Fundamentals of Aerodynamics Sixth Edition PDF

John Anderson Fundamentals of Aerodynamics S I XTH E D ITION Fundamentals of Aerodynamics Sixth Edition John D. Anderson, Jr. McGRAW-HILL SERIES IN AERONAUTICAL AND AEROSPACE ENGINEERING T he Wright brothers invented the first practical ai...

John Anderson Fundamentals of Aerodynamics S I XTH E D ITION Fundamentals of Aerodynamics Sixth Edition John D. Anderson, Jr. McGRAW-HILL SERIES IN AERONAUTICAL AND AEROSPACE ENGINEERING T he Wright brothers invented the first practical airplane in the first decade of the twentieth century. Along with this came the rise of aeronautical engineering as an exciting, new, distinct discipline. College courses in aeronautical engineering were offered as early as 1914 at the University of Michigan and at MIT. Michigan was the first university to establish an aero- nautics department with a four-year degree-granting program in 1916; by 1926 it had graduated over one hundred students. The need for substantive textbooks in various areas of aeronautical engineering became critical. Rising to this demand, McGraw-Hill became one of the first publishers of aeronautical engineering text- books, starting with Airplane Design and Construction by Ottorino Pomilio in 1919, and the classic and definitive text Airplane Design: Aerodynamics by the iconic Edward P. Warner in 1927. Warner’s book was a watershed in aeronautical engineering textbooks. Since then, McGraw-Hill has become the time-honored publisher of books in aeronautical engineering. With the advent of high-speed flight after World War II and the space program in 1957, aeronautical and aerospace engineering grew to new heights. There was, however, a hiatus that occurred in the 1970s when aerospace engineering went through a transition, and virtually no new books in the field were published for almost a decade by anybody. McGraw-Hill broke this hiatus with the foresight of its Chief Engineering Editor, B.J. Clark, who was instrumental in the publication of Introduction to Flight by John Anderson. First published in 1978, Introduction to Flight is now in its 8th edition. Clark’s bold decision was followed by McGraw-Hill riding the crest of a new wave of students and activity in aerospace engineering, and it opened the flood-gates for new textbooks in the field. In 1988, McGraw-Hill initiated its formal series in Aeronautical and Aerospace Engineering, gathering together under one roof all its existing texts in the field, and soliciting new manuscripts. This author is proud to have been made the consulting editor for this series, and to have contributed some of the titles. Starting with eight books in 1988, the series now embraces 24 books cov- ering a broad range of discipline in the field. With this, McGraw-Hill continues its tradition, started in 1919, as the premier publisher of important textbooks in aeronautical and aerospace engineering. John D. Anderson, Jr. Fundamentals of Aerodynamics Sixth Edition John D. Anderson, Jr. Curator of Aerodynamics National Air and Space Museum Smithsonian Institution and Professor Emeritus University of Maryland FUNDAMENTALS OF AERODYNAMICS, SIXTH EDITION Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright © 2017 by McGraw-Hill Education. All rights reserved. Printed in the United States of America. Previous editions © 2011, 2007, and 2001. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. 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Donnelley All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. Library of Congress Cataloging-in-Publication Data Names: Anderson, John D., Jr. (John David), 1937- author. Title: Fundamentals of aerodynamics / John D. Anderson, Jr. Description: Sixth edition. | New York, NY : McGraw-Hill Education, | Series: McGraw-Hill series in aeronautical and aerospace engineering | Includes bibliographical references and index. Identifiers: LCCN 2015040997| ISBN 9781259129919 (alk. paper) | ISBN 1259129918 (alk paper) Subjects: LCSH: Aerodynamics. Classification: LCC TL570.A677 2017 | DDC 629.132/3–dc23 LC record available at http://lccn.loc.gov/2015040997 The Internet addresses listed in the text were accurate at the time of publication. The inclusion of a website does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites. mheducation.com/highered  ABOUT THE AUTHOR John D. Anderson, Jr., was born in Lancaster, Pennsylvania, on October 1, 1937. He attended the University of Florida, graduating in 1959 with high honors and a bachelor of aeronautical engineering degree. From 1959 to 1962, he was a lieutenant and task scientist at the Aerospace Research Laboratory at Wright- Patterson Air Force Base. From 1962 to 1966, he attended the Ohio State Univer- sity under the National Science Foundation and NASA Fellowships, graduating with a Ph.D. in aeronautical and astronautical engineering. In 1966, he joined the U.S. Naval Ordnance Laboratory as Chief of the Hypersonics Group. In 1973, he became Chairman of the Department of Aerospace Engineering at the Uni- versity of Maryland, and since 1980 has been professor of Aerospace Engineer- ing at the University of Maryland. In 1982, he was designated a Distinguished Scholar/Teacher by the University. During 1986–1987, while on sabbatical from the University, Dr. Anderson occupied the Charles Lindbergh Chair at the Na- tional Air and Space Museum of the Smithsonian Institution. He continued with the Air and Space Museum one day each week as their Special Assistant for Aero- dynamics, doing research and writing on the history of aerodynamics. In addition to his position as professor of aerospace engineering, in 1993, he was made a full faculty member of the Committee for the History and Philosophy of Science and in 1996 an affiliate member of the History Department at the University of Maryland. In 1996, he became the Glenn L. Martin Distinguished Professor for Education in Aerospace Engineering. In 1999, he retired from the University of Maryland and was appointed Professor Emeritus. He is currently the Curator for Aerodynamics at the National Air and Space Museum, Smithsonian Institution. Dr. Anderson has published 11 books: Gasdynamic Lasers: An Introduction, Academic Press (1976), and under McGraw-Hill, Introduction to Flight (1978, 1984, 1989, 2000, 2005, 2008, 2012, 2016), Modern Compressible Flow (1982, 1990, 2003), Fundamentals of Aerodynamics (1984, 1991, 2001, 2007, 2011), Hypersonic and High Temperature Gas Dynamics (1989), Computational Fluid Dynamics: The Basics with Applications (1995), Aircraft Performance and De- sign (1999), A History of Aerodynamics and Its Impact on Flying Machines, Cambridge University Press (1997 hardback, 1998 paperback), The Airplane: A History of Its Technology, American Institute of Aeronautics and Astronautics (2003), Inventing Flight, Johns Hopkins University Press (2004), and X-15, The World’s Fastest Rocket Plane and the Pilots Who Ushered in the Space Age, with co-author Richard Passman, Zenith Press in conjunction with the Smithsonian Institution (2014). He is the author of over 120 papers on radiative gasdynam- ics, reentry aerothermodynamics, gasdynamic and chemical lasers, computational fluid dynamics, applied aerodynamics, hypersonic flow, and the history of aero- nautics. Dr. Anderson is a member of the National Academy of Engineering, and v vi About the Author is in Who’s Who in America. He is an Honorary Fellow of the American Institute of Aeronautics and Astronautics (AIAA). He is also a fellow of the Royal Aero- nautical Society, London. He is a member of Tau Beta Pi, Sigma Tau, Phi Kappa Phi, Phi Eta Sigma, The American Society for Engineering Education, the History of Science Society, and the Society for the History of Technology. In 1988, he was elected as Vice President of the AIAA for Education. In 1989, he was awarded the John Leland Atwood Award jointly by the American Society for Engineering Education and the American Institute of Aeronautics and Astronautics “for the lasting influence of his recent contributions to aerospace engineering education.” In 1995, he was awarded the AIAA Pendray Aerospace Literature Award “for writing undergraduate and graduate textbooks in aerospace engineering which have received worldwide acclaim for their readability and clarity of presentation, including historical content.” In 1996, he was elected Vice President of the AIAA for Publications. He has recently been honored by the AIAA with its 2000 von Karman Lectureship in Astronautics. From 1987 to the present, Dr. Anderson has been the senior consulting editor on the McGraw-Hill Series in Aeronautical and Astronautical Engineering.  CONTENTS Preface to the Sixth Edition XV 1.13 Historical Note: The Illusive Center of Pressure 89 1.14 Historical Note: Aerodynamic PART 1 Fundamental Principles 1 Coefficients 93 1.15 Summary 97 1.16 Integrated Work Challenge: Forward-Facing Axial Aerodynamic Force on an Airfoil— Chapter 1 Can It Happen and, If So, How? 98 Aerodynamics: Some Introductory 1.17 Problems 101 Thoughts 3 1.1 Importance of Aerodynamics: Historical Chapter 2 Examples 5 Aerodynamics: Some Fundamental Principles 1.2 Aerodynamics: Classification and Practical and Equations 105 Objectives 11 2.1 Introduction and Road Map 106 1.3 Road Map for This Chapter 15 2.2 Review of Vector Relations 107 1.4 Some Fundamental Aerodynamic Variables 15 2.2.1 Some Vector Algebra 108 1.4.1 Units 18 2.2.2 Typical Orthogonal Coordinate Systems 109 1.5 Aerodynamic Forces and Moments 19 2.2.3 Scalar and Vector Fields 112 1.6 Center of Pressure 32 2.2.4 Scalar and Vector Products 112 1.7 Dimensional Analysis: The Buckingham 2.2.5 Gradient of a Scalar Field 113 Pi Theorem 34 2.2.6 Divergence of a Vector Field 115 1.8 Flow Similarity 41 2.2.7 Curl of a Vector Field 116 1.9 Fluid Statics: Buoyancy Force 52 2.2.8 Line Integrals 116 1.10 Types of Flow 62 2.2.9 Surface Integrals 117 1.10.1 Continuum Versus Free Molecule Flow 62 2.2.10 Volume Integrals 118 1.10.2 Inviscid Versus Viscous Flow 62 2.2.11 Relations Between Line, Surface, and Volume Integrals 119 1.10.3 Incompressible Versus Compressible Flows 64 2.2.12 Summary 119 1.10.4 Mach Number Regimes 64 2.3 Models of the Fluid: Control Volumes and 1.11 Viscous Flow: Introduction to Boundary Fluid Elements 119 Layers 68 2.3.1 Finite Control Volume Approach 120 1.12 Applied Aerodynamics: The Aerodynamic 2.3.2 Infinitesimal Fluid Element Coefficients—Their Magnitudes and Approach 121 Variations 75 2.3.3 Molecular Approach 121 vii viii Contents 2.3.4 Physical Meaning of the Divergence 3.4 Pitot Tube: Measurement of Airspeed 226 of Velocity 122 3.5 Pressure Coefficient 235 2.3.5 Specification of the Flow Field 123 3.6 Condition on Velocity for Incompressible 2.4 Continuity Equation 127 Flow 237 2.5 Momentum Equation 132 3.7 Governing Equation for Irrotational, 2.6 An Application of the Momentum Equation: Incompressible Flow: Laplace’s Drag of a Two-Dimensional Body 137 Equation 238 2.6.1 Comment 146 3.7.1 Infinity Boundary Conditions 241 2.7 Energy Equation 146 3.7.2 Wall Boundary Conditions 241 2.8 Interim Summary 151 3.8 Interim Summary 242 2.9 Substantial Derivative 152 3.9 Uniform Flow: Our First Elementary 2.10 Fundamental Equations in Terms of the Flow 243 Substantial Derivative 158 3.10 Source Flow: Our Second Elementary 2.11 Pathlines, Streamlines, and Streaklines Flow 245 of a Flow 160 3.11 Combination of a Uniform Flow with a 2.12 Angular Velocity, Vorticity, and Strain 165 Source and Sink 249 2.13 Circulation 176 3.12 Doublet Flow: Our Third Elementary 2.14 Stream Function 179 Flow 253 2.15 Velocity Potential 183 3.13 Nonlifting Flow over a Circular Cylinder 255 2.16 Relationship Between the Stream Function and Velocity Potential 186 3.14 Vortex Flow: Our Fourth Elementary Flow 264 2.17 How Do We Solve the Equations? 187 3.15 Lifting Flow over a Cylinder 268 2.17.1 Theoretical (Analytical) Solutions 187 3.16 The Kutta-Joukowski Theorem and the 2.17.2 Numerical Solutions—Computational Generation of Lift 282 Fluid Dynamics (CFD) 189 3.17 Nonlifting Flows over Arbitrary Bodies: 2.17.3 The Bigger Picture 196 The Numerical Source Panel Method 284 2.18 Summary 196 3.18 Applied Aerodynamics: The Flow over a 2.19 Problems 200 Circular Cylinder—The Real Case 294 3.19 Historical Note: Bernoulli and Euler—The PART 2 Origins of Theoretical Fluid Dynamics 302 Inviscid, Incompressible Flow 203 3.20 Historical Note: d’Alembert and His Paradox 307 Chapter 3 3.21 Summary 308 Fundamentals of Inviscid, Incompressible 3.22 Integrated Work Challenge: Relation Flow 205 Between Aerodynamic Drag and the Loss of 3.1 Introduction and Road Map 206 Total Pressure in the Flow Field 311 3.2 Bernoulli’s Equation 209 3.23 Integrated Work Challenge: Conceptual 3.3 Incompressible Flow in a Duct: The Venturi Design of a Subsonic Wind Tunnel 314 and Low-Speed Wind Tunnel 213 3.24 Problems 318  Contents ix Chapter 4 Chapter 5 Incompressible Flow over Airfoils 321 Incompressible Flow over Finite Wings 423 4.1 Introduction 323 5.1 Introduction: Downwash and Induced 4.2 Airfoil Nomenclature 326 Drag 427 4.3 Airfoil Characteristics 328 5.2 The Vortex Filament, the Biot-Savart Law, 4.4 Philosophy of Theoretical Solutions for and Helmholtz’s Theorems 432 Low-Speed Flow over Airfoils: The 5.3 Prandtl’s Classical Lifting-Line Vortex Sheet 333 Theory 436 4.5 The Kutta Condition 338 5.3.1 Elliptical Lift Distribution 442 4.5.1 Without Friction Could We 5.3.2 General Lift Distribution 447 Have Lift? 342 5.3.3 Effect of Aspect Ratio 450 4.6 Kelvin’s Circulation Theorem and the 5.3.4 Physical Significance 456 Starting Vortex 342 5.4 A Numerical Nonlinear Lifting-Line 4.7 Classical Thin Airfoil Theory: The Method 465 Symmetric Airfoil 346 5.5 The Lifting-Surface Theory and the Vortex 4.8 The Cambered Airfoil 356 Lattice Numerical Method 469 4.9 The Aerodynamic Center: Additional 5.6 Applied Aerodynamics: The Delta Considerations 365 Wing 476 4.10 Lifting Flows over Arbitrary Bodies: The 5.7 Historical Note: Lanchester and Vortex Panel Numerical Method 369 Prandtl—The Early Development of 4.11 Modern Low-Speed Airfoils 375 Finite-Wing Theory 488 4.12 Viscous Flow: Airfoil Drag 379 5.8 Historical Note: Prandtl—The Man 492 4.12.1 Estimating Skin-Friction Drag: 5.9 Summary 495 Laminar Flow 380 5.10 Problems 496 4.12.2 Estimating Skin-Friction Drag: Turbulent Flow 382 4.12.3 Transition 384 Chapter 6 4.12.4 Flow Separation 389 Three-Dimensional Incompressible Flow 499 4.12.5 Comment 394 6.1 Introduction 499 4.13 Applied Aerodynamics: The Flow over an 6.2 Three-Dimensional Source 500 Airfoil—The Real Case 395 6.3 Three-Dimensional Doublet 502 4.14 Historical Note: Early Airplane Design and 6.4 Flow over a Sphere 504 the Role of Airfoil Thickness 406 6.4.1 Comment on the Three-Dimensional 4.15 Historical Note: Kutta, Joukowski, and the Relieving Effect 506 Circulation Theory of Lift 411 6.5 General Three-Dimensional Flows: Panel 4.16 Summary 413 Techniques 507 4.17 Integrated Work Challenge: Wall Effects on 6.6 Applied Aerodynamics: The Flow over a Measurements Made in Subsonic Wind Sphere—The Real Case 509 Tunnels 415 4.18 Problems 419 x Contents 6.7 Applied Aerodynamics: Airplane Lift 8.3 Speed of Sound 567 and Drag 512 8.3.1 Comments 575 6.7.1 Airplane Lift 512 8.4 Special Forms of the Energy Equation 576 6.7.2 Airplane Drag 514 8.5 When Is a Flow Compressible? 584 6.7.3 Application of Computational Fluid 8.6 Calculation of Normal Shock-Wave Dynamics for the Calculation of Lift and Properties 587 Drag 519 8.6.1 Comment on the Use of Tables to Solve 6.8 Summary 523 Compressible Flow Problems 602 6.9 Problems 524 8.7 Measurement of Velocity in a Compressible Flow 603 8.7.1 Subsonic Compressible Flow 603 PART 3 Inviscid, Compressible Flow 525 8.8 8.7.2 Supersonic Flow 604 Summary 608 8.9 Problems 611 Chapter 7 Compressible Flow: Some Preliminary Chapter 9 Aspects 527 Oblique Shock and Expansion Waves 613 7.1 Introduction 528 9.1 Introduction 614 7.2 A Brief Review of Thermodynamics 530 9.2 Oblique Shock Relations 620 7.2.1 Perfect Gas 530 9.3 Supersonic Flow over Wedges and 7.2.2 Internal Energy and Enthalpy 530 Cones 634 7.2.3 First Law of Thermodynamics 535 9.3.1 A Comment on Supersonic Lift and Drag 7.2.4 Entropy and the Second Law of Coefficients 637 Thermodynamics 536 9.4 Shock Interactions and Reflections 638 7.2.5 Isentropic Relations 538 9.5 Detached Shock Wave in Front of a Blunt 7.3 Definition of Compressibility 542 Body 644 7.4 Governing Equations for Inviscid, 9.5.1 Comment on the Flow Field Behind a Compressible Flow 543 Curved Shock Wave: Entropy Gradients 7.5 Definition of Total (Stagnation) and Vorticity 648 Conditions 545 9.6 Prandtl-Meyer Expansion Waves 648 7.6 Some Aspects of Supersonic Flow: Shock 9.7 Shock-Expansion Theory: Applications to Waves 552 Supersonic Airfoils 660 7.7 Summary 556 9.8 A Comment on Lift and Drag 7.8 Problems 558 Coefficients 664 9.9 The X-15 and Its Wedge Tail 664 Chapter 8 9.10 Viscous Flow: Shock-Wave/ Normal Shock Waves and Related Topics 561 Boundary-Layer Interaction 669 9.11 Historical Note: Ernst Mach—A 8.1 Introduction 562 Biographical Sketch 671 8.2 The Basic Normal Shock Equations 563  Contents xi 9.12 Summary 674 11.6 Critical Mach Number 756 9.13 Integrated Work Challenge: Relation 11.6.1 A Comment on the Location of Minimum Between Supersonic Wave Drag and Pressure (Maximum Velocity) 765 Entropy Increase—Is There a 11.7 Drag-Divergence Mach Number: The Relation? 675 Sound Barrier 765 9.14 Integrated Work Challenge: The Sonic 11.8 The Area Rule 773 Boom 678 11.9 The Supercritical Airfoil 775 9.15 Problems 681 11.10 CFD Applications: Transonic Airfoils and Wings 777 Chapter 10 11.11 Applied Aerodynamics: The Blended Wing Body 782 Compressible Flow Through Nozzles, Diffusers, and Wind Tunnels 689 11.12 Historical Note: High-Speed Airfoils—Early Research and 10.1 Introduction 690 Development 788 10.2 Governing Equations for 11.13 Historical Note: The Origin of the Quasi-One-Dimensional Flow 692 Swept-Wing Concept 792 10.3 Nozzle Flows 701 11.14 Historical Note: Richard T. 10.3.1 More on Mass Flow 715 Whitcomb—Architect of the Area Rule 10.4 Diffusers 716 and the Supercritical Wing 801 10.5 Supersonic Wind Tunnels 718 11.15 Summary 802 10.6 Viscous Flow: Shock-Wave/ 11.16 Integrated Work Challenge: Transonic Boundary-Layer Interaction Inside Testing by the Wing-Flow Method 804 Nozzles 724 11.17 Problems 808 10.7 Summary 726 10.8 Integrated Work Challenge: Chapter 12 Conceptual Design of a Supersonic Linearized Supersonic Flow 811 Wind Tunnel 727 10.9 Problems 736 12.1 Introduction 812 12.2 Derivation of the Linearized Supersonic Pressure Coefficient Formula 812 Chapter 11 12.3 Application to Supersonic Airfoils 816 Subsonic Compressible Flow over Airfoils: 12.4 Viscous Flow: Supersonic Airfoil Linear Theory 739 Drag 822 11.1 Introduction 740 12.5 Summary 825 11.2 The Velocity Potential Equation 742 12.6 Problems 826 11.3 The Linearized Velocity Potential Equation 745 Chapter 13 11.4 Prandtl-Glauert Compressibility Introduction to Numerical Techniques for Correction 750 Nonlinear Supersonic Flow 829 11.5 Improved Compressibility 13.1 Introduction: Philosophy of Computational Corrections 755 Fluid Dynamics 830 xii Contents 13.2 Elements of the Method of 14.8 Hypersonic Viscous Flow: Aerodynamic Characteristics 832 Heating 901 13.2.1 Internal Points 838 14.8.1 Aerodynamic Heating and Hypersonic 13.2.2 Wall Points 839 Flow—The Connection 901 13.3 Supersonic Nozzle Design 840 14.8.2 Blunt Versus Slender Bodies in Hypersonic Flow 903 13.4 Elements of Finite-Difference Methods 843 14.8.3 Aerodynamic Heating to a Blunt Body 906 13.4.1 Predictor Step 849 14.9 Applied Hypersonic Aerodynamics: 13.4.2 Corrector Step 849 Hypersonic Waveriders 908 13.5 The Time-Dependent Technique: 14.9.1 Viscous-Optimized Waveriders 914 Application to Supersonic Blunt Bodies 850 14.10 Summary 921 13.5.1 Predictor Step 854 14.11 Problems 922 13.5.2 Corrector Step 854 13.6 Flow over Cones 858 13.6.1 Physical Aspects of Conical Flow 859 PART 4 Viscous Flow 923 13.6.2 Quantitative Formulation 860 13.6.3 Numerical Procedure 865 Chapter 15 13.6.4 Physical Aspects of Supersonic Flow Introduction to the Fundamental Principles and over Cones 866 Equations of Viscous Flow 925 13.7 Summary 869 15.1 Introduction 926 13.8 Problem 870 15.2 Qualitative Aspects of Viscous Flow 927 15.3 Viscosity and Thermal Conduction 935 Chapter 14 15.4 The Navier-Stokes Equations 940 Elements of Hypersonic Flow 871 15.5 The Viscous Flow Energy Equation 944 15.6 Similarity Parameters 948 14.1 Introduction 872 15.7 Solutions of Viscous Flows: A Preliminary 14.2 Qualitative Aspects of Hypersonic Discussion 952 Flow 873 15.8 Summary 955 14.3 Newtonian Theory 877 15.9 Problems 957 14.4 The Lift and Drag of Wings at Hypersonic Speeds: Newtonian Results for a Flat Plate at Angle of Attack 881 Chapter 16 14.4.1 Accuracy Considerations 888 A Special Case: Couette Flow 959 14.5 Hypersonic Shock-Wave Relations and 16.1 Introduction 959 Another Look at Newtonian Theory 892 16.2 Couette Flow: General Discussion 960 14.6 Mach Number Independence 896 16.3 Incompressible (Constant Property) Couette 14.7 Hypersonics and Computational Fluid Flow 964 Dynamics 898 16.3.1 Negligible Viscous Dissipation 970  Contents xiii 16.3.2 Equal Wall Temperatures 971 18.6 Boundary Layers over Arbitrary Bodies: 16.3.3 Adiabatic Wall Conditions (Adiabatic Finite-Difference Solution 1043 Wall Temperature) 973 18.6.1 Finite-Difference Method 1044 16.3.4 Recovery Factor 976 18.7 Summary 1049 16.3.5 Reynolds Analogy 977 18.8 Problems 1050 16.3.6 Interim Summary 978 16.4 Compressible Couette Flow 980 Chapter 19 16.4.1 Shooting Method 982 Turbulent Boundary Layers 1051 16.4.2 Time-Dependent Finite-Difference Method 984 19.1 Introduction 1052 16.4.3 Results for Compressible Couette 19.2 Results for Turbulent Boundary Layers on Flow 988 a Flat Plate 1052 16.4.4 Some Analytical Considerations 990 19.2.1 Reference Temperature Method for 16.5 Summary 995 Turbulent Flow 1054 19.2.2 The Meador-Smart Reference Temperature Method for Turbulent Chapter 17 Flow 1056 19.2.3 Prediction of Airfoil Drag 1057 Introduction to Boundary Layers 997 19.3 Turbulence Modeling 1057 17.1 Introduction 998 19.3.1 The Baldwin-Lomax Model 1058 17.2 Boundary-Layer Properties 1000 19.4 Final Comments 1060 17.3 The Boundary-Layer Equations 1006 19.5 Summary 1061 17.4 How Do We Solve the Boundary-Layer 19.6 Problems 1062 Equations? 1009 17.5 Summary 1011 Chapter 20 Navier-Stokes Solutions: Chapter 18 Some Examples 1063 Laminar Boundary Layers 1013 20.1 Introduction 1064 18.1 Introduction 1013 20.2 The Approach 1064 18.2 Incompressible Flow over a Flat Plate: 20.3 Examples of Some Solutions 1065 The Blasius Solution 1014 20.3.1 Flow over a Rearward-Facing Step 1065 18.3 Compressible Flow over a Flat Plate 1021 20.3.2 Flow over an Airfoil 1065 18.3.1 A Comment on Drag Variation with 20.3.3 Flow over a Complete Airplane 1068 Velocity 1032 20.3.4 Shock-Wave/Boundary-Layer 18.4 The Reference Temperature Method 1033 Interaction 1069 18.4.1 Recent Advances: The Meador-Smart 20.3.5 Flow over an Airfoil with a Reference Temperature Protuberance 1070 Method 1036 20.4 The Issue of Accuracy for the Prediction of 18.5 Stagnation Point Aerodynamic Skin Friction Drag 1072 Heating 1037 20.5 Summary 1077 xiv Contents Appendix A Appendix E Isentropic Flow Properties 1079 Standard Atmosphere, English Engineering Units 1103 Appendix B Normal Shock Properties 1085 References 1111 Appendix C Index 1117 Prandtl-Meyer Function and Mach Angle 1089 Appendix D Standard Atmosphere, SI Units 1093  PREFACE TO THE SIXTH EDITION T his book follows in the same tradition as the previous editions: it is for students—to be read, understood, and enjoyed. It is consciously written in a clear, informal, and direct style to talk to the reader and gain his or her immediate interest in the challenging and yet beautiful discipline of aerodynamics. The explanation of each topic is carefully constructed to make sense to the reader. Moreover, the structure of each chapter is highly organized in order to keep the reader aware of where we are, where we were, and where we are going. Too frequently the student of aerodynamics loses sight of what is trying to be accomplished; to avoid this, I attempt to keep the reader informed of my intent at all times. For example, preview boxes are introduced at the beginning of each chapter. These short sections, literally set in boxes, inform the reader in plain language what to expect from each chapter and why the material is important and exciting. They are primarily motivational; they help to encourage the reader to actually enjoy reading the chapter, therefore enhancing the educational process. In addition, each chapter contains a road map—a block diagram designed to keep the reader well aware of the proper flow of ideas and concepts. The use of preview boxes and chapter road maps are unique features of this book. Also, to help organize the reader’s thoughts, there are special summary sections at the end of most chapters. The material in this book is at the level of college juniors and seniors in aerospace or mechanical engineering. It assumes no prior knowledge of fluid dynamics in general, or aerodynamics in particular. It does assume a familiarity with differential and integral calculus, as well as the usual physics background common to most students of science and engineering. Also, the language of vector analysis is used liberally; a compact review of the necessary elements of vector algebra and vector calculus is given in Chapter 2 in such a fashion that it can either educate or refresh the reader, whatever may be the case for each individual. This book is designed for a one-year course in aerodynamics. Chapters 1 to 6 constitute a solid semester emphasizing inviscid, incompressible flow. Chapters 7 to 14 occupy a second semester dealing with inviscid, compressible flow. Finally, Chapters 15 to 20 introduce some basic elements of viscous flow, mainly to serve as a contrast to and comparison with the inviscid flows treated throughout the bulk of the text. Specific sections on viscous flow, however, have been added much earlier in the book in order to give the reader some idea of how the inviscid results are tempered by the influence of friction. This is done by adding self-contained viscous flow sections at the end of various chapters, written and placed in such a way that they do not interfere with the flow of the inviscid flow discussion, but are there to complement the discussion. For example, at the end of Chapter 4 on xv xvi Preface to the Sixth Edition incompressible inviscid flow over airfoils, there is a viscous flow section that deals with the prediction of skin friction drag on such airfoils. A similar viscous flow section at the end of Chapter 12 deals with friction drag on high-speed airfoils. At the end of the chapters on shock waves and nozzle flows, there are viscous flow sections on shock wave/boundary-layer interactions. And so forth. Other features of this book are: 1. An introduction to computational fluid dynamics as an integral part of the study of aerodynamics. Computational fluid dynamics (CFD) has recently become a third dimension in aerodynamics, complementing the previously existing dimension of pure experiment and pure theory. It is absolutely necessary that the modern student of aerodynamics be introduced to some of the basic ideas of CFD—he or she will most certainly come face to face with either its “machinery” or its results after entering the professional ranks of practicing aerodynamicists. Hence, such subjects as the source and vortex panel techniques, the method of characteristics, and explicit finite-difference solutions are introduced and discussed as they naturally arise during the course of our discussion. In particular, Chapter 13 is devoted exclusively to numerical techniques, couched at a level suitable to an introductory aerodynamics text. 2. A chapter is devoted entirely to hypersonic flow. Although hypersonics is at one extreme end of the flight spectrum, it has current important applications to the design of hypervelocity missiles, planetary entry vehicles, and modern hypersonic atmospheric cruise vehicles. Therefore, hypersonic flow deserves some attention in any modern presentation of aerodynamics. This is the purpose of Chapter 14. 3. Historical notes are placed at the end of many of the chapters. This follows in the tradition of some of my previous textbooks, Introduction to Flight: Its Engineering and History, 8th Edition (McGraw-Hill, 2016) and Modern Compressible Flow: With Historical Perspecive, 3rd Edition (McGraw-Hill, 2003). Although aerodynamics is a rapidly evolving subject, its foundations are deeply rooted in the history of science and technology. It is important for the modern student of aerodynamics to have an appreciation for the historical origin of the tools of the trade. Therefore, this book addresses such questions as who Bernoulli, Euler, d’Alembert, Kutta, Joukowski, and Prandtl were; how the circulation theory of lift developed; and what excitement surrounded the early development of high-speed aerodynamics. I wish to thank various members of the staff of the National Air and Space Museum of the Smithsonian Institution for opening their extensive files for some of the historical research behind these history sections. Also, a constant biographical reference was the Dictionary of Scientific Biography, edited by C. C. Gillespie, Charles Schribner’s Sons, New York, 1980. This is a 16-volume set of books that is a valuable source of biographic information on the leading scientists in history.  Preface to the Sixth Edition xvii 4. Design boxes are scattered throughout the book. These design boxes are special sections for the purpose of discussing design aspects associated with the fundamental material covered throughout the book. These sections are literally placed in boxes to set them apart from the mainline text. Modern engineering education is placing more emphasis on design, and the design boxes in this book are in this spirit. They are a means of making the fundamental material more relevant and making the whole process of learning aerodynamics more fun. Due to the extremely favorable comments from readers and users of the first five editions, virtually all the content of the earlier editions has been carried over intact to the present sixth edition. In this edition, however, a completely new edu- cational tool has been introduced in some of the chapters in order to enhance and expand the reader’s learning process. Throughout the previous editions, numer- ous worked examples have been included at the end of many of the sections to illustrate and reinforce the ideas and methods discussed in that particular section. These are still included in the present sixth edition. However, added at the end of a number of the chapters in this sixth edition, a major challenge is given to the reader that integrates and uses thoughts and equations drawn from the chapter as a whole. These new sections are called END OF CHAPTER INTEGRATED WORK CHALLENGES. They are listed next: 1. Chapter 1: A forward-facing axial aerodynamic force on an airfoil sounds not possible, but it can actually happen. What are the conditions under which it can happen? Also, the history of when such a forward-facing force was first observed is discussed. 2. Chapter 2: Using the momentum equation, develop the relation between drag on an aerodynamic body and the loss of total pressure in the flow field. 3. Chapter 3: Perform a conceptual design of a low-speed subsonic wind tunnel. 4. Chapter 4: Find a way to account for the effects of wind tunnel walls on the measurements made on an aerodynamic body in a low-speed wind tunnel. 5. Chapter 7: Obtain and discuss a relation between supersonic wave drag on a body and the entropy increase in the flow. 6. Chapter 9: Consider the sonic boom generated from a body in supersonic flight. What is it? How is it created? How can its strength be reduced? 7. Chapter 10: Perform a conceptual design of a supersonic wind tunnel. 8. Chapter 11: At the end of World War II, in the face of the lack of reliable transonic wind tunnels and the extreme theoretical difficulty solving the nonlinear mathematical equations that govern transonic flow, the NACA developed an innovative experimental method for obtaining transonic aerodynamic data. Called the “wing-flow technique,” it involved mounting a small airfoil wing model vertically on the surface of the wing of a P-51 xviii Preface to the Sixth Edition fighter airplane at a location inside the bubble of locally supersonic flow formed on the P-51 wing when the airplane exceeded its critical Mach number. Design this apparatus, taking into account the size of the test model, the flow conditions over the test model, the optimum locations on the P-51 wing, etc. Also, the history of the wing-flow techniques will be given. The answers to these Integrated Work Challenges are given right there in the text so that the reader can gain instant gratification after working them out, just like the other worked examples; the answers are just more complex with a more widespread educational value. New homework problems have been added to McGraw-Hill’s online learning environment, Connect®. These question banks will include all end-of-chapter problems from the textbook and additional problems unique to Connect. All the new additional material not withstanding, the main thrust of this book remains the presentation of the fundamentals of aerodynamics; the new material is simply intended to enhance and support this thrust. I repeat that the book is organized along classical lines, dealing with inviscid incompressible flow, inviscid compressible flow, and viscous flow in sequence. My experience in teaching this material to undergraduates finds that it nicely divides into a two-semester course with Parts 1 and 2 in the first semester and Parts 3 and 4 in the second semester. Also, I have taught the entire book in a fast-paced, first-semester graduate course intended to introduce the fundamentals of aerodynamics to new graduate students who have not had this material as part of their undergraduate education. The book works well in such a mode. I would like to thank the McGraw-Hill editorial and production staff for their excellent help in producing this book, especially Jolynn Kilburg and Thomas Scaife, PhD, in Dubuque. Our photo researcher, David Tietz, was invaluable in searching out new and replacement photographs for the new edition to sat- isfy new McGraw-Hill guidelines; I don’t know what I would have done with- out him. Also, special thanks go to my long-time friend and associate, Sue Cunningham, whose expertise as a scientific typist is beyond comparison and who has typed all my book manuscripts for me, including this one, with great care and precision. I want to thank my students over the years for many stimulating discussions on the subject of aerodynamics, discussions that have influenced the development of this book. Special thanks go to three institutions: (1) The University of Maryland for providing a challenging intellectual atmosphere in which I have basked for the past 42 years; (2) The National Air and Space Museum of the Smithsonian Institution for opening the world of the history of the technology of flight for me; and (3) the Anderson household—Sarah-Allen, Katherine, and Elizabeth—who have been patient and understanding over the years while their husband and father was in his ivory tower. Also, I pay respect to the new generation, which includes my two beautiful granddaughters, Keegan and Tierney Glabus, who represent the future. To them, I dedicate this book.  Preface to the Sixth Edition xix As a final comment, aerodynamics is a subject of intellectual beauty, com- posed and drawn by many great minds over the centuries. Fundamentals of Aero- dynamics is intended to portray and convey this beauty. Do you feel challenged and interested by these thoughts? If so, then read on, and enjoy! John D. Anderson, Jr.  P A R T 1 Fundamental Principles I n Part 1, we cover some of the basic principles that apply to aerodynamics in general. These are the pillars on which all of aerodynamics is based. 1  C H A P T E R 1 Aerodynamics: Some Introductory Thoughts The term “aerodynamics” is generally used for problems arising from flight and other topics involving the flow of air. Ludwig Prandtl, 1949 Aerodynamics: The dynamics of gases, especially atmospheric interactions with moving objects. The American Heritage Dictionary of the English Language, 1969 PREVIEW BOX Why learn about aerodynamics? For an answer, just take a look at the following five photographs showing a progression of airplanes over the past 70 years. The Douglas DC-3 (Figure 1.1), one of the most famous aircraft of all time, is a low-speed subsonic trans- port designed during the 1930s. Without a knowl- edge of low-speed aerodynamics, this aircraft would have never existed. The Boeing 707 (Figure 1.2) opened high-speed subsonic flight to millions of pas- sengers beginning in the late 1950s. Without a knowl- edge of high-speed subsonic aerodynamics, most of us would still be relegated to ground transportation. Figure 1.1 Douglas DC-3 (NASA). 3 4 PA RT 1 Fundamental Principles Figure 1.2 Boeing 707 (© Everett Collection Figure 1.5 Lockheed-Martin F-22 (U.S. Air Force Historical/Alamy). Photo/Staff Sgt. Vernon Young Jr.). Figure 1.3 Bell X-1 (Library of Congress [LC-USZ6-1658]). Figure 1.6 Blended wing body (NASA). point-designed to fly at twice the speed of sound, accomplished in the 1950s. The Lockheed-Martin F-22 (Figure 1.5) is a modern fighter aircraft designed for sustained supersonic flight. Without a knowledge of supersonic aerodynamics, these supersonic air- planes would not exist. Finally, an example of an innovative new vehicle concept for high-speed sub- sonic flight is the blended wing body shown in Figure 1.6. At the time of writing, the blended-wing-body promises to carry from 400 to 800 passengers over Figure 1.4 Lockheed F-104 (Library of Congress long distances with almost 30 percent less fuel per [LC-USZ62-94416]). seat-mile than a conventional jet transport. This would be a “renaissance” in long-haul transport. The salient The Bell X-1 (Figure 1.3) became the first piloted air- design aspects of this exciting new concept are dis- plane to fly faster than sound, a feat accomplished cussed in Section 11.10. The airplanes in Figures 1.1– with Captain Chuck Yeager at the controls on Oc- 1.6 are six good reasons to learn about aerodynamics. tober 14, 1947. Without a knowledge of transonic The major purpose of this book is to help you do this. aerodynamics (near, at, and just above the speed of As you continue to read this and subsequent chapters, sound), neither the X-1, nor any other airplane, would you will progressively learn about low-speed aerody- have ever broken the sound barrier. The Lockheed namics, high-speed subsonic aerodynamics, transonic F-104 (Figure 1.4) was the first supersonic airplane aerodynamics, supersonic aerodynamics, and more.  C H A PTER 1 Aerodynamics: Some Introductory Thoughts 5 Airplanes are by no means the only application emmersed in a flowing fluid—and exert an aerody- of aerodynamics. The air flow over an automobile, namic force on the object? We will find out here. The the gas flow through the internal combustion engine resultant aerodynamic force is frequently resolved powering an automobile, weather and storm predic- into two components defined as lift and drag; but tion, the flow through a windmill, the production of rather than dealing with the lift and drag forces them- thrust by gas turbine jet engines and rocket engines, selves, aerodynamicists deal instead with lift and drag and the movement of air through building heater and coefficients. What is so magic about lift and drag air-conditioning systems are just a few other exam- coefficients? We will see. What is a Reynolds number? ples of the application of aerodynamics. The material Mach number? Inviscid flow? Viscous flow? These in this book is powerful stuff—important stuff. Have rather mysterious sounding terms will be demystified fun reading and learning about aerodynamics. in the present chapter. They and others constitute the To learn a new subject, you simply have to start language of aerodynamics, and as we all know, to at the beginning. This chapter is the beginning of our do anything useful you have to know the language. study of aerodynamics; it weaves together a series Visualize this chapter as a beginning language lesson, of introductory thoughts, definitions, and concepts necessary to go on to the exciting aerodynamic appli- essential to our discussions in subsequent chapters. cations in later chapters. There is a certain enjoyment For example, how does nature reach out and grab and satisfaction in learning a new language. Take this hold of an airplane in flight—or any other object chapter in that spirit, and move on. 1.1 IMPORTANCE OF AERODYNAMICS: HISTORICAL EXAMPLES On August 8, 1588, the waters of the English Channel churned with the gyrations of hundreds of warships. The great Spanish Armada had arrived to carry out an invasion of Elizabethan England and was met head-on by the English fleet under the command of Sir Francis Drake. The Spanish ships were large and heavy; they were packed with soldiers and carried formidable cannons that fired 50 lb round shot that could devastate any ship of that era. In contrast, the English ships were smaller and lighter; they carried no soldiers and were armed with lighter, shorter-range cannons. The balance of power in Europe hinged on the outcome of this naval encounter. King Philip II of Catholic Spain was attempting to squash Protestant England’s rising influence in the political and religious affairs of Europe; in turn, Queen Elizabeth I was attempting to defend the very existence of England as a sovereign state. In fact, on that crucial day in 1588, when the English floated six fire ships into the Spanish formation and then drove headlong into the ensuing confusion, the future history of Europe was in the balance. In the final outcome, the heavier, sluggish, Spanish ships were no match for the faster, more maneuverable, English craft, and by that evening the Spanish Armada lay in disarray, no longer a threat to England. This naval battle is of particular importance because it was the first in history to be fought by ships on both sides powered completely by sail (in contrast to earlier combinations of oars and sail), and it taught the world that political power was going to be synonymous with naval power. In turn, naval power was going to depend greatly on the speed and 6 PA RT 1 Fundamental Principles Figure 1.7 Isaac Newton’s model of fluid flow in the year 1687. This model was widely adopted in the seventeenth and eighteenth centuries but was later found to be conceptually inaccurate for most fluid flows. maneuverability of ships. To increase the speed of a ship, it is important to reduce the resistance created by the water flow around the ship’s hull. Suddenly, the drag on ship hulls became an engineering problem of great interest, thus giving impetus to the study of fluid mechanics. This impetus hit its stride almost a century later, when, in 1687, Isaac Newton (1642–1727) published his famous Principia, in which the entire second book was devoted to fluid mechanics. Newton encountered the same difficulty as others before him, namely, that the analysis of fluid flow is conceptually more difficult than the dynamics of solid bodies. A solid body is usually geometrically well defined, and its motion is therefore relatively easy to describe. On the other hand, a fluid is a “squishy” substance, and in Newton’s time it was difficult to decide even how to qualitatively model its motion, let alone obtain quantitative relationships. Newton considered a fluid flow as a uniform, rectilinear stream of particles, much like a cloud of pellets from a shotgun blast. As sketched in Figure 1.7, Newton assumed that upon striking a surface inclined at an angle θ to the stream, the particles would transfer their normal momentum to the surface but their tangential momentum would be preserved. Hence, after collision with the surface, the particles would then move along the surface. This led to an expression for the hydrodynamic force on the surface which varies as sin2 θ. This is Newton’s famous sine-squared law (described in detail in Chapter 14). Although its accuracy left much to be desired, its simplicity led to wide application in naval architecture. Later, in 1777, a series of experiments was carried out by Jean LeRond d’Alembert (1717–1783), under the support of the French government, in order to measure the resistance of ships in canals. The results showed that “the rule that for oblique planes resistance varies with the sine square of the angle of incidence holds good only for angles between 50 and 90◦ and must be abandoned for lesser angles.” Also, in 1781, Leonhard Euler (1707–1783) pointed out the physical inconsistency of Newton’s model (Figure 1.7) consisting of a rectilinear stream of particles impacting without warning on a surface. In contrast to this  C H A PTER 1 Aerodynamics: Some Introductory Thoughts 7 model, Euler noted that the fluid moving toward a body “before reaching the latter, bends its direction and its velocity so that when it reaches the body it flows past it along the surface, and exercises no other force on the body except the pressure corresponding to the single points of contact.” Euler went on to present a formula for resistance that attempted to take into account the shear stress distribution along the surface, as well as the pressure distribution. This expression became proportional to sin2 θ for large incidence angles, whereas it was proportional to sin θ at small incidence angles. Euler noted that such a variation was in reasonable agreement with the ship-hull experiments carried out by d’Alembert. This early work in fluid dynamics has now been superseded by modern con- cepts and techniques. (However, amazingly enough, Newton’s sine-squared law has found new application in very high-speed aerodynamics, to be discussed in Chapter 14.) The major point here is that the rapid rise in the importance of naval architecture after the sixteenth century made fluid dynamics an important science, occupying the minds of Newton, d’Alembert, and Euler, among many others. Today, the modern ideas of fluid dynamics, presented in this book, are still driven in part by the importance of reducing hull drag on ships. Consider a second historical example. The scene shifts to Kill Devil Hills, 4 mi south of Kitty Hawk, North Carolina. It is summer of 1901, and Wilbur and Orville Wright are struggling with their second major glider design, the first being a stunning failure the previous year. The airfoil shape and wing design of their glider are based on aerodynamic data published in the 1890s by the great German aviation pioneer Otto Lilienthal (1848–1896) and by Samuel Pierpont Langley (1934–1906), secretary of the Smithsonian Institution—the most presti- gious scientific position in the United States at that time. Because their first glider in 1900 produced no meaningful lift, the Wright brothers have increased the wing area from 165 to 290 ft2 and have increased the wing camber (a measure of the airfoil curvature—the larger the camber, the more “arched” is the thin airfoil shape) by almost a factor of 2. But something is still wrong. In Wilbur’s words, the glider’s “lifting capacity seemed scarcely one-third of the calculated amount.” Frustration sets in. The glider is not performing even close to their expectations, although it is designed on the basis of the best available aerodynamic data. On August 20, the Wright brothers despairingly pack themselves aboard a train going back to Dayton, Ohio. On the ride back, Wilbur mutters that “nobody will fly for a thousand years.” However, one of the hallmarks of the Wrights is perseverance, and within weeks of returning to Dayton, they decide on a complete departure from their previous approach. Wilbur later wrote that “having set out with abso- lute faith in the existing scientific data, we were driven to doubt one thing after another, until finally after two years of experiment, we cast it all aside, and de- cided to rely entirely upon our own investigations.” Since their 1901 glider was of poor aerodynamic design, the Wrights set about determining what constitutes good aerodynamic design. In the fall of 1901, they design and build a 6 ft long, 16 in square wind tunnel powered by a two-bladed fan connected to a gasoline engine. A replica of the Wrights’ tunnel is shown in Figure 1.8a. In their wind tunnel they test over 200 different wing and airfoil shapes, including flat plates, 8 PA RT 1 Fundamental Principles (a) (b) Figure 1.8 (a) Replica of the wind tunnel designed, built, and used by the Wright brothers in Dayton, Ohio, during 1901–1902. (b) Wing models tested by the Wright brothers in their wind tunnel during 1901–1902. ((a) NASA; (b) Courtesy of John Anderson). curved plates, rounded leading edges, rectangular and curved planforms, and var- ious monoplane and multiplane configurations. A sample of their test models is shown in Figure 1.8b. The aerodynamic data are taken logically and carefully. Armed with their new aerodynamic information, the Wrights design a new glider in the spring of 1902. The airfoil is much more efficient; the camber is reduced considerably, and the location of the maximum rise of the airfoil is moved closer to the front of the wing. The most obvious change, however, is that the ratio of the length of the wing (wingspan) to the distance from the front to the rear of the airfoil (chord length) is increased from 3 to 6. The success of this glider during  C H A PTER 1 Aerodynamics: Some Introductory Thoughts 9 the summer and fall of 1902 is astounding; Orville and Wilbur accumulate over a thousand flights during this period. In contrast to the previous year, the Wrights return to Dayton flushed with success and devote all their subsequent efforts to powered flight. The rest is history. The major point here is that good aerodynamics was vital to the ultimate success of the Wright brothers and, of course, to all subsequent successful airplane designs up to the present day. The importance of aerodynamics to successful manned flight goes without saying, and a major thrust of this book is to present the aerodynamic fundamentals that govern such flight. Consider a third historical example of the importance of aerodynamics, this time as it relates to rockets and space flight. High-speed, supersonic flight had become a dominant feature of aerodynamics by the end of World War II. By this time, aerodynamicists appreciated the advantages of using slender, pointed body shapes to reduce the drag of supersonic vehicles. The more pointed and slender the body, the weaker the shock wave attached to the nose, and hence the smaller the wave drag. Consequently, the German V-2 rocket used during the last stages of World War II had a pointed nose, and all short-range rocket vehicles flown during the next decade followed suit. Then, in 1953, the first hydrogen bomb was exploded by the United States. This immediately spurred the development of long-range intercontinental ballistic missiles (ICBMs) to deliver such bombs. These vehicles were designed to fly outside the region of the earth’s atmosphere for distances of 5000 mi or more and to reenter the atmosphere at suborbital speeds of from 20,000 to 22,000 ft/s. At such high velocities, the aerodynamic heating of the reentry vehicle becomes severe, and this heating problem dominated the minds of high-speed aerodynamicists. Their first thinking was conventional—a sharp- pointed, slender reentry body. Efforts to minimize aerodynamic heating centered on the maintenance of laminar boundary layer flow on the vehicle’s surface; such laminar flow produces far less heating than turbulent flow (discussed in Chapters 15 and 19). However, nature much prefers turbulent flow, and reentry vehicles are no exception. Therefore, the pointed-nose reentry body was doomed to failure because it would burn up in the atmosphere before reaching the earth’s surface. However, in 1951, one of those major breakthroughs that come very infre- quently in engineering was created by H. Julian Allen at the NACA (National Advisory Committee for Aeronautics) Ames Aeronautical Laboratory—he in- troduced the concept of the blunt reentry body. His thinking was paced by the following concepts. At the beginning of reentry, near the outer edge of the atmo- sphere, the vehicle has a large amount of kinetic energy due to its high velocity and a large amount of potential energy due to its high altitude. However, by the time the vehicle reaches the surface of the earth, its velocity is relatively small and its altitude is zero; hence, it has virtually no kinetic or potential energy. Where has all the energy gone? The answer is that it has gone into (1) heating the body and (2) heating the airflow around the body. This is illustrated in Figure 1.9. Here, the shock wave from the nose of the vehicle heats the airflow around the vehicle; at the same time, the vehicle is heated by the intense frictional dissipation within the boundary layer on the surface. Allen reasoned that if more of the total reentry 10 PA RT 1 Fundamental Principles ve wa e o ck ir av Sh ats a kw he Hot boundary oc Sh layer Very high– Heat transfer into body from speed flow boundary layer Figure 1.9 Energy of reentry goes into heating both the body and the air around the body. energy could be dumped into the airflow, then less would be available to be trans- ferred to the vehicle itself in the form of heating. In turn, the way to increase the heating of the airflow is to create a stronger shock wave at the nose (i.e., to use a blunt-nosed body). The contrast between slender and blunt reentry bodies is illustrated in Figure 1.10. This was a stunning conclusion—to minimize aerody- namic heating, you actually want a blunt rather than a slender body. The result was so important that it was bottled up in a secret government document. Moreover, because it was so foreign to contemporary intuition, the blunt-reentry-body con- cept was accepted only gradually by the technical community. Over the next few years, additional aerodynamic analyses and experiments confirmed the validity of blunt reentry bodies. By 1955, Allen was publicly recognized for his work, receiving the Sylvanus Albert Reed Award of the Institute of the Aeronautical Sciences (now the American Institute of Aeronautics and Astronautics). Finally, in 1958, his work was made available to the public in the pioneering document NACA Report 1381 entitled “A Study of the Motion and Aerodynamic Heating of Ballistic Missiles Entering the Earth’s Atmosphere at High Supersonic Speeds.” Since Harvey Allen’s early work, all successful reentry bodies, from the first Atlas ICBM to the manned Apollo lunar capsule, have been blunt. Incidentally, Allen went on to distinguish himself in many other areas, becoming the director of the NASA Ames Research Center in 1965, and retiring in 1970. His work on the blunt reentry body is an excellent example of the importance of aerodynamics to space vehicle design. In summary, the purpose of this section has been to underscore the importance of aerodynamics in historical context. The goal of this book is to introduce the fundamentals of aerodynamics and to give the reader a much deeper insight to many technical applications in addition to the few described above. Aerodynamics is also a subject of intellectual beauty, composed and drawn by many great minds over the centuries. If you are challenged and interested by these thoughts, or even the least bit curious, then read on.  C H A PTER 1 Aerodynamics: Some Introductory Thoughts 11 Figure 1.10 Contrast of aerodynamic heating for slender and blunt reentry vehicles. (a) Slender reentry body. (b) Blunt reentry body. 1.2 AERODYNAMICS: CLASSIFICATION AND PRACTICAL OBJECTIVES A distinction between solids, liquids, and gases can be made in a simplistic sense as follows. Put a solid object inside a larger, closed container. The solid object will not change; its shape and boundaries will remain the same. Now put a liquid inside the container. The liquid will change its shape to conform to that of the container and will take on the same boundaries as the container up to the maximum depth of the liquid. Now put a gas inside the container. The gas will completely fill the container, taking on the same boundaries as the container. The word fluid is used to denote either a liquid or a gas. A more technical distinction between a solid and a fluid can be made as follows. When a force is applied tangentially to the surface of a solid, the solid will experience a finite deformation, and the tangential force per unit area—the shear stress—will usually be proportional to the amount of deformation. In contrast, when a tangential shear 12 PA RT 1 Fundamental Principles stress is applied to the surface of a fluid, the fluid will experience a continuously increasing deformation, and the shear stress usually will be proportional to the rate of change of the deformation. The most fundamental distinction between solids, liquids, and gases is at the atomic and molecular level. In a solid, the molecules are packed so closely together that their nuclei and electrons form a rigid geometric structure, “glued” together by powerful intermolecular forces. In a liquid, the spacing between molecules is larger, and although intermolecular forces are still strong, they allow enough movement of the molecules to give the liquid its “fluidity.” In a gas, the spacing between molecules is much larger (for air at standard conditions, the spacing between molecules is, on the average, about 10 times the molecular diameter). Hence, the influence of intermolecular forces is much weaker, and the motion of the molecules occurs rather freely throughout the gas. This movement of molecules in both gases and liquids leads to similar physical characteristics, the characteristics of a fluid—quite different from those of a solid. Therefore, it makes sense to classify the study of the dynamics of both liquids and gases under the same general heading, called fluid dynamics. On the other hand, certain differences exist between the flow of liquids and the flow of gases; also, different species of gases (say, N2 , He, etc.) have different properties. Therefore, fluid dynamics is subdivided into three areas as follows: Hydrodynamics—flow of liquids Gas dynamics—flow of gases Aerodynamics—flow of air These areas are by no means mutually exclusive; there are many similarities and identical phenomena between them. Also, the word “aerodynamics” has taken on a popular usage that sometimes covers the other two areas. As a result, this author tends to interpret the word aerodynamics very liberally, and its use throughout this book does not always limit our discussions just to air. Aerodynamics is an applied science with many practical applications in engi- neering. No matter how elegant an aerodynamic theory may be, or how mathemat- ically complex a numerical solution may be, or how sophisticated an aerodynamic experiment may be, all such efforts are usually aimed at one or more of the fol- lowing practical objectives: 1. The prediction of forces and moments on, and heat transfer to, bodies moving through a fluid (usually air). For example, we are concerned with the generation of lift, drag, and moments on airfoils, wings, fuselages, engine nacelles, and most importantly, whole airplane configurations. We want to estimate the wind force on buildings, ships, and other surface vehicles. We are concerned with the hydrodynamic forces on surface ships, submarines, and torpedoes. We need to be able to calculate the aerodynamic heating of flight vehicles ranging from the supersonic transport to a planetary probe entering the atmosphere of Jupiter. These are but a few examples.  C H A PTER 1 Aerodynamics: Some Introductory Thoughts 13 Figure 1.11 A CO2 -N2 gas-dynamic laser, circa 1969 (Courtesy of John Anderson). 2. Determination of flows moving internally through ducts. We wish to calculate and measure the flow properties inside rocket and air-breathing jet engines and to calculate the engine thrust. We need to know the flow conditions in the test section of a wind tunnel. We must know how much fluid can flow through pipes under various conditions. A recent, very interesting application of aerodynamics is high-energy chemical and gas-dynamic lasers (see Reference 1), which are nothing more than specialized wind tunnels that can produce extremely powerful laser beams. Figure 1.11 is a photograph of an early gas-dynamic laser designed in the late 1960s. The applications in item 1 come under the heading of external aerodynamics since they deal with external flows over a body. In contrast, the applications in item 2 involve internal aerodynamics because they deal with flows internally within ducts. In external aerodynamics, in addition to forces, moments, and aerodynamic heating associated with a body, we are frequently interested in the details of the flow field away from the body. For example, the communication blackout experienced by the space shuttle during a portion of its reentry trajectory is due to a concentration of free electrons in the hot shock layer around the body. We need to calculate the variation of electron density throughout such flow fields. Another example is the propagation of shock waves in a supersonic flow; for instance, does 14 PA RT 1 Fundamental Principles the shock wave from the wing of a supersonic airplane impinge upon and interfere with the tail surfaces? Yet another example is the flow associated with the strong vortices trailing downstream from the wing tips of large subsonic airplanes such as the Boeing 747. What are the properties of these vortices, and how do they affect smaller aircraft which happen to fly through them? The above is just a sample of the myriad applications of aerodynamics. One purpose of this book is to provide the reader with the technical background nec- essary to fully understand the nature of such practical aerodynamic applications. Figure 1.12 Road map for Chapter 1.  C H A PTER 1 Aerodynamics: Some Introductory Thoughts 15 1.3 ROAD MAP FOR THIS CHAPTER When learning a new subject, it is important for you to know where you are, where you are going, and how you can get there. Therefore, at the beginning of each chapter in this book, a road map will be given to help guide you through the material of that chapter and to help you obtain a perspective as to how the material fits within the general framework of aerodynamics. For example, a road map for Chapter 1 is given in Figure 1.12. You will want to frequently refer back to these road maps as you progress through the individual chapters. When you reach the end of each chapter, look back over the road map to see where you started, where you are now, and what you learned in between. 1.4 SOME FUNDAMENTAL AERODYNAMIC VARIABLES A prerequisite to understanding physical science and engineering is simply learn- ing the vocabulary used to describe concepts and phenomena. Aerodynamics is no exception. Throughout this book, and throughout your working career, you will be adding to your technical vocabulary list. Let us start by defining four of the most frequently used words in aerodynamics: pressure, density, temperature, and flow velocity.1 Consider a surface immersed in a fluid. The surface can be a real, solid surface such as the wall of a duct or the surface of a body; it can also be a free surface which we simply imagine drawn somewhere in the middle of a fluid. Also, keep in mind that the molecules of the fluid are constantly in motion. Pressure is the normal force per unit area exerted on a surface due to the time rate of change of momentum of the gas molecules impacting on (or crossing) that surface. It is important to note that even though pressure is defined as force “per unit area,” you do not need a surface that is exactly 1 ft2 or 1 m2 to talk about pressure. In fact, pressure is usually defined at a point in the fluid or a point on a solid surface and can vary from one point to another. To see this more clearly, consider a point B in a volume of fluid. Let dA = elemental area at B dF = force on one side of dA due to pressure Then, the pressure at point B in the fluid is defined as   dF p = lim dA → 0 dA The pressure p is the limiting form of the force per unit area, where the area of interest has shrunk to nearly zero at the point B.2 Clearly, you can see that pressure 1 A basic introduction to these quantities is given on pages 56–61 of Reference 2. 2 Strictly speaking, dA can never achieve the limit of zero, because there would be no molecules at point B in that case. The above limit should be interpreted as dA approaching a very small value, near zero in terms of our macroscopic thinking, but sufficiently larger than the average spacing between molecules on a microscopic basis. 16 PA RT 1 Fundamental Principles is a point property and can have a different value from one point to another in the fluid. Another important aerodynamic variable is density, defined as the mass per unit volume. Analogous to our discussion on pressure, the definition of density does not require an actual volume of 1 ft3 or 1 m3. Rather, it is a point property that can vary from point to point in the fluid. Again, consider a point B in the fluid. Let dv = elemental volume around B dm = mass of fluid inside dv Then, the density at point B is dm ρ = lim dv → 0 dv Therefore, the density ρ is the limiting form of the mass per unit volume, where the volume of interest has shrunk to nearly zero around point B. (Note that dv cannot achieve the value of zero for the reason discussed in the footnote concerning dA in the definition of pressure.) Temperature takes on an important role in high-speed aerodynamics (intro- duced in Chapter 7). The temperature T of a gas is directly proportional to the average kinetic energy of the molecules of the fluid. In fact, if KE is the mean molecular kinetic energy, then temperature is given by KE = 32 kT , where k is the Boltzmann constant. Hence, we can qualitatively visualize a high-temperature gas as one in which the molecules and atoms are randomly rattling about at high speeds, whereas in a low-temperature gas, the random motion of the molecules is relatively slow. Temperature is also a point property, which can vary from point to point in the gas. The principal focus of aerodynamics is fluids in motion. Hence, flow velocity is an extremely important consideration. The concept of the velocity of a fluid is slightly more subtle than that of a solid body in motion. Consider a solid object in translational motion, say, moving at 30 m/s. Then all parts of the solid are simultaneously translating at the same 30 m/s velocity. In contrast, a fluid is a “squishy” substance, and for a fluid in motion, one part of the fluid may be traveling at a different velocity from another part. Hence, we have to adopt a certain perspective, as follows. Consider the flow of air over an airfoil, as shown in Figure 1.13. Lock your eyes on a specific, infinitesimally small element of mass Figure 1.13 Illustration of flow velocity and streamlines.  C H A PTER 1 Aerodynamics: Some Introductory Thoughts 17 in the gas, called a fluid element, and watch this element move with time. Both the speed and direction of this fluid element can vary as it moves from point to point in the gas. Now fix your eyes on a specific fixed point in space, say, point B in Figure 1.13. Flow velocity can now be defined as follows: The velocity of a flowing gas at any fixed point B in space is the velocity of an infinitesimally small fluid element as it sweeps through B. The flow velocity V has both magnitude and direction; hence, it is a vector quantity. This is in contrast to p, ρ, and T , which are scalar variables. The scalar magnitude of V is frequently used and is denoted by V. Again, we emphasize that velocity is a point property and can vary from point to point in the flow. Referring again to Figure 1.13, a moving fluid element traces out a fixed path in space. As long as the flow is steady (i.e., as long as it does not fluctuate with time), this path is called a streamline of the flow. Drawing the streamlines of the flow field is an important way of visualizing the motion of the gas; we will frequently be sketching the streamlines of the flow about various objects. A more rigorous discussion of streamlines is given in Chapter 2. Finally, we note that friction can play a role internally in a flow. Consider two adjacent streamlines a and b as sketched in Figure 1.14. The streamlines are an infinitesimal distance, dy, apart. At point 1 on streamline b the flow velocity is V ; at point 2 on streamline a the flow velocity is slightly higher, V + dV. You can imagine that streamline a is rubbing against streamline b and, due to friction, exerts a force of magnitude dF f on streamline b acting tangentially toward the right. Furthermore, imagine this force acting on an elemental area dA, where dA is perpendicular to the y axis and tangent to the streamline b at point 1. The local shear stress, τ , at point 1 is   dF f τ = lim dA → 0 dA The shear stress τ is the limiting form of the magnitude of the frictional force per unit area, where the area of interest is perpendicular to the y axis and has shrunk V + dV a y V dFf b 2 dy 1 Figure 1.14 Generation of frictional force due to a velocity gradient in a flow. 18 PA RT 1 Fundamental Principles to nearly zero at point 1. Shear stress acts tangentially along the streamline. For the type of gases and liquids of interest in aerodynamic applications, the value of the shear stress at a point on a streamline is proportional to the spatial rate of change of velocity normal to the streamline at that point (i.e., for the flow illustrated in Figure 1.14, τ ∝ dV/dy). The constant of proportionality is defined as the viscosity coefficient, μ. Hence, dV τ =μ dy where dV/dy is the velocity gradient. In reality, μ is not really a constant; it is a function of the temperature of the fluid. We will discuss these matters in more detail in Section 1.11. From the above equation, we deduce that in regions of a flow field where the velocity gradients are small, τ is small and the influence of friction locally in the flow is small. On the other hand, in regions where the velocity gradients are large, τ is large and the influence of friction locally in the flow can be substantial. 1.4.1 Units Two consistent sets of units will be used throughout this book, SI units (Systeme International d’Unites) and the English engineering system of units. The basic units of force, mass, length, time, and absolute temperature in these two systems are given in Table 1.1. For example, units of pressure and shear stress are lb/ft2 or N/m2 , units of density are slug/ft3 or kg/m3 , and units of velocity are ft/s or m/s. When a consistent set of units is used, physical relationships are written without the need for conversion factors in the basic formulas; they are written in the pure form intended by nature. Consistent units will always be used in this book. For an extensive discussion on units and the significance of consistent units versus nonconsistent units, see pages 65–70 of Reference 2. The SI system of units (metric units) is the standard system of units throughout most of the world today. In contrast, for more than two centuries the English engineering system (or some variant) was the primary system of units in the United States and England. This situation is changing rapidly, especially in the aerospace industry in the United States and England. Nevertheless, a familiarity with both systems of units is still important today. For example, even though most engineering work in the future will deal with the SI units, there exists a huge bulk of Table 1.1 Force Mass Length Time Temp. SI Units Newton kilogram meter second Kelvin (N) (kg) (m) (s) (K) English pounds slug feet second deg. Rankine Engineering (lb) (ft) (s) (◦ R) Units  C H A PTER 1 Aerodynamics: Some Introductory Thoughts 19 present and past engineering literature written in the English engineering system, literature that will be used well into the future. The modern engineering student must be bilingual in these units, and must feel comfortable with both systems. For this reason, although many of the worked examples and end-of-the-chapter problems in this book are in the SI units, some are in the English engineering system of units. You are encouraged to join this bilingual spirit and to work to make yourself comfortable in both systems. 1.5 AERODYNAMIC FORCES AND MOMENTS At first glance, the generation of the aerodynamic force on a giant Boeing 747 may seem complex, especially in light of the complicated three-dimensional flow field over the wings, fuselage, engine nacelles, tail, etc. Similarly, the aerody- namic resistance on an automobile traveling at 55 mi/h on the highway involves a complex interaction of the body, the air, and the ground. However, in these and all other cases, the aerodynamic forces and moments on the body are due to only two basic sources: 1. Pressure distribution over the body surface 2. Shear stress distribution over the body surface No matter how complex the body shape may be, the aerodynamic forces and moments on the body are due entirely to the above two basic sources. The only mechanisms nature has for communicating a force to a body moving through a fluid are pressure and shear stress distributions on the body surface. Both pressure p and shear stress τ have dimensions of force per unit area (pounds per square foot or newtons per square meter). As sketched in Figure 1.15, p acts normal to the surface, and τ acts tangential to the surface. Shear stress is due to the “tugging action” on the surface, which is caused by friction between the body and the air (and is studied in great detail in Chapters 15 to 20). The net effect of the p and τ distributions integrated over the complete body surface is a resultant aerodynamic force R and moment M on the body, as sketched in Figure 1.16. In turn, the resultant R can be split into components, two sets of Figure 1.15 Illustration of pressure and shear stress on an aerodynamic surface. 20 PA RT 1 Fundamental Principles Figure 1.16 Resultant aerodynamic force and moment on the body. Figure 1.17 Resultant aerodynamic force and the components into which it splits. which are shown in Figure 1.17. In Figure 1.17, V∞ is the relative wind, defined as the flow velocity far ahead of the body. The flow far away from the body is called the freestream, and hence V∞ is also called the freestream velocity. In Figure 1.17, by definition, L ≡ lift ≡ component of R perpendicular to V∞ D ≡ drag ≡ component of R parallel to V∞ The chord c is the linear distance from the leading edge to the trailing edge of the body. Sometimes, R is split into components perpendicular and parallel to the chord, as also shown in Figure 1.17. By definition, N ≡ normal force ≡ component of R perpendicular to c A ≡ axial force ≡ component of R parallel to c The angle of attack α is defined as the angle between c and V∞. Hence, α is also the angle between L and N and between D and A. The geometrical relation between these two sets of components is, from Figure 1.17, L = N cos α − A sin α (1.1) D = N sin α + A cos α (1.2)  C H A PTER 1 Aerodynamics: Some Introductory Thoughts 21 Figure 1.18 Nomenclature for the integration of pressure and shear stress distributions over a two-dimensional body surface. Let us examine in more detail the integration of the pressure and shear stress distributions to obtain the aerodynamic forces and moments. Consider the two- dimensional body sketched in Figure 1.18. The chord line is drawn horizontally, and hence the relative wind is inclined relative to the horizontal by the angle of attack α. An x y coordinate system is oriented parallel and perpendicular, respec- tively, to the chord. The distance from the leading edge measured along the body surface to an arbitrary point A on the upper surface is su ; similarly, the distance to an arbitrary point B on the lower surface is sl. The pressure and shear stress on the upper surface are denoted by pu and τu , both pu and τu are functions of su. Similarly, pl and τl are the corresponding quantities on the lower surface and are functions of sl. At a given point, the pressure is normal to the surface and is oriented at an angle θ relative to the perpendicular; shear stress is tangential to the surface and is oriented at the same angle θ relative to the horizontal. In Figure 1.18, the sign convention for θ is positive when measured clockwise from the vertical line to the direction of p and from the horizontal line to the direction of τ. In Figure 1.18, all thetas are shown in their positive direction. Now con- sider the two-dimensional shape in Figure 1.18 as a cross section of an infinitely long cylinder of uniform section. A unit span of such a cylinder is shown in Figure 1.19. Consider an elemental surface area dS of this cylinder, where dS = (ds)(1) as shown by the shaded area in Figure 1.19. We are interested in the contribution to the total normal force N  and the total axial force A due to the pressure and shear stress on the elemental area dS. The primes on N  and A denote force per unit span. Examining both Figures 1.18 and 1.19, we see that the elemental normal and axial forces acting on the elemental surface dS on the 22 PA RT 1 Fundamental Principles p ␶ s l ds y x ␣ V Figure 1.19 Aerodynamic force on an element of the body surface. upper body surface are d Nu = − pu dsu cos θ − τu dsu sin θ (1.3) dAu = − pu dsu sin θ + τu dsu cos θ (1.4) On the lower body surface, we have d Nl = pl dsl cos θ − τl dsl sin θ (1.5) dAl = pl dsl sin θ + τl dsl cos θ (1.6) In Equations (1.3) to (1.6), the positive directions of N  and A are those shown in Figure 1.17. In these equations, the positive clockwise convention for θ must be followed. For example, consider again Figure 1.18. Near the leading edge of the body, where the slope of the upper body surface is positive, τ is inclined upward, and hence it gives a positive contribution to N . For an upward inclined τ , θ would be counterclockwise, hence negative. Therefore, in Equation (1.3), sin θ would be negative, making the shear stress term (the last term) a positive value, as it should be in this instance. Hence, Equations (1.3) to (1.6) hold in general (for both the forward and rearward portions of the body) as long as th