《数值传热学》研究生课程教学资源（课件讲稿）Chapter 4 Discretized Schemes of Diffusion and Convection Equation（1/2，4.1-4.4）

热流科学与工程西步文源大堂E教育部重点实验室4.2.2CD discretizationof1-Ddiffusion-convectionequation1.Integrationof1-DmodelequationAdopting the linear profile, integration over a CVyields:p[(pu)。+(pu),]+ w[1=Φ-(pu)m(pu)w+2(8x)(8x)(x)aEawapThus:i+1i-1WEapp=aepe+awwA(ar)u(ar)CFD-NHT-EHTΦ11/48CENTER

11/48 4.2.2 CD discretization of 1-D diffusion-convection equation 1. Integration of 1-D model equation Adopting the linear profile, 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

热流科学与工程西步文源大堂E教育部重点实验室2.RelationshipbetweencoefficientsRewriting ap as follows:(Sx)wWox(pu)+(pu)w-(pu), +pu)。-(pu)。+(pu)。(8x)(Sx)wWI+[(pu)。-(pu)]=ag+aw+[(pu)。-(pu)]pu)pu(8x)wW(8x)erDefining diffusionaeawEDConductance:Sxpu=FInterface flowrate:CFD-NHT-EHTΦ12/48CENTER

12/48 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=

热流科学与工程西步文源大堂E教育部重点实验室The discretized form of 1-D steady diffusion andconvectionequation is:1app=aepe +awdw ap=D.DHL+dAW122ap=ae +aw +(F-F.)If in the iterative process the mass conservation is satisfiedthenF-Fw=0In order to guarantee the convergence of iterative process.it is always required:ap=ae+awHence,itis demanded that atany iteration level massmust be conserved, i.e., mass conservation should be satisfied中CFD-NHT-EH13/48CENTER

13/48 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 that at any iteration level mass must be conserved, i.e., mass conservation should be satisfied!

热流科学与工程西步文源大堂E教育部重点实验室3.Analysis of discretized diffu-conv.eq.by CDFrom apΦ, = a,Φe +awΦw it can be obtained:,=d +D.-F)g+(D +F,)0nUni.gridae+aw(D. -IE)+(F+DConst property21EHpe+(1+=P)gue+（112Rp(D + D) / D= pu(8x)P is the grid Peclet number,P :With the given Pe and w Φ, can be determined.ΦCFD-NHT-EHT14/48CENTER

14/48 3. Analysis of discretized diffu-conv. 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     

热流科学与工程西步文源大堂E教育部重点实验室P. = 2Given dw =100,Φe = 200for P =0,1,2,4200the calculated results arePA=0shown in the figure精确解150Physically and accordingto the analytical solution-211100extp-dodr -doexp50ofLthe value ofshould be0P = 4largerthan zero.pulPe=2PCFD-NHT-EHTG15/48CENTER

15/48 Given 100, 200   W E   Physically and according to the analytical solution the calculated results are shown in the figure. for P 0,1,2,4   0 0 exp( ) 1 L exp( ) 1 uL x L uL              P 4   P 2   the value of should be larger than zero.  2 uL Pe P     