FLIGHT THEORY AND AERODYNAMICS FLIGHT THEORY AND AERODYNAMICS A Practical Guide for Operational Safety THIRD EDITION Charles E. Dole James E. Lewis Joseph R. Badick Brian A. Johnson This book is printed on acid-free paper. Copyright © 2017 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada 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 permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. 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For general information about our other products and services, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com. Library of Congress Cataloging-in-Publication Data: Names: Dole, Charles E. (Charles Edward), 1916– author. | Lewis, James E., 1946– author. | Badick, Joseph R. (Joseph Robert), 1952– author. | Johnson, Brian A. (Brian Andrew), 1975– author. Title: Flight theory and aerodynamics : a practical guide for operational safety / Charles E. Dole, James E. Lewis, Joseph R. Badick, Brian A. Johnson. Description: Third edition. | Hoboken, New Jersey : John Wiley & Sons Inc., [2017] | Includes bibliographical references and index. Identifiers: LCCN 2016025499 | ISBN 9781119233404 (hardback) | ISBN 9781119233411 (epub) | ISBN 9781119233428 (epdf) Subjects: LCSH: Aerodynamics—Handbooks, manuals, etc. | Airplanes—Piloting—Handbooks, manuals, etc. | Aeronautics—Safety measures—Handbooks, manuals, etc. | BISAC: TECHNOLOGY & ENGINEERING / Mechanical. Classification: LCC TL570 .D56 2017 | DDC 629.132—dc23 LC record available at https://lccn.loc.gov/2016025499 Cover Design: Wiley Cover Images: Blue sky © Fabian Rothe/Getty Images, Inc.; Airplane frontview © Laksone/iStockphoto; Isometric flying plane © aurin/iStockphoto Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Contents Preface About the Authors 1 Introduction xi xiii 1 The Flight Environment, 1 Basic Quantities, 1 Forces, 2 Mass, 3 Scalar and Vector Quantities, 4 Moments, 5 Equilibrium Conditions, 6 Newton’s Laws of Motion, 6 Linear Motion, 7 Rotational Motion, 8 Work, 8 Energy, 8 Power, 9 Friction, 9 Symbols, 10 Equations, 11 Problems, 12 2 Atmosphere, Altitude, and Airspeed Measurement 13 Properties of the Atmosphere, 13 ICAO Standard Atmosphere, 15 Altitude Measurement, 16 Continuity Equation, 19 Bernoulli’s Equation, 19 Airspeed Measurement, 22 Symbols, 26 Equations, 27 Problems, 27 3 Structures, Airfoils, and Aerodynamic Forces 31 Aircraft Structures, 31 Airfoils, 37 Development of Forces on Airfoils, 42 Aerodynamic Force, 44 v vi CONTENTS Aerodynamic Pitching Moments, 45 Aerodynamic Center, 46 Symbols, 46 Problems, 47 4 Lift 49 Introduction to Lift, 49 Angle of Attack Indicator, 49 Boundary Layer Theory, 51 Reynolds Number, 53 Adverse Pressure Gradient, 54 Airflow Separation, 55 Stall, 56 Aerodynamic Force Equations, 57 Lift Equation, 58 Airfoil Lift Characteristics, 60 High Coefficient of Lift Devices, 61 Lift During Flight Manuevers, 65 Symbols, 67 Equations, 67 Problems, 68 5 Drag 71 Drag Equation, 71 Induced Drag, 71 Ground Effect, 77 Laminar Flow Airfoils, 81 Parasite Drag, 82 Total Drag, 85 Lift to Drag Ratio, 87 Drag Reduction, 88 Symbols, 90 Equations, 91 Problems, 91 6 Jet Aircraft Basic Performance Thrust-Producing Aircraft, 95 Principles of Propulsion, 96 Thrust-Available Turbojet Aircraft, 100 Specific Fuel Consumption, 101 Fuel Flow, 102 Thrust-Available–Thrust-Required Curves, 103 Items of Aircraft Performance, 104 Symbols, 113 Equations, 113 Problems, 114 95 CONTENTS 7 Jet Aircraft Applied Performance vii 117 Variations in the Thrust-Required Curve, 117 Variations of Aircraft Performance, 121 Equations, 125 Problems, 125 8 Propeller Aircraft: Basic Performance 129 Power Available, 129 Principles of Propulsion, 131 Power-Required Curves, 133 Items of Aircraft Performance, 139 Symbols, 145 Equations, 146 Problems, 146 9 Propeller Aircraft: Applied Performance 149 Variations in the Power-Required Curve, 149 Variations in Aircraft Performance, 153 Equations, 157 Problems, 157 10 Takeoff Performance 161 Definitions Important to Takeoff Planning, 161 Aborted Takeoffs, 164 Linear Motion, 166 Factors Affecting Takeoff Performance, 168 Improper Liftoff, 171 Symbols, 174 Equations, 175 Problems, 175 11 Landing Performance Prelanding Performance, 179 Improper Landing Performance, 185 Landing Deceleration, Velocity, and Distance, Landing Equations, 194 Hazards of Hydroplaning, 197 Symbols, 199 Equations, 199 Problems, 200 12 Slow-Speed Flight Stalls, 203 Region of Reversed Command, 210 Spins, 212 179 190 203 viii CONTENTS Low-Level Wind Shear, 216 Aircraft Performance in Low-Level Wind Shear, Effect of Ice and Frost, 221 Wake Turbulence, 222 Problems, 224 218 13 Maneuvering Performance General Turning Performance, Equations, 242 Problems, 243 227 227 14 Longitudinal Stability and Control 245 Definitions, 245 Oscillatory Motion, 246 Airplane Reference Axes, 248 Static Longitudinal Stability, 248 Dynamic Longitudinal Stability, 260 Pitching Tendencies in a Stall, 261 Longitudinal Control, 264 Symbols, 266 Equations, 266 Problems, 266 15 Directional and Lateral Stability and Control 269 Directional Stability and Control, 269 Static Directional Stability, 269 Directional Control, 276 Multi-Engine Flight Principles, 280 Lateral Stability and Control, 284 Static Lateral Stability, 284 Lateral Control, 288 Dynamic Directional and Lateral Coupled Effects, 288 Symbols, 293 Equations, 293 Problems, 293 16 High-Speed Flight The Speed of Sound, 295 High-Subsonic Flight, 297 Design Features for High-Subsonic Flight, 298 Transonic Flight, 301 Supersonic Flight, 305 Symbols, 316 Equations, 316 Problems, 316 295 CONTENTS 17 Rotary-Wing Flight Theory ix 319 Momentum Theory of Lift, 320 Airfoil Selection, 320 Forces on Rotor System, 321 Thrust Development, 323 Hovering Flight, 324 Ground Effect, 326 Rotor Systems, 328 Dissymmetry of Lift in Forward Flight, 330 High Forward Speed Problems, 333 Helicopter Control, 334 Helicopter Power-Required Curves, 336 Power Settling, Settling with Power, and Vortex Ring State, 338 Autorotation, 340 Dynamic Rollover, 341 Problems, 343 Answers to Problems 345 References 349 Index 353 Preface The third edition of Flight Theory and Aerodynamics was revised to further enhance the book’s use as an introductory text for colleges and universities offering an aeronautical program. The publisher conducted a survey with aviation schools to determine what was needed in an updated text. The result is this third edition that meets not only classroom requirements but also practical application. All seventeen chapters have some level of updating and additional content. The revision retains mathematical proofs, but also seeks to provide a non-mathematical discussion of aerodynamics geared toward a more practical application of flight theory. As such, it is a how to handbook as well as one about the theory of flying. It was written for all participants in the aviation industry: Pilots, aviation maintenance technicians, aircraft dispatchers, air traffic controllers, loadmasters, flight engineers, flight attendants, meteorologists, avionics technicians, aviation managers, as all have a vested interest in both safety and operational efficiency. Updates in the third edition: • • • • • New sequence of chapters for better flow of topics Extensive upgrade to the helicopter chapter, including discussion of other types of rotorcraft Added modern graphics, including correlation with current FAA publications Added detail in subject matter emphasizing practical application Additional terms and abbreviations The authors would like to thank our contacts at Wiley for their support throughout this revision as well as the support of our colleagues and families. In particular the authors would like to thank Steven. A. Saunders for his technical contribution to this revision, employing over 50 years of military, airline, and general aviation experience in the process. Finally, the authors would like to gratefully acknowledge the previous work of Charles E. Dole and James E. Lewis for their contribution to improving aviation safety throughout the aviation industry. Joseph R. Badick Brian A. Johnson xi About the Authors A former marine, the late CHARLES E. DOLE taught flight safety for twenty-eight years to officers of the U.S. Air Force, Army, and Navy, as well as at the University of Southern California. The late JAMES E. LEWIS was an associate professor of Aeronautical Science at Embry Riddle Aeronautical University in Florida, former aeronautical engineer for the Columbus Aircraft Division of Rockwell International, and retired Ohio National Guard military pilot. JOSEPH R. BADICK has over forty years of flight experience in single, multi-engine, land/seaplane aircraft. Rated in commercial rotor-craft and gliders, with the highest rating of (A.T.P.) Airline Transport Pilot. A licensed airframe and powerplant mechanic, with inspection authorization (I.A.), he has installed numerous aircraft aerodynamic performance (S.T.C’s) Supplemental Type Certificates, with test flight checks. He holds a Ph.D. (ABD) in Business from Northcentral University of Arizona and a Master’s degree in Aeronautical Science. He was a Naval Officer for 30 years as an Aeronautical Engineer Duty Officer (AEDO), involved in all aspects of aircraft maintenance, logistics, acquisition, and test/evaluation. Currently he is a professor of aviation at a community college in the Career Pilot/Aviation Management degree programs. BRIAN A. JOHNSON is a former airline and corporate pilot who holds a multi-engine Airline Transport Pilot certificate, in addition to Commercial pilot single-engine land/sea privileges. He is an active instrument and multi-engine Gold Seal flight instructor with an advanced ground instructor rating. He holds a Master’s degree in Aeronautical Science from Embry-Riddle Aeronautical University and currently serves in a faculty position for a two-year Career Pilot/Aviation Management degree program, in addition to serving as an adjunct faculty member in the Aeronautical Science department of a major aeronautical university. xiii FLIGHT THEORY AND AERODYNAMICS 1 Introduction A basic understanding of the physical laws of nature that affect aircraft in flight and on the ground is a prerequisite for the study of aerodynamics. Modern aircraft have become more sophisticated, and more automated, using advanced materials in their construction, requiring pilots to renew their understanding of the natural forces encountered during flight. Understanding how pilots control and counteract these forces better prepares pilots and engineers for the art of flying, and for harnessing the fundamental physical laws that guide them. Perhaps your goal is to be a pilot, who will “slip the surly bonds of earth,” as John Gillespie Magee wrote in his classic poem “High Flight.” Or maybe you aspire to build or maintain aircraft as a skilled technician. Or possibly you wish to serve in another vital role in the aviation industry, such as manager, dispatcher, meteorologist, engineer, teacher, or another capacity. Whichever area you might be considering, this textbook will attempt to build on previous material you have learned, and hopefully will prepare you for a successful aviation career. THE FLIGHT ENVIRONMENT This chapter begins with a review of the basic principles of physics and concludes with a summary of linear motion, mechanical energy, and power. A working knowledge of these areas, and how they relate to basic aerodynamics, is vital as we move past the rudimentary “four forces of flight” and introduce thrust and power-producing aircraft, lift and drag curves, stability and control, maneuvering performance, slow-speed flight, and other topics. Up to this point you have seen that there are four basic forces acting on an aircraft in flight: lift, weight, thrust, and drag. Now we must understand how these forces change as an aircraft accelerates down the runway, or descends on final approach to a runway and gently touches down even when traveling twice the speed of a car on the highway. Once an aircraft has safely made it into the air, what effect does weight have on its ability to climb, and should the aircraft climb up to the flight levels or stay lower and take “advantage” of the denser air closer to the ground? By developing an understanding of the aerodynamics of flight, how design, weight, load factors, and gravity affect an aircraft during flight maneuvers from stalls to high speed flight, the pilot learns how to control the balance between these forces. This textbook will help clarify these issues, among others, hopefully leaving you with a better understanding of the flight environment. BASIC QUANTITIES An introduction to aerodynamics must begin with a review of physics, and in particular, the branch of physics that will be presented here is called mechanics. We will examine the fundamental physical laws governing the forces acting on an aircraft in flight, and what effect these natural laws and forces have on the performance characteristics of aircraft. To control an aircraft, whether it is an airplane, helicopter, glider, or balloon, the pilot must understand the principles involved and learn to use or counteract these natural forces. 1 2 INTRODUCTION We will start with the concepts of work, energy, power, and friction, and then build upon them as we move forward in future chapters. Because the metric system of measurement has not yet been widely accepted in the United States, the English system of measurement is used in this book. The fundamental units are Force Distance Time pounds (lb) feet (ft) seconds (sec) From the fundamental units, other quantities can be derived: Velocity (distance/time) Area (distance squared) Pressure (force/unit area) Acceleration (change in velocity) ft/sec (fps) square ft (ft2 ) lb/ft2 (psf) ft/sec/sec (fps2 ) Aircraft measure airspeed in knots (nautical miles per hour) or in Mach number (the ratio of true airspeed to the speed of sound). Rates of climb and descent are measured in feet per minute, so quantities other than those above are used in some cases. Some useful conversion factors are listed below: Multiply by knots fps miles per hour (mph) fps mph knots nautical miles (nm) nm sm knots 1.69 0.5925 1.47 0.6818 0.8690 1.15 6076 1.15 0.869 101.3 to get feet per second (fps) knots fps mph knots mph feet (ft) statute miles (sm) nm feet per minute (fpm) FORCES A force is a push or a pull tending to change the state of motion of a body. Typical forces acting on an aircraft in steady flight are shown in Fig. 1.1. Figure 1.2 shows the resolution of the aerodynamic forces during AERODYNAMIC FORCE (AF) THRUST WEIGHT Fig. 1.1. Forces on an airplane in steady flight. 3 MASS Lift Thrust Drag Weight Fig. 1.2. Resolved forces on an airplane in steady flight. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 straight-and-level, unaccelerated flight and is separated into four components. The component that is 90∘ to the flight path and acts toward the top of the airplane is called lift. The component that is parallel to the flight path and acts toward the rear of the airplane is called drag; while the opposing forward force is thrust and is usually created by the engine. Weight opposes lift and as we will see is a function of the mass of the aircraft and gravity. MASS Mass is a measure of the amount of material contained in a body. Weight, on the other hand, is a force caused by the gravitational attraction of the earth (g = 32.2 ft∕s2 ), moon, sun, or other heavenly bodies. Weight will vary, depending on where the body is located in space. Mass will not vary with position. Weight (W) = Mass (m) × Acceleration of gravity (g) W = mg Rearranging gives m= This mass unit is called the slug. (1.1) lb ⋅ sec2 W lb = g ft∕sec2 ft 4 INTRODUCTION SCALAR AND VECTOR QUANTITIES A quantity that has size or magnitude only is called a scalar quantity. The quantities of mass, time, and temperature are examples of scalar quantities. A quantity that has both magnitude and direction is called a vector quantity. Forces, accelerations, and velocities are examples of vector quantities. Speed is a scalar, but if we consider the direction of the speed, then it is a vector quantity called velocity. If we say an aircraft traveled 100 nm, the distance is a scalar, but if we say an aircraft traveled 100 nm on a heading of 360∘ , the distance is a vector quantity. Scalar Addition Scalar quantities can be added (or subtracted) by simple arithmetic. For example, if you have 5 gallons of gas in your car’s tank and you stop at a gas station and top off your tank with 9 gallons more, your tank now holds 14 gallons. Vector Addition Vector addition is more complicated than scalar addition. Vector quantities are conveniently shown by arrows. The length of the arrow represents the magnitude of the quantity, and the orientation of the arrow represents the directional property of the quantity. For example, if we consider the top of this page as representing north and we want to show the velocity of an aircraft flying east at an airspeed of 300 knots, the velocity vector is as shown in Fig. 1.3. If there is a 30-knot wind from the north, the wind vector is as shown in Fig. 1.4. To find the aircraft’s flight path, groundspeed, and drift angle, we add these two vectors as follows. Place the tail of the wind vector at the arrow of the aircraft vector and draw a straight line from the tail of the aircraft vector to the arrow of the wind vector. This resultant vector represents the path of the aircraft over the ground. The length of the resultant vector represents the groundspeed, and the angle between the aircraft vector and the resultant vector is the drift angle (Fig. 1.5). The groundspeed is the hypotenuse of the right triangle and is found by use of the Pythagorean theorem 2 + Vw2 : Vr2 = Va∕c √ Groundspeed = Vr = (300)2 + (30)2 = 301.5 knots Va /c = 300k Fig. 1.3. Vector of an eastbound aircraft. Vw = 30k Fig. 1.4. Vector of a north wind. Va/c Vw Vr Fig. 1.5. Vector addition. MOMENTS 5 V a /c 𝛾 CLIMB ANGLE HORIZONTAL Fig. 1.6. Vector of an aircraft in a climb. V a /c Vv Vh Fig. 1.7. Vectors of groundspeed and rate of climb. The drift angle is the angle whose tangent is Vw ∕Va∕c = 30∕300 = 0.1, which is 5.7∘ to the right (south) of the aircraft heading. Vector Resolution It is often desirable to replace a given vector by two or more other vectors. This is called vector resolution. The resulting vectors are called component vectors of the original vector and, if added vectorially, they will produce the original vector. For example, if an aircraft is in a steady climb, at an airspeed of 200 knots, and the flight path makes a 30∘ angle with the horizontal, the groundspeed and rate of climb can be found by vector resolution. The flight path and velocity are shown by vector Va∕c in Fig. 1.6. In Fig. 1.7 to resolve the vector Va∕c into a component Vh parallel to the horizontal, which will represent the groundspeed, and a vertical component, Vv , which will represent the rate of climb, we simply draw a straight line vertically upward from the horizontal to the tip of the arrow Va∕c . This vertical line represents the rate of climb and the horizontal line represents the groundspeed of the aircraft. If the airspeed Va∕c is 200 knots and the climb angle is 30∘ , mathematically the values are Vh = Va∕c cos 30∘ = 200(0.866) = 173.2 knots (Groundspeed) Vv = Va∕c sin 30∘ = 200(0.500) = 100 knots or 10,130 fpm (Rate of climb) MOMENTS If a mechanic tightens a nut by applying a force to a wrench, a twisting action, called a moment, is created about the center of the bolt. This particular type of moment is called torque (pronounced “tork”). Moments, M, are measured by multiplying the amount of the applied force, F, by the moment arm, L: Moment = force × arm or M = FL (1.2) 6 INTRODUCTION The moment arm is the perpendicular distance from the line of action of the applied force to the center of rotation. Moments are measured as foot-pounds (ft-lb) or as inch-pounds (in.-lb). If a mechanic uses a 10-in.-long wrench and applies 25 lb of force, the torque on the nut is 250 in.-lb. The aircraft moments that are of particular interest to pilots include pitching moments, yawing moments, and rolling moments. If you have ever completed a weight and balance computation for an aircraft you have calculated a moment, where weight was the force and the arm was the inches from datum. Pitching moments, for example, occur when an aircraft’s elevator is moved. Air loads on the elevator, multiplied by the distance to the aircraft’s center of gravity (CG), create pitching moments, which cause the nose to pitch up or down. As you can see from Eq. 1.2, if a force remains the same but the arm is increased, the greater the moment. Several forces may act on an aircraft at the same time, and each will produce its own moment about the aircraft’s CG. Some of these moments may oppose others in direction. It is therefore necessary to classify each moment, not only by its magnitude, but also by its direction of rotation. One such classification could be by clockwise or counterclockwise rotation. In the case of pitching moments, a nose-up or nose-down classification seems appropriate. Mathematically, it is desirable that moments be classified as positive (+) or negative (−). For example, if a clockwise moment is considered to be a + moment, then a counterclockwise moment must be considered to be a − moment. By definition, aircraft nose-up pitching moments are considered to be + moments. EQUILIBRIUM CONDITIONS Webster defines equilibrium as “a state of balance or equality between opposing forces.” A body must meet two requirements to be in a state of equilibrium: 1. There must be no unbalanced forces acting on the body. This is written as the mathematical formula ΣF = 0, where Σ (cap sigma) is the Greek symbol for “sum of.” Figures 1.1 and 1.2 illustrate situations where this condition is satisfied (lift = weight, thrust = drag, etc.) 2. There must be no unbalanced moments acting on the body. Mathematically, ΣM = 0 (Fig. 1.8). Moments at the fulcrum in Fig. 1.8 are 50 ft-lb clockwise and 50 ft-lb counterclockwise. So, ΣM = 0. To satisfy the first condition of equilibrium, the fulcrum must press against the seesaw with a force of 15 lb. So, ΣF = 0. NEWTON’S LAWS OF MOTION Sir Isaac Newton summarized three generalizations about force and motion. These are known as the laws of motion. 10 lb 5 lb 5 ft 10 ft 15 lb Fig. 1.8. Seesaw in equilibrium. LINEAR MOTION 7 Newton’s First Law In simple language, the first law states that a body at rest will remain at rest and a body in motion will remain in motion, in a straight line, unless acted upon by an unbalanced force. The first law implies that bodies have a property called inertia. Inertia may be defined as the property of a body that results in its maintaining its velocity unchanged unless it interacts with an unbalanced force, as with an aircraft at rest on a ramp without unbalanced forces acting upon it. The measure of inertia is what is technically known as mass. Newton’s Second Law The second law states that if a body is acted on by an unbalanced force, the body will accelerate in the direction of the force and the acceleration will be directly proportional to the force and inversely proportional to the mass of the body. Acceleration is the change in motion (speed) of a body in a unit of time, consider an aircraft accelerating down the runway, or decelerating after touchdown. The amount of the acceleration a, is directly proportional to the unbalanced force, F, and is inversely proportional to the mass, m, of the body. These two effects can be expressed by the simple equation a= F m or, more commonly, F = ma (1.3) Newton’s Third Law The third law states that for every action force there is an equal and opposite reaction force. Note that for this law to have any meaning, there must be an interaction between the force and a body. For example, the gases produced by burning fuel in a rocket engine are accelerated through the rocket nozzle. The equal and opposite force acts on the interior walls of the combustion chamber, and the rocket is accelerated in the opposite direction. As a propeller aircraft pushes air backwards from the propeller, the aircraft moves forward. LINEAR MOTION Newton’s laws of motion express relationships among force, mass, and acceleration, but they stop short of discussing velocity, time, and distance. These are covered here. In the interest of simplicity, we assume here that acceleration is constant. Then, Acceleration a = V − V0 Change in velocity ΔV = = Change in time Δt t − t0 where Δ (cap delta) means “change in” V = velocity at time t V0 = velocity at time t0 If we start the time at t0 = 0 and rearrange the above, then V = V0 + at (1.4) If we start the time at t0 = 0 and V0 = 0 (brakes locked before takeoff roll) and rearrange the above where V can be any velocity given (for example, liftoff velocity), then t= V a 8 INTRODUCTION The distance s traveled in a certain time is s = Vav t The average velocity Vav is Vav = Therefore, s= V0 + at + V0 t 2 V + V0 2 or 1 s = V0 t + at2 2 (1.5) Solving Eqs. 1.4 and 1.5 simultaneously and eliminating t, we can derive a third equation: s= V 2 − V02 2a (1.6) Equations 1.3, 1.4, and 1.5 are useful in calculating takeoff and landing factors. They are studied in some detail in Chapters 10 and 11. ROTATIONAL MOTION Without derivation, some of the relationships among tangential (tip) velocity, Vt ; radius of rotation, r; revolutions per minute, rpm; centripetal forces, CF; weight of rotating parts, W; and acceleration of gravity, g, are shown below. The centripetal force is that force that causes an airplane to turn. The apparent force that is equal and opposite to this is called the centrifugal force. r(rpm) (fps) 9.55 WVt2 (lb) CF = gr Vt = CF = W r(rpm)2 2930 (1.7) (1.8) (1.9) WORK In physics, work has a meaning different from the popular definition. You can push against a solid wall until you are exhausted but, unless the wall moves, you are not doing any work. Work requires that a force must move an object in the direction of the force. Another way of saying this is that only the component of the force in the direction of movement does any work: Work = Force × Distance Work is measured in ft-lb. ENERGY Energy is the ability to do work. There are many kinds of energy: solar, chemical, heat, nuclear, and others. The type of energy that is of interest to us in aviation is mechanical energy. FRICTION 9 There are two kinds of mechanical energy. The first is called potential energy of position, or more simply potential energy, PE. No movement is involved in calculating PE. A good example of this kind of energy is water stored behind a dam. If released, the water would be able to do work, such as running a generator. As a fighter aircraft zooms to a zenith point it builds PE; once it starts to accelerate downward it converts PE to KE. PE equals the weight, W, of an object multiplied by the height, h, of the object above some base plane: PE = Wh (ft-lb) (1.10) The second kind of mechanical energy is called kinetic energy, KE. As the name implies, kinetic energy requires movement of an object. It is a function of the mass, m, of the object and its velocity, V: KE = 12 mV 2 (ft-lb) (1.11) The total mechanical energy, TE, of an object is the sum of its PE and KE: TE = PE + KE (1.12) The law of conservation of energy states that the total energy remains constant. Both potential and kinetic energy can change in value, but the total energy must remain the same: Energy cannot be created or destroyed, but can change in form. POWER In our discussion of work and energy we have not mentioned time. Power is defined as “the rate of doing work” or work/time. We know: Work = force × distance and Speed = distance∕time Power = work force × distance = = force × speed time time (ft-lb∕sec) James Watt defined the term horsepower (HP) as 550 ft-lb/sec: Horsepower = Force × Speed 550 If the speed is measured in knots, Vk , and the force is the thrust, T, of a jet engine, then HP = Thrust × Vk TVk = 325 325 (1.13) Equation 1.13 is very useful in comparing thrust-producing aircraft (turbojets) with power-producing aircraft (propeller aircraft and helicopters). FRICTION If two surfaces are in contact with each other, then a force develops between them when an attempt is made to move them relative to each other. This force is called friction. Generally, we think of friction as something 10 INTRODUCTION 0.8 DRY CONCRETE COEFFICIENT OF 0.6 FRICTION μ F/ N CONCRETE LIGHT RAIN 0.4 HEAVY RAIN 0.2 SMOOTH, CLEAR ICE ROLLING WHEEL LOCKED WHEEL 0 10 20 30 40 50 60 70 80 PERCENT WHEEL SLIP 90 100 Fig. 1.9. Coefficients of friction for airplane tires on a runway. to be avoided because it wastes energy and causes parts to wear. In our discussion on drag, we will discuss the parasite drag on an airplane in flight and the thrust or power to overcome that force. Friction is not always our enemy, however, for without it there would be no traction between an aircraft’s tires and the runway. Once an aircraft lands, lift is reduced and a portion of the weight is converted to frictional force. Depending on the aircraft type, aerodynamic braking, thrust reversers, and spoilers will be used to assist the brakes and shorten the landing, or rejected takeoff distance. Several factors are involved in determining friction effects on aircraft during takeoff and landing operations. Among these are runway surfacing material, condition of the runway, tire material and tread, and the amount of brake slippage. All of these variables determine a coefficient of friction 𝜇 (mu). The actual braking force, Fb , is the product of this coefficient 𝜇 (Greek symbol mu) and the normal force, N, between the tires and the runway: Fb = 𝜇N (lb) Figure 1.9 shows typical values of the coefficient of friction for various conditions. SYMBOLS a CF E KE PE TE F Fb g Acceleration (ft/sec2 ) Centrifugal force (lb) Energy (ft-lb) Kinetic energy Potential energy Total energy Force (lb) Braking force Acceleration of gravity (ft/sec2 ) (1.14) EQUATIONS h HP L m M N r rpm s T t V Vk V0 Vt W 𝜇 (mu) Height (ft) Horsepower Moment arm (ft or in.) Mass (slugs, lb-sec2 /ft) Moment (ft-lb or in.-lb) Normal force (lb) Radius (ft) Revolutions per minute Distance (ft) Thrust (lb) Time (sec) Speed (ft/sec) Speed (knots) Initial speed Tangential (tip) speed Weight (lb) Coefficient of friction (dimensionless) EQUATIONS 1.1 W = mg 1.2 M = FL 1.3 F = ma 1.4 V = V0 + at 1.5 s = V0 t + 12 at2 1.6 s= 1.7 1.8 V 2 − V02 2a r(rpm) Vt = 9.55 WVt2 CF = gr 1.10 W r(rpm)2 2930 PE = Wh 1.11 KE = 12 mV 2 1.12 TE = PE + KE TVk HP = 325 Fb = 𝜇N 1.9 1.13 1.14 CF = 11 12 INTRODUCTION PROBLEMS Note: Answers to problems are given at the end of the book. 1. An airplane weighs 16,000 lb. The local gravitational acceleration g is 32 fps2 . What is the mass of the airplane? 2. The airplane in Problem 1 accelerates down the takeoff runway with a net force of 6000 lb. Find the acceleration of the airplane. 3. An airplane is towing a glider to altitude. The tow rope is 20∘ below the horizontal and has a tension force of 300 lb exerted on it by the airplane. Find the horizontal drag of the glider and the amount of lift that the rope is providing to the glider. Sin 20∘ = 0.342; cos 20∘ = 0.940. 4. A jet airplane is climbing at a constant airspeed in no-wind conditions. The plane is directly over a point on the ground that is 4 statute miles from the takeoff point and the altimeter reads 15,840 ft. Find the tangent of the plane’s climb angle and the distance that it has flown through the air. 20 lb 10 lb 24 ft s F 5. Find the distance s and the force F on the seesaw fulcrum shown in the figure. Assume that the system is in equilibrium. 6. The airplane in Problem 2 starts from a brakes-locked position on the runway. The airplane takes off at an airspeed of 200 fps. Find the time for the aircraft to reach takeoff speed. 7. Under no-wind conditions, what takeoff roll is required for the aircraft in Problem 6? 8. Upon reaching a velocity of 100 fps, the pilot of the airplane in Problem 6 decides to abort the takeoff and applies brakes and stops the airplane in 1000 ft. Find the airplane’s deceleration. 9. A helicopter has a rotor diameter of 30 ft and it is being operated in a hover at 286.5 rpm. Find the tip speed Vt of the rotor. 10. An airplane weighs 16,000 lb and is flying at 5000 ft altitude and at an airspeed of 200 fps. Find (a) the potential energy, (b) the kinetic energy, and (c) the total energy. Assuming no extra drag on the airplane, if the pilot dove until the airspeed was 400 fps, what would the altitude be? 11. An aircraft’s turbojet engine produces 10,000 lb of thrust at 162.5 knots true airspeed. What is the equivalent power that it is producing? 12. An aircraft weighs 24,000 lb and has 75% of its weight on the main (braking) wheels. If the coefficient of friction is 0.7, find the braking force Fb on the airplane. 2 Atmosphere, Altitude, and Airspeed Measurement PROPERTIES OF THE ATMOSPHERE The aerodynamic forces and moments acting on an aircraft in flight are due, in great part, to the properties of the air mass in which the aircraft is flying. By volume the atmosphere is composed of approximately 78% nitrogen, 21% oxygen, and 1% other gases. The most important properties of air that affect aerodynamic behavior are its static pressure, temperature, density, and viscosity. Static Pressure The static pressure of the air, P, is simply the weight per unit area of the air above the level under consideration. For instance, the weight of a column of air with a cross-sectional area of 1 ft2 and extending upward from sea level through the atmosphere is 2116 lb. The sea level static pressure is, therefore, 2116 psf (or 14.7 psi). Static pressure is reduced as altitude is increased because there is less air weight above. At 18,000 ft altitude the static pressure is about half that at sea level. Another commonly used measure of static pressure is inches of mercury. On a standard sea level day the air’s static pressure will support a column of mercury (Hg) that is 29.92 in. high (Fig. 2.1). Weather reports use a third method of measuring static pressure called millibars, standard pressure here is 1,013.2 mb. In addition to these rather confusing systems, there are the metric measurements in use throughout most of the world. For the discussion of performance problems later in this textbook, we will primarily use the measurement of static pressure in inches of mercury. In aerodynamics it is convenient to use pressure ratios, rather than actual pressures. Thus the units of measurement are canceled out: P (2.1) Pressure ratio 𝛿 (delta) = P0 where P0 is the sea level standard static pressure (2116 psf or 29.92 in. Hg). Thus, a pressure ratio of 0.5 means that the ambient pressure is one-half of the standard sea level value. At 18,000 ft, on a standard day, the pressure ratio is 0.4992. Temperature The commonly used measures of temperature are the Fahrenheit, F, and Celsius, C (formerly called centigrade) scales. Aviation weather reports for pilots, as well as performance calculation tables, will usually report the temperature in ∘ C. Neither of these scales has absolute zero as a base, so neither can be used in calculations. Absolute temperature must be used instead. Absolute zero in the Fahrenheit system is −460∘ and in the Celsius system is −273∘ . To convert from the Fahrenheit system to the absolute system, called Rankine, R, add 460 to the ∘ F. To convert from the Celsius system to the absolute system, called Kelvin, K, add 273 to the ∘ C. The 13 14 ATMOSPHERE, ALTITUDE, AND AIRSPEED MEASUREMENT Standard Sea Level Pressure 29.92 Hg Inches of Mercury 30 Millibars 1016 25 847 20 677 15 508 10 339 Standard Sea Level Pressure 1013 mb 5 1 70 0 170 Atmospheric Pressure 0 0 Fig. 2.1. Standard pressure. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 symbol for absolute temperature is T and the symbol for sea level standard temperature is T0 : T0 = 519∘ R (59∘ F) = 288∘ K (15∘ C) By using temperature ratios, instead of actual temperatures, the units cancel. The temperature ratio is the Greek letter theta, 𝜃: T (2.2) 𝜃= T0 At sea level, on a standard day, 𝜃0∘ = 1.0. Temperature decreases with an increase in altitude until the tropopause is reached (36,089 ft on a standard day). It then remains constant until an altitude of about 82,023 ft. The temperature at the tropopause is −69.7∘ F and 𝜃 = 0.7519. Density Density is the most important property of air in the study of aerodynamics, and is directly impacted by pressure, temperature, and humidity changes. Since air can be compressed and expanded, the lower the pressure, the less dense the air; density is directly proportional to pressure. Increasing the temperature of the air (particles have greater kinetic energy) also decreases the density of the air, so in this case density and temperature have an inverse relationship. Less dense, thinner air has a lower air density and is said to be a higher density altitude ICAO STANDARD ATMOSPHERE 15 (decreasing aircraft performance); more dense, thicker air is said to be a lower density altitude (greater aircraft performance). Pressure decreases as you climb in altitude, and usually temperature decreases as well to a certain point. Two exceptions occur with temperature decrease with increasing altitude: one is an inversion layer, and the other is in the upper region of the atmosphere near the tropopause where the temperature remains constant and may even rise with increasing altitude periodically. The discussion above would indicate that greater altitude (less dense air) and colder temperature (more dense air) would result in a conflict in regards to density. But usually the effect of a decrease in pressure with altitude overcomes any improvement in performance the colder, dense air may have and a lower density altitude is the rule of thumb the higher in altitude an aircraft climbs. The effect of moisture content on performance will be largely ignored in this textbook because most textbooks treat the effect of humidity as being negligible for practical purposes, but it is important to understand that water vapor is lighter than air, so moist air is lighter than dry air. As the amount of water vapor increases, the density of the air decreases, resulting in a higher density altitude (decrease in aircraft performance). Density is the mass of the air per unit of volume. The symbol for density is 𝜌 (rho): 𝜌= Mass Unit volume (slugs∕ft3 ) Standard sea level density is 𝜌0 = 0.002377 slugs/ft3 . Density decreases with an increase in altitude. At 22,000 ft, the density is 0.001183 slugs/ft3 (about one-half of sea level density). It is desirable in aerodynamics to use density ratios instead of the actual values of density. The symbol for density ratio is 𝜎 (sigma): 𝜌 𝜎= (2.3) 𝜌0 The universal gas law shows that density is directly proportional to pressure and inversely proportional to absolute temperature: P 𝜌= (2.4) RT Forming a ratio gives P∕P0 P∕RT 𝜌 = = 𝜌0 P0 ∕RT0 T∕T0 R is the gas constant and cancels, so the density ratio, or sigma, is a function of pressure and temperature: 𝜎= 𝛿 𝜃 (2.5) Viscosity Viscosity can be simply defined as the internal friction of a fluid caused by molecular attraction that makes it resist its tendency to flow. The viscosity of the air is important when discussing airflow in the region very close to the surface of an aircraft. This region is called the boundary layer. We discuss viscosity in more detail when we take up the subject of boundary layer theory. ICAO STANDARD ATMOSPHERE To provide a basis for comparing aircraft performance in different parts of the world and under varying atmospheric conditions, the performance data must be reduced to a set of standard conditions. These are defined by the International Civil Aviation Organization (ICAO) and are compiled in a standard atmosphere 16 ATMOSPHERE, ALTITUDE, AND AIRSPEED MEASUREMENT Table 2.1. Standard Atmosphere Table Altitude (ft) Density Ratio, 𝜎 √ 𝝈 Pressure Ratio, 𝛿 Temperature (∘ F) Temperature Ratio, 𝜃 Speed of Sound (knots) Kinematic Viscosity v (ft2 /sec) 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 15,000 20,000 25,000 30,000 35,000 36,089a 40,000 45,000 50,000 1.0000 0.9711 0.9428 0.9151 0.8881 0.8617 0.8359 0.8106 0.7860 0.7620 0.7385 0.6292 0.5328 0.4481 0.3741 0.3099 0.2971 0.2462 0.1936 0.1522 1.0000 0.9854 0.9710 0.9566 0.9424 0.9283 0.9143 0.9004 0.8866 0.8729 0.8593 0.7932 0.7299 0.6694 0.6117 0.5567 0.5450 0.4962 0.4400 0.3002 1.0000 0.9644 0.9298 0.8962 0.8637 0.8320 0.8014 0.7716 0.7428 0.7148 0.6877 0.5643 0.4595 0.3711 0.2970 0.2353 0.2234 0.1851 0.1455 0.1145 59.00 55.43 51.87 48.30 44.74 41.17 37.60 34.04 30.47 26.90 23.34 5.51 −12.32 −30.15 −47.98 −65.82 −69.70 −69.70 −69.70 −69.70 1.0000 0.9931 0.9862 0.9794 0.9725 0.9656 0.9587 0.9519 0.9450 0.9381 0.9312 0.8969 0.8625 0.8281 0.7937 0.7594 0.7519 0.7519 0.7519 0.7519 661.7 659.5 657.2 654.9 652.6 650.3 647.9 645.6 643.3 640.9 638.6 626.7 614.6 602.2 589.5 576.6 573.8 573.8 573.8 573.8 .000158 .000161 .000165 .000169 .000174 .000178 .000182 .000187 .000192 .000197 .000202 .000229 .000262 .000302 .000349 .000405 .000419 .000506 .000643 .000818 a The tropopause. table. An abbreviated table is shown here as Table 2.1. Columns in the table show standard day density, density ratio, pressure, pressure ratio, temperature, temperature ratio, and speed of sound at various altitudes. ALTITUDE MEASUREMENT When a pilot uses the term altitude, the reference is usually to altitude above sea level as read on the altimeter, but it is important that the distinction is made to understand what types of altitude exist. When meteorologists refer to the height of the cloud layer above an airfield they are usually referring to the altitude above the field elevation. When air traffic control refers to an altitude at FL180 and above, they are referring to pressure altitude. Understanding what “altitudes” are important at different periods of flight, and the effect of temperature, pressure, and moisture on those altitudes, is imperative for safe flight. Indicated Altitude Indicated altitude is the altitude that is read directly from the altimeter and is uncorrected for any errors. In the United States below FL180 the altimeter is set to the current altimeter setting of the field you are departing from or arriving to, or is given by air traffic control for the current area you are flying in. In the U.S., when flying at or above 18,000 feet, altitude is measured in Flight Levels (e.g., FL180 for 18,000 feet). At FL180 the indicated ALTITUDE MEASUREMENT 17 altitude will be equal to pressure altitude as the altimeter setting is set to 29.92′′ , standard pressure, or QNE. The altitude at which the crew changes to 29.92 is called the transition altitude (TA). When the crew descends for landing, the altitude at which they return the altimeter setting to local barometric pressure corrected to sea level (QNH) is called the transition level (TL). (Remember it this way: 29.92 is selected at the TA, and the “A” stands for aloft, as in climbing or cruise. When returning to land, the TL is set on descent, and “L” stands for low, or landing.) When QNE is lower than 29.92, the lowest useable flight level is no longer FL180. The lowest useable FL is obtained from the aeronautical publications. For instance, in the US, if the pressure in the area of operations is between 29.91 and 29.42 inches, the lowest useable enroute altitude is FL185. It should also be noted that the TA and TL outside the United States will not always be 18,000. ICAO members set their own values. Incidentally, QFE is the reference pressure set in the altimeter if the pilot wishes to know the elevation above the airfield. When the aircraft is on the airfield, the altimeter reads zero. QFE is seldom used as it would be of limited value when away from the immediate vicinity of the airfield. Calibrated altitude is indicated altitude corrected for instrumentation errors. True Altitude True altitude is the actual altitude above mean sea level and is referenced as mean sea level (MSL). On most aeronautical charts MSL altitudes are published for man-made objects such as towers and buildings, as well as for terrain, since this is the altitude closest to the altitude read off the altimeter. An important note is that true altitude will only be the same as indicated altitude when flying in standard conditions, which is very rare. When flying in conditions colder than standard, the altimeter will read a higher altitude then you are flying, so true altitude will be lower than indicated altitude. The same dangerous situation can develop when you are flying from a high pressure area to a low pressure area and the altimeter is not corrected for the local altimeter setting. Your altimeter will interpret the lower pressure as a higher altitude and your true altitude will again be lower than your indicated altitude. From the variations in true altitude versus indicated altitude, the saying was developed “high to low, or hot to cold, look out below.” Of course, this assumes that the altimeter is never reset to local pressure for an entire flight covering a long distance with varying temperatures and pressures. Absolute Altitude Absolute altitude is the vertical altitude above the ground (AGL), and can be measured with devices like a radar altimeter. Of course your absolute altitude is more critical the closer to the ground you are flying, so even when not equipped with a radar altimeter a pilot should be aware of their AGL altitude. When conducting an instrument approach in inclement weather, knowledge of your AGL altitude is vital to the safe completion of the approach or execution of a missed approach. Your height above airport (HAA), height above touchdown zone (HAT), and decision height (DH) are all AGL altitudes and should be briefed before the approach. Pressure and Density Altitude Regarding aircraft performance, two types of altitude are of most interest to a pilot: pressure altitude and density altitude. Pressure altitude is that altitude in the standard atmosphere corresponding to a certain static pressure. Pressure altitude is the vertical distance above a standard datum plane where atmospheric pressure is 29.92". In the United States, at FL180 and above the altimeter is always set to 29.92" unless abnormally low pressure exists in the area. Pressure altitude is used in performance calculations to compute true airspeed, density altitude, and takeoff and landing data. Figure 2.2 indicates a convenient way to determine pressure altitude when unable to set 29.92" in the altimeter. 18 ATMOSPHERE, ALTITUDE, AND AIRSPEED MEASUREMENT Method for Determining Pressure Altitude Altitude Altimeter correction setting 28.0 1,825 1,725 1,630 1,535 1,435 1,340 1,245 1,150 1,050 28.1 28.2 Add 28.3 28.4 28.5 28.6 28.7 28.8 28.9 29.0 29.3 29.4 29.5 29.6 29.7 955 865 770 675 580 485 390 300 205 110 20 0 29.8 29.9 –75 –165 –255 –350 Subtract Field elevation is sea level 29.1 29.2 29.92 30.0 30.1 30.2 30.3 30.4 30.5 30.6 30.7 30.8 30.9 31.0 Alternate Method for Determining Pressure Altitude –440 –530 –620 –710 –805 –895 To field elevation To get pressure altitude From field elevation –965 Fig. 2.2. Field elevation versus pressure altitude. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 When calculating the pressure ratio we will use the standard pressure of 2,116 psf. If the pressure at a certain altitude is 1,455 psf, then the pressure ratio is: 𝛿= 1455 = 0.6876 2116 Entering Table 2.1 with this value, we find the corresponding pressure altitude of 10,000 ft. BERNOULLI’S EQUATION 19 Density altitude is found by correcting pressure altitude for nonstandard temperature conditions (Fig. 2.3). Pressure altitude and density altitude are the same when conditions are standard. Once pressure altitude has been determined, the density altitude is calculated using outside air temperature. If the temperature is below standard, then the density altitude is lower than pressure altitude and aircraft performance is improved. If the outside air temperature is warmer than standard, the density altitude is higher than pressure altitude and aircraft performance is degraded. When using Table 2.1 instead, if the air has a density ratio of 0.6292, the density ratio column shows that this value corresponds to a density altitude of 15,000 ft. As previously discussed, density altitude influences aircraft performance; the higher the density altitude, the lower aircraft performance. Low air density equals a higher density altitude; high air density equals a lower density altitude. Therefore, aircraft performance charts are provided for various density altitudes. CONTINUITY EQUATION Consider the flow of air through a pipe of varying cross section, as shown in Fig. 2.4. There is no flow through the sides of the pipe. Air flows only through the ends. The mass of air entering the pipe, in a given unit of time, equals the mass of air leaving the pipe, in the same unit of time. The mass flow through the pipe must remain constant. The mass flow at each station is equal. Constant mass flow is called steady-state flow. The mass airflow is equal to the volume of air multiplied by the density of the air. The volume of air, at any station, is equal to the velocity of the air multiplied by the cross-sectional area of that station. The mass airflow symbol Q is the product of the density, the area, and the velocity: Q = 𝜌AV (2.6) The continuity equation states that the mass airflow is a constant: 𝜌1 A1 V1 = 𝜌2 A2 V2 = 𝜌3 A3 V3 = constant (2.7) The continuity equation is valid for steady-state flow, both in subsonic and supersonic flow. For subsonic flow the air is considered to be incompressible, and its density remains constant. The density symbols can then be eliminated; thus, for subsonic flow, A1 V1 = A2 V2 = A3 V3 = constant (2.8) Velocity is inversely proportional to cross-sectional area or as cross-sectional area decreases, velocity increases. BERNOULLI’S EQUATION The continuity equation explains the relationship between velocity and cross-sectional area. It does not explain differences in static pressure of the air passing through a pipe of varying cross sections. Bernoulli, using the principle of conservation of energy, developed a concept that explains the behavior of pressures in gases. Consider the flow of air through a Venturi tube, as shown in Fig. 2.5. The energy of an airstream is in two forms: It has a potential energy, which is its static pressure, and a kinetic energy, which is its dynamic pressure. The total pressure of the airstream is the sum of the static pressure and the dynamic pressure. The total pressure (fe et ) e tu d al ti re 0 6, 7,000 Pr es su 0 0 00 00 7,0 8,000 ture 0 6,000 5, 00 Density altitude (feet) mpera ard te 9,000 Stand 10,000 10 ,0 00 11,000 00 12,000 9, 00 13,000 8, 14,000 13 ,0 00 15,000 14 ,0 00 ATMOSPHERE, ALTITUDE, AND AIRSPEED MEASUREMENT 12 11 ,0 ,0 00 00 20 4, 00 0 5,000 3, 00 0 4,000 2, 00 0 3,000 F 0° 10° 20° 00 ,0 –2 Sea level C –20° –10° –1 ,0 00 Se a 1,000 le ve l 1, 00 0 2,000 30° 40° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90°100° Outside air temperature (OAT) Fig. 2.3. Density altitude chart. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 BERNOULLI’S EQUATION STATION 1 STATION 2 STATION 3 A1 V1 A2 V2 A 3 V3 21 Fig. 2.4. Flow of air through a pipe. 4 6 4 VELOCITY 2 0 10 6 4 2 8 0 6 4 VELOCITY PRESSURE 8 10 2 0 6 4 PRESSURE 8 2 10 0 6 4 VELOCITY 8 10 2 0 10 6 PRESSURE 8 2 8 0 10 Fig. 2.5. Pressure change in a Venturi tube. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 remains constant, according to the law of conservation of energy. Thus, an increase in one form of pressure must result in a decrease in the other. Static pressure is an easily understood concept (see the discussion earlier in this chapter). Dynamic pressure, q, is similar to kinetic energy in mechanics and is expressed by q = 12 𝜌V 2 (psf) (2.9) where V is measured in feet per second. Pilots are much more familiar with velocity measured in knots instead of in feet per second, so a new equation for dynamic pressure, q, is used in this book. Its derivation is shown here: Density ratio, 𝜎 = 𝜌 𝜌 = 𝜌0 0.002377 or 𝜌 = 0.002377𝜎 22 ATMOSPHERE, ALTITUDE, AND AIRSPEED MEASUREMENT V = 100 k V = 200 k V = 100 k P = 2014 psf P = 2116 psf P = 2116 psf Fig. 2.6. Velocities and pressures on an airfoil superimposed on a Venturi tube. Vfps = 1.69Vk (Vfps )2 = 2.856(Vk )2 Substituting in Eq. 2.9 yields q= 𝜎Vk2 295 (psf) (2.10) Bernoulli’s equation can now be expressed as Total pressure, H = Static pressure, P + Dynamic pressure, q: H =P+ 𝜎Vk2 295 (psf) (2.11) To visualize how lift is developed on a cambered airfoil, draw a line down the middle of a Venturi tube. Discard the upper half of the figure and superimpose an airfoil on the necked down section of the tube (Fig. 2.6). Note that the static pressure over the airfoil is less than that ahead of it and behind it, so as dynamic pressure increases static pressure decreases. AIRSPEED MEASUREMENT If a symmetrically shaped object is placed in a moving airstream (Fig. 2.7), the flow pattern will be as shown. Some airflow will pass over the object and some will flow beneath it, but at the point at the nose of the object, the flow will be stopped completely. This point is called the stagnation point. Since the air velocity at this point is zero, the dynamic pressure is also zero. The stagnation pressure is, therefore, all static pressure and must be equal to the total pressure, H, of the airstream. In Fig. 2.8 the free stream values of velocity and pressure are used to measure the indicated airspeed of an aircraft. The pitot tube is shown as the total pressure port and must be pointed into the relative wind for accurate readings. The air entering the pitot tube comes to a complete stop and thus the static pressure, we will refer to as P2 , in the tube is equal to the total free stream pressure, H. This pressure is ducted into a diaphragm inside the airspeed indicator. The static pressure port can be made as a part of the point tube or, in more expensive indicators, it can be at a distance from the pitot tube. It should be located at a point where the local air velocity is exactly equal to the airplane velocity. The static port is made so that none of the velocity enters the port. The port measures only static pressure, P1 for our discussion, and none of the dynamic pressure. The static pressure is ducted into the chamber surrounding the diaphragm within the inside of the airspeed indicator. Now we have static pressure (P2) inside the diaphragm that is equal to total pressure (H), and then static pressure (P1) measured from the static port outside the diaphragm. The difference between the pressure inside the diaphragm and outside the diaphragm is the differential pressure that deflects the flexible diaphragm that is geared to the airspeed pointer. The airspeed indicator is calibrated to read airspeed. Figure 2.9 shows a modern pitot-static system associated with an air data computer (ADC). AIRSPEED MEASUREMENT FORWARD STAGNATION POINT 23 AFT STAGNATION POINT Fig. 2.7. Flow around a symmetrical object. 2 250 40 ALINEED 200 KNCTS 150 60 80 100 .5 0 100 3 9 UP VERTICAL SPEED THOUSAND PT PER MIN 4 .5 DOWN 1 2 3 Drain hole 8 FAST 0 ALT CM MAIN OR SPEE THOU 7 2 29.6 29.9 30.0 3 6 5 4 Pitot pressure chamber Baffle plate Ram air Static hole Pitot tube Drain hole Heater (100 watts) Static hole Heater (35 watts) Static port ON OFF Static chamber Pitot heater switch Alternate static source Fig. 2.8. Schematic of a pitot–static airspeed indicator. U.S. Department of Transportation Federal Aviation Administration, Instrument Flying Handbook, 2012 Indicated Airspeed Indicated airspeed (IAS) is the reading of the airspeed indicator dial. If there are any errors in the instrument, they will be shown on an instrument error card located near the instrument. Position error results if the static pressure port is not located on the aircraft where the local air velocity is exactly equal to the free stream velocity of the airplane. If this error is present, it will be included in the instrument error chart. Calibrated Airspeed Calibrated airspeed (CAS) is obtained when the necessary corrections have been made to the IAS for installation error and instrument error. In fast, high-altitude aircraft, the air entering the pitot tube is subjected to a ram effect, which causes the diaphragm to be deflected too far. The resulting airspeed indication is too high and must be corrected. 24 ATMOSPHERE, ALTITUDE, AND AIRSPEED MEASUREMENT Auto transformer AOA probe Total air temperature probe Copilot’s pitot pressure TAT indicator Pilot’s pitot pressure Mach Copilot’s mach/airspeed Static pressure Pilot’s mach/airspeed 2 3 1 Static pressure 2 Pilot’s altimeter Digital air data computer Static air temp gauge TAS Airspeed mach altitude Auto throttle Copilot’s altimeter True airspeed indicator Altitude rate TAT Transponder FLT management computer units 1 & 2 Inertial reference units 1, 2, & 3 FLT control augmentation computer Pitot pressure Static pressure Electrical connection Direction of data flow Altitude encoding Altitude error DADC data bus DADC data bus Computed airspeed Altitude Altitude error airspeed Computed airspeed altitude Flight director system 1 & 2 Fuel temperature indicator Autopilot system Flight recorder/locator 1 Pilot’s altimeter provides altitude signal to flight recorder/locator if in reset mode 2 Servo-corrected altitude 3 In reset mode, copilot uses pilot’s static source Fig. 2.9. Air data computer and pitot-static sensing. U.S. Department of Transportation Federal Aviation Administration, Aviation Maintenance Technician Handbook: Airframe, Volume 2, 2012 AIRSPEED MEASUREMENT 25 M = 1.00 ,00 –20 EAS = CAS + ∆VC 0 55, 000 50, 000 40, 45,00 000 0 35 ,00 0 30 25 ,00 0 ,0 00 –30 60 AIRSPEED COMPRESSIBILITY CORRECTION ∆VC, KNOTS COMPRESSIBILITY CORRECTION –10 00 ,0 20 00 ,0 15 00 0 10, 5,000 0 100 200 300 400 PRESSURE ALTITUDE SEA LEVEL FT. 500 600 CALIBRATED AIRSPEED, KNOTS Fig. 2.10. Compressibility correction chart. Figure 2.10 shows a compressibility correction chart. A rule of thumb is that, if flying above 10,000 ft and 200 knots, the compressibility correction should be made. Unlike the instrument and position error charts, which vary with different aircraft, this chart is good for any aircraft. Equivalent Airspeed Equivalent airspeed (EAS) results when the CAS has been corrected for compressibility effects. One further correction must be made to obtain true airspeed. The airspeed indicator measures dynamic pressure and is calibrated for sea level standard day density. √ As altitude increases, the density ratio decreases and a correction must be made. The correction factor is 𝜎. EAS is not a significant factor in airspeed computations when aircraft fly at relatively low speeds and altitudes (i.e., below 300 KCAS and at altitudes below 20,000 feet PA). True Airspeed True airspeed (TAS) is obtained when EAS has been corrected for density ratio: EAS TAS = √ 𝜎 (2.12) √ √ Values of 𝜎 can be found in the ICAO Standard Altitude Chart (Table 2.1); values of 1∕ 𝜎 (called “SMOE”) can be found in Fig. 2.11. Understanding the relationship between the speeds above, and the calculation of each one, can be facilitated by remembering “ICE-T.” IAS is read off the airspeed indicator, CAS is IAS corrected for installation/position errors, EAS is CAS corrected for compressibility, and finally TAS is EAS corrected for temperature and pressure. 26 ATMOSPHERE, ALTITUDE, AND AIRSPEED MEASUREMENT 25,000 0 0 ,00 28 27,00 ,000 00 26 25,0 15,000 0 15, STANDARD ATMOSPHERE TEMPERATURE 10,000 00 000 14, ,000 13 000 12, ,000 11 00 0 10, 0 5000 500 0 –5000 –80 0 900 0 800 0 700 0 600 –70 –60 –50 –40 –30 –20 –10 PRESSURE ALTITUDE FT. 0 400 00 30 0 200 0 100 EL LEV 00 A SE –10 00 –20 00 –30 000 –4 00 –50 6000 7000 – – 0 +10 +20 +30 TEMPERATURE–°C 1.521 1.494 1.468 1.443 1.418 1.394 1.370 1.347 1.325 1.303 1.282 1.261 1.240 1.221 1.201 1.182 1.164 1.146 1.128 1.111 1.094 1.077 1.061 1.045 1.030 1.015 1.000 .985 .970 .955 .939 .923 DENSITY ALTITUDE CHART 1 = TAS EAS σ DENSITY ALTITUDE–FT. 20,000 000 24, 000 23, 000 22, 00 0 21, 000 19, 00 000 0 8 20, 1 , ,000 17 00 0 16, +40 +50 +60 Fig. 2.11. Altitude and EAS to TAS correction chart. Groundspeed Groundspeed (GS) is the actual speed of the aircraft over the ground, either calculated manually or more commonly nowadays read off the GPS (Global Positioning Satellite) navigational unit. The GS increases with a tailwind and decreases with a headwind, and is TAS adjusted for the wind. SYMBOLS A AGL CAS ∘C DA EAS ∘F GS H IAS ∘K MSL P Area (ft2 ) Above ground level Calibrated airspeed (knots) Celsius temperature (deg.) Density altitude Equivalent airspeed Fahrenheit temperature Ground speed Total pressure(head)(psf) Indicated airspeed Kelvin temperature Mean sea level Static pressure PROBLEMS PA P0 q R ∘R TAS T T0 V Vk 𝛿 (delta) 𝜃 (theta) 𝜌 (rho) 𝜎 (sigma) Pressure altitude Sea level standard pressure Dynamic pressure Universal gas constant Rankine temperature True airspeed Absolute temperature Sea level standard temperature Velocity (fps) Velocity (knots) Pressure ratio Temperature ratio Density Density ratio EQUATIONS (2.1) 𝛿= (2.2) 𝜃= (2.3) P P0 T T0 𝜌 𝜎= 𝜌0 (2.6) P RT 𝛿 𝜎= 𝜃 Q = 𝜌AV (2.7) 𝜌1 A1 V1 = 𝜌2 A2 V2 (2.8) A1 V1 = A2 V2 (2.9) q= (2.10) q= (2.11) H =P+ (2.4) (2.5) (2.12) 𝜌= 1 2 𝜌V 2 𝜎Vk2 295 (psf) (psf) 𝜎Vk2 295 EAS TAS = √ 𝜎 PROBLEMS 1. An increase in static air pressure a. affects air density by decreasing the density. b. does not affect air density. c. affects air density by increasing the density. 27 28 ATMOSPHERE, ALTITUDE, AND AIRSPEED MEASUREMENT 2. A decrease in temperature a. affects air density by decreasing the density. b. does not affect air density. c. affects air density by increasing the density. 3. Pressure ratio is a. ambient pressure divided by sea level standard pressure measured in the same units. b. ambient pressure in millibars divided by 29.92. c. ambient pressure in pounds per square inch divided by 2116. d. sea level standard pressure in inches of mercury divided by 29.92. 4. Density ratio, 𝜎 (sigma) is a. equal to pressure ratio divided by temperature ratio. b. measured in slugs per cubic foot. c. equal to the ambient density divided by sea level standard pressure. d. None of the above 5. Bernoulli’s equation for subsonic flow states that a. if the velocity of an airstream within a tube is increased, the static pressure of the air increases. b. if the area of a tube decreases, the static pressure of the air increases. c. if the velocity of an airstream within a tube increases, the static pressure of the air decreases, but the sum of the static pressure and the velocity remains constant. d. None of the above 6. Dynamic pressure of an airstream is a. directly proportional to the square of the velocity. b. directly proportional to the air density. c. Neither (a) nor (b) d. Both (a) and (b) 7. In this book, we use the formula for dynamic pressure, q = 𝜎Vk2 ∕295, rather than the more conventional formula, q = 12 𝜌V 2 , because a. V in our formula is measured in knots. b. density ratio is easier to handle (mathematically) than the actual density (slugs per cubic foot). c. Both (a) and (b) d. Neither (a) nor (b) 8. The corrections that must be made to indicated airspeed (IAS) to obtain calibrated airspeed (CAS) are a. position error and compressibility error. b. instrument error and position error. c. instrument error and density error. d. position error and density error. 9. The correction that must be made to CAS to obtain equivalent airspeed (EAS) is called compressibility error, which a. is always a negative value. PROBLEMS 29 b. can be ignored at high altitude. c. can be ignored at high airspeed. d. can be either a positive or a negative value. 10. The correction from EAS to true airspeed (TAS) is dependent on a. temperature ratio alone. b. density ratio alone. c. pressure ratio alone. d. None of the above 11. An airplane is operating from an airfield that has a barometric pressure of 27.82 in. Hg and a runway temperature of 100∘ F. Calculate (or find in Table 2.1) the following: a. Pressure ratio b. Pressure altitude c. Temperature ratio d. Density ratio e. Density altitude 12. Fill in the values below for the stations in the drawing. 1 2 3 At station 1 At station 2 At station 3 A1 = 10 ft2 V1 = 100 knots P1 = 2030 psf 𝜎 = 0.968 H=? A2 = 5 ft2 V2 = ? P2 = ? V3 = 80 knots A3 = ? P3 = ? q2 = ? q3 = ? 13. Using Table 2.1, calculate the dynamic pressure, q at 5000 ft density altitude and 200 knots TAS. 14. The airspeed indicator of an airplane reads 355 knots. There are no instrument or position errors. If the airplane is flying at a pressure altitude of 25,000 ft, find the equivalent airspeed (EAS). 15. Find the true airspeed (TAS) of the airplane in Problem 14 if the outside air temperature is −40∘ C. 3 Structures, Airfoils, and Aerodynamic Forces AIRCRAFT STRUCTURES Up to this point you may have considered only the wing an important structure of an aircraft within the concept of aerodynamics. But in fact the entire structure of the airplane plays a role in the efficiency of an aircraft in flight, and identifying how, and to what extent, each part of an airplane structure plays a role is an important first step. A review of the more prominent structures discussed in aerodynamics is covered here first, because their direct role on lift and drag provide the foundation for more complicated discussions in the future. Flight control systems are commonly separated into two areas, primary and secondary systems. Primary control systems include the ailerons, elevator (stabilator), and rudder and, depending on the aircraft type and aircraft speed, give the pilot a “feel” of how the aircraft is performing. Secondary systems such as trim systems, flaps, spoilers, and leading edge devices are used in relieving control pressures for the pilots, assisting primary control surfaces in high-speed flight, or improving the performance characteristics of the aircraft in general. Aircraft are flown in various configurations of gear and flaps, but for this textbook we will commonly refer to clean and dirty as the two reference configurations. In a clean configuration, the gear is retracted (when applicable), and the flaps and other high-lift devices are retracted. In the dirty configuration, the gear is considered down and locked, and the high-lift devices are fully deployed. Figure 3.1 contains a sample of the airfoils found on modern aircraft, and many that we will discuss throughout the subsequent chapters. Primary Flight Controls Ailerons The ailerons are usually located on the trailing edge of each wing closer to the outboard area by the tip. Ailerons control roll about the longitudinal axis as they move opposite of each other when the pilot banks left or right with the yoke or stick, with the “up” aileron on the downward moving wing. Most ailerons are connected by a mechanical means to the aircraft yoke through cables, bell cranks, pulleys, and/or push-pull tubes. Some jet aircraft have additional ailerons located in the midwing area for high-speed maneuvering, to reduce roll rate. There are many styles of ailerons used throughout the fixed-wing aircraft industry today, including differential ailerons, Frise-type ailerons, and flaperons. Differential ailerons work by deflecting the up aileron more than the down aileron, increasing drag on the downward wing counteracting adverse yaw (Fig. 3.2). Frise-type ailerons project the leading edge of the raised aileron into the wind, increasing drag on the lowered wing to once again minimize the effects of adverse yaw (Fig. 3.3). Flaperons combine the control surfaces of the ailerons with the function of the trailing edge flaps, as they can be lowered like flaps but still control bank angle like traditional ailerons. Chapter 15 will discuss these items, as well as adverse yaw, in more depth regarding aircraft stability. 31 32 STRUCTURES, AIRFOILS, AND AERODYNAMIC FORCES Rudder Inboard wing Elevator Tab Outboard wing Stabilizer Foreflap Midflap Inboard flap Airflap Flight spoilers Ground spoiler Outboard flap Aileron Tab Landing Flaps Aileron Leading edge slats Inboard wing Leading edge flaps 737 Control Surfaces Outboard wing Control tab Inboard aileron Stabilizer Leading edge flaps Control tab Elevator Aileron Upper rudder Takeoff Flaps Anti-balance tabs Lower rudder Vortex generators Pilot tubes Ground spoilers Inboard flaps Flight spoilers Outboard flap Balance tab Outboard aileron Foreflap Midflap Altflap Inboard wing Outboard wing Leading edge slat Leading edge flap Slats Fence 727 Control Surfaces Aileron Flaps Restracted Fig. 3.1. Modern transport category control surfaces. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 Aileron deflected u p Differential aileron Aileron deflected down Fig. 3.2. Differential ailerons. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 AIRCRAFT STRUCTURES 33 Neutral Raised Drag Lowered Fig. 3.3. Frise-type ailerons. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 Elevator/Stabilator An elevator or stabilator controls pitch about the lateral axis, allowing for varying angles of attack during flight. An elevator is attached to the trailing edge of the horizontal stabilizer, which is usually fixed to the empennage, sometimes with an angle of incidence built in. A stabilator is a one-piece horizontal stabilizer where the whole unit moves around a pivot point in order to allow the pilot to once again control the angle of attack by adjusting the tail down force resulting in pitch variations of the nose of the aircraft. The elevator is controlled by the pilot through various mechanical linkages, when the pilot pulls aft on the stick the elevator goes up and when the pilot pushes forward the elevator goes down. As we discussed in Chapter 2, the tail down force provides a moment that moves the nose of the aircraft around the aircraft’s center of gravity. In the example of an up elevator, when the pilot pulls aft on the stick, a larger “camber” is created on the tail and thus a greater aerodynamic force is created (Fig. 3.4). A stabilator essentially works like the elevator, but due to the fact the entire rear horizontal piece is movable, more force is created when the pilot moves the stick fore and aft and sensitivity is increased. This leads to greater chances of the pilot overcontrolling the aircraft so components like an antiservo tab and balance weight are added to reduce the sensitivity. Some larger aircraft incorporate an adjustable horizontal stabilizer controlled by a jackscrew through a wheel in the cockpit or a motor. Though an elevator is still located on the trailing edge, the usually fixed horizontal 34 STRUCTURES, AIRFOILS, AND AERODYNAMIC FORCES Tail down Control column Aft Nose up Up elevator CG Downward aerodynamic force Fig. 3.4. Elevator movement. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 Adjustable stabilizer Nose down Nose up Jackscrew Pivot Trim motor or trim cable Fig. 3.5. Adjustable horizontal stabilizer. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 stabilizer is adjustable, in this case allowing the pilot to move the stabilizer to reduce control pressures on the stick (Fig. 3.5). Rudder The rudder controls yaw, or movement of the aircraft about its vertical axis. Similar to the smaller, movable elevator attached to the trailing edge of the horizontal stabilizer, the rudder is attached to the rear of the fixed vertical stabilizer. As with other components we discussed, the rudder is connected to the rudder pedals in the cockpit via various mechanical linkages; pressing on the left or right rudder pedal moves the rudder left or right AIRCRAFT STRUCTURES 35 Yaw Left rudder forward Left rudder Ae ro dynam ic force Fig. 3.6. Rudder movement. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 respectfully. Most of the time the rudder is used to maintain coordinated flight, especially when banking the aircraft. As will be discussed later in this chapter, moving the rudder creates a larger camber on the vertical stabilizer, which in turn creates a greater sideways force. This force then moves the nose of the aircraft left and right, or yaw, around the vertical axis (Fig. 3.6). Secondary Flight Controls Secondary flight control systems usually consist of wing flaps, leading edge devices, spoilers, and trim systems. These controls often support or supplement the primary controls, and their importance to understanding aerodynamic principles cannot be overstated. Flaps The flaps are the most common high-lift devices used on aircraft, and their contribution to the amount of lift an airfoil can produce will be discussed in more detail in Chapter 4. For our discussion here we will review their location on the aircraft, as well as the basic flap designs on aircraft today. Some flaps are located on the trailing edge of the wing, usually inboard close to the fuselage, and are referred to as trailing edge flaps. These surfaces contribute to the camber of the wing airfoil in most cases, as well as to the area of the wing in other cases. By increasing the angle of attack of the wing, and in some cases the area of the wing, they allow the aircraft to fly at lower speeds, which may be needed for takeoff and landing or for times of increased maneuverability. For this discussion we will look at four types of flaps: plain, split, slotted, and fowler; and as you can see there are designs where the benefit of one design is combined with another (Fig. 3.7). 36 STRUCTURES, AIRFOILS, AND AERODYNAMIC FORCES Basic section Plain flap Split flap Slotted flap Fowler flap Slotted Fowler flap Fig. 3.7. Common flap designs. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 AIRFOILS 37 The plain flap is the simplest design and usually used to describe the advantages and purpose of flaps. As the flaps are deployed, the camber of the wing increases and the CL (discussed in Chapter 4) increases accordingly with the angle of attack. The farther the flaps are deployed, the greater the lift and the resulting drag. A split flap is deployed from underneath the wing, and results in more drag initially than the plain flap due to the disruption of the flow of air around the bottom and top of the wing. When the flaps are slotted, at high angles of attack high energy air is allowed to move through the slot and energize the air on top of the deployed flap. This allows for an increase in CL at lower speeds, allowing an aircraft to operate out of shorter landing strips or with obstacles surrounding the airport. The highly energized air also delays boundary layer separation, which lowers the stalling speed, improving performance at slow speeds. More on this topic will be discussed throughout this textbook. Fowler flaps are commonly found on larger transport category aircraft, as they are heavier than the other flap designs and incorporate more complex systems to operate. Fowler flaps slide out and back from the wing, which offers the benefit of not only increasing the camber of the wing but also of the wing area. Fowler flaps also double as slotted flaps in that they allow higher-energy air from beneath the wing to flow over the deployed flap area. Cessna high-wing, single-engine aircraft are the best example of the use of Fowler flaps on light aircraft. Leading Edge Devices Leading edge devices are discussed in more depth in Chapter 4 so only a brief review will be presented here. As with trailing edge flaps, leading edge devices like movable slots, leading edge flaps (Krueger flaps), and leading edge cuffs work to increase the CL over a wing, allowing the aircraft to fly at slower speeds for takeoff and landing. In some cases the camber of the wing is also increased, which allows for an increase in the angle of attack and increased performance. As with trailing edge flaps, the first setting results in more lift with little increase in drag, while fully deployed settings usually result in mostly drag with little lift. Spoilers Spoilers are used on many different types of aircraft as high-drag devices used to increase drag and reduce lift. They “spoil” the smooth air around a wing, and may be used in flight, or on the ground when landing or during a rejected takeoff. Some aircraft use the spoilers in flight for roll control at higher airspeeds when the action of the ailerons would be too much of a force. Sometimes spoilers are used as speed brakes to reduce lift on both wings, which allows the aircraft to descend without increasing airspeed. Trim Systems Trim systems are designed to alleviate the pressures on the primary flight controls as experienced by the pilot during aircraft operation. Usually located on the trailing edge of these devices, the pilot (or autopilot) operates the respective trim system in order to position the flight control where minimum pressures are exerted in the system. The two most common trim systems are trim tabs and antiservo tabs. A trim tab is usually found on the trailing edge of the elevator or rudder, and unlike the antiservo tab, the trim tab moves in the opposite direction to which the primary control moves (Fig. 3.8). An antiservo tab is found on the trailing edge of a stabilator, and moves in the same direction as the primary control to which it is attached (Fig. 3.9). As previously mentioned, the antiservo tab serves to reduce the sensitivity of the stabilator during pitch to the pilot, as well as reducing control pressure as needed. AIRFOILS An airfoil or, more properly, an airfoil section, is commonly shown as a vertical slice of a wing (see Fig. 3.10). In discussing airfoils in this chapter, the planform (or horizontal plane) of the wing is ignored as it will be discussed later. Keep in mind during this discussion that an airfoil is also found on the vertical stabilizer, 38 STRUCTURES, AIRFOILS, AND AERODYNAMIC FORCES Nose-down trim Elevator Trim tab Tab up — elevator down Nose-up trim Elevator Trim tab Tab down — elevator up Fig. 3.8. Trim tabs. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 horizontal stabilizer, and rotor blades. Wingtip effects, sweepback, taper, wash/out or wash/in, and other design features are not considered. No singular airfoil design is perfect for every flight situation, wind tunnel tests, and computer-generated designs dictate what airfoil is best for the mission of the aircraft. Airfoil Terminology The terminology used to discuss an airfoil is shown in Fig. 3.11: 1. 2. 3. 4. Chord line is a straight line connecting the leading edge and the trailing edge of the airfoil. Chord is the length of the chord line. All airfoil dimensions are measured in terms of the chord. Mean camber line is a line drawn equidistant between the upper surface and the lower surfaces. Maximum camber is the maximum distance between the mean camber line and the chord line. The location of maximum camber is important in determining the aerodynamic characteristics of the airfoil. 5. Maximum thickness is the maximum distance between the upper and lower surfaces, and its location of maximum thickness will also be important when determining aerodynamic characteristics. 6. Leading edge radius is a measure of the sharpness of the leading edge. It may vary from zero for a knife-edge supersonic airfoil to about 2% (of the chord) for rather blunt leading-edge airfoils. AIRFOILS 39 Stabilator Pivot point Antiservo tab Fig. 3.9. Antiservo tab. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 AIRFOIL PLANE PARALLEL TO LONGITUDINAL AND VERTICAL AXES RW Fig. 3.10. Airfoil section. Definitions Flight Path Velocity The speed and direction of a body passing through the air. Relative Wind (RW) The speed and direction of the air impinging on a body passing through it. It is equal and opposite in direction to the flight path velocity. Angle of Attack (AOA or 𝛂, pronounced alpha) The acute angle between the relative wind and the chord line of an airfoil. Aerodynamic Force (AF) The net resulting static pressure multiplied by the planform area of an airfoil. 40 STRUCTURES, AIRFOILS, AND AERODYNAMIC FORCES 5 5 MAXIMUM THICKNESS LOCATION OF MAX. THICKNESS 6 UPPER SURFACE 3 4 MAXIMUM LEADING EDGE RADIUS MEAN CAMBER LINE CAMBER CHORDLINE LOWER SURFACE LEADING EDGE 2 1 TRAILING EDGE CHORD 4 LOCATION OF MAXIMUM CAMBER Fig. 3.11. Airfoil terminology. Lift The component L of the aerodynamic force that is perpendicular to the relative wind. Drag The component D of the aerodynamic force that is parallel to the relative wind. Center of Pressure (CP) The point on the chord line where the aerodynamic force acts. Laminar Flow or Streamlined Flow Smooth airflow with little transfer of momentum between parallel layers. Turbulent Flow Airflow where the streamlines break up and there is much mixing of the layers. Geometry Variables of Airfoils There are four main variables in the geometry of an airfoil: 1. 2. 3. 4. Shape of the mean camber line Thickness Location of maximum thickness Leading-edge radius If the mean camber line coincides with the chord line, the airfoil is said to be symmetrical. In symmetrical airfoils, the upper and lower surfaces have the same shape and are equidistant from the chord line. Figure 3.12 shows examples of early airfoil design to more modern, supersonic designs. Classification of Airfoils Most airfoil development in the United States was done by the National Advisory Committee for Aeronautics (NACA) starting in 1929. NACA was the forerunner of the National Aeronautics and Space Administration (NASA). The first series of airfoils investigated was the “four-digit” series. The first digit gives the amount of camber, in percentage of chord. The second digit gives the position of maximum camber, in tenths of chord, and the last two give the maximum thickness, in percentage of chord. For example, a NACA 2415 airfoil has a AIRFOILS 41 Early airfoil Later airfoil Clark ‘Y’ airfoil (Subsonic) Laminar flow airfoil (Subsonic) Circular arc airfoil (Supersonic) Double wedge airfoil (Supersonic) Fig. 3.12. Examples of airfoil design. U.S. Department of Transportation Federal Aviation Administration, Pilot’s Handbook of Aeronautical Knowledge, 2008 maximum camber of 2% C, located at 40% C (measured from the leading edge), and has a maximum thickness of 15% C. A NACA 0012 airfoil is a symmetrical airfoil (has zero camber) and has a thickness of 12% C. Further development led to the “five-digit” series, the “1-series,” and, with the advent of higher speeds, to the so-called laminar flow airfoils. The NACA’s 23000 series created in 1935 were very popular and are still in use today. The laminar flow airfoils are the “6-series” and “7-series” airfoils and result from moving the maximum thickness back and reducing the leading-edge radius. Two things happen with this treatment. First, the point of minimum pressure is moved backward, thus increasing the distance from the leading edge that laminar (smooth) airflow exists, which reduces drag. Second, the critical Mach number is increased, thus allowing the airspeed of the aircraft to be increased without encountering compressibility problems. In the 6-series, the first digit indicates the series and the second gives the location of minimum pressure in tenths of chord. The third digit represents the design lift coefficient in tenths, and the last two digits (as in all NACA airfoils) show the thickness in percentage of chord. For example, NACA 64-212 is a 6-series airfoil with minimum pressure at 40% C, a design lift coefficient of 0.2, and a thickness of 12% C. Sketches of NACA subsonic airfoil series are shown in Fig. 3.13. A modern design used worldwide on corporate, military, and air transport aircraft is the supercritical airfoil, which is flatter on top and more rounded on the bottom than a conventional wing. The upper trailing edge has a downward curve to restore lift lost by the flattening of the upper surface. The benefit of this design in the high-speed realm of flight, as well as other supersonic airfoils, are discussed in Chapter 16. 42 STRUCTURES, AIRFOILS, AND AERODYNAMIC FORCES NACA 0015 0 0.2 0.4 NACA 4415 0.6 0.8 1.0 0 0.2 NACA 23015 0 0.2 0.4 0.2 0.4 0.6 0.6 0.6 0.8 1.0 NACA 16−015 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 0.8 1.0 NACA 747A015 NACA 641212 0 0.4 0.8 1.0 0 0.2 0.4 0.6 Fig. 3.13. NACA airfoils (NACA data). DEVELOPMENT OF FORCES ON AIRFOILS Leonardo da Vinci stated the cardinal principle of wind tunnel testing nearly 400 years before the Wright brothers achieved powered flight. Near the beginning of the sixteenth century, da Vinci said: the action of the medium upon a body is the same whether the body moves in a quiescent medium, or whether the particles of the medium impinge with the same velocity upon the quiescent body. This principle allows us to consider only relative motion of the airfoil and the air surrounding it. We may use such terms as “airfoil passing through the air” and “air passing over the airfoil” interchangeably. Pressure Disturbances on Airfoils If an airfoil is subjected to a moving airflow, velocity and pressure changes take place that create pressure disturbances in the airflow surrounding it. These disturbances originate at the airfoil surface and propagate in all directions at the speed of sound. If the flight path velocity is subsonic, the pressure disturbances that are moving ahead of the airfoil affect the airflow approaching the airfoil (Fig. 3.14). Velocity and Static Pressure Changes about an Airfoil The air approaching the leading edge of an airfoil is first slowed down and then speeds up again as it passes over or beneath the airfoil. Figure 3.15 compares two local velocities with the flight path velocity V1 and with each other. As the velocity changes, so does the dynamic pressure “q” and, according to Bernoulli’s principle, so does the static pressure “P.” Air near the stagnation point has slowed down, so the static pressure in this region is higher than the ambient static pressure. Air that is passing above and below the airfoil, and thus has speeded up to a value higher than the flight path velocity, will produce static pressures that are lower than ambient static pressure. So as “q” increases, “P” static pressure decreases and a greater pressure differential is realized. At a point near maximum thickness, maximum velocity and minimum static pressure will occur. Because air has viscosity, some of its energy will be lost to friction and a “wake” of low-velocity, turbulent air exists near the trailing edge, resulting in a small, high-pressure area. Figure 3.16 shows a symmetrical airfoil (a) at DEVELOPMENT OF FORCES ON AIRFOILS FLOW DIRECTION CHANGES WELL AHEAD OF LEADING EDGE Fig. 3.14. Effect of pressure disturbances on airflow around an airfoil. v1 < v3 v1 < v2 v2 > v3 v2 v1 v3 Fig. 3.15. Velocity changes around an airfoil. UPWASH INCREASED LOCAL VELOCITY DOWNWASH DECREASED LOCAL VELOCITY POSITIVE LIFT ZERO LIFT (a) (b) Fig. 3.16. Static pressure on an airfoil (a) at zero AOA, and (b) at a positive AOA. 43 44 STRUCTURES, AIRFOILS, AND AERODYNAMIC FORCES zero AOA and the resulting pressure distribution (b) at a positive AOA and its pressure distribution. Arrows pointing away from the airfoil indicate static pressures that are below ambient static pressure; arrows pointing toward the airfoil indicate pressures higher than ambient. AERODYNAMIC FORCE Aerodynamic force (AF) is the resultant of all static pressures acting on an airfoil in an airflow multiplied by the planform area that is affected by the pressure. The line of action of the AF passes through the chord line at a point called the center of pressure (CP). It is convenient to consider that the forces acting on an aircraft, or on an airfoil, do so in some rectangular coordinate system. One such system could be defined by the longitudinal and vertical axes of an aircraft. Another could be defined by axes parallel to and perpendicular to the earth’s surface. A third rectangular coordinate system is defined by the relative wind direction and an axis perpendicular to it. This last system is chosen to define lift and drag forces. Aerodynamic force (AF) is resolved into two components: one parallel to the relative wind, called drag (D), and the other perpendicular to the relative wind, called lift (L). Figure 3.17 shows the resolution of AF into its components L and D. Pressure Distribution on a Rotating Cylinder A stationary (nonrotating) cylinder is located in a wind tunnel as shown in Fig. 3.18a. The cylinder is equipped with static pressure taps. These measure the local static pressure with respect to the ambient static pressure in the test chamber. When the tunnel is started, the airflow approaches the cylinder from the left as shown by the D L AF RW CENTER OR PRESSURE (CP) Fig. 3.17. Components of aerodynamic force. ZERO NET LIFT RELATIVE WIND (a) POSITIVE LIFT (b) Fig. 3.18. Pressure forces on (a) nonrotating cylinder and (b) rotating cylinder. AERODYNAMIC PITCHING MOMENTS 45 relative wind vector. Arrows pointing toward the cylinder show pressures that are higher (+) than ambient static pressure; arrows pointing away from the cylinder show pressures that are less (−) than ambient static pressure. Figure 3.18a shows that the upward forces are resisted by the downward forces and no net vertical force (lift) is developed by the cylinder. Now consider if the wind tunnel is stopped, and the cylinder begins to rotate in a motionless fluid. We begin to see the factors of viscosity and friction at work. The more viscous the fluid, the more it is resistant to flow, and since air has viscosity properties it will resist flow. Similar to a wing, the surface of the cylinder has some “roughness” to it, so as the cylinder turns some molecules adhere to the surface. The closer to the surface of the cylinder (airfoil), the greater the possibility the molecules are drawn in a clockwise direction by viscosity, so now substituting air we see the velocity increase in the direction of rotation above the cylinder. This circular movement of the air is called circulation. Finally let us consider a rotating cylinder in a moving fluid as the cylinder continues rotating in the clockwise direction when the wind tunnel is once again started (Fig. 3.18b). The air passing over the top of the cylinder will be speeded up by circulation, while the air passing over the bottom of the cylinder will be retarded. According to Bernoulli’s equation, the static pressure on the top will be reduced and the static pressure on the bottom will be increased, similar to an airfoil with a positive angle of attack. The new pressure distribution will be as shown in Fig. 3.18b, where a low-pressure area produces an upward force. This is called the Magnus effect, named after Gustav Magnus, who discovered it in 1852. It explains why you slice (or hook) your golf ball or why a good pitcher can throw a curve. AERODYNAMIC PITCHING MOMENTS Consider the pressure distribution about a symmetrical airfoil at zero angle of attack (AOA) (Fig. 3.19a). The large arrows show the sum of the low pressures on the top and bottom of the airfoil. They are at the center of pressure (CP) of their respective surfaces. The CP on the top of the airfoil and the CP on the bottom are located at the same point on the chord line. The large arrows indicate that the entire pressure on the top and bottom surfaces is acting at the CP. Because these two forces are equal and opposite in direction, no net lift is generated. Note also that the lines of action of these forces coincide, so there is no unbalance of moments about any point on the airfoil. Figure 3.19b shows the pressure distribution about a symmetrical airfoil at a positive angle of attack (AOA). There is now an unbalance in the upper surface and lower surface lift vectors, and positive lift is being developed. However, the two lift vectors still have the same line of action, passing through the CP. There can be no moment developed about the CP. We can conclude that symmetrical airfoils do not generate pitching moments at any AOA. It is also true that the CP does not move with a change in AOA for a symmetric airfoil. UPPER SURFACE LIFT UPPER SURFACE LIFT LOWER SURFACE LIFT (a) LOWER SURFACE LIFT (b) Fig. 3.19. Pitching moments on a symmetrical airfoil (a) at zero AOA and (b) at positive AOA. 46 STRUCTURES, AIRFOILS, AND AERODYNAMIC FORCES UPPER SURFACE LIFT UPPER SURFACE LIFT LOWER SURFACE LIFT LOWER SURFACE LIFT (a) (b) Fig. 3.20. Pitching moments on a cambered airfoil: (a) zero lift, (b) developing lift. Now consider a cambered airfoil operating at an AOA where it is developing no net lift (Fig. 3.20a). Upper-surface lift and lower-surface lift are numerically equal, but their lines of action do not coincide. A nose-down pitching moment develops from this situation. When the cambered airfoil develops positive lift (Fig. 3.20b), the nose-down pitching moment still exists. By reversing the camber it is possible to create an airfoil that has a nose-up pitching moment. Delta-wing aircraft have a reversed camber trailing edge to control the pitching moments. AERODYNAMIC CENTER For cambered airfoils the CP moves along the chord line when the AOA changes. As the AOA increases, the CP moves forward and vice versa. This movement makes calculations involving stability and stress analysis very difficult. There is a point on a cambered airfoil where the pitching moment is a constant with changing AOA, if the velocity is constant. This point is called the aerodynamic center (AC). The AC, unlike the CP, does not move with changes in AOA. If we consider the lift and drag forces as acting at the AC, the calculations will be greatly simplified. The location of the AC varies slightly, depending on airfoil shape. Subsonically, it is between 23 and 27% of the chord back from the leading edge. Supersonically, the AC shifts to the 50% chord. In summary, the pitching moment at the AC does not change when the angle of attack changes (at constant velocity) and all changes in lift effectively occur at the AC. As an airfoil experiences greater velocity its AC commonly moves towards the trailing edge, with the AC near 25% chord subsonically and at 50% supersonically. SYMBOLS AC AF AOA CP D L RW 𝛼 (alpha) Aerodynamic center Aerodynamic force (lb) Angle of attack (degrees) Center of pressure Drag (lb) Lift (lb) Relative wind Angle of attack (degrees) PROBLEMS PROBLEMS 5 3 4 1 2 1. Number 3 on the drawing shows a. the chord line. b. the maximum camber. c. the thickness. d. the mean camber line. 2. Number 4 on the drawing shows a. the chord line. b. the thickness. c. the maximum camber. d. the mean camber line. 3. Number 5 on the drawing shows a. the maximum camber. b. the mean camber line. c. the upper-surface curvature. d. the maximum thickness. 4. The Magnus effect explains why a. a bowling ball curves. b. a pitched baseball curves. c. a ...
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