Ethirajan Rathakrishnan Theoretical Aerodynamics
THEORETICAL AERODYNAMICS
THEORETICAL AERODYNAMICS Ethirajan Rathakrishnan Indian Institute of Technology Kanpur, India
This edition first published 2013 2013 John Wiley & Sons Singapore Pte. Ltd. Registered office John 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 to reuse 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 form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as expressly permitted by law, without either the prior written permission of the Publisher, or authorization through payment of the appropriate photocopy 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: enquiry@wiley.com. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Rathakrishnan, E. Theoretical aerodynamics / Ethirajan Rathakrishnan. pages cm Includes bibliographical references and index. ISBN 978-1-118-47934-6 (cloth) 1. Aerodynamics. I. Title. TL570.R33 2013 629.132 3 dc23 2012049232 Typeset in 9/11pt Times by Thomson Digital, Noida, India
This book is dedicated to my parents, Mr Thammanur Shunmugam Ethirajan and Mrs Aandaal Ethirajan Ethirajan Rathakrishnan
Contents About the Author Preface xv xvii 1 Basics 1 1.1 Introduction 1 1.2 Lift and Drag 1 1.3 Monoplane Aircraft 4 1.3.1 Types of Monoplane 5 1.4 Biplane 5 1.4.1 Advantages and Disadvantages 6 1.5 Triplane 6 1.5.1 Chord of a Profile 7 1.5.2 Chord of an Aerofoil 8 1.6 Aspect Ratio 9 1.7 Camber 10 1.8 Incidence 11 1.9 Aerodynamic Force 12 1.10 Scale Effect 15 1.11 Force and Moment Coefficients 17 1.12 The Boundary Layer 18 1.13 Summary 20 Exercise Problems 21 Reference 22 2 Essence of Fluid Mechanics 23 2.1 Introduction 23 2.2 Properties of Fluids 23 2.2.1 Pressure 23 2.2.2 Temperature 24 2.2.3 Density 24 2.2.4 Viscosity 25 2.2.5 Absolute Coefficient of Viscosity 25 2.2.6 Kinematic Viscosity Coefficient 27 2.2.7 Thermal Conductivity of Air 27 2.2.8 Compressibility 28 2.3 Thermodynamic Properties 28 2.3.1 Specific Heat 28 2.3.2 The Ratio of Specific Heats 29
viii Contents 2.4 Surface Tension 30 2.5 Analysis of Fluid Flow 31 2.5.1 Local and Material Rates of Change 32 2.5.2 Graphical Description of Fluid Motion 33 2.6 Basic and Subsidiary Laws 34 2.6.1 System and Control Volume 34 2.6.2 Integral and Differential Analysis 35 2.6.3 State Equation 35 2.7 Kinematics of Fluid Flow 35 2.7.1 Boundary Layer Thickness 37 2.7.2 Displacement Thickness 38 2.7.3 Transition Point 39 2.7.4 Separation Point 39 2.7.5 Rotational and Irrotational Motion 40 2.8 Streamlines 41 2.8.1 Relationship between Stream Function and Velocity Potential 41 2.9 Potential Flow 42 2.9.1 Two-dimensional Source and Sink 43 2.9.2 Simple Vortex 45 2.9.3 Source-Sink Pair 46 2.9.4 Doublet 46 2.10 Combination of Simple Flows 49 2.10.1 Flow Past a Half-Body 49 2.11 Flow Past a Circular Cylinder without Circulation 57 2.11.1 Flow Past a Circular Cylinder with Circulation 59 2.12 Viscous Flows 63 2.12.1 Drag of Bodies 65 2.12.2 Turbulence 70 2.12.3 Flow through Pipes 75 2.13 Compressible Flows 78 2.13.1 Perfect Gas 79 2.13.2 Velocity of Sound 80 2.13.3 Mach Number 80 2.13.4 Flow with Area Change 80 2.13.5 Normal Shock Relations 82 2.13.6 Oblique Shock Relations 83 2.13.7 Flow with Friction 84 2.13.8 Flow with Simple T 0 -Change 86 2.14 Summary 87 Exercise Problems 97 References 102 3 Conformal Transformation 103 3.1 Introduction 103 3.2 Basic Principles 103 3.2.1 Length Ratios between the Corresponding Elements in the Physical and Transformed Planes 106 3.2.2 Velocity Ratios between the Corresponding Elements in the Physical and Transformed Planes 106 3.2.3 Singularities 107
Contents ix 3.3 Complex Numbers 107 3.3.1 Differentiation of a Complex Function 110 3.4 Summary 112 Exercise Problems 113 4 Transformation of Flow Pattern 115 4.1 Introduction 115 4.2 Methods for Performing Transformation 115 4.2.1 By Analytical Means 116 4.3 Examples of Simple Transformation 119 4.4 Kutta Joukowski Transformation 122 4.5 Transformation of Circle to Straight Line 123 4.6 Transformation of Circle to Ellipse 124 4.7 Transformation of Circle to Symmetrical Aerofoil 125 4.7.1 Thickness to Chord Ratio of Symmetrical Aerofoil 127 4.7.2 Shape of the Trailing Edge 129 4.8 Transformation of a Circle to a Cambered Aerofoil 129 4.8.1 Thickness-to-Chord Ratio of the Cambered Aerofoil 132 4.8.2 Camber 134 4.9 Transformation of Circle to Circular Arc 134 4.9.1 Camber of Circular Arc 137 4.10 Joukowski Hypothesis 137 4.10.1 The Kutta Condition Applied to Aerofoils 139 4.10.2 The Kutta Condition in Aerodynamics 140 4.11 Lift of Joukowski Aerofoil Section 141 4.12 The Velocity and Pressure Distributions on the Joukowski Aerofoil 144 4.13 The Exact Joukowski Transformation Process and Its Numerical Solution 146 4.14 The Velocity and Pressure Distribution 147 4.15 Aerofoil Characteristics 155 4.15.1 Parameters Governing the Aerodynamic Forces 157 4.16 Aerofoil Geometry 157 4.16.1 Aerofoil Nomenclature 157 4.16.2 NASA Aerofoils 161 4.16.3 Leading-Edge Radius and Chord Line 161 4.16.4 Mean Camber Line 161 4.16.5 Thickness Distribution 162 4.16.6 Trailing-Edge Angle 162 4.17 Wing Geometrical Parameters 162 4.18 Aerodynamic Force and Moment Coefficients 166 4.18.1 Moment Coefficient 169 4.19 Summary 171 Exercise Problems 180 Reference 181 5 Vortex Theory 183 5.1 Introduction 183 5.2 Vorticity Equation in Rectangular Coordinates 184 5.2.1 Vorticity Equation in Polar Coordinates 186
x Contents 5.3 Circulation 188 5.4 Line (point) Vortex 192 5.5 Laws of Vortex Motion 194 5.6 Helmholtz s Theorems 195 5.7 Vortex Theorems 196 5.7.1 Stoke s Theorem 200 5.8 Calculation of u R, the Velocity due to Rotational Flow 204 5.9 Biot-Savart Law 207 5.9.1 A Linear Vortex of Finite Length 210 5.9.2 Semi-Infinite Vortex 211 5.9.3 Infinite Vortex 211 5.9.4 Helmholtz s Second Vortex Theorem 216 5.9.5 Helmholtz s Third Vortex Theorem 220 5.9.6 Helmholtz s Fourth Vortex Theorem 220 5.10 Vortex Motion 220 5.11 Forced Vortex 223 5.12 Free Vortex 224 5.12.1 Free Spiral Vortex 226 5.13 Compound Vortex 229 5.14 Physical Meaning of Circulation 230 5.15 Rectilinear Vortices 235 5.15.1 Circular Vortex 236 5.16 Velocity Distribution 237 5.17 Size of a Circular Vortex 239 5.18 Point Rectilinear Vortex 239 5.19 Vortex Pair 240 5.20 Image of a Vortex in a Plane 241 5.21 Vortex between Parallel Plates 242 5.22 Force on a Vortex 244 5.23 Mutual action of Two Vortices 244 5.24 Energy due to a Pair of Vortices 244 5.25 Line Vortex 247 5.26 Summary 248 Exercise Problems 254 References 256 6 Thin Aerofoil Theory 257 6.1 Introduction 257 6.2 General Thin Aerofoil Theory 258 6.3 Solution of the General Equation 261 6.3.1 Thin Symmetrical Flat Plate Aerofoil 262 6.3.2 The Aerodynamic Coefficients for a Flat Plate 265 6.4 The Circular Arc Aerofoil 269 6.4.1 Lift, Pitching Moment, and the Center of Pressure Location for Circular Arc Aerofoil 271 6.5 The General Thin Aerofoil Section 275 6.6 Lift, Pitching Moment and Center of Pressure Coefficients for a Thin Aerofoil 278 6.7 Flapped Aerofoil 283 6.7.1 Hinge Moment Coefficient 286
Contents xi 6.7.2 Jet Flap 288 6.7.3 Effect of Operating a Flap 288 6.8 Summary 289 Exercise Problems 294 References 295 7 Panel Method 297 7.1 Introduction 297 7.2 Source Panel Method 297 7.2.1 Coefficient of Pressure 300 7.3 The Vortex Panel Method 302 7.3.1 Application of Vortex Panel Method 302 7.4 Pressure Distribution around a Circular Cylinder by Source Panel Method 305 7.5 Using Panel Methods 309 7.5.1 Limitations of Panel Method 309 7.5.2 Advanced Panel Methods 309 7.6 Summary 329 Exercise Problems 330 Reference 330 8 Finite Aerofoil Theory 331 8.1 Introduction 331 8.2 Relationship between Spanwise Loading and Trailing Vorticity 331 8.3 Downwash 332 8.4 Characteristics of a Simple Symmetrical Loading Elliptic Distribution 335 8.4.1 Lift for an Elliptic Distribution 336 8.4.2 Downwash for an Elliptic Distribution 336 8.4.3 Drag D v due to Downwash for Elliptical Distribution 338 8.5 Aerofoil Characteristic with a More General Distribution 339 8.5.1 The Downwash for Modified Elliptic Loading 341 8.6 The Vortex Drag for Modified Loading 343 8.6.1 Condition for Vortex Drag Minimum 345 8.7 Lancaster Prandtl Lifting Line Theory 347 8.7.1 The Lift 349 8.7.2 Induced Drag 350 8.8 Effect of Downwash on Incidence 353 8.9 The Integral Equation for the Circulation 355 8.10 Elliptic Loading 356 8.10.1 Lift and Drag for Elliptical Loading 357 8.10.2 Lift Curve Slope for Elliptical Loading 359 8.10.3 Change of Aspect Ratio with Incidence 359 8.10.4 Problem II 360 8.10.5 The Lift for Elliptic Loading 363 8.10.6 The Downwash Velocity for Elliptic Loading 366 8.10.7 The Induced Drag for Elliptic Loading 366 8.10.8 Induced Drag Minimum 369 8.10.9 Lift and Drag Calculation by Impulse Method 370
xii Contents 8.10.10 The Rectangular Aerofoil 371 8.10.11 Cylindrical Rectangular Aerofoil 372 8.11 Aerodynamic Characteristics of Asymmetric Loading 372 8.11.1 Lift on the Aerofoil 372 8.11.2 Downwash 372 8.11.3 Vortex Drag 373 8.11.4 Rolling Moment 374 8.11.5 Yawing Moment 376 8.12 Lifting Surface Theory 378 8.12.1 Velocity Induced by a Lifting Line Element 378 8.12.2 Munk s Theorem of Stagger 381 8.12.3 The Induced Lift 382 8.12.4 Blenk s Method 383 8.12.5 Rectangular Aerofoil 384 8.12.6 Calculation of the Downwash Velocity 385 8.13 Aerofoils of Small Aspect Ratio 387 8.13.1 The Integral Equation 388 8.13.2 Zero Aspect Ratio 390 8.13.3 The Acceleration Potential 390 8.14 Lifting Surface 391 8.15 Summary 394 Exercise Problems 401 9 Compressible Flows 405 9.1 Introduction 405 9.2 Thermodynamics of Compressible Flows 405 9.3 Isentropic Flow 409 9.4 Discharge from a Reservoir 411 9.5 Compressible Flow Equations 413 9.6 Crocco s Theorem 414 9.6.1 Basic Solutions of Laplace s Equation 418 9.7 The General Potential Equation for Three-Dimensional Flow 418 9.8 Linearization of the Potential Equation 420 9.8.1 Small Perturbation Theory 420 9.9 Potential Equation for Bodies of Revolution 423 9.9.1 Solution of Nonlinear Potential Equation 425 9.10 Boundary Conditions 425 9.10.1 Bodies of Revolution 427 9.11 Pressure Coefficient 428 9.11.1 Bodies of Revolution 429 9.12 Similarity Rule 429 9.13 Two-Dimensional Flow: Prandtl-Glauert Rule for Subsonic Flow 429 9.13.1 The Prandtl-Glauert Transformations 429 9.13.2 The Direct Problem-Version I 431 9.13.3 The Indirect Problem (Case of Equal Potentials): P-G Transformation Version II 434 9.13.4 The Streamline Analogy (Version III): Gothert s Rule 435 9.14 Prandtl-Glauert Rule for Supersonic Flow: Versions I and II 436 9.14.1 Subsonic Flow 436 9.14.2 Supersonic Flow 436
Contents xiii 9.15 The von Karman Rule for Transonic Flow 439 9.15.1 Use of Karman Rule 440 9.16 Hypersonic Similarity 442 9.17 Three-Dimensional Flow: The Gothert Rule 444 9.17.1 The General Similarity Rule 444 9.17.2 Gothert Rule 446 9.17.3 Application to Wings of Finite Span 447 9.17.4 Application to Bodies of Revolution and Fuselage 448 9.17.5 The Prandtl-Glauert Rule 450 9.17.6 The von Karman Rule for Transonic Flow 454 9.18 Moving Disturbance 455 9.18.1 Small Disturbance 456 9.18.2 Finite Disturbance 457 9.19 Normal Shock Waves 457 9.19.1 Equations of Motion for a Normal Shock Wave 457 9.19.2 The Normal Shock Relations for a Perfect Gas 458 9.20 Change of Total Pressure across a Shock 462 9.21 Oblique Shock and Expansion Waves 463 9.21.1 Oblique Shock Relations 464 9.21.2 Relation between β and θ 466 9.21.3 Supersonic Flow over a Wedge 469 9.21.4 Weak Oblique Shocks 471 9.21.5 Supersonic Compression 473 9.21.6 Supersonic Expansion by Turning 475 9.21.7 The Prandtl-Meyer Function 477 9.21.8 Shock-Expansion Theory 477 9.22 Thin Aerofoil Theory 479 9.22.1 Application of Thin Aerofoil Theory 480 9.23 Two-Dimensional Compressible Flows 485 9.24 General Linear Solution for Supersonic Flow 486 9.24.1 Existence of Characteristics in a Physical Problem 488 9.24.2 Equation for the Streamlines from Kinematic Flow Condition 489 9.25 Flow over a Wave-Shaped Wall 491 9.25.1 Incompressible Flow 491 9.25.2 Compressible Subsonic Flow 492 9.25.3 Supersonic Flow 493 9.25.4 Pressure Coefficient 494 9.26 Summary 495 Exercise Problems 509 References 512 10 Simple Flights 513 10.1 Introduction 513 10.2 Linear Flight 513 10.3 Stalling 514 10.4 Gliding 516 10.5 Straight Horizontal Flight 518 10.6 Sudden Increase of Incidence 520 10.7 Straight Side-Slip 521
xiv Contents 10.8 Banked Turn 522 10.9 Phugoid Motion 523 10.10 The Phugoid Oscillation 525 10.11 Summary 529 Exercise Problems 531 Further Readings 533 Index 535
About the Author Ethirajan Rathakrishnan is Professor of Aerospace Engineering at the Indian Institute of Technology Kanpur, India. He is well-known internationally for his research in the area of high-speed jets. The limit for the passive control of jets, called Rathakrishnan Limit, is his contribution to the field of jet research, and the concept of breathing blunt nose (BBN), which reduces the positive pressure at the nose and increases the low-pressure at the base simultaneously, is his contribution to drag reduction at hypersonic speeds. He has published a large number of research articles in many reputed international 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); Fluid Mechanics: An Introduction, 3rd ed. (PHI Learning, New Delhi, 2012); Gas Tables, 3rd ed. (Universities Press, 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 Gas Dynamics (John Wiley, New Jersey, USA, 2010); and Elements of Heat Transfer, (CRC Press, Taylor & Francis Group, Boca Raton, USA, 2012).
Preface This book has been developed to serve as a text for theoretical aerodynamics at the introductory level for both undergraduate courses and for an advanced course at graduate level. The basic aim of this book is to provide 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 aerodynamic theory 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 undergraduate level. The essence of fluid mechanics, conformal transformation and vortex theory, being the basics for the subject of theoretical aerodynamics, are given in separate chapters. A considerable number of solved examples are given in these chapters to fix the concepts introduced and a large number of exercise problems along with answers are listed at the end of these chapters to test the understanding of the material studied. To make readers comfortable with the basic features of aircraft geometry and its flight, vital parts of aircraft and the preliminary aspects of its flight are discussed in the first and final chapters. The entire spectrum of theoretical aerodynamics is presented in this book, with necessary explanations on every aspect. The material covered in this book is so designed that any beginner can follow it comfortably. The topics covered are broad based, starting from the basic principles and progressing towards the physics of 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, the basic aspects of the fluid flow and vortices are reviewed in order to establish a firm basis for the subject of aerodynamic theory. Following this, conformal transformation of flows is introduced with the elementary aspects and then gradually proceeding to the vital aspects and application of Joukowski transformation which 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 generation and its effect on lift and drag are discussed in depth. The chapter on thin aerofoil theory discusses the performance of aerofoils, highlighting the application and limitations of the thin aerofoils. The chapter on panel methods presents the source and vortex panel techniques meant for solving the flow around nonlifting and lifting bodies, respectively. The chapter on finite wing theory presents the performance of wings of finite aspect ratio, where the horseshoe vortex, made up of the bound vortex and tip vortices, plays a dominant role. The procedure for calculating the lift, drag and pitching moment for symmetrical and cambered profiles is discussed in detail. The consequence of the velocity induced by the vortex system is presented in detail, along with solved examples at appropriate places. The chapter on compressible flows covers the basics and application aspects in detail for both subsonic and supersonic regimes of the flow. The similarity consideration covering the Parandtl-Glauert I and II rules and Gothert rule are presented in detail. The basic governing equation and its simplification with small perturbation assumption is covered systematically. Shocks and expansion waves and their influence on the flow field are discussed in depth. Following this the shock-expansion theory and thin aerofoil theory and their application to calculate the lift and drag are presented.
xviii Preface In the final chapter, some basic flights are introduced briefly, covering the level flight, gliding and climbing 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 the voluminous subject. This book is the outgrowth of lectures presented over a number of years, both at undergraduate and graduate level. The student, or reader, is assumed to have a background in the basic courses of fluid mechanics. 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 can be useful for scientists and engineers working in the field of aerodynamics in industries and research laboratories. My sincere thanks to my undergraduate and graduate students in India and abroad, who are directly and indirectly responsible for the development of this book. I would like to express my sincere thanks to Yasumasa Watanabe, doctoral student of Aerospace Engineering, the University of Tokyo, Japan, for his help in making some solved examples along with computer codes. I thank Shashank Khurana, doctoral student of Aerospace Engineering, the University of Tokyo, Japan, for critically checking the manuscript of this book. Indeed, incorporation of the suggestions given by Shashank greatly enhanced the clarity of manuscript of this book. I thank my doctoral students Mrinal Kaushik and Arun Kumar, for checking the manuscript and the solutions manual, and for giving some useful suggestions. For instructors only, a companion Solutions Manual is available from John Wiley and contains typed solutions to all the end-of-chapter problems can be found at www.wiley.com/go/rathakrishnan. The financial support extended by the Continuing Education Centre of the Indian Institute of Technology Kanpur, for the preparation of the manuscript is gratefully acknowledged. Ethirajan Rathakrishnan
1 Basics 1.1 Introduction Aerodynamics is the science concerned with the motion of air and bodies moving through air. In other words, aerodynamics is a branch of dynamics concerned with the study of motion of air, particularly when it interacts with a moving object. The forces acting on bodies moving through the air are termed aerodynamic forces. Air is a fluid, and in accordance with Archimedes principle, an aircraft will be buoyed up by a force equal to the weight of air displaced by it. The buoyancy force F b will act vertically upwards. The weight W of the aircraft is a force which acts vertically downwards; thus the magnitude of the net force acting on an aircraft, even when it is not moving, is (W F b ). The force (W F b ) will act irrespective 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 is maintained 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, when it is at a steady flight (unaccelerated motion). Therefore, there must be an additional force F ad, say, such that the vectorial sum of the forces acting on the aircraft is: T + (W F b ) + F ad = 0 Force F ad 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 imagine that the aircraft is at rest with the air streaming past it. In this case, the air velocity over the aircraft will 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 an aircraft using wind tunnels, this principle is adopted, that is, the aircraft model is fixed in the wind tunnel test-section and the air is made to flow over the model. In our discussions we shall always refer to the direction of V as the direction of aircraft motion, and the direction of V as the direction of airstream or relative wind. 1.2 Lift and Drag The aerodynamic force F ad can be resolved into two component forces, one at right angles to V and the other 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.
2 Theoretical Aerodynamics L D F ad V θ T (W F b ) Figure 1.1 Forces acting on an aircraft in horizontal flight. component opposite to V is called drag D. Ifθ is the angle between L and F ad,wehave: L = F ad cos θ D = F ad 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 gliding angle θ is said to be streamlined. This explicitly implies that when θ is large the aircraft can not be regarded as a streamlined body. This may make us wonder about the nature of the aircraft geometry, whether it is streamlined 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 declared as a streamlined body. What is the genesis for this drastic conflict? These doubts will be cleared if we get 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 the total 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, and the skin friction drag is insignificant. Therefore, the basis for declaring a body as streamlined or bluff is the relative magnitudes of skin friction and wake drag components and not just the geometry of the body shape alone. Indeed, sometimes the shape 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, as shown is Figure 1.2(b), is a typical bluff body. This clearly demonstrates that the streamlined and bluff nature of a body is dictated by the combined effect of the body geometry and its orientation to the flow direction. Therefore, even though an aircraft is usually regarded as a streamlined body, it can behave as a bluff body when the gliding angle θ is large, causing the formation of large wake, leading to a large value of wake drag. That is why it is stated that, for small values of gliding angle θ an aircraft is said
Basics 3 (a) (b) 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 with small gliding angle in most portion of their mission and hence are referred to as streamlined bodies. All fighter 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 freestream flowis3, 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, F ad, and lift, L, is: L = F ad cos θ. The aerodynamic force becomes: F ad = L cos θ = 14715 cos 3 = 14735.2N. The relation between the aerodynamic force, F ad, and drag, D, is: D = F ad sin θ.
4 Theoretical Aerodynamics Therefore, the drag becomes: The lift to drag ratio of the aircraft is: D = 14735.2 sin 3 = 771.2N. 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 or triplane. 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 consists of two wings; the port (left) and starboard (right) wings, which together constitute the aerofoil. The tail plane 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 which means 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 the vertical tail, shown in Figure 1.3, are control surfaces. When the ailerons and rudder are in their neutral positions, 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 then the 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 of symmetry, as illustrated in Figure 1.4. Thus, the wing tips can be a point or a line or an area, according to the design of the aerofoil. The distance between the wing tips is called the span. The section of a wing by a plane parallel to the plane of symmetry is called a profile. The shape and general orientation of the profile 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. V Starboard wing Fuselage Fin Rudder z Engine Port wing Flap Tail plane Aileron Elevator y x Figure 1.3 Main features of a monoplane aircraft.
Basics 5 Span Tip Port wing Starboard wing Tip b b Plane of symmetry 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 Flyer used a biplane design, as did most aircraft in the early years of aviation. While a biplane wing structure has a structural advantage, it generates more drag than a similar monoplane wing. Improved structural techniques and materials and the quest for greater speed made the biplane configuration obsolete for most purposes 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 to produce twice as much lift as a single wing of similar size and shape because both the upper and lower wings are working on nearly the same portion of the atmosphere. For example, in a wing of aspect ratio 6, 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.
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 is raised above the fuselage with an arrangement of cabane struts, although other arrangements have been used. Almost all biplanes also have a third horizontal surface, the tailplane, to control the pitch, or angle of attack of the aircraft (although there have been a few exceptions). Either or both of the main wings can support flaps or ailerons to assist lateral rotation and speed control; usually the ailerons are mounted on the upper wing, and flaps (if used) on the lower wing. Often there is bracing between the upper and lower wings, 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 a similarly sized monoplane of similar wingspan. Biplanes will therefore typically have a shorter wingspan than a similar monoplane, which tends to afford greater maneuverability. The struts and wire bracing of a 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 interferes with the aerodynamics of the other. For a given wing area the biplane generates more drag and produces less lift than a monoplane. Now, one may ask what is the specific difference between a biplane and monoplane? The answer is as follows. A biplane has two (bi) sets of wings, and a monoplane has one (mono) set of wings. The two sets of wings on a biplane add lift, and also drag, allowing it to fly slower. The one set of wings on a monoplane do not add as much lift or drag, making it fly faster, and as a result, all fast planes are monoplanes, and most planes these days are monoplanes. 1.5 Triplane A triplane is a fixed-wing aircraft equipped with three vertically-stacked wing planes. Tailplanes and canard fore-planes are not normally included in this count, although they may occasionally be. A typical example for triplane is the Fokker Dr. I of World War I, shown in Figure 1.7.
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 arrangement has a narrower wing chord than a biplane of similar span and area. This gives each wing plane a slender appearance with a higher aspect ratio, making it more efficient and giving increased lift. This potentially offers a faster rate of climb and tighter turning radius, both of which are important in a fighter plane. The Sopwith Triplane was a successful example, having the same wing span as the equivalent biplane, the Sopwith Pup. Alternatively, a triplane has a reduced span compared with a biplane of given wing area and aspect ratio, leading to a more compact and lightweight structure. This potentially offers better maneuverability for 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 shorter span 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 is partially offset by the increased depth of the overall structure, allowing a more efficient construction. The Caproni 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 and drag of the structural bracing, and the aerodynamic inefficiency inherent in the stacked wing layout. As biplane 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 cabane struts. 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, as illustrated in Figure 1.8. The chord has direction, position, and length. The main requisite is that in each case the chord should be precisely defined, because the chord enters into the constants such as the lift and drag coefficients, which describe the aerodynamic properties of the profile. For the profile shown in Figure 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 is the chord. For the profile in Figure 1.8(c), the line joining the tips of leading and trailing edges is the chord.
8 Theoretical Aerodynamics chord c (a) Leading and trailing edges are circular arcs. chord c (b) Circular arc leading edge and sharp trailing edge. chord c (c) Faired leading edge and sharp trailing edge. 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 tangent to its lower surface (that is, the tangent which touches the profile at two distinct points), as shown in Figure 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 the span, as shown in Figure 1.5), the chord of the aerofoil is taken to be the chord of the profile in which the plane of symmetry cuts the aerofoil. In all other cases, the chord of the aerofoil is defined as the mean or average 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 x y o z Figure 1.10 A wing with Cartesian coordinates.
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. This means, 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 this profile, θ be the inclination of the chord to the xy plane, and (x, y, z) be the coordinates of the quarter point 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 when y is given, all quantities characterizing the profile, namely, the mean chord, its position and inclination to the flow, are functions of y. The chord of an aerofoil is defined by averaging the distance between the leading and trailing edges of the profiles at different locations along the span. Thus, if c m is the length of the mean chord, (x m, 0,z m ) its quarter point, and θ m its inclination, we take the average or mean chord as: c m = 1 2b θ m = 1 2b x m = 1 2b z m = 1 2b +b b +b b +b b +b These mean values completely define the chord of the aerofoil in length (c m ), direction (θ m ), and position (x m,z m ). b cdy θdy xdy zdy. 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 Figure 1.10. 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). The projection in this case is a rectangular area S, say, which is called the plan area of the wing. The plan area is different from the total surface area of the wing. The simplest cylindrical wing would be a rectangular plate, 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 the aspect ratio is defined as: A representative value of aspect ratio is 6. = (2b)2 S.
10 Theoretical Aerodynamics Example 1.2 The semi-span of a rectangular wing of planform area 8.4 m 2 is 3.5 m. Determine the aspect ratio of the wing. Solution Given, S = 8.4 m 2 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 = 1.2m. 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) from the chord of the profile, as illustrated in Figure 1.11. Let z u and z l be the ordinates on the upper and lower parts of the profile, respectively, for the same value of x. Let c be the chord, and the x-axis coincide with the chord. Now, the upper and lower camber are defined as: Upper camber = (z u) max c Lower camber = (z l) max, c z P u A P P l M Chord, c H A Camber line Chord Camber H (a) (b) Figure 1.11 Illustration of camber, camberline and chord of aerofoil profile.