4.1.2Twowaysforconstructing(构建)schemes1.Taylor expansion-providing the FD form at a pointTaking CD as an example:adE e - = d- -_-, O(Ax22△x2△xax2. CV integration-providing average value within thedomainBy assuming a profile for the interface variablePiecewise linearaddx -_d-AxUniform gridsAax(dg + Φp)/2 -(dp + dm)/2 _ 9 -w, O(Ax2 )△x2Ax6/49
6/49 4.1.2 Two ways for constructing(构建) schemes 1. Taylor expansion－providing the FD form at a point Taking CD as an example: 1 1 2 ) ( ) 2 2 E W i i P O x x x x + − − − = = ， 2. CV integration－providing average value within the domain 1 e w dx x x Piecewise linear ( ) / 2 ( ) / 2 2 E P P W E W x x + − + − = = e w x − = Uniform grids By assuming a profile for the interface variable 2 ，O x ( )
4.1.3Relationshipbetweenthetwoways1. For the same scheme they have the same order of the T.E.2. For the same scheme, the coefficients of the 1st termin TE. are different The absolutevalue of FVM isusually less than that of FD3. Taylor expansion provides the FD form at a point while CVintegration gives the average value by integration within thedomaind.-d1adadΦe - dwdi+1 -Φ4AxaxAx2△x2△xaxWFVMformFD form7/49
7/49 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 − = 1 1 ) 2 2 E W i i P x x x + − − − = = ； FD form FVM form
4.2CDandUDschemesoftheconvectionterm4.2.1Analytical solution of 1-Dmodelequation4.2.2CDdiscretizationof1-Ddiffusion-convectionequation4.2.3 Up wind scheme of convectionterm1.Definition of CVintegration2.Compactform3.Discretization equation withUDof convectionandCDofdiffusion8/49
8/49 4.2.1 Analytical solution of 1-D model equation 4.2 CD and UD schemes 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
4.2CD and UD of convection term4.2.1 Analytical solution of 1-D model eqwithoutsourceterm(diffusionandconvectioneq.)dd(pup)ddPhysical properties andvelocityareknownconstantsdxdxdxx= O, Φ=; x= L, Φ=ΦThe analytical solution of this ordinary differentequation:pulexp(Pe=)-1-exptΦ-d。exp(pux/)-1exp(puL/T)-1Φ -exp(puL/)-1exp(Pe)-19/49
9/49 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 − − =
Solution Analysis人dtPe = 0, pure diffusion, lineardistribution,5With increasing Pe, distribution1curve becomes more and more0convex downward (下);When Pe=10, in the most region5from x=0-LPe≥10Φ=8Only when x is very close to L, @Lincreases dramatically andwhen x=L, Φ = Φ, .10/49
10/49 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