《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WhenCourantnumberislessthan1,severeerroroccurs，whicherases（抹平）thesharppeak(抹平尖峰）and magnifythebase（放大基底）gradually.Sucherroriscalledstreamwisefalsediffusion（流向假扩散）4.5.3Errorscausedbyobliqueintersection（倾斜交叉）F=0I±0Two gas streamsTwithdifferenttempera-HotgasturesmeeteachotherAssuming zero gas difLy7fusivities. If the flowdirection is obliquelyCold gasClapboardwith respect to the gridlines, big numericalerrorswill beintroducedGasflowwithandnon-OGammaCFD-NHT-EHT中11/52CENTER

11/52 When Courant number is less than 1 ,severe error occurs，which erases （抹平）the sharp peak(抹平尖 峰) and magnify the base (放大基底) gradually. Such error is called streamwise false diffusion (流向假扩散). 4.5.3 Errors caused by oblique intersection (倾斜交叉)   0 Gas flow with 0 and non-0 Gamma Two gas streams with different tempera￾tures meet each other. Assuming zero gas dif￾fusivities. If the flow direction is obliquely with respect to the grid lines, big numerical errors will be introduced.   0

热流科学与工程西步文源大堂G教育部重点实验室1.Case1:withx-ycoordinateseitherparalleorperpendiculartoflowdirectionAdopting FUD, then A(P D = 1 ; For the CV. P:100NU>0,I=0100ag = D, +[-F,0DEW100100s100U>0,I=0aw=Dw+[Fw,0-W0V=0,I=00a =D, +[-F,000V=0,I=0as = D,+[F,0Upstream velocity U0ap = +aw++s=aw so Φp =pw!Thus we have:Theupstreamtemperatureiskept downstream!中CFD-NHT-EH12/52CENTER

12/52 1. Case 1: with x-y coordinates either parallel or perpendicular to flow direction , 0 E e a D F    U    0, 0 0 , 0 W W W a D F   U    0, 0 Fw , 0 N n a D F    V    0, 0 , 0 S s a D F   V    0, 0 0 0 Upstream velocity U E so !   P W  The upstream temperature is kept downstream! Adopting FUD, then A P( ) 1 ; For the CV. P:   Thus we have: a a a a a P E W N S   + + W  a

热流科学与工程亚步文源大堂E教育部重点实验室2.Case 2:x-y coordinates intersect the on comingflowwith45degree2From upstream velocityU, u=12Again FUD is adopted, then for CV. P:u>0,F=0010096.8889.0677.86ae = D+-F,093.7510081.2566.65u>0,=0aw = Dw +[F,,010087.5668.755034.38WW31.2518.750.753EV>0,I=0an=D, +[-F,012.56.2550 B250S000as = D, +[F,0v>0,=0U(b)sdw +dsap=aw+as,ΦpFw=F,app=awdw+asps+0+0,2中CFD-NHT-EHT13/52CENTER

13/52 2. Case 2: x-y coordinates intersect the on coming flow with 45 degree Again FUD is adopted，then for CV. P： 2 , 2 From upstream velocity U ，u v U   u    0, 0 0 u    0, 0 Fw v    0, 0 Fs 0 v    0, 0 , 0 0, F F a a a w s P P W W S S         2 , ! P W S W S P a a a        U , 0 E e e a D F    , 0 W W w a D F   , 0 N n n a D F    , 0 S s s a D F  

热流科学与工程西步文源大学E教育部重点实验室Fluid temperatures across the diagonal become smoothand continuous. This is caused by the cross-diffusion.Discussion: For case 1 where velocity is parallel to xcoordinatethe FUD scheme also produces false diffusionbut compared with convection it can not be exhibited(展现): the zero diffusivity corresponds to an extremely largePeclet number, i.e., convection is so strong that falsediffusion can not be exhibited. When chances come (有机会时) it will take action. Example 1 of this section is sucha situation.4.5.4 Errors caused by non-constant source termS non-constant,d(pud)ddp+Sdistribuiton isGiven:dxdxdxspecifiedx= O,Φ=; x= L,Φ=ΦCFD-NHT-EHTΦ14/52CENTER

14/52 Fluid temperatures across the diagonal become smooth and continuous. This is caused by the cross-diffusion. Discussion：For case 1 where velocity is parallel to x coordinate，the FUD scheme also produces false diffusion, but compared with convection it can not be exhibited(展 现): the zero diffusivity corresponds to an extremely large Peclet number，i.e., convection is so strong that false diffusion can not be exhibited. When chances come (有机 会时) it will take action. Example 1 of this section is such a situation. 4.5.4 Errors caused by non-constant source term ( ) ( ) , d u d d S dx dx dx       0 0, ; , L x x L         S non-constant, distribuiton is specified. Given：

热流科学与工程西步文源大堂E教育部重点实验室Forcaseswithsuchnon-constantsourceterm neither one of the five 3-point schemescangetaccuratesolution.Taking hybrid scheme as an example. When gridPecletnumberislessthan2,numericalresults agreewith analytical solution quite well; However, whengridPecletnumberislargerthan2,deviationsbecomelarge. Its coefficient is defined by:a =D,A(PeD+[-F,0 , aw =D,A(PAD+ Fw,0 A(PeD=[o,1-0.5PelAssuming that variation of Peclet number isimplementedviachanging diffusioncoefficient whileflowrateis remained unchanged thenwhen中CFD-NHT-EHT15/52CENTER

15/52 For cases with such non-constant source term neither one of the five 3-point schemes can get accurate solution. Taking hybrid scheme as an example. When grid Peclet number is less than 2，numerical results agree with analytical solution quite well; However, when grid Peclet number is larger than 2 ,deviations become large. Its coefficient is defined by: ( ) ,0 , E e e a D A P F     ( ) 0,1 0.5 A P P   e e   Assuming that variation of Peclet number is implemented via changing diffusion coefficient while flow rate is remained unchanged then when ( ) ,0 W w w w a D A P F   