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# Finite Element Method for Fluid Dynamics

**ProfessorOC Zienkiewicz, Professor R.L. Taylor, P. Nithiarasu**

ISBN **0750663227**

Pages **435**

Description

Dealing with general problems in fluid mechanics, convection diffusion, compressible and incompressible laminar and turbulent flow, shallow water flows and waves, this is the leading text and reference for engineers working with fluid dynamics in fields including aerospace engineering, vehicle design, thermal engineering and many other engineering applications. The new edition is a complete fluids text and reference in its own right. Along with its companion volumes it forms part of the indispensable Finite Element Method series.

New material in this edition includes sub-grid scale modelling; artificial compressibility; full new chapters on turbulent flows, free surface flows and porous medium flows; expanded shallow water flows plus long, medium and short waves; and advances in parallel computing.

Contents

1 Introduction to the equations of fluid dynamics and the finite element approximation 1.1 General remarks and classification of fluid dynamics problems discussed in this book 1.2 The governing equations of fluid dynamics 1.3 Inviscid, incompressible flow 1.4 Incompressible (or nearly incompressible) flows 1.5 Numerical solutions: weak forms, weighted residual and finite element approximation 1.6 Concluding remarks 1.7 Exercises References 2 Convection dominated problems ???finite element approximations to the convection–diffusion-reaction equation 2.1 Introduction 2.2 The steady-state problem in one dimension 2.3 The steady-state problem in two (or three) dimensions 2.4 Steady state -- concluding remarks 2.5 Transients -- introductory remarks 2.6 Characteristic-based methods 2.7 Taylor--Galerkin procedures for scalar variables 2.8 Steady-state condition 2.9 Non-linear waves and shocks 2.10 Treatment of pure convection 2.11 Boundary conditions for convection--diffusion 2.12 Summary and concluding remarks 2.13 Exercises References 3 The characteristic-based split (CBS) algorithm. A general procedure for compressible and incompressible flow 3.1 Introduction viii Contents 3.2 Non-dimensional form of the governing equations 3.3 Characteristic-based split (CBS) algorithm 3.4 Explicit, semi-implicit and nearly implicit forms 3.5 Artificial compressibility and dual time stepping 3.6 ‘Circumvention???of the Babu¡ska--Brezzi (BB) restrictions 3.7 A single-step version 3.8 Boundary conditions 3.9 The performance of two and single step algorithms on an inviscid problem 3.10 Concluding remarks References 4 Incompressible Newtonian laminar flows 4.1 Introduction and the basic equations 4.2 Use of the CBS algorithm for incompressible flows 4.3 Adaptive mesh refinement 4.4 Adaptive mesh generation for transient problems 4.5 Slow flows -- mixed and penalty formulations 4.6 Concluding remarks References 5 Incompressible non-Newtonian flows 5.1 Introduction 5.2 Non-Newtonian flows - metal and polymer forming 5.3 Viscoelastic flows 5.4 Direct displacement approach to transient metal forming 5.5 Concluding remarks References 6 Free surface and buoyancy driven flows 6.1 Introduction 6.2 Free surface flows 6.3 Buoyancy driven flows 6.4 Concluding remarks References 7 Compressible high-speed gas flow 7.1 Introduction 7.2 The governing equations 7.3 Boundary conditions -- subsonic and supersonic flow 7.4 Numerical approximations and the CBS algorithm 7.5 Shock capture 7.6 Variable smoothing 7.7 Some preliminary examples for the Euler equation 7.8 Adaptive refinement and shock capture in Euler problems 7.9 Three-dimensional inviscid examples in steady state 7.10 Transient two- and three-dimensional problems Contents ix 7.11 Viscous problems in two dimensions 7.12 Three-dimensional viscous problems 7.13 Boundary layer--inviscid Euler solution coupling 7.14 Concluding remarks References 8 Turbulent flows 8.1 Introduction 8.2 Treatment of incompressible turbulent flows 8.3 Treatment of compressible flows 8.4 Large eddy simulation 8.5 Detached Eddy Simulation (DES) and Monotonically Integrated LES (MILES) 8.6 Direct Numerical Simulation (DNS) 8.7 Summary References 9 Flow through porous media 9.1 Introduction 9.2 A generalized porous medium flow approach 9.3 Discretization procedure 9.4 Non-isothermal flows 9.5 Forced convection 9.6 Natural convection 9.7 Summary References 10 Shallow water problems 10.1 Introduction 10.2 The basis of the shallow water equations 10.3 Numerical approximation 10.4 Examples of application 10.5 Drying areas 10.6 Shallow water transport 10.7 Concluding remarks References 11 Long and medium waves 11.1 Introduction and equations 11.2 Waves in closed domains - finite element models 11.3 Difficulties in modelling surface waves 11.4 Bed friction and other effects 11.5 The short-wave problem 11.6 Waves in unbounded domains (exterior surface wave problems) 11.7 Unbounded problems 11.8 Local Non-Reflecting Boundary Conditions (NRBCs) 11.9 Infinite elements 11.10 Mapped periodic (unconjugated) infinite elements x Contents 11.11 Ellipsoidal type infinite elements of Burnett and Holford 11.12 Wave envelope (or conjugated) infinite elements 11.13 Accuracy of infinite elements 11.14 Trefftz type infinite elements 11.15 Convection and wave refraction 11.16 Transient problems 11.17 Linking to exterior solutions (or DtN mapping) 11.18 Three-dimensional effects in surface waves 11.19 Concluding remarks References 12 Short waves 12.1 Introduction 12.2 Background 12.3 Errors in wave modelling 12.4 Recent developments in short wave modelling 12.5 Transient solution of electromagnetic scattering problems 12.6 Finite elements incorporating wave shapes 12.7 Refraction 12.8 Spectral finite elements for waves 12.9 Discontinuous Galerkin finite elements (DGFE) 12.10 Concluding remarks References 13 Computer implementation of the CBS algorithm 13.1 Introduction 13.2 The data input module 13.3 Solution module 13.4 Output module References Appendix A Non-conservative form of Navier–Stokes equations Appendix B Self-adjoint differential equations Appendix C Postprocessing Appendix D Integration formulae Appendix E Convection–diffusion equations: vector-valued variables Appendix F Edge-based finite element formulation Appendix G Multigrid method Appendix H Boundary layer–inviscid flow coupling Appendix I Mass-weighted averaged turbulence transport equations Author Index Subject Index