《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（1/2，4.1-4.4）

热流科学与工程西步文源大学E教育部重点实验室Numerical Heat Transfer数值传热学）Chapter4DiscretizedSchemesofDiffusionandConvectionEquation(1)aInstructorTao.Wen-OuanKeyLaboratoryofThermo-FluidScience&EngineeringInt.JointResearchLaboratoryof Thermal Science&EngineeringXi'an Jiaotong UniversityXi'an,2022-Sept-28CFD-NHT-EHTG1/48CENTER

1/48 Key Laboratory of Thermo-Fluid Science & Engineering Int. Joint Research Laboratory of Thermal Science & Engineering Xi’an Jiaotong University Xi’an, 2022-Sept-28 Numerical Heat Transfer (数值传热学) Chapter 4 Discretized Schemes of Diffusion and Convection Equation (1) Instructor Tao, Wen-Quan

热流科学与工程西老义毛大堂G教育部重点实验室Chapter4Discretizeddiffusion-convectionequation4.1 Two ways of discretization of convection term4.2CDandUDoftheconvectionterm4.3Hybridandpower-lawschemes4.4Characteristicsoffivethree-pointschemes4.5 Discussion on false diffusion4.6Methodsfor overcoming oralleviatingeffectsoffalsediffusion4.7 Discretization of multi-dimensional problem andB.C.treatment中CFD-NHT-EHT2/48CENTER

2/48 4.1 Two ways of discretization of convection term 4.2 CD and UD of the convection term 4.3 Hybrid and power-law schemes Chapter 4 Discretized diffusion－convection equation 4.4 Characteristics of five three-point schemes 4.5 Discussion on false diffusion 4.6 Methods for overcoming or alleviating effects of false diffusion 4.7 Discretization of multi-dimensional problem and B.C. treatment

热流科学与工程西步文源大学G教育部重点实验室4.1 Two ways of discretization of convection term4.1.1Importanceofdiscretizedschemeofconvectionterm1.Accuracy2.Stability3.Economics4.1.2Two ways forconstructing discretizationschemesofconvectiveterm4.1.3RelationshipbetweenthetwowaysΦCFD-NHT-EHT3/48CENTER

3/48 4.1.1 Importance of discretized scheme of convection term 4.1.2 Two ways for constructing discretization schemes of convective term 1. Accuracy 2. Stability 3. Economics 4.1 Two ways of discretization of convection term 4.1.3 Relationship between the two ways

热流科学与工程西老义毛大堂G教育部重点实验室4.1Twowaysofdiscretizationofconvectionterm4.1.1Importanceofdiscretizationscheme（离散格式）Mathematically convective term is only of 1st orderderivative, while its physical meaning (strong directional)makes its discretization one of the hot spots (热点)ofnumericalsimulation:1.Itaffectsthenumericalaccuracy(精确性)Whenaschemeoftheconvectiontermwith1st-orderisused the solution involves severe numerical error2.It affects the numerical stability(稳定性）The schemes ofCD，TUD(三阶迎风) and QUICK areonly conditionally stable3.Itaffectsnumerical economics（经济性）中FO-NHTCE4/48CENTER

4/48 4.1 Two ways of discretization of convection term 4.1.1 Importance of discretization scheme (离散格式) Mathematically convective term is only of 1 st order derivative, while its physical meaning ( strong directional) makes its discretization one of the hot spots (热点) of numerical simulation： 1. It affects the numerical accuracy(精确性). When a scheme of the convection term with 1 st -order is used the solution involves severe numerical error. 2. It affects the numerical stability(稳定性). The schemes of CD，TUD(三阶迎风) and QUICK are only conditionally stable. 3. It affects numerical economics (经济性)

热流科学与工程西步文源大堂G教育部重点实验室4.1.2Twowaysforconstructing(构建)schemes1.Taylor expansion-providing the FD form at a pointTaking CD as an example:adE-w = +-_-, O(Ax22△x2△xax2. CV integration-providing average value within thedomainBy assuming a profile for the interface variablePiecewise linear.oddx -d-dAxUniform gridsAxax(虹 +Φp)/2 -(Φp +dw)/2 _ d -d, O(Ax2 )Ax2△xCFD-NHT-EHTΦ5/48CENTER

5/48 4.1.2 Two ways for constructing(构建) schemes 1. Taylor expansion－providing the FD form at a point Taking CD as an example: 1 1 2 ) ( ) 2 2 E W i i P O x x x x                 ， 2. CV integration－providing average value within the domain 1 e w dx x x     Piecewise linear ( )/ 2 ( )/ 2 2 E P P W E W x x               e w x     Uniform grids By assuming a profile for the interface variable 2 ，O x ( ) 

热流科学与工程西步文源大堂G教育部重点实验室4.1.3Relationshipbetweenthetwo ways1. For the same scheme they have the same order of the T.E2. For the same scheme, the coefficients of the 1st termin TE. are different The absolutevalue of FVM isusually less than that of FD.3. Taylor expansion provides the FD form at a point while CVintegration gives the average value by integration within thedomainpe-d2oOxxAxΦCFD-NHT-EHT6/48CENTER

6/48 4.1.3 Relationship between the two ways 1. For the same scheme they have the same order of the T.E. 2. For the same scheme, the coefficients of the 1st term in T.E. are different. The absolute value of FVM is usually less than that of FD. 3. Taylor expansion provides the FD form at a point while CV integration gives the average value by integration within the domain 1 e w dx x x     e w x    

热流科学与工程西步文源大堂G教育部重点实验室4.2cDand UD of theconvectionterm4.2.1 Analytical solution of 1-D modelequation4.2.2cD discretizationof1-Ddiffusion-convectionequation4.2.3Upwindschemeofconvectionterm1.Definition of CV integration2.Compactform3.Discretization eguation withUD of convectionandCDofdiffusionΦCFD-NHT-EHT7/48CENTER

7/48 4.2.1 Analytical solution of 1-D model equation 4.2 CD and UD of the convection term 4.2.2 CD discretization of 1-D diffusion-convection equation 4.2.3 Up wind scheme of convection term 1. Definition of CV integration 2. Compact form 3. Discretization equation with UD of convection and CD of diffusion

热流科学与工程西步文源大堂G教育部重点实验室4.2CD and UD of convection term4.2.1Analytical solutionof1-Dmodeleq.withoutsourceterm(diffusionandconvectioneq.)dd(pup)ddPhysical properties andvelocityareknownconstantsdxdxdxx=0, Φ=; x=L, Φ=ΦThe analytical solution of this ordinary differentequation:pulexp(Pe=)-1expΦ-exp(pux/T)-1exp(puL/)-1d -dexp(puL/T)-1exp(Pe)-1CFD-NHT-EHTΦ8/48CENTER

8/48 4.2 CD and UD of convection term 4.2.1 Analytical solution of 1-D model eq. without source term (diffusion and convection eq.) ( ) ( ), d u d d dx dx dx      Physical properties and velocity are known constants 0 0, ; , L x x L         The analytical solution of this ordinary different equation： 0 0 exp( ) 1 exp( / ) 1 exp( / ) 1 exp( / ) 1 L uL x ux L uL uL                     exp( ) 1 exp( ) 1 x Pe L Pe   

热流科学与工程西步文源大堂G教育部重点实验室Solution AnalysisdtPe = 0, pure diffusion, linearDistribution;5With increasing Pe, distribution~1curve becomes more and moreconvex downward (下凸);When Pe =10, in the most region5from x=0-LPe≥10Φ=d80XOnly when x is very close to L, @Lincreases dramatically andwhen x=L ,Φ = ΦL .CFD-NHT-EHTΦ9/48CENTER

9/48 Solution Analysis Pe＝0，pure diffusion, linear Distribution; With increasing Pe，distribution curve becomes more and more convex downward (下凸)； When Pe＝10，in the most region from x=0-L    0 when x=L ， .    L Only when x is very close to L， increases dramatically and 

热流科学与工程西步文源大堂E教育部重点实验室The above variation trend with Peclet number isconsistent(协调的)with the physical meaning of PepulConvectionpuPe/ LrDiffusionWhen Pe is small-Diffusion dominated, lineardistribution ;When Pe is large-Convection dominated, i.e.,upwind(上游)effect dominated,upwind information istransported downstream, and when Pe ≥ 100, axialconduction can be totally neglected.It is reguired in some sense that the discretizedscheme of the convective term has some similar physicalcharacteristics.ΦCFD-NHT-EH10/48CENTER

10/48 The above variation trend with Peclet number is consistent(协调的) with the physical meaning of Pe When Pe is small－Diffusion dominated，linear distribution ； Convection Diffusion It is required in some sense that the discretized scheme of the convective term has some similar physical characteristics. / uL u Pe L       When Pe is large－Convection dominated，i.e., upwind(上游) effect dominated， upwind information is transported downstream, and when Pe 100, axial conduction can be totally neglected. 