Fundamentals of Computational Fluid Dynamics - The Finite Volume Method

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