Basic Aerodynamics: Incompressible Flow

431 Pages·2011·10.18 MB·English

by  Gary A. Flandro

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 pro-
pulsion, 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.
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
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© Cambridge University Press 2012
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
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
Additional resources for this publication at www. cambridge.org/fl andro
Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external
or third-party Internet Web sites referred to in this publication and does not guarantee that any content
on such Web sites is, or will remain, accurate or appropriate.
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 under-
graduate 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
v
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 Defi ning 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 calcu-
lations 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 obs erve 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 prob-
lems using panel methods or nonlinear problems using fi nite-difference methods. Sample
ix
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 aerodynam ics 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 experi-
ence in aeronautics and associated disciplines. We wish you success in your study of aero-
dynamics 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
1 Basic Aerodynamics
This was the last fi rst in aviation, we had always said, a milestone, and that made it
unique. Would we do it again? No one can do it again. And that is the best thing about it.
Jeana Yeager and Dick Rutan
“Voyager” 1987
1.1 Introduction
Aerodynamics is the study of the fl ow of air around and within a moving object. Its
main objective is understanding the creation of forces by the interaction of the gas
motion with the surfaces of an object. Aerodynamics is closely related to hydrody-
namics and gasdynamics, which represent the motion of liquid and compressible-gas
fl ows, respectively.
Aerodynamics is the essence of fl ight and has been the focus of intensive
research for about a century. Although this might seem to be a rather long period of
development, it is really quite short considering the time span usually required for
the formulation and full solution of basic scientifi c problems. In this relatively short
time, mankind has advanced from the fi rst gliding and primitive-powered airplane
fl ights to interplanetary spacefl ight.
Perhaps the most important motivation for this rapid development is the
challenge to the human spirit represented by manned fl ight. However, practical
needs also have strongly affected these endeavors, and we often fi nd periods of
rapid growth in aerodynamic knowledge associated with the solution of compel-
ling problems in transportation, military applications, industry, and even sporting
competitions. Figure 1.1 illustrates this growth in terms of the maximum speed
attained by manned aircraft. Speed is a key measure of performance in almost
every aspect of fl ight. It is of obvious vital importance in commercial fl ight as
well as in military operations.
Even a casual study of the history of aviation yields considerable insight into the
pressures that have motivated periods of almost explosive growth in the technology
of fl ight. Much of the increase in speed during the 1940s and several subsequent
decades was motivated strongly by military considerations. However, notice that two
radical departures (shown as dashed lines in Fig. 1.1) from the curve for conven-
tional airplanes occur in the 1920s and in the two decades between 1940 and 1960.
1