Adaptive filtering for the lattice Boltzmann method

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Adaptive filtering for the lattice Boltzmann method

Simon Marie, Xavier Gloerfelt

To cite this version:

Adaptive

filtering

for

the

lattice

Boltzmann

method

Simon Marié

a

,

c

,

∗

,

Xavier Gloerfelt

b

,

c

a_{ConservatoireNationaldesArtsetMétiers,France} b_{EcoleNationaleSupérieuredesArtsetMétiers,France} c_{LaboratoireDynFluid,France}

a

b

s

t

r

a

c

t

Keywords:

LatticeBoltzmannmethod Adaptivefiltering Numericalstability Taylor–Greenvortex

Inthisstudy, anew selectivefiltering techniqueis proposed for the LatticeBoltzmann Method. This technique is based on an adaptive implementation of the selective filter coefficient

σ

.Theproposed model makes the lattercoefficientdependent onthe shear stressinordertorestricttheuseofthespatialfilteringtechniqueinshearedstressregion wherenumericalinstabilitiesmayoccur.Differentparametersaretestedon2Dtest-cases sensitivetonumericalstabilityandona3DdecayingTaylor–Green vortex.Theresultsare comparedtotheclassicalstaticfilteringtechniqueandtotheuseofastandard subgrid-scalemodelandgive significantimprovementsinparticularforlow-orderfilterconsistent withtheLBMstencil.

1. Introduction

TheLatticeBoltzmannMethod[1,2](LBM)isnowadaysrecognizedasafastandreliablemethodtosimulatethedynamics ofweaklycompressibleflows.Somestudies[3–6]haveshownthecapabilitiesofLBMtoperformcomplexandmulti-physical simulationsfromturbulentflowstoaeroacousticapplicationsthankstothelowdissipationerrorintroducedbythemethod. Asacounterpart,LBMsuffersfromnumericalinstabilitieswhenReynoldsnumberbecomeshigh.

The originsofLBMinstabilitieshavebeenactivelystudiedandremainsanopen subject[7–10].The mainconsequence oftheinherentLBMinstabilityistocreatedivergingoscillationsmainlycharacterizedbyhighfrequencieswhichpropagate in the whole domain. These numerical instability waves are often generated by unadapted initial conditions,geometric singularitiesorinregionwherehighgradientsareobserved.Inindustrialapplications,severalofthesenumericalconstraints arepresent,thuscomputationsoftenbecomedramaticallyunstable.

Numerousstudieshaveproposedstabilizationtechniquesbasedondifferentapproachessuchasmultiplerelaxationtimes

[11],regularizationtechniques[12],energyconserving[13],entropicmodelsorpositivityenforcing[14].Thevastmajority ofthesemodelsconsist inatheoreticalmodificationoftheLBMschemeandgive relevantinformation abouttheunstable nature of LBM. As a counterpart, lots of stabilizing strategies have a global effect on the viscosity thus modifying the effectiveReynoldsnumberorincreasing theglobaldissipationofthemethod.Thisdissipationinherenttotheconstruction ofmorestableschemescanimpact theevaluationofpressurefluctuationswhoseaccuracyiscrucial(e.g.foraeroacoustic simulations). Inthatcasealocalstrategy wouldbe preferredinordertodistinguishspatial zoneswherea stabilizationis requiredtothoseinwhichthestandardLBMschemecanbeapplied.

*

Correspondingauthor.

Ricotetal.[15]proposed touseselectivespatialfilters[16]tostabilizethemethodbyincreasingthedissipation inthe highwavenumberrangewheretheLBMinstabilitiesoccur,keepingalowdissipationatsmallwavenumbers.Thisapproach can be applied to aeroacoustic simulations by maintaining an acceptable level ofdissipation error atlow wavenumbers. However, this method basically applies to the whole domain andbecomes unavailing outside of sheared regions where numericalinstabilitieshavelesschancestodevelop.Furthermore,theuseofhigh-order filtersincreasesthestencilofLBM whichislowbynatureandleadstoalossoflocalityofthemethodwhichispenalizingformassivelyparallelcomputations. Finally,fromadynamicalpointofview,theselectivespatialfiltershaveneverbeentestedontransitionsituationwherethe timeevolutionpredictionisofmajorimportancefortheaccuracyoftheresults.Therefore,theneedforalocalandadaptive stabilization procedure is relevant and should be carried out, in particular in the framework of the Lattice Boltzmann Method.

Theideaofthepresentstudyistoproposeanimprovedfilteringstrategyrestrictedtohighlyshearedregions[9]keeping weaklyshearedonesfreeofartificialdissipation.Thechoiceoftheshearstress asasegregationparameterishighly moti-vatedbythelargeeddysimulationframeworkwheretheshearstresssensitivityisofcrucialimportanceintheconstruction ofsubgrid-scalemodels[17,18].Moreover,thisstudycanbeincludedintheframeworkofotherlocalapproacheswhere ad-ditionalnumericaltreatmentisdoneinrestrictedzonesofinterest[19–21].Thereforethepresentstudyaimsatintroducing ashear-stressselectivityintheapplicationofspatialfiltering.Theproposedstrategyisdevelopedintheframeworkofthe LatticeBoltzmannMethodandappliedtounsteadytest-caseshighlysensitivetonumericaldissipation.

AfterabriefpresentationoftheLatticeBoltzmannmodelinsection2,thenewfilteringstrategyisdescribedinsection3

andvalidatedinsections4,5and6on2Dand3Dtest-caseswithsomecomparisonstothetraditionalfilteringtechniques. Finally,section7isdedicatedtocomputationalcostissues.

2. LatticeBoltzmannmethod

TheLatticeBoltzmannmethod[1]isdescribedbythefollowingalgorithm:

gα

(

x

+

cα



t

,

t

+ 

t

)

=

gα

(

x

,

t

)

−

1

τ

g

[

gα

(

x

,

t

)

−

geqα

(

x

,

t

)

]

(1)

where gα aredistributionfunctionscomputedonaregularvelocitylatticecα ,collidingandrelaxingtoalocalequilibrium

gα witheq arelaxingparameter

τ

g

=

ν

˜

c2₀

+

1

2 where

ν

andc

˜

0 arethenondimensional viscosityandspeedofsound respec-tively. Inthis study, we use the D2Q 9 and D3Q 19 models for 2D and 3D simulations. The classical parameters of the modelaredefinedasfollows:

geq_α

(

x

,

t

)

=

ρω

α

(

1

+

u

.

cα

˜

c2 0

+

(

u

.

cα

)

2 2c

˜

4 0

−

|

u

|

2 2c

˜

2 0

)

(2)

ω

α

=

⎧

⎪

⎨

⎪

⎩

D2Q 9 4 9

,

1 9

,

1 36

,

α

=

0

,

α

=

1

..

4

,

α

=

5

..

8 D3Q 19 1 3

,

1 18

,

1 36

,

α

=

0

,

α

=

1

..

6

,

α

=

7

..

18

˜

c2₀

=

1 3 (3)



t

=

c

˜

0



x c0 (4) Themacroscopicquantities

ρ

andu canbecomputedfromthedistributionfunctionswiththediscretemoments:

ρ

=



α fα (5)

ρ

u

=



α cαfα (6)

Thepressureisrecoveredbytherelation:

p

= ˜

c2₀

ρ

(7)

Based on theseparameters, itcan be shown [2]that LBM simulates thecompressible Navier–Stokesequations inthe limitoflowMachnumberswithasecond-orderaccuracyinspaceandtime.

3. Adaptiveselectivespatialfilters

Table 1

Coefficientsoftheselectivefilters:dn=d−n.

d0 d1 d2 d3 d4

3-point 1/2 −1/4

5-point 6/16 −4/16 1/16

9-point (optimized[16]) 0.243527493120 −0.204788880640 0.120007591680 −0.045211119360 0.008228661760

Fig. 1. Shear stress sensitivity.



Q

(

x

)

 =

Q

(

x

)

−

σ

D



j=1 N



n=−N dnQ

(

x

+

n



xj

)

(8) where

σ

isacoefficientbetween0 and1,oftentakento0

.

1,dn arecoefficientsdependingonthefilterorderandD isthe numberofspatialdimensions.Inthisstudy,classical3-point,5-pointstencilandoptimized9-pointstencilfiltersare used

[16].ThecoefficientsofthefiltersaregiveninTable 1.

Different filteredquantitycan bechosen accordingtotheLBM schemewiththefollowing possibilitieswithincreasing computationalcost:

1. Filteringmoments:

ρ

,u (D

+

1 tables); 2. Filteringdistributionfunctions: gα (Nv tables); 3. Filteringcollisionoperator:

−

_τ1

g

(

gα

−

g eq

α

)

(Nv tables),

where D isthenumberofphysicaldimensionsandNv isthenumberofdiscretevelocities.Inordertolimitcomputational cost,thefirstsolutionwillbepreferredinthisstudyandtheinfluenceofthefilteredquantitywillbediscussedinsection6. Inthewavenumberspace, Ricotetal.[15] haveshownbyalinearstability analysisthattheexplicitfilteringintroduces anadditionaldissipationlinkedtothecoefficient:

κ

(

k

)

=

1

−

σ

F

(

k

)

(9)

where

F(

k

)

isthetransferfunctionoftheexplicitfilterasafunctionofthewavenumbervectork.Thentheglobalefficiency ofsuchafilterisledbyboth

F(

k

)

and

σ

.Theideaofthisstudyistomakethecoefficient

σ

ofrelation(8)dependenton theshearstress.Forinstance,letusconsider

σ

d

(

x

)

tobeoftheform:

σ

d

(

x

)

=

σ

0



1

−

e−(|S(x)|/S0)2



2

(10) where

|

S

|

=

2Si jSi j and

σ

0 isthestaticfilteramplitude. S0isareferenceshearstressamountdefiningasensitivityfrom

whichthe dynamicalfilterstarts tobeactive. Thequantity

|

S

|

isevaluatedintheLatticeBoltzmannframework,fromthe secondordermoment:

τ

i j

=

2

ρν

Si j

= −



α c_α,icα,j

(

gα

−

geqα

)

(11) whichgives:

|

S

| =

Qf 2

ρν

(12) with Qf

=

2Pi jPi j and Pi j

=

α cα,icα,j

(

gα

−

geqα

)

.Relation (12) isoften usedfor theimplementation ofsubgrid-scale

modelsintheLatticeBoltzmannMethod[22,23]anddoesnotrequireanyderivativecomputations.

Thenceforth, a crucial point relies in the estimation of the sensitivity shear stress S0. When the shear stress is low

(

|

S

|

<

S0), the filter has no effect (

σ

d

∼

0) and when the shear stress rises to higher values (

|

S

|

>

S0), the filter acts

normally (

σ

d

∼

σ

0). Then S0 can be seen asa shear sensitivityparameter andis of major importance in thedynamical

filtering efficiency.If S0 ischosen smaller thanthe minimumamount ofshearstress (Smin), the filtercoefficient

σ

d will be close to

σ

0 in the whole domain and the adaptive filterwill behave like a classical static filter. Conversely, if S0 is

chosenhigherthanthemaximumamountofshearstress(Smax),

σ

d willbeverylowinthewholedomainandtheadaptive filter willhavealmost noeffect.Consequently, thepresentmethodology becomes efficientforintermediate valuesof S0:

Table 2

Referencenamesusedforthesimulations.

Simulation name Fs F0

ad Fad1 Fad2

Type Static Adaptive Adaptive Adaptive

S0 0 ξmax(|S|) ξSmax ξSmax

Smax × Computed Imposed by(13) Imposed by(16)

In thisstudy S0 is evaluated in termsof the maximum amountof shear stress: S0

= ξ

Smax where

ξ

is a selectivity parameterclosetounityandSmax canbeevaluatedbytwodifferentways:

1. Computedvalue:evaluationofmax

(

|

S

|)

ateachtimestepbasedoninstantaneousormeanfield.

2. Imposedvalue:Imposedconstantvalueatinitializationbasedonphysical,numericalorempiricalcriteria.

Thefirst typeofestimationhasa straightforwardimplementationandimpliesthat themaximumvalue of

σ

d remains closetoaconstantintime:

σ

d

=

σ

0



1

−

e−(1/ξ2)



2 _{andcouldbeusediftheaimistocontroltheoveralldissipationinduced}

bythefilter.

Thesecondtypeofestimationimposesafixedamountofshearstressfromwhichthefilterwillbeactive.Inthat case, thetimeevolutionoftheeffectivestresswouldleadtoatimeevolutionofthefiltercoefficient

σ

d.A firstbasiccriterionfor theestimationofSmaxcanbebasedontheratiobetweenavelocityscaleU andalengthscale

δ

:

Smax

=

U

δ

(13)

Equation(13)isevaluatedatinitializationstepbasedontheprescribedsimulationparameters.

Anotherwaytoevaluate Smax canbebasedonanumericalcriterionassessingthepositivityofthedistributionfunction

gα asanumericalstabilitycriterion [24].Indeed,assuming thatthequantities gα mustbepositive,anupperlimitcanbe foundforrelation(12)bysubstituting gα withzeroandconsideringthesecond-ordermoment:



α cα,icα,jgeqα

=

ρ

uiuj (14) whichyields:

|

S

| =

Qf 2

ρν

=



2

_αcα,icα,jgαeq

αcα,icα,jgeqα 2

ρν

≤

√

2uiuj 2

ν

(15)

Then Smax ensuring the stability condition can be written from (15)by only keeping theleading termof thelinearized

uiuj: Smax

=

√

2U2 0 2

ν

∼

ReδU0

δ

(16)

where

δ

isthecharacteristiclength scale.Itshould benoticed that,byconstruction, relation(16)isgiveninlattice units andimplicitlycontainsthemeshsizeinformation.Indeed,indimensionalunits,relation(16)wouldbemultipliedby1

/

t

whichdependson



x (seerelation (4)). Thus forcoarse grids,relation (16)willgive smaller valueof Smax thanforfine grids,whichisconsistentwithstabilityissues.

Inthefollowing,theproposedadaptivefilteringprocedureisstudiedonillustrativetestcaseswithdifferentevaluations ofSmaxfollowingthenomenclaturepresentedinTable 2.

First,aconvected2Dvortexisusedtocharacterizetheinfluenceofthepresentfilteringstrategyonthelocaldissipation, thentheflowpastasquarecylinderisinvestigatedtodemonstratethestabilizingcapabilitiesoftheadaptivefilteringand finally,thesimulationofa3D decayingTaylor–Greenvortexisperformedtodemonstratetheeffectivenessofthepresent filteringonafully3Dturbulentflow.

4. Applicationtoa2Dconvectedvortex

4.1. Testcaseimplementation

Fig. 2. (a)Evolutionofthemaximalamplitudeofthevortexkineticenergyateachcrossingofthedomain.(b)Timetraceofthenormalizedkineticenergy atthecenter pointlocationafterthelastcrossingofthedomain.(—):Withoutfiltering,( ):Fs,( ):F0

ad,( ):F 1 ad,( ):F

2

ad.Thinlinestothicklines areforξ=0.5,ξ=1.0 andξ=1.5.

Inthepurposeofcharacterizingtheamountofnumericaldissipationinducedby thepresentmethodology,the compu-tations areperformedforan inviscid vortexbysettingtherelaxationtime to0

.

5 (e.g.

ν

=

0). TheMach numberis setto

M_∞

=

0

.

1 andtheinitialfieldisdefinedasfollows:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

ρ

=

1 u

=

U_∞

+

ε

U_∞

(

y

−

y0

)

exp



−

ln

(

2

)

b2p



(

x

−

x0

)

2

+ (

y

−

y0

)

2



v

= −

ε

U_∞

(

x

−

x0

)

exp



−

ln

(

2

)

b2p



(

x

−

x0

)

2

+ (

y

−

y0

)

2



(17)

with

(

x0

,

y0

)

= (

nx

/

2

,

ny

/

2

)

,

ε

=

10−3 andbp

=

20. Thegrid

(

nx

,

ny

)

is setto 256

×

128 points withperiodic boundary conditionsandthenumberoftime-stepsischosensoastoachieveatleasttenvortexcrossingsofthedomain.

4.2. Resultsanddiscussions

For this test case, only the low-order 3-point filter is considered and the static filter coefficient is set to

σ

0

=

0

.

1.

Different estimationsof S0 aretestedsummarized inTable 2.The F_ad0 estimationisbasedon thecomputation of Smax

=

max

(

|

S

|)

ateachtimestep, F_ad1 imposesavalueof Smax basedon(13)withU

=

ε

U0 and

δ

=

bp andF_ad2 imposes Smaxfrom

(16).

Fig. 2showsthetimeevolutionofthekineticenergycomputedatthecenter pointofthedomainforvariousvaluesof

ξ

. Theclassicalfilteringprocedurewithaconstantcoefficient

σ

=

0

.

1 isalsoaddedforcomparison.

Whenthefilteractswiththesameamountinthewholedomain(classicalfilter)theintroduceddissipationhasdamped themajor partoftheinitialkinetic energyconfirming thattheclassicallow-orderfilteristoodissipativewhenappliedin thewholedomain.Fig. 3representstheisocontoursof

σ

d atagiventimestepfordifferentestimationsof S0 anddifferent

values of

ξ

.As expected, non-zero valuesof

σ

d are concentrated nearthe vortexboundary andreach nullvaluesin the vortex center.The results indicate less dissipation fromthe F2

ad estimation based on relation (16) becausethe imposed value ofSmaxisneverreachedand

σ

d remainssmallallalongthevortexconvectionforallconsideredvaluesof

ξ

.Indeed, theconvectedvortexisnumericallystableandinstabilitywavesarenotobservedforthistest casewhichimpliesthat the filterisalmostneveractive.

Forlowvaluesof

ξ

theshearselectivityoftheadaptivefilterisweakandthebehavior isclosetotheoneobtainedwith theclassicalfilters.The F_ad1 estimationinduceshighvaluesof

σ

d atinitializationduetothesimilaritybetweentheimposed

Fig. 3. ( )Isocontoursofdensityand( )isocontoursofσd=0.1σ0after13200 timesteps.(a–c)Fad1 withξ=0.5,1.0 and1.5.(d–f)F 0

adwithξ=0.5, 1.0 and1.5.

Fig. 4. Timetraceofthemaximumvalueofthecoefficientσd.(—):Withoutfiltering,( ):Fs,( ):Fad0,( ):F 1 ad,( ):F

2

ad.Thinlinestothicklinesare forξ=0.5,ξ=1.0 andξ=1.5.

Theseobservations show that thenumericaldissipation introduced by thefilter canbe controlled by theshear stress selectivity.The F_ad0 andF_ad1 estimationsleadtodissipatedresultsbecauseoftheunder-estimationof Smax whereasthe F_ad2 estimationshowsagoodabilitytoswitchoffthefilterifinstabilitywavesarenotdetected.Theninthefollowing,onlythe

F2

adestimationwillberetainedandwillnowbestudiedinthecontextofnumericalstabilityissues. 5. Applicationtotheflowpastasquarecylinder

5.1. Testcaseimplementation

Inthissectionweareconsideringtheflowpasta2Dsquarecylinderwhichisknowntoexhibitssomestrongnumerical instabilitiesfor Reynoldsnumbers largerthan 500.The computationsare performedona uniform500

×

200 pointsgrid witha 10

×

10 points solid square located at

(

xs

,

ys

)

= (

100

,

100

)

.Wall boundary conditionsare implemented withthe classicalbounce-backmethodtoensureanullvelocityatthewall,andperiodicboundaryconditionsareusedatthedomain boundarywitha spongezone atthe outletinordertodamp outgoingstructures. Thesponge zonehasa Gaussianshape andisdefinedby:

ν

ν

0

=

α

exp



−

ln

(

2

)

m2

(

x

−

p

)

2



(18) Forthisstudy,theparameters havebeenfixedto,

α

=

200,m

=

50 and p

=

nx

−

75 meaningthattheviscositystartto increasefrom

ν

0 to200

ν

0 inthelast 100 gridpoints ofthedomain.TheReynoldsbasedonthesquare sideis ReD

=

800 andtheMachnumberissettoM_∞

=

0

.

25.Fortheseparameters,theclassicalLBM-BGKmethodbecomes highlyunstable andleadstocollapsingcomputations.Consequentlysomeartificialdissipationmustbeaddedtodampinstabilitywaves.

Here, thepresentfiltering strategyis testedandcompared tothe classical staticfilterforthe 3-point stencilfilter.In thisstudy,thequantitiesatthewallarenot filteredbutthe 3-pointfilterallows theclose-neighboringpoints ofthewall quantitiestobefilterednormally.However,asdiscussedin[21],theuseofexplicitfilteringnearthewallisconditionedby arelativelyhighwall-resolution.Forthepresenttest-case,thewallresolutionissetintentionallycoarseinordertoexhibit numericalinstability.Thenthepresentadaptivetechniqueisexpectedtomodifythenear-wallquantitieswhereahighshear stressisdetectedyieldingahighvalueforthefiltercoefficient.

Table 3

ComparisonoftheaveragedragcoefficientCmean

d andrmslift co-efficientCrms

l fordifferentvaluesof xi. Cmean d C rms l ξ=0.5 0.825 0.08 ξ=1.0 0.831 0.09 ξ=1.5 0.837 0.10

Fig. 5. Isocontoursofvorticityfrom−0.02 to0.02 inlatticeunitattime=2000 timesteps.Dashedcontoursdenotesnegativevalues.(a)3-pointclassic filter,(b)3-pointadaptivefilterwithξ=0.5,(c)ξ=1.0 (d)ξ=1.5.

5.2. Resultsanddiscussions

The isocontoursofvorticitydisplayed inFig. 5showthat theclassicallow-orderfilteristoodissipativeandleadstoa laminarflowwithoutvortexshedding.Thisconfirmsthatthesekindoflow-orderfiltershouldnotbeappliedforunsteady simulations. Incontrast,thepresentadaptivefilteringtechniqueexhibitsunsteadyflowwithidentifiedvortexshedding in thecylinderwake.Allthevalueof

ξ

givesimilarwakeswithaslightdelayintheestablishmentofthevortexsheddingfor the lowestvalue of

ξ

dueto atoohighdissipationcloseto thewall.Theseobservationsare confirmedbylooking atthe drag andliftcoefficientofTable 3whichareslightlyunderestimatedas

ξ

isdecreasedduetotheuseofnear-wallfiltering withlowresolution.

Byplottingthetime evolutionof

σ

d ata wallpoint anda wakepointinFig. 6,we cansee thatthehighestvaluesof

σ

d arereachednearthewallwheretheshearstressismaximum.Forthelow valueof

ξ

,

σ

d isfluctuatingaround 60% of

σ

0 nearthewall whereas forthe highvalue of

ξ

,

σ

d isfluctuating around 1% of

σ

0.From the signal ofthewake point,

it could be inferred that

σ

d remains smallerthan 1% of

σ

0 forboth value of

ξ

exceptfor

ξ

=

1

.

5 after 2000 timesteps.

For thisvalue of

ξ

,the simulation isclose to the stability limit andoscillations are visible forhighvalues of theshear stress.Thisphenomenonisobservedinthevicinityofcontra-rotativevorticeswhicharegettingveryclosetoeachother.An instabilitywaveiscreatedbutthefiltercoefficientisnothighenoughtodampthisinstability.However,whentheinstability isdeveloping,highershearstressisdetectedandthefiltercoefficientisincreasedto80% of

σ

0 infewtimesteps.Thenthe

instabilitywaveisdampedandthefiltercoefficientdecreasestoverylowvalues.

Theseobservationsconfirmtheadaptivenatureofthepresentedfilteringstrategyandshowthatlow-orderfilterscanbe applied tounsteadyflows iftheyare restrictedtolocalizedzonesbasedona shearcriterion.Moreover, theestimationof

Smax usedforthistestcasegivesrelevantresultsintermsofstabilitycontrol anddemonstratestheabilityoftheadaptive filterstobeanefficientstabilizationprocedurefortheLatticeBoltzmannmethod.

6. Applicationtoa3DTaylor–Greenvortex

6.1. Testcaseimplementation

Inordertostudytheeffectofthepresentfilteringtechniqueona3Dturbulentconfiguration,thedecayingTaylor–Green vortex (TGV) is used.It isa fundamental test caseused asprototypefor vortexstretching andproductionof small-scale eddiesandthereforeallowsthestudyofthedynamicsoftransitiontoturbulence.Thistest-casehasbeenwidelyused to studythedissipationerrorsofnumericalschemes[25].Inparticular,Aubardetal.[26]haverecentlyusedthistest-caseto confronttheselectivefilteringtechniquestotheuseofsubgrid-scalemodels.

Fig. 6. Timetraceoffiltercoefficientσdat(a)(xc,yc)= (100,105)and(b)(xc,yc)= (230,85).( :Fs),( Fad2:ξ=0.5),(- -:ξ=1.0)and( :ξ=1.5).

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

p

=

p_∞

+

ρ

∞U 2 ∞

16

[

cos

(

2z

)

+

2

][

cos

(

2x

)

+

cos

(

2 y

)

]

u

=

U_∞sin

(

x

)

cos

(

y

)

cos

(

z

)

v

= −

U_∞cos

(

x

)

sin

(

y

)

cos

(

z

)

w

=

0

(19)

Inordertoreducenumericaloscillationsatthebeginningofthesimulation,thedistributionfunctionsgα areinitialized to their equilibriumstate withan additionalnon-equilibrium partbased on theChapman–Enskog micro-scale expansion

[27].

Forthis study,the simulations are performedon a 2

π

-periodic cubic domain witha Reynolds numberof Re

=

1600 basedonaphysicalcharacteristiclengthofLre f

=

1m:

Re

=

U∞Lre f

ν

_∞

=

˜

U_∞Lre f



x

ν

˜

_∞ (20)

where

˜.

quantitiesdenote latticeunitquantities.

The Machnumberis takento M_∞

=

0

.

085 and

ρ

∞

=

1 fixing theother parameters to U

˜

_∞

=

0

.

049 and p

˜

_∞

=

1

/

3 in latticeunit.TherelaxationparameterissetwiththeReynoldsnumberto

τ

g

=

˜

U_∞Lre f



xc

˜

2₀Re

+

1 2.

ThefiltercoefficientisherecomputedwithF_ad2 estimationofSmax basedonrelation(16)anddifferentvaluesof

ξ

are tested.Differentfiltertypesarealsocomparedtoanalyzetheinfluenceofthefilteringorderontheflowdynamics.Forthe validationofthistestcase,thespectraldatafromBrachetetal.[28]areusedandcomparedtoourreferencesimulationon a2563_grid.

6.2. Resultsanddiscussion

A reference simulation is performedon a 2563 _{grid without any filtering technique. Fig. 7 displays the evolution of}

Q -criterioninthedomain.TheclassicalbehavioroftheTaylor–Greenvortexisobserved,theinitialfieldgivesrisetolarge vorticeswhicharethenstretchedandleadtotheproductionofsmall-scaleeddiesanddecayingturbulence.

Inthefollowing,aseriesoftestsareperformedtocharacterizethepresentfilteringstrategy. First,theinfluenceofthe gridresolution onthenon-filteredscheme ispresented,thentheinfluence ofparameters

σ

0 andS0 in equation(10)are

Fig. 7. Isosurface of the Q-criterion colored by kinetic energy at time t=0, t=4, t=10, t=16 for Re=1600 on a 2563_grid.

Fig. 8. Non-dimensional time evolution of the dissipation rate. (•spectral data), (... 643_{), (- - 96}3_{), ( 128}3_{) and ( 256}3_).

6.2.1. Gridsensitivity

Thelattice Boltzmannmethodisasecond-orderaccuracyschemeinspaceandtime.Intheliterature,fewstudieshave beenpublishedonthevalidationofthe3DTaylor–GreenvortexwithLBMsimulations[29].Itisthusinterestingtoseethe capabilitiesofthestandardL B G K schemetosimulatethedynamicsofa3Ddecayingvortex.Forthatpurpose,theevolution ofthekinetic energydissipationrate



= −∂

tk is scrutinizedforvariousresolutionsfrom643 to2563 andcomparedwith thespectralsimulationofBrachetetal.[28]inFig. 8.Inthefollowing,thetimeisnormalizedwithrespecttothequantity

Lre f

/

U∞.Thedissipationrateiscomputedwithasecond-ordercenteredfinite-differenceapproximation.

TheTGVdynamicalevolutionischaracterizedbythreemainstepsvisibleinthetimetraceof



.First,theinitiallaminar state is transitioningto turbulenceuntil thestretched vortextubes break downinto smallscales aroundt

=

5.Then the dissipation rate is rising to a sharp peak near t

=

9 corresponding to the fully turbulent state which is then decaying similarlytoanisotropicandhomogeneousturbulence.

The resultsoftheclassicalLBM simulationswithout anyfilteringtechnique are displayedinFig. 8.Thereference sim-ulation on a 2563 _{grid isseen to be invery goodagreement with the spectral results. The 128}3 _{grid gives satisfactory}

too early. The 643 and 963 simulations give rise to numericalinstability at differenttime. The 963 grid simulation ex-hibitsa relatively goodtransitiontoturbulencebutcollapses justbefore thepeak ofdissipation isreachedaroundt

=

8. Finally, the 643 _{grid simulation collapses earlier around t}

₌

_{5, when the stretched vortex tubes break down into small}

scales.

Thus,theLBMschemeshowsagoodabilitytosimulatetheTaylor–Greenvortexdynamicswhenusingfinegridsbutis limitedbyits inherentinstabilityforcoarsergrids andhighReynoldsnumbers.Therefore theuseofastabilizingstrategy becomesacrucialpointtoinvestigateturbulentsimulationswiththeLatticeBoltzmannMethod.

6.2.2. Influenceof

σ

0and

ξ

Asdescribedinsection3,thedynamicalfilteringstrategyissensitivetotwomainparameterswhichare

σ

0 and

ξ

.The

firstonedeterminesthefilteringamplitude whentheshearstressishighandcouldbe seenastheefficiencyofthefilter. Thelatteroneplaystheroleofshearstressselectivityandisessentialforcontrollingthedissipationamountofthefiltering techniqueasdescribedintheprevioussections.Thesensitivitytotheseparametersisinvestigatedona963 _{gridforvarious}

valuesof

σ

0,

ξ

andthedifferentselectivefilterspresentedinTable 1.

Afirstsensitivityanalysisisperformedonthe

σ

0parameterwithafixedvalueof

ξ

=

1.ResultsarereportedinFig. 9-left

and exhibit relatively similar behavior. A high value of

σ

0 induces more damped results, in particular after the vortex

breakdownneart

=

5 whenthe filterorderislow.Ontheotherhand,betterresultsare obtainedforlow

σ

0 values.This

improvementisparticularlyvisibleforthe3-pointfilterwhichcanfairlyreproducethevortexbreakdownaroundt

=

5 when

σ

0 is setto 0

.

01.Lower valuesof

σ

0 have beentestedbutleadto unstablesimulations. Thissuggest thatthe dynamical

filteringstrategy should be appliedforvaluesof

σ

0 aslow aspossible inthelimit ofstability. BylookingatFig. 9-right

wherethetime evolutionof

σ

d isrepresented,onecanseethatthevalueof

σ

0 hasadirectimpactonthetimeevolution

of

σ

d. Indeed,when

σ

0 changes,the filteringamount ismodified andso isthe shearstress whichmodifiesthe local

σ

d value. Moreover,forhigher valuesof

σ

0 the maximumvalue of

σ

d is neverreachedsuggestingthat thefilteringamount induces lowershearstressandinturnreduces

σ

d.Ontheotherhand,when

σ

0 islow,thefilteringamountallowslarger

shearstressvaluesthenincreasing

σ

d closetoitsmaximumvaluefixed by

σ

0.However,theseconsiderationsdependalso

ontheshearstressselectivitywhichisdrivenbythevalueof S0.

Thesecond sensitivityanalysisisperformedon theparameter

ξ

whichcontrols S0 andfora fixedvalue of

σ

0

=

0

.

05.

ResultsarereportedinFig. 10-leftwhichshowsthattheselected

ξ

valueshaveanimportantimpactontheresults.A lower sensitivityinduces an earlierfilteringactivity whichdampsthe initiallaminarstate andlead toa wrong dissipation rate evolution.Indeed,resultsofFig. 10-rightindicatethat

σ

dstartstoincreasefromanon-zerovalueforlowsensitivity param-eters.Incontrast,whenthesensitivityishigh,thefilterdoesnotactinlowshearedregionandisactivatedonlywhenlarge vortices havebrokenup andsmall-scales structuresare developingtoturbulence.Moreoveritcould beseenfromFig. 10

thattheeffectofthefilterorderisreducedwhenincreasingthesensitivity.Indeed,3-pointfilterresultsareclosetothose ofthe9-pointfilterwhensensitivityishigh. Thisisanimportantoutcomeintermsofcomputationalcost suggestingthat thefilterordercouldbe reducedwhenincreasingsensitivity.Theseconsiderations canberecastintheturbulence frame-workwherelarge-scalestructuresareknowntobemoreenergetic.Theglobaldissipationofthepresentadaptivefiltering canbedescribedbyequation(9)withacoefficient

σ

(

k

)

dependentofthewavenumber.Indeed,theshearstressamountis expectedtobe higherforlarge andenergeticstructuresinaturbulent flowsuch asTaylor–Greenvortex.Thenthe coeffi-cient

σ

(

k

)

actsasaselectivityenhancementthroughtheterm

σ

(

k

)F(

k

)

.Forthepresentstudy,thefrequencyrepartition of

σ

(

k

)

isimposedbytheflowthrough Si j butitispossibletodesignspecific

σ

(

k

)

forLESpurposes.

Thesetestshavealsoemphasized theimportantrole playedby thecouple

(

σ

0

,

ξ )

onthe simulationofa3D decaying

Taylor–Greenvortex.Inthefollowingsimulations,thiscouplewillbesetto

(

0

.

05

,

1

.)

.

6.2.3. Influenceofthefilteredquantity

As discussed insection 3,the filtering procedurecan be applied to three differentquantities which are basically the distributionfunctions gα,thewholecollisionoperator

−

_τ1

g

(

gα

−

g eq

α

)

orthefirstmoments

ρ

and

ρ

u.Thefilteringofthe gα quantitiesisdoneafterthepropagationstep,thefilteringofthecollisionoperatorisdonebetweenthecollisionandthe propagationstepsandthefilteringofthemacroscopicquantitiesisdoneaftertheircomputations.

Fig. 11comparestheinfluenceofthefilteredquantityonthedissipationofthekineticenergyforthepresenttest-case. Theobserveddifferencesindicateaweakdependenceonthefilterorderandthebetterresultsareobtainedforthefiltered moments.Moreover, thislatterchoice inducesa lower costbecauseonly fourquantitiesare filteredinthree-dimensional simulationswhereasthetwootherchoicesrequirethefilteringofnineteenquantities.Therefore,thefollowingcomputations willbeperformedwithfilteredmoments.

6.2.4. Comparisonwithstaticfilters

Thestaticfilteringtechniquecouldbeseenasaparticularcaseofthedynamicalonewhenthesensitivityissettozero (S0

=

0). Therefore,staticfiltersacteverywhere withthesameamount,withpotential dampingofimportantstructuresin

thedynamical evolution.Withtheabove discussion,thestaticfilteringtechnique isthus expectedtogive over-dissipated results.ThisisconfirmedbylookingatFig. 12whichdisplaysthecomparisonofstaticanddynamicalfilteringona963_grid

Fig. 9. Influenceoftheσ0parameter.Left:Non-dimensionaltimeevolutionofthedissipationrate.Right:Timetraceofthemaximumvalueofthe nor-malizedcoefficientmax[σd(x)]/σ0.(a):3-point,(b)5-pointand(c) 9-pointfilters.( Referencesimulationon2563)( σ0=0.01),( σ0=0.05)and (- -σ0=0.1).

A striking result is observed for 3-point stencil filter which gives completely wrong behavior with a static strategy whereas results closeto higherorder filtersare observed forthe dynamical strategy. Thisis particularlyapparent in the transitionregionwherethe3-point dynamicalfiltergivesbetterresultsthanthestatic5-pointfilterandsimilarresultsas thestatic9-pointfilter.Moreover,thedynamicalfilteringappearstobetterpredictthedissipationpeakaroundt

=

9 than the staticfilters.The abovediscussions thus indicatethat theresultscanbe severelyimprovedby increasing thefiltering sensitivityanddecreasingthe

σ

0.Thisbehavior suggeststhatthewavenumberselectivityofthespatialfiltersplaysaminor

Fig. 10. Influenceoftheparameterξ.Left:Non-dimensionaltimeevolutionofthedissipationrate.Right:Timetraceofthemaximumvalueofthe nor-malizedcoefficientmax[σd(x)]/σ0. (a):3-point,(b)5-pointand(c) 9-pointfilters.( Referencesimulationon 2563)( ξ=0.5), (- -ξ=1.0)and ( ξ=1.5).

6.2.5. ComparisonwithSGSmodels

TheLBMimplementationofsubgrid-scalemodelisveryclosetotheoneofthepresentdynamicalfiltering.Relation(12)

isusedtoestimatetheeddyviscosityandthustherelaxationtime

τ

g.Thecomparisonbetweentheimplementationofthe classicalSmagorinskymodelandthepresentmethodology ispresentedinFig. 13 withtheSmagorinskymodelassociated toa constant Cs

=

0

.

1 and Cs

=

0

.

18. Theresultsindicatethat theSGSimplementationappears tobe toodissipativeand describespoorlythelaminar-turbulenttransitionregion.SimilarobservationsaredetailedinAubardetal.[26]wherevarious SGSstrategieshavebeencomparedandfoundtobenotsuitableforthistransitiontestcase.

The present dynamical strategy is found to better predict the transition region by filtering only the sheared region. ThemaindifferencebetweenthepresentimplementationandtheSGSmethodologyreliesintheamountoffilteringwhen shearstress islower thanthe imposedsensitivity(S

<

S0). Indeedforthose valuesthefilterhavenoimpact andletthe

turbulent structures free of numerical dissipation. Contrariwise, the SGS implementation is directly proportional to the amountofshearstressandtheeddy viscosityhasanon-zero valueforshearstress amountcloseto S0 thusimposingan

Fig. 11. Comparisonofthefilteredquantityona963_domainwithσ

0=0.05 andξ=1.0:( Referencesimulationon2563.Red,blueandgreencurves refer to3-point, 5-pointand9-pointfilters respectively.) filteredcollisionoperator, filtereddistributionfunctions, filteredmoments.(For interpretationofthecolorsinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

Fig. 12. Comparisonbetweenstaticfilters(solidlines)anddynamicalfilters(dashedlines)ona963_{domain(left)anda128}3_{domain(right).Red,blue} andgreencurvesrefer to3-point,5-pointand9-pointfiltersrespectively.(Forinterpretationofthecolorsinthisfigure,thereaderisreferredtotheweb versionofthisarticle.)

Fig. 13. Comparisonbetweendynamicalfilteringstrategyandsubgrid-scalemodelona963_{domain(left)anda128}3_{domain(right). –– referencesimulation,} ––presentfilteringwith3-pointfilter,−−SGSwithCs=0.18, SGSwithCs=0.1.

the SGSapproachisnota boundedprocedure andtheeddy viscositycan reacharbitraryhighvalue whenhighlysheared regions areencounteredwhereasthepresentedstrategyislimitedbythe

σ

0 valuerestrictingthenumericaldissipation to

a limitedamount.Finally,fromageneralpointofview,theSGSstrategy couldbe seenasthe presentdynamical filtering techniquewithalowshearstresssensitivityandshouldbelinkedtotheresultsofsection 6.2.2.

7. Computationalcost

Table 4

Computationalcostsofthepresentfilteringstrategy.

Models D2Q9 D3Q19

Standard 0.147 0.516

Filters Classical filters Present filters Classical filters Present filters

3-point 0.178 (+21%) 0.185 (+26%) 0.564 (+09%) 0.588 (+14%) 5-point 0.184 (+25%) 0.190 (+29%) 0.571 (+10%) 0.598 (+16%) 9-point 0.192 (+30%) 0.199 (+35%) 0.586 (+14%) 0.614 (+19%)

the3Dcomputationaltimesrefertothetest-caseofsection6fora1283_{grid.Inthissection,thepresentfilteringtechnique}

isappliedonthemomentsofthedistributionfunctions.ThepresentinhousecodeisasetofPythonmoduleswithatime loopwrappedinFortran-90(f2py).Allthecomputationaltimereferstoasinglestandardprocessor(IntelXeonCPUW3565@ 3.20GHz)andaregivenin

μ

s

/

point

/

iterations.

First,theresultsindicatethattheovercostduetofilteringishigherfor2Dsimulationsthanfor3Dsimulations.Thiscan be explainedby the numberofdiscrete velocitiesinvolved inD2Q9andD3Q19 models.For2D simulations the collision andpropagationstepsaredoneon9 velocitiesandthefilteringofthemomentsaredoneon3 quantitieswhichrepresents 33% of the velocity number. For 3D simulations, the filtering is done on 4 quantities which represents 21% of the 19 velocities.Then,theovercost ofthepresentfilteringtechnique duetothecomputation ofrelation(10),isrelatively small withanadditionaltimeof5% in3Dcomparedtoclassicalfilters.ThisovercostmustbeseenintheHPCframeworkwhere communicationtimeisofmajorimportance. Indeed,it hasbeenshownintheprevious sectionsthat low-stenciladaptive filterresults were comparable andmore stable than highstencil classical filter. Theseresults demonstrate the abilityof thepresentfilteringtechniquetobeusedformassivelyparallelcomputationswheretheovercostofcomputing

σ

d willbe dampedbythegainofscalabilityinducedbylowcommunications.

8. Conclusion

AdynamicalfilteringtechniqueforthelatticeBoltzmannmethodhasbeenpresentedandtestedonrepresentative test-casesin2D and3D. Ithasbeenshownthat theuseofselectivespatial filterswitha coefficientbasedon theamountof shear stress ledto improvedstability andlimited dissipation.In particular, ithasbeen emphasized that theshear stress selectivity wasrestrictingthe actionofthe filterstolocalized zonesthusreducing the globalamount ofnumerical dissi-pation.The choiceofshearstress selectivityhasbeenmotivatedbythelattice Boltzmannframework forwhichtheshear stressisarelevantquantitythatcanbeaccessedwithaminimumamountofadditionalcomputationaltime.Theresultsare particularlyimprovedwhenthefilterorderislow,especiallyforthecomparisonwithstaticselectivefilterssuggestingthat thewavenumberselectivityisdominatedbytheoneoftheshearstress.Thecomparisonofthepresentedmethodologyto theclassicalsubgrid-scale modelmethodologyonalaminar-turbulent transitiontestcasesuchastheTaylor–Greenvortex havealsoledtopromisingresultshighlightingtheimportanceofshear-stressselectivityinthepredictionoftheturbulence dynamical evolution.From acomputational costpoint ofview,thepresentedstrategy haveshowninterestingcapabilities whenusinglow-orderfiltersthus reducingtheeffectivestenciltotheone oftheLBMwhichisconsistent withmassively parallel simulations.Then the dynamical filteringstrategy should beseen asan enhanced stabilizationprocedure forthe latticeBoltzmannmethodwheretheamountofnumericaldissipationislocallycontrolledinspace.Thenextstepwillbeto introducea

σ

d coefficientoptimizedinwavenumberspaceforLESpurposes.

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