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A new approach for three-phase flows
Jean-Marc Hérard
To cite this version:
Jean-Marc Hérard. A new approach for three-phase flows. 4th AIAA Theoretical Fluid Me- chanics Meeting, American Institute Aeronautics and Astronautics, Jun 2005, Toronto, Canada.
�10.2514/6.2005-4939�. �hal-01582727�
Jean-Mar Herard
EDF-DRD,78401Chatou edex,Frane
Wepresenthereanewmodeltodesribe three-eldpatternsor three-phaseows. The
basi ideas rely on the ounterpart of the two-uid two-pressure model whih has been
introdued in the DDT framework, and more reently extended to water-vapour simula-
tions. We show the systemis hyperboli without any onstraining ondition on the ow
patterns. A detailed investigation of the struture of the Riemann problem is ahieved.
Regular solutions ofthe whole arein agreementwith physial requirements onvoid fra-
tions, densities and internal energies for a rather wide lass of equations of state. Even
more, this approah enables to perform omputations of standard single pressure three-
phase owmodels, using relaxation tehniques and oarse meshes. A few omputational
results onrmthe stability ofthe wholeapproah.
I. Introdution
Some simulations in the framework of pressurized powerreators in the nulear energy require using
two-uidmodels,andsomeothersevenaskforathreeelddesriptionofthewholeow(see 1,2
). Thismay
happenforinstane whenpreditingthemotionofliquiddispersed dropletsinsideaontinuousgasphase,
whilesomegas-liquidinterfaeismovingintheore. Otherappliationsinvolvingagaseousphaseandtwo
distintliquids(forinstane oilandwater)also urgeforthedevelopmentofthreeeld models.
Somemodelsandtoolshavealreadybeenproposed,whihbasiallyrelyonthetwo-uidsingle-pressure
formalism. These either assume that liquiddropletsveloities andveloitiesin thesurrounding gasphase
areequal,orretaindierentveloitiesbutassumeinanyasealoalpressureequilibriumbetweenthethree
omponents. A straightforward onsequene is that these models suer from the same deienies than
standardtwo-uidmodels. Morelearly,thelossofhyperboliityoftheonvetivesubsetimpliesthatom-
putationsonsuÆientlyne gridsrathereasilyenter "elliptiin time"regions;asaonsequene,eventhe
most"stable"upwindingshemesleadtoablowupoftheodewhenreningthemesh,thoughaounting
forstabilizingdrag eets(see 3
forinstane forsuhanumerialexperiene).
An alternativewayto dealwith these ows onsists in gettingrid of thepressureequilibrium between
phases. Thiswas rstintrodued intheframeworkoftheDDT (see, 4{15
among others),andmorereently
appliedtowater-vapourpreditions(see 16{18
). Oneofthemainadvantageswiththelatterapproahisthat
itinherits from the hyperbolistruture of Navier-Stokesequations-whihseems quitereasonable-onthe
onehand;moreover,theoverallentropyinequalityprovidessomebetterunderstandingofvariousinterfaial
transfer terms. For all these reasons, it seems appealing to examine whether one might derive a similar
framework to opewith three-phase orthree-eld ow strutures. Suh atrialis disussed in this paper.
An underlying idea is that the interfae between phases remains innitely thin when submitted to pure
onvetivepatterns.
ResearhEngineer,DepartementMFTT,6quaiWatier,andAssoiateResearhDiretor,CNRS,LATP,AIAAMember
eets. Themain properties of thewhole set will be examined, inludinga disussion onthe solutionsof
the one dimensional Riemann problem. These properties enable to ompute the whole set with help of
roughshemes(Rusanovsheme)ormoreaurateapproximateRiemannsolverssuhastheoneintrodued
in.
19,20
A few omputationalresultsillustrate thewhole,whih issuefrom theomputation of aRiemann
problem.
II. Governing equations
Thedensity,veloity, pressure,internalenergy andtotalenergywithin phasek willbedenoted
k ,U
k ,
P
k ,e
k
=e
k (P
k
;
k
)andE
k
=0:5
k U
k U
k +
k e
k
respetively. Thevolumetrifrationofphaselabelledkis
denedas
k
,andthethreemustomplywiththeonstraint:
1
=1
2
3
Thegoverningset ofequationsis:
(I+D(W)) W
t +
F(W)
x
+C(W) G(W)
x
=S(W)+
x (E(W)
W
x
) (1)
It requires aninitial onditionW(x;0)=W
0
(x) andsuitable boundary onditions. Thestatevariable W,
theuxesF(W),G(W)andthesouretermsS(W)lieinR 11
. Weset:
W t
=(
2
;
3
;
1
1
;
2
2
;
3
3
;
1
1 U
1
;
2
2 U
2
;
3
3 U
3
;
1 E
1
;
2 E
2
;
3 E
3 )
and,notingm
k
=
k
k :
F(W) t
=(0;0;m
1 U
1
;m
2 U
2
;m
3 U
3
;
1 (
1 U
2
1 +P
1 );
2 (
2 U
2
2 +P
2 );
3 (
3 U
2
3 +P
3 );
1 U
1 (E
1 +P
1 );
2 U
2 (E
2 +P
2 );
3 U
3 (E
3 +P
3 ))
SeondranktensorsC(W);D(W);E(W)lieinR 1 111
. Thenon-onservativeonvetivetermsare:
8
>
<
>
: D(W)
W
t
=(0;0;0;0;0;0;0;0; P
2
2
t P
3
3
t
;P
2
2
t
;P
3
3
t )
C(W) G(W)
x
=(U
1
2
x
;U
1
3
x
;0;0;0;P
2
2
x +P
3
3
x
; P
2
2
x
; P
3
3
x
;0;0;0)
(2)
Visoustermsshould atleastaountforthefollowingontributions(thermaluxesmightbeinluded):
E(W) W
x
=(0;0;0;0;0;
1
1 U
1
x
;
2
2 U
2
x
;
3
3 U
3
x
;
1
1 U
1 U
1
x
;
2
2 U
2 U
2
x
;
3
3 U
3 U
3
x
) (3)
Soure termsS(W)aountformasstransferterms,drag eets, energyloss,andother ontributions. To
simplify ourpresentation,weonlyretain heretheeetofpressurerelaxationanddrageets. Thus:
S(W)=(
2
;
3
;0;0;0;S
U1
;S
U2
;S
U3
;U
1 S
U1
;U
1 S
U2
;U
1 S
U3
) (4)
Wealsoset
1
=
2
3
andwereallthatthemomentuminterfaialtransfertermsmustomplywith:
S
U1
(W)+S
U2
(W)+S
U3
(W)=0
Wefousrstonthehomogeneousproblemassoiatedwiththelefthandsideof (1). Wedeneasusual
speientropiess
k
andspeeds
k
intermsofthedensity
k
andtheinternalenergye
k :
(
k )
2
= kPk
k
=( P
k
(
k )
2 e
k (P
k
;
k )
k )(
e
k (P
k
;
k )
P
k )
1
k P
k s
k (P
k
;
k )
P
k
+
k s
k (P
k
;
k )
k
=0
Property1 :
1:1Thehomogeneoussystemassoiatedwiththelefthandsideof (1)haseigenvalues:
1;2;3
=U
1 ,
4
=U
2 ,
5
=U
3 ,
6
=U
1
1 ,
7
=U
1 +
1 ,
8
=U
2
2 ,
9
=U
2 +
2 ,
10
=U
3
3 ,
11
=
U
3 +
3
. Assoiatedrighteigenvetorsspan thewhole spaeR 11
unlessU
1
=U
k +
k orU
1
=U
k
k , for
k=2;3.
1:2Fieldsassoiatedwitheigenvalues
k
withkin(1;2;3;4;5)areLinearlyDegenerate;othereldsare
Genuinely NonLinear.
Thelist ofRiemann invariantsthroughLDelds assoiatedwith k=4;5andGNL eldsmaybeom-
putedquiteeasilyusingvariable: Z t
=(
2
;
3
;s
1
;s
2
;s
3
;U
1
;U
2
;U
3
;P
1
;P
2
;P
3
)(seeappendiesA,B,Cin 21
).
Property2 :
2:1ThelattersystemadmitsthefollowingRiemanninvariantsthroughthe1 2 3LDwave:
I 1 2 3
1
(W)=m
2 (U
2 U
1
) I
1 2 3
2
(W)=m
3 (U
3 U
1 )
I 1 2 3
3
(W)=s
2 I
1 2 3
4
(W)=s
3 I
1 2 3
5
(W)=U
1
I 1 2 3
6
(W)=
1 P
1 +
2 P
2 +
3 P
3 +m
2 (U
1 U
2 )
2
+m
3 (U
1 U
3 )
2
I 1 2 3
7
(W)=2e
2 +2
P2
2 +(U
1 U
2 )
2
I 1 2 3
8
(W)=2e
3 +2
P3
3 +(U
1 U
3 )
2
2:2Wenote ( )=
r l
. Apartfrom the1 2 3LDwave,thefollowingexatjump onditionshold
fork=1;2;3,throughanydisontinuityseparatingstatesl;rmovingwithspeed:
(
k )=0
(m
k (U
k
))=0
(m
k U
k (U
k
)+
k P
k )=0
(
k E
k (U
k
)+
k P
k U
k )=0
Weneedtodene:
a
k
=(s
k )
1
( s
k (P
k
;
k )
P
k )(
e
k (P
k
;
k )
P
k )
1
(5)
and:
k
=Log(s
k
),butalsothepair(;F
)suhthat : = m
1
1 m
2
2 m
3
3
,and: F
= m
1
1 U
1
m
2
2 U
2 m
3
3 U
3
. DragtermsS
U
k
(W)andsoureterms
k
(W)in(1)omplywith:
0a
2 (U
1 U
2 )S
U
2
(W)+a
3 (U
1 U
3 )S
U
3
(W) (6)
0a
1 (
1 P
1 +
2 P
2 +
3 P
3
) (7)
Condition(7)reads:
2 (P
1 P
2 )+
3 (P
1 P
3 )0
1 2 3 1
Property3 :
Closures in agreement with the above mentionned onstraints (6),(7) ensure that the following entropy
inequalityholdsforregularsolutionsof (1):
t +
F
x
0 (8)
Wefromnowwillassumethattheonditions(6),(7)arefullled. Foronvenienywewillhoosehere:
2
=f
1 2 (W)
1
2 (P
2 P
1 )=(P
1 +P
2 +P
3
) (9)
3
=f
1 3 (W)
1
3 (P
3 P
1 )=(P
1 +P
2 +P
3
) (10)
wherepositivesalarfuntionsf
k l
(W)denoteboundedfrequenies. Itiseasytohekthat:
1 P
1 +
2 P
2 +
3 P
3
>0
Evenmore,thegoverningequation of =
1
2
3
guaranteesthat regularsolutions
k
(x;t)remainin the
admissiblerange[0;1℄. Wewill relyonstandardlosuresoftheform(see 2
forinstane):
S
U2 (W)=
2 (W)(U
1 U
2
) (11)
S
U
3 (W)=
3 (W)(U
1 U
3
) (12)
wherethesalarfuntions
2 (W),
3
(W)should remainpositive. Hene(6)and(7)hold.
Property4 :
We assume perfet gas state law within eah phase (k = 1;2;3). We onsider a single wave assoiated
with
m
,separatingstatesl;r. Ifthe initialonditionssatisfy: (
k )
L;R
(1
k )
L;R
6=0,fork =1;2;3the
onnetionofstatesthroughthiswaveensuresthatallstatesareinagreementwith: 0
k ,0m
k ,0P
k .
Atually,theproofisalmostobviouswhenfousingonasingleeldonnetedwitheigenvalue
k where
k=4to11. Turningthentothe1;2;3-eld,themainguidelines(seeappendixEin 21
)arethesameasin.
17
Detailsonsomesuitableforms ofmassandenergytransfertermsanbefoundinappendixFin.
21
IV. Numerial approah
Thewholeenablestointrodueafrationalstepapproahinagreementwiththeoverallentropyinequality,
whih isagain the ounterpart ofthe onedesribed in.
17
Wethus simply omputeapproximationsof the
onvetivesubset:
(I+D(W)) W
t +
F(W)
x
+C(W) G(W)
x
=0 (13)
andthenaountforsouretermsandvisoustermsupdatingvaluesthroughthestep:
(I+D(W)) W
t
=S(W)+
x (E(W)
W
x
) (14)
This frational step method is in agreement with the whole entropy inequality. When negleting visous
ontributions,theseondoneturnstoanordinarydierentialsystem.
Our basiapproahto ompute onvetivetermsrelies on theGodunov approah.
22,23
More preisely
here,weusethe shemesintrodued in 17
to omputeapproximationsof thesystem(13). This is ahieved
withhelpofeithertheRusanovshemeortheapproximateGodunovshemeVFRoe-nv. Inordertoope
with thestandardstep (13)whih requiresdisretizing onvetiveeets, arather eÆientwayonsists in
usingtheapproximateGodunovshemeintrodued in 19
withthespeivariable:
Z t
=(
2
;
3
;s
1
;s
2
;s
3
;U
1
;U
2
;U
3
;P
1
;P
2
;P
3
) (15)
(see 20
whih details themain advantages of suh ahoie). Computations belowhavebeenobtainedwith
theformersheme,whilefulllingstandardCFLonditions.
One mustbearefulwhen providing approximationsofsystem(14). Otherwise,thestabilityof loally
equal-pressureregionsmaybeviolated. Theonnetionwiththeshemeintroduedin 24
isobvious.
Thisapproahhasanotheradvantage,sineitalsoenablestoopewiththeinstantaneouspressureequi-
libriumassumption. Thisisusefultoomputemodelssuhasthosedesribedin 2
forinstane. Owingtothe
entropystruture(seeappendixDin 21
),onemayatuallyintroduethepressurerelaxationstepinvolvedin
(14)asatooltoomputethesinglepressuremodelsdetailedin.
2
Thisistheounterpartofwhathasbeen
ahievedin thetwo-phaseframework(see 25,26
or 3
forinstane).
V. A few omputations of the shok tube apparatus
We restrit here to some simple omputations of the shok tube apparatus. We use a uniform mesh
with 10000ellsand set CFL=0:49 in order to avoidthe interation of waveswithin the ells. In these
omputations,allsouretermsandvisoustermshavebeennegleted,inordertoassessthestabilityofthe
wholeonvetivesubset.
We assume that the perfet gas law holds within eah phase:
k e
k
= (
k 1)P
k
, setting
1
= 7=5,
2
= 1:05 and
3
= 1:01. Initial onditions are : (
2 )
L
= 0:4, (
3 )
L
= 0:5, (
2 )
R
= 0:5, (
3 )
R
= 0:4,
(U
1 )
L
= 100, (
1 )
L
= 1, (P
1 )
L
= 10 5
, (U
1 )
R
= 100, (
1 )
R
= 8, (P
1 )
R
= 10 5
, (U
2 )
L
= 100, (
2 )
L
= 1,
(P
2 )
L
=10 5
, (U
2 )
R
=100, (
2 )
R
= 8, (P
2 )
R
=10 5
, (U
3 )
L
= 100,(
3 )
L
=1, (P
3 )
L
=10 5
, (U
3 )
R
=100,
(
3 )
R
= 8, (P
3 )
R
= 10 5
, for the rst ase (g. (1-3)), and: (
2 )
L
= 0:4, (
3 )
L
= 0:5, (
2 )
R
= 0:5,
(
3 )
R
=0:4, (U
1 )
L
=0,(
1 )
L
=1,(P
1 )
L
=10 5
, (U
1 )
R
=0,(
1 )
R
=8,(P
1 )
R
=10 4
, (U
2 )
L
=0,(
2 )
L
=1,
(P
2 )
L
=10 5
,(U
2 )
R
=0,(
2 )
R
=8,(P
2 )
R
=10 4
,(U
3 )
L
=0,(
3 )
L
=1,(P
3 )
L
=10 5
, (U
3 )
R
=0,(
3 )
R
=8,
(P
3 )
R
=10 4
,fortheseondtest.
0 2000 4000 6000 8000 10000
0.4 0.42 0.44 0.46 0.48 0.5
Void fractions alpha2 (squares), alpha3 CFL=0.49 _ 10000 cells
Figure1. Voidfrations
2 ,
3
0 2000 4000 6000 8000 10000
0 0.1 0.2 0.3 0.4 0.5
Partial masses m1 (circles), m2 (squares), m3 CFL=0.49 _ 10000 cells
Figure 2. Partial masses m1, m2,
m3
0 2000 4000 6000 8000 10000
90000 95000 1e+05 1.05e+05 1.1e+05
Pressures P1 (circles), P2 (squares), P3 CFL=0.49 _ 10000 cells
Figure3. PressuresP
1 ,P
2 ,P
3
0 2000 4000 6000 8000 10000 0
100 200 300 400
Velocities U1 (circles), U2 (squares), U3 Shock tube _ CFL=0.49 _ 10000 cells
Figure4. VeloitiesU
1 ,U
2 ,U
3
0 2000 4000 6000 8000 10000
0 0.1 0.2 0.3 0.4 0.5
Partial masses m1 (circles), m2 (squares), m3 Shock tube _ CFL=0.49 _ 10000 cells
Figure 5. Partial masses m
1 , m
2 ,
m
3
0 2000 4000 6000 8000 10000
0 20000 40000 60000 80000 1e+05
Pressures P1 (circles), P2 (squares), P3 Shock tube _ CFL=0.49 _ 10000 cells
Figure6. PressuresP
1 ,P
2 ,P
3
VI. Conlusion
Thisnewmodelbenetsfromimportantproperties. Fromaphysialpointofview,aninterestingpointis
thatitpreservesthepositivityof(expeted)positivequantities: voidfrations,massfrationsandinternal
energies,atleastwhenrestritingtosuÆientlysimpleEOS.Itsmathematialpropertiesenableusto on-
strut nonlinearstable numerialmethods, and thus toexplore highly unsteadyow patterns. Conditions
toobtainaexisteneanduniquenessoftheexatsolutionoftheonedimensional Riemannproblemannot
beobtainedeasily. A spei diÆultyis linked withthe possible oureneof theresonanephenomena.
Anotherpoint,whihseemsworthbeingnoted,isthattheounterpartoftheaverage"andidate"interfae
veloityV
I
=(m
1 U
1 +m
2 U
2 )=(m
1 +m
2
)nolongerarisesin thethree-eldframework.
Another part of our urrent work onerns the omparison with standard single pressure three-eld
models, when restriting to oarse meshes. This is ahieved using relaxationtehniques, following ideas
from.
3,24{30
Aknowledgments
ThisworkhasreeivednanialsupportfromtheNEPTUNEprojet,whihhasbeenlaunhedbyEDF
(Eletriitede Frane) and CEA (Frenh Atomi Ageny), and benets from a omplementary part from
IRSNandFRAMATOME-ANP.OlivierHurisseisalsoaknowledgedforhishelp.
Referenes
1
S.JayantiandM.Valette,Preditionofdryoutandpostdry-outheattransferathighpressureusingaone-dimensional
three-eldmodel,Int.J.of HeatandMassTransfer,2004,vol.47-22,pp.4895-4910.
2
M.Valette and S.Jayanti, Annulardispersedow alulationswitha two-phasethreeeld model,European Two
phaseFlow GroupMeeting,Norway,2003,internalCEAreportDTP/SMTH/LMDS/2003-085.
3
J.M.
H
erardandO.Hurisse,Asimplemethodtoomputestandardtwo-uidmodels,submitted,2005.
4
N.Andrianov and G.Warneke, TheRiemannproblemfor the BaerNunziatotwo-phaseowmodel,J.of Comp.
Physis.,vol.195,pp434{464,2004
5
M.R.Baer and J.W.Nunziato, Atwophasemixturetheory forthe deagrationto detonationtransition(DDT)in
reativegranularmaterials,Int.J.MultiphaseFlow,1986,vol.12-6,pp.861{889.
6
F.CoquelandS.Cordier,CEMRACSenalulsientique1999,MATAPLI62,pp.27-58,2000.
7
S.Gavrilyuk,H.GouinandY.V.Perepehko,Avariationalpriniplefortwouidmodels,C.R.Aad.Si.Paris,
1997,vol.IIb-324,pp.483-490.
8
S.GavrilyukandR.Saurel,Mathematialandnumerialmodellingoftwophaseompressibleowswithmiroinertia,
J.ofComp.Phys.,2002,vol.175,pp.326-360.
