（严重的）When Courant number is less than 1, severeerroroccurs，whicherases（抹平）thesharppeak（抹平尖峰）andmagnifythebase（放大基底）gradually.Sucherroriscalledstreamwisefalsediffusion（流向假扩散）4.5.3Errorscausedbyobliqueintersection（倾斜交叉）I±0T=0Two gas streamsTwithdifferenttempera-HotgasturesmeeteachotherAssuming zero gas dif-1yfusivities. If the flowdirection is obliquelyColdgasCiapboardwith respect to the gridlines, big numericalerrorswillbeintroducedGasflowwithOandnon-OGamma11/55
11/55 When Courant number is less than 1 , severe （严重的） 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 temperatures meet each other. Assuming zero gas diffusivities. If the flow direction is obliquely with respect to the grid lines, big numerical errors will be introduced. 0
1. Case 1:with x-y coordinates either parallel orperpendicular（垂直的）to flow directionAdopting FUD, then A(P D = 1 ; For the CV. P:U>0,F=0ag =D +[-F,0100N100U>0,I=0aw=Dw+Fw,0PEW100A100s100Similar expressions can be written for10northand southnodes0V=0,F=0a = D, +[-F,0福00V=0,r=0Upstream velocity Uas =D, +[F,0福ap =E+aw+o+as=aw so Φp =dw!Thus we have:The upstream temperature is kept downstream!12/55
12/55 1. Case 1: with x-y coordinates either parallel or perpendicular（垂直的） 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 Similar expressions can be written for north and south nodes:
(与·交叉)2.Case 2:x-y coordinates intersectthe oncomingflowwith45degree2From upstream velocityU, u=v2Again FUD is adopted, then for CV. P:u>0,F=010096.8889.06077.86ag= De+-F,093.7510081.2566.65u>0,=0aw = Dw +[Fv, 010087.5668.755034.38WW31.2518.7515Ev>0,F=0an =D, +[-F,012.56.2550B250S000v>0,T=0as = D, +F,,0U(b)Fsw +dsFw=F,,appp=awpw+asds +0+0,0ap=aw+as,213/55
13/55 2. Case 2: x-y coordinates intersect （与.交叉）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 = +
Fluid temperatures across the diagonal become smooth andcontinuous.This is caused by the cross-diffusion （交叉扩散）Discussion:For case 1 where velocity is parallel to x coordinate,theFUD scheme alsoproducesfalse diffusion,but compared withconvection it can not be exhibited(展现): the zero diffusivitycorresponds to an extremely large Peclet number, i.e., convection isso strong that false diffusion can not be exhibited. When chancescome (有机会时)it will take action.Example 1 of this section is sucha situation.4.5.4Errors caused by non-constant sourcetermS non-constant,d(pup)-d(rdSGiven:dxits distribuiton isdxdxspecified（规定了）x=O,Φ=;x= L,Φ=Φ14/55
14/55 Fluid temperatures across the diagonal become smooth and continuous. This is caused by the cross-diffusion（交叉扩散）. 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, its distribuiton is specified (规定了). Given：
Forcaseswithsuchnon-constantsourcetermneitherone of thefive3-pointschemescangetaccuratesolution.Taking hybrid scheme as an example. When grid Pecletnumber is less than 2, numerical results agree with analyticalsolution quite well; However, when grid Peclet number is largerthan 2 ,deviations become large. Its coefficient is defined bya =D,A(PAeD+-F,0 , aw =D,A(PA,D+ Fw,0 A(|PAe)=[o,1-0.5|PelAssuming that variation of Peclet number is implemented(实施）via changingdiffusion coefficient whileflowrateisremained unchanged then when15/55
15/55 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 (实施) via changing diffusion coefficient while flow rate is remained unchanged then when ( ) ,0 W w w w a D A P F = +