热流科学与工程西步文源大堂E教育部重点实验室2. ADI-line iteration is widely adopted in the numericalsolution of flow and heat transfer problemThe ABEqs. generated on structured grid system can besolved by ADI---each line has the same number of unknownsMM1j411L1ML2i1LiCFD-NHT-EHTG16/56CENTER
16/56 The ABEqs. generated on structured grid system can be solved by ADI-each line has the same number of unknowns 2. ADI-line iteration is widely adopted in the numerical solution of flow and heat transfer problem
热流科学与工程西步文源大堂E教育部重点实验室5.2.4ADl-iteration（交替方向迭代）isidenticaltoADI-implicit（交替方向隐式）ADI-iteration is identical to the ADI-Implicit of solvingmultidimensional unsteady problem for one time sten(1)ADI-Jakob iteration can be expressed扫描方向as:(2)b(k+1)b(k+1/2)(k+1/2)(k+1/2) +awpwk) +asd(k) +b]F(k+1/2)h(k+1) +asds(k+I)=a(k+)(k+1/2) +b)la1YEWThis expression is very similartoPeaceman-RachfordADlmplicitmethodfor2Dtransientproblem:CFD-NHT-EHTΦ17/56CENTER
17/56 ADI-Jakob iteration can be expressed as：： ( 1/ 2) ( 1/ 2) ( 1/ 2) ( ) ( ) [ ] k k k k k P P E E W W N N S S a a a a a b ( 1/ 2) k b ( 1) ( 1) ( 1) ( 1/ 2) ( 1/ 2) [ ] k k k k k P P N N S S E E W W a a a a a b This expression is very similar to Peaceman-Rachford ADImplicit method for 2 D transient problem： ( 1) k b ADI-iteration is identical to the ADI-Implicit of solving multidimensional unsteady problem for one time step 5.2.4 ADI-iteration (交替方向迭代）is identical to ADI-implicit (交替方向隐式）
热流科学与工程西步文源大堂E教育部重点实验室2-D Peaceman-Rachford methodtlDividing △t into two sub-periods.oplicitFAt/2n+1In the 1st sub-period △t /2x-direction is implicit, y-directionAt/2Atis explicit;-implicitIn the 2nd △t / 2 y-direction2-D AD Implicitis implicit, and x is explicit@(k+1/2)Letrepresent temporary values at middle time.2/k.represent CDfor2nd-order x-direction0derivative at time level k : then we have中CFD-NHT-EHT18/56CENTER
18/56 2-D Peaceman-Rachford method Let represent temporary values at middle time ( 1/ 2) k Dividing t into two sub-periods. In the 1st sub-period t / 2 x-direction is implicit, y-direction is explicit; In the 2nd t / 2 y-direction is implicit, and x is explicit. 2 , k x i j represent CD for 2nd-order x-direction derivative at time level k ; then we have: t 2 2-D AD Implicit
热流科学与工程西步文源大堂E教育部重点实验室+k+1/2kkd1st sub-Di6k+1/2(1)+$@k)2period:△t / 2dk+1/22nd sub-(2)1k+1/2+82gk+l)a(s"i.jperiod:△t / 2+k+1/2k+1/2+k+1/2182@k+1/2+1i-1.1Substituting the expression into Eq.(1):Ar2adtataAtaA+1/2k+1/2(1-+(124xAx241AxADImplicitapae,aybas,an(k+//2) +awdw(k+1/2)6(k+1/2)(+asg(* + b]ADIterationa.a1PEThe same comparison can be done for the second sub-time step;Thus one-time step forward of transient problem is eguivalent toonecycleiterationfor steadyproblemΦFD-N19/56CENTER
19/56 1/ 2 , , 2 2 1 / , 2 , ( ) / 2 k i j k x y i j k i j k a i j t 1st subperiod： (1) 2 nd subperiod： 1/ 2 , 2 1/ 2 2 , 1 , 1 , ( ) / 2 k i j k i j k i j k x i j y a t (2) Substituting the expression into Eq.(1): 1/ 2 1/ 2 1/ 2 2 2 2 2 , 1, 1, , 1 , 1 , (1 ) ( )( ) ( )( ) (1 ) 2 2 k k k k k k i j i j i j i j i j i j a t a t a t a t x x x x P a , E W a a , S N a a b The same comparison can be done for the second sub-time step; Thus one-time step forward of transient problem is equivalent to one cycle iteration for steady problem. 1/2 1/2 1/2 1/2 2 +1, , -1, , 2 -2 = + k k k i j i j i j x k i j x ( 1/ 2) ( 1/ 2) ( 1/ 2) ( ) ( ) [ ] k k k k k P P E E W W N N S S a a a a a b ADImplicit ADIteration