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HAL Id: hal-00762841

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viscous two-phase flows with real gas effects

Remi Abgrall, Maria Giovanna Rodio, Pietro Marco Congedo

To cite this version:

Remi Abgrall, Maria Giovanna Rodio, Pietro Marco Congedo. Towards an efficient algorithm for the simulation of viscous two-phase flows with real gas effects. [Research Report] RR-8173, INRIA. 2012.

�hal-00762841�

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IS S N 0 2 4 9 -6 3 9 9 IS R N IN R IA /R R -- 8 1 7 3 -- F R + E N G

RESEARCH REPORT N° 8173

December 8, 2012 Project-Team Bacchus

Towards an efficient algorithm for the

simulation of viscous

two-phase flows with real gas effects

Remi Abgrall, Maria Giovanna Rodio, Pietro Marco Congedo

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RESEARCH CENTRE BORDEAUX – SUD-OUEST 351, Cours de la Libération Bâtiment A 29

33405 Talence Cedex

simulation of visous two-phase ows with real

gas eets

RemiAbgrall,Maria Giovanna Rodio, Pietro Maro Congedo

Projet-TeamBahus

ResearhReport n° 8173Deember8,201226pages

Abstrat: A disrete equation method (DEM) for the simulation of ompressiblemultiphase

owsinludingvisousandreal-gaseetsisillustrated. Areduedveequationmodelisobtained

startingfrom thesemi-disrete numerial approximationof the two-phasevisous model. Then,

asimplealgorithm is proposed in order to use two dierentequations of stateat the vaporand

liquid-vapor onditions. Simulation results are validated with well-known results in literature.

Potentialitiesinimprovingthequalityofthenumerialpreditionbyusingamoreomplexequation

ofstatearethusdrawn.

Key-words: DEM method, two-phase ows, Peng-Robinson equation of state, Stiened Gas

equationofstate,shoktube.

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ompte des eets de gaz réel

Résumé : Une méthode de type DEM pour la simulation des éoulements multiphasiques

ompressiblesinluantleseets devisositéet de gazréel est proposée. Un modèle réduità5

équationsestobtenuenpartantd'uneapproximationnumériquedumodèlevisqueuxdiphasique.

Ensuite, un algorithme est dérit qui permet d'utiliser deux équations d'état diérentes pour

prendreenompte lesonditionsdelavapeuret dumélangeliquide-vapeur. Lesrésultatsdela

simulationsontvalidésavedessolutionsderéféreneonnuesenlittérature. L'intérêtàutiliser

deséquationsd'état plusomplexeset préisesestmontréaveplusieursas-tests.

Mots-lés : Méthode DEM, éoulements diphasiques, équation d'état de Peng-Robinson,

équationd'étatdetypeStiened,tubeàho.

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1 Introdution

Modelingtwo-phaseowsisofprimaryimportaneforengineeringappliations. Twoaspetsare

fundamental: (i) how to model theinterfae betweentwo uidswith dierent thermodynami

propertiesand (ii)toharaterizethemehanismsourring attheinterfaeaswellasinzones

where the volume frations are not uniform. Several methods have been proposed to model

theinterfaetimeevolutionandreonstrutionasinLagrangianmethods,ArbitraryLagrangian-

Eulerian methods (ALE), theLevel set method,et. [1, 2,3℄orto dealwith theinterfae asa

diusionzone, forwhihonsistentthermodynamiequationsshouldbeonsidered [4,5,6, 7℄.

Instead of the traditional approahes to multiphase modeling, where an averaged system

of (ill-posed) partial dierential equations (PDEs) disretized to form anumerial sheme are

onsidered,thedisreteequationmethod(DEM)resultsin well-posedhyperbolisystems. This

allows a lear treatment of non-onservative terms (terms involving interfaial variables and

volume fration gradients) permitting the solution of interfae problems without onservation

errors. This method displaysseveraladvantages,suh asanaurateomputation oftransient

owsasthemodelisunonditionallyhyperboli,boundaryonditionssolvedwithasimpleand

auratetreatment,anaurateomputation ofnon-equilibrium owsaswell asowsevolving

inpartialortotalequilibrium. WiththeDEM,eahphaseisompressibleandbehavesaording

toaonvexequationofstate(EOS).Inmanyworksofinterfaeproblem,theStienedGas (SG)

EOS wasusually used [8, 6, 9, 10℄. As explained in Saurel et al. [11, 12℄, this EOS allowsan

expliitmathematialalulationsofimportantowrelationthankstoitssimpleanalytialform.

Moreover,in masstransferproblem itassuresthepositivity ofspeedofsound inthetwo-phase

region,under thesaturationurve.

When omplex uids are onsidered, suh as ryogeni, moleularly omplex and so on,

the use of simplex EOS an produe impreise estimation of the thermodynami properties,

thus deteriorating theauray of thepredition. Inreasing the omplexityof the model and

alibratingtheaddingparameterswithrespettotheavailableexperimental dataonstitutesa

validoptionforsavingthegoodpreditionofthemodel. Nevertheless,itouldbeveryhallenging

beauseofthenumerialdiultiesfortheimplementationofmoreomplexmathematialmodel

andbeauseofthelargeunertaintiesthat generallyaetedtheexperimental data.

Aneortfordevelopingamorepreditivetoolformultiphaseompressibleowsisunderway

in Bahus Team (INRIA-Bordeaux). Within this projet, several advanements have been

performed,i.e. onsideringamoreompletesystemsofequationsinludingvisosity[13℄,working

on thethermodynami modeling ofomplexuids [14, 15℄, and developingstohasti methods

for unertainty quantiation in ompressibleows[14, 16℄. Theaim of this paperis to show

howthenumerialsolverbasedonaDEM formulationhasbeenmodiedforinludingvisous

eetsandamoreomplexequationofstateforthevaporregion. Themethodusedinthispaper

is theDEM(see [6℄)fortheresolutionof areduedveequation model withthehypothesisof

pressure and veloity equilibrium [17℄, without mass and heat transfer. This method results

in a well-posed hyperboli systems, allowing an expliit treatment of non onservativeterms,

without onservation error. The DEM method diretly obtains awell-posed disrete equation

systemfromthesingle-phaseonservationlaws,produinganumerialshemewhihaurately

omputesuxesforarbitrarynumberof phases. TheDEM methodhasbeenextensivelytested

inseveraltestasesreproduingunsteadyandwavepropagationows[6,17,18℄. Inthispaper,

twothermodynamimodelsareonsidered,i.e. theSGEOSandthePeng-Robinson(PR)EOS.

WhileSGallowspreservingthehyperboliityofthesystemalsoinspinodalzone,real-gaseets

are takeninto aountby usingthe moreomplexPR equation. Thehigher robustnessofthe

PRequationwhenoupledwithCFDsolverswithrespettomoreomplexandpotentiallymore

aurate multi-parameter equations of state has been disussed in [19, 20℄. In this paper, no

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masstransfereetistakenintoaount,thusthePRequationanbeusedonlytodesribethe

vaporbehavior,while onlytheSGmodelisusedfordesribingtheliquid.

This paperis organizedasfollows. Insetion 2, adesriptionof the reduedve equation

model with visous eets is given. Wederiveasemi-disrete numerial approximationof the

two-phasemodel. Then,weperformanaverageproedureofthedisreteapproximationandan

extensionto theseond-order. Finally,the asymptotiexpansionis analyzedto obtainasemi-

disreteapproximationforareduedve-equationmodel. Then,in setion3.1,wedesribethe

SGandPREOSforthepureuidandthenwederivethethermodynamipropertiesofmixture,

supposingtheuseofSGEOSforallphaseandofPREOSonlyforthevaporphase. Thesetion

4isdividedintwoparts: intherstone,theimplementationoftheomplexequationofstateis

validated by reproduingaquasisingle-uidshoktube,i.e. onsideringaveryreduedliquid

fration.Then,intheseondpart,theodeisvalidatedagainstsomewell-knowntwo-phasetest-

asesinliterature. Finally,theinuene ofusing amoreomplexequationof stateisanalyzed

byonsidering severaloperatingonditionsintheproximityofthesaturationurve.

2 Problem Statement

Thepresentapproahisbasedonave-equationmodelwithpressureandveloityequilibrium,

obtainedafteranasymptotianalysisinwhihtherelaxationtermsdisappears. Themethodfor

theresolutionofavisous veequation modelhasbeenamplydevelopedin [13℄. Inthiswork,

we give the main lines of the sheme. In this setion, let us start by illustrating the redued

model,thenthenumerialshemeandnallythethermodynamilosureofthesystem.

2.1 Five equation model

Thestarting point of the present analysis is avisous sevenequation model omposed by the

onservativeequationforeahphaseandbytheharateristifuntion

X k

ofthephase

Σ k

. The

funtion

X k (x, t)

isequalto

1

if

x

liesin theuid

Σ k

attime tand0otherwise. An averaging

proeduresimilar tothat usedbyDrew[21℄isapplied tothis equation,obtainingthefollowing

model:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

∂ α ¯ k

∂t = − E (σ · ▽ α ¯ k ) + µ r (P k − P k )

∂ α ¯ k ρ ¯ k

∂t + ▽ ( ¯ α k ρ ¯ k ~v k ) = E (ρ I (~v I k − σ) · ▽ X k )

∂ α ¯ k ρ ¯ k ~v k

∂t + ▽ ( ¯ α k ρ ¯ k ~v k ⊗ ~v k + ¯ α k P ¯ k I ) = E

k I ~v I k (~v k I − σ) + P I I ) · ▽ X k + + ▽ · ( ¯ α k τ ¯ k ) − E (τ I · ▽ X k ) + λ(~v k − ~v k )

∂ α ¯ k ρ ¯ k E ¯ k

∂t + ▽ ( ¯ α k (¯ ρ k E ¯ k ~v k + ¯ P k I ~v k ) =E

(ρ I E I (~v k I − σ) + P I I ) · ▽ X k + + λ~v k I (~v k

− ~v k ) + µ r P I (P k − P k

) + ▽ · ( ¯ α k (¯ τ k · ~v I k )) − E ((τ I ~v k ) · ▽ X k )

(1)

where thetwophasesare indiatedwith

k

and

k

;

ρ k

,

α k

,

~v k

and

P k

arethe uiddensity,

thevolume fration,the veloityvetorand the pressurefor eah phase

k

, respetively;

E k

is

thetotal energyof eah phasedened as

E k = ¯ e k + 1 2 u ¯ k · u ¯ k

.

P I

and

v I

arethe pressureand

veloityattheinterfaeoftheomponent

k

,respetively,dened as:

P I k = Z k P k

+Z k P k

Z k +Z k + sign

∂α k

∂x , (v k ∗

− v k ) Z k +Z k

v k I = Z k v k +Z k

v k

Z k +Z k + sign

∂α k

∂x ,

P k +P k Z k +Z k

(2)

(8)

where

Z k

representsthe aoustiimpedane, i.e.

Z = ρc

, where

c

isthe speed ofsound. The

oeients

λ

and

µ r

aretherelaxationveloityparameterandthedynamiompationvisosity,

respetively. Theyareassoiatedtotherelaxationtermsthatappearinthesystemtoreprodue

relaxationproess behind shokand pressurewavesin thetwo-phase ow, induingapressure

andveloityequilibrium.

Writing

λ = 1 ε

and

µ r = 1 ε

,thesystemanbeformulatedinasimpliednotation,asfollows:

∂U

∂t + ∂F (U )

∂x = ∂U

∂t + A(U ) ∂U

∂x = R(U )

ε

(3)

where

U

is thevetorof onservativevariables,

F(U )

theux vetorand

R(U )

the relaxation

term. The redued model, i.e. supposing

P k = P k

and

v k = v k

, an beobtained using an

asymptoti analysis that allows to nd the solutions

S

, a subset of

R N

of system (3) when

ε → 0 +

:

W =

U ∈ R N : R(U ) = 0

(4)

As demonstratedby Murroneand Guillard [9℄ (seealso [22, 17℄), foreah solution

U ∈ W

, we

an omputea parametrization

M

suh that, let

u = (α 1 , ρ 1 , u, P, α 2 , ρ 2 ) t ∈ R 5

the vetorof

primitivesvariables,theparametrization

M : u → M (u)

isdened asfollows:

u → U = M (u) =

 α 1

ρ 1

u P α 2

ρ 2

u P

.

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Thereduedmodelof(3)isthusobtainedbynegletingthetermsoforder

ε

:

∂u

∂t + P A(M (u))dM u

∂u

∂x = O(ε).

(6)

Theterm

dM u

representstheJaobianmatrixthatformsabasisofker

R (M (u))

,moreover

P

is

theprojetionoverker

R (M (u))

inthediretionof

Rng(M (u))

,thatistherangeof

R (M (u))

.

Inpartiular,theprojetion

P

isobtainedasinversionofthematrix

S = [dM u 1 , ..., dM u n , I 1 , I 2 ]

,

where

I 1 , I 2

isabasisof

Rng(R (M (u)))

. Thereduedmodelisthefollowing:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

∂ α ¯ 1

∂t = − ~v · ▽ α 1 + K ▽ (~v)

∂α 1 ρ 1

∂t + ▽ (α 1 ρ 1 ~v) =0

∂α 2 ρ 2

∂t + ▽ (α 2 ρ 2 ~v) =0

∂ρ~v

∂t + ▽ (ρ k ~v ⊗ ~v + P ) = ▽ τ

∂E

∂t + ▽ (E + P ) = ▽ τ · ~v

(7)

where

K = α 1 α 2 (ρ 2 c 2 2 − ρ 1 c 2 1 )/(α 1 ρ 2 c 2 2 + α 2 ρ 1 c 2 1 )

.

AsexplainedinAbgrall[17 ℄,unfortunately,inthesystem(7),anononservativeprodutionappearsin

thevaporvolumefrationequation. Thediulty,fromamathematialpointofview,isthatthisterm

hasnomeaningintheasewhere

v

and

K

aresimultaneouslydisontinuous. Inthisase,itisdiult

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toderiveanumerialsheme. However,in[17 ℄,ashemethatallowtooveromethisproblemhasbeen

proposedanditisusedinthiswork. Itisexplainedinthefollowingsetion.

Inthesystem(7),

ρ = α 1 ρ 1 +α 2 ρ 2

,

P

and

E = α 1 ρ 1 e 1 +α 2 ρ 2 e 2

arethemixturedensity,pressureand

energy,respetively. Tolose thesystem(7), weneedanEOSfor eahpureuidandforthe mixture,

permittingtodeneall theneededthermodynamiproperties. IntheaseofSGEOS,thevalueofthe

mixturepressureiseasilyderivedfromthemixtureenergyandthedensityforeahphase[23 ℄. Onthe

ontrary,whenthePREOSisused,thereis noexpliit relationbetweenthepressure andthe energy,

thusmakingmoreomplextheomputationofthemixturepressure.

2.2 Numerial Method

Here, we illustrate how obtaining a semi-disrete numerial approximation of the two-phase system

inludingthevisousterm(Eq. 1),followingthesameproedureof[6℄.

WedesribethisshemeintheframeworkofnitevolumedisretizationwithaGodunovsolver,but

theproedureanbeeasily adaptedto othersolvers. Let usdeneourtime-spaedomain. Attime

t

,

theomputationaldomain

isdividedintoaonstant numberofells

C i =]x i−1/2 , x i+1/2 [

. Inaddition,

eahell

C i

isdividedintoarandomsubdivision

x i − 1/2 = ξ 0 < ξ 1 < ... < ξ N(ω) = x i+1/2

(where

ω

isa

randomparameter).

Ineahsubell

]ξ l , ξ l+1 [

,

X k

isonstantandsoonlyonephase

Σ k

anexist. Negletingthesoure

terms

S

ofEq.2,theintegralformofNavier-Stokesequationsforthespae-timedomain

C i × [t, t + s]

is

equalto

Z t+s t

Z

C i

X ∂U

∂t + Z t+s

t

Z

C i

X ∂(F − F v )

∂x dx dt = 0

(8)

TheGodunovshemeisnolongerappliedonthemeshells,butonthemodiedandnon-uniformells

onstrutedaording tothepositionoftheinterfae.

Thevariable

σ i+1/2 = σ(U i , U i+1 )

denotes thespeedof theinterfaebetween thetwoells

C i

and

C i+1

.Remarkthatitisequaltozero,ifthesamephaseispresentintotheells,otherwise

σ i+1/2

oinides

withthespeedofpropagationofthe interfaeintheRiemannsolution

(x, t) → v r ( x t ; U i , U i+1 )

. Thus,

assumingthat betweenthetimes

t n

and

t n+1 = (t + s)

,the interfae

x i+1/2

movesatveloity

σ i+1/2

,

theell

C i

isnotxed,butit evolvesin

C ¯ i =](x i−1/2 + sσ i−1/2 ), (x i+1/2 + sσ i+1/2 )[

(seeFig. 1). The

ellmay beeithersmaller orlarger thanthe originalones

C i

,depending onthesigns ofthe veloities

σ i+1/2

. Wedenotewith

F (U L , U R )

theGodunovnumerialuxbetweenthestates

U L

and

U R

,andwith

F lag (U L , U R )

theuxarosstheontatdisontinuitybetweenthestates

U L

and

U R

(seeFig. 2). Asa

onsequene,wehave:

F lag (U L , U R ) = F (U LR + ) − σ(U L , U R )U LR + = F (U LR ) − σ(U L , U R )U LR

ConsideringFig. 1,theprevious integral(Eq.8),denedonthemeshell

C i

,anbedividedintothree

ontributions,i.e. theintegralsonabb' fortheleftboundary,theintegralonb'fortherightboundary

andtheLagrangianinternalells.

ThusEq.8anberewrittenas

Z

abb

X ∂U

∂t + ∂(F − F v )

∂x

dx dt + (I)

+

N(̟) − 2

X

l=2

Z t+s t

Z ξ l+1 +sσ(U l i ,U l+1 i ) ξ l +sσ(U i l 1 ,U i l )

X ∂U

∂t + ∂(F − F v )

∂x

dx dt + (II)

+ Z

dcc

X ∂U

∂t + ∂(F − F v )

∂x

dx dt = 0 (III )

Inludingtheharateristi funtion

X

inderivativetermsand applyingthe Godunov sheme, we

obtainforeahterm,thesolutionattime

t + s

. Fortheboundaryterms(I)and(III),itisimportantto

remarkthatthedomainwhere omputingthe integral isatriangle. Attime

t

wehave onlyonepoint

(a)andthusthespatialintegraliszero.

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Simulation ofvisoustwo-phaseows withreal gaseets 7

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1*(2"3/*4 0!, (!)-)(0,-/#0/( 5"*(0/6* /* 3,-/7)0/7, 0,-8# )*3 )992:/*4 0!, ;63"*67 #(!,8,< ., 6+0)/* 56- ,)(!

0,-8< 0!, #62"0/6* )0 0/8, & =6- 0!, +6"*3)-: 0,-8# >1? )*3 >111?< /0 /# /896-0)*0 06 -,8)-@ 0!)0 0!, 368)/*

.!,-, (689"0/*4 0!, /*0,4-)2 /# ) 0-/)*42,& A0 0/8, ., !)7, 6*2: 6*, 96/*0 >)? )*3 0!"# 0!, #9)0/)2 /*0,4-)2 /# B,-6&

C,8,8+,-/*4 0!)0 0!, (!)-)(0,-/#0/( 5"*(0/6* 6+,:# 06 0!, 56226./*4 9-69,-0:D

>EF?

!"#$% &' (#)*!+!,!-. -/ 0-12#343!-.45 *-14!.6

!"#$% 7' 89% +4$!-#, ,343%, !. 39% :!%14.. 2$-)5%1 )%3;%%. ,343%, 4.* 6

<

Figure1: Subdivision ofomputationaldomain.

!"# $%&' ()* +, -,.-/00,* )#

1*(2"3/*4 0!, (!)-)(0,-/#0/( 5"*(0/6* /* 3,-/7)0/7, 0,-8# )*3 )992:/*4 0!, ;63"*67 #(!,8,< ., 6+0)/* 56- ,)(!

0,-8< 0!, #62"0/6* )0 0/8, & =6- 0!, +6"*3)-: 0,-8# >1? )*3 >111?< /0 /# /896-0)*0 06 -,8)-@ 0!)0 0!, 368)/*

.!,-, (689"0/*4 0!, /*0,4-)2 /# ) 0-/)*42,& A0 0/8, ., !)7, 6*2: 6*, 96/*0 >)? )*3 0!"# 0!, #9)0/)2 /*0,4-)2 /# B,-6&

C,8,8+,-/*4 0!)0 0!, (!)-)(0,-/#0/( 5"*(0/6* 6+,:# 06 0!, 56226./*4 9-69,-0:D

>EF?

!"#$% &' (#)*!+!,!-. -/ 0-12#343!-.45 *-14!.6

!"#$% 7' 89% +4$!-#, ,343%, !. 39% :!%14.. 2$-)5%1 )%3;%%. ,343%, 4.* 6

<

Figure2: Thevariousstatesin theRiemannproblembetweenstates

U L

and

U R

.

Rememberingthattheharateristifuntionobeystothefollowing property:

N(̟) − 1

X

l=0

Z t+s t

Z ξ l+1 ξ l

∂X

∂t dx dt +

N(̟) − 1

X

l=0

Z t+s t

Z ξ l+1 ξ l

σ ∂X

∂x dx dt = 0,

(10)

weanwritefor(I)

Z

abb

X ∂U

∂t + ∂ (F − F v )

∂x

dx dt = Z

abb

∂XU

∂t + ∂X(F − F v )

∂x

dxdt − Z

abb

U ∂X

∂t + (F − F v ) ∂X

∂x

dxdt = Z x i − 1/2 +sσ + (U i + − 1 ,U i ,)

x i − 1/2

X (x, t + s)U(x, t + s) dx − sX(x i − 1/2 , t)F (U i−1/2 ) +

− sF lag (U i−1 + , U i )[X] j=0 + sX ( x i−1/2 , t ) F v ( U i−1/2 ) + sF vI ( U +

i−1 , U

i )[ X ] j=0 ,

where

U i ± 1/2

isthesolutionoftheRiemannproblem. Inthiswork,nomasstransferisonsidered,

thenwhenajump isonsidered,thevisousontributioniszerobeausethereisnottransferof

(11)

visousinformation. Similarly,theterm(III)anbemanipulatedasfollows

Z

dcc

X ∂U

∂t + ∂(F − F v )

∂x

dx dt = Z

dcc

∂XU

∂t + ∂X(F − F v )

∂x

dxdt − Z

dcc

U ∂X

∂t + (F − F v ) ∂X

∂x

dxdt = Z x i+1/2 +sσ (U i + ,U i+1 ,)

x i+1/2

X(x, t + s)U (x, t + s) dx + sX(x i+1/2 , t)F (U i+1/2 ) + +sF lag (U i + , U i+1 )[X] j=N(w) − sX ( x i+1/2 , t ) F v ( U i+1/2 ) − sF vI ( U +

i , U i+1 )[ X ] j=N(w) .

Conerningtheinternal terms(II),theintegralbeomes

Z t+s t

Z ξ j+1 +sσ(U i l ,U i l+1 ) ξ j +sσ(U i j 1 ,U i j )

X ∂U

∂t + ∂(F − F v )

∂x

dx dt = Z ξ j+1 +sσ(U i l ,U i l+1 )

ξ j +sσ(U i j 1 ,U i j )

X(x, t + s)U (x, t + s) dx − Z ξ j+1

ξ j

X (x, t)U (x, t) dx +

− s

F lag (U i j , U i j+1 )[X ] j − F lag (U i j 1 , U i j )[X ] j − 1 , + + s

F v ( U j

i , U j+1

i )[ X ] j − F v ( U j 1

i , U j

i )[ X ] j−1 , .

Summingupallterms,dividingfor

s

andtakingthelimitwhen

s → 0

,weobtainthesemi-disrete sheme

∂t 1

△x

Z x i+1/2 x i−1/2

X (x, t)U (x, t) dx

! + + 1

△x X(x i+1/2 , t)F (U i+1/2 ) − X(x i − 1/2 , t)F (U i − 1/2 ) +

− 1

△ x X ( x i+1/2 , t ) F v ( U i+1/2 ) − X ( x i−1/2 , t ) F v ( U i−1/2 )

=

= 1

△x

N(w) − 1

X

j=1

F lag (U i j , U i j+1 )[X] j − F lag (U i j−1 , U i j )[X] j − 1 , + + 1

△x

F lag (U i , U i + − 1 )[X] j=0 − F lag (U i + , U i+1 )[X] j=N(w) +

N(w) − 1

X

j=1

F v ( U j

i , U j+1

i )[ X ] j − F v ( U j−1

i , U j

i )[ X ] j − 1 , +

− 1

△ x F v ( U +

i , U

i+1 )[ X ] j=0 − F v ( U +

i , U

i+1 )[ X ] j=N(w)

.

(11)

Wemayassumethattwoadjaentsubellontainsdierentphases,sothat

N(w) − 1

X

j=1

F lag (U i j , U i j+1 )[X ] j − F lag (U i j 1 , U i j )[X ] j − 1

=

= N (ω) int

F lag (U i 2 , U i 1 ) − F lag (U i 1 , U i 2 )

(12)

and

(12)

N(w) − 1

X

j=1

F v ( U j

i , U j+1

i )[ X ] j − F v ( U j 1

i , U j

i )[ X ] j−1 ,

=

(13)

= N (ω) int F v (U i 2 , U i 1 ) − F v (U i 1 , U i 2 ) .

2.3 Averaging proedure

Now,itispossibletoapplytothedisreteequationssystem(11),thesameaveragingproedure

usedforthesystemofpartialdierentialequationsof(1). Taking themathematialexpetany

ofthesemi-disretesheme(Eq.11)forwhihwehave:

∂t 1

△ x

Z x i+1/2 x i − 1/2

X (x, t)U (x, t) dx

!

= ∂α (1) i U i (1)

∂t

(14)

andintroduingthenotationfortheaveragenumberofinternalinterfaes,

λ i = E

N (ω) int

△x

,

theshemeanberewrittenas:

∂α (1) i U i (1)

∂t + 1

△x E

X(x i+1/2 , t)F (U i+1/2 ) − X (x i − 1/2 , t)F(U i 1/2 ) +

− 1

△ x E

X ( x i+1/2 , t ) F v ( U

i+1/2 ) − X ( x i

− 1/2 , t ) F v ( U

i − 1/2 )

= + 1

△ x E F lag (U i , U i + 1 )[X ] j=0 − F lag (U i + , U i+1 )[X ] j=N (w)

+

− 1

△ x E F vI ( U +

i , U

i − 1 )[ X ] j=0

− F vI ( U +

i , U

i+1 )[ X ] j=N(w)

+λ i F lag (U i 2 , U i 1 ) − F lag (U i 1 , U i 2 )

− λ i (F v (U i 2 , U i 1 ) − F v (U i 1 , U i 2 )

.

(15)

Thedevelopmentofbothonservativeandnon-onservativestermshavebeenfullydesribed

in[6,17,13℄,thenhereonlynalexpressionsofthesetermsaregivenforaseondordersheme.

Weonsidertheellboundary

i + 1/2

andfousontheuxesavailableforuid

Σ 1

. Onthisell

boundary, four instanes may our onthe base of thephase present in the ell

x i

and in the

ell

x i+1

(seeTab. 1). Thus, weandenetheuxindiator

β i+1/2 (l,r) = sign(σ(U i l , U i+1 r )) =

1 if σ(U i l , U i+1 r ) ≥ 0,

− 1 if σ(U i l , U i+1 r ) < 0

and theorrespondingprobabilityto havethe samephaseortwodierent phasesinto theleft

andrightellofellboundary

i + 1/2 P i+1/2 (Σ 1 , Σ 1 ) := P

X(x i+1/2 ) = 1 and X(x + i+1/2 ) = 1

= min

α (1) i , α (1) i+1 , P i+1/2 (Σ 1 , Σ 2 ) := P

X(x i+1/2 ) = 1 and X (x + i+1/2 ) = 0

= max

α (1) i − α (1) i+1 , 0 , P i+1/2 (Σ 2 , Σ 1 ) := P

X(x i+1/2 ) = 0 and X (x + i+1/2 ) = 1

= max

α (2) i − α (2) i+1 , 0 , P i+1/2 (Σ 1 , Σ 2 ) := P

X(x i+1/2 ) = 0 and X(x + i+1/2 ) = 0

= min

α (2) i , α (2) i+1 .

RRn°8173

(13)

FollowingtheMUSCLapproah,weproposethefollowingextensionto aseond-orderapproxi-

mationofthesheme. Thesystem15anbewrittenasfollows:

α (1) i U i (1) n+1

α (1) i U i (1) n

∆t

+ E (XF ) i+1/2 − E (XF ) i 1/2

∆x − E (XF v ) i+1/2 − E (XF v ) i 1/2

∆x =

= E

F lag ∂X

∂x

i,bound

+ E

F lag ∂X

∂x

i,relax

+

−E

F v ∂X

∂x

i,bound

− E

F v ∂X

∂x

i,relax

.

(16)

Inthefollowing,

U i ± 1/2,l (resp.U i ± 1/2,r )

arethevetorofonservativevariablesontheleft(resp.

right)oftheboundaryell

x i ± 1/2

aftertheMUSCLextrapolation,usingaminmodlimiter. Thus, alltermsinthepreditor-orretorshemeforamultiphaseows,takethefollowingform

ˆ Conservativeterms

E (XF ) i+1/2 = P 1+1/2 (Σ 1 , Σ 1 )F(U i+1/2,l (1),n+1/2 , U i+1/2,r (1),n+1/2 ) +P 1+1/2 (Σ 1 , Σ 2 )

β (1,2) i+1/2 +

F (U i+1/2,l (1),n+1/2 , U i+1/2,r (2),n+1/2 )+

+P 1+1/2 (Σ 2 , Σ 1 )

−β i+1/2 (2,1) +

F(U i+1/2,l (2),n+1/2 , U i+1/2,r (1),n+1/2 )

(17)

E (XF ) i 1/2 = P 1 − 1/2 (Σ 1 , Σ 1 )F(U i (1),n+1/2 1/2,l , U i (1),n+1/2 1/2,r ) +P 1 − 1/2 (Σ 1 , Σ 2 )

β (1,2) i 1/2 +

F (U i (1),n+1/2 1/2,l , U i (2),n+1/2 1/2,r )+

+P 1 − 1/2 (Σ 2 , Σ 1 )

− β i (2,1) 1/2 +

F(U i (2),n+1/2 1/2,l , U i (1),n+1/2 1/2,r )

(18)

E (XF v ) i+1/2 = P 1+1/2 (Σ 1 , Σ 1 )F v (U i+1/2,l (1) , U i+1/2,r (1) ) +P 1+1/2 (Σ 1 , Σ 2 )

β (1,2) i+1/2 +

F v (U i+1/2,l (1) , U i+1/2,r (2) )+

+P 1+1/2 (Σ 2 , Σ 1 )

− β i+1/2 (2,1) +

F v (U i+1/2,l (2) , U i+1/2,r (1) )

(19)

E (XF v ) i 1/2 = P 1 − 1/2 (Σ 1 , Σ 1 )F v (U i (1) 1/2,l , U i (1) 1/2,r ) +P 1 − 1/2 (Σ 1 , Σ 2 )

β (1,2) i 1/2 +

F v (U i (1) 1/2,l , U i (2) 1/2,r )+

+P 1 1/2 (Σ 2 , Σ 1 )

−β i (2,1) 1/2 +

F v (U i (2) 1/2,l , U i (1) 1/2,r )

(20)

(14)

ˆ Non-onservative terms

E

F lag ∂X

∂x

i

= E

F lag ∂X

∂x

bound

+ E

F lag ∂X

∂x

relax

=

= +P 1+1/2 (Σ 1 , Σ 2 )

β i+1/2 (1,2),n +

F lag (U i+1/2,l (1),n , U i+1/2,r (2),n )+

− P 1+1/2 (Σ 2 , Σ 1 )

β (2,1) i+1/2 +

F lag (U i+1/2,l (2),n , U i+1/2,r (1),n )+

−P 1 − 1/2 (Σ 1 , Σ 2 )

β i (1,2) 1/2 +

F lag (U i (1),n 1/2,l , U i (2),n 1/2,r )+

+P 1 − 1/2 (Σ 2 , Σ 1 )

β i (2,1) 1/2 +

F lag (U i (2) 1/2,l , U i (1) 1/2,r )+

+

l=1

X

N − 1

max 0, ∆α 1 i

F lag (U i 2 , U i 1 ) − max

l=1

X

N − 1

0, ∆α 2 i

F lag (U i 1 , U i 2 )

! ,

(21)

E

F v

∂X

∂x

i

= E

F v

∂X

∂x

bound

+ E

F v

∂X

∂x

relax

=

= P 1+1/2 (Σ 1 , Σ 2 )

β (1,2) i+1/2

F vI (U i+1/2,l (1) , U i+1/2,r (2) )+

−P 1+1/2 (Σ 2 , Σ 1 )

β (2,1) i+1/2

F vI (U i+1/2,l (2) , U i+1/2,r (1) )+

−P 1 − 1/2 (Σ 1 , Σ 2 )

β i (1,2) 1/2 +

F vI (U i (1) 1/2,l , U i (2) 1/2,r )+

+P 1 − 1/2 (Σ 2 , Σ 1 )

β i (2,1) 1/2 +

F vI (U i (2) 1/2,l , U i (1) 1/2,r )+

+max 0, ∆α 1 i

F v (U i 2 , U i 1 ) − max 0, ∆α 2 i

F v (U i 1 , U i 2 ) ,

(22)

where

∆α 1 i = α 1 i+1/2,l − α 1 i+1/2,r

and

∆α 2 i = α 2 i+1/2,l − α 2 i+1/2,r

are thelimitedslopeof

α 1

and

α 2

intheell

C i

. Thevisous uxes

F v

and

F vI

arereonstruted byabakwarddiereneon thebaseofowdiretion. Weonsiderthat theellboundary

i + 1/2

andthevisousuxesfor

themomentumandenergyequationsare:

ˆ momentum

F v (U i+1/2,l (k) , U i+1/2,r (k) ) =

 

 

4

3 µ u i+ 1 2 u

(k) (r) i − 1

(3/2)dx if σ ≥ 0

4

3 µ u i+ 1 2 u

(k) (l) i+2

(3/2)dx if σ < 0

(23)

ˆ energy

F v (U i+1/2,l (k) , U i+1/2,r (k) ) =

 

 

4 3 µu i+ 1

2

u (k)

i+ 1 2 − u (k) (r)

i−1

(3/2)dx if σ ≥ 0

4 3 µu i+ 1

2

u (k)

i+ 1 2 − u (k) (l)

i+2

3/2dx if σ < 0

(15)

ˆ momentum

F vI (U i+1/2,l (k) , U i+1/2,r (k) ) =

4 3 µ u

(k)

I(r) i+1 − u (k) I(r)

i

dx if σ ≥ 0

4

3 µ u I(r) i u

(k) I(r) i+1

dx if σ < 0

(24)

ˆ energy

F vI (U i+1/2,l (k) , U i+1/2,r (k) ) =

 

 

4 3 µu (k) I

i+ 1 2

u (k) I(r)

i+1 − u (k) I(r)

i

dx if σ ≥ 0

4 3 µu (k) I

i+ 1 2

u (k) I(r)

i − u (k) I(r)

i+1

dx if σ < 0

where

u (k) i+ 1 2 =

( u (k) (r) i if σ ≥ 0 u (k) (l) i+1 if σ < 0 u (k) I

i+ 1 2

=

 u (k) (r)

Ii if σ ≥ 0 u (k) (l)

Ii+1 if σ < 0

where

l

and

r

representthe leftand rightside. Toobtainthepreditor sheme, thesolutionis

alulatedattime

t = n + 1/2

andintheorretorshemethesolutionisalulatedatthetime

t = n + 1

,usingthesolutionomputedatthepreditorstep.

3 Asymptoti analysis of the numerial sheme

TheshemepresentedinEq. 16isasemi-disretenumerialapproximationofthesevenequation

model for modeling a two-phase ow. Now, the aim is to obtain the semi-disrete sheme for

the redued ve equations model in a 1D onguration, where visous eets are onsidered,

followingtheproedure desribedin[13℄. Ifweset

ε i = 1/λ i

, thedisrete shemefortheseven

equationsmodelanbeformallyrewrittenas

∂W

∂t + G

△ x = R(W )

ε i .

(25)

where

W = (α (1) , α (1) U (1) , α (2) , α (2) U (2) )

,

G

is thesumofonvetiveuxes, visousuxes and

bound lagrangianuxes,and

R(W )

aretherelaxationterms. Asexplainedin setion2.2,letus

onsider

ε → 0

searhingforthesolutionssuh thattherelaxationtermsoulddisappear:

W = n

W such that R(W ) = 0 ⇒ E F lag ∂X ∂x

i,relax = 0. o

(26)

(16)

FlowPatterns LeftandRightStates FluxIndiator

Σ 1 , Σ 2 U i (1) , U i+1 (2)

β i+1/2 (1,2) +

Σ 1 , Σ 1 U i (1) , U i+1 (1) 1

Σ 2 , Σ 1 U i (2) , U i+1 (1)

β i+1/2 (2,1) +

Σ 2 , Σ 2 U i (2) , U i+1 (2) 0

Σ 1 , Σ 2 F v (U i (1) , U i+1 (2) )

β i+1/2 (1,2) +

Σ 1 , Σ 1 F v (U i (1) , U i+1 (1) ) 1 Σ 2 , Σ 1 F v (U i (2) , U i+1 (1) ) −

β i+1/2 (2,1) +

Σ 2 , Σ 2 F v (U i (2) , U i+1 (2) ) 0

InterfaeFlux

Σ 1 − Σ 2 F lag (U i (1) , U i+1 (2) )

β i+1/2 (1,2)

Σ 1 − Σ 1 F lag (U i (1) , U i+1 (1) ) 0 Σ 2 − Σ 1 F lag (U i (2) , U i+1 (1) ) −

β i+1/2 (2,1) Σ 2 − Σ 2 F lag (U i (2) , U i+1 (2) ) 0

Σ 1 − Σ 2 F v I (U i (1) , U i+1 (2) )

β i+1/2 (1,2) +

Σ 1 − Σ 1 F v I (U i (1) , U i+1 (1) ) 0 Σ 2 − Σ 1 F v I (U i (2) , U i+1 (1) ) −

β i+1/2 (2,1) +

Σ 2 − Σ 2 F v I (U i (2) , U i+1 (2) ) 0

Table1: Theowongurationsattheellboundary

i + 1/2

.

(17)

Hene,thenalshemefortheveequationmodelinonservativevariables isequalto:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

∂ α ¯ 2

∂t = F V 2 + α ¯ 1 α ¯ 2

¯

α 2 ρ ¯ 1 a 2 1 + ¯ α 1 ρ ¯ 2 a 2 2

( SE 2

¯ α 2 ρ ¯ 2 β 2

− u 2 SU 2

¯ α 2 ρ ¯ 2 β 2

+

u 2 2

2 − ε 2 − ρ 2 κ 2

¯ α 2 ρ ¯ 2 β 2

M 2 + ρ 2 2 κ 2 F V 2

¯ α 2 ρ ¯ 2 β 2

− SE 1

¯ α 1 ρ ¯ 1 β 1

+ u 1 SU 1

¯ α 1 ρ ¯ 1 β 1

u 1 2

2 − ε 1 − ρ 1 κ 1

¯ α 1 ρ ¯ 1 β 1

M 1 − ρ 2 1 κ 1 F V 1

¯ α 1 ρ ¯ 1 β 1

)

∂ α k ¯ ρ k

∂t = M k

∂ρu

∂t = SU 1 + SU 2

∂ρE

∂t = SE 1 + SE 2

(27)

whereFV,

M k

,SU andSE aretheuxesof vaporvolumefration equation,onservativemass equation,momentumequationandonservativeenergyequation, respetivelyandwhere:

β i k = ∂ǫ k i

∂P i k

ρ k i

and κ k i = ∂ǫ k i

∂ρ k i

P i k

3.1 Thermodynami losure

Aswehavepreviouslymentioned, wedealwith pureuid andartiialmixture zone, thus the

EOSmustbeabletodesribetheowbothinpureuidsandmixturezones.

In this setion, rst we desribe two EOSs, i.e. the Stiened Gas (SG) EOS and the Peng-

Robinson(PR) EOS. Then, webuild themixture EOS using rst theSG EOS for eah phase

andafterthePRandtheSGforthegasandtheliquidphase,respetively.

3.1.1 Stiened GasEOS forpure uid

The Stiened Gas EOS is usually used for shok dynamis and its robustness for simulating

two-phaseowwithorwithoutmasstransferhasbeenamplydemonstrated[6,9,10,17,23℄. It

anbewritten asfollows:

P(ρ, e) = (γ − 1)(e − q)ρ − γP ,

(28)

e(ρ, T ) = T c v + P ρc v

+ q

(29)

h(T ) = γc v T,

(30)

where

p

,

ρ

and

e

are the pressure, the density and the energy, respetively. The politropi oeient

γ

istheonstantratioofspeiheatapaities

γ = c p /c v

,

P

isaonstantreferene

pressureandq is theenergyof theuid at agiven referenestate. Moreover,T,

c v

and

h

are

thetemperature,thespeiheatatonstantvolumeandtheenthalpy,respetively. Thespeed

ofsound,dened as

c 2 = ( ∂P ∂ρ ) s

anbeomputedasfollows:

c 2 = γ P + P

ρ = (γ − 1)c p T

(31)

where

c 2

remainsstritlypositive(for

γ > 1

). Itensuresthehyperboliityofthesystemandthe existeneofaonvexmathematialentropy[24℄.

(18)

3.1.2 Peng-Robinson (PR) EOS for pure uid

PengandRobinson(1976)proposedaubiEoSofvanderWaalstypeintheform:

p = RT

v − b − a

v 2 + 2bv − b 2 .

(32)

where

p

and

v

denoterespetivelytheuidpressureanditsspeivolume,

a

and

b

aresubstane-

spei parameters related to the uid ritial-point properties

p c

and

T c

and representative of attrative and repulsive moleular fores. To ahieve high auray for saturation-pressure

estimatesofpureuids,thetemperature-dependentparameter

a

inEq. (32)isexpressedas

a = 0.457235R 2 T c 2 /p c

· α (T ) ,

(33)

while

b = 0.077796RT c /p c .

(34)

Theseparametersarenotompletelyindependent,sineisothermallinesinthep-v planeshould

satisfy the thermodynami stability onditionsof zero urvature and zeroslopeat theritial

point. Suhonditionsallowomputingtheritialompressibilityfator

Z c = (p c v c )/(RT c )

as

thesolutionofaubiequation. Theorretionfator

α

inEq. (33)isgivenby

α (T r ) =

1 + K 1 − T r 0.5 2

,

(35)

with

K = 0.378893 + 1.4897153ω − 0.17131848ω 2 + 0.0196554ω 3 .

(36)

The parameter

ω

is the uid aentrifator. The other needed information to omplete the thermodynamimodel,namelytheideal-gasisohorispeiheatoftheuid,isapproximated

throughapowerlaw,i.e.,

c v, ∞ (T ) = c v, ∞ (T c ) T

T c

n

(37)

with n a uid-dependent parameter. From thermodynami rules, the energy equation an be

expressedas:

e = e c + c v, ∞ (T c )

(n + 1)T c n (T n+1 − T c n+1 ) − a 8 0.5 b

α(T ) − T dα(T ) dT

ln

V +b(1+2 0.5 ) V +b(1 − 2 0.5 )

(38)

where

e c

isanenergyreferenevalue. Thespeedofsoundanbeexpressedasfollows:

a 2 = ∂P

∂ρ

T

+ 1 ρ 2

T c v

∂P

∂T 2

ρ

.

(39)

3.1.3 SG EOS based mixture

The EOS for the mixture anbe easily obtainedusing the EOS of the single phases. In this

setion,letusonsiderthemixture obtainedsupposing aSGEOSforeahphase. Thestarting

pointisthemixtureenergyequation:

ρE = α 1 ρ 1 e 1 + α 2 ρ 2 e 2 .

(40)

(19)

The energy of eah phase,

e k

, an be replaed by the Eq.(28), obtaining the mixture total

energyasafuntion ofthephasepressure. Under pressureequilibrium,weobtainthefollowing

expressionforthepressuremixture:

P (ρ, e, α k ) = ρ(E − α 1 ρ ρ 1 q 1α 2 ρ ρ 2 q 2 ) − α

1 γ 1 P ∞,1

γ 1 − 1 + α 2 γ γ 2 2 P ∞,2 1

α 1

γ 1 − 1 + γ α 2 2 1

(41)

Inthis paper,theterm

q

issupposed equaltozeroforeahphase.

3.1.4 PR-SG EOS based mixture

In this setion, let us onsider a mixture, obtained using the SG EOS for the liquid and the

PREOS forthe gas. In theaseof SGEOS (seesetion(3.1.1)),weshowed howthepressure

mixturean be easilyobtainedfromthe energyandthedensityofeahphase. IfaPREOS is

onsidered,itisnotpossibletoformulateexpliitlythepressureasafuntion oftheenergyand

thedensity. Then, theproedure shown in setion (3.1.3) forthe SG EOS annotbeused in

thisase.

Underpressureequilibrium, thefollowingsystemoftwoequationsisobtained:

P 1 (T 1 , ρ 1 ) = P 2 (T 2 , ρ 2 )

ρE = α 1 ρ 1 e 1 + α 2 ρ 2 e 2 .

(42)

where

P 1

representsthepressurestateomputedforthephase1desribedbythePREOS,and

P 2

thepressureofthe phase2desribed bySG EOS. Remarkthat in this ase, theunknowns

are

T 1

and

T 2

. Replaing

P 1

asafuntion of

T 1

and

ρ 1

usingEq. (32)and

P 2

usingEq. (28)in

therstequation ofthesystem(42),the liquidtemperature

T 2

anbeexpressed asafuntion

ofthegastemperature

T 1

:

T 2 =

"

T 1 R 1 1 ρ 1 − b 1

− α(T 1 )a 1 1

ρ 2 1 + 2b ρ 1 1 − b 2 1 + P ,2

# 1 (γ 2 − 1)ρ 2 c v, ∞ 2

(43)

Replaing theenergy

e k

of eah phaseusing Eq. (38) for

e 1

and using Eq. (29)for

e 2

in the

mixtureenergyequation(seondequationofthesystem(42)),weobtain:

ρE = α 1 ρ 1

n

e c + (n+1)T c v, (T c n )

c (T 1 n+1 − T c n+1 ) − a

2 √ 2b

α(T 1 ) − T 1 dα(T ) dT

ln

V +b(1+ √ 2) V +b(1 − √

2 ) o

+α 2 ρ 2

T 2 c v, ∞ 2 + P ,2

ρ 2 c v, ∞ 2

+

(44)

Now,byreplaing

T 2

intheEq. (44)usingtheEq. (43),itispossibletoderivearelationbetween

thegas temperature,

T 1

, andthe mixture energy. Thisis afuntion

E = E(T 1 )

that depends

exlusivelyby

T 1

. SolvingiterativelyEq. (44)byusingaNewton-Raphsonmethod,thevalueof thegastemperature

T 1

anbeomputed.

One

T 1

isobtained,themixturepressureanbeeasilyomputedusingEq. (32).

4 Results

This setion illustrates various results. First,the implementation of the PR equation of state

is validated by running a monophasi shok tube where the liquid fration is supposed very

redued, and by omparing with respet to well-known results in literature. In this ase, the

workinguid istheFC70uid,permittingtosimulate ararefationshokwaveinthetubefor

(20)

somespei onditionsofpressureandtemperature. Theaimofthis testaseistwofold,i)to

validatetheimplementationofPREOS andtohektherobustnessoftheproposed numerial

sheme. Seondly, a two-phase shok tube is onsidered where the interest of using a more

omplex equation of state with respet to a simpler one is determined for several operating

onditions.

4.1 Validation

Let us onsider the test ase presented by Fergason et al. [25℄, where a rarefation shok is

observed in a single-phase shok tube onguration. This non-lassial phenomenon has been

observednumeriallyin literature[26,15, 27℄, evenifan experimental onrmationof therar-

efationshokwavestilldoesnotexist. OnlyanaurateEOS,suhasthePREOS,andesribe

a sopartiular gas thermodynami behavior. Forthis reason, this test-ase representsa good

validationforhekingtheEOSimplementation.

The shok tube is lled out with only one uid, the FC70, but for numerial reasons, eah

hamber ontainsa veryweak volume of fration of water (

α l = 10 8

). The left side is at a

pressureof

10.766 × 10 5

Pa,withadensityequalto

ρ = 470.398 kg/m 3

, while therightoneis

at apressure of

8.635 × 10 5

Pawith adensity equalto

ρ = 248.991 kg/m 3

. The diaphragmis

loatedat

x = 2.5

m(thetubeislong5m). TheresultsobtainedwithDEMhavebeenvalidated

withthenumerialresultsobtainedwiththeNZDG ode(see[15℄formoredetails),omparing

theprolesat atime of

t = 2.3

ms. TheTable2providestheuid propertiesof FC70andthe

orrespondingPREOSparameters,i.e. theuidaentrifator

ω

andthe

n

oeient(seeEq.

37),takenfrom[28℄.

In Figure 3, the evolution of dimensionless (omputed with respet to the ritial point)

pressure,density,veloityandMahalongthetubeaxisareshown. Anon-lassialdisontinuity

waveeld displaying ararefationshok waveon

x = 1.8

m (seeFigure 3) and aompression

shokwaveon

x = 3.5

mareobserved. TheresultsobtainedwiththeDEModeandtheNZDG

ode[15℄displayaperfetagreement.

Name

M (Kg/mol) T c (K) P c (atm) v c (m 3 /kg) ω

n

F C 70

0.821 608.2 10.2 1.8544

×10 3

0.7584 0.4930

Table2:Moleularweight

M

,ritialtemperature

T c

,ritialpressure

P c

,ritialspeivolume

v c

, aentrifator

ω

and

n

oeient

4.2 Dodeane test-ases

Inthissetion,letusonsiderashoktubelledoutwithliquiddodeaneontheleftandwith

vapor dodeane on the right without mass transfer. In the rst test-ase, already shown in

[11℄, thevaporoperatingonditionsarefarfrom thesaturationurve. Inorderto evaluatethe

importane of using a moreomplexEOS, three others test aseshave been onsidered,using

thesameoperatingonditionsfortheliquiddodeane,but withthevaporoperatingonditions

losertothesaturationurve. Thesethreeonditions,reportedintheAmagatdiagramingure

4,havebeennamedasTC2,TC3,andTC4.

4.2.1 Originaldodeanetest-ase

Theshoktubeis lledoutwithliquid dodeane ontheleft andvapordodeaneon theright,

butfornumerialreasons,eahhamberontainsaweakvolumeoftheotheruid(

α k = 10 8

).

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