Aerodynamics for Engineering Students

Aerodynamics for Engineering Students

Aerodynamics for Engineering Students Sixth Edition E.L. Houghton P.W. Carpenter Steven H. Collicott Daniel T. Valentine AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Butterworth-Heinemann is an imprint of Elsevier

Butterworth-Heinemann is an imprint of Elsevier 225 Wyman Street, Waltham, MA 02451, USA The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK c 2013 Elsevier, Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. MATLAB R is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book s use or discussion of MATLAB R software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB R software. Library of Congress Cataloging-in-Publication Data Aerodynamics for engineering students / E.L. Houghton... [et al.]. 6th ed. p. cm. ISBN: 978-0-08-096632-8 (pbk.) 1. Aerodynamics. 2. Airplanes Design and construction. I. Houghton, E. L. (Edward Lewis) TL570.H64 2012 629.132'5 dc23 2011047033 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. For information on all Butterworth-Heinemann publications visit our Web site at www.elsevierdirect.com Printed in the United States 12 13 14 15 16 17 18 10 9 8 7 6 5 4 3 2 1

Contents Preface... xv CHAPTER 1 Basic Concepts and Definitions... 1 1.1 Introduction... 1 1.1.1 Basic Concepts... 2 1.1.2 Measures of Dynamical Properties... 4 1.2 Units and Dimensions... 5 1.2.1 Fundamental Dimensions and Units... 6 1.2.2 Fractions and Multiples... 7 1.2.3 Units of Other Physical Quantities... 7 1.2.4 Imperial Units... 9 1.3 Relevant Properties... 12 1.3.1 Forms of Matter... 12 1.3.2 Fluids... 13 1.3.3 Pressure... 14 1.3.4 Temperature... 16 1.3.5 Density... 16 1.3.6 Viscosity... 17 1.3.7 Speed of Sound and Bulk Elasticity... 19 1.3.8 Thermodynamic Properties... 20 1.4 Aeronautical Definitions... 25 1.4.1 Airfoil Geometry... 25 1.4.2 Wing Geometry... 27 1.5 Dimensional Analysis... 29 1.5.1 Fundamental Principles... 29 1.5.2 Dimensional Analysis Applied to Aerodynamic Force.. 32 1.6 Basic Aerodynamics... 38 1.6.1 Aerodynamic Force and Moment... 38 1.6.2 Force and Moment Coefficients... 40 1.6.3 Pressure Distribution on an Airfoil... 41 1.6.4 Pitching Moment... 43 1.6.5 Types of Drag... 47 1.6.6 Estimation of Lift, Drag, and Pitching Moment Coefficients from the Pressure Distribution... 51 1.6.7 Induced Drag... 55 v

vi Contents 1.6.8 Lift-Dependent Drag... 58 1.6.9 Airfoil Characteristics... 58 1.7 Exercises... 65 CHAPTER 2 Fundamental Equations of Fluid Mechanics... 69 2.1 Introduction... 69 2.1.1 Selection of Coordinates... 70 2.1.2 A Comparison of Steady and Unsteady Flow... 71 2.2 One-Dimensional Flow: The Basic Equations... 73 2.2.1 One-Dimensional Flow: The Basic Equations of Conservation... 73 2.2.2 Comments on the Momentum and Energy Equations... 80 2.3 Measurement of Air Speed... 81 2.3.1 Pitˆot-Static Tube... 81 2.3.2 Pressure Coefficient... 82 2.3.3 Air-Speed Indicator: Indicated and Equivalent Air Speeds... 83 2.3.4 Incompressibility Assumption... 84 2.4 Two-Dimensional Flow... 87 2.4.1 Component Velocities... 88 2.4.2 Equation of Continuity or Conservation of Mass... 91 2.4.3 Equation of Continuity in Polar Coordinates... 93 2.5 Stream Function and Streamline... 94 2.5.1 Stream Function ψ... 94 2.5.2 Streamline... 96 2.5.3 Velocity Components in Terms of ψ... 97 2.6 Momentum Equation... 100 2.6.1 Euler Equations... 105 2.7 Rates of Strain, Rotational Flow, and Vorticity... 105 2.7.1 Distortion of Fluid Element in Flow Field... 106 2.7.2 Rate of Shear Strain... 107 2.7.3 Rate of Direct Strain... 108 2.7.4 Vorticity... 109 2.7.5 Vorticity in Polar Coordinates... 109 2.7.6 Rotational and Irrotational Flow... 110 2.7.7 Circulation... 110 2.8 Navier-Stokes Equations... 113 2.8.1 Relationship between Rates of Strain and Viscous Stresses... 113 2.8.2 Derivation of the Navier-Stokes Equations... 115

Contents vii 2.9 Properties of the Navier-Stokes Equations... 117 2.10 Exact Solutions of the Navier-Stokes Equations... 121 2.10.1 Couette Flow: Simple Shear Flow... 122 2.10.2 Plane Poiseuille Flow: Pressure-Driven Channel Flow. 122 2.10.3 Hiemenz Flow: Two-Dimensional Stagnation-Point Flow... 124 2.11 Prandtl s Boundary-Layer Equations... 128 2.11.1 Development of the Boundary Layer... 130 2.11.2 Boundary-Layer Thickness... 132 2.11.3 Nondimensional Profile... 132 2.11.4 Laminar and Turbulent Flows... 133 2.11.5 Growth along a Flat Surface... 134 2.11.6 Effects of an External Pressure Gradient... 136 2.12 Boundary-Layer Equations... 137 2.12.1 Derivation of the Laminar Boundary-Layer Equations. 138 2.12.2 Boundary-Layer Thickness for Laminar and Turbulent Flows... 143 2.12.3 Boundary-Layer/Potential-Flow Model of Airfoils and Wings... 144 2.13 Exercises... 144 CHAPTER 3 Potential Flow... 149 3.1 Two-Dimensional Flows... 149 3.1.1 The Velocity Potential... 153 3.1.2 The Equipotential... 155 3.1.3 Velocity Components in Terms of φ... 156 3.2 Standard Flows in Terms of ψ and φ... 157 3.2.1 Uniform Flow... 158 3.2.2 Two-Dimensional Flow from a Source (or towards a Sink)... 160 3.2.3 Doublet Located at (x, y)=(0, 0)... 162 3.2.4 Line (Point) Vortex... 163 3.2.5 Solid Boundaries and Image Systems... 165 3.2.6 Rankine Leading Edge... 168 3.2.7 Rankine Oval... 171 3.2.8 Circular Cylinder with Circulation in a Cross Flow... 175 3.2.9 Joukowski Airfoil and the Circular Cylinder... 182 3.3 Axisymmetric Flows (Inviscid and Incompressible Flows)... 183 3.3.1 Cylindrical Coordinate System... 184 3.3.2 Spherical Coordinates... 185

viii Contents 3.3.3 Axisymmetric Flow from a Point Source (or towards a Point Sink)... 186 3.3.4 Point Source and Sink in a Uniform Axisymmetric Flow... 187 3.3.5 The Point Doublet and the Potential Flow around a Sphere... 189 3.3.6 Flow around Slender Bodies... 192 3.4 Computational (Panel) Methods... 195 3.5 Exercises... 203 CHAPTER 4 Two-Dimensional Wing Theory... 209 4.1 Introduction... 209 4.1.1 The Kutta Condition... 211 4.1.2 Circulation and Vorticity... 213 4.1.3 Circulation and Lift (The Kutta Joukowski Theorem)... 218 4.2 The Development of Airfoil Theory... 220 4.3 General Thin-Airfoil Theory... 223 4.4 Solution to the General Equation... 228 4.4.1 Thin Symmetrical Flat-Plate Airfoil... 229 4.4.2 General Thin-Airfoil Section... 231 4.5 The Flapped Airfoil... 235 4.5.1 Hinge Moment Coefficient... 237 4.6 The Jet Flap... 240 4.7 Normal Force and Pitching Moment Derivatives Due to Pitching... 240 4.7.1 (Z q )(M q ) Wing Contributions... 241 4.8 Particular Camber Lines... 245 4.8.1 Cubic Camber Lines... 245 4.8.2 NACA Four-Digit Wing Sections... 249 4.9 The Thickness Problem for Thin-Airfoil Theory... 251 4.9.1 Thickness Problem for Thin Airfoils... 252 4.10 Computational (Panel) Methods for Two-Dimensional Lifting Flows... 256 4.11 Exercises... 265 CHAPTER 5 Wing Theory... 269 5.1 The Vortex System... 270 5.1.1 Starting Vortex... 270 5.1.2 Trailing Vortex System... 271

Contents ix 5.1.3 Bound Vortex System... 272 5.1.4 Horseshoe Vortex... 273 5.2 Laws of Vortex Motion... 273 5.2.1 Helmholtz s Theorems... 275 5.2.2 The Biot-Savart Law... 276 5.2.3 Variation of Velocity in Vortex Flow... 279 5.3 The Wing as a Simplified Horseshoe Vortex... 281 5.3.1 Influence of Downwash on the Tailplane... 285 5.3.2 Ground Effects... 286 5.4 Vortex Sheets... 289 5.4.1 Use of Vortex Sheets to Model the Lifting Effects of a Wing... 290 5.5 Relationship between Spanwise Loading and Trailing Vorticity... 294 5.5.1 Induced Velocity (Downwash)... 296 5.5.2 The Consequences of Downwash Trailing Vortex Drag... 299 5.5.3 Characteristics of Simple Symmetric Loading Elliptic Distribution... 302 5.5.4 General (Series) Distribution of Lift... 306 5.5.5 Aerodynamic Characteristics for Symmetrical General Loading... 309 5.6 Determination of Load Distribution on a Given Wing... 315 5.6.1 General Theory for Wings of High Aspect Ratio... 316 5.6.2 General Solution to Prandtl s Integral Equation... 318 5.6.3 Load Distribution for Minimum Drag... 323 5.7 Swept and Delta Wings... 325 5.7.1 Yawed Wings of Infinite Span... 326 5.7.2 Swept Wings of Finite Span... 327 5.7.3 Wings of Small Aspect Ratio... 330 5.8 Computational (Panel) Methods for Wings... 337 5.9 Exercises... 342 CHAPTER 6 Compressible Flow... 349 6.1 Introduction... 350 6.2 Isentropic One-Dimensional Flow... 352 6.2.1 Pressure, Density, and Temperature Ratios along a Streamline in Isentropic Flow... 355 6.2.2 Ratio of Areas at Different Sections of the Stream Tube in Isentropic Flow... 359

x Contents 6.2.3 Velocity along an Isentropic Stream Tube... 361 6.2.4 Variation of Mass Flow with Pressure... 363 6.3 One-Dimensional Flow: Weak Waves... 375 6.3.1 Speed of Sound (Acoustic Speed)... 376 6.4 One-Dimensional Flow: Plane Normal Shock Waves... 380 6.4.1 One-Dimensional Properties of Normal Shock Waves... 381 6.4.2 Pressure-Density Relations across the Shock... 381 6.4.3 Static Pressure Jump across a Normal Shock... 383 6.4.4 Density Jump across the Normal Shock... 384 6.4.5 Temperature Rise across the Normal Shock... 385 6.4.6 Entropy Change across the Normal Shock... 385 6.4.7 Mach Number Change across the Normal Shock... 386 6.4.8 Velocity Change across the Normal Shock... 386 6.4.9 Total Pressure Change across the Normal Shock... 388 6.4.10 Pitˆot Tube Equation... 389 6.4.11 Converging-Diverging Nozzle Operations... 391 6.5 Mach Waves and Shock Waves in Two-Dimensional Flow... 395 6.6 Mach Waves... 396 6.6.1 Mach Wave Reflection... 404 6.6.2 Mach Wave Interference... 407 6.7 Shock Waves... 407 6.7.1 Plane Oblique Shock Relations... 407 6.7.2 Shock Polar... 412 6.7.3 Two-Dimensional Supersonic Flow Past a Wedge... 419 6.8 Exercises... 422 CHAPTER 7 Airfoils and Wings in Compressible Flow... 427 7.1 Wings in Compressible Flow... 427 7.1.1 Transonic Flow: The Critical Mach Number... 427 7.1.2 Subcritical Flow: The Small-Perturbation Theory (Prandtl-Glauert Rule)... 431 7.1.3 Supersonic Linearized Theory (Ackeret s Rule)... 446 7.1.4 Other Aspects of Supersonic Wings... 470 7.2 Exercises... 476 CHAPTER 8 Viscous Flow and Boundary Layers... 479 8.1 Introduction... 479 8.2 Boundary-Layer Theory... 484 8.2.1 Blasius s Solution... 485 8.2.2 Definitions of Boundary-Layer Thickness... 487

Contents xi 8.2.3 Skin-Friction Drag... 491 8.2.4 Laminar Boundary-Layer Thickness along a Flat Plate... 495 8.2.5 Solving the General Case... 496 8.3 Boundary-Layer Separation... 498 8.3.1 Separation Bubbles... 500 8.4 Flow Past Cylinders and Spheres... 501 8.4.1 Turbulence on Spheres... 507 8.4.2 Golf Balls... 509 8.4.3 Cricket Balls... 509 8.5 The Momentum-Integral Equation... 511 8.5.1 An Approximate Velocity Profile for the Laminar Boundary Layer... 515 8.6 Approximate Methods for a Boundary Layer on a Flat Plate with Zero Pressure Gradient... 519 8.6.1 Simplified Form of the Momentum-Integral Equation... 519 8.6.2 Rate of Growth of a Laminar Boundary Layer on a Flat Plate... 520 8.6.3 Drag Coefficient for a Flat Plate of Streamwise Length L with a Wholly Laminar Boundary Layer... 520 8.6.4 Turbulent Velocity Profile... 521 8.6.5 Rate of Growth of a Turbulent Boundary Layer on a Flat Plate... 523 8.6.6 Drag Coefficient for a Flat Plate with a Wholly Turbulent Boundary Layer... 527 8.6.7 Conditions at Transition... 528 8.6.8 Mixed Boundary-Layer Flow on a Flat Plate with Zero Pressure Gradient... 529 8.7 Additional Examples of the Momentum-Integral Equation... 534 8.8 Laminar-Turbulent Transition... 538 8.9 The Physics of Turbulent Boundary Layers... 545 8.9.1 Reynolds Averaging and Turbulent Stress... 545 8.9.2 Boundary-Layer Equations for Turbulent Flows... 548 8.9.3 Eddy Viscosity... 549 8.9.4 Prandtl s Mixing-Length Theory of Turbulence... 553 8.9.5 Regimes of Turbulent Wall Flow... 554 8.9.6 Formulae for Local Skin-Friction Coefficient and Drag... 556

xii Contents 8.9.7 Distribution of Reynolds Stresses and Turbulent Kinetic Energy across the Boundary Layer... 558 8.9.8 Turbulence Structures in the Near-Wall Region... 558 8.10 Computational Methods... 565 8.10.1 Methods Based on the Momentum-Integral Equation.. 565 8.10.2 Transition Prediction... 569 8.10.3 Computational Solution for the Laminar Boundary-Layer Equations... 570 8.10.4 Computational Solution for Turbulent Boundary Layers... 575 8.10.5 Zero-Equation Methods... 576 8.10.6 k ε: A Typical Two-Equation Method... 577 8.10.7 Large-Eddy Simulation... 579 8.11 Estimation of Profile Drag from the Velocity Profile in a Wake... 581 8.11.1 Momentum-Integral Expression for the Drag of a Two-Dimensional Body... 581 8.11.2 Jones s Wake Traverse Method for Determining Profile Drag... 582 8.11.3 Growth Rate of a Two-Dimensional Wake Using the General Momentum-Integral Equation... 584 8.12 Some Boundary-Layer Effects in Supersonic Flow... 587 8.12.1 Near-Normal Shock Interaction with the Laminar Boundary Layer... 588 8.12.2 Near-Normal Shock Interaction with the Turbulent Boundary Layer... 589 8.12.3 Shock-Wave/Boundary-Layer Interaction in Supersonic Flow... 590 8.13 Exercises... 598 CHAPTER 9 Flow Control and Wing Design... 601 9.1 Introduction... 601 9.2 Maximizing Lift for Single-Element Airfoils... 602 9.3 Multi-Element Airfoils... 608 9.3.1 The Slat Effect... 611 9.3.2 The Flap Effect... 612 9.3.3 Off-the-Surface Recovery... 612 9.3.4 Fresh Boundary-Layer Effect... 614 9.3.5 The Gurney Flap... 615 9.3.6 Movable Flaps: Artificial Bird Feathers... 619

Contents xiii 9.4 Boundary Layer Control Prevention to Separation... 621 9.4.1 Boundary-Layer Suction... 622 9.4.2 Control by Tangential Blowing... 623 9.4.3 Other Methods of Separation Control... 630 9.5 Reduction of Skin-Friction Drag... 631 9.5.1 Laminar Flow Control by Boundary-Layer Suction... 631 9.5.2 Compliant Walls: Artificial Dolphin Skins... 633 9.5.3 Riblets... 636 9.6 Reduction of Form Drag... 638 9.7 Reduction of Induced Drag... 639 9.8 Reduction of Wave Drag... 642 CHAPTER 10 Propulsion Devices... 645 10.1 Froude s Momentum Theory of Propulsion... 645 10.2 Airscrew Coefficients... 652 10.2.1 Thrust Coefficient... 652 10.2.2 Torque Coefficient... 654 10.2.3 Efficiency... 654 10.2.4 Power Coefficient... 654 10.2.5 Activity Factor... 655 10.3 Airscrew Pitch... 659 10.3.1 Geometric Pitch... 659 10.3.2 Effect of Geometric Pitch on Airscrew Performance... 660 10.3.3 Experimental Mean Pitch... 662 10.4 Blade-Element Theory... 662 10.4.1 Vortex System of an Airscrew... 662 10.4.2 Performance of a Blade Element... 664 10.5 The Momentum Theory Applied to the Helicopter Rotor... 671 10.5.1 Actuator Disc in Hovering Flight... 671 10.5.2 Vertical Climbing Flight... 672 10.5.3 Slow, Powered Descending Flight... 673 10.5.4 Translational Helicopter Flight... 673 10.6 The Rocket Motor... 675 10.6.1 Free Motion of a Rocket-Propelled Body... 677 10.7 The Hovercraft... 682 10.8 Exercises... 685 Appendix A: Symbols and Notation... 689 Appendix B... 695

xiv Contents Appendix C: A Solution of Integrals of the Type of Glauert s Integral... 701 Appendix D: Conversion of Imperial Units to Systéme International (SI) Units... 705 Bibliography... 707 Index... 715

Preface This volume is intended for engineering students in introductory aerodynamics courses and as a reference useful for reviewing foundational topics for graduate courses. The sequence of subject development in this edition begins with definitions and concepts and then moves on to incompressible flow, low speed airfoil and wing theories, compressible flow, high speed wing theories, viscous flow, boundary layers, transition and turbulence, wing design, and concludes with propellers and propulsion. Reinforcing or teaching first the units, dimensions, and properties of the physical quantities used in aerodynamics addresses concepts that are perhaps both the simplest and the most critical. Common aeronautical definitions are covered before lessons on the aerodynamic forces involved and how the forces drive our definitions of airfoil characteristics. The fundamental fluid dynamics required for the development of aerodynamic studies and the analysis of flows within and around solid boundaries for air at subsonic speeds is explored in depth in the next two chapters. Classical airfoil and wing theories for the estimation of aerodynamic characteristics in these regimes are then developed. Attention is then turned to the aerodynamics of high speed air flows in Chapters 6 and 7. The laws governing the behavior of the physical properties of air are applied to the transonic and supersonic flow speeds and the aerodynamics of the abrupt changes in the flow characteristics at these speeds, shock waves, are explained. Then compressible flow theories are applied to explain the significant effects on wings in transonic and supersonic flight and to develop appropriate aerodynamic characteristics. Viscosity is a key physical quantity of air and its significance in aerodynamic situations is next considered in depth. The powerful concept of the boundary layer and the development of properties of various flows when adjacent to solid boundaries create a body of reliable methods for estimating the fluid forces due to viscosity. In aerodynamics, these forces are notably skin friction and profile drag. Chapters on wing design and flow control, and propellers and propulsion, respectively, bring together disparate aspects of the previous chapters as appropriate. This permits discussion of some practical and individual applications of aerodynamics. Obviously aerodynamic design today relies extensively on computational methods. This is reflected in part in this volume by the introduction, where appropriate, of descriptions and discussions of relevant computational techniques. However, this text is aimed at providing the fundamental fluid dynamics or aerodynamics background necessary for students to move successfully into a dedicated course on computation methods or experimental methods. As such, experience in computational techniques or experimental techniques are not required for a complete understanding of the aerodynamics in this book. The authors urge students onward to such advanced courses and exciting careers in aerodynamics. xv

xvi Preface ADDITIONAL RESOURCES A set of.m files for the MATLAB routines in the book are available by visiting the book s companion site, www.elsevierdirect.com and searching on houghton. Instructors using the text for a course may access the solutions manual and image bank by visiting www.textbooks.elsevier.com and following the online registration instructions. ACKNOWLEDGEMENTS The authors thank the following faculty, who provided feedback on this project through survey responses, review of proposal, and/or review of chapters: Alina Alexeenko Purdue University S. Firasat Ali Tuskegee University David Bridges Mississippi State University Russell M. Cummings California Polytechnic State University Paul Dawson Boise State University Simon W. Evans, Ph.D Worcester Polytechnic Institute Richard S. Figliola Clemson University Timothy W. Fox California State University Northridge Ashok Gopalarathnam North Carolina State University Dr. Mark W. Johnson University of Liverpool Brian Landrum, Ph.D University of Alabama in Huntsville Gary L. Solbrekken University of Missouri Mohammad E. Taslim Northeastern University Valana Wells Arizona State University Professors Collicott and Valentine are grateful for the opportunity to continue the work of Professors Houghton and Carpenter and thank Joe Hayton, Publisher, for the invitation to do so. In addition, the professional efforts of Mike Joyce, Editorial Program Manager, Heather Tighe, Production Manager, and Kristen Davis, Designer are instrumental in the creation of this sixth edition. The products of one s efforts are of course the culmination of all of one s experiences with others. Foremost amongst the people who are to be thanked most warmly for support are our families. Collicott and Valentine thank Jennifer, Sarah, and Rachel and Mary, Clara, and Zach T., respectively, for their love and for the countless joys that they bring to us. Our Professors and students over the decades are major contributors to our aerodynamics knowledge and we are thankful for them. The authors share their deep gratitude for God s boundless love and grace for all.