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and discussed in where a spectrally accurate approach is applied to the direct numerical simulation DNS of high Reynolds number flows

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25 pages
Abstract 1. Introduction (and discussed in [16]) where a spectrally accurate approach is applied to the direct numerical simulation (DNS) of high Reynolds number flows. Here we essentially focus on the spectral vanishing viscosity (SVV) method, which appears to be an e?cient stabilization technique possessing the property to preserve the spectral accuracy. It was initially * High Reynolds number flows are di?cult to compute, especially when using spectrally accurate nu- merical schemes. This directly results from the fact that spectral approximations are much less numerically di?usive than low-order ones, so that the non-artificially dissipated energy accumulates at the high spatial frequencies and finally leads to the divergence of the computations. One way to overcome this di?culty is to use stabilization techniques, but then the spectral accuracy of the algorithm is generally destroyed. This is, e.g., particularly obvious for approaches which essentially consist in adding some O?hr? hyper-viscous term, in the spirit of [14]. For a long time filtering techniques have also been proposed to overcome the stability problem. In the frame of spectral element approximations it is however essential to preserve the inter-el- ement continuity, as discussed in [2]. One of the most recent advances in this field has been proposed in [6] The possibility of using the spectral vanishing viscosity method for the spectral element computation of high Reynolds number incompressible flows is investigated.

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JournalofComputationalPhysics196(2004)680–704www.elsevier.com/locate/jcpStabilizedspectralelementcomputationsofhighReynoldsnumberincompressibleflowsChuanjuXua,RichardPasquettib,*aDepartmentofMathematics,XiamenUniversity,Xiamen,PRChinabLab.J.A.Dieudonne,UMRCNRS6621,UniversitedeNice-SophiaAntipolis,Nice,FranceReceived16June2003;receivedinrevisedform14November2003;accepted15November2003AbstractThepossibilityofusingthespectralvanishingviscositymethodforthespectralelementcomputationofhighReynoldsnumberincompressibleflowsisinvestigated.Anexponentiallyaccuratestabilizedformulationisproposedandthenappliedtothecomputationofthe2Dwakeofacylinder.Suchaformulationcanbeeasilyimplementedinexistingspectralelementsolvers,sinceonlymodifyingthecomputationoftheviscoustermwhilepreservingthesymmetryofthecorrespondingbilinearform.2003ElsevierInc.Allrightsreserved.1.IntroductionHighReynoldsnumberflowsaredifficulttocompute,especiallywhenusingspectrallyaccuratenu-mericalschemes.Thisdirectlyresultsfromthefactthatspectralapproximationsaremuchlessnumericallydiffusivethanlow-orderones,sothatthenon-artificiallydissipatedenergyaccumulatesatthehighspatialfrequenciesandfinallyleadstothedivergenceofthecomputations.Onewaytoovercomethisdifficultyistousestabilizationtechniques,butthenthespectralaccuracyofthealgorithmisgenerallydestroyed.Thisis,e.g.,particularlyobviousforapproacheswhichessentiallyconsistinaddingsomeOðhrÞhyper-viscousterm,inthespiritof[14].Foralongtimefilteringtechniqueshavealsobeenproposedtoovercomethestabilityproblem.Intheframeofspectralelementapproximationsitishoweveressentialtopreservetheinter-el-ementcontinuity,asdiscussedin[2].Oneofthemostrecentadvancesinthisfieldhasbeenproposedin[6](anddiscussedin[16])whereaspectrallyaccurateapproachisappliedtothedirectnumericalsimulation(DNS)ofhighReynoldsnumberflows.Hereweessentiallyfocusonthespectralvanishingviscosity(SVV)method,whichappearstobeanefficientstabilizationtechniquepossessingthepropertytopreservethespectralaccuracy.Itwasinitially*Correspondingauthor.E-mailaddress:rpas@math.unice.fr(R.Pasquetti).0021-9991/$-seefrontmatter2003ElsevierInc.Allrightsreserved.doi:10.1016/j.jcp.2003.11.009
C.Xu,R.Pasquetti/JournalofComputationalPhysics196(2004)680–704681developedfortheresolutionofhyperbolicequationsusingstandardFourierspectralmethods[21].Thenon-periodiccasewasthenconsideredintheframeofthespectralLegendreapproximation[13].Furtherrefinementshavebeenrecentlysuggested,throughtheuseofaspectralhyper-viscous(ratherthanaviscous)termorthroughtheredefinitionofthestabilizationterm[8].Recently,ithasalsobeensuggestedtousetheSVVmethod,initsfirstformulation,forthelarge-eddysimulationofturbulentflows[10,15].Inthispaper,ourgoalistocheckthecapabilitiesoftheSVVmethod,intermsofaccuracyandstability,whenitisimplementedinaNavier–Stokesspectralelementsolver.FirstweshowhowtoimplementtheSVVmethodintheframeofaspectralelementapproximation.Thefactthatcomplexmultidimensionalgeometriesandvectorvaluedfunctionsareconcernedmakethispointnon-trivial,sothatonecannotyetconsiderthatastandardwaytoimplementtheSVVmethodalreadyexists.Moreover,wesuggestusinganapproximateformwhichcanbeefficientlyimplementedinanyspectralelementsolver.Theadvantageofsuchanapproximateformisthatthecomputationalcostperiteration(time-stepifadirectsolverisused)isroughlythesamewithandwithoutSVVstabilization.Secondweconsideranellipticequationsolvedbyasteepanalyticalsolutionandshowthatthecon-vergenceresultsobtainedwiththespectralelementapproximationarecoherentwiththoseobtainedinthe1Dperiodiccase,whenusingFourierexpansions.AdetailedstudyoftheinfluenceoftheSVVtuningparametersontheconvergenceratesisprovided.Thenweconsidertheso-called‘‘Kovasznayflow’’,whichisanexactsolutionoftheincompressibleNavier–Stokesequations,andagaincheckthecapabilitiesoftheSVVmethod.Third,inordertonumericallydemonstratethestabilizationpropertyofthemethod,wecompute2Dwakesofacylinder,atReynoldsnumbersuptoRe¼1000,i.e.,muchhigherthanthecriticalvalueas-sociatedwiththe2D–3Dtransition(Re190).Finally,weconcludebyemphasizingtheinterestintheSVV-stabilizedspectralelementmethod(SEM)forthelarge-eddysimulation(LES)ofturbulentflows.2.StabilizedspectralelementformulationTheflowofanincompressibleNewtonianfluidisgovernedbythe‘‘incompressibleNavier–Stokesequations’’.ForanunsteadyflowinadomainXtheyread:Dtumr2uþrp¼sinXRþ;ru¼0inXRþ;ð1Þwhereu,pandsdenotethevelocity,pressureandsourceterm,respectively,Dtuthematerial(Lagrangian)derivativeofuwithrespecttotimetandmthedimensionlessviscosity(theinverseoftheReynoldsnumber).TheunsteadyNavier–Stokesequationsmustbeassociatedtoappropriateinitialandboundaryconditions,e.g.uðt¼0Þ¼0(fluidatrest)andujC¼0(no-slipconditionattheboundaryCofX),inordertosetupawell-posedproblemthatonecanthentrytosolvenumerically.If(i)complexgeometriesareconsideredand(ii)highaccuracyisdesired,thentheSEMiswellsuited(see,e.g.[11]).ThespectralelementapproximationoftheweakformoftheincompressibleNavier–Stokesequationsyieldsthefollowingsemi-discretevariationalproblem,tobesolvedateachtime-stepafterthetime-discretization:FinduN2XNandpN2MNsuchthatðDtuN;vNÞþmðruN;rvNÞðrvN;pNÞ¼ðsN;vNÞ8vN2XN;ð2ÞðruN;qNÞ¼08qN2MN;whereuN,pNandsNdenotethespectralelementapproximationsofu,pandsandwhereð;isusedtodenotethestandardL2ðXÞinnerproduct,withoutdifferenceifscalar,vectorialortensorialfunctionsare
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