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1. Direct numerical simulations of fundamental turbulent flows

1.1 HighresolutionDNSofturbulentchannelflowIn order to study the small-scale statistics of high-Reynolds

numberwall-bounded turbulence,we performed a directnumericalsimulation(DNS)of turbulentchannelflow(TCF)of an incompressible fluid obeying the Navier-Stokes (NS) equations,inacomputationalboxwithstreamwise(x) and span-wise (z) periodicities (Lx=2πh and Lz=πh), ataReynoldsnumber Reτ =2560 based on the friction velocity uτ and the channelhalf-widthh. We achieved the sustained performance of 6.1Tflops(11.7%ofthepeakperformance)intheDNSofTCFon2048x1536x2048gridpointsusing64nodesofES2.Thenumbersofgridpointsaswellasthecomputationaldomainsizein the x and zdirectionsaretwiceaslargeasourpreviousDNS.

TheDNS isbasedon theFourier-spectralmethod inx- and z-directions,andtheChebyshev-taumethodinthewall-normaly-direction.Thealiaserrorsareremovedbythe3/2rule.TheNSequationsareexpressedintermsofthewall-normalvorticitycomponentsand theLaplacianofwall-normalvelocity.Timeadvancement is accomplished by a third-order Runge-Kutta methodfortheconvectiontermandthefirst-orderimplicitEulermethod for the viscous terms. Our DNS has been advanced up to t = 2.2 tw,wheretwisthewash-outtime.Wehavetoperformthe DNS until t > 10tw toobtainmorereliablestatistics,andtoelucidatethefiniteboxsizeeffectonthepossibleuniversalityinthe small-scale statistics.

Figure 1 shows the comparison between compensatedlongitudinal spectra E11(kx) and E33(kz) obtained in our previous DNS in the smaller domain; E11 is for the streamwise (x)

Direct Numerical Simulations of Fundamental Turbulent Flows with the World's Largest Number of Grid-points and Application to Modeling of Engineering Turbulent Flows

Project Representative

Yukio Kaneda GraduateSchoolofEngineering,NagoyaUniversity

Authors

Yukio KanedaTakashiIshiharaKaoruIwamotoTetsuroTamura

YasuoKawaguchiTakahiroTsukahara

GraduateSchoolofEngineering,NagoyaUniversity

GraduateSchoolofEngineering,NagoyaUniversity

MechanicalSystemsEngineering,TokyoUniversityofAgricultureandTechnology

InterdisciplinaryGraduateSchoolofScienceandEngineering,TokyoInstituteof

Technology

DepartmentofMechanicalEngineering,TokyoUniversityofScience

DepartmentofMechanicalEngineering,TokyoUniversityofScience

Weperformedhigh-resolutiondirectnumericalsimulations(DNSs)ofcanonicalturbulentflowsontheEarthSimulator2.Theyinclude(i)turbulentchannelflow,(ii)quasi-staticMHDturbulenceinanimposedmagneticfield,and(iii)turbulentboundarylayeronsinusoidalwavywalls.TheDNSsprovideinvaluabledataforthefollowingstudies,respectively;(1)thelocalanisotropyinsmall-scalestatisticsinthelog-lawlayerofturbulentchannelflow,(2)anisotropyandintermittencyofquasi-staticMHDturbulenceinanimposedmagneticfield,and(3)theeffectofthewavelengthofthesinusoidalwavywallupontheturbulentstatistics.BytheanalysisoftheDNSdata,itwasshownthat(1)thelocalanisotropydecreaseswiththedistancefromtheboundarywall,(2)themagneticfieldenhancestheintermittency,and(3)thepressuredragdecreaseswiththewavelengthofthesinusoidalwavywall.Wealsoperformedthefollowingturbulencesimulationsforenvironmentalandindustrialapplications;(i)LargeEddySimulationofturbulentboundarylayeroverhomogenousvegetationfieldusingahybridLES-RANSmodelwhichcanrepresentappropriatelyandefficiently theroughnessconditionongroundsurface,and(ii)DNSof turbulentflowsofnon-Newtoniansurfactantsolutioninachannelwithrectangularorifices.By(ii),wecouldestimatethecharacteristicsoftheheattransferassociatedwiththedragreduction.

Keywords: High-resolutionDNS, turbulentchannel flow,MHDturbulence, turbulentboundary layer, roughwall,LES,urbanturbulentboundarylayer,non-Newtonianfluid,dragreduction

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velocity component and E33isforthespan-wise(z) component. Theclassicalvalue for theKolmogorovconstantC1 = 0.5 is alsoshown inFig.1. It is shown that there isawavenumberrangeinwhicheachspectrumisnotfarfromtheK41spectra.ThedifferencebetweenE11(kx) and E33(kz) isseeninFig.1 todecreasewiththeincreaseoftheTaylorscaleReynoldsnumberReλ (Note that Reλisafunctionofthedistancefromthewall).

1.2 DNS of quasi-static magnetohydrodynamic (MHD) turbulence

Low-magnetic-Reynolds-numbermagnetohydrodynamic(MHD)turbulenceinanimposedmagneticfieldwidelyexistsin industrialapplications,suchaselectro-magneticprocessing

of materials in metallurgical industry. When the magnetic Reynoldsnumber issufficientlysmall,wecanapply theso-calledquasi-staticapproximationtotheMHDturbulence.

Quasi-staticMHDturbulenceischaracterizedbymultiscaleanisotropy and intermittency. Wavelet representation is an efficientwaytoanalyzesuchintermittentdata,sincewaveletsarewell localized inspace, scaleanddirection.Toquantifyintermittencyinanisotropicturbulence,Boset al. [1] introduced scale- and direction-dependent statistics using three-dimensional orthonormaldiscretewavelets.

In this study,we performed high-resolutionDNSs ofincompressible quasi-staticMHD turbulence at the twointeractionparameters,i.e.N=1and3,with5123 grid points on theESandexamined theanisotropyand intermittencyof theobtainedDNSfields,usingthescale-anddirection-dependentstatistics [1].

We found that for the N=3,theimposedmagneticfieldplaysa major role on the increase of intermittency in the direction paralleltothemagneticfield.Thedetailsofthecurrentstudyareshownin[2].

2. DNS of turbulent boundary layer on rough wallsTurbulentboundarylayeronroughplatesisoneofthemost

important problems in fundamental turbulent heat transfer research,practicalengineeringapplicationsandenvironmentalprocesses.DNSof turbulentboundary layeron roughwallshasbeenbarelyperformedcomparedwith thatofotherwall-boundedturbulencesuchasturbulentchannelflows.

In this study, direct numerical simulation of turbulentboundary layerwithseveral sinusoidalwavywallshasbeenperformedinordertoinvestigatetheeffectofthewavelength

Fig.1 Comparisonbetweencompensated longitudinalspectraE11(kx) and E33(kz),aty

+=100,200,400and600intheTCFobtainedbytheDNSwith1024x1536x1024gridpointsandReτ =2560. ηisKolmogorov's lengthscale.Solidlineis theK41spectrumk5/3E(k)/ε2/3 = C1withC1 = 0.5.

Fig.2 Computationaldomainsforturbulentboundarylayeronseveralsinusoidalwavywalls.

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ofthesinusoidalwavywall,λ,upontheturbulentstatistics.Theamplitudeof thesinusoidalwavywall,a,waskeptconstantinwallunits,andthewavelengthwasset tobea /λ=0.011,0.022and0.033.Forthespatiallydevelopingboundarylayersonsinusoidalwavywalls,weprovidedadriversectionwithaflatwallandananalysissectionwithasinusoidalwavywallasshowninFig.2.Turbulent inflowconditionsfor thedriversection are generated by rescaling the turbulent boundary layer atsomedistancedownstreamoftheinflowandbyreintroducingtherecycledmeanprofileandfluctuationfield.This techniquefollowsthoseofKongetal.[3]andLundetal.[4].Turbulentinflowconditionsfor theanalysissection,on theotherhand,aregeneratedbyexactlycopyingaturbulentfieldofthedriversection.Theparallelandvectorizationefficienciesare98.43%and99.50%,respectively.

Theaverageof thewall shearstresshardlychangeswithdecreasingthewavelength,whilstthefrictioncoefficient,whichwasdefinedasthesummationof thewallshearstressandthepressuredrag,was increasedwithdecreasingthewavelengthowingtotheincreaseofthepressuredrag(notshownhere).

3. Application of LES of turbulent flows to urban environmental and strong wind disaster problemsBasedonthefundamentalknowledgeofturbulentflows,we

extendtheLEStechniquestoatmosphericphenomenaappearingasanenvironmentalaswellasastrongwinddisasterproblemwhicharestronglyrelatedwiththehumansociety.

Firstly,LargeEddySimulation(LES)ofaturbulentboundaryflowoverhomogenousvegetationfieldwasperformedusinghybridLES-RANSmodelwhichcan representappropriatelyandefficientlytheroughnessconditionongroundsurface[5].InLESofboundarylayerflowsovervegetationfields, leavesandplantsaretoothintoresolvethembythesufficientnumberofgridpoints.So, theeffectof thoseleavesandplantsontheflowmustbe treatedwithanartificialmodel.The turbulenceclosuremodelforplantcanopyflowsusedherewasproposedbyHiraokaandOhashi[6],whichisformulatedbasedonRANS(k-ɛ) turbulencemodel.Thecomputationaldomain is2.5kmlongx0.4kmwideanditshorizontalresolutionis5mlongx2.5mwide.Thequasi-periodicboundaryconditionisemployedinstreamwisedirection(Fig.3).Theboundarylayerthicknessreachestoapproximately500mandthemeanvelocityprofile

Fig.4 Meanvelocityandturbulenceintensityprofiles.

Fig.5 Contourofstreamwisevelocity.

(a) RANS region

(b) LES region

Fig.3 Computationalregionandconceptofthequasi-periodic boundary condition for a turbulent boundarylayerflowovervegetationfield.

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and turbulence intensity reasonably fit to those obtained by thepreviousexperimentalstudy(Fig.4). Itcanbeseen thatthe turbulence structure in LES region is very small and fine comparedtothatinRANSregion(Fig.5).

Recent architectural buildings have a variety of shapes basedonuniquedesignerconcepts,and thecurvedsurfacesare frequentlyusedforbuildingwall.Here,asa typicalanda fundamentalcase in suchbuildings,acircularcylinder isfocusedon.Theflowcharacteristicsaroundacircularcylinderin realistic high Reynolds number region are investigated by use oftheLESmodel.Asaresult,thepresentLESmodelsucceededin accurately simulating the drastic change of aerodynamic coefficient (Fig.6).Also,detailsof the flowstructuresnearseparatingandreattachingregionareclarifiedbyvisualizationofthecomputeddata(Fig.7).

4. DNS of the turbulence in non-Newtonian surfactant solutionTheeffectofpolymerorsurfactantadditiveson turbulent

flowhas receivedmuch attention frombothpractical andscientificperspectivessincethediscoveryinthe1940s,thatis,small additive concentration can lead to significant reduction indragof50%orgreater[7].Thisphenomenonisofpracticalimportance and has recently been implemented in several industrialsystemstosaveenergy.Ingeneral, thesolutionusedasaworkingfluidforsuchadrag-reducingflowisaviscoelastic(non-Newtonian) liquid.Thepropertiesof the liquidsolutionmeasuredeveninsimpleshearorextensionalflowsareknowntoexhibitappreciablydifferentfromthoseofthepuresolvent.Thegoalofthepresentworkistobetterunderstandthephysicsofviscoelastic turbulent flow,particularly in thecontextofcomplicatedflowgeometry.

ManyDNS of polymer-induced drag reductionwereperformed for various types of canonical flows, such as

Fig.6 Drasticchange indragcoefficientat theveryhighReynolds numbers.

Fig.7 Three-dimensional flow structures nearseparating and reattaching region.

Fig.8 Instantaneousflowandthermalfieldinviscoelasticturbulentflowthrougharectangularrib (at x=0with0.1δ thickness):green isosurface,vortex identifiedby thesecondinvariantofstraintensor;contours, temperatureorwallheatflux.Thermalboundaryconditionistheconstanttemperaturedifferencebetweentwowalls(y=0and2δ),theperiodic condition is applied in x and z.

Fig.9 LocalNusselt-numberprofile as afunction of x, fordifferentPrandtlnumbers.Alsoshownareestimatedvalues in the case of a smooth channel at the same bulk Reynolds number.

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isotropicturbulence,shear-driventurbulence,channelflow,andboundarylayer.Althoughflowsthroughcomplicatedgeometrieshave been studied by a number of researchers for the laminar regimeforbothNewtonianandviscoelastic fluids [8], thoseunder turbulentconditionshavereceivedmuchlessattention,exceptthatMakinoet al. [9]carriedoutDNSoftheturbulent'Newtonian'flowinachannelwithperiodicrectangularorifices.Further,itshouldbenotedthat,totheauthors'knowledge,therehasneverbeenanyDNSofturbulentviscoelasticflowwiththeorifice partly due to the Hadamard instability in viscoelastic-flowcalculations[10].Inthepresentstudy,weexecutedDNSofaviscoelasticfluidinthesamegeometrywiththatofMakinoet al. [9],usingacompositeflux-limiter(minmod)schemetothe convective term in the Giesekus-model constitutive equation andapplyingfinergridsrelativetoexistingsimilarworks.

Majordifferencesbetweenthepresentstudyandpublishedworks on smooth channels are related to the streamwisevariationoftheflowstateandthemainareaswhereturbulenceisproduced.Therefore,theinstantaneousvortexstructuresandtherelevantmomentumandheattransportswithinthestrongshearlayerjustdownstreamoftheorificewillbeexplored.Figure8presentsaninstantaneoussnapshotofeddies,withemphasisontheorificedownstream,andsurfacedistributionsoftemperatureandwallheatflux,revealinghighheatfluxesonthewallsurfaceunderthevortexmotions.It is interestingtonotethatthewallheat fluxalsobecomes intermittentlyhigh fardownstreamoftheorifice,namely,atx=8-10,wherenoapparenteddyisobserved. Such heat-transfer enhancement leads to an increase oflocalNusseltnumbercomparedwiththeNewtoniancase,ascanbeseeninFig.9.Therefore,wedrawaconclusionthatthisgeometrygives rise to thedissimilaritybetweenmomentumand heat transports and the advantage of the heat transfer comparedwiththesmoothchannel.However,thepresentbulkReynolds number (Rem~1200)wasconsiderably lower thanthosecorrespondingconditionsunderwhichdragreductioninpracticalflowsystemsisobservedwithdiluteadditivesolutions.It isnecessarytofurthercalculateviscoelasticflowsathigherReynolds numbers andwith awide range of rheologicalproperties.

References[1] W.Bos,L.Liechtenstein,andK.Schneider,"Directional

and sca le -dependen t s t a t i s t i c s o f quas i - s t a t i c magnetohydrodynamic turbulence,"Phys.Rev.E76,046310,2007.

[2] N.Okamoto,K.Yoshimatsu,K.Schneider,andM.Farge,"Directionalandscale-dependentstatisticsofquasi-staticmagnetohydrodynamicturbulence,"ESAIM:Proceedings(accepted).

[3] Kong,H.,Choi,H., andLee, J.S., "Directnumericalsimulationof turbulent thermalboundary layers,"Phys.Fluids,12,2666,2000.

[4] Lund,T.S.,Wu,X.,andSquires,K.D.,"Generationofturbulent inflowdataforspatially-developingboundarylayersimulations,"J.Comput.Phys.,140(2),1998,233-258.

[5] F.Hamba,"AHybridRANS/LESSimulationofTurbulentChannelFlow,"Theoret.Comput.FluidDynamics,16,387-403,2003.

[6] H.HiraokaandM.Ohashi, "Ak-ɛ turbulenceclosuremodelforplantcanopyflows,"Proc.ofthe4thInt.Symp.onComp.WindEng.,693-696,2006.

[7] A.Gyr andH.W.Bewersdorff,Drag reduction of turbulent flows by additives,KluwerAcademicPublisher,Dordrecht,Netherlands,1995.

[8] P.J.Oliveira,"Asymmetric flowsofviscoelastic fluidsin symmetric planar expansiongeometries," J.Non-NewtonianFluidMech.,114,33-63,2003.

[9] S.Makino,K. Iwamoto,andH.Kawamura,"Turbulentstructuresandstatistics in turbulentchannel flowwithtwo-dimensionalslits,"Int.J.Heat&FluidFlow,29,602-611,2008.

[10] D.D. Joseph,Fluid dynamics of viscoelastic liquids,Springer-Verlag,NewYork.,1990.

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乱流の世界最大規模直接数値計算とモデリングによる応用計算

プロジェクト責任者

金田　行雄　　名古屋大学　大学院工学研究科

著者金田　行雄　　名古屋大学　大学院工学研究科 石原　　卓　　名古屋大学　大学院工学研究科

岩本　　薫　　東京農工大学　工学府

田村　哲郎　　東京工業大学　大学院総合理工学研究科 川口　靖夫　　東京理科大学　理工学部

塚原　隆裕　　東京理科大学　理工学部

地球シミュレータ（ES2）を用いて、乱流の規範的（カノニカル）な問題の大規模直接数値計算（DNS）を実施した。 具体的には（i） 世界最大レイノルズ数の平行平板間乱流 （ii） 磁場中の準定常 MHD 乱流、（iii） 正弦波状壁面上の乱流境界層の DNS である。これらの DNS は各々、（1） 高レイノルズ数壁乱流の対数領域における局所非等方性、（2） 磁場中の準定常 MHD 乱流における非等方性と間欠性、（3） 正弦波状壁面の波長が乱流等計量に与える影響、について研究するための貴重なデータを提供するものである。データ解析により、（1） 局所的な非等方性が壁から離れるに従って小さくなること、（2） 磁場が MHD 乱流場中の間欠性を強める働きがあること、（3） 正弦波の波長が小さくなるに従い圧力抵抗が増加することを見いだした。

我々はまた、これまでに得られた乱流統計の基礎的な知見に基づき、環境や工業的な応用問題として、以下の大規模数値計算を実施した。具体的には、（i） 適切かつ有効に粗面の条件を与えることが可能な LES と RANS のハイブリッドモデルを用いた計算により、一様な植生上で発達する乱流境界層の非定常解析を行った。また、（ii） 直方体型のオリフィスのあるチャネル中の非ニュートン流体の乱流の DNS により、抵抗低減に伴う熱伝達率特性の評価に成功した。

キーワード : 大規模直接数値計算 , 平行二平板間乱流 , MHD乱流 , 乱流境界層 , 粗面 , LES, 都市型大気乱流境界層 , 界面活性剤 , 抵抗低減