropertyofrandomsurfa esx=h(y;z;t). Our
estimationisasfollows:
(S=S
0
)=(1+jgradhj 2
) 1=2
jgradhj(jgradhj 2
) 1=2
Z
k 2
F(k)dk =;(3)
where 2
= (x x) 2
= R
F(k)dk =2D
t
t is the dispersion, is the mi ro-s ale of
the length of the random surfa e and F(k) is the spe trum of the amelet surfa e
disturban es. We see, that 2
is de ned by large s ale and (S=S
0
) by small s ale
disturban esofthe ameletsurfa e. InISP amesthespe trumF(k;t))asmallwave
numberk partis non-steady(in reasing oflargewrinkles byturbulen e)and at the
sametime alarge wavenumberpart is steady(equilibrium between generation and
onsumptionofsmallwrinkles).
The mi ro-s ale, due to theassumed equilibrium, is a fun tion of large s ale
turbulen e hara teristi sL, u 0
, amelet parameters Æ
f , U
f
and time, t. Applying
then the-theorem of adimensional analysis yields =Æ
f
=f
1 (u
0
t=Æ
f
;u 0
=U
f
;L=Æ
f ).
Taking intoa ountexpressions(2)andusingthe onditionof(S=S
0
)stationary,we
obtainthat=Æ
f
=(u 0
t=Æ
f )
1=2
f
2
(Da)(u 0
t=Æ
f )
1=2
f
2
(1)(u 0
t=Æ
f )
1=2
. Hen eusing
(3),theexpressionfortheaveraged amelet sheetareareads:
(S=S
0
)(Da) 3=4
(u 0
=U
f )
3=2
(L=Æ
f )
1=2
>>1: (4)
We see that assumption of physi al equilibriums made possible to express
parameters of U
f , Æ
f
and (S=S
0
), ontrolled mainly by small-s ale turbulen e, in
termsoflarge-s aleturbulent hara teristi s,asshownbyformulas(2)and(4).
. Expressions (2) and (4) and well known formula for the hemi al time
h
= =U 2
l
give nally theexpression of the turbulent ombustion velo ity for the
ISP ames:
U
t
=U
f (S=S
0 )=Au
0
(Da) 1=4
=Au 03=4
U 1=2
1=4
L 1=4
; (5)
where A 1 is an empiri al parameter. It is worth emphasizing that all powers
havebeenderivedfrom the physi al modeland they don't ontainany quantitative
empiri alinformation.
The hemistry dependen e of U
t
that is given by (5) (U
t
1=4
h
) is mu h
weaker than for laminar ombustion (U
l
1=2
h
). The reasonis mentionedabove
partial amelet ombustionself- ompensationme hanism: in reasingofU
f
de reasing
in a ordan e with (4) (S=S
0
) and vi e-versa. Comparison of this predi tion with
empiri al orrelationsforU
t
andtherangeofappli abilityof(5)arepresentedin[8℄.
Wehavetomentionthat,forthehypotheti al aseofthi kenedbutnotwrinkled ame
at L<< Æ
l
proposed by Damkohler[10℄ theturbulent ame speed isU
t
(D
t
=
h .
This amehaslaminar ame hemistrydependen e,i. e. morestrongthanISP ame.
Thisregimeis ontainedasalimiting aseinthenewPetersmodel[13℄.
It should be parti ularly emphasized that the thi kened amelet in our model
havenoquasi-laminarstru ture: ina ordan ewithanalysisoftemperaturepulsation
balan epresentedin [5℄thetemperaturepulsationsinside thethi kened amelet are
high,i. e. instantaneousrea tionsheetstronglywrinkledinsidethe amelet. Itseems
thatthismodel losely orrespondtothethinrea tionzoneregime[13℄. Inouranalysis
weignore thetemperaturepulsationinside a amelet. From amethodologi alpoint
ofviewitissimilartoignorethedissipationratepulsationsintheKolmogorovtheory
ofthe ne-s aleturbulen e[3℄.
Turbulent ame losure (TFC) ombustionmodel:
TFC ombustion model is based on the kinemati s equation (1) modi ed of
6= onst[7℄. This equation is asfollows (the onventional and Favreaveragingare
symbolized aand~a=a=,):
(~ )=(t)+r(~u~ )=r(D
t
r~ )+(
u U
t
)jr~ j: (6)
Eq. (6)des ribesonlyISP amesanddoesnot ontainlimiting aseof1-Dstationary
ames, it strongly simpli es the ombustion model and the same time does not
impose alimitation onitspra ti alappli ationsforsimulation ombustionat strong
turbulen e.
In TFC model the theoreti al expression (5) for U
t
as a fun tion of the
physi o- hemi alproperties of the ombustible mixture and turbulentparameters is
substitutedintheEq. (6). Inotherwordsweintrodu edire tlyinthemodelequation
thepropertiesof the ISP ames (their ombustion velo ityand width dependen es)
that why we name it the turbulent ame losure (TFC) equation. TFC model
equation simulated together with uid dynami s Reynolds equations and "k ""
turbulen e (at simulations it was assumed =
u
and lo al u 0
and L expressed
in terms of k and "). This set of equations simulated only large-s ale pro esses
but in a ordan e with the foregoing it des ribes the small-s ale oupling between
turbulen e, mole ular transport and hemistry. It isintera tion betweenturbulen e
and mole ular vis osity (dissipation rate), between turbulen e and instantaneous
rea tionzone( ameletparameters),betweenturbulen eand ameletsheet(itsarea).
To des ribe experimental bending for U
t
at regimes lose to a blow-out boundary
(Lipatnikovproposal)Braymodel[14℄wasusedofthestret he e tintermsof amelet
riti alvelo itygradientbasedinfa tontheassumptionofuniversalequilibriumPDF
forinstantaneousdissipation(i. e. forinstantaneous hara teristi velo itygradients
in small s ales ontrolling amelet extin tion). It wasin fa t the forth equilibrium
me hanismusedinour ombustionsimulations.
Good agreement with a great body of Karpov experimental data in spheri al
bombs with arti ial turbulen e for di erent fuels, air ex ess oeÆ ients and
turbulen e[15℄ (i. e. at large variation of kinemati s, mole ular transport and
turbulentpropertiesof mixtures)is good indire teviden e that their oupling using
idea of physi al equilibrium is fruitful. The omparison with Moreaudata on high
velo ity ombustion ina hannel[16℄ on rmexisten eof ISP ameswith in reasing
brushwidth.
W and the ounter-gradienttransport in the TFC model:
In the equation (6) the transport and the sour e terms are not real transport
and sour e termsof theun losed equation, i. e. r(D
t
r~ ) 6= r(u" "), and
(
u U
t
)jr~ j6=W. InTFCmodelequationtransporttermhasgradientnature,while
thetransporttermr(u" ")hasinmanyturbulentpremixed ames ounter-gradient
nature. It means that in the model equation transport term we in lude only the
gradientphysi aldi usionpart(todes ribe ameswithin reasing amebrushwidth)
whereasthe ounter-gradientpartwasin ludedinthemodelsour eterm(
u U
t )jr~ j.
So though TFC model equation des ribe physi al distributions of ~ and onne ted
with it, T and on entrationof spe ies, for extra tionof thephysi alsour e term
W =
u U
f
from themodel sour etermitis ne essary,asitwould seenbelow, to
haveadditionallysomehydrodynami modelfortheprogressvariabletransportterm.
4. Kinemati sequation. The ounter-gradient transport phenomenon.
Themodelingkinemati sequationfortheFavreaveragedprogressvariableinthe ase
of amesintheISPregimeisgivenby:
~
t +u~
~
x
=
x
D
t
~
x
+
u U
t
~
x
; (7)
where~ istheaveragedprogressvariable(~ =0and~ =1 orrespondrespe tivelyto
the old rea tantsandhot produ ts),U
t
isthe turbulent ombustion velo ity,D
t is
thephysi al turbulent di usion oeÆ ient. The model sour e termin this equation
determinespropagationofthe amewithvelo ityU
t
withrespe ttotheunburnt uid
mixturewhile thegradientdi usiontermdeterminesthethi keningof theturbulent
amebrusha ordingtotheturbulentdispersionlaw.
Thisequationmodelstheexa tbut un losedequation:
~
t +u~
~
x
=
( g
u 00
00
)
x
+ f
W; (8)
where g
u 00
00
is the progress variable ux that as a rule is generally attributed
a ounter-gradientnature and f
W is the averagedrate of produ ts formation. As
wehavementionedintheintrodu tionthereisno ontradi tionbetweenthegradient
transportterminthemodelequation (7)andtheasarule ounter-gradienttransport
termintheexa tun losedequation (8). Thisbe ausethetransportterminthemodel
equation( 7) ontainsonlytheturbulent di usion omponent ofthetotal transport
term g
u 00
00
,while these ond gas-dynami s onve tive omponent onne ted with
the di erential a eleration of hot and old volumes in a nonuniform pressure eld
(thepressure-driventransport)isintegratedtogetherwiththea tualsour eterm f
W
inthemodelsour eterm
u U
t
j~ =xjin equation(7). Obviously
u Z
+1
U
t
~
x i
dx=
Z
+1
f
Wdx; (9)
i. e. thispro eduredoesnot hangetheintegral ombustionintensity.
Thisinterpretationofthemodelequation(7) wasformulatedbyZimontin [16℄
answering to a question from F. Williams about the possibility to predi t ounter-
gradient transport within the framework of this model. The methodology for the
analysis of the ounter-gradienttransport phenomenon basedon this interpretation
wassubsequently developed in [7, 5℄ and now in this paperin moreadvan ed form.
Itshouldalsoberemarkedthat Lipatnikovwastheonewho rstrealizedin[23℄that
theinterpretationof theturbulentdi usion termin themodel equation(7) usedas
approximationofthetransport terminun losedequation (8)andthemodelsour e
termasapproximationofrealsour e f
W isnon orre t. A"joint losure"took pla e
in fa t while developingthe model givenby equation (7), su h that thesumof the
totaltransporttermandthesour etermin (8)isinsteadapproximatedbythesum
ofturbulentdi usionandmodelsour eterminthemodelequation(7). Atthesame
timeLipatnikovideapresentedin[17℄aboutthepossibilitytomodel ounter-gradient
transportisquitedi erentfromtheonedevelopedhere.
Itissigni anttorememberthatequation(7)(andthefollowingequations(13)
and (17))are valid onlyfor ameswith in reasing ame brushwidth ontrolled by
turbulent di usion, i. e. in the ase of amesin theISP regime of ombustion. In
turbulent ames omposed by laminar amelets this regime takespla e in the ase
of u 0
>> s
L
and for time t <<
t (u
0
=s
L )
2
when turbulent transport by pulsation
velo ities u 0
prevails over the transport onne ted with amelets lo al propagation
withvelo itys
L
. Inthe aseofturbulent ames omposed bythi kened amelets at
R e
t
>> 1and Da >>1 with amelet velo ity U
f
>s
L
this regime takespla e for
t<
t Da[7℄.
Forlargertimest>
t (u
0
=s
L )
2
(ort>
t
Da)turbulentpremixed amespropagate
a ordingtothe1-Dstationarypropagating ombustionfrontandinsteadofthemodel
equationwe anwriteexa tkinemati s equation oftherunning wave(in oordinate
systemwhere thewaveismotionless)
u~
~
x
=
u U
st
t
~
x
; (10)
wherein a ordan ewiththeideasofDamkohler[18℄andSh helkin[19℄, inthe ase
ofstrongturbulen ewehaveU st
t
u 0
,i.e. inthe aseofu 0
>>s
L (oru
0
>>u
F )the
turbulent ombustionvelo itydoesnotdependson hemistryandmole ularproperties
ofthefuel/airmixture.
Comparisonofequations (8)and (10)yields
u U
st
t
~
x
= f
W
(
g
u" ")
x
(11)
demonstrating that thetransport and sour e termsin theexa tkinemati equation
are lubbedunderasingletermgiveninourmodelequation(7)by
u U
t
jd~ =dxj,i.e.
theyareintimately oupledforthe1-Dstationary ame.
Inthe ase of ombustion o urring in the amelet regime (and therefore with
negligible probability to nd burning mixture) the total transport term an be
expressedas:
g
u 00
00
= ~ (1 )~ (u
b u
u
); (12)
whereu
u andu
b
are onditionedaveragedvelo itiesofunburnedandburnedvolumes.
Simpleestimationshowsthatfor>>1(strongheatrelease)andÆ st
>>L,theratio
oftheturbulentdi usion uxtothepressure-driven uxis(uL=Æ
t )=(U
t
)<<1,
i. e. the ounter-gradienttransport omponentandthea tualsour etermare lubbed
under thesingle term
u U
t
~ =x. Inthe aseof theISP ombustion regimewhere
ameshavein reasing amebrushwidthandÆ
t
<Æ st
t
wehaveextra tedtheturbulent
di usion omponent( ontrollingin ourmodelthebrushwidth) intoaseparateterm
andleft theremaininggas-dynami stransportandthe hemi alsour e lubbedinto
themodelsour eterm.
A ording to these ideaswhi h were originally proposed in [5,7℄), theprogress
variable ux an be split into two ontributions: one of gradient nature generated
by real turbulent di usion transport whi h has the property to in rease the ame
brushthi kness(a ordingtotheturbulentdispersionlaw)andthese ondof ounter-
gradient nature whi h is of onve tive type and is generated by the pressure drop
a rossthe turbulent amebrush. The rst omponent ofthetransport is ontrolled
bypositiveturbulentdi usion oeÆ ientthat anbeestimatedusingusualturbulen e
models andthe se ond omponent an beestimated from somegas-dynami smodel
thatisstrongly onne tedwiththenonuniformaveragedpressuredistribution(inour
aseofopen ames,withtheaveragedpressuredropa rossthe ameandresultingin
the ounter-gradientgas-dynami al omponent).
The possibility of su h de omposition an be better understood observing the
exa texpressionfor theprogressvariable ux (12). Thisexpression showsthat the
turbulent s alar ux g
u 00
00
has gradient ( ounter-gradient)nature when u
b
<u
u
(u
b
> u
u
). In absen e of a negative pressure gradient whi h an a elerate the
produ tsmorethantherea tants,thee e tofturbulentdispersionwillbetogenerate
penetration of burnt mixture inside the unburnt one at the old boundary and
penetrationofunburntmixtureinside theburntoneat thehotboundary. A ording
to this des ription the onditional velo ities will be su h that u
b
< u
u
. In ase
presen eofapressuredropa rosstheturbulent amebrush,thelightburntgasesare
a eleratedmorethan theunburntones. This mayredu e thegradienttransport or
eventransformitintothe ounter-gradientphenomenonwhenu
b
>u
u .
The onne tion of the ounter-gradient transport with the averaged pressure
gradientis wellknown. Inre entDNSwork,ithas beenshownawell-de ned e e t
ofpressurepulsations
(BRAY ite?).
Thise e tis onne tedwiththeinstantaneousdi eren eof onditionalaveraged
pressuresof unburnedp
u
and burned gasesp
b
asthe boundary betweenthese gases
isamoving ameletsthat resultsp
u
>p
b
. Therefore,forexample,evenatp= onst
there is some ounter-gradient omponent. In our simulationsbelow we ignore this
e e t.
Itmustbeobservedthat,be auseofthe onve tivenatureofthepressure-driven
transport, while developing the TFC ombustion model we have assumed that it
doesn't have the apability to hange the thi kness of the ame boundary layer, a
propertywhi histypi alofturbulentdi usionphenomena.
Subtra tingnowequation(7)from (8) weobtainthefollowingrelation:
u U
t
~
x
= f
W +
x
g
u 00
00
D
t
~
x
(13)
whi h formally showsthat the model sour etermin (7) a ounts atthe sametime
for real heat release and the pressure-driven ( ounter-gradient in our ase of open
transport term from ( 13) two approa hes may be used: a) give a model for the
averageheatrelease f
W,b)modeldire tlythe ounter-gradienttransport omponent
(inthis aserelation(13)willgiveanexpressionfor f
W).
In[7℄ approa ha) hasbeenfollowed. It wasassumed that the a tual averaged
hemi al sour e term in equation ( 8) is proportional to the probability to nd the
amelet at a given position p
f
(x); this probability is related to the probability of
ndingprodu tsP
b
(x)at thegivenpositionbytherelation:
P
b (x)=
Z
x
1 p
f
()d)p
fl (x)=
P
b (x)
x
(14)
Notealsothefollowingexpressionfortheaveragedprogressvariable(as ameletsare
thin,theprobabilitytohave0< <1hasbeennegle ted):
=P
b
1+(1 P
b )0=P
b
(15)
Thereforewe anwrite:
f
W = onst
x
(16)
where the onstant is equalto
u U
t
as anbe shown by integrating equation ( 13)
from 1to+1.
Using(16)in(13)thefollowingexpressionforthese ondordervelo ity-progress
variable orrelation is obtained (where it has assumed d =dx > 0 without loss of
generality):
( g
u 00
00
)
x
=
u U
t
x
(~ )+
x
D
t
~
x
(17)
whi h integratedfrom 1toxyields:
g
u 00
00
=
u U
t
(~ )+D
t
~
x
(18)
Thisrelationexpli itlygiveseviden ethatthese ondorderFavre orrelationbetween
theprogressvariableandvelo ity u tuationsis omposedoftwo ontributions:
a) realturbulenttransport(modeledherewithaneddydi usivityassumption)whi h
isresponsibleforthethi keningoftheISP amebrush;
b) a ontributionwhi hisproportionalto theintegralofthedi eren ebetweenthe
model
u U
t
d~ =dx and thereal hemi al sour e term f
W =
u U
t
d =dx. This
ontribution anbeexpressedwithsimplealgebrai manipulationsas:
u U
t
(~ )=
u U
t
~ (1 )~
1
1+ ~( 1)
(19)
It should be again emphasized that thepoint of viewadopted hereis that the
turbulent ame brush in reases (here this is assumed to be an e e t of turbulent
dispersion), independently on the nature of the progress variable ux (of gradient
or ounter-gradienttype). Nevertheless it sometimes stated that in ase of a total
s alar ux of ounter-gradienttypethe amebrushbe omesthinner whereasin ase
ofgradienttypeitbe omesthi ker[11℄.
Wethinkthatthispointofviewisobtainedasresultofinterpretingtheprogress
variable ux in turbulent premixed ames only as a turbulent di usion ux of
passive on entration. Theterm" ountegradientturbulentdi usion"isinfa t often
asso iatedwiththephenomenonof ounter-gradienttransportwhi h,westresshere,
isnot onne tedwithanegativevaluefortheturbulentdi usion oeÆ ient.
We believe that in open ames a de reasing ame brush width is highly
improbable. Numerousexperimentsshowinfa tthatthe amebrushwidthin reases
with distan ealong the ame similarly to adi using wakedespite of the ommonly
observed ounter-gradient nature of the transport in ames and in prin iple this
thi kness in reases until a "statisti ally steady" nite value whi h is unattainable
in pra ti al ombustion devi es. At the same time, in presen e of strong external
pressuregradient( ombustion inanozzle,forexample) thequalitativepi ture ould
bemore ompli ated,butwedonotanalyzethis asehere.
Asalreadymentionedthenature of theprogressvariabletransport (gradientor
ounter-gradient) anbedeterminedbyanalyzingthedi eren ebetweentheaverage
onditional velo ities u
u and u
b
respe tivelyin the oldand hot gasesas shown by
relation(12). Solvingthesystemofequationsgivenby(12)andthemass onservation
equationinthetwounknownu
u andu
b :
~(1 ~ )(u
b u
u )=
u U
t
(~ )+D
t
~
x
(20)
(1 ~ )u
u + ~u
b
=[1+( 1)~ ℄U
t
(21)
weobtainthefollowingexpressionforthe onditionalvelo ities:
u
u
= D
t
1 ~ d~
dn +U
t
(22)
u
b
= D
t
~
d~
dn +U
t
(23)
These relations show that ea h of the onditional velo ities is omposed by two
ontributions: onerelatedtoturbulentdispersionandtheotherto onve tion.
In the ase of a steady ame (t ! 1, onstant thi kness and velo ity) the
ame is des ribed by the equation (10), i.e. in 22 and 23 we must put D
t
= 0,
resulting in the two onditional velo ities being onstant a ross the ame brush,
u
u
=U st
t
=u( 1)and u
b
=U st
t
=u(+1). Thisalsoimplies onstant onditional
pressures(p
u
=p( 1),p
b
=p(+1))asitwillbeshowninthenextparagraph. Su h
value of velo ities orrespond in fa t to the assumption that old and hot volumes
move inside the ame without mutual for e intera tion. Obviously this strongly
overestimates the ounter-gradient transport (for typi al onditions approximately
three times, see below) and orresponds to the possible upper boundary of this
phenomenon. Su h model for estimationof the ounter-gradientphenomenon based
ontheexpression(16)fora tual ombustionratesandfurtherusingoftheequation
(13)forestimationofthetransport g
u 00
00
wasdevelopedin[7℄whereitwasusedfor
estimationofthe ounter-gradientphenomenonin wellknownstandardMoreauhigh
velo itypremixed ombustionexperimentaldataina hannel. Wewill allbelowthis
model"aroughmodel"or"upperestimationmodel".
More realisti modeling must take into a ount this intera tion, i. e. the
a elerationofrelativelyslow oldvolumesbymorefasthotvolumesandvi eversa.
Su hgas-dynami smodelthatestimatesdire tlythepressure-driven ounter-gradient
omponent (des ribed in ( 13) by an expression between the bra kets is developed
in the next se tion. Forestimation of a tual ombustion rate itis ne essary to use
the equation ( 13). We will all this model for estimation of the ounter-gradient
phenomenon and a tual ombustion rates"an a uratemodel"or"a gas-dynami s
5. A gas-dynami smodel of the ounter-gradienttransport phenomenon
In this se tion we analyze 1-D stationary premixed ames. These results then
will be applied to open ames with in reasing ame brush width. The basis of
it is an assumption that at real relatively slow in reasing of the brush width the
pressuregradienta rossopen amesbringingaboutthe ounter-gradientphenomenon
an be assumed the same as in 1-D stationary ames. Good a ord between
resultsof numeri al simulationsof this phenomenon and experimental experimental
data demonstrated below on rm the truth of this hypothesis at least as a rst
approximation.
A ording to the idea presented in se tion 4, the progress variable transport
in open turbulent premixed ames is omposed by two fundamental ontributions:
real gradient turbulent di usion whi h is responsible for the growth of ame brush
thi knessand ounter-gradienttransportrelatedtothepressuredropa rossthe ame
brushdue to heat release. Inthis se tion wewill presentagasdymami s model for
thepressure driven omponentof the progressvariable transport in the aseof 1-D
stationary ameswhi hwillbeappliedtorealopen amewithin reasing amebrush
width.
Thebasisforthisanalysisisrepresentedbyequations(10)-(12). Forestimating
thepressure-driven omponentweassumeherethatthe onditionalvelo itiesin(12)
are ontrolledonlybygas-dynami s,i. e. weignorethee e tofturbulentdispersion.
Inthis asetheterm g
u 00
00
in ludesonlythe ounter-gradienttransport omponent,
andinfa tthemodelsour e(11)oftheequation(10)term ontainsthea tualsour e
termandthepressure-driven ounter-gradient omponent. Theresultsobtainedunder
these onditions anthereforebeextended to ISP amesbyusing equation (7). In
this equationin fa t thegradienttransport termis ontrolled byturbulentdi usion
and themodel sour eterm ontainsthe a tual sour e termand the pressure-driven
ounter-gradient omponent.
Considerthereforethe onservationequationsformass,momentumandrea tants
totalpressure(iso-entropi onditionforrea tants). Theequationshavebeenwritten
hereforU st
t
=1m=sand
u
=1kg=m 3
(orequivalentlyin normalizedform):
(1 )u
u +
u
b
=1 (24)
p+(1 )[u 2
u +u
02
u
℄
+ 1
[u
2
b +u
02
b
℄ =p
1
+1+u 02
1
(25)
p
u +
1
2 [u
2
u +u
02
u
℄ =p
1 +
1
2 (1+u
02
1
) (26)
wheretheaveragepressurepisgivenbyp=(1 )p
u + p
b
. Inthe aseintrodu ed
in the previous paragraph we have seen that f
W =
u U
t dP
b
=dn implies uniform
onditionalvelo itiesa rosstheturbulent amebrushandequaltou
u
=1andu
b
=
in the present ase. In this ase equation (24) be omes redundant and thesystem
thereforeredu estotwoequationin thetwo onditionalpressuresasunknowns.
If we subtra t equation ( 26) from equation ( 25) and require the u tuating
termsto an el,weendupwiththefollowingtworelations:
(p
b p
u )+
1
2
u 2
u +
1
u
b 2
= 1
2
(27)
2
u 0
u +
u
0
b
=
2 u
02
u
1
(28)
These ondoftheserelationsgivesaturbulentvelo ity u tuationatthehotboundary
whi h is equal to =2u 02
u 1
. This result is onsistent with several experimental
works where the turbulent velo ity u tuationhasbeen foundin reasing a rossthe
turbulent amebrush. Forexample,Moreau[26℄ has foundin oblique high velo ity
planarmethane/air amesatequivalen eratio0:8thattheaxialturbulent u tuating
velo ityin reasesinsidethe ame,rea hingavalueaboutthreetimesthatmeasured
inthe old ow.
Intheparti ular aseof u
u
=1and u
b
=analyzedin thepreviousse tionwe
havep
u p
b
=( 1). Obviously thisis an hypotheti alunrealisti ase. We will
insteadanalyzeherethe asewithstronginterra tionbetweenhotand oldvolumes
with an assumption p
u
= p
b
= p, i. e. with ignoring the small di eren e in the
presuurebeforeandbehind ameletsheetthatismovingboundarybetween oldand
hotvolumesaswealreadymentionedabove.
Underthisassumptionwehavefromequation(27):
( 1
2 )u
2
u +
1
u
2
b
= 1
2
(29)
The system given by ( 29) and ( 24) an be solved with simple algebrai
manipulations. Thisyieldsthefollowingexpressionsforu
u andu
b :
u
b
= +
p
4 + 2
2
; u
u
=
1 u
b =
1
(30)
=
0:5
(1 ) 2
2
2
+
; = 2
0:5
(1 ) 2
; =
0:5
(1 ) 2
0:5(31)
Theexpression obtainednowforthe onditionalvelo ities givestheopportunity
to al ulatetheaverageheatreleasewithintheturbulent amebrushusingequation
( 13). If we assume =x to be distributed as a Gaussian fun tion (validation of
this hypothesis has been performed in [8℄ ontaining very areful measurement of
thetemperaturepro lesinpremixed amesandalsoin[32℄ ontaininganalysissome
literatureexperimental data that demonstratethe omplementary errorfun tion for
thepro les):
x
= 1
p
2 2
e
(x a(t)) 2
=2 2
(32)
introdu ing the non-dimensionalspatial variable = (x a(t))=
p
2 2
, the average
heatreleasenondimensionalisedwith
u U
t
= p
2 2
isgivenby:
f
W
u U
t
= p
2 2
= 1
p
e
2
~
(1 ~ ) u
b u
u
U
t
(33)
Inthehypotheti al aseof onditionalvelo ities onstanta rossthe amebrush,this
expressionyieldstheresultalreadyintrodu edatparagraph,i.e. f
W =
u U
t
j =xj.
Figure 1showsthedistributionofthe onditionalvelo itiesnon-dimensionalisedwith
U
t
and non-dimensionalheat release, = and ~ =. The heat releaseis shifted
toward the front part of ame brush with respe t to the model sour e term, the
maximumshift obtainedfor themodel basedon uniformdistribution of onditional
pressuresand velo ities a ross the turbulent ame brush(when u
b u
u
attains its
for e intera tion between high velo ity hot and low velo ity old volumes that we
analyzedassumingp
u
=p
b
theheatreleaseislessshiftedtowardthefrontpartofthe
amebrush.
It is interesting to note that su h situation has been observed experimentally
by [10℄ and [15℄ who analyzed a large dataset of experimental data on turbulent
premixed ameofvarioustypes(v-, oni al,stagnation and swirlstabilized ames).
The experiments from [10, 15℄ will be onsidered in paragraph 7 for a qualitative
validationoftheresultobtainedwiththepresentmodel.
6. Modeling of the turbulent ame speed
Theturbulent amespeedU
t
ofa1-D ameisgivenbytheprodu tbetweenthelo al
propagation velo ity U
f
and the surfa e areafor unit of ame ross se tional area
(S=S
0
) of the thin amelet, i. e. U
t
= U
f (S=S
0
). These two quantities havebeen
estimated by asymptoti analysis at largeturbulent ReynoldsR e
t
and Danumbers
basedontheKolmogorovmethodology: theequilibrium ne-s aleturbulen efromthe
inertialinterval ontrols thethi kened amelet ombustion velo ity giving U
f
> s
L
and amelet widthÆ
f
>Æ
L
,theequilibrium smalls ale amelet wrinkles ontrolthe
pra ti ally onstant amesurfa eareaand nallynon-equilibriumlarge-s alewrinkles
ontrolthe amebrushwidthwhi hin reasessimilarlytoanon-rea tivemixinglayer
[4,7℄.
Inthismodelthemi ro-turbulenttransportofheator on entrations(
f 'D
f )
withinthe ameletsthi knessisdes ribedbythewell-knownRi hardsondi usionlaw
f ' "
1=3
Æ 4=3
f
where " is the averaged rate of turbulent kineti energy dissipation.
Thefollowingexpressionshavebeenobtained:
U
f
u 0
Da 1=2
; Æ
f
LDa 3=2
;
f
D
t Da
2
)
(S=S
0 )Da
3=4
(u 0
=U
f )
3=2
(L=Æ
f )
1=2
: (34)
A ordingtothe rstoftheserelationsthelo alpropagationvelo ityofthethi kened
amelet in reases with de reasingDamkohler number (for example,redu ing of the
hemistry rate). This an be explained onsidering that aredu tion in Damkohler
numberprodu esanin reaseinthe ameletthi kness(these ondrelation). Itresults
thereforeanin reasein themi ro-turbulenttransport oeÆ ient(the third relation)
be ause ofsmall vorti esin theinertialrangefalling inside the amelet whi h being
dominant over the redu tion in hemistry rate produ es an overall in rease in the
lo al propagation velo ity. The fourth expression shows that there is onne tion
between the amelet area and the amelet velo ity: an in rease in the amelet
velo ity(orin reasingofthe ameletwidth)produ eade reasein the amelet sheet
area. Thephysi alreasonofthise e tisself-smoothingofthewrinkledsheet butits
movement(orduetoin reasingofthe ameletwidth): anin reaseinU
f
(orde rease
inÆ
f
)" onsumes"thesmallestexistingwrinklesanditde reasesthearea. This self-
ompensationme hanismisresponsiblefortherelativelyweak hemistrydependen e
ofU
t
shownbyexperiments.
The nal expressionfortheturbulent ombustionvelo ityisasfollows:
U
t
=U
f (S=S
0 )=Au
0
Da 1=4
=Au 03=4
s 1=2
L
1=4
L 1=4
; (35)
where A 1 is an empiri al oeÆ ient, is the mole ular heat transfer oeÆ ient
that(referred totheunburnedmixture inthesimulations,i. e. =
u
). It isworth
