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