Implicit large eddy simulation of compressible turbulence flow with PnTm − BVD scheme

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Implicit large eddy simulation of compressible turbulence flow with PnTm – BVD scheme

Xi Deng, Zhen-Hua Jiang, Feng Xiao, Chao Yan

To cite this version:

Xi Deng, Zhen-Hua Jiang, Feng Xiao, Chao Yan. Implicit large eddy simulation of compressible

turbulence flow with PnTm – BVD scheme. Applied Mathematical Modelling, Elsevier, 2020, 77,

pp.17-31. �10.1016/j.apm.2019.07.022�. �hal-03235122�

Implicit large eddy simulation of compressible turbulence flow with P _n T _m − BVD scheme

Xi Deng

^a,c

, Zhen-Hua Jiang

^b,∗

, Feng Xiao

^a

, Chao Yan

^b

aDepartment of Mechanical Engineering, Tokyo Institute of Technology, 2-12-1(i6-29) Ookayama, Meguro-ku, Tokyo 152–8550, Japan

bCollege of Aeronautics Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, PR China

cAix Marseille University, CNRS, Centrale Marseille, Marseille M2P2, France

a r t i c l e i n f o

Article history:

Received 17 December 2018 Revised 21 May 2019 Accepted 11 July 2019 Available online 18 July 2019 Keywords:

Shock capturing Compressible turbulence Very high order schemes P nT m−BVD scheme WENO scheme Implicit LES

a b s t r a c t

Implicit large eddy simulation (ILES) of compressible turbulence with shock capturing schemesrequires wideinvestigationsand numericalexperiments.Inthisstudy, anewly proposedPnTm−BVD(polynomialofn-degreeandTHINCfunctionofm-levelreconstruc- tionbased on BVD algorithm)shock capturing schemeis introduced to simulate com- pressibleturbulenceflowwithILES.Thenewschemeisdesignedbyemployinghigh-order linear-weightpolynomialsandTHINC(TangentofHyperbolaforINterfaceCapturing)func- tionswithadaptive steepnessas the reconstruction candidates.The finalreconstruction functionineachcellisdeterminedwith amulti-stageBVD(BoundaryVariationDimin- ishing) algorithmso as to effectively control numerical oscillation and dissipation. Nu- mericaltestsinvolvingshockwavesandbroadbandturbulenceareconductedincompari- sonwithWENO(WeightedEssentiallyNon-oscillatory)schemeswhicharewidelyusedin ILES.TheresultsdemonstrateperformingILES withPnTm−BVDschemeisabletoobtain higherresolutionandmorefaithfulresultsthanWENOdoes.Importantly,thesuperiority ofPnTm−BVDbecomesmorenotableinhighwave-numberregion.Thusthispaperpro- videsand verifiesanewschemewhichispromisinginprovidinghigh-resolutionresults forreal-caseILESofcompressibleturbulenceflow.

1. Introduction

Havingbeendevotedtremendouseffortsinthepastdecades,computationalfluiddynamicshavereachedamaturestage formanyengineeringapplications.However,therearestillsomedifficultissuesthatareofgreatapplicationimportancebut remainunsolvedyet.Numericalsimulationofcompressibleturbulenceisamongthem.Thenumericalschemesdesignedfor theshock-turbulencesimulationrequireschemesbeinglow-dissipativetoresolve small-scalestructure butalsobeingable toobtainstablesolutioninvolvingstrongshock.

Thepopularlyusedpractical methods,such ashigh-resolutionshockcapturingTVD(TotalVariationDiminishing) [1,2], ENO(EssentiallyNon-oscillatory)[3–5]andWENO(WeightedEssentiallyNon-oscillatory)[6,7]methodsarefoundtobetoo dissipativetoadequately resolvesmallflowstructuresthat playessentialroleinturbulentinteraction processes[8].Intro- ducing centralflux which ispreferable forturbulenceflow simulations, hybridWENO/central schemes[9–13]are able to

∗ Corresponding author.

E-mail address: jiangzhenhua@buaa.edu.cn (Z.-H. Jiang).

resolvebroadbandturbulence.However,theperformanceofhybridschemesusuallyrelyonshocksensors.Artificialviscos- ity/diffusivitymethods[14–17]generallygivebetterresolutionthanWENOschemes.Still,theintroducedartificialviscosity mayalsooverwhelmsmall-scalestructures.

The excessivenumerical dissipationof highordershock-capturingschemes comes intrinsically fromthedilemma that onehastocompromisebetweenhigh-orderinterpolationandspuriousoscillationinthecurrentparadigmbasedonpolyno- mialreconstruction andnonlinearlimitingprojection.KnownastheGodunovbarrier,anyreconstructionusingpolynomial higherthan2ndordertendstogeneratenumericaloscillationsinpresenceofdiscontinuityorlargegradient.Ononehand, ahigh-orderpolynomialispreferredforsmoothsolution,ontheotherhandalowerorder(orsmoother)interpolantissug- gestedinthevicinityofdiscontinuoussolution.Itisrealizedautomaticallythroughanonlinearswitchingbasedonsmooth- nessmeasurement.Beingthemostrepresentativenumericalmethodofthissort,theWENOreconstructionisdesignedsoas toretrievethehighestpossibleconvergencerateinthesmoothregionwhiledegradetoaflatterorlower-orderinterpolant around discontinuoussolution.In spiteofhuge amountofwork devoted todevise nonlinearlimiting formulationsbased onWENOconcept, theexisting schemesshow atleastthefollowingtwo drawbacks.(1)Thenonlinearly weightedrecon- structionfunctionscanhardlyretrievetheunlimitedpolynomialsthatpossessthedesirednumericalpropertiesforsmooth solutions,and(2)forhigh-orderreconstructionsusingpolynomialswithnonlinearweightsmightnotbeadequatelyrobust noreffectivelysuppressnumericaloscillationsinthepresenceofstrongdiscontinuities.

Besides themostrepresentative WENOmethodology, other shock-capturingschemes equippedwithnonlinearlimiters havealsobeenproposedin[18–21].Forexample,thehigh-orderMonotonicity-Preserving(MP)schemewhichcaneffectively suppressnumericaloscillationsacrossthediscontinuityandpreservetheaccuracyatextremaisfirstdesignedin[20].The MP schemeis further improvedby the work [22,23] to optimizethe dispersion anddissipation propertyof MP scheme.

AlthoughtheMPschemesshowsomeadvantagesoverWENOschemes,theMPlimitermaybestilltriggeredinhighwave- numberregion,whichunderminestheaccuracyofunderlyinglow-dissipationinterpolation.Beingadifferentapproachfrom finite volume method (FVM) orfinite difference method (FDM), discontinuous Galerkin(DG) method [24–26] introduces localhighorderreconstructionfunctionsbyaddingdegreesoffreedominthelocalcellelement.Limitingprocessesarealso demandedtorealizeshockcapturing[27–29].Designingsuchalimitingprocessisstillahottopicinthecommunityofhigh orderschemes[30].

Although shock capturing schemes have inherent numerical dissipation ofwhich sources usually come fromupwind biasedinterpolationandlimitingprocesses,shockcapturingschemesarestillusedincompressibleturbulencesimulations throughanapproachcalledimplicitlargeeddysimulation(ILES)[31].Originatedfromtheobservationsin[32],ILESisone of simple and efficient wayto simulatecompressible turbulence flow. Different from direct numericalsimulation (DNS) which requires an expensive conditionto solve Navier-Stokes (NS) equation and also differentfrom explicitLES method [33] which requires additionalsub-grid scale (SGS)model resulted fromfiltering operation of NS equation, ILESmethod solvesNSequationwithdissipatednon-oscillatoryfinitevolume(NFV)numericalschemes.InILES,theembeddednumerical dissipationinNFVschemesisusedinreplaceoftheexplicitSGSmodels.Ithasbeenshowninthework[34]thatILEScan giveresultsclosetothosefromstrictDNSorexplictLES.However,asindicatedin[35,36],ILESmethodsneedtobecarefully validatedandunderstandingofthesemethods stillreliesonperforming numericalexperiments.Forexample,theworkof [37]showsthat numerical dissipationof WENOschemes can playa similar roleas a verydisspativeSGSmodel.Inthework [38],performanceofdifferenthighorderWENOschemeonILEShasbeeninvestigatedoncompressibleturbulenceflowand turbulent combustion.In[36],ILEShas beenstudiedwithvery highorderWENOschemes (upto eleventhorder),which indicatesthatahigherorderschemeprovidesamorepragmaticapproachthanusingalowerorderscheme.

We haverecentlyproposed anewreconstruction approachforGodunov finitevolume method,so-calledBVDmethod, to resolveboth smooth anddiscontinuous solutionswith highfidelity. ABVD methodadaptively chooses theinterpolant fromtheBVD-admissiblereconstructionfunctionssothatthedifferencebetweenthereconstructedvaluesatcellboundary isminimized,whichtheneffectivelyreducesthenumericaldissipation.UndertheBVDconcept,anewclassofschemescan bedesigned,usingproperBVDalgorithmsandBVD-admissiblereconstructionfunctions[39–42].Ourpreviousworksshow that theBVDschemes areabletosubstantially improvesthenumericalsolutions,particularlyfordiscontinuousandsmall flow structures.Forexample,ithas beenshownin[41]thatBVD algorithm canprevent spurious phenomenawhich may appearwithstiff detonation waves.In ourrecentwork[43,44],high-orderunlimitedpolynomialsandTHINCfunction are usedasthecandidate reconstructionfunctionsbasedon amulti-stageBVD algorithm.The resultant schemescanretrieve thenumericaldispersion anddissipation propertiesoftheunlimitedpolynomialsforallwave numbers,thusdemonstrate superior solution quality forsmooth solutions.Meanwhile, the spurious oscillationcan be effectively eliminated through theBVD algorithmthatproperlychoosestheTHINCfunctionforreconstructionintheneighboringcellsofadiscontinuity.

The extensive numerical verification forscalar hyperbolic conservation law and inviscid Euler equations reveal that the BVDschemesusingunlimitedhigh-orderpolynomialshavemoreappealingnumericalpropertiesincomparisonwithother existinghighresolutionschemesthatmakeuseofnonlinearlimitingprojections.Thesemethodsthushaveagreatpotential tosimulateturbulentflowswithILES.Inthispaper,weimplementtheBVDschemestocompressibleNavier-Stokesequations toevaluatetheirperformanceinILESofcompressibleturbulenceflows.

The rest of this paper is organized as follows. Section 2 provides an outline of Godunov finite volume formulation.

Section 3 describesthemulti-stage BVD schemethat use theunlimited high-order polynomialsandTHINC asthecandi- datereconstructionfunction.Section4presentsnumericaltestsfortheproposedschemes.Specialattentionispaidtoaset ofbenchmarktestsofcompressibleNavierStokesequationsforturbulentflowsofdifferentReynoldsnumbers.Itisobserved

thatperforming ILESwiththe presentBVD schemesare superiortotheWENOschemeintermsofreproducing turbulent statisticproperties.Weendthepaperwithsomeconcludingremarks,inSection5.

2. Finitevolumemethod

Tosimulatecompressibleviscousflow,thecompressibleNavier-Stokesequationsforacaloricallyperfectgasaresolved:

∂ρ ∂

^{t +}

∇

^{·m=0, (1)}

∂

^m

∂

^{t +}

∇

·

(

^mu+p

δ )

=

∇

·

τ

, (2)

∂

^E

∂

^{t +}

∇

^·

(

^uE+pu

)

⁼

∇

^·

(

^u·

τ

^−q

)

, (3)

where

ρ

^{isthedensity,misthemomentum,uisthevelocity,pisthepressure,}

δ

^{istheunit tensor,E isthetotalenergy,}

τ

^{istheviscousstress tensorandqistheheatflux.Toclosuretheaboveequationsystem, thefollowingequation ofstate}

isapplied:

p=

( γ

⁻¹

) ρ

^{e (4)}

inwhich

γ

^{istheratioofspecificheatsandeisthespecificinternalenergy.AccordingtoStokes’hypothesisforaNewtonian}

fluid,theviscousstresstensoriscalculatedas

τ

=2

μ

^S−2

3

μ ( ∇

·u

)

⁽⁵⁾

wherestrain ratetensorSisdefinedasS=¹₂(

∇

^u+(

∇

^u)^T)^and

μ

^{isthedynamicshearviscosity.FollowingFourier’slaw,}

theheat fluxisdefinedasq=−k

∇

^{T wherek isthethermal} conductivityandT isthe temperature.It hasbeenlong as- sumedthat theaccuracyof theconvectiveterms iscriticalforresolving theturbulence[45].Thus,the mainchallenge to simulateshock-turbulenceproblemistointroduceahighaccuracyschemetodiscretizetheEulerequation systemswhich isrecoveredfromNavier-Stokesbyignoring viscouseffect.TheEulerequation systemscanbe furtheranalyzedbysolving theone-dimensionalscalarmodelequationas

∂

^q

∂

^{t +}

∂

^f

(

^q

)

∂

^{x =0, (6)}

whereqisthesolutionvariableswhichcanbeincharacteristicspaceandf(q)isthefluxfunction.

Considering1D scalar modelequation ofEq.(6), astandard finite volume methodis applied to obtainthe numerical solutions.WedividethecomputationaldomainintoN non-overlappingcellelements, Ii:x∈[x_i−1/2,x_i+1/2],i=1,2,...,N, withauniformgridspacingh=^x=x_i+1/2−x_i−1/2.thevolume-integratedaveragevalueq¯_i(^t)^incellIiisdefinedas

¯

q_i

(

^t

)

≈ 1

^x

_xi+1/2

xi−1/2

q

(

^x,t

)

^dx. (7)

Thesemi-discreteversionofEq.(6)inthefinitevolumeformcanbeexpressedasanordinarydifferentialequation(ODE)

∂

^q¯

(

^t

)

∂

^{t =−}

1

^x

(

^f˜i+1/2−f˜i−1/2

)

, (8)

wherethenumericalfluxes f˜atcellboundariescanbecomputedbyaRiemannsolver

f˜i+1/2=f_i^Riemann_+1/2

(

^qLi+1/2,q^R_i+1/2

)

⁽⁹⁾

aslongasthereconstructedleft-sidevalueq^L_i+1/2andright-sidevalueq^R_i+1/2atcellboundariesareprovided.Essentially,the Riemannfluxcanbewritteninacanonicalformas

f_i^Riemann_+1/2

(

^qLi+1/2,q^R_i+1/2

)

=1 2

f

(

^qLi+1/2

)

+f

(

^qRi+1/2

)

−

|

^ai+1/2

|

2

q^R_i+1/2−q^L_i+1/2

)

, (10)

wherea_i+1/2 standsfor the characteristic speed of the hyperbolic conservation law.The remaining main task is how to calculateq^L_i+1/2andq^R_i+1/2 throughthereconstructionprocess.

3. PnTm−BVD:anewreconstructionscheme

In this subsection, we give the details of how to calculate q^L_i+1/2 and q^R_i+1/2 using the newly proposed PnTm−BVD schemesin [43]. Theseschemes are designedby employinglinearweight polynomial ofn-degreeand THINCfunction of m-levelasthecandidateinterpolants.Thefinal reconstructionfunctionineachcellisselectedfromthesecandidateinter- polantswiththeBVDalgorithm.Next,weintroducethecandidateinterpolantsbeforethedescriptionoftheBVDalgorithm.

3.1. CandidateinterpolantP_n:linearupwindschemeofn-degreepolynomial

Afinitevolumeschemeof(ⁿ+1)^thordercanbeconstructedfromaspatialapproximationforthesolutioninthetarget cellIiwithapolynomialq˜^Pn_i (^x)^{ofdegreen.Then}+1unknowncoefficientsofthepolynomialaredeterminedbyrequiring that q˜^Pn_i (^x)^{hasthesamecell averageoneachcell overan}appropriately selectedstencilS=

{

ⁱ−n⁻,...,i+n⁺

}

^withn−+ n⁺=n,whichisexpressedas

1

^x

_xj+1/2

x_j−1/2

q˜^Pn_i

(

^x

)

^dx=q¯j, j=i−n⁻,i−n⁻+1,...,i+n⁺. (11)

Toconstruct2r−1orderupwind-biased(UW)finitevolumeschemesasdetailedin[6,7,46],thestencilisdefinedwithn⁻= n⁺=r−1.Theunknowncoefficientsofpolynomialof2r−2degreecanbethencalculatedfrom(11).Withthepolynomial

˜

q^Pn_i (^x),highorderapproximationforreconstructedvaluesatthecellboundariescanbeobtainedby q^L_i^,Pn

+¹2 =q˜^Pn_i

(

^xi+¹2

)

^{and qR}_i−^,Pn1

2 =q˜^Pn_i

(

^xi−¹2

)

. (12)

The analysis of [6,7,46] shows that in smooth region, the approximation with polynomial (12) can achieve 2r−1 order accuracy.Inthiswork,weapplyschemesfromfifthorder(r=3)toninthorder(r=5)byusingpolynomialsof4th,6th,and 8thdegreeasunderlyingschemeforsmooth solution.Theexplicitformulasofq^L_i+^,Pn_1/2 andq^R_i−^,Pn_1/2 for(n+1)th-orderscheme aregivenasfollows:

• 5th-orderscheme q^L,P4_i+1/2 = 1

30q¯_i−2−13

60q¯_i−1+47 60q¯_i+ 9

20q¯_i+1− 1 20q¯_i+2, q^R_i−^,P4_1/2 = 1

30q¯_i+2−13

60q¯_i+1+47 60q¯_i+ 9

20q¯_i−1− 1

20q¯_i−2. (13)

• 7th-orderscheme q^L,P6_i+1/2 =− 1

140q¯_i−3+ 5

84q¯_i−2− 101

420q¯_i−1+319

420q¯_i+107

210q¯_i+1− 19

210q¯_i+2+ 1 105q¯_i+3, q^R_i−^,P6_1/2 =− 1

140q¯_i+3+ 5

84q¯_i+2− 101

420q¯_i+1+319

420q¯_i+107

210q¯_i−1− 19

210q¯_i−2+ 1

105q¯_i−3. (14)

• 9th-orderscheme q^L_i+^,P8_1/2 = 1

630q¯_i−4− 41

2520q¯_i−3+ 199

2520q¯_i−2− 641

2520q¯_i−1+1879 2520q¯_i +275

504q¯_i+1− 61

504q¯_i+2+ 11

501q¯_i+3− 1 504q¯_i+4, q^R_i−^,P8_1/2 = 1

630q¯i+4− 41

2520q¯i+3+ 199

2520q¯i+2− 641

2520q¯i+1+1879 2520q¯i

+275

504q¯i−1− 61

504q¯i−2+ 11

501q¯i−3− 1

504q¯i−4. (15)

3.1.1. CandidateinterpolantTm:non-polynomialTHINCfunctionwithm-levelsteepness

AnothercandidateinterpolationfunctioninPnTm−BVDschememakesuseoftheTHINCinterpolationwhichisadiffer- entiableandmonotoneSigmoidfunction[47,48].ThepiecewiseTHINCreconstructionfunctioniswrittenas

˜

q^T_i

(

^x

)

=q¯_min+q¯max

2

1+

θ

^tanh

β

x−xi−1/2

x_i+1/2−x_i−1/2 −x˜_i

, (16)

whereq¯_min=min(^q¯i−1,q¯_i+1),q¯max=max(^q¯i−1,q¯_i+1)−q¯_minand

θ

=sgn(^q¯i+1−q¯_i−1)^{.Thejumpthicknessiscontrolledbythe} parameter

β

^{,i.e.asmallvalueof}

β

^{leadstoasmoothprofilewhilealargeoneleadstoasharpjump-like}distribution.The unknownx˜_i,whichrepresentsthelocationofthejumpcenter,iscomputedfromconstraintconditionq¯_i=¹_x x^x_iⁱ₋₁⁺¹_/^/₂²q˜^T_i(^x)^dx. Since thevalue givenby hyperbolic tangent function tanh(x) lays in theregion of[−1,1], thevalue ofTHINC recon- structionfunctionq˜^T_i(^x)^{isrigorouslyboundedbyq¯}i−1andq¯_i+1.Giventhereconstructionfunction q˜^T_i(^x),wecalculatethe boundaryvaluesq^L_i+^,T_1/2 andq^R_i−^,T_1/2 byq^L_i+^,T_1/2=q˜^T_i(^xi+1/2)^andqRi−^,T1/2=q˜^T_i(^xi−1/2)^,respectively.

Following[42],inthiswork weuseTHINCfunctionswith

β

^{ofm-leveltorepresentdifferentsteepness, torealizenon-}

oscillatoryandless-dissipativereconstructionsadaptivelyforvariousflowstructures.ATHINCreconstructionfunctionq˜^T_i^k(^x) with

β

kgivesthereconstructedvaluesq^L,T_i+1^k_/2 andq^R,T_i−1^k_/2,(k=1,2,...,m).Wewillusemuptothreeinpresentstudy.

3.2.TheBVDalgorithm

InthePnTm−BVDschemes,reconstructionfunctionisdeterminedfromthecandidateinterpolantswithk-stage(^k=m) BVDalgorithmsoastominimizetheTBVofthetargetcell.WedenotethereconstructionfunctioninthetargetcellIiafter thekthstageBVDasq˜^<_i^k>(^x)^.

Thek-stageBVDalgorithmisformulatedasfollows:

(I): Initialstage(^k=0)^:

(I-I)Asthefirststep,usethelinearhigh-orderupwindschemeasthebasereconstruction schemeandinitializethe reconstructedfunctionasq˜^<_i^0>(^x)=q˜^Pn_i ^(x).

(II) TheintermediateBVDstage(^k=1,...,m−1)^:

(II-I)Setq˜^<k>_i (^x)=q˜^<_i^k−1>(^x)

(II-II)CalculatetheTBVvaluesfortargetcellIifromthereconstructionofq˜^<k>_i (^x)

TBV_i^<k>=

q^L_i−^,<₁^k_/^>₂−q^R_i−^,<_1/^k>₂

+

q^L_i+^,<₁^k_/^>₂ −q^R_i+^,<₁^k_/2^>

(17)

andfromtheTHINCfunctionq˜^T_i^k(^x)^{withasteepness}

β

kas

TBV_i^Tk=

q^L_i−1^,Tk_/2−q^R_i−1^,Tk_/2

₊

q^L_i+1^,Tk_/2−q^R_i+1^,Tk_/2

_. (18) (II-III)Modifythereconstructionfunctionforcellsi−1,iandi+1accordingtothefollowingBVDalgorithm

˜

q^<k>_j

(

^x

)

=q˜^T_j^k

(

^x

)

, j=i−1,i,i+1; if TBV_i^Tk<TBV_i^<k>. (19)

(III) ThefinalBVDstage(^k=m)^:

(III-I)Giventhereconstructionfunctionsq˜^<_i^m−1>(^x)^{fromabovestage, computetheTBVusingthe}reconstructed cell boundaryvaluesfrompreviousstageby

TBV_i^<m−1>=

q^L_i−1^,<m_/2^−1>−q^R_i−1^,<m_/2^−1>

₊

q^L_i+1^,<m_/2^−1>−q^R_i+1^,<_/^m₂^−1>

_, (20)

andtheTBVforTHINCfunctionof

β

^mby

TBV_i^Tm=

q^L_i−^,T₁^m_/2−q^R_i−^,T₁^m_/2

₊

q^L_i+^,T₁^m_/2−q^R_i+^,T₁^m_/2

_. (21) (III-II)DeterminethefinalreconstructionfunctionforcellIiusingtheBVDalgorithmas

˜

q^<_i^m>

(

^x

)

⁼

q˜^T_i^m; if TBV_i^Tm<TBV_i^<m−1>,

˜

q^<m_i ^−1>; otherwise . (22)

(III-III)Computethereconstructedvaluesontheleft-sideofx_i+1

2 andtheright-sideofx_i−1

2 respectivelyby q^L_i

+¹2=q˜^<_i^m>

(

^xi+¹2

)

^{and qR}_i−¹

2 =q˜^<_i^m>

(

^xi−¹2

)

. (23)

Inthisstudy,weproposeandtestfifthorderschemewithP₄T₂−BVD,seventhorderschemewithP₆T₃−BVDandninth orderschemewithP₈T₃−BVD.Accordingtoprevious studyin[42],inall testsofthepresentstudyweuse

β

1=1.1 and

β

2=1.8forP₄T₂−BVD,and

β

1=1.2,

β

2=1.1and

β

3=1.8forPnT₃−BVD(ⁿ=6,8)^schemes.

Remark 1. In current schemes, simple upwind-biased interpolation functions are employed as underlying high order schemes. However, less-dissipative interpolations optimized as [23,49] are also possible to be employed as underlying schemes.

4. Numericalexperiments

Inthissection,we shownumericalresults forbenchmark teststoverifytheperformance ofPnTm−BVD insimulating compressibleturbulence.Wewillfirstlyanalyzethespectralpropertyofschemesandsolvenon-broadband1DEulerequa- tionsinSections4.2and4.3.Thenthesimulationresultsofthree-dimensionalbroadbandcompressibleturbulenceproblems willbegiven.AllnumericalresultsarecomparedwiththoseofthemappedWENO(WENOM)[50]schemes.Althoughthere areotherimprovedWENOschemessuchasWENOZ[51],itisshownin[38]thattheperformancesofWENOMandWENOZ arealmostequivalent.TheCFLnumber0.4isusedinourtestsunlessspecificallynoted.

w

Imag

0.5 1 1.5 2 2.5 3

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0

5th order UW 7th order UW 9th order UW P4T2 P6T3 P8T3 WENOM5 WENOM7 WENOM9 Exact

w

Real

0.5 1 1.5 2 2.5 3

0 0.5 1 1.5 2 2.5 3

5th order UW 7th order UW 9th order UW P4T2 P6T3 P8T3 WENOM5 WENOM7 WENOM9 Exact

Fig. 1. Approximate dispersion and dissipation properties for different schemes. Imaginary parts of modified wavenumber are shown in the left panel, while real parts are shown in the right. Comparisons between linear upwind schemes, WENOM and P nT m−BVD schemes.

4.1. Spectralproperty

Weanalyzedthespectralpropertyoftheproposedschemebyusingtheapproximatedispersionrelation(ADR)method describedin[52].Thenumericaldissipationanddispersionofaschemecanbeevaluated respectivelyfromtheimaginary andrealpartsofthemodifiedwavenumber.

Thespectralpropertiesofthehighorderlinearupwindschemes,WENOMandPnTm−BVDschemesareshowninFig.1. TheWENOMschemesshowsignificantdiscrepancieswiththelinearschemeforhighwavenumberregime.Itispartlyowed tothe factthat theWENOsmoothnessindicatorstendto mis-interpretamonghigh-wavenumberstructures anddisconti- nuities.Thus thenumerical dissipation introduced by the wronglytriggered WENOnonlinear adaptation maysmear out thehigh-wavenumberstructures whicharecommoninturbulenceflow. Onthecontrary,thenewly proposedPnTm−BVD schemes have almost the same spectral property astheir underlying linearschemes, which surpasses WENO methodol- ogy. Thus the P_nT_m−BVD can distinguish high-wavenumber structures fromtrue discontinuities, andpreserve the low- dissipation property of highorder upwind schemes. The spectral property indicates that PnTm−BVD schemes are more preferablethanWENOMinsimulatingsmoothflowregion.

4.2. InviscidstrongLax’sproblem

In shock-turbulencesimulations,schemes should beable to dealwiththe possibilityofexisting strong discontinuities such asshock wavesand detonationwaves [53–58]. Thus, we conduct aLax problemto test the robustnessof different schemesoncapturingstrongdiscontinuities.Theinitialconditionisprescribedas

( ρ

0, u0, p0

)

=

(

^{1.0, 0.0, 1000.0}

)

^{, 0≤x≤0.5}

(

¹.0, 0.0, 0.01

)

, otherwise . (24)

Withinitialhighpressureratio,aright-movingMach198shockandacontactisgenerated.Thecomputationlastsuntiltime t=0.012with200 cellelements.ThenumericalsolutionsfromdifferentschemesfordensityfieldareplottedinFig.2.In each figures,the zoomedregions aroundthe highlycompressed regionare alsopresented.WecompareP_nT_m−BVD with WENOMschemes atthesameorder. Itis obviousthatPnTm−BVD schemesproduce morefaithfulsolution withreduced errorsinbothdissipationandovershooting.Arounddiscontinuities,WENOMschemesresultinmorediffusiveprofiles.More seriously,WENOMschemesproduceovershootvaluesinhighlycompressedregion.AsorderofWENOMincreased,theover- shootandnumericaloscillationsbecomemoreobvious.Thistestjustifiestheproposedschemeasarobustsolverforstrong discontinuitieswith highaccuracy. We alsoobserve that PnTm−BVD schemes ofdifferent orders producesimilar results across thecontactwheretheTHINC function isselectedby theBVD algorithm.It isnoteworthythat the numberofcells requiredtocapturethediscontinuityisindependentofgridresolutions.

4.3. Inviscidshockdensitywaveinteractionproblem

Inorder toverifytheperformance ofthe schemesincapturingshocks aswellassmooth solutions ofdifferentscales, we simulatedthecaseproposed in[59],whichserves asagood andsimpletest bedforthe simulationsof compressible

x

rho

0 0.2 0.4 0.6 0.8

1 2 3 4 5 6

exact P4T2 WENOM5

(a) 5th order scheme

x

rho

0 0.2 0.4 0.6 0.8

1 2 3 4 5 6

exact P6T3 WENOM7

(b) 7th order scheme

x

rho

0 0.2 0.4 0.6 0.8

1 2 3 4 5 6

exact P6T3 WENOM7

(c) 9th order scheme

Fig. 2. Numerical results of strong Lax’s problem with Mach 198 for density field at time t = 0 . 012 with 200 cells. Comparisons are made between WENOM and P nT m−BVD at the same order.

x

rho

-4 -2 0 2 4

1 1.2 1.4 1.6

exact P6T3 WENOM7

(a) 7th order scheme

x

rho

-4 -2 0 2 4

1 1.2 1.4 1.6

exact P8T3 WENOM9

(b) 9th order scheme

Fig. 3. Numerical results of shock density wave interaction problem. Comparisons between 7th order WENOM and P 6T 3are shown in the left panel, and between 9th order WENOM and P 8T 3and P 10T 3in the right panel.

turbulenceinvolvingshockwaves.Inthiscase,ashockwave interactswithdensitydisturbancesandgeneratesaflowfield containingwavesofhigherwavenumberanddiscontinuities.Theinitialconditionisspecifiedsimilarlyas[59]

( ρ

^{0, u0, p0}

)

=

(

^{1.515695, 0.523346, 1.805}

)

^{, ifx≤ −4.5,}

(

^1+0.1sin

(

¹²

π

^x

)

, 0, 1

)

^{, otherwise. (25)}

Thecomputationwascarriedout uptot=5.0.Thenumericalsolutionswith500cellswere showninFigs. 3and4 with zoomed region,where comparisons betweenWENOM andPnTm−BVD atdifferent orderare included. It is obviousthat thePnTm−BVDschemesresolvebetterdensityperturbationsandcapturemoreaccuratelythepeakofthewaves.Also,the resolutionofhigh-frequencywavesisimprovedwiththeorderoftheunderlyingpolynomialincreased. Thistestconfirms that PnTm−BVD schemes can simultaneously resolveflows containingboth smooth and discontinuousregions withhigh accuracy.

4.4.InviscidTaylor-Greenvortex

The numerical dissipation of the schemes are further evaluated through an inviscid Taylor-Green vortex problem [8,60,61].Sameas[8],theinitialconditionisgivenby

x

rho

-2 -1.5 -1 -0.5 0 0.5 1

1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7

exact P6T3 WENOM7

(a) 7th order scheme

x

rho

-2 -1.5 -1 -0.5 0 0.5 1

1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7

exact P8T3 WENOM9

(b) 9th order scheme

Fig. 4. The same as Fig. 3 but the zoomed region is shown.

ρ

=1,

u=sin

(

^x

)

^cos

(

^y

)

^cos

(

^z

)

,

v

=−cos

(

^x

)

^sin

(

^y

)

^cos

(

^z

)

, w=0,

p=100+[cos

(

^2x

)

+cos

(

^2y

)

^][cos

(

^2z

)

+2]−2

16 (26)

wherethepressureishighenoughtomake theproblemnearlyincompressible. Thedomainis[0,2

π

^{)3 withuniformgrid}

size of2

π

^{/64 in each direction. Periodicboundary conditionisprescribed for all} boundaries. Comparisonsare madebe- tween thenewly proposed P_nT_m−BVD andWENOMschemes.Firstly, the temporalevolution ofthemean kineticenergy andenstrophyispresentedintheFig.5.Inordertoprovideaquantitativecomparisonbetweendifferentschemes,wecal- culateaccuracymetricsfortheTaylor-Greenvortexfollowing[61].Themeankineticenergynormalizedbyitsinitialvalueat t=5andthemeanenstrophynormalizedbyitsinitialvalueatt=3.5arecalculatedrespectively.Theresultsfromdifferent schemearepresentedinTable1.TheseresultsofinviscidTaylor-GreenvortexshowthatPnTm−BVD schemeshaveoverall better accuracythanWENOMschemes.Althoughthe numericaldissipation islimitedby upwind-biasedinterpolationsfor allschemes,thePnTm−BVDarestillmuchlessdissipativethanWENOM.Itshouldbenotedthatschemeswhichintroduce non-dissipativecentralfluxcanexactlypreservethediscretetotalkineticenergy[8,60,62].

4.5. Taylor-GreenvortexproblemwithRe=1600

The three-dimensional periodic Taylor-Green vortex is simulated in this subsection. Taylor-Green vortex serves as a canonical problemwhichcontains keyphysical processesin turbulencesuch asturbulent transition, turbulentdecay and theenergydissipationprocess.Thusitisusuallyemployedtoevaluatethemethodforturbulentflowsimulation.Theprob- lemconsistsofacubicvolumeofsidelength2

π

^{Lwithananalyticinitialconditiondefinedby}

u=U0sin

(

^x/L

)

^cos

(

^y/L

)

^cos

(

^z/L

)

^,

v

=−U0cos

(

^x/L

)

^sin

(

^y/L

)

^cos

(

^z/L

)

^, w=0,

p=p0+

ρ

0U₀²[cos

(

^2x/L

)

+cos

(

^2y/L

)

^][cos

(

^2z/L

)

+2]

16 (27)

whereL=1.0,U₀=1.0,

ρ

0=1.0and p₀=100.0.The initial temperatureis300k.The Machnumberis setto M₀=^U_c0⁰ = 0.08andtheReynoldsnumberissettoRe₀=^ρ0_μ^U0L=1600byadjustingtheviscosity.Weusethecharacteristicconvective time definedast=U^L₀.Acoarsemeshconsistingof64³ elementsisemployed inourcalculation.Inordertoevaluatethe

t

Kinetic Energy

0 2 4 6 8 10

0.4 0.6 0.8 1

Ref.

P4T2 P6T3 P8T3 WENOM5 WENOM7 WENOM9

a

t

Enstrophy

0 2 4 6 8 10

0 2 4 6 8 10 12

Ref.

P4T2 P6T3 P8T3 WENOM5 WENOM7 WENOM9

b

Fig. 5. Evolution of mean kinetic energy and enstrophy for the inviscid Taylor-Green vortex problem. Comparisons between P nT m−BVD and WENOM schemes are presented. The evolution for the mean kinetic energy is shown in (a) and for the enstrophy is shown in (b).

Table 1

Accuracy metrics for the Taylor-Green vortex. Comparisons between different schemes and the semi-analytical result of [61] . WENOM5 WENOM7 WENOM9 P 4T 2−BVD P 6T 3−BVD P 8T 3−BVD Ref. [61]

T-G energy t = 5 0.922 0.957 0.971 0.950 0.960 0.972 1

T-G enstrophy t = 3 . 5 3.082 3.153 3.170 3.083 3.160 3.207 3.46

performance ofschemes, we introduce the diagnostic quantities includethe integratedkinetic energy E_k and thekinetic energydissipationrate

k,whicharedefinedas

E_k= 1

^V

1

2

ρ (

^u2+

v

²+w²

)

^dV,

(

^Ek

)

=−dEk

dt (28)

Weconduct thesimulationwiththe newlyproposed P_nT_m−BVDschemes andtheWENOMschemes of5th,7thand9th order.

Firstly,we presentthetemporalevolution ofintegratedkinetic energysimulatedwithdifferentschemesinFig.6. The referencesolution isobtainedbyhigh-resolution DNSsimulationona meshof512³ [45].Asa whole,thesolutionof5th orderWENOMscheme(blackdashedline)largely deviateswiththereferencesolution.The inherentnumericaldissipation of5thorderWENOMleadstorapiddecayofkineticenergy.Byincreasingtheorder,7thand9thorderWENOM(greyand purpledashed line)schemes producesolution closerto thereferenceone. ComparedwithWENOM,P_nT_m−BVD schemes show better performance. Forexample,5thorder P₄T₂−BVD (redline) scheme producingcompetitive solution with7th orderWENOM.7thorderP₆T₃−BVDand9thorderP₈T₃−BVD(greenandblueline)obtainthemostaccurateresultscom- paredtothereferencesolutionwithP8T3−BVDproducesalittlebetterone.Itshouldbenotedthatduring10<t<20,the kineticenergydecayspeedsofWENOM7andWENOM9becomeslowerthanthereferenceoneandresultinoverestimated kineticenergyatfinaltime.However,7thorderand9thorderBVDschemesaccuratelypredictthekineticenergyevenafter longtemporalevolution.

Tofurther evaluate the performance of schemes, the evolution of kinetic energy dissipation ratewith P_nT_m−BVD is plottedinFig. 7.Withorderincreased, the predictionof theevolution becomesmoreaccurate interms ofthe evolution trendandthepeakofthedissipationrate.ThecomparisonsbetweenWENOMschemesofdifferentordersarepresentedin Fig.8.It isobviousthat BVD schemes produceoverall betterresults.For example,inFig.8 (b)the5thorder P₄T₂−BVD producescompetitivesolutioncomparedwith7thorderWENOM,andinFig.8(c)the7thorderP₆T₃−BVDevenobtainsa littlebettersolutioncomparedwith9thorderWENOMregardingtopredictionofthepeakofthedissipationrate.

Toshowtheinfluenceofdifferentwavenumberonthedecayoftheturbulentstructures,thespectraofturbulentkinetic energyatt=9isshowninFig.9withthereferencespectralsolution[63].Again,PnTm−BVDschemesshowbetterresults compared with WENOM. The advantages of PnTm−BVD schemes become more obvious asthe wave number increases.

As showninFig. 9(b),all ofWENOM schemes areinferior to the PnTm−BVD schemes in highwavenumber region.In a

t

Kinetic Energy

0 5 10 15 20

0.02 0.04 0.06 0.08 0.1 0.12

Ref.

P4T2 P6T3 P8T3 WENOM5 WENOM7 WENOM9

a

t

Kinetic Energy

10 15 20

0.02 0.04 0.06 0.08

Ref.

P4T2 P6T3 P8T3 WENOM5 WENOM7 WENOM9

b

Fig. 6. Evolution of mean kinetic energy for the Taylor-Green vortex problem. Comparisons between P nT m−BVD and WENOM schemes are presented. The evolution for the whole time band is shown in (a) and the zoomed region for the later time is shown in (b).

t

Di ssi p at io n Rat e

0 5 10 15

0 0.002 0.004 0.006 0.008 0.01 0.012

0.014 Ref.

P4T2 P6T3 P8T3

Fig. 7. Temporal evolution of kinetic energy dissipation rate for the Taylor-Green vortex problem. Results are calculated by P nT m−BVD of different orders.

conclusion ofaboveanalysis, PnTm−BVDschemes producehighresolutionandhighfidelitysolutionfortheTaylor-Green vortexsimulation.

4.6. Compressibleisotropicturbulence

Thedecayingcompressibleisotropicturbulencewitheddyshocklets[8]issimulatedinthissubsection.Inthisproblem, weakshockwaveswilldevelopfromtheturbulentmotionsifasufficientlyhighturbulentMachnumberisprovided.Thus thecoexistofshockwavesaswellasturbulencerequirestheshock-capturingschemeshouldbeabletohandleshockletsand