《数值传热学》研究生课程教学资源（课件讲稿）Chapter 7 Mathematical and Physical Characteristics of Discretized Equations（2/2，7.3-7.5）

热流科学与工程西步文源大堂G教育部重点实验室12 (up)i+1 -(up),-1-dnui+di+1 -u-id-12>2△x2△x△t11I.12- (up)i+1 -(up)i-1n+l(dr-d")Ax = -△t 一2Increment(增值) of Φ within t and [,l2]Is it equal to the net amount of @ entering the spaceregion by convection within the same time period?Analyzing should be made for the right hand termsof the equation to see whether this is true: (up)i+1 -(up)i-1t>Z[(up);- -(up)]2211ΦCFD-NHT-EHT6/41CENTER

6/41 1 2 1 2 1 1 1 1 1 2 I n n i i I I i i i i I t u u x                   2 2 1 1 1 1 1 ( ) ( ) ( ) 2 I I n n i i i i I I u u x t               2 2 1 1 1 1 1 1 ( ) ( ) [( ) ( ) ] 2 2 I I i i i i I I u u t t u u                Analyzing should be made for the right hand terms of the equation to see whether this is true: 2 1 1 1 ( ) ( ) 2 I i i I u u x         Is it equal to the net amount of entering the space region by convection within the same time period?  Increment(增值) of within and 1 2  t [ , ] l l

热流科学与工程西步文源大堂G教育部重点实验室12directly summing up: forZ[(ug)-- -(up);]For the termthe left end, we have:1i=I(ud)i-1udI,+1(ud),iudi=I, +1(ub01,+3i=I +2udi=I, +3+4i=I +410otin(up),- +(ud)n13AXCED-NHT-EHTG49/4112CENTER

1 i I  1 1 ( )I u  1 1 ( )I  u  1 i I  1 1 ( )I u 1 2 ( )I  u  1 i I   2 1 1 ( )I u  1 3 ( )I  u  1 i I   3 1 2 ( )I u  1 4 ( )I  u  1 i I   4 1 3 ( )I u  . . . . directly summing up: for the left end, we have： 1 1 1 ) ( ) I I （u u     2 1 For the term [( ) ( ) ] 1 1 I i i I  u u      49/41

热流科学与工程西步文源大堂G教育部重点实验室OutinFor the right end:r121l1Ax）12ud)-i=I, -3-2(ud)oi=I,-2-3I-(up)i=l, -10i=I2(ud) 1,+l10[(ud), +(u)1,+1]12△tThen:[(up)i-- -(ud)i+]2/△t([(up)r- +(up), ]-[(up), +(up)r+)2Left end of domainRight end of domainCFD-NHT-EHTG8/41CENTER

8/41 2 i I  3 2 4 ( )I u  2 2 ( )I  u  2 i I   2 2 3 ( )I u  2 1 ( )I  u  2 2 ( )I 2 u  i I  1 2 ( )I  u 2 i I  2 1 ( )I u  2 1 ( )I  u  2 1 1 1 [( ) ( ) ] 2 I i i I t u u        1 1 2 2 1 1 {[( ) ( ) ] [( ) ( ) ]} 2 I I I I t u u u u            For the right end： . . Left end of domain Right end of domain 2 2 1 [( ) ( ) ] I I Then:   u u   

热流科学与工程西步文源大堂E教育部重点实验室AtFurther:[(up)-1 +(up), ]-[(up)r2 +(up)1til) =2ud)CD-uniform grid(up)12 +(up)1+1+uAt2outinTTI-1-1LAr2= △t(Φ flowin - Φ flowout)Thus the central difference discretization of theconvective term possesses conservative featureCFD-NHT-EHTΦ9/41CENTER

9/41 1 1 2 1 2 1 {[( ) ( ) ] [( ) ( ) ]} 2 I I I I t u u u u            1 1 2 1 2 1 ( ) ( ) ( ) ( ) {[ ] [ ]} 2 2 I I I I u u u u t              t flowin flowout ( )   CD-uniform grid Thus the central difference discretization of the convective term possesses conservative feature. I1 -1 I2+1 Further:

热流科学与工程西步文源大堂G教育部重点实验室7.3.3 Conditions forguaranteeing conservation1.Governing equation should be conservativead.od0For non-conservativeform:-11atOxΦi+1 - Φi-1@Its FTCS scheme is-u△t2△xBy direct summation, the above results do not possessconservation because of no cancellation (抵消) can be madefor the product terms. Only when u and have the samesubscript , the cancellation of inner terms can be done2.Dependent variable and its 1st derivative arecontinuous at interfaceΦCFD-NHT-EHT10/41CENTER

10/41 7.3.3 Conditions for guaranteeing conservation 1.Governing equation should be conservative u 0 t x         For non-conservative form: 1 1 1 2 n n i i i i i u t x              Its FTCS scheme is 2. Dependent variable and its 1st derivative are continuous at interface By direct summation, the above results do not possess conservation because of no cancellation (抵消) can be made for the product terms. Only when have the same subscript , the cancellation of inner terms can be done. u and 