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Page 1: Ethirajan Rathakrishnan Theoretical Aerodynamics · Rathakrishnan Theoretical Aerodynamics Theoretical ... Theoretical Aerodynamics is a user-friendly text for a full course on ...

Ethirajan Rathakrishnan

Rathakrishnan

TheoreticalAerodynamics

TheoreticalAerodynamics

TheoreticalAerodynamics

DO NOT PRINT PANTONE 032 RED GUIDELINES. FOR PROOFING ONLY.

Ethirajan Rathakrishnan Indian Institute of Technology Kanpur, India

Theoretical Aerodynamics is a user-friendly text for a full course on theoretical aerodynamics. The author systematically introduces aerofoil theory, its design features and performance aspects,beginning with the basics required, and then gradually proceeding to a higher level. The mathematicsinvolved is presented so that it can be followed comfortably, even by those who are not strong inmathematics. The examples are designed to fix the theory studied in an effective manner. Throughoutthe book, the physics behind the processes are clearly explained. Each chapter begins with anintroduction and ends with a summary and exercises.

This book is intended for graduate and advanced undergraduate students of Aerospace Engineering, as well as researchers and designers working in the area of aerofoil and blade design.

• Provides a complete overview of the technical terms, vortex theory, lifting line theory, and numerical methods

• Presented in an easy-to-read style making full use of figures and illustrations to enhanceunderstanding, and moves from simpler to more advanced topics

• Includes a complete section on fluid mechanics and thermodynamics, essential background topics for aerodynamic theory

• Blends mathematical and physical concepts of design and performance aspects of lifting surfaces, and introduces the reader to thin aerofoil theory, panel method, and finite aerofoil theory

• Includes a Solutions Manual for end-of-chapter exercises, and Lecture slides on the book’sCompanion Website

www.wiley.com/go/rathakrishnan

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THEORETICALAERODYNAMICS

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THEORETICALAERODYNAMICS

Ethirajan RathakrishnanIndian Institute of Technology Kanpur, India

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This edition first published 2013© 2013 John Wiley & Sons Singapore Pte. Ltd.

Registered officeJohn Wiley & Sons Singapore Pte. Ltd., 1 Fusionopolis Walk, #07-01 Solaris South Tower, Singapore 138628

For details of our global editorial offices, for customer services and for information about how to apply for permission toreuse the copyright material in this book please see our website at www.wiley.com.

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any formor by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as expressly permitted bylaw, without either the prior written permission of the Publisher, or authorization through payment of the appropriatephotocopy fee to the Copyright Clearance Center. Requests for permission should be addressed to the Publisher, John Wiley& Sons Singapore Pte. Ltd., 1 Fusionopolis Walk, #07-01 Solaris South Tower, Singapore 138628, tel: 65-66438000,fax: 65-66438008, email: [email protected].

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available inelectronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and productnames used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. ThePublisher is not associated with any product or vendor mentioned in this book. This publication is designed to provideaccurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that thePublisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, theservices of a competent professional should be sought.

Library of Congress Cataloging-in-Publication Data

Rathakrishnan, E.Theoretical aerodynamics / Ethirajan Rathakrishnan.

pages cmIncludes bibliographical references and index.ISBN 978-1-118-47934-6 (cloth)

1. Aerodynamics. I. Title.TL570.R33 2013629.132′3–dc23

2012049232

Typeset in 9/11pt Times by Thomson Digital, Noida, India

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This book is dedicated to my parents,

Mr Thammanur Shunmugam Ethirajanand

Mrs Aandaal Ethirajan

Ethirajan Rathakrishnan

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ContentsAbout the Author xv

Preface xvii

1 Basics 11.1 Introduction 11.2 Lift and Drag 11.3 Monoplane Aircraft 4

1.3.1 Types of Monoplane 51.4 Biplane 5

1.4.1 Advantages and Disadvantages 61.5 Triplane 6

1.5.1 Chord of a Profile 71.5.2 Chord of an Aerofoil 8

1.6 Aspect Ratio 91.7 Camber 101.8 Incidence 111.9 Aerodynamic Force 121.10 Scale Effect 151.11 Force and Moment Coefficients 171.12 The Boundary Layer 181.13 Summary 20Exercise Problems 21Reference 22

2 Essence of Fluid Mechanics 232.1 Introduction 232.2 Properties of Fluids 23

2.2.1 Pressure 232.2.2 Temperature 242.2.3 Density 242.2.4 Viscosity 252.2.5 Absolute Coefficient of Viscosity 252.2.6 Kinematic Viscosity Coefficient 272.2.7 Thermal Conductivity of Air 272.2.8 Compressibility 28

2.3 Thermodynamic Properties 282.3.1 Specific Heat 282.3.2 The Ratio of Specific Heats 29

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2.4 Surface Tension 302.5 Analysis of Fluid Flow 31

2.5.1 Local and Material Rates of Change 322.5.2 Graphical Description of Fluid Motion 33

2.6 Basic and Subsidiary Laws 342.6.1 System and Control Volume 342.6.2 Integral and Differential Analysis 352.6.3 State Equation 35

2.7 Kinematics of Fluid Flow 352.7.1 Boundary Layer Thickness 372.7.2 Displacement Thickness 382.7.3 Transition Point 392.7.4 Separation Point 392.7.5 Rotational and Irrotational Motion 40

2.8 Streamlines 412.8.1 Relationship between Stream Function and Velocity Potential 41

2.9 Potential Flow 422.9.1 Two-dimensional Source and Sink 432.9.2 Simple Vortex 452.9.3 Source-Sink Pair 462.9.4 Doublet 46

2.10 Combination of Simple Flows 492.10.1 Flow Past a Half-Body 49

2.11 Flow Past a Circular Cylinder without Circulation 572.11.1 Flow Past a Circular Cylinder with Circulation 59

2.12 Viscous Flows 632.12.1 Drag of Bodies 652.12.2 Turbulence 702.12.3 Flow through Pipes 75

2.13 Compressible Flows 782.13.1 Perfect Gas 792.13.2 Velocity of Sound 802.13.3 Mach Number 802.13.4 Flow with Area Change 802.13.5 Normal Shock Relations 822.13.6 Oblique Shock Relations 832.13.7 Flow with Friction 842.13.8 Flow with Simple T0-Change 86

2.14 Summary 87Exercise Problems 97References 102

3 Conformal Transformation 1033.1 Introduction 1033.2 Basic Principles 103

3.2.1 Length Ratios between the Corresponding Elements in thePhysical and Transformed Planes 106

3.2.2 Velocity Ratios between the Corresponding Elements in thePhysical and Transformed Planes 106

3.2.3 Singularities 107

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Contents ix

3.3 Complex Numbers 1073.3.1 Differentiation of a Complex Function 110

3.4 Summary 112Exercise Problems 113

4 Transformation of Flow Pattern 1154.1 Introduction 1154.2 Methods for Performing Transformation 115

4.2.1 By Analytical Means 1164.3 Examples of Simple Transformation 1194.4 Kutta−Joukowski Transformation 1224.5 Transformation of Circle to Straight Line 1234.6 Transformation of Circle to Ellipse 1244.7 Transformation of Circle to Symmetrical Aerofoil 125

4.7.1 Thickness to Chord Ratio of Symmetrical Aerofoil 1274.7.2 Shape of the Trailing Edge 129

4.8 Transformation of a Circle to a Cambered Aerofoil 1294.8.1 Thickness-to-Chord Ratio of the Cambered Aerofoil 1324.8.2 Camber 134

4.9 Transformation of Circle to Circular Arc 1344.9.1 Camber of Circular Arc 137

4.10 Joukowski Hypothesis 1374.10.1 The Kutta Condition Applied to Aerofoils 1394.10.2 The Kutta Condition in Aerodynamics 140

4.11 Lift of Joukowski Aerofoil Section 1414.12 The Velocity and Pressure Distributions on the Joukowski Aerofoil 1444.13 The Exact Joukowski Transformation Process and Its Numerical Solution 1464.14 The Velocity and Pressure Distribution 1474.15 Aerofoil Characteristics 155

4.15.1 Parameters Governing the Aerodynamic Forces 1574.16 Aerofoil Geometry 157

4.16.1 Aerofoil Nomenclature 1574.16.2 NASA Aerofoils 1614.16.3 Leading-Edge Radius and Chord Line 1614.16.4 Mean Camber Line 1614.16.5 Thickness Distribution 1624.16.6 Trailing-Edge Angle 162

4.17 Wing Geometrical Parameters 1624.18 Aerodynamic Force and Moment Coefficients 166

4.18.1 Moment Coefficient 1694.19 Summary 171Exercise Problems 180Reference 181

5 Vortex Theory 1835.1 Introduction 1835.2 Vorticity Equation in Rectangular Coordinates 184

5.2.1 Vorticity Equation in Polar Coordinates 186

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5.3 Circulation 1885.4 Line (point) Vortex 1925.5 Laws of Vortex Motion 1945.6 Helmholtz’s Theorems 1955.7 Vortex Theorems 196

5.7.1 Stoke’s Theorem 2005.8 Calculation of uR, the Velocity due to Rotational Flow 2045.9 Biot-Savart Law 207

5.9.1 A Linear Vortex of Finite Length 2105.9.2 Semi-Infinite Vortex 2115.9.3 Infinite Vortex 2115.9.4 Helmholtz’s Second Vortex Theorem 2165.9.5 Helmholtz’s Third Vortex Theorem 2205.9.6 Helmholtz’s Fourth Vortex Theorem 220

5.10 Vortex Motion 2205.11 Forced Vortex 2235.12 Free Vortex 224

5.12.1 Free Spiral Vortex 2265.13 Compound Vortex 2295.14 Physical Meaning of Circulation 2305.15 Rectilinear Vortices 235

5.15.1 Circular Vortex 2365.16 Velocity Distribution 2375.17 Size of a Circular Vortex 2395.18 Point Rectilinear Vortex 2395.19 Vortex Pair 2405.20 Image of a Vortex in a Plane 2415.21 Vortex between Parallel Plates 2425.22 Force on a Vortex 2445.23 Mutual action of Two Vortices 2445.24 Energy due to a Pair of Vortices 2445.25 Line Vortex 2475.26 Summary 248Exercise Problems 254References 256

6 Thin Aerofoil Theory 2576.1 Introduction 2576.2 General Thin Aerofoil Theory 2586.3 Solution of the General Equation 261

6.3.1 Thin Symmetrical Flat Plate Aerofoil 2626.3.2 The Aerodynamic Coefficients for a Flat Plate 265

6.4 The Circular Arc Aerofoil 2696.4.1 Lift, Pitching Moment, and the Center of Pressure Location

for Circular Arc Aerofoil 2716.5 The General Thin Aerofoil Section 2756.6 Lift, Pitching Moment and Center of Pressure Coefficients for a Thin Aerofoil 2786.7 Flapped Aerofoil 283

6.7.1 Hinge Moment Coefficient 286

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Contents xi

6.7.2 Jet Flap 2886.7.3 Effect of Operating a Flap 288

6.8 Summary 289Exercise Problems 294References 295

7 Panel Method 2977.1 Introduction 2977.2 Source Panel Method 297

7.2.1 Coefficient of Pressure 3007.3 The Vortex Panel Method 302

7.3.1 Application of Vortex Panel Method 3027.4 Pressure Distribution around a Circular Cylinder by Source Panel Method 3057.5 Using Panel Methods 309

7.5.1 Limitations of Panel Method 3097.5.2 Advanced Panel Methods 309

7.6 Summary 329Exercise Problems 330Reference 330

8 Finite Aerofoil Theory 3318.1 Introduction 3318.2 Relationship between Spanwise Loading and Trailing Vorticity 3318.3 Downwash 3328.4 Characteristics of a Simple Symmetrical Loading – Elliptic Distribution 335

8.4.1 Lift for an Elliptic Distribution 3368.4.2 Downwash for an Elliptic Distribution 3368.4.3 Drag Dv due to Downwash for Elliptical Distribution 338

8.5 Aerofoil Characteristic with a More General Distribution 3398.5.1 The Downwash for Modified Elliptic Loading 341

8.6 The Vortex Drag for Modified Loading 3438.6.1 Condition for Vortex Drag Minimum 345

8.7 Lancaster – Prandtl Lifting Line Theory 3478.7.1 The Lift 3498.7.2 Induced Drag 350

8.8 Effect of Downwash on Incidence 3538.9 The Integral Equation for the Circulation 3558.10 Elliptic Loading 356

8.10.1 Lift and Drag for Elliptical Loading 3578.10.2 Lift Curve Slope for Elliptical Loading 3598.10.3 Change of Aspect Ratio with Incidence 3598.10.4 Problem II 3608.10.5 The Lift for Elliptic Loading 3638.10.6 The Downwash Velocity for Elliptic Loading 3668.10.7 The Induced Drag for Elliptic Loading 3668.10.8 Induced Drag Minimum 3698.10.9 Lift and Drag Calculation by Impulse Method 370

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xii Contents

8.10.10 The Rectangular Aerofoil 3718.10.11 Cylindrical Rectangular Aerofoil 372

8.11 Aerodynamic Characteristics of Asymmetric Loading 3728.11.1 Lift on the Aerofoil 3728.11.2 Downwash 3728.11.3 Vortex Drag 3738.11.4 Rolling Moment 3748.11.5 Yawing Moment 376

8.12 Lifting Surface Theory 3788.12.1 Velocity Induced by a Lifting Line Element 3788.12.2 Munk’s Theorem of Stagger 3818.12.3 The Induced Lift 3828.12.4 Blenk’s Method 3838.12.5 Rectangular Aerofoil 3848.12.6 Calculation of the Downwash Velocity 385

8.13 Aerofoils of Small Aspect Ratio 3878.13.1 The Integral Equation 3888.13.2 Zero Aspect Ratio 3908.13.3 The Acceleration Potential 390

8.14 Lifting Surface 3918.15 Summary 394Exercise Problems 401

9 Compressible Flows 4059.1 Introduction 4059.2 Thermodynamics of Compressible Flows 4059.3 Isentropic Flow 4099.4 Discharge from a Reservoir 4119.5 Compressible Flow Equations 4139.6 Crocco’s Theorem 414

9.6.1 Basic Solutions of Laplace’s Equation 4189.7 The General Potential Equation for Three-Dimensional Flow 4189.8 Linearization of the Potential Equation 420

9.8.1 Small Perturbation Theory 4209.9 Potential Equation for Bodies of Revolution 423

9.9.1 Solution of Nonlinear Potential Equation 4259.10 Boundary Conditions 425

9.10.1 Bodies of Revolution 4279.11 Pressure Coefficient 428

9.11.1 Bodies of Revolution 4299.12 Similarity Rule 4299.13 Two-Dimensional Flow: Prandtl-Glauert Rule for Subsonic Flow 429

9.13.1 The Prandtl-Glauert Transformations 4299.13.2 The Direct Problem-Version I 4319.13.3 The Indirect Problem (Case of Equal Potentials): P-G

Transformation – Version II 4349.13.4 The Streamline Analogy (Version III): Gothert’s Rule 435

9.14 Prandtl-Glauert Rule for Supersonic Flow: Versions I and II 4369.14.1 Subsonic Flow 4369.14.2 Supersonic Flow 436

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9.15 The von Karman Rule for Transonic Flow 4399.15.1 Use of Karman Rule 440

9.16 Hypersonic Similarity 4429.17 Three-Dimensional Flow: The Gothert Rule 444

9.17.1 The General Similarity Rule 4449.17.2 Gothert Rule 4469.17.3 Application to Wings of Finite Span 4479.17.4 Application to Bodies of Revolution and Fuselage 4489.17.5 The Prandtl-Glauert Rule 4509.17.6 The von Karman Rule for Transonic Flow 454

9.18 Moving Disturbance 4559.18.1 Small Disturbance 4569.18.2 Finite Disturbance 457

9.19 Normal Shock Waves 4579.19.1 Equations of Motion for a Normal Shock Wave 4579.19.2 The Normal Shock Relations for a Perfect Gas 458

9.20 Change of Total Pressure across a Shock 4629.21 Oblique Shock and Expansion Waves 463

9.21.1 Oblique Shock Relations 4649.21.2 Relation between β and θ 4669.21.3 Supersonic Flow over a Wedge 4699.21.4 Weak Oblique Shocks 4719.21.5 Supersonic Compression 4739.21.6 Supersonic Expansion by Turning 4759.21.7 The Prandtl-Meyer Function 4779.21.8 Shock-Expansion Theory 477

9.22 Thin Aerofoil Theory 4799.22.1 Application of Thin Aerofoil Theory 480

9.23 Two-Dimensional Compressible Flows 4859.24 General Linear Solution for Supersonic Flow 486

9.24.1 Existence of Characteristics in a Physical Problem 4889.24.2 Equation for the Streamlines from Kinematic Flow Condition 489

9.25 Flow over a Wave-Shaped Wall 4919.25.1 Incompressible Flow 4919.25.2 Compressible Subsonic Flow 4929.25.3 Supersonic Flow 4939.25.4 Pressure Coefficient 494

9.26 Summary 495Exercise Problems 509References 512

10 Simple Flights 51310.1 Introduction 51310.2 Linear Flight 51310.3 Stalling 51410.4 Gliding 51610.5 Straight Horizontal Flight 51810.6 Sudden Increase of Incidence 52010.7 Straight Side-Slip 521

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10.8 Banked Turn 52210.9 Phugoid Motion 52310.10 The Phugoid Oscillation 52510.11 Summary 529Exercise Problems 531

Further Readings 533

Index 535

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About the AuthorEthirajan Rathakrishnan is Professor of Aerospace Engineering at the Indian Institute of TechnologyKanpur, India. He is well-known internationally for his research in the area of high-speed jets. Thelimit for the passive control of jets, called Rathakrishnan Limit, is his contribution to the field of jetresearch, and the concept of breathing blunt nose (BBN), which reduces the positive pressure at thenose and increases the low-pressure at the base simultaneously, is his contribution to drag reductionat hypersonic speeds. He has published a large number of research articles in many reputed interna-tional journals. He is a fellow of many professional societies, including the Royal Aeronautical Society.Professor Rathakrishnan serves as editor-in-chief of the International Review of Aerospace Engineering(IREASE) Journal. He has authored nine other books: Gas Dynamics, 4th ed. (PHI Learning, New Delhi,2012); Fundamentals of Engineering Thermodynamics, 2nd ed. (PHI Learning, New Delhi, 2005); FluidMechanics: An Introduction, 3rd ed. (PHI Learning, New Delhi, 2012); Gas Tables, 3rd ed. (UniversitiesPress, Hyderabad, India, 2012); Instrumentation, Measurements, and Experiments in Fluids (CRC Press,Taylor & Francis Group, Boca Raton, USA, 2007); Theory of Compressible Flows (Maruzen Co., Ltd.,Tokyo, Japan, 2008); Gas Dynamics Work Book (Praise Worthy Prize, Napoli, Italy, 2010); Applied GasDynamics (John Wiley, New Jersey, USA, 2010); and Elements of Heat Transfer, (CRC Press, Taylor &Francis Group, Boca Raton, USA, 2012).

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PrefaceThis book has been developed to serve as a text for theoretical aerodynamics at the introductory level forboth undergraduate courses and for an advanced course at graduate level. The basic aim of this book is toprovide a complete text covering both the basic and applied aspects of aerodynamic theory for students,engineers, and applied physicists. The philosophy followed in this book is that the subject of aerodynamictheory is covered by combining the theoretical analysis, physical features and application aspects.

The fundamentals of fluid dynamics and gas dynamics are covered as it is treated at the undergraduatelevel. The essence of fluid mechanics, conformal transformation and vortex theory, being the basicsfor the subject of theoretical aerodynamics, are given in separate chapters. A considerable number ofsolved examples are given in these chapters to fix the concepts introduced and a large number of exerciseproblems along with answers are listed at the end of these chapters to test the understanding of the materialstudied.

To make readers comfortable with the basic features of aircraft geometry and its flight, vital parts ofaircraft and the preliminary aspects of its flight are discussed in the first and final chapters. The entirespectrum of theoretical aerodynamics is presented in this book, with necessary explanations on everyaspect. The material covered in this book is so designed that any beginner can follow it comfortably. Thetopics covered are broad based, starting from the basic principles and progressing towards the physicsof the flow which governs the flow process.

The book is organized in a logical manner and the topics are discussed in a systematic way. First, thebasic aspects of the fluid flow and vortices are reviewed in order to establish a firm basis for the subject ofaerodynamic theory. Following this, conformal transformation of flows is introduced with the elementaryaspects and then gradually proceeding to the vital aspects and application of Joukowski transformationwhich transforms a circle in the physical plane to lift generating profiles such as symmetrical aerofoil,circular arc and cambered aerofoil in the tranformed plane. Following the transformation, vortex genera-tion and its effect on lift and drag are discussed in depth. The chapter on thin aerofoil theory discusses theperformance of aerofoils, highlighting the application and limitations of the thin aerofoils. The chapteron panel methods presents the source and vortex panel techniques meant for solving the flow aroundnonlifting and lifting bodies, respectively.

The chapter on finite wing theory presents the performance of wings of finite aspect ratio, where thehorseshoe vortex, made up of the bound vortex and tip vortices, plays a dominant role. The procedurefor calculating the lift, drag and pitching moment for symmetrical and cambered profiles is discussed indetail. The consequence of the velocity induced by the vortex system is presented in detail, along withsolved examples at appropriate places.

The chapter on compressible flows covers the basics and application aspects in detail for both subsonicand supersonic regimes of the flow. The similarity consideration covering the Parandtl-Glauert I andII rules and Gothert rule are presented in detail. The basic governing equation and its simplificationwith small perturbation assumption is covered systematically. Shocks and expansion waves and theirinfluence on the flow field are discussed in depth. Following this the shock-expansion theory and thinaerofoil theory and their application to calculate the lift and drag are presented.

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xviii Preface

In the final chapter, some basic flights are introduced briefly, covering the level flight, gliding andclimbing modes of flight. A brief coverage of phugoid motion is also presented.

The selected references given at the end are, it is hoped, a useful guide for further study of thevoluminous subject.

This book is the outgrowth of lectures presented over a number of years, both at undergraduate andgraduate level. The student, or reader, is assumed to have a background in the basic courses of fluidmechanics. Advanced undergraduate students should be able to handle the subject material comfortably.Sufficient details have been included so that the text can be used for self study. Thus, the book canbe useful for scientists and engineers working in the field of aerodynamics in industries and researchlaboratories.

My sincere thanks to my undergraduate and graduate students in India and abroad, who are directlyand indirectly responsible for the development of this book.

I would like to express my sincere thanks to Yasumasa Watanabe, doctoral student of AerospaceEngineering, the University of Tokyo, Japan, for his help in making some solved examples along withcomputer codes. I thank Shashank Khurana, doctoral student of Aerospace Engineering, the University ofTokyo, Japan, for critically checking the manuscript of this book. Indeed, incorporation of the suggestionsgiven by Shashank greatly enhanced the clarity of manuscript of this book. I thank my doctoral studentsMrinal Kaushik and Arun Kumar, for checking the manuscript and the solutions manual, and for givingsome useful suggestions.

For instructors only, a companion Solutions Manual is available from John Wiley and contains typedsolutions to all the end-of-chapter problems can be found at www.wiley.com/go/rathakrishnan. Thefinancial support extended by the Continuing Education Centre of the Indian Institute of TechnologyKanpur, for the preparation of the manuscript is gratefully acknowledged.

Ethirajan Rathakrishnan

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1Basics

1.1 Introduction

Aerodynamics is the science concerned with the motion of air and bodies moving through air. In otherwords, aerodynamics is a branch of dynamics concerned with the study of motion of air, particularlywhen it interacts with a moving object. The forces acting on bodies moving through the air are termedaerodynamic forces. Air is a fluid, and in accordance with Archimedes principle, an aircraft will bebuoyed up by a force equal to the weight of air displaced by it. The buoyancy force Fb will act verticallyupwards. The weight W of the aircraft is a force which acts vertically downwards; thus the magnitude ofthe net force acting on an aircraft, even when it is not moving, is (W − Fb). The force (W − Fb) will actirrespective of whether the aircraft is at rest or in motion.

Now, let us consider an aircraft flying with constant speed V through still air, as shown in Figure 1.1,that is, any motion of air is solely due to the motion of the aircraft. Let this motion of the aircraft ismaintained by a tractive force T exerted by the engines.

Newton’s first law of motion asserts that the resultant force acting on the aircraft must be zero, whenit is at a steady flight (unaccelerated motion). Therefore, there must be an additional force Fad, say, suchthat the vectorial sum of the forces acting on the aircraft is:

T + (W − Fb) + Fad = 0

Force Fad is called the aerodynamic force exerted on the aircraft. In this definition of aerodynamic force,the aircraft is considered to be moving with constant velocity V in stagnant air. Instead, we may imaginethat the aircraft is at rest with the air streaming past it. In this case, the air velocity over the aircraftwill be −V . It is important to note that the aerodynamic force is theoretically the same in both cases;therefore we may adopt whichever point of view is convenient for us. In the measurement of forces on anaircraft using wind tunnels, this principle is adopted, that is, the aircraft model is fixed in the wind tunneltest-section and the air is made to flow over the model. In our discussions we shall always refer to thedirection of V as the direction of aircraft motion, and the direction of −V as the direction of airstreamor relative wind.

1.2 Lift and Drag

The aerodynamic force Fad can be resolved into two component forces, one at right angles to V and theother opposite to V , as shown in Figure 1.1. The force component normal to V is called lift L and the

Theoretical Aerodynamics, First Edition. Ethirajan Rathakrishnan.© 2013 John Wiley & Sons Singapore Pte. Ltd. Published 2013 by John Wiley & Sons Singapore Pte. Ltd.

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2 Theoretical Aerodynamics

(W − Fb)

Fad

θ

T

V

LD

Figure 1.1 Forces acting on an aircraft in horizontal flight.

component opposite to V is called drag D. If θ is the angle between L and Fad, we have:

L = Fad cos θ

D = Fad sin θ

tan θ = D

L.

The angle θ is called the glide angle. For keeping the drag at low value, the gliding angle has to be small.An aircraft with a small gliding angle is said to be streamlined.

At this stage, it is essential to realize that the lift and drag are related to vertical and horizontal directions.To fix this idea, the lift and drag are formally defined as follows:

“Lift is the component of the aerodynamic force perpendicular to the direction of motion.”“Drag is the component of the aerodynamic force opposite to the direction of motion.”

Note: It is important to understand the physical meaning of the statement, “an aircraft with a small glidingangle θ is said to be streamlined.” This explicitly implies that when θ is large the aircraft can not be regardedas a streamlined body. This may make us wonder about the nature of the aircraft geometry, whether it isstreamlined or bluff. In our basic courses, we learned that all high-speed vehicles are streamlined bodies.According to this concept, an aircraft should be a streamlined body. But at large θ it can not be declaredas a streamlined body. What is the genesis for this drastic conflict? These doubts will be cleared if weget the correct meaning of the bluff and streamlined geometries. In fluid dynamics, we learn that:

“a streamlined body is that for which the skin friction drag accounts for the major portion of thetotal drag, and the wake drag is very small.”“A bluff body is that for which the wake drag accounts for the major portion of the total drag, andthe skin friction drag is insignificant.”

Therefore, the basis for declaring a body as streamlined or bluff is the relative magnitudes of skin frictionand wake drag components and not just the geometry of the body shape alone. Indeed, sometimes theshape of the body can be misleading in this issue. For instance, a thin flat plate kept parallel to the flow,as shown is Figure 1.2(a), is a perfectly streamlined body, but the same plate kept normal to the flow, asshown is Figure 1.2(b), is a typical bluff body. This clearly demonstrates that the streamlined and bluffnature of a body is dictated by the combined effect of the body geometry and its orientation to the flowdirection. Therefore, even though an aircraft is usually regarded as a streamlined body, it can behave asa bluff body when the gliding angle θ is large, causing the formation of large wake, leading to a largevalue of wake drag. That is why it is stated that, “for small values of gliding angle θ an aircraft is said

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Basics 3

(b)(a)

Figure 1.2 A flat plate (a) parallel to the flow, (b) normal to the flow.

to be streamlined.” Also, it is essential to realize that all commercial aircraft are usually operated withsmall gliding angle in most portion of their mission and hence are referred to as streamlined bodies. Allfighter aircraft, on the other hand, are designed for maneuvers such as free fall, pull out and pull up,during which they behave as bluff bodies.

Example 1.1

An aircraft of mass 1500 kg is in steady level flight. If the wing incidence with respect to the freestreamflow is 3◦, determine the lift to drag ratio of the aircraft.

Solution

Given, m = 1500 kg and θ = 3◦.In level flight the weight of the aircraft is supported by the lift. Therefore, the lift is:

L = W = mg

= 1500 × 9.81

= 14715 N.

The relation between the aerodynamic force, Fad, and lift, L, is:

L = Fad cos θ.

The aerodynamic force becomes:

Fad = L

cos θ

= 14715

cos 3◦= 14735.2 N.

The relation between the aerodynamic force, Fad, and drag, D, is:

D = Fad sin θ.

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4 Theoretical Aerodynamics

Therefore, the drag becomes:

D = 14735.2 × sin 3◦

= 771.2 N.

The lift to drag ratio of the aircraft is:

L

D= 14715

771.2

= 19 .

Note: The lift to drag ratio L/D is termed aerodynamic efficiency.

1.3 Monoplane Aircraft

A monoplane is a fixed-wing aircraft with one main set of wing surfaces, in contrast to a biplane ortriplane. Since the late 1930s it has been the most common form for a fixed wing aircraft.

The main features of a monoplane aircraft are shown in Figure 1.3. The main lifting system consistsof two wings; the port (left) and starboard (right) wings, which together constitute the aerofoil. The tailplane also exerts lift. According to the design, the aerofoil may or may not be interrupted by the fuselage.The designer subsequently allow for the effect of the fuselage as a perturbation (a French word whichmeans disturbance) of the properties of the aerofoil. For the present discussion, let us ignore the fuselage,and treat the wing (aerofoil) as one continuous surface.

The ailerons on the right and left wings, the elevators on the horizontal tail, and the rudder on thevertical tail, shown in Figure 1.3, are control surfaces. When the ailerons and rudder are in their neutralpositions, the aircraft has a median plane of symmetry which divides the whole aircraft into two parts,each of which is the optical image of the other in this plane, considered as a mirror. The wings are thenthe portions of the aerofoil on either side of the plane of symmetry, as shown in Figure 1.4.

The wing tips consist of those points of the wings, which are at the farthest distance from the plane ofsymmetry, as illustrated in Figure 1.4. Thus, the wing tips can be a point or a line or an area, accordingto the design of the aerofoil. The distance between the wing tips is called the span. The section of a wingby a plane parallel to the plane of symmetry is called a profile. The shape and general orientation of theprofile will usually depend on its distance from the plane of symmetry. In the case of a cylindrical wing,shown in Figure 1.5, the profiles are the same at every location along the span.

Starbo

ardwing

V

Tail plane

Aileron

Elevator

Rudder

Fin

Engine

Fuselage

y

z

x

Flap

Portwin

g

Figure 1.3 Main features of a monoplane aircraft.

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Basics 5

b

TipStarboard wingTip

Plane of symmetry

Port wing

Span

b

Figure 1.4 Typical geometry of an aircraft wing.

Profile

Figure 1.5 A cylindrical wing.

1.3.1 Types of Monoplane

The main distinction between types of monoplane is where the wings attach to the fuselage:

Low-wing: the wing lower surface is level with (or below) the bottom of the fuselage.Mid-wing: the wing is mounted mid-way up the fuselage.High-wing: the wing upper surface is level with or above the top of the fuselage.Shoulder wing: the wing is mounted above the fuselage middle.Parasol-wing: the wing is located above the fuselage and is not directly connected to it, structural support

being typically provided by a system of struts, and, especially in the case of older aircraft, wire bracing.

1.4 Biplane

A biplane is a fixed-wing aircraft with two superimposed main wings. The Wright brothers’ Wright Flyerused a biplane design, as did most aircraft in the early years of aviation. While a biplane wing structurehas a structural advantage, it generates more drag than a similar monoplane wing. Improved structuraltechniques and materials and the quest for greater speed made the biplane configuration obsolete for mostpurposes by the late 1930s.

In a biplane aircraft, two wings are placed one above the other, as in the Boeing Stearman E75 (PT-13D)biplane of 1944 shown in Figure 1.6. Both wings provide part of the lift, although they are not able toproduce twice as much lift as a single wing of similar size and shape because both the upper and lowerwings are working on nearly the same portion of the atmosphere. For example, in a wing of aspect ratio6, and a wing separation distance of one chord length, the biplane configuration can produce about 20%more lift than a single wing of the same planform.

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6 Theoretical Aerodynamics

Figure 1.6 Boeing Stearman E75 (PT-13D) biplane of 1944.

In the biplane configuration, the lower wing is usually attached to the fuselage, while the upper wing israised above the fuselage with an arrangement of cabane struts, although other arrangements have beenused. Almost all biplanes also have a third horizontal surface, the tailplane, to control the pitch, or angleof attack of the aircraft (although there have been a few exceptions). Either or both of the main wings cansupport flaps or ailerons to assist lateral rotation and speed control; usually the ailerons are mounted onthe upper wing, and flaps (if used) on the lower wing. Often there is bracing between the upper and lowerwings, in the form of wires (tension members) and slender inter-plane struts (compression members)positioned symmetrically on either side of the fuselage.

1.4.1 Advantages and Disadvantages

Aircraft built with two main wings (or three in a triplane) can usually lift up to 20% more than can asimilarly sized monoplane of similar wingspan. Biplanes will therefore typically have a shorter wingspanthan a similar monoplane, which tends to afford greater maneuverability. The struts and wire bracing ofa typical biplane form a box girder that permits a light but very strong wing structure.

On the other hand, there are many disadvantages to the configuration. Each wing negatively interfereswith the aerodynamics of the other. For a given wing area the biplane generates more drag and producesless lift than a monoplane.

Now, one may ask what is the specific difference between a biplane and monoplane? The answer is asfollows.

A biplane has two (bi) sets of wings, and a monoplane has one (mono) set of wings. The two sets ofwings on a biplane add lift, and also drag, allowing it to fly slower. The one set of wings on a monoplanedo not add as much lift or drag, making it fly faster, and as a result, all fast planes are monoplanes, andmost planes these days are monoplanes.

1.5 Triplane

A triplane is a fixed-wing aircraft equipped with three vertically-stacked wing planes. Tailplanes andcanard fore-planes are not normally included in this count, although they may occasionally be. A typicalexample for triplane is the Fokker Dr. I of World War I, shown in Figure 1.7.

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Basics 7

Figure 1.7 Fokker Dr. I of World War I.

The triplane arrangement may be compared with the biplane in a number of ways. A triplane arrange-ment has a narrower wing chord than a biplane of similar span and area. This gives each wing plane aslender appearance with a higher aspect ratio, making it more efficient and giving increased lift. Thispotentially offers a faster rate of climb and tighter turning radius, both of which are important in a fighterplane. The Sopwith Triplane was a successful example, having the same wing span as the equivalentbiplane, the Sopwith Pup.

Alternatively, a triplane has a reduced span compared with a biplane of given wing area and aspectratio, leading to a more compact and lightweight structure. This potentially offers better maneuverabilityfor a fighter plane, and higher load capacity with more practical ground handling for a large aircraft type.

The famous Fokker Dr.I triplane was a balance between the two approaches, having moderately shorterspan and moderately higher aspect ratio than the equivalent biplane, the Fokker D.VI.

Yet a third comparison may be made between a biplane and triplane having the same wing planform—the triplane’s third wing provides increased wing area, giving much increased lift. The extra weight ispartially offset by the increased depth of the overall structure, allowing a more efficient construction. TheCaproni Ca.4 series had some success with this approach.

These advantages are offset, to a greater or lesser extent in any given design, by the extra weight anddrag of the structural bracing, and the aerodynamic inefficiency inherent in the stacked wing layout. Asbiplane design advanced, it became clear that the disadvantages of the triplane outweighed the advantages.

Typically the lower set of wings are approximately level with the underside of the aircraft’s fuselage,the middle set level with the top of the fuselage, and the top set supported above the fuselage on cabanestruts.

1.5.1 Chord of a Profile

A chord of any profile is generally defined as an arbitrarily fixed line drawn in the plane of the profile, asillustrated in Figure 1.8. The chord has direction, position, and length. The main requisite is that in eachcase the chord should be precisely defined, because the chord enters into the constants such as the liftand drag coefficients, which describe the aerodynamic properties of the profile. For the profile shown inFigure 1.8(a), the chord is the line joining the center of the circle at the leading and trailing edges.

For the profile in Figure 1.8(b), the line joining the center of the circle at the nose and the tip of the tail isthe chord. For the profile in Figure 1.8(c), the line joining the tips of leading and trailing edges is the chord.

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chord c

chord c

(a) Leading and trailing edges are circular arcs.

(b) Circular arc leading edge and sharp trailing edge.

(c) Faired leading edge and sharp trailing edge.

chord c

Figure 1.8 Illustration of chord for different shapes of leading and trailing edges.

Chord c

Figure 1.9 Chord of a profile.

A definition which is convenient is: the chord is the projection of the profile on the double tangentto its lower surface (that is, the tangent which touches the profile at two distinct points), as shown inFigure 1.9. But this definition fails if there is no such double tangent.

1.5.2 Chord of an Aerofoil

For a cylindrical aerofoil (that is, a wing for which the profiles are the same at every location along thespan, as shown in Figure 1.5), the chord of the aerofoil is taken to be the chord of the profile in which theplane of symmetry cuts the aerofoil. In all other cases, the chord of the aerofoil is defined as the mean oraverage chord located in the plane of symmetry.

Let us consider a wing with rectangular Cartesian coordinate axes, as shown in Figure 1.10. The x-axis,or longitudinal axis, is in the direction of motion, and is in the plane of symmetry; the y-axis, or lateral

z

y

o

x

Figure 1.10 A wing with Cartesian coordinates.

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Basics 9

axis, is normal to the plane of symmetry and along the (straight) trailing edge. The z-axis, or normal axis,is perpendicular to the other two axes in the sense that the three axes form a right-handed system. Thismeans, in particular, that in a straight horizontal flight the z-axis will be directed vertically downwards.Consider a profile whose distance from the plane of symmetry is |y|. Let c be the chord length of thisprofile, θ be the inclination of the chord to the xy plane, and (x, y, z) be the coordinates of the quarterpoint of the chord, that is, the point of the chord at a distance c/4 from the leading edge of the profile.This point is usually referred to as the quarter chord point. Since the profile is completely defined wheny is given, all quantities characterizing the profile, namely, the mean chord, its position and inclinationto the flow, are functions of y.

The chord of an aerofoil is defined by averaging the distance between the leading and trailing edges ofthe profiles at different locations along the span. Thus, if cm is the length of the mean chord, (xm, 0, zm)its quarter point, and θm its inclination, we take the average or mean chord as:

cm = 1

2b

∫ +b

−b

c dy

θm = 1

2b

∫ +b

−b

θ dy

xm = 1

2b

∫ +b

−b

x dy

zm = 1

2b

∫ +b

−b

z dy.

These mean values completely define the chord of the aerofoil in length (cm), direction (θm), and position(xm, zm).

1.6 Aspect Ratio

Aspect ratio of a wing is the ratio of its span 2b to chord c. Consider a cylindrical wing shown in Figure1.10. Imagine this to be projected on to the plane (xy-plane), which contains the chords of all the sections(this plane is perpendicular to the plane of symmetry (xz-plane) and contains the chord of the wing). Theprojection in this case is a rectangular area S, say, which is called the plan area of the wing. The plan areais different from the total surface area of the wing. The simplest cylindrical wing would be a rectangularplate, and the plan area would then be half of the total surface area.

The aspect ratio of the cylindrical wing is then defined by:

= 2b

c= (2b)2

S,

where S = span × chord = 2b × c.In the case of a wing which is not cylindrical, the plan area is defined as the area of the projection on

the plane through the chord of the wing (mean chord) perpendicular to the plane of symmetry, and theaspect ratio is defined as:

= (2b)2

S.

A representative value of aspect ratio is 6.

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10 Theoretical Aerodynamics

Example 1.2

The semi-span of a rectangular wing of planform area 8.4 m2 is 3.5 m. Determine the aspect ratio of thewing.

Solution

Given, S = 8.4 m2 and b = 3.5 m.The planform area of a wing is S = span × chord. Therefore, the wing chord becomes:

c = S

2b

= 8.4

2 × 3.5= 1.2 m.

The aspect ratio of the wing is:

= Span

Chord

= 2 × 3.5

1.2

= 5.83 .

1.7 Camber

Camber is the maximum deviation of the camber line (which is the bisector of the profile thickness) fromthe chord of the profile, as illustrated in Figure 1.11.

Let zu and zl be the ordinates on the upper and lower parts of the profile, respectively, for the samevalue of x. Let c be the chord, and the x-axis coincide with the chord. Now, the upper and lower camberare defined as:

Upper camber = (zu)max

c

Lower camber = (zl)max

c,

AA

HH

P

Pl

Pu

(a) (b)

Chord, c

Camber line

CamberM

Chord

z

Figure 1.11 Illustration of camber, camberline and chord of aerofoil profile.