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Aerodynamics forEngineering Students


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Aerodynamics forEngineering Students

Sixth Edition

E.L. Houghton

P.W. Carpenter

Steven H. Collicott

Daniel T. Valentine

AMSTERDAM • BOSTON • HEIDELBERG • LONDONNEW YORK • OXFORD • PARIS • SAN DIEGO

SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Butterworth-Heinemann is an imprint of Elsevier


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Butterworth-Heinemann is an imprint of Elsevier225 Wyman Street, Waltham, MA 02451, USAThe Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK

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Practitioners and researchers must always rely on their own experience and knowledge in evaluating and usingany information, methods, compounds, or experiments described herein. In using such information or methodsthey should be mindful of their own safety and the safety of others, including parties for whom they have aprofessional responsibility.

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MATLABr is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does notwarrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLABr softwareor related products does not constitute endorsement or sponsorship by The MathWorks of a particularpedagogical approach or particular use of the MATLABr software.

Library of Congress Cataloging-in-Publication DataAerodynamics for engineering students / E.L. Houghton . . . [et al.]. – 6th ed.

p. cm.ISBN: 978-0-08-096632-8 (pbk.)1. Aerodynamics. 2. Airplanes–Design and construction. I. Houghton, E. L. (Edward Lewis)

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Preface

This volume is intended for engineering students in introductory aerodynamicscourses and as a reference useful for reviewing foundational topics for graduatecourses.

The sequence of subject development in this edition begins with definitions andconcepts and then moves on to incompressible flow, low speed airfoil and wingtheories, compressible flow, high speed wing theories, viscous flow, boundary layers,transition and turbulence, wing design, and concludes with propellers and propulsion.

Reinforcing or teaching first the units, dimensions, and properties of the physicalquantities used in aerodynamics addresses concepts that are perhaps both the simplestand the most critical. Common aeronautical definitions are covered before lessonson the aerodynamic forces involved and how the forces drive our definitions of air-foil characteristics. The fundamental fluid dynamics required for the development ofaerodynamic studies and the analysis of flows within and around solid boundaries forair at subsonic speeds is explored in depth in the next two chapters. Classical airfoiland wing theories for the estimation of aerodynamic characteristics in these regimesare then developed.

Attention is then turned to the aerodynamics of high speed air flows in Chapters 6and 7. The laws governing the behavior of the physical properties of air are appliedto the transonic and supersonic flow speeds and the aerodynamics of the abruptchanges in the flow characteristics at these speeds, shock waves, are explained.Then compressible flow theories are applied to explain the significant effects onwings in transonic and supersonic flight and to develop appropriate aerodynamiccharacteristics. Viscosity is a key physical quantity of air and its significance in aero-dynamic situations is next considered in depth. The powerful concept of the boundarylayer and the development of properties of various flows when adjacent to solidboundaries create a body of reliable methods for estimating the fluid forces due toviscosity. In aerodynamics, these forces are notably skin friction and profile drag.Chapters on wing design and flow control, and propellers and propulsion, respec-tively, bring together disparate aspects of the previous chapters as appropriate. Thispermits discussion of some practical and individual applications of aerodynamics.

Obviously aerodynamic design today relies extensively on computational meth-ods. This is reflected in part in this volume by the introduction, where appropriate, ofdescriptions and discussions of relevant computational techniques. However, this textis aimed at providing the fundamental fluid dynamics or aerodynamics backgroundnecessary for students to move successfully into a dedicated course on computationmethods or experimental methods. As such, experience in computational techniquesor experimental techniques are not required for a complete understanding of the aero-dynamics in this book. The authors urge students onward to such advanced coursesand exciting careers in aerodynamics.

xv


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xvi Preface

ADDITIONAL RESOURCESA set of .m files for the MATLAB routines in the book are available by visitingthe book’s companion site, www.elsevierdirect.com and searching on ‘houghton.’Instructors using the text for a course may access the solutions manual and imagebank by visiting www.textbooks.elsevier.com and following the online registrationinstructions.

ACKNOWLEDGEMENTSThe authors thank the following faculty, who provided feedback on this projectthrough survey responses, review of proposal, and/or review of chapters:

Alina Alexeenko Purdue UniversityS. Firasat Ali Tuskegee UniversityDavid Bridges Mississippi State UniversityRussell M. Cummings California Polytechnic State UniversityPaul Dawson Boise State UniversitySimon W. Evans, Ph.D Worcester Polytechnic InstituteRichard S. Figliola Clemson UniversityTimothy W. Fox California State University NorthridgeAshok Gopalarathnam North Carolina State UniversityDr. Mark W. Johnson University of LiverpoolBrian Landrum, Ph.D University of Alabama in HuntsvilleGary L. Solbrekken University of MissouriMohammad E. Taslim Northeastern UniversityValana Wells Arizona State University

Professors Collicott and Valentine are grateful for the opportunity to continue thework of Professors Houghton and Carpenter and thank Joe Hayton, Publisher, forthe invitation to do so. In addition, the professional efforts of Mike Joyce, EditorialProgram Manager, Heather Tighe, Production Manager, and Kristen Davis, Designerare instrumental in the creation of this sixth edition.

The products of one’s efforts are of course the culmination of all of one’s experi-ences with others. Foremost amongst the people who are to be thanked most warmlyfor support are our families. Collicott and Valentine thank Jennifer, Sarah, and Racheland Mary, Clara, and Zach T., respectively, for their love and for the countless joysthat they bring to us. Our Professors and students over the decades are major con-tributors to our aerodynamics knowledge and we are thankful for them. The authorsshare their deep gratitude for God’s boundless love and grace for all.

http://www.elsevierdirect.comhttp://www.textbooks.elsevier.com


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CHAPTER

Basic Concepts andDefinitions 1

“To work intelligently” (Orville and Wilbur Wright)“one needs to know the effects of variationsincorporated in the surfaces. . . . The pressures on squaresare different from those on rectangles, circles, triangles, or

ellipses. . . .The shape of the edge also makes a difference.”

from The Structure of the Plane – Muriel Rukeyser

LEARNING OBJECT IVES

• Review the fundamental principles of fluid mechanics and thermodynamicsrequired to investigate the aerodynamics of airfoils, wings, and airplanes.

• Recall the concepts of units and dimension and how they are applied to solvingand understanding engineering problems.

• Learn about the geometric features of airfoils, wings, and airplanes and how thenames for these features are used in aerodynamics communications.

• Explore the aerodynamic forces and moments that act on airfoils, wings, andairplanes and learn how we describe these loads quantitatively in dimensionalform and as coefficients.

1.1 INTRODUCTIONThe study of aerodynamics requires a number of basic definitions, including anunambiguous nomenclature and an understanding of the relevant physical proper-ties, related mechanics, and appropriate mathematics. Of course, these notions arecommon to other disciplines, and it is the purpose of this chapter to identify andexplain those that are basic and pertinent to aerodynamics and that are to be used inthe remainder of the volume.

Aerodynamics for Engineering Students. DOI: 10.1016/B978-0-08-096632-8.00001-1c© 2013 Elsevier Ltd. All rights reserved.

1

http://dx.doi.org/10.1016/B978-0-08-096632-8.00001-1


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2 CHAPTER 1 Basic Concepts and Definitions

1.1.1 Basic ConceptsThis text is an introductory investigation of aerodynamics for engineering students.1

Hence, we are interested in theory to the extent that it can be practically appliedto solve engineering problems related to the design and analysis of aerodynamicobjects.

The design of vehicles such as airplanes has advanced to the level where werequire the wealth of experience gained in the investigation of flight over the past100 years. We plan to investigate the clever approximations made by the few wholearned how to apply mathematical ideas that led to productive methods and usefulformulas to predict the dynamical behavior of “aerodynamic” shapes. We need tolearn the strengths and, more important, the limitations of the methodologies anddiscoveries that came before us.

Although we have extensive archives of recorded experience in aeronautics, thereare still many opportunities for advancement. For example, significant advancementscan be achieved in the state of the art in design analysis. As we develop ideasrelated to the physics of flight and the engineering of flight vehicles, we will learnthe strengths and limitations of existing procedures and existing computational tools(commercially available or otherwise). We will learn how airfoils and wings performand how we approach the designs of these objects by analytical procedures.

The fluid of primary interest is air, which is a gas at standard atmospheric con-ditions. We assume that air’s dynamics can be effectively modeled in terms of thecontinuum fluid dynamics of an incompressible or simple-compressible fluid. Air isa fluid whose local thermodynamic state we assume is described either by its massdensity ρ = constant, or by the ideal gas law. In other words, we assume air behavesas either an incompressible or a simple-compressible medium, respectively. The con-cepts of a continuum, an incompressible substance, and a simple-compressible gaswill be elaborated on in Chapter 2.

The equation of state, known as the ideal gas law, relates two thermodynamicproperties to other properties and, in particular, the pressure. It is

p= ρRT

where p is the thermodynamic pressure, ρ is mass density, T is absolute (thermo-dynamic) temperature, and R= 287J/(kg K) or R= 1716 ft-lb (slug◦R)−1. Pressureand temperature are relatively easy to measure. For example, “standard” barometricpressure at sea level is p= 101,325Pascals, where a Pascal (Pa) is 1N/m2. In Impe-rial units this is 14.675 psi, where psi is lb/in2 and 1 psi is equal to 6895 Pa (note that

1It has long been common in engineering schools for an elementary, macroscopic thermodynamicscourse to be completed prior to a compressible-flow course. The portions of this text that discuss com-pressible flow assume that such a course precedes this one, and thus the discussions assume someelementary experience with concepts such as internal energy and enthalpy.


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1.1 Introduction 3

14.675 psi is equal to 2113.2 lb/ft2). The standard temperature is 288.15 K (or 15◦C,where absolute zero equal to −273.15◦C is used). In Imperial units this is 519◦R (or59◦F, where absolute zero equal to−459.67◦F is used). Substituting into the ideal gaslaw, we get for the standard density ρ = 1.225 kg/m3 in SI units (and ρ = 0.00237slugs/ft3 in Imperial units). This is the density of air at sea level given in the table ofdata for atmospheric air; the table for standard atmospheric conditions is provided inAppendix B.

The thermodynamic properties of pressure, temperature, and density are assumedto be the properties of a mass-point particle of air at a location x= (x,y,z) in spaceat a particular instant in time, t. We assume the measurement volume to be suf-ficiently small to be considered a mathematical point. We also assume that it issufficiently large so that these properties have meaning from the perspective of equi-librium thermodynamics. And we further assume that the properties are the same asthose described in a course on classical equilibrium thermodynamics. Therefore, weassume that local thermodynamic equilibrium prevails within the mass-point parti-cle at x and t regardless of how fast the thermodynamic state changes as the particlemoves from one location in space to another. This is an acceptable assumption for ourmacroscopic purposes because molecular processes are typically faster than changesin the flow field we are interested in from a macroscopic point of view. In addition,we invoke the continuum hypothesis, which states that we can define all flow proper-ties as continuous functions of position and time and that these functions are smooth,that is, their derivatives are continuous. This allows us to apply differential integralcalculus to solve partial differential equations that successfully model the flow fieldsof interest in this course. In other words, predictions based on the theory reported inthis text have been experimentally verified.

To develop the theory, the fundamental principles of classical mechanics areassumed. They are

• Conservation of mass• Newton’s second law of motion• First law of thermodynamics• Second law of thermodynamics

The principle of conservation of mass defines a mass-point particle, which is a fixed-mass particle. Thus the principle also defines mass density ρ, which is mass per unitvolume. If a mass-point particle conserves mass, as we have postulated, then densitychanges can only occur if the volume of the particle changes, because the dimensionof mass density is M/L3, where M is mass and L is length. The SI unit of density isthus kg/m3.

Newton’s second law defines the concept of force in terms of acceleration (“F=ma”). The acceleration of a mass-point particle is the change in its velocity withrespect to a change in time. Let the velocity vector u= (u,v,w); this is the velocity ofa mass-point particle at a point in space, x= (x,y,z), at a particular instant in time t.


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The acceleration of this mass-point particle is

a=DuDt=∂u∂t+u · ∇u

This is known as the substantial derivative of the velocity vector. Since we are inter-ested in the properties at fixed points in space in a coordinate system attached tothe object of interest (i.e., the “laboratory” coordinates), there are two parts to mass-point particle acceleration. The first is the local change in velocity with respect totime. The second takes into account the convective acceleration associated with achange in velocity of the mass-point particle from its location upstream of the pointof interest to the observation point x at time t.

We will also be interested in the spatial and temporal changes in any propertyf of a mass-point particle of fluid. These changes are described by the substantialderivative as follows:

Df

Dt=∂f

∂t+u · ∇f

This equation describes the changes in any material property f of a mass point ata particular location in space at a particular instant in time. This is in a laboratoryreference frame, the so-called Eulerian viewpoint.

The next step in conceptual development of a theory is to connect the changesin flow properties with the forces, moments, and energy exchange that cause thesechanges to happen. We do this by first adopting the Newtonian simple-compressibleviscous fluid model for real fluids (e.g., water and air), which is described in detailin Chapter 2. Moreover, we will apply the simpler, yet quite useful, Euler’s perfectfluid model, also described in Chapter 2. It is quite fortunate that the latter model hassignificant practical use in the design analysis of aerodynamic objects.

Before we proceed to Chapter 2 and look at the development of the equationsof motion and the simplifications we will apply to potential flows in Chapters 3, 4and 5, we review some useful mathematical tools, define the geometry of the wing,and provide an overview of wing performance in the next three sections, respectively.

1.1.2 Measures of Dynamical PropertiesThe mathematical concepts presented in this section and applied in this text describethe dynamic behavior of a thermo-mechanical fluid. In other words, we neglect elec-tromagnetic, relativistic, and quantum effects on dynamics. Also, as already pointedout, we take the view that the properties are continuous functions of location in spaceand time. The discussion of units and dimensions here are thus limited to the mea-sures of flow properties of fluids (liquids and gases) near the surface of the Earthunder standard conditions.

The units and dimensions of all physical properties and the relevant proper-ties of fluids are recalled, and after a review of the aeronautical definitions of

MachinniiaecHighlight



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1.2 Units and Dimensions 5

wing and airfoil geometry, the remainder of the chapter discusses aerodynamicforce.

The origins of aerodynamic force and how it is manifest on wings and other aero-nautical bodies, and the theories that permit its evaluation and design, are to be foundin the following chapters. In this chapter the lift, drag, side-wind components, andassociated moments of aerodynamic force are conventionally identified, the applica-tion of dimensional theory establishing their coefficient form. The significance ofthe pressure distribution around an aerodynamic body and the estimation of lift,drag, and pitching moment on the body in flight completes the basic concepts anddefinitions.

1.2 UNITS AND DIMENSIONSMeasurement and calculation require a system of units in which quantities are mea-sured and expressed. Aerospace is a global industry, and to be best prepared for aglobal career, engineers need to be able to work in both systems in use today. Evenwhen one works for a company with a strict standard for use of one set of units,customers, suppliers, and contractors may be better versed in another, and it is theengineer’s job to efficiently reconcile the various documents or specifications with-out introducing conversion errors. Consider, too, the physics behind the units. Thatis, one knows that for linear motion, force equals the product of mass and accelera-tion. The units one uses do not change the physics but change only our quantitativedescriptions of the physics. When confused about units, focus on the process or statebeing described and step through the analysis, tracking units the entire way.

In the United States, “Imperial” or “English” units remain common. Distance(within the scale of an aerodynamic design) is described in inches or feet. Mass isdescribed by either the slug or the pound-mass (lbm). Weight is described by pounds(lb) or by the equivalent unit with a redundant name, the pound-force (lbf). Largedistances—for example, the range of an aircraft—are described in miles or nauticalmiles. Speed is feet per second, miles per hour, or knots, where one knot is onenautical mile per hour. Multimillion dollar aircraft are still marketed and sold usingknots and nautical miles (try a web search on “777 range”), so these units are notobsolete.

In other parts of the world, and in K-12 education in the United States, the domi-nant system of units is the Système International d’Unités, commonly abbreviated as“SI units.” It is used throughout this book, except in a very few places as speciallynoted.

It is essential to distinguish between “dimension” and “unit.” For example, thedimension “length” expresses the qualitative concept of linear displacement, or dis-tance between two points, as an abstract idea, without reference to actual quantitativemeasurement. The term “unit” indicates a specified amount of a quantity. Thus ameter is a unit of length, being an actual “amount” of linear displacement, and so is




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a mile. The meter and mile are different units, since each contains a different amountof length, but both describe length and therefore are identical dimensions.2

Expressing this in symbolic form:

x meters = [L] (a quantity of x meters has the dimension of length)x miles = [L] (a quantity of x miles has the dimension of length)x meters 6= x miles (x miles and x meters are unequal quantities of length)[x meters] = [x miles] (the dimension of x meters is the same as the dimension ofx miles).

1.2.1 Fundamental Dimensions and UnitsThere are five fundamental dimensions in terms of which the dimensions of all otherphysical quantities may be expressed. They are mass [M], length [L], time [T], tem-perature [θ ], and charge.3 (Charge is not used in this text so is not discussed further.)A consistent set of units is formed by specifying a unit of particular value for each ofthese dimensions. In aeronautical engineering the accepted units are, respectively, thekilogram, the meter, the second, and the Kelvin or degree Celsius. These are identi-cal with the units of the same names in common use and are defined by internationalagreement.

It is convenient and conventional to represent the names of these units byabbreviations:

kg—kilogram, slugs for slugs, and lbm for pound-massm—meter and ft for feets—second◦C—degree Celsius and ◦F for degree FahrenheitK—Kelvin and R for Rankine (but also for the gas constant)

The degree Celsius is one one-hundredth part of the temperature rise involvedwhen pure water at freezing temperature is heated to boiling temperature at standardpressure. In the Celsius scale, pure water at standard pressure freezes at 0◦C(32◦F)and boils at 100◦C(212◦F).

The unit Kelvin (K) is identical in size to the degree Celsius (◦C), but the Kelvinscale of temperature is measured from the absolute zero of temperature, which isapproximately –273◦C. Thus a temperature in K is equal to a temperature in ◦C plus273.15. Similarly, degrees Rankine equals ◦F plus 459.69.

2Quite often “dimension” appears in the form “a dimension of 8 meters,” meaning a specified length.This is thus closely related to the engineer’s “unit,” and implies linear extension only. Another commonexample of the use of “dimension” is in “three-dimensional geometry,” implying three linear extensionsin different directions. References in later chapters to two-dimensional flow, for example, illustrate this.The meaning here must not be confused with either of these uses.3Some authorities express temperature in terms of length and time. This introduces complications thatare briefly considered in Section 1.3.8.


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1.2.2 Fractions and MultiplesSometimes, the fundamental units just defined are inconveniently large or inconve-niently small for a particular case. If so, the quantity can be expressed as a fractionor multiple of the fundamental unit. Such multiples and fractions are denoted by aprefix appended to the unit symbol. The prefixes most used in aerodynamics are:

M (mega)—1 millionk (kilo)—1 thousandm (milli)—1-thousandth partµ (micro)—1-millionth partn (nano)—1-billionth part

Thus

1 MW = 1,000,000 W1 mm = 0.001 m1 µm = 0.001 mm

A prefix attached to a unit makes a new unit so, for example,

1mm2 = 1(mm)2 = 10−6m2(

not 10−3m2)

For some purposes, the hour or the minute can be used as the unit of time.For Imperial units, everyday scientific notation is used rather than suffixes or

prefixes. One exception is stress or pressure of thousands of pounds per square inch,known as kpsi. Additionally, length may switch from feet to inches or miles. It iscommon to use fractional inches, but the student engineer needs to be aware thatthe implied precision in a fraction increases rapidly. For example, 1/2= 0.5, but1/32= 0.03125.

1.2.3 Units of Other Physical QuantitiesHaving defined the four fundamental dimensions and their units, it is possible toestablish units of all other physical quantities (see Table 1.1). Speed, for example, isdefined as the distance traveled in unit time. It therefore has the dimension LT−1 andis measured in meters per second (ms−1). It is sometimes desirable to use kilometersper hour or knots (nautical miles per hour; see Appendix D) as units of speed; caremust be exercised to avoid errors of consistency.

To find the dimensions and units of more complex quantities, we use the principleof dimensional homogeneity. This simply means that, in any valid physical equation,the dimensions of both sides must be the same. Thus, for example, if (mass)n appearson the left-hand side of the equation, it must also appear on the right-hand side;similarly for length, time, and temperature.

Thus, to find the dimensions of force, we use Newton’s second law of motion

Force = mass × acceleration


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Table 1.1 Units and Dimensions

Quantity Dimension Unit (abbreviation)

Length L Meter (m) or feet (ft)

Mass M Kilogram (kg) or slug or pound-mass(lbm)

Time T Second (s)

Temperature θ Degree Celsius (◦C) or Fahrenheit (◦F) orKelvin (K) or Rankine (R)

Area L2 Square meter (m2) or square foot (ft2)

Volume L3 Cubic meter (m3) or cubic foot (ft3)

Speed LT−1 Meters per second (m s−1) or feet persecond (ft s−1)

Acceleration LT−2 Meters per second per second (m s−2) orfeet per second squared (ft s−2)

Angle 1 Radian or degree (◦) (radian is expressedas a ratio and is therefore dimensionless)

Angular velocity T−1 Radians per second (s−1)

Angular acceleration T−2 Radians per second per second (s−2)

Frequency T−1 Cycles per second, Hertz (s−1, Hz)

Density ML−3 Kilograms per cubic meter (kg m−3) orslugs per cubic foot (slug ft−3) orpound-mass per cubic foot (lbm ft−3)

Force MLT−2 Newton (N) or pound (lb)

Stress ML−1T−2 Newtons per square meter or Pascal(N m−2 or Pa) or pounds per square inch(psi) or pounds per square foot (psf)

Strain 1 None (expressed as a nondimensionalratio)

Pressure ML−1T−2 Newtons per square meter or Pascal(N m−2 or Pa) or pounds per square inch(psi) or pounds per square foot (psf)

Energy work ML2T−2 Joule (J) or foot-pounds (ft lb)

Power ML2T−3 Watt (W) or horsepower (Hp)

Moment ML2T−2 Newton meter (Nm) or foot-pounds, (ft lb)

Absolute viscosity ML−1T−1 Kilograms per meter per second orPoiseuilles (kg m−1 s−1 or PI) or slugs perfoot per second (slug ft−1 s−1)

Kinematic viscosity L2T−1 Meters squared per second (m2 s−1) or feetsquared per second (ft2 s−1)

Bulk elasticity ML−1T−2 Newtons per square meter or Pascal(N m−2 or Pa) or pounds per square inch(psi) or pounds per square foot (psf).


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where acceleration is speed ÷ time. Expressed dimensionally, this is

Force= [M]×

[L

T÷T

]=

[MLT−2

]Writing in the appropriate units, it is seen that a force is measured in units of kgms−2.Since, however, the unit of force is given the name Newton (abbreviated usually toN), it follows that

1N= 1kgms−2

It should be noted that there can be confusion between the use of m bothfor “milli” and for “meter.” This is avoided by use of a space. Thus ms denotesmillisecond while m s denotes the product of meter and second.

The concept of dimension forms the basis of dimensional analysis, which is usedto develop important and fundamental physical laws. Its treatment is postponed toSection 1.5.

1.2.4 Imperial UnitsEngineers in some parts of the world, the United States in particular, use a set of unitsbased on the Imperial systems4 in which the fundamental units are

Mass—slugLength—footTime—secondTemperature—degree Fahrenheit or Rankine

AERODYNAMICS AROUND USUnits in UseStudents have long struggled with learning to use units correctly. The danger is that it is simple tocreate large quantitative errors when even an experienced engineer is hurrying through an analysistask. That the SI system is supposedly “easier” to use than the Imperial system is irrelevant if you aregoing to commit errors—the only difference between the systems will be how large those errors are.There is no doubt that all humans are fallible. The relevant question is: How can one work with unitscorrectly all of the time? Your pursuit of excellence may be aided by a short review of the basics ofunit conversion.

Engineers are wise to remember that the mathematical symbol stating equality between twoquantities (the “equals” sign) relates not just numerical equivalence but dimensional equivalence aswell. That 5280 feet equals 1 mile is a statement of equivalence between two descriptions of thesame distance. The same is true for 100 cm= 1 m. But what about equalities between systems ofunits, such as how many centimeters equal one inch? We all learn that 2.54 cm= 1 in, but is this

4Since many valuable texts and papers exist using Imperial units, this book contains, as Appendix D, atable of factors for converting from the Imperial to the SI system.


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exact? Yes (see the NIST handbook discussed in a moment for details). Does 1 m= 39.37 in? No.(Can you show this to be the case?)

However, for anything that actually flies, error exists because we can only build and measure withfinite precision. Thus many unit conversions—such as 1 m= 39.37 in, 1.15078 mi= 1nm (that is,nautical mile, not nanometers), or 1 slug = 32.174 lb—are sufficiently precise for use throughoutaerodynamics. The modern student or engineer can find authoritative values at the web site of theUnited States’ National Institute of Science and Technology (NIST), such as in Appendix C of NISTHandbook 44-2010.

Consider a wing chord of 3.50 meters, which we need to convert to inches. How should weproceed? A quick search on your smartphone tells you 1m∼= 39.37 in. Yet, for the sake of education,let’s work through this from the exact relation, 2.54 cm = 1 in. First, convert meters to centimetersusing (100 cm/1 m)= 1:

3.50 m= 3.50 m × 1

3.50 m= 3.50 m×

(100 cm

1 m

)3.50 m= 350 cm

Note that the units of meter in the original description, 3.5 m, cancel the units of meter in thedenominator of the ratio, leaving units of centimeters on the right. Knowing that 2.54 cm= 1 in, wecan form another unit (as in magnitude of one) ratio, 1= (1 in/2.54 cm). Multiplying 350 cm by 1does not change the length:

3.50 m= 350 cm× 1

3.50 m= 350 cm×

(1 in

2.54 cm

)3.50 m=

350

2.54in≈ 137.795 in≈ 138.0 in

Remember to review the concept of significant digits, such as in an introductory physics text. Again,a length on the left side of the equality requires a length on the right side, plus proper numericalcomputation.

Most common among questions about units that confound students each semester involves therelationships between units of mass and force or weight and the physical difference between massand weight. The latter isn’t a question of units at all but a fundamental physics question that requiresimmediate answer for the student to excel. High school physics teachers teach the differencebetween mass and weight, but, through no fault of theirs, students generally seem to require severalyears of thinking about the issue before it is clarified. Consider an astronaut launching into orbit.While on the ground, you know that the astronaut’s weight W is computed from his or her massM= 80 kg and the Earth-surface value of the acceleration due to gravity, go = 9.81m/s2,

W =Mg= 80 kg× 9.8ms−2 = 784kgms−2 = 784N

where N denotes Newtons. This equation does not equate mass and weight but says that weight ishow we describe the effect of Earth’s gravity on a mass. Mass is an intrinsic property of the countlesssubatomic particles that make up the atoms that make up the molecules of the astronaut. When theastronaut is on the ground, her particles all have mass and her mass is the sum of the masses of allthose particles. When the astronaut is launched into orbit, she may feel weightless because of thecentripetal acceleration of the orbital path, but all of the particles have the same masses they had onthe Earth’s surface, and the sum of those is still the mass of the astronaut.

Note that Newton’s first law, which, as we know, is force equals the product of mass andacceleration (F=ma), relates the units of mass and weight. For example, using the numerical values


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from the weight example, we know that

784N= (80kg)×(

9.8ms−2)

Isolate the units on the left and the numbers on the right:

N

kgms−2=

80× 9.8

784

or

N

kgms−2= 1

This shows the common definition in a format that students often lose sight of when working ontasks. That is, if

N

kgms−2= 1

then any occurrence of Newton may be replaced by the product of kilogram, meter, and inverseseconds squared. Thus, while weight (or force) and mass are different, their units are related, and wecan use this relationship in analysis.

This elementary discussion can help the student in working with force (or weight) and mass.Weight is one specific type of force—the action of gravity on mass. Weight is not mass; 1 kilogramweighs 9.8 Newtons, but it does not equal 9.8 Newtons. One slug weighs 32.174 pounds, but doesnot equal 32.174 pounds because weight (force) and mass are two different things. You cannotequate slugs and pounds any more than you can equate meters and Coulombs.

The kg-m-s and slug-ft-s systems of units are identical in their use in aerodynamics (use anelectrodynamics text for lessons when working in that field). In other words,

N

kgms−2= 1

and

lb

slugft s−2= 1

so

N

kgms−2=

lb

slugft s−2

One slug does equal 32.174 pound-mass (lbm) because the pound-mass is a unit of mass. Whereveryou see the slug, you can replace it with 32.174 lbm. Thus,

N

kgms−2=

lb

slugft s−2= 1=

lbf

32.174lbmfts−2

where pounds are now called pound-force, or lbf, presumably to make a clear distinction betweenlbf and lbm.

There are good and bad units practices around us. For example, one may see an equation forpressure drop in a system given as “The pressure drop across the device, 1p, is given by1p= 17.34Q2 for Q in gallons per minute and pressure in psi.” The choice in how to present the


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units information may be fine for a technician who will be sizing specific items or determining if aspecific design will meet a specification. Where this statement is poor practice is in engineeringreports where you will document and communicate your results to your project team and to thepeople who will follow you. Consider that today you can fly on commercial airliners whose designsare older than you are (Boeing 737 and 767 in this country and the 727 elsewhere in the world).Engineering knowledge is passed on over the decades in reports, not by word of mouth. A properway to report the previous pressure loss relation for a device would be to include the units in theequation:

“The pressure drop across the device, 1p, is given by

1p=

(17.34psi

gpm2

)Q2”

Then anyone who uses the equation in their own analysis or computer model will be able to convertrapidly into the units that they wish. It is a good habit in communication to keep the units in theequation, not in an accompanying sentence, especially in this age of computer cut-and-paste.

The authors agree with what is likely running through a student’s head, that these different unitsystems are a strange way to run an industry. History has led us to where we are. The student aimingfor success in the global aerospace industry, in either atmospheric flight or spaceflight, will be wiseto practice working problems in all set of units. Focus on the physics involved and write down all theunits in your analysis.

1.3 RELEVANT PROPERTIESAny fluid that we wish to describe exists in some state of matter. For example, if weare working with a flow of nitrogen, is it gaseous nitrogen or liquid nitrogen? Forwhatever the state the fluid is in, we need a collection of “tools” to use to describethe thermodynamic state of the fluid at a point, over time, and throughout a field.An unambiguous description of the thermodynamic state of the fluid is important ofcourse to a mathematical model of a flow and is vital to effective engineering commu-nication. Thus in this section we develop the tools to use to form these unambiguousdescriptions.

1.3.1 Forms of MatterMatter may exist in three principal forms—solid, liquid, or gas—corresponding inthat order to decreasing rigidity of the bonds between the molecules the matter com-prises. A special form of a gas, a plasma, has properties different from those of anormal gas; although belonging to the third group, it can be regarded justifiably asa separate, distinct form of matter that is relevant to the highest-speed aerodynamicssuch as flows over spacecraft reentering the atmosphere.

In a solid the intermolecular bonds are very rigid, maintaining the molecules inwhat is virtually a fixed spatial relationship. Thus a solid has a fixed volume and


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1.3 Relevant Properties 13

shape. This is seen clearly in crystals, in which the molecules or atoms are arrangedin a definite, uniform pattern, giving all crystals of that substance the same geometricshape.

A liquid has weaker bonds between its molecules. The distances between themolecules are fairly rigidly controlled, but the arrangement in space is free. There-fore, liquid has a closely defined volume but no definite shape, and may accommodateitself to the shape of its container within the limits imposed by its volume.

A gas has very weak bonding between the molecules and therefore has neitherdefinite shape nor definite volume, but rather will fill the vessel containing it.

A plasma is a special form of gas in which the atoms are ionized—that is, theyhave lost or gained one or more electrons and therefore have an electrical charge.Any electrons that have been stripped from the atoms are wandering free within theplasma and have a negative electrical charge. If the number of ionized atoms and freeelectrons is such that the total positive and negative charges are approximately equal,so that the gas as a whole has little or no charge, it is termed a plasma. In astronauticsplasma is of particular interest for the reentry of rockets, satellites, and space vehiclesinto the atmosphere.

1.3.2 FluidsA fluid is a liquid or a gas. Equations of motion for a fluid do not depend on liq-uid or gas, but the equation of state will differ. The basic feature of a fluid is thatit can flow—this is the essence of any definition of it. However, flow applies tosubstances that are not true fluids—for example a fine powder piled on a slopingsurface will flow. For example, flour poured in a column onto a flat surface willform a roughly conical pile, with a large angle of repose, whereas water, which is atrue fluid, poured onto a fully wetted surface will spread uniformly over it. Equally,a powder may be heaped in a spoon or bowl, whereas a liquid will always form alevel surface. Any definition of a fluid must allow for these facts, so a fluid maybe defined as “matter capable of flowing, and either finding its own level (if a liq-uid), or filling the whole of its container (if a gas).” Once we restrict ourselves toan ideal gas, such as for steady, level atmospheric flight, distinctions between airas a “Newtonian fluid” and fine particulates are clear. A Newtonian fluid is one inwhich shear stress is proportional to rate of shearing strain; this is never found inparticulates.

Experiment shows that an extremely fine powder, in which the particles are notmuch larger than molecular size, finds its own level and may thus come underthe common definition of a liquid. Also, a phenomenon well known in the trans-port of sands, gravels, and so forth, is that these substances find their own levelif they are agitated by vibration or the passage of air jets through the particles.These are special cases, however, and do not detract from the authority of thedefinition of a fluid as a substance that flows or (tautologically) that possessesfluidity.


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1.3.3 PressureAt any point in a fluid, whether liquid or gas, there is a pressure. If a body is placed ina fluid, its surface is bombarded by a large number of molecules moving at random.Under normal conditions the collisions on a small area of surface are so frequent thatthey cannot be distinguished as individual impacts but appear as a steady force on thearea. The intensity of this “molecular bombardment” is its static pressure.

Very frequently the static pressure is referred to simply as pressure. The termstatic is rather misleading as it does not imply that the fluid is at rest.

For large bodies moving or at rest in the fluid (e.g., air), the pressure is not uniformover the surface, and this gives rise to aerodynamic or aerostatic force, respectively.

Since a pressure is force per unit area, it has the dimensions

[Force]÷ [area]= [MLT−2]÷ [L2]= [ML−1T−2]

and is expressed in units of Newtons per square meter or in Pascals (N m−2 or Pa).Pressure is also commonly specified in pounds per square inch (psi) or pounds persquare foot (psf). It can also be of use to consider the above equation multiplied bylength over length:

[Force] ∗ [Length]/([Area] ∗ [Length])= [ML2T−2]/[L3]= [Energy]/[Volume]

Thus, besides the most common view of it as a force per area, pressure also has unitsof energy per volume.

Pressure in Fluid at RestConsider a small cubic element containing fluid at rest in a larger bulk of fluid alsoat rest. The faces of the cube, assumed conceptually to be made of some thin flexiblematerial, are subject to continual bombardment by the molecules of the fluid and thusexperience a force. The force on any face may be resolved into two components, oneacting perpendicular to the face and the other along it (i.e., tangential to it). Con-sider the tangential components only; there are three significantly different possiblearrangements (Fig. 1.1). System (a) would cause the element to rotate, and thus thefluid would not be at rest; system (b) would cause the element to move (upward andto the right for the case shown), and, once again, the fluid would not be at rest. Sincea fluid cannot resist shear stress but only rate of change in shear strain (Sections 1.3.6and 2.7.2), system (c) would cause the element to distort, the degree of distortionincreasing with time, and the fluid would not remain at rest. The conclusion is that afluid at rest cannot sustain tangential stresses.

Pascal’s LawConsider the right prism of length δz in the direction into the page and cross-sectionABC, the angle ABC being a right angle (Fig. 1.2). The prism is constructed of mate-rial of the same density as a bulk of fluid in which the prism floats at rest with theface BC horizontal.


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(a) (b) (c)

FIGURE 1.1

Fictitious systems of tangential forces in static fluid.

B

A

C

p2

p1

p3

δx

α

FIGURE 1.2

Prism for Pascal’s Law.

Pressures p1, p2, and p3 act on the faces shown and, as just proved, act in thedirection perpendicular to the respective face. Other pressures act on the end faces ofthe prism, but are ignored in the present problem. In addition to these pressures, theweight W of the prism acts vertically downward. Consider the forces acting on thewedge that is in equilibrium and at rest.

Resolving forces horizontally,

p1(δx tanα)δy− p2(δxsecα)δysinα = 0

Dividing by δx δy tan α, this becomes

p1− p2 = 0

that is,

p1 = p2 (1.1)


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Resolving forces vertically,

p3δxδy− p2(δxsecα)δycosα−W = 0 (1.2)

Now

W = ρg(δx)2 tanαδy/2

Therefore, substituting this in Eq. (1.2) and dividing by δx δy,

p3− p2−1

2ρg tanαδy= 0

If now the prism is imagined to become infinitely small, so that δx→ 0, the thirdterm tends to zero, leaving

p3− p2 = 0

Thus, finally,

p1 = p2 = p3 (1.3)

Having become infinitely small, the prism is in effect a point, so this analysisshows that, at a point, the three pressures considered are equal. In addition, the angleα is purely arbitrary and can take any value, while the whole prism can be rotatedthrough a complete circle about a vertical axis without affecting the result. It may beconcluded, then, that the pressure acting at a point in a fluid at rest is the same in alldirections.

1.3.4 TemperatureIn any form of matter the molecules are in motion relative to each other. In gases themotion is random movement of magnitude ranging from approximately 60 nm undernormal conditions to some tens of millimeters at very low pressures. The distance offree movement of a molecule of gas is the distance it can travel before colliding withanother molecule or the walls of the container. The mean value of this distance for allmolecules in a gas is called the length of the mean molecular free path.

By virtue of this motion, the molecules possess kinetic energy, and this energy issensed as the temperature of the solid, liquid, or gas. In the case of a gas in motionit is called the static temperature or, more usually, just the temperature. Temperaturehas the dimension [θ ] and the units K, ◦C, ◦F, or ◦R (Section 1.1). In practically allcalculations in aerodynamics, temperature is measured in K or ◦R (i.e., from absolutezero).

1.3.5 DensityThe density of a material is a measure of the amount of the material contained in agiven volume. In a fluid the density may vary from point to point. Consider the fluidcontained in a small region of volume δV centered at some point in the fluid, and let


Houghton — Ch01-9780080966328 — 2012/2/3 — 21:13 — Page 17 — #17


the mass of fluid within this spherical region be δm. Then the density of the fluid atthe point on which the volume is centered is defined by

Density ρ = limδv→0

δm

δV(1.4)

The dimensions of density are thus ML−3, and density is measured in units of kilo-gram per cubic meter (kg m−3). At standard temperature and pressure (288 K, 101,−325 N m−2), the density of dry air is 1.2256 kg m−3.

Difficulties arise in rigorously applying the definition given a real fluid composedof discrete molecules, since the volume, when taken to the limit, either will or willnot contain part of a molecule. If it does contain a molecule, the value obtained forthe density will be fictitiously high. If it does not contain a molecule, the resultantvalue will be zero. This difficulty is generally avoided in the range of temperaturesand pressures normally encountered in aerodynamics because the molecular nature ofa gas may for many purposes—in fact, for nearly every terrestrial flight application—be ignored and the assumption made that the fluid is a continuum—that is, it does notconsist of discrete particles. This “continuum assumption” suffices because the meanfree path of the molecular motion is much less than the smallest length scale on thevehicle for almost every atmospheric flight regime.

1.3.6 ViscosityViscosity is regarded as the tendency of a fluid to resist sliding between layers or,more rigorously (as explained later) a rate of change in shear strain. There is verylittle resistance to the movement of a knife blade edge-on through air, but to producethe same motion through thick oil requires much more effort. This is because theviscosity of oil is high compared with that of air.

Dynamic ViscosityThe dynamic (more properly, coefficient of dynamic, or absolute) viscosity is a directmeasure of the viscosity of a fluid. Consider two parallel flat plates placed a distanceh apart, with the space between them filled with fluid. One plate is held fixed, andthe other is moved in its own plane at a speed V (see Fig. 1.3). The fluid immediatelyadjacent to each plate will move with it (i.e., there is no slip). Thus the fluid in contactwith the lower plate will be at rest while that in contact with the upper plate will bemoving with speed V . Between the plates the speed of the fluid will vary linearly, asshown in Fig. 1.3, in the absence of other influences. As a direct result of viscosity,a force F has to be applied to each plate to maintain the motion, the fluid tending toretard the moving plate and drag the fixed plate to the right. If the area of fluid incontact with each plate is A, the shear stress is F/A. The rate of shear strain causedby the upper plate sliding over the lower is V/h.


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V F

F

h

FIGURE 1.3

Simple flow geometry to create a uniform sear.

These quantities are connected by Newton’s equation, which serves to define thedynamic viscosity µ:

F

A= µ

(V

h

)(1.5)

Hence

[ML−1T−2]= [µ][LT−1L−1]= [µ][T−1]

Thus

[µ]= [ML−1T−1]

and the units of µ are therefore kg m−1 s−1; in the SI system the name Poiseuille (Pl)has been given to this combination of fundamental units. At 0◦C (273 K) the dynamicviscosity for dry air is 1.714× 10−5 kg m−1 s−1.

Note that while the relationship of Eq. (1.5) with constant µ applies nicely toaerodynamics, it does not apply to all fluids. For an important class of fluids, whichincludes blood, some oils, and some paints, µ is not constant but is a function ofV/h—that is, the rate at which the fluid is shearing. Numerous classes of “non-Newtonian fluids” are important in fields outside of aerodynamics, and the eagerstudent can explore these best with good knowledge of Newtonian fluid behavior asdiscussed in this book.

Kinematic ViscosityThe kinematic viscosity (or, more properly, the coefficient of kinematic viscosity) isa convenient form in which the viscosity of a fluid may be expressed. It is formed bycombining the density ρ and the dynamic viscosity µ according to the equation

v=µ

ρ

and has the dimensions L2T−1 and the units m2 s−1. It may be regarded as a mea-sure of the relative magnitudes of fluid viscosity and inertia and has the practical


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advantage, in calculations, of replacing two values representing µ and ρ with a singlevalue.

1.3.7 Speed of Sound and Bulk ElasticityBulk elasticity is a measure of how much a fluid (or solid) will be compressed by theapplication of external pressure. If a certain small volume V of fluid is subjected toa rise in pressure δp, this reduces the volume by an amount –δV . In other words, itproduces a volumetric strain of –δV/V . Accordingly, bulk elasticity is defined as

K =−δp

δV/V=−V

dp

dV(1.6a)

The volumetric strain is the ratio of two volumes and is evidently dimensionless, sothe dimensions of K are the same as those for pressure: ML−1T−2. The SI unit isNm−2 (or Pa) and the Imperial unit is psi. When written in terms of density of the airrather than volume, Eq. (1.6a) becomes

K = ρdp

dρ

The propagation of sound waves involves alternating compression and expansionof the medium. Accordingly, bulk elasticity is closely related to the speed of sound aas follows:

a=

√K

ρ(1.6b)

Let the mass of the small volume of fluid be M; then by definition the density ρ =M/V . By differentiating this definition, keeping M constant, we obtain

dρ =−M

V2dV =−ρ

dV

V

Therefore, combining this with Eqs. (1.6a) and (1.6b), it can be seen that

a=

√dp

dρ(1.6c)

The propagation of sound in a perfect gas is regarded as a lossless process; that is,no energy is lost and the wave process lacks heat transfer to or from the surroundingfluid. Accordingly (see the passage on Entropy to come), the pressure and density arerelated by Eq. (1.24), so for a perfect gas, for which P= ρRT ,

a=

√γ p

ρ=√γRT (1.6d)


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where γ is the ratio of the specific heats and R is the specific gas constant for thatgas. Eq. (1.6d) is the formula normally used to determine the speed of sound in gasesfor aerodynamics applications.

1.3.8 Thermodynamic PropertiesHeat, like work, is a form of energy transfer. Consequently, it has the same dimen-sions as energy (i.e., ML2T−2) and is measured in Joules (J) or foot-pounds(ft-lb).

Specific HeatThe specific heat of a material is the amount necessary to raise the temperature of aunit mass of the material by one degree. Thus it has the dimensions L2T−2θ−1 andis measured in SI units of J kg−1 K−1. Imperial units of ft-lb slug−1 ◦F−1 or ft-lbslug−1 ◦R−1 are most common.

There are countless ways in which gas may be heated. Two important and dis-tinct ways are at constant volume and at constant pressure. These define importantthermodynamic properties of the gas.

Specific Heat at Constant VolumeIf a unit mass of the gas is enclosed in a cylinder sealed by a piston, and the pistonis locked in position, the volume of the gas cannot change. It is assumed that thecylinder and piston do not receive any of the heat. The specific heat of the gas underthese conditions is the specific heat at constant volume cV . For dry air at normalaerodynamic temperatures, cV = 718 J kg−1 K−1 = 4290 ft-lb slug−1 ◦R−1.

Internal energy (E) is a measure of the kinetic energy of the molecules that makeup the gas, so

internal energy per unit mass E = cVT

or more generally

cV =

[∂E

∂T

]ρ

(1.7)

Specific Heat at Constant PressureAssume that the piston just referred to is now freed and acted on by a constant force.The pressure of the gas is that necessary to resist the force and is therefore constantas well. The application of heat to the gas causes its temperature to rise, which leadsto an increase in its volume in order to maintain the constant pressure. Thus thegas does mechanical work against the force, so it is necessary to supply the heatrequired to increase its temperature (as in the case at constant volume) as well as heatequivalent to the mechanical work done against the force. This total amount is calledthe specific heat at constant pressure cp and is defined as that amount required to raise


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the temperature of a unit mass of the gas by one degree, the pressure of the gas beingkept constant while heating. Therefore, cp is always greater than cV . For dry air atnormal aerodynamic temperatures, cp = 1005 J kg−1 K−1 = 6006 ft-lb slug−1 ◦R−1.

The sum of internal energy per unit mass and pressure energy per unit mass isknown as enthalpy (h per unit mass) (discussed momentamly). Thus

h= cpT

or more generally

cp =

[∂h

∂T

]p

(1.8)

Ratio of Specific HeatsThe ratio of specific heats is a property important in high-speed flows and is definedby the equation

γ =cpcV

(1.9)

(The value of γ for air depends on the temperature, but for much of practical aero-dynamics it may be regarded as constant at about 1.403. This value is often in turnapproximated to γ = 1.4, which is in fact the theoretical value for an ideal diatomicgas.)

EnthalpyThe enthalpy h of a unit mass of gas is the sum of the internal energy E and thepressure energy p × 1/ρ. Thus

h= E+ p/ρ (1.10)

Enthalpy may be a new term to many students, but it is simply a tool for keeping trackof a sum of two energies. It is not an exotic new property, but it is an energy. Fromthe definition of specific heat at constant volume, Eq. (1.7), Eq. (1.10) becomes

h= cVT + p/ρ

Again from the definition in Eq. (1.8), Eq. (1.10) gives

cpT = cVT + p/ρ (1.11)

Now the pressure, density, and temperature are related in the equation of state, whichfor perfect gases takes the form

p/(ρT) = constant = R (1.12)


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Substituting for p/ρ in Eq. (1.11) yields the relationship

cp− cV = R (1.13)

The specific gas constant R is thus the amount of mechanical work obtained byheating the unit mass of a gas through a unit temperature rise at constant pressure. Itfollows that R is measured in units of J kg−1 K−1. For air over the range of temper-atures and pressures normally encountered in aerodynamics, R has the value 287.26J kg−1 K−1, or 1716.6 ft-lb slug−1 R−1.

Introducing the ratio of specific heats (Eq. 1.9), the following expressions areobtained:

cp =γ

γ − 1R and cV =

R

γ − 1(1.14)

Replacing cVT by [l/(γ – 1)]p/ρ in Eq. (1.11) readily gives the enthalpy as

cpT =γ

γ − 1

p

ρ(1.15)

It is often convenient to link the enthalpy or total heat to the other energy ofmotion. This would be kinetic energy K̄ per unit mass of gas moving with meanvelocity V:

K̄ =V2

2(1.16)

Thus the total energy flux in the absence of external, tangential surface forces andheat conduction becomes

V2

2+ cpT = cpT0 = constant (1.17)

where, with cp invariant, T0 is the absolute temperature when the gas is at rest.The quantity cpT0 is referred to as total or stagnation enthalpy. This quantity is animportant parameter of the equation for the conservation of energy.

Applying the first law of thermodynamics to the flow of non-heat-conductinginviscid fluids gives

d(cVT)

dt+ p

d(1/ρ)

dt= 0 (1.18)

Further, if the flow is unidirectional and cVT = E, Eq. (1.18) becomes, on can-celling dt,

dE+ pd

(1

ρ

)= 0 (1.19)


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However, differentiating Eq. (1.10) gives

dh= dE+ pd

(1

ρ

)+

1

ρdp (1.20)

Combining Eqs. (1.19) and (1.20), we get

dh=1

ρdp (1.21)

but

dh= cpdT =cpR

d

(p

ρ

)=

γ

γ − 1

[1

ρdp+ pd

(1

ρ

)](1.22)

which, together with Eq. (1.21), gives the identity

dp

p+ γρd

(1

ρ

)= 0 (1.23)

Integrating gives

ln p+ γ ln

(1

ρ

)= constant

or

p= kργ (1.24)

which is the isentropic relationship between pressure and density.Note that this result is obtained from the equation of state for a perfect gas and

from the equation of conservation of energy of the flow of a non-heat-conductinginviscid fluid. Such a flow behaves isentropically and, notwithstanding the apparentlyrestrictive nature of the assumptions made, can be used as a model for a great manyaerodynamic applications.

EntropyEntropy is a function of state that follows from, and indicates the working of, thesecond law of thermodynamics, which is concerned with the direction of any processinvolving heat and energy. Any increase in the entropy of the fluid as it experiences aprocess is a measure of the energy no longer available to the system. Negative entropychange is possible when work is performed on a system or heat is removed. Zeroentropy change indicates an ideal or completely adiabatic and reversible process, andwe call such a constant entropy process an isentropic process.


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By definition, specific entropy (S)5 (Joules per kilogram per Kelvin) is given bythe integral

S=∫

dQ

T(1.25)

for any reversible process, with the integration extending from some datum condition;however, as we saw earlier, it is the change in entropy that is important:

dS=dQ

T(1.26)

In this and the previous equation, dQ is a heat transfer to a unit mass of gas from anexternal source. This addition will change the internal energy and do work.

Thus, for a reversible process,

dQ= dE+ pd

(1

ρ

)dS=

dQ

T=

cVdT

T+

pd(1/ρ)

T(1.27)

but p/T = Rρ ; therefore,

dS=cVdT

T+

Rd(1/ρ)

1/ρ(1.28)

Integrating Eq. (1.28) from datum conditions to conditions given by suffix 1,

S1 = cV lnT1TD+R ln

ρD

ρ1

Likewise,

S2 = cV lnT2TD+R ln

ρD

ρ2

The entropy change from condition 1 to condition 2 is given by

1S= S2− S1 = cV lnT2T1+R ln

ρ1

ρ2(1.29)

With the use of Eq. (1.14) this is more usually rearranged to a nondimensional form:

1S

cV= ln

T2T1+ (γ − 1) ln

ρ1

ρ2(1.30)

5Note that here the unconventional symbol S is used for specific entropy to avoid confusion with lengthsymbols.


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1.4 Aeronautical Definitions 25

or to the exponential form:

e1S/cV =T2T1

(ρ1

ρ2

)γ−1(1.31)

Alternatively, for example, using the equation of state,

e1S/cV =

(T2T1

)γ (p1p2

)γ−1(1.32)

These latter expressions are useful in particular problems.

1.4 AERONAUTICAL DEFINITIONS1.4.1 Airfoil GeometryIf a horizontal wing is cut by a vertical plane parallel to the centerline, the shape ofthe resulting section is usually like that Fig. 1.4. This is an airfoil section, which forsubsonic use almost always has a rounded leading edge (early stealth designs being

Symmetrical fairing(a)

Camber line

(b)

Upper surface

Lower surfaceCambered aerofoil

(c)

Chord line

y

x

yU

yU= ys+ ycyL= ys− yc

yc

yL

ys

ys

FIGURE 1.4

Airfoil (wing section) geometry and definitions.


Houghton — Ch01-9780080966328 — 2012/2/3 — 21:13 — Page 26 — #26


the primary exceptions). The thickness increases smoothly to a maximum that usuallyoccurs between one-quarter and halfway along the profile and thereafter tapers offtoward the rear of the section.

If the leading edge is rounded, it is described by a planar curve and therefore hasa definite radius of curvature. It is here that the curvature of the airfoil shape is thegreatest aside from the trailing edge. The trailing edge may be sharp or may also havea very small radius of curvature or bluntness.

Consider a circle that is larger than and contains the airfoil. As its diameter isreduced, the circle will, for some diameter, contact the airfoil at two points only.These are the leading and trailing edges, and the diameter that connects them is thechord line. The length of the chord line is the airfoil chord, denoted c.

The point where the chord line intersects the front (or nose) of the section is usedas the origin of a pair of axes: the x-axis is the chord line; the y-axis is perpendicularto the chord line, positive in the upward direction. The shape of the section is thenusually given as a table of values of x and corresponding values of y. These sectionordinates are usually expressed as percentages of the chord.

CamberAt any distance along the chord from the nose, a point may be marked midwaybetween the upper and lower surfaces. The locus of all such points, usually curved,is the median line of the section and is called the camber line (here the word “line”is sloppy; it does not mean that the camber curve is straight, but it is used throughoutthe industry). The maximum height of the camber line above the chord line is denotedδ, and the quantity δ/c is called the maximum camber of the section. Airfoil sectionshave cambers usually in the range from 0% (a symmetrical section) to 5%, althoughmuch larger cambers are used in cascades (e.g., turbine blades).

Seldom can a camber line be expressed in simple geometric or algebraic forms,although a few simple curves, such as circular arcs or parabolas, have been used.

Thickness DistributionHaving found the median, or camber, line, the distances from it to the upper and lowersurfaces may be measured at any value of x. These distances are, by the definition ofthe camber line, equal. They may be measured at all points along the chord and thenplotted against x from a straight line. The result is a symmetrical shape, called thethickness distribution or symmetrical fairing.

An important parameter of the thickness distribution is the maximum thicknesst, which, when expressed as a fraction of the chord, is called the thickness-to-chordratio and commonly expressed as a percentage. Current values vary tremendouslyas aircraft now fly in many scales, from extreme low-Reynolds-number micro-airvehicles to massive airliners, along with super-cruise fighters and hypersonic flighttest vehicles. However, airfoils with greater than about 18% thickness are rare.

The position along the chord at which maximum thickness occurs is anotherimportant characteristic of the thickness distribution. Values usually lie between 30%and 60% of the chord from the leading edge. For some older sections the value is


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1.4 Aeronautical Definitions 27

about 25% of the chord, whereas for some more extreme sections it is more than60% of the chord behind the leading edge.

Any airfoil section may be regarded as a thickness distribution plotted arounda camber line. American and British conventions differ in the exact derivation ofan airfoil section from a given camber line and thickness distribution. The Britishconvention is to plot the camber line and then plot the thickness ordinates from this,perpendicular to the chord line. Thus the thickness distribution is in effect sheareduntil its median line, initially straight, has been distorted to coincide with the givencamber line. The American convention is to plot the thickness ordinates perpendic-ular to the curved camber line, so the thickness distribution is regarded as bent untilits median line coincides with the given camber line.

Since the camber-line curvature is generally very small, the difference in airfoilsection shape given by these two conventions is also very small.

1.4.2 Wing GeometryThe planform of a wing is its shape seen on a plan (top) view of the aircraft.Fig. 1.5 illustrates this and defines the symbols for the various planform-geometryparameters. Note that the root ends of the leading and trailing edges have been con-nected across the fuselage by straight lines. An alternative to this is to produce theleading and trailing edges, if straight, to the aircraft centerline.

WingspanThe wingspan is the dimension b, the distance between the two wingtips. The distanceb/2 from each tip to the centerline is the wing semi-span.

Leading

edge

Trailing edge

Fus

elag

e si

de

Fus

elag

e si

de Directionof flight

Roo

t

CL

Win

g tip

s

b = 2s

c0

cTΛTE

ΛLE

s

X

X

FIGURE 1.5

Wing planform geometry.


Houghton — Ch01-9780080966328 — 2012/2/3 — 21:13 — Page 28 — #28


ChordsThe two lengths cT and c0 are the tip and root chords, respectively; with the alter-native convention, the root chord is the distance between the intersections with thefuselage centerline of the leading and trailing edges produced. The ratio cT/c0 is thetaper ratio λ. Sometimes the reciprocal of this, c0/cT, is taken as the taper ratio. Formost wings, cT/c0 < 1.

Wing AreaThe plan-area of the wing including its continuation in the fuselage is the gross wingarea SG. The unqualified term wing area S usually means this gross wing area. Theplan-area of the exposed wing (i.e., excluding the continuation in the fuselage) is thenet wing area SN.

Mean ChordsA useful parameter is the standard mean chord or the geometric mean chord, denotedc̄ and defined by c̄= SG/b or SN/b. It should be clear whether SG or SN is used. Thedefinition may also be written as

c̄=

b/2∫−b/2

cdy

b/2∫−b/2

dy

where y is distance measured from the centerline toward the starboard tip (right-handto the pilot). “Standard mean chord” is often abbreviated as “SMC.”

Another mean chord is the aerodynamic mean chord (AMC) which is denoted c̄Aor ¯̄c and is defined by

c̄=

b/2∫−b/2

c2 dy

b/2∫−b/2

cdy

Aspect RatioAspect ratio is a measure of the narrowness of the wing planform. It is denoted ARand is given by

AR=span

area=

b2

c

Note that only for a rectangular wing does AR= b/c.

Wing SweepThe sweep angle of a wing is that between a line drawn along the span at a constantfraction of the chord from the leading edge, and a line perpendicular to the centerline.It is usually denoted 3. Sweep-back is commonly measured on the leading edge


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1.5 Dimensional Analysis 29

Γ Γ

FIGURE 1.6

Dihedral angle.

(3LE), on the quarter-chord line (i.e., the line one-quarter of the chord behind theleading edge (31/4)), or on the trailing edge (3TE).

Dihedral AngleIf an airplane is viewed from directly ahead, it is seen that the wings are generallynot in a single geometric plane but instead inclined to each other at a small angle.Imagine lines drawn on the wings along the locus of the intersections between thechord lines and the section noses, as in Fig. 1.6. Then the angle 20 is the dihedralangle of the wings. If the wings are inclined upward, they are said to have dihedral;if inclined downward, they have anhedral.

Incidence, Twist, Wash-out, and Wash-inWhen an airplane is in flight, the chord lines of the various wing sections are notnormally parallel to the direction of flight. The angle between the chord line of agiven airfoil section and the direction of flight or of the undisturbed stream is thegeometric angle of attack α.

Carrying this concept of incidence to the twist of a wing, it may be said that, ifthe geometric angles of attack of all sections are not the same, the wing is twisted.If the angle of attack increases towards the tip, the wing has wash-in; if it decreasestowards the tip, the wing has wash-out.

1.5 DIMENSIONAL ANALYSIS1.5.1 Fundamental PrinciplesThe theory of dimensional homogeneity has more uses t

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