CRS4
Centre for Advan ed Studies, Resear h
and Development in Sardinia
Cagliari - Italy
CRS4-CCT
Transport equations for an in ompressible
rea tive ow with two separated phases.
VERSION 1.6
July 9, 2003
Prepared by:
V. Moreauand V. Battaglia
Clean Combustion Te hnology Area
eling in the approximation leading to the sole use of a progress variable to
hara terize themixture state. First, we shortlydes ribe the TFC modelto
setthe problemati . Then were all the onstitutive instantaneousequations
andtheiraveraged ounterparts. Weexamineindetailsthelimit aseinwhi h
the rea tion takes pla e in an in nitelythin sheet. This situation isformaly
identi alto themodellingoftwo separated uidsex hangingmatterthrough
their interfa e. In this ase, their is an exa t dependen e of the so- alled
ounter-gradient transport termon thesour eterm. The form of thedepen-
den eprovesthe outer-gradientnatureofthetermwhi hwasuptonowonly
intuited. It also shows that their is fundamentalyonly one un losed termin
theaveraged progressvariable equation. We examinethe losure assumption
ofthesour eterminthisframeworkand naturallyre-derivetheTFCmodel.
We propose the basi idea for a slight improvement of the TFC model that
takes into a ount the nitespeedof the in reasingbrushwidth. The treat-
ment ofthelimit aseisdonebyuseofgeneralizedfun tions. Theyarequite
deli ate to manipulate and their is no strong familiarity with them in the
ombustionmodelling ommunityleadingtothee e tivediÆ ultyinjudging
the orre tnessof theresults. Forthisreason, were-derivethe orresponding
resultsinthegeneral aseof regular fun tions.
1 Introdu tion 2
2 System of equations 2
2.1 Instantaneousequations . . . 2
2.2 Averaged equations . . . 3
3 Derivations for the bimodal in ompressible ase 4 3.1 Additionalequations . . . 4
3.2 Un losed expression . . . 6
3.3 Closedexpression . . . 6
3.4 Closureof theprogressvariable equationsour eterm. . . 8
4 Extension to the general ase 9 4.1 Formulain thegeneral ase . . . 10
4.2 Results inthegeneral ase . . . 11
5 Con lusion 13
A Additional formula 14
1 Introdu tion
TheTurbulentFlameClosure(TFC)model[1 ℄requirestheuseofatransport
equation for the progress variable giving the state of progress, normalised
between 0 and 1, of the ombustion pro ess. The un losed equation for the
progressvariablereads:
t
(~ )+r:(~u~ )= r:(
g
u 00
00
)+
~
_
!; (1)
whereisthegasdensity,tisthetimeand uisthevelo ity. Theterm g
u 00
00
is the progress variable turbulent transport and
~
_
! is the hemi al sour e
term.
Both Reynolds averages (denoted by over-line) and Favre averages (de-
noted by tilde) su h as ~ are used, with " = ( ).~ For the amelets
me hanism ofpremixed ombustion,we assumethatP
u +P
b
=1asP
i
<<1,
where P
u , P
b
and P
i
are the probabilities of unburned mixture, ombustion
produ tsandintermediate ompositions. Inthis asetheaverage densityand
the progress variable have the form =
u P
u +
b P
b
, and ~ = (
b
= )P
b . It
is fundamentally more orre t and a general pra ti e to use an equation in
termsof ,~ theFavreaveragedprogressvariable,insteadoftheprobabilityP
b .
Physi almeaning of e
isquite lear. It isinfa tthe oeÆ ientof ombustion
ompleteness: e = 0 refers to rea tants, it in reases a ross the ame (inside
premixed ame 0<
e
<1) and e
=1refersto produ ts.
The losedequation oming fromtheTFC modelis asfollows:
t
(~ )+r:(~u~ )=r:(D
t
r~ )+(
u U
t
)jr~ j; (2)
Forthemotivationofsu ha losure,seeforexample[2 ℄anditsreferen es.
Here, we want to show that in the limit ase of fast hemistry where the
fun tion degenerates into the hara teristi fun tion of the burned phase,
then themean transportequation of ~ an be derived analyti ally withonly
one termrequiringa losureassumption. Wealso deriveina pre isewaythe
orre t expression of the transport term of the momentum equation. This
termis losedinthedire tionofr~ ,buta losureassumptionisstillrequired
inthedire tions parallelto theiso-surfa es of~ .
2 System of equations
2.1 Instantaneous equations
We onsiderthefollowingset ofinstantaneousequations.
Mass onservation:
+r(u) = 0; (3)
Progress variable equation:
t
+ur = _
S
; (4)
this form be omes unde ned if u is dis ontinuous where is also dis ontin-
uous, so we add the ontinuity equation to get a onservative form whi h
ir umnavigatesthisproblem:
t
( )+r(u ) = _
S: (5)
The simplestpossibilityforthesour eterm _
S is:
_
S =
? U
f
jr j; (6)
des ribing a variable mass onsumption velo ity rate su h that the velo -
ity variations dueto the density variations are exa tly ompensatedto keep
un hanged the pro le of (
?
is a onstant). The meaning of U
f
hanges
a ording to thevaluetaken by
? .
Momentum equation:
t
(u)+r(uu)+rP rru = 0: (7)
2.2 Averaged equations
The averaging ofthe formersetleadsto the followingequations.
Mass onservation:
t
+r(~u) = 0; (8)
Progress variable equation:
t
(~ )+r(~u~ ) = _
S+r(~u~ ) ru (9)
The simplestpossibilityforthesour eterm _
S is:
_
S =
? U
f
0
jr j; (10)
orintermof theFavre averaging:
_
S =
? U
f
0
2
u
b
jr~ j; (11)
where
0
isageometri almultiplieroftheinterfa elengthovertheminimum
(straight)interfa e length.
Wewillseethattheintrodu tionofthisexpressionintheprogressvariable
equation for the bi-modal ase leads to the same equation as for the TFC
model but without di usion term. This means that the di usion term is a
property of the sour e term, or in other words a property of the interfa e
dynami s. Thisalso meansthatthismodellingassumption isinsuÆ ient.
Momentum equation:
(~u)+r(~u~u)+rP rr~u = r(~uu)~ ruu: (12)
3 Derivations for the bimodal in ompress-
ible ase
We onsiderthelimit aseofin nitefast hemistrywheretheprogressvariable
an be only either zero or one and therefore oin ide with the hara teristi
fun tion of the burned side. We also onsiderthat either the unburned and
theburned sidesarein ompressible.
3.1 Additional equations
In the bimodal approximation of the ombustion, the ow is omposed of
the juxtaposition of two in ompressible ows, one is the unburned or fresh
mixture(subs riptu),theotheristheprodu tsorburnedmixture(subs ript
b). The region where lies the mixtures is de ned by the value of a s alar,
named withvaluezero forthefresh mixtureand onefor theprodu ts. The
variable is some kindof limiting ase of the lassi alprogress variable. We
express hereafter these on ept in mathemati al form. The problem is that
the variables u
b and u
u
(but also the onstant
b and
u
) have no physi al
meaningrespe tivelyintheunburntandburntregion. But,be ause wewant
to write formula valid in all the domain onsidered we must extend u
b and
u
u
to the whole domain. The extension an be arbitrary butwe want to be
authorisedto usethestandardderivationrulesliker(u
b )=u
b
r + ru
taking into a ount that is a dis ontinuous fun tion. For this reason, we
hoose to onsiderregular extensions of u
b and u
u
, preservingat least their
ontinuity and the ontinuity of their divergen e. Similarly,
b and
u are
extended to a onstant value all over the domain. Note that we an not
extend thedivergen e free propertyof u
b and u
u
to theentire domain when
the interfa e has losed ontours, but we will not need it. Nevertheless, by
ontinuity,u
b and u
u
aredivergen efree also ontheinterfa e. Thisproperty
wasalready eviden edin[℄.
In mathemati al form,it an beexpressed as:
ru
b
= 0 (13)
(1 )ru
u
= 0 (14)
Density,pressure, velo ityand momentum:
=
b
+(1 )
u
(15)
P = P
b
+(1 )P
u
(16)
u = u
b
+(1 )u
u
(17)
u = u +(1 ) u (18)
uu =
b u
b u
b
+(1 )
u u
u u
u
(19)
Note that whilethese rst two equationsare infa tde nitions,thethree
othersareproperties. Spe i allyu is thevelo ityasso iatedto .
These de nitions lead to additional properties that an prove useful for
thederivation of theun losedterms. For example:
=
b
(20)
u =
b u
b
(21)
= 2
(22)
(1 ) = 0 (23)
and some meanvariable properties(u
b and u
u
are onditionalaveraging):
= (
~
b +
1 ~
u )
1
(24)
~ = (
1
1
u )(
1
b 1
u
) (25)
=
b
+(1 )
u
(26)
~ =
b
(27)
(1 ~ ) =
u
(1 ) (28)
u =
b u
b
+(1 )
u u
u
(29)
~
u = ~ u
b
+(1 ~ )u
u
(30)
~
u = u
b
(1 ~ )(u
b u
u
) (31)
~
u = u
u + (u~
b u
u
) (32)
u = (
~ u
b
b +
(1 )u~
u
u
) (33)
= u~+(
1
b 1
u
)~ (1 ~ )(u
b u
u
) (34)
uu =
b u
b u
b
+(1 )
u u
u u
u
(35)
u =
b u
b
(36)
We add some properties spe i of the in ompressibility, with eventual
un ertainty linked to the variables value at the interfa e avoided thanks to
ourprolongation hoi e foru
b and u
u :
ru
b
= 0 (37)
ru
u
= 0 (38)
ru~ = (u
b u
u
)r~ (from31) (39)
3.2 Un losed expression
S alartransport(see formulainappendix):
u =
b u
b
(40)
= ~ u
b
(41)
= ~ [~u+(1 )(u~
b u
u
)℄ (42)
u = ~ ~u+~ (1 )(u~
b u
u
) (43)
Momentum transport:
uu =
b u
b u
b
+(1 )
u u
u u
u
(44)
u
b u
b
= u
b u
b +Dt
b
(45)
u
u u
u
= u
u u
u +Dt
u
(46)
Dt =
b Dt
b
+(1 )
u Dt
u
(47)
uu =
b u
b u
b
+(1 )
u u
u u
u
+Dt (48)
= ~ u
b u
b
+(1 ~ )u
u u
u
+Dt (49)
Noting that:
~
u~u = [u
b
(1 ~ )(u
b u
u )℄[u
u +~ (u
b u
u
)℄ (50)
= u~
b u
b
+(1 ~ )u
u u
u
~
(1 ~ )(u
b u
u )(u
b u
u
) (51)
it omes:
uu = ~u~u+~ (1 ~ )(u
b u
u )(u
b u
u
)+Dt (52)
3.3 Closed expression
We do not need to know the previous uxes but onlytheir divergen e. The
problemisto eliminatethetermsusingtheun-resolvedvariableu
b u
u . The
methodistousethestrongdependen eoftheprogressvariableequation and
mass onservation equation. Starting from this later one (equation 8), we
have:
t
+u~r = ru:~ (53)
we usethat is fun tionof ~to expresstheLHSinterms of :~
~
(
t
(~ )+u~r~ ) = ru;~ (54)
Multiplyingby and re-usingthe ontinuityequation, we have:
~
[
t
(~ )+r(~u~ )℄ = 2
ru;~ (55)
and usingtheexpressionof ru,~ we get:
~
[
t
(~ )+r(~u~ )℄ = 2
(u
b u
u
)r~ : (56)
Onanotherhand,usingtheprogressvariableequation 9and equation43,
we have:
t
(~ )+r(~u~ ) = _
S r[~ (1 ~ )(u
b u
u
)℄ (57)
and usingthein ompressibilityof u
b u
u :
t
(~ )+r(~u~ ) = _
S (u
b u
u
)r[~ (1 ~ )℄ (58)
t
(~ )+r(~u~ ) = _
S
~
[~ (1 )℄(u~
b u
u
)r~ (59)
Equations 56 and 59 form a system of two equations in the unknown
t
(~ )+r(~u~ ) and (u
b u
u
)r~ whi hsolutionis:
t
(~ )+r(~u~ ) = 2
f
2
~
~
[~ (1 ~ )℄g 1
_
S (60)
(u
b u
u
)r~ = f
2
~
~
[~ (1 )℄g~ 1
~
_
S (61)
Expanding thederivatives, al ulation nallygives:
t
(~ )+r(~u~ ) =
b
u
2
_
S (62)
(u
b u
u
)r~ =
u
b
2
_
S (63)
The term r(u ~u~ ) whi h was expe ted to have a ounter-gradient
nature [2℄ an be now des ribed in termof themean (positive)sour e term.
It omesfrom43 and 106:
r(~u~ ) u ) = [
~ 2
b
(1 )~ 2
u
℄(
u
b )
_
S: (64)
This formula demonstrates the ounter-gradient, or usingan alternative ter-
minology,theanti-di usive nature of thisterm. Ine e t, it lowers thevalue
of ~ when ~ is smalland in reases thevalue of ~ when ~ is loseto one, thus
sharpeningtheinterfa e.
Momentum transport:
The momentum transport needs a losure assumption for the expression
of(u
b u
u
)inthedire tionsparallelto theiso-surfa esof ~whileitisalready
knowninthedire tionofr~ . Ifwe makethesimplestassumptionthat(u
b
u
u
) and r~ are parallel. That is:
(u
b u
u
)^r~ = 0; (65)
orequivalently
(u
b u
u
)r~ = ju
b u
u
jjr~ j; (66)
sothatuuis losed. Thishypothesislooksveryreasonablefortheappli ation
inmind,atleastwhere~ isnottoo losetozeroorone. Underthisassumption,
we have:
u
b u
u
=
u
b
2
_
Sn~ (67)
wheren~= r~
jr~ j
andusingequation52negle tingthedi usionterm,it omes:
uu u~u~ = (
u
b )
2
3
~ (1 ~ )
_
S 2
~
n~n: (68)
3.4 Closure of theprogress variable equationsour e
term
Theanalysisofthebimodal aseleavesonlyonefundamentallyun losedterm,
the sour e term. The \turbulent transport" whi h is usually un losed in a
more general ase result to be algebrai ally related to the losure of the un-
losedsour eterm, _
S. Ithasalreadybeenseenthatthe\turbulenttransport"
isananti-di usivetermr(u ~u~ )isananti-di usiveterminour aseand
a t as a multiplier by
b
u
2
of _
S so that the determination of _
S ompletely
losesthe equation.
Forthe appli ation foreseen (turbulent ombustion with ame an horag-
ing),the main hara teristi of the ame is an instantaneousstrongly orru-
gatedthin sheet in ludedinan in reasing brushwidth.
The representation of _
S in theform
? U
f
0
2
u
b
jr~ jtakesinto a ount
only the highly orrugated aspe t of the instantaneous ame but not the
in reasing brushwidth. In thisdis ussion, we will onsider that ea h aspe t
an be onsideredseparately and that _
S isthe sumof two ontributions,the
rst one, _
S
tr
, hara terising the global ame speed by means of a standard
transport term and the se ond, _
S
diff
hara terising the in reasing brush by
meansof adi usingterm.
_
S =
_
S
tr +
_
S
diff
: (69)
The modelling an nowbe reformulatedas:
jr j =
0
jr j+ di usione e ts; (70)
leadingto:
_
S
tr
= (
? U
f )
2
jr~ j: (71)
Forthemodellingof
0
,refertotheTFCmodel. Someimprovementmay
beforeseen bygiving a dependen eof thistermon ~ .
In theTFCmodel,thedi usivetermismodelledasa standardturbulent
di usiontermwhoseexpressioninframe of ourderivation is:
_
S
tr
=
2
u
b r(D
t
r~ ): (72)
On e again,refer to theTFC model[1℄ fortheexpressionof D
t .
Thisexpressionfor _
S
tr
hasto be onsideredasaroughapproximation. In
e e t, being ase ond order terminspa eina rst orderequation intime, it
gives a paraboli nature to the equation. In other word, thedi usion takes
pla e at in nite speed, whi h is largely exaggerated in this ontext where
the di usion should takes pla e at nite speed roughly proportional to the
turbulent pulsation. To solve this problem, one has basi ally two solutions,
bothaimed at giving an hyperboli hara ter to theequation. The rst one
isto to keeptheformerdi usive termand add aterminthe equation whi h
is se ond order in time term. In fa t, whemn the mean velo ity is not zero,
thismethodleadstothein lusionofseveralse ondordertermsgivingamu h
more ompli atedequation,notlikelytobeimplementedin ommer ialCFD
tools. The se ond methodis to generatea rst order inspa edi usive term.
The di usive termmaybeof a form similarto:
_
S
diff
=
2
u
b
(1 2~ )U
diff
jr~ j (73)
whereU
diff
isorder theturbulentpulsationand oin idewiththeexpan-
sion velo ityof the ame brushwidth.
4 Extension to the general ase
Theformerresultshavebeenobtainedinthelimit aseofin nitelyfast hem-
istrywhere theprogress variable takes onlythevalueszero and one and has
probability zero to have an intermediate value. The advantage is that a lot
of relations are highly simpli ed. Mainly, the mean of the progress variable
oin ide with the probabilityof being in the burned gas. The drawba ks of
thisapproa his thatwe areobligedto deal withgeneralised fun tionswhose
orre t manipulation and understanding is quite deli ate and not so widely
widespread. One way to demonstrate that the formerderivationshave been
done properly is to obtain the same result asthe limit for thin ame of the
general ase, thereforemanipulatingonlyregularfun tions. The omputation
is mu h more diÆ ult to inferin the general ase, but aswe know the nal
Inthegeneral ase, we splittheprobabilityspa e inthree omplementary
domains:
P
b
istheprobabilityof being intheburned regionwhere =1,
P
u
istheprobabilityof being intheunburnedregionwhere =0,
P
i
istheprobabilityofbeingintheintermediateregionwhere0< <1.
Indi es b,uand i willrefer respe tivelyto the onditional probabilitiesof
beingintheburned,unburned andintermediateregions.
4.1 Formula in the general ase
Be ause we onsider an in ompressible gas and the rea tion is supposed to
take pla e onlyinside theintermediaryregion, we have:
ru
b
= 0; (74)
ru
u
= 0; (75)
b
=
b
; (76)
u
=
u
: (77)
Intheformerderivations,westronglyusedthefa tthatand ~arelinked
throughsimplealgebrai relations. Aswe willusethese relations,they must
keept true in the general ase. This is obtainby requiringan aÆne relation
between and, thatis =a+bwherea andb are onstant. Thisleads
to thefollowingrelation, infa tde ningtheprogress variable :
= (
b +
1
u )
1
; (78)
leadingto:
= (
~
b +
1 ~
u )
1
; (79)
~ = (
1
1
u )(
1
b 1
u
): (80)
The aÆne relationleadsto:
~ = ( 1
b 1
u )
1
(1
u
); (81)
1 = ( 1
b 1
u )~ +
u
; (82)
=
u
u
b
b
~ ; (83)
whi h are three formulations of the same property. As a onsequen e, the
We expressthemean variablesinterms of probability:
=
b P
b +
u P
u +
i P
i
; (84)
=
b P
b +
i P
i
; (85)
u =
b u
b P
b +
u u
u P
u +u
i P
i
; (86)
u =
b u
b P
b + u
i P
i
: (87)
Astheprogressvariableisbetweenzeroand one,theprobabilitiessumto
one:
P
b +P
u +P
i
= 1: (88)
Makingelementarysubstitutionsinequations85to 87,we an expressP
u
and P
b
interms ofthemean variablesand P
i :
P
b
=
b
i
b P
i
(89)
P
u
=
u
i
i
u P
i
: (90)
4.2 Results in the general ase
We are now able to generate the mean variables in a new usable form, by
eliminatingP
b andP
u
from equation 85 to 87:
u = u
b
i P
i u
b +u
u
u
u +
i P
i u
u
i P
i u
u +u
i P
i (91)
u = ~ (u
b u
u )+u
u +"
1 (P
i
): (92)
This givesanew expressionforu:~
~
u = ~ (u
b u
u )+u
u +"
2 (P
i
) (93)
and forru:~
ru~ = (u
b u
u
)r~ +"
3 (P
i
); (94)
whi h isthe rst expe tedresult.
Now we dealwith uand ~ ~u.
u = u
b
i P
i u
b + u
i P
i
; (95)
u = ~ u
b +"
4 (P
i
); (96)
and
u ~ ~u = ~ u
b
~ [~ (u
b u
u )+u
u
℄+"
5 (P
i
) (97)
u ~ ~u = ~ (1 )(u~
b u
u )+"
5 (P
i
); (98)
whi h isthese ond expe tedresult.
To model he sour e term, we have used a relation between the Favre
and Reynolds average of the progress variable. So, we have to derive its
ounterpart inthegeneral ase.
= P
b +
i P
i
(99)
=
b
i
b P
i +
i P
i
(100)
=
~
b +"
6 (P
i
): (101)
All the results in the bimodal ase are therefor validin the general ase
upto afun tionof orderP
i
and lettingP
i
go tozero, wegetba ktheresults
ofthe bimodal ase.
5 Con lusion
Starting from the onstitutive instantaneous equations and their averaged
ounterparts,wehaveexaminedindetailsthelimit aseinwhi htherea tion
takes pla e in an in nitely thin sheet. This situation is formaly identi al
to the modelling of two separated uids ex hanging matter through their
interfa e. In this ase, we have shown that their is an exa t dependen e of
the so- alled ounter-gradient transport term on the sour e term. The form
of the dependen e proves the outer-gradient nature of the term whi h was
up to now only intuited. It also shows that their is fundamentaly only one
un losedtermintheaveraged progress variable equation. We haveexamined
the losure assumption of the sour e term in this framework and naturally
re-derived the TFC model. We have proposed the basi idea for a slight
improvement of the TFC model that takes into a ount the nite speed of
thein reasing brushwidth. These resultshave been extendedto thegeneral
ase, arrivingto thesame on lusionupto atermoforderP
i
,theprobability
to be insidethe amelet. The extensionhasbeendonewithoutrequiringthe
useof generalizedfun tions.
A Additional formula
From thefun tionaldependen e of on ~ ,it omes:
r = (
1
b 1
u )
2
r~ (102)
t
= (
1
b 1
u )
2
t
~
(103)
r(~ ) = 1
u (
1
b 1
u )
1
r (104)
= 1
u
2
r~ (105)
r[~ (1 ~ )℄ = [
~ 2
b
(1 ~ ) 2
u
℄
2
r~ (106)
Referen es
[1℄ LipatnikovA. N.Zimont V. L.Karpov V.P. Test of theengineering pre-
mixed ombustion model. In Twenty-Sixth Symposion (International) on
Combustion,pages 249{261. the CombustionInstitute,1996.
[2℄ Vladimir L. Zimont and Fernando Biagioli. Gradient , ounter-gradient
transport and theirtransition in turbulent premixed ames. In Combus-
tion Theory and Modeling,volumevol. 6,pages 79{101. 2002.
