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Fluid Dynamics Book

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Fluid Dynamics Book Fluid dynamics An introduction to the fundamentals Brian D. Storey Olin College © 2018 Brian D. Storey All rights reserved Contents 1 Introduction page 1 1.1 Scope of the text 3 1.2 What is a fluid? 4 1.3 Simple fluid properties 4 1.4 Other properties 5 1.5 What is a continuum? 7 1.6 Mathematical prerequisites 9 2 Dimensional analysis 11 2.1 Units 11 2.2 Buckingham Pi Theorem 12 2.3 Making equations dimensionless 25 2.4 Summary 29 3 The empirical approach 31 3.1 Drag revisited 32 3.2 Drag coefficients 35 3.3 Pipe flow 38 3.4 Summary 42 4 Vector calculus notation and review 43 4.1 Scalar and vector functions 43 4.2 Gradient, divergence, and curl 45 4.3 Normal vectors and flow through a surface 47 4.4 Volume integrals 48 4.5 Surface integrals 49 4.6 Surfaces and volumes in 2D 51 4.7 Divergence theorem 52 iv Contents 4.8 Summary 53 5 Diffusive mass transfer 55 5.1 Physical picture of mass diffusion 56 5.2 Continuum concentration 59 5.3 Equation in 1D 65 5.4 Role of simulation 73 5.5 Summary 74 6 Convective mass transfer 75 6.1 Convective mass transfer 75 6.2 Material point of view - be one with the fluid 76 6.3 The material derivative 77 6.4 Liebniz and Reynold's transport theorem 81 6.5 Conservation of mass: the fluid 82 6.6 Dimensionless formulation 86 7 Conservation of momentum in a fluid 89 7.1 Transport theorem revisited 90 7.2 Linear momentum 91 7.3 F = m a 94 7.4 So what's a tensor? 101 8 The Navier Stokes equations 103 8.1 Euler's equation 104 8.2 Newtonian fluid & incompressible flow 106 8.3 Boundary conditions 109 8.4 Computing stress from flows 110 8.5 Comments on kinematics 111 8.6 Non-dimensionalization 117 8.7 The Reynolds number 118 8.8 Summary 119 9 Solutions to the Navier-Stokes equations 123 9.1 Flow between parallel plates - Poiseuille flow 124 9.2 Flow driven by a wall - Couette flow 129 9.3 Flow down a ramp 131 9.4 Combined Poiseuille and Couette flow 132 9.5 Slider bearing 134 9.6 Impulsively started Couette flow 138 9.7 Cylindrical Couette flow 140 Contents v 9.8 Poiseuille flow in a pipe 143 9.9 Comments on the stability of solutions 145 9.10 Computational Fluid Dynamics (CFD) 148 10 Inviscid flow, Euler's equation and Bernoulli 151 10.1 Flow along a streamline: Bernoulli 152 10.2 General Bernoulli 154 10.3 Euler equations across a streamline 156 10.4 Vorticity 161 10.5 Irrotational flow 167 10.6 Lift on an airfoil 169 11 Boundary layers 173 11.1 Impulsively started plate 174 11.2 Boundary layer equations and laminar solution 176 11.3 Turbulent boundary layers 182 11.4 Boundary layer separation 183 11.5 Observations of drag on a sphere 186 12 Turbulence 191 12.1 Can we simulate turbulence? 191 12.2 Kolmogorov theory and the energy cascade 192 12.3 Numerical simulation of turbulence (CFD) 195 13 Control volume analysis 199 13.1 Control volume formulation 200 13.2 Examples 202 13.3 Some difficulties 212 Bibliography 215 1 Introduction The text lays the foundations for the study of fluid dynamics. The first question a student might ask is why should we be interested in fluid dynamics? Let me address the question through some examples of applications where these topics are relevant. Biology: Our bodies are mostly water. Fluids are essential to life as we know it. The understanding of the transport of oxygen and nutrients throughout the body by fluids (at the level of lungs, heart, and blood vessels as well a the cellular one) requires a basic understanding of the topics herein. Locomotion of many species from fish to birds to small microorganisms is determined by fluid flow. Geophysical: The atmosphere and oceans are fluids. The currents, the winds, the gulf stream, and the jet stream carry energy and matter around the planet. Our basic understanding of the weather, the climate, and the environment can only begin with a basic understanding of fluid dynamics. The Earth's magnetic field is generated by convection of the mantle deep within the Earth. Energy: Most of our current energy needs are met by the combustion of fuel; the basic processes of combustion depend upon the details of the fluid mixing of fuel and air and and the resulting heat release. Advances in understanding of the details of these processes has led to a number of developments over the years which made combustion cleaner and significantly reduced the number of pollutants emitted from cars. Many of the renewable sources of energy such as wind, geothermal, ocean waves and tidal seems obvious to state that fluid flow plays a prominent role. It is fair to claim that an understanding of all energy systems must begin with the basic understanding of topics in this text. 2 Introduction Climate The computer models used by climate scientists begin with the basic equations we will develop here. Since we can only run a sin- gle irreversible real-time experiment on our planet's climate, computer models are needed to understand what is likely to happen to the cli- mate in the coming centuries. One cannot appreciate the complexity, accuracy, and shortcomings of these computer models without under- standing the basics of fluid transport. Infrastructure and transportation: Fluid dynamics also plays a role in our basic civil infrastructure; water delivery to our homes, water resources management, sewage, building heating and cooling. Losses due to aerodynamic drag on boats, cars, and planes set the efficiency and energy use of most transportation systems. Engineering: Many fields of engineering need some understanding of flow transfer. Aerospace engineers need to understand fluid flow to get planes to fly. Automotive engineers need to understand the pro- cesses of combustion, the lift and drag forces on the vehicle, as well as how to keep the engine cool. Civil engineers need to understand water management in cities. Computer engineers need to understand that dissipation of heat is one of the main limitations in making faster computer chips. As argued before, our bodies are mostly water, thus it seems obvious that bio-engineers need to understand fluids as well. Astronomy: Despite the extraordinarily large scale, the formation of galaxies, planets, and stars can be studied with the equations of fluid dynamics. We cannot test experimental galaxies in a lab, thus fluid dynamics simulations is an integral part of answering questions about the formation of objects in our universe. Mathematics: Fluid dynamics has also provided a rich field of study for mathematicians. The basic equations of fluid dynamics, the Navier Stokes equations, are notoriously difficult to solve. If you can provide a general solution, you can win a million dollars (see the Millennium Prize problems sponsored by the Clay Mathematics Institute). There are many cases where in an attempt to solve a problem in fluid dynamics has lead to new and powerful mathematical techniques. Computational fluid dynamics, solving the Navier Stokes equations numerically, has been one of the most powerful drivers in developing supercomputers and other advances in high performance computing. In short, while my viewpoint is biased, the foundations in this book are critical to essentially all scientists and engineers. 1.1 Scope of the text 3 1.1 Scope of the text Now that I have built up the number of applications this field is rele- vant to, we must scale things back. A study this comprehensive would overwhelm most mortals over their lifetime. We only begin our study of these topics and this course will provide only a basic background. This text will mostly provide the theoretical and mathematical foun- dations. The book is not meant to stand-alone but should be part of course which also involves a balance of working examples, conducting some simple hands-on experiments, and learning how computer pack- ages can help solve some of these problems. It is expected that this text will provide the (somewhat) rigorous background and derivations. You should read and work through many of these results yourself. Even though the text is not too long, many of the chapters are very dense in terms of content. In my course we will review some of these high points in class, but I will leave it to you to really read through the details. The details can sometimes be tedious, and until you have some working knowledge (perhaps by the end of the course), it is hard to see why one would care about them. These notes should hopefully provide you a view that the mathematical foundations for the field are relatively rigorous. In many cases, we can write the equations down in a few short lines and we could in theory solve any problem. We will soon see that matters are not so simple and that while the equations are known, their solution can be extraordinarily challenging. Finally, there are a number of good books on these topics. I will never provide some insight that was not thought of before. My treatment will rely on a number of excellent books through which I learned the topic. So why bother to write my own notes? The basic answer is that professors can't help themselves. The way everyone else explains stuff is never quite the way you want it. However, the excellent books that I use the most and those who influenced my view are, • Fluid Mechanics by Kundu and Cohen (2004).
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