A First Course in Computational Fluid Dynamics (Cambridge Texts in Applied Mathematics)

This book has been in preparation for over a decade. Hassan Aref and I had been making substantial additions and revisions each year, in our desire to reach the perfect book for a first course in Computational Fluid Dynamics (CFD). I sincerely wish that we had completed the book a few years ago, so that Hassan was there when the book was published. Unfortunately this was not the case. September 9th 2011 was a tragic day for fluid mechanics. We lost an intellectual leader, a fearless pioneer and for me an inspirational mentor. It is quite amaz- ing how my academic career intersected with Hassan’s over the years. He was my Masters Thesis advisor at Brown University. But when Hassan left for the University of California at San Diego, I decided to stay and finish my PhD at Brown. A few years later, when I was an assistant professor at the Theoretical and Applied Mechanics Department at the University of Illinois at Urbana– Champaign, he was appointed as the head of the department. This is when he asked me to join him in this project of writing a non-traditional introductory book on CFD. The project was delayed when Hassan moved to Virginia Tech as the Dean and I moved to the University of Florida as the Chair of the Me- chanical and Aerospace Engineering Department. His passing away a few years ago made me all the more determined to finish the book as a way to honor his memory. I am very glad that the book is now finished and can be a monument to his far-sighted vision. Decades ago when CFD was in its infancy he foresaw how powerful computers and numerical methods would dominate the field of fluid mechanics. Contents......Page 5 Preface......Page 8 1.1 The Nature of CFD......Page 12 1.2 Overview of the Book......Page 14 1.3 Algorithm, Numerical Method, Implementation and Simulation......Page 16 1.4 Models and Methods......Page 20 1.5 Round-off Error......Page 26 1.6 A Hierarchy of Computation......Page 29 1.7 Ramification, Turbulence and the Complexity of Fluid Flows......Page 37 1.8 The Development of Computer Hardware......Page 42 1.9 Some Remarks on Software......Page 44 2.1 Numerical Methods as Mappings......Page 46 2.2 Fixed Points: Stability, Instability and Superstability......Page 48 2.3 Stability of Exact Solution, Implicit and Explicit Approximations......Page 51 2.4 More on Mappings......Page 54 2.5 Random Number Generation......Page 61 2.6 Newton’s Method in the Complex Plane......Page 63 2.7 Mappings and Fluid Flows......Page 67 3.1 Some Conventional Wisdom......Page 80 3.2 Explicit Euler and Implicit Euler Schemes......Page 82 3.3 Runge–Kutta Methods......Page 88 3.4 Adams–Bashforth–Moulton Methods......Page 95 3.5 Other Methods and Considerations......Page 101 3.6 Bashforth’s Problem: Sessile Drop on a Flat Plate......Page 103 3.7 Flow Due to a Collapsing Bubble......Page 106 3.8 Motion of a Solid in Ideal Fluid......Page 114 3.9 The Point-Vortex Equations......Page 117 3.10 Vortex Sheet Roll-up......Page 122 4 Spatial Discretization......Page 129 4.1 Forward, Backward and Central Difference......Page 130 4.2 Matrix Derivative Operators......Page 138 4.3 Compact Differences......Page 149 4.4 Non-uniform Discretization......Page 152 4.5 Numerical Interpolation......Page 155 4.6 Numerical Integration......Page 160 5 Boundary Value and Eigenvalue ODEs......Page 177 5.1 Linear Boundary Value Problems......Page 178 5.2 Nonlinear Boundary Value Problems......Page 186 5.3 Boundary Value Problems in Viscous Flow......Page 195 5.4 Eigenvalue Problems......Page 199 5.5 Hydrodynamic Instability......Page 203 6.1 Introduction......Page 221 6.2 Fourier Approximation......Page 222 6.3 Polynomial Approximation......Page 229 6.4 Galerkin, Tau, Collocation and Pseudo-spectral Methods......Page 237 6.5 Some Examples......Page 251 7.1 Definitions and Preliminaries......Page 255 7.2 The Advection Equation......Page 261 7.3 The Diffusion Equation......Page 270 7.4 The Advection–Diffusion Equation......Page 273 7.5 Godunov’s Theorem......Page 278 7.6 More on Stability: Non-periodic Boundary Conditions......Page 281 7.7 Burgers’ Equation......Page 290 7.8 Implicit Time-differencing......Page 294 7.9 Direct Solution with Matrix Representation......Page 298 8.1 Multi-dimensions......Page 304 8.2 Navier–Stokes Equations......Page 320 8.3 Navier–Stokes Equations in Spectral Form......Page 339 8.4 Finite Volume Formulation......Page 345 8.5 CFD for Complex Geometries......Page 348 8.6 Sharp Interface Cartesian Grid Method......Page 367 8.7 Immersed Boundary Method......Page 371 References......Page 386 Index......Page 401 Fluid mechanics is a branch of classical physics that has a rich tradition in applied mathematics and numerical methods. It is at work virtually everywhere, from nature to technology. This broad and fundamental coverage of computational fluid dynamics (CFD) begins with a presentation of basic numerical methods and flows into a rigorous introduction to the subject. A heavy emphasis is placed on the exploration of fluid mechanical physics through CFD, making this book an ideal text for any new course that simultaneously covers intermediate fluid mechanics and computation. Ample examples, problems and computer exercises are provided to allow students to test their understanding of a variety of numerical methods for solving flow physics problems, including the point-vortex method, numerical methods for hydrodynamic stability analysis, spectral methods and traditional CFD topics. A broad and fundamental coverage of computational fluid dynamics (CFD), explaining basic numerical methods and offering a rigorous introduction to CFD. This book emphasises the exploration of fluid science through computation and is an ideal text for any course in this area.