10.2.2 Discretized eguation of three kinds of variables1.General scalarvariable(标量)ΦFor all other scalar variables (including T, etc)a(p d)2 + div(p'U)= div(F gradg)+ S,atapp=aepe+aww+anp+asps+baw=DwA(|PwD+[F,0ae = D,A(PeD+-F,0ap=ag+aw+a+as+ap-S,vPAVb= S.AV +apdp20△tA(PD=0,(1-0.1P)Power-law scheme:16/68
16/68 10.2.2 Discretized equation of three kinds of variables ( ) , 0 E e e e a = + − D F A P ( ) ,0 W w w w a D A P F = + 1. General scalar variable (标量) P P E E W W N N S S a a a a a b = + + + + 0 P E W N S P P a a a a a a S V = + + + + − 0 P V a t = 0 c P P b S V a = + * * * ( ) div U div grad S ( ) ( ) t + = + For all other scalar variables (including T, etc) Power-law scheme: 5 A P P ( ) 0,(1 0.1 ) = −
(p*uA)。_pu (Sx)F.pudxPD.(TA / Sx)。I.rFortemperature:F=ap'=pc,2.Pressure & pressure correction equationThe derivation is based on mass conservation a(pu;)/ax; = 0[(pu)。-(pu) wJAy +[(pv), -(pv), JAx = 0(1) For SIMPLER, substituting u。=Ue +(A/a.)(Pp- Pe)into discretized mass conservation equationapPp=aePe+awPw+anP+asPs+bb=(pAi).-(pAi)。+(pA),-(pA)ae =(pAd)。,ap =Zanl17/68
17/68 e e e F P D = = For temperature： = 2. Pressure & pressure correction equation The derivation is based on mass conservation * = c p into discretized mass conservation equation: P P E E W W N N S S a p a p a p a p a p b = + + + + ( ) ( ) ( ) ( ) w e s n b Au Au A A = − + − v v ( ) * * e = e e e e u x u x = ( / ) e e uA A x （ ） （1） For SIMPLER, substituting u u A a p p e e e P E = +e ( ) - ( ) ( ) , E P nb e a = = Ad a a
(2)Substituting ue =uinto massOconservationequation:apPp=aePw'+awPw'+aPn'+asPs+bExcept b term, ap and ae,w,N,s are the same as p-equation.b=(pAu).-(pAu),+(pAv),-(pAvRemarks:The adopted mass conservation equationdoesnot include p/at andvelocity correctionneglects density effect. Thus, this code only applicabletoincompressibleflow18/68
18/68 conservation equation: P P E W W W N N S S a p a p a p a p a p b = + + + + ( ) ( ) ( ) ( ) * * * * w e s n b Au Au Av Av = − + − Except b term， , , , and P E W N S a a are the same as pequation. Remarks：The adopted mass conservation equation does not include and velocity correction neglects density effect. Thus, this code only applicable to incompressible flow. t ( ) e e e P E e A u u p p a Substituting = + − （ into mass 2）
3. Momentum equation (taking u as example)Governing equation:a(pu)op+S+div(piu)=div(ugradu)atox;Discretized equation:aeue =aeeuee +avuw +aneune +aseuse +b+ A (Pp-Pe)Volume of CV:NEuFor general @(8y).*neAV=Ax·Ay(8y).(8y),scalar variableAyEEEWeeu.wAV-Sx·AyFor u:(&r)Forv:AV-Ax·SySu.SEse19/68
19/68 Governing equation： ( ) ( ) ( ) u i u p div uu div gradu S t x + = − + Discretized equation： ae e ee ee w w ne ne se se e u a u a u a u a u b A p p = + + + + + − ( P E ) For general scalar variable = V x y For u : = V x y For v : = V x y w u ee u se u ne u Volume of CV: 3. Momentum equation (taking as example) u e u
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20/68 u-CV v-CV main-CV