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