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This page intentionally left blank GeneralizedRiemannProblems inComputationalFluidDynamics Numericalsimulationofcompressible,inviscid,time-dependentflowisamajor branchofcomputationalfluiddynamics.Itsprimarygoalistoobtainaccurate representationofthetimeevolutionofcomplexflowpatterns,involvinginter- actionsofshocks,interfaces,andrarefactionwaves.ThegeneralizedRiemann problem(GRP)algorithm,developedbytheauthorsforthispurpose,provides aunifying“shell”thatcomprisessomeofthemostcommonlyusednumerical schemes for such flows. This monograph gives a systematic presentation of the GRP methodology, starting from the underlying mathematical principles, throughbasicschemeanalysisandschemeextensions(suchasreactingflowor two-dimensionalflowsinvolvingmovingorstationaryboundaries).Anarrayof instructiveexamplesillustratestherangeofapplications,extendingfrom(sim- ple) scalar equations to computational fluid dynamics. Background material frommathematicalanalysisandfluiddynamicsisprovided,makingthebook accessible to both researchers and graduate students of applied mathematics, science,andengineering. MataniaBen-ArtziisaprofessorofmathematicsattheInstituteofMathemat- ics,TheHebrewUniversityofJerusalem.Hisareaofinterestismathematical physics,wherepuremathematicalanalysis,theoryofpartialdifferentialequa- tions,andnumericalanalysisareappliedinthestudyofvariousfundamental differentialequationsofphysics. Joseph Falcovitz is a research Fellow at the Institute of Mathematics, The HebrewUniversityofJerusalem.Heretiredin1990fromtheRafaelBallistic Center, where he specialized in the development of computational methods of compressible flows and the simulation of terminal ballistics, blast wave phenomena,andfluid–structureinteraction. CAMBRIDGEMONOGRAPHSON APPLIEDANDCOMPUTATIONAL MATHEMATICS SeriesEditors P.G.CIARLET,A.ISERLES,R.V.KOHN,M.H.WRIGHT 11 Generalized Riemann Problems in Computational Fluid Dynamics The Cambridge Monographs on Applied and Computational Mathematics reflectsthecrucialroleofmathematicalandcomputationaltechniquesincon- temporaryscience.Theseriespublishesexpositionsonallaspectsofapplicable andnumericalmathematics,withanemphasisonnewdevelopmentsinthisfast- movingareaofresearch. State-of-the-art methods and algorithms as well as modern mathematical descriptionsofphysicalandmechanicalideasarepresentedinamannersuited tograduateresearchstudentsandprofessionalsalike.Soundpedagogicalpre- sentationisaprerequisite.Itisintendedthatbooksintheserieswillserveto informanewgenerationofresearchers. Alsointhisseries: APracticalGuidetoPseudospectralMethods,BengtFornberg DynamicalSystemsandNumericalAnalysis,A.M.StuartandA.R.Humphries LevelSetMethods,J.A.Sethian TheNumericalSolutionofIntegralEquationsoftheSecondKind, KendallE.Atkinson OrthogonalRationalFunctions,AdhemarBultheel,PabloGonzalez-Vera, ErikHendriksen,andOlavNjastad Generalized Riemann Problems in Computational Fluid Dynamics MATANIABEN-ARTZIandJOSEPHFALCOVITZ TheHebrewUniversityofJerusalem    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521772969 © Cambridge University Press 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 -  isbn-13 978-0-511-06675-7 eBook (NetLibrary) -  isbn-10 0-511-06675-9 eBook (NetLibrary) -  isbn-13 978-0-521-77296-9 hardback -  isbn-10 0-521-77296-6 hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents ListofFigures pagexi Preface xv 1 Introduction 1 I BASICTHEORY 5 2 ScalarConservationLaws 7 2.1 TheoreticalBackground 7 2.2 BasicConceptsofNumericalApproximation 25 3 TheGRPMethodforScalarConservationLaws 36 3.1 FromGodunovtotheGRPMethod 36 3.2 1-DSampleProblems 49 3.2.1 TheLinearConservationLaw 49 3.2.2 TheBurgersNonlinearConservationLaw 55 3.3 2-DSampleProblems 63 4 SystemsofConservationLaws 81 4.1 NonlinearHyperbolicSystemsinOneSpaceDimension 81 4.2 EulerEquationsofQuasi-1-D,Compressible, InviscidFlow 101 5 TheGeneralizedRiemannProblem(GRP)for CompressibleFluidDynamics 135 5.1 TheGRPforQuasi-1-D,Compressible,InviscidFlow 135 5.2 TheGRPNumericalMethodforQuasi-1-D, Compressible,InviscidFlow 169 vii viii Contents 6 AnalyticalandNumericalTreatmentofFluid DynamicalProblems 184 6.1 TheShockTubeProblem 184 6.2 WaveInteractions 189 6.2.1 Shock–ContactInteraction 192 6.2.2 Shock–ShockInteraction 195 6.2.3 Shock–CRWInteraction 203 6.2.4 CRW–ContactInteraction 207 6.3 SphericallyConvergingFlowofColdGas 218 6.4 TheFlowInducedbyanExpandingSphere 219 6.5 Converging–DivergingNozzleFlow 222 II NUMERICALIMPLEMENTATION 233 7 FromtheGRPAlgorithmtoScientificComputing 235 7.1 GeneralDiscussion 235 7.2 Strang’sOperator-SplittingMethod 237 7.3 Two-DimensionalFlowinCartesianCoordinates 244 8 GeometricExtensions 251 8.1 GridsThatMoveinTime 251 8.2 SingularityTracking 252 8.3 MovingBoundaryTracking(MBT) 255 8.3.1 BasicSetup 257 8.3.2 SurveyoftheFullMBTAlgorithm 264 8.3.3 AnExample:ShockLiftingofan EllipticDisk 266 9 APhysicalExtension:ReactingFlow 269 9.1 TheEquationsofCompressibleReactingFlow 271 9.2 TheChapman–Jouguet(C–J)Model 276 9.3 TheZ–N–D(Zeldovich–vonNeumann–Do¨ring) Solution 281 9.4 TheLinearGRPfortheReacting-FlowSystem 286 9.5 TheGRPSchemeforReactingFlow 298 10 WaveInteractioninaDuct–AComparativeStudy 305 A EntropyConditionsforScalarConservationLaws 313 B ConvergenceoftheGodunovScheme 320