Introduction to Computational Fluid Dynamics

Cover
About the Authors
Preface
Acknowledgements
Contents
Part I: Finite Difference Method for Partial Differential Equations
	Chapter 1: Introduction and Mathematical Preliminaries
		1.1 Introduction
		1.2 Typical Partial Differential Equations in Fluid Dynamics
		1.3 Types of Second-order Equations
			1.3.1 Characteristics of Second-Order Equations
		1.4 Well-posed Problems
			1.4.1 Examples of Well-Posed Problems
			1.4.2 An Ill-Posed Problem
		1.5 Properties of Linear and Quasilinear Equations
			1.5.1 Qualitative Properties of Partial Differential Equations
		1.6 Physical Character of Subsonic and Supersonic Flows
		1.7 Second-order Wave Equations
			1.7.1 Cauchy Problem for the Wave Equation
			1.7.2 Domain of Dependence and Range of Influence
		1.8 System of First-order Equations
			1.8.1 Classification and Types of First-Order Systems
			1.8.2 Conservation Form and Conservation-Law Form
		1.9 Weak Solutions
		1.10 Summary
		1.11 Key Terms
	Chapter 2: Finite Difference and Finite Volume Discretisations
		2.1 Introduction
		2.2 Finite Difference Discretisation
		2.3 Discretisation of Derivatives
		2.4 Consistency, Convergence, and Stability
		2.5 Finite Volume Discretisation
			2.5.1 Cell-Centred Scheme
		2.6 Face Area and Cell Volume
			2.6.1 Equivalence Between Finite Difference and Finite Volume Methods
		2.7 Summary
		2.8 Key Terms
		2.9 Exercise 2
	Chapter 3: Equations of Parabolic Type
		3.1 Introduction
		3.2 Finite Difference Scheme for Heat Conduction Equation
			3.2.1 FTCS Scheme: Truncation Error and Consistency
			3.2.2 Modified Equation
			3.2.3 FTCS Scheme: Convergence
			3.2.4 FTCS Scheme: Stability
			3.2.5 Derivative Boundary Conditions
		3.3 Crank-Nicholson Implicit Scheme
		3.4 Analogy with Schemes for Ordinary Differential Equations
			3.4.1 Thomas Algorithm for Tridiagonal Systems
			3.4.2 Crank-Nicholson Scheme: Truncation Error, Consistency, and Convergence
			3.4.3 Dissipative and Dispersive Errors
			3.4.4 Stability of the Crank-Nicholson Scheme
		3.5 A Note on Implicit Methods
		3.6 Leap-frog and DuFort-Frankel Schemes
			3.6.1 Truncation Error of the DuFort-Frankel Scheme
			3.6.2 Stability of DuFort-Frankel Scheme
		3.7 Operator Notation
		3.8 The Alternating Direction Implicit (ADI) Method
			3.8.1 ADI Scheme
			3.8.2 Splitting and Approximate Factorisation
			3.8.3 Stability of the ADI Scheme
			3.8.4 Program 3.1: adi.f
		3.9 Summary
		3.10 Key Terms
		3.11 Exercise 3
	Chapter 4: Equations of Hyperbolic Type
		4.1 Introduction
		4.2 Explicit Schemes
			4.2.1 FTCS Scheme
			4.2.2 FTFS Scheme
			4.2.3 Upwind Scheme: First Order
			4.2.4 Upwind Scheme: Modified Equation
			4.2.5 The Lax Scheme
			4.2.6 Consistency of Lax Scheme
			4.2.7 Lax Scheme: Modified Equation
			4.2.8 The Leap-Frog Scheme
		4.3 Lax-Wendroff Scheme and Variants
			4.3.1 Lax-Wendroff Scheme: Modified Equation
			4.3.2 Two-Step Lax-Wendroff Scheme
			4.3.3 The MacCormack Scheme
			4.3.4 Upwind Scheme: Warming-Beam
		4.4 Implicit Schemes
		4.5 More on Upwind Schemes
		4.6 Scalar Conservation Law: Lax-Wendroff and Related Schemes
			4.6.1 Program 4.1: Ixmc.f
			4.6.2 Implicit Schemes for Scalar Conservation Law
		4.7 Hyperbolic System of Conservation Laws
			4.7.1 System of Conservation Laws
		4.8 Second-order Wave Equation
			4.8.1 Stability of the Leap-Frog Scheme for the Wave Equation
			4.8.2 An Implicit Scheme for the Second-Order Wave Equation
			4.8.3 Stability of the Implicit Scheme
		4.9 Method of Characteristics for Second-order Hyperbolic Equations
		4.10 Model Convection-Diffusion Equation
			4.10.1 Steady Convection-Diffusion Equation
			4.10.2 Linear Convection-Diffusion Equation: FTCS Scheme
			4.10.3 First-Order Upwind Scheme for Convection-Diffusion Equation
			4.10.4 Burgers Equation
		4.11 Summary
		4.12 Key Terms
		4.13 Exercise 4
	Chapter 5: Equations of Elliptic Type
		5.1 Introduction
		5.2 The Laplace Equation in Two Dimension
		5.3 Iterative Methods for Solution of Linear Algebraic Systems
			5.3.1 The Jacobi and the Gauss-Seidel Schemes
		5.4 Solution of the Pentadiagonal System
			5.4.1 Program 5.1: sor.f
		5.5 Approximate Factorisation Schemes
			5.5.1 Analysis of Line Gauss-Seidel Scheme for the Laplace Equation
			5.5.2 Time-Dependent Analogy
			5.5.3 Program 5.2: afl.f
		5.6 Grid Generation Example
		5.7 Body-fitted Grid Generation Using Elliptic-type Equations
			5.7.1 Solution of the Algebraic Equations by AFI Scheme
		5.8 Some Observations of AF Schemes
		5.9 Multi-grid Method
			5.9.1 Program 5.3: mgc.f
		5.10 Summary
		5.11 Key Terms
		5.12 Exercise 5
	Chapter 6: Equations of Mixed Elliptic-Hyperbolic Type
		6.1 Introduction
		6.2 Tricomi Equation
		6.3 Transonic Computations Based on TSP Model
			6.3.1 Finite Difference Discretisation
			6.3.2 Implementation of Boundary Conditions
			6.3.3 Iterative Solution of the Discretised Equations
			6.3.4 Artificial Viscosity and Conservative Schemes
			6.3.5 Computational Results
			6.3.6 Program 6.1 tsc.f
		6.4 Summary
		6.5 Key Terms
		6.6 Exercise 6
Part II: Computational Fluid Dynamics
	Chapter 7: The Basic Equations of Fluid Dynamics
		7.1 Introduction
		7.2 Basic Conservation Principles
		7.3 Unsteady Navier-Stokes Equations in Integral Form
		7.4 Navier-Stokes Equations in Differential Form
			7.4.1 Compressible Two-Dimensional Equations in Vector Form
			7.4.2 Incompressible Navier-Stokes Equations in Cartesian Coordinates
			7.4.3 Dimensionless Form of the Basic Equations
			7.4.4 Incompressible Two-Dimensional Equations: Dimensionless Form
			7.4.5 Observations on the Basic Equations
		7.5 Boundary Conditions for Navier-Stokes Equations
		7.6 Reynolds Averaged Navier-Stokes Equations
		7.7 Boundary-layer, Thin-layer and Associated Approximations
		7.8 Euler Equations for Inviscid Flows
			7.8.1 Certain Observations on Euler and Navier-Stokes Equations
		7.9 Boundary Conditions for Euler Equations
			7.9.1 Far-field Boundary Conditions for Euler Equations
		7.10 The Full Potential Equation
			7.10.1 Potential Equation in Conservative Form
			7.10.2 Boundary Conditions for the Full Potential Equation
			7.10.3 Transonic Small Perturbation Model
			7.10.4 Oswatitsch Reduction
			7.10.5 Cole’s and Other Forms of the TSP Equation
		7.11 Inviscid Incompressible Irrotational Flow
		7.12 Summary
		7.13 Key Terms
	Chapter 8: Grid Generation
		8.1 Introduction
		8.2 Co-ordinate Transformation
		8.3 Differential Equation Methods
		8.4 Algebraic Methods
			8.4.1 Calculation of the Arc Length
			8.4.2 Desired Arc Length Distribution
			8.4.3 Calculation of the Angle θ on the Aerofoil and Cut
			8.4.4 Calculation of ymin and nmax
			8.4.5 Δn - Distribution on the Aerofoil and the Cut
			8.4.6 Mesh Spacing in n-Direction
			8.4.7 Calculation of x and y at Nodal Points
			8.4.8 Cubic Spline
		8.5 Transfinite Interpolation Methods
		8.6 Unstructured Grid Generation
		8.7 Mesh Adaptation
			8.7.1 Moving Mesh
			8.7.2 Mesh Enrichment
		8.8 Summary
		8.9 Key Terms
		8.10 Exercise 8
	Chapter 9: Inviscid Incompressible Flow
		9.1 Introduction
		9.2 Potential Flow Problem
		9.3 Panel Methods
			9.3.1 AMO Smith Method for a Lifting Airfoil
			9.3.2 Influence Coefficients
		9.4 Panel Methods (Continued)
			9.4.1 Mathematical Preliminaries for Morino-Kuo Method
			9.4.2 Flow Past an Aerofoil
			9.4.3 A Constant-Potential Panel Method
			9.4.4 Morino-Kuo Method
				9.4.4.1 Pressure coefficient, forces, and moments
			9.4.5 Program 9.1: Morinoprogram.c
			9.4.6 Discretisation Error in Panel Methods
		9.5 More on Panel Methods
		9.6 Panel Methods for Subsonic and Supersonic Flows
		9.7 Summary
		9.8 Key Terms
		9.9 Exercise 9
	Chapter 10: Inviscid Compressible Flow
		10.1 Introduction
			10.1.1 Transonic Controversy
		10.2 Small-perturbation Flow
			10.2.1 Subsonic Flow Past a Thin Profile
			10.2.2 Supersonic Small-Perturbation Flow
		10.3 Numerical Solution of the Full Potential Equation
			10.3.1 Rotated Difference Scheme
			10.3.2 Conservative Schemes for the Potential Equation
		10.4 Full Potential Solution in Generalised Coordinates
			10.4.1 Spatial Differencing and Artificial Viscosity
			10.4.2 AF2 Iteration Scheme
			10.4.3 Boundary Conditions
			10.4.4 Computational Results of Full-Potential Solution
		10.5 Observations on the Full Potential Model
		10.6 Euler Model
			10.6.1 Governing Equations in Two Dimension
			10.6.2 Numerical Methods for the Euler Model
			10.6.3 Explicit and Implicit Schemes
			10.6.4 Review of Acceleration Techniques
			10.6.5 Finite Volume Discretisation
			10.6.6 Artificial Dissipation
		10.7 Boundary Conditions
			10.7.1 Time Stepping Scheme
			10.7.2 Acceleration Techniques
		10.8 Computed Examples Based on the Euler Model
		10.9 Supersonic Flow Field Computation
			10.9.1 Examples of Supersonic Flow Computation
		10.10 Summary
		10.11 Key Terms
		10.12 Exercise 10
	Chapter 11: Boundary Layer Flow
		11.1 Introduction
		11.2 The Boundary Layer: Physical Considerations
			11.2.1 Separation of the Boundary Layer from the Surface
			11.2.2 Turbulence
			1 1.2.3 Measures of Boundary Layer Thickness
		11.3 The Boundary Layer Equations
			1 1.3.1 Assumptions of the Boundary Layer Theory
			11.3.2 The Boundary Layer Equations for Laminar Flow
				1 1.3.2.1 Non-dimensionalisation of the governing equations
				11.3.2.2 Order of magnitude analysis
				11.3.2.3 Obtaining the laminar boundary layer equations
			11.3.3 The Boundary Layer Equations for Turbulent Flow
			11.3.4 Handling the Reynolds Stresses: Turbulence Modelling
			11.3.5 Mathematical Nature of the Boundary Layer Equation (Boundary Conditions)
		11.4 Computations of the Laminar Boundary Layer
			11.4.1 Objectives
			11.4.2 Similarity Transformation and the Falkner-Skan Equation
			11.4.3 Laminar Boundary Layer on a Flat Plate
				11.4.3.1 Solution by the “shooting method”
				11.4.3.2 Displacement thickness and skin friction coefficient
				11.4.3.3 What of the displacement effect?
			11.4.4 Non-Similar Solutions of the Boundary Layer Equation
			11.4.5 The Keller Box Scheme
		11.5 Turbulent Boundary Layers
		11.6 Summary
		11.7 Key Terms
		11.8 Exercise 11
	Chapter 12: Viscous Incompressible Flow
		12.1 Introduction
		12.2 Incompressible Flow Computation
		12.3 Stream-function Vorticity Approach
			12.3.1 Pressure Poisson Equation
			12.3.2 Boundary Conditions for Stream-Function and Vorticity
			12.3.3 Method of Solution
		12.4 Primitive Variables Approach
			12.4.1 Discretisation Using Staggered Grid
		12.5 The MAC Method
			12.5.1 The Pressure Poisson Equation
			12.5.2 Stability Restriction
			12.5.3 Boundary Conditions in Primitive Variables
		12.6 Solution Scheme
			12.6.1 Variants of the MAC Method
			12.6.2 Treatment of Convective Terms
		12.7 Case Study: Separated Flow in a Constricted Channel
			12.7.1 The Problem and Method of Solution
			12.7.2 Boundary Conditions
			12.7.3 Initial Condition
			12.7.4 Co-ordinate Transformation
			12.7.5 Numerical Solution
				12.7.5.1 Type of grid used
			12.7.6 Results and Discussion
				12.7.6.1 Wall vorticity
				12.7.6.2 Streamlines and vorticity contours
			12.7.7 Conclusion: Case Study
		12.8 Turbulent Flow
			12.8.1 Physical Characteristics of Turbulent Flow
			12.8.2 Incompressible Reynolds Averaged Navier-Stokes Equations
				12.8.2.1 Properties of averaging
			12.8.3 Closure Problem and Turbulence Modelling
			12.8.4 Boussinesq Hypothesis
			12.8.5 Eddy Viscosity Models
			12.8.6 Zero-Equation Models
			12.8.7 K-e Model
		12.9 Summary
		12.10 Key Terms
		12.11 Exercise 12
	Chapter 13: Viscous Compressible Flow
		13.1 Introduction
		13.2 Dynamic Similarity
		13.3 RANS (Reynolds Averaged Compressible Navier-Stokes) Equations
		13.4 Turbulence Modelling
			13.4.1 Algebraic Turbulence Models
			13.4.2 Other Models
		13.5 Boundary Conditions
		13.6 Basic Computational Methods for Compressible Flow
		13.7 Finite Volume Computation in 2D
		13.8 Solution Procedure
		13.9 Computational Results
			13.9.1 Flow Over a Flat Plate
			13.9.2 Viscous Flow Past NACA0012 Aerofoil
			13.9.3 Viscous Transonic Flow Past Other Aerofoils
			13.9.4 Internal Flow Through Nozzle
			13.9.5 Turbulent Flow Through Cascades
			13.9.6 Viscous Flow Past Aerofoil-Flap Configuration
		13.10 Summary
		13.11 Key Terms
		13.12 Exercise 13
Appendix A: Glossary
	A.l Glossary
Appendix B: Ready-made Softwares for CFD
	B.l Introduction
	B.2 Software Packages for CFD
		B.2.1 Commercial CFD Codes
		B.2.2 Free/public domain/shareware CFD Codes
Appendix C: Programs in the \'C\' Language
	C.l Program 3.1: ADI.C
	C.2 Program 4.1: LXMC.C
	C.3 Program 5.1: SOR.C
	C.4 Program 5.2: AFI.C
	C.5 Program 5.3: MGC.C
	C.6 Program 6.1: TSP.C
Appendix D: Answers and Hints to Solutions
	D.1 Chapter 2
	D.2 Chapter 3
	D.3 Chapter 4
	D.4 Chapter 5
	D.5 Chapter 6
	D.6 Chapter 10
	D.7 Chapter 12
Bibliography
Index