《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学G教育部重点实验室Numerical HeatTransfer数值传热学）Chapter7MathematicalandPhysical CharacteristicsofDiscretizedEguations(Chapter3ofTextbook）QInstructorTao,Wen-QuanKeyLaboratoryofThermo-FluidScience&EngineeringInt.JointResearchLaboratoryofThermalScience&EngineeringXi'an Jiaotong UniversityInnovativeHarborofWestChina,Xian2022-Dec.-01CFD-NHT-EHTΦ1/41CENTER

1/41 Instructor Tao, Wen-Quan Key Laboratory of Thermo-Fluid Science & Engineering Int. Joint Research Laboratory of Thermal Science & Engineering Xi’an Jiaotong University Innovative Harbor of West China, Xian 2022-Dec.-01 Numerical Heat Transfer (数值传热学) Chapter 7 Mathematical and Physical Characteristics of Discretized Equations (Chapter 3 of Textbook)

热流科学与工程西步文源大堂G教育部重点实验室Contents7.1 Consistence,Convergence and Stability ofDiscretized Equations7.2von Neumann Method forAnalysingStabilityof Initial Problems7.3Conservationof Discretized Equations7.4TransportivePropertyofDiscretizedEquations7.5Sign-preservationPrincipleforAnalyzingConvectiveStabilityΦCFD-NHT-EHT2/41CENTER

2/41 7.1 Consistence, Convergence and Stability of Discretized Equations 7.3 Conservation of Discretized Equations Contents 7.4 Transportive Property of Discretized Equations 7.5 Sign-preservation Principle for Analyzing Convective Stability 7.2 von Neumann Method for Analysing Stability of Initial Problems

热流科学与工程西步文源大堂E教育部重点实验室7.3 Conservation of Discretized Equations7.3.1 Definition and analyzing model7.3.2 Direct summation method7.3.3 Conditions for guaranteeing conservationofdiscretizedequations7.3.4 Discussion-expected but not necessary（期待而非必须）ΦCFD-NHT-EHT3/41CENTER

3/41 7.3 Conservation of Discretized Equations 7.3.1 Definition and analyzing model 7.3.2 Direct summation method 7.3.3 Conditions for guaranteeing conservation of discretized equations 7.3.4 Discussion－expected but not necessary (期待而非必须)

热流科学与工程西步文源大学E教育部重点实验室7.3ConservationofDiscretizedEquations7.3.1 Definition and analyzing model1. DefinitionIf the summation of a certain number of discretizedequationsoverafinitevolume（有限大小体积）satisfiesconservationrequirementthesediscretizedeguationsaresaidtopossessconservation（离散方程具有守恒性）2.Analyzing model---advection equationIt is easy to show that CD of diffusion term possessesconservation.Discussion is onlyperformed fortheeguationwhich only has transient term and convective term（advectionequation，平流方程）ΦHFO-NHTCE4/41CENTER

4/41 7.3 Conservation of Discretized Equations 7.3.1 Definition and analyzing model 1. Definition 2. Analyzing model-advection equation It is easy to show that CD of diffusion term possesses conservation. Discussion is only performed for the equation which only has transient term and convective term (advection equation, 平流方程 ). If the summation of a certain number of discretized equations over a finite volume （有限大小体积）satisfies conservation requirement , these discretized equations are said to possess conservation (离散方程具有守恒性)

热流科学与工程西步文源大堂E教育部重点实验室ada(ud)0(Conservative)Advectionataxequationadad0(Non-conservative)uatax7.3.2Directsummationmethod(直接求和法)Summing up FTCS scheme of advection eg. ofconservative form over the region of [l, l, J :n+1 --d"__uidi+-u-d-Time level of the2Axspatial termsAtare not showninout31211ArCFD-NHT-EHTΦ5/41CENTER

5/41 ( ) 0 u t x         （Conservative） u 0 t x         （Non-conservative） 7.3.2 Direct summation method (直接求和法) Summing up FTCS scheme of advection eq. of conservative form over the region of [ , ] l l 1 2 ： 1 1 1 1 1 2 n n i i i i i i u u t x                Time level of the spatial terms are not shown Advection equation

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 