Fundamentals of Computational Fluid Dynamics. The Finite Volume Method

Description

Foreword Preface Acknowledgements Contents Nomenclature Superscripts Subscripts Greek Letters 1 Introduction 1.1 Preliminaries 1.2 Available Tools for the Engineer 1.3 Classes of Numerical Methods Available 1.4 Objectives and Scope of This Book 1.5 Applications of Computational Fluid Dynamics Reference 2 Conservation Equations—Physical and Mathematical Aspects 2.1 Introduction 2.2 Models Formulation Levels 2.3 Conservation Equations 2.3.1 Mass Conservation Equation 2.3.2 Linear Momentum Conservation Equations 2.3.3 Energy Conservation Equation 2.4 Elliptic, Parabolic and Hyperbolic Problems 2.4.1 Preliminaries 2.4.2 Parabolic and Hyperbolic Problems 2.4.3 Elliptic Problems 2.5 True and Distorted Transient 2.6 Conclusions 2.7 Exercises References 3 The Finite Volume Method 3.1 Introduction 3.2 The Task of a Numerical Method 3.3 Why Finite Volume Methods is a Good Choice 3.4 Few Words About the Conservative Property 3.5 Cell-Center and Cell-Vertex Methods 3.6 One Dimensional Transient Heat Diffusion 3.7 Explicit, Implicit and Fully Implicit Formulations 3.7.1 Explicit Formulation 3.7.2 Fully Implicit Formulation 3.7.3 Implicit Formulation 3.8 Linearization of the Source Term 3.9 Boundary Conditions 3.9.1 Balances for the Boundary Volumes 3.9.2 Using Fictitious Volumes 3.9.3 About Boundary Conditions in Cell-Vertex 3.10 Discretization of the 3D Diffusion Equation 3.11 Structure of the Matrix of Coefficients 3.12 Handling Non-linearities 3.13 Relevant Issues When Discretizing the Equations 3.13.1 Positivity of Coefficients 3.13.2 Fluxes Continuity at Interfaces 3.13.3 Linearization of Source Term with SP negative 3.13.4 Truncation Errors 3.13.5 Consistency, Stability and Convergence 3.14 Conclusions 3.15 Exercises References 4 Solution of the Linear System 4.1 Introduction 4.2 Iterative Methods 4.2.1 Jacobi 4.2.2 Gauss-Seidel 4.2.3 SOR-Successive Over Relaxation 4.2.4 Alternating Direction Implicit Methods 4.2.5 Incomplete LU Decomposition 4.2.6 A Note on Convergence of Iterative Methods 4.2.7 Multigrid Method 4.3 Conclusions 4.4 Exercises References 5 Advection and Diffusion—Interpolation Functions 5.1 Introduction 5.2 The General Equation 5.3 The Difficulty of the Advective-Dominant Problem 5.4 Interpolation Functions for φ 5.4.1 The Physics Behind the Interpolation Functions 5.4.2 One Dimensional Interpolation Functions 5.4.3 Numerical or False Diffusion 5.4.4 Two and Three-Dimensional Interpolation Functions 5.5 Conclusions 5.6 Exercises References 6 Three-Dimensional Advection/Diffusion of φ 6.1 Introduction 6.2 Integration of the 3D Equation for φ 6.3 Explicit Formulation 6.3.1 True Transient 6.3.2 Distorted Transient 6.4 Fully Implicit Formulation 6.5 Conclusions 6.6 Exercises 7 Finding the Velocity Field—Pressure/Velocity Couplings 7.1 Introduction 7.2 System of Equations 7.2.1 About Segregated and Simultaneous Solution 7.3 Segregated Formulation. Incompressibility 7.4 Variable Arrangement on the Grid 7.4.1 Co-located Grid Arrangement 7.4.2 Staggered Grid Arrangement 7.5 Co-located PV Coupling (CPVC) Methods 7.5.1 Rhie and Chow-Like Methods 7.5.2 PIS—Physical Influence Scheme 7.6 Segregated PV Coupling (SPVC) Methods 7.6.1 Chorin’s Method 7.6.2 SIMPLE—Semi Implicit Linked Equations 7.6.3 SIMPLER—Simple-Revisited 7.6.4 PRIME—Pressure Implicit Momentum Explicit 7.6.5 SIMPLEC—Simple Consistent 7.6.6 PISO—Pressure Implicit with Split Operator 7.6.7 SIMPLEC for Co-located Grids 7.6.8 PRIME for Co-located Grids 7.7 Boundary Conditions for p and p 7.8 Simultaneous Solution and the Couplings 7.9 A Note on Boundary Conditions 7.9.1 Impermeable Boundary—φ Prescribed 7.9.2 Impermeable Boundary—Flux of φ Prescribed 7.9.3 Inflow and Outflow Boundary Conditions 7.9.4 General Comments About Boundary Conditions 7.9.5 Incompressible Flows 7.9.6 Compressible Flows 7.10 Conclusions 7.11 Exercises References 8 All Speed Flows Calculation—Coupling P to [V - ρ] 8.1 Introduction 8.2 Pressure–Velocity and Pressure-Density Coupling 8.2.1 Linearization of the Mass Flow 8.3 Two-Dimensional All Speed Flow Discretization 8.3.1 Velocity Relations as Function of p- SIMPLEC 8.3.2 Density Relations as Function of p- SIMPLEC 8.3.3 Velocity/Density Relations as Function of p-PRIME 8.4 Conclusions 8.5 Exercises References 9 Two and Three-Dimensional Parabolic Flows 9.1 Introduction 9.2 Two-Dimensional Parabolic Flows 9.2.1 External Two-Dimensional Parabolic Flows 9.2.2 Internal Two-Dimensional Parabolic Flows 9.3 Three-dimensional Parabolic Flows 9.3.1 External Three-Dimensional Parabolic Flows 9.3.2 Internal Three-Dimensional Parabolic Flows 9.4 Conclusions 9.5 Exercises References 10 General Recommendations for Conceiving and Testing Your Code 10.1 Introduction 10.2 Writing Your Code 10.2.1 Generalities 10.2.2 Coding Languages 10.2.3 Tools to Aid the Development 10.3 Running Your Application 10.3.1 Compiling 10.3.2 Size of the Mesh 10.3.3 Convergence Criteria 10.4 Choosing Test Problems—Finding Errors 10.4.1 Heat Conduction—2D Steady State 10.4.2 Transient Heat Conduction—One Dimensional 10.4.3 One Dimensional Advection/Diffusion 10.4.4 Two-Dimensional Advection/Diffusion 10.4.5 Entrance Flow Between Parallel Plates 10.5 Observing Details of the Solution 10.5.1 Symmetry of the Solution 10.5.2 The Coefficients 10.5.3 Testing the Solver of the Linear System 10.6 Conclusions References 11 Introducing General Grids Discretization 11.1 Introduction 11.2 Structured and Non-structured Grids 11.3 The Concept of Element 11.4 Construction of the Control Volume 11.5 Conclusions 12 Coordinate Transformation—General Curvilinear Coordinate Systems 12.1 Introduction 12.2 Global Coordinate Transformation 12.2.1 General 12.2.2 Length Along a Coordinate Axis 12.2.3 Areas (or Volumes) in the Curvilinear System 12.2.4 Basis Vectors 12.2.5 Vector Representation in the Curvilinear System 12.2.6 Mass Flow Calculation 12.2.7 Example of a Nonorthogonal Transformation 12.2.8 Calculation of the Metrics of a Transformation 12.3 Nature of the Discrete Transformation 12.3.1 Preliminaries 12.3.2 The Nature of the Transformation 12.4 Equations Written in the Curvilinear System 12.5 Discretization of the Transformed Equations 12.6 Comments on the Solution of the Equation System 12.6.1 Simultaneous Solution 12.6.2 Segregated Solution 12.7 Boundary Conditions 12.7.1 No-Flow Boundary (ρU = 0). φ Prescribed 12.7.2 No-Flow Boundary (ρU = 0). Flux of φ Prescribed 12.7.3 Bounday With Mass Flow (ρU =0). Mass Entering With ρU Known 12.7.4 Boundary With Mass Flow (ρU =0). Mass Leaving With ρU Unknown 12.8 Conclusions 12.9 Exercises References 13 Unstructured Grids 13.1 Introduction 13.2 Cell-Center Methods 13.2.1 Conventional Finite Volume Method 13.2.2 Voronoi Diagrams 13.3 EbFVM—Element-based Finite Volume Method 13.3.1 Geometrical Entities 13.3.2 Local Coordinates. Shape Functions 13.3.3 Determination of (φ)ip 13.3.4 Determination of φip 13.3.5 Family of Positive Advection Schemes 13.3.6 Integration of the Conservation Equations 13.3.7 Assembling Strategies 13.3.8 Boundary Conditions 13.4 Conclusions 13.5 Exercises References 14 Pressure Instabilities: From Navier–Stokes to Poroelasticity 14.1 Introduction 14.2 Pressure Instabilities 14.2.1 Remedy 1 14.2.2 Remedy 2 14.3 Conclusions References 15 Applications 15.1 Introduction 15.2 Aerodynamics 15.2.1 All Speed Flow Over a Blunt Body 15.2.2 Ice Accretion on Aerodynamic Profiles 15.3 Porous Media Flows 15.4 Conclusions References Index