Fundamentals of Compressible Fluid Mechanics

Fundamentals of Compressible Fluid Mechanics Genick Bar Meir, Ph.. 1107 16 th Ave S. E. Minneapolis, MN 55414-411 email:barmeir@gmail.com Copyright 006, 005, and 004 by Genick Bar-Meir See the file copying.fdl or copyright.tex for copying conditions. Version (0.4.4. aka 0.4.4.1j May 1, 007)

We are like dwarfs sitting on the shoulders of giants from The Metalogicon by John in 1159

CONTENTS GNU Free ocumentation License...................... xvii 1. APPLICABILITY AN EFINITIONS................ xviii. VERBATIM COPYING........................ xix 3. COPYING IN QUANTITY....................... xix 4. MOIFICATIONS........................... xx 5. COMBINING OCUMENTS..................... xxii 6. COLLECTIONS OF OCUMENTS................. xxii 7. AGGREGATION WITH INEPENENT WORKS......... xxiii 8. TRANSLATION............................ xxiii 9. TERMINATION............................ xxiii 10. FUTURE REVISIONS OF THIS LICENSE............. xxiii AENUM: 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, 006.......................... xxxv Version 0.4................................... 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. Why Compressible Flow is Important?................. 1.3 Historical Background.......................... 1.3.1 Early evelopments....................... 4 1.3. 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 Fundamentals of Basic Fluid Mechanics 5.1 Introduction................................ 5. Fluid Properties.............................. 5.3 Control Volume.............................. 5.4 Reynold s Transport Theorem...................... 5 3 Speed of Sound 7 3.1 Motivation................................. 7 3. Introduction................................ 7 3.3 Speed of sound in ideal and perfect gases............... 9 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. Relationships for Small Mach Number............. 44 4. Isentropic Converging-iverging Flow in Cross Section........ 45 4..1 The Properties in the Adiabatic Nozzle............. 46 4.. Isentropic Flow Examples.................... 50

CONTENTS v 4..3 Mass Flow Rate (Number)................... 53 4.3 Isentropic Tables............................. 6 4.3.1 Isentropic Isothermal Flow Nozzle............... 63 4.3. General Relationship....................... 63 4.4 The Impulse Function.......................... 70 4.4.1 Impulse in Isentropic Adiabatic Nozzle............ 70 4.4. 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. Formal Model........................... 84 5.1.3 Prandtl s Condition........................ 88 5. Operating Equations and Analysis................... 89 5..1 The Limitations of the Shock Wave............... 90 5.. Small Perturbation Solution................... 90 5..3 Shock Thickness......................... 91 5.3 The Moving Shocks........................... 91 5.3.1 Shock Result from a Sudden and Complete Stop....... 94 5.3. 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 ynamics.......... 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 ucts.......................... 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 uct Areas 13 6.1 Nozzle efficiency............................. 19 6. iffuser Efficiency............................ 19 7 Nozzle Flow With External Forces 135 7.1 Isentropic Nozzle (Q = 0)........................ 136 7. Isothermal Nozzle (T = constant)................... 136 8 Isothermal Flow 137 8.1 The Control Volume Analysis/Governing equations.......... 138 8. imensionless Representation..................... 138 8.3 The Entrance Limitation of Supersonic Branch............ 14 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............................ 15 9 Fanno Flow 155 9.1 Introduction................................ 155 9. 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 9.9.1 Variations of The Tube Length ( 4fL ) Effects.......... 171 9.9. The Pressure Ratio, P P 1, effects................. 176 9.9.3 Entrance Mach number, M 1, effects.............. 178 9.10 The Approximation of the Fanno flow by Isothermal Flow....... 185 9.11 More Examples of Fanno Flow..................... 186 9.1 The Table for Fanno Flow........................ 187 10 RAYLEIGH FLOW 189 10.1 Introduction................................ 189 10. Governing Equation........................... 190 10.3 Rayleigh Flow Tables........................... 193 10.4 Examples For Rayleigh Flow...................... 196 11 Evacuating and Filling a Semi Rigid Chambers 01 11.1 Governing Equations and Assumptions................ 0 11. General Model and Non-dimensioned................. 04 11..1 Isentropic Process........................ 05 11.. Isothermal Process in The Chamber.............. 06 11..3 A Note on the Entrance Mach number............. 06 11.3 Rigid Tank with Nozzle.......................... 07 11.3.1 Adiabatic Isentropic Nozzle Attached.............. 07 11.3. Isothermal Nozzle Attached................... 09 11.4 Rapid evacuating of a rigid tank.................... 09 11.4.1 With Fanno Flow......................... 09 11.4. Filling Process.......................... 11 11.4.3 The Isothermal Process..................... 1 11.4.4 Simple Semi Rigid Chamber.................. 13 11.4.5 The Simple General Case................... 13 11.5 Advance Topics.............................. 15

CONTENTS vii 1 Evacuating/Filing Chambers under External Volume Control 17 1.1 General Model.............................. 17 1.1.1 Rapid Process.......................... 18 1.1. Examples............................. 1 1.1.3 irect Connection........................ 1 1. Summary................................. 13 Topics in Unsteady one imensional gas dynamics 5 14 Oblique-Shock 7 14.1 Preface to Oblique Shock........................ 7 14. Introduction................................ 8 14..1 Introduction to Oblique Shock.................. 8 14.. Introduction to Prandtl Meyer Function............ 8 14..3 Introduction to Zero Inclination................. 9 14.3 Oblique Shock.............................. 9 14.4 Solution of Mach Angle......................... 3 14.4.1 Upstream Mach Number, M 1, and eflection Angle, δ.... 3 14.4. When No Oblique Shock Exist or When > 0........ 35 14.4.3 Upstream Mach Number, M 1, and Shock Angle, θ...... 41 14.4.4 Given Two Angles, δ and θ................... 4 14.4.5 Flow in a Semi Shape.................... 44 14.4.6 Small δ Weak Oblique shock................. 44 14.4.7 Close and Far Views of the Oblique Shock.......... 45 14.4.8 Maximum Value of Oblique shock................ 45 14.4.9 etached shock......................... 46 14.4.10Issues Related to the Maximum eflection Angle....... 47 14.4.11Oblique Shock Examples.................... 49 14.4.1Application of Oblique Shock.................. 51 14.4.13Optimization of Suction Section esign............ 6 14.5 Summary................................. 6 14.6 Appendix: Oblique Shock Stability Analysis.............. 63 15 Prandtl-Meyer Function 65 15.1 Introduction................................ 65 15. Geometrical Explanation......................... 66 15..1 Alternative Approach to Governing Equations......... 67 15.. Comparison And Limitations between the Two Approaches. 70 15.3 The Maximum Turning Angle...................... 71 15.4 The Working Equations for the Prandtl-Meyer Function....... 71 15.5 d Alembert s Paradox.......................... 7 15.6 Flat Body with an Angle of Attack.................... 73 15.7 Examples For Prandtl Meyer Function................ 74 15.8 Combination of the Oblique Shock and Isentropic Expansion.... 76

viii CONTENTS 16 Topics in Steady state Two imensional flow 79 A Computer Program 81 A.1 About the Program............................ 81 A. Usage................................... 81 A.3 Program listings............................. 84 Index 85 Subjects index................................. 85 Authors index.................................. 87

LIST OF FIGURES 1.1 The shock as connection of Fanno and Rayleigh lines after Stodola, Steam and Gas Turbine......................... 7 1. The schematic of delavel s turbine after Stodola, Steam and Gas Turbine.................................. 9 1.3 The measured pressure in a nozzle taken from Stodola 197 Steam and Gas Turbines............................. 11 1.4 Flow rate as a function of the back pressure taken from Stodola 197 Steam and Gas Turbines........................ 1 1.5 Portrait of Galileo Galilei......................... 16 1.6 Photo of Ernest Mach.......................... 17 1.7 The photo of thebullet in a supersonic flow not taken in a wind tunnel 17 1.8 Photo of Lord Rayleigh.......................... 18 1.9 Portrait of Rankine............................ 19 1.10 The photo of Gino Fanno approximately in 1950........... 0 1.11 Photo of Prandtl............................. 1 1.1 The photo of Ernst Rudolf George Eckert with the author s family.. 3.1 A very slow moving piston in a still gas................. 8 3. Stationary sound wave and gas moves relative to the pulse..... 8 3.3 The Compressibility Chart........................ 3 4.1 Flow of a compressible substance (gas) through a converging diverging nozzle.................................... 41 4. Perfect gas flows through a tube.................... 43 4.3 The stagnation properties as a function of the Mach number, k = 1.4 44 4.4 Control volume inside a converging-diverging nozzle.......... 46 ix

x LIST OF FIGURES 4.5 The relationship between the cross section and the Mach number on the subsonic branch......................... 50 4.6 Various ratios as a function of Mach number for isothermal Nozzle. 66 4.7 The comparison of nozzle flow..................... 67 4.8 Comparison of the pressure and temperature drop as a function of the normalized length (two scales)................... 68 4.9 Schematic to explain the significances of the Impulse function.... 71 4.10 Schematic of a flow of a compressible substance (gas) thorough a converging nozzle for example (4.7).................. 7 5.1 A shock wave inside a tube, but it can also be viewed as a one dimensional shock wave......................... 81 5. The intersection of Fanno flow and Rayleigh flow produces two solutions for the shock wave......................... 83 5.3 The exit Mach number and the stagnation pressure ratio as a function of upstream Mach number...................... 87 5.4 The ratios of the static properties of the two sides of the shock.... 89 5.5 Comparison between stationary shock and moving shock in ducts. 91 5.6 Comparison between a stationary shock and a moving shock in a stationary medium in ducts........................ 94 5.7 The moving shock Mach numbers as a result of a sudden and complete stop.................................. 95 5.8 A shock moves into a still medium as a result of a sudden and complete opening of a valve......................... 96 5.9 The number of iterations to achieve convergence............ 97 5.10 The maximum of downstream Mach number as a function of the specific heat, k............................... 99 5.11 A shock moves into a moving medium as a result of a sudden and complete open valve............................ 10 5.1 The results of the partial opening of the valve.............. 103 5.13 A shock as a result of a sudden and partially a valve closing or a narrowing the passage to the flow................... 103 5.14 Schematic of a piston pushing air in a tube............... 107 5.15 Figure for Example (5.8)........................ 109 5.16 The shock tube schematic with a pressure diagram.......... 110 5.17 Figure for Example (5.10)........................ 114 5.18 The results for Example (5.10)..................... 115 6.1 The flow in the nozzle with different back pressures.......... 13 6. A nozzle with normal shock....................... 14 6.3 escription to clarify the definition of diffuser efficiency........ 130 6.4 Schematic of a supersonic tunnel in a continuous region (and also for example (6.3)............................. 130 8.1 Control volume for isothermal flow................... 137

LIST OF FIGURES xi 8. escription of the pressure, temperature relationships as a function of the Mach number for isothermal flow................ 143 8.3 The Mach number at the entrance to a tube under isothermal flow model as a function 4fL......................... 153 9.1 Control volume of the gas flow in a constant cross section...... 155 9. Various parameters in Fanno flow as a function of Mach number.. 163 9.3 Schematic of Example (9.1)....................... 164 9.4 The schematic of Example (9.).................... 166 9.5 The maximum length as a function of specific heat, k......... 170 9.6 The effects of increase of 4fL on the Fanno line........... 171 9.7 The development properties in of converging nozzle......... 17 9.8 The Mach numbers at entrance and exit of tube and mass flow rate for Fanno Flow as a function of the 4fL................ 173 9.9 M 1 as a function M for various 4fL.................. 174 9.10 M 1 as a function M for different 4fL for supersonic entrance velocity.175 9.11 The pressure distribution as a function of 4fL 4fL for a short.... 177 9.1 The pressure distribution as a function of 4fL 4fL for a long.... 178 9.13 The effects of pressure variations on Mach number profile as a function of 4fL 4fL when the total resistance = 0.3 for Fanno Flow... 179 9.14 Fanno Flow Mach number as a function of 4fL 4fL when the total = 0.3..................................... 180 9.15 Schematic of a long tube in supersonic branch........... 181 9.16 The extra tube length as a function of the shock location, 4fL supersonic branch............................... 181 9.17 The maximum entrance Mach number, M 1 to the tube as a function of 4fL supersonic branch........................ 18 9.18 The entrance Mach number as a function of dimensionless resistance and comparison with Isothermal Flow.............. 185 10.1 The control volume of Rayleigh Flow.................. 189 10. The Temperature Entropy iagram For Rayleigh Line......... 191 10.3 The basic functions of Rayleigh Flow (k=1.4)............. 195 11.1 The two different classifications of models that explain the filling or evacuating of a single chamber..................... 01 11. A schematic of two possible connections of the tube to a single chamber.................................. 0 11.3 A schematic of the control volumes used in this model........ 0 11.4 The pressure assumptions in the chamber and tube entrance.... 03 11.5 The reduced time as a function of the modified reduced pressure.. 10 11.6 The reduced time as a function of the modified reduced pressure.. 1 1.1 The control volume of the Cylinder.................. 18

xii LIST OF FIGURES 1. The pressure ratio as a function of the dimensionless time for chokeless condition............................... 3 1.3 The pressure ratio as a function of the dimensionless time for choked condition................................. 4 1.4 The pressure ratio as a function of the dimensionless time..... 4 14.1 A view of a straight normal shock as a limited case for oblique shock 7 14. The regions where oblique shock or Prandtl Meyer function exist. Notice that both have a maximum point and a no solution zone, which is around zero. However, Prandtl-Meyer function approaches closer to a zero deflection angle..................... 8 14.3 A typical oblique shock schematic................... 9 14.4 Flow around spherically blunted 30 cone-cylinder with Mach number.0. It can be noticed that the normal shock, the strong shock, and the weak shock coexist....................... 35 14.5 The view of a large inclination angle from different points in the fluid field..................................... 36 14.6 The various coefficients of three different Mach numbers to demonstrate that is zero........................... 39 14.7 The Mach waves that are supposed to be generated at zero inclination.40 14.8 The calculation of (possible error), shock angle, and exit Mach number for M 1 = 3............................ 41 14.9 The possible range of solutions for different parameters for given upstream Mach numbers........................ 43 14.10Schematic of finite wedge with zero angle of attack.......... 44 14.11A local and a far view of the oblique shock............... 45 14.1The schematic for a round tip bullet in a supersonic flow...... 47 14.13The schematic for a symmetrical suction section with Mach reflection 48 14.14The detached shock in a complicated configuration sometimes referred to as Mach reflection....................... 48 14.15Oblique shock occurs around a cone. This photo is courtesy of r. Grigory Toker, a Research Professor at Cuernavaco University of Mexico. According to his measurement, the cone half angle is 15 and the Mach number is........................ 50 14.16Maximum values of the properties in an oblique shock........ 51 14.17Two variations of inlet suction for supersonic flow........... 51 14.18Schematic for Example (14.4)..................... 5 14.19Schematic for Example (14.5)...................... 53 14.0Schematic of two angles turn with two weak shocks......... 54 14.1Typical examples of unstable and stable situations.......... 63 14.The schematic of stability analysis for oblique shock.......... 64 15.1 The definition of the angle for the Prandtl Meyer function....... 65 15. The angles of the Mach line triangle.................. 65 15.3 The schematic of the turning flow................... 66

LIST OF FIGURES xiii 15.4 The schematic of the coordinate based on the mathematical description.................................... 67 15.5 Expansion of Prandtl-Meyer function when it exceeds the maximum angle................................... 7 15.7 A simplified diamond shape to illustrate the supersonic d Alembert s Paradox.................................. 7 15.6 The angle as a function of the Mach number............. 73 15.8 The definition of the angle for the Prandtl Meyer function....... 73 15.9 The schematic of Example 15.1..................... 74 15.10The schematic for the reversed question of example (15.)..... 75 A.1 Schematic diagram that explains the structure of the program.... 8

xiv LIST OF FIGURES

LIST OF TABLES 3.1 Water speed of sound from different sources............. 35 3. Liquids speed of sound, after Aldred, John, Manual of Sound Recording, London: Fountain Press, 197................... 36 3.3 Solids speed of sound, after Aldred, John, Manual of Sound Recording, London:Fountain Press, 197................... 37 4.1 Fliegner s number and other paramters as function of Mach number 58 4.1 Fliegner s number and other paramters as function of Mach number (continue)................................. 59 4.1 Fliegner s number and other paramters as function of Mach number (continue)................................. 60 4. Isentropic Table k = 1.4......................... 6 4. Isentropic Table k=1.4 (continue).................... 63 4.3 Isothermal Table............................ 73 4.3 Isothermal Table (continue)....................... 74 5.1 The shock wave table for k = 1.4.................... 115 5.1 The shock wave table for k = 1.4 (continue).............. 116 5.1 The shock wave table for k = 1.4 (continue).............. 117 5. Table for a Reflective Shock from a suddenly closed end (k=1.4).. 117 5. Table for Reflective Shock from suddenly closed valve (end) (k=1.4)(continue)118 5.3 Table for shock propagating from suddenly opened valve (k=1.4).. 118 5.3 Table for shock propagating from suddenly opened valve (k=1.4).. 119 5.4 Table for shock propagating from a suddenly opened valve (k=1.3) 119 5.4 Table for shock propagating from a suddenly opened valve (k=1.3) 10 5.4 Table for shock propagating from a suddenly opened valve (k=1.3) 11 xv

xvi LIST OF TABLES 8.1 The Isothermal Flow basic parameters................ 147 8.4 The flow parameters for unchoked flow................ 15 9.1 Fanno Flow Standard basic Table................... 187 9.1 Fanno Flow Standard basic Table (continue)............. 188 10.1 Rayleigh Flow k=1.4.......................... 193 10.1 Rayleigh Flow k=1.4 (continue)..................... 194 10.1 Rayleigh Flow k=1.4 (continue)..................... 195 14.1 Table of maximum values of the oblique Shock k=1.4........ 45 14.1 Maximum values of oblique shock (continue) k=1.4......... 46

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POTTO PROJECT LICENSE Potto Project License This document may be redistributed provided a pointer appears in a prominent place showing clearly where the original version was published and/or was obtained. The original version of this document may be found at http://www. potto.org/copyright.html This document is derived from open content license http://opencontent. org/opl.shtml LICENSE Terms and Conditions for Copying, istributing, and Modifying 1. isclaimer of warranty of the original author You may copy and distribute exact replicas of this document as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty of the original author; keep intact all the copyright notices that refer to this document. You may at your discretion charge a fee for the media and/or handling involved in creating a unique copy of this document. You may offer instructional support for this document and software exchange for a fee. You may at your option offer warranty in exchange for a fee.. Modification and distribution of modified material You may modify your copy or copies of this document and the attached software or any portion of it. You may distribute such modifications, all the material based on this original content or work, under the terms of Section 1 above. 3. Your Name and Communication With You If you wish to modify this text or software in any way, you must document the nature of those modifications in the Credits section along with your name, and information concerning how you may be contacted. You must have a reasonable way to contact you. 4. No Endorsement The names POTTO Project and Fundamentals of Compressible Fluid Mechanics or the author of this document must not be used to endorse or promote products derived from this text (book or software) without prior written permission. 5. erived Name(s) Products derived from this software may not be called POTTO Project, or alleged association with this author nor may POTTO or POTTO Project appear in their name, without prior written permission of the r. Genick Bar- Meir. xxv

xxvi LIST OF TABLES 6. Applicability of this license You are not required to accept this License, since you have not signed it. However, nothing else grants you permission to copy, distribute or modify these materials. These actions are prohibited by law if you do not accept this License. Therefore, by distributing or translating these materials, or by deriving works here from, you indicate your acceptance of this License to do so, and all its terms and conditions for copying, distributing or translating these materials. 7. No Warranty Because these materials are licensed free of charge, there is no warranty for the manuscript, to the extent permitted by applicable law. Except when otherwise stated in writing the copyright holders and/or other parties provide these manuscripts AS IS without warranty of any kind, either expressed or implied, including, but not limited to, the implied warranties of merchantability and fitness for a particular purpose. The entire risk of use of this manuscript is with you. Should this manuscript prove faulty, inaccurate, or otherwise unacceptable you assume the cost of all necessary repair or correction. 8. No Liability In no event unless required by applicable law or agreed to in writing will any copyright holder, or any other party who may mirror and/or redistribute these materials as permitted above, be liable to you for damages, including any general, special, incidental or consequential damages arising out of the use or inability to use this manuscript, even if such holder or other party has been advised of the possibility of such damages. 9. Jurisdiction These terms and conditions are governed by and will be interpreted in accordance with the state of POTTO Project residence law and any disputes relating to these terms and conditions will be exclusively subject to the jurisdiction of the courts of POTTO Project residence. Currently, the POTTO Project residence is the state of Minnesota. The various provisions of these terms and conditions are severable and if any provision is held to be invalid or unenforceable by any court of competent jurisdiction then such invalidity or unenforceability shall not affect the remaining provisions. If these terms and conditions are not accepted in full, you use the book and or the software must be terminated immediately.

CONTRIBUTOR LIST How to contribute to this book As a copylefted work, this book is open to revision and expansion by any interested parties. The only catch is that credit must be given where credit is due. This is a copyrighted work: it is not in the public domain! If you wish to cite portions of this book in a work of your own, you must follow the same guidelines as for any other GL copyrighted work. Credits All entries arranged in alphabetical order of surname. Major contributions are listed by individual name with some detail on the nature of the contribution(s), date, contact info, etc. Minor contributions (typo corrections, etc.) are listed by name only for reasons of brevity. Please understand that when I classify a contribution as minor, it is in no way inferior to the effort or value of a major contribution, just smaller in the sense of less text changed. Any and all contributions are gratefully accepted. I am indebted to all those who have given freely of their own knowledge, time, and resources to make this a better book! ate(s) of contribution(s): 004 to present Nature of contribution: Original author. Contact at: barmeir@gmail.com John Martones ate(s) of contribution(s): June 005 xxvii

xxviii LIST OF TABLES Nature of contribution: HTML formatting, some error corrections. Grigory Toker ate(s) of contribution(s): August 005 Nature of contribution: Provided pictures of the oblique shock for oblique shcok chapter. Ralph Menikoff ate(s) of contribution(s): July 005 Nature of contribution: Some discussion about the solution to oblique shock and about the Maximum eflection of the oblique shock. Your name here ate(s) of contribution(s): Month and year of contribution Nature of contribution: Insert text here, describing how you contributed to the book. Contact at: my email@provider.net Typo corrections and other minor contributions H. Gohrah, Ph.., September 005, some LaTeX issues. Roy Tate November 006, Suggestions on improving english and gramer.