Regularized characteristic boundary condition for the Lattice Boltzmann methods at high Reynolds number flows

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Regularized characteristic boundary condition for the

Lattice Boltzmann methods at high Reynolds number

flows

Gauthier Wissocq, Nicolas Gourdain, Orestis Malaspinas, Alexandre

Eyssartier

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Eprints ID: 17413

To cite this version:

Wissocq, Gauthier and Gourdain, Nicolas and Malaspinas, Orestis and

Eyssartier, Alexandre Regularized characteristic boundary condition for the Lattice

Boltzmann methods at high Reynolds number flows. (2017) Journal of Computational

Physics, vol. 331. pp. 1-18. ISSN 0021-9991

Regularized

characteristic

boundary

conditions

for

the

Lattice-Boltzmann

methods

at

high

Reynolds

number

flows

Gauthier Wissocq

a

,

b

,

c

,

∗

,

Nicolas Gourdain

a

,

Orestis Malaspinas

d

,

Alexandre Eyssartier

b

a_{ISAE,Dpt.ofAerodynamics,EnergeticsandPropulsion,Toulouse,France} b_{Altran,DOME,Blagnac,France}

c_{CentreEuropéendeRechercheetdeFormationAvancéeenCalculScientifique(CERFACS),CFDTeam,42avenueGaspardCoriolis,}

31057 ToulouseCedex01,France

d_{SPC- CentreUniversitaired’Informatique,UniversitédeGenève7,routedeDrize,CH-1227Switzerland}

a

b

s

t

r

a

c

t

Keywords:

LatticeBoltzmannmethod Characteristicboundaryconditions LODI

HighReynoldsnumberflows

ThispaperreportstheinvestigationsdonetoadapttheCharacteristicBoundaryConditions (CBC) tothe Lattice-Boltzmann formalismfor highReynoldsnumber applications.Three CBC formalismsare implementedand tested inanopensource LBMcode: the baseline local one-dimension inviscid (BL-LODI)approach, its extension including the effects of the transverse terms (CBC-2D) and a local streamline approach in which the problem is reformulatedin theincidentwaveframework (LS-LODI). Thenall implementationsof theCBC methodsare testedforavarietyoftestcases,rangingfromcanonicalproblems (such as 2D plane and spherical waves and 2D vortices) to a 2D NACA profile at highReynoldsnumber(Re=105_{),representativeofaeronauticapplications.The LS-LODI}

approachprovidesthebestresultsforpureacousticswaves(planeand sphericalwaves). However,itisnotwellsuitedtotheoutflowofaconvectedvortexforwhichtheCBC-2D associatedwitharelaxationondensityandtransversewavesprovidesthebestresults.As regardsnumericalstability,aregularizedadaptationisnecessarytosimulatehighReynolds number flows. The so-called regularized FD (Finite Difference) adaptation, a modified regularized approach where the off-equilibrium part of the stress tensor is computed thanksto afinite differencescheme, is the onlytested adaptation that can handlethe highReynoldscomputation.

Introduction

Abetter understandingofturbulentunsteady flowsis anecessarysteptowards abreakthrough inthedesignof mod-ern aircraft andpropulsivesystems. Dueto the difficulty ofpredicting turbulence withcomplex geometry,the flow that developsintheseenginesremains difficulttopredict.Atthistime,themostpopularmethodtomodeltheeffectof turbu-lence isstill theReynoldsAveragedNavier–Stokes (RANS)approach. Howeverthere issome evidencethat thisformalism isnot accurateenough,especially whenadescriptionoftime-dependent turbulentflows isdesired(highincidenceangle, laminar-to-turbulenttransition,etc.)[1].Withtheincreaseincomputingpower,LargeEddySimulation(LES)appliedtothe

*

Correspondingauthor.

E-mailaddress:[email protected](G. Wissocq).

Navier–Stokes equationsemergesasa promisingtechnique toimproveboth knowledgeofcomplexphysics andreliability offlow solverpredictions[1].Itisstillthe mostpopularandmatureapproachtodescribe thebehaviorofturbulentflow incomplexgeometries(e.g. aircraftandgasturbines).However,theresolutionoftheNSequationsrequirestoaddartificial dissipation toensurenumericalstability [1].Theconsequenceisan over-dissipationwhichaffectstheflowandlimitsthe capabilitytotransportflowpatterns(liketurbulence)onalongdistance.Insomespecificcases,likeaero-acoustic(far-field noise),NScanthusfacesomedifficultiestopredicttheflow.

In thiscontext, there isan increasing interestin the fluid dynamicscommunityfor emerging methods,based on the LatticeBoltzmannapproach[2–4].TheLattice-BoltzmannMethod(LBM)hasalreadydemonstrateditspotentialforcomplex geometries,thankstoimmersedboundaryconditions(thatallowtheuseofcartesiangrids)andlowdissipationproperties requiredforcapturingthesmallacousticpressurefluctuations[5,6].LBMalsoprovidestheadvantageofaneasy paralleliza-tion,makingitwellsuitedforHigh-PerformanceComputing[7].However,themostwidelyusedLattice-Boltzmannmodels stillsufferfromweaknesseslikealackofrobustnessforhighMachnumberflows(M

>

0

.

4),alimitationtolow compress-ible isothermalflows [2]andtheuseofartificialboundaryconditions(Dirichlet/Neumann typescan leadtothereflection ofoutgoingacousticwavesthathaveasignificantinfluenceontheflowfield [8–10]).Whiletheuseofartificialboundary conditionsisalsocriticalforNSmethods,itismoreproblematicforLBMduetothelowdissipationofthemethod.

A potentialwayto avoidunphysicalacousticreflections attheboundaryis tousea “sponge layer”inside the compu-tational domain,on which artificial dissipation (by upwinding) is introduced or physicalviscosity is increased(viscosity sponge zones).Acoustic waves (physicalornot) arethus dampedin such azone, whichallows to eliminateor limit nu-merical reflections[9].Thissolutionhashoweverimportantdrawbacks.Firstthecalibrationofspongelayersisdifficult,as a balancemustbefoundbetweenabrutalincrease oftheviscosity(thatwillgenerateacousticreflections)andatoolow dissipation that will not be effective. Such sponge layersalso havean impact on thecomputational cost, sincea part of thedomainisdedicatedtoslowlyincreasingtheviscosity.Last,someboundaryconditionscannotbetreatedwithasponge layer,forinstanceaninletwithturbulenceinjection.

For NS methods, asuccessful approach isthe use ofnon-reflective boundary conditions basedon a treatment of the characteristicwavesofthelocalflow[11–13].However,theextensionofthisapproachtoLBMisnotstraightforwardgiven thedifficultytofindabridgebetweentheLBMthatdescribestheworldatthemesoscopiclevel(apopulationofparticles) andtheNSworldbasedonamacroscopicdescriptionoftheflow.Butsomeprogresshasrecentlybeenmadeonthe adap-tationofcharacteristicboundaryconditionstotheLBMformalism.IzquierdoandFueyo[14]usedapressureantibounceback boundarycondition[15]adaptedtothemultiplerelaxationtime(MRT)collisionscheme[16]toimposetheDirichletdensity andvelocityconditionsgivenbythelocalone-dimensionalinviscid(LODI)equations,whichprovidednon-reflectiveoutflow boundaryconditionsforone-dimensionalwaves.Morerecently,Junget al.[17]extendedthepreviousworktoincludethe effectsoftransverseandviscoustermsintheCharacteristic BoundaryConditions(CBC) andshowedgoodperformancefor vortex outflow.Meanwhile,Heubes et al.[18]adapted thesolutiongivenby amodified-Thompson approachby imposing thecorrespondingequilibriumpopulationsandSchlaffer[19]assessedamodifiedZou/Heboundarycondition[20].

Still, previousresearchesare limitedtolow Reynoldsnumberapplications whileLattet al.showedthatincreasing the Reynolds numbercan havedrasticimpacton thenumericalstabilityoftheLBM boundarycondition [21].Evenwhenthe MRT collisionisused,asin[14,15],thenumericalstabilityofthecharacteristicboundaryconditionshasnot been demon-strated. Theaimofthisstudyisthustodevelop anumericallystableadaptationoftheCBCtothe LBMformalism taking advantage of theregularized collision scheme [22], whichhas proved tobe numerically stableathigh Reynoldsnumber flows [23],andthecorrespondingregularizedboundaryconditions[21].DifferentkindsofCBCwillalsobeevaluated.This articleisstructuredasfollows.ThefirstsectiondescribestheLBMframework.Then,thesecondsectionpresentsthreekinds ofCBCsandthreepossibleadaptationstotheLBMformalismfor2D problemsinthelowcompressibleisothermalcase. In thethirdsection,thesemodelsareassessedforsimplecases:normal,obliqueandsphericalwavesandaconvectedvortex at Re

=

103_{.Finally,themethodisassessedonahighReynoldsnumberapplication:aNACA0015airfoilatRe}

₌

₁₀5_.

1. Numericalmethodandgoverningequations

LatticeBoltzmannframeworkforisothermalflows

AdescriptionoftheLattice-BoltzmannMethodcanbefoundin[2–4].Thegoverningequationsdescribetheevolutionof theprobabilitydensityoffindingasetofparticleswithagivenmicroscopicvelocityatagivenlocation:

fi

(

x

+

ci



t

,

t

+ t

)

=

fi

(

x

,

t

)

+ 

i

(

x

,

t

)

(1) for0

≤

i

<

q,whereciisadiscretesetofq velocities, fi

(

x

,

t

)

isthediscretesingleparticledistributionfunction correspond-ingtociand



iisanoperatorrepresentingtheinternalcollisionsofpairsofparticles.

Macroscopic values such asdensity,

ρ

, and the flow velocity, u,can be deduced from the set of probability density functions fi

(

x

,

t

)

,suchas:

Fig. 1. Schematic plot of velocity directions of the D2Q9 model (left) and the D3Q27 model (right).

SomeofthemostpopularchoicesforthesetofvelocitiesareD2Q9andD3Q27lattices,respectively9velocitiesin2D and27 velocitiesin 3D (see Fig. 1). For bothof theselattices, the soundspeed in lattice units(normalized by theratio betweenthespatialresolutionandthetimestep



x

/

t)isgivenbycs

=

1

/

√

3[3].

Thecollisionoperator



iisusuallymodelledwiththeBhatnagar–Gross–Krook(BGK)approximation[24],whichconsists inarelaxation,witharelaxationtime

τ

,ofeverypopulationtothecorrespondingequilibriumprobabilitydensityfunction

f_i(eq):



i

= −

1

τ



fi

(

x

,

t

)

−

f_i(eq)

(

x

,

t

)



.

(3)

Theequilibriumdistributionfunction f_i(eq)isalocalfunctionthatonlydependsondensityandvelocityintheisothermal case.ItcanbecomputedthankstoasecondorderdevelopmentoftheMaxwell–Boltzmannequilibriumfunction[25]:

f_i(eq)

=

wi

ρ



1

+

ci

·

u c2 s

+



ci

·

u 2c2 s



2

−

u2 2c2 s



,

(4)

wherewiarethegaussianweightsofthelattice.

AChapman–Enskogexpansion,basedontheassumptionthat fiisgivenbythesumoftheequilibriumdistributionplus asmallperturbation f_i(1)

fi

=

f_i(eq)

+

f_i(1)

,

with f_i(1)



f_i(eq)

,

(5)

can be appliedto (1)inorder torecover theexact Navier–Stokesequation forquasi-incompressible flows inthe limit of long-wavelength[26].The pressureis thusgiven by p

=

c2s

ρ

andthe kinematicviscosityis linked tothe BGK relaxation parameterthrough

ν

=

c2_s

(

τ

−

1

2

).

(6)

TheChapman–Enskogexpansionalsorelatesthesecondordertensor



(1)definedas



(1)

=



i

cicifi(1)

,

(7)

withthestrainratetensorS

= (∇

u

+ (∇

u

)

T

)/

2 throughtherelation



(1)

= −

2c2_s

ρτ

S

.

(8)

Inturntotheleadingorder f_i(1)canbeapproximatedby

f_i(1)

∼

₌

wi 2c2 s Qi

: 

(1)

,

(9) whereQi

≡

cici

−

c2sI

.ThecolonsymbolstandsforthedoublecontractionoperatorandI istheidentitymatrix.

ThePalabosopen-sourcelibrary

The LBM flow solver used in this work is the Palabos1 open-source library. The Palabos library is a framework for general-purpose CFDwitha kernelbased onthelattice Boltzmannmethod.The useofC++ codemakes it easy toinstall and to run on every machine. It is thus possible to set up fluid flow simulations withrelative ease and to extendthe open-source library with newmethods andmodels, which is of paramount importance for the implementation of new boundaryconditions.Thenumericalschemeisdividedintwosteps:

•

AcollisionstepwheretheBGKmodelisapplied:

fi

(

x

,

t

+

1 2

)

=

fi

(

x

,

t

)

+

1

τ



f_i(eq)

(

x

,

t

)

−

fi

(

x

,

t

)



,

(10)

with f_i(eq) computedusingthemacroscopicvaluesattime t and fi canberegularized inordertoincreasenumerical stabilityforhighReynoldsnumberflows.

•

Astreamingstep:

fi

(

x

+

ci

,

t

+

1)

=

fi

(

x

,

t

+

1

2

).

(11)

Thestreamingstepconsistsinanadvectionofeachdiscretepopulationtotheneighbornodelocatedinthedirectionof thecorrespondingdiscretevelocity.Sinceaboundarynodehaslessneighborsthananinternalnode(lessthan9neighbors in2D or27neighborsin3D),some populations aremissingatthe boundaryaftereachiteration.Thesepopulations need to be reconstructed, which is the purpose of the implementation of boundary conditions in LBM. Up to now, different methods canbeused inPalabos,such asregularized BC[27] orZou/HeBC [20]to implementopen boundaries.However, noneofthem canbe usedasthey standforan outflowboundarycondition andtheuse ofspongezonesisnecessary to avoidnon-physical reflections.Thenext sectionswillaimatdevelopingamorenaturalboundarycondition thatminimize acousticreflectionsforanoutflowboundarytype,basedontheCharacteristicBoundaryConditions(CBC).

2. AdaptationofcharacteristicboundaryconditionstotheLBMformalism

OneofthemostpopularmethodsintheNScommunityforsubsonicnon-reflective outflowboundaryconditionsisthe CBC method [11]. The adaptation of the CBC to the LBM formalism is presented here foran isothermal flow, in lattice units(normalizedby



x and



t). Acousticwavesthuspropagateattheconstantlattice soundspeedcs.Intheisothermal case, pressure is defined as p

=

c2_s

ρ

. Three CBC methods will be introduced below: the local one-dimensional inviscid (LODI) approximation, a 2D extension of the LODI approximation including transverse waves and a last method called local-streamlineLODI.

LODIapproximation(baselineLODI)

Letusconsideradomainoutletlocatedatx

=

L,asdescribedinFig. 2.Adiagonalization ofthex-derivativetermsinthe Navier–Stokesequationallowstodefinefivewaves

L

ithatpropagaterespectivelyatvelocityu

−

cs,u,u,u andu

+

cs,where

u isthex-component(streamwise)ofthenon-dimensionalmacroscopicvelocityu

= [

u

,

v

,

w

]

.Thesewavesarerepresented inFig. 2ontheinlet(x

=

0)andoutlet(x

=

L)ofacomputationaldomain.

At theoutlet(x

=

L inFig. 2),

L

2,

L

3,

L

4 and

L

5 leavethe computationaldomainandare obtainedwiththegeneral

expressionofcharacteristicwaves:

L

2

=

u



c2_s

∂

ρ

∂

x

−

∂

p

∂

x



=

0, (12)

L

3

=

u

∂

v

∂

x

,

(13)

L

4

=

u

∂

w

∂

x

,

(14)

L

5

= (

u

+

c

)



∂

p

∂

x

+

ρ

cs

∂

u

∂

x



.

(15)

Letusnoticethat

L

2,theentropywave,isnullintheisothermalcase.The x-derivativetermscanbecomputedusingthe

interiorpointsbyone-sidedfinitedifference.Thetreatmentof

L

1isdifferent:sinceitcomesfromtheoutside,itcannotbe

computedusingtheinteriorpoints.Theperfectlynon-reflectingcaseisobtainedbyfixing

L

1

=

0,whichensureseliminating

Fig. 2. Wavesleavingandenteringthecomputationaldomainthroughaninletplane(x=0)andanoutletplane(x=L)forasubsonicflow[11]inlattice units.

theincomingwave.However,thisisknowntobeunstablebecauseoflackofcontroloftheoutletflowvariables.Asimple waytoensurewell-posednessistoset

L

1

=

K1

(

p

−

p∞

),

(16)

where K1

=

σ

(

1

−

M2

)

cs

/

L, p∞ isthetargetpressureattheoutlet,

σ

isaconstant, M isthemaximumMachnumberin theflowandL isacharacteristicsizeofthedomain[11].

Thetime-derivativeoftheprimitivevariablescanbecomputedinfunctionofthewaveamplitudesbyexaminingaLODI problem:

∂

ρ

∂

t

+

1 c2s



L

2

+

1 2

(

L

5

+

L

1

)



=

0, (17)

∂

p

∂

t

+

1 2

(

L

5

+

L

1

)

=

0, (18)

∂

u

∂

t

+

1 2

ρ

cs

(

L

5

−

L

1

)

=

0, (19)

∂

v

∂

t

+

L

3

=

0, (20)

∂

w

∂

t

+

L

4

=

0. (21)

Intheisothermal case,(17)and(18)areequivalent. Finally,withatemporaldiscretizationusingan explicitsecond-order scheme, the physical values that must be imposed at the next time step in order to avoid acoustic reflections can be computed.ThisimplementationwillbereferredtoasthebaselineLODI(BL-LODI)intherestofthepaper.

LODIapproximationincludingtransverseterms

Fig. 3. Schematic of characteristic wave projections in two coordinate systems: a) geometry based frame R; b) local streamline based frameR.˜

Thetransversewavescanbecomputedasfollows:

T

1

= −

[ut

· ∇

tp

+

p

∇t

·

ut

−

ρ

csut

· ∇

tu]

,

(26)

T

3

= −



ut

· ∇

tv

+

1

ρ

∂

p

∂

y



,

(27)

T

4

= −



ut

· ∇

tw

+

1

ρ

∂

p

∂

z



,

(28)

T

5

= −

[ut

· ∇

tp

+

p

∇t

·

ut

+

ρ

csut

· ∇

tu]

,

(29)

whereut

= [

v

,

w

]

and

∇

t

= [∂

y

,

∂

z

]

.Thesecondtransversewave

T

2isnotintroducedheresinceitisnullintheisothermal

case,aswellas

L

2.Thenon-reflective outflowboundaryconditionneedsnowtobesetas:

L

1

=

K1

(

p

−

p∞

)

−

K2

(

T

1

−

T

1,exact

)

+

T

1

,

(30)

with K1

=

σ

(

1

−

M2

)

cs

/

L, K2shouldbeequaltotheMachnumberofthemeanflowand

T

1,exact isadesiredsteadyvalue of

T

1 [28].Inthe restofthepaper,thismethodwill benamed2D-CBCintheperfectlynon-reflectingcase(K1

=

K2

=

0)

and2D-CBCrelaxedwhenarelaxationisdoneon

L

1.

LocalstreamlineLODI

AnotherpotentialsolutionistocomputetheLODIequationinthelocalstreamlinebasedframe

R

˜

(Fig. 3)[29].Inorder tocomputethenewcharacteristicwaves

L

˜

i,thenon-dimensionalvelocityvectorisprojectedintothenewreferenceframe withthedifficultytocompute

˜

x-derivativetermsfromthelatticediscretization.A simpleapproximationistoset

∂ ˜

φ

∂

x

˜

=

∂ ˜

φ

∂

x

,

(31)

and thus to compute it by a first-orderupwind schemeusing the lattice discretization.This implementation ofthe CBC conditionwillbereferredtoasthelocalstreamlineLODI(LS-LODI).

AdaptationtotheLatticeBoltzmannscheme

Fig. 4. Schematic plot of the population set at the boundary of a D2Q9 model. After the stream phase, f1, f2and f3are missing.

AdaptationwithZou/Heboundaryconditions

Letusconsideranoutflowboundarylocatedatx

=

L onaD2Q9lattice(Fig. 4).Afterstreaming,threeincoming popula-tionsaremissing: f1, f2and f3.Thezerothandfirst-orderhydrodynamicmomentsare:

ρ

=

f1

+

f2

+

f3

+

ρ

0

+

ρ

+

,

(32)

ρ

u

=

ρ

₊

− (

f1

+

f2

+

f3

),

(33)

where

ρ

+

=

f5

+

f6

+

f7 and

ρ

0

=

f0

+

f4

+

f8.Theseequationscanbecombined,byeliminatingthemissingpopulations

( f1

+

f2

+

f3),toobtain

ρ

=

1

1

−

u

(

ρ

0

+

2

ρ

+

),

(34)

where

ρ

andu arestill non-dimensional.Thus,

ρ

andu arelinked withrelation(34),whichproves thatit isimpossible to impose both

ρ

b andub at theboundary withoutmodifying any ofthe knownpopulations. The samerelation can be

obtainedforevery1D, 2Dor3Dlatticeasfarasthereisonlyonelevelofvelocity.Thisrelationconcernseveryboundary conditionofthefirstfamily(thosepreservingtheknown populations)andwillbeusedlater.

Letusnotegithecorrectedpopulationsattheboundary.Asproposedin[20],themissingpopulationscanbecomputed asfollows: g1

=

f₁(eq)

+

f₅(neq)

+

1 2

(

f (neq) 4

−

f (neq) 8

),

(35) g2

=

f2(eq)

+

f (neq) 6

,

(36) g3

=

f₃(eq)

+

f₇(neq)

−

1 2

(

f (neq) 4

−

f (neq) 8

),

(37)

whileeveryotherpopulationsarekeptunchanged:

gi

=

fi

,

i

=

0,4,5,6,7,8, (38)

with f_i(neq)

=

fi

−

f_i(eq) and where the equilibriumpopulations are computedwith the physical values imposed at the boundary,

ρ

b andub

= (

ub

,

vb

)

. As shown in [20], corrections (35), (36)and (37) then allow to impose the first-order momentattheboundary

ρ

bub.However,densityandvelocityarestilllinkedwith(34).Thisisnotaproblemforadirichlet

Zou/Heboundaryconditionwhereonlyonephysicalvalue(either

ρ

oru)isimposedandtheother oneiscomputedwith (34).Onthecontrary,foranon-reflective outflow,bothofthemneedtobeimposedastheresultgivenbytheCBCmethod, sothattheonlyconditionsetby theuseristhevalueofacharacteristicwave.Itisthennecessarytomodifyatleastone knownpopulation.Schlaffersuggestscorrectingeverypopulationsinordertoimposethecorrectdensity[19].Thesolution proposedhereistoaddacorrectiononthepopulationassociatedtoanullvelocityonly:

g0

=

f0

+

ρ

b

−

1

1

−

u

(

ρ

0

+

2

ρ

+

),

(39)

whichensuresthevalueof

ρ

b.Thischoiceismotivatedbythefactthatthisaddedcorrectionwillonlyaffectthecollision phaseandwillnotbestreamedintothecomputationaldomain.

Tosumupthismethod,g1, g2, g3 and g0 arecomputedwithrespectively(35),(36),(37)and(39)whileother

popula-tionsarekeptunchangedafterstreaming:

gi

=

fi

,

i

=

4,5,6,7,8. (40)

Adaptationwiththeregularizedmethod

The Zou/He boundary condition provides the advantage of a very good precision in the definition of the boundary physical values.Unfortunately,thissolution suffersfromlackof stability forlargeReynolds numbers.Forexample,it has been shown that Zou/He boundary conditions become unstable at Re

>

100 for a given resolution N

=

200 nodes per characteristiclengthina2Dchannelflow[21].Anotherpossibleadaptationistousetheregularizedboundaryconditionin ordertoimposethephysicalvaluescomputedbytheCBCtheory.Thissolutionisyetlessaccuratebutwellmorestable,as shownbyLattet al.[21].

More detailsaboutthe regularizedmethodforboundary conditionscanbe foundforexamplein [21].The purposeof thissectionistoexplainhowthisparticularboundaryconditionisusedtoimpose

ρ

bandub onaflatboundary.

Theleadingorderofthepopulations fi canbeexpressed(seeendof(5)and(9))as

fi

=

fi(eq)

(

ρ

b

,

ub

)

+

fi(1)

(

(1)

).

(41)

On aboundary node thedensity

ρ

b andvelocity ub areimposed. Thereforein ordertobe able tousethislast equation oneneedsawaytocompute



(1).ThisisachievedbyusingthefactthatQi isasymmetrictensorwithrespecttoi which meansthatQi

=

Qopp(i),where

opp

(

i

)

= {

j

|

ci

= −

cj

}.

(42)

Attheleadingorder,thispropertyleadsto

f_i(1)

=

f_opp(1)_(i)

.

(43)

Theknown f_i(1) canbestraightforwardlycomputedbythefollowingformula

f_i(1)

=

fi

−

f_i(eq)

(

ρ

b

,

ub

).

(44)

Withthelasttwoequations,thesetof f_i(1)iscomplete(theyareallknown)andcanbeusedtocompute



(1)through(7). Then using (9) one recomputes regularized f_i(1) populations and the total populations fi are computed with the rela-tion(41).Thismethodisvalidinboth2Dand3DandwillbecalledRegularizedBounceback(orRegularizedBB)adaptation inthenextsections.

Another possibility isto compute



(1) _{by a second orderfinite difference scheme thanks to (8) andrecompute f}(1)

using(9).ThismethodwillbecalledtheRegularizedFDadaptation.

Summaryofthemethod

The non-reflecting outflow boundary condition using CBCtheory for LBM can be summarized asfollows,considering everythingisknownat(non-dimensional)time t:

1. Computationofthephysicalvaluesthatmustbeimposedatt

+

1 toavoidnon-physical reflectionsbytheCBCtheory usingeitherBL-LODI,CBC-2DorLS-LODImethod.Thesevaluesarestoredtobeusedinthelaststep.

2. Collisionstep.

3. Streamingstep:somepopulationsaremissingattheboundary.

4. Correctionofthesetofpopulationsattheboundarysothatthephysicalvaluesstoredinthefirststepareimposedby usingtheZou/Headaptationtheso-calledregularizedBouncebackadaptationortheRegularizedFDadaptation. 3. Applicationtoacademiccases

In this section, the CBCapproach forLBM isassessed onsimple 2D cases: a normal plane wave, a plane wave with differentincidenceangles,asphericaldensitywaveandaconvectedvortex.

2Dnormalplanewave

Fig. 5. Density waves before and after reflection of the(u+cs)wave on the right boundary (non-reflecting outflow).

Table 1

Reflectionratesforthe2Dnormalplanewave. Variable ρ u v

Reflection rate 1.2% 1.1% <10−5_%

withx0

=

110 and 2R2c

=

20 inlatticeunits(i.e. innumberofcells).The Reynoldsnumber,computedwiththehorizontal non-dimensionalinitialvelocityu0,thecharacteristicsizeoftheboxinnumberofvoxelsandtheviscosityinlatticeunits,

is equalto 100.The boundaryconditions are: vertical periodicity,reflecting inleton the left andperfectlynon-reflecting outflowontheright.Forthispure1Dcase, thechoiceoftheCBCcondition (baseline,localstreamlineLODIorLODIwith transverseterms)hasnoeffectontheabsorptionrate.Moreover,allthethreeadaptationsprovidethesameresultsat10−6

andno stability issueshave beenencountered withthe Zou/Headaptation forthis testcase. Thus, the resultspresented belowhavebeenobtainedwithbaselineLODIandZou/Headaptation.Thistestcasehasbeenchosensothatthereflection rateforevery macroscopicvalue (

ρ

,u and v)can be computed, since twopressure andaxial velocitywaves propagates atspeed

(

u

−

cs

)

and

(

u

+

cs

)

andone transversevelocity wavepropagatesatspeed u. Thecomputeddensitywavesare represented inFig. 5 attwo different time steps:before reflectionofthe

(

u

+

cs

)

wave onthe non-reflective outletand shortlyafterreflection.

Avery low reflected amplitudecan be distinguished. The

(

u

−

cs

)

wavepropagatesto theleft ofthe domainwithout encountering anyboundary atthe two observed time steps. It is only affected by viscous dissipation and is used as a reference amplitude in the computation of the reflection rates ofdensity and axial velocity. Forthe transverse velocity wave, one cancompute the reflectionrateasthe ratiobetweenthe reflectedwave amplitude andtheamplitude shortly beforereflection.TheobtainedresultsarepresentedinTable 1.

As often with CBC, the treatment is more difficult forthe pressure wave, but the results obtainedhere are in good agreementcomparedtowhatisfoundintheliterature[14,18].

Itisnoticedthat

ρ

=

1 isnotcorrectlyrecoveredattheoutletafterreflection,astheboundaryhasbeensetasperfectly non-reflecting(

L

1

=

0).Inordertoimposethecorrectboundarycondition,a relaxationshouldbe implemented,asineq.

(16).However,inthatcase,thereflectionratewillincreaseasshownin[11].

2Dplanewavewithincidence

At t

=

0,thecomputational domain isatrest (

ρ

0

=

1, u0

=

0)exceptforan oblique lineon whichthe densityisset

to

ρ

0

=

1

.

1 in order togeneratea plane wave withan incidence

α

.The reflectioncoefficient ismeasured by computing

the ratioofmaximal amplitudesindensitywaves only,asthisis themostcriticalhydrodynamic variable. Thethree im-plementationsaretestedforthiscase:BL-LODI,LS-LODIandCBC-2Dintheperfectlynon-reflectingcase.Again,theresults obtainedwiththeZou/Headaptationonlywillbepresentedhere,asthesameresultsat10−6 havebeenobtainedwiththe RegularizedBBandRegularizedDFadaptations.

Fig. 6. Schematicplotoftheinitializationoftheplanewavetestcasewithanincidenceangleα.Sphericalwavesinstantaneouslyappearat thetwo extremitiestodistorttheplanewave.

Fig. 7. Reflectioncoefficient(%)withrespecttotheincidenceoftheplanewave:(a)baselineLODI,(b)localstreamlineLODI,(c)CBC-2D[17]and(d) Mod-Thompson3/4[18].

wave reachesM afteratimeh

/

cs.ThepointM reachesthenon-reflectiveoutletboundaryconditionatatime htan

(

α

)/

cs. Theconditionforthispointofthewavetoreachtheoutletbeforebeingdistortedis:

h cs

tan(

α

) <

h

cs

⇔

tan(

α

) <

1, (45)

whichmeansthanonlyincidenceanglesbelow45◦canbecomputedwithsuchatestcase.Thisproblemisavoidedin[18] by imposing an‘exact’solution obtainedonalarger computationaldomain untilthedesiredwave reachesthe boundary. TheexactsolutionisthenswitchedtotheCBCcondition.Ithasnotbeentestedinthispaperinordertoavoidtheeventual acousticwavesgeneratedbyabrutalchangeintheboundarycondition.

The reflectioncoefficientof thedensitywave withrespect tothe incidenceangleisrepresentedin Fig. 7forBL-LODI, LS-LODI andCBC-2D.Theresultsarecomparedwithwhatisobtainedforthesametest casewiththemodifiedThompson methodwithacoefficient

γ

=

3

/

4 asintroducedin[18].

As expected,forthebaseline approach(a),thereflectioncoefficient increasesastheincidenceangleincreases.Forthe modified Thompsonapproach,theresultsareclosetowhatcanbefoundin[18]: thereflectionrateslightlyincreasesand reaches 5% at40◦ ofincidence.On thecontrary,when theCBCmethodis extendedwiththe effectsoftransverse waves as in[28,17],the reflectionrate decreases until 30◦ and then beginsto increase. In the caseof a localstreamline LODI implementation,thecoefficientremainsstableataround2% ofreflectedwave.

2DSphericalwave

Thecomputationaldomain,asquareof600

×

600 cellsisinitiatedwithagaussiandensityinordertogenerateaspherical wave,asfollows:

ρ

=

1

+

0.1

∗

exp



−

((

x

−

x0

)

2

+ (

y

−

y0

)

2

)

2R2c



,

withRc

=

3

.

2,x0

=

520 andy0

=

300 nodes.Theboundaryconditionsareperiodicintheverticaldirection,aCBCcondition

Fig. 8. Schematicplotofthecomputationaldomain–solidline:CBCcases(a),(b)and(c),dottedline:referencecase(d).Dimensionsaregiveninnumber ofcells.

Fig. 9. Density wave after reflection of a spherical wave at the outlet (400 iterations): (a) baseline LODI, (b) CBC-2D[28,17]and (c) local streamline CBC.

terms,(c) local streamlineLODI and(d)referencecasewhere thedomain isenlarged inthe horizontaldirectionso that thesphericalwave isnotaffectedbythe boundary(Fig. 8).As fortheprevious testcases,theLBM adaptationoftheCBC conditionhadnoimpactontheresults.

Thelocalreflectioncoefficientforsuchasphericalwavecanbecomputedbythefollowingformula:

r

=

R

−

Are f

|

I

|

,

(46)

where I istheamplitude oftheoriginaldensitywave runningtowardstheboundarycondition, R istheamplitudeofthe densitywaveafterreflectionattheoutletandAre f istheamplitudeofthedensitywaveatthesamelatticenodecompared tothereferencecase(d)(noreflection).

2Dconvectedvortex

A2Dvortexisconvectedfromlefttorightandexitsthecomputationaldomain,asquareof600

×

600 gridpoints.A par-ticularattentionmustbepaidtotheinitializationoftheLamb–Oseenvortex[31]whichhastobeadaptedtotheisothermal caseinordertoavoidspuriouswavesgeneratedbytheadaptationofawronginitialdensity.Theinitialconditions,inlattice units,areimposedasfollows:

u

=

u0

− β

u0

(

y

−

y0

)

Rc exp



−

r2 2Rc



,

(47) v

= βu

0

(

x

−

x0

)

Rc exp



−

r2 2Rc



,

(48)

ρ

=



1

−

(β

u0

)

2 2Cv exp



−

r2 2



1/(γ−1)

,

(49)

where u0

=

0

.

1 in latticeunits,

β

=

0

.

5,x0

=

y0

=

300 nodes(the vortexis initiallycenteredon thebox), Rc

=

20 nodes andr2

= (

x

−

x0

)

2

+ (

y

−

y0

)

2.Withthe BGK collisionoperator witha single relaxationtime,the simulatedgas hasthe

followingconstants[32]:

γ

=

D

+

2 D

,

(50) Cv

=

D 2c 2 s

,

(51)

where D

=

2 isthedimension oftheproblem.The specificheatcapacityatconstantvolume Cv appearsinsteadofCp in (49)becauseofan errorintheheatfluxobtainedintheNavier–StokesequationsaftertheChapman–Enskogdevelopment forathermal lattices[32].The Reynoldsnumberbasedon u0 andthesize ofthecomputational domainisequal to1000

and the Regularized BGK scheme is chosen forthe collision step. This convected vortex test case is a well known test for boundaryconditionsasit oftenrevealsspurious distortions atboundariesduetothepresence oftransverse termsin the Navier–Stokes equation [33].As previously, top andbottom boundaryconditions are periodic, theleft condition isa regularizedinletandfourdifferentCBCconditionswillbeevaluatedattherightboundary:

(a)baselineLODIwithK1

=

0,

(b)localstreamlineLODIwithK1

=

0,

(c)CBC-2Dwith

L

1

=

T

1,

(d)CBC-2Drelaxedincludingtransversetermswith:

L

1

=

σ

(1

−

M2

₎

_c3 s Rc

(

ρ

−

1)

+ (

1

−

M

)

T

1

,

where M

=

0

.

2 and

σ

=

0

.

9.

ForlowReynoldsnumbers(Re

<

1000),simulations—notshownhere—providedthesameresultsat10−6_.However,

for Re

=

1000, the Zou/He adaptation was no more stable forthis test case, contrary to both regularized methods. The resultspresentedherehavebeenobtainedwiththeRegularizedBBadaptation.

Figs. 10 and11showisovaluesofnon-dimensional longitudinalvelocityforthefourstudiedcases.First,itcanbenoticed that, contrary towhat was observed inthe previous test cases,the useof localstreamlineboundary condition doesnot allowtoreducenon-physical reflectionscomparedtobaselineLODI.A possibleexplanationwouldbethat,insidethevortex, local streamlines are nearly perpendicular to thedirection of propagationof the localwave, whereas they were aligned inthe caseofapure acousticwave.The LODIequationsarethusappliedina wrongframe whichgeneratesnon-physical reflections.Results are slightlybetterfortheBL-LODI forwhichthelocalframe isthe goodone foratleastlocalnormal waves.TheadditionofthetransversewavesintheCBC-2Dboundaryallowsthevortextokeepacorrectshapeatleastuntil 1000iterations.However,oncethefirsthalfofthevortexhasreachedtheoutlet,itisdistortedasintheBL-LODIcase.The additionofrelaxationcoefficientsallowsthevortextokeepitsshapeatthelastiteration,evenifitisabitdistorted.Itcan benoticedthattheconfigurationappearstobesymmetricforboundaries(a)and(c),whichisnotthecasefor(b)and(d). Indeed,withtheLS-LODI,thestreamlinesusedinthechangeofframearenotsymmetricandinthelastcase,thedensity frame appearstobe asymmetricwhichhasaninfluenceon

L

1 throughtherelaxedpressure,andthusonthelongitudinal

velocityfields.

Stabilityanalysis

Fig. 10. Isovaluesoflongitudinalvelocity(u−u0)/u0 (min=-0.285,max=0.285)oftheconvectedvortexleavingthecomputationaldomainatfourtime

steps:400,700,1000and1300iterations.(a)BaselineLODIand(b)LS-LODI.A smallpartofthecomputationaldomainisrepresented.

Fig. 12 compares the stability of the Zou/He adaptation and the Regularized BB adaptation. As predicted from [21], the classical regularized boundary condition is more stable than the Zou/He adaptation. However, a huge resolution is still requiredin orderto reachhighReynolds numbers.Onthe contrary,the RegularizedFDadaptation hasshownto be unconditionally numericallystable:ineveryconfigurationsoftheconvectedvortextestcase,thefirstnumericalinstabilities camefromtheinsideofthedomainandnottheCBCboundarycondition.

4. ApplicationtoaNACA0015profile athighReynoldsnumber

Theobjectiveofthissection istodemonstratetherobustness oftheCBCsinacaserelevantforhighReynolds aerody-namicsapplications.TheconfigurationisaNACA0015profile,ina8C

×

8C domain(withC

=

1 m thechordoftheprofile). The Mach number is set to 0.04 and the Reynoldsnumber is set to Re

=

105_{. The lattice dimension is}

_

_x

₌

₁

_/

_{400 m,}

corresponding to a time step



t

=

4

.

25

×

10−6 _{s. Each simulation isrun for200000 time steps, but onlythe last 100}

out-Fig. 11. Isovaluesoflongitudinalvelocity(u−u0)/u0(min=-0.285,max=0.285)oftheconvectedvortexleavingthecomputationaldomainatfourtime

steps:400,700,1000and1300iterations.(c)CBC-2Dand(d)CBC-2Drelaxed.A smallpartofthecomputationaldomainisrepresented.

Fig. 12. NumericalstabilityoftheZou/HeandRegularizedBBadaptationoftheCBCforthe2Dconvectedvortexcase.ThemaximumReynoldsnumber whichcanbereachedbeforenumericalinstabilitiesappearisplotted.RegularizedFDisnotpresentedheresinceithasshownnonumericalinstability.

Fig. 13. SimulationoftheflowaroundaNACA0015profileatRe=105_{:time-averagedflowfieldcoloredwiththepressurefluctuation p}

R M S (reference

configuration,16C×8C domain,onlythelefthalfofthedomainisrepresented).

Fig. 14. SimulationoftheflowaroundaNACA0015profileatRe=105_{:time-averagedflowfieldcoloredwiththepressurefluctuationp}

R M S.(a)BL-LODI,

(b)LS-LODI,(c)CBC-2D.

Fig. 15. ComparisonsoftheCBCapproacheswiththe referencesolutionfor thetime-averagedfluctuations atx=7.5C :a)Pressurefluctuation pR M S,

b) AxialvelocityfluctuationuR M Sandc)TransversevelocityfluctuationvR M S.

•

theinitialacousticwavegeneratedbythepresenceoftheairfoilleavesthedomainwithminimumreflection,

•

thevorticesgeneratedbytheboundarylayerseparationarecorrectlyconvectedbeyondtheoutletplanewithminimum spuriouswavereflection.

Thedifficultyforsuchatestcaseliesinthechaoticnatureoftheflow,sincethetrajectoryofthevorticesisaffectedby smallperturbations.Becauseofthisphenomenon,thevisualizationoferrorfields,computedasthedifferencebetweenthe referencesolution andeach testedCBC, wouldnotbe compellantsince vorticesare not superposedin eachcomputation. A betterchoiceistoplottherootmeansquareofthepressurefield,definedaspR M S

=

p2

_/

_{p,tounderlinethebehavior}

ofeachCBCduringthewholecomputationasinFig. 14.

It can be noticed that the baseline LODIapproach (a) allows toevacuate the initial acoustic wave andthe convected vortices withminimalreflection. Theonlydifferenceswiththereference pR M S field arethesmallperturbations observed attheoutletandabackgroundnoisewhichcanbe duetothereflectionoftheinitial acousticwave.Thelocalstreamline LODIapproach(b)islessefficient:thebackgroundnoiseismorepronouncedandthedissymetryobservedintheprevious academictestcasesisstillpresent, whichincreasespressurefluctuationsattheoutlet.IntheCBC-2Dcase(c), themap of pressure fluctuationsseems moresmoothandthevorticesdoesnotseemdoproducespurious reflections.Buttheoverall

pR M S is increased in the whole domain, which can be due to a drift of the mean pressure because of the absence of relaxationintheboundarycondition.

case.TheCBCthatprovidesthebestresultsintermofaccuracyforbothpressureandvelocityfluctuationsistheBL-LODI approach.TheLS-LODI givessatisfactory resultsoutsidethewakebutitoverestimatespR M S by afactor2inthewake(at

y

/

C =4)becauseofthedissymetry inthevortices reflections.TheCBC-2Dapproachoverestimates pR M S outsidethewake byafactor4butitgivesgoodresultsinthewake,similartothoseobtainwiththeBL-LODIapproach.Similarconclusions canbedrawnforaxialandtransversevelocityfluctuations,exceptthatallmethodspredictthecorrectvelocityfluctuations inthewake,includingtheLS-LODIapproach.

5. Summaryandconclusion

An implementation of a non-reflective outflow boundary condition, which does not need additional absorbing layers nor extendeddomains, has beenproposed fora lattice Boltzmann solver.The methods presented hereare basedon the Characteristic Boundary Conditions (CBC) with the classical LODI approach (BL-LODI), its extension to transverse waves (CBC-2D)andthe LODIapproach inthelocal streamlinebasedframe (LS-LODI). Threewaysof computingmissing popu-lations inorder to impose theCBC physicalvalues havebeen introduced. Thefirst one is based onthe classical Zou/He boundary condition, while the other ones are based on the more stable regularized boundary condition: the so-called “RegularizedBB”adaptation,wherethe off-equilibriumpartofthe stresstensorisevaluatedthanksto abouncebackrule, andthe“RegularizedFD”adaptation,whereitiscomputedthankstoanupwindfinitedifferencescheme.Allthese meth-ods provided very good results in the test case of a normal plane wave, where computed reflection rates were about 1%. Testcases of a plane wave with incidence and a spherical wave showed that the reflectionrate increases with the incidence anglefor the BL-LODIadaptation. When adding theeffect of transverse terms,the reflected wave is consider-ably reduced but begins to increase for incidence angles greater than 30◦. For these pure acoustic waves, the LS-LODI adaptation provided the best results since the reflection rate remained below 5% whatever the incidence angle. How-ever, this method is not adapted for a convected vortex, for which the CBC-2D adaptation with relaxation on density and transverse waves provided the best results andlead to only slight distortions of the velocity fields. As regards the numerical stability of the implemented CBC, the regularized adaptations have shown to be well more stable than the Zou/Heone. Theregularized FDadaptationassociatedwitha regularizedBGK schemeallowed torun theNACA0015case at highReynolds number (Re

=

105) in 2D withN=400 cells per chord length, whichwas not possible withZou/He or Regularized BB adaptations. Thus, the use of the regularized FDadaptation is well advised for high Reynolds computa-tions.

Acknowledgements

TheauthorsaregratefultotheCalmipcomputingcenteroftheFederalUniversityofToulouse(Projectaccount number P1425) for providing all resources that have been used in this work. The authors would also thank J.F.Boussuge from CERFACSforhishelponpost-processingandthediscussionaboutthemethod.

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