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Applied Mathematics: Body and Soul Johan Hoffman Claes Johnson Computational Turbulent Incompressible Flow Applied Mathematics: Body and Soul 4 ABC Johan Hoffman Claes Johnson School of Computer Science School of Engineering Sciences and Communication Royal Institute of Technology-KTH Royal Institute of Technology - KTH 10044 Stockholm, Sweden 10044 Stockholm, Sweden e-mail:[email protected] e-mail:[email protected] Mathematics Subject Classification (2000): 35Q30, 76D03, 76D05, 65M60 LibraryofCongressControlNumber:2006936082 ISBN-10 3-540-46531-6 SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-46531-7SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2007 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. TypesettingbytheauthorsusingaSpringerLATEXmacropackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:11580607 46/SPi 543210 To our families Preface Applied Mathematics: Body&Soul is a mathematics education reform program including a series of books, together with associated educational mate- rial and open source software freely available from the project web page at www.bodysoulmath.org. Body&Soul reflects the revolutionary new possibilities of mathematical modeling opened by the modern computer in the form ofComputational Cal- culus (CC), which is now changing the paradigm of mathematical modeling in science and technology with new methods, questions and answers, as a modern form of the classical calculus of Leibniz and Newton. The Body&Soul series of books presents CC in a synthesis of computational mathematics (Body) and analytical mathematics (Soul) including applications. Volumes 1-3 [36] give a modern version of calculus and linear algebra including computation starting at a basic undergraduate level, and subsequent volumes on a graduate level cover different areas of applications with focus on computational methods: • Volume 4: Computational Turbulent Incompressible Flow. • Volume 5: Computational Thermodynamics. • Volume 6: Computational Dynamical Systems. The present book is Volume 4, with Volumes 5 and 6 to appear in 2007 and further volumes on solid mechanics and electro-magnetics being planned. A gentle introduction to the Body&Soul series is given in [63]. The overall goal of the Body&Soul project may be formulated as theAu- tomation of Computational Mathematical Modeling (ACMM) involving the key steps of automation of (i) discretization, (ii) optimization and (iii) modeling. The objective of ACMM is to open for massive use of CC in science, engineering, medicine, and other areas of application. ACMM is realized in the FEniCS project (www.fenics.org), which may be seen to represent the top software part of Body&Soul. The automation of discretization (i) involves automatic translation of a given differential equation in standard mathematical notation into a discrete VIII Preface system of equations, which can be automatically solved using numerical linear algebra to produce an approximate solution of the differential equation. The translation is performed using adaptive stabilized finite element methods, which we refer to as General Galerkin or G2 with the adaptivity based on a posteriori error estimation by duality and the stabilization repre- senting a weighted least squares control of the residual. The automation of optimization (ii) is performed similarly starting from thedifferentialequationsexpressingstationarityofanassociatedLagrangian. Finally, one can couple modeling to optimization by seeking from an Ansatz a model with best fit to given data. The present Vol 4 may be viewed as a test of the functionality of the general technique for ACMM based on G2. In this book we apply G2 imple- mentedinFEniCStothespecificproblemofsolvingtheincompressibleEuler and Navier–Stokes (NS) equations computationally. The challenge includes computational simulation of turbulent flow, since solutions of the Euler and NS equations in general are turbulent, and thus the challenge in particular includes the open problem of computational turbulence modeling. We show in the book that G2 passes this test successfully: By direct application of G2 to the Euler and NS equations, we can on a PC compute quantities of interest in turbulent flow in the form of mean values such as drag and lift, up to tolerances of interest. G2 does not require any user speci- fied turbulence model or wall model for turbulent boundary layers; by the direct application of G2 to the Euler or NS equations, we avoid introducing Reynolds stresses in averaged NS equations requiring turbulence models. In- stead the weighted least squares stabilization of G2 automatically introduces sufficient turbulent dissipation on the finest computational scales and thus actsasanautomaticturbulencemodelincludingfrictionboundaryconditions as wall model. Furthermore, the adaptivity of G2 ensures that the flow is automatically resolved by the mesh where needed. G2 thus opens for the Au- tomation of Computational Fluid Dynamics, which could be an alternative title of this book. Applying G2 to the Euler and NS equations opens a vast area for exploration, which we demonstrate by resolving several scientific mysteries, including d’Alembert’s paradox of zero drag in inviscid flow, the 2nd Law of thermodynamicsandtransitiontoturbulence.Wealsouncoverseveralsecrets of fluid dynamics including secrets of ball sports, flying, sailing and racing. In particular we are led to a new computational foundation of thermodynamics based on deterministic microscopical mechanics producing deterministic mean value outputs coupled with indeterminate pointwise outputs, in which the 2nd Law is a consequence of the 1st Law. The new foundation of thermodynamics is not based on microscopical statistics as the statistical mechanics foundation pioneered by Boltzmann, and thus offers a rational scientific basis of thermodynamics based on computation, without the mystery ofthe2ndLawinthestatisticalapproach.Webelievethenewcomputational approach also may give insight to physics following the idea that Nature in Preface IX one way or the other is performing an analog computation when evolving in time from one moment to the next. We initiate the development of the new foundation in this volume and expand in Vol 5. We are also led to a new computational approach to basic mathematical questions concerning existence and uniqueness of solutions of the Euler and NS equations, for which analytical methods have not shown to be productive. In particular we show the usefulness of the new concepts of approximate weak solutions and weak uniqueness, through which we may mathematically describe turbulent solutions with non-unique point values but unique mean values. In short, we show that G2 opens to new insights into both mathematics, physics and mechanics with an amazingly rich range of possible applications. The main message of this book thus is that of a breakthrough: Using G2 one can simulate turbulent flow on a standard PC with a 2 GHz processor and 1-2 Gb memory computing on adaptive meshes with 105 −106 mesh points in space (but not less). We thus show that G2 simulation leads not only to images and movies, which are fun (and instructive) to watch, but also to new insights into the rich physical world of turbulence as well as the mathematics of turbulence. The book is a test not only of the functionality of G2/FEniCS for simulation of turbulent flow, but also of the functionality of the Body&Soul educational program: The book is at the research front of computational turbulence, while it can be digested with the CC basis of Body&Soul Vol 1-3. If we are correct, and experience will tell, then masters programs in computational science and engineering based on Body&Soul may reach the very forefront of research, and in particular give a flying start for PhD studies. This is made possible by the amazing power of CC using only basic tools of calculus combined with computing. We hope the reader will have a good productive time reading the book and also trying out the G2 FEniCS software on old and new challenges. For inspirationavastmaterialofG2simulationsofturbulentflowsisavailableon the web page of the book at www.bodysoulmath.org. Theauthorswouldliketothanktheparticipantsofthe2006GeiloWinter SchoolinComputationalMathematics,whoofferedvaluablecommentsonthe manuscript, and who helped in tracking down some of the mistakes. The first author would like to acknowledge the joint work with Prof. Jonathan Goodman at the Courant Institute in developing the mesh smooth- ing algorithm of Section 32.5. The main source of mathematicians pictures is the MacTutor History of Mathematicsarchive,otherpicturesaretakenfromwhatisassumedtobethe public domain, or otherwise the sources are stated in the picture captions. Stockholm and Go¨teborg, Johan Hoffman April 2006 Claes Johnson Contents Part I Overview 1 Main Objective............................................ 3 1.1 Computational Turbulent Incompressible Flow ........... 3 2 Mysteries and Secrets ..................................... 29 2.1 Mysteries ............................................ 29 2.2 Secrets .............................................. 30 3 Turbulent Flow and History of Aviation ................... 33 3.1 Leonardo da Vinci, Newton and d’Alembert.............. 33 3.2 Cayley and Lilienthal.................................. 34 3.3 Kutta, Zhukovsky and the Wright Brothers .............. 34 4 The Euler Equations....................................... 39 4.1 Foundation of Fluid Dynamics.......................... 39 4.2 Derivation of the Euler Equations....................... 40 4.3 The Euler Equations as a Continuum Model ............. 41 4.4 Incompressible Flow................................... 42 5 The Incompressible Euler and Navier–Stokes Equations ... 43 5.1 The Incompressible Euler Equations..................... 43 5.2 The Incompressible Navier–Stokes Equations ............. 44 5.3 What is Viscosity? .................................... 44 5.4 What is Heat Conductivity?............................ 46 5.5 Friction Boundary Conditions .......................... 46 5.6 Einstein’s Ideal ....................................... 46 5.7 Euler and NS as Dynamical Systems .................... 47 XII Contents 6 Triumph and Failure of Mathematics ...................... 49 6.1 Triumph: Celestial Mechanics .......................... 49 6.2 Failure: Potential Flow ................................ 50 7 Laminar and Turbulent Flow .............................. 51 7.1 Reynolds ............................................ 51 7.2 Applications and Reynolds Numbers .................... 53 8 Computational Turbulence ................................ 57 8.1 Are Turbulent Flows Computable? ...................... 57 8.2 Typical Outputs: Drag and Lift......................... 58 8.3 What about Boundary Layers? ......................... 59 8.4 Approximate Weak Solutions: G2 ....................... 59 8.5 G2 Error Control and Stability ......................... 60 8.6 What about Mathematics of Euler and NS? .............. 60 8.7 When is a Flow Turbulent? ............................ 61 8.8 G2 vs Physics ........................................ 61 8.9 Computability and Predictability ....................... 62 9 A First Study of Stability.................................. 65 9.1 The Linearized Euler Equations ........................ 65 9.2 Flow in a Corner or at Separation....................... 66 9.3 Couette Flow......................................... 69 9.4 Resolution of Sommerfeld’s Mystery..................... 70 9.5 Reflections on Stability and Perspectives................. 70 10 d’Alembert’s Mystery and Bernoulli’s Law ................ 73 10.1 Introduction ......................................... 73 10.2 Bernoulli, Euler, Ideal Fluids and Potential Solutions...... 74 10.3 d’Alembert’s Mystery ................................. 74 10.4 A Vector Calculus Identity............................. 75 10.5 Bernoulli’s Law....................................... 75 10.6 Potential Flow around a Circular Cylinder ............... 76 10.7 Zero Drag/Lift of Potential Flow........................ 76 10.8 Ideal Fluids and Vorticity.............................. 78 10.9 d’Alembert’s Computation of Zero Drag/Lift ............. 78 10.10 A Reformulation of the Momentum Equation............. 79 11 Prandtl’s Resolution of d’Alembert’s Mystery ............. 81 11.1 Quotation from a Standard Source ...................... 81 11.2 Quotation from Prandtl’s 1904 report ................... 82 11.3 Discussion of Prandtl’s Resolution ...................... 83