Fluid Mechanics and Its Applications F. Moukalled L. Mangani M. Darwish The Finite Volume Method in Computational Fluid Dynamics An Advanced Introduction with OpenFOAM® and Matlab® Fluid Mechanics and Its Applications Volume 113 Series editor André Thess, German Aerospace Center, Institute of Engineering Thermodynamics, Stuttgart, Germany Founding Editor René Moreau, Ecole Nationale Supérieure d’Hydraulique de Grenoble, Saint Martin d’Hères Cedex, France Aims and Scope of the Series The purpose of this series is to focus on subjects in which fluid mechanics plays a fundamental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows and numerical modeling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advancement. Fluids have the ability to transport matter and its properties as well astotransmitforce,thereforefluidmechanicsisasubjectthatisparticularlyopento crossfertilizationwithothersciencesanddisciplinesofengineering.Thesubjectof fluidmechanicswillbehighlyrelevantindomainssuchaschemical,metallurgical, biological and ecological engineering. This series is particularly open to such new multidisciplinary domains. Themedianlevelofpresentationisthefirstyeargraduatestudent.Sometextsare monographs defining the current state of a field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. More information about this series at http://www.springer.com/series/5980 F. Moukalled L. Mangani • M. Darwish The Finite Volume Method in Computational Fluid Dynamics An Advanced Introduction ® ® with OpenFOAM and Matlab 123 F. Moukalled M.Darwish Department ofMechanical Engineering Department ofMechanical Engineering American University of Beirut American University of Beirut Beirut Beirut Lebanon Lebanon L. Mangani Engineering andArchitecture Lucerne University of AppliedScience andArts Horw Switzerland ISSN 0926-5112 ISSN 2215-0056 (electronic) Fluid MechanicsandIts Applications ISBN978-3-319-16873-9 ISBN978-3-319-16874-6 (eBook) DOI 10.1007/978-3-319-16874-6 LibraryofCongressControlNumber:2015939213 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia (www.springer.com) Preface The impetus to write this book came about from three sources: The first source was the bi-yearly computational fluid dynamics (CFD) course, whichhasbeenofferedoverthelast15yearsattheAmerican University ofBeirut (AUB) by both Drs. Darwish and Moukalled to senior and graduate mechanical engineeringstudents,acoursethatfocusesonthefinitevolumemethod(FVM)and CFD applications. The second source grew over the years to become more significant as it was noticedthatgraduateshavestartedworkingonincreasinglymorefocusedareasand topics in CFD while becoming less cognizant of the general algorithmic expertise thatearlierstudentsdeveloped.Itbecameclearthatthereisaneednotonlytocover the basis of the numerics at the core of CFD codes but also to discuss the imple- mentation issues to ensure that all students receive a robust understanding of the techniques they are working on. Finally, the collaborative work in advanced numerics with Prof. Dr. Mangani from HSLU, Lucerne, Switzerland, which started during the Ph.D. supervision of M.Buchmyer(Ph.D.)fromTUGraz,providedalltheincentivetoclarifyanddetail ® much of the numerical basis of the algorithms used in OpenFOAM . Tothisend,itwasdecidedthatthebookwouldcombineamixofnumericaland implementation details allowing the reader, if she/he desires, to fully understand and implement a robust and versatile CFD code based on the FVM. This ambitious task was possible only by selecting from the various numerical methods in each of the topics covered in the book a handful set with which the authors are intimately familiar. The result is a book that covers intimately all the topics necessary for the development of a robust CFD code for the simulation of fluid flow at all speeds within the framework of the collocated unstructured finite volume method. Thebookwasalsowrittenwiththeclassroominmindasreflectedbytheuseof copious illustrations; the provision of many exercises covering numerics, pro- ® gramming, and applications; the availability of an academic code (in MATLAB ) that imbeds much of the numerics presented in the book; and finally the various ® programs and routines in OpenFOAM . v vi Preface The hope is that as you read through this book, you will share with us the excitement and intense interest that we have grown to have for this subject. Beirut F. Moukalled Horw L. Mangani Beirut M. Darwish January 2015 Acknowledgments It took nearly two years to complete this book, but much of what went in it was learned over a much longer period from interaction with numerous people in conferences and academic visits, from answering pertinent questions in our CFD coursesandfromourresearchwork.Howevertheenablerforallthatisforemostthe patience and kindness of our families. We also wish to acknowledge the support provided to us by our respective institutions American University of Beirut Beirut, Lebanon Lucerne University of Applied Science and Arts vii Contents Part I Foundation 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 What Is Computational Fluid Dynamics (CFD). . . . . . . . . . . 3 1.2 What Is the Finite Volume Method. . . . . . . . . . . . . . . . . . . 4 1.3 This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 Foundation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.2 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.3 Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.4 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Review of Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Vectors and Vector Operations . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 The Dot Product of Two Vectors . . . . . . . . . . . . . 11 2.2.2 Vector Magnitude . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.3 The Unit Direction Vector . . . . . . . . . . . . . . . . . . 12 2.2.4 The Cross Product of Two Vectors . . . . . . . . . . . . 12 2.2.5 The Scalar Triple Product. . . . . . . . . . . . . . . . . . . 14 2.2.6 Gradient of a Scalar and Directional Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.7 Operations on the Nabla Operator . . . . . . . . . . . . . 17 2.2.8 Additional Vector Operations . . . . . . . . . . . . . . . . 19 2.3 Matrices and Matrix Operations . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 Square Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.2 Using Matrices to Describe Systems of Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.3 The Determinant of a Square Matrix . . . . . . . . . . . 23 2.3.4 Eigenvectors and Eigenvalues. . . . . . . . . . . . . . . . 26 2.3.5 A Symmetric Positive-Definite Matrix . . . . . . . . . . 27 ix