On the stability of a relative velocity lattice Boltzmann scheme for compressible Navier-Stokes equations

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Discrete simulation of fluid dynamics

On the stability of a relative velocity lattice Boltzmann scheme for compressible Navier–Stokes equations

Stabilité d’un schéma de Boltzmann sur réseau à vitesse relative appliqué aux équations de Navier–Stokes

François Dubois

^a,b,∗

, Tony Février

^b

, Benjamin Graille

^b

aCNAMParis,Laboratoiredemécaniquedesstructuresetdessystèmescouplés,292,rueSaint-Martin,75141Pariscedex03,France bUniversitéParis-Sud,Laboratoiredemathématiques,UMRCNRS8628,91405Orsaycedex,France

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

Articlehistory:

Received6December2014 Accepted25March2015 Availableonline7September2015

Keywords:

RelativevelocitylatticeBoltzmannscheme d’Humièresscheme

Cascadedautomaton Moments

Stability

Mots-clés :

SchémadeBoltzmannàvitesserelative Schémaded’Humières

Schémacascade Moments Stabilité

Thispaperstudies thestabilitypropertiesofatwo-dimensionalrelativevelocityscheme for the Navier–Stokesequations. Thisschemeinspired bythe cascaded schemehasthe particularitytorelaxinaframemovingwithavelocityfieldfunctionofspaceandtime.

ItsstabilityisstudiedfirstinalinearcontextthenonthenonlineartestcaseoftheKelvin–

Helmholtzinstability.Thelinkwiththechoiceofthemomentsisputinevidence.Theset ofmomentsofthecascadedschemeimprovesthestabilityofthed’Humièresschemefor smallviscosities.Onthecontrary,arelativevelocityschemewiththeusualsetofmoments deterioratesthestability.

©^{2015PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.}

r é s um é

Cet article étudie la stabilité d’un schéma de Boltzmann sur réseau à vitesse relative appliquéauxéquationsdeNavier–Stokesbidimensionnelles.Ceschémaestuneextension duschémaencascadeets’yramènedansunrepèreassociéàunchampdevitessefonction del’espaceetdutemps.Sastabilitéestd’abordétudiéedansuncadrelinéairepuispour le castest non linéaire de l’instabilité de Kelvin–Helmholtz. L’importancedu choix des momentsestmiseenévidence.Lechoixdemomentsduschémaencascadeaméliorela stabilité dela variante de d’Humièresdu schémade Boltzmannsurréseau dans le cas depetitesviscosités.Aucontraire, unrégimede vitesserelativeaveclejeu habitueldes momentsdétériorelastabilité.

©^{2015PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.}

*

Correspondingauthor.

E-mailaddresses:[email protected](F. Dubois),[email protected](T. Février),[email protected](B. Graille).

http://dx.doi.org/10.1016/j.crme.2015.07.010

1631-0721/©^{2015PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.}

0. Introduction

Thelattice BoltzmannschemeshavebeensuccessfullyusedforthesimulationofthecompressibleNavier–Stokesequa- tions intwoorthreedimensions[1–4].Thismethodaimstomimic themicroscopicbehaviourinordertosimulatesome macroscopicproblems.Thealgorithmconsistsinevaluatingsomeparticledistributions.Theparticles,movingfromnodeto nodeofalattice,undergoaphaseofcollisionandaphaseoftransport.Differentcollisionoperatorshavebeenproposedfor the simulationofthe Navier–Stokesequations.Thesimplestone isthesingle relaxationtime operator[5,1–3] alsocalled BGK.Analternativecalledthemultiplerelaxationtimes(MRT)operator[4,6]hasbeenproposed.Duringthecollision,some moments,linearcombinationsoftheparticledistributions,relaxtowardstheequilibriumwithaprioridifferentvelocities.

Itcontainstheparticularitytooffermoredegreesoffreedomtofixthedifferentparametersastheviscosities.Themultiple relaxationtimesapproachisthusmoreflexiblethantheBGK.Bothschemeshavebeenwellstudiedparticularlyintermsof stability [6].Theystill encountersome instabilityfeaturesastheviscositiestendtozerothat limitshighReynoldsnumber simulations.

In2006,acascadedschemeimprovingthestability forlowviscositieshasbeenpresented[7].Itsrelaxationoccursina framemovingwiththefluidvelocity.Tounderstandthepositivefeaturesofthisscheme,ageneralnotionofrelativevelocity schemeswasdefined[9].Theirrelaxationismadeforasetofmomentsdependingonavelocityfieldfunctionofspaceand time thatisthevelocityfluidforthecascadedscheme[9]andzeroforthed’Humièresscheme[4].Theserelativevelocity schemes are not restricted to the simulation of the Navier–Stokes equations: they are definedfor an arbitrary number of conservation laws.Their consistency hasalready beenstudied for one and two conservationlaws [8,9] butthe same reasoningholdsforanarbitrarynumberofconservationlaws.Thepurposeisnottocomparedifferentschemesintermsof stabilitybuttostudythestabilityoftheclassoftherelativevelocityschemesaccordingtothechoiceofsomeparameters.

The purposeof thiscontributionis topresentsome numericalstability results ofthetwo dimensional nine velocities (D2Q9)relativevelocityschemeforthecompressibleNavier–Stokesequations.Wewanttocharacterizetheinfluenceofthe relativevelocityandthelinkwiththemomentschoice:thepolynomialsdefiningthemomentsofthecascadedschemeare differentfromtheusualonesandmayactonthestability.Inafirstpart,werecallthebasisoftherelativevelocityschemes.

We then presenttherelative velocity D2Q9 we are interestedin. Thesecond partexhibitsthe resultsofstability,first in a linearcontext(L² vonNeumannnotion) andthen foranon-lineartestcase, theKelvin–Helmholtzinstability.Itputs in evidencethelinkbetweentherelativevelocity,thechoiceofthepolynomialsdefiningthemomentsandthestability.

1. Descriptionofthescheme

We firstintroducetherelativevelocityschemeforanarbitrarynumberofdimensionsandvelocities.Wethen particu- larizeittothecaseoftwodimensionsandninevelocities.

1.1. TherelativevelocityD_dQ_qscheme

ThissectionpresentsthederivationoftherelativevelocitylatticeBoltzmannschemesintroducedin[9]andinspiredby the cascadedscheme[7].LetL ^{bea Cartesian latticein ddimensionswitha typicalmesh size} x. Thetime step t is linked tothespacestepbytheacousticscalingt=x/λforλ∈R^{thevelocityscale.Weintroduce V}=(v0,. . . ,vq−¹)a setofq velocitiesofR^{d.ThisdefinestheschemecalledD}dQq.Weassumethat foreachnode x ofthelattice L^,andeach vj inV,thepointx+^vjt isstillanodeofL.TheDdQqschemecomputesaparticledistribution f=(f0,. . . ,fq−¹)onthe lattice L^{atdiscretevaluesoftime.Aniterationoftheschemeconsistsintwophases:therelaxationthatisnonlinearand} localinspace,andthelineartransportsolvedexactlybyacharacteristicmethod.

Therelaxationphasereadsmoreeasilyinamomentsbasisusingthed’Humièresframework[4].A velocityfieldu(x,t) that depends onspaceand time beinggiven, we define thechange ofbasis matrix M(^u), usually called“matrix ofmo- ments”,by

M

(

^u

)

kj

=

^Pk

(

v_j

−

^u

),

0

^k

,

j

^q

−

¹

,

where (P0,. . . ,Pq−¹) are some polynomialsof R[^X1,. . . ,Xd]^{. Thismatrixof moments,supposed to be} invertible,defines themomentsm(u)=(m0(u),. . . ,mq−¹(u))bytherelation

m

(

u

) =

^M

(

u

)

f

,

(1)

where m_k(^u)is thekthmoment.Ifthe vector f represents thecoordinates ofthe state inthecanonical basis, then the vectorm(^u)representsthesamestateinthenewbasisobtainedbythelineartransformationgivenby M(^u).

The collision phase is viewed as the relaxation of the particle distributions towards an equilibriumdistribution f^eq independent ofthemoments andofthevelocity fieldparameter^{u. Thisallowsusto definethe momentsat}equilibrium m^eq(u)=(m^eq₀ (u),. . . ,m^eq_q−1(u))by

m^eq

(

u

) =

M

(

u

)

f^eq

.

(2)

Therelativevelocityschemesuseadiagonalrelaxationphaseinthisshiftedmomentsbasis

Fig. 1.TheD2Q9velocities.

m_k

(

^u

) =

^mk

(

^u

)+

^sk

(

m^eq_k

(

^u

) −

^mk

(

^u

)),

0

^k

^q

−

¹

,

(3) where sk is the relaxation parameter associated with the kth moment for 0^kq−^{1. Some of these moments are} conserved by the relaxation: they are associated with relaxationparameters equal to zero. The equilibrium distribution functionattheequilibrium f^eqisafunctionoftheconservedmomentsdeterminedbythepartialdifferentialequationsto beapproximated.Theinverseofthematrixofmomentsisusedtoreturntothedistributions

f

=

^M−1

(

u

)

m

(

u

).

(4)

Thetransportphasespreadstheparticledistributionsontheneighbouringnodes

f_j

(

x

,

t

+

t

) =

^f_j

(

x

−

^vj

t

,

t

),

0

^j

^q

−

¹

.

Thisframeworkembeddesthed’Humièresschemefor^{uequalto0andthecascadedschemeforuequaltothefluidvelocity} andaparticularsetofmoments[9].Inthefollowing,whenu isspecified,we calltheassociatedrelativevelocity scheme theschemerelativetou.

1.2. Thestudyframework:therelativevelocityD₂Q₉scheme

Thepurposeofthissection istointroducetheschemewhosestability propertiesareinvestigated:therelativevelocity D2Q9 schemewithtwoconservationlawsonthedensityandthemomentum

ρ =

j

fj

,

q^α

=

j

v^α_j fj

,

1

α

d

.

forthecompressibleNavier–Stokesequations.Weexposeitsfeaturesandthedifferentdegreesoffreedomusedtocheckits stability.Weputaparticularattentiononthedefinitionofthemoments.

Forthistwo-dimensionalscheme,ninevelocitiesareinvolved:theyaredefinedby

v

= { (

0

,

0

), (λ,

0

), (

0

, λ), ( − λ,

0

), (

0

, − λ), (λ, λ), ( − λ, λ), ( − λ, − λ), (λ, − λ) } ,

withλ∈R^{thevelocityscale.Thesevelocitiesarealso}representedinFig. 1.

Weneed todealwiththesetofthemoments andtheequilibriumtocompletelycharacterizethescheme.Historically theD2Q9schemehasbeenmainlyusedwiththefollowingsetofmoments

1

,

X

,

Y

,

X²

+

^Y2

,

X²

−

^Y2

,

X Y

,

X

(

X²

+

^Y2

),

Y

(

X²

+

^Y2

), (

X²

+

^Y2

)

²

,

(5) orits orthogonalized analogueforthe simulation ofthe Navier–Stokes equations[6].They havebeen chosen becauseof theirphysicalmeaning:theyinvolvethedensity,themomentum,theenergy,thediagonalandoff-diagonalcomponentsof thestresstensor,theheatfluxandthesquareoftheenergy.Nevertheless,theD₂Q₉cascadedscheme[7],whichseemsto improvethestabilityatlowviscosities,hasbroughttolightanothersetofmomentsgivenby

1

,

X

,

Y

,

X²

+

^Y2

,

X²

−

^Y2

,

X Y

,

X Y²

,

Y X²

,

X²Y²

.

(6) Thisschemehasbeenwrittenasarelativevelocityschemeforthemoments(6)andthefluidvelocity[9].Therelaxations of(5)and(6)areequivalent inthed’Humièresframework correspondingtou=^{0(Section2.2).However, thisisnot true}

anymorewhenu isdifferentfrom0.Thispointnaturallyleadstothequestion:doesthesetofmomentshaveaninfluence onthestabilitypropertiesoftherelativevelocityscheme?Givingsomeexperimentalrudimentsofanansweristhepurpose ofthisstudy.

That’swhyweintroducetwosetsofmomentstunedbyaparameter

α

∈R^:

1

,

X

,

Y

,

X²

+

^Y2

,

X²

−

^Y2

,

X Y

,

X

( α

X²

+

^Y2

),

Y

(

X²

+ α

Y²

), α

2

(

X⁴

+

^Y4

) +

^X2Y2

,

(7) and

1

,

X

,

Y

,

X²

+

^Y2

,

X²

−

^Y2

,

X Y

,

X Y²

+ α (

X²

+

^Y2

),

Y X²

+ α (

X²

+

^Y2

),

X²Y²

.

(8) The moments(7)generalizethoseoftheD₂Q₉ cascadedschemecorrespondingto

α

=^{0[9]givenby(6)andtheones} associated with

α

=^{1 defined by (5). The} introduction of

α

results fromthe will not to restrict the studyto two sets of moments.This alsoallows usto understandthe impact onthe stability ofthe X³ componentwhen

α

moves from0 to 1.Thechoiceofthemoments(8),evenifitseemsstrange becauseitmixessomesecond- andthird-orderpolynomials, improvestheunderstandingofthedifferencesofstabilitybetween(5)and(6)fortherelativevelocityD2Q9 scheme.Taking

α

₌0 alsorecoversthecascadedmoments.

Theequilibriummayalsohaveaninfluenceonthestability.That’swhy,denotingu=^q/

ρ

thefluidvelocity,weintroduce

f₁^eq_,j

( ρ ,

u

) = ρω

j

1

+

^u

.

v_j

c²₀

+ (

u

.

v_j

)

² 2c⁴₀

− |

^u

|

²

2c²₀

,

0

^j

⁸

,

(9)

and

f₂^eq_,j

( ρ ,

u

) = ρω

j

1

+

^u

.

v_j

c²₀

+ (

u

.

v_j

)

² 2c⁴₀

− |

u

|

²

2c²₀

+ (

u

.

v_j

)

³

6c⁶₀

− |

^u

|

²

(

u

.

v_j

)

2c⁴₀

+

^dj

(

u^x

)

²

(

u^y

)

² c⁴₀

,

0

^j

⁸

,

(10)

where d₀= −¹/4,d_j=¹/2, j=¹,. . . ,4, d_j= −^{1, j}=⁵,. . . ,8,respectivelycorresponding to the second-ordertruncated equilibrium [3] andto a fourth-order equilibrium [10,11] particularly used for the D₂Q₉ cascaded scheme.The product equilibriumcorrespondstothefourth-ordertruncationoftheMaxwellian equilibrium.Bothequilibriaallowustosimulate the compressible Navier–Stokes equations whatever the velocity field u. Indeed, the second-order equivalent equations of the relative velocity schemes are independent of u [9] and the Navier–Stokes equations are recovered by the D2Q9 d’Humièresscheme(u=^{0)atthesecondorderforsmallMachnumbers[12].Letusnotethatthe}simulationsofSection3 have beenalsomadefortheincompressibleanalogueof(9)and(10)usedin [6]: thesametrends asthose presentedin Section3areobtained.

We chooseto work withseveraltwo-relaxation-timeschemes (TRT)tounderstand therole ofeach polynomialof the moments:theonegivenby

s

= (

se

,

s_ν

,

s_ν

,

se

,

se

,

se

),

(11)

calledTRT1 andtheonegivenby

s

= (

s_e

,

s_e

,

s_e

,

s_p

,

s_p

,

s_e

),

(12)

calledTRT2 wherese,sν,sp∈R^{.NotethattheTRT}1 andtheTRT2 differfromtheTRTschemesdefinedin[13]andbasedon thesymmetryofthelattice.Ifalltherelaxationparametersareidentical,werecovertheBGKscheme[4].

Fourdegreesoffreedomaretunableinthissection:themoments,thevectoroftherelaxationparameterss,theveloc- ity fieldu andtheequilibrium. The link betweentheseparameters andtheir influence onthe stability is studiedin the following.

2. Experimentalstudyoflinearstability

Inthissection,westudythelinearvonNeumannL² stabilityoftherelativevelocityD2Q9schemedefinedinSection1.2.

Theinfluenceofthemomentsaccordingtothechoiceofthevelocityfieldu isthekeypointofthesection.Ourfirstinterest goestothemoments(6)and(5),respectivelycorrespondingto

α

₌0 and

α

₌1 in(7),becausetheyareusuallychosenby thecommunity[4,6,7,14].Thefirstsubsectioncomparesthosetwosetsaccordingtothevelocityfieldparameter.Weshow thattaking^u=^{u,thevelocityofthefluidimprovesthestabilityifthemoments(6)arechosen,}deterioratesitiftheset(5) is taken.The secondsubsectionanswers thefollowingquestion:whatisthebetter choiceofmoments(of

α

) intermsof stability?Arangeof

α

isproposedand

α

=^{0 isshowedtobethemoststablechoice.}

Table 1

HigheststableV=(V^x,0)inλunitsforthed’Humièresscheme(u=^0),withα=^{0 or}α=^1,ofequilibrium(9).se=²−^2−mandsν=²−^2−n.

n m

0 1 2 3 4 5 6 7

0 0.42 0.41 0.34 0.26 0.20 0.15 0.11 0.08

1 0.42 0.41 0.36 0.30 0.23 0.18 0.13 0.09

2 0.31 0.34 0.34 0.32 0.28 0.23 0.17 0.13

3 0.21 0.28 0.32 0.30 0.25 0.22 0.18 0.15

4 0.14 0.21 0.28 0.26 0.22 0.18 0.16 0.13

5 0.10 0.16 0.22 0.23 0.20 0.17 0.13 0.11

6 0.07 0.12 0.17 0.20 0.18 0.16 0.12 0.11

7 0.05 0.08 0.12 0.17 0.16 0.15 0.11 0.11

2.1. Methodology:thevonNeumannstability

ThestudyoftherelativevelocityD₂Q₉ schemeisbasedonthe L² vonNeumannstability.Thisnotionbeingadaptedto linearcontexts,welinearizetheequilibria(9)and(10)aroundavelocity V= |^V|^eiθ∈R^2,θ∈R^{.Thusthereexistsamatrix} E sothat

f^eq

=

^{E f}

.

Using(1),(3),(2),(4)thelinearizedrelaxationphaseoftherelativevelocityschemesreads

f

= (

I

+

M

(

u

)

⁻¹D M

(

u

)(

E

−

I

))

f

,

where D=^diag(s)isthediagonalmatrixoftherelaxationparameters.Thisexpressionholdsforeachnodex ofthelattice, the relaxation being local in space. One can deduce the expression of the distribution after an iteration thanks to the transportphase

fj

(

x

,

t

+

t

) = [(

I

+

M

(

u

)

⁻¹D M

(

u

)(

E

−

I

))

f

]

j

(

x

−

vj

t

,

t

),

x

∈

L

,

t

∈ R.

In theFourier space, the transport operatorbecomes localin spaceandis represented by thediagonal matrix A whose diagonalcomponentsaregivenbye^itk.vj,0^{j8.Wecanthendefinethe}amplificationmatrixL(^u)=^L(^u,V,k,s,

α

)= A(I+^M(u)⁻¹D M(u)(E−^I)),fork,V,u∈R^2,s∈R^9,

α

∈R^{, V}∈R² characterizingatimeiterationoftheschemeinthe Fourierspace

f

(

k

,

t

+

t

) =

^L

(

u

)

f

(

k

,

t

),

t

∈ R,

wheref istheFouriertransformof f.Wewanttodeterminethequantity

max

{|

V

|,

max

k∈R² r

(

L

(

u

))

1

},

(13)

forsome parameterss,u,

α

,adirectionoflinearizationθ∈R^andr(L(u))thespectralradiusofL(u).Itcharacterizesthe setofthelinearizationvelocitiesV forwhichtheschemeverifiesthenecessaryconditionofL²stabilitymax

k∈R²r(L(u))^1.

2.2. Comparisonbetweenthed’Humièresschemeandtheschemerelativetothelinearizationvelocity

Weshowthattheschemesrelativetou=^{V canimproveor}deterioratethelinearstabilitycomparedtothed’Humières scheme.Thestabilitybehaviourdependsstronglyonthechoiceofthemoments.

Wecomparetheschemesrelativeto^u=^0andu=^{V forthetwosetsofmoments(5)and(6):wehave}

α

=⁰,1 in(7).

Wehererestricttothesecond-ordertruncatedequilibrium(9)linearizedaroundV.Thevariableofcomparisonisthelargest stablevelocity V (13)foralinearizationdirectionθ equalto0.WechoosetodealwiththeTRT1 (11)forse=²−^{2−m and} sν=2−2⁻ⁿ wherem,n,∈N,0m,n7.Theparametersseandsν respectivelytunethebulkandtheshearviscositiesof theNavier–Stokesequations.Thischoiceofparametersallowsustostudythezeroviscositylimitbyincreasingmor/andn.

Table 1dealswiththed’Humièresschemeforbothsetsofmoments,thevaluesforthosetwosetsbeingidentical.Tables 2 and3giveanalogousresultsfortheschemerelativetou=^{V:theycorrespond}respectivelytothemomentswith

α

=⁰⁽⁶⁾ and

α

=^1(5).

Wenoticetheimportanceofthechoiceofthemomentsfortheschemesrelativetou=^{V:stabilityareasarethebiggest} for

α

=^{0 (6)(Table 2) andthesmallestfor}

α

=^{1(5)(Table 3) whateverthechoiceof s.The d’Humièresscheme(}u=^0, Table 1)hassmallerstabilityareasthantheschemerelativetou=^{V with}

α

₌0 andbiggerthantheonewith

α

₌1.

Theschemerelativeto^u=^{V with}

α

=^{0 providesthemostimportantgaincomparedtothed’Humières schemewhen} seorsν iscloseto 2 andtheotherisfarfrom2 (for onesmallandonelargeviscosity). Insteadtheseareasare themost

Table 2

HigheststableV=(V^x,0)inλunitsfortheschemerelativetou=^{V with}α=^{0 of}equilibrium(9).se=²−^2−mandsν=²−^2−n.

n m

0 1 2 3 4 5 6 7

0 0.42 0.42 0.40 0.37 0.33 0.28 0.24 0.21

1 0.42 0.41 0.37 0.34 0.33 0.30 0.27 0.23

2 0.42 0.36 0.34 0.33 0.29 0.25 0.22 0.19

3 0.36 0.34 0.33 0.30 0.25 0.21 0.18 0.16

4 0.32 0.32 0.30 0.26 0.22 0.18 0.16 0.13

5 0.29 0.30 0.27 0.24 0.21 0.17 0.13 0.11

6 0.26 0.28 0.24 0.21 0.18 0.16 0.12 0.11

7 0.23 0.26 0.22 0.19 0.16 0.15 0.11 0.11

Table 3

HigheststableV=(V^x,0)inλunitsfortheschemerelativetou=V withα=1 ofequilibrium(9).se=2−2^−mandsν=2−2⁻ⁿ.

n m

0 1 2 3 4 5 6 7

0 0.42 0.42 0.27 0.18 0.12 0.08 0.06 0.04

1 0.42 0.41 0.37 0.31 0.20 0.13 0.09 0.06

2 0.24 0.36 0.34 0.32 0.27 0.21 0.14 0.09

3 0.15 0.29 0.33 0.30 0.25 0.21 0.18 0.14

4 0.10 0.19 0.29 0.26 0.22 0.18 0.16 0.13

5 0.07 0.12 0.20 0.29 0.21 0.17 0.13 0.11

6 0.05 0.09 0.14 0.20 0.18 0.16 0.12 0.11

7 0.03 0.06 0.09 0.14 0.16 0.15 0.11 0.11

Table 4

HigheststableV=(V^x,0)inλunitsforthed’Humièresscheme(u=0),withα=0,ofequilibrium(10).se=2−2^−mandsν=2−2⁻ⁿ.

n m

0 1 2 3 4 5 6 7

0 0.42 0.42 0.39 0.32 0.24 0.16 0.11 0.07

1 0.42 0.42 0.41 0.38 0.31 0.20 0.14 0.09

2 0.42 0.42 0.41 0.40 0.38 0.30 0.20 0.14

3 0.26 0.41 0.40 0.39 0.37 0.32 0.28 0.20

4 0.16 0.28 0.38 0.36 0.33 0.29 0.24 0.21

5 0.10 0.18 0.29 0.33 0.31 0.26 0.21 0.19

6 0.07 0.12 0.19 0.30 0.29 0.24 0.20 0.18

7 0.05 0.08 0.13 0.20 0.28 0.23 0.19 0.17

deterioratedwhen

α

isequalto1.When se andsν are close,theschemepresentsstability areasnearly independentofu.

Thecasese=^sν correspondstotheBGKscheme:thevelocityfieldudoesnotplayanyrolesince(2)isverified.

Finally,forthe d’Humièresscheme,the resultsareindependent ofthechoice ofthemoments. Indeed,thethird-order moment X(X²+^Y2) is equivalent to λ²X+^{X Y2 on the velocity network [8]: its relaxation is then equivalent to the} relaxationof X Y², X beingaconservedcomponent. Thesamereasoningholdsforthesymmetricalmoment.Themoment (X²+^Y2)² is equal to X²Y²+λ²(X²+^Y2) on the velocity network. Its relaxation isequivalent to relax X²Y² because

X²+^{Y2and X2Y2 arebothintheeigenspacerelatedtos}efortheTRT1.

Allthesetrendsareindependentofthedirectionoflinearizationθ:theresultsaresimilarforθ=

π

/8,

π

/4,

π

/3.

We now do the same job for the product equilibrium(10) restricting the studyto the choice

α

₌0. Note that the combinationofthesemomentsandthisequilibriumcorrespondstotheD2Q9cascadedscheme[11,9].TheTRT1 (11),tuned byseandsν,andthedirectionθ=^{0 arechosen.Table 4isaboutthed’Humièresscheme,Table 5}correspondstothescheme relativeto^u=^V.

The resultsare analogousto theonesassociatedwithequilibrium(9).Thescheme relativeto^u=^{V hasbigger linear} stabilityareasthanthed’Humièresscheme(u=^0)when

α

=^{0.Thegainismoreimportantwhentherelaxationparameters} arefarfromeachother.Thevelocityfieldimpactislightenedwhenseandsν areclose.

We finallyassesstheinfluenceoftheequilibriumonlinearstability.Whateverthechoiceofu ands,equilibrium(10) providesbiggerstabilityareasthanthetruncatedequilibrium(9).Particularly,theBGKschemeassociatedwith(10)ismore stablethantheonecorrespondingto(9).

As aconclusion,themostimportantfactofthestudyisthefollowing:theschemerelativeto^u=^{V for}

α

=^{0 ismore} stablethanfor

α

=^{1.Insteadchoosingaschemerelativetou}=^{V withan}“inappropriate”setofmomentscandeteriorate stability.

Table 5

HigheststableV=(V^x,0)inλunitsfortheschemerelativetou=^{V with}α=^{0 of}equilibrium(10).se=²−^2−mandsν=²−^2−n.

n m

0 1 2 3 4 5 6 7

0 0.42 0.42 0.42 0.42 0.35 0.30 0.26 0.23

1 0.42 0.42 0.42 0.41 0.39 0.35 0.32 0.28

2 0.42 0.41 0.41 0.40 0.40 0.35 0.31 0.29

3 0.41 0.41 0.40 0.39 0.36 0.30 0.27 0.23

4 0.40 0.40 0.39 0.36 0.33 0.28 0.23 0.20

5 0.35 0.37 0.36 0.33 0.31 0.26 0.21 0.18

6 0.31 0.33 0.33 0.31 0.29 0.25 0.20 0.17

7 0.28 0.30 0.31 0.29 0.27 0.24 0.19 0.17

Fig. 2.DrawofV^xasafunctionofαforthed’Humièresschemewiththemoments(7).Left:TRT1,se=2−2^−mandsν=2−2⁻ⁿ.Right:TRT2,se=2−2^−m andsp=²−^2−n.

2.3. Influenceofthechoiceofthemomentsonthestability

Theprevioussectionhasstudiedthestabilityoftherelativevelocityandd’Humièresschemesfortwochoicesof

α

.This parameter seems to be crucialforthe relative velocity schemes. Thepurpose ofthis section isto see moreprecisely its influenceonstability.Itstudiesthestabilitypropertiesoftheschemesrelativeto^u=^0andu=^{V forabiggerrangeof}

α

. Weshownumericallyandjustifythat

α

=0 constitutesthebetterchoiceofmoments.

Weareinterested inthestabilityoftherelativevelocity schemesforboth setsofmoments(7)and(8):thediscussion carriesonthechoiceoftheparameter

α

_∈Rcharacterizingthesemoments.Bothequilibrialeading tothesametrends,we focusonthetruncatedone(9)linearizedaround V=(V^x,0)∈R^{2.Twosetsofrelaxationparameterss areused:theTRT}1 (11)andtheTRT2(12)wherese=²−^2−mandsν=^sp=²−^{2−n with}(m,n)=(0,3),(3,0),(0,7),(7,0),(7,7).Thequantity (13)isdrawnasafunctionof

α

in[−¹,1]^{.Anegativevalueof(13)meansthattheschemeisunstableforallV.}

Wefirstfocusonthed’Humièresschemecorrespondingtou=^0.Figs. 2 and3representrespectivelythedrawsassoci- atedwiththemoments(7)and(8).Oneachfigure,theleft drawisassociatedwiththeTRT₁ andtherightone withthe TRT2.

Forthemoments(7),the drawsareindependentof

α

whatevertheTRTchosenands.Forthemoments(8),thedraw corresponding to the TRT₁ is independent of

α

unlike theTRT₂. The figure associated withthe TRT₂ induces to choose

α

=^0:itcorrespondstothemaximumofthecurveandthestabilityareadecreasesas|

α

|^{increases.Asexpected,thedraw} form=ⁿ=7 correspondingtoaBGK schemeisconstantin

α

.Wenoticethat

α

=^{0 belongstothesetof}

α

maximizing thestabilitywhateverthedraw.

Wecanexhibittheoriginofthedependenceorindependenceon

α

.Let’sconsiderthemoments(7).Forthed’Humières scheme,therelaxationofthesemomentsisindependentof

α

.Thelastthreemomentsof(7)are:

α

X³

+

^{X Y2}

, α

Y³

+

^X2Y

, α

2

(

X⁴

+

^Y4

) +

^X2Y2

.

Knowingthat X³=λ²X onthevelocityset[8],theschemeisunchangedifwereplacethemby

λ

²

α

X

+

^{X Y2}

, λ

²

α

Y

+

^X2Y

, λ

²

α

2

(

X²

+

^Y2

) +

^X2Y2

.

Fig. 3.DrawofV^xasafunctionofαforthed’Humièresschemewiththemoments(8).Left:TRT1,se=2−2^−mandsν=2−2⁻ⁿ.Right:TRT2,se=2−2^−m andsp=²−^2−n.

Fig. 4.DrawofV^xasafunctionofαfortheschemerelativetou=V withthemoments(7).Left:TRT1,se=2−2^−mandsν=2−2⁻ⁿ.Right:TRT2, se=²−^2−mandsp=²−^2−n.

Relaxing themoments (7)isthenequivalent to relaxthe samemomentsfor

α

=^{0.Indeed, X andY areassociatedwith} someconservedmomentsand X²+^{Y2 hasthesamerelaxationparameters}easthefourthordermoment.Itisthusconsis- tentforthisdrawstobeindependentof

α

.

We nowfocusonthemoments(8):theparameter

α

appearsonlyinthethirdordermoments.FortheTRT1, X²+^Y2 andthethird-orderpolynomialsarerelaxedwiththesamerelaxationparameters_e.Choosing

X Y²

+ α (

X²

+

^Y2

),

X²Y

+ α (

X²

+

^Y2

),

isthenequivalenttochoose

X Y²

,

X²Y

,

and the scheme doesnot depend on

α

as the left draw of Fig. 3 shows it. Forthe TRT2, X²+^{Y2 and the} third-order momentsarerelaxedwithdifferentrelaxationparameters:itisexpectedtohaveadependenceon

α

,exceptedfortheBGK case(m=ⁿ=^{7)involvingonlyonerelaxationparameter.}

Wenowdothesamejobfortheschemerelativetou=^{V.Fig. 4isassociatedwiththemoments(7)andFig. 5withthe} moments(8).

The stability ofthe schemerelativetou=^{V dependson}

α

whateverthemoments (Fig. 4). Themaximumisreached for

α

=^{0 whateverthechoiceofs.Forthemoments(8),thestabilityoftheTRT}1isnotlinkedto

α

(Fig. 5ontheleftside).

Instead,thisparameterisinfluentialfortheTRT2(Fig. 5ontherightside):

α

=^{0 still}correspondstotheoptimum.

WeinterpretFig. 4ascorrespondingtothemoments(7).Because X³=λ²X andY³=λ²Y onthevelocityset,relaxing therelativemomentsassociatedwith(7)isequivalenttorelax

Fig. 5.DrawofV^xasafunctionofαfortheschemerelativetou=^{V withthemoments(8).Left:TRT}1,se=²−^{2−m ands}ν=²−^{2−n.Right:TRT}2, se=²−^2−mandsp=²−^2−n.

1

,

X

,

Y

,

X²

+

^Y2

,

X²

−

^Y2

,

X Y

,

P6

(

u

, α ),

P7

(

u

, α ),

P8

(

u

, α ),

where

P6

(

u

, α ) =

X Y²

+ α (−

3

u^xX²

+ (λ

²

−

3

(

u^x

)

²

)

X

+

u^x

(λ

²

− (

u^x

)

²

)),

P₇

(

u

, α ) =

^X2Y

+ α (−

³

u^yY²

+ (λ

²

−

³

(

u^y

)

²

)

Y

+

u^y

(λ

²

− (

u^y

)

²

)),

(14) and

P₈

(

u

, α ) =

^X2Y2

+ α

2

(λ

²

+

⁶

(

u^x

)

²

)

X²

+ (λ

²

+

⁶

(

u^y

)

²

)

Y²

+

2

u^x

(−λ

²

+

4

(

u^x

)

²

)

X

+

2

u^y

(−λ

²

+

4

(

u^y

)

²

)

Y

−

3

(

u^x

)

²

(λ

²

− (

u^x

)

²

) −

3

(

u^y

)

²

(λ

²

− (

u^y

)

²

)

.

Letusobservetheequivalent classofthethird-ordermomentgivenby (14).Thedependenceon

α

ofthestabilitycomes fromtheterm−³

α

^{uxX2.Indeed,relaxing(14)isequivalenttorelax X Y2}−³

α

^{uxX2 sincethemoments}correspondingtothe polynomials1 and X areconservedbythecollision.Onthecontrary,themomentassociatedwith X² isnotconserved.For theTRT1,itisalinearcombinationofthemoments X²+^{Y2 and X2}−^{Y2 associatedwithdifferentrelaxationparameters} seandsν.FortheTRT2,itisassociatedwithse,whereas P6 correspondstosp.

Theseremarksjustifytheintroductionofthemoments(8)tostudytheinfluenceofthenon-conservedcomponents X² andY².Fig. 5,implying themoments(8),givessimilarresultsasitsanalogousfor^u=^{0(Fig. 3):thesame}interpretation is still valid. Note that for

α

=^{0, the areas are bigger with}u=^{V (Fig. 5) than with}u=^{0 (Fig. 3). This confirms the} observationsoftheSection2.2.

3. NumericalstabilityfortheKelvin–Helmholtztestcase

Thepurposeofthissection istoconfirm onanon-lineartestcasetheprevious linearstability results:thistest caseis theKelvin–Helmholtzinstability[15,16].

We comparesixversionsoftherelative velocity D2Q9 scheme tostudytheinfluence ofthemoments, ofthevelocity fieldu,andoftheequilibrium.Weconsidertheschemeassociatedwith

α

=^{0 relativeto}u=^0andu=^{u(thefluidveloc-} ity)fortheequilibria(9)and(10).Wecompareittothechoice

α

₌1 fortherelativevelocities0anduwithequilibrium(9).

Wechoose notto considertheproductequilibrium(10)for

α

=^1,thisequilibriumbeingintroduced forthemomentsof thecascadedscheme[11].WeworkwiththeTRT1definedby(11):unlessotherwisespecified,se etsν arefixedby

μ = λ

²

σ

e

3

, ν = λ

²

σ

ν 3

,

where

σ

e=¹/s_e−¹/2 and

σ

ν=¹/sν−¹/2,sothattheviscosities

μ

and

ν

aresetto0.0366 and10⁻⁴. WetestthestabilityoftheschemebyincreasingthevelocityU definingtheinitialshearlayers

Fig. 6.(Colour online.) Vorticity draw att=0.6.

Fig. 7.(Colour online.) Vorticity draw att=1.

u^x

(

x

,

y

,

0

) =

Utanh

(

k

(

y

−

¹₄

))

ify

¹₂

Utanh

(

k

(

³₄

−

^y

))

ify

>

¹₂

, (

x

,

y

) ∈ [

⁰

,

1

]

²

,

u^y

(

x

,

y

,

0

) =

^U

δ

sin

(

2

π (

x

+

¹₄

)), (

x

,

y

) ∈ [

⁰

,

1

]

²

.

This velocity U is chosen asMa/√

3 forMa∈R ^{theMach number.The parametersk and} δcontrolling thewidth ofthe shearlayersandthemagnitudeoftheinitialdataaresetto80 and0.05.

We first validate the vorticity draws obtainedin [17,16,15] using the scheme relative to the fluid velocity u for the second-ordertruncatedequilibrium(9).Thisvorticityisdefinedby

ω ₌ ∂

_xu_y

− ∂

_yu_x

.

Forthissimulation,thedomainisconstitutedof128×128 points,theMachnumberisfixedat0.04 (λischosenasin[16]

sothatU=^1).Figs. 6 and7arethevorticityplotsattimet=⁰.6 andt=^1.

We nowpresentastability analysisdepending onthedifferentparameters forλ=^{1.Weexpect toconfirm thelinear} stability results.The schemeis consideredstableifit hasnotbrokenafter2000iterations. Table 6contains themaximal stableMachnumberMafordifferentmeshesat0.01 close.Table 7presentsthegreaterReynoldsnumberRe=¹/

ν

stable at 1000 closefordifferent meshesandMa=⁰.09.Since wediscusson theReynolds number,the viscosity

ν

becomes a parameter.

We obtain results consistent withthe linearstability study.First, choosing a schemerelative tou=^{u has a positive} effect if

α

₌0, negative if

α

₌1.We must choosethe moments ofthe D2Q9 cascaded schemeto improvethe stability.

This improvement occurswhatever theequilibrium andthe mesh: the stability limit sν=^{2 isstable (Table 7) andhigh} Machnumbersarereachedforthisscheme(Table 6).Second,thed’Humièresschemeisindependentof

α

asforthelinear stability study.Its stabilityareaissmallerthantheschemerelativetou=^{u when}

α

=^{0,greaterwhen}

α

=^{1.Finally,the}