第8讲PPT-(The-Finite-Volume-Method-for-Unsteady-Flow

Chapter8Chapter8TheFiniteVolumeMethodforTheFiniteVolumeMethodforTheFiniteVolumeMethodforTheFiniteVolumeMethodforUnsteadyFlowsUnsteadyFlows1Introduction()()()dididSφφφ∂+Γ+()()()divudivgradStφρφρφφ+=Γ+∂ttttΔΔ⎛⎞⎛⎞IntegraionoveracontrolvolumeandatimestepΔt()()ttttCVttAdtdVnudAdttρφρφ+Δ+Δ⎛⎞⎛⎞∂+•⎜⎟⎜⎟∂⎝⎠⎝⎠∫∫∫∫()ttttngraddAdtSdVdtφφ+Δ+Δ⎝⎠⎝⎠⎛⎞=•Γ+⎜⎟⎝⎠∫∫∫∫2tAtCVφ⎝⎠∫∫∫∫One-dimensionalUnsteadyheatOne-dimensionalUnsteadyheatconduction⎛⎞TTckSttxρ∂∂∂⎛⎞=+⎜⎟∂∂∂⎝⎠ttttttTT+Δ+Δ+Δ∂∂∂⎛⎞tttttttCVtCVtCVTTcdVdtkdVdtsdVdttxxρ+Δ+Δ+Δ∂∂∂⎛⎞=+⎜⎟∂∂∂⎝⎠∫∫∫∫∫∫ettttttTTTcdtdVkAkAdtSVdtρ+Δ+Δ+Δ⎡⎤⎡⎤∂∂∂⎛⎞⎛⎞=+Δ⎢⎥⎜⎟⎜⎟⎢⎥∫∫∫∫3ewwtttcdtdVkAkAdtSVdttxxρ=−+Δ⎢⎥⎜⎟⎜⎟⎢⎥∂∂∂⎝⎠⎝⎠⎣⎦⎣⎦∫∫∫∫One-dimensionalUnsteadyheatOne-dimensionalUnsteadyheatconduction()0ttPPTcdtdVcTTVρρ+Δ⎡⎤∂=−Δ⎢⎥∫∫first-orderbackwarddiffih()PPCVttρρ⎢⎥∂⎣⎦∫∫T0ttttilltdifferencingschemeTp0—temperatureattimeleveltTp—temperatureattimelevelt+Δt()0ttPWEPPPewTTTTcTTVkAkAdtxxρδδ+Δ⎡⎤⎛⎞⎛⎞−−−Δ=−⎢⎥⎜⎟⎜⎟⎝⎠⎝⎠⎣⎦∫()PEWPtttxxSVdtδδ+Δ⎝⎠⎝⎠⎣⎦+Δ∫∫4tSVdt+Δ∫One-dimensionalUnsteadyheatOne-dimensionalUnsteadyheatconductionNeedtomakeanassumptionaboutthevariationofTP,TE,dTandTWOneapproachistouseweightingparameterOneapproachistouseweightingparameter()01ttITdtTTtθθ+Δ⎡⎤+Δ⎣⎦∫()01TPPPtITdtTTtθθ⎡⎤==+−Δ⎣⎦∫5One-dimensionalUnsteadyheatOne-dimensionalUnsteadyheatconduction()0ttttPWEPPPewTTTTcTTVkAkAdtSVdtxxρδδ+Δ+Δ⎡⎤⎛⎞⎛⎞−−−Δ=−+Δ⎢⎥⎜⎟⎜⎟⎝⎠⎝⎠⎣⎦∫∫PEWPttxxδδ⎝⎠⎝⎠⎣⎦∫∫()()()()()000001eEPwPWeEPwPWPPPEWPPEWPkTTkTTkTTkTTTTcxSxtxxxxρθθδδδδ⎡⎤−−−−⎡⎤⎛⎞−⎢⎥Δ=−+−−+Δ⎜⎟⎢⎥Δ⎢⎥⎝⎠⎣⎦⎣⎦⎣⎦()()00kkkkx⎡⎤⎛⎞Δ⎡⎤⎡⎤⎢⎥()()0011ewewPEEWWPEWPPEWPkkkkxcTTTTTtxxxxkkxρθθθθθδδδδ⎡⎤⎛⎞Δ⎡⎤⎡⎤++=+−++−⎢⎥⎜⎟⎣⎦⎣⎦Δ⎝⎠⎣⎦⎡⎤Δ6()()011ewPPEWPkkxcTSxtxxρθθδδ⎡⎤Δ+−−−−+Δ⎢⎥Δ⎣⎦One-dimensionalUnsteadyheatOne-dimensionalUnsteadyheatconduction()()0011PPθθθθ⎡⎤⎡⎤=+−++−⎣⎦⎣⎦()()0011PWEPaaaTbθθ⎡⎤+−−−−+⎣⎦()0PWEPaaaaθ=++0xΔ0PxactρΔ=Δ0explicit01implicitθθ=71fullyimplicit0.5Crank-Nicolsonθθ==Explicitscheme00anduPPbSSTθ==+()0000PPWWEEPWEPPuaTaTaTaaaSTS⎡⎤=++−+−+⎣⎦0PPxaactρΔ==ΔTime-marchingmethodFirstorderbackwardFirst-orderbackwardRequirementofconvergenceandboundednessRequirementofconvergenceandboundedness00PWEaaa−−()22xxkΔΔ8PEWPxxxδδ==Δ()2or2xxkctctxkρρΔΔΔΔΔCrank-Nicolsonscheme0.5θ=00TTTTS⎡⎤⎡⎤+⎡⎤⎛⎞000022222⎡⎤⎡⎤+⎡⎤+⎛⎞=++−+−+⎢⎥⎜⎟⎢⎥⎢⎥⎝⎠⎣⎦⎣⎦⎣⎦11()01122PWEPPaaaaS=++−0xΔCentraldifferencingSecond-orderaccuracy0PxactρΔ=Δy0EWaaa+⎡⎤⎢⎥2xtcρΔΔekwk()01SSTT⎡⎤EabWa92Pa⎢⎥⎣⎦tckρΔPExδWPxδ()02upPPSSTT⎡⎤++⎢⎥⎣⎦Thefullyimplicitscheme1anduPPbSSTθ==+00PPWWEEPPuaTaTaTaTS=+++00PPWEpaaaaS=++−0xΔ0PxactρΔ=ΔUnconditionallystablefirst-orderaccuracyy10Examples200at0Tt==0at0,0Txtx∂==∂x∂0at,0TxLt==63L=2cmk=10W/m/Kandc=1010J/m/Kρ×已知：L2cm,k10W/m/K,andc1010J/m/Kρ×已知：11Temperaturedistributionat(i)t=40s;(ii)t=80s;(iii)t=120s?ExamplesTTckρ∂∂∂⎛⎞=⎜⎟⎝⎠txxρ⎜⎟∂∂∂⎝⎠AnalyticalAnalyticalsolution()()()()121,14expcos20021nnnnTxttxnαλλπ+∞=−=−−∑1n()21nπλ−=kcαρ=122nLλ=kcαρNumericalsolutionTTckρ∂∂∂⎛⎞=⎜⎟∂∂∂⎝⎠txxρ⎜⎟∂∂∂⎝⎠Explicitmethodp0dbθFornode2,3,400anduPPbSSTθ==+()0000aTaTaTaaaSTS⎡⎤=++−+−+⎣⎦()PPWWEEPWEPPuaTaTaTaaaSTS⎡⎤=++++⎣⎦0xΔ130PPaactρ==ΔNumericalsolutionFornode1()()0000PPEPTTkcxTTρ−⎡⎤Δ=−−⎢⎥ΔΔ⎣⎦()EPtxρ⎢⎥ΔΔ⎣⎦Fornode5()()()00002PPBPPWTTkkcxTTTTtxxρ−⎡⎤⎡⎤Δ=−−−⎢⎥⎢⎥ΔΔΔ⎣⎦⎣⎦14⎣⎦⎣⎦Numericalsolution()0000PPWWEEPWEPuaTaTaTaaaTS⎡⎤=++−++⎣⎦()PPWWEEPWEPu⎣⎦0PPxaacρΔ==PPaactρΔ()()22610100.00482210cxtskρΔ××Δ==×selectt=2s1025000004kΔ==Δ60.0040.0041010200002xxctρΔΔ=××=Δ15Numericalsolution00000Node1:20025175Node2-4:2002525150PEPPWEPTTTTTTT=+=++00Node5:20025125PWPTTT=+2tΔ2tsΔ=16NumericalsolutionExplicitmethodTimestep2s&8sInaccurateandunrealisticltisolution17Numericalsolutionimplicitmethodimplicitmethod()()00PPEPTTkcxTTtxρ−⎡⎤Δ=−−⎢⎥ΔΔ⎣⎦Fornode1txΔΔ⎣⎦()()()02PPBPPWTTkkcxTTTTtxxρ−⎡⎤⎡⎤Δ=−−−⎢⎥⎢⎥ΔΔΔ⎣⎦⎣⎦Fornode52txxΔΔΔ⎣⎦⎣⎦00PPWWEEPPuaTaTaTaTS=+++Discretisedequations0PWEPpaaaaS=++−xΔ180PxactρΔ=ΔNumericalsolution1025000004kx==Δ0PENode1:225T=25T+200TP60.0040.0041010200002xxctρΔΔ=××=Δ0PE0PNode2-4:225T=25T+25T+200TNode5:275T=25T+200T50WPWPBT+011022525000200252502500200TTTT−⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥−−⎢⎥⎢⎥⎢⎥22033044252502500200025250250200002525025200TTTTTT⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥=−−⎢⎥⎢⎥⎢⎥−−⎢⎥⎢⎥⎢⎥4405500025275200TT⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥−⎣⎦⎣⎦⎣⎦Iliitthdithltiftthtill19Implicitmethod→requiresthesolutionofasystemateachtimelevelExplicitmethod→straightforwardevaluationofasinglealgebraicequationNumericalsolution20Comparisonofnumericalresultswiththeanalyticalsolution(implicitmethod)Numericalsolution21ComparisonofimplicitandexplicitsolutionsforΔt=8sImplicitmethodfortwo-andImplicitmethodfortwo-andthree-dimensionalproblemsThefullyimplicitmethodisrecommendedforgeneralpurposeCFDcomputationsduetoitsstability.ckkkStxxyyzzφφφφρ⎛⎞∂∂∂∂∂∂∂⎛⎞⎛⎞=+++⎜⎟⎜⎟⎜⎟∂∂∂∂∂∂∂⎝⎠⎝⎠⎝⎠computationsduetoitsstability.txxyyzz∂∂∂∂∂∂∂⎝⎠⎝⎠⎝⎠00PPWWEESSNNBBTTPPuaaaaaaaaSφφφφφφφφ=+++++++0PWESNBTPpaaaaaaaaS=++++++−0PVactρΔ=Δ22()uPPbSSφ=+DiscretisationoftransientDiscretisationoftransientconvection-diffusionequationsDiscretisationoftheconvectiontermisthesameasinthesteady-state()()()divudivgradSφρφρφφ∂+=Γ+()()()divudivgradStφρφρφφ+Γ+∂()()()()uvwtρφρφρφρφ∂∂∂∂+++∂∂∂∂txyzSxxyyzzφφφ∂∂∂∂⎛⎞∂∂∂∂∂∂⎛⎞⎛⎞=Γ+Γ+Γ+⎜⎟⎜⎟⎜⎟∂∂∂∂∂∂⎝⎠⎝⎠⎝⎠Implicit/h