《数值传热学》研究生课程教学资源（课件讲稿）Chapter 6 Primitive Variable Methods for Elliptic Flow and Heat Transfer（3/3，6.7-6.8）

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）Chapter6PrimitiveVariableMethodsforEllipticFlowandHeatTransfer(3)oInstructorTao,Wen-QuanKeyLaboratoryofThermo-FluidScience&EngineeringInt.JointResearchLaboratoryofThermalScience&EngineeringXi'anJiaotongUniversityInnovativeHarborofWestChina,Xian2022-OCT.-26CFD-NHT-EHTΦ1/34CENTER

1/34 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.-26 Numerical Heat Transfer (数值传热学) Chapter 6 Primitive Variable Methods for Elliptic Flow and Heat Transfer（3）

热流科学与工程西步文源大堂G教育部重点实验室6.7Boundary condition treatmentsforopen system6.7.1Selectionsforoutletboundary6.7.2Treatment of outlet boundary conditionwithoutrecirculation1. Local one-way2. Fully developed6.7.3 Treatment of outlet boundary condition withrecirculation1. Example with recirculation ; 2. Suggestion6.7.4Methodsfor outlet normal velocity satisfyingtotalmassconservation1. Two cases;2. Application中CFD-NHT-EH'2/34CENTER

2/34 6.7.1 Selections for outlet boundary 6.7.2 Treatment of outlet boundary condition without recirculation 6.7.3 Treatment of outlet boundary condition with recirculation 1. Local one-way 2. Fully developed 1. Two cases； 2. Application 1. Example with recirculation ; 2. Suggestion 6.7 Boundary condition treatments for open system 6.7.4 Methods for outlet normal velocity satisfying total mass conservation

热流科学与工程西步文源大堂G教育部重点实验室6.7 Boundary condition treatments for open system6.7.1Selectionsforoutletboundaryposition1. At the location without recirculation (回流)--Suggested byPatankar2 .At the location with recirculation---special attentionshouldbepaidforboundaryconditiontreatment6.7.2Treatment ofB.C.without recirculationUi,Mi1. Local one-way assumptionui+IM1ui,M（局部单向化假设）+1-M2Di,M2CFD-NHT-EHTG3/34CENTER

3/34 6.7 Boundary condition treatments for open system 6.7.1 Selections for outlet boundary position 1. At the location without recirculation (回流)- Suggested by Patankar 2 .At the location with recirculation-special attention should be paid for boundary condition treatment 6.7.2 Treatment of B.C. without recirculation 1. Local one-way assumption (局部单向化假设) , 2 ( ) 0 N i M a 

热流科学与工程西步文源大堂E教育部重点实验室adDi,Mi2. Fully developed,MI=0ui+1M1ui,Man(1) Updating (更新)boundary valueIsM中,m -d.M2 = 0Pi,MI=gDUi,M2i.M2(8y)b(2) ASTM:adTaking= as givenheatflux conditionanFor both methods, the outlet normal velocity mustsatisfy thetotal mass conservation condition6.7.3Treatment ofoutletboundaryconditionwithrecirculation1.Necessity（必要性）for such selectionRequired from somepractical problemsΦCFD-NHT-EHT4/34CENTER

4/34 2. Fully developed , 1 ) 0 i M n    (1) Updating (更新)boundary value , 1 , 2 0 ( ) i M i M B y      *   i M i M , 1 , 2  (2) ASTM： Taking , 1 ) 0 i M n    as given heat flux condition For both methods，the outlet normal velocity must satisfy the total mass conservation condition. 6.7.3 Treatment of outlet boundary condition with recirculation 1. Necessity (必要性) for such selection Required from some practical problems

热流科学与工程亚步文源大堂E教育部重点实验室AccordingtoPatankar,the outletboundary of thesudden expansion case must be positioned at the locationwithout recirculation ("Good" position) . It should notbe positioned at the “Bad" location, otherwise the resultsare meaningless.This suggestion not only needs more computermemory but also is not possible for some situations.Openflow systemGooGoodBadBadΦCFD-NHT-EHT5/34CENTER

5/34 According to Patankar，the outlet boundary of the sudden expansion case must be positioned at the location without recirculation (“Good” position）. It should not be positioned at the “Bad” location, otherwise the results are meaningless. Open flow system Bad Good This suggestion not only needs more computer memory but also is not possible for some situations

热流科学与工程西步文通大学G教育部重点实验室If theneglectofthediffusion at anoutflowboundaryappears,forsomcreason, to be serious, then we should conclude that the analyst has placedAparticular badthe outflow boundary at an inappropriate location,A repositioning of theboundary would normally make the outflow treatment acceptableA partic-choice of anularly bad choice of an outflow-boundary location is the one in which thcre is an"inflow"overapartof it.Anexampleof this is shown inFig.5.12.Forsu-bad choice of the boundary,no meaningful solution can be obtained.outflow-boundaryThis may be a convenient place to review the boundary-conditionpractices for convection-diffusion problems.Whenever there is no fluid flowlocation is the oneacross the boundary of the calculation domain,theboundary flux is purelyadiffusion flux,and the practices described in Chapter 4apply.For thosepartsin which there is anof the boundary where the fluid flows into the domain, usually the values ofareknown.(Theproblemisnotproperlyspecifiedif wedonotknowthevalue of that a fluid stream brings with it.)The parts of the boundary"inflow" over awhere the fluid leaves the calculation domain form the outflow boundary,which we have already discussed.part of it. ...Forsuch a bad choiceof the boundary, nomeaningfulsolution can beobtained.(1980)BadGoodFigure 5.12 Good and bad choices of the location of the outflow boundary.CFD-NHT-EHG6/34CENTER

6/34 A particular bad choice of an outflow-boundary location is the one in which there is an “inflow” over a part of it. . For such a bad choice of the boundary, no meaningful solution can be obtained .(1980)

热流科学与工程西步文通大学C教育部重点实验室出口边券Cooling of plateTV screenCFD-NHT-EHTG7/34CENTER

7/34 Cooling of plate TV screen

热流科学与工程亚步文源大堂E教育部重点实验室2. Suggestion(1) Outlet normal velocity---treated according to localmassconservation(2) Outlet parallel velocity---treated by homogeneousNeumanncondition（齐次诺曼条件）Total mass conservationU,T.Local massVi.M1conservation个出口边界MIM212福i+1.M2.M25Li,M2CFD-NHT-EHTG8/34CENTER

8/34 (1) Outlet normal velocity-treated according to local mass conservation 2. Suggestion (2) Outlet parallel velocity-treated by homogeneous Neumann condition (齐次诺曼条件) Total mass conservation Local mass conservation

热流科学与工程西步文源大堂E教育部重点实验室VUi+l,M2 -Ui,M2 = 0 -Vi,M1 -Vi,M2i,M1Ui.MAxAyAyAylu2Vi,M1i+1,M2i.M2+1.M2△xThe resulted Vi,M1 has to bei,M2Axcorrected by total mass conservationcondition.auU.M=U.M2=0Tangential velocity6.7.4 Methods for outlet normal velocity to satisfytotalmassconservation1. Two situations1) Outlet without recirculation(1) Relative changes of outlet normal velocity =constant中HFO-NHTCEH9/34CENTER

9/34 , 1 , 2 1, 2 , 2 0 i M i M i M i M v v u u y x        * * * , 1 , 2 1, 2 , 2 ( ) i M i M i M i M y v v u u x       The resulted has to be corrected by total mass conservation condition. i M, 1 v 6.7.4 Methods for outlet normal velocity to satisfy total mass conservation 1. Two situations 1) Outlet without recirculation (1) Relative changes of outlet normal velocity =constant Vi M, 1 , 1 ) 0 i M U y   Tangential velocity  * U U i M i M , 1 , 2  Ui M, 1 x y

热流科学与工程亚步文源大堂E教育部重点实验室Di,MiVi,M1-Vi,m2 = k = constu+1M144.5Vi,M2Vi,M1 = Vi,M2(1+k)= f vi,M2 *f is determined according to total mass conservation :L2L2ZpimilimAx, =Zpvi,M2Ax, = FLOWINi=2i=2FLOWINTViMi = fVL2Vi.M2ZPi,miV,m2Ax,It is taken as the boundary condition for next iterationΦCFD-NHT-EHT10/34CENTER

10/34 , 1 , 2 , 2 i M i m i M v v k const v    , 1 , 2 2 , (1 ) i M i M i M v    v k f v f is determined according to total mass conservation : 2 2 , 1 , 1 , 1 , 2 2 2 L L i M i M i i M i M i i i   v x v x FL f OWIN         2 , 1 , 2 2 L i M i M i FLOWIN f  v x    , 1 * i M i M, 2 v   f v It is taken as the boundary condition for next iteration