An efficient lattice Boltzmann method for compressible aerodynamics on D3Q19 lattice

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An efficient lattice Boltzmann method for compressible aerodynamics on D3Q19 lattice

S. Guo, Yongliang Feng, Jérôme Jacob, F. Renard, Pierre Sagaut

To cite this version:

S. Guo, Yongliang Feng, Jérôme Jacob, F. Renard, Pierre Sagaut. An efficient lattice Boltzmann

method for compressible aerodynamics on D3Q19 lattice. Journal of Computational Physics, Elsevier,

2020, 418, pp.109570. �10.1016/j.jcp.2020.109570�. �hal-03232070�

An efficient lattice Boltzmann method for compressible aerodynamics on D3Q19 lattice

S. Guo

^a

, Y. Feng

^a

^,∗ , J. Jacob

^a

, F. Renard

^b

, P. Sagaut

^a

aAixMarseilleUniv,CNRS,CentraleMarseille,M2P2UMR7340,13451Marseille,France bCERFACS,Toulouse,France

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

Articlehistory:

Received24July2019

Receivedinrevisedform27April2020 Accepted13May2020

Availableonline20May2020

Keywords:

D3Q19lattice HighMach HybridthermalLBM Shocksensor Recursiveregularized

AnefficientlatticeBoltzmann(LB)modelrelyingonahybridrecursiveregularization(HRR) collisionoperator onD3Q19stencil is proposedfor thesimulation ofthree-dimensional high-speed compressible flows in both subsonic and supersonic regimes. An improved thermal equilibrium distribution function on D3Q19 lattice is derived to reduce the complexity ofcorrecting terms.Asimpleshock capturingschemeandanupwindbiased discretizationofcorrectiontermsareimplementedforsupersonicflowswithshocks.Mass and momentumequations are recoveredby anefficient streaming, collisionand forcing process onD3Q19 lattice.Thenanon-conservativeformulation ofthe entropyevolution equationis used,that issolvedusing afinitevolume method.The proposedmethodis assessed considering the simulation of i) 2D isentropic vortexconvection, ii) 3D non- isothermal acoustic pulse, iii)2D supersonicflow overabump,iv) 3D shock explosion inabox,v)2Dvortexinteractionwithshockwave,vi)2Dlaminarflowsoveraflatplate atMaof0.5,1.0and1.5.

1. Introduction

AccurateandefficientsolutionsofEulerandNavier-Stokesequationsforfullycompressibleflowshavebeensubjectto intensive researchin aerodynamics,combustionand aeroacoustic.Theyare veryimportant formanyfieldsofapplication includingaerospaceengineering,combustion-basedpropulsionandacousticnoisesimulations.

AsanalternativeapproachtosimulatefluidflowsbasedontheBoltzmannequation,thelatticeBoltzmannmethod(LBM) hasbeenprovedtobeverywell suitedforthesimulationofnearlyincompressible,athermalflows [1–6].Motivatedbyits advantagesformassivelyparallelcomputingaswell asits abilitytohandleverycomplexgeometries,thereare significant researcheffortsdevotedtoextendingLBMtothermalandsubsonictosupersonicapplications.

InachievingthatgoalofconstructingLBmodelsforfullycompressibleflowsathigherMachnumbers,mostofexisting attempts are restrictedto two-dimensional space andrely on finitevolume/difference/element discretization [7–12]. The superiorityofthoseLBmodelsremainsan openquestioncomparedwiththegaskineticschemes[13] anddiscreteveloci- tiesmethods[11,14–16].FocusingontheclassicalLBmethodswhicharecharacterizedbythecanonicalstream-and-collide algorithmthat amountsto aStrang-splittingapproach [17],it isrequiredtoconsiderhigh-order momentsof densitydis- tributionfunctionstorecoverthemacroscopicenergyconservationequationforthermallatticeBoltzmannmodel[18–23],

*

Correspondingauthor.

E-mailaddress:[email protected](Y. Feng).

Table 1

Listofstream-collide LBmodelsforcompressibleaerodynamics. Theabbreviations:fractionalBGK(FBGK);en- tropicLB(ELB);ELBwithshiftedframe(ELB*);multiplerelaxationtime(MRT);fractionalMRT(FMRT);single distributionfunction(SDF);doubleDF(DDF);regularizedBGK(RBGK);recursiveregularizedBGK(RR);hybridRR BGK(HRR);Rayleigh-Taylor(RT);boundarylayer(BL);supersonic(sup.).

Representative refs. Discrete velocities Collision model Approach on energy Test cases

Alexander et al. [18] D2Q13 BGK SDF Couette flow

Shan et al. [19] D2Q17 etc. BGK SDF

Scagliarini et al. [20] D2Q21/V37 BGK SDF RT instability

Philippi et al. [21] D2V37 BGK SDF

Yu and Zhao [42] D2V17 BGK athermal sup. wedge

Yan et al. [43] D2V25 BGK SDF sup. cylinder

Prasianakis et al. [44] D2Q9 ELB SDF 1D shock tube

Li et al. [45] D2Q9 MRT DDF acoustic

Li et al. [46] D2V37 MRT SDF shock tube

Coreixas et al [25] D2V37 RR SDF shock tube

Mattila et al. [26] D2V37 RR SDF shock tube

Saadat et al. [47] D2Q9 ELB* DDF shock vortex

Feng et al. [40] D2Q9 HRR hybrid subsonic BL

Chen et al. [48] D3V40 BGK SDF shear wave

McNamara et al. [49] D3V27 FMRT SDF RB convection

Nie et al. [34] D3Q39 FBGK Hybrid transonic airfoil

Li et al. [50] D3Q39 RBGK Hybrid transonic airfoil

Frapolli et al. [22] D3Q343 ELB DDF sup. airfoil

Shan [51] D3Q103 BGK SDF

Fares et al. [52,53] D3Q39 BGK SDF nozzle, airliner

Latt et al. [54] D3Q39 BGK DDF sup. airfoil

usually relyingon extended-neighbor latticesets(D1Q7,D2Q37,D3Q343etc.)leading toLBmethodsreferred toasmulti- speed models [18–22,24–30].Thisapproachseemsratherexpensiveforindustrialapplicationsduetothehighnumberof discrete velocitiesandtheincreasedcomplexityfortheimplementationofboundaryconditionsandlocalgrid refinement.

A brief overviewofvarious 2D and3D stream-and-collide-type LB models forcompressible aerodynamics isreported in Table1.

The thermal LB models basedon the nearest-neighbor latticesets for highMach flows have attractedmanyresearch interestsrecently.Atwo-dimensionalthermallatticeBoltzmannmodelinthemannerofdouble-distributionfunction(DDF) was extended to high-subsonic flows using finite volume approach to improve numerical stability in [31]. A consistent two-distribution function thermal lattice Boltzmannwithentropy stabilizeron compressibleflows was proposed in [32].

In higherdimensions, aLB modelusingreductionof soundspeed [33] andhybridfinite differenceentropyequation was constructed for transonic andsupersonic flows on D3Q39 lattice set [34,35], which isthe closest one with3D standard lattices(i.e. D3Q19, D3Q27).A three-dimensionalDDF thermallatticeBoltzmann modelforthermalflows was developed forsimulationofshockexplosionina3Dbox [36].Recentbreakthroughshavebeenmadeonregularlatticesextendingthe hybrid RecursiveRegularizedschemeproposed in[37] to2D compressiblethermalflows[38,39] andthen to2D highMa flows[40,41].Inthelatterarticles,theenergyistakenintoaccountviaanentropyevolutionequationwritteninquasi-linear form,thatissolvedusingastabilizedfinitevolumeapproach.

Due tothethree-dimensionalintrinsicnature ofrealistic compressibleflows,itisnaturaltoseeka stableandefficient three-dimensional LB model which is able to simulate fullycompressible flows usingthe minimumnumber of discrete velocities. In contrast to D3Q27 model,the D3Q19 model is computationally less expensivebut hasmore flaws dealing withthebreakdownofGalileaninvariance,especiallyforhighMachnumberflows.Theobjectiveofthepresentpaperisto further extendtheapproach proposedin[40] to 3Dhigh-speedcompressibleflows,includingflows withshockwaves,on theD3Q19lattice.WeaimtoconstructanefficientLBmodelfor3Dcompressibleflowsusingonly19discretevelocitiesand 20 degreesoffreedom per cell(19 distributionfunctionsplusone thermodynamicscalar quantityforenergy),which can be considered asminimumnumberoffreedomper cell forcompressibleaerodynamicswithinthe frameworkofLBM. To thisend, boththeexpressionoftheequilibriumfunctionandtheGalilean-invariance-correctionterms arerevisited,along thestablediscretizationofthecorrectiontermsandentropyequation,aswellastheappropriateshockcapturingapproach.

The paperis organized asfollows:In section2, theproposed thermallattice Botlzmannmethodwitha newequilibrium distributionfunctiononD3Q19latticeisintroduced.Section 3presentstheoreticalanalysisandimplementationofentropy equation forenergyconservationlaw.Afterthat,theshockcapturingapproachandstablediscretizationofcorrectionterm in highMachflows isanalyzed anddiscussedin Sec.4.Section 5 givesthedetails oftheimplementation ofthepresent hybridLBmethodandboundaryconditions.Then,theresultsobtainedconsidering sixclassicaltestcasesarepresentedin Sec.6toassesstheproposedmethod.Finally,section7drawsconclusionsandperspectives.

Table 2

DiscretevelocitiesandcorrespondingweightsofD3Q27andD3Q19.

D3Q27 D3Q19

[ciα,wi] (0, 0, 0) 8/27 (0, 0, 0) 1/3 cyc(±^{1, 0, 0) 2/27 cyc(}±^{1, 0, 0) 1/18} cyc(±^1,±^{1, 0) 1/54 cyc(}±^1,±^{1, 0) 1/36} (±^1,±^1,±^{1) 1/216}

2. HybridthermallatticeBoltzmannmethodonD3Q19lattice

Inthepresenthybridthermallattice Boltzmannmethod,massandmomentumequationsarerecoveredby anefficient streaming,collisiononD3Q19lattice.Thenathermodynamic scalarquantity forenergyissolvedbyusingafinitevolume method.

2.1. ThermallatticeBoltzmannnmodel

ThethermallatticeBoltzmannmodelinthisstudyaimsatsolvingdiscretevelocityBoltzmannBGKequationindiscrete physical space and time to obtain macroscopic density

ρ

and velocity uα. The lattice Boltzmann equation with hybrid recursiveregularizationcollisionmodelisusedinthisstudy,thatcanbeexpressedas [40]

fi

(

x_α

+

^ciα

δ

t

,

t

+ δ

t

) =

^f_i^eq

(

x_α

,

t

) + (

1

−

¹

τ ^{) R (}

^f

neq i

) + δ

t

2

ψ

i

(

x_α

,

t

)

(1)

IntheLBalgorithm,themacroscopicdensity

ρ

andmomentum

ρ

uα arecomputedandupdatedas

ρ =

i

f_i

,

(2a)

ρ

u_α

=

i

c_iαf_i

+ δ

_t 2

i

c_iα

ψ

_i (2b)

InEq.(1),

τ = τ /δ

t

+

¹

/

2 isnon-dimensionalrelaxationtimeand

R(

^f_i^neq

)

representsrecursivelyregularizedoff-equilibrium part ofthe density distribution functions f_i^neq

=

^fi

−

^fieq

+

^δ₂^t

ψ

i. The Galilean-invariance correction term

ψ

i is given in formofaforceand fieq

isthethermodynamicsequilibriumdistributionfunction.Theequilibriumdistributionfunctionand correctiontermonD3Q19latticeareintroducedanddiscussedinthefollowingsection.

2.2. ImprovedequilibriumdistributionfunctiononD3Q19lattice

The setup ofdiscrete equilibriumdistribution functionsandassociated velocitiesis one amongthekey issues forthe design ofthecompressible LBmodels. Followingthe approachdiscussed in[19,40], theequilibrium distributionfunction

f^eq couldbegiveninthethird-orderGrad-HermiteexpansionofMaxwell-Boltzmanndistributionas

f_i^eq,Q

=

^wi

⎡

⎣ ρ ₊

^ciα

c²_s

ρ

u_α

+ ^H

(2) iαβ

2c⁴_s

A

⁽_α⁰_β⁾

+ ^H

(3) iαβγ 6c⁶_s

A

⁽_α⁰_β⁾_γ

⎤

⎦ ,

(3)

where w_i isthe i_th weight coefficient associatedto discrete velocity c_iα, c_s is lattice soundspeed andthediscrete Her- mitepolynomialsaregivenas

H

⁽_iα²⁾_β

=

^ciαciβ

−

^c2s,

H

⁽_i³_α⁾_βγ

=

^ciαciβciγ

−

^c2s

[

^ci

δ ]

αβγ and

[

^ci

δ ]

αβγ

=

^ciα

δ

βγ

+

^ciβ

δ

_αγ

+

^ciγ

δ

_αβ. The second and third-order terms in the equilibrium distribution function are

A

⁽_α⁰_β⁾

= ρ

uαuβ

+ ρ

c²_s

(θ −

¹

)δ

_αβ,

A

⁽_α⁰_β⁾_γ

= ρ

uαu_βuγ

+ ρ

c²_s

(θ −

¹

)[

^u

δ]

αβγ with

[

^u

δ]

αβγ

=

^uα

δ

βγ

+

^uβ

δ

αγ

+

^uγ

δ

αβ.

θ

is defined according to temperature T as

θ =

^{R T}

/

c²_s.

Theparameter Q denotesthenumberofdiscretevelocities.Forthethree-dimensionallattices, therearetwo common discrete velocity models, namelythe nineteen-velocity model (D3Q19)and the twenty-seven-velocitymodel (D3Q27) as showninTable2,whichareassociatedtotheequilibriumfunctions f_i^eq,19and f_i^eq,27,respectively.

ConsideringtheequilibriumdistributionfunctionofEq. (3),thedeviationtermsonthethird-ordermomentduetothe defectofsymmetrycanbecalculatedasfollows

_αβγ

=

^eq_αβ^,M_γ

−

^eq_αβ^,Q_γ (4)

where

αβγ denotesdeviationtermsonthethird-ordermoment.

^eq_αβ^,M_γ and

^eq_αβ^,Q_γ arerespectivelythethird-ordermoment computedbyexactMaxwelliandistributionandbyQ-sitevelocitiesdiscreteequilibriumdistributionfunctionasfollows

Table 3

Thethird-ordermomentof3Dlatticemodels.

^eq_αβγ D3Q27 (3) D3Q19 (3) D3Q19r (8) Maxwellian (7)

^eqxxx ρux ρux ρux ρuxuxux+3pux

xxx ρux(θ−1+u²x) ρux(θ−1+u²x) ρux(θ−1+u²x) 0

^eqxxy ρuxuxuy+^puy θ+1

2 ρc²_suy+ρuy(u²_x−^u2^2z) ρuxuxuy+^puy ρuxuxuy+^puy

_xxy 0 ρuy(^θ−₆¹+^u2^2z) 0 0

^eqxyz ρuxuyuz 0 0 ρuxuyuz

_xyz 0 ρuxuyuz ρuxuyuz 0

^eq_αβ^,M_γ

=

c_αc_βc_γf^eq,Md³c

=

^p

[

^u

δ]

αβγ

+ ρ

u_αu_βu_γ

,

(5)

^eq_αβ^,Q_γ

=

Q

−¹ i=⁰

c_iαc_iβc_iγf_i^eq,Q (6)

wherethepressurep

= ρ

c²_s

θ

satisfiestheequationofstateofperfectgas. f^eq,M isMaxwell-Boltzmanndistributionfunction, whichiswrittenas

f^eq,M

= ρ

1

2

π

R T

3 2

exp

− (ξ

_α

−

^uα

)

² 2R T

(7)

where R isthegasconstant. Thethird-ordermoments

^eq_αβ^,27_γ and

^eq_αβ^,19_γ computedusingD3Q27andD3Q19latticesfol- lowingEq. (3) and

^eq_αβ^,M_γ computedusingtheexactMaxwellianequilibriumdistributionfunctionsaswellastheassociated deviationtermsof

αβγ aresummarizedinTable3.Itisobservedfromthetablethatonly

xxx,

y y y,

zzz isnon-zeroin D3Q27lattice set,whichrenderthecorrectiontermmuchsimplercomparedwiththeoneobtainedontheD3Q19lattice [40].ConsideringthesymmetryofGaussian-Hermitemoment [55,56],thefollowingnon-isothermalequilibriumdistribution functiononD3Q19lattice(denotedasD3Q19r)isintroducedtoreducethedefectofthethird-ordermoment

f_i^eq,19r

=

^wi

⎡

⎣ ρ ₊

^ciα

c²_s

ρ

u_α

+ ^H

(2) iαβ

2c⁴_s

A

⁽_α⁰_β⁾

+ [ HA ]

⁽_i,³_Q^,0_19r⁾ 6c⁶_s

⎤

⎦

(8)

=

^wi

ρ +

^ciα

ρ

u_α c²_s

+

^A

(0) αβ

H

⁽_iα²⁾_β

2c⁴_s

+

¹ 6c⁶_s

3

( H

⁽_i,³_xxy⁾

+ H

⁽_i,³_yzz⁾

)(

A⁽_xxy⁰⁾

+

A⁽_yzz⁰⁾

) + ( H

⁽_i³_,xxy⁾

− H

⁽_i,³_yzz⁾

)(

A⁽_xxy⁰⁾

−

A⁽_yzz⁰⁾

) +

³

( H

⁽_i,³_xzz⁾

+ H

⁽_i,³_{xy y}⁾

)(

A⁽_xzz⁰⁾

+

^A(xy y⁰⁾

) + ( H

⁽_i³_,xzz⁾

− H

⁽_i,³_{xy y}⁾

)(

A⁽_xzz⁰⁾

−

^A(xy y⁰⁾

) +

3

( H

⁽_i,³_{y yz}⁾

+ H

⁽_i,³_xxz⁾

)(

A⁽_{y yz}⁰⁾

+

A⁽_xxz⁰⁾

) + ( H

⁽_i,³_{y yz}⁾

− H

⁽_i³_,xxz⁾

)(

A⁽_{y yz}⁰⁾

−

A⁽_xxz⁰⁾

)

Thethird-ordermoments

^eq_αβ^,19r_γ stemmingfromthenewequilibriumfunctionfortheD3Q19latticegivenbyEq. (8) are alsosummarizedinTable3.Asshowninthetable,thethird-ordermomenttermsrelatedtoc_ixc_ixc_ixandc_ixc_iyc_iz remain thesameasfortheoriginal f_i^eq,19equilibriumfunctionwhenconsidering f_i^eq,19r,buttheexacttermsforc_ixc_iyc_iy arefully recoveredby therevisedequilibriumdistributionfunction.Thedeviationterms

xyzand

xxxonc_ixc_iyc_iz andc_ixc_ixc_ix are balanced by introducinga correctingterm

ψ

i,which canbe appliedasaforcing terminlattice BoltzmannBGKequation (1).

ψ

i

= −

^wi

H

⁽_iα²⁾_β 2c⁴_s

∂

∂

x_γ

_αβγ

,

i

c_iαc_iβ

ψ

i

= − ∂

∂

x_γ

_αβγ (9)

Thecorrectionterm

ψ

ionD3Q19latticeisgiveninthefollowingsimpleform:

ψ

i

=

^wi 2c⁴_s

H

⁽_ixx²⁾

∂

∂

x

[ ρ

ux

(

1

− θ −

u²_x

)] − H

⁽_iyz²⁾

∂

∂

x

( ρ

uxuyuz

) + H

⁽_{iy y}²⁾

∂

∂

y

[ ρ

uy

(

1

− θ −

^u2y

) ] − H

⁽_ixz²⁾

∂

∂

y

( ρ

uxuyuz

) + H

⁽_izz²⁾

∂

∂

z

[ ρ

uz

(

1

− θ −

^u2z

) ] − H

⁽_ixy²⁾

∂

∂

z

( ρ

uxuyuz

)

(10)

FollowingtheChapman-Enskogmultiscaletechnique(seeAppendixA),themassandmomentumconservationequations canberecoveredasfollows

∂ ρ

∂

t

+ ∂

∂

x_α

( ρ

u_α

) =

0

,

(11a)

∂

∂

t

( ρ

u_α

) + ∂

∂

x_β

( ρ

u_αu_β

+

^p

δ

_αβ

) = ∂

∂

x_β

(

_αβ

),

(11b)

where

αβ isstresstensor,

_αβ

= μ ^∂

^uβ

∂

x_α

+ ∂

u_α

∂

x_β

−

² 3

∂

u_γ

∂

x_γ

δ

_αβ

(12)

μ

is dynamic viscosity, which is relatedwith non-dimensional relaxationtime through

μ ₌

p

( τ ₋

0

.

5

)δ

t in the present thermallatticeBoltzmannmodel.

2.3. AssociatedrecursiveregularizationonD3Q19lattice

Therecursivelyregularizedoff-equilibriumdistributionfunctiononD2Q9ororiginalD3Q19modelasgiveninRef. [40]

isexpressedasfollows

R (

f_i^neq

) =

^wi

_H

⁽²⁾

iαβ

2c⁴_s

A

⁽_α¹_β⁾

+ ^H

(3) iαβγ 6c⁶_s

A

⁽_α¹_β⁾_γ

(13) where

A

⁽_α¹_β⁾

=

ic_iαc_iβf_i^neqand

A

⁽_α¹_β⁾_γ

≈

^uα

A

⁽_β¹_γ⁾

+

^uβ

A

⁽γ α¹⁾

+

^uγ

A

⁽_α¹_β^).Consideringthesamesymmetryfeatureofcoefficient of Hermite polynomialsforoff-equilibrium, the associated off-equilibrium on improvedD3Q19 modelis now recursively regularizedasfollows

R (

f_i^neq

) =

^wi

A⁽_α¹_β⁾

H

⁽_i²_α⁾_β 2c⁴_s

+

¹

6c⁶_s

3

( H

⁽_i,³_xxy⁾

+ H

⁽_i,³_yzz⁾

)(

A⁽_xxy¹⁾

+

A⁽_yzz¹⁾

) + ( H

⁽_i,³_xxy⁾

− H

⁽_i,³_yzz⁾

)(

A⁽_xxy¹⁾

−

A⁽_yzz¹⁾

) +

³

( H

⁽_i,³_xzz⁾

+ H

⁽_i³_{,xy y}⁾

)(

A⁽_xzz¹⁾

+

^A(xy y¹⁾

) + ( H

⁽_i,³_xzz⁾

− H

⁽_i,³_{xy y}⁾

)(

A⁽_xzz¹⁾

−

^A(xy y¹⁾

) +

³

( H

⁽_i,³_{y yz}⁾

+ H

⁽_i,³_xxz⁾

)(

A⁽_{y yz}¹⁾

+

^A(xxz¹⁾

) + ( H

⁽_i,³_{y yz}⁾

− H

⁽_i,³_xxz⁾

)(

A⁽_{y yz}¹⁾

−

^A(xxz¹⁾

)

(14)

Itisworthnotingthatthecoefficientofthesecondviscositybeing-2/3isnaturallyrecoveredbyD3Q19latticemodelin contrasttoD2Q9model[40].Inthesamemannerusedin[37,40],aparameterizedhybridrecursiveregularizedprocedure is adopted to suppress non-hydrodynamic modes by using

A

⁽_α¹_β^,HRR)

= σ _A

⁽_α¹_β⁾

₊ (

1

− σ ) A

⁽_α¹_β^,FD).

σ _∈

[0

,

1] is an arbitrary weightingcoefficient.

A

⁽_α¹_β^{,FD)isestimatedbyits}Chapman-Enskogsolutionwhichisapproximatedbyasecond-orderfinite- differencescheme[40].

3. Entropybasedenergyconservationequation

In the presentstudy, the thermodynamic quantity forenergy conservation is solved by a finite volume method. The numerical stability of the solution isintrinsically tiedto the control of the discrete energy of thesolution, which must remainbounded.AsdiscussedinthecaseofNavier-Stokessolutions,e.g.[57],thesolutionshouldpreservethe secondlaw ofthermodynamicforentropyevolution.Therefore,theexplicitcontrol oftheentropyappearsasanaturalwaytoenforce bothnumerical stabilityandphysicalevolutionofthe entropyatthesametime. Thisleadseveralgroupstowork onthe entropicvariablesto solvebothEulerandNavier-Stokesequations,e.g. [58], orto designnumericalmethodthatpreserve entropy insmooth flows [59–66], or toderive new LBM schemes basedon the preservationof an entropic principle for manufacturedentropyvariable[22,24,67–69].

Asamatteroffact,themassconservationisnotexplicitlyandstrictlyguaranteedinLBmethods,somethingthatcanbe shownconsidering theChapman-Enskogexpansion [70]. Thislackofexactmass conservationinLBMmust betakeninto

account whendesigningan hybridmethod.Forinstance,an errorterm ρ implicitlyexists inmassconservationequation recoveredbyLBM

∂ ρ

∂

t

+ ∂

∂

x_α

( ρ

u_α

) =

_ρ (15)

where

_ρ scales as

O (

Kn^{2 ∂3(}ρu_α)

∂x_α∂xβ∂xβ

)

by considering third-orderterms in the Chapman-Enskog expansion [70]. Here, the Knudsen number Kn isassumedto be a smallparameter, which istrue forthe class offlows considered in thepresent paper.Itisworth keepinginmindthatahierarchyofhydrodynamicmodelscanbederived fromtheBoltzmannequation thankstotheChapman-Enskogexpansion(byincreasingorderintermsofsmallparameterexpansion:Euler,Navier-Stokes, Burnett,super-Burnett...)showingthatthedeparturefromEulerandNavier-Stokesequationsisaphysicalphenomenon.

Inordertoanalyzetheimpactof ρ ontheentropyevolution,adensity

ρ

c thatsatisfiestheexactmassconservationis introduced,i.e.

∂ ρ

c

∂

t

+ ∂

∂

x_α

( ρ

cu_α

) =

0 (16)

Now omittingviscous productiontermsforthesakeofsimplicitywithoutrestrictingthegenerality ofthedemonstration, andstartingfromnon-conservativeformoftheentropyequation

∂

s

∂

t

+

^uα

∂

s

∂

x_α

=

^{0 (17)}

thefollowingusualconservativeformofentropyevolutionequationisobtainedbycombiningEqs.(16) and(17):

∂ ρ

_cs

∂

t

+ ∂ ρ

_cu_αs

∂

x_α

=

⁰

,

(18)

wherenospurioussourcetermappears,thankstotheexactmassconservation.

Thisequationmustbemodifiedtoaccountforthe ρ term.Startingfromtheconservativeformofentropyequation

∂ ρ

s

∂

t

+ ∂ ρ

u_αs

∂

x_α

=

⁰

,

(19)

andaccountingforEqs. (15), (16) and(19),thefollowingconservativeformoftheentropyequationisobtained

∂ ρ

cs

∂

t

+ ∂ ρ

cu_αs

∂

x_α

= − ρ

c

ρ

_ρs

.

(20)

Consequently,theconservativeformofentropy-basedenergyconservationcouldinducespuriousentropysource,whilethis termisnotpresentwhenthenon-conservativeformulationisused.Itisworthnotingthatthesameconclusionholdswhen considering theuseofother thermodynamicalquantitiessuchasinternalenergyortotalenergy.Anumericalvalidationof thistheoreticalanalysisfornon-conservativeformentropyequationiscarriedoutinSec.6.1.Therefore,anon-conservative formofentropyequationisusedinthepaper.

∂

s

∂

t

+

u_α

∂

s

∂

x_α

=

¹

ρ

T

∂

∂

x_α

(λ ∂

T

∂

x_α

) +

¹

ρ

T

_αβ

∂

u_α

∂

xβ

,

(21)

wheres

=

^cvln

(

p

/ ρ

^γ

)

istheentropywithcvbeingspecificheatcapacityatconstantvolumeandspecificheatratio

γ

.

λ

is heatconductivity.

AnexplicitEulerschemeisadoptedastemporalintegration.Theconvectivefluxisconstructedusingthird-orderMUSCL scheme [71] and vanAlbada limiter [72], while the classical second-order central difference scheme is adopted for the diffusiontermandtermofviscousdissipation.

4. Shockcapturingtechniqueandimproveddiscretizationofcorrectionterm 4.1. Shockcapturingtechnique

Thecomputationofflowcontainingshockwavesisanextremelydifficulttaskbecausesuchflowsresultinsharp,discon- tinuouschangesinflowvariablessuchaspressure,temperature,density,andvelocityacrosstheshock.Inordertocapture the shockand increase numericalstability, the shocksensorand associatedartificial viscosityused in Jameson-Schmidt- Turkel(JST)scheme [73] isadoptedinthepresentLBmodeltodetecttheshockwavefornon-smoothcompressibleflows, whichisgivenas

ε

_α

₌ κ

^pi−1

⁻

^2pi

⁺

^pi+1

p_i−1

+

^2pi

+

^pi+1

⁽²²⁾

whereiistheindexofCartesiangrid.

κ

isafreeparameterintroducedtotunetheartificialviscosity.Thevalueof

κ

isset tounityinallthesimulationsofthisstudy.Accordingly,aneffectiverelaxationtimecanbesummedas

τ

e

= μ

p

+

^max

[ ε

x

, ε

y

, ε

z

] δ

t (23)

Thenthenon-dimensionalrelaxationtime

τ = τ

e

/δ

t

+

⁰

.

5 isactuallyemployedinthefollowingtestcaseswithshocks.Itis worthnotingthatthevariables,e.g.density

ρ

,entropyscanbealternativelyusedinEq. (22).

4.2. Upwind-biaseddiscretizationofcorrectionterm

The discretization of correctionsterm in Eq.(10) requires specific treatment in supersonicflows withrespect to the subsonicflowcasetoensurenumericalstability,consideringthatthetruncationerrorofitsfinitedifferenceapproximation maypotentiallyviolentGalileaninvariant.Thecorrectiontermisderivedfromthird-ordermomentofequilibriumdistribu- tionfunction,whichinvolvesconvectivefluxesoftheenergyequation.Althoughtheenergyequationisnotdirectlysolved bytheLBapproach,thediscreteenergyoftheLBsolutionobeysan evolutionequationthat canbederivedfromtheLBM scheme,exactlyasdoneforthediscreteenergyofNavier-Stokescomputationalsolution,e.g.,[74,75,61,62,57,76–80,63].Asa consequence,thethermalandkineticenergyoscillationscanimplicitlyweakenthenumeralstability.Therefore,adissipative termisintroduced inthemomentumequation inordertopreventspurious growthofthediscreteenergyofthesolution, whichappearsasasinkterminthediscreteenergyequation.ThedetailedChapman-Enskoganalysisoftheenergyequation ispresentedinAppendixA.

Inordertoimprovethenumericalstabilitybycontrollingthediscreteenergyofthesolution,alow-dissipativesecond- orderupwindschemeisadoptedinthisstudytodiscretizedthecorrectionterminthemomentumequation,e.g.,foru

>

0

∂

∂

x

=

¹

4

x

(

i−2

−

⁵

i−1

+

³

i

+

i+1

)

(24)

whereahalf centraldifferenceandahalf second-orderupwinddifferenceisusedinthisupwindschemetoapproximate thepartialdifferenceoperatorinthecorrectionterm.Itisincontrasttoisotropiccentraldiscretizationinthecompressible LBmodelforsubsonicflows[40].

5. Implementationofthemethod

ThefullprocedureofthehybridLBmethodforcompressibleaerodynamicsissummarizedasfollows

1. Initializethemacroscopicvariables

ρ

,uα,T,saswellasdensitydistributionfunctionandcorrectionterm

ψ

i. 2. Implementboundarytreatmentformacroscopicvariablesanddensitydistribution.i) Themacroscopicvelocities

ρ

B,uB,

TB and pB ontheboundarynodesaredirectlycomputedfortheplanarboundariesorestimatedfromtheir neighbor fluidnodesusingShepard’sInverseDistanceWeighting(IDW)methodforcurvedboundaries,followingboundarycon- ditionscommonlyused inclassicalNavier-Stokessolvers. ii)thedensitydistributionfunction iscomputedby afinite differencereconstruction approach[81,40].iii) Theboundarycondition forentropyequation,entropy sB,iscalculated using

ρ

B and p_B by thermodynamicclosure.Implementationdetailsarethesameasin[40] andwillnot berepeated here.

3. PerformcollisionandstreamingprocedurebyEq.(1) atthen+1timestep.

4. Updatedensity

ρ

andvelocityuα atthen+1timestepbyusingEq.(2).

5. Solveentropyequation(21) bythefinitevolumemethodandupdateT andpofthen+1timestepbyusingthermody- namicclosure.

6. Validation:numericalresultsanddiscussion

Theproposedmethodisassessedconsideringsixcasesdealingwithcompressiblesubsonictosupersonicflows:

1. 2Disentropicvortexconvection, 2. 3Dnon-isothermalacousticpulse, 3. 2Dsupersonicflowoverabump, 4. 3Dshockexplosioninabox,

5. 2Dvortexinteractionwithshockwave,

6. 2DlaminarflowwithMa=(0.5,1.0,1.5)overaflatplate.

Inthesesimulations,theinviscidflowsaretreatedasquasi-inviscid,withavery smallnon-dimensionalviscosity

μ =

^10−15. The valueof

σ

plays arole ofhyper-viscosity inthe HRR-LBMmodelandthehyper-viscosity decreases by increasing

σ

, whichwasillustratedandinvestigatedin[37,40].Asamatteroffact,theglobalsecond-orderaccuracyofthemethodwith

σ =

⁰

.

5 iswell confirmed inthe caseofisentropicvortex convectionin Sec. 6. Thus, therecommended minimumvalue

σ

is

σ =

⁰

.

5 inthein-viscidcompressibleflows,

σ =

⁰

.

7 intheviscoussupersonicflows,and

σ =

⁰

.

9 intheviscoushigh

Fig. 1.Thedistributionsofdensityandvelocityonthemid-lineatt=^{50T (}x=⁰.025).(Forinterpretationofthecolorsinthefigure(s),thereaderis referredtothewebversionofthisarticle.)

Fig. 2.Convergence rate study of the proposed LB. The relative error is computed throughL2error on density att=^50T.

subsonicflows.Allofthe2Dand3DcomputationalexamplespresentedinthenextsectionwereobtainedwithintheD3Q19 frameworkintheProLBsolver[82].

6.1. Isentropicvortexconvection

TheproblemofisentropicvortexconvectionisfirstconsideredtoassesstheproposedHRR-LBmethodandtoinvestigate the influence of the formulationof the entropyequation. The size ofthe computational domain is

[

⁰

,

10

] × [

⁰

,

10

]

^{. The} uniformfree-streamparametersare

ρ

_∞

=

¹

,

u_∞

=

¹

.

0

,

v_∞

=

⁰

,

p_∞

=

¹

,

Ma_∞

=

⁰

.

84515.Attheinitialtime,thefollowing disturbanceisaddedtotheabovefree-stream:

ρ =

1

− ( γ −

1

)

b² 8

γ π

^{2 e}

1−^r2

_γ−¹₁

,

p

= ρ

^γ

,

(25)

u

=

^u∞

−

^b 2

π

^e

1 2

1−r²

(

y

−

^yc

) ,

(26)

v

=

^v∞

+

^b 2

π

^e

1 2

1−r²

(

x

−

^xc

) ,

(27)

whereb

=

⁰

.

5

,

x_c

=

⁵

,

y_c

=

^{5 andr}

=

(

x

−

^xc

)

²

+ (

y

−

^yc

)

²

1/2

.Inthissimulation,theHRRweightingparameteris

σ =

⁰

.

5.

Fig.1showsthedistributions ofdensityandvelocityalong thesymmetry lineofthedomainafter50flow-through-times, i.e.t

=

^50T.These distributions areobtainedusing agrid size equalto

x

=

⁰

.

025. From thefigure, itcan be seenthat presentresultsmatchverywellwiththeanalyticalsolution.

Inordertostudytheglobalspatialaccuracyofthepresentmethod,differentmeshsizesarenowconsidered,i.e.

x

=

0

.

05

,

0

.

1

,

0

.

2.Fig.2displaystheconvergencerateoftheL2-normoftheerroroftheproposedLBmethod.Forthefinegrid

Fig. 3.Thedensitycontours(from0.993to0.999with20levels).Left:resultsobtainedconsideringtheconservativeformoftheentropyequation;middle:

resultsobtainedusingthepresentLBmethodwithaconservativeentropyequation,butinwhichanauxiliarydensityfieldcomputedbysolvingtheexact massconservationequationwithafinitevolumesolverisused;right:presentLBmethodwithnon-conservativeentropyequation.

resolutions,itisobservedthatthepresentmethodexhibitsasuper-convergencepropertywithaslopecloseto3,asalready reportedforsome LBmethods[83].Theseresults assesstheaccuracyofpresentcompressibleHRR-LBMmethodequipped with3rd-orderMUSCLschemefortheentropyconvectionterm.

Asmentionedinsection3,theconservativeformofentropy-basedenergyconservationequationcouldinduceerroneous entropysources.Thispointwillnowbeillustratedbyadequatenumericalexperiments.Moreprecisely,threemethodswill becompared:

•

^HRR-LBMsupplementedbyanequationfor

ρ

swritteninconservativeform

•

^HRR-LBMsupplementedbyan auxiliaryequationfor

ρ

cswritteninconservativeform,wherethe“exact”density

ρ

c is computedsolvingan additionalequationby aclassicalfinitevolume method.Thismethodisintroducedhereonlyto assessthetheoreticalanalysisgivenabove,andisnotproposedasaregularmethodforapplications.

•

^HRR-LBMsupplementedbyanequationforswritteninnon-conservativeform

ResultsobtainedwiththesethreeapproachesaredisplayedinFig.3.Itcanbeseenfromtheseresultsthatthespurious entropyproductioncorrupts theevolutionofdensityveryquickly.However, boththeconservativeentropy withcorrected densityand non-conservative entropyexhibit very goodsolutions and numericalstability. Thispoint is further validated looking atthe time historyof theentropy field displayed in Fig.4. Itis observed that thenon-conservative formof en-

Fig. 4.Time history of the entropy obtained by the different forms of entropy equation.

Fig. 5.Thepressure,u-velocity,densityandtemperaturefieldsattimetend=¹.0 obtainedusingpurerecursive-regularizationcollisionmodel(parameter σ=^{1)ongridresolutionof}x=⁰.02.

tropy equation yields very satisfactory stableresults withconstant entropy, while the conservative formleads to a very rapid growthofentropy,duetospurious sourcetermsarising fromthelackofconsistencywiththeLBM massconserva- tion.

Fig. 6.Comparison of profiles along x-direction middle line. (Lines: present LBM, symbols: reference solution obtained by grid spacing ofx=0.0002).

6.2. Non-isothermalGaussianpulse

Here, we considera three-dimensional thermalacousticwave travelinginradial directionincluding temperatureevo- lution.The aim of thistest caseis to verifythe preservation ofisotropy by the presentD3Q19 modeland the coupling betweenvelocity,pressureandtemperature.Theinitialconditionisgivenby:

ρ =

1

.

0

,

p

=

1

+

exp

(−

kr²

),

u

=

0

,

v

=

0

,

w

=

0 (28) wherer

=

(

x

−

^xc

)

²

+ (

y

−

^yc

)

²

+ (

z

−

^zc

)

²

1/2

isthedistancefromthepulsecenter

(

xc

,

yc

,

zc

) = (

0

,

0

,

0

)

.Theperturbation parameteristakenequaltok

=

^40.The computationaldomainsize is

[−

¹

,

1

] × [−

¹

,

1

] × [−

¹

,

1

]

^{,withperiodicboundary} conditions.Thepresentsimulationsareperformedon100

×

¹⁰⁰

×

^{100 (}

x

=

⁰

.

02)and200

×

²⁰⁰

×

^{200 (}

x

=

⁰

.

01)grids with

δ

t of0.00667and0.00333,respectively.

Thepressure,u-velocity,densityandtemperaturefieldsattime tend

=

¹

.

0 obtained usingpurerecursive-regularization collision model(parameter

σ =

^{1) ongrid resolution of}

x

=

⁰

.

02 arepresented inFig. 5.It is found that theisotropic evolutioniswellpreservedbythecompressibleD3Q19LBmodel.

Fig.6displaystheprofilesofpressure,horizontalvelocity, densityandtemperatureobtainedbytheproposedLBmodel usingHRR parameter

σ =

^{0and1 on gridresolution of}

x

=

⁰

.

02 and 0

.

01 at time t_end

=

¹

.

0.A very goodagreement withthereferencesolutionisobtained,showingthecapabilityofthepresentmethodtocapturethermodynamiccouplings andnonlinearwavepropagation.Thecomputationofthereferencesolutionwasperformedusingasecond-orderTVDfinite volume schemewiththe Osher type flux using10⁴ cells in1D [84,85]. It isobserved that the smallvalue ofparameter

σ

inducesafinitedissipatingoncoarsegridresolution.Thehyper-viscousfeatureoftheHRRcollisionmodelisconsistent withthetwo-dimensionalLBmodel [40].

Fig. 7.Pressure, density and Mach number fields obtained by the present D3Q19 LB model on grid sizex=⁰.005.

6.3. Inviscidsupersonicflowovera4%circularbump

Thesecondcaseisaveryclassicalonetoinvestigatetheaccuracyandrobustnessofthenumericalmethodwhendealing with shockwave, i.e. theinviscid steadysupersonicflow in a channel with a bump atinlet Mach numberequal to 1.4.

The inviscid fluidbehavior ismimickedinthe presentLBM by enforcingavery smallmolecularviscosityalong withslip boundary conditionsatsolid wall(thereforepreventingthegrowthofboundarylayers).The domainofthe channelisx

∈ [−

¹

.

5

,

1

.

5

] ×

^y

∈ [

⁰

,

1

]

^{.Thebumpissetonthebottomofthechannelfromx}

= −

⁰

.

5 to0

.

5.Theheightofthebumpis0

.

04.

Theinletboundaryofthedomainisasupersonicinflowwiththefixedvalues

ρ

_∞

=

¹

.

0

,

p_∞

=

¹

/ γ ,

T_∞

=

^p

/ ρ ,

u_∞

=

1

.

4.Thesupersonicoutflowconditionsareimplementedontheoppositeboundary.Otherboundariesarefree-slipadiabatic walls. Inthesimulations,thepresentLBmodelisassessedon600

×

²⁰⁰

×

^{1 (}

x

=

⁰

.

005) and300

×

¹⁰⁰

×

^{1 (}

x

=

⁰

.

01) gridswiththetimestep

δ

t being0.00166and0.00333,respectively.ThevalueofHRRparameter

σ

issetto0.9.

Fig.7showsthepressure,densityandMachnumberfieldsobtainedonthefinegrid.It canbeseenthatthereflection andinteractionoftheshockwavesarevery wellcapturedwithoutnonphysicalwiggles.Asa quantitativecomparison,the MachnumberdistributionsonthebottomandtopwallsareplottedinFig.8.Thereferenceresultsmarkedbythesymbols were reported in Ref. [86]. The reference solution was obtained using the artificial compression method (ACM) witha second-orderaccuracyon90

×

30 nonuniformbody-fittedgrids.Inthecomparison,itcanbefoundthatboththeresultson coarseandfinegridsareinexcellentagreementwiththereferenceresults.

ThepressureandMachnumberprofilesalonghorizontal(y

=

⁰

.

5)andvertical(x

=

^{0)mid-lineareshowninFig.9.The} referenceresultsareobtainedusingtheRoeschemeandJSTschemeimplementedinasecond-orderaccuratefinitevolume solver(FVM)fortheEulerequations.BoththesolutionsoftheLBMandtheFVMarecomputedontheuniformgridspacing

x

=

⁰

.

005.Fromthisfigure,itcanbeseenthattheresultsobtainedbythepresentLBMareveryclosetothosebytheFVM.

6.4. Sphericalexplosionina3Denclosedbox

A spherical explosion ina 3D enclosed box is considered to assess capability of the presentmodel to capture three dimensionalcomplexshockwaves.Thenumericaltestisanunsteadycompressibleflow.Theinitialconditionofthisproblem areillustratedinFig.10.

Fig. 8.Mach number distributions obtained by the present D3Q19 LB model. (Lines: present results, symbols: reference solution in Ref. [86].)

Fig. 9.Machnumberandpressuredistributionsalongmid-line.ThereferencesolutioniscomputedusingaclassicalEulersolverbasedontheFiniteVolume Methodwiththesecond-orderRoeschemeandJSTscheme.

Fig. 10.Initial condition for spherical explosion in 3D enclosed box.

Fig. 11.Instantaneous density isosurface ofρ=1.8 in the 3D enclosed box.

Thecomputationaldomainisaunitcubeandalltheboundaryconditionsarefree-slipadiabaticwalls.Thecomputational gridresolutionis100

×

¹⁰⁰

×

^{100 andtimesteptakenequalto}

δ

t

=

⁰

.

00333.TheHRRparameter

σ

issetto0.7.

The densityiso-surfaces

ρ ₌

1

.

8 obtained att

=

^{0.125, 0.25, 0.375 and 0.5 are shown in Fig. 11. The} instantaneous evolutionagreeswellwiththereferencesolutionpresentedin[87].

Inordertofurthercompareourresultswithothersolutions,thedensitycontoursinthez

=

⁰

.

4 planeatt

=

⁰

.

5 obtained bypresentLBmodelandbythefinitevolumesolverarepresentedinFig.12.Itcanbefoundthatthecomplexflowfeatures such asthe shockwave interactionsarewell capturedbythepresentmodel.Thedensitycontours computedviaLBMare similar withthoseoftheFVMsolution.Forthequantitativecomparison,thedensity,temperatureandMa numberprofiles obtained by the above two methods at t

=

⁰

.

5 are shown inFig. 13. It is seen that the present LB results are in good agreementwiththeresultsofFVM.

6.5. Shock-vortexinteractions

This test case deals with the interaction ofa stationarynormal shockwave witha single vortex.The shockwave is definedbyanupstreamMachnumberMs.Therightandleftstatesofthenormalshockareasfollows:

ρR=1.0 ρL= (γ+¹)M²_s 2+(γ−¹)M²s

ρR pR=¹.0 pL=( ²γ

γ+^1M2s−γ−1 γ+¹)pR TR=pR/ρR TL=pL/ρL

uR= −^Ms uL= −²+(γ−1)M²_s (γ+1)M²s

Ms vR=^{0 vL}=⁰

Fig. 12.Densitycontoursina2Dslicethroughz=0.4 att=0.5.(a)thepresentsolutionobtainedbyHRR-LBmodelon100×100×100 grids,(b)the referencesolutioncomputedbyafinitevolumesolverwiththesecond-orderRoeschemeonthesamegridresolution.Thecontoursaredisplayedfrom0.1 to2.6with40levelsinthebothfigures.

Fig. 13.Density,temperatureandManumberprofilealongcrosslinesina2Dslicethroughz=⁰.4 att=⁰.5.ThepresentsolutionisobtainedbyHRR-LB modelon100×¹⁰⁰×^{100 grids,andthereferencesolutioncomputedbyafinitevolumesolverwiththe}second-orderRoeschemeonthesamegrid resolution.

Theinitialdensity,pressure,tangentialandradialvelocitiesofthevortexareexpressedby

ρ

θ

(

r

) = [

¹

− γ −

¹

2 M²_vrexp

(

1

−

^r2

)]

^γ−11

,

p

(

r

) =

¹

γ ρ

^γ

(

r

).

(29)

u_θ

(

r

) =

^Mvrexp

[ (

1

−

^r2

)/

2

] ,

ur

(

r

) =

⁰

,

(30)

wherethedistancefromthevortexcorerisnon-dimensionalizedbythevortexradiusR.

Theaboveflowfieldofvortexisaddedtotheupstreamoftheshockwaveatinitialtime.Thefollowingflowparameters areusedinthetest:

Ms

=

¹

.

2

,

Mv

=

⁰

.

25

,

Re

=

⁸⁰⁰

,

R

=

¹

, γ ₌

1

.

4 (31)

TheReynoldsnumberisdefinedby Re

= ρ

RaRR

/ μ

withaR beingthesoundspeedoftheupstreamoftheshock.Acompu- tationaldomain[-20R,8R]

×

^{[-12R,12R]isconsideredinthe}simulation.Initiallythesinglevortexislocatedatx

=

^{2R and} y

=

^{0,andtheplanarshockwave isspecifiedatx}

=

^{0 by imposingdensity,velocityandpressurevariables}corresponding totheaboveleftandrightstatesofthenormalshock.

Inthissimulation,thegridresolutionis1120

×

⁹⁶⁰

×

^{1,thetimestepissetto}

δ

t

=

⁰

.

00833 andtheHRRparameter

σ

is takenequalto0.7.ThepressurefieldsatdifferenttimeareshowninFig.14.Here,thesoundpressureis

p

= (

p

−

^pL

)/

pL. Thesoundpressurecontoursarefrom-0.48to0.16with60levels.Itcanbefoundthattheseresultsareverysimilarwith thesoundpressurefieldsgiveninRef. [88].

Fig. 14.InstantaneoussoundpressurecontoursobtainedbyHRR-LBmodelon1120×⁹⁶⁰×^{1 grids.Thesoundpressure}p=(p−^pL)/pL contoursare from-0.48to0.16with60levels.

TheradialdistributionsofthesoundpressureareplottedinFig.15.Inthefigure,risthedistancefromthecenterofthe vortexwithafixedangle

θ = −

^{45◦.Fig.16isthe}circumferentialdistributionsofthesoundpressureatt

=

^{6T.Inthosetwo} figures,theresultsrepresentedbysymbolsarefromRef. [88].Thosereferenceresultswereobtainedusingafinitedifference methodwithasixth-order-accuratecompactschemeinspaceandthefourth-orderRunge-Kuttaschemefortime-integration on1044

×

1170 non-uniformgrids.ItcanbeobservedthattheresultsobtainedbyLBareinverygoodagreementwiththe referenceresults.

6.6. Compressiblelaminarflowoverflatplate

Thecompressiblelaminarflowoverflatplatehasbeeninvestigatednumericallyandtheoreticallyovertheyears [89,90].

Here, we usethis problemtoassess thecapability ofthe presentmethodon handlingthe viscous effects.Consideringa compressibleflowwithupstreamMach numberMa_∞

=

⁰

.

5

,

1

.

0

,

1

.

5 over aplateoflength L,thecomputational domain is x

∈ [−

⁰

.

25

,

1

] ×

^y

∈ [

⁰

,

h

] ×

^z

∈ [

⁰

,

x

]

^{. Here,}

x denotes the mesh size which is

x

=

¹

.

25

×

^{10−3. h is setto 1}

.

25

Fig. 15.Radialdistributionsofthesoundpressurep.risthedistancefromthecenterofthevortexwithafixedangleθ= −45^◦.Thesolidlinesrepresent thepresentLBsolutionandthesymbolsdenotethereferencesolutioninRef. [88].

Fig. 16.Circumferentialdistributionsofthesoundpressurepatt=^{6T.ThesolidlinesrepresentthepresentLBsolutionandthesymbolsdenotethe} referencesolutioninRef. [88].

forMachnumber1

.

5 and1

.

0.ForMach0

.

5,onetakesh

=

⁰

.

25.The HRRparameterissetto

σ ₌

0

.

9 andthetimestep

δ

t

=

⁴

.

166

×

^{10−4.Theboundaryconditionsaresetasfollows:}

•

^{SymmetryBConat y}

=

⁰

, −

⁰

.

25

≤

^x

<

0.

•

^{No-slip,adiabaticBCony}

=

⁰

,

0

≤

^x

≤

^1.

•

^InflowBCatx

=

^{0 with}

ρ

in

=

^1,and

(

u

,

v

)

in

= (

u_∞

,

0

)

.

•

^{Subsonicoutflowaty}

=

^handx

=

^{1 withpressure p}out

=

¹

/ γ

forMa_∞

=

⁰

.

5.

•

^{Supersonicoutflowat y}

=

^handx

=

^{1 forMa}∞

=

¹

.

0

,

1

.

5.

Inorder toobtain asteadylaminarsolution,the Reynoldsnumberistaken as10⁴.Avariable temperature-dependent viscositygivenby

ρμ = ρ

_∞

μ

_∞isusedinthissimulation.ThePrandtlnumberistakenequalto1.0.TheseviscosityandP r aresetaccordingtothereferencesimilaritysolutionsofcompressibleboundarylayer [91].

Fig.17displaysthecontoursofMachnumberobtainedbythepresentLBmodelatMa

=

¹

.

5.Fig.18showstheprofiles of horizontal velocity, temperature, densityand skin friction obtained by the presentLB model. In the figure,

η

is the dimensionlesscoordinatewithIllingworthtransformwhichisdefinedastheone inreference [40]. Thereferencesolution oftemperatureanddensityareobtainedusingthewayin[40]. Cw istheChapman-Rubesinparameter whichisequalto 1undertheconditionof

ρμ ₌ ρ

_∞

μ

_∞.Fromthefigure,itcanbe foundthatthesimulatedvelocityprofilesandreference solutionareinaverygoodagreementatMa_∞

=

⁰

.

5

,

1

.

0

,

1

.

5.Thiskindofgoodagreementcanalsobefoundbetweenthe skinfrictionpredictedbytheLBmethodandtheonesoftheBlasius solution.Forthedensityandtemperature,theresults

Fig. 17.Mach number field of laminar boundary layer problem atMa_∞=¹.5.

Fig. 18.Profileofhorizontalvelocity,temperature,densityandskinfrictionobtainedbytheLBmethodonlaminarflowoverflatplateatMa∞=0.5,1.0 and1.5.ThesolidlinesrepresentBlasiussolutionatthecorrespondingManumber.