Gas Dynamics - Fundamentals of Compressible Fluid Mechanics...

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Fundamentals of Compressible Fluid Mechanics Genick Bar–Meir, Ph. D. 1107 16 th Ave S. E. Minneapolis, MN 55414-2411 email: “[email protected] Copyright © 2007, 2006, 2005, and 2004 by Genick Bar-Meir See the file copying.fdl or copyright.tex for copying conditions. Version (0.4.8.2 February 5, 2008)
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‘We are like dwarfs sitting on the shoulders of giants” from The Metalogicon by John in 1159
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CONTENTS GNU Free Documentation License . . . . . . . . . . . . . . . . . . . . . . xv 1. APPLICABILITY AND DEFINITIONS . . . . . . . . . . . . . . . . xvi 2. VERBATIM COPYING . . . . . . . . . . . . . . . . . . . . . . . . xvii 3. COPYING IN QUANTITY . . . . . . . . . . . . . . . . . . . . . . . xviii 4. MODIFICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 5. COMBINING DOCUMENTS . . . . . . . . . . . . . . . . . . . . . xx 6. COLLECTIONS OF DOCUMENTS . . . . . . . . . . . . . . . . . xxi 7. AGGREGATION WITH INDEPENDENT WORKS . . . . . . . . . xxi 8. TRANSLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi 9. TERMINATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi 10. FUTURE REVISIONS OF THIS LICENSE . . . . . . . . . . . . . xxii ADDENDUM: How to use this License for your documents . . . . . . xxii How to contribute to this book . . . . . . . . . . . . . . . . . . . . . . . . xxiii Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii John Martones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii Grigory Toker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv Ralph Menikoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv Domitien Rataaforret . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv Your name here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv Typo corrections and other ”minor” contributions . . . . . . . . . . . xxiv Version 0.4.8 Jan. 23, 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi Version 0.4.3 Sep. 15, 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi Version 0.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxii Version 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii Version 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii Version 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix Version 0.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl iii
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iv CONTENTS Version 0.4.1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliv Stagnation effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliv Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliv Normal Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliv Isothermal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliv Fanno Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv Rayleigh Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv Evacuation and filling semi rigid Chambers . . . . . . . . . . . . . . xlv Evacuating and filling chambers under external forces . . . . . . . . xlv Oblique Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv Prandtl–Meyer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv Transient problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv 1 Introduction 1 1.1 What is Compressible Flow ? . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Why Compressible Flow is Important? . . . . . . . . . . . . . . . . . 2 1.3 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3.1 Early Developments . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.2 The shock wave puzzle . . . . . . . . . . . . . . . . . . . . . 5 1.3.3 Choking Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.4 External flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.5 Filling and Evacuating Gaseous Chambers . . . . . . . . . . 15 1.3.6 Biographies of Major Figures . . . . . . . . . . . . . . . . . . 15 2 Fundamentals of Basic Fluid Mechanics 25 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Control Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Reynold’s Transport Theorem . . . . . . . . . . . . . . . . . . . . . . 25 3 Speed of Sound 27 3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 Speed of sound in ideal and perfect gases . . . . . . . . . . . . . . . 29 3.4 Speed of Sound in Real Gas . . . . . . . . . . . . . . . . . . . . . . 31 3.5 Speed of Sound in Almost Incompressible Liquid . . . . . . . . . . . 35 3.6 Speed of Sound in Solids . . . . . . . . . . . . . . . . . . . . . . . . 36 3.7 Sound Speed in Two Phase Medium . . . . . . . . . . . . . . . . . . 37 4 Isentropic Flow 41 4.1 Stagnation State for Ideal Gas Model . . . . . . . . . . . . . . . . . . 41 4.1.1 General Relationship . . . . . . . . . . . . . . . . . . . . . . . 41 4.1.2 Relationships for Small Mach Number . . . . . . . . . . . . . 44 4.2 Isentropic Converging-Diverging Flow in Cross Section . . . . . . . . 45
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CONTENTS v 4.2.1 The Properties in the Adiabatic Nozzle . . . . . . . . . . . . . 46 4.2.2 Isentropic Flow Examples . . . . . . . . . . . . . . . . . . . . 50 4.2.3 Mass Flow Rate (Number) . . . . . . . . . . . . . . . . . . . 53 4.3 Isentropic Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3.1 Isentropic Isothermal Flow Nozzle . . . . . . . . . . . . . . . 64 4.3.2 General Relationship . . . . . . . . . . . . . . . . . . . . . . . 64 4.4 The Impulse Function . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4.1 Impulse in Isentropic Adiabatic Nozzle . . . . . . . . . . . . 71 4.4.2 The Impulse Function in Isothermal Nozzle . . . . . . . . . . 73 4.5 Isothermal Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.6 The effects of Real Gases . . . . . . . . . . . . . . . . . . . . . . . . 75 5 Normal Shock 81 5.1 Solution of the Governing Equations . . . . . . . . . . . . . . . . . . 84 5.1.1 Informal Model . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.1.2 Formal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.1.3 Prandtl’s Condition . . . . . . . . . . . . . . . . . . . . . . . . 88 5.2 Operating Equations and Analysis . . . . . . . . . . . . . . . . . . . 89 5.2.1 The Limitations of the Shock Wave . . . . . . . . . . . . . . . 90 5.2.2 Small Perturbation Solution . . . . . . . . . . . . . . . . . . . 90 5.2.3 Shock Thickness . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2.4 Shock or Wave Drag . . . . . . . . . . . . . . . . . . . . . . . 91 5.3 The Moving Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3.1 Shock or Wave Drag Result from a Moving Shock . . . . . . 95 5.3.2
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