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1. Direct numerical simulations of fundamental turbulent flows

1.1HighresolutionDNSofturbulentchannelflowTo investigate the small-scale statistics inwall-bounded

high-Reynolds-number turbulence,weperformedaseriesofdirectnumericalsimulations(DNS)ofturbulentchannelflows(TCF)with the frictionReynoldsnumberReτ up to 5120. In theDNS,wesolved the incompressibleNavier-Stokes (NS)equationsusingtheFourier-spectralmethodinthestreamwise(x)andspanwize(z)directionsand theChebyshev-taumethod inthewall-normal(y)direction.Thecomputationaldomain isπh×2h×πh/2 (smallbox)and2πh×2h×πh (largerbox) for theDNSwithReτupto2560andisπh×2h×πh/2(smallbox)fortheDNSwithReτ=5120.Inthefiscalyearof2012,wegenerateda

TCFdatabasewithReτ=5120usingtheTCFcodethatachievesaperformanceof6.1Tflops(11.7%ofthepeakperformance)on64nodesofES2.TheTCFwiththelargestvalueofReτwassimulateduntil time tas largeas14wash-out-times, tostudythesmall-scalestatisticsata statistically steadystateof theturbulence.Table1givesasummaryofourDNSdatabase.

Dataanalysisshowedthat thesmall-scalestatistics inTCFareinsensitivetowhethertheboxsizeissmall(S)orlarge(L),at least inourcases.TheDNSwithhighvaluesofReτexhibita rangewhere themeanstream-wisevelocityU fitswell tothelog-lawandtheenergydissipationrate isnearlyinverselyproportional to thedistance fromthewall.Thewidthof therange(inthewallunitnormalizedbyuτandh) increaseswiththe values of Reτ.InCase5S,themaximumTaylormicroscale

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

Takashi Ishihara GraduateSchoolofEngineering,NagoyaUniversity

Authors

Takashi IshiharaYukioKanedaKaoruIwamotoTetsuro TamuraYasuoKawaguchiTakahiro Tsukahara

GraduateSchoolofEngineering,NagoyaUniversity

Aichi Institute of Technology

MechanicalSystemsEngineering,TokyoUniversityofAgricultureandTechnology

InterdisciplinaryGraduateSchoolofScienceandEngineering,TokyoInstituteofTechnology

DepartmentofMechanicalEngineering,TokyoUniversityofScience

DepartmentofMechanicalEngineering,TokyoUniversityofScience

High-resolutiondirectnumericalsimulations(DNS)ofcanonical turbulencewereperformedontheEarthSimulator2.Theyinclude(i)high-Reynolds-numberturbulentchannelflowswiththefrictionReynoldsnumberupto5120and(ii)turbulentboundarylayersonsinusoidalwavywalls.Theyprovideuswithinvaluabledetailedinformationontherelatedturbulencephenomena.TheanalysesoftheDNSdatashowthefollowing.(1)Thespectrumofthecrosscorrelationbetweenthestreamwiseandwall-normalfluctuatingvelocitycomponents in thelog-lawregionagreeswellwiththecorrespondingspectrumobtainedin theinertialsub-rangeofhomogeneousshearturbulence.(2)IntheTBLwithasinusoidalwavywall,thedissimilaritybetweenmomentumandmasstransfersappears.Wealsoperformedtheturbulencesimulationsforenvironmentalandindustrialapplications;(iii)Large-scaleLESofturbulentflowsforstrongwinddisastermitigationand(iv)DNSofturbulentflowofnon-Newtoniansurfactantsolutionpassingcomplicatedgeometry.Theresultsof thesesimulationsshowthefollowing. (3)Byimposing the turbulentwindwithspecificcharacteristics, theLEScanbeusedfortheestimationofthewind-resistantperformanceofthespecifiedbuildingthat is locatedwithinmanybuildingsarrayeddensely.(4)Inachannelflowwithbluffbodiesoffiniteplatesforaviscoelasticfluid,thefrictional-dragreductionoccursjustbehindthebluffbody,becausetheKelvin-Helmholtzeddiesaredampenedsignificantly.

Keywords: High-resolutionDNS,turbulentchannelflow,turbulentboundarylayer,roughwall,LES, urbanturbulentboundarylayer,non-Newtonianfluid,dragreduction

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ReynoldsnumberRλ=370wasattainedaty+~1200inthelog-law

range.Intherange,thespectrumE12(k1)ofthecrosscorrelationbetween thestreamwiseandwall-normalfluctuatingvelocitycomponents fitswell to the -7/3power law in the inertialsub-range,wherek1 is thewavenumber in the streamwisedirection. Itspre-factorwasshown tobe ingoodagreementwith laboratoryexperimentsof turbulentboundary layerbySaddoughietal [1]andTsuji [2],andDNSofhomogeneousturbulentshearflowbyIshiharaetal[3].

2. DNS of turbulent boundary layer on rough wallsTurbulentboundarylayerflowoveraroughwall isoneof

canonical flows,while is important in fundamental turbulentresearches,practicalengineeringapplicationsandenvironmentalproblems.TheDNScodeoptimized forES2 isemployed toinvestigatetheeffectofasinusoidalwavywallsurfacewhichis a simplemodel of the roughness.The amplitudeof thesinusoidalwavywall,a,iskeptconstantinwallunits,whereasdifferentwavelengthsλare investigatedforλ /2a=12.5,15,22.5and45.

For the spatially developing boundary layer flowoverthe sinusoidalwavywall,weprovide adriver and amaincomputationaldomainsas shown inFig.1.Thedriverpartprovidesaninflowconditionofthemainpart,wheretherecyclemethod[4]isused.Themainparthasthesinusoidalwavywall.Theparallelandvectorizationefficienciesof thepresentDNScodeare98.43%and99.50%, respectively. In thisyear,wefocusonthedissimilaritybetweenthemomentumandthemass

transfers. Figure2visualizes thedistributionof thewallshearstress

and theSherwood number.TheSherwood number is thedimensionlessnumberofthemasstransfer.Thehighwallshearstressappearsonthetopofthewavywall.Ontheotherhand,theSherwoodnumber ishighnotonlyonthetop,butalsointhevalleyofthewavywall.Thisindicatesthatthedissimilaritybetweenthemomentumandthemasstransfersappears.

3. Large-scale LES of turbulent flows for strong wind disaster mitigationForthemitigationofstrongwinddisaster, largescaleLES

(Largeeddysimulation)hasbeenperformedusing theactualurbanmodel.Theactual shapeofbuildingand structure isindividuallyreproducedin thebroadareaofacity.Thewindcharacteristicsathighaltitudeaboveacityarespatiallyandtemporallyanalyzedanditsrelationwiththesurfaceroughnessiselucidated.By imposing the turbulentwindwith specificcharacteristics,LEShasbeencarriedoutfor theestimationofthewind-resistantperformanceofthespecifiedbuildingwhichis locatedwithinmanybuildingsarrayeddensely.Detailsofwindforcesactingon thesurfaceofbuildingwerediscussedwith indicating theoccurrenceof thepeakpressureat localareabythesurroundingbuildingorsomespecialgeometryofbuildingitself.

Figure3illustratesthenumericalmodelforLESofthewindflowsamong the tallbuildings in theactualcity.The targetbuildingfortheestimationofwindpressuresissurroundedby

Fig.1Computationaldomainsforturbulentboundarylayerflowandthesinusoidalwavywall.

Table1 ParametersusedintheseriesoftheDNSofTCF.histhechannelhalfwidth;Lx,Lz,fundamentalperiodiclengthsinthestreamwise(x)andspanwise(y)directions,respectively;Nx,Ny,Nz,thenumberofgridpointsinthex,y,z−directions;Δx

+andΔz+,themeshsize(normalizedbyuτandh)inthex,z−directions,andΔyc

+isthewall-normalmeshsizeatthechannelcenter.

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theothermanytallbuildings.For investigatingdetailsof thecomplex flowsamongmanybuildingsand theirwakeswithsmallerscale,theshapeofeachbuildingisdirectlyreproduced.Figure 4 shows the pressure distributions around the tallbuildingsurroundedbyother tallbuildingsatwinddirectionfromsouth.Somespecialcharacteristicswithbiasofthespatialdistributionofpressureareactuallyrecognizedaswecanfindin theexperimentaldata.ThisLESmodel is alsovalidatedbycomparingwith fieldmeasurementdataofwind-inducedoscillations.Furthermore,we investigatedetailsof a localflowpatternsamongbuildings(Fig.5)andprovideadominantroleoftheshapeandthedirectionofthespecifiedbuildingtodeterminethewindforces inviewof themitigationofstrongwinddisaster.

Recent architectural buildingshave avarietyof shapesbasedonuniquedesignerconcepts,and thecurvedsurfacesare frequentlyusedforbuildingwall.Here,asa typicalanda fundamentalcase in suchbuildings,acircularcylinder isfocusedon.TheflowcharacteristicsaroundacircularcylinderinrealistichighReynoldsnumberregionareinvestigatedbyuseoftheLESmodel.Asaresult,thepresentLESmodelsucceededin simulating the aerodynamic characteristics and flowcharacteristicsatthecriticalReynoldsnumbers(Figs.6and7).Thepresentcomputationsclarifiedandthecharacteristicsofthelocalliftcoefficientsassociatedwiththethree-dimensionalitiesofthewakestructuresinthespan-wisedirectionatthecriticalReynoldsnumbers.

4. DNS of turbulent heat transfer of non-Newtonian surfactant solution passing complicated geometryTurbulentdragreductionbysurfactantadditives in liquid

fluid flow isof importanceandprofitable forsavingenergyin the fluid transportation, such asoil-pipeline circuits ordistrictheatingandcooling(DHC)recirculationsystems[5].Thesurfactant solutions thatgive rise to thedrag reductionreveal viscoelasticity (and become non-Newtonian).Theviscoelastic-fluid flow through complex geometries hasattractedfundamentalscientificinterestandrelatedtonumerouspracticalapplications,suchasflowsassociatedwithchemical,

Fig.2Distributionofthewallshearstress(left)andtheSherwoodnumber(right)onthewavywallsurface(λ/2a=12.5).

Fig.3LESmodelofwindflowsinthecity.

Fig.4Pressuredistributionsandflowpatternsaroundthetallbuilding(Southwind).

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pharmaceutical, foodprocessing,andbiomedicalengineering,where theanalysisanddesigningfor theirpipe-flowsystemsaremoredifficultthanforitsNewtoniancounterpart.However,complexfeaturesandbehavioroftheturbulentviscoelasticflowthroughcomplicatedgeometriesarestillunclear.Thepresentstudyisaimedtoaddressthisissue.

BasedonGiesekus’viscoelastic-fluidmodel [6],DNSofturbulentflowofaviscoelasticfluidpassingfiniteplaneshasbeencarriedout.Thecomputationalmethodsweusedherewerebasically samewith those inTsukaharaetal. [7].Thegoalof thiswork is tobetterunderstand the fluid-dynamicscharacteristics of the viscoelastic turbulent flow behindbluff bodies.Majordifferencesbetween thepresent studyandpublishedworkson smoothchannelsare related to thestreamwisevariationof the flowstateand theexistenceofsecondary-flowstructuresintheformoflarge-scalelongitudinalvortices.Therefore,theinstantaneousvortexstructuresandtherelevantmomentumtransferandfrictional-dragreductionwithinthestrongshearlayerjustdownstreamoftheorificeshouldbeexplored.

Figure 8 shows instantaneous flow field both of theNewtonian fluid and the viscoelastic fluid,where vortexstructures are visualized by the second invariant of thedeformation tensor. In theNewtonian flow,manyvorticesare inducedjustbehindtheplateandtheydecaygraduallyaspropagatingdownstream.Theviscoelastic flow reveal lessvorticesand,inregionsfarfromtheplate,relativelylarge-scaleeddiesarevisible.Somevorticesthatelongateinthestreamwisedirection (i.e., longitudinalvortices) appeardominantly. Iffocusingonthestrongshearlayerjustdownstreamoftheplateedge,wecanobservethespanwiseprimaryKelvin-Helmholtz(K-H)vortices inFig.8(a),butalmostabsent inFig.8(b)sothat small-scaleeddiesdonotoccur.This isconsistentwithourpreviousstudyontheorificeflow[7],whichreportedtheK-Hvorticesdecayedquicklydownstreamof theorifice inviscoelastic fluid.Figure9 shows the reduction rateof thestreamwisedragforce,andFig.10showsthelocalskinfrictioncoefficientateitherz* = z/δ=0(plate-freeregion)or3.2(theplatecenter).From those results,wemaydrawconclusions

Fig.5Flowpatternsaroundthetallbuilding(Northwestwind).

Fig.7 Time histories of lift coefficients of a circular cylinder(Re=2×105).

Fig.6Wakestructuresofacircularcylinder(Re=2×105).

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thata largedragreductionoccursat therib locationandthatasignificantreductionoccurs insomeextentbehindtheribs,wherewall-normalandspanwisevorticesaresuppressed.

References[1] S.G.SaddoughiandS.V.Veeravalli,“Local isotropyin

turbulentboundary layersathighReynoldsnumber,”J.FluidMech.,268,333-372(1994).

[2] Y.Tsuji, “Large-scaleanisotropyeffecton small-scalestatisticsoverroughwallturbulentboundarylayers,”Phys.Fluids,15(12),3816-3828(2003).

[3] T. Ishihara,K.Yoshida, andY.Kaneda, “Anisotropicvelocity correlation spectrum at small scales in a homogeneous turbulent shear flow,”Phys.Rev.Lett.,88(15),154501(2002).

[4] T.S.Lund,X.Wu,andK.D.Squires, “Generationofturbulent inflowdata for spatially-developingboundarylayer simulations,” J.Comput.Phys.,140 (2),233-258,(1998).

[5] H.Takeuchi,“DemonstrationtestofenergyconservationofcentralairconditioningsystemattheSapporocityofficebuilding−Reductionofpumppowerbyflowdragreductionusingsurfactant,”Synthesiology−Englishedition,vol.4,136-143,(2012).

[6] H.Giesekus,“Asimpleconstitutiveequationforpolymerfluidsbasedon the conceptofdeformation-dependenttensorialmobility,” Journal ofNon-NewtonianFluidMechanics,vol.11,69-109,(1982).

[7] T.Tsukahara,T.Kawase,andY.Kawaguchi,“DNSofviscoelasticturbulentchannelflowwithrectangularorificeatlowReynoldsnumber,”InternationalJournalofHeatandFluidFlow,vol.32,529-538,(2011).

Fig.8 Instantaneousvortexstructuresaroundtheribonthebottomwall: iso-surfacesofthesecondinvariantofthedeformationtensor.Themainstreamdirectionisformtop-lefttobottom-right.

(a)Newtonianfluid (b)Viscoelasticfluid

Fig.10Streamwisedistributionofthelocalskinfrictioncoefficient.Thefiniteplateislocatedatx* = 6.4.

Fig.9 Spanwisedistributionof thepercentdrag reductionaveragedinthestreamwisedirection.Thefiniteplate isspannedinz* = 1.6–3.2.

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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

石原　　卓　　名古屋大学　大学院工学研究科

著者石原　　卓　　名古屋大学 大学院工学研究科

金田　行雄　　愛知工業大学

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、（i）高レイノルズ数（壁摩擦速度に基づくレイノルズ数 5120）の平行平板間乱流、 （ii）正弦波状壁面上の乱流境界層を含む、カノニカルな問題の大規模直接数値シミュレーション（DNS）を実施した。これらの DNS は , 関連する乱流現象に対して詳細で有益な情報を与えるものである。これらの DNS で得られたデータを解析することにより（1）高レイノルズ数壁乱流の対数則領域で得られる流れ方向の速度揺らぎと壁垂直方向の速度揺らぎの相関スペクトルが単純剪断乱流の慣性小領域で得られるスペクトルとよく一致すること、および、（2）正弦波状の壁における壁乱流の壁面摩擦と汚染物質拡散の非相似性を示すことを見出した。また、我々は環境や工学的な応用問題に対する乱流数値計算として、（iii）実際の都市を対象とした、環境・防災問題の低減化をめざした高解像度大規模乱流のラージ・エディ・シミュレーション（LES）、（iv）リブ列を有する複雑流路内乱流の DNS 実施と粘弾性流体の熱流動特性評価を実施した。これらにより、（3）LES において都市上空における強い風の時空間構造の特性を有する流れを用いることで、都市部における建物群の中にある当該建築物の耐風性能評価が可能であること、および、（4）リブ列を有する平行平板間の粘弾性流体乱流においては、同じ境界条件におけるニュートン流体の流れと異なり、リブ後方でケルビン・ヘルムホルツ不安定によって発生する渦が著しく抑制され、その結果として抵抗低減が起きることが判明した。

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