A few results on the modelling of multiphase flows...Vincent Guillemaud (PhD thesis, CEA-LATP, nov. 2003-oct. 2006), Philippe Helluy (Strasbourg University, IRMA), through DFG-CNRS

IntroductionTwo-phase flows

Some related work:Numerical Schemes

Conclusion and perspectives/ References

A few results on the modelling of multiphase flows

Jean-Marc Hérard ?

?EDF R&D, MFEE6 quai Watier, 78400 Chatou. France.

[email protected]

Applied Mathematics In SavoieChambery, June 19-22 2012

Jean-Marc Hérard A few results on the modelling of multiphase flows




Outline

1 IntroductionIntroductionCollaborationsTeamsConstraints

2 Two-phase flows3 Some related work:

Three-phase flowsTwo-phase flow in a porous mediumSynthesis

4 Numerical SchemesFractional step methodConvective termsInterfacial transfer terms]

5 Conclusion and perspectives/ ReferencesConclusion and perspectivesReferences





IntroductionCollaborationsTeamsConstraints

Outline











Introduction

A review on recent results on multiphase flow modelling and simulation. Thiswork has benefited from financial support by:

EDF, Research Direction (MFEE)

CNRS (LATP, Aix Marseille University) from 1999 till 2005,

NEPTUNE project (EDF, CEA, AREVA and IRSN) since 2002,

joint DFG-CNRS project (FOR 563), since 2005:"Micro-macro modelling and simulation of liquid-vapour flows"(see: http://www.mathematik.uni-freiburg.de/IAM/dfg-cnrs/)

SITAR project (EDF).

It mainly concerns:

Modelling and analysis of two-phase flows, three-phase flows, ...

Numerical schemes for the simulation of solutions of these PDEs,

Thorough verification of schemes, especially in unsteady situations.

Focus on multiphase flow modelling -rather than numerical simulation-.






Introduction







It mainly concerns:










Introduction







It mainly concerns:










Introduction







It mainly concerns:










Outline











Main collaborations:

Main collaborations:

Frédéric Coquel (CNRS, LJLL and X-CMAP),

Thierry Gallouët ( Aix-marseille University,LATP),

Laetitia Girault (PhD thesis, CIFRE EDF R&D-LATP, MFEE, jan.2007-jan. 2010),

Vincent Guillemaud (PhD thesis, CEA-LATP, nov. 2003-oct. 2006),

Philippe Helluy (Strasbourg University, IRMA),through DFG-CNRS FOR 563 project"Micro-macro modelling and simulation of liquid-vapour flows",

Olivier Hurisse (EDF R&D, MFEE),

Yujie Liu (PhD thesis, CIFRE EDF R&D-LATP, AMA, sept. 2010-...),

Khaled Saleh (PhD thesis, CIFRE EDF R&D-LJLL, MFEE, sept.2009-...),

Nicolas Seguin (Pierre et Marie Curie University, and formally during hisPhD thesis, EDF R&D-LATP, MFTT, dec. 1999-august 2002)






Outline











Several teams have been working on that topic:

Following early papers by Ransom and Baer-Nunziato, several teams andresearchers have been investigating that topic of multiphase flow modellingand simulation of two-phase compressible flows:

J. Glimm and co-workers,

A.K. Kapila and co-workers,

In Europe, R. Abgrall, S. Gavrilyuk, H. Guillard, R. Saurel, E.F Toro, G.Wahrnecke (1997-...), through various research projects SMASH(IUSTI-INRIA) and governmental initiatives (among which DFG-CNRSFOR 563),

Many other actors : Gonthier, Schwendemann, Wahle, ...


















































Outline











Main constraints:

We must comply with the following constraints:

The non-viscous model should be hyperbolic and admit unique jumpconditions,

The non-viscous model should be dissipative, and the whole modelshould be governed by a physically relevant entropy inequality,

Source termes should agree with the entropy inequality,

Positive values are expected for statistical fractions, mass fractions,internal energies, within each phase.

Obviously the latter specifications are necessary but not necessarilysufficient. On the whole the model should provide meaningful solutions forunsteady flows with shocks.






Main constraints:












Main constraints:












Main constraints:












Main constraints:











Main variables:

Two-phase flows:

Starting with a statistical description of a mixture of two compressible phases,independent first-order moments are:

mean fraction of phase αk =< ξk >, such that : α1(x , t) + α2(x , t) = 1,

mean densities ρk =< ξkρk > /αk ,

mean velocities Uk =< ξkρk Uk > / < ξkρk >,

mean pressures πk =< ξk Pk > /αk .

We note mk =< ξkρk > the mass fraction within phase k. Mean EOS shouldbe provided by user in terms of mean variables within phase k:ek =< ξkρk ek > / < ξkρk >= εk (πk , ρk ).

Thus, if we neglect second-order and higher order correlations, and if wenote Ek = ρk (Uk )

2/2 + ρk ek the total mean energy within phase k, we obtaina six-equation open set of equations (see Ishii, Drew, Glimm, ...):





Main variables:

Two-phase flows:













Main variables:

Two-phase flows:













Main variables:

Two-phase flows:













Main variables:

Two-phase flows:













Open set of equations for two-phase flows:

Open set of PDEs:8>>><>>>:∂t (mk ) + ∂x

“mk Uk

”= JM

k

∂t

“mk Uk

”+ ∂x

“mk U2

k + αk πk

”= JQ

k

∂t

“αk Ek

”+ ∂x

“αk Uk (Ek + πk )

”= JE

k

In addition, the governing equation for < ξk > is:

∂t (< ξk >) + VΣ∂x (< ξk >) = κk

noting : κk =< (< uσ > −uσ)∂x (ξk ) >.

Hence, which closure laws should be given for interfacial transfer termsJM

k , JQk , J

Ek , κk and interface quantity VΣ =< uσ > ?





Open set of equations for two-phase flows:

Open set of PDEs:8>>><>>>:∂t (mk ) + ∂x

“mk Uk

”= JM

k

∂t

“mk Uk

”+ ∂x

“mk U2

k + αk πk

”= JQ

k

∂t

“αk Ek

”+ ∂x

“αk Uk (Ek + πk )

”= JE

k

In addition, the governing equation for < ξk > is:

∂t (< ξk >) + VΣ∂x (< ξk >) = κk

noting : κk =< (< uσ > −uσ)∂x (ξk ) >.

Hence, which closure laws should be given for interfacial transfer termsJM

k , JQk , J

Ek , κk and interface quantity VΣ =< uσ > ?





A few notations and definitions:

We set :

ρk (ck )2 = (∂πk (εk (πk , ρk )))

−1(πk

ρk− ρk∂ρk (εk (πk , ρk )))

We define the mean phasic entropy Sk (πk , ρk ) complying with :

(ck )2∂πk

“Sk

”+ ∂ρk

“Sk

”= 0

and the mean mixture entropy :

S = (m1S1 + m2S2)/(m1 + m2)

eventually, we introduce:

1/Tk = ∂πk

“Sk

”/∂πk (εk )

and Gibbs potentials :

µk = εk (πk , ρk ) +πk

ρk− Tk Sk





Closure laws

Getting rid of external forces, the interfacial transfer terms satisfy:8>><>>:JM

1 + JM2 = 0

JQ1 + JQ

2 = 0JE

1 + JE2 = 0

κ1 + κ2 = 0

By enforcing that these are such that:

the entropy inequality:

0 ≤ ∂t

“(m1 + m2)S

”+ ∂x

“(m1S1U1 + m2S2U2

”is fulfilled;

the compatibility of source terms with the entropy inequality is achieved,

relevant closure laws may be proposed.Remark : without any ambiguity, we omit the a notations in the sequel.





Closure laws


1 + JM2 = 0

JQ1 + JQ

2 = 0JE

1 + JE2 = 0

κ1 + κ2 = 0



0 ≤ ∂t

“(m1 + m2)S

”+ ∂x

“(m1S1U1 + m2S2U2

”is fulfilled;







Closure laws


1 + JM2 = 0

JQ1 + JQ

2 = 0JE

1 + JE2 = 0

κ1 + κ2 = 0



0 ≤ ∂t

“(m1 + m2)S

”+ ∂x

“(m1S1U1 + m2S2U2

”is fulfilled;







Closure laws

We consider interfacial transfer terms of the form:8<:JM,0

k = Γk (W )

JQ,0k = Ik (W ) + (U1 + U2)Γk (W )/2

JE,0k = Ψk (W ) + (U1 + U2)Ik (W )/2 + Γk (W )U1U2/28<:

JM,1k = 0

JQ,1k = Π(W )∂x (αk )

JE,1k = −Π(W )∂t (αk )

where : 8>>><>>>:Γk (W ) = (−1)k m1m2

m1+m2T0/µ0(µ1/T1 − µ2/T2)/τM(W )

Ik (W ) = (−1)k m1m2m1+m2

(U1 − U2)/τU(W )

Ψk (W ) = (−1)k m1m2m1+m2

e0/T0(T1 − T2)/τT (W )

κk (W ) = (−1)kα1α2(P2 − P1)/π0/τP(W )

These involve four relaxation time scales τM(W ), τU(W ), τP(W ), τT (W ).





Closure laws

We consider interfacial transfer terms of the form:8<:JM,0

k = Γk (W )

JQ,0k = Ik (W ) + (U1 + U2)Γk (W )/2

JE,0k = Ψk (W ) + (U1 + U2)Ik (W )/2 + Γk (W )U1U2/28<:

JM,1k = 0

JQ,1k = Π(W )∂x (αk )

JE,1k = −Π(W )∂t (αk )

where : 8>>><>>>:Γk (W ) = (−1)k m1m2

m1+m2T0/µ0(µ1/T1 − µ2/T2)/τM(W )

Ik (W ) = (−1)k m1m2m1+m2

(U1 − U2)/τU(W )

Ψk (W ) = (−1)k m1m2m1+m2

e0/T0(T1 − T2)/τT (W )

κk (W ) = (−1)kα1α2(P2 − P1)/π0/τP(W )

These involve four relaxation time scales τM(W ), τU(W ), τP(W ), τT (W ).





Closure law for Π(W )

The contribution Π(W ) is now explicitly known in terms of VΣ. Assuming that:

VΣ = β(W )U1 + (1− β(W ))U2

we get at once:Π(W ) = γ(W )P1 + (1− γ(W ))P2

noting :

γ(W ) =(1− β(W ))/T1

(1− β(W ))/T1 + β(W )/T2

Obviously, if β(W ) ∈ [0, 1], then: γ(W ) ∈ [0, 1]. The "consistency" conditionfor Π(W ) is gained. At this level, we note that the couple (Π(W ),VΣ) is a"staggered" one: provided VΣ behaves as Uk , then Π(W ) is close to P3−k .

It remains to determine relevant choices for β(W ) that will completelydetermine VΣ.







VΣ = β(W )U1 + (1− β(W ))U2


noting :

γ(W ) =(1− β(W ))/T1

(1− β(W ))/T1 + β(W )/T2









VΣ = β(W )U1 + (1− β(W ))U2


noting :

γ(W ) =(1− β(W ))/T1

(1− β(W ))/T1 + β(W )/T2







Closure law for VΣ

Main Hypothesis:It is assumed that no rarefaction wave may develop for the statistical fractionαk in the homogeneous convective system. A straightforward consequence isthat the field associated with λ = VΣ should be Linearly Degenerate.

Proposition 1:Assume that β(W ) = 0, or β(W ) = 1, or β(W ) = m1/(m1 + m2) ; then thefield associated with λ = VΣ is LD.

We have now a closed set of seven EDP governing the main variable:

(α1,m1,m2,m1U1,m2U2, α1E1, (1− α1)E2)

.





Closure law for VΣ

Main Hypothesis:It is assumed that no rarefaction wave may develop for the statistical fractionαk in the homogeneous convective system. A straightforward consequence isthat the field associated with λ = VΣ should be Linearly Degenerate.

Proposition 1:Assume that β(W ) = 0, or β(W ) = 1, or β(W ) = m1/(m1 + m2) ; then thefield associated with λ = VΣ is LD.

We have now a closed set of seven EDP governing the main variable:

(α1,m1,m2,m1U1,m2U2, α1E1, (1− α1)E2)

.





A brief summary

The two-phase flow model is given by:8>>>><>>>>:∂t (αk ) + VΣ∂x (αk ) = κk (W )∂t (mk ) + ∂x (mk Uk ) = Γk (W )

∂t (mk Uk ) + ∂x`mk U2

k + αk Pk´− Π(W )∂x (αk ) = Ik (W ) + (U1 + U2)Γk (W )/2

∂t (αk Ek ) + ∂x (αk Uk (Ek + Pk )) + Π(W )∂t (αk ) = Ψk (W )...+ (U1 + U2)Ik (W )/2 + Γk (W )U1U2/2

Recall that mk = αkρk , and that closure laws for :

κk (W ), Γk (W ), Ik (W ),Ψk (W ),

and for the couple (VΣ,Π(W )) are defined in the previous slides. Diffusioncontributions, if considered, are classical.

Before going further on, we wish to emphasize a few points:





A few remarks

Focusing on the previous system, we note that:This formalism has been introduced previously in 2001 in [P1,P2], andthen extended to the three-phase flow, porous, and dense gas-particleframework in [P4-P7];When β(W ) = 1 we retrieve the Baer-Nunziato model, introduced forthe DDT applications (IJMF, 1984);Closure laws proposed by Glimm and coworkers (LANL, Stony BrookUniversity) are somewhat distinct, especially when focusing on thecouple Π,VΣ (see for instance J. Glimm, D. Saltz, D.H. Sharp, Physletters, 1996, or: J. Fluid Mech. 1999), but the target applications arealso different !;An extension of the framework for VΣ was proposed recently ([R5]);We retrieve formally the classical six-equation model when τP = 0;Closure laws for interfacial mass, momentum and energy transfer are apriori the same for these seven-equation models and six-equationmodels. The entropy inequality is the same in both cases, which is arather remarkable point.





A few remarks






A few remarks






A few remarks






A few remarks






A few remarks






Main properties

The two-phase flow model is such that:An entropy inequality governs smooth solutions, the mixture entropybeing :

mS = m1S1 + m2S2

The two-phase flow model guarantees positive values for αk and partialmasses mk ;The equilibrium corresponds with avelocity/pressure/temperature/potential equilibrium between phases;The LHS of the set of PDEs is hyperbolic: it admits seven realeigenvalues and the set of right eigenvectors spans the whole space R7,unless:

Uk − VΣ

ck= ±1

Jump conditions are unique when VΣ is chosen among previousclosure laws.





Main properties


mS = m1S1 + m2S2


Uk − VΣ

ck= ±1






Main properties


mS = m1S1 + m2S2


Uk − VΣ

ck= ±1






Main properties


mS = m1S1 + m2S2


Uk − VΣ

ck= ±1






Main properties


mS = m1S1 + m2S2


Uk − VΣ

ck= ±1






Main properties


mS = m1S1 + m2S2


Uk − VΣ

ck= ±1






Some related work:

Owing to this formalism, we may develop algorithms and codes in order tocompute unsteady solutions including shock patterns. It also seemsappealing to examine the following items:

The building of a framework for three-phase flow models. This work hasbeen motivated by: (i) the CATHARE project, which aims at providing athree-field model for some particular situations, including the refloodingof the core after a LOCA ; (ii) the interfacial coupling of a three-fluidmodel with the former two-fluid model. See : [P4-P5] and [R4];Moreover, for forthcoming applications, the investigation of the porousextension of the current fluid model was urged by strategic motivations.See : [P6] and : [P8-P9];The PhD thesis by Julien Nussbaum (ISL et IRMA, Université deStrasbourg), sponsored by the DFG-CNRS FOR 563 project, alsoproposed an extension to the framework of dense granular gas-particleflows. See : [P7];Eventually, a few points have been examined, that concern the"turbulent" extension (see [C1,P12]), considering basic and crudeclosures such as those proposed by Spallart for single-phase flows.





Some related work:







Some related work:







Some related work:







Some related work:








Outline











A three-phase flow model

Similar guidelinesVariables αk , ρk , Pk , and Uk respectively denote statistical fractions,densities, pressures and velocities in phase k . The open set of equationsmay be written as:8>><>>:

∂t (αk ) + VΣ∂x (αk ) = φk (W )∂t (αkρk ) + ∂x (αkρk Uk ) = 0∂t (αkρk Uk ) + ∂x

`αkρk U2

k + αk Pk´

+P3

l=1,l 6=k πkl(W )∂x (αl) = Ik (W )

∂t (αk Ek ) + ∂x (αk Uk (Ek + Pk ))−P3

l=1,l 6=k πkl(W )∂t (αl) = Vm(W )Ik (W ) + ψk (W )

where: Ek = ρk ek (Pk , ρk ) + ρkU2

k2 denotes the total energy, for k = 1, 2, 3.

The basic question is: which closure laws are relevant for:

Ik (W ), ψk (W ), φk (W ),Vm(W ), πkl(W ),VΣ(W )






A three-phase flow model

Similar guidelinesVariables αk , ρk , Pk , and Uk respectively denote statistical fractions,densities, pressures and velocities in phase k . The open set of equationsmay be written as:8>><>>:

∂t (αk ) + VΣ∂x (αk ) = φk (W )∂t (αkρk ) + ∂x (αkρk Uk ) = 0∂t (αkρk Uk ) + ∂x

`αkρk U2

k + αk Pk´

+P3

l=1,l 6=k πkl(W )∂x (αl) = Ik (W )

∂t (αk Ek ) + ∂x (αk Uk (Ek + Pk ))−P3

l=1,l 6=k πkl(W )∂t (αl) = Vm(W )Ik (W ) + ψk (W )

where: Ek = ρk ek (Pk , ρk ) + ρkU2

k2 denotes the total energy, for k = 1, 2, 3.

The basic question is: which closure laws are relevant for:

Ik (W ), ψk (W ), φk (W ),Vm(W ), πkl(W ),VΣ(W )






Constraints

Interfacial transfer terms must comply with:8>>>>><>>>>>:

P3k=1 αk = 1P3k=1 φk (W ) = 0P3k=1 Ik (W ) = 0P3k=1 ψk (W ) = 0P3k=1

P3l=1,l 6=k πkl(W )∂x (αl) = 0

We note again partial masses as mk = αkρk in phase k .

Remark: we do not retain mass transfer in the following.






Constraints



P3l=1,l 6=k πkl(W )∂x (αl) = 0








Constraints



P3l=1,l 6=k πkl(W )∂x (αl) = 0








Closure laws

Entropy inequality:Defining: (η, fη) the entropy-entropy flux pair:

η = m1S1 + m2S2 + m3S3

fη = m1S1U1 + m2S2U2 + m3S3U3

We may check that smooth solutions of the previous system fulfill thefollowing inequality:

∂t (η) + ∂x (fη) = RHSη .

where the phasic entropy Sk (Pk , ρk ) satisfies:

c2k∂Pk (Sk ) + ∂ρk (Sk ) = 0 .






Closure laws

Exact form of the right-hand side RHSη

RHSη =P3

k=1 akψk

+P3

k=1 ak Ik (Vm − Uk )

+P3

k=1 akP3

l=1,l 6=k (πkl − Pk )φl

+P3

k=1 ak (Uk − VΣ)P

l=1,l 6=k (πkl − Pk )∂x (αl)

noting :

ak = ∂Pk (sk ) /∂Pk (ek )

The constraint : 0 ≤ RHSη enables to pick up admissible closures.We notice that :

U1 − U3 = (U1 − U2) + (U2 − U3)

and we postulate once more the "consistent" form:

VΣ =3X

k=1

βk Uk ,

whereP

k βk = 1.Jean-Marc Hérard A few results on the modelling of multiphase flows





Closure laws

Exact form of the right-hand side RHSη

RHSη =P3

k=1 akψk

+P3

k=1 ak Ik (Vm − Uk )

+P3

k=1 akP3

l=1,l 6=k (πkl − Pk )φl

+P3

k=1 ak (Uk − VΣ)P

l=1,l 6=k (πkl − Pk )∂x (αl)

noting :

ak = ∂Pk (sk ) /∂Pk (ek )

The constraint : 0 ≤ RHSη enables to pick up admissible closures.We notice that :

U1 − U3 = (U1 − U2) + (U2 − U3)

and we postulate once more the "consistent" form:

VΣ =3X

k=1

βk Uk ,

whereP

k βk = 1.Jean-Marc Hérard A few results on the modelling of multiphase flows





Closure laws for πkl

Closure laws for coefficient functions πkl :

We set :3X

k=1

(Uk − VΣ)ak (3X

l=1,l 6=k

(Pk − πkl)∂x (αl)) = 0

We define : Z = (π12, π21, π13, π31, π23, π32), and :

C t = (a3β1P3 − a2β1P2, a2β3P2 + (1− β3)a3P3,a3β1P3 + (1− β1)a1P1, a1β3P1 + (1− β3)a3P3, 0, 0)

Thus previous constraints imply :

BZ = C

where B denotes the following matrix:







B =

0BBBBBB@a1(β1 − 1) 0 a1(1− β1) 0 −a2β1 a3β1

−a1β3 0 a1β3 0 a2β3 a3(1− β3)0 a2β1 a1(1− β1) a3β1 −a2β1 00 −a2β3 a1β3 a3(1− β3) a2β3 01 −1 0 −1 0 11 0 −1 0 −1 1

1CCCCCCA .

The determinant of B is non-zero since :

det(B) = −(a1a2β3 + a1a3β2 + a2a3β1)2

Remarks:

Obviously πkl are built as functions of VΣ, and not reversely, which is notso obvious when investigating two-phase flows.

The extension to a greater number of phases is straightforward...buttedious !







B =

0BBBBBB@a1(β1 − 1) 0 a1(1− β1) 0 −a2β1 a3β1


1CCCCCCA .


det(B) = −(a1a2β3 + a1a3β2 + a2a3β1)2

Remarks:









B =

0BBBBBB@a1(β1 − 1) 0 a1(1− β1) 0 −a2β1 a3β1


1CCCCCCA .


det(B) = −(a1a2β3 + a1a3β2 + a2a3β1)2

Remarks:









If we assume that:

β1 = 1, β2 = β3 = 0

we get at once the unique solution:

π13 = π31 = π32 = P3

π12 = π21 = π23 = P2

This is more or less the counterpart of the BN model for three-phase flows.






Closure laws for φk

Entropy consistent closure laws for source terms φk :

φ2 = α2(f1−2(W )α1(P2 − P1) + f2−3(W )α3(P2 − P3))/(|P1|+ |P2|+ |P3|)φ3 = α3(f1−3(W )α1(P3 − P1) + f2−3(W )α2(P3 − P2))/(|P1|+ |P2|+ |P3|)φ1 + φ2 + φ3 = 0

Time scales fk−l are positive. We retrieve formally two-phase closure lawswhen α3 = 0.

Setting π = α1α2α3, we also note that:

∂t (π) + U1∂x (π) = π(Xk<l

fk−l(W )(αk − αl)(Pl − Pk ))(|Pk |+ |Pk′ |+ |Pk”|)−1






Closure laws for φk

Entropy consistent closure laws for source terms φk :

φ2 = α2(f1−2(W )α1(P2 − P1) + f2−3(W )α3(P2 − P3))/(|P1|+ |P2|+ |P3|)φ3 = α3(f1−3(W )α1(P3 − P1) + f2−3(W )α2(P3 − P2))/(|P1|+ |P2|+ |P3|)φ1 + φ2 + φ3 = 0

Time scales fk−l are positive. We retrieve formally two-phase closure lawswhen α3 = 0.

Setting π = α1α2α3, we also note that:

∂t (π) + U1∂x (π) = π(Xk<l

fk−l(W )(αk − αl)(Pl − Pk ))(|Pk |+ |Pk′ |+ |Pk”|)−1






Closure laws for ψk

Closure laws for source terms ψk :

ψ2 = K2(a2 − a1)ψ3 = K3(a3 − a1)ψ1 + ψ2 + ψ3 = 0

provided positive values of K2 and K3. Of course, the sole constraint is:

0 ≤3X

k=1

akψk

Heat transfer without mass transfer may also occur between all phases.






Closure laws for ψk

Closure laws for source terms ψk :

ψ2 = K2(a2 − a1)ψ3 = K3(a3 − a1)ψ1 + ψ2 + ψ3 = 0

provided positive values of K2 and K3. Of course, the sole constraint is:

0 ≤3X

k=1

akψk

Heat transfer without mass transfer may also occur between all phases.






Closure laws for Ik

Entropy consistent closure laws for source terms Ik :

I2 = m2(U1 − U2)/τU12

I3 = m3(U1 − U3)/τU13

I1 + I2 + I3 = 0

Other closure laws are admissible, that may include a drag between allphases:

I2 = m2((U1 − U2)/τU12 + (U3 − U2)/τ

U23)

I3 = m3((U1 − U3)/τU13 + (U2 − U3)/τ

U32)

I1 + I2 + I3 = 0

assuming that a constraint holds for time scales: m3a3τU23 = m2a2τ

U32.






Closure laws for Ik

Entropy consistent closure laws for source terms Ik :

I2 = m2(U1 − U2)/τU12

I3 = m3(U1 − U3)/τU13

I1 + I2 + I3 = 0

Other closure laws are admissible, that may include a drag between allphases:

I2 = m2((U1 − U2)/τU12 + (U3 − U2)/τ

U23)

I3 = m3((U1 − U3)/τU13 + (U2 − U3)/τ

U32)

I1 + I2 + I3 = 0

assuming that a constraint holds for time scales: m3a3τU23 = m2a2τ

U32.






Closure laws for the interfacial velocity VΣ

We make a similar assumption as before on the structure of the statisticalfraction wave associated with λ = VΣ (LD property).

Admissible interface velocities:

Velocities VΣ that guarantee teh LD property are :

VΣ = Uk

VΣ =m1U1 + m2U2 + m3U3

m1 + m2 + m3

Hence we retrieve the counterpart of results in the two-phase flow framework,see P1,P2










VΣ = Uk

VΣ =m1U1 + m2U2 + m3U3

m1 + m2 + m3











VΣ = Uk

VΣ =m1U1 + m2U2 + m3U3

m1 + m2 + m3







Outline











Two-phase flows in a porous medium

Keeping previous notations, and noting ε(x) ∈]0, 1] the porosity, thetwo-phase flow model is:

8>>>>>>>><>>>>>>>>:

∂t (ε) = 0∂t (αk ) + VΣ∂x (αk ) = κk (W )∂t (εmk ) + ∂x (εmk Uk ) = εΓk (W )

∂t (εmk Uk ) + ∂x`εmk U2

k´

+ εαk∂x (Pk ) + ε(Pk − Π(W ))∂x (αk ) =...+ ε(Ik (W ) + (U1 + U2)Γk (W )/2)

∂t (εαk Ek ) + ∂x (εαk Uk (Ek + Pk )) + εΠ(W )∂t (αk ) = εΨk (W )...+ ε(U1 + U2)Ik (W )/2 + εΓk (W )U1U2/2

Remark: this model does not take into account sudden variations of theporosity (in the mean momentum equations).






Entropy inequality

Entropy inequality:

We introduce (η, fη) the entropy-entropy flux pair:

η = ε(m1S1 + m2S2)

fη = ε(m1S1U1 + m2S2U2)

Then smooth solutions comply with the following inequality:

∂t (η) + ∂x (fη) ≥ 0 .

Previous closure laws for

κk (W ), Γk (W ), Ik (W ),Ψk (W ),VΣ,Π(W )

preserve the latter entropy inequality.






Outline











Some comments pertaining to these extensions:

Three-phase flows:The classical formalism yields the expected results, in agreement withour early specifications: (i) hyperbolicity, (ii) entropy inequality, (iii)existence and uniqueness of jump conditions...However the simulationis difficult, mainly due to the occurence or three (or even four, dependingon the closure law for VΣ) LD fields, which slow down the rate ofconvergence of upwinding schemes (h1/2 for first-order schemes or h2/3

for second-order schemes);Flows in a porous medium:Difficulties arise when sudden changes of ε occur (this will be illustratedafterwards);Granular flows:"No" specific difficulties, but once more a real need for fine meshes,otherwise numerical approximations are meaningless;Turbulence:Some preliminar results have been obtained, while restricting to crudeclosure laws, but these inherit from non-uniqueness of solutions, due tonon-linear non-conservative products (see for instance the work byAudebert and Coquel, Gavrilyuk and Saurel, ...) .





































Fractional step methodConvective termsInterfacial transfer terms

Outline











Fractional step method

We use an entropy-preserving fractional step method, which involves twosteps.

Evolution step (in order to account for convective terms):8>><>>:∂t (αk ) + VΣ∂x (αk ) = 0∂t (mk ) + ∂x (mk Uk ) = 0∂t (mk Uk ) + ∂x

`mk U2

k + αk Pk´− Π(W )∂x (αk ) = 0

∂t (αk Ek ) + ∂x (αk Uk (Ek + Pk )) + Π(W )∂t (αk ) = 0

Relaxation step (in order to account for source terms):8>><>>:∂t (αk ) = κk (W )∂t (mk ) = Γk (W )∂t (mk Uk ) = Ik (W ) + (U1 + U2)Γk (W )/2∂t (αk Ek ) + Π(W )∂t (αk ) = Ψk (W ) + (U1 + U2)Ik (W )/2 + Γk (W )U1U2/2









`mk U2

k + αk Pk´− Π(W )∂x (αk ) = 0











`mk U2

k + αk Pk´− Π(W )∂x (αk ) = 0








Outline











Convection schemes

We use classical upwind Finite Volume schemes in order to get stableapproximations of the evolution step. It remains to check whether theseapproximations converge towards the correct solution.One building block is grounded on the structure of the solutions to theRiemann problem.Thus we have a classical approach:

Look for exact solutions to the Riemann problem;

Construct expected "first-order" schemes;

Compute and verify CV towards the true solution wrt the mesh size;

Extend to higher-order schemes and check the rate of CV(using classically RK2 + Minmod reconstruction on the symetrizingvariable);

Use the structure of the solution to the 1DRPb in order to constructboundary conditions.






Convection schemes












Convection schemes












Convection schemes












Convection schemes












Convection schemes












Structure of the solution to the 1DRPb:

The 1DRPb is such that:

Eigenvalues are :

λ1 = VΣ λ2 = U1 λ3 = U2 λ4,5 = U1 ± c1 λ6,7 = U2 ± c2

Fields associated with λ1−3 are LD, others are GNL; RI in 2− 7 fields areclassical, whereas those associated with the statistical fraction wave λ1

are specific;

αk only varies through the VFW corresponding to VΣ;

Through GNL fields, and assuming that no resonance phenomenonoccurs, jump conditions are also single-phase jump conditions:8<:

−σ[ρk ] + [ρk Uk ] = 0−σ[ρk Uk ] + [ρk U2

k + Pk ] = 0−σ[Ek ] + [Uk (Ek + Pk )] = 0






Structure of the solution to the 1DRPb:

The 1DRPb is such that:

Eigenvalues are :

λ1 = VΣ λ2 = U1 λ3 = U2 λ4,5 = U1 ± c1 λ6,7 = U2 ± c2

Fields associated with λ1−3 are LD, others are GNL; RI in 2− 7 fields areclassical, whereas those associated with the statistical fraction wave λ1

are specific;

αk only varies through the VFW corresponding to VΣ;

Through GNL fields, and assuming that no resonance phenomenonoccurs, jump conditions are also single-phase jump conditions:8<:

−σ[ρk ] + [ρk Uk ] = 0−σ[ρk Uk ] + [ρk U2

k + Pk ] = 0−σ[Ek ] + [Uk (Ek + Pk )] = 0






A few convection schemes:

Several schemes have been proposed and verified:

A non-conservative version of Rusanov scheme : simple enough, and itenables to obtain correct CV (h1/2 for first-order schemes and h2/3 forsecond-order schemes) (see [P8,P11,C6]);

An approximate Godunov scheme VFRoe-ncv, using symetrizingvariable Y = (αk ,Uk ,Pk ,Sk ) (see [P2]);

Fractional step methods (see [C6]);

Relaxation schemes (with or without FSM, see the PhD thesis of K.Saleh).

Other schemes have been proposed in the literature, among which : anapproximate Godunov scheme by Schwendemann-Wahle-Kapila 2006, aHLL-type scheme by Tokareva-Toro 2010, ... Other approaches aim atbuilding accurate schemes that deal with slow and fast waves separately (seethe work by Chalons, Coquel, Kokh, Spillane,...).






















































Outline











Relaxation schemes

A 3D, parallel code has been developed by O. Hurisse in MFEE, in whichrelaxation processes are dealt with sequentially, in such a way that thestability constraint is the CFL condition (see R3,P11). Difficulties appearwhen complex EOS are considered.

For instance, the implicit pressure relaxation step becomes rather uneasy :8<:∂t (αk ) = κk (W )∂t (mk ) = ∂t (mk Uk ) = 0∂t (αk Ek ) + Π(W )∂t (αk ) = 0

Conditions to get existence and uniqueness of discrete solutions withrelevant values of pressures and fractions αk ∈ [0, 1] may be obtained insome particular cases (see [R3,P7,P11]).

Taking heat and mass transfer (for complex EOS) into account is also adifficult task.






Verification of schemes and simulations

Verifying evolution and relaxation steps:

CV of some upwinding schemes when computing a two-fluid model (see[P8]);

CV of some upwinding schemes when computing a two-fluid modelin a porous medium;

CV of some schemes when computing the relaxation step (see [P11]).

Two simulations:

2D unsteady simulation of a heated wall with the two-fluid approachUnsteady simulation of a heated pipe with the two-fluid approach






A Riemann problem in a porous medium

0.0001 0.001 0.01 0.1 1

0.0001

0.001

0.01

0.1

U1

P2

α1

ρ2P

1ρ1

U2

Ch1/2

Convergence in L1 norm for WBR scheme. Coarse mesh: 100 cells; finermesh: 800000 cells, setting: CFL = 1/2.






A Riemann problem in a porous medium

0.001 0.01 0.1 1

0.001

0.01

0.1

U2

P2

U1 α1

ρ2

P1ρ1 Ch

1/2

Convergence in L1 norm for MR scheme. Coarse mesh: 100 cells; finermesh: 400000 cells, setting: CFL = 1/2.






2D unsteady simulation of a heated wall with the two-fluid approach

Figure: Heated wall: contours of the liquid void fraction αl (left) and liquid pressure Pl(right) at time T = 0.0237.






Unsteady simulation of a heated pipe with the two-fluid approach

0 100 200 300 4000,992

0,993

0,994

0,995

0,996

0,997

0,998

0,999

0 100 200 300 4000,99984

0,99986

0,99988

0,9999

0,99992

0,99994

0,99996

0 100 200 300 400912

913

914

915

916

917

918

919

0 100 200 300 40019,6

19,7

19,8

19,9

20

20,1

0 100 200 300 4003,98e+06

4e+06

4,02e+06

4,04e+06

4,06e+06

4,08e+06

4,1e+06

0 100 200 300 400

5

5,2

5,4

5,6

Figure: Heated pipe with mass transfer: profiles along the main axis of: αl (top left),ml/(ml + mv ) (top right), ρl (medium left), ρv (medium right), Pl (bottom left), Ul(bottom right), at time T = 5. Plain line: weak interphase heat transfer (τT = 10−1),dashed line: high interphase heat transfer (τT = 10−5).





Conclusion and perspectivesReferences

Outline











Conclusion and perspectives-Models

Conclusion and perspectives

Modeling, analysis and simulation: a synthesis

The basic specifications enable to build multiphase flow models in orderto handle unsteady simulations including mass, momentum and energytransfer,

The analysis of the Riemann problem provides solutions and tools thatenable the building of stable schemes and their verification, eitherfocusing on two-phase flows or three-phase flows, in a fluid or a porousmedium;

The relaxation step is crucial and no so easy to cope with, depending onthe EOS.






Conclusion and perspectives-Models


Modeling, analysis and simulation: a synthesis

The basic specifications enable to build multiphase flow models in orderto handle unsteady simulations including mass, momentum and energytransfer,

The analysis of the Riemann problem provides solutions and tools thatenable the building of stable schemes and their verification, eitherfocusing on two-phase flows or three-phase flows, in a fluid or a porousmedium;

The relaxation step is crucial and no so easy to cope with, depending onthe EOS.







A few current difficulties:

Modeling:

- Hierarchy of models wrt relaxation time scales associated withU,P,T , µ and possible applications to the interfacial unsteady coupling,(see [R1, R2, R4, C3, C4]),- A challenging point concerns the transition from a dilute (αv << 1) to adense situation (αl << 1) , see [R5];

Analysis:

Existence and uniqueness of solutions of the 1DRPb (when resonanceoccurs);








Modeling:


Analysis:









Modeling:


Analysis:









Simulation:

- Construction of more accurate schemes : some work in progress (PhDthesis of Yujie Liu -AMA, SITAR project- and PhD thesis of Khaled Saleh-MFEE, Neptune project-),- Schemes that are well suited for low Mach applications (collaborationwith F. Coquel),- Coupled solvers for the relaxation step (towards a U,P,T , µequilibrium),- Improvement of the porous formulation and associated schemes (largeporosity inhomogeneities);

Better understanding of physical patterns:

Relaxation time scales








Simulation:











Simulation:









A relaxation scheme for BN model -with courtesy of Khaled Saleh-

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-0.4 -0.2 0 0.2 0.4

Phase 1 velocity u1

RusanovRelaxation

Exact solution

10-3

10-2

10-1

10-5 10-4 10-3 10-2

E(∆

x)

∆x

Error in L1-norm

α1α1ρ1

α1ρ1u1α2ρ2

α2ρ2u2sqrt(∆x)

10-3

10-2

10-1

100 101 102 103 104 105

L1 -Err

or

CPU-time (s)

α1

RelaxationRusanov

Velocity profile (top-left) and convergence in L1 norm (top-right) for therelaxation scheme. L1 norm of the error for α wrt CPU time, including acomparison with Rusanov scheme (bottom).






Outline











Publications, Reports and CommunicationsP1- Closure laws for two-fluid two-pressure model, F. Coquel, T. Gallouët, J.M. H., N.Seguin, C.R. Acad. Sci. Paris, vol.I-334, p.927-932,2002.P2- Numerical modelling of two-phase flows using the two-fluid two-pressure approach, T. Gallouët, J.M. H., N.Seguin , M3AS, vol.14(5),p.663-700, 2004.P3- A simple method to compute standard two-fluid models, J.M. H., O. Hurisse, Int. J. Comp. Fluid Dyn., vol.19(7), p.475-482, 2005.P4- An hyperbolic three-phase flow model, J.M. H., C.R. Acad. Sci. Paris, vol.I-342, p.779-784, 2006.P5- A three-phase flow model, J.M. H., Math. Comp. Model., vol.45, p.732-755,2007.P6- Un modèle hyperbolique diphasique bifluide en milieu poreux, J.M. H., C.R. Mecanique, vol.336, p.650-655, 2008.P7- Hyperbolic relaxation model for granular flow, T. Gallouët, P. Helluy, J.M. H., J. Nussbaum, Math Mod. Num. Anal., vol.44, p.371-400,2010.P8- A two-fluid hyperbolic model in a porous medium, L. Girault, J.M. H., Math Mod. Num. Anal., vol.44, p.1319-1348, 2010.P9- Multidimensional computations of a two-fluid hyperbolic model in a porous medium, L. Girault, J.M. H., Int. J. Finite Volumes vol. 7(1),p. 1-33, 2010.P10- Multidimensional two-phase flow modelling applied to interior balistics, J. Nussbaum, P. Helluy, J.M. H., B. Baschung, J. Applied Mech., 2011.P11- A fractional step method to compute a class of compressible gas-liquid flows, J.M. H., O. Hurisse, Computers and Fluids, vol.55,p.57-69, 2012.P12- A turbulent two-phase flow model, J.M. H., submitted.R1- Couplage interfacial d’un modèle homogène et d’un modèle bifluide, J.M. H., O. Hurisse,EDF report H-I81-2006-04691-FR, 2006.R2- Couplage interfacial du code Neptune-CFD et d’un code HRM, J.M. H., O. Hurisse, J. Lavieville EDF report H-I81-2007-0863-FR, 2007.R3- Schémas d’intégration du terme source de relaxation des pressions phasiques d’un modèle bifluide hyperbolique, J.M. H., O.Hurisse,EDF report H-I81-2009-01514-FR, 2009.R4- Proposition pour un couplage interfacial instationnaire entre un modèle triphasique et un modle diphasique, L. Girault, J.M. H., EDFreport H-I81-2010-0355-FR, 2010.R5- Une classe de modèles diphasiques bifluides avec changement de régime, J.M. H., EDF report H-I81-2010-0486-FR, 2010.C1- Numerical modelling of turbulent two-phase flows using the two-fluid approach, J.M. H., http://www.aiaa.org/, AIAA paper 2003-4113,2003.C2- A relaxation scheme to compute three-phase flow models, J.M. H., http://www.aiaa.org/, AIAA paper 2007-4455, 2007.C3- Boundary conditions for the coupling of two-phase flow models, J.M. H., O. Hurisse, http://www.aiaa.org/, AIAA paper 2007-4458, 2007.C4- The unsteady coupling of multiphase flows, J.M. H., http://www.aiaa.org/, AIAA paper, 2012.C5- Computing two-fluid models of compressible water-vapour flows with mass transfer, J.M. H., O. Hurisse, http://www.aiaa.org/, AIAApaper, 2012.C6- On the computation of the Baer-Nunziato model, F. Crouzet, F. Daude, P. Galon, P. Helluy, J.M. H., Y. Liu http://www.aiaa.org/, AIAApaper, 2012.


Documents

A few results on the modelling of multiphase flows...Vincent Guillemaud (PhD thesis, CEA-LATP, nov. 2003-oct. 2006), Philippe Helluy (Strasbourg University, IRMA), through DFG-CNRS