《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（1/3，6.1-6.3）

热流科学与工程西步文医大学E教育部重点实验室Numerical HeatTransfer（数值传热学）Chapter6PrimitiveVariableMethodsforEllipticFlowandHeatTransfer(1)InstructorTao,Wen-QuanKeyLaboratoryofThermo-FluidScience&EngineeringInt.JointResearchLaboratoryofThermalScience&EngineeringXi'anJiaotongUniversityInnovativeHarborofWestChina,Xian2022-0ct-19CFD-NHT-EHTΦ1/37CENTER

1/37 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-19 Numerical Heat Transfer (数值传热学) Chapter 6 Primitive Variable Methods for Elliptic Flow and Heat Transfer（1）

热流科学与工程西步文源大堂G教育部重点实验室Contents6.1Sourcetermsinmomentumequationsand twokeyissuesinnumericallysolvingmomentumeguation6.2Staggeredgridsystemanddiscretizationofmomentumequation6.3PressurecorrectionmethodsforN-Sequation6.4ApproximationsinSiMPLEalgorithm6.5DiscussiononSIMPLEalgorithmandcriteriaforconvergence6.6DevelopmentsofSiMPLEalgorithm6.7Boundaryconditiontreatmentsforopensystem6.8Fluidflow&heat transferinaclosedsystem中CFD-NHT-EH'2/37CENTER

2/37 6.1 Source terms in momentum equations and two key issues in numerically solving momentum equation 6.2 Staggered grid system and discretization of momentum equation 6.3 Pressure correction methods for N-S equation 6.4 Approximations in SIMPLE algorithm 6.5 Discussion on SIMPLE algorithm and criteria for convergence 6.6 Developments of SIMPLE algorithm 6.7 Boundary condition treatments for open system 6.8 Fluid flow & heat transfer in a closed system Contents

热流科学与工程西步文通大堂G教育部重点实验室6.1Source terms in momentum eguationsand twokeyissuesinnumericallysolvingmomentumequation6.1.1 Introduction6.1.2Sourceinmomentumequations6.1.3Twokeyissuesinsolvingflowfield1.The conventional methods may lead to oscillatingpressure field2.Pressure has no governing equation-To improve anassumed pressure field a specially designed algorithmis neededCFD-NHT-EHT中3/37CENTER

3/37 6.1 Source terms in momentum equations and two key issues in numerically solving momentum equation 6.1.2 Source in momentum equations 6.1.3 Two key issues in solving flow field 1. The conventional methods may lead to oscillating pressure field 2. Pressure has no governing equation-To improve an assumed pressure field a specially designed algorithm is needed 6.1.1 Introduction

热流科学与工程西步文通大学E教育部重点实验室6.1 Sourcetermsin momentum equations and two keyissuesinnumericallysolvingmomentumequation6.1.1Introduction1 . Two kinds of most often encountered engineeringflows: boundary layer type and recirculation type00性底店边界障怕满止点CFD-NHT-EHTG速区4/37CENTER

4/37 1 . Two kinds of most often encountered engineering flows: boundary layer type and recirculation type 6.1 Source terms in momentum equations and two key issues in numerically solving momentum equation 6.1.1 Introduction

热流科学与工程西步文源大堂G教育部重点实验室2. Flow field solution is the most important step for solvingconvective heat transfer problems.3.Numerical approaches for solution of incompressibleIn such approaches no specialflow field :algorithm is needed. The onlySimultaneouslyrequirementis an extremelysolving（同时求解large computerresourcedifferent dependentPrimitive variable methodvariables(u, y, w, p, T)（原始变量法，u,vw,p）Pressure correction methodisthemostwidely used oneSegregated solutions（分离式求解）ofNon-primitivevariablemethodVortex-streamfunctionmethoddifferent dependent（涡量流函数法）isthemostvariableswidely used one (Chapter8 ofthe textbook)CFD-NHT-EHTG5/37CENTER

5/37 3 . Numerical approaches for solution of incompressible flow field： 2. Flow field solution is the most important step for solving convective heat transfer problems. In such approaches no special algorithm is needed. The only requirement is an extremely large computer resource. Simultaneously solving (同时求解) different dependent variables (u, v, w, p, T). Segregated solutions (分离式求解）of different dependent variables Primitive variable method (原始变量法，u,v,w,p）, Pressure correction method is the most widely used one Non-primitive variable method. Vortex-stream function method (涡量流函数法) is the most widely used one (Chapter 8 of the textbook)

热流科学与工程西步文源大学G教育部重点实验室6.1.2SourcetermsinmomentumequationsThe general governing equation is:a(pd) + div(pUp)= div(Fagradp)+ SaatComparing N-S equations in the three coordinateswith the above general governing equation, the relatedsource terms can be obtained,where bothphysical sourceterm (such as gravitation) and numerical source term areincluded;Treatment of source term is very important innumerical simulationof momentum equations.ΦCFD-NHT-EHT6/37CENTER

6/37 Comparing N-S equations in the three coordinates with the above general governing equation, the related source terms can be obtained, where both physical source term (such as gravitation) and numerical source term are included; The general governing equation is: ( ) div U div grad S ( ) ( ) t             6.1.2 Source terms in momentum equations Treatment of source term is very important in numerical simulation of momentum equations

热流科学与工程西步文源大堂E教育部重点实验室Table 6-1 (Text book)Sourcetermsof2-Dincompressibleflow(n = const. No gravitation)Coordinatesu-equationv-equationyl0Cartesian00uIVAxi--uyusymmetric02tcylindricalvupu2209au2nau_q大puynuPolar12r2ae72a6r2r中CFD-NHT-EH7/37CENTER

7/37 Source terms of 2-D incompressible flow ( const. No gravitation)   v u Table 6-1 (Text book)

热流科学与工程西步文源大学G教育部重点实验室6.1.3Twokeyissuesinsolvingincompressibleflowfield1.Conventional discretization method forpressuregradient in momentum equation may lead to oscillatingpressure field.Conventionally, one grid system is used to store all kinds ofinformation. If we store pressure , velocity, temperature, etc. atthe same grids, then the discretized momentum equations cannotdetectun-reasonablepressurefieldFor example. At node ithe 1-D steady momentum equationdrdd'udu_ dpoundr?xi-2i-1ii+1i+2dxdxCFD-NHT-EHTΦ8/37CENTER

8/37 6.1.3 Two key issues in solving incompressible flow field 1. Conventional discretization method for pressure gradient in momentum equation may lead to oscillating pressure field. Conventionally, one grid system is used to store all kinds of information. If we store pressure , velocity, temperature, etc. at the same grids, then the discretized momentum equations can not detect un-reasonable pressure field. 2 2 du dp d u u dx dx dx      For example. At node i the 1-D steady momentum equation

热流科学与工程西步文源大堂G教育部重点实验室canbediscretizedbvFDMasfollows:, ui+ -2u, +ui-I; O(△x2)Ui+I -ui-l --- Pi+I - Pi-l ++npu,(8x)228x28xCDCDCDDiscussion: this is the discretized momentum equationfor node i, but it does not contain the pressure at node i.while includesthepressuredifferencebetweentwonodespositioned two-steps apart, leading to following result: thediscretized momentum equation can not detect anunreasonable pressure solution! Because it is the pressuregradient rather than pressure itself that occurs in themomentumequationPressure difference over two steps is called 2- Sxpressure difference中CFD-NHT-EHT9/37CENTER

9/37 1 1 1 1 1 1 2 2 2 2 ( ) i i i i i i i i u u p p u u u u x x x                   ； CD CD CD Discussion：this is the discretized momentum equation for node i, but it does not contain the pressure at node i, while includes the pressure difference between two nodes positioned two-steps apart, leading to following result: the discretized momentum equation can not detect an unreasonable pressure solution！Because it is the pressure gradient rather than pressure itself that occurs in the momentum equation. Pressure difference over two steps is called pressure difference. 2  x can be discretized by FDM as follows： 2 O( ) x

热流科学与工程西步文源大堂E教育部重点实验室2i-2i-1ii+1i+2力True solutionrCFD-NHT-EHTG10/37CENTER

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