P1:JZP 0521853265pre CB908/Date 0521853265 May27,2005 0:54 INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS IntroductiontoComputationalFluidDynamicsintroducesalltheprimarycom- ponentsforlearningandpracticingcomputationalfluiddynamics(CFD).The bookiswrittenforfinalyearundergraduatesand/orgraduatestudentsinme- chanical, chemical, and aeronautical engineering who have undergone basic coursesinthermodynamics,fluidmechanics,andheatandmasstransfer.Chap- ters cover discretisation of equations for transport of mass, momentum, and energyonCartesian,structuredcurvilinear,andunstructuredmeshes;solution ofdiscretisedequations,numericalgridgeneration,andconvergenceenhance- ment.Thebookfollowsaconsistentphilosophyofcontrol-volumeformulation ofthefundamentallawsoffluidmotionandenergytransferandintroducesa novel notion of “smoothing pressure correction” for solution of flow equa- tions on collocated grids within the framework of the well-known SIMPLE algorithm. Thereareover50solvedproblemsinthetextandover130end-of-chapter problems.Practicingindustryprofessionalswillalsofindthisbookusefulfor continuingeducationandrefreshercourses. ProfessorAnilW.Dateobtainedhisbachelor’sdegreeinmechanicalengineer- ingfromBombayUniversity;hismaster’sdegreeinthermo-fluidsfromUMIST Manchester, UK; and his doctorate in heat transfer from Imperial College of Science and Technology, London. He has been a member of the Thermo- Fluids-Engineering group of the Mechanical Engineering Department at IIT Bombaysince1973.Overthepastthirtyyears,hehastaughtcoursesatboth undergraduateandpostgraduatelevelinthermodynamics,energyconversion, heatandmasstransfer,andcombustion.Hehasbeenengagedinresearchand consultinginthermo-fluidsengineeringandisanactivereviewerofresearch proposalsandpapersforvariousnationalandinternationalbodiesandjournals. He has been Editor for India of the Journal of Enhanced Heat Transfer and has contributed research papers to several international journals in the field. HehasbeenavisitingscientistatCornellUniversityandavisitingprofessor attheUniversityofKarlsruhe,Germany.Hehasdeliveredseminarlecturesat universitiesinAustralia,HongKong,Sweden,Germany,UK,USA,andIndia. Professor Date derives great satisfaction from applying thermo-fluid science torural-technologyproblemsinIndiaandhastaughtcoursesinscience,tech- nology, and society and in appropriate technology at IIT Bombay. Professor DateisaFellowoftheIndianNationalAcademyofEngineering(FNAE). i P1:JZP 0521853265pre CB908/Date 0521853265 May27,2005 0:54 ii P1:JZP 0521853265pre CB908/Date 0521853265 May27,2005 0:54 Introduction to Computational Fluid Dynamics ANIL W. DATE IndianInstituteofTechnology,Bombay iii cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press TheEdinburghBuilding,Cambridgecb22ru,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Informationo nthi stitle :www.cambri dge.org/9780521853262 © Cambridge University Press 2005 Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionof relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Firstpublishedinprintformat 2005 isbn-13 978-0-511-13052-6 eBook (NetLibrary) isbn-10 0-511-13052-x eBook (NetLibrary) isbn-13 978-0-521-85326-2 hardback isbn-10 0-521-85326-5 hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurls forexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnot guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. P1:JZP 0521853265pre CB908/Date 0521853265 May27,2005 0:54 Dedicatedtothememoryof Aai,Kaka,andWalmik v P1:JZP 0521853265pre CB908/Date 0521853265 May27,2005 0:54 vi P1:JZP 0521853265pre CB908/Date 0521853265 May27,2005 0:54 Contents Nomenclature pagexiii Preface xvii 1. Introduction..........................................1 1.1 CFDActivity 1 1.2 TransportEquations 2 1.3 NumericalVersusAnalyticalSolutions 5 1.4 MainTask 6 1.5 ANoteonNavier–StokesEquations 9 1.6 OutlineoftheBook 12 Exercises 13 2. 1D Heat Conduction................................17 2.1 Introduction 17 2.2 1DConductionEquation 17 2.3 GridLayout 19 2.4 Discretisation 20 2.4.1 TSEMethod 21 2.4.2 IOCVMethod 23 2.5 StabilityandConvergence 24 2.5.1 ExplicitProcedureψ =0 25 2.5.2 PartiallyImplicitProcedure0<ψ <1 28 2.5.3 ImplicitProcedureψ =1 29 2.6 MakingChoices 31 2.7 DealingwithNonlinearities 32 2.7.1 NonlinearSources 33 2.7.2 NonlinearCoefficients 33 vii P1:JZP 0521853265pre CB908/Date 0521853265 May27,2005 0:54 viii CONTENTS 2.7.3 BoundaryConditions 35 2.7.4 Underrelaxation 37 2.8 MethodsofSolution 38 2.8.1 Gauss–SeidelMethod 38 2.8.2 TridiagonalMatrixAlgorithm 38 2.8.3 Applications 40 2.9 ProblemsfromRelatedFields 45 Exercises 47 3.1DConduction–Convection ........................55 3.1 Introduction 55 3.2 ExactSolution 55 3.3 Discretisation 57 3.4 UpwindDifferenceScheme 59 3.5 ComparisonofCDS,UDS,andExactSolution 60 3.6 NumericalFalseDiffusion 61 3.7 HybridandPower-LawSchemes 63 3.8 TotalVariationDiminishingScheme 63 3.9 StabilityoftheUnsteadyEquation 65 3.9.1 ExactSolution 65 3.9.2 ExplicitFinite-DifferenceForm 66 3.9.3 ImplicitFinite-DifferenceForm 67 Exercises 68 4. 2D Boundary Layers................................71 4.1 GoverningEquations 71 4.2 AdaptiveGrid 73 4.3 Transformationto(x,ω)Coordinates 74 4.4 Discretisation 76 4.5 Determinationofω, y,andr 79 4.6 BoundaryConditions 80 4.6.1 Symmetry 80 4.6.2 Wall 81 4.6.3 FreeStream 82 4.7 SourceTerms 85 4.7.1 PressureGradient 85 4.7.2 Q(cid:2)(cid:2)(cid:2) and R 86 k 4.8 TreatmentofTurbulentFlows 87 4.8.1 MixingLengthModel 88 4.8.2 e−(cid:3) Model 89 4.8.3 Free-ShearFlows 91 P1:JZP 0521853265pre CB908/Date 0521853265 May27,2005 0:54 CONTENTS ix 4.9 OverallProcedure 91 4.9.1 CalculationSequence 91 4.9.2 InitialConditions 92 4.9.3 ChoiceofStepSizeandIterations 93 4.10 Applications 93 Exercises 101 5.2DConvection –CartesianGrids .................105 5.1 Introduction 105 5.1.1 MainTask 105 5.1.2 SolutionStrategy 106 5.2 SIMPLE–CollocatedGrids 109 5.2.1 MainIdea 109 5.2.2 Discretisation 110 5.2.3 Pressure-CorrectionEquation 112 5.2.4 FurtherSimplification 117 5.2.5 OverallCalculationProcedure 119 5.3 MethodofSolution 120 5.3.1 IterativeSolvers 120 5.3.2 EvaluationofResiduals 121 5.3.3 Underrelaxation 122 5.3.4 BoundaryConditionsfor(cid:4) 123 5.3.5 BoundaryConditionfor p(cid:2) 125 m 5.3.6 NodeTagging 126 5.4 TreatmentofTurbulentFlows 128 5.4.1 LREModel 128 5.4.2 HREModel 129 5.5 NotionofSmoothingPressureCorrection 133 5.6 Applications 139 Exercises 152 6.2DConvection –ComplexDomains ..............161 6.1 Introduction 161 6.1.1 CurvilinearGrids 161 6.1.2 UnstructuredGrids 162 6.2 CurvilinearGrids 164 6.2.1 CoordinateTransformation 164 6.2.2 TransportEquation 165 6.2.3 InterpretationofTerms 166 6.2.4 Discretisation 168 6.2.5 Pressure-CorrectionEquation 170