热流科学与工程西步文源大学E教育部重点实验室Numerical HeatTransfer数值传热学）Chapter4DiscretizedSchemesofDiffusionandConvectionEquation(2)InstructorTao,Wen-QuanKeyLaboratoryofThermo-FluidScience&EngineeringInt.JointResearchLaboratoryofThermalScience&EngineeringXi'an Jiaotong UniversityInnovativeHarborofWestChina,Xian2022-0ct-12CFD-NHT-EHTΦ1/52CENTER
1/52 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-Oct-12 Numerical Heat Transfer (数值传热学) Chapter 4 Discretized Schemes of Diffusion and Convection Equation (2)
热流科学与工程西老义毛大堂G教育部重点实验室Chapter 4Discretizeddiffusion-convection eguation4.1 Two waysof discretization of convection term4.2CDandUDoftheconvectionterm4.3 Hybrid and power-law schemes4.4Characteristics of fivethree-pointschemes4.5Discussiononfalsediffusion4.6Methodsforovercomingoralleviating effectsoffalsediffusion4.7Discretizationofmulti-dimensional problemandB.C.treatmentΦCFD-NHT-EHT2/51CENTER
2/51 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堂G教育部重点实验室4.5 Discussion on false diffusion4.5.1 Meaning and reasons of false diffusion1.Original meaning2.Extendedmeaning3.Taylorexpansionanalysis4.5.2Examplesofseverefalsediffusioncausedby1st-orderscheme4.5.3Errorscausedbyobliqueintersection（倾斜交叉）ofgridlines4.5.4Falsediffusioncausedbynon-constantsourceterm4.5.5TwofamousexamplesCFD-NHT-EHT中3/52CENTER
3/52 4.5.1 Meaning and reasons of false diffusion 4.5.2 Examples of severe false diffusion caused by 1 st-order scheme 1.Original meaning 2.Extended meaning 3.Taylor expansion analysis 4.5 Discussion on false diffusion 4.5.3 Errors caused by oblique intersection (倾斜 交叉) of grid lines 4.5.4 False diffusion caused by non-constant source term 4.5.5 Two famous examples
热流科学与工程西步文源大堂E教育部重点实验室4.5 Discussion on false diffusion4.5.1Meaningand reasonsoffalsediffusionFalse diffusion（假扩散),also called numerical viscosity（数值黏性），isanimportantnumericalcharacterofthediscretizedconvective scheme1.Original meaningNumerical errors caused by discretized scheme with 1 storderaccuracyis calledfalse diffusion;The 1st term in the TE of such scheme contains 2nd orderderivative, thus the diffusion action is somewhat magnifiedat the sense of second-order accuracy, hence the numericalerror is called “false diffusion"4/52CFD-NHT-EHT中CENTER
4/52 4.5 Discussion on false diffusion 4.5.1 Meaning and reasons of false diffusion False diffusion (假扩散), also called numerical viscosity (数值黏性)，is an important numerical character of the discretized convective scheme. 1. Original meaning Numerical errors caused by discretized scheme with 1st order accuracy is called false diffusion； The 1st term in the TE of such scheme contains 2nd order derivative, thus the diffusion action is somewhat magnified at the sense of second-order accuracy, hence the numerical error is called “false diffusion
热流科学与工程亚步文源大堂E教育部重点实验室Taking 1-D unsteady advection equation as an exampleThe two 1st-order derivatives are discretized by 1st-orderaccuracyschemesn+11st-oder schemehd" -Φ"ladad?-u>0△tAxataxExpanding Φ-, nat (i,n) by Taylor series, andsubstituting into the above equation:一uΦCFD-NHT-EHT5/52CENTER
5/52 Taking 1-D unsteady advection equation as an example. The two 1st -order derivatives are discretized by 1st -order accuracy schemes. u t x 1 st-oder scheme 1 1 n n n n i i i i u t x u >0 Expanding 1 1 , n n i i at (i,n) by Taylor series，and 2 3 2 3 , 2 3 , , 1 1 ) ) . 2 ) 6 n n i i n i n i n i t t t t t t t , 2 3 2 3 2 3 , , 1 1 [ ) ) . ) 2 6 n n i i i n i n i n x x x x x u x x substituting into the above equation：