Eq.(1)Theconvergencerateof1600c_=2353J/(kgC)the new G.E. is significantly-c,3723J/ (kg.C)1200 +c,=3143J/(kgc)higher than that of theOa+c=1086J/(kg℃)800previous format-c,=1219J/(kgC)400Eq(5)Resultsfrom0.Eq.(5)40602080100Neirid2） Example 2 ---Conjugated problems(耦合问题)(ax)(8x)(8x)e+(8x)e12pEHarmonic mean(ox).- (or),t11/68
11/68 Eq.(1) The convergence rate of the new G.E. is significantly higher than that of the previous format. 2）Example 2 -Conjugated problems(耦合问题) Results from Eq.(5) (5) ( ) ( ) ( ) e e e e E P x x x + − = + Harmonic mean
Harmonic mean should be used to ensure continuity ofheat flux. In the original G.E., diffusivity is nominal, F=a / cn; Thus for conjugated problem where fluid and solidare coupled, we should assume c, are the same for the fluidand solid:(8x)(Sx)(8x)Harmonicmeanfor=αAp元eMEconjugatedproblem(8x)(8x)(8x)(8x)(Sx)(Sx)a/cpf元e/cofAp/cp=c,For steady problems, such treatment is OK, while fortransient problem, it does not work. For the new G.E., thediffusivity = a, so there is no such problem at all.12/68
12/68 ( ) ( ) ( ) / / / e e e pf E pf P pf x x x c c c + − = + Harmonic mean for conjugated problem For steady problems, such treatment is OK, while for transient problem, it does not work. For the new G.E., the diffusivity = , so there is no such problem at all. Harmonic mean should be used to ensure continuity of heat flux. In the original G.E., diffusivity is nominal, = ; Thus for conjugated problem where fluid and solid are coupled, we should assume cp are the same for the fluid and solid: / p c ( ) ( ) ( ) e e e e E P x x x + − = + ( ) ( ) ( ) e e e e E P x x x + − = = + = /cp
10.2NumericalMethodsAdoptedinTeachingCodeandDiscretizationEquations10.2.1 Major numerical methods adopted1.Primitivevariablemethod（原始变量法):Dependentvariables are u,y,p ;2.Practice B of domain discretization: first determine theinterfaces positions then the node positions;3.Finite volume method (FVM) for discretization: to ensureconservation;4. Staggered (交错) grids: three grid systems for u,v and p)13/68
13/68 10.2.1 Major numerical methods adopted 1.Primitive variable method (原始变量法): Dependent variables are u v p , , ; 2.Practice B of domain discretization: first determine the interfaces positions then the node positions； 3.Finite volume method (FVM) for discretization: to ensure conservation; 4. Staggered (交错) grids: three grid systems for u v p , and ; 10.2 Numerical Methods Adopted in Teaching Code and Discretization Equations
5.Power-law(乘方格式）forconvection-diffusiondiscretizationBut easy to be replaced by CD, FUD or HS: For higher-orderschemes,adopting defer correction(延迟修正)method;6. Linearization for source term:S=Sc+Spdp,Sp≤07. Harmonic mean for interface diffusivity:(8x)。(8x)e_ (0x)et.元。Mp8.Fully implicit for transient problems: space derivativesare determined by the end instant of a time step:9. Boundary conditions (BCs) treated as 1st kind: adoptingadditional source term method(ASTM) for 2nd and 3rd kinds BCs14/68
14/68 5.Power-law(乘方格式) for convection-diffusion discretization: But easy to be replaced by CD, FUD or HS; For higher-order schemes, adopting defer correction(延迟修正) method; , 0 C P P P S S S S = + 6. Linearization for source term: 7. Harmonic mean for interface diffusivity: 8. Fully implicit for transient problems: space derivatives are determined by the end instant of a time step; 9. Boundary conditions (BCs) treated as 1 st kind: adopting additional source term method(ASTM) for 2 nd and 3 rd kinds BCs ( ) ( ) ( ) e e e e E P x x x + − = +
10. SIMPLER algorithm to treat the coupling betweenvelocityandpressurea(pu)op + div(pu)= div(μgradg)(NoEq.forPressure)axat11. Iterative method for solving discretized equations:appp=aepe+awpw+and+asps+b1) Iterative method for solving algebraic equations(inner iteration) ;2) Iterative method for nonlinear Eqs. (outer iteration);12.AlternativeDirection Iteration(ADI)with blockcorrectionto solve algebraic equations15/68
15/68 10. SIMPLER algorithm to treat the coupling between velocity and pressure 12. Alternative Direction Iteration (ADI) with block correction to solve algebraic equations. 11. Iterative method for solving discretized equations： 1) Iterative method for solving algebraic equations (inner iteration) ； 2) Iterative method for nonlinear Eqs. (outer iteration); P P E E W W N N S S a a a a a b = + + + + ( ) ( ) ( ) u p u x div U div grad t = − + (No Eq. for Pressure)