《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