4 FINITE ELEMENT METHODS FOR FLUIDS FINITE ELEMENT

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  • 4 FINITE ELEMENT METHODS FOR FLUIDS

    FINITE ELEMENT METHODS FOR FLUIDS.

    O. Pironneau

    (Universite Pierre et Marie Curie & INRIA)

    (To appear in 1988 (Wiley))

    MacDraw, MacWrite, Macintosh are trade marks of Apple Computer Co.TEXis a trade mark of the American Math. Society.TEXtures is trade mark of Blue Sky Research Co.EasyTeX and MacFEM are copywrite softwares of Numerica Co. (23 Bd

    de Brandebourg BP 215, 94203 Ivry/Seine, France)

  • 5

    TABLE OF CONTENTS

    Foreword, Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7,9

    Introduction

    1. Partial differential equations of fluid mechanics.1.1 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 The general equations of Newtonian fluids: conservation of mass,

    momentum, energy and the state equation . . . . . . . . . . . . . . . . . . . . . . 151.3 Inviscid flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4 Incompressible and weakly compressible flows . . . . . . . . . . . . . . . . . . . . 201.5 Irrotational flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.6 The Stokes problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.7 Choice of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.8 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    2. Irrotational and weakly irrotational flows2.1 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .272.2 Incompressible potential flows: introduction,variational formulation

    and discretisation, solution of the linear systems with the conjugategradient method, nozzle computation, lift of airfoils . . . . . . . . . . . . . 13

    2.3 Compressible subsonic potential flows: variational formulation, dis-cretisation, solution by the conjugate gradient method . . . . . . . . . . .13

    2.4 Transonic potential flows: generalities, numerical considerations 132.5 Potential vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Rotational corrections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

    3. : Convection - diffusion phenomena3.1 Introduction: boundary layers, instability of centered schemes . . . .133.2 Stationary convections: introduction, schemes based on the method

    of characteristics, upwind schemes, the method of streamline diffusion,Upwinding by discontinuities, upwinding by cells. . . . . . . . . . . . . . . . 13

    3.3 Convection diffusion 1: introduction, theoretical results on the con-vection - diffusion equation, time approximation. . . . . . . . . . . . . . . . . 13

    3.4 Convection diffusion 2: discretisation of the total derivative, the Lax- Wendroff/Taylor-Galerkin scheme, streamline upwinding (SUPG).13

    4. : The Stokes problem4.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Functional setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134.3 Discretisation : the P1bubble/P1 element, the P1isoP2/P1 element,

    the P2/P1 element, the non conforming P1 / P0 element , the nonconforming P2 bubble / P1 element, comparison . . . . . . . . . . . . . . . . 13

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    4.4 Numerical solution of the linear systems : generalities, solution ofthe saddle point problem by conjugate gradient . . . . . . . . . . . . . . . . . .13

    4.5 Error estimates : abstract setting, general theorems, verification ofthe inf-sup condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    4.6 Other boundary conditions: slip boundary conditions, boundary con-ditions on the pressure and on the rotational.

    5. The Navier-Stokes equations5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 Existence, uniqueness, regularity: the variational problem, existence,

    uniqueness, regularity, behavior at infinity, The case = 0 . . . . . . 135.3 Discretisation in space: generalities, error estimates. . . . . . . . . . . . 135.4 Discretisation in time: semi-explicit discretisation, implicit and semi-

    implicit discretisations, solution of the nonlinear system . . . . . . . . . 135.5 Discretisation of the total derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.6 Simulation of turbulent flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

    6. Euler, compressible Navier-Stokes and the shallow water equa-tions.6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2 The compressible Euler equations: statement of the problem, 2D tests,

    existence, a few centered schemes, some upwind schemes. . . . . . . . .136.3 The compressible Navier-Stokes equations: introduction, an exam-

    ple, an implicit scheme for weakly compressible flows, Euler-basedschemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

    6.4 The Shallow water equations: introduction, a velocity-depth scheme,A flux-depth scheme, comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

    6.5 Conclusion

    References : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    Appendix : A finite element program for fluids on the Macintosh . . . . . . 13

    Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

  • 7

    ... The Unseen grew visible to student eyes,Explained was the immense Inconscients scheme,

    Audacious lines were traced upon the void;The Infinite was reduced to square and cube.

    Sri Aurobindo Savitri. Book 2, Canto XI

    FOREWORD.

    This book is written from the notes of a course given by the author at theUniversite Pierre et Marie Curie (Paris 6) in 1985, 86 and 87 at the Masterlevel. This course addresses students having a good knowledge of basic nu-merical analysis, a general idea about variational techniques and finite elementmethods for partial differential equations and if possible a little knowledge offluid mechanics; its purpose is to prepare them to do research in numericalanalysis applied to problems in fluid mechanics. Such research starts, veryoften with practical training in a laboratory; this book is therefore chiefly ori-ented towards the production of programs; in other words its aim is to givethe reader the basic knowledge about the formulation, the methods to analyzeand to resolve the problems of fluid mechanics with a view to simulating themnumerically on computer; at the same time the most important error estimatesavailable are given.

    Unfortunately, the field of Computational Fluid Dynamics has become avast area and each chapter of this book alone could be made a separate Masterscourse: so it became necessary to restrict the discussion to the techniqueswhich are used in the laboratories known to the author, i.e. INRIA, DassaultIndustries, LNH/EDF-Chatou, ONERA...Moreover, the selected methods area reflection of his career (and his antiquated notions ?). Briefly, in a word, thisbook is not a review of all existing methods.

    In spite of this incompleteness the author wishes to thank his close collabo-rators whose work has been presented in this book and hopes that it would not

  • 8 FINITE ELEMENT METHODS FOR FLUIDS

    hurt them to have their results appear in so partial a work: MM C. Bernardi,J.A. Desideri, F. Eldabaghi, S. Gallic, V. Girault, R. Glowinski, F. Hecht, C.Pares, J. Periaux, P.A. Raviart, Ph. Rostand, and J.H. Saiac. The authorwishes to thanks his colleagues too, whose figures are reproduced here: MrsC. Begue and M.O. Bristeau, MM B. Cardot, J.M. Dupuy, F. Eldabaghi, J.Hasbani, F. Hecht, B. Mantel, C. Pares, Ph. Rostand, J.H. Saiac,V. Schoenand B. Stoufflet as well as the following institutions AMD-BA, PSA, SNCF andSTCAN with special thanks to AMD-BA for some of the color pictures. Theauthor also wishes to thank warmly MM P.G. Ciarlet and J.L. Lions for theirguidance for the publication of this work, G. Arumugam and R. Knowles fortheir assistance in the translation into English and finally Mrs C. Demars fortyping the script on MacWriteTM . This book has been typeset with TEX; thetranslation from MacWrite to TeXtureTM has been made with EasyTeXTM .

    New Delhi Sept 4, 1987

  • 9

    NOTATION :

    References:A number in brackets, ex (1), refers to the equation (1) of the current

    chapter . The notation (1.1) refers to the equation (1) of the chapter 1.The numbers in square brackets like [2] indicate a reference in the bibliog-

    raphy.

    Vectors, matrices, scalar products ...Unless stated otherwise the repeated indices are summed. Thus,

    u.v = uivi

    denotes a scalar product of u with v.For all vectors u, v and for all tensors of order 2 A,B we denote :

    (u v)ij = uivj ; trA = Aii; A : B = tr(ABT ), |A| = (trAAT ) 12

    Differential operators :The partial derivatives with respect to t or xj are denoted respectively

    v,t =v

    tvi,j =

    vixj

    ; v,j =v

    xj

    The classical grad, div, curl and laplacian operators are denoted respectively

    p, .v, v, v