Finite Element and Finite Volume Methods for Heat Transfer and Fluid Dynamics

Cover
Half-title
Endorsement
Title page
Copyright information
Contents
Preface
Dedication
Symbols
Part I Preliminaries
	1
Mathematical Preliminaries
		1.1 Introduction
		1.2 Mathematical Models
			1.2.1 Preliminary Comments
			1.2.2 Types of Differential Equations
			1.2.3 Examples of Mathematical Models
			1.2.4 Numerical Solution of First-Order Ordinary Differential Equations
			1.2.5 Partial Differential Equations and their Classification
		1.3 Numerical Methods
			1.3.1 Introduction
			1.3.2 The Finite Difference Method
			1.3.3 The Finite Volume Method
			1.3.4 The Finite Element Method
		1.4 Errors and Convergence
			1.4.1 Types of Errors
			1.4.2 Numerical Convergence
			1.4.3 Order of Accuracy and Grid Convergence Index
		1.5 Veracity of Numerical Solutions
			1.5.1 Verification and Validation
			1.5.2 Manufactured Solutions for Verification
		1.6 Present Study
		Problems
	2
Equations of Heat Transfer and Fluid Mechanics
		2.1 Introduction
		2.2 Elements of Vectors and Tensors
			2.2.1 Introduction
			2.2.2 Coordinate Systems and Summation Convention
			2.2.3 Calculus of Vectors and Tensors
		2.3 Governing Equations of a Continuous Medium
			2.3.1 Descriptions of Motion
			2.3.2 Material Time Derivative
			2.3.3 Velocity Gradient Tensor
			2.3.4 Conservation of Mass
			2.3.5 Reynolds Transport Theorem
			2.3.6 Conservation of Momenta
			2.3.7 Conservation of Energy
			2.3.8 Equation of State
			2.3.9 Constitutive Equations
		2.4 Summary
		Problems
	3
Solution Methods for Algebraic Equations
		3.1 Introduction
		3.2 Linearization of Nonlinear Equations
			3.2.1 Introduction
			3.2.2 The Picard Iteration Method
			3.2.3 The Newton Iteration Method
		3.3 Solution of Linear Equations
			3.3.1 Introduction
			3.3.2 Direct Methods
			3.3.3 Iterative Methods
			3.3.4 Iterative Methods for the Finite Volume Method
		Problems
Part II The Finite Element Method
	4
The Finite Element Method: Steady-State Heat Transfer
		4.1 The Basic Idea
		4.2 One-Dimensional Problems
			4.2.1 Model Differential Equation
			4.2.2 Division of the Whole into Parts
			4.2.3 Approximation over the Element
			4.2.4 Derivation of the Weak Form
			4.2.5 Approximation Functions
			4.2.6 Finite Element Model
			4.2.7 Axisymmetric Problems
			4.2.8 Numerical Examples
		4.3 Two-Dimensional Problems
			4.3.1 Model Differential Equation
			4.3.2 Finite Element Approximation
			4.3.3 Weak Form
			4.3.4 Finite Element Model
			4.3.5 Axisymmetric Problems
			4.3.6 Approximation Functions and Evaluation of Coefficients for Linear Elements
			4.3.7 Higher-Order Finite Elements
			4.3.8 Assembly of Elements
			4.3.9 Numerical Examples
		4.4 Summary
		Problems
	5
The Finite Element Method: Unsteady Heat Transfer
		5.1 Introduction
		5.2 One-Dimensional Problems
			5.2.1 Model Equation
			5.2.2 Steps in Finite Element Model Development
			5.2.3 Weak Form
			5.2.4 Semidiscrete Finite Element Model
			5.2.5 Time Approximations
			5.2.6 Fully Discretized Finite Element Equations
		5.3 Two-Dimensional Problems
			5.3.1 Model Equation
			5.3.2 Weak Form
			5.3.3 Semidiscrete Finite Element Model
			5.3.4 Fully Discretized Model
		5.4 Explicit and Implicit Formulations and Mass Lumping
		5.5 Numerical Examples
			5.5.1 One-Dimensional Problems
			5.5.2 Two-Dimensional Example
		5.6 Summary
		Problems
	6
Finite Element Analysis of Viscous Incompressible Flows
		6.1 Governing Equations
		6.2 Velocity–Pressure Finite Element Model
			6.2.1 Weak-Form Development
			6.2.2 Semidiscretized Finite Element Model
			6.2.3 Fully Discretized Equations
		6.3 Penalty Finite Element Model
			6.3.1 Weak Forms
			6.3.2 Finite Element Model
			6.3.3 Postcomputation
			6.3.4 Numerical Examples
		6.4 Nonlinear Penalty Finite Element Model
			6.4.1 Weak Forms and the Finite Element Model
			6.4.2 Tangent Matrix for the Penalty Finite Element Model
			6.4.3 Numerical Examples
		6.5 Summary
		Problems
Part III The Finite Volume Method
	7
The Finite Volume Method: Diffusion Problems
		7.1 Introduction
		7.2 One-Dimensional Problems
			7.2.1 Governing Equations
			7.2.2 Grid Generation
			7.2.3 Development of Discretization Equations
			7.2.4 Neumann Boundary Condition: Prescribed Flux
			7.2.5 Mixed Boundary Condition: Convective Heat Flux
			7.2.6 Interface Properties
			7.2.7 Numerical Examples
			7.2.8 Axisymmetric Problems
		7.3 Two-Dimensional Diffusion
			7.3.1 Model Equation
			7.3.2 Grid Generation
			7.3.3 Discretization of the Model Equation
			7.3.4 Discrete Equations for Control Volumes and Nodes on the Boundary
		7.4 Unsteady Problems
			7.4.1 One-Dimensional Problems
			7.4.2 Two-Dimensional Problems
			7.4.3 Numerical Examples
		7.5 Summary
		Problems
	8
The Finite Volume Method: Advection–Diffusion Problems
		8.1 Introduction
		8.2 Discretization of the Advection–Diffusion Flux
			8.2.1 General Discussion
			8.2.2 A General Two-Node Formulation
			8.2.3 Central Difference Approximation
			8.2.4 Upwind Scheme
			8.2.5 Exponential Scheme
			8.2.6 Hybrid Scheme
			8.2.7 Power–Law Scheme
			8.2.8 A Three-Node Formulation: QUICK Scheme
			8.2.9 A Numerical Example
		8.3 Numerical Diffusion
		8.4 Steady Two-Dimensional Problems
		8.5 Summary
	9
Finite Volume Methods for Viscous Incompressible Flows
		9.1 Governing Equations
		9.2 The Velocity–Pressure Formulation
			9.2.1 Introduction
			9.2.2 Discretized Equations
			9.2.3 Residuals and Declaring Convergence
			9.2.4 Boundary Conditions
			9.2.5 Treatment of Source Terms
		9.3 Collocated-Grid Method
			9.3.1 General Introduction
			9.3.2 Calculation of Control Volume Face Velocities
			9.3.3 Correction of Velocity and Pressure Fields by Enforcing the Incompressibility Condition
		9.4 Numerical Examples
		9.5 Treatment of Solid Obstacles in Flow Paths
			9.5.1 Preliminary Comments
			9.5.2 Domain Decomposition Method
			9.5.3 High-Viscosity Method
			9.5.4 Dominant-Source-Term Method
		9.6 Vorticity–Stream Function Equations
			9.6.1 Governing Equations in Terms of Vorticity and Stream Function
			9.6.2 Poisson’s Equation for Pressure
		9.7 Summary
		Problems
	10
Advanced Topics
		10.1 Introduction
			10.1.1 General Remarks
			10.1.2 Periodic and Buoyancy-Driven Flows
			10.1.3 Non-Newtonian Fluids
			10.1.4 Solution Methods
		10.2 Periodically Fully Developed Flows
			10.2.1 Introduction
			10.2.2 Governing Equations
			10.2.3 Thermally Fully Developed Flows
			10.2.4 Uniform Heat Flux Condition
			10.2.5 Uniform Wall Temperature Condition
			10.2.6 Cyclic Tri-Diagonal Matrix Algorithm
		10.3 Natural Convection
			10.3.1 Governing Equations
			10.3.2 Discretized Equations
		10.4 Multigrid Algorithms
			10.4.1 Preliminary Comments
			10.4.2 Coarse-Grid Equations
			10.4.3 Grid-Transfer Operators
			10.4.4 Multigrid Cycles
		10.5 Summary
References
Index