Theoretical Aerodynamics E-Book

Ethirajan Rathakrishnan

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 719

Veröffentlichungsjahr: 2013

Beschreibung

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 higher level. The mathematics involved is presented so that it can be followed comfortably, even by those who are not strong in mathematics. The examples are designed to fix the theory studied in an effective manner. Throughout the book, the physics behind the processes are clearly explained. Each chapter begins with an introduction 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 enhance understanding, and moves well simpler to more advanced topics * Includes a complete section on fluid mechanics and thermodynamics, essential background topics to the theory of aerodynamics * Blends the mathematical and physical concepts of design and performance aspects of lifting surfaces, and introduces the reader to the thin aerofoil theory, panel method, and finite aerofoil theory * Includes a Solutions Manual for end-of-chapter exercises, and Lecture slides on the book's Companion Website

Leseprobe

Contents

Cover

Title Page

Copyright

Dedication

About the Author

Preface

Chapter 1: Basics

1.1 Introduction

1.2 Lift and Drag

1.3 Monoplane Aircraft

1.4 Biplane

1.5 Triplane

1.6 Aspect Ratio

1.7 Camber

1.8 Incidence

1.9 Aerodynamic Force

1.10 Scale Effect

1.11 Force and Moment Coefficients

1.12 The Boundary Layer

1.13 Summary

Reference

Chapter 2: Essence of Fluid Mechanics

2.1 Introduction

2.2 Properties of Fluids

2.3 Thermodynamic Properties

2.4 Surface Tension

2.5 Analysis of Fluid Flow

2.6 Basic and Subsidiary Laws

2.7 Kinematics of Fluid Flow

2.8 Streamlines

2.9 Potential Flow

2.10 Combination of Simple Flows

2.11 Flow Past a Circular Cylinder without Circulation

2.12 Viscous Flows

2.13 Compressible Flows

2.14 Summary

References

Chapter 3: Conformal Transformation

3.1 Introduction

3.2 Basic Principles

3.3 Complex Numbers

3.4 Summary

Exercise Problems

Chapter 4: Transformation of Flow Pattern

4.1 Introduction

4.2 Methods for Performing Transformation

4.3 Examples of Simple Transformation

4.4 Kutta−Joukowski Transformation

4.5 Transformation of Circle to Straight Line

4.6 Transformation of Circle to Ellipse

4.7 Transformation of Circle to Symmetrical Aerofoil

4.8 Transformation of a Circle to a Cambered Aerofoil

4.9 Transformation of Circle to Circular Arc

4.10 Joukowski Hypothesis

4.11 Lift of Joukowski Aerofoil Section

4.12 The Velocity and Pressure Distributions on the Joukowski Aerofoil

4.13 The Exact Joukowski Transformation Process and Its Numerical Solution

4.14 The Velocity and Pressure Distribution

4.15 Aerofoil Characteristics

4.16 Aerofoil Geometry

4.17 Wing Geometrical Parameters

4.18 Aerodynamic Force and Moment Coefficients

4.19 Summary

Exercise Problems

References

Chapter 5: Vortex Theory

5.1 Introduction

5.2 Vorticity Equation in Rectangular Coordinates

5.3 Circulation

5.4 Line (point) Vortex

5.5 Laws of Vortex Motion

5.6 Helmholtz's Theorems

5.7 Vortex Theorems

5.8 Calculation of uR, the Velocity due to Rotational Flow

5.9 Biot-Savart Law

5.10 Vortex Motion

5.11 Forced Vortex

5.12 Free Vortex

5.13 Compound Vortex

5.14 Physical Meaning of Circulation

5.15 Rectilinear Vortices

5.16 Velocity Distribution

5.17 Size of a Circular Vortex

5.18 Point Rectilinear Vortex

5.19 Vortex Pair

5.20 Image of a Vortex in a Plane

5.21 Vortex between Parallel Plates

5.22 Force on a Vortex

5.23 Mutual action of Two Vortices

5.24 Energy due to a Pair of Vortices

5.25 Line Vortex

5.26 Summary

Exercise Problems

References

Chapter 6: Thin Aerofoil Theory

6.1 Introduction

6.2 General Thin Aerofoil Theory

6.3 Solution of the General Equation

6.4 The Circular Arc Aerofoil

6.5 The General Thin Aerofoil Section

6.6 Lift, Pitching Moment and Center of Pressure Coefficients for a Thin Aerofoil

6.7 Flapped Aerofoil

6.8 Summary

Exercise Problems

References

Chapter 7: Panel Method

7.1 Introduction

7.2 Source Panel Method

7.3 The Vortex Panel Method

7.4 Pressure Distribution around a Circular Cylinder by Source Panel Method

7.5 Using Panel Methods

7.6 Summary

References

Chapter 8: Finite Aerofoil Theory

8.1 Introduction

8.2 Relationship between Spanwise Loading and Trailing Vorticity

8.3 Downwash

8.4 Characteristics of a Simple Symmetrical Loading –Elliptic Distribution

8.5 Aerofoil Characteristic with a More General Distribution

8.6 The Vortex Drag for Modified Loading

8.7 Lancaster –Prandtl Lifting Line Theory

8.8 Effect of Downwash on Incidence

8.9 The Integral Equation for the Circulation

8.10 Elliptic Loading

8.11 Aerodynamic Characteristics of Asymmetric Loading

8.12 Lifting Surface Theory

8.13 Aerofoils of Small Aspect Ratio

8.14 Lifting Surface

8.15 Summary

Exercise Problems

Chapter 9: Compressible Flows

9.1 Introduction

9.2 Thermodynamics of Compressible Flows

9.3 Isentropic Flow

9.4 Discharge from a Reservoir

9.5 Compressible Flow Equations

9.6 Crocco's Theorem

9.7 The General Potential Equation for Three-Dimensional Flow

9.8 Linearization of the Potential Equation

9.9 Potential Equation for Bodies of Revolution

9.10 Boundary Conditions

9.11 Pressure Coefficient

9.12 Similarity Rule

9.13 Two-Dimensional Flow: Prandtl-Glauert Rule for Subsonic Flow

9.14 Prandtl-Glauert Rule for Supersonic Flow: Versions I and II

9.15 The von Karman Rule for Transonic Flow

9.16 Hypersonic Similarity

9.17 Three-Dimensional Flow: The Gothert Rule

9.18 Moving Disturbance

9.19 Normal Shock Waves

9.20 Change of Total Pressure across a Shock

9.21 Oblique Shock and Expansion Waves

9.22 Thin Aerofoil Theory

9.23 Two-Dimensional Compressible Flows

9.24 General Linear Solution for Supersonic Flow

9.25 Flow over a Wave-Shaped Wall

9.26 Summary

Exercise Problems

References

Chapter 10: Simple Flights

10.1 Introduction

10.2 Linear Flight

10.3 Stalling

10.4 Gliding

10.5 Straight Horizontal Flight

10.6 Sudden Increase of Incidence

10.7 Straight Side-Slip

10.8 Banked Turn

10.9 Phugoid Motion

10.10 The Phugoid Oscillation

10.11 Summary

Exercise Problems

Further Readings

Index

This edition first published 2013

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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

This book is dedicated to my parents,

Mr Thammanur Shunmugam Ethirajan

and

Mrs Aandaal Ethirajan

Ethirajan Rathakrishnan

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.

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 Fb 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 . The force 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.

Figure 1.1 Forces acting on an aircraft in horizontal flight.

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:

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 liftL and the component opposite to V is called dragD. If θ is the angle between L and F ad, we have:

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 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.

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