《数值传热学》研究生课程教学资源（课件讲稿）Chapter 4 Discretized Schemes of Diffusion and Convection Equation（2/2，4.5-4.7）

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：

热流科学与工程西步文源大堂E教育部重点实验室At a?adaduAx a"pd+O(△x2,△t)2axat2Oxatwhere the transient 2nd derivative can be re-written as follows:aadada2saaadadauuuuax?at?axaxatataxaxatatsubstituting into above equationapaduaxu+O(△x2,△t2)uataxThus at the sense of 2nd-order accuracy abovediscretized equation simulates a convective-diffusiveprocess，rather than an advection process（平流，纯对流）ΦCED-NHT-EHT6/52CENTER

6/52 2 2 2 , , , 2 2 2 , ) ) ) ( , ) ) 2 2 i n i n i n i n t u x u O t t x x t x                      where the transient 2nd derivative can be re-written as follows: 2 2 ( ) t t t          ( ) u t x         u ( ) x t      2 2 2 u u u ( ) x x x             substituting into above equation 2 2 2 , , 2 , ) ) [ (1 )]( ) 2 ( , ) i n i n i n u x u t u t x t x O x x                    Thus at the sense of 2nd -order accuracy above discretized equation simulates a convective-diffusive process , rather than an advection process（平流，纯对流）

热流科学与工程西步文源大堂E教育部重点实验室u\tOnly whenO this error disappears.Axutis called Courant number, in memory of aAxGerman mathematician CourantapaduxuAt+O(Ax?, At?)21at2axAxdxRemark: We only study the false diffusion at the sense of2nd-order accuracy; i.e., if inspecting(审视) at the 2nd-orderaccuracythe abovediscretized equation actuallysimulatesaconvection-diffusion process. For most engineering problems2nd-oder accuracy solutions are satisfied.CFD-NHT-EHTΦ7/52CENTER

7/52 Only when 1 0 this error disappears. u t x     u t x   is called Courant number，in memory of a 2 2 2 , , 2 , ) ) [ (1 )]( ) 2 ( , ) i n i n i n u x u t u t x t x O x x                    Remark：We only study the false diffusion at the sense of 2 nd -order accuracy；i.e., if inspecting(审视) at the 2nd-order accuracy the above discretized equation actually simulates a convection-diffusion process. For most engineering problems 2 nd -oder accuracy solutions are satisfied. German mathematician Courant

热流科学与工程西步文源大堂G教育部重点实验室2.Extended meaningIn most existing literatures almost all numerical errorsare called false diffusion,which includes:(1) 1st-order accuracy schemes of the 1st order derivatives(original meaning);(2)Oblique intersection(倾斜交叉）offlowdirectionwith grid lines;(3) The effects of non-constant source term which arenot considered in the discretized schemes.4.5.2Examplescausedby1st-orderaccuracyschemes1.1-D steady convection-diffusion problemWhenconvectiontermisdiscretizedbyFUDdiffusion term by CD, numerical solutions will severelycFD-NHT-EHr deviate from analytical solutionsd8/52

8/52 2. Extended meaning In most existing literatures almost all numerical errors are called false diffusion，which includes： (1) 1 st -order accuracy schemes of the 1st order derivatives (original meaning)； (2) Oblique intersection(倾斜交叉) of flow direction with grid lines； (3) The effects of non-constant source term which are not considered in the discretized schemes. 4.5.2 Examples caused by 1st-order accuracy schemes 1. 1-D steady convection-diffusion problem When convection term is discretized by FUD， diffusion term by CD, numerical solutions will severely deviate from analytical solutions:

热流科学与工程西步文源大堂E教育部重点实验室1.0OFUDP.=20,PA=4$-中oFUD: Physically-CDPL-90exactplausible solution0.5FUD: severeerrorCD: oscillatingsolution0.50.51.0/L2.1-Dunsteadyadvectionproblem (Noye,1976)adad0≤ x≤1, u=0.1, Φ(0,t)=Φ(l,t) = 0ataxIn the range of x e[O,O.1] initial distribution is antriangle, others are zero. The two derivatives are discretizedCFD-NHT-EHTΦ9/52CENTER

9/52 2. 1-D unsteady advection problem (Noye,1976) u ,   (0, ) (1, ) 0 t t   t x         0 1, 0.1,    x u triangle，others are zero. CD: oscillating solution FUD: Physically plausible solution FUD: severe error In the range of x[0,0.1] initial distribution is an FUD CD The two derivatives are discretized

热流科学与工程西步文通大堂E教育部重点实验室the 1st -order accuracy schemes. The results are as follows虹t=4Caused by falset=41.0FI c=1.0diffusion of the1storderaccuracyu=0.1scheme1=0dc=0.81.of.Initialcondition00.20.40.60.81.03a.u=0.10.5t=80叫0.20.40.60.81.07411.0c=1.0Au=0.1Caused byfalse0.5diffusion of theist order accuracyc=0.8scheme00.20.40.60.81.0rCFD-NHT-EHTG10/52CENTER

10/52 Initial condition t=4 t=8 Caused by false diffusion of the 1 st order accuracy scheme Caused by false diffusion of the 1 st order accuracy scheme the 1st –order accuracy schemes. The results are as follows