《数值传热学》研究生课程教学资源（课件讲稿）Chapter 1 Introduction Numerical Heat Transfer

热流科学与工程西步文医大学C教育部重点实验室NumericalHeatTransfer（数值传热学）Chapter1 IntroductionInstructorTao,Wen-OuanKeyLaboratoryofThermo-FluidScience&EngineeringInt.JointResearchLaboratoryof Thermal Science&EngineeringXi'anJiaotongUniversityInnovativeHarborofWestChina,Xian2022-Sept-17CFD-NHT-EHTG1/57CENTER

1/57 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-Sept-17 Numerical Heat Transfer (数值传热学) Chapter 1 Introduction

热流科学与工程西步文通大堂E教育部重点实验室ContentsofChapter1（数学描述）of1.1Mathematicalformulationheattransferandfluidflow (HT&FF)problems1.2Basic conceptsof NHT,itsimportance andapplicationexamples1.3Mathematical and physical classificationofHT&FFproblemsanditseffectsonnumericalsolution中CFD-NHT-EHT2/57CENTER

2/57 1.1 Mathematical formulation （数学描述）of heat transfer and fluid flow (HT & FF) problems 1.2 Basic concepts of NHT, its importance and application examples 1.3 Mathematical and physical classification of HT & FF problems and its effects on numerical solution Contents of Chapter 1

热流科学与工程西步文源大堂E教育部重点实验室1.1 Mathematicalformulationofheat transferandfluidflow(HT&FF)problemsand their1.1.1Governing equations(控制方程）general form1.Massconservation2.Momentum conservation3.Energy conservation4. General form1.1.2 Conditions for unique solution （唯一解)1.1.3Exampleof mathematicalformulation中CFD-NHT-EHT3/57CENTER

3/57 1.1 Mathematical formulation of heat transfer and fluid flow (HT & FF) problems 1.1.1 Governing equations （控制方程）and their general form 1.1.2 Conditions for unique solution（唯一解） 1.1.3 Example of mathematical formulation 1. Mass conservation 2. Momentum conservation 3. Energy conservation 4. General form

热流科学与工程西步文源大堂G教育部重点实验室1.1 Mathematical formulation of heat transfer andfluidflow(HT&FF)problemsAllmacro-scale（宏观）HT&FFproblemsaregoverned by three conservation laws: mass, momentumandenergyconservationlaw（守恒定律）The differences between different problems are in:conditions for the unique solution（唯一解）：initial（初始的）&boundaryconditions，physicalpropertiesandsourceterms1.1.1Governing equations and their general form1.Mass conservation(pu)a(pv)a(pw)p0OzataxayCFD-NHT-EHTG4/57CENTER

4/57 1.1 Mathematical formulation of heat transfer and fluid flow (HT & FF) problems All macro-scale （宏观）HT & FF problems are governed by three conservation laws：mass, momentum and energy conservation law (守恒定律). The differences between different problems are in: conditions for the unique solution（唯一解）：initial （初始的）& boundary conditions, physical properties and source terms. 1.1.1 Governing equations and their general form 1. Mass conservation ( ) ( ) ( ) 0 u v w t x y z                

热流科学与工程西步文源大学E教育部重点实验室diy"is themathematicaldysymbol for divergence(散度)a(pw)apa(pv)a(pu)div(pu)+ div(pU)= 0azaxdyatFor incompressible fluid（不可压缩流体）OuOvowdiv(U) = 0CaxOzdyflow without source and sink（没有源与汇的流动）。CFD-NHT-EHTΦ5/57CENTER

5/57 div U ( ) 0 t       For incompressible fluid（不可压缩流体）： ( ) 0 ; 0 u v w div U x y z           flow without source and sink (没有源与汇的流动)。 ( ) ( ) ( ) ( )= u v w div U x y z             ( ) 0, 0 u v w div U x y z           “div” is the mathematical symbol for divergence （散度）

热流科学与工程亚步文源大堂G教育部重点实验室2.Momentum conservationApplying the 2nd law of Newton (F=ma) to theelementalcontrolvolume（控制容积）in the three-dimensional coordinates:r[IncreasingrateofmomentumoftheCV]=[Summationofexternal（外部）forcesapplying on the CVAdopting Stokes assumption: stress is linearly proportionalto strain(应力与应变成线性关系），Wehavefollowingresultfor component u in x-direction :CFD-NHT-EHT中6/57CENTER

6/57 Applying the 2nd law of Newton （F=ma) to the elemental control volume（控制容积） in the three￾dimensional coordinates: 2. Momentum conservation [Increasing rate of momentum of the CV] = [Summation of external（外部） forces applying on the CV] Adopting Stokes assumption：stress is linearly proportional to strain(应力与应变成线性关系)，We have following result for component u in x-direction：

热流科学与工程西步文源大堂教育部重点实验室SourcetermaOua(pu)a(puu).(puv).a(puw)op(AdivU +2nataxaxOxaxoyozConvectionDiffusionTransienttermtermtermaaouovouowpFnTOzOzayaxayOxDiffusiontermn dynamic viscosity , α fluid 2nd molecular viscosity.2For gas, π=-=n3CFD-NHT-EHTΦ7/57CENTER

7/57  dynamic viscosity ， ( ) ( ) ( ) ( ) ( 2 ) [ ( )] [ ( )] x u uu uv uw p u divU t x y z x x x v u u w F y x y z z x                                                 fluid 2nd molecular viscosity. For gas, 2 =- 3   Transient term Convection term Diffusion term Diffusion term Source term

热流科学与工程西步文源大堂E教育部重点实验室It can be shown (see the notes) that the above eguationcan be reformulated as（改写为）following general form ofNavier-Stokes equation for u component:a(pu)+ div(puU)= div(ngradu)+SatDiffusionTransientConvectionSourceterm扩散项term源项term非稳态项term对流项u, y, w ----velocity components in three directions, respectivelydependent variable（因变量）to be solved;U ----fluid velocity vector; U=ui +vj+ wkS,----source term.中CFD-NHT-EHT8/57CENTER

8/57 ( ) ( ) ( ) u div U div grad S u u u t         It can be shown (see the notes) that the above equation can be reformulated as （改写为）following general form of Navier-Stokes equation for u component: Transient term 非稳态项 Convection term对流项 Diffusion term扩散项 Source term源项 u, v, w -velocity components in three directions, respectively, dependent variable（因变量） to be solved； -fluid velocity vector; Su -source term. U U ui v j wk =  

热流科学与工程亚步文源大堂CE教育部重点实验室Sourcetermin x-direction:ForincompressiblefluidaaaaowauOvap(adivU)+'pFnaxOzaxaxayaxaxOxSimilarly:aaaOuOvOwaop(adivU)+ pF,-(nn(noyoyaxOzayayayayaaaavowaOuap(adivU)+ pF(nnOzOzayOzOzOzazaxFor incompressible fluid with constant properties the sourceterm does not contain velocity-related partΦCFD-NHT-EHT9/57CENTER

9/57 Source term in x-direction： ( ) ( ) ( ) ( ) u x u v w p S divU F x x x y z x x x                            ( ) ( ) ( ) ( ) v y u v w p S divU F x y y y z y y y                            ( ) ( ) ( ) ( ) w z u v w p S divU F x z z y z z z z                            Similarly： For incompressible fluid with constant properties the source term does not contain velocity-related part. For incompressible fluid

热流科学与工程西步文源大堂G教育部重点实验室3.Energy conservationIncreasingrateof internal energyintheCVl-[Netheat going into the CVJ+[Work conducted by bodyforcesand surfaceforcesIntroducing Fourier's law of heat conduction andneglecting the work conducted by forces; Introducingenthalpy （焰) h = C,T,assuming C,= constant,We have:a(pT+ div(pTU)= div(一grad(T) + STat2anaT-aTaT4nKgrad(TOzaxayPrcc,nC,nD中CFD-NHT-EHT10/57CENTER

10/57 3. Energy conservation [Increasing rate of internal energy in the CV]= [Net heat going into the CV]+[Work conducted by body forces and surface forces] Introducing Fourier’s law of heat conduction and neglecting the work conducted by forces；Introducing enthalpy（焓） ( ) ( ) ( ( )) T p div U di T T T c v grad S t         ( ) Pr p p p c c c           ( ) Pr p p p c c c           ( ) Pr p p p c c c           ( ) Pr p p p c c c           h c T  p , assuming p c  constant, ( ) T T T grad T i j k x y z          We have: