相变传热与流体流动数值分析课件.ppt

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1、Chapter2TheFiniteVolumeMethodforDiffusionProblems§2.1Finitevolumemethod(FVM)forone-dimensionalsteadystatediffusion（2.1）Firstconsidersteadydiffusionofapropertyφinaone-dimensionaldomain,definedinafollowingFigure.Theprocessisgovernedby:whereΓisthediffusioncoefficient;Sisthesourceterm;φAandφBa

2、reboundaryvaluesofφatpointAandB.whereAisthecross-sectionalareaofthecontrolvolumeface;ΔVisthevolume;istheaveragevalueofsourceSoverthecontrolvolume.andStep2Discretization（2.2）ThefirststepintheFVMistodividethedomainintodiscretecontrolvolumesStep1:Gridgeneration(2.3)(2.4)将上面式子带入离散后方程（2.2）可得：or

3、Thisequationcanberearrangedaswhere源项也可以做线性化处理：可认为是因变量的函数。在计算域中的每个节点均可获得上面的代数方程，将所有的代数方程和边界条件联立求解即可。解线性代数方程组的方法很多，均可用于求解此问题。Step3SolutionofequationsExample1.Considertheproblemofsource-freeheatconductioninaninsulatedrodwhoseendsaremaintainedatconstanttemperaturesof100℃and500℃respectively.The

4、one-dimensionalproblemsketchedintheFigureisgovernedbyCalculatethesteadystatetemperaturedistributionintherod.Thermalconductivitykequals1000W/m/K,cross-sectionalareaAis1010-3(1)0.5mTB=500TA=100Area(A)BSolution:Wecandividethelengthoftherodintofiveequalcontrolvolumeasshowninthisfigure.Thisgiv

5、esδx=0.1m.Thegridconsistsoffivenodes.Foreachoneofnodes2,3and4temperaturevaluestotheeastandwestareavailableasnodalvalues.Consequently,discretisedequationsoftheform(2.4)canbereadilywrittenforcontrolvolumesurroundingthesenodes:(2)Thethermalconductivity(),nodespacing(δx)andcross-sectionalarea(

6、)areconstants.Thereforethediscretisedequationfornodalpoints2,3,4is:(3)andarezerosinthiscasesincethereisnosourceterminthegoverningEquation(1).Nodes1and5areboundarynodes,andthereforerequirespecialattention.Integrationofequation(1)overthecontrolvolumesurroundingpoint1giveswith(4)=>(5)=>aWaEaP

7、SpSu0Ak/dxaW+aE-Sp-2Ak/dx-2AkTA/dxWithThecontrolvolumesurroundingnode5canbetreatedinasimilarmanner.(2.6)Itsdiscretisedequationisgivenby=>(2.7)=>(2.8)Thissetofequationscanbere-arrangedas:withTheresultingsetofalgebraicequationsforthisexampleisthesolutioncanobtai