The numerical computation of turbulent flows

COMPUTERMETHODSINAPPLIEDMECHANICSANRENGINEERING3(1974)269-2890NORTH-HOLLANDPUBLISHINGCOMPANYTHENUMERICALCOMPUTATIONOFTURBULENTFLOWSB.E.LAUNDERandD.B.SPALDINGImperialCollegeofScienceandTechnology,DepartmentofMechanicalEngineering,~x~i~~t~o~Road,London,S,W.7,UKReceived13August1973Thepaperreviewstheproblemofmakingnumericalpredictionsofturbulentflow.Itadvocatesthatcomputatio-naleconomy,rangeofappii~b~ityandphysicalrealismarebestservedatpresentbyturbulencemodelsinwhichthemagnitudesoftwoturbulencequantities,theturbulencekineticenergykanditsdissipationratee,arecalculatedfromtransportequationssolvedsimultaneouslywiththosegoverningthemeanflowbehaviour.Thewidthofappli-cabihtyofthemodelisdemonstratedbyreferencetonumericalcomputationsofninesubstantiallydifferentkindsofturbulentflow.NomenclatureVanDriest’sconstantCurtetnumberdefinedby(3.1-1)CoefficientsinapproximatedturbulenttransportequationsSpecificheatatconstantpressureDiffusioncoefficientforquantity(pRateofdiffusivetransportofReynoldsstressConstantinnear-walldescriptionofvelocityprofile(-9)Functionaldefinedby(2.2-6)Turbulencekineticenergyuiuj/2LengthofenergycontainingeddiesFluctuatingcomponentofstaticpressureHeatfluxRadiusReynoldsnumberinpipeflowbasedonbulkvelocityandpipediameterRateofredistributionofReynoldsstressthroughpressurefluctuationsTurbulentReynoldsnumberk2/veTemperatureFluctuatingcomponentofvelocityindirectionxiMeancomponentofvelocityindirectionXiStreamwisevelocitynondimen~onalizedbyT,JPMeanstreamwisevelocityonaxisChangeinmeanvelocityacrossshearflow‘Vorticity’fluctuationssquaredCartesianspacecoordinate270B.E.Launder,D.B.Spaldirrg,ThenumericalcomputationofturbulentflowsY3:RadialwidthofmixingregionCoordinatenormaltowallGreekSymbolsERateofdissipationofturbulenceenergyKvonKarman’sconstantappearingin(2.1-11)pMolecularviscosity&TurbulentviscosityVKinematicviscosity4AgeneralizeddependentvariablePDensity0,EffectiveturbulentPrandtlnumberOtiEffectiveturbulentPrandtlnumberfortransportofQ,‘hMolecularPrandtlnumber7ShearstressSubscriptsijkSubscriptsdenotingCartesiancoordinatedirectionsiInnersurface0OutersurfacePValueatanodeadjacenttothewallWWallvalueSuperscript+Denotesquantitynon-dimensionalizedbymeansofv,r,,andp1.Introduction1.1.TheProblemTurbulentflows,whichareofgreatpracticalimportance,arethreedimensionalandtime-depen-dent.Computermethodsofsolvingthedifferenti~equationsoffluiddynastarewelladvancedevenforthree-dimensionaltime-dependentflows.Thenwhyisitthattherearenocomputermodelsofturbulentflowwhichdofulljusticetothefluiddynamicsandwhichcanbeappliedtopracticalproblems?Theansweristhatthenecessarycomputerstorageexceedsbymanyordersofmagnitudewhatiscurrentlyavailable,tosaynothingofthecomputertime,forimportantconstituentsofthetur-bulencephenomenontakeplaceineddiesoftheorderofamillimeterinsize,whilethewholeflowdomainmayextendovermetersorkilometers.Agridfineenoughtoallowaccuratedescriptionofaturbulentflowwouldthereforerequireanimmenseandtotallyimpracticalnumberofnodes.Yetthepracticalneedforcomputationofturbulentflowsispressing;tomeetit,“turbulencemodels”havebeeninvented.Theseconsistofsetsofdifferentialequations,andassociatedalge-braicequationsandconstants,thesolutionsofwhich,inconjunctionwiththoseoftheNavier-Stokesequations,closelysimulatethebehaviourofrealturbulentfluids.B.E.Launder,D.B.&&ding,Fhenumericalcomputationof~r~ulentflows271Agoodturbulencemodelhasextensiveuniversality,andisnottoocomplextodeveloporuse.Universalityimpliesthatasinglesetofempiricalconstantsorfunctions,insertedintotheequa-tions,providesclosesimulationofalargevarietyoftypesofflow.Complexityismeasuredbythenumberofdifferentialequationswhichthemodelcontains,andthenumberoftheempiricalcon-stantsandfunctionswhicharerequiredtocompletethem;increaseinthefirstcomplicatesthetaskofusingthemodel,increaseinthesecondthatofdevelopingit.Satisfactorycalculationproceduresandcomputersarenowavailableforsolvingdifferentiaiequations,onthescaleofthemeanmotion,forquitelargenumbers(e.g.20)ofsimultaneousequations.Themainobstaclestomodeldevelopmentarethereforethedifficultyofselectingwhichsetofdifferentialequationsismostcapableofprovidinguniversality,andthedifficultyofthenproviding,fromexperimentalknowledge,therequiredconstantsandfunctions.Inthepresentpaper,theauthorsdescriberecentworkonthedevelopmentofaparticulartur-bulencemodel,thatinwhichtwodifferentialequationsaresolved,thedependentvariablesofwhicharetheturbulenceenergykandthedissipationrateofturbulenceenergyE.Emphasisisgiventoaspectsofthemodelhavingimportanceforflowsadjacenttosolidwalls.Thisisofcoursenottheonlyavailableturbulencemodel.Othershavebeenreviewedinrecentworksbytheauthors[1,2]andothers(Harlow[3]andMellorandHerring(41).Amongsuchmodelsare:-Prandtl’s[51mixing-lengthmodel;theone-differential-equationmodelsofPrandtl[6],Bradshaw,FerrissandAtwell[7]andNeeandKovasznay[8];thetwo-differential-equationmodelsofKolmogorov[91,HarlowandNakayamaf101,Spalding[111,andJonesandLaunder[12I;andthemorecomplexmodelsofChou[131,Rotta[141,Davidov[15I,KolovandinandVatutin[161,HanjaliC[171andHan