An Overview of Undergrad Fluids Notes developed for MAE 222 Written and compiled by: Clayton P. Byers Princeton University, 2016 Foreword These notes are a compilation and edit of my precept notes I developed for MAE 222 in Spring 2016 at Princeton University. The goal here was to combine them into a sort of study guide that can be used by future students. They assume the reader is either famil- iar with, or concurrently studying, basic fluids mechanics. These notes will not stand alone as a way to learn fluids from scratch! Some sections go into fine detail about derivations, which aren't necessary for the basic undergrad education. Other sections as- sume you have some base level of knowledge, and may be complicated if you haven't seen the material before. They will serve well for an undergrad reviewing the material for your final, or even for a graduate student in the early preparation for your general examination. The flow and order of the material follows the general outline in which it was presented in the course. In general, each section should have a number of fully worked examples to demonstrate the core concepts. I go through each step in the math and logic to help develop your understanding of the application of the material. I have tried to edit the content so that it is consistent in the internal equation and figure references, but there might be an equation or two that accidentally got cross-referenced. If you see any mistakes in notation (or more egregious errors in the content!) please let me know: claytonb (at) princeton (dot) edu. This material was inspired by Prof. Marcus Hultmark's class lectures, Prof. Alexander Smits' book \A Physical Introduction to Fluid Mechanics," and the book \Engineering Fluid Mechanics" by Crowe, Elger, Roberson, and Williams. You'll find similarities to those texts or classes for good reason, but my goal was to present them in a new light or with a different approach. I hope you find this useful and enlightening! Intro to Fluids Notes 1 C. P. Byers 2016 Contents 1 Introductory Concepts5 1.1 Dimensional homogeneity . .5 1.2 The unique characteristics of pressure in fluids . .5 1.3 Example: breaking a barrel . .7 1.4 Example: an airplane door . .8 1.5 Statics and moments . 10 1.5.1 Example: a dam wall with one fluid . 10 1.5.2 Example: a dam wall with two fluids . 13 1.6 Rigid body motion summary . 15 1.7 Specific weight and specific gravity . 18 1.8 Buoyancy . 18 1.8.1 Example: floating some aluminum . 19 1.8.2 Example: floating in a different fluid . 21 1.9 Control Volumes . 22 1.9.1 Example: accumulating mass . 22 1.9.2 Example: non-uniform flow . 23 1.10 Material Derivative . 25 2 Continuity and Momentum 27 2.1 Continuity equation . 27 2.1.1 Differential form of the continuity equation (bonus material) . 29 2.1.2 Example: continuity on a nozzle . 30 2.2 Momentum Balance . 32 2.2.1 Example: force on a nozzle . 32 2.2.2 Example: Balance with angled velocities . 34 3 The Momentum Equation 38 3.1 Integral & Differential momentum equation . 38 3.1.1 Example: momentum practice . 41 3.1.2 Example: differential momentum practice . 45 3.2 Bernoulli's equation . 46 3.2.1 Rigorous derivation of Bernoulli's equation (bonus material) . 47 3.2.2 Pressure and streamlines . 49 3.3 Example: flow over a bump . 49 Intro to Fluids Notes 2 C. P. Byers 2016 CONTENTS 4 Introducing Potential Flow 53 4.1 Some flow definitions and characteristics . 53 4.1.1 Vorticity . 53 4.1.2 Velocity Potential . 54 4.1.3 Stream functions . 55 4.2 Example: a potential vortex . 56 4.3 The power of the velocity potential . 59 4.4 Example: superposition . 60 5 Non-Dimensionalization 63 5.1 Non-dimensionalization . 63 5.1.1 Similarity and non-dimensional parameters . 63 5.1.2 Non-dimensional equations . 66 5.1.3 Example: non-dimensionalizing the momentum equation . 66 5.1.4 Example: flow in a channel . 68 5.2 Prandtl's Boundary Layer . 69 5.2.1 x-momentum equation . 69 5.2.2 y-momentum equation . 72 5.2.3 Combining the x- and y-momentum equations . 73 5.2.4 Bonus: similarity solution . 74 5.3 Bonus: Energy Equation . 75 6 Analysis of the Boundary Layer 77 6.1 Summary of the boundary layer . 77 6.2 Zero pressure gradient . 78 6.3 Displacement thickness . 78 6.3.1 Displacement thickness and the zero pressure gradient assumption . 80 6.4 Momentum thickness . 81 6.4.1 Momentum thickness and drag . 84 6.5 Example: approximate boundary layer . 84 6.6 Example: finding the skin friction . 85 7 Drag, Separation, and Shedding 89 7.1 Drag force from friction . 89 7.1.1 Example: drag on a plate . 90 7.2 Separation of the boundary layer . 91 7.2.1 Favorable pressure gradient . 93 7.2.2 Zero pressure gradient . 93 7.2.3 Adverse pressure gradient . 94 7.3 Resistance to separation . 94 7.4 Case: a curved surface . 94 7.5 Strouhal number . 95 7.5.1 Example: Tacoma Narrows Bridge . 96 7.5.2 Example: car antenna . 98 Intro to Fluids Notes 3 C. P. Byers 2016 CONTENTS 8 Internal Flows and Losses 99 8.1 Internal flows . 99 8.1.1 Continuity . 100 8.1.2 Inertial terms . 101 8.1.3 The x-momentum equation for a channel (and pipe) . 101 8.1.4 The pressure gradient . 102 8.2 The velocity profile in a pipe . 103 8.3 The friction factor in a laminar pipe . 105 8.4 Example: a vertical pipe . 106 8.5 Example: force to hold a pipe . 109 8.6 Losses in pipe flow . 110 8.7 Example: flow from a reservoir . 113 9 Supercritical and Supersonic Flows 115 9.1 Comparison of Mach number and Froude number . 115 9.2 Hydraulic jumps . 116 9.2.1 Example: hydraulic jumps on a dam . 118 9.2.2 Example: jump in a channel . 118 9.3 Compressible flow relations . 120 9.3.1 Example: space shuttle re-entry . 121 9.4 Normal shocks . 122 9.4.1 Example: revisiting the shuttle . 123 Intro to Fluids Notes 4 C. P. Byers 2016 Chapter 1 Introductory Concepts 1.1 Dimensional homogeneity This concept is one of the most important concepts in science and engineering, yet is often glazed over and not actively discussed as early as (I think) it should be. Whenever we write an equation, we aren't just looking to plug in number, but are also dealing with dimensions or quantities of some sort. Something as simple as Newton's Second Law, P F~ = m~a has no meaning or usefulness unless we use the proper quantities and dimensions. m ![mass] a fundamental dimension length a ! [acceleration] time2 mass * length F ![force] time2 From this information, we can see that both sides of the equation balance in terms of fundamental dimensions - we have [mass ∗ length]=time2 on both sides. However, we must also make sure we use consistent units. It doesn't make sense to have meters on one side and feet on the other. All equations we work with (which are based on some physical interaction in the world around us) will be dimensionally homogeneous. This seems simplistic, but keep it in mind and always take a look at it to make sure you're approaching your equation correctly. In fluids, we can sometimes get information in units like psi but need P a. The easiest way to make sure you're doing things right is to write down the units when doing math. Not only will it help you keep track of what you're doing, but it may clarify where a mistake could be. 1.2 The unique characteristics of pressure in fluids We like to make analogies of pressure with weight balancing on an area. However, this can be misleading, and requires a little more thought to grasp how pressure really works. Intro to Fluids Notes 5 C. P. Byers 2016 CHAPTER 1. INTRODUCTORY CONCEPTS Remember that we classify pressure as a \scalar," which means it has a value, but no direction. This is analogous to temperature - does it have a direction? Heat transfer does, but temperature itself does not. It simply has a value at a location. Pressure is the same, while the force we feel (or calculate) due to pressure isn't prescribed by the pressure field, but instead by the physical boundaries in which pressure.