Spectral methods: evolution to complex geometries and applications to fluid dynamics PDF
Preview Spectral methods: evolution to complex geometries and applications to fluid dynamics
Scientific Computation EditorialBoard J.-J.Chattot,Davis,CA,USA P.Colella,Berkeley,CA,USA W.E,Princeton,NJ,USA R.Glowinski,Houston,TX,USA M.Holt,Berkeley,CA,USA Y.Hussaini,Tallahassee,FL,USA P.Joly,LeChesnay,France H.B.Keller,Pasadena,CA,USA J.E.Marsden,Pasadena,CA,USA D.I.Meiron,Pasadena,CA,USA O.Pironneau,Paris,France A.Quarteroni,Lausanne,Switzerland andPolitecnicoofMilan,Italy J.Rappaz,Lausanne,Switzerland R.Rosner,Chicago,IL,USA P.Sagaut,Paris,France J.H.Seinfeld,Pasadena,CA,USA A.Szepessy,Stockholm,Sweden M.F.Wheeler,Austin,TX,USA C. Canuto M.Y. Hussaini A. Quarteroni T.A. Zang Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics WithFiguresandTables 123 ClaudioCanuto M.YousuffHussaini DipartimentodiMatematica SchoolofComputationalScience PolitecnicodiTorino FloridaStateUniversity CorsoDucadegliAbruzzi, Tallahassee,FL-,USA Torino,Italy e-mail:[email protected] e-mail:[email protected] AlfioQuarteroni ThomasA.Zang,Jr. SB-IACS-CMCS,EPFL NASALangleyResearchCenter* Station MailStop Lausanne,Switzerland Hampton,VA-,USA and e-mail:[email protected] MOX,PolitecnicodiMilano PiazzaLeonardodaVinci, Milano,Italy e-mail:alfio.quarteroni@epfl.ch * This does not constitute an endorsement of this work by either the U.S. Government or the NASA LangleyResearchCenter. Coverfigures:Upperleft:StructuraldynamicsanalysisoftheRomanColosseum:spectralelementcalcula- tionofunitarydilatation(Fig..).Upperright:Computationalsurfacemeshinthevicinityofanengine forasixth-orderspectraldiscontinuousGalerkincomputationofnoisepropagationandscatteringoff anaircraftsurface(Fig..).Lowerleft:VelocitycontoursforaTaylor-Greenvortexsimulationusing aFourierspectralmethod(Fig..).Lowerright:Spectralelementapproximationofaneigenfunctionof theLaplaceoperatorwithDirichletboundaryconditionsinasquare(Fig..). 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Typesettingandproduction:LE-TEXJelonek,Schmidt&Vo¨cklerGbR,Leipzig,Germany Coverdesign:WMXDesignGmbH,Heidelberg SPIN: //YL– Printedonacid-freepaper Preface TwodecadesagowhenwewroteSpectral Methods in Fluid Dynamics (1988), thesubjectwasstillfairlynovel.Motivatedbythemanyfavorablecomments we have received and the continuing interest in that book (which will be referred to as CHQZ1), and yet desiring to present a more modern perspec- tive, we embarkedon the project which resulted in our recent book (Canuto et al. (2006), referred to as CHQZ2) and the present new book (referred to as CHQZ3). Our objectives with these two new books are to modernize our thoroughdiscussionofclassicalspectralmethods,accountingforadvancesin the theory and more extensive application experience in the fluid dynamics arena,while summarizingthe currentstate ofmultidomainspectralmethods from the perspective of classicalspectral methods. While the two new books drawextensivelyfromportionsofourearliertext,CHQZ1,muchofCHQZ2, and most of CHQZ3 is new. The added content has necessitated our pub- lishing this new work as two separate books. The rationale for the division of the material between the books is that we furnished in the first new book a comprehensive discussion of the fundamental aspects of classical spectral methods in single domains. This second new book focuses on applications to fluid dynamics and on multidomain spectral methods. The historical evolution of spectral methods from their initial (now clas- sical) versions to the contemporary multidomain versions has been covered in some detail in the Preface of CHQZ2. In short, both the theory and the algorithmsofclassicalspectralmethodsforsmoothproblemswerereasonably mature already in the mid-1980s, and singular progress has been made over the past two decades in extending spectral methods to arbitrarygeometries, enabling what some would consider the mathematical nirvana of a method of arbitrarily high-order capable of application to problems on an arbitrary geometry. In this respect, the trajectory of spectral methods over the past 20 years has been approaching that of hp finite-element methods. This pro- cess of migration from single-domain to multidomain spectral methods has requiredtheinjectionofnovelmathematicaltoolsandstimulatedoriginalin- vestigationdirections.Mathematicshashadaprofoundimpactonthecorrect design and interpretation of the methods, and in some cases it has inspired the development of discontinuous spectral methods even for problems with continuous solutions. On the other hand, since in general a geometrically VI Preface complex computational domain is split into polygonal or polyhedral subdo- mains (or elements), tensor-productdomains are no longera prerequisite for spectral methods, with the development of spectral bases on such elements as triangles and tetrahedra. Moreover, the splitting into many subdomains hasmotivatedtheuseofmoderatepolynomialdegreesineverysubdomain— small from the perspective of classical spectral methods, but large from the perspectivesoffinite-volumeandfinite-elementmethods.Inspite ofthis ma- jor change of perspective, the new multidomain spectral methods still enjoy some of the most distinguishing (and desirable) features of “classical” spec- tral methods—Gaussian integration formulas, low dispersion, and ease of preconditioning by low-order discretization matrices. The main contents of the current text are as follows: Chapter 1 surveys the relevantequations offluid dynamics,including the historicalbackground and the underlying assumptions of the mathematical models. Chapter 2 presentsspectralalgorithmsforsolutionoftheboundary-layerequationsand for analyses of linear and nonlinear stability of fluid flows. Chapter 3, on single-domain algorithms for incompressible flows, focuses its discussion of spectral methods for problems with at most one nonperiodic direction on those algorithms that remained in reasonably extensive use post-1990. Fur- thermore, it surveys the basic spatial discretization schemes for problems with two or more nonperiodic directions; these schemes are the foundation of many multidomain spectral methods for incompressible flows. Chapter 4, on classicalspectral methods for hyperbolic systems and compressible flows, emphasizes algorithms for enforcing boundary conditions, methods for com- puting smooth compressible flows, and spectrally accurate solution of dis- continuous compressible flows using the shock-fitting approach. Chapter 5 introduces the main strategies for constructing spectral approximations in complex domains, in particular, the spectral element method, the mortar el- ement method, the spectral discontinuous Galerkin method, as well as the more traditional patching collocation method. Their theoretical properties are analyzed and their algebraic aspects are investigated. Some of the per- spectives and results coincide with those of the hp version of finite-element methods (apart from the difference in notation). We arrive at them from the “spectralmethod”road,i.e.,beingcoherentwiththespectraltechnology within each element while reducing the size of the elements (the alternative roadstartsfromthe h versionofFEMandconservesthe typicalglobal/local interplaywhile increasingthe polynomialdegreeineachelement).Chapter6 illustrates solution strategies based on domain decomposition techniques for the spectral discretizations investigated in Chap. 5. Both Schur-based and Schwarz-basediterativesolversandpreconditionersareconsidered,andtheir computational advantages (in particular, their property of scalability with respect to the number of subdomains) are illustrated. Chapter 7 caps our discussion of multidomain spectral methods with its survey of the modern Preface VII approachtocomputingincompressibleflowsingeneralgeometriesusinghigh- order, spectral discretizations. HavingCHQZ2athandisnotnecessaryforreadingCHQZ3,asallessen- tial formulas are repeated here (in Chap. 8) in a short “primer for classical spectralmethods”,andwehavealsoincludedthekeyelementsofAppendices A–DfromCHQZ2.(Theseappendicescovermaterialon(A)somebasicnota- tionsandtheoremsfromfunctionalanalysis,(B)theFastFourierTransform, (C) iterative methods, and (D) temporal discretizations.) However, CHQZ2 does contain considerable material necessary for a better understanding of the subject,andwe do makecopiousreferencesto specific andmoredetailed material in CHQZ2. The complete Table of Contents from CHQZ2 is listed rightafterthatofthepresentbook,asaconveniencetothosereaderswishing to locate specific backgroundmaterial.Also,the newAppendix D.3 contains some useful technical details for, but peripheral to, some of the algorithms discussed in the text. In contrast to other recent expositions on the subject, such as the books by Karniadakis and Sherwin (2005) and Deville et al. (2002), this work cer- tainly contains fewer implementation details of multidomain spectral algo- rithms. On the other hand, we accompany the algorithmic description with the theoretical framework necessary for the a priori prediction of their per- formance,whichinturnsupportsinformedchoicesfortheiruseandprovides somecriteriaforcodeverification.Wetrustthatourcopiousnumericalexam- ples,andespeciallythe large-scaleapplications,provideconvincingempirical evidence that spectral methods have indeed evolved sufficiently that even in complex geometries they provide efficient, highly accurate numerical simula- tions of diverse physical phenomena. We hope that this effort, together with the other books and the ever-expanding literature on the subject, will foster the wider penetration of higher-order ideas and methods in computational engineering and science. The level of diffusion and appreciation that is by now granted to higher-order methods in certain communities, such as the structural mechanics one, is certainly a positive trend. Whereas with our first text we made a valiant effort to provide compre- hensivecoverageofallavailablespectralmethods(atleastforfluiddynamics applications) and to provide a bibliography that encompassed all extant ref- erences to spectralmethods, here we acknowledgethe practicalimpossibility ofsuchanambitioninthefaceofalltheworkthathassincetranspiredinthe field.Westillaimtoprovidecomprehensivecoverageofgeneralmethodology. However,ourdiscussionofparticularalgorithmsis necessarilyrepresentative ratherthan complete.We focus onthose single-domainalgorithmsthathave seen the widest use in fluid dynamics applications and on a broad represen- tation of multidomain algorithms. However, our knowledge in this area is certainly not exhaustive, and others would no doubt have made somewhat different choices. In our citations we enforce a strong preference for archival publications.Werecognizethatmanydevelopmentsappearedearlier(insome VIII Preface casesmanyyearsearlier)inpre-printsorconferencepublications,butweonly cite non-archival sources when no archival reference is available. Themanynumericalexamplesproducedexpresslyforthisbookhavebeen run on desktop computers in 64-bit arithmetic unless otherwise noted. Nowadays, considerable software for spectral methods is freely available ontheWeb,rangingfromlibrariesofbasicspectraloperationsallthe wayto complete spectral codes for Navier–Stokes(and other) complex applications. Due to the highly dynamic nature of these postings, we have chosen not to list them in the text (except to acknowledge codes that we have used here for numerical examples), but to maintain a reasonably current list of such sources on the Web site (http://www.dimat.polito.it/chqz/) for this and the companiontext. There is alwaysthe possibility that this site itself may need to be moved due to unforeseen circumstances; in this case, one should check the Springer site for the link to the detailed Web site of the book. The authors are grateful to Dr. Wolf Beiglbo¨ck, Dr. Ramon Khanna and the Springer staff for their encouragement and facilitation of this project. ThetechnicalsupportofPaolaGervasioandMarcoDiscacciatiinperforming numericaltests,preparingfiguresandtables,typing,andeditingmuchofthe manuscript was indispensable. The authors are pleased to acknowledge the many discussions and helpful comments on the manuscript that have been providedbycolleaguessuchasErikBurman,PaolaGervasio,DavidKopriva, MujeebMalik,RobertMoser,LucaPavarino,DavidPruett,JamieQuirk,and AndreaToselli.WeappreciatetheassistanceprovidedbyDavidPruett,David Kopriva,andBenjaminStammforseveralofthenumericalexamples.Thanks are also due to Stefano Berrone, Dilek Dustegor, Giuseppe Ghib`o, Nicola Parolini, and Svetlana Poroseva for providing additional technical support. We are also indebted to the many individuals who have graciously given us permissiontoreprintfiguresfromtheirwork.Theauthorsaregratefultothe Politecnico di Torino, Florida State University (esp. Lawrence Abele), the Ecole Polytechnique F´ed´erale de Lausanne, the Politecnico di Milano, and the NASA Langley Research Center for their facilitation of this endeavor. Finally, we are most appreciative of the support and understanding we have receivedfrom our wives (Manuelita,Khamar,Fulvia, and Ann) and children (Arianna, Susanna, Moin, Nadia, Marzia, and Silvia) during this project. Torino, Italy Claudio Canuto Tallahassee, Florida M. Yousuff Hussaini Lausanne, Switzerland and Milan, Italy Alfio Quarteroni Hampton, Virginia Thomas A. Zang June, 2007 Contents 1. Fundamentals of Fluid Dynamics ......................... 1 1.1 Introduction ........................................... 1 1.2 Fluid Dynamics Background ............................. 1 1.2.1 Phases of Matter ................................. 2 1.2.2 Thermodynamic Relationships ..................... 3 1.2.3 Historical Perspective............................. 6 1.3 Compressible Fluid Dynamics Equations .................. 7 1.3.1 Compressible Navier–Stokes Equations .............. 8 1.3.2 Nondimensionalization ............................ 12 1.3.3 Navier–Stokes Equations with Turbulence Models .... 13 1.3.4 Compressible Euler Equations...................... 17 1.3.5 Compressible Potential Equation ................... 17 1.3.6 Compressible Boundary-LayerEquations ............ 19 1.3.7 Compressible Stokes Limit......................... 20 1.3.8 Low Mach Number Compressible Limit ............. 21 1.4 Incompressible Fluid Dynamics Equations ................. 21 1.4.1 Incompressible Navier–Stokes Equations............. 21 1.4.2 Incompressible Navier–Stokes Equations with Turbulence Models........................... 22 1.4.3 Vorticity–Streamfunction Equations ................ 25 1.4.4 Vorticity–Velocity Equations....................... 26 1.4.5 Incompressible Boundary-LayerEquations........... 27 1.5 Linear Stability of ParallelFlows ......................... 27 1.5.1 Incompressible Linear Stability..................... 29 1.5.2 Compressible Linear Stability ...................... 31 1.6 Stability Equations for Nonparallel Flows.................. 36 2. Single-Domain Methods for Stability Analysis ............ 39 2.1 Introduction ........................................... 39 2.2 Boundary-LayerFlows .................................. 41 2.2.1 Incompressible Boundary-LayerFlows............... 41 2.2.2 Compressible Boundary-LayerFlows................ 48 2.3 Linear Stability of Incompressible ParallelFlows ........... 52 2.3.1 Spectral Approximations for Plane Poiseuille Flow.... 52 X Contents 2.3.2 Numerical Examples for Plane Poiseuille Flow ....... 57 2.3.3 Some Other Incompressible Linear Stability Problems. 61 2.4 Linear Stability of Compressible ParallelFlows............. 64 2.5 Nonparallel Linear Stability.............................. 69 2.5.1 Linear Parabolized Stability Equations.............. 69 2.5.2 Two-Dimensional Global Stability Analysis .......... 71 2.6 Transient Growth Analysis .............................. 72 2.7 Nonlinear Stability ..................................... 75 2.7.1 Quasi-Steady Finite-Amplitude Solutions............ 75 2.7.2 Secondary Instability Theory ...................... 77 2.7.3 Nonlinear Parabolized Stability Equations ........... 81 3. Single-Domain Methods for Incompressible Flows ........ 83 3.1 Introduction ........................................... 83 3.2 Conservation Properties and Time-Discretization ........... 86 3.2.1 Conservation Properties........................... 86 The Rotation Form ............................... 88 The Skew-Symmetric Form ........................ 90 Convection and Divergence Forms .................. 92 3.2.2 General Guidelines for Time-Discretization .......... 92 3.2.3 Coupled Methods ................................ 93 Fully Implicit Schemes ............................ 93 Semi-Implicit Schemes ............................ 93 3.2.4 Splitting Methods ................................ 95 3.2.5 Other Integration Methods ........................ 96 Operator Integration Factors....................... 96 Characteristics Methods ........................... 97 3.3 Homogeneous Flows .................................... 98 3.3.1 Fourier Galerkin Approximation for Isotropic Turbulence........................... 98 3.3.2 De-aliasing Using Transform Methods............... 99 3.3.3 Pseudospectraland Collocation Methods ............ 103 3.3.4 Rogallo Transformation for Homogeneous Turbulence . 106 3.3.5 Large-Eddy Simulation of Isotropic Turbulence....... 108 3.3.6 The Taylor–GreenVortex Example: Stability, Accuracy and Aliasing.................... 110 3.4 Flows with One Inhomogeneous Direction ................. 121 3.4.1 Coupled Methods ................................ 123 Kleiser–Schumann Algorithm ...................... 124 Normal Velocity–Normal Vorticity Algorithms........ 127 3.4.2 Galerkin Methods Using Divergence-Free Bases ...... 131 3.4.3 Splitting Methods ................................ 133 Chebyshev Staggered Grid ......................... 133 Zang–Hussaini Algorithm.......................... 135 3.4.4 Other Confined Flows............................. 138
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