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`

`

`Mechanics of Flight
`Mechanics of Flight
`
`

`

`

`

`Mechanics of Flight
`
`Warren F. Phillips
`Professor
`Mechanical and Aerospace Engineering
`Utah State University
`
`John Wiley & Sons, Inc.
`
`

`

`The cover was created by the author from photographs by John Martin and Barry Santana.
`
`This book is printed on acid-free paper. @
`
`Copyright 0 2004 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, Inc., 222 Rosewood Drive, Danvers,
`MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests
`to the Publisher for permission should be addressed to the Permissions Department, John Wiley &
`Sons, Inc., 11 l River Street, Hoboken, NJ 07030, (201) 748-601 1, fax (201) 748-6008, e-mail:
`permcoordinator@wiley.com.
`
`Limit of LiabilitylDisclaimer of Warranty: While the publisher and author have used their best
`efforts in preparing this book, they make no representations or warranties with respect to the
`accuracy or completeness of the contents of this book and specifically disclaim any implied
`warranties of merchantability or fitness for a particular purpose. No warranty may be created or
`extended by sales representatives or written sales materials. The advice and strategies contained
`herein may not be suitable for your situation. You should consult with a professional where
`appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other
`commercial damages, including but not limited to special, incidental, consequential, or other
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`
`For general information on our other products and services or for technical support, please contact
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`
`Library of Congress Cataloging-in-Publication Data:
`Phillips, Warren F .
`Mechanics of flight I Warren F. Phillips.
`p. cm.
`Includes bibliographical references and index.
`ISBN 0-471-33458-8 (Cloth)
`1. Aerodynamics. 2. Flight. 3. Flight control. I. Title.
`TL570.P46 2004
`629.132'3--dc2 1
`
`Printed in the United States of America
`
`

`

`Dedicated to Hannah
`in the hope that she may one day
`get as much from reading this
`book as I did from writing it
`
`

`

`

`

`CONTENTS
`
`Preface
`
`Acknowledgments
`
`xiii
`
`1. Overview of Aerodynamics
`Introduction and Notation
`Fluid Statics and the Atmosphere
`The Boundary Layer Concept
`Inviscid Aerodynamics
`Review of Elementary Potential Flows
`Incompressible Flow over Airfoils
`Trailing-Edge Flaps and Section Flap Effectiveness
`Incompressible Flow over Finite Wings
`Flow over Multiple Lifting Surfaces
`Inviscid Compressible Aerodynamics
`Compressible Subsonic Flow
`Supersonic Flow
`Problems
`
`2. Overview of Propulsion
`Introduction
`The Propeller
`Propeller Blade Theory
`Propeller Momentum Theory
`Off-Axis Forces and Moments Developed by a Propeller
`Turbojet Engines: The Thrust Equation
`Turbojet Engines: Cycle Analysis
`The Turbojet Engine with Afterburner
`Turbofan Engines
`Concluding Remarks
`Problems
`
`Aircraft Performance
`3.1.
`Introduction
`3.2.
`Thrust Required
`3.3.
`Power Required
`Rate of Climb and Power Available
`3.4.
`Fuel Consumption and Endurance
`3.5.
`3.6.
`Fuel Consumption and Range
`Power Failure and Gliding Flight
`3.7.
`Airspeed, Wing Loading, and Stall
`3.8.
`
`vii
`
`

`

`viii Contents
`
`The Steady Coordinated Turn
`3.9.
`Takeoff and Landing Performance
`3.10.
`3.1 1. Accelerating Climb and Balanced Field Length
`3.12.
`Problems
`
`4. Longitudinal Static Stability and Trim
`339
`339
`4.1.
`Fundamentals of Static Equilibrium and Stability
`343
`4.2.
`Pitch Stability of a Cambered Wing
`346
`4.3.
`Simplified Pitch Stability Analysis for a Wing-Tail Combination
`362
`4.4.
`Stick-Fixed Neutral Point and Static Margin
`4.5.
`Estimating the Downwash Angle on an Aft Tail
`37 1
`4.6.
`Simplified Pitch Stability Analysis for a Wing-Canard Combination 381
`4.7.
`Effects of Drag and Vertical Offset
`396
`4.8.
`Contribution of the Fuselage and Nacelles
`416
`4.9.
`Contribution of Running Propellers
`420
`4.10.
`Contribution of Jet Engines
`426
`4.1 1.
`Problems
`430
`
`5. Lateral Static Stability and Trim
`5.1.
`Introduction
`5.2.
`Yaw Stability and Trim
`5.3.
`Estimating the Sidewash Gradient on a Vertical Tail
`5.4.
`Estimating the Lift Slope for a Vertical Tail
`5.5.
`Roll Stability and Dihedral Effect
`5.6.
`Roll Control and Trim Requirements
`5.7.
`Longitudinal-Lateral Coupling
`5.8.
`Control Surface Sign Conventions
`5.9.
`Problems
`
`6. Aircraft Controls and Maneuverability
`Longitudinal Control and Maneuverability
`6.1.
`6.2.
`Effects of Structural Flexibility
`6.3.
`Control Force and Trim Tabs
`6.4.
`Stick-Free Neutral and Maneuver Points
`6.5.
`Ground Effect, Elevator Sizing, and CG Limits
`6.6.
`Stall Recovery
`6.7.
`Lateral Control and Maneuverability
`6.8.
`Aileron Reversal
`6.9.
`Other Control Surface Configurations
`6.10. Airplane Spin
`6.1 1.
`Problems
`
`7. Aircraft Equations of Motion
`7.1.
`Introduction
`Newton's Second Law for Rigid-Body Dynamics
`7.2.
`
`

`

`Contents
`
`i~
`
`Position and Orientation: The Euler Angle Formulation
`Rigid-Body 6-DOF Equations of Motion
`Linearized Equations of Motion
`Force and Moment Derivatives
`Nondimensional Linearized Equations of Motion
`Transformation of Stability Axes
`Inertial and Gyroscopic Coupling
`Problems
`
`8. Linearized Longitudinal Dynamics
`Fundamentals of Dynamics: Eigenproblems
`8.1.
`Longitudinal Motion: The Linearized Coupled Equations
`8.2.
`8.3.
`Short-Period Approximation
`8.4.
`Long-Period Approximation
`8.5.
`Pure Pitching Motion
`8.6.
`Summary
`8.7.
`Problems
`
`9. Linearized Lateral Dynamics
`9.1.
`Introduction
`Lateral Motion: The Linearized Coupled Equations
`9.2.
`9.3.
`Roll Approximation
`Spiral Approximation
`9.4.
`9.5.
`Dutch Roll Approximation
`9.6.
`Pure Rolling Motion
`9.7.
`Pure Yawing Motion
`9.8.
`Longitudinal-Lateral Coupling
`9.9.
`Nonlinear Effects
`9.10.
`Summary
`9.1 1.
`Problems
`
`10. Aircraft Handling Qualities and Control Response
`10.1.
`Introduction
`10.2.
`Pilot Opinion
`10.3. Dynamic Handling Quality Prediction
`10.4. Response to Control Inputs
`10.5. Nonlinear Effects and Longitudinal-Lateral Coupling
`10.6.
`Problems
`
`11. Aircraft Flight Simulation
`1 1.1.
`Introduction
`1 1.2.
`Euler Angle Formulations
`1 1.3. Direction-Cosine Formulation
`1 1.4.
`Euler Axis Formulation
`The Euler-Rodrigues Quaternion Formulation
`1 1.5.
`
`

`

`X Contents
`
`Quaternion Algebra
`Relations between the Quaternion and Other Attitude Descriptors
`Applying Rotational Constraints to the Quaternion Formulation
`Closed-Form Quaternion Solution for Constant Rotation
`Numerical Integration of the Quaternion Formulation
`Summary of the Flat-Earth Quaternion Formulation
`Aircraft Position in Geographic Coordinates
`Problems
`
`Bibliography
`
`Appendixes
`A
`Standard Atmosphere, SI Units
`B
`Standard Atmosphere, English Units
`
`Index
`
`943
`
`

`

`Preface
`
`This book was written for aeronautical, aerospace, andlor mechanical engineers. The
`book is intended primarily as a textbook for an engineering class that is taught as an
`upper-division undergraduate or lower-division graduate class. Such an engineering class
`is typically required in aeronautical engineering programs and is a popular elective in
`many astronautical and mechanical engineering programs.
`Collectively the engineering topics covered in this book are usually referred to as
`Fight mechanics. These topics build directly upon a related set of engineering topics that
`are typically grouped under the title of aerodynamics. Together the topics of
`aerodynamics and flight mechanics form the foundation of aeronautics, which is the
`science that deals with the design and operation of aircraft. Aerodynamics is the science
`of predicting and controlling the forces and moments that act on a craft moving through
`the atmosphere. Flight mechanics is the science of predicting and controlling the motion
`that results from the aerodynamic forces and moments acting on an aircraft.
`This textbook was written with the assumption that the reader has completed the
`lower-division coursework that is required in any engineering program. Critical topics
`include calculus, differential equations, computer programming, numerical methods,
`statics, dynamics, and fluid mechanics. It is also assumed that the student has previously
`completed an introductory course in aerodynamics. The book does include an overview
`of aerodynamics. Thus, it would be possible for a discerning student who has completed
`the lower-division engineering coursework to comprehend the material in this book with
`no previous experience in aerodynamics. However, in the author's experience, most
`students require at least a full semester to comprehend the fundamentals of aerodynamics.
`Such comprehension is an essential prerequisite to the study of flight mechanics.
`One important feature of this book is the inclusion of many worked example
`problems, which are designed to help the student understand the process of applying the
`principles of flight mechanics to the solution of engineering problems. Another unique
`feature is the inclusion of a detailed presentation of the quaternion formulation for six-
`degree-of-freedom flight simulation. Efficient numerical methods for integration of the
`quaternion formulation are also presented and discussed. This material is presented in
`sufficient detail to allow undergraduate students to write their own code. The quaternion
`formulation is not typically presented in other textbooks on atmospheric flight mechanics.
`In fact, many textbooks that deal with atmospheric flight mechanics do not even mention
`the quaternion formulation. Here it is shown that this formulation of the aircraft
`equations of motion, which has typically been implemented to eliminate a singularity
`associated with the Euler angle formulation, is far superior to the other commonly used
`formulations based on computational efficiency alone. Most practicing engineers
`working in the field of flight dynamics should find the chapter on flight simulation to be a
`valuable reference.
`This book contains considerably more material than can be covered in a single one-
`semester class. This provides significant benefit for both the instructor and the student.
`The additional coverage gives the instructor some flexibility in choosing the material to
`
`

`

`xii Preface
`
`be included in a particular one-semester course. The instructor may choose to cover all
`of the topics offered but omit some of the more advanced material on each topic. On the
`other hand, the instructor may wish to cover fewer topics in greater detail. Most of the
`material in this book can be covered in a two-semester undergraduate sequence. A
`convenient division of this material is to cover aircraft performance together with static
`stability and control during the first semester and then extend the treatment to include
`linearized flight dynamics and flight simulation in the second semester. From the
`student's point of view, the information contained in this book, beyond that which is
`covered in their formal coursework, provides an excellent source of reference material for
`future independent study.
`There are two overall philosophical approaches that can be used to present the
`material associated with atmospheric flight mechanics. One approach is to start with a
`development of the general six-degree-of-freedom aircraft equations of motion and then
`treat everything else as a special case of this general formulation. The other approach is
`to begin with simple problems that are formulated and solved starting from fundamental
`principles and then gradually work up to the more complex problems, leading eventually
`to the development of the general equations of motion. The latter approach is used in this
`book. This approach necessitates some repetition. As a result, the experienced reader
`may find some of the developments in the earlier chapters to be somewhat tedious.
`However, while the approach used here is not the most concise, the author believes that
`most students will find it more understandable. In general, the reader will find that
`throughout this book, conciseness is often sacrificed in favor of student convenience and
`understanding.
`In Chapters 3 through 6, some of the basic principles of flight mechanics are
`explored by example, starting with simple problems such as steady level flight and
`building to more complex problems, including turns and spins. In Chapter 7, the more
`general rigid-body equations of motion are developed from fundamental principles.
`Since this chapter could serve as a starting point for a second-semester course in flight
`mechanics, it includes a review of coordinate systems and notation that were introduced
`gradually throughout Chapters 3 through 6. Chapter 7 was deliberately written to be
`independent of the presentation in Chapters 3 through 6. Thus, students or practicing
`engineers who are already familiar with the principles of static stability and control could
`start with Chapter 7 to begin a study of aircraft dynamics.
`
`Warren F. Phillips
`
`

`

`Acknowledgments
`
`First, I would like to thank my good friend and counsel, Dr. Barry W. Santana, for the
`broad base of support and assistance that he provided during the writing of this book. In
`addition to reading drafts of the text and providing many insightful suggestions, he
`dedicated a great deal of time and travel to shooting and processing photographs
`I particularly appreciate the many hours of work that Barry
`especially for this book.
`invested in developing the index.
`While thanks are due a great number of my students for their feedback and many
`stimulating discussions, I am particularly grateful to Nicholas R. Alley for his input on
`the book and his assistance as coauthor of the solutions manual. I would also like to
`thank Dr. Glenn Gebert and Dr. Fred Lutze for reviewing a complete draft of this book
`and providing many helpful suggestions.
`I wish to extend special thanks to Drs. John D. Anderson, John J. Bertin, Bernard
`Etkin, Barnes W. McCormick, Robert C. Nelson, Courtland D. Perkins, and Ludwig
`Prandtl, who have through their writings been my teachers and mentors.
`If I have
`contributed in some small way to this very exciting field, it is only because of what I
`learned from the writings of these great men.
`Most of all, I want to thank my wife, Barbara, for her love, encouragement, and
`assistance, without which 1 could not have completed this work. I am especially grateful
`for the countless hours that she spent proofreading the many drafts that eventually led to
`the finished version of this book. I love you.
`
`Warren F. Phillips
`
`xiii
`
`

`

`

`

`Mechanics of Flight
`MechanicsofFlight
`
`

`

`

`

`Chapter 1
`Overview of Aerodynamics
`
`1 .I. Introduction and Notation
`Flight mechanics is the science of predicting and controlling aircraft motion. From
`Newton's second law we know that the motion of any body depends on the forces and
`moments acting on the body. The forces and moments exerted on an aircraft in flight are
`the aerodynamic forces and moments acting on the aircraft's skin, the propulsive forces
`and moments created by the aircraft's engine or engines, and the gravitational force
`between the aircraft and the Earth. Since aerodynamic forces and moments are central to
`the study of aircraft motion, an understanding of the fundamentals of aerodynamics is a
`prerequisite to the study of flight mechanics. In this text it will be assumed that the
`reader has gained this prerequisite knowledge, either through the completion of at least
`one engineering course on aerodynamics or through independent study. In this chapter
`we review briefly some of the more important concepts that the reader should understand
`before proceeding with the material in the remainder of the book.
`The aerodynamic forces and moments acting on any body moving through the
`atmosphere originate from only two sources,
`
`1. The pressure distribution over the body surface.
`2. The shear stress distribution over the body surface.
`
`A resultant aerodynamic force, Fa, and a resultant aerodynamic moment, M,, are the
`net effects of the pressure and shear stress distributions integrated over the entire surface
`of the body. To express these two vectors in terms of components, we must define a
`coordinate system. While several different coordinate systems will be used in our study
`of flight mechanics, the coordinate system commonly used in the study of aerodynamics
`is referred to here as Cartesian aerodynamic coordinates. When considering flow over a
`body such as an airfoil, wing, or airplane, the x-axis of this particular coordinate system is
`aligned with the body axis or chord line, pointing in the general direction of relative air-
`flow. The origin is located at the front of the body or leading edge. The y-axis is chosen
`normal to the x-axis in an upward direction. Choosing a conventional right-handed
`coordinate system requires the z-axis to be pointing in the spanwise direction from right
`to left, as shown in Fig. 1.1.1. Here, the components of the resultant aerodynamic force
`and moment, described in this particular coordinate system, are denoted as
`
`F, = Ai, + N i , + B i z
`
`where i,, i,, and i, are the unit vectors in the x-, y-, and z-directions, respectively. The
`terminology that describes these components is
`
`

`

`2 Chapter 1 Overview of Aerodynamics
`
`Figure 1.1.1. Cartesian aerodynamic coordinate system used in the study of aerodynamics.
`
`A = aftward axial force = x-component of Fa (parallel to the chord)
`N = upward normal force = y-component of Fa (normal to the chord and span)
`B = leftward side force = z-component of Fa (parallel with the span)
`f? = rolling moment (positive right wing down)
`n = yawing moment (positive nose right)
`m = pitching moment (positive nose up)
`
`The traditional definitions for the moments in roll, pitch, and yaw do not follow the right-
`hand rule in this coordinate system. It is often convenient to split the resultant aero-
`dynamic force into only two components,
`
`D = drag = the component of F, parallel to V, (D = F, . i, )
`(L = I Fa - Di, I)
`L = lift = the component of Fa perpendicular to V,
`
`is the unit
`
`is the freestream velocity or relative wind far from the body and i,
`where V,
`vector in the direction of the freestream.
`For two-dimensional flow, it is often advantageous to define the section force and
`section moment to be the force and moment per unit span. For these definitions the
`notation used in this book will be
`-
`D = section drag = drag force per unit span (parallel to V, )
`-
`L = section lift = lift force per unit span (perpendicular to V, )
`-
`-
`A = section axial force = axial force per unit span (parallel to chord)
`N = section normal force = normal force per unit span (perpendicular to chord)
`i7I = section moment = pitching moment per unit span (positive nose up)
`
`where the chord is a line extending from the leading edge to the trailing edge of the body.
`The chord length, c, is the length of this chord line.
`
`

`

`The aerodynamic forces and moments are usually expressed in terms of dimension-
`less force and moment coefficients. For example,
`
`1.1. Introduction and Notation 3
`
`C, = drag coefficient =
`
`D
`pm vm2s
`CL = lift coefficient = , L
`
`. p J 3
`
`C, = axial force coefficient =
`
`A
`; p, v,Zs
`N
`C, = normal force coefficient =
`; p m v 3
`m
`
`C, = pitching moment coefficient - ; p , ~ , ' ~ c
`
`where p, is the freestream density, S is the reference area, and c is the reference length.
`For a streamlined body such as a wing, S is the planform area and c is the mean chord
`length. For a bluff body, the frontal area is used as the reference. For two-dimensional
`flow, the section aerodynamic coefficients per unit span are defined:
`-
`-
`D
`C, = section drag coefficient = - ; p,v&
`-
`C, = section lift coefficient = - : pmv,zc
`C , = section axial force coefficient = ---- ; pmv&
`-
`N
`C, = section normal force coefficient = -
`ipmV2~
`- m
`-
`C, = section moment coefficient = , pmv,2c2
`
`?.,
`
`L
`
`A
`
`The resultant aerodynamic force acting on a 2-D airfoil section is completely
`specified in terms of either lift and drag or axial and normal force. These two equivalent
`descriptions of the resultant aerodynamic force are related to each other through the angle
`of attack, as shown in Fig. I . 1.2,
`
`a = angle of attack = the angle from V,
`
`to the chord line (positive nose up)
`
`If the normal and axial coefficients are known, the lift and drag coefficients can be found
`from the relations
`
`

`

`4 Chapter 1 Overview of Aerodynamics
`4~
`
` ,aA
`Leading ~ d ~ e \
`
`Edge
`
`/ Vm
`
`Chord Length, c
`
`4
`
`Figure 1.1.2. Section forces and moment.
`
`-
`-
`-
`CL = CN cos w - CA sin w
`
`(1.1.1)
`
`and when the lift and drag coefficients are known, the normal and axial coefficients are
`found from
`
`Because the lift is typically much larger than the drag, from Eq. (1.1.4) we see why the
`axial force is sometimes negative even though the angle of attack is small.
`The resultant aerodynamic force and moment acting on a body must have the same
`effect as the distributed loads. Thus, the resultant moment will depend on where the
`resultant force is placed on the body. For example, let x be the coordinate measured
`along the chord line of an airfoil, from the leading edge toward the trailing edge. If we
`place the resultant force and moment on the chord line, the value of the resultant moment
`will depend on the x-location of the resultant force. The resultant moment about some
`arbitrary point on the chord line a distance x from the leading edge, hi,, is related to the
`resultant moment about the leading edge, El,, according to
`
`or in terms of dimensionless coefficients,
`-
`x -
`,.,
`C,n, = C,, - -CN
`C
`
`

`

`1 .I. Introduction and Notation 5
`
`Two particular locations along the chord line are of special interest:
`
`x, = center of pressure E point about which resultant moment is zero
`x,, = aerodynamic center = point about which resultant moment is independent of a
`
`Using the definition of center of pressure in Eq. (1.1.5), the section pitching moment
`coefficient relative to the leading edge can be written as
`
`or after solving for the center of pressure,
`
`Using the aerodynamic center in Eq. ( 1 . 1 3 , we can write
`
`Equation (1.1.7) must hold for any angle of attack and any value of x. Thus, at the angle
`of attack that gives a normal force coefficient of zero, the moment coefficient about any
`point on the chord line is equal to the moment coefficient about the aerodynamic center,
`which by definition does not vary with angle of attack. Thus, the section pitching
`moment coefficient relative to the aerodynamic center can be expressed as
`
`Using Eq. (1.1.8) in Eq. (1.1.7) and solving for the location of the aerodynamic center
`results in
`
`Thus, to determine the aerodynamic center of an airfoil section, one can evaluate the
`normal force and pitching moment coefficients for any point on the chord line, as a
`function of angle of attack. The moment coefficient is then plotted as a function of the
`normal force coefficient. The moment axis intercept is the moment coefficient about the
`aerodynamic center. For any nonzero normal force coefficient, the location of the aero-
`dynamic center can be determined according to Eq. (1.1.9) from knowledge of the normal
`force coefficient and the moment coefficient about some arbitrary point.
`
`

`

`6 Chapter 1 Overview of Aerodynamics
`
`1.2. Fluid Statics and the Atmosphere
`The lift and drag acting on a moving aircraft are strong functions of air density. Thus, to
`predict aircraft motion, we must be able to determine the density of the air at the aircraft's
`altitude. For this purpose, the atmosphere can be regarded as a static fluid.
`In a static fluid, the change in pressure,^, with respect to geometric altitude, H, is
`
`where p is the fluid density and g is the acceleration of gravity. For all practical
`purposes, the atmosphere can be assumed to behave as an ideal gas, which gives
`
`where R is the ideal gas constant for the gas and T is the absolute temperature. Thus, the
`pressure variation in the atmosphere is governed by
`
`From Newton's law of gravitation, g varies with altitude according to the relation
`
`where go is the standard acceleration of gravity at sea level, go = 9.806645 m/s2, and RE is
`the radius of the Earth at sea level, RE= 6,356,766 m. Using Newton's law of gravitation
`in Eq. (1.2.3) gives
`
`The integration of Eq. (1.2.4) is simplified by introducing the change of variables
`
`With this change of variables, Eq. (1.2.4) can be written as
`
`The new variable that is defined in Eq. (1.2.5), Z, is called the geopotential altitude. The
`difference between the geopotential and geometric altitudes is small in the first several
`thousand feet above sea level. However, at higher altitudes the difference is significant.
`
`

`

`1.2. Fluid Statics and the Atmosphere 7
`
`Since the temperature and pressure variations in the atmosphere are different from
`day to day, aircraft instruments make use of the concept of a standard atmosphere. Many
`different "standard" atmospheres have been defined. However, most are essentially
`indistinguishable below 100,000 feet, which encloses the domain of most aircraft.
`The standard atmosphere commonly used in the calibration of aircraft instruments is
`divided into several regions, each of which is assumed to have a constant temperature
`gradient with respect to geopotential altitude. The temperature gradients defined for this
`standard atmosphere are given in Table 1.2.1. This information completely defines the
`temperature throughout the standard atmosphere. Other atmosphere definitions are quite
`commonly used for special area environments. These include the polar atmosphere, the
`desert atmosphere, and the 10 percent hot day.
`Since the temperature gradient is constant in each of the altitude ranges defined in
`Table 1.2.1, the temperature in each range is linear,
`
`where Z, is the minimum geopotential altitude in the range and T/ is the temperature
`gradient for the range, defined as the change in temperature with respect to geopotential
`altitude, dT/dZ. The negative of this temperature gradient is commonly referred to as the
`lapse rate. Applying Eq. (1.2.7), the integration of Eq. (1.2.6) subject to the boundary
`condition, p(Zi) = p , , yields
`
`From Eqs. (1.2.7) and (1.2.8), combined with the information given in Table 1.2.1, the
`temperature and pressure throughout the atmosphere can be determined from the ideal
`
`Geo~otential Altitude Range
`
`Initial Temperature
`
`Temperature Gradient
`
`Table 1.2.1. Temperature gradients with respect to geopotential altitude for the standard atmos-
`phere commonly used in the calibration of aircraft instruments.
`
`

`

`8 Chapter 1 Overview of Aerodynamics
`
`gas constant and the temperature and pressure at standard sea level. The gas constant for
`the standard atmosphere is defined to be 287.0528 N d k g - K . The standard atmosphere
`is defined to have a temperature at sea level of 288.150 K, and standard atmospheric
`pressure at sea level is defined as 101,325 ~ / m ' .
`Once the temperature and pressure are known at any altitude the density can be
`determined from Eq. (1.2.2) and the speed of sound can be computed from
`
`where yis the ratio of constant pressure to constant volume specific heat, which is 1.4 for
`the standard atmosphere. Remember that since a newton is 1 kgm/s2, the ideal gas
`constant for the standard atmosphere can also be written as 287.0528 m2/s2-K.
`
`EXAMPLE 1.2.1. Compute the absolute temperature, pressure, density, and
`speed of sound for the standard atmosphere defined in Table 1.2.1 at a geometric
`altitude of 30,000 meters.
`
`Solution. We start by determining geopotential altitude from the known geometric
`altitude. From Eq. (1.2.5)
`
`Since this is in the third of altitude ranges given in Table 1.2.1, we must first
`integrate across the first two ranges. At the boundary between the first and the
`second range (ZI = 1 1 km), from Eqs. (1.2.7) and (1.2.8) we obtain
`
`and at the boundary between the second and the third range (Z2 = 20 km),
`
`P2 = PI exp -
`
`[
`
`Z1)] = 22,632 ~
`
`
`
`l m ~exp 9.806645 (20 - 1 1)x lo3
`
`
`
`287.0528(2 16.650) I
`
`

`

`1.2. Fluid Statics and the Atmosphere 9
`
`Integrating from this point to the specified geopotential altitude of 29,859 meters,
`Eqs. (1.2.7) and (1.2.8) yield
`
`From Eq. (1.2.2), the density is
`
`and from Eq. (1.2.9), the speed of sound is
`
`EXAMPLE 1.2.2. Compute the absolute temperature, pressure, density, and
`speed of sound, in English engineering units, for the standard atmosphere that is
`defined in Table 1.2.1 at a geometric altitude of 100,000 feet.
`
`Solution. Since this standard atmosphere is defined in SI units, we start by
`converting the given geometric altitude to meters:
`
`H = 100,000ft (0.304800dft) = 30,480 m
`
`Following the procedure used in Example 1.2.1, we obtain
`
`Z = 30,335 ml0.3048 d f t = 99,523 ft
`
`
`
`p = 1 , 1 1 4 . 3 ~ 1 m ~ (0.02088543 1bf/ft2/N/m2) = 23.272 1bf/ft2 = 0.16161 psi
`
`

`

`10 Chapter 1 Overview of Aerodynamics
`
`Atmospheric temperature and pressure vary substantially with time and location on
`the Earth. Nevertheless, for the correlation of test data taken at different times and
`locations and for the calibration of aircraft flight instruments, it is important to have an
`agreed-upon standard for atmospheric properties. From the assumption of an ideal gas,
`pressure, density, and the speed of sound can be determined directly from a defined
`temperature profile. The standard atmosphere, based on the temperature profile defined
`in Table 1.2.1 and tabulated in Appendices A and B, is commonly used for the purposes
`mentioned above. Slightly different temperature profiles have also been used to define
`other standard atmospheres, but the variation is small below 100,000 feet.
`
`1.3. The Boundary Layer Concept
`As pointed out in Sec. 1.1, there are only two types of aerodynamic forces acting on a
`body moving through the atmosphere: pressure forces and viscous shcar forces. The
`Reynolds number provides a measure of the relative magnitude of the pressure forces in
`relation to the viscous shear forces. For the airspeeds typically encountered in flight,
`Reynolds numbers are quite high and viscous forces are usually small compared to
`pressure forces. This does not mean that viscous forces can be neglected. However, it
`does allow us to apply the simplifying concept of boundary layer theory.
`For flow over a streamlined body at low angle of attack and high Reynolds number,
`the effects of viscosity are essentially confined to a thin layer adjacent to the surface of
`the body, as shown in Fig. 1.3.1. Outside the boundary layer, the shear forces can be
`neglected and since the boundary layer is thin, the change in pressure across the thickness
`of this layer is insignificant. With this flow model, the pressure forces can be determined
`from the inviscid flow outside the boundary layer, and the shear forces can be obtained
`from a solution to the boundary layer equations.
`While boundary layer theory provides a tremendous simplification over the complete
`Navier-Stokes equations, solutions to the boundary layer equations are far from trivial,
`especially for the complex geometry that is often encountered in an aircraft. A thorough
`review of boundary layer.theory is beyond the intended scope of this chapter and is not
`prerequisite to an understanding of the fundamental principles of flight mechanics.
`However, there are some important results of boundary layer theory that the reader
`should know and understand.
`
`thin viscous boundary layer
`
`s