The above variation trend with Peclet number isconsistent(协调的)with the physical meaning of PepulConvectionpuPer/ LDiffusionWhen 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.11/49
11/49 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.
4.2.2CD discretization of 1-D diffusion-convectionequation1.Integration of 1-D modelequationAdopting the linear profile for both convection and diffusion terms ,integration over a CV yields:ppl(pu)+ou),] +dwpu)(Sx)(8x)(8x)(Sx)。2awaEapThus:i+1i-1WEappp=aepe+awpu(&x)(ar)12/49
12/49 4.2.2 CD discretization of 1-D diffusion-convection equation 1. Integration of 1-D model equation Adopting the linear profile for both convection and diffusion terms , integration over a CV yields: 1 1 1 1 [ ( ) ( ) ] [ ( ) ] [ ( ) ] 2 ( ) 2 ( ) ( ) 2 ( ) 2 P E e w e w e w e W w e w e w u u u u x x x x + − + = − + + E a W a P P E E W W a a a = + Thus： P a
2.Relationship between coefficientsRewriting ap as follows:Not equal to ae+aw(Sx)(Sx)wWT(pu),+(pu).-(pu), +(pu)。-(pu)。 +(pu)。+(Sx)。(Sx)wW+[(pu)。-(pu)] =a +aw +[(pu)。-(pu)]pu(8x)(8x)VawaerDefining diffusionEDConductance:Sxpu=FInterfaceflowrate:13/49
13/49 2. Relationship between coefficients Rewriting aP as follows： 1 1 ( ) ( ) 2 ( ) 2 ( ) e w P e w e w W a u u x x = + − + = 1 1 ( ) ( ) 2 ( ) 2 ( ) ) ( ) ( ) ( ( ) e e w w e w e w W e w u u x u u x u u − + + + − + − = 1 1 ( ) ( ) [( ) ] 2 ( ) 2 ( ) ( ) e w e w e w W e w u u x x u u − + + + + = − [( ) ( ) ] E W e w a a u u + + − E a W a D , x = Defining diffusion Conductance: Interface flow rate: u F= Not equal to aE+aW
The discretized form of 1-D steady diffusion and convectionequation is:11appp=aede+awdwag=De-EDH+22ap=ae +aw +(F-F.)If in the iterative process the mass conservation is satisfied thenF-F,=0In order to guarantee the convergence of iterative process.it is always required:ap=ae+awHence, it is demanded(要求)that at any iteration level massmust be conserved, i.e., mass conservation should be satisfied!14/49
14/49 The discretized form of 1-D steady diffusion and convection equation is: P P E E W W a a a = + 1 2 a D F E e e = − 1 2 a D F W w w = + ( ) P E W e w a a a F F = + + − If in the iterative process the mass conservation is satisfied then 0 F F e w − = In order to guarantee the convergence of iterative process, it is always required: P E W a a a = + Hence, it is demanded(要求) that at any iteration level mass must be conserved, i.e., mass conservation should be satisfied!
3.Analysis of discretized diffusion-convection eg.by CDFrom appp = aeΦe +awΦw it can be obtained:Uni.grid, = Lte aru _ D. -F)0a +(D +F,) ae+awConst.property)+(D(DF-(1+P)F2.2 DD2(D+D)/ D2Pa i the grid Peclet number, Pe = Pu(ox)TWith the given e and dw, Φ, can be determined.15/49
15/49 3. Analysis of discretized diffusion-convection eq. by CD P P E E W W From a a a = + it can be obtained: 1 1 ( ) ( ) 2 2 1 1 ( ) ( ) 2 2 e e E w w W E E W W P E W e e w w D F D F a a a a D F D F − + + + = = + − + + Uni.grid Const. property 1 1 (1 ) (1 ) 2 2 ( )/ E W P F F D D D D D − + + = + 1 1 (1 ) (1 ) 2 2 2 − + + P P E W P is the grid Peclet number , With the given and E W， P can be determined. u x ( ) P =