第5讲PPT-(The-Finite-Volume-Method-for-Convection-Di

1Chapter5TheFiniteVolumeMethodforTheFiniteVolumeMethodforConvectionConvection--DiffusionProblemsDiffusionProblems2()()()divudivgradStρφρφφ∂+=Γ+∂rateofchangewithtimeconvectiondiffusionsourcetermIntroduction()0divgradSφφΓ+=Purediffusioninsteadystate()()0CVCVACVdivgraddVSdVngraddASdVφφφφΓ+=Γ+=∫∫∫∫iControlvolumeintegration3Introduction()()divudivgradSρφφ=Γ+NetconvectivefluxNetdiffusivefluxsourceterm.().()AACVnudAngraddASdVφρφφ=Γ+∫∫∫Integrationoveracontrolvolume1.ConvectionVSDiffusion←Direction2.Iscentraldifferenceschemesuitableforconvection?4()()dddudxdxdxφρφ=Γ()0dudxρ=一维稳态对流-扩散问题命名法则与第四章相同()()()()ewewuAuAAAxxφφρφρφ∂∂−=Γ−Γ∂∂()()0ewuAuAρρ−=25一维稳态对流-扩散问题定义：Fuρ=DxδΓ=则控制体边界上的值可表示为：()wwFuρ=()eeFuρ=δΓ=eePEDxδΓ=Centraldifferencefordiffusionterm6一维稳态对流-扩散问题如果认为()()eewweEPwPWFFDDφφφφφφ−=−−−0ewFF−=weAAA==且对扩散项采用中心差分，则有()()()()ewewuAuAAAxxφφρφρφ∂∂−=Γ−Γ∂∂()()0ewuAuAρρ−=求解时，假设速度场已知！7中心差分(Centraldifferencingscheme)从扩散项得到启发，也采用中心差分()/2ePEφφφ=+()/2wWPφφφ=+()()eewweEPwPWFFDDφφφφφφ−=−−−()()()()22ewPEWPeEPwPWFFDDφφφφφφφφ+−+=−−−代入(5.12)8对公式(5.12)进行整理中心差分(Centraldifferencingscheme)[()()]()()2222wewewePwWeEFFFFDDDDφφφ−++=++−[()()()]()()2222weweweewPwWeEFFFFDDFFDDφφφ++−+−=++−PPWWEEaaaφφφ=+与扩散问题得形式差不多，就是系数里面多了一项F39例题（中心差分）()()dddudxdxdxφρφ=Γ(5.3)Case1:u=0.1m/s;5gridnodesCase2:u=2.5m/s;5gridnodesCase3:u=2.5m/s;20gridnodes已知条件：L=1.0m;Γ=0.1kg/m/s;ρ=1.0kg/m310例题（中心差分）ewFFF==()()()2ePEAAeEPAPAFFDDφφφφφφφ+−=−−−()//2ADxδ=Γ2,3,4点的离散就是公式(5.12)对于1点：(5.16)/Dxδ=ΓFuρ=ewDDD==00exp(/)1exp(/)1LuxuLφφρφφρ−Γ−=−Γ−11例题（中心差分）ABFFF==PPWWEEuaaaSφφφ=++()PWEewpaaaFFS=++−−()()()2wBBPWBBPwPWFFDDφφφφφφφ−+=−−−(5.17)对于5点：()//22BDxDδ=Γ=统一成标准形式：12Case1123451.550.450001.10.551.00.4500000.551.00.4500000.551.00.4500000.551.450φφφφφ−⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥−−⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥=−−⎢⎥⎢⎥⎢⎥−−⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥−⎣⎦⎣⎦⎣⎦0.1/;0.1,/0.1/0.20.5umsFuDxρδ====Γ==1.45-0.90.9ΦB00.5551.0000.450.5541.0000.450.5531.0000.450.5521.55-1.11.1ΦA0.4501aP=aW+aE-SPSPSuaEaWNode413Case1123450.94210.80060.62760.41630.1579φφφφφ⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦()()2.7183exp1.7183xxφ−=ComparisonofnumericalandanalyticalsolutionsforCase1解析解数值解14Case22.5/;2.5,/0.1/0.20.5umsFuDxρδ====Γ==()()101exp2517.2010xxφ−=+×解析解0.251.5-1.5ΦB01.7551.000-0.751.7541.000-0.751.7531.000-0.751.7522.75-3.53.5ΦA-0.7501aP=aW+aE-SPSPSuaEaWNode15Case2ComparisonofnumericalandanalyticalsolutionsforCase2Oscillation“wiggles”16Case32.5/;2.5,/0.1/0.052.0umsFuDxρδ====Γ==网格加密，F不变，D变大4.75-1.55ΦB03.25204.00000.753.252-197.25-6.56.5ΦA0.7501aP=aW+aE-SPSPSuaEaWNodeTheeffectofF/D(from5to1.25)!517Propertiesofdiscretisationschemes1.上面的例子说明，对流项采用中心差分时，如果F/D过大，也就是对流用作远远大于扩散的作用，则数值解可能出现震荡或出现“奇点”；2.增大D的办法包括加密网格（即减小分母δx），也就是说，理论上讲只要网格足够密就没有问题。然而，这是不符合实际情况的；如何在比较疏或者不太密的网格下求得最好的结果？？？差分方式必须满足：守恒性(conservativeness)有界性(boundedness)传输性(transportiveness)18Conservativeness()22322/eexφφδΓ−通过面进入节点的通量在两个相邻cell的共用face上面，通量的表述必须一致！以下图为例w2e2()22212/wwxφφδΓ−通过面离开节点的通量19Conservativenessw2e2通过计算区域总的通量122334324332212143()()()()()[][][]()[]eAewewBwBAqxxxxxqqqxφφφφφφφφφφδδδδδφφδ−−−−−Γ−+Γ−Γ+Γ−Γ−+−Γ=−内部face上的通量互相抵消，最后只剩下qB-qA，计算区域内变量Φ守恒20Conservativeness不守恒的例子e2w2确定面e2的值利用节点1,2,3的二次多项式确定面w2的值利用节点2,3,4的二次多项式如果两条曲线在e2(w2)出的斜率不同，则通量不能互相抵消，也就是不守恒！此例并不说明二次多项式不能用作差分方式的一种；Fluent中QUICK就是二次多项式，它是守恒的！621Boundedness()PWEewPaaaFFS=++−−pnbPaaS=−∑'PPPaaS=−Scarborough(1958)年指出，利用迭代的方法求解离散后的代数方程组收敛的充分条件是：1.满足上式的矩阵是diagonallydominant;2.为了实现diagonallydominant，SP需为负，及源项离散时需要注意！1atallnodes1atonenodeatleastnbPaa≤Σ⎧⎨′⎩UPPSVSSφΔ=+22Boundedness有界性的物理意义是：在一个没有源项的问题中，内部节点的值不可能大于最大边界值，也不可能小于最小边界值！有界性要求：1.每个节点的系数为正（物理意义是什么）；2.源项的SP为负。反观例题5.1中的case2，aE系数为负，不满足要求不满足有界性会怎样？Divergence！Overshoot&Undershoot！23Transportiveness/FuuxPeDxρρδδ====ΓΓ对流扩散(1)Noconvectionandpurediffusion(Pe=0);(2)Nodiffusionandpureconvection(Pe→∞)ItisimportantthattherelationshipbetweenthemagnitudeofthePecletnumberandthedirectionalityoftheinfluencing,knownasthetransportiveness,isborneoutinthediscretisationscheme.24对流-扩散问题中心差分的评估Transportiveness◊——useconsistentexpressionstoevaluateconvectiveanddiffusivefluxesatthecontrolvolumefacesBoundednessμ——aP=aW+aE+(Fe-Fw)satisfytheScarbroughcriterion——aE=De-Fe/2，ifaE0,thenDeFe/2,i.e.Fe/De=Pe2.0Transportivenessμ——usingallitsneighborstocalculatetheconvectiveanddiffusiveflux——doesnotrecognizetheflowofdirectionorthestrengthofconvectionrelativetodiffusion725对流-扩散问题中心差分的讨论Pe=F/D2.0Fluidproperties:ρandΓFlowproperties:uGrids:δX/FuuxPeDxρρδδ====ΓΓ对流扩散UpwindHybridPower-lawQUICK中心差分从泰勒展开截断误差分析具有二阶精度26Theupwinddifferencingscheme()()eewweEPwPWFFDDφφφφφφ−=−−−中心差分的最大缺陷是不能考虑流动方向的影响！andwWePφφφφ==()()ePwWeEPwPWFFDDφφφφφφ−=−−−()()weePφφφ++=++[()()]()wweewPφφφ+++−=++andwPeEφφφφ==()()eEwPeEPwPWFFDDφφφφφφ−=−−−[()()]()weeewPwWeeEDDFFFDDFφφφ+−+−=+−27TheupwinddifferencingschemePPWWEEaaaφφφ=+()PWEewaaaFF=++−(5.31)Neighbourcoefficientsneighborcoefficientsoftheupwinddifferencingmethodthatcoversbothflowdirections28Theupwinddifferencingscheme()()dddudxdxdxφρφ=ΓCase1:u=0.1m/s;5gridnodesCase2:u=2.5m/s;5gridnodes已知条件：L=1.0m;Γ=0.1kg/m/s;ρ=1.0kg/m3829TheupwinddifferencingschemeewFFFuρ===PPWWEEaaaφφφ=+()PWEewaaaFF=++−对于