Development of a third-order accurate vorticity confinement scheme

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Development of a third-order accurate vorticity confinement scheme

Michel Costes, Ilias Petropoulos, Paola Cinnella

To cite this version:

Michel Costes, Ilias Petropoulos, Paola Cinnella. Development of a third-order accu- rate vorticity confinement scheme. Computers and Fluids, Elsevier, 2016, 136, pp.132-151.

�10.1016/j.compfluid.2016.05.025�. �hal-02877518�

Development of a third-order accurate vorticity confinement scheme

M. Costes^a,∗,I. Petropoulos^a,P. Cinnella^b

aONERA, The French Aerospace Lab, 8 rue des Vertugadins, F-92190 Meudon, France

bDynFluid Lab., Arts et Métiers ParisTech, 151 Boulevard de l’Hopital, F-75013 Paris, France

Keywords:

Vorticity confinement High-order schemes Linear advection equation Euler/RANS equations

Third-order VC2 confinement scheme

a b s t ra c t

Anew3rd-orderVorticityConfinementschemeispresentedasanextensionoftheoriginalVC2scheme developedbySteinhoff fortheresolutionofthefluiddynamicequations.Thetheoreticaldevelopments are explained, and the method is tested.The results obtainedshow that the new scheme combines theaccuracyoftheunderlyinghigherorderschemeand theconfinementcapabilityoftheoriginalVC2 method.

1. Introduction

The computation of vortices and wakes by CFD is a difficult problem.Alargepartofthemethodsappliedinthepastforwake simulationsusesaLagrangian approach[1,2],whichallows aper- fectconservationofthevortexsheets.However,mostofthemare restrictedtoinviscid,incompressibleflows,andhavedifficultiesto dealwiththemergingofvorticalstructures.TheEulerianapproach ismore general from all these aspects, butit suffers from other weaknesses.Numericalschemesforcompressibleflowsneedtobe dissipativefor stabilityreasons, thus wakes andvortices are dif- fusedmuchfasterin thecomputationsthan whatactually occurs inreality.Asignificant amountof workhasbeen done toreduce theseflaws,consideringeitherautomaticmeshrefinementinorder toconcentrate the meshpointsin thevicinity of thevortical re- gionsofinterest,ortheapplicationofhigher-orderdiscretizations.

Meshadaptionisgenerallyperformed withunstructuredgrids or with Cartesian grids and the Chimera overset grids method [3–

6]. Grid adaption can be based on physical criteria or on error estimates,whichmay requirethe solution ofan adjoint problem [7–10]. Higher-order space discretizations are an appealing alter- nativetodecreasethedissipationofnumericalschemes.However, thederivation ofhigh-order spacediscretizationsongeneralgrids iscomplexandvery oftentrulyhighaccuracyisimplementedon Cartesiangridsonly [11–13].Whatevertheapproach,asignificant

∗ Corresponding author. Fax: +33146734146.

E-mail addresses: [email protected] (M. Costes), [email protected] (I.

additionalCPUcostisneeded,andtheartificialspreadingofwakes cannotbetotallyremoved.

Alternative techniques in the Eulerian framework include the VorticityConfinement methodof Steinhoff[14–16],which proved tobeveryefficientforwakeconservation.Suchamethodhasbeen investigatedatONERA in thepast [17–20].We are moreparticu- larly interestedhereinthesecondVC schemeproposedby Stein- hoff, knownas VC2 [21,22]. In spiteof its capability to maintain concentrated vorticity inthe numerical simulations at a reduced extracostwithrespecttofinemeshcomputations,theoriginalVC formulationis only 1st-order accurate and theinternal profile of the confinedvortices is rapidlygoverned by the VC term. There- foreitisoflittleinteresttousethismethodwhenahigher-order numericalschemeisapplied,althoughthecapabilitiesofVCcould be beneficial even in this kind of simulation. This is more par- ticularly the case when considering turbulent or separated flows for whichvorticity plays a major role. The developmentof a VC methodadapted tohigher-orderdiscretizations isthereforeofin- terest.Furthermore,suchahigher-orderVCmethodmayallowthe level ofnegativedissipation applied in vorticalregions to be de- creasedaccordingtothelowernumericaldissipationintroducedby higher-orderschemes.

In[23],higher-orderconfinementschemes weredevelopedfor the lineartransport equation. The main outcome ofthis studyis that all schemes asymptotically converge towards the same con- finedsolutionwhatevertheorderoftheconfinementscheme,but therateof convergencetowardsthe asymptoticsolutiondepends onthe orderofthescheme.The resultingschemes thuscombine confinementpropertyandhighaccuracy. Thetopicofthepresent paperis the extension of the 3rd-orderVC schemedeveloped in Petropoulos), [email protected] (P.

Cinnella).

[23] for the scalar linear advection equation to the Euler/RANS equations.Apreliminarystudyincludingapplicationstohelicopter- relatedproblemswaspresentedin[24].

Inthefirstpartofthispaper,thetheoreticaldevelopmentsare presented, starting with a brief reminder of the basic 3rd-order confinementschemeforthelinear1Dtransportequation,andthen describing its extension to thesystemof governingequations for gasdynamics.Inafollowingpart,thenewapproachistestedover simple test cases. First, a convergence study is performed for a steady2Disentropicvortexinordertoevaluatetheactualaccuracy of thevarious schemesused, withandwithout vorticity confine- ment. Thenthe advectionofan inviscid 2D vortexover longdis- tancesisconsidered.Thehigh-orderVCschemeiscomparedwith theoriginalconfinementscheme.Finally,a2-Dairfoil/vortexinter- action is studied to investigatethe capabilitiesof 3rd-order con- finementalongsidehigh-orderschemestocapturethephysicsofa morerealisticapplication.

2. PresentationoftheVCscheme

2.1. VCmethodforthe1Dlineartransportequation

TheVCmethodologywasdevelopedbySteinhoff etal.basedon thetheory ofnonlinearsolitary waves.Here weadoptadifferent approach,leadingtosimilarresults.Letusconsiderthesimplecase ofthe1Dtransportequation:

∂^u

∂^{t +c}∂^u

∂^{x =0 (1)}

withc >0 theconstanttransport speedof aquantity u≥0.We start fromastandard 1st-orderupwinddiscretizationon aCarte- siangridofconstantspacestep^{xandtimestept:}

uⁿ_j⁺¹=uⁿ_j−σ δ^un_j−1/2 (2) where uⁿ_j=u(^jx,n^t) ^{isthe numericalsolution,}δ^uj−1/2=u_j− u_j−1,andσ=^c_x^t denotestheCFLnumber,withσ <1forstabil- ity.

The original VC2 term uses a 2nd-difference ofthe harmonic mean between two successive grid values to correct the highly- diffusive1st-orderdiscretizationoftheaboveequation.Introducing theharmonicmeanofthesolutionattwoadjacentgridpoints,ex- pressedas:h(^uj,u_j−1)= _u^2ujuj−1

j+u_j−1 foru_ju_j−1>0andh(^uj,u_j−1)=0 otherwise,the1st-orderVCdiscretizationof(1)writes:

uⁿ_j⁺¹=uⁿ_j−σ δ^unj−1/2+ε σ⁽σ−1)

2 δ^2h(^unj,uⁿ_j−1) ⁽³⁾

whereδ^2h=δ(δ^h)^andε^{isarealconstantcalledtheconfinement}

parameter. The flux difference correction due tothe VC2 termis thusexpressedby:

ε¹⁻σ

2 δ^2h(^unj,uⁿ_j−1) ⁽⁴⁾

Derivingtheequivalentpartialdifferentialequationfromthelinear differencesof(3)andleavingtheharmonicmeantermunmodified leadsto amixeddifferential/differenceequation representativeof thenumericalproblemwhichisactuallysolved:

∂^u

∂^{t +c}∂^u

∂^{x −(1−}σ)^cx 2

∂^2u

∂^{x2 −}ε δ^2h(u,_x^T₂^−u)

=0 (5)

Here,T₋u(^x)=u(^x−^x)^{isthebackwardshiftingoperatorsothat} h(^u,T₋u) ^{isa symbolic}representationof theharmonic meanbe- tween twosuccessivenodevaluesofu.Whenu=0,aTaylorex- pansionofthe2nddifferenceoftheharmonicmeancanbedone, givingasleadingtermofthetruncationerrorof(3):

(ε⁻¹)^cx1−σ

2

∂^2u

∂^{x2 (6)}

Sinceσ < 1,forε > 1negativedissipation isintroduced on the right hand side of (1) and the corresponding scheme, although 1st-orderaccurate,hasthecapabilitytoconserveindefinitelynon- trivialsolutionswhicharetransportedatthecorrectspeedbythe numericalscheme.Asshownin[23],thesesolutionshaveapulse shapeintheformsech(^k(^x−ct))^{andtheyareobtainedbybalanc-} ingthenumericaldiffusionofthefirst-orderschemeandthenon- linearconfinementtermatthediscretelevel.Asaresultofeq.(5), asymptoticpulsesolutionssatisfytheequality:

∂^2u

∂^{x2 −}ε δ^2h(u,_x^T₂^{−u) =0 (7)}

A sufficient condition for obtaining asymptotic solutions to the linear transport equation with confinement can be obtained by approximating ^∂2u

∂^x2 on the same stencil as the harmonic mean difference in (7). Precisely, setting ^δ2μu_x^j−1₂ ^/2, where μ^uj−1/2=

1 2

u_j−1+u_j

is the arithmetic averaging operator, this condition reducestothesimplerelationbetweenthearithmeticandthehar- monicmean:

μ^uj−1/2=ε^h(^uj−1,u_j) ⁽⁸⁾

Anontrivialandnonsingularsolutionto(8)isgivenby:

u_j=sech

kx_j

= 1 cosh

kx_j ⁽⁹⁾

where k is a positive real parameter such that ε=cosh^k₂^x. As shown in [23], the counterpart of the numerical solution (9) is u=_cosh(^1axx)^{,where a=kx. Because}ε=cosh^a₂ or,equivalently, a=2ln(ε+

ε²−1)^{it isclear that,fora prescribed} ε^,confined

solutions dependon themesh sizebecause thesignal is concen- tratedoverthesamenumberofcells, whateverthediscretization.

Ontheotherhand,itispossibletocompute theconfinementpa- rameter to keep a pulse solution close to the exact solution. In ordertoverifythisassumption,it ispossibletocompute thenu- mericalsolutionof(3)withtheinitial conditionu(^x)=_cosh(¹_2.09x), corresponding toan asymptoticsolution withε=1.6for^x=1. The computational domain is a segment of length L=100 with periodicityboundary conditionsat the left andrightends of the domain. The CFL number is set to σ=0.611, a typical value al- readyusedin[23] and[20].Thesimulation isperformedforvar- ious mesh refinements. Hereafter we consider ^x=1, ^x=1/4 and^x=1/16.Forthe refined grids, theconfinement parameter correspondingtoε=cosh^a₂ isε=1.03andε=1.002for^x=1/4 and^x=1/16respectively.Thesolutionsobtainedafterthesignal hastraveledadistancect=6110arecomparedwiththeexactone inFig.1,usingthevalueofεcorresponding tothemeshcellsize ofthesimulation.Thegoodpreservationoftheinitialconditionin thecomputationcanbenotedforall meshcellsizes.Additionally, solutionsfortwootherinitialconditionscorrespondingtoaGaus- sianfunctionwithtwovaluesofthestandarddeviation,2and1/2, were computedusing ^x=1/2 andε=1.14. Forthe purposeof comparison,all initial conditions hadthe same integral versus x, equaltothatofthehyperbolicsecantinitialcondition.Theresults attimect=6110arecomparedwiththesolutionusingthehyper- bolicsecantasinitialconditioninFig.2.Forreadability,theresults havebeenshiftedbyx=5fromoneanother,andthecorrespond- ingexactsolutionsaswell.Itisclearthatwhatevertheinitialcon- dition,all pulse solutions convergetowards the same asymptotic solution which is approximately described by the hyperbolic se- cantu=_cosh(^1axx)^{.Thisisconfirmedbythetime evolutionofthe} discreteenergyplottedinFig.3:allinitialconditionsconvergeto- wardsa constantenergylevel,equaltothat ofthehyperbolicse- cantinitialcondition.

x

u

10 20 30

0 0.5 1

=1.6, x=1

=1.03, x=1/4

=1.002, x=1/16 exact

Fig. 1. Asymptotic solution at time c t = 6110 with confinement parameter adapted to grid spacing.

x

u

15 20 25 30 35

0 0.5 1

1.5 sech

gaussian s.d.=2 gaussian s.d.=0.5 exact

Fig. 2. Asymptotic solution at time c t = 6110 for various initial conditions.

Finally,it isimportanttopointout that ifthe VC2term(4)is replacedbyanylinearsecond-differenceapproximationinEq.(3), numerical stabilityrequires ε=1. Since the leading termin the TaylorseriesdevelopmentisidenticalbetweenthenonlinearVC2 andanylinear2nd-difference, (6) showsthat the linearschemes are at least 2nd-order accurate. According to the stencil used, theLax-Wendroff,Warming-BeamandFrommschemecanbe ob- tained. Then the leading term in the truncation error has a 3rd derivative andis dispersive. Further, all these linearschemes are dissipativewithaleadingdissipativetermgivenbythe4thderiva- tiveintheirTaylorexpansion.Asaresult,theyasymptoticallycon- vergetowardsaconstantsolutionforwhichtheinitialinformation isalmosttotallylost.

The VC2 scheme wasextended to higher odd orders in [23], usingthesame ideasunderlyingthefirst-order approach.Anon- linearharmonicmeanapproximationoftheopposite ofthelead- ingdissipativetermofthetruncationerrorofanodd-orderlinear schemeisaddedtothelinearpartofthescheme.Atthe3rd-order ofaccuracy,thediscretizationbecomes:

uⁿ_j⁺¹=uⁿ_j−σδ^unj−1/2+σ (σ−1)

2! δ^2unj−(σ+1)σ (σ−1) 3! δ^3unj−1/2

+ε (σ⁺¹)σ (σ−1)(σ−2)

4! δ^4h(^unj,uⁿ_j−1) ⁽¹⁰⁾

ct xuj2

10^-2 10^-1 10⁰ 10¹ 10^-1

10⁰ 10¹

sech

gaussian s.d.=2 gaussian s.d.=0.5

Fig. 3. Time evolution of energy norm for various initial conditions.

The3rd-orderVCfluxdifferencecorrectionisnow:

−ε⁽¹⁺σ)(¹−σ)(²−σ)

4! δ^4h(^unj,uⁿ_j−1) ⁽¹¹⁾

Accordingly,theleadingterminthetruncationerroris:

−(ε⁻¹)^cx3(¹+σ)(¹−σ)(²−σ) 4!

∂^4u

∂^{x4 (12)}

Whenε >1,wehaveagainnegativedissipationaddedtotheright hand side of(1), ensuring the confinementproperty of this3rd- order accurate scheme.In [23],we showed that the VC schemes extendedtoanyoddorderparecharacterizedbythesameasymp- toticpulsesolutionswhich canbe transportedexactlyatthedis- cretelevel.Thisiseasilyseenby derivingtheequivalentequation similarlyforthepth-orderVCscheme:

∂^u

∂^{t +c}∂^u

∂^x+(−1)

p+1 2 O(^xp)

∂^p+1u

∂^{xp+1 −}ε δ^p+1h(_x^u,_p+1^T−u)

=0 (13) Approximatingthelinear derivative withthesamedifference op- erator asthe oneused fortheharmonicmeanleadsto thesame sufficientconditionaseq.(8),andthustothesameasymptoticso- lution.Theinterestofthehigher-orderschemesisthatanyinitial solution converges to this asymptotic solution more slowly than the 1st-order VC, so that the benefits of higher-accuracy and of confinementcanbecombined.Thisisshownintheexamplebelow usingthefollowing5th-orderdiscretizationwithconfinement:

uⁿ_j⁺¹=uⁿ_j−σδ^unj−1/2+σ (σ−1)

2! δ^2unj−(σ+1)· · ·(σ−1) 3! δ^3unj−1/2

+ (σ+1)· · ·(σ−2)

4! δ^4unj−(σ+2)· · ·(σ−2) 5! δ^5unj−1/2

+ε (σ⁺²)· · ·(σ−3)

6! δ^6h(^unj,uⁿ_j−1) ⁽¹⁴⁾ The5th-orderschemewasrunforthesameinitialconditionscon- sideredbefore(hyperbolicsecant, Gaussianswithstandarddevia- tion of2 and1/2). As showninFigs. 4 and5,the numericalso- lutionscorrespondingtotheGaussian initialconditionsevolveto- wardthe hyperbolicsecant asymptoticsolution(andthus deviate from the exact Gaussian solution) after a much longer travelling distance(ct>10⁴)thanthe1st-orderVCscheme,sothattheorig- inalshapeisstillreasonablywellconservedforct=6110forthe lessimpulsive initial condition. Thisis alsoconfirmed by inspec- tionofthediscreteenergyofthesolutionpresentedinFig.6.The