Flight Stability
`and Automatic Control
`•:rr•n•m•r•n•••·:r1111::1111111111111ammz ---•r-••---•1•::m•.-a-.v.:Wlllmsl!Cllilll12•: m1--•1w•:•m -----nt•JU-----•
`
`SECOND EDITION
`
`Dr. Robert C. Nelson
`Deparrmenr nf Aero.'ipace and A-lec:hanicuJ Engineering
`Unh·t.•rsiry of Notre D,une
`
`WCB
`McGraw-HIii
`
`Boston, Massachusetts Burr Ridge. Illinois Dubuque, Iowa
`Madison, Wisconsin New York, Ne,v York San Francisco. California
`St. Louis1 Missouri
`
`DJI-1018
`IPR2023-01104
`
`

`

`WCB/McGraw-Hill
`1\ Division of The McGra1v-Hill Con1pa11ies
`
`tJ
`
`FLl(Hfr STABILITY AND AUTOMATIC CONTROL
`l.ntc111ational Editions 1998
`
`Exclusive rights by McGraw-Hill Book Co - Singapore. for n1nnufacture and export. This
`book cannot be re-exported from the country to which it is consigned hy McGrav,-Hil I.
`
`Copyright •:s.) 1998 by The Mc(ira,v-ijill Companies, Inc. All rights reserved. Previous
`editions •.t.1 1989. Except as permitted .under tht United State~ Copyright Act of 1976.
`no part of this publication may be reproduced or distributed in any fum1 or by any means.
`or stored in a data base or retrieval syscem. \Vithout the prior \Vrlttcn pem,ission of
`the publisher.
`
`6 7 8 9 10 SLP P?\lfP 20 9 8 7 6 5 4 3 2
`
`Library of Congress Cataloging-in-Publication Data
`
`Nelson. Robert C., 1942-
`Flight stability and automatic control / Robert C. Nelson. - 2nd ed.
`en,.
`p.
`Includes bibliographical references and index.
`ISBN 0-07-046273-9
`2. Airplanes-Control syste1ns.
`1. Stability of airplanes.
`3, Airplanes-Automatic control.
`I. Title.
`TL574.S7N45
`1998
`629. l 32'36-dc2 l
`
`97-26109
`CIP
`
`~,~'
`,X.
`, ..
`.,~ ~
`When ordering this title, use ISBN 0-07-115838-l (./)
`I .
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`hhtp:Jv.,,v,v.mhcollege.com
`
`Printed in Singapore
`
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`

`

`ABOUT THE AUTHOR
`
`ROBERT C_. NELSON received his B. S. and M. S. degrees in Aerospace Engi(cid:173)
`neering fron1 the University of Notre Daine and his Ph.D. in Aerospace Engi(cid:173)
`neering from the Pennsylvania State University. Prior to joining Notre Dame.
`Dr. Nelson was an instructor of Aerospace Engineering at the Pennsylvania State
`University and an engineer for the Air Force Flight Dynarnics Laboratory at
`Wright-Patterson Air Force Base. Fairborn, Ohio. While employed at AFFDL. he
`worked on an advanced development program to develop the technology for an air
`to air short range bon1ber defense missile. For his contrjbution to this effort he
`received a Technical Achievement award from the Air Force Systems Command.
`Tn 1975, Dr. Nelson joined the faculty at Notre Dame and has been active in
`research dealing with the aerodynamics and flight dynamic!-i of hoth aircraft and
`missiles. His present research interests include the aerodynanlics of slender bodies
`at large angles of attack, flow visun lization techniques, delta wing .aerodynamics,
`and aircraft stability and control. He has ,Yritten over I 00 articles and papers on
`his rese.arch. Dr. Nelson is the chairman of the Departm·ent of Aerospace and
`Mechanical Engineering at Notre .D,lmc. He has also been active. as a consultant to
`government and industrial organizations. He is a Registered Professional Engineer
`and a Fellow of the American Institute of Aeronautics and Astronautics (AIAA).
`He served as the general chairn1an of the AIAA At111ospheric Flight Mechanics
`Conference in 1982 and was the chairman of the AIAA Atmospheric Flight Me(cid:173)
`chanics Technical Cotnmi ttee from May 1983-1985. Dr. Nelson also served as a
`men1ber of the AIAA Applied Acrndynan,ics Technical Comn1ittee from 1986 to
`1989. Other professional activities include participation as a lecturer and course
`coordinator of four short courses and one home study course sponsored by the
`AIAA ( 1982. l 984, 1989, 1995 ). He also has been an AGARD lecturer ( 1991,
`1993! 1995, 1997}. ln 19.91, Dr. N·e1son received Lhe John Leland At\vood Award
`from the AlAA and ASEE. This award is given annually for contributions to
`Aerospace Engineering Education.
`
`

`

`

`

`PREFACE
`
`An understanding of flight stability and control played an important role in the
`u ltin1ate success of the earliest aircraft designs. ln later years the design of auto(cid:173)
`matic controls ushered in the rapid development of commercial and military air(cid:173)
`craft. Today, both military and civilian aircraft rely heavily on automatic control
`systems to provide artificial stabilization and autopilots to aid pilots in navigating
`and landing their aircraft in adverse weather conditions. The goal of Lhis book is
`to present an integrated treatment of the basic elements of aircraft stability, flight
`control, and autopilot design.
`
`NEW TO 'l'HIS EDI'l'ION
`
`In the second edition, 1 have attempted to improve the first six chapters from the
`first edition. These chapters cover the topics of static stability, Hjght control,
`aircraft dynamics and flying qualities. This is accomplished by including more
`worked-out example problems, additional problems at the end of each chapter ..
`and new material to provide additional insight on the subject. The major change in
`the text is the addition of an expanded section on automatic control theory and
`its application to flight control sy~tem design.
`
`CONTENTS
`
`This book is intended as a textbook for a course in aircraft flight dynamics for
`senior undergraduate or first year graduate students. The material presented in(cid:173)
`cludes static stability, aircraft equations of motion, dyna1nic stability, flying or
`handling qualities, automatic control theory. and application of control theory to
`the synthesis of automatic flight control systems. Chapter 1 revie\vs some basic
`concepts of aerodynamics, properties of the atmosphere, several of the primary
`flight instruments, and nomenclature. ln Chapter 2 the concepts of airplane static
`stability and control are presented. The design features that can be incorporated
`into an aircraft design to provide static stability and sufficient control power are
`discussed. The rigid body aircraft equations of motion are developed along with
`techniques to model the aerodynamic forces and moments acting on the airplane in
`Chapter 3. The aerodynamic forces and moments are modeled using the concept
`of aerodynamic stability derivatives. Methods for estimating the derivatives are
`presented in Chapter3 along with a detailed exa1nple calculation of the longitudinal
`derivatives of a STOL transport. The dynantic characteristics of an airplane for free
`and forced response are presented in Chapters 4 and 5. Chapter 4 discusses the
`
`V
`
`

`

`vi Preface
`
`longitudinal dynamics while Chapter 5 presents the lateral dynantlcs. ln both
`chapters the relationship between the rigid body 1notions and the pilot's opinion of
`the ease or difficulty of flying the airplane is explained. Handling or flying qualities
`are those control and dyna1nic characteristics that govern how \veil a pilot can fly
`a particular control ta~k. Chapter 6 discusses the solution of the equations of
`1notion for either arbitrary control input or atmospheric disturbances. Chapters
`7-10 include the major changes incorporated into the second edition of this book.
`Chapter 7 provides a review of classical control concepts and discusses control
`system synthesis and design. The root locus method is used to design control
`syste1ns to meet given tin1e and frequency domain performance specifications.
`Classical control techniquc.s are used to design automatic control systems for vari(cid:173)
`ous· night applications in Chapter 8. Automatic control systems arc presented that
`can be used to maintain an airplane·s bank angle, pitch orientation, altitude, and
`speed. 1n addition a qualitative description of a fully automated landing system is
`presented. In Chapter 9 .. tJ1e concepls of modern control theory and design tech(cid:173)
`niques are reviewed. By using state teedhack design, it is theoretically possible for
`the designer to locate the roots of the closed loop systen1 so that any desired
`performance can be achieved. The practical constraints of arbitrary root placen1ent
`are discussed along with the necessary requirements to successfully implement
`state feedback control. FinaJly in Chapter 10 modern control design methods are
`applied to the design of aircraft automatic flight control systems.
`
`LEARNING rfOOL.~
`
`To help in understanding the concepts presented in the text 1 have included a
`number of worked-out cxan1p]c problems throughout the book, and at the end of
`each chapter one will find a problern set. Son1e of the example problems and
`selected problerns at the end of later chapters require computer solutions. Commer(cid:173)
`cially available co1nputer aided design soft\.vare is used for selected example prob(cid:173)
`le1ns and assigned problenu;. Problen1s that require the use of a computer are
`clearly identified in the problem sets. A 111ajor feature of the textbook is that the
`material is introduced by ,vay of sin1ple exercises. For exa1nple, dynamic stability
`is pre~cntcd Jirst by restricted single degree of freedon1 1notions. This approach
`permits the reader to gain some experience in the mathcn1atical representation and
`physical understanding or aircraft response bctore the more complicated rnultiple
`degree of freedom 111otions are anal yzcd. A similar approach is used in developing
`the control syste1n designs. For example, a roil autopilot to n1aintain a wings level
`attitude is 111odclcd using the simplest n1alhcmatical tbrmulation to represent the
`aircraft and control systen1 ele1nents. Follo\ving this approach the students can be
`introduced to the design process without undue mathematical c;omplexity. Several
`appendices have also been included to provide additional data on airplane aerody(cid:173)
`nan1ic, mass. and geometric characteristics as ,vel1 us revie\v n1at.erial of some of
`the n1athernatical and analysis techniques used in the text.
`
`

`

`Ackno,vledge1nents vii
`
`ACKNOWl"'EDGEMENTS
`
`I-am indehted to all the students who used the early drafts of this book. Their many
`suggestions and patience as the book evolved is greatly appreciated. 1 wou Id like
`to express my thanks for the many useful comn1ents and suggestions provided
`by co11eagues who reviewed this text during the course of its development, espe(cid:173)
`cially to:
`
`Donald T. Ward
`Texas A & M University
`Andrew S. Aren~ Jr.
`Oklahoma State University
`Georgia Institute of Technology
`C.H. Chuang
`Frederick H. Lutze
`Virginia Polytechnic Institute and State University
`University of Maryland
`Roberto Celi
`Finally. l would like to express my appreciation to Marilyn Walker for her
`patience in typing the many versions of this n1anuscript.
`
`Robert C. Nelson
`
`

`

`CONTENTS
`
`-
`
`_
`
`- -
`
`, -
`
`- • · a
`
`• 4 •
`
`"
`
`-
`
`-
`
`•
`
`,
`
`-
`
`,o
`
`■
`
`-
`
`•
`
`"
`
`•
`
`I
`
`_
`
`"
`
`Preface
`
`I
`
`Introduction
`1.1 Atmospheric Flight Mechanics
`1.2 Basic Definitions
`1.2.1 Fluid l 1.2.2 Pressure I 1.2.3 Te111perature I
`J.2.4 J)ensity I 1.2.5 ViscoJity I /.2.6 The Mach Nu11Jber
`and tire Speed of Sound
`J.3 Aerostatics
`I .3.1 Variation of Pre.tsure 111 a Static: 1'7.uid
`1.4 Develop1ncnt of Bernoulli's Equation
`1.4. I lncontpressible Bernoulli Equation I 1 .4.2 Bernoulli'.,·
`Equation for a Conipressible Fluid
`1.5 The Atmosphere
`1.6 Aerodynamic Nomenclature
`1..7 Aircraft Instru1nents
`1.7.J Air Data S).•sre,ns I /.7.2· Airspeed Indicator I
`I. 7.3 Altilneter I 1. 7.4 Rate of Clilnb /J1dicator I
`1. 7.5 Machnieter I I. 7.6 A,tgle of Attack ln.dic:atars
`1.8 Summary
`Problems
`References
`
`2.3
`
`2 Static Stability and Control
`2.1 Historical Perspective
`2.2
`Introduction
`2.2. l Static Stability I 2.2. 2 D\'11a11iic Stabilit,•
`.
`.
`.
`Static Stability and Control
`2.3. 1 Definitio11 o.f J.,ongitlidinal Static Stability I
`2.3.2 Contribution of Aircra.ft Con1pone11ts I 2.3.3 Wing
`Contribution I 2.3.4 Tail Contribution-Aft Tail I
`2.3.5 Canard-Forvvard Tail .Surface I 2.3.6 Fuselage
`Co11tributlon I 2.3. 7 Ptnver Effects I 2.3.8 Stick Fixed
`Neutral Point
`2.4 Longitudinal Control
`2.4. J Elevator Effectiveness I 2.4.2 Elevator Angle to
`Trin1 I 2.4.3 Flight Measure111.ent of XNP I 2.4.4 Eh~l'alor
`Hinge Mnntent
`
`.
`Xl
`
`1
`1
`
`3
`
`7
`
`9
`
`12
`19
`22
`
`32
`32
`33
`
`35
`35
`39
`
`42
`
`62
`
`.
`lX.
`
`

`

`x Contents
`
`2.5
`
`Stick Forces
`2.5. I Tri111 Tabs I 2.5.2 Stick 1'i'.1rce Gradients
`2.6 Definition of Directional Stability
`2.6.J ContrUnttion qf" Aircraft Co111po11e11t.,·
`2.7 Directiona 1 Control
`2.8 Roll Stability
`2.9 Ro 11 Control
`2.10 Su1nmary
`Problems
`References
`
`3 Aircraft Equations of Motion
`3.1
`Introduction
`3.2 Derivation of Rigid Body Equations of Motion
`3.3 Orientation and Position of the Airplane
`3.4 Gravitational and Thrust. Forces
`3.5
`Small-D.isturbance Theory
`3.6 Aerodynan1ic Force and Motnent Representation
`3.6. 1 Deriratives Due to the Change in Forn·urd
`Speed I 3.6.2 Derivatives Due to the Pitching
`Velocity, q I 3.6.3 Derivati,•es Due to the Tilne Rate of
`Change of the Angle of Attack I 3.6.4 Derivative Due
`to tht Rolling Rare, p I 3.6.5 Derivative Due to the
`Ytnving Rate, r
`Summary
`Problems.
`References
`
`3.7
`
`4 Longitudinal Motion (Stick Fixed)
`4.1 Historical Perspective
`4.2
`Second-Order Differential Equations
`4.3
`Pure Pitching Mot ion
`4.4
`Stick Fixed Longitudinal Motion
`4.4. l State \la,;able Representation of the Equations
`of Motion
`4.S Longitudinal Approximations
`4.5. J Short-Period Approxi,na.tion
`4.6 The Influence of Stability Derivatives on the
`Lonititudinal Modes of Motion
`Flying Qualities
`4. 7. l Pi/01 Opinion
`
`4.7
`
`.
`
`I,,;
`
`70
`
`73
`
`77
`78
`81
`84
`85
`95
`
`96
`96
`97
`10]
`103
`104
`108
`
`127
`128
`130
`
`13.1
`131
`133
`139
`147
`
`152
`
`162
`164
`
`

`

`Contents xi
`
`s
`
`6
`
`4.8
`4.9
`
`Flight Simulation
`Summary
`Problems
`References
`
`Lateral Motion (Stiel, Fixed)
`S.1
`Introduction
`S.2
`Pure Rolling Motion
`5.2. 1 Wing Rock I 5.2.2 Roll Control Reversal
`5.3
`Pure Ya,ving Motion
`S.4 Lateral-Directiona 1 Equations of Motion
`5.4. I Spiral Approxilnation I 5.4.2 Roll
`Approxi,nation. I 5.4.3 Dutch Roll Appoxi,nation
`s.s Lateral Flying Qualities
`5.6
`Inertial Coupling
`S.7
`Summary
`Problems
`References
`
`Aircraft Response to Control or Atmospheric Inputs
`6.1
`Introduction
`6.2 Equations of Motion in a Nonunifor1n Atmosphere
`6.3 Pure Vertical or Plunging Motion
`6.4 Atmospheric Turbulence
`6.5 Harmonic Analysis
`6. 5.1 Turbulence Models
`6.6 Wintl Shear
`Sumn1ary
`6.7
`Problems
`References
`
`7 Automatic Control Theory(cid:173)
`The Classical Approach
`7.1
`Introduction
`7.2 Routh's Criterion
`7.3 Root Locus Technique
`7.3. I Addition of Poles_ and Zeros
`7.4 Frequency Domain Techniques
`7.5 Tin1e-Domain and Frequency-Domain Specifications
`7.5.1 Gain and Pha.,·e /tllargin Jro111 Root Locul· I
`7.5.2 Higher-Order Systenzs
`
`169
`171
`174
`179
`
`181
`18]
`
`182
`
`1.88
`193
`
`203
`205
`206
`206
`210
`
`212
`212
`215
`218
`225
`227
`
`229
`232
`233
`234
`
`235.
`235
`2J8
`243
`
`250
`251
`
`

`

`xii Contents
`
`7.6
`Steady-State E1Tor
`7.7 Control System Design
`7. 7. I C,nnpensation I 7. 7.2 Forward-Palh
`Con1pe11satiDn I 7. 7.3 Feedback-Path Co11ipe11sation
`PID Controller
`Sun11nary
`Problems
`References
`
`7.8
`7.9
`
`8 Application of Classical Contro.l Theory to Aircraft
`Autopilot Design
`8.1
`Introduction
`8.2 Aircraft Transfer Functions
`8.2. I Shorr-l'eriod Dyna,nics I 8.2.2 Long Period or
`Phugoid Dyna,nic.\· I 8.2.3 Roll Dynun1ic.~ I 8.2.4 Dutch
`Roll Approxinzation
`8.3 Control Surface Actuator
`8.4 Displacen1ent Autopilot
`8.4. 1 Pitch l)isplace111ent Autopilot I 8.4.2 Roll Attitude
`Autopilot I 8.4.3 Altitude Hold C~onrrol Sy.\'tc1n I
`8.4.4 Velocity Hold C'o11trol Sy.,·ten1
`Stability Augmentation
`Instru1nent Landing
`Sun1mary
`Problerns
`References
`
`8.5
`8.6
`8.7
`
`9 Modern Control Theory
`Introduction
`9.1
`State-Space Modeling
`9.2
`9.2.1 Sttue Transition Matrix I 9.2.2 Nu,nerical Svluti.011
`vf State Equu1ions
`9.3 Canonical ·rransformations
`9.3. J Real Distinct J~Jge11\'alues I 9.3.2 Repeated
`Ei~en,•ulues I 9.3.3 Co,nplex EiKeni•alues
`9.4 Controllability and Observability
`9.5 State Feedback Design
`9.5. 1 Nu,nerical Me1hvcl for DetenninillR Feedback
`Gai,u I 9.5.2 Multiple lnput•Output Sy:.tent I
`9.5.3 Eigenvalue P/ace,nent
`State Variable Reconstruction: The State Observer
`
`9.6
`
`258
`262
`
`271
`274
`275
`280
`
`281
`281
`283
`
`288
`292
`
`312
`314
`318
`319
`322
`
`323
`
`323
`
`324
`
`335
`
`344
`347
`
`355
`
`

`

`Contents
`
`x·111
`
`9.7 Optimal State-Space Control System Design
`9.8
`Summary
`Problems
`References
`
`10
`
`Application of Modern Control Theory to Aircraft
`.Autopilot Design
`10.1
`Introduction
`10.2 Stability Augmentation
`I 0.2.1 Longitudinal Stability Augm.entatio11 I
`10.2.2 Lateral S1ability Aug1ne11tation
`10.3 Autopilot Design
`10.4 State Observer
`10.5 Optin1al Control
`10.6 Summary
`Problems
`References
`
`359
`362
`362
`366
`
`367
`367
`367
`
`379
`383
`386
`391
`391
`394
`
`395
`395
`
`398
`
`420
`
`429
`
`435
`
`Appendices
`Atmospheric Tables (ICAO Standard Atmosphere)
`A
`B
`Geornetric, Mass, and Aerodynamic Characteristics of
`Sele-cted Airplanes
`Mathematical Review of Laplace Transforms and
`Matrix Al0 ebra
`Review of Control System Analysis Techniques
`
`C
`
`D
`
`0
`
`Index
`
`

`

`

`

`CHAJ>TER 1
`
`z::c:::
`
`S MX
`
`e-:e--_•.
`
`Introduction
`
`"For so,ne _vears I have been qf.flicted u1ith 1he belief tht,t flight is possible
`to rnan. "
`
`Wilbur Wright, May l.3, 1900
`
`1.1
`ATMOSPHERIC FLIGHT MECHANICS
`
`Atmospheric flight 1nechanics is a broad heading that encon1passes three major
`disciplines; nru11ely. perfor1nance, flight dynan1ics, and aeroelasticity. In the past
`each of these subject~ was treated independently of the others. Hovlever, because
`of the structural flexibility of 1nodern airplanes, the interplay runong the disciplines
`no longer can be ignored. For exarnple, if the flight loads cause significant structural
`deforn,alion of the aircraft. one can expect changes in the airplane's aerodynamic
`and stability characteristics that· \\'ill influence its performance and dynamic
`behavior.
`Airplane perforn1ance deals with the determination of perfonnance character(cid:173)
`istics such as range, endurance, rate of climb~ and takeoff and landing distance as
`wel I as Hight path optimization. To evaluate these performance characteristics, one
`normally treats the airplane as a point n1ass acted on by gravity, lift, drag, and
`thrust. The accuracy of the performance calculations depends on ho,v accurately
`the lift, drag, and thrust can be determined.
`Flight dynamics is concerned with the n1otion of an airplane due to internally
`or externally generated disturbances. We particularly are interested in the vehicle's
`stability and control capabilities. To describe adequately the rigid-body n1otion of
`an airplane one needs to consider the complete equations of motion with six
`degrees of freedom. Again. this will require accurate estimates of the aerodynamic
`forces and moments acting on Lhc airplane.
`The final subje.ct included under the heading of atn1ospheric flight mechanics
`is acroelasticity. Aeroelasticity deals with hoth static and dynamic aeroelastic
`phenomena. Basically. aeroelasticity is concerned with phenomena associated with
`interactions bet.ween inertial. elastic, and aerodynamic forces. Problems that arise
`for a flexible aircraft include control reversal. wing divergence, and control surface
`flutter. to nan1e just a few.
`
`

`

`2 CI-IAPTF.R 1: Introduction
`
`AtJ\111nccd Teonclog!o:1
`ln1:0rpora~ed 1,1 1ho X•29A.
`
`nn 11uparcrllltul wlng
`
`FIGURE 1.1
`Advanced lechuologies inc.:orporated in the X-29A aircraft.
`
`This book is divided into three parts: The first part deals with the properties of
`the atmosphere, static stability and control concepts, development of aircraft equa(cid:173)
`tions of motion, and aerodynamic modeling of the airplane: the second part exam(cid:173)
`ines aircraft motions due to control inputs or atmospheric d.isturbances; the third
`part is devoted to aircraft autopilots. Although no specific chapters are devoted
`entirely to performance or aeroelasticity, an effort is made to show the reader, al
`least in a qualitative way. ho,v perfor1nance specifications and aeroelastic phenom(cid:173)
`ena influence aircraft stability and control characteristics.
`The interplay an1ong the three disciplines that make up atmospheric flight
`mechanics is best illustrated by the experimental high-performance airplane shown
`in Figure 1.1. The X-29A aircraft incorporates the latest advanced technologies in
`controls, structures, and acrodynantlcs. These technologies will provide substantial
`perfor1nance improven1ents over more conventional fighter designs. Such a design
`could not be developed ,vithout paying close attention to the interplay a1nong
`performance. aeroelasticity, stability .. and control. In fact, the evolution of thi8
`raclical design \Vas developed using trade-off studies bet,veen the various disciplines
`to justify the expected performance improvements.
`The forces and moments acting on an airplane depend on the properties of the
`atmosphere through which it is flying. In the following sections ,ve will review some
`basic concepts of fluid mechanics that will help us appreciate the atmospheric
`properties essential to our understanding of airplane flight mechanics. ln addition
`we will discuss some of the important aircraft instru,nents that provide flight
`inforn1ation to the pilot.
`
`

`

`1.2 Busic Dcfiniti ons 3
`
`1.2
`BASIC DEFINITIONS
`
`The aerodynamic forces and moments generated on an airplane are due to its
`geometric shape, attitude to the flow, airspeed, and the properties of the ambient
`air mass through which it is flying. Air is a fluid and as such possesses certain fluid
`properties. The properties we are interested in are the pressure, temperature,
`density, viscosity, and speed of sound of air at the flight altitude.
`
`1.2.1 Fluid
`
`A fluid can be thought of as any substance that flows. To have such a property, the
`Huid must deform continuously ,vhen acted on by a shearing force. A shear force
`is a force tangent to the surface of the fluid element. No shear stresses are present
`in the fluid when it is at rest. A fluid can transmit forces normal to any chosen
`direction. The normal force and the normal stress are the pressure force and
`pressure, respectively.
`Both liquids and gases can be considered fluids. Liquids under most conditions
`do not change their weight per unit of volume appreciably and can be considered
`incompressible for most engineering applications. Gases, on the other hand, change
`their weight or mass per unit of volume appreciably under the influences of pressure
`or temperature and therefore must be considered compressible.
`
`1.2.2 Pressure
`
`Pressure is the normal force per unit area acting on the fluid. The average pressure
`is calculated by dividing the normal force to the surface by the surface.area:
`
`F P=(cid:173)
`A
`
`( 1.1)
`
`The static pressure in the atmosphere is nothing more than the weight per unit
`of area of the air above the elevation being considered. The ratio of the pressure P
`at altitude to sea-level standard pressure P0 is given the symbol 8:
`
`8 = !_
`Po
`
`( 1.2)
`
`The relationship between pressure, density p, and temperature 1· is given by the
`equation of state
`
`P = pRT
`where R is a constant, the magnitude depending on the gas being considered.
`For air, R has a value 287 J/(kg°K) or 1718 ft 2/(s 20R). Atmospheric air follows the
`
`( 1.3)
`
`

`

`4 CHAPTER l: Introduction
`
`equation of state provided that the temperature is not too high and that air can be
`treated as a continuum.
`
`12.3 Temperature
`
`In aeronautics the ten1perature of air is an extre1nely i1nportant parameter in that
`it affects the properties of air such as density and viscosity. Temperature is an
`abstract concept but can be thought of as a measure of the motion of 1nolecular
`particles within a substance. The concept of te111perature also serves as a means of
`determining the direction in which heal energy will flow when two objects of
`di ffcrcnt temperatures come into contact. Heat energy will Row fron1 the higher
`temperature object to that al lower temperarurc.
`As we will show later the temperature of the atmosphere varie~ significantly
`with altitude. The ratio of the an1bient lc1nperature at altitude, T., to a sea-level
`standard value. Tc.1 is denoted by the symbol fJ:
`
`0
`where the ten1peratures are 111easured using the absolute Kelvin or Rankine scales.
`
`T
`H = 1·
`
`( 1.4)
`
`1.2.4 Density
`
`The density of a substance is defined as the mass per unit of volun1e:
`
`ivlass
`p = Unit of volun1e
`From the equation of state., it can be seen that the density of a gas is directly
`proportionnl to the pressure and inversely proportional to the. absolute tempera(cid:173)
`ture. The ratio of an1bient air density p lo standard sea-leve] air density p0 occurs
`in 1nany aeronautical formulas and is given the designation <T:
`<T = p/po
`
`( 1.5)
`
`(] .6)
`
`1.2.5 Viscosity
`
`Viscosity can he thought of as the internal friction of a fluid. Both liquids and gases
`possess viscosity, with liquids being 1nuch more viscous Lhan gases. As an aid in
`visualizing the concept of viscosity, consider the following simple experiment.
`Consider the motion of the fluid between two parallel plates separated by the
`distance h. If one plate is held fixed \vhilc the other plate is being pulled with a
`t:onslant velocity u. then the velocity distribution of the fluid between the plates will
`be linear as shown in r◄ igure 1.2.
`To produce the constant velocity motion of the upper plate, a tangential force
`n1ust be applied to the plate. 1~he magnitude of the force must be equal to the
`
`

`

`1.2 Basic Definitions 5
`
`--
`"""""""'...W.:..-.iJ.li ......... ...i..:...,;;=--,._.a__,_.........,..........,i~"...c,''-,;.I.: '..,;,i•i...,:;;f•·.______,. _ _ _ ....,
`
`Moving plate
`
`u
`
`F
`
`·,,
`
`.t
`
`.
`
`App.ar~nt
`ve·toclty
`.profile
`
`Viscous
`·11·uid
`
`Fixed plate
`
`FIGURE 1.2
`Shear stress between two plate~.
`
`friction torces in the fluid. It has been established from experiments that the force
`per unit of area of the plate is proportional to the velocity of the moving plate and
`inversely proportional to the distance between the plates. Expressed mathemati(cid:173)
`cally we have
`
`"
`T oc -
`h
`
`( 1.7)
`
`where T is the force per unit area, which is called the shear stress.
`A more general for1n of Equation ( 1. 7) can be written by replacing u/h with
`the derivative du/dy. The proportionality factor is denoted by M, the coefficient of
`absolute viscosity, which is obtained experimentally.
`
`( 1.8)
`
`Equation ( 1.·8) is known as Newton's law of friction.
`For gases, the absolute viscosity depends only on the temperature, with in(cid:173)
`creasing te1nperature causing an increase in viscosity. To estimate the change in
`viscosity with the temperature, several empirical formulations commonly are used.
`The simplest formula is Rayleigh's, which is
`!!:J_ = ('!J.)3/4
`To
`I.Lo
`
`( 1.9)
`
`where the temperatures are on the absolute scale and the subscript O denotes the
`reference condition.
`An alternate expression for calculating the variation of absolute viscosity with
`temperature was developed by Sutherland. The empirical formula developed by
`Sutherland is valid provided the pressure is greater than 0.1 atmosphere and is
`M1 = (T1):-.;2. 10 + Si
`T, + Si
`T0
`ILrl
`
`( 1.10)
`
`where S 1 is a constant. When the temperatures are expressed in the Rankine scale,
`S 1 = l.98°R; when Lhc te1nperatures are expressed in the Kelvin scale, S 1 = 110°K.
`The ratio of. the absolute viscosity to the density of the fluid is a parameter that
`appears frequently and has been identified with the syrnbol 11; it is called the
`
`

`

`6 CHAP·reR 1: Introduction
`
`kinematic viscosity:
`
`(1.11)
`
`An important dimensionless quantity, known as the Reynolds number, is defined as
`
`R = I!)!!_ = Y!_
`IL
`V
`
`e
`
`(1.12)
`
`\Vhere I is a characteristic length and Vis the nuid velocity.
`The Reynolds number can be thought of as the ratio of the inertial to viscous
`forces of the fluid.
`
`1.2.6 The Mach Number and the Speed of Sound
`
`The ratio of an airplane's speed V to the local speed of sound a is an extremely
`important parameter, called the Mach number after the Austrian physicist Ernst
`Mach. The mathematical definition of Mach number is
`V M=(cid:173)
`a
`
`(1.13)
`
`As an airplane moves through the air. it creates pressure disturbances that propa(cid:173)
`gate away fro1n the airplane in all directions with the speed of sound. If the airplane
`is flying at a Mach number less than 1, the pressure disturbances travel faster
`than the airplane and influence the air ahead of the airplane. An example of this
`phenomenon is the upwash field created in front of a wing. However, for flight at
`Mach numbers greater than 1 the pressure disturbances move more slowly than
`the airplane and, therefore, the flow ahead of the airplane has no warning of the
`oncoming aircraft.
`The aerodynamic characteristics of an airplane depend on the flow regime
`around the airplane. As the flight Mach number is increased. the flow around the
`airplane can be completely subsonic, a mixture of subsonic and supersonic flow. or
`completely supersonic. The flight Mach number is used to classify the various flow
`regimes. An approximate classification of the How regin1es follows:
`0 < M < 0.5
`0.5 < M < 0.8
`0.8 < M < 1.2
`1.2 < M < 5
`5 <M
`To have accurate aerodynamic predictions at M > 0.5 compressibility effects must
`be included.
`The lo<.:al speed of sound n1ust be known to detern1ine the Mach number. The
`speed of sound can be shown to be related to the absolute arnbient temperature by
`
`Incompressible subsonic How
`Co1npressible subsonic flow
`Transonic flow
`Supersonic flow
`Hypersonic Ho,v
`
`

`

`1.3 Aerostatics 7
`
`the following expression:
`
`a = (-yRT)'n.
`where 'Y is the ratio of specific heats and R is the gas constant. The ambient
`temperature will be shown in a later section to be a function of altitude.
`
`(1.14)
`
`1.3
`AEROSTATICS
`
`Aerostatics deals with the state of a gas at rest. It follows fro1n the definition given
`for a fluid that all forces acting on the rluid must be normal to any cross-section
`within the fluid. Unlike a solid. a fluid at rest cannot support a shearing force. A
`consequence of this is that the pressure in a fluid at rest is independent of direction.
`That is to say that at any point the pressure is the same in all directions. This
`fundamental concept owes its origin to Pascal. a French scientist ( 1623-1662).
`
`1.3.1 Variation of Pressure in a Static Fluid
`
`Consider the small vertical column of fluid shown in Figure 1.3. Because the fluid
`is at resl, the forces in both the vertical and horizontal directions n1ust su1n to 0.
`The forces in the vertical direction are due to the pressure forces and the weight of
`the fluid column. The force balance in the vertical direction is given by
`PA = ( P + dP )A + pg A dh
`d.P = -pg dh
`
`(1.15)
`
`(1.16)
`
`or
`
`P+ dP
`
`FIGURE 1.3
`Element of Huid at rest.
`
`dh
`
`I
`I
`I
`I
`
`I
`µgdhdxdy
`I
`I
`I
`
`EB : I
`---.-..., ---
`
`-
`A= dx dy ' - .........
`
`p
`
`-dx------
`
`

`

`8 CHAPTER l: Introduction
`
`Equation ( 1.16) tells us how the pressure varies with elevation above some refer(cid:173)
`ence level in a fluid. As the elevation is increased, the pressure will decrease.
`Therefore, the pressure in a static fluid is equal to the weight of the column of fluid
`above the point of interest.
`One of the simplest means of measuring pressure is by a fluid manometer.
`Figure 1.4 shows two types of manometers. The first manometer consists of a
`U-shaped tube containing a liquid. When pressures of different magnitudes are
`applied across the manometer the fluid will rise on the side of the lo\ver pressure
`and fall on the side of the higher pressure. By writing a force balance for each side.
`one can show that
`
`(1.17)
`
`which yie]ds a relationship for the pressure difference in terms of the change in
`height .of the liqui<.l column:
`
`( 1.18)
`
`The second sketch shows a simple mercury barometer. The barometer can be
`thought of as a modi fled U-tube manometer. One leg of the tube is closed off and
`evacuated. The pressure at the top of this leg is O and atmospheric pressure acts on
`the open leg. The atmospheric pressure therefore is equal to. the heighl of the
`mercury column; that is,
`
`P1A1m = pgh
`
`( 1.19)
`
`In practice the atmospheric pressure is commonly expressed as