Elements of Computational Fluid Dynamics
M. Ramakrishna
Department of Aerospace Engineering
Indian Institute of Technology Madras
Chennai 600 036 India
A Golden Jubilee Publication
,Published 2011
Elements of Computational Fluid Dynamics
Ramakrishna Mokkapati, PhD
Department of Aerospace Engineering
Indian Institute of Technology Madras
Chennai 600 036
ISBN 978-93-80689-05-0
Revision Number: PRN20160217
c 2011 by Ramakrishna Mokkapati,
Department of Aerospace Engineering
Indian Institute of Technology Madras
Chennai 600 036
, To
everyone who has taught me
, Contents
Preface 6
Chapter 1. Introduction 10
1.1. What is Computational Fluid Dynamics? 10
1.2. Modelling the Universe 11
1.3. How do we develop models? 13
1.4. Modelling on the Computer 18
1.5. Important ideas from this chapter 24
Chapter 2. Representations on the Computer 25
2.1. Representing Numbers on the Computer 25
2.2. Representing Matrices and Arrays on the Computer 31
2.3. Representing Intervals and Functions on the Computer 34
2.4. Functions as a Basis: Box Functions 40
2.5. Linear Approximations: Hat Functions 48
2.6. Higher Order Approximations 59
2.7. Local Error Estimates of Approximations 64
2.8. Representing Derivatives - Finite Differences 67
2.9. Differential Equations 77
2.10. Grid Generation I 78
2.11. Important ideas from this chapter 80
Chapter 3. Simple Problems 81
3.1. Laplace’s Equation 81
3.2. Convergence of Iterative Schemes 92
3.3. Properties Of Solutions To Laplace’s Equation 97
3.4. Accelerating Convergence 98
3.5. Neumann Boundary Conditions 105
3.6. First Order Wave Equation 106
3.7. Numerical Solution to Wave Equation: Stability Analysis 119
3.8. Numerical Solution to Wave Equation:Consistency 125
3.9. Numerical Solution to Wave Equation:Dissipation, Dispersion 128
3.10. Solution to Heat Equation 137
3.11. A Sampling of Techniques 141
3.12. Boundary Conditions 145
3.13. A Generalised First Order Wave Equation 148
3.14. The “Delta” form 152
3.15. The One-Dimensional Second Order Wave Equation 156
3.16. Important ideas from this chapter 158
3
M. Ramakrishna
Department of Aerospace Engineering
Indian Institute of Technology Madras
Chennai 600 036 India
A Golden Jubilee Publication
,Published 2011
Elements of Computational Fluid Dynamics
Ramakrishna Mokkapati, PhD
Department of Aerospace Engineering
Indian Institute of Technology Madras
Chennai 600 036
ISBN 978-93-80689-05-0
Revision Number: PRN20160217
c 2011 by Ramakrishna Mokkapati,
Department of Aerospace Engineering
Indian Institute of Technology Madras
Chennai 600 036
, To
everyone who has taught me
, Contents
Preface 6
Chapter 1. Introduction 10
1.1. What is Computational Fluid Dynamics? 10
1.2. Modelling the Universe 11
1.3. How do we develop models? 13
1.4. Modelling on the Computer 18
1.5. Important ideas from this chapter 24
Chapter 2. Representations on the Computer 25
2.1. Representing Numbers on the Computer 25
2.2. Representing Matrices and Arrays on the Computer 31
2.3. Representing Intervals and Functions on the Computer 34
2.4. Functions as a Basis: Box Functions 40
2.5. Linear Approximations: Hat Functions 48
2.6. Higher Order Approximations 59
2.7. Local Error Estimates of Approximations 64
2.8. Representing Derivatives - Finite Differences 67
2.9. Differential Equations 77
2.10. Grid Generation I 78
2.11. Important ideas from this chapter 80
Chapter 3. Simple Problems 81
3.1. Laplace’s Equation 81
3.2. Convergence of Iterative Schemes 92
3.3. Properties Of Solutions To Laplace’s Equation 97
3.4. Accelerating Convergence 98
3.5. Neumann Boundary Conditions 105
3.6. First Order Wave Equation 106
3.7. Numerical Solution to Wave Equation: Stability Analysis 119
3.8. Numerical Solution to Wave Equation:Consistency 125
3.9. Numerical Solution to Wave Equation:Dissipation, Dispersion 128
3.10. Solution to Heat Equation 137
3.11. A Sampling of Techniques 141
3.12. Boundary Conditions 145
3.13. A Generalised First Order Wave Equation 148
3.14. The “Delta” form 152
3.15. The One-Dimensional Second Order Wave Equation 156
3.16. Important ideas from this chapter 158
3
