热流科学与工程西步文源大堂G教育部重点实验室4.1.3Relationshipbetweenthetwo ways1. For the same scheme they have the same order of the T.E2. For the same scheme, the coefficients of the 1st termin TE. are different The absolutevalue of FVM isusually less than that of FD.3. Taylor expansion provides the FD form at a point while CVintegration gives the average value by integration within thedomainpe-d2oOxxAxΦCFD-NHT-EHT6/48CENTER
6/48 4.1.3 Relationship between the two ways 1. For the same scheme they have the same order of the T.E. 2. For the same scheme, the coefficients of the 1st term in T.E. are different. The absolute value of FVM is usually less than that of FD. 3. Taylor expansion provides the FD form at a point while CV integration gives the average value by integration within the domain 1 e w dx x x e w x
热流科学与工程西步文源大堂G教育部重点实验室4.2cDand UD of theconvectionterm4.2.1 Analytical solution of 1-D modelequation4.2.2cD discretizationof1-Ddiffusion-convectionequation4.2.3Upwindschemeofconvectionterm1.Definition of CV integration2.Compactform3.Discretization eguation withUD of convectionandCDofdiffusionΦCFD-NHT-EHT7/48CENTER
7/48 4.2.1 Analytical solution of 1-D model equation 4.2 CD and UD of the convection term 4.2.2 CD discretization of 1-D diffusion-convection equation 4.2.3 Up wind scheme of convection term 1. Definition of CV integration 2. Compact form 3. Discretization equation with UD of convection and CD of diffusion
热流科学与工程西步文源大堂G教育部重点实验室4.2CD and UD of convection term4.2.1Analytical solutionof1-Dmodeleq.withoutsourceterm(diffusionandconvectioneq.)dd(pup)ddPhysical properties andvelocityareknownconstantsdxdxdxx=0, Φ=; x=L, Φ=ΦThe analytical solution of this ordinary differentequation:pulexp(Pe=)-1expΦ-exp(pux/T)-1exp(puL/)-1d -dexp(puL/T)-1exp(Pe)-1CFD-NHT-EHTΦ8/48CENTER
8/48 4.2 CD and UD of convection term 4.2.1 Analytical solution of 1-D model eq. without source term (diffusion and convection eq.) ( ) ( ), d u d d dx dx dx Physical properties and velocity are known constants 0 0, ; , L x x L The analytical solution of this ordinary different equation： 0 0 exp( ) 1 exp( / ) 1 exp( / ) 1 exp( / ) 1 L uL x ux L uL uL exp( ) 1 exp( ) 1 x Pe L Pe
热流科学与工程西步文源大堂G教育部重点实验室Solution AnalysisdtPe = 0, pure diffusion, linearDistribution;5With increasing Pe, distribution~1curve becomes more and moreconvex downward (下凸);When Pe =10, in the most region5from x=0-LPe≥10Φ=d80XOnly when x is very close to L, @Lincreases dramatically andwhen x=L ,Φ = ΦL .CFD-NHT-EHTΦ9/48CENTER
9/48 Solution Analysis Pe＝0，pure diffusion, linear Distribution; With increasing Pe，distribution curve becomes more and more convex downward (下凸)； When Pe＝10，in the most region from x=0-L 0 when x=L ， . L Only when x is very close to L， increases dramatically and
热流科学与工程西步文源大堂E教育部重点实验室The above variation trend with Peclet number isconsistent(协调的)with the physical meaning of PepulConvectionpuPe/ LrDiffusionWhen Pe is small-Diffusion dominated, lineardistribution ;When Pe is large-Convection dominated, i.e.,upwind(上游)effect dominated,upwind information istransported downstream, and when Pe ≥ 100, axialconduction can be totally neglected.It is reguired in some sense that the discretizedscheme of the convective term has some similar physicalcharacteristics.ΦCFD-NHT-EH10/48CENTER
10/48 The above variation trend with Peclet number is consistent(协调的) with the physical meaning of Pe When Pe is small－Diffusion dominated，linear distribution ； Convection Diffusion It is required in some sense that the discretized scheme of the convective term has some similar physical characteristics. / uL u Pe L When Pe is large－Convection dominated，i.e., upwind(上游) effect dominated， upwind information is transported downstream, and when Pe 100, axial conduction can be totally neglected.