热流科学与工程西步文源大学E教育部重点实验室Numerical Heat Transfer数值传热学）Chapter4DiscretizedSchemesofDiffusionandConvectionEquation(1)aInstructorTao.Wen-OuanKeyLaboratoryofThermo-FluidScience&EngineeringInt.JointResearchLaboratoryof Thermal Science&EngineeringXi'an Jiaotong UniversityXi'an,2022-Sept-28CFD-NHT-EHTG1/48CENTER
1/48 Key Laboratory of Thermo-Fluid Science & Engineering Int. Joint Research Laboratory of Thermal Science & Engineering Xi’an Jiaotong University Xi’an, 2022-Sept-28 Numerical Heat Transfer (数值传热学) Chapter 4 Discretized Schemes of Diffusion and Convection Equation (1) Instructor Tao, Wen-Quan
热流科学与工程西老义毛大堂G教育部重点实验室Chapter4Discretizeddiffusion-convectionequation4.1 Two ways of discretization of convection term4.2CDandUDoftheconvectionterm4.3Hybridandpower-lawschemes4.4Characteristicsoffivethree-pointschemes4.5 Discussion on false diffusion4.6Methodsfor overcoming oralleviatingeffectsoffalsediffusion4.7 Discretization of multi-dimensional problem andB.C.treatment中CFD-NHT-EHT2/48CENTER
2/48 4.1 Two ways of discretization of convection term 4.2 CD and UD of the convection term 4.3 Hybrid and power-law schemes Chapter 4 Discretized diffusion－convection equation 4.4 Characteristics of five three-point schemes 4.5 Discussion on false diffusion 4.6 Methods for overcoming or alleviating effects of false diffusion 4.7 Discretization of multi-dimensional problem and B.C. treatment
热流科学与工程西步文源大学G教育部重点实验室4.1 Two ways of discretization of convection term4.1.1Importanceofdiscretizedschemeofconvectionterm1.Accuracy2.Stability3.Economics4.1.2Two ways forconstructing discretizationschemesofconvectiveterm4.1.3RelationshipbetweenthetwowaysΦCFD-NHT-EHT3/48CENTER
3/48 4.1.1 Importance of discretized scheme of convection term 4.1.2 Two ways for constructing discretization schemes of convective term 1. Accuracy 2. Stability 3. Economics 4.1 Two ways of discretization of convection term 4.1.3 Relationship between the two ways
热流科学与工程西老义毛大堂G教育部重点实验室4.1Twowaysofdiscretizationofconvectionterm4.1.1Importanceofdiscretizationscheme（离散格式）Mathematically convective term is only of 1st orderderivative, while its physical meaning (strong directional)makes its discretization one of the hot spots (热点)ofnumericalsimulation:1.Itaffectsthenumericalaccuracy(精确性)Whenaschemeoftheconvectiontermwith1st-orderisused the solution involves severe numerical error2.It affects the numerical stability(稳定性）The schemes ofCD，TUD(三阶迎风) and QUICK areonly conditionally stable3.Itaffectsnumerical economics（经济性）中FO-NHTCE4/48CENTER
4/48 4.1 Two ways of discretization of convection term 4.1.1 Importance of discretization scheme (离散格式) Mathematically convective term is only of 1 st order derivative, while its physical meaning ( strong directional) makes its discretization one of the hot spots (热点) of numerical simulation： 1. It affects the numerical accuracy(精确性). When a scheme of the convection term with 1 st -order is used the solution involves severe numerical error. 2. It affects the numerical stability(稳定性). The schemes of CD，TUD(三阶迎风) and QUICK are only conditionally stable. 3. It affects numerical economics (经济性)
热流科学与工程西步文源大堂G教育部重点实验室4.1.2Twowaysforconstructing(构建)schemes1.Taylor expansion-providing the FD form at a pointTaking CD as an example:adE-w = +-_-, O(Ax22△x2△xax2. CV integration-providing average value within thedomainBy assuming a profile for the interface variablePiecewise linear.oddx -d-dAxUniform gridsAxax(虹 +Φp)/2 -(Φp +dw)/2 _ d -d, O(Ax2 )Ax2△xCFD-NHT-EHTΦ5/48CENTER
5/48 4.1.2 Two ways for constructing(构建) schemes 1. Taylor expansion－providing the FD form at a point Taking CD as an example: 1 1 2 ) ( ) 2 2 E W i i P O x x x x ， 2. CV integration－providing average value within the domain 1 e w dx x x Piecewise linear ( )/ 2 ( )/ 2 2 E P P W E W x x e w x Uniform grids By assuming a profile for the interface variable 2 ，O x ( )