Fundamentals of Compressible Fluid Mechanics Genick Bar–Meir, Ph. D. 1107 16th Ave S. E. Minneapolis, MN 55414-2411 email:[email protected] Copyright © 2006, 2005, and 2004 by Genick Bar-Meir See the file copying.fdl or copyright.tex for copying conditions. Version (0.4.4.2 aka 0.4.4.1j May 21, 2007) `We are like dwarfs sitting on the shoulders of giants” from The Metalogicon by John in 1159 CONTENTS GNU Free Documentation License . xvii 1. APPLICABILITY AND DEFINITIONS . xviii 2. VERBATIM COPYING . xix 3. COPYING IN QUANTITY . xix 4. MODIFICATIONS . xx 5. COMBINING DOCUMENTS . xxii 6. COLLECTIONS OF DOCUMENTS . xxii 7. AGGREGATION WITH INDEPENDENT WORKS . xxiii 8. TRANSLATION . xxiii 9. TERMINATION . xxiii 10. FUTURE REVISIONS OF THIS LICENSE . xxiii ADDENDUM: How to use this License for your documents . xxiv Potto Project License . xxv How to contribute to this book . xxvii Credits . xxvii John Martones . xxvii Grigory Toker . xxviii Ralph Menikoff . xxviii Your name here . xxviii Typo corrections and other ”minor” contributions . xxviii Version 0.4.3 Sep. 15, 2006 . xxxv Version 0.4.2 . xxxv Version 0.4 . xxxvi Version 0.3 . xxxvi Version 4.3 . xli Version 4.1.7 . xlii Speed of Sound . xlvi iii iv CONTENTS Stagnation effects . xlvi Nozzle . xlvi Normal Shock . xlvi Isothermal Flow . xlvi Fanno Flow . xlvii Rayleigh Flow . xlvii Evacuation and filling semi rigid Chambers . xlvii Evacuating and filling chambers under external forces . xlvii Oblique Shock . xlvii Prandtl–Meyer . xlvii Transient problem . xlvii 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 4.2.1 The Properties in the Adiabatic Nozzle . 46 4.2.2 Isentropic Flow Examples . 50 CONTENTS v 4.2.3 Mass Flow Rate (Number) . 53 4.3 Isentropic Tables . 62 4.3.1 Isentropic Isothermal Flow Nozzle . 63 4.3.2 General Relationship . 63 4.4 The Impulse Function . 70 4.4.1 Impulse in Isentropic Adiabatic Nozzle . 70 4.4.2 The Impulse Function in Isothermal Nozzle . 73 4.5 Isothermal Table . 73 4.6 The effects of Real Gases . 74 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 5.3 The Moving Shocks . 91 5.3.1 Shock Result from a Sudden and Complete Stop . 94 5.3.2 Moving Shock into Stationary Medium (Suddenly Open Valve) 96 5.3.3 Partially Open Valve . 101 5.3.4 Partially Closed Valve . 103 5.3.5 Worked–out Examples for Shock Dynamics . 104 5.4 Shock Tube . 109 5.5 Shock with Real Gases . 113 5.6 Shock in Wet Steam . 113 5.7 Normal Shock in Ducts . 113 5.8 More Examples for Moving Shocks . 114 5.9 Tables of Normal Shocks, k = 1:4 Ideal Gas . 115 6 Normal Shock in Variable Duct Areas 123 6.1 Nozzle efficiency . 129 6.2 Diffuser Efficiency . 129 7 Nozzle Flow With External Forces 135 7.1 Isentropic Nozzle (Q = 0) . 136 7.2 Isothermal Nozzle (T = constant) . 136 8 Isothermal Flow 137 8.1 The Control Volume Analysis/Governing equations . 138 8.2 Dimensionless Representation . 138 8.3 The Entrance Limitation of Supersonic Branch . 142 8.4 Comparison with Incompressible Flow . 143 vi CONTENTS 8.5 Supersonic Branch . 145 8.6 Figures and Tables . 146 8.7 Isothermal Flow Examples . 147 8.8 Unchoked situation . 152 9 Fanno Flow 155 9.1 Introduction . 155 9.2 Model . 156 9.3 Non–dimensionalization of the equations . 157 9.4 The Mechanics and Why the Flow is Choked? . 160 9.5 The working equations . 161 9.6 Examples of Fanno Flow . 164 9.7 Supersonic Branch . 169 9.8 Maximum length for the supersonic flow . 169 9.9 Working Conditions . 170 4fL 9.9.1 Variations of The Tube Length ( D ) Effects . 171 9.9.2 The Pressure Ratio, P2 , effects . 176 P1 9.9.3 Entrance Mach number, M1, effects . 178 9.10 The Approximation of the Fanno flow by Isothermal Flow . 185 9.11 More Examples of Fanno Flow . 186 9.12 The Table for Fanno Flow . 187 10 RAYLEIGH FLOW 189 10.1 Introduction . 189 10.2 Governing Equation . 190 10.3 Rayleigh Flow Tables . 193 10.4 Examples For Rayleigh Flow . ..