The finite element method. Volume 3, Fluid dynamics

In the years since the fourth edition of this seminal work was published, active research has developed the Finite Element Method into the pre-eminent tool for the modelling of physical systems. Written by the pre-eminent professors in their fields, this new edition of the Finite Element Method maintains the comprehensive style of the earlier editions and authoritatively incorporates the latest developments of this dynamic field. Expanded to three volumes the book now covers the basis of the method and its application to advanced solid mechanics and also advanced fluid dynamics. Volume Three: Fluid Dynamics is intended for readers studying fluid mechanics at a higher level. Although it is an ideal companion volume to Volume One: The Basis, this advanced text also functions as a "stand-alone" volume, accessible to those who have been introduced to the Finite Element Method through a different route. Volume 1 of the Finite Element Method provides a complete introduction to the method and is essential reading for undergraduates, postgraduates and professional engineers. Volume 2 concentrates on non-linear solid and structural mechanics and is ideal for postgraduate and professional engineers working in this discipline. Coverage of the whole range of fluid dynamics - including incompressible slow viscous flow, high-speed supersonic flows, shallow water flow, ocean waves, and metal and plastic forming.Up-to-date material on the Characteristic Galerkin Method.New methodologies for dealing with supersonic and hypersonic behaviours. Preface to Volume 3 12 Acknowledgements 13 1 Introduction and the equations of fluid dynamics 14 1.1 General remarks and classification of fluid mechanics problems discussed in the book 14 1.2 The governing equations of fluid dynamics 17 1.3 Incompressible (or nearly incompressible) flows 23 1.4 Concluding remarks 25 2 Convection dominated problems - finite element appriximations to the convection-diffusion equation 26 2.1 Introduction 26 2.2 the steady-state problem in one dimension 28 2.3 The steady-state problem in two (or three) dimensions 39 2.4 Steady state - concluding remarks 43 2.5 Transients - introductory remarks 45 2.6 Characteristic-based methods 48 2.7 Taylor-Galerkin procedures for scalar variables 60 2.8 Steady-state condition 61 2.9 Non-linear waves and shocks 61 2.10 Vector-valued variables 65 2.11 Summary and concluding 72 3 A general algorithm for compressible and incompressible flows - the characteristic-based split (CBS) algorithm 77 3.1 Introduction 77 3.2 Characteristic-based split (CBS) algorithm 80 3.3 Explicit, semi-implicit and nearly implicit forms 89 3.4 'Circumventing' the Babuska-Brezzi (BB) restrictions 91 3.5 A single-step version 93 3.6 Boundary conditions 94 3.7 The performance of two- and single-step algorithms on an inviscid problems 98 3.8 Concluding remarks 100 4 Incompressible laminar flow - newtonian and non-newtonian fluids 104 4.1 Introduction and the basic equations 104 4.2 Inviscid, incompressible flow (potential flow) 106 4.3 Use of the CBS algorithm for incompressible or nearly incompressible flows 110 4.4 Boundary-exit conditions 113 4.5 Adaptive mesh refinement 115 4.6 Adaptive mesh generation for transient problems 126 4.7 Importance of stabilizing convective terms 126 4.8 Slow flows - mixed and penalty formulations 126 4.9 Non-newtonian flows - metal and polymer forming 131 4.10 Direct displacement approach to transient metal forming 145 4.11 Concluding remarks 146 5 Free surfaces, buoyancy and turbulent incompressible flows 156 5.1 Introduction 156 5.2 Free surface flows 157 5.3 Buoyancy driven flows 166 5.4 Turbulent flows 174 6 Compressible high-speed gas flow 182 6.1 Introduction 182 6.2 The governing equations 183 6.3 Boundary conditions - subsonic and supersonic flow 184 6.4 Numerical approximations and the CBS algorithm 186 6.5 Shock capture 187 6.6 Some preliminary examples for the Euler equation 189 6.7 Adaptive refinement and shock capture in Euler problems 193 6.8 Three-dimensional inviscid examples in steady state 201 6.9 Transient two and three-dimensional problems 208 6.10 Viscous problems in two dimensions 210 6.11 Three-dimensional viscous problems 220 6.12 Boundary layer-inviscid Euler solution coupling 222 6.13 Concluding remarks 225 7 Shallow-water problems 231 7.1 Introduction 231 7.2 The basis of the shallow-water equations 232 7.3 Numerical approximation 236 7.4 Examples of application 237 7.5 Drying areas 249 7.6 Shallow-water transport 250 8 Waves 255 8.1 Introduction and equations 255 8.2 Waves in closed domains - finite element models 256 8.3 Difficulties in modelling surface waves 258 8.4 Bed friction and other effects 258 8.5 The short-wave problem 258 8.6 Waves in unbounded domains (exterior surface wave problems) 263 8.7 Unbounded problems 266 8.8 Boundary dampers 266 8.9 Linking to exterior solutions 268 8.10 Infinite elements 272 8.11 Mapped periodic infinite elements 273 8.12 Ellipsoidal type infinite elements of Burnnet and Holford 274 8.13 Wave envelope infinite elements 275 8.14 Accuracy of infinite elements 277 8.15 Transient problems 278 8.16 Three-dimensional effects in surface waves 279 9 Computer implementation of the CBS algorithm 287 9.1 Introduction 287 9.2 The data input module 288 9.3 Solution module 291 9.4 Output module 302 9.5 Possible extensions to CBSflow 302 Appendix A Non-conservative form of Navier-Stokes equations 304 Appendix B Discontinuous Galerkin methods in the solution of the convection-diffusion equation 306 Appendix C Edge-based finite element forumlation 311 Appendix D Multigrid methods 313 Appendix E Boundary layer-inviscid flow coupling 315 Author index 320 Subject index 328 Referex Preface to Volume 3......Page 12 Acknowledgements......Page 13 1.1 General remarks and classification of fluid mechanics problems discussed in the book......Page 14 1.2 The governing equations of fluid dynamics......Page 17 1.3 Incompressible (or nearly incompressible) flows......Page 23 1.4 Concluding remarks......Page 25 2.1 Introduction......Page 26 2.2 the steady-state problem in one dimension......Page 28 2.3 The steady-state problem in two (or three) dimensions......Page 39 2.4 Steady state - concluding remarks......Page 43 2.5 Transients - introductory remarks......Page 45 2.6 Characteristic-based methods......Page 48 2.7 Taylor-Galerkin procedures for scalar variables......Page 60 2.9 Non-linear waves and shocks......Page 61 2.10 Vector-valued variables......Page 65 2.11 Summary and concluding......Page 72 3.1 Introduction......Page 77 3.2 Characteristic-based split (CBS) algorithm......Page 80 3.3 Explicit, semi-implicit and nearly implicit forms......Page 89 3.4 'Circumventing' the Babuska-Brezzi (BB) restrictions......Page 91 3.5 A single-step version......Page 93 3.6 Boundary conditions......Page 94 3.7 The performance of two- and single-step algorithms on an inviscid problems......Page 98 3.8 Concluding remarks......Page 100 4.1 Introduction and the basic equations......Page 104 4.2 Inviscid, incompressible flow (potential flow)......Page 106 4.3 Use of the CBS algorithm for incompressible or nearly incompressible flows......Page 110 4.4 Boundary-exit conditions......Page 113 4.5 Adaptive mesh refinement......Page 115 4.8 Slow flows - mixed and penalty formulations......Page 126 4.9 Non-newtonian flows - metal and polymer forming......Page 131 4.10 Direct displacement approach to transient metal forming......Page 145 4.11 Concluding remarks......Page 146 5.1 Introduction......Page 156 5.2 Free surface flows......Page 157 5.3 Buoyancy driven flows......Page 166 5.4 Turbulent flows......Page 174 6.1 Introduction......Page 182 6.2 The governing equations......Page 183 6.3 Boundary conditions - subsonic and supersonic flow......Page 184 6.4 Numerical approximations and the CBS algorithm......Page 186 6.5 Shock capture......Page 187 6.6 Some preliminary examples for the Euler equation......Page 189 6.7 Adaptive refinement and shock capture in Euler problems......Page 193 6.8 Three-dimensional inviscid examples in steady state......Page 201 6.9 Transient two and three-dimensional problems......Page 208 6.10 Viscous problems in two dimensions......Page 210 6.11 Three-dimensional viscous problems......Page 220 6.12 Boundary layer-inviscid Euler solution coupling......Page 222 6.13 Concluding remarks......Page 225 7.1 Introduction......Page 231 7.2 The basis of the shallow-water equations......Page 232 7.3 Numerical approximation......Page 236 7.4 Examples of application......Page 237 7.5 Drying areas......Page 249 7.6 Shallow-water transport......Page 250 8.1 Introduction and equations......Page 255 8.2 Waves in closed domains - finite element models......Page 256 8.5 The short-wave problem......Page 258 8.6 Waves in unbounded domains (exterior surface wave problems)......Page 263 8.8 Boundary dampers......Page 266 8.9 Linking to exterior solutions......Page 268 8.10 Infinite elements......Page 272 8.11 Mapped periodic infinite elements......Page 273 8.12 Ellipsoidal type infinite elements of Burnnet and Holford......Page 274 8.13 Wave envelope infinite elements......Page 275 8.14 Accuracy of infinite elements......Page 277 8.15 Transient problems......Page 278 8.16 Three-dimensional effects in surface waves......Page 279 9.1 Introduction......Page 287 9.2 The data input module......Page 288 9.3 Solution module......Page 291 9.5 Possible extensions to CBSflow......Page 302 Appendix A Non-conservative form of Navier-Stokes equations......Page 304 Appendix B Discontinuous Galerkin methods in the solution of the convection-diffusion equation......Page 306 Appendix C Edge-based finite element forumlation......Page 311 Appendix D Multigrid methods......Page 313 Appendix E Boundary layer-inviscid flow coupling......Page 315 Author index......Page 320 Subject index......Page 328 Volume 2 considers more advanced problems in solid mechanics. This volume can be used by investigators familiar with the finite element method in general terms and introduces them here to the subject of specialized topics in solid mechanics. Two main subject areas in solid mechanics are covered: non-linear problems and plates and shells. This volume concludes with a chapter on Computer Procedures, in which we describe application of the basic program presented in v. 1 to solve non-linear problems Volume 1 provides the general basis applicable to linear problems in many fields whilst the second and third volumes are devoted to more advanced topics in solid and fluid mechanics, respectively. This volume will help to introduce postgraduate students, researchers and practioners to the modern concepts of finite element methods. It stresses the relationship between the finite element method and the more classic finite difference and boundary solution methods .. Written by the preeminent professors in their fields, this new edition of the Finite Element Method maintains the comprehensive style of the earlier editions and authoritatively incorporates the latest developments of this dynamic field. Volume 1: The Basis is intended as a broad overview of the Finite Element Method; it provides a complete introduction to the method. Volume 3 is devoted entirely to fluid mechanics and uses in the main the methods introduced in Volume 1. However, it enlarges these to deal with the non-self-adjoint problems of convection which are essential to fluid mechanics problems