An interface condition to compute compressible flows in variable cross section ducts

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Numerical analysis/Mathematical problems in mechanics

An interface condition to compute compressible flows in variable cross section ducts

Conditions d’interface pour le calcul d’écoulements compressibles en conduite à section variable

Jean-Marc Hérard

^a,b

, Jonathan Jung

^c,d,e

aEDFR&D,MFEE,6,quaiWatier,78400,Chatou,France

bI2M,UMRCNRS7353,39,rueJoliot-Curie,13453 Marseille,France

cLMA-IPRA,UniversitédePauetdespaysdel’Adour,UMRCNRS5142,avenuedel’Université,64013Pau,France dINRIABordeauxSudOuest,CagireTeam,351,coursdelaLibération,33405 Talencecedex,France

eLJLL&LRCManon,4,placeJussieu,75005,Paris,France

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received19June2015

Acceptedafterrevision4November2015 Availableonline4February2016 PresentedbyOlivierPironneau

We proposeanimprovedinterfaceconditioninordertoaccountforheadlosses inpipe whensomediscontinuouscrosssectionsoccur.

©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r é s um é

Nousproposonsd’améliorerlaconditiond’interfaceafindeprendreencomptelapertede chargepourunécoulementenconduiteàsectionvariablediscontinue.

©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

We examinein thispapersome possible improvementofthe well-balancedstrategy inorder tocompute approxima- tionsofsolutionstoEuler-typemodelsinone-dimensionalgeometriescorrespondingbasicallytopipeconfigurations.From a practical point of view, these methods are very usefulin order to build stable and efficient algorithms. If

ρ

,u,p re- spectivelydenote thedensity,the velocityandthe pressureofthe fluid,andnotingasusual

(p,

ρ

)the internalenergy, E=

ρ

(p,

ρ

)+^u2/2

thetotalenergy,andA(x)the–steady–crosssectionofthepipe,thetargetsetofequationsis

⎧ ⎨

⎩

∂

t

(

A

ρ ) + ∂

x

(

A

ρ

u

) =

⁰

,

∂

_t

(

A

ρ

u

) + ∂

_x

A

ρ

u²

+

^Ap

−

^p

∂

_xA

=

^A

(

x

)

S^r_u

,

∂

t

(

A E

) + ∂

x

(

Au

(

E

+

^p

)) =

^A

(

x

)

u S^r_u

,

(1)

E-mailaddresses:[email protected](J.-M. Hérard),[email protected](J. Jung).

http://dx.doi.org/10.1016/j.crma.2015.10.026

1631-073X/©^{2015Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

andtheassociatedunknownisW=(A

ρ

,A

ρ

u,A E)^T.Thetermsontheright-handsideS^r_U standfortheregularheadlosses, which accountforviscous effectsonthe wallsofthe pipe.Thissetofequationsismeant whenthecross-section A(x)is regular (at least C²). In practice,this information is provided by the user. When convergent or divergent regions occur in the pipe, andwhen the meshesthat are used are rather coarse, the well-balancedstrategy naturally arises forthose who use FiniteVolumemethods. Inthat case, thecross section isassumedto be constantwithin each cell, andanother equation is added to (1), which is ∂tA=^{0. The idea is very appealing, and many authors have been} investigatingthat particular topic, bothat thecontinuous anddiscrete levels,among whichwe mayatleastcite [2,3,8–10,12]. Thoughthe crosssection A(x)isknown,thenewunknownisnow W˜ =(A,A

ρ

,A

ρ

u,A E)^T,andtermsoftheformψ(W)∂xA arenow partofthe convectivesubset.Meanwhile, asteadywave associatedwithλ=^0,correspondingtothe “constraint”∂tA=⁰ arises. The one-dimensionalRiemannproblemassociatedwithsystem(1) complementedwith∂tA=^{0 and}discontinuous initialconditionsmaythenbeinvestigated.Asaconsequence,thereisthetemptationtousethisapproachnotonlytodeal withregular crosssections,butalsoto applyitwhen discontinuitiesare presentinthe pipe,whichisobviouslyofmajor interest for practical computations in an industrial framework. In that case, the classical well-balanced strategy simply assumes thatinterface conditionsthat should beenforced atthesteadyinterface separatingtwo cells withdistinctcross sectionscorrespondtotheRiemanninvariantsofthesteadywave.InthecaseofEulerequations,thismeansthatquantities Q=^A

ρ

u,Q H whereH=(E+^p)/

ρ

andtheentropys(p,

ρ

)shouldbepreservedacrossthesteadyinterface.Well-Balanced (WB) schemesrelyingonthisapproachcanthenbebuilt,andsomeofthembehavequitewellowingtotheirstabilityand abilitytomaintainwell-balancedinitialconditions,evenwhenthecrosssectionisdiscontinuous.ItevenseemsthattheWB techniqueismandatoryinordertoachievenumericalconvergencetowardsthecorrectsolutions(seeforinstance[6,11]).

Nonetheless,amajordrawbackishiddeninthebasicformulation.Thisisduetothefactthatforthecaseofdiscontinu- ous crosssections,thephysicalsingularheadlosseshavenotbeenaccountedfor.Asaresult,thecomparisonofthe“true”

multi-dimensionalsolutionandoftheone-dimensionalwell-balancedcomputationalresultsisratherpoor;thisisdescribed in[7]forinstance.Oneretrievestheexpectedresultsthat arethefollowing:the quantities Q =^A

ρ

u,Q H should remain constant acrossinterfacessupportingdiscontinuous crosssections,butthe entropydoesnot. Thisisdirectlylinkedto the factthatsomesingularcontributionhasbeenomittedinthemomentumequation.

In order to cure that deficiency, at least two strategies may be proposed. A first one simply consistsin a modified one-dimensionalapproachthatwouldrelyonan integralformulationofthemulti-dimensionalsetofgoverningequations.

This scheme (called 1D+ ^{in the sequel) hasbeen partially} investigated in[1], andthisseems to be a ratherpromising technique in order to handle this difficulty. Anotherstrategy that is appealing in based on similar ideas, and it simply consistsinenforcinganewsetofinterfaceconditionsatthesteadyinterfaceassociatedwiththecross-sectiondiscontinuity.

The basicrequirementisthat thesemodifiedinterface conditionsshouldaccountforsingular headlossesduetopressure effects,atleastinaweakway.Thisispreciselythemainobjectiveofthepresentcontribution.

Thus the paperisorganized asfollows: we firstlyintroduce thenewinterface condition, we secondly explain howto compute left andrightstates satisfyingthe newinterface condition, we thirdly introduce thenumerical schemeandwe finallydosomenumericalvalidations.

2. Anewformulationforflowsinvariablecrosssectionducts 2.1. Thenewinterfacecondition

WeomittheregularheadlossesinSystem(1),whichmeansthatS^r_u=^{0 inSystem(1).}Integratingthemulti-dimensional Eulerequationsonthephysicaldomain(seeFigurebelow),wemaynowproposethreenaturalcandidatesforthesteady interfaceconditions

⎧ ⎨

⎩

[Q]⁺₋

=

0

,

[Q u

+

^Ap]+₋

−

^p[A]+₋

=

⁰

,

[Q H]⁺₋

=

⁰

,

(2)

withp

=

p⁻

,

ifA⁻

>

A⁺

,

p⁺

,

otherwise

,

⁽³⁾

wherewenoteby·^+(resp.·^{−)thevalueofthequantityontheright(resp.theleft)sideofthe}discontinuousinterface.The choice ofthewallpressure pisbasedonthetwo-dimensionalapproach.Indeed,ifwe haveacontractionwith A⁻>A⁺ (resp.anexpansionwith A⁻<A⁺),thepressureontherestrictedwallistheleft(resp.right)pressure.

2.2. Computationofleftandrightstatessatisfyingtheinterfacecondition(2)

Weassumethatthefluidsatisfiesaperfectgasequationofstates:p=(

γ

₋1)

ρ

. W⁺=^W+

W⁻

:weassumethatW⁻isknownandwecomputeW⁺suchthattheinterfacecondition(2)issatisfied.For givenvalueofW⁻,wecancompute Q⁻=^A−

ρ

⁻u⁻andH⁻=_γ^γ_−1ρ^p−−+^(u−₂^{)2.Thefirsttwoconditionsin(2)give}

u⁺

=

^Q−

A⁺

ρ

^{+ and p}

+

=

^p−

+

^Q− min

A⁻

,

A⁺

u⁻

−

^Q− A⁺

ρ

⁺

^.

Injectingthesetwoquantitiesinthethirdconditionof(2),wededucethat

ρ

⁺isarootofthesecond-orderpolynomial:

P₋

(

X

) =

H⁻X²

− γ γ −

1

p⁻

+

^Q− min

A⁻

,

A⁺

u⁻

X

+

Q⁻

₂

A⁺

γ

γ −

1 1 min

A⁻

,

A⁺

−

¹ 2A⁺

=

0

.

(4)

W⁻=^W−(W⁺): we now assume that W⁺ isknown andwe compute W⁻ asa function of W⁺. Given W⁺, we com- pute Q⁺=^A+

ρ

⁺u⁺andH⁺=_γ^γ_−1ρ^p+++^(u+₂^{)2.Wegetatonce:}

u⁻

=

^Q+

A⁻

ρ

^{− and p}

−

=

^p+

+

^Q+ min

A⁻

,

A⁺

u⁺

−

^Q+ A⁻

ρ

⁻

where

ρ

⁻ isarootof

P₊

(

X

) =

H⁺X²

− γ γ −

1

p⁺

+

^Q+ min

A⁻

,

A⁺

u⁺

X

+

Q⁺

2

A⁻

γ

γ −

1 1 min

A⁻

,

A⁺

−

¹ 2A⁻

=

0

.

(5)

Ifequations(4)and(5)havetwodistinct realsolutions,we choosetheonethat hasthesamesonicityasthestateon theothersideoftheinterface.Ifthereisonlyonesolution,wechoosethisuniquesolution.Whenthereisnorealsolution, thealgorithmisstopped.

2.3. Numericalscheme

Wepresentaschemeforapproximatingsolutionstosystem(1)on[^xmin,xmax]^{,relyingontheargumentsintheprevious} sections.Given an uniform time step t anda mesh size x= ^xmax−_N^{xmin, setting x}j=^xmin+^jx,j=⁰· · ·^{N, andt}n= nt,n∈N^{,we denoteby Wn}_j ^the approximationof thevalues W(x_j,tⁿ) oftheexact solutionto system(1).The WB–HJ schemeisdefinedby

Wⁿ_j⁺¹

=

^Wn_j

−

t

x

F^Rus

Wⁿ_j

,

Wⁿ_j+1

₋

−

^FRus

Wⁿ_j−1

₊

,

Wⁿ_j

(6)

where

Wⁿ_{j+1 −}

= W⁻

Wⁿ_j+1

,

Wⁿ_{j−1 +}

= W⁺

Wⁿ_j−1

are computed in Section 2.2, and the numerical flux F^Rus(W_l,W_r)istheRusanovflux

F^Rus

(

W_l

,

Wr

) =

^F

(

W_l

) +

^F

(

Wr

)

2

−

^max

( |

^ul

| +

^cl

, |

^ur

| +

^cr

)

2

(

Wr

−

W_l

)

(7)

with F(W)=(Q,Q u+^Ap,Q H)^T. The WB–HJscheme is an adaptation ofthe scheme ofKröner andThanh [12], called WB–KT inthesequel,forwhichtheclassicalinterfaceconditionon

Q,s=_ρ^pγ,H

isreplacedbythenewinterfacecondi- tion(2).

Proposition1.TheschemeWB–HJiswell-balanced:ifforall j, Q⁰_j =^{Q0, I0}j=^{I0and H0}j=^{H0where I}:=^{Q u}+^A(p−^p),then forall j andforall n,

Qⁿ_j

=

^Q0_j

=

^Q0

,

Iⁿ_j

=

^I0_j

=

^{I0 and Hn}_j

=

^H0_j

=

^H0

.

(8) 3. Numericalresults

Foralltest cases,we chooseaperfectgas equationofstate: p=(

γ

₋1)

ρ

, where

γ

issetto 7/5.The time stept satisfiestheCFLconditionwheretheCFLconstanthasbeensetto 0.4.

3.1. Astationaryflowforthenewinterfacecondition(2)

We start witha test casewhereinitial conditions are Well-Balancedin thesense of (8) forthe newinterface condi- tion(2).Inpractice,weconsiderthefollowingvalues

(

A

, ρ ,

u

,

p

) (

x

,

t

=

⁰

) =

(

2

,

4

.

711

, −

¹⁰

.

614

,

38

.

588

) ,

ifx

<

0

,

(

1

,

10

, −

¹⁰

,

100

) ,

otherwise. (9)

Fig. 1.Astationaryflowforthenewinterfacecondition(2):densityρ (topleft),meandischargeQ (topmiddle),totalenthalpyH(topright),physical entropys(bottomleft)andthevariationofthenewinvariantI=^{Q u}+^A(p−^p)(bottomright)attimet=⁰.2 sforaninitialconditionsatisfyingthe newinterfacecondition(2).

Fig. 2.AclassicalRiemannproblem:densityρ(left),velocityu(middle)andL¹normoftheerrorforthedensity(right).Thecoarserandfinermeshes usedforthecomputationoftheerrorcontainN=200 andN=15000 regularcells.

The computational domain is [−⁰.5,0.5]^{, the mesh contains 4000 nodes. In Fig. 1, we plot the density, the mean dis-} charge Q, the total enthalpy H, the physical entropy s and the variation of the new invariant I=^{Q u}+^A(p−^p) at time t=⁰.025 s. Acrossthe interface x=^{0, Q and H arepreservedby thethree}schemes: 1D+^{[1],WB–HJandWB–KT.}

Thedifferencebetweentheselasttwoschemesisontheinvariant I:WB–HJpreservesI,whilesispreservedbytheWB–KT scheme.

3.2. AclassicalRiemannproblem

OneadvantageoftheWB–HJschemeagainstthediscrete 1D+^{scheme[1]isthatwecancompareourresulttotheexact} solutiontoaclassicalRiemannproblem.Theexactsolutioniscomputedwiththeclassicalapproachwhereweusethethree invariants(Q,I,H)atthesteadycontactdiscontinuity.WeconsidertheRiemann probleminvolvingashockwaveon the leftoftheinterfacediscontinuity

(

A

, ρ ,

u

,

p

) (

x

,

t

=

⁰

) =

(

2

,

2

.

970

,−

⁹

.

093

,

20

) ,

ifx

<

0

,

(

1

,

10

, −

¹⁰

,

100

) ,

otherwise. (10)

Thefinal timeofcomputation ist=⁰.03 s.InFig. 2,weplotthedensity

ρ

andthevelocity uobtainedwith 4000 nodes.

The approximatedsolutionisclosetotheexactsolution.Wealsoperformaconvergencestudyonthedensityvariable.It maybecheckedthattheconvergencerateisverycloseto 0.8 inL¹norm.

Fig. 3.A shock wave hitting a section contraction: densityρ(left), velocityu(middle) and pressurep(right).

3.3. Ashockwavehittingasectioncontraction Weconsiderthefollowinginitialdata

(

A

, ρ ,

u

,

p

) (

x

,

t

=

0

) =

⎧ ⎨

⎩

0

.

2

,

0

.

167

,

100

,

1

.

5

×

¹⁰⁴

,

ifx

< −

⁰

.

1

,

0

.

2

,

0

.

125

,

0

,

10⁴

,

if

−

⁰

.

1

≤

^x

<

0

,

0

.

1

,

0

.

125

,

0

,

10⁴

,

otherwise.

(11)

Ashockwavestartingfromx= −⁰.1 travelstotherightandhitsthesectioncontractionlocatedatx=^{0.Partoftheshock} waveisreflectedandpartistransmittedthroughtheright.Wecomparetheresultsobtainedwiththethreeone-dimensional schemes(WB–HJ,WB–KTand 1D⁺)with 4000 nodestotheresultsobtainedwithatwodimensionalsetofEulerequations onaveryfinemeshcontaining 216000 cells(1200 inx-direction),usingRusanovscheme.Thetwo-dimensionalresultsare averagedinthe y-direction. Thecomputation wasdone usingthetoolbox CDMATH[4,5].InFig. 3,weplotthedensity

ρ

, thevelocityu andthepressure pattimet=¹²×^{10−4 s.Wetakethe}2D-averagingsolutionasthereferencesolution.The WB–HJ schemegivesbetter resultsthan WB–KT and 1D+ ^{schemes. It remains tostudythe Riemannproblemassociated} withsystem(1)together withthe newinvariant (Q,I,H) atthesteadyinterface. Itis alsoplannedto develop asimple solver[3]associatedwiththenewinterface condition(2).Thisnewsolvercouldbe morerobust thantheWB–HJscheme.

Finally,weemphasizethatthepresentinterfaceconditioncanbeeasilyextendedtootherconservativesystems.

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