Basic Aerodynamics: Incompressible Flow Gary A

Cambridge University Press 978-0-521-80582-7 - Basic Aerodynamics: Incompressible Flow Gary A. Flandro, Howard M. McMahon and Robert L. Roach Frontmatter More information

BASIC AERODYNAMICS

In the rapidly advancing fi eld of fl ight aerodynamics, it is important for students to completely master the fundamentals. This textbook, written by renowned experts, clearly presents the basic concepts underlying aerodynamic prediction methodology. These concepts are closely linked to physical principles so that they may be more readily retained and their limits of applicability fully appreciated. The ultimate goal is to provide the student with the necessary tools to confi dently approach and solve practical fl ight-vehicle design problems of current and future interest. The text is designed for use in a course in aerodynamics at the advanced undergraduate or graduate level. A comprehensive set of exercise problems is included at the end of each chapter.

Gary A. Flandro is the Boling Chair (Emeritus) of Mechanical and Aerospace Engineering at the University of Tennessee Space Institute. He is also Vice President and Chief Engineer of Gloyer-Taylor Laboratories, LLC. He is a Fellow of the American Institute of Aeronautics and Astronautics. His research interests include acoustics, aerodynamics, rocket propulsion, ﬂ ight mechanics and performance, hypersonic aerodynamics, propulsion, and vehicle design. Dr. Flandro received the National Aeronautics and Space Administration Exceptional Achievement Medal (1998) and the Fédération Aéronautique Internationale Diamond Soaring Badge (1979) for his work.

Dr. Howard McMahon was a Professor Emeritus of Aerospace Engi- neering at the Georgia Institute of Technology. Following his graduate work at Santa Clara and doctoral research at the California Institute of Technology, he worked for CARDE (the Canadian Armament and Research Development Establishment) near Quebec City, with top rocketry researchers in Canada, including Gerald Bull. His desire to teach and expand his research work led him to the Georgia Institute of Technology, where he guided undergraduate and graduate students for 26 years. His particular area of focus was in aerodynamics, and he spent many years guiding research in the university wind tunnel with pro- jects involving rotary aviation, compressible ﬂ ow, and ﬂ uid mechanics. Following his retirement from the university in 1990, he continued to col- laborate with both teaching and research faculty until his death in 2008.

Robert L. Roach currently teaches mathematics and science at the Kfar Hayarok School in Ramat Hasharon, Israel. He was formerly an Assistant Professor of Aerospace Engineering at the University of Tennessee Space Institute. He also taught aerodynamics, rocket propulsion, and mathematics at the Georgia Institute of Technology and at the U. S. Air Force Institute of Technology. He is an Associate Fellow of the American Institute of Aeronautics and Astronautics. His research interests include numerical solutions of the canonical equations of engineering, propulsion, and many aspects of solar energy.

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-80582-7 - Basic Aerodynamics: Incompressible Flow Gary A. Flandro, Howard M. McMahon and Robert L. Roach Frontmatter More information

Cambridge Aerospace Series

Editors: Wei Shyy and Michael J. Rycroft

1. J. M. Rolfe and K. J. Staples (eds.): Flight Simulation 2. P. Berlin: The Geostationary Applications Satellite 3. M. J. T. Smith: Aircraft Noise 4. N. X. Vinh: Flight Mechanics of High-Performance Aircraft 5. W. A. Mair and D. L. Birdsall: Aircraft Performance 6. M. J. Abzug and E. E. Larrabee: Airplane Stability and Control 7. M. J. Sidi: Spacecraft Dynamics and Control 8. J. D. Anderson: A History of Aerodynamics 9. A. M. Cruise, J. A. Bowles, C. V. Goodall, and T. J. Patrick: Principles of Space Instrument Design 10. G. A. Khoury (ed.): Airship Technology, 2nd Edition 11. J. P. Fielding: Introduction to Aircraft Design 12. J. G. Leishman: Principles of Helicopter Aerodynamics, 2nd Edition 13. J. Katz and A. Plotkin: Low-Speed Aerodynamics, 2nd Edition 14. M. J. Abzug and E. E. Larrabee: Airplane Stability and Control: A History of the Technologies that Made Aviation Possible, 2nd Edition 15. D. H. Hodges and G. A. Pierce: Introduction to Structural Dynamics and Aeroelasticity, 2nd Edition 16. W. Fehse: Automatic Rendezvous and Docking of Spacecraft 17. R. D. Flack: Fundamentals of Jet Propulsion with Applications 18. E. A. Baskharone: Principles of Turbomachinery in Air-Breathing Engines 19. D. D. Knight: Numerical Methods for High-Speed Flows 20. C. A. Wagner, T. Hüttl, and P. Sagaut (eds.): Large-Eddy Simulation for Acoustics 21. D. D. Joseph, T. Funada, and J. Wang: Potential Flows of Viscous and Viscoelastic Fluids 22. W. Shyy, Y. Lian, H. Liu, J. Tang, and D. Viieru: Aerodynamics of Low Reynolds Number Flyers 23. J. H. Saleh: Analyses for Durability and System Design Lifetime 24. B. K. Donaldson: Analysis of Aircraft Structures, 2nd Edition 25. C. Segal: The Scramjet Engine: Processes and Characteristics 26. J. F. Doyle: Guided Explorations of the Mechanics of Solids and Structures 27. A. K. Kundu: Aircraft Design 28. M. I. Friswell, J. E. T. Penny, S. D. Garvey, and A. W. Lees: Dynamics of Rotating Machines 29. B. A. Conway (ed): Spacecraft Trajectory Optimization 30. R. J. Adrian and J. Westerweel: Particle Image Velocimetry 31. G. A. Flandro, H. M. McMahon, and R. L. Roach: Basic Aerodynamics 32. H. Babinsky, and J. K. Harvey: Shock Wave–Boundary-Layer Interactions 33. C. K. W. Tam: Computational Aeroacoustics: A Wave Number Approach

Basic Aerodynamics

Incompressible Flow

Gary A. Flandro University of Tennessee Space Institute

Howard M. McMahon Georgia Institute of Technology

Robert L. Roach formerly University of Tennessee Space Institute

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Tokyo, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521805827

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First published 2012

Printed in the United States of America

A catalog record for this publication is available from the British Library.

Library of Congress Cataloging in Publication data Flandro, G. A. (Gary A.), 1934– Basic aerodynamics : incompressible ﬂ ow / Gary A. Flandro, Howard M. McMahon, Robert L. Roach. p. cm. – (Cambridge Aerospace Series) Includes bibliographical references and index. ISBN 978-0-521-80582-7 (hardback) 1. Air ﬂ ow – Textbooks. 2. Aerodynamics – Textbooks. I. McMahon, Howard M. (Howard Martin), 1927– II. Roach, Robert L. III. Title. IV. Series. TL574.F5F53 2011 629.132'3–dc23 2011022458

ISBN 978-0-521-80582-7 Hardback

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For Howard

Dr. Howard McMahon spent 26 years of his career as a professor and researcher at the Georgia Institute of Technology in Atlanta. Having been born just a few weeks after Charles Lindbergh’s fl ight across the Atlantic, he moved through the age of biplanes to the modern era of jets and rockets, working fi rst as a researcher and followed by his years as a teaching professor. His retirement from education allowed him to focus his energy on the completion of this textbook in collaboration with Dr. Gary Flandro and former student Dr. Bob Roach. Though he did not live to see the benefi ts of this book for aerospace students, we, his wife and children, are happy and proud that his work will be recognized.

Dr. McMahon’s tenure at Georgia Tech occurred during a time of rapid change within the university. His work in the classroom and as a departmental colleague and leader helped to deﬁ ne the Aerospace Engineering department’s identity and solidify the school’s recognition for excellence and high standards. Dr. McMahon’s strengths were his attention to meticulous detail and understanding of people. He directed the undergraduate ﬂ uid dynamics laboratories and guided students through cooperative work- study programs, placing them in research and laboratory settings throughout the country.

Whether at the university or at home, Dr. McMahon’s ofﬁ ce was an obvious destin- ation for anyone who needed advice on how to handle a difﬁ cult problem. He blended a knowledge and love of people with an ability to always make the complex seem simple. He shared his talents with much energy and passion during his tenure at the university, having taught more than 4000 aerospace engineers. We know that through the gift of this textbook many more people will learn from his wisdom and insights. As family members, we were fortunate to have that experience throughout our own lives.

With loving memory,

The McMahon Family June 2011

Contents

Preface ...... page ix 1 Basic Aerodynamics ...... 1 1.1 Introduction ...... 1 1.2 The Fundamental Problem of Aerodynamics ...... 9 1.3 Plan for Study of Aerodynamics...... 12 2 Physics of Fluids ...... 15 2.1 Aerodynamic Forces ...... 15 2.2 Aerodynamic Variables...... 22 2.3 Mathematical Description of Fluid Flows ...... 35 2.4 Behavior of Gases at Rest: Fluid Statics ...... 39 2.5 Summary...... 44 3 Equations of Aerodynamics ...... 48 3.1 Introduction ...... 48 3.2 Approach ...... 48 3.3 Physical Laws for Motion of a System ...... 50 3.4 Physical Laws in Control-Volume Form ...... 54 3.5 Physical Laws in Differential-Equation Form ...... 80 3.6 Properties of the Deﬁning Equations ...... 100 3.7 Boundary Conditions...... 100 3.8 Solution Procedures ...... 101 3.9 Summary...... 104 4 Fundamentals of Steady, Incompressible, Inviscid Flows...... 110 4.1 Introduction ...... 110 4.2 Basic Building Blocks...... 112 4.3 Special Solutions of the Conservation Equations ...... 129 4.4 Solving the Conservation Equations ...... 138 4.5 Elementary Solutions...... 141 4.6 Superposition of Elementary Solutions...... 150 4.7 The Kutta–Joukouski Theorem ...... 161 4.8 The Kutta Condition ...... 162 4.9 The Starting Vortex: Kelvin’s Theorem ...... 164 4.10 Summary...... 165 5 Two-Dimensional Airfoils...... 169 5.1 Introduction ...... 169 5.2 The Joukowski Airfoil ...... 173 5.3 The NACA Series of Airfoils ...... 177 5.4 Thin-Airfoil Theory...... 179 5.5 Thin Airfoil with a Flap...... 204 5.6 Distributed Singularity (Panel) Numerical Methods ...... 205 5.7 Inverse Methods of Solution...... 212 5.8 Summary...... 214

vii

viii Contents

6 Incompressible Flow about Wings of Finite Span...... 218 6.1 Introduction ...... 218 6.2 The Biot-Savart Law ...... 223 6.3 Prandtl Lifting-Line Theory ...... 226 6.4 Wing-Panel Methods ...... 252 6.5 Comments on Wing-Analysis Methods ...... 264 6.6 Aerodynamic Strip Theory...... 264 6.7 Ground Effect...... 266 6.8 Winglets ...... 268 6.9 Vortex Lift ...... 269 6.10 Strakes and Canards ...... 274 7 Axisymmetric, Incompressible Flow around a Body of Revolution. . . 281 7.1 Introduction ...... 281 7.2 Axisymmetric Continuity and Momentum Equations...... 283 7.3 Deﬁ ning Equation for the Velocity Potential...... 287 7.4 Deﬁ ning Equation for the Stream Function ...... 288 7.5 Three-Dimensional Point Source at the Origin of Coordinates . . . 289 7.6 Incompressible Flow around a Sphere ...... 291 7.7 Elementary Solutions for the Stream Function in Axisymmetric Flow ...... 292 7.8 Superposition of Uniform Flow and a Source Flow ...... 294 7.9 Flow Past a Rankine Body ...... 295 7.10 Flow Past a General Body of Revolution...... 296 7.11 Numerical Methods...... 298 7.12 Numerical Methods for the Complete Airplane ...... 305 8 Viscous Incompressible Flow ...... 309 8.1 Introduction ...... 309 8.2 Navier–Stokes Equations ...... 312 8.3 Exact Solutions of the Navier–Stokes Equations ...... 321 8.4 Role of the Reynolds Number...... 325 8.5 The Prandtl Boundary-Layer Equations ...... 328 8.6 Incompressible Boundary-Layer Theory ...... 334 8.7 Results from the Solution of the Blasius Equation ...... 347 8.8 Boundary Layer with a Streamwise Pressure Gradient ...... 353 8.9 Free-Shear Layers, Wakes, and Jets ...... 374 8.10 Transition to Turbulence ...... 376 8.11 Turbulent Flow ...... 380 9 Incompressible Aerodynamics: Summary ...... 393 9.1 Introduction ...... 393 9.2 Prediction of Lift and Drag on a Flight Vehicle ...... 395 9.3 Aircraft-Performance Calculations ...... 405 9.4 Extension to High-Speed Flight ...... 416 Index ...... 419

Preface

This textbook presents the fundamentals of aerodynamic analysis. Major emphasis is on in viscid fl ows whenever this simplifi cation is appropriate, but viscous effects also are discussed in more detail than is usually found in a textbook at this level. There is continual attention to practical applications of the material. For example, the concluding chapter demonstrates how aerodynamic analysis can be used to predict and improve the performance of fl ight vehicles. The ma terial is suitable for a semester course on aerodynamics or fl uid mechanics at the junior/ senior undergraduate level and for fi rst-year graduate students. It is assumed that the student has a sound background in calculus, vector analysis, mechanics, and basic thermodynamics and physics. Access to a digital computer is required and an understanding of computer program- ming is desirable but not necessary. Computational methods are introduced as required to solve complex problems that cannot be handled analytically. The objective of this textbook is to present in a clear and orderly manner the basic concepts underlying aerodynamic-prediction methodology. The ultimate goal is for the student to be able to use confi dently various solution methods in the analysis of practical problems of current and future interest. Today, it is important for the student to master the basics because technology is advancing at such a rate that a more directed or specifi c approach often is rapidly outdated. In this book, the basic concepts are linked closely to physical principles so that they may be under- stood and retained and the limits of applicability of the concepts can be appreciated. Numerous example problems stress solution methods and the order of magnitude of key parameters. A comprehensive set of problems for home study is included at the end of each chapter. Physical insights are developed primarily by constructing analytical solutions to important aerodynamics problems. In doing this, we follow the example set by Theodore von Kàrmàn and we subscribe to Dr. Küchemann’s concept of “ingenious abstractions and approximations.” However, after graduation, the student in the workplace will encounter many numerical-analysis techniques and solutions. Thus, the textbook introduces the fundamentals of modern numerical methods (as they are used in aerodynamics and fl uid mechanics) as well. Physical understanding plays a valuable role in computational analysis because it provides an important check on the expected ranges of magnitude of numerical solutions that are generated by these techniques. A feature of this textbook is a companion Web site (www.cambridge.org/fl andro) that con- tains numerical-analysis codes of three types: (1) codes for performing routine algebraic calculations for evaluating atmospheric properties or compressible fl ow properties, which are often found only in tables or charts; (2) menu-driven codes that allow the student to observe the effects of parametric variations on solutions that are developed in the text; and (3) numerical- analysis codes for complex fl ow problems. The latter codes arise when solving linear problems using panel methods or nonlinear problems using fi nite-difference methods. Sample

x Preface

applications of these codes are presented as needed to illustrate their use in addressing realistic aerodynamics problems. The authors express their gratitude to members of the aerodynamics faculties at Georgia Institute of Technology (GIT) and the University of Tennessee Space Institute (UTSI) for many helpful discussions during the writing of this textbook. In addition, Professors Jagoda (GIT) and Collins (UTSI) kindly used draft copies of certain chapters in their classes to provide valuable feedback. We are indebted to Professors Harper and Hubbartt of GIT for allowing the use of materials developed in their classroom notes. Finally, the fi rst two authors thank their teachers and research advisors for insight into the inner workings of fl uid mechanics and aerodynamics attained dur ing their graduate studies in aeronautics at the California Institute of Technology. Giants such as Clark Millikan, Hans Liepmann, Lester Lees, Frank Marble, and Anatol Roshko deserve special mention for their infl uence on our understanding of this subject. The three authors of this book represent more than 90 years of teaching and practical experience in aeronautics and associated disciplines. We wish you success in your study of aerodynamics and hope that it is as fulfi lling to you as it has been to us.

Gary A. Flandro Howard M. McMahon Robert L. Roach