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1900 1910 1920 1930 1940 1950 1960 1970
Year
Figure 1.31 Typical flight altitudes over the years.
Singled out in Figure 1.30 are the winners of the Schneider Cup races between
1913 and 1931 (with a moratorium during World War I). The Schneider Cup
races were started in 1913 by Jacques Schneider of France as a stimulus to the
development of highspeed float planes. They prompted some early but advanced
development of highspeed aircraft. The winners are shown by the dashed line in
Figure 1.30, for comparison with standard aircraft of the day. Indeed, the winner
of the last Schneider race in 1931 was the Supermarine S.6B, a forerunner of the
famous Spitfire of World War II. Of course, today the maximum speed of flight
has been pushed to the extreme value of 36,000 ft/s, escape velocity from the
earth, by the Apollo lunar spacecraft.
As a companion to speed, the maximum altitudes of typical manned aircraft
are shown in Figure 1.31 as a function of chronological time. The same push to
higher values is evident; so far, the record is the moon in 1969.
In the present chapter, we have only been able to note briefly several
important events and people in the historical development of aeronautics. More
over, there are many other places, people, and accomplishments which we simply
could not mention in the interest of brevity. Therefore, the reader is urged to
48 INTRODUCTION TO FLIGHT
consult the short bibliography at the end of this chapter for additional modern
reading on the history of aeronautics.
1.12 CHAPTER SUMMARY
You are about to embark on a study of aerospace engineering. This chapter has
presented a short historical sketch of some of the heritage behind modern
aerospace engineering. The major steppingstones to controlled, manned,
heavierthanair, powered flight are summarized as follows:
1. Leonardo da Vinci conceives the ornithopter and leaves over 500 sketches of
his design, drawn during 1486 to 1490. However, this approach to manned
flight proves to be unsuccessful over the ensuing centuries.
2. The Montgolfier hotair balloon floats over Paris on November 21, 1783. For
the first time in history, a human being is lifted and carried through the air
for a sustained period of time.
3. A redletter date in the progress of aeronautics is 1799. In that year, Sir
George Cayley in England engraves on a silver disk his concept of a fuselage,
a fixed wing, and horizontal and vertical tails. He is the first person to
propose separate mechanisms for the generation of lift and propulsion. He is
the grandparent of the concept of the modern airplane.
4. The first two powered hops in history are achieved by the Frenchman Felix
Du Temple in 1874 and the Russian Alexander F. Mozhaiski in 1884.
However, they do not represent truly controlled, sustained flight.
5. Otto Lilienthal designs the first fully successful gliders in history. During the
period 18911896, he achieves over 2500 successful glider flights. If he had
not been killed in a glider crash in 1896, Lilienthal might have achieved
powered flight before the Wright brothers.
6. Samuel Pierpont Langley, secretary of the Smithsonian Institution, achieves
the first sustained heavierthanair, unmanned, powered flight in history with
his smallscale Aerodrome in 1896. However, his attempts at manned flight
are unsuccessful, the last one failing on December 8, 1903just nine days
before the Wright brothers' stunning success.
7. Another redletter date in the history of aeronautics, indeed in the history of
humanity, is December 17, 1903. On that day, at Kill Devil Hills in North
Carolina, Orville and Wilbur Wright achieve the first controlled, sustained,
powered, heavierthanair, manned flight in history. This flight is to revo
lutionize life during the twentieth century.
8. The development of aeronautics takes off exponentially after the Wright
brothers' public demonstrations in Europe and the United States in 1908. The
ongoing work of Glenn Curtiss and the Wrights and the continued influence
of Langley's early work form an important "aeronautical triangle" in the
development of aeronautics before World War I.
THE FIRST AERONAUTICAL ENGINEERS 49
Throughout the remainder of this book, various historical notes will appear, to
continue to describe the heritage of aerospace engineering as its technology
advances over the twentieth century. It is hoped that such historical notes will add
a new dimension to your developing understanding of this technology.
BIBLIOGRAPHY
Angelucci, E., Airplanes from the Dawn of Flight to the Present Day, McGrawHill, New York, 1973.
Combs, H., Kill Devil Hill, Houghton Mifflin, Boston, 1979.
Crouch, T. D., A Dream of Wings, W.W. Norton, New York, 1981.
GibbsSmith, C. H., Sir George Cayley's Aeronautics 17961855, Her Majesty's Stationery Office,
London, 1962.
___ , The Invention of the Aeroplane (17991909), Faber, London, 1966.
___ , Aviation: An Historical Survey from its Origins to the End of World War Tl, Her Majesty's
Stationery Office, London, 1970.
___ , Flight Through the Ages, Crowell, New York, 1974.
The following are a series of small booklets prepared for the British Science Museum by C. H.
GibbsSmith, published by Her Majesty's Stationery Office, London:
The Wrights Brothers, 1963
The World's First Aeroplane Flights, 1965
Leonardo da Vinci's Aeronautics, 1967
A Brief History of Flying, 1967
Sir George Cayley, 1968
Josephy, A. M., and Gordon, A., The American Heritage History of Flight, Simon and Schuster, New
York, 1962.
McFarland, Marvin W. (ed.), The Papers of Wilbur and Orville Wright, McGrawHill, New York,
1953.
Roseberry, C.R., Glenn Curtiss: Pioneer of Flight, Doubleday, Garden City, NY, 1972.
Taylor, J. W.R., and Munson, K., History of Aviation, Crown, New York, 1972.
CHAPTER
TWO
FUNDAMENTAL THOUGHTS
Engineering: "The application of scientific principles to practical ends." From the Latin word
"ingenium," meaning inborn talent and skill, ingenious.
The American Heritage Dictionary of the English Language, 1969
The language of engineering and physical science is a logical collection and
assimilation of symbols, definitions, formulas, and concepts. To the average
person in the street, this language is frequently esoteric and incomprehensible. In
fact, when you become a practicing engineer, do not expect to converse with your
spouse across the dinner table about your great technical accomplishments of the
day. Chances are that he or she will not understand what you are talking about.
The language is intended to convey physical thoughts. It is our way of describing
the phenomena of nature as observed in the world around us. It is a language that
has evolved over at least 2500 years. For example, in 400 B.C., the Greek
philosopher Democritus introduced the word and concept of the "atom," the
smallest bit of matter that could not be cut. The purpose of this chapter is to
introduce some of the everyday language used by aerospace engineers; in turn,
this language will be extended and applied throughout the remainder of this book.
2.1 FUNDAMENTAL PHYSICAL QUANTITIES
OF A FLOWING GAS
As you read through this book, you will soon begin to appreciate that the flow of
air over the surface of an airplane is the basic source of the lifting or sustaining
force that allows a heavierthanair machine to fly. In fact, the shape of an
50
FUNDAMENTAL THOUGHTS 51
airplane is designed to encourage the airflow over the surface to produce a lifting
force in the most efficient manner possible. (You will also begin to appreciate that
the design of an airplane is in reality a compromise between many different
requirements, the production of aerodynamic lift being just one.) The science that
deals with the flow of air (or for that matter, the flow of any gas) is called
aerodynamics, and the person who practices this science is called an aerodynami
cist. The study of the flow of gases is important in many other aerospace
applications, e.g., the design of rocket and jet engines, propellers, vehicles
entering planetary atmospheres from space, wind tunnels, and rocket and pro
jectile configurations. Even the motion of the global atmosphere, and the flow of
effluents through smokestacks, fall within the realm of aerodynamics. The appli
cations are almost limitless.
Four fundamental quantities in the language of aerodynamics are pressure,
density, temperature, and velocity. Let us look at each one.
A Pressure
When you hold your hand outside the window of a moving automobile, with your
palm perpendicular to the incoming airstream, you can feel the air pressure
exerting a force and tending to push your hand rearward, in the direction of the
airflow. The force per unit area on your palm is defined as the pressure. The
pressure exists basically because air molecules (oxygen and nitrogen molecules)
are striking the surface of your hand and transferring some of their momentum to
the surface. More precisely:
Pressure is the normal force per unit area exerted on a surface due to the time
rate of change of momentum of the gas molecules impacting on that surface.
It is important to note that, even though pressure is defined as force per unit
area, e.g., newtons/meter
2
or pounds/foot2, you do not need a surface that is
actually 1 m
2
or 1 ft
2
to talk about pressure. In fact, pressure is usually defined at
a point in the gas or a point on a surface and can vary from one point to another.
We can use the language of differential calculus to see this more clearly.
Referring to Figure 2.1, consider a point B in a volume of gas. Let
dA = an incremental area around B
dF = force on one side of dA due to pressure
Then, the pressure p at point B in the gas is defined as
(2.1)
Equation (2.1) says that, in reality, the pressure p is the limiting form of the force
per unit area where the area of interest has shrunk to zero around the point B. In
52 INTRODUCTION TO FLIGHT
~ F
~ A
Gas
Figure 2.1 Definition of pressure.
this formalism, it is easy to see that p is a point property and can have a different
value from one point to another in the gas.
Pressure is one of the most fundamental and important variables in
aerodynamics, as we shall soon see. Common units of pressure are newtons/
meter
2
, dynes/centimeter
2
, pounds/foot
2
, and atmospheres. Abbreviations for
these quantities are N/m
2
, dyn/cm
2
, lb/ft
2
, and atm, respectively. See Appendix
C for a list of common abbreviations for physical units.
B Density
The density of a substance (including a gas) is the mass of that substance per
unit volume.
Density will be designated by the symbol p. For example, consider air in a room
that has a volume of 250 m
3
• If the mass of the air in the room is 306.25 kg and is
evenly distributed throughout the space, then p = 306.25 kg/250 m
3
= 1.225
kg/m
3
and is the same at every point in the room.
Analogous to the previous discussion on pressure, the definition of density
does not require an actual volume of 1 m
3
or 1 ft
3
• Rather, p is a point property
and can be defined as follows. Referring to Figure 2.2, consider point B inside a
volume of gas. Let
dv = an elemental volume around the point B
dm = the mass of gas inside dv
Then, p at point B is
p = lim( ~ :
dv + 0
B[!f)
dv
Volume of gas
Figure 2.2 Definition of density.
(2.2)
FUNDAMENTAL THOUGHTS 53
Therefore, p is the mass per unit volume where the volume of interest has shrunk
to zero around point B. The value of p can vary from point to point in the gas.
Common units of density are kg/m
3
, slug/ft
3
, gm/cm
3
, and lbm/ft
3
• (The pound
mass, lbm, will be discussed in a subsequent section.)
C Temperature
Consider a gas as a collection of molecules and atoms. These particles are in
constant motion, moving through space and occasionally colliding with one
another. Since each particle has motion, it also has kinetic energy. If we watch the
motion of a single particle over a long period of time during which it experiences
numerous collisions with its neighboring particles, then we can meaningfully
define the average kinetic energy of the particle over this long duration. If the
particle is moving rapidly, it has a higher average kinetic energy than if it were
moving slowly. The temperature T of the gas is directly proportional to the
average molecular kinetic energy. In fact, we can define T as follows:
Temperature is a measure of the average kinetic energy of the particles in the
gas. If KE is the mean molecular kinetic energy, then temperature is given by
KE= tkT, where k is the Boltzmann constant.
The value of k is 1.38 X 10
23
J/K, where J is an abbreviation for joules.
Hence, we can qualitatively visualize a hightemperature gas as one in which
the particles are randomly rattling about at high speeds, whereas in a lowtemper
ature gas, the random motion of the particles is relatively slow. Temperature is an
important quantity in dealing with the aerodynamics of supersonic and hyper
sonic flight, as we shall soon see. Common units of temperature are the kelvin
(K), degree Celsius (
0
C), degree Rankine (°R), and degree Fahrenheit (°F).
D Flow Velocity and Streamlines
The concept of speed is commonplace: it represents the distance traveled by some
object per unit time. For example, we all know what is meant by traveling at a
speed of 55 mi/h down the highway. However, the concept of the velocity of a
flowing gas is somewhat more subtle. First of all, velocity connotes direction as
well as speed. The automobile is moving at a velocity of 55 mi/h due north in a
horizontal p(ane. To designate velocity, we must quote both speed and direction.
For a flowing gas, we must further recognize that each region of the gas does not
necessarily have the same velocity; i.e., the speed and direction of the gas may
vary from point to point in the flow. Hence, flow velocity, along with p, p, and T,
is a point property.
To see this more clearly, consider the flow of air over an airfoil or the flow of
combustion gases through a rocket engine, as sketched in Figure 2.3. To orient
yourself, lock your eyes on a specific, infinitesimally small element of mass in the
gas, and watch this element move with time. Both the speed and direction of this
54 INTRODUCTION TO FLIGHT
~ v
~
Rocket engine
Flow over an airfoil
Figure 2.3 Flow velocity and streamlines.
element (usually called a fluid element) can vary as it moves from point to point
in the gas. Now, fix your eyes on a specific fixed point in the gas flow, say point B
in Figure 2.3. We can now define flow velocity as follows:
The velocity at any fixed point B in a flowing gas is the velocity of an
infinitesimally small fluid element as it sweeps through B.
Again, emphasis is made that velocity is a point property and can vary from point
to point in the flow.
Figure 2.4 Smoke photograph of the lowspeed flow over a Lissaman 7769 airfoil at 10° angle of
attack. The Reynolds number based on chord is 150,000. This is the airfoil used on the Gossamer
Condor manpowered aircraft. (The photograph was taken in one of the Notre Dame University smoke
tunnels hv Dr. T. J. Mueller. Professor of Aerospace Engineering at Notre Dame, and is shown here
through his courtesy.)
FUNDAMENTAL THOUGHTS 55
Referring again to Figure 2.3, we note that as long as the flow is steady (as
long as it does not fluctuate with time), a moving fluid element is seen to trace out
a fixed path in space. This path taken by a moving fluid element is called a
streamline of the flow. Drawing the streamlines of the flow field is an important
way of visualizing the motion of the gas; we will frequently be sketching the
streamlines of the flow about various objects. For example, the streamlines of the
flow about an airfoil are sketched in Figure 2.3 and clearly show the direction of
motion of the gas. Figure 2.4 is an actual photograph of streamlines over an
airfoil model in a lowspeed subsonic wind tunnel. The streamlines are made
visible by injection of filaments of smoke upstream of the model; these smoke
filaments follow the streamlines in the flow. Using another flow field visualization
technique, Figure 2.5 shows a photograph of a flow where the surface streamlines
are made visible by coating the model with a mixture of white pigment in mineral
oil. Clearly, the visualization of flow streamlines is a useful aid in the study of
aerodynamics.
Figure 2.5 An oil streak photograph showing the surface streamline pattern for a fin mounted on a
flat plate in supersonic flow. The parabolic curve in front of the fin is due to the bow shock wave and
flow separation ahead of the fin. Note how clearly the streamlines can be seen in this complex flow
pattern. Flow is from right to left. The Mach number is 5 and the Reynolds number is 6.7 x 10
6
.
(Courtesy of Allen E. Winkelmann, University of Maryland, and the Naval Surface Weapons Center.)
56 INTRODUCTION TO FLIGHT
2.2 THE SOURCE OF ALL AERODYNAMIC FORCES
We have just discussed the four basic aerodynamic flow quantities: p, p, T, and
V, where V is velocity which has both magnitude and direction; i.e., velocity is a
vector quantity. A knowledge of p, p, T, and V at each point of a flow fully
defines the flow field. For example, if we were concerned with the flow about a
sharppointed cone as shown in Figure 2.6, we could imagine a cartesian xyz
threedimensional space, where the velocity far ahead of the cone, V
00
, is in the x
direction and the cone axis is also along the x direction. The specification of the
following quantities then fully defines the flow field:
p=p(x,y,z)
p = p(x,y,z)
T = T(x, y, z)
V=V(x,y,z)
(In practice, the flow field about a rightcircular cone is more conveniently
described in terms of cylindrical coordinates, but we are concerned only with the
general ideas here.)
Theoretical and experimental aerodynamicists labor to calculate and measure
flow fields of many types. But why? What practical information does knowledge
of the flow field yield with regard to airplane design or to the shape of a rocket
engine? A substantial part of the first five chapters of this book endeavors to
answer these questions. However, the roots of the answers lie in the following
discussion.
Probably the most practical consequence of the flow of air over an object is
that the object experiences a force, an aerodynamic force, such as your hand feels
outside the open window of a moving car. Subsequent chapters will discuss the
nature and consequences of such aerodynamic forces. The purpose here is to state
that the aerodynamic force exerted by the airflow on the surface of an airplane,
Voo
Flow
y
p=p(x,y,z) .
p = p(x,y,z)}
r: T(x,y,z) Flowf1eld
V V(x,y,z)
Figure 2.6 Specification of a How field.
FUNDAMENTAL THOUGHTS 57
Figure 2. 7 Pressure and shearstress
distributions.
missile, etc., stems from only two simple natural sources:
1. Pressure distribution on the surface
2. Shear stress (friction) on the surface
We have already discussed pressure. Referring to Figure 2.7, we see pressure
exerted by the gas on the solid surface of an object always acts normal to the
surface, as shown by the directions of the arrows. The lengths of the arrows
denote the magnitude of the pressure at each local point on the surface. Note that
the surface pressure varies with location. The net unbalance of the varying
pressure distribution over the surface creates a force, an aerodynamic force. The
second source, shear stress acting on the surface, is due to the frictional effect of
the flow "rubbing" against the surface as it moves around the body. The shear
stress 'Tw is defined as the force per unit area acting tangentially on the surface due
to friction, as shown in Figure 2.7. It is also a point property; it also varies along
the surface; and the net unbalance of the surface shear stress distribution also
creates an aerodynamic force on the body. No matter how complex the flow field,
and no matter how complex the shape of the body, the only way nature has of
communicating an aerodynamic force to a solid object or surf ace is through the
pressure and shear stress distributions which exist on the surf ace. These are the basic
fundamental sources of all aerodynamic forces.
Finally, we can state that a primary function of theoretical and experimental
aerodynamics is to predict and measure the aerodynamic forces on a body. In
many cases, this implies prediction and measurement of p and 'Tw along a given
surface. Furthermore, a prediction of p and 'Tw on the surface frequently requires
knowledge of the complete flow field around the body. This helps to answer our
earlier question as to what practical information is yielded by knowledge of the
flow field.
2.3 EQUATION OF STATE FOR A PERFECT GAS
Air under normal conditions of temperature and pressure, such as that encoun
tered in subsonic and supersonic flight through the atmosphere, behaves very
58 INTRODUCTION TO FLIGHT
much like a perfect gas. The definition of a perfect gas can best be seen by
returning to the molecular picture. A gas is a collection of particles (molecules,
atoms, electrons, etc.) in random motion, where each particle is, on the average, a
long distance away from its neighboring particles. Each molecule has an inter
molecular force field about it, a ramification of the complex interaction of the
electromagnetic properties of the electrons and nucleus. The intermolecular force
field of a given particle extends a comparatively long distance through space and
changes from a strong repulsive force at close range to a weak attractive force at
long range. The intermolecular force field of a given particle reaches out and is
felt by the neighboring particles. If the neighboring particles are far away, they
feel only the tail of the weak attractive force; hence the motion of the neighboring
particles is only negligibly affected. On the other hand, if they are close, their
motion can be greatly affected by the intermolecular force field. Since the pressure
and temperature of a gas are tangible quantities derived from the motion of the
particles, then p and T are directly influenced by intermolecular forces, especially
when the molecules are packed closely together (i.e., at high densities). This leads
to the definition of a perfect gas:
A perfect gas is one in which intermolecular forces are negligible.
Clearly, from the above discussion, a gas in nature where the particles are
widely separated (low densities) approaches the definition of a perfect gas. The air
in the room about you is one such case; each particle is separated, on the average,
by more than 10 molecular diameters from any other. Hence, air at standard
conditions can be readily approximated by a perfect gas. Such is also the case for
the flow of air about ordinary flight vehicles at subsonic and supersonic speeds.
Therefore, in this book, we will always deal with a perfect gas for our aerodynamic
calculations.
The relation among p, p, and T for a gas is called the equation of state. For a
perfect gas, the equation of state is
(2.3)
where R is the specific gas constant, the value of which varies from one type of
gas to another. For normal air we have
287 J = 1716 ft . lb
R = (kg)(K) (slug)(
0
R)
It is interesting that the deviation of an actual gas in nature from perfect gas
behavior can be expressed approximately by the modified Berthelot equation of
state:
__!!__ =
1
+ ap _ bp
pRT T T3
where a and b are constants of the gas. Thus, the deviations become smaller as p
FUNDAMENTAL THOUGHTS 59
decreases and T increases. This makes sense, because if p is high, the molecules
are packed closely together, intermolecular forces become important, and the gas
behaves less like a perfect gas. On the other hand, if T is high, the molecules
move faster. Thus their average separation is larger, intermolecular forces become
less significant in comparison to the inertia forces of each molecule, and the gas
behaves more like a perfect gas.
Also, it should be noted that when the air in the room around you is heated
to temperatures above 2500 K, the oxygen molecules begin to dissociate (tear
apart) into oxygen atoms; at temperatures above 4000 K, the nitrogen begins to
dissociate. For these temperatures, air becomes a chemically reacting gas, where
its chemical composition becomes a function of both p and T; i.e., it is no longer
normal air. As a result, R in Eq. (2.3) becomes a variable, R = R(p, T), simply
because the gas composition is changing. The perfect gas equation of state, Eq.
(2.3), is still valid for such a case, except that R is no longer a constant. This
situation is encountered in very high speed flight, e.g., the atmospheric entry of
the Apollo capsule, in which case the temperatures in some regions of the flow
field reach 11,000 K.
Again, in this book, we will always assume that air is a perfect gas, obeying
Eq. (2.3), with a constant R = 287 J/(kg)(K) or 1716 ft· lb/(slug)(
0
R).
2.4 DISCUSSION ON UNITS
Physical units are vital to the language of engineering. In the final analysis, the
end result of most engineering calculations or measurements is a number which
represents some physical quantity, e.g., pressure, velocity, or force. The number is
given in terms of combinations of units, for example, 10
5
N/m
2
, 300 m/s, or 5
N, where the newton, meter, and second are examples of units. (See Appendix C.)
Historically, various branches of engineering have evolved and favored sys
tems of units which seemed to most conveniently fit their needs. These various
sets of "engineering" units usually differ among themselves and are different from
the metric system preferred for years by physicists and chemists. In the modern
world of technology, where science and engineering interface on almost all fronts,
such duplicity and variety of units has become an unnecessary burden. Metric
units are now the accepted norm in both science and engineering in most
countries outside the United States. More importantly, in 1960 the Eleventh
General Conference on Weights and Measures defined and officially established
the Systeme International d 'Unitesthe SI system of unitswhich was adopted
as the preferred system of units by 36 participating countries, including the
United States. Since then, the United States has made progress toward the
voluntary implementation of SI units in engineering. For example, several NASA
(National Aeronautics and Space Administration) laboratories have made SI
units virtually mandatory for all results reported in technical reports, although
engineering units can be shown as a duplicate set. The AIAA (American Institute
of Aeronautics and Astronautics) has made a policy of encouraging SI units for
60 INTRODUCTION TO FLIGHT
all papers reported in their technical journals. At the time of this writing, the
United States Congress is studying legislation to require total conversion to SI
units for all uses of weights and measures. It is apparent that in a few decades, the
United States, along with the rest of the world, will be using SI units almost
exclusively.
For these reasons, students who prepare themselves for practicing engineering
in the last quarter of the twentieth century must do "double duty" with regard to
familiarization with units. They must be familiar with the old engineering units in
order to use the vast bulk of existing technical literature quoted in such units. At
the same time they must be intimately familiar with the SI system for all future
work and publications; i.e., our next generation of engineers must be bilingual
with regard to units.
In order to promote fluency in both the engineering and SI units, this book
will incorporate both sets. However, you should adopt the habit of working all
your problems using the SI units from the start, and when results are requested in
engineering units, convert your final SI results to the desired units. This will give
you every chance to develop a "feel" for SI units. It is important that you develop
a natural feeling for SI units; e.g., you should feel as at home with pressures in
N/m
2
as you probably already do with lb/in
2
(psi). A mark of successful
experienced engineers is their feel for correct magnitudes of physical quantities in
familiar units. Make the SI your familiar units.
For all practical purposes, the SI system is a metric system based on the
meter, kilogram, second, and kelvin as basic units of length, mass, time, and
temperature. It is a coherent, or consistent, set of units. Such consistent sets of
units allow physical relationships to be written without the need for "conversion
factors" in the basic formulas. For example, in a consistent set of units, Newton's
second law can be written
F= m X a
Force= mass X acceleration
In the SI system,
F= ma
(1 newton) = (1 kilogram)(l meter/second
2
) (2.4)
The newton is a force defined such that it accelerates a mass of 1 kilogram by 1
meter/second squared.
The English engineering system of units is another consistent set of units.
Here the basic units of mass, length, time, and temperature are the slug, foot,
second, and Rankine, respectively. In this system,
F= ma
(1 pound)= (1 slug)(l footjsecond
2
) (2.5)
The pound is a force defined such that it accelerates a mass of 1 slug by 1
FUNDAMENTAL THOUGHTS 61
foot/second
2
• Note that in both systems, Newton's second law is written simply
as F = ma, with no conversion factor on the righthand side.
In contrast, a nonconsistent set of units defines force and mass such that
Newton's second law must be written with a conversion factor, or constant, as
F =(l/gJ
i t
x m X a
i i
Force Conversion Mass Acceleration
factor
A nonconsistent set of units frequently used in the past by mechanical engineers
includes the pound or pound force, pound mass, foot, and second, whereby
gc = 32.2 (lbm)(ft)/(s
2
)(lb
1
)
F =(l/gJ m x a
(2.6)
i i i i
lb/ (1/32.2) Jbm ft/s
2
In this system, the unit of mass is the pound mass lbm. By comparing Eqs.
(2.5) and (2.6), we see that 1 slug = 32.2 lbm. A slug is a large hunk of mass,
whereas the lbm is considerably smaller, by a factor of 32.2. This is illustrated
in Figure 2.8.
Already, you can sense how confusing the various sets of units can become,
especially nonconsistent units. Confusion on the use of Eq. (2.6) with the gc
factor has resulted in many heat exchangers and boilers being designed 32.2 times
either too large or too small. On the other hand, the use of a consistent set of
units, such as in Eqs. (2.4) and (2.5), which have no conversion factor, eliminates
such confusion. This is one of the beauties of the SI system.
For this reason, we will always deal with a consistent set of units in this book.
We will use both the SI units from Eq. (2.4) and the English engineering units
from Eq. (2.5). As stated before, you will frequently encounter the engineering
units in the existing literature, whereas you will be seeing the SI units with
increasing frequency in the future literature; i.e., you must become bilingual. To
summarize, we will deal with the English engineering system unitslb, slug, ft, s,
0
Rand the Systeme International (SI) unitsN, kg, m, s, K.
1 slug
I slug= 3 2.2 lbm
Figure 2.8 Comparison between the slug and pound mass.
62 INTRODUCTION TO FLIGHT
Therefore, returning to the equation of state, Eq. (2.3), where p = pRT, the
units are
English Engineering
System SI
p lb/ft
2
N/m
2
p slugs/ft
3
kg/m
3
T
OR
K
R (for air) 1716 ft · lb/(slug)(
0
R) 287 J /(kg)(K)
There are two final points about units that you should note. First, the units of
a physical quantity can frequently be expressed in more than one combination
simply by appealing to Newton's second law. From Newton's law, the relation
between N, kg, m, and s is
F= ma
N =kg· m/s
2
Thus, a quantity such as R = 287 J /(kg)(K) can also be expressed in the
equivalent way as
28
1
287 N . m 287 kg · m m = 287 m
2
R =
7
(kg)(K) = (kg)(K) = s
2
 (kg)(K) (s
2
)(K)
Thus, R can also be expressed in the equivalent terms of velocity squared divided
by temperature. In the same vein,
ft . lb = 1716 ft
2
R = 1716 (slug)(OR) (s2)(0R)
Secondly, in the equation of state, Eq. (2.3), T is always the absolute
temperature, where zero degrees is the absolutely lowest temperature possible.
Both Kand
0
R are absolute temperature scales, where 0 °R = 0 K = temperature
at which almost all molecular translational motion theoretically stops. On the
other hand, the familiar Fahrenheit (F) and Celsius (C) scales are not absolute
scales. Indeed,
0°F = 460 °R
0°C = 273 K = 32°F
For example, 90°F is the same as 460 + 90 = 550°R,
and 10°C is the same as 273 + 10 = 283 K.
Please remember: T in Eq. (2.3) must be the absolute temperature, either Kelvin
or Rankine.
FUNDAMENTAL THOUGHTS 63
2.5 SPECIFIC VOLUME
Density p is the mass per unit volume. The inverse of this quantity is also
frequently used in aerodynamics. It is called the specific volume v and is defined
as the volume per unit mass. By definition,
1
v=
p
Hence, from the equation of state
we also have
1
p = pRT= RT
v
I pv =RT'
Units for v are m
3
/kg and ft
3
/slug.
(2.7)
Example 2.1 The air pressure and density at a point on the wing of a Boeing 747 are l.10X10
5
N/m
2
and 1.20 kg/m3, respectively. What is the temperature at that point?
SOLUTION From Eq. (2.3), p = pRT, hence T = p /pR
T= l.lOXlOsN/m2 =1319KI
( 1.20 kg/m
3
)[287 J /(kg)(K))
Example 2.2 The highpressure air storage tank for a supersonic wind tunnel has a volume of
1000 ft
3
. If air is stored at a pressure of 30 atm and a temperature of 530 °R, what is the mass of
gas stored in the tank in slugs? In lbm?
SOLUTION
1 atm = 2116 lb/ft
2
Hence, p = (30)(2116) lb/ft
2
=6.348X10
4
lb/ft
2
. Also, from Eq. (2.3), p = pRT, hence p =
p/RT.
= 6.348Xl04 lb/ft2 =6.98x102 slu /ft3
p [1716 ft·lb/(slug){
0
R)]{530 R) g
This is the density, which is mass per unit volume. The total mass M in the tank of volume Vis
M = pV = ( 6.98x10
2
slug/ft
3
){1000ft
3
)=169.8 slugs I
Recall that 1 slug= 32.2 lbm.
Hence M = { 69.8)(32.2) = 12248 lbm I
Example 2.3 Air flowing at high speed in a wind tunnel has pressure and temperature equal to
0.3 atm and 100°C, respectively. What is the air density? What is the specific volume?
64 INTRODUCTION TO FLIGHT
SOLUTION
1 atm=l.Olx10
5
N/m
2
Hence p = (0.3)(1.01x10
5
) = 0.303x10
5
N/m
2
Note that T = 100°C is not an absolute temperature.
Hence T= 100+273 =173 K
From Eq. (2.3), p = pRT, hence p = p /RT.
0.303X10
5
N/m2 ~ k 3 1
p= (287J/(kg)(K)](173K) = "
610
g/m
v=.!.=
1
=l 1.64m
3
/kgl
p 0.610 . .
Note: It is worthwhile to remember that
1 atm = 2116 lb/ft
2
1 atm = 1.01X10
5
N/m
2
2.6 HISTORICAL NOTE: THE NACA AND NASA
Let us pick up the string of aeronautical engineering history from Chap. 1. After
Orville and Wilbur Wright's dramatic public demonstrations in the United States
and Europe in 1908, there was a virtual explosion in aviation developments. In
turn, this rapid progress had to be fed by new technical research in aerodynamics,
propulsion, structures, and flight control. It is important to realize that then, as
well as today, aeronautical research was sometimes expensive, always demanding
in terms of intellectual talent, and usually in need of large testing facilities. Such
research in many cases was either beyond the financial resources of, or seemed too
out of the ordinary for, private industry. Thus, the fundamental research so
necessary to fertilize and pace the development of aeronautics in the twentieth
century had to be established and nurtured by national governments. It is
interesting to note that George Cayley himself (see Chap. 1) as far back as 1817
called for "public subscription" to underwrite the expense of the development of
airships. Responding about 80 years later, the British Government set up a school
for ballooning and military kite flying at Farnborough, England. By 1910, the
Royal Aircraft Factory was in operation at Farnborough with the noted Geoffrey
de Havilland as its first airplane designer and test pilot. This was the first major
governmental aeronautical facility in history. This operation was soon to evolve
into the Royal Aircraft Establishment (RAE), which today is still conducting
viable aeronautical research for the British Government.
In the United States, aircraft development as well as aeronautical research
languished after 1910. During the next decade, the United States embarrassingly
fell far behind Europe in aeronautical progress. This set the stage for the U.S.
Government to establish a formal mechanism for pulling itself out of its
FUNDAMENTAL THOUGHTS 65
aeronautical "dark ages." On March 3, 1915, by an act of Congress, the National
Advisory Committee for Aeronautics (NACA) was created, with an initial ap
propriation of $5000 per year for 5 years. This was at first a true committee,
consisting of 12 distinguished members knowledgeable about aeronautics. Among
the charter members in 1915 were Professor Joseph S. Ames of Johns Hopkins
University (later to become President of Johns Hopkins) and Professor William
F. Durand of Stanford University, both of whom were to make major impressions
on aeronautical research in the first half century of powered flight. This advisory
committee, the NACA, was originally to meet annually in Washington, D.C., on
"the Thursday after the third Monday of October of each year," with any special
meetings to be called by the chair. Its purpose was to advise the government on
aeronautical research and development and to bring some cohesion to such
activities in the United States.
The committee immediately noted that a single advisory group of 12 mem
bers was not sufficient to breathe life into U.S. aeronautics. Their insight is
apparent in the letter of submittal for the first annual report of the NACA in
1915, which contained the following passage:
There are many practical problems in aeronautics now in too indefinite a form to enable their
solution to be undertaken. The committee is of the opinion that one of the first and most
important steps to be taken in connection with the committee's work is the provision and
equipment of a flying field together with aeroplanes and suitable testing gear for determining the
forces acting on fullsized machines in constrained and in free flight, and to this end the estimates
submitted contemplate the development of such a technical and operating staff, with the proper
equipment for the conduct of fullsized experiments.
It is evident that there will ultimately be required a wellequipped laboratory specially
suited to the solving of those problems which are sure to develop, but since the equipment of
such a laboratory as could be laid down at this time might well prove unsuited to the needs of the
early future, it is believed that such provision should be the result of gradual development.
So the first action of this advisory committee was to call for major govern
mental facilities for aeronautical research and development. The clouds of war in
EuropeWorld War I had already started a year earliermade their recom
mendations even more imperative. In 1917, when the United States entered the
conflict, actions followed the committee's words. We find the following entry in
the third annual NACA report: "To carry on the highly scientific and special
investigations contemplated in the act establishing the committee, and which
have, since the outbreak of the war, assumed greater importance, and for which
facilities do not already exist, or exist in only a limited degree, the committee has
contracted for a research laboratory to be erected on the Signal Corps Experimen
tal Station, Langley Field, Hampton, Virginia." The report goes on to describe a
single, twostory laboratory building with physical, chemical, and structural
testing laboratories. The building contract was for $80,900; actual construction
began in 1917. Two wind tunnels and an engine test stand were contemplated "in
the near future." The selection of a site 4 mi north of Hampton, Virginia, was
based on general health conditions and the problems of accessibility to Washing
66 INTRODUCTION TO FLIGHT
ton and the larger industrial centers of the East, protection from naval attack,
climatic conditions, and cost of the site.
Thus, the Langley Memorial Aeronautical Research Laboratory was born. It
was to remain the only NACA laboratory and the only major U.S. aeronautical
laboratory of any type for the next 20 years. Named after Samuel Pierpont
Langley (see Sec. 1.7), it pioneered in windtunnel and flight research. Of
particular note is the airfoil and wing research performed at Langley during the
1920s and 1930s. We will return to the subject of airfoils in Chap. 5, at which
time the reader should note that the airfoil data included in Appendix D were
obtained at Langley. With the work which poured out of the Langley laboratory,
the United States took the lead in aeronautical development. High on the list of
accomplishments, along with the systematic testing of airfoils, was the develop
ment of the NACA engine cowl (see Sec. 6.19), an aerodynamic fairing built
around radial piston engines which dramatically reduced the aerodynamic drag of
such engines.
In 1936, Dr. George Lewis, who was then NACA Director of Aeronautical
Research (a position he held from 1924 to 1947), toured major European
laboratories. He noted that the NACA's lead in aeronautical research was quickly
disappearing, especially in light of advances being made in Germany. As World
War II drew close, the NACA clearly recognized the need for two new laboratory
operations: an advanced aerodynamics laboratory to probe into the mysteries of
highspeed (even supersonic) flight, and a major enginetesting laboratory. These
needs eventually led to the construction of the Ames Aeronautical Laboratory at
Moffett Field, near Mountain View, California (authorized in 1939) and the Lewis
Engine Research Laboratory at Cleveland, Ohio (authorized in 1941). Along with
Langley, these two new NACA laboratories again helped to spearhead the United
States to the forefront of aeronautical research and development in the 1940s and
1950s.
The dawn of the space age occurred on October 4, 1957, when Russia
launched Sputnik I, the first artificial satellite to orbit the earth. Taking its
somewhat embarrassed technical pride in hand, the United States moved quickly
to compete in the race for space. On July 29, 1958, by another act of Congress
(Public Law 85568), the National Aeronautics and Space Administration (NASA)
was born. At this same moment, the NACA came to an end. Its programs, people,
and facilities were instantly transferred to NASA, lock, stock, and barrel. How
ever, NASA was a larger organization than just the old NACA; it absorbed in
addition numerous Air Force, Navy, and Army projects for space. Within two
years of its birth, NASA was authorized four new major installations: an existing
Army facility at Huntsville, Alabama, renamed the George C. Marshall Space
Flight Center; the Goddard Space Flight Center at Greenbelt, Maryland; the
Manned Spacecraft Center (now the Johnson Spacecraft Center) in Houston,
Texas; and the Launch Operations Center (now the John F. Kennedy Space
Center) at Cape Canaveral, Florida. These, in addition to the existing but slightly
renamed Langley, Ames, and Lewis research centers, were the backbone of
NASA. Thus, the aeronautical expertise of the NACA now formed the seeds for
FUNDAMENTAL THOUGHTS 67
NASA, shortly thereafter to become one of the world's most important forces in
space technology.
This capsule summary of the roots of the NACA and NASA is included in
the present chapter on fundamental thoughts because it is virtually impossible for
a student or practitioner of aerospace engineering in the United States not to be
influenced or guided by NACA or NASA data and results. The extended
discussion on airfoils in Chap. 5 is a case in point. Thus, the NACA and NASA
are "fundamental" to the discipline of aerospace engineering, and it is important
to have some impression of the historical roots and tradition of these organiza
tions. Hopefully, this short historical note provides such an impression. A much
better impression can be obtained by taking a journey through the NACA and
NASA technical reports in the library, going all the way back to the first NACA
report in 1915. In so doing, a panorama of aeronautical and space research
through the years will unfold in front of you.
2.7 CHAPTER SUMMARY
Some of the major ideas in this chapter are listed below.
1. The language of aerodynamics involves pressure, density, temperature, and
velocity. In turn, the illustration of the velocity field can be enhanced by
drawing streamlines for a given flow.
2. The source of all aerodynamic forces on a body is the pressure distribution
and the shear stress distribution over the surface.
3. A perfect gas is one where intermolecular forces can be neglected. For a
perfect gas, the equation of state which relates p, p, and T is
p = pRT (2.3)
where R is the specific gas constant.
4. In order to avoid confusion, errors, and a number of unnecessary "conversion
factors" in the basic equations, always use consistent units. In this book, the
SI system (newton, kilogram, meter, second) and the English engineering
system (pound, slug, foot, second) will be used.
BIBLIOGRAPHY
Gray, George W., Frontiers of Flight, Knopf, New York, 1948.
Hartman, E. P., Adventures in Research: A History of Ames Research Center 19401965, NASA
SP4302, 1970.
Mechtly, E. A The International System of Units, NASA SP7012, 1969.
68 INTRODUCTION TO FLIGHT
PROBLEMS
2.1 Consider the lowspeed flight of the space shuttle as it is nearing a landing. If the air pressure and
temperature at the nose of the shuttle are 1.2 atm and 300 K, respectively, what are the density and
specific volume?
2.2 Consider 1 kg of helium at 500 K. Assuming that the total internal energy of helium is due to the
mean kinetic energy of each atom summed over all the atoms, calculate the internal energy of this gas.
Note: The molecular weight of helium is 4. Recall from chemistry that the molecular weight is the
mass per mole of gas; that is, 1 mo! of helium contains 4 kg of mass. Also, 1 mo! of any gas contains
6.02 x 10
26
molecules or atoms (Avogadro's number).
2.3 Calculate the weight of air (in pounds) contained within a room 20 ft long, 15 ft wide, and 8 ft
high. Assume standard atmospheric pressure and temperature of 2116 lb/ft
2
and 59°P, respectively.
2.4 Comparing with the case of Prob. 2.3, calculate the percentage change in the total weight of air in
the room when the air temperature is reduced to 10°P (a very cold winter day), assuming the
pressure remains the same at 2116 lb/ft
2
•
2.5 If 1500 lbm of air is pumped into a previously empty 900ft
3
storage tank and the air temperature
in the tank is uniformly 70°P, what is the air pressure in the tank in atmospheres?
2.6 In the above problem, assume the rate at which air is being pumped into the tank is 0.5 lbm/s.
Consider the instant in time at which there is 1000 lbm of air in the tank. Assume the air temperature
is uniformly 50°P at this instant and is increasing at the rate of 1 °P /min. Calculate the rate of
change of pressure at this instant.
2.7 Assume that, at a point on the wing of the Concorde supersonic transport, the air temperature is
10°C and the pressure is 1.7 x 10
4
N/m2. Calculate the density at this point.
2.8 At a point in the test section of a supersonic wind tunnel, the air pressure and temperature are
0.5 x 10
5
N/m
2
and 240 K, respectively. Calculate the specific volume.
2.9 Consider a flat surface in an aerodynamic flow (say a flat side wall of a wind tunnel). The
dimensions of this surface are 3 ft in the flow direction (the x direction) and 1 ft perpendicular to the
flow direction (the y direction). Assume that the pressure distribution (in lb/ft
2
) is given by
p = 2116  lOx and is independent of y. Assume also that the shearstress distribution (in lb/ft
2
) is
given by Tw = 90/(x + 9)
1
1
2
and is independent of y. In the above expressions, x is in feet, and
x = 0 at the front of the surface. Calculate the magnitude and direction of the net aerodynamic force
on the surface.
CHAPTER
THREE
THE STANDARD ATMOSPHERE
Sometimes gentle, sometimes capricious, sometimes awful, never the same for two moments together;
almost human in its passions, almost spiritual in its tenderness, almost divine in its infinity.
John Ruskin, The Sky
Aerospace vehicles can be divided into two basic categories: atmospheric vehicles
such as airpianes and helicopters, which always fly within the sensible atmo
sphere, and space vehicles such as satellites, the Apollo lunar vehicle, and deep
space probes, which operate outside the sensible atmosphere. However, space
vehicles do encounter the earth's atmosphere during their blastoffs from the
earth's surface and again during their reentries and recoveries after completion of
their missions. If the vehicle is a planetary probe, then it may encounter the
atmospheres of Venus, Mars, Jupiter, etc. Therefore, during the design and
performance of any aerospace vehicle, the properties of the atmosphere must be
taken into account.
The earth's atmosphere is a dynamically changing system, constantly in a
state of flux. The pressure and temperature of the atmosphere depend on altitude,
location on the globe (longitude and latitude), time of day, season, and even solar
sunspot activity. To take all these variations into account when considering the
design and performance of flight vehicles is impractical. Therefore, a standard'
atmosphere is defined in order to relate flight tests, windtunnel results, and
general airplane design and performance to a common reference. The standard
atmosphere gives mean values of pressure, temperature, density, and other
properties as functions of altitude; these values are obtained from experimental
balloon and soundingrocket measurements combined with a mathematical model
69
70 INTRODUCTION TO FLIGHT
of the atmosphere. To a reasonable degree, the standard atmosphere reflects
average atmospheric conditions, but this is not its main importance. Rather, its
main function is to provide tables of common reference conditions that can be
used in an organized fashion by aerospace engineers everywhere. The purpose of
this chapter is to give you some feeling for what the standard atmosphere is all
about and how it can be used for aerospace vehicle analyses.
It should be mentioned that several different standard atmospheres exist,
compiled by different agencies at different times, each using slightly different
experimental data in their models. For all practical purposes, the differences are
insignificant below 30 km (100,000 ft), which is the domain of contemporary
airplanes. A standard atmosphere in common use is the 1959 ARDC Model
Atmosphere. (ARDC stands for the U.S. Air Force's previous Air Research and
Development Command, which is now the Air Force Systems Command.) The
atmospheric tables used in this book are 1:aken from the 1959 ARDC Model
Atmosphere.
3.1 DEFINITION OF ALTITUDE
Intuitively, we all know the meaning of altitude. We think of it as the distance
above the ground. But like so many other general terms, it must be more precisely
defined for quantitative use in engineering. In fact, in the following sections we
will define and use six different altitudes: absolute, geometric, geopotential,
pressure, temperature, and density altitudes.
First, imagine that we are at Daytona Beach, Florida, where the ground is
sea level. If we could fly straight up in a helicopter and drop a tape measure to
the ground, the measurement on the tape would be, by definition, the geometric
altitude he, i.e., the geometric height above sea level.
Now, if we would bore a hole through the ground to the center of the earth
and extend our tape measure until it hit the center, then the measurement on the
tape would be, by definition, the absolute altitude ha. If r is the radius of the
earth, then ha =he+ r. This is illustrated in Figure 3.1.
The absolute altitude is important, especially for space flight, because the
local acceleration of gravity, g, varies with h.
1
• From Newton's law of gravitation,
g varies inversely as the square of the distance from the center of the earth.
Letting g
0
be the gravitational acceleration at sea level, the local gravitational
acceleration g at a given absolute altitude h., is:
(3.1)
The variation of g with altitude must be taken into account when you are dealing
with mathematical models of the atmosphere, as follows.
THE STANDARD ATMOSPHERE 71
Figure 3.1 Definition of altitude.
3.2 THE HYDROSTATIC EQUATION
We will now begin to piece together a model which will allow us to calculate
variations of p, p, and T as functions of altitude. The foundation of this model is
the hydrostatic equation, which is nothing more than a force balance on an
element of fluid at rest. Consider the small stationary fluid element of air shown
in Figure 3.2. We take for convenience an element with rectangular faces, where
the top and bottom faces have sides of unit length and the side faces have an
infinitesimally small height dh G· On the bottom face, the pressure p is felt, which
gives rise to an upward force of p X 1 X 1 exerted on the fluid element. The top
face is slightly higher in altitude (by the distance dh d, and because pressure
varies with altitude, the pressure on the top face will be slightly different from
that on the bottom face, differing by the infinitesimally small value dp. Hence, on
the top face, the pressure p + dp is felt. It gives rise to a downward force of
p + dp
p
Figure 3.2 Force diagram for the hydro
static equation.
72 INTRODUCTION TO FLIGHT
( p + dp )(1 )(1) on the fluid element. Moreover, the volume of the fluid element is
(1)(1) dha = dha, and since p is the mass per unit volume, then the mass of the
fluid element is simply p(l)(l) dha = p dha. If the local acceleration of gravity is
g, then the weight of the fluid element is gp dha, as shown in Figure 3.2. The
three forces shown in Figure 3.2, pressure forces on the top and bottom and the
weight must balance because the fluid element is not moving. Hence
Thus
p=(p+dp)+pgdhG
I dp = pgdhG I (3.2)
Equation (3.2) is the hydrostatic equation and applies to any fluid of density p,
e.g., water in the ocean as well as air in the atmosphere.
Strictly speaking, Eq. (3.2) is a differential equation; i.e., it relates an
infinitesimally small change in pressure dp to a corresponding infinitesimally
small change in altitude dha, where in the language of differential calculus, dp
and dh G are differentials. Also, note that g is a variable in Eq. (3.2); g depends
on ha as given by Eq. (3.1).
To be made useful, Eq. (3.2) should be integrated to give us what we want,
namely, the variation of pressure with altitude, p = p(ha). To simplify the
integration, we will make the assumption that g is constant through the atmo
sphere, equal to its value at sea level, g
0
. This is something of a historical
convention in aeronautics. At the altitudes encountered during the earlier devel
opment of human flight (less than 15 km or 50,000 ft), the variation of g is
negligible. Hence, we can write Eq. (3.2) as
(3.3)
However, to make Eqs. (3.2) and (3.3) numerically identical, the altitude h in Eq.
(3.3) must be slightly different from that of ha in Eq. (3.2), to compensate for the
fact that g is slightly different from g
0
• Suddenly, we have defined a new altitude
h, which is called the geopotential altitude and which differs from the geometric
altitude. For the practical mind, geopotential altitude is a "fictitious" altitude,
defined by Eq. (3.3) for ease of future calculations. However, many standard
atmosphere tables quote their results in terms of geopotential altitude, and care
must be taken to make the distinction. Again, geopotential altitude can be
thought of as that fictitious altitude which is physically compatible with the
assumption of g = const = g
0
.
3.3 RELATION BETWEEN GEOPOTENTIAL
AND GEOMETRIC ALTITUDES
We still seek the variation of p with geometric altitude, p = p(ha). However, our
calculations using Eq. (3.3) will give, instead, p = p ( h ). Therefore, we need to
THE STANDARD ATMOSPHERE 73
relate h to hG, as follows. Dividing Eq. (3.3) by (3.2), we obtain
or (3.4)
Substitute Eq. (3.1) into (3.4):
(3.5)
By convention, we set both h and hG equal to zero at sea level. Now, consider a
given point in the atmosphere. This point is at a certain geometric altitude hG,
and associated with it is a certain value of h (different from he). Integrating Eq.
(3.5) between sea level and the given point, we have
Thus
h  ,2   ,2  + 
(
1 ) he ( 1 1 )
r +he
0
r +he r
=r2(r+r+he)
(r + he)r
(3.6)
where h is geopotential altitude and he is geometric altitude. This is the desired
relation between the two altitudes. When we obtain relations such as p = p(h),
we can use Eq. (3.6) to subsequently relate p to he.
A quick calculation using Eq. (3.6) shows that there is little difference
between h and he for low altitudes. For such a case, he« r, r/(r +he)::::::: l,
hence h ::::::: hG. Putting in numbers, r = 6.356766 X 10
6
m (at a latitude of 45°),
and if he = 7 km (about 23,000 ft), then the corresponding value of h is, from
Eq. (3.6), h = 6.9923 km, about 0.1 of 1 percent difference! Only at altitudes
above 65 km (213,000 ft) does the difference exceed 1 percent. (It should be noted
that 65 km is an altitude where aerodynamic heating of NASA's space shuttle
becomes important during reentry into the earth's atmosphere from space.)
74 INTRODUCTION TO FLIGHT
3.4 DEFINITION OF THE STANDARD ATMOSPHERE
We are now in a position to obtain p, T, and p as functions of h for the standard
atmosphere. The keystone of the standard atmosphere is a defined variation of T
with altitude, based on experimental evidence. This variation is shown in Figure
3.3. Note that it consists of a series of straight lines, some vertical (called the
constanttemperature, or isothermal, regions) and some inclined (called the gradi
ent regions). Given T = T(h) as defined by Figure 3.3, then p = p(h) and
p = p( h) follow from the laws of physics, as shown below.
First, consider again Eq. (3.3):
dp = pg
0
dh
Divide by the equation of state, Eq. (2.3):
dp = _ pg
0
dh
p pRT
_kdh
RT
(3.7)
Consider first the isothermal (constanttemperature) layers of the standard atmo
sphere, as given by the vertical lines in Figure 3.3 and sketched in Figure 3.4. The
temperature, pressure, and density at the base of the isothermal layer shown in
100
80
60
40
20
200
a3 = 4.5 X 10
3
K/m
53
a1 = 6.5 X 10
3
K/m
240
Temperature, K
280 320
Figure 3.3 Temperature distribution in the standard atmosphere.
THE STANDARD ATMOSPHERE 75
Figure 3.4 are T
1
, p
1
, and p
1
, respectively. The base is located at a given
geopotential altitude h
1
. Now consider a given point in the isothermal layer above
the base, where the altitude is h. The pressure p at h can be obtained by
integrating Eq. (3.7) between h
1
and h.
!
Pdp = _ kJh dh
Pip RT hi
(3.8)
Note that g
0
, R, and Tare constants that can be taken outside the integral. (This
clearly demonstrates the simplification obtained by assuming that g = g
0
= const,
and therefore dealing with geopotential altitude h in the analysis.) Performing the
integration in Eq. (3.8), we obtain
or
From the equation of state:
Thus
ln.E._ =  k(h  h )
P1 RT i
.E_ = e(go/RT)(hhi)
P1
..£._ = pT = .P_
P1 P1T1 P1
.P_ = e(g
0
/RT)(hh
1
)
P1
(3.9)
(3.10)
Equations (3.9) and (3.10) give the variation of p and p versus geopotential
altitude for the isothermal layers of the standard atmosphere.
Considering the gradient layers, as sketched in Figure 3.5, we find the
temperature variation is linear and is geometrically given as
Isothermal layer
h T, p, p
h1
I
I
I
I

Ti, p1, P1 Figure 3.4 Isothermal layer.
76 INTRODUCTION TO FLIGHT
where a is a specified constant for each layer obtained from the defined
temperature variation in Figure 3.3. The value of a is sometimes called the lapse
rate for the gradient layers.
Thus
dT
a= dh
1
dh = dT
a
Substitute this result into Eq. (3.7):
dp
p
g
0
dT

aR T
(3.11)
Integrating between the base of the gradient layer (shown in Figure 3.5) and some
point at altitude h, also in the gradient layer, Eq. (3.11) yields
JP dp = _ 19_ 1TdT
p, p aR T, T
p go T
In= ln
P1 aR T
1
Thus
From the equation of state
h
Gradient region
1
I
I
I
I
I
I

T, p, P
Figure 3.5 Gradient layer.
a
Base of gradient region
I hi
I
I
I

T1, PI• PI
(3.12)
THE STANDARD ATMOSPHERE 77
Hence, Eq. (3.12) becomes
or (3.13)
Recall that the variation of T is linear with altitude and is given by the specified
relation
(3.14)
Equation (3.14) gives T = T(h) for the gradient layers; when it is plugged into
Eq. (3.12), we obtain p = p(h); similarly from Eq. (3.13) we obtain p = p(h).
Now we can see how the standard atmosphere is pieced together. Looking at
Figure 3.3, start at sea level ( h = 0), where standard sea level values of pressure,
density, and temperature, p,, p
5
, and T,, respectively, are
Ps = 1.01325 X 10
5
N/m
2
= 2116.2 lb/ft
2
Ps = 1.2250 kg/m
3
= 0.002377 slug/ft
3
T, = 288.16 K = 518.69°R
These are the base values for the first gradient region. Use Eq. (3.14) to obtain
values of T as a function of h until T = 216.66 K, which occurs at h = 11.0 km.
With these values of T use Eqs. (3.12) and (3.13) to obtain the corresponding
values of p and p in the first gradient layer. Next, starting at h = 11.0 km as the
base of the first isothermal region (see Figure 3.3), use Eqs. (3.9) and (3.10) to
calculate values of p and p versus h, until h = 25 km, which is the base of the
next gradient region. In this manner, with Figure 3.3 and Eqs. (3.9), (3.10), and
(3.12) to (3.14), a table of values for the standard atmosphere can be constructed.
Such a table is given in Appendix A for SI units and Appendix B for English
engineering units. Look at these tables carefully and become familiar with them.
They are the standard atmosphere. The first column gives the geometric altitude,
and the second column gives the corresponding geopotential altitude obtained
from Eq. (3.6). The third through fifth columns give the corresponding standard
values of temperature, pressure, and density, respectively, for each altitude,
obtained from the discussion above.
We emphasize again that the standard atmosphere is a reference atmosphere
only and certainly does not predict the actual atmospheric properties that may
exist at a given time and place. For example, Appendix A says that at an altitude
(geometric) of 3 km, p = 0.70121 x 10
5
N/m
2
, T = 268.67 K, and p = 0.90926
78 INTRODUCTION TO FLIGHT
kg/m
3
• In reality, situated where you are, if you could right now levitate yourself
to 3 km above sea level, you would most likely feel a p, T, and p different from
the above values obtained from Appendix A. The standard atmosphere allows us
only to reduce test data and calculations to a convenient, agreedupon reference,
as will be seen in subsequent sections of this book.
Example 3.1 Calculate the standard atmosphere values of T, p, and p at a geopotential altitude
of 14 km.
SOLUTION Remember that T is a defined variation for the standard atmosphere. Hence, we can
immediately refer to Figure 3.3 and find that at h = 14 km,
T= 216.66 K
To obtain p and p, we must use Eqs. (3.9) to (3.14), piecing together the different regions from
sea level up to the given altitude with which we are concerned. Beginning at sea level, the first
region (from Figure 3.3) is a gradient region from h = 0 to h = 11.0 km. The lapse rate is
= dT = 216.66288.16 = _
6 5
K/k
a dh ll.00 · m
or a= 0.0065 K/m
Therefore, using Eqs. (3.12) and (3.13), which are for a gradient region and where the base of the
region is sea level (hence p
1
=1.01X10
5
N/m
2
and p
1
=1.23 kg/m3 ), we find at h = 11.0 km,
_ ( I_)go/aR _
5
(
216
.
66
)9.8/0.0065(287)
P  P1 T1  (1.01X10 ) 288.16
where g
0
= 9.8 m/s
2
in the SI units. Hence, p (at h = 11.0 km)= 2.26x10
4
N/m
2
.
(
T )(g0 /aR+l)
P =p1 T1
 ( 216.66 )[9.8/0.0065(287)+1)
p  (1.
23
) 288.16
p = 0.367 kg/m
3
at h =ll.O km
The above values of p and p now form the base values for the first isothermal region (see Figure
3.3). The equations for the isothermal region are Eqs. (3.9) and (3.10), where now Pl = 2.26X10
4
N/m
2
and p
1
=0.367 kg/m3. For h =14 km, h  h
1
=1411=3 km= 3000 m. From Eq. (3.9),
p =Pl e(go/RT)(hh1l = (2.26X104) e[9.8/287(216.66))(3000)
Ip =1.41 X10
4
N/m
2
I
From Eq. (3.10),
Hence p = p
1
 = 0.367 = 0.23 kg/m:
(
p ) 1.41x10
4
I I
P1 2.26X10
4
THE STANDARD ATMOSPHERE 79
These values check, within roundoff error, with the values given in Appendix A. Note: This
example demonstrates how the numbers in Appendixes A and B are obtained!
3.5 PRESSURE, TEMPERATURE, AND DENSITY ALTITUDES
With the tables of Appendixes A and B in hand, we can now define three new
"altitudes" pressure, temperature, and density altitudes. This is best done by
example. Imagine that you are in an airplane flying at some real, geometric
altitude. The value of your actual altitude is immaterial for this discussion.
However, at this altitude, you measure the actual outside air pressure to be
6.16 x 10
4
N/m
2
• From Appendix A, you find that the standard altitude that
corresponds to a pressure of 6.16 X 10
4
N/m
2
is 4 km. Therefore, by definition,
you say that you are flying at a pressure altitude of 4 km. Simultaneously, you
measure the actual outside air temperature to be 265.4 K. From Appendix A, you
find that the standard altitude that corresponds to a temperature of 265.4 K is 3.5
km. Therefore, by definition, you say that you are flying at a temperature altitude
of 3.5 km. Thus, you are simultaneously flying at a pressure altitude of 4 km and
a temperature altitude of 3.5 km while your actual geometric altitude is yet a
different value. The definition of density altitude is made in the same vein. These
quantitiespressure, temperature, and density altitudesare just convenient
numbers that, via Appendix A or B, are related to the actual p, T, and p for the
actual altitude at which you are flying.
Example 3.2 If an airplane is flying at an altitude where the actual pressure and temperature are
4.72X10
4
N/m
2
and 255.7 K, respectively, what are the pressure, temperature, and density
altitudes?
SOLUTION For the pressure altitude, look in Appendix A for the standard altitude value
corresponding top= 4.72X10
4
N/m
2
. This is 6000 m. Hence
I Pressure altitude= 6000 m = 6 km I
For the temperature altitude, look in Appendix A for the standard altitude value corresponding
to T = 255. 7 K. This is 5000 m. Hence
I Temperature altitude = 5000 m = 5 km I
For the density altitude, we must first calculate p from the equation of state:
= ..J!_ =
4
·
72
Xl0
4
= O 643 k m
3
P RT 287(255.7) . g/
Looking in Appendix A and interpolating between 6.2 km and 6.3 km, we find that the standard
altitude value corresponding to p = 0.643 kg/m3 is about 6.240 m. Hence
I Density altitude= 6240 m = 6.24 km I
80 INTRODUCTION TO FLIGHT
It should be noted that temperature altitude is not a unique value. The above answer for
temperature altitude could equally well be 5.0, 38.2, or 59.5 km because of the multivalued nature
of the altitudeversustemperature function. In this section, only the lowest value of temperature
altitude is used.
Example 3.3 The flight test data for a given airplane refer to a levelflight maximumvelocity run
made at an altitude which simultaneously corresponded to a pressure altitude of 30,000 ft and a
density altitude of 28,500 ft. Calculate the temperature of the air at the altitude at which the
airplane was flying for the test.
SOLUTION From Appendix B
For pressure altitude= 30,000 ft: p = 629.66 lb/ft
2
For density altitude = 28,500 ft: p = 0.9408 x 10
3
slug/ft
3
These are the values of p and p that simultaneously existed at the altitude at which the airplane
was flying. Therefore, from the equation of state
 = 390°R T 
 _P  629.66 B
pR (0.94082 x 10
3
)(1716)
3.6 HISTORICAL NOTE: THE STANDARD ATMOSPHERE
With the advent of ballooning in 1783 (see Chap. 1), people suddenly became
interested in acquiring a greater understanding of the properties of the atmo
sphere above ground level. However, a compelling reason for such knowledge did
not arise until the coming of heavierthanair flight in the twentieth century. As
we shall see in subsequent chapters, the flight performance of aircraft is depen
dent upon such properties as pressure and density of the air. Thus, a knowledge
of these properties, or at least some agreedupon standard for worldwide refer
ence, is absolutely necessary for intelligent aeronautical engineering.
The situation in 1915 was summarized by C. F. Marvin, Chief of the U.S.
Weather Bureau and chairman of an NACA subcommittee to investigate and
report upon the existing status of atmospheric data and knowledge. In his
"Preliminary Report on the Problem of the Atmosphere in Relation to
Aeronautics," NACA Report No. 4, 1915, Marvin writes:
The Weather Bureau is already in possession of an immense amount of data concerning
atmospheric conditions, including wind movements at the earth's surface. This information is no
doubt of distinct value to aeronautical operations, but it needs to be collected and put in form to
meet the requirements of aviation.
The following four years saw such efforts to collect and organize atmospheric
data for use by aeronautical engineers. In 1920, the Frenchman A. Toussaint,
director of the Aerodynamic Laboratory at SaintCyrl'Ecole, France, suggested
the following formula for the temperature decrease with height:
T = 15  0.0065h
THE STANDARD ATMOSPHERE 81
where T is in degrees Celsius and h is the geopotential altitude in meters.
Toussaint's formula was formally adopted by France and Italy with the Draft of
InterAllied Agreement on Law Adopted for the Decrease of Temperature with
Increase of Altitude, issued by the Ministere de la Guerre, Aeronautique Militaire,
Section Technique, in March 1920. One year later, England followed suit. The
United States was close behind. Since Marvin's report in 1915, the U.S. Weather
Bureau had compiled measurements of the temperature distribution and found
Toussaint's formula to be a reasonable representation of the observed mean
annual values. Therefore, at its executive committee meeting of December 17,
1921, the NACA adopted Toussaint's formula for airplane performance testing,
with the statement: "The subcommittee on aerodynamics recommends that for
the sake of uniform practice in different countries that Toussaint's formula be
adopted in determining the standard atmosphere up to 10 km (33,000 ft) .... "
Much of the technical data base that supported Toussaint's formula was
reported in NACA Report No. 147, "Standard Atmosphere," by Willis Ray
Gregg in 1922. Based on freeflight tests at McCook Field in Dayton, Ohio, and
at Langley Field in Hampton, Virginia, and on other flights at Washington, D.C.,
as well as artillery data from Aberdeen, Maryland, and Dahlgren, Virginia, and
soundingballoon observations at Fort Omaha, Nebraska, and at St. Louis,
Missouri, Gregg was able to compile a table of mean annual atmospheric
properties. An example of his results is as follows:
Mean Annual Temperature
Temperature in from Toussaint's
Altitude, United States, Formula,
m K K
0 284.5 288
1,000 281.0 281.5
2,000 277.0 275.0
5,000 260.0 255.5
10,000 228.5 223.0
Clearly, Toussaint's formula provided a simple and reasonable representation of
the mean annual results in the United States. This was the primary message in
Gregg's report in 1922. However, the report neither gave extensive tables nor
attempted to provide a document for engineering use.
Thus, it fell to Walter S. Diehl (who later became a wellknown aerodynami
cist and airplane designer as a captain in the Naval Bureau of Aeronautics) to
provide the first practical tables for a standard atmosphere for aeronautical use.
In 1925, in NACA Report No. TR 218, entitled (again) "Standard Atmosphere,"
Diehl presented extensive tables of standard atmospheric properties in both
metric and English units. The tables were in increments of 50 m up to an altitude
of 10 km and then in increments of 100 m up to 20 km. In English units, the
tables were in increments of 100 ft up to 32,000 ft and then in increments of 200
ft up to a maximum altitude of 65,000 ft. Considering the aircraft of that day (see
82 INTRODUCTION TO FLIGHT
Figure 1.31), these tables were certainly sufficient. Moreover, starting from
Toussaint's formula for T up to 10,769 m, then assuming T = const =  55°C
above 10,769 m, Diehl obtained p and p in precisely the same fashion as
described in the previous sections of this chapter.
The 1940s saw the beginning of serious rocket flights, with the German V2
and the initiation of sounding rockets. Moreover, airplanes were flying higher
than ever. Then, with the advent of intercontinental ballistic missiles in the 1950s
and space flight in the 1960s, altitudes became quoted in terms of hundreds of
miles rather than feet. Therefore, new tables of the standard atmosphere were
created, mainly extending the old tables to higher altitudes. Popular among the
various tables is the ARDC 1959 Standard Atmosphere, which is used in this
book and is given in Appendixes A and B. For all practical purposes, the old and
new tables agree for altitudes of most interest. Indeed, it is interesting to compare
values, as shown below:
T from T from
Altitude, Diehl, 1925, ARDC, 1959,
m K K
0 288 288.16
1,000 281.5 281.66
2,000 275.0 275.16
5,000 255.5 255.69
10,000 223.0 223.26
10,800 218.0 218.03
11,100 218.0 216.66
20,000 218.0 216.66
So Diehl's standard atmosphere from 1925, at least up to 20 km, is just as
good as the values today.
3.7 CHAPTER.SUMMARY
Some of the major ideas of this chapter are listed below.
1. The standard atmosphere is defined in order to relate flight tests, windtunnel
results, and general airplane design and performance to a common reference.
2. The definitions of the standard atmospheric properties are based on a given
temperature variation with altitude, representing a mean of experimental
data. In turn, the pressure and density variations with altitude are obtained
from this empirical temperature variation by using the laws of physics. One of
these laws is the hydrostatic equation:
dp = pgdhc (3.2)
3. In the isothermal regions of the standard atmosphere, the pressure and
density variations are given by
}!_ = _E_ = e(g0 /RT)(hhi)
Pi Pi
(3.9) and (3.10)
THE STANDARD ATMOSPHERE 83
4. In the gradient regions of the standard atmosphere, the pressure and density
variations are given by
:l = ( )go/aR
(3.12)
.E_ = (I_ )[(g
0
/aR)+ 1]
P1 T1
(3.13)
where T = T
1
+ a(h  h
1
) and a is the given lapse rate.
5. The pressure altitude is that altitude in the standard atmosphere which
corresponds to the actual ambient pressure encountered in. flight or labora
tory experiments. For example, if the ambient pressure of a flow, no matter
where it is or what it is doing, is 393.12 lb/ft
2
, the flow is said to correspond
to a pressure altitude of 40,000 ft (see Appendix B). The same idea can be
used to define density and temperature altitudes.
BIBLIOGRAPHY
Minzner, R. A., Champion, K. S. W., and Pond, H. L., The ARDC Model Atmosphere, 1959, Air Force
Cambridge Research Center Report No. TR59267, U.S. Air Force, Bedford, MA, 1959.
PROBLEMS
3.1 At 12 km in the standard atmosphere, the pressure, density, and temperature are 1.9399 x 10
4
N/m2, 3.1194 X 10
1
kg/m3, and 216.66 K, respectively. Using these values, calculate the standard
atmospheric values of pressure, density, and temperature at an altitude of 18 km, and check with the
standard altitude tables.
3.2 Consider an airplane flying at some real altitude. The outside pressure and temperature are
2.65 X 10
4
N/m2 and 220 K, respectively. What are the pressure and density altitudes?
3.3 During a flight test of a new airplane the pilot radios to the ground that she is in level flight at a
standard altitude of 35,000 ft. What is the ambient air pressure far aliead of the airplane?
3.4 Consider an airplane flying at a pressure altitude of 33,500 ft and a density altitude of 32,000 ft.
Calculate the outside air temperature.
3.5 At what value of the geometric altitude is the difference h  hG equal to 2 percent of h?
3.6 Using Toussaint's formula, calculate the pressure at a geopotential altitude of 5 km.
3.7 The atmosphere of Jupiter is essentially made up of hydrogen, H
2
. For H
2
, the specific gas
constant is 4157 J/(kg)(K). The acceleration of gravity of Jupiter is 24.9 m/s
2
. Assuming an
isothermal atmosphere with a temperature of 150 K, and assuming that Jupiter has a definable
surface, calculate the altitude above that surface where the pressure is onehalf the surface pressure.
3.8 An F15 supersonic fighter aircraft is in a rapid climb. At the instant it passes through a standard
altitude of 25,000 ft, its time rate of change of altitude is 500 ftjs, which by definition is the rate of
climb, discussed in Chap. 6. Corresponding to this rate of climb at 25,000 ft is a time rate of change of
ambient presure. Calculate this rate of change of pressure in units of lb/(ft
2
)(s).
CHAPTER
FOUR
BASIC AERODYNAMICS
Mathematics up to the present day have been quite useless to us in regard to flying.
From the fourteenth Annual Report of
the Aeronautical Society of Great Britain, 1879
Mathematical theories from the happy hunting grounds of pure mathematicians are found suitable to
describe the airflow produced by aircraft with such excellent accuracy that they can be applied
directly to airplane design.
Theodore von Karman, 1954
Consider an airplane flying at an altitude of 3 km (9840 ft) at a velocity of 112
m/s (367 ftjs or 251 mi/h). At a given point on the wing, the pressure and
airflow velocity are specific values, dictated by the laws of nature. One of the
objectives of the science of aerodynamics is to decipher these laws and to give us
methods to calculate the flow properties. In turn, such information allows us to
calculate practical quantities, such as the lift and drag on the airplane. Another
example is the flow through a rocket engine of a given size and shape. If this
engine is sitting on the launch pad at Cape Canaveral and given amounts of fuel
and oxidizer are ignited in the combustion chamber, the flow velocity and
pressure at the nozzle exit are again specific values dictated by the laws of nature.
The basic principles of aerodynamics allow us to calculate the exit flow velocity
and pressure, which in turn allow us to calculate the thrust. For such as
these, the study of aerodynamics is vital to the overall understanding of flight.
The purpose of this chapter is to provide an introduction to the basic laws and
concepts of aerodynamics and to show how they are applied to solving practical
problems.
84
',
'
/
/
\
I
I
I
I
I
I
I
I
I
I
I
I
I
I
4.1 THE CONTINUITY EQUATION
BASIC AERODYNAMICS 85
Figure 4.1 Stream tube with mass conserva
tion.
The laws of aerodynamics are formulated by applying to a flowing gas several
basic principles from physics. For example,
Physical principle: Mass can be neither created nor destroyed.*
To apply this principle to a flowing gas, consider an imaginary circle drawn
perpendicular to the flow direction, as shown in Figure 4.1. Now look at all the
streamlines that go through the circumference of the circle. These streamlines
form a tube, called a stream tube. As we move along with the gas confined inside
the stream tube, we see that the crosssectional area of the tube may change, say
in moving from point I to point 2 in Figure 4.1. However, as long as the flow is
steady (invariant with time), the mass that flows through the cross section at point
I must be the same as the mass that flows through the cross section at point 2,
because by the definition of a streamline, there can be no flow across streamlines.
The mass flowing through the stream tube is confined by the streamlines of the
boundary, much as the flow of water through a flexible garden hose is confined by
the wall of the hose.
Let A
1
be the crosssectional area of the stream tube at point 1. Let V
1
be the
flow velocity at point 1. Now, at a given instant in time, consider all the fluid
elements that are momentarily in the plane of A
1
. After a lapse of time dt, these
same fluid elements all move a distance Vi dt; as shown in Figure 4.1. In so doing,
the elements have swept out a volume ( A
1
Vi dt) downstream of point 1. The mass
of gas, dm, in this volume is equal to the density times the volume, i.e.,
(4.1}
This is the mass of gas that has swept through area A
1
during the time interval dt.
Definition: The mass flow m through area A is the mass crossing A per unit
time.
* Of course, Einstein has shown that e = mc
2
and hence mass is truly not conserved in situations
where energy is released. However, for any noticeable change in mass to occur, the energy release must
be tremendous, such as occurs in a nuclear reaction. We are generally not concerned with such a case
in practical aerodynamics.
86 INTRODUCTION TO FLIGHT
Therefore, from Eq. (4.1), for area A
1
,
dm
Mass flow= ;ft= rh
1
= p
1
A
1
V
1
kg/s or slugs/s
Also, the mass flow through A
2
, bounded by the same streamlines that go
through the circumference of A
1
, is obtained in the same fashion, as
Since mass can be neither created nor destroyed, we have rh
1
= m
2
• Hence
(4.2)
This is the continuity equation for steady fluid flow. It is a simple algebraic
equation which relates the values of density, velocity, and area at one section of
the stream tube to the same quantities at any other section.
4.2 INCOMPRESSIBLE AND COMPRESSIBLE FLOW
Before we proceed any further, it is necessary to point out that all matter in real
life is compressible to some greater or lesser extent. That is, if we take an element
of matter and squeeze on it hard enough with some pressure, the volume of the
element of matter will decrease. However, its mass will stay the same. This is
shown schematically in Figure 4.2. As a result, the density p of the element
changes as it is squeezed. The amount by which p changes depends on the nature
of the material of the element and how hard we squeeze it, i.e., the magnitude of
the pressure. If the material is solid, such as steel, the change in volume is
insignificantly small and p is constant for all practical purposes. If the material is
a liquid, such as water, the change in volume is also very small and again p is
essentially constant. (Try pushing a tightly fitting lid into a container of liquid,
and you will find out just how "solid" the liquid can be.) On the other hand, if
the material is a gas, the volume can readily change and p can be a variable.
P2 >Pl
m
P2 = m/v2
Figure 4.2 Illustration of compressibility.
BASIC AERODYNAMICS 87
The preceding discussion allows us to characterize two classes of aerodynamic
flow: compressible flow and incompressible flow.
1. Compressible flowflow in which the density of the fluid elements can change
from point to point. Referring to Eq. ( 4.2), we see if the flow is compressible,
p
1
if. p
2
• The variability of density in aerodynamic flows is particularly
important at high speeds, such as for highperformance subsonic aircraft, all
supersonic vehicles, or rocket engines. Indeed, all reallife flows, strictly
speaking, are compressible. However, there are some circumstances in which
the density changes only slightly. These circumstances lead to the second
definition, as follows.
2. Incompressible flowflow in which the density of the fluid elements is always
constant. Referring to Eq. (4.2), we see if the flow is incompressible, p
1
= p
2
,
hence
(4.3)
Incompressible flow is a myth. It can never actually occur in nature, as discussed
above. However, for those flows in which the actual variation of p is negligibly
small, it is convenient to make the assumption that p is constant in order to
simplify our analysis. (Indeed, it is an everyday activity of engineering and
physical science to make idealized assumptions about real physical systems in
order to make such systems amenable to analysis. However, care must always be
taken not to apply results obtained from such idealizations to those real problems
where the assumptions are grossly inaccurate or inappropriate.) The assumption
of incompressible flow is an excellent approximation for the flow of liquids, such
as water or oil. Moreover, the lowspeed flow of air, where V < 100 m/s (or
V < 225 mi/h) can also be assumed to be incompressible to a close approxima
tion. A glance at Figure 1.27 shows that such velocities were the domain of almost
all airplanes from the Wright Flyer (1903) to the late 1930s. Hence, the early
development of aerodynamics always dealt with incompressible flows, and for this
reason there exists a huge body of incompressible flow literature with its atten
dant technology. At the end of this chapter, we will be able to prove why airflow
at velocities less than 100 m/s can be safely assumed to be incompressible.
In solving and examining aerodynamic flows, you will constantly be faced
with making distinctions between incompressible and compressible flows. It is
important to start that habit now, because there are some striking quantitative
and qualitative differences between the two types of flow.
As a parenthetical comment, for incompressible flow, Eq. (4.3) explains why
all common garden hose nozzles are convergent shapes, such as shown in Figure
4.3. From Eq. (4.3),
If A
2
is less than A
1
, then the velocity increases as the water flows through the
nozzle, as desired. The same principle is used in the design of nozzles for subsonic
88 INTRODUCTION TO FLIGHT
Ai <A1
Hence,
V2 >Vi
Figure 4.3 Incompressible flow in a conver
gent duct.
wind tunnels built for aerodynamic testing, as will be discussed in a subsequent
section.
Example 4.1 Consider a convergent duct with an inlet area A
1
= 5 m
2
. Air enters this duct with a
velocity V
1
= 10 m/s and leaves the duct exit with a velocity V
2
= 30 m/s. What is the area of the
duct exit?
SOLUTION Since the flow velocities are less than 100 m/s, we can assume incompressible flow.
From Eq. (4.3)
4.3 THE MOMENTUM EQUATION
The continuity equation, Eq. (4.2), is only part of the story. For example, it says
nothing about the pressure in the flow; yet, we know, just from intuition, that
pressure is an important flow variable. Indeed, differences in pressure from one
point to another in the flow create forces that act on the fluid elements and cause
them to move. Hence, there must be some relation between pressure and velocity,
and that relation will be derived in this section.
Again, we first state a fundamental law of physics, namely, Newton's second
law.
Physical principle: Force = mass X acceleration
or F= ma ( 4.4)
To apply this principle to a flowing gas, consider an infinitesimally small fluid
element moving along a streamline with velocity V, as shown in Figure 4.4. At
some given instant, the element is located at point P. The element is moving in
the x direction, where the x axis is oriented parallel to the streamline at point P.
The y and z axes are mutually perpendicular to x. The fluid element is very
small, infinitesimally small. However, looking at it through a magnifying glass, we
see the picture shown at the upper right of Figure 4.4. Question: What is the force
BASIC AERODYNAMICS 89
: :Jl
   t :  ~ l ___    ~
dx dz
y
Figure 4.4 Force diagram for the momentum equation.
p + (dp )dx
dx
on this element? Physically, the force is a combination of three phenomena:
1. Pressure acting in a normal direction on all six faces of the element
2. Frictional shear acting tangentially on all six faces of the element
3. Gravity acting on the mass inside the element
For the time being, we will ignore the presence of frictional forces; moreover, the
gravity force is generally a small contribution to the total force. Therefore, we will
assume that the only source of a force on the fluid element is pressure.
To calculate this force, let the dimensions of the fluid element be dx, dy, and
dz, as shown in Figure 4.4. Consider the left and right faces, which are perpendic
ular to the x axis. The pressure on the left face is p. The area of the left face is
dy dz, hence the force on the left face is p ( dy dz). This force is in the positive x
direction. Now recall that pressure varies from point to point in the flow. Hence,
there is some change in pressure per unit length, symbolized by the derivative
dp / dx. Thus, if we move away from the left face by a distance dx along the x
axis, the change in pressure is ( dp / dx) dx. Consequently, the pressure on the
right face is p + ( dp / dx) dx. The area of the right face is also dy dz, hence the
force on the right face is [ p + ( dp / dx) dx ]( dy dz). This force acts in the negative
x direction, as shown in Figure 4.4. The net force in the x direction, F, is the sum
of the two:
or
F = pdydz ( p + 1x dx) dydz
dp
F =  (dxdydz)
dx
(4.5)
Equation ( 4.5) gives the force on the fluid element due to pressure. Because of the
convenience of choosing the x axis in the flow direction, the pressures on
90 INTRODUCTION TO FLIGHT
the faces parallel to the streamlines do not affect the motion of the element
along the streamline.
The mass of the fluid element is the density p multiplied by the volume
dxdydz:
m = p(dxdydz) (4.6)
Also, the acceleration a of the fluid element is, by definition of acceleration (rate
of change of velocity), a= dV/dt. Noting that, also by definition, V = dx/dt, we
can write
a = dV = dV dx = dV V
dt dx dt dx
(4.7)
Equations (4.5) to (4.7) give the force, mass, and acceleration, respectively,
that go into Newton's second law, Eq. (4.4):
F= ma
dp dV
 dx (dxdydz) = p(dxdydz)V dx
or
\ dp = pvdv\
(4.8)
Equation ( 4.8) is Euler's equation. Basically, it relates rate of change of
momentum to the force; hence it can also be designated as the momentum
equation. It is important to keep in mind the assumptions utilized in obtaining Eq.
(4.8); we neglected friction and gravity. For flow which is frictionless, aerody
namicists sometimes use another term, inviscid flow. Equation (4.8) is the
momentum equation for inviscid (frictionless) flow. Moreover, the flow field is
assumed to be steady, i.e., invariant with respect to time.
Please note that Eq. (4.8) relates pressure and velocity (in reality, it relates a
change in pressure, dp, to a change in velocity, dV). Equation (4.8) is a
differential equation, and hence it describes the phenomena in an infinitesimally
small neighborhood around the given point P in Figure 4.4. Now consider two
points, 1 and 2, far removed from each other in the flow but on the same
streamline. In order to relate p
1
and V
1
at point 1 to p
2
and V
2
at the other,
farremoved point 2, Eq. (4.8) must be integrated between points 1 and 2. This
integration is different depending on whether the flow is compressible or incom
pressible. Euler's equation itself, Eq. (4.8), holds for both cases. For compressible
flow, p in Eq. (4.8) is a variable; for incompressible flow, p is a constant.
First, consider the case of incompressible flow. Let points 1 and 2 be located
2
Figure 4.5 Two points at different loca
tions along a streamline.
BASIC AERODYNAMICS 91
along a given streamline, such as that shown over an airfoil in Figure 4.5. From
Eq. (4.8),
dp + pVdV= 0
where p = const. Integrating between points 1 and 2, we obtain
f
P2 f "'i
dp + p VdV = 0
P1 V1
v/ v/
P2 +PT= P1 +PT
(4.9a)
v2
p + p 2 = const along streamline
( 4.9b)
Either form, Eq. ( 4.9a) or ( 4.9b ), is called Bernoulli's equation. Historically,
Bernoulli's equation is one of the most fundamental equations in fluid mechanics.
The following important points should be noted:
1. Equations (4.9a) and (4.9b) hold only for inviscid (frictionless), incom
pressible flow.
2. Equations (4.9a) and (4.9b) relate properties between different points along a
streamline.
3. For a compressible flow, Eq. (4.8) must be used, with p treated as a variable.
Bernoulli's equation must not be used for compressible flow.
4. Remember that Eqs. (4.8) and (4.9a) and (4.9b) say that F = ma for a fluid
flow. They are essentially Newton's second law applied to fluid dynamics.
Returning to Figure 4.5, if all the streamlines have the same value of p and V
far upstream (far to the left in Figure 4.5), then the constant in Bernoulli's
equation is the same for all streamlines. This would be the case, for example, if the
flow far upstream was uniform flow, such as that encountered in flight through
the atmosphere and in the test sections of welldesigned wind tunnels. In such
cases, Eqs. (4.9a) and (4.9b) are not limited to the same streamline. Instead,
points 1 and 2 can be anywhere in the flow, even on different streamlines.
For the case of compressible flow also, Euler's equation, Eq. (4.8), can be
integrated between points 1 and 2; however, because p is a variable, we must in
principle have some extra information on how p varies with V before the
integration can be carried out. This information can be obtained; however,
there is an alternate, more convenient route to treating many practical problems
in compressible flow that does not explicitly require the use of the momentum
equation. Hence, in this case, we will not pursue the integration of Eq. ( 4.8)
further.
92 INTRODUCTION TO FLIGHT
4.4 A COMMENT
It is important to make a philosophical distinction between the nature of the
equation of state, Eq. (2.3), and the flow equations of continuity, Eq. (4.2), and
momentum, such as Eq. (4.9a). The equation of state relates p, T, and p to each
other at the same point; in contrast, the flow equations relate p and V (as in the
continuity equation) and p and V (as in Bernoulli's equation) at one point in the
flow to the same quantities at another point in the flow. There is a basic difference
here, and it is well to keep it in mind when setting up the solution of aerodynamic
problems.
Example 4.2 Consider an airfoil (the cross section of a wing as shown in Figure 4.5) in a flow of
air, where far ahead (upstream) of the airfoil, the pressure, velocity, and density are 2116 lb/ft
2
,
100 mijh, and 0.002377 slug/ft
3
, respectively. At a given point A on the airfoil, the pressure is
2070 lb/ft
2
• What is the velocity at point A?
SOLUTION First, we must deal in consistent units; V
1
=100 mi/h is not in consistent units.
However, a convenient relation to remember is that 60 mijh = 88 ft/s. Hence, V
1
= 100(88/60)
= 146. 7 ftjs. This velocity is low enough that we can assume incompressible flow. Hence,
Bernoulli's equation, Eq. (4.9), is valid.
pV/ pV/
P1 + 2 =PA + 2
Thus
V =[2(p1PA) +v,2]112
,,, p I
= [ 2(2116 2070) + (146 )2]
112
0.002377 ·
7
I VA= 245.4 ftjs I
Example 4.3 Consider the same convergent duct and conditions as in Example 4.1. If the air
pressure and temperature at the inlet are p
1
= 1.2x10
5
N/m
2
and T
1
= 330 K, calculate the
pressure at the exit.
SoLUTION First, we must obtain the density. From the equation of state,
= _.!!J_ = 1.2X105 = 1 27 k /m3
Pi RTi_ 287(330) . g
Still assuming incompressible flow, Eq. (4.9) gives
pV12 pVz2
Pi +
2
 = P2 +2
I p2 =l.195xl0
5
N/m
2
I
Note: In accelerating from 10 to 30 m/s, the air pressure decreases only a small amount, less
than 0.45%. This is a characteristic of very low velocity airflow.
BASIC AERODYNAMICS 93
4.5 ELEMENTARY THERMODYNAMICS
As stated earlier, when the airflow ve1ocity exceeds 100 m/s, the flow can no
longer be treated as incompressible. Later on, we shall restate this criterion in
terms of the Mach number, which is the ratio of the flow velocity to the speed of
sound, and we will show that the flow must be treated as compressible when the
Mach number exceeds 0.3. This is the situation with the vast majority of current
aerodynamic applications; hence, the study of compressible flow is of extreme
importance.
A highspeed flow of gas is also a highenergy flow. The kinetic energy of the
fluid elements in a highspeed flow is large and must be taken into account. When
highspeed flows are slowed down, the consequent reduction in kinetic energy
appears as a substantial increase in temperature. As a result, highspeed flows,
compressibility, and vast energy changes are all related. Thus, to study com
pressible flows, we must first examine some of the fundamentals of energy
changes in a gas and the consequent response of pressure and temperature to
these energy changes. Such fundamentals are the essence of the science of
thermodynamics.
Here, the assumption is made that the reader is not familiar with thermody
namics. Therefore, the purpose of this section is to introduce those ideas and
results of thermodynamics which are absolutely necessary for our further analysis
of highspeed, compressible flows.
The pillar of thermodynamics is a relationship called the first law, which is an
empirical observation of natural phenomena. It can be developed as follows.
Consider a fixed mass of gas (for convenience, say a unit mass) contained within a
flexible boundary, as shown in Figure 4.6. This mass is called the system, and
everything outside the boundary is the surroundings. Now, as in Chap. 2, consider
the gas making up the system to be composed of individual molecules moving
about with random motion. The energy of this molecular motion, summed over
all the molecules in the system, is called the internal energy of the system. Let e
denote the internal energy per unit mass of gas. The only means by which e can
be increased (or decreased) are the following:
1. Heat added to (or taken away from) the system. This heat comes from the
surroundings and is added to the system across the boundary. Let 8q be an
incremental amount of heat added per unit mass.
Surroundings
oq
Figure 4.6 System of unit mass.
94 INTRODUCTION TO FLIGHT
2. Work done on (or by) the system. This work can be manifested by the
boundary of the system being pushed in (work done on the system) or pushed
out (work done by the system). Let 8w be an incremental amount of work
done on the system per unit mass.
Also, let de be the corresponding change in internal energy per unit mass.
Then, simply on the basis of common sense, confirmed by laboratory results, we
can write
(4.10)
Equation ( 4.10) is termed the first law of thermodynamics. It is an energy equation
which states that the change in internal energy is equal to the sum of the heat
added to and the work done on the system. (Note in the above that 8 and d both
represent infinitesimally small quantities; however, d is a "perfect differential"
and 8 is not.)
Equation (4.10) is very fundamental; however, it is not in a practical form for
use in aerodynamics, which speaks in terms of pressures, velocities, etc. To obtain
more useful forms of the first law, we must first derive an expression for 8w in
terms of p and v (specific volume), as follows. Consider the system sketched in
Figure 4.7. Let dA be an incremental surface area of the boundary. Assume that
work .'.lW is being done on the system by dA being pushed in a small distances,
as also shown in Figure 4.7. Since work is defined as force times distance, we have
.'.lW = (force)( distance)
.'.lW = (pdA)s (4.11)
Now assume that many elemental surface areas of the type shown in Figure 4.7
are distributed over the total surface area A of the boundary. Also, assume that
all the elemental surfaces are being simultaneously displaced a small distance s
into the system. Then, the total work 8w done on the unit mass of gas inside the
system is the sum (integral) of each elemental surface over the whole boundary,
that is, from Eq. (4.11),
A
Boundary
surface
8w = J (pdA)s = J psdA
A A
(4.12)
Figure 4.7 Work being done on the
system by pressure.
BASIC AERODYNAMICS 95
Assume p is constant everywhere in the system (which, in thermodynamic terms,
contributes to a state of thermodynamic equilibrium). Then, from Eq. (4.12),
8w = p j sdA
A
(4.13)
The integral fAs dA has physical meaning. Geometrically, it is the change in
volume of the unit mass of gas inside the system, created by the boundary surface
being displaced inward. Let du be the change in volume. Since the boundary is
pushing in, the volume decreases (du is a negative quantity) and work is done on
the gas (hence 8w is a positive quantity in our development). Thus
l sdA =du
Substituting Eq. (4.14) into Eq. (4.13), we obtain
J sw = pdu J
(4.14)
(4.15)
Equation (4.15) gives the relation for work done strictly in terms of the thermody
namic variables p and u.
When Eq. (4.15) is substituted into Eq. (4.10), the first law becomes
Jsq=de+pduJ (4.16)
Equation (4.16) is an alternate form of the first law of thermodynamics.
It is convenient to define a new quantity called enthalpy h as
h = e + pu = e + RT (4.17)
where pu = RT, assuming a perfect gas. Then, differentiating the definition, Eq.
(4.17), we find
dh =de+ pdu + udp
Substituting Eq. (4.18) into (4.16), we obtain
8q= de + p du = ( dh  p du  u dp) + p du
J sq = dh  u dp I
Equation ( 4.19) is yet another alternate form of the first law.
(4.18)
(4.19)
Before we go further, remember that a substantial part of science and
engineering is simply the language. In this section, we are presenting some of the
language of thermodynamics essential to our future aerodynamic applications.
We continue developing this language.
Figures 4.6 and 4.7 illustrate systems to which heat 8q is added and on
which work 8w is done. At the same time, 8q and 8w may cause the pressure,
temperature, and density of the system to change. The way (or means) by which
changes of the thermodynamic variables (p, T, p, u) of a system take place is
96 INTRODUCTION TO FLIGHT
O
oqHeat
v added
Rigid boundary
(such as a hollow
sphere of
constant volume)
Constantvolume process
/
Assume the piston
is moving in just
the right way to
keep p constant
oq
Constantpressure process
Figure 4.8 Illustration of constantvolume and constantpressure processes.
called a process. For example, a constantvolume process is illustrated at the left
of Figure 4.8. Here, the system is a gas inside a rigid boundary, such as a hollow
steel sphere, and therefore the volume of the system always remains constant. If
an amount of heat 8q is added to this system, p and T will change. Thus, by
definition, such changes take place at constant volume; this is a constantvolume
process. Another example is given at the right of Figure 4.8. Here, the system is a
gas inside a cylinderpiston arrangement. Consider that heat 8q is added to the
system and at the same time assume the piston is moved in just exactly the right
way to maintain a constant pressure inside the system. When 8q is added to this
system, T and v (hence p) will change. By definition, such changes take place at
constant pressure; this is a constantpressure process. There are many different
kinds of processes treated in thermodynamics. The above are only two examples.
The last concept to be introduced in this section is that of specific heat.
Consider a system to which a small amount of heat 8q is added. The addition of
8q will cause a small change in temperature dT of the system. By definition,
specific heat is the heat added per unit change in temperature of the system. Let c
denote specific heat. Thus,
 8q
c=
dT
However, with this definition, c is multivalued. That is, for a fixed quantity 8q,
the resulting value of dT can be different, depending on the type of process in
which 8q is added. In turn, the value of c depends on the type of process.
Therefore, we can in principle define more precisely a different specific heat for
each type of process. We will be interested in only two types of specific heat, one
at constant volume and the other at constant pressure, as follows.
If the heat 8q is added at constant volume and it causes a change in
temperature dT, the specific heat at constant volume cv is defined as
 ( 8q)
C v = dT at constant volume
or 8q = cv dT (constant volume) (4.20)
On the other hand, if 8q is added at constant pressure and it causes a change in
BASIC AERODYNAMICS 97
temperature dT (whose value is different from the dT above), the specific heat at
constant pressure c P is defined as
c = ~ )
p  dT at constant pressure
or 8q = c P dT (constant pressure) (4.21)
The above definitions of cv and cP, when combined with the first law, yield
useful relations for internal energy e and enthalpy h as follows. First, consider a
constantvolume process, where by definition du = 0. Thus, from the alternate
form of the first law, Eq. (4.16),
l3q = de + p du = de + 0 = de
Substituting the definition of cv, Eq. (4.20), into Eq. (4.22), we get
I de= cvdT I
(4.22)
(4.23)
Assuming that cv is a constant, which is very reasonable for air at normal
conditions, and letting e = 0 when T = 0, Eq. (4.23) may be integrated to
I e = cvT I (4.24)
Next, consider a constantpressure process, where by definition dp = 0. From the
alternate form of the first law, Eq. (4.19),
8q = dh  udp = dh  0 = dh (4.25)
Substituting the definition of cP, Eq. (4.21), into Eq. (4.25), we find
I dh = cPdT I
(4.26)
Again, assuming that c P is constant and letting h = 0 at T = 0, we see that Eq.
( 4.26) yields
( 4.27)
Equations (4.23) to (4.27) are very important relationships. They have been
derived from the first law, into which the definitions of specific heat have been
inserted. Look at them! They relate thermodynamic variables only ( e to T and h
to T); work and heat do not appear in these equations. In fact, Eqs. (4.23) to
(4.27) are quite general. Even though we used examples of constant volume and
constant pressure to obtain them, they hold in general as long as the gas is a
perfect gas (no intermolecular forces). Hence, for any process,
de= cvdT
dh = cPdT
e = cvT
h = cPT
This generalization of Eqs. (4.23) to (4.27) to any process may not seem logical
98 INTRODUCTION TO FLIGHT
and may be hard to accept; nevertheless, it is valid, as can be shown by good
thermodynamic arguments beyond the scope of this book. For the remainder of
our discussions, we will make frequent use of these equations to relate internal
energy and enthalpy to temperature.
4.6 ISENTROPIC FLOW
We are almost ready to return to our consideration of aerodynamics. However,
there is one more concept we must introduce, a concept that bridges both
thermodynamics and compressible aerodynamics, namely, that of isentropic flow.
First, consider thre.: more definitions:
1. Adiabatic processa process in which no heat is added or taken away,
8q = 0
2. Reversible processa process in which no frictional or other dissipative
effects occur
3. Isentropic processa process which is both adiabatic and reversible
Thus, an isentropic process is one in which there is neither heat exchange nor any
effect due to friction. (The source of the word "isentropic" comes from another
defined thermodynamic variable called entropy. The entropy is constant for an
isentropic process. A discussion of entropy is not vital to our discussion here;
therefore, no further elaboration will be given.)
Isentropic processes are very important in aerodynamics. For example,
consider the flow of air over the airfoil shown in Figure 4.5. Imagine a fluid
element moving along one of the streamlines. There is no heat being added or
taken away from this fluid element; heat exchange mechanisms such as heating by
a flame, cooling in a refrigerator, or intense radiation absorption are all ruled out
by the nature of the physical problem we are considering. Thus, the flow of the
fluid element along the streamline is adiabatic. At the same time, the shearing
stress exerted on the surface of the fluid element due to friction is generally quite
small and can be neglected (except very near the surface, as will be discussed
later). Thus, the flow is also frictionless. [Recall that this same assumption was
used in obtaining the momentum equation, Eq. (4.8).] Hence, the flow of the fluid
element is both adiabatic and reversible (frictionless); i.e., the flow is isentropic.
Other aerodynamic flows can also be treated as isentropic, e.g., the flows through
windtunnel nozzles and rocket engines.
Note that, even though the flow is adiabatic, the temperature need not be
constant. Indeed, the temperature of the fluid element can vary from point to
point in an adiabatic, compressible flow. This is because the volume of the fluid
element (of fixed mass) changes as it moves through regions of different density
along the streamline; when the volume varies, work is done [Eq. (4.15)], hence the
internal energy changes [Eq. (4.10)], and hence the temperature changes [Eq.
(4.23)]. This argument holds for compressible flows, where the density is variable.
BASIC AERODYNAMICS 99
On the other hand, for incompressible flow, where p = const, the volume of the
fluid element of fixed mass does not change as it moves along a streamline; hence
no work is done and no change in temperature occurs. If the flow over the airfoil
in Figure 4.5 were incompressible, the entire flow field would be at constant
temperature. For this reason, temperature is not an important quantity for
frictionless incompressible flow. Moreover, our present discussion of isentropic
flows is relevant to compressible flows only, as explained below.
An isentropic process is more than just another definition. It provides us with
several important relationships between the thermodynamic variables T, p, and p
at two different points (say points 1 and 2 in Figure 4.5) along a given streamline.
These relations are obtained as follows. Since the flow is isentropic (adiabatic and
reversible), l3q = 0. Thus, from Eq. (4.16),
l3q = de + p du = 0
 pdu =de
Substitute Eq. (4.23) into (4.28):
pdu = cvdT
(4.28)
(4.29)
In the same manner, using the fact that l3q = 0 in Eq. (4.19), we also obtain
l3q=dhudp=O
udp = dh
Substitute Eq. (4.26) into (4.30):
Divide Eq. (4.29) by (4.31):
or
dp _ ( cP) du
   
p CV u
(4.30)
(4.31)
(4.32)
The ratio of specific heats cp/cv appears so frequently m compressible flow
equations that it is given a symbol all its own, usually y; cp/cv = y. For air at
normal conditions, which apply to the applications treated in this book, both c P
and cv are constants, and hence y = const = 1.4 (for air). cp/cv = y = 1.4 (for
air at normal conditions). Thus, Eq. (4.32) can be written as
dp du
 = y (4.33)
p u
Referring to Figure 4.5, integrate Eq. (4.33) between points 1 and 2:
f P2dp = y ~
P1 p V1
P2 U2
In= yln
P1 U1
(4.34)
100 INTRODUCTION TO FLIGHT
Since v
1
= l/p
1
and v
2
= l/p
2
, Eq. (4.34) becomes
isentropic flow {4.35)
From the equation of state, we have p = p/(RT). Thus, Eq. (4.35) yields
or
Pi = ( T2) rl<rll
P1 T1
isentropic flow (4.36)
Combining Eqs. ( 4.35) and ( 4.36), we obtain
isentropic flow {4.37)
The relationships given in Eq. ( 4.37) are powerful. They provide important
information for p, T, and p between two different points on a streamline in an
isentropic flow. Moreover, if the streamlines all emanate from a uniform flow far
upstream (far to the left in Figure 4.5), then Eq. (4.37) holds for any two points in
the flow, not necessarily those on the same streamline.
Emphasis is again made that the isentropic flow relations, Eq. ( 4.37), are
relevant to compressible flows only. By contrast, the assumption of incom
pressible flow (remember, incompressible flow is a myth, anyway) is not con
sistent with the same physics that went into the development of Eq. (4.37). To
analyze incompressible flows, we need only the continuity equation [say, Eq. (4.3)]
and the momentum equation [Bernoulli's equation, Eqs. (4.9a) and (4.9b)]. To
analyze compressible flows, we need the continuity equation, Eq. (4.2), the
momentum equation [Euler's equation, Eq. (4.8)], and another soontobederived
relation called the energy equation. If the compressible flow is isentropic, then Eq.
( 4.37) can be used to replace either the momentum or the energy equation. Since
Eq. (4.37) is a simpler, more useful algebraic relation than Euler's equation, Eq.
( 4.8), which is a differential equation, we will frequently use Eq. ( 4.37) in place of
Eq. (4.8) for the analysis of compressible flows in this book.
As mentioned above, to complete the development of the fundamental
relations for the analysis of compressible flow, we must now consider the energy
equation.
BASIC AERODYNAMICS 101
Example 4.4 Aµ airplane is flying at standard sealevel conditions. The temperature at a point on
the wing is 250 K. What is the pressure at this point?
SOLUTION The air pressure and temperature, p
1
and Ti, far upstream of the wing correspond to
standard sea level. Hence, p
1
=1.01X10
5
N/m
2
and T
1
= 288.16 K. Assume the flow is
isentropic (hence compressible). Then, the relation between points 1 and 2 is obtained from Eq.
(4.37):
(
T )1!<1ll
P2 2
P1 = T1
 ( Tz )1!<1ll  s
P2  P1 T1  (1.01X10 ) 288.16
I p 2 =6.14x10
4
N/m
2
I
4.7 THE ENERGY EQUATION
Recall that our approach to the derivation of the fundamental equations for fluid
flow is to state a fundamental principle and then to proceed to cast that principle
in terms of the flow variables p, T, p, and V. Also recall that compressible flow,
highspeed flow, and massive changes in energy also go hand in hand. Therefore,
our last fundamental physical principle that we must take into account is as
follows.
Physical principle: Energy can be neither created nor destroyed. It can only
change in form.
In quantitative form, this principle is nothing more than the first law of
thermodynamics, Eq. (4.10). To apply this law to fluid flow, consider again a fluid
element moving along a streamline, as shown in Figure 4.4. Let us apply the first
law of thermodynamics,
l3q + l3w = de
to this fluid element. Recall that an alternate form of the first law is Eq. (4.19),
l3q = dh  udp
Again, we consider an adiabatic flow, where l3q = 0. Hence, from Eq. (4.19),
dh  udp = 0 (4.38)
Recalling Euler's equation, Eq. (4.8),
dp = pVdV
we can combine Eqs. (4.38) ap.d (4.8) to obtain
dh + upVdV = 0
However, u = l/p, hence Eq. (4.39) becomes
dh + VdV = 0
(4.39)
(4.40)
102 INTRODUCTION TO FLIGHT
Integrating Eq. (4.40) between two points along the streamline, we obtain
t
2
dh + fv
2
VdV= 0
h1 V1
v2 v2
h1 + T = h2 +}
v2
h + T = const
(4.41)
Equation (4.41) is the energy equation for frictionless, adiabatic flow. It can be
written in terms of T by using Eq. (4.27), h = cPT. Hence, Eq. (4.41) becomes
(4.42)
Equation (4.42) relates the temperature and velocity at two different points along
a streamline. Again, if all the streamlines emanate from a uniform flow far
upstream, then Eq. (4.42) holds for any two points in the flow, not necessarily on
the same streamline. Moreover, Eq. (4.42) is just as powerful and necessary for
the analysis of compressible flow as is Eq. (4.37).
Example 4.5 A supersonic wind tunnel is sketched in Figure 4.22. The air temperature and
pressure in the reservoir of the wind tunnel are T
0
= 1000 K and Po = 10 atm, respectively. The
static temperatures at the throat and exit are T* = 833 K and Te= 300 K, respectively. The mass
flow through the nozzle is 0.5 kg/s. For air, cP = 1008 J /(kg)(K). Calculate:
(a) Velocity at the throat, V*
(b) Velocity at the exit, Ve
(c) Area of the throat, A*
(d) Area of the exit, Ae
SOLUTION Since the problem deals with temperatures and velocities, the energy equation seems
useful.
(a) From Eq. (4.42), written between the reservoir and the throat,
However, in the reservoir, V
0
"" 0. Hence,
=/2(1008)(1000833) =I 580m/s I
BASIC AERODYNAMICS 103
(b) From Eq. (4.42) written between the reservoir and the exit,
V. =j2cP (T
0
T,)
= /2(1008)(1000 300) = / 1188 m/s /
(c) The basic equation dealing with mass flow and area is the continuity equation, Eq. (4.2).
Note that the velocities are certainly large enough that the flow is compressible, hence Eq. (4.2),
rather than Eq. ( 4.3), is appropriate.
or
m = p*A*V*
A = ~
p*V*
In the above, m is given and V* is known from part (a). However, p* must be obtained before
we can calculate A* as desired. To obtain p*, note that, from the equation of state,
= __f!Q_ = (10)(1.01x10s) = 3.52 k /m3
Po RT
0
(287)(1000) g
Assuming the nozzle flow is isentropic, which is a very good approximation for the real case,
from Eq. (4.37), we get
Thus
(
T* )l/(yll ( 833 )1/(1.4ll
p* =Po To = (3.52) WOO = 2.23 kg/m
3
•  ~  0.5 13 4 2  2 j
A  p*V*  (
2
_
23
)(
580
)  .87X10 m 3.87cm
(d) Finding Ae is similar to the above solution for A*
m = PeAeVe
where, for isentropic flow,
(
Te )l/(yll ( 300 )1/(1.4ll 3
Pe= Po To = (3.52) lOOO = 0.174 kg/m
Thus A = ~ =
0
·
5
= / 24.2 X 10
4
m
2
= 24.2cm
2
I
e PeV. (0.174)(1188) . .
4.8 SUMMARY OF EQUATIONS
We have just finished applying some very basic physical principles to obtain
equations for the analysis of flowing gases. The reader is cautioned not to be
confused by the multiplicity of equations; they are useful, indeed necessary, tools
to examine and solve various aerodynamic problems of interest. It is important
for an engineer or scientist to look at such equations and see not just a
mathematical relationship, but primarily a physical relationship. These equations
104 INTRODUCTION TO FLIGHT
talk! For example, Eq. (4.2) says that mass is conserved; Eq. (4.42) says that
energy is conserved for an adiabatic, frictionless flow; etc. Never lose sight of the
physical implications and limitations of these equations.
To help set these equations in your mind, here is a compact summary of our
results so far:
1. For the steady incompressible flow of a frictionless fluid in a stream tube of
varying area, p and V are the meaningful flow variables; p and T are
constants throughout the flow. To solve for p and V, use
A1V1 = A2V2
P1 + !pV/ = P2 + !pV/
continuity
Bernoulli's equation
2. For steady isentropic (adiabatic and frictionless) compressible flow in a
stream tube of varying area, p, p, T, and V are all variables. They are
obtained from
cPT
1
+ ! V/ = cPT
2
+ ! V
2
2
P1 = P1RT1
P2 = P2RT2
continuity
isentropic relations
energy
equation of state
Let us now apply these relations to study some basic aerodynamic phenomena
and problems.
4.9 THE SPEED OF SOUND
Sound waves travel through the air at a definite speedthe speed of sound. This
is obvious from natural observation; a lightning bolt is observed in the distance,
and the thunder is heard at some later instant. In many aerodynamic problems,
the speed of sound plays a pivotal role. How do we calculate the speed of sound?
What does it depend on: pressure, temperature, density, or some combination
thereof? Why is it so important? Answers to these questions are discussed in this
section.
First let us derive a formula to calculate the speed of sound. Consider a
sound wave moving into a stagnant gas, as shown in Figure 4.9. This sound wave
is created by some source, say a small firecracker in the corner of a room. The air
in the room is motionless and has density p, pressure p, and temperature T. If
you are standing in the middle of the room, the sound wave sweeps by you at
velocity a in m/s, ftjs, etc. The sound wave itself is a thin region of disturbance
Stagnant gas
p
p
T
a
p + dp
p + dp
T + dT
Sound wave moving
to the left with
velocity a in to
a stagnant gas
Source of
sound wave
BASIC AERODYNAMICS 105
Figure 4.9 Model of a sound wave
moving into a stagnant gas.
in the air, across which the pressure, temperature, and density change slightly.
(The change in pressure is what activates your eardrum and allows you to hear the
sound wave.) Imagine that you now hop on the sound wave and move with it. As
you are sitting on the moving wave, look to the left in Figure 4.9, i.e., look in the
direction in which the wave is moving. From your vantage point on the wave, the
sound wave seems to stand still and the air in front of the wave appears to be
corning at you with velocity a; i.e., you see the picture shown in Figure 4.10,
where the sound wave is standing still and the air ahead of the wave is moving
toward the wave with velocity a. Now, return to Figure 4.9 for a moment. Sitting
on top of and riding with the moving wave, look to the right, i.e., look behind the
wave. From your vantage point, the air appears to be moving away from you.
This appearance is sketched in Figure 4.10, where the wave is standing still. Here,
the air behind the motionless wave is moving to the right, away from the wave.
However, in passing through the wave, the pressure, temperature, and density of
the air are slightly changed by the amounts dp, dT, and dp. From our previous
discussions, you would then expect the air speed a to change slightly, say by an
amount da. Thus, the air behind the wave is moving away from the wave with
velocity a+ da, as shown in Figure 4.10. Figures 4.9 and 4.10 are completely
analogous pictures; only their perspectives are different. Figure 4.9 is what you
see by standing in the middle of the room and watching the wave go by; Figure
4.10 is what you see by riding on top of the wave and watching the air go by.
Both pictures are equivalent. However, Figure 4.10 is easier to work with, so we
will concentrate on it.
Let us apply our fundamental equations to the gas flow shown in Figure 4.10.
Our objective will be to obtain an equation for a, where a is the speed of the
Ahead of wave
p
p
T
a
2
Motionless sound wave
Figure 4.10 Model with the sound wave stationary.
p + dp
a+ da
p + dp
T+dT
Behind the wave
106 INTRODUCTION TO FLIGHT
sound wave, the speed of sound. Let points 1 and 2 be ahead of and behind the
wave, respectively, as shown in Figure 4.10. Applying the continuity equation, Eq.
( 4.2), we find
P1A1V1 = P2A2V2
or pA
1
a = (p + dp)A
2
(a + da) (4.43)
Here, A
1
and A
2
are the areas of a stream tube running through the wave. Just
looking at the picture shown in Figure 4.10, there is no geometric reason why the
stream tube should change area in passing through the wave. Indeed, it does not;
the area of the stream tube is constant, hence A = A
1
= A
2
= const. (This is an
example of a type of flow called onedimensional, or constantarea, flow.) Thus,
Eq. ( 4.43) becomes
pa= (p + dp)(a + da)
or pa= pa+ adp + pda + dpda (4.44)
The product of two small quantities dp da is very small in comparison to the
other terms in Eq. (4.44) and hence can be ignored. Thus, from Eq. (4.44),
da
a= p (4.45)
dp
Now, apply the momentum equation in the form of Euler's equation, Eq. (4.8):
dp = pada
da=dp
pa
(4.46) or
Substitute Eq. (4.46) into (4.45):
p dp
a=
dp pa
or
2 dp
a=
dp
( 4.47)
On a physical basis, the flow through a sound wave involves no heat addition, and
the effect of friction is negligible. Hence, the flow through a sound wave is
isentropic. Thus, from Eq. (4.47), the speed of sound is given by
a = ( X ) isentropic
(4.48)
Equation (4.48) is fundamental and important. However, it does not give us a
straightforward formula for computing a number for a. We must proceed further.
For isentropic flow, Eq. (4.37) gives
or (4.49)
BASIC AERODYNAMICS 107
Equation ( 4.49) says that the ratio p / p Y is the same constant value at every point
in an isentropic flow. Thus, we can write everywhere
(4.50)
Hence
(
dp) = _!!_cpY = cypr1
d P isentropic d P
(4.51)
Substituting for c in Eq. (4.51) the ratio of Eq. (4.50), we obtain
(
dp ) = .!!.._ yp y  1 = y p
dp isentropic pY P
(4.52)
Substitute Eq. ( 4.52) into ( 4.48):
(4.53)
However, for a perfect gas, p and p are related through the equation of state;
p = pRT, hence p / p = RT. Substituting this result into Eq. ( 4.53) yields
(4.54)
Equations (4.48), (4.53), and (4.54) are important results for the speed of
sound; however, Eq. (4.54) is the most useful. It also demonstrates a fundamental
result, that the speed of sound in a perfect gas depends only on the temperature of
the gas. This simple result may appear surprising at first. However, it is to be
expected on a physical basis, as follows. The propagation of a sound wave
through a gas takes place via molecular collisions. For example, consider again a
small firecracker in the corner of the room. When the firecracker is set off, some
of its energy is transferred to the neighboring gas molecules in the air, thus
increasing their kinetic energy. In turn, these energetic gas molecules are moving
randomly about, colliding with some of their neighboring molecules and transfer
ring some of their extra energy to these new molecules. Thus, the energy of a
sound wave is transmitted through the air by molecules which collide with each
other. Each molecule is moving at a different velocity, but summed over a large
number of molecules, a mean or average molecular velocity can be defined.
Therefore, looking at the collection of molecules as a whole, we see that the sound
energy released by the firecrackers will be transferred through the air at some
thing approximating this mean molecular velocity. Recall from Chap. 2 that
temperature is a measure of the mean molecular kinetic energy, hence of the mean
molecular velocity; then temperature should also be a measure of the speed of a
sound wave transmitted by molecular collisions. Eq. (4.54) proves this to be a
fact.
For example, consider air at standard sealevel temperature T,; = 288.16 K.
From Eq. (4.54), the speed of sound is a = JyRT = /1.4(287)(288.16) = 340.3
108 INTRODUCTION TO FLIGHT
m/s. From the results of the kinetic theory of ases, the mean molecular velocity
can be obtained as V = J(8/7T )RT = (8/7T )287(288.16) = 458.9 m/s. Thus,
the speed of sound is of the same order of magnitude as the mean molecular
velocity and is smaller by about 26 percent.
Emphasis is made that the speed of sound is a point property of the flow, in
the same vein as T is a point property as described in Chap. 2. It is also a
thermodynamic property of the gas, defined by Eqs. (4.48) to (4.54). In general,
the value of the speed of sound varies from point to point in the flow.
The speed of sound leads to another, vital definition for highspeed gas flows,
namely, the Mach number. Consider a point B in a flow field. The flow velocity at
B is V, and the speed of sound is a. By definition, the Mach number Mat point
B is the flow velocity divided by the speed of sound:
(4.55)
We will find that M is one of the most powerful quantities in aerodynamics. We
can immediately use it to define three different regimes of aerodynamic flows:
1. If M < 1, the flow is subsonic.
2. If M = 1, the flow is sonic.
3. If M > 1, the flow is supersonic.
Each of these regimes is characterized by its own special phenomena, as will
be discussed in subsequent sections. In addition, two other specialized
aerodynamic regimes are commonly defined, namely, transonic flow, where M
generally ranges from slightly less than to slightly greater than 1 (for example,
0.8 :::; M :::; 1.2), and hypersonic flow, where generally M > 5. The definitions of
subsonic, sonic, and supersonic flows in terms of M as given above are precise;
the definitions of transonic and hypersonic flows in terms of M are a bit more
imprecise and really refer to sets of specific aerodynamic phenomena, rather than
to just the value of M. This distinction will be made more clear in subsequent
sections.
Example 4.6 A jet transport is flying at a standard altitude of 30,000 ft with a velocity of 550
mijh. What is its Mach number?
SOLUTION From the standard atmosphere table, Appendix B, at 30,000 ft, T
00
= 411.86°R.
Hence, from Eq. (4.54),
a
00
= JyRT = y'l.4(1716)( 411.86) = 995 ftjs
The airplane velocity is V
00
= 550 mijh; however, in consistent units, remembering that 88
ftjs = 60 mijh, we find that
V
00
= 550 (88/60) = 807 ftjs
From Eq. (4.55)
BASIC AERODYNAMICS 109
Example 4.7 In the nozzle flow described in Example 4.5. calculate the Mach number of the flow
at the throat, M*, and at the exit, Me.
SoLUTION From Example 4.5, at the throat, V* = 580 m/s and T* = 833 K. Hence, from Eq.
(4.54),
a*= /yRT* = y'l.4(287)(833) = 580 m/s
From Eq. (4.55)
M* = V* = 580 = lil
a* 580 LJ
Note: The flow is sonic at the throat. We will soon prove that the Mach number at the throat is
always sonic in supersonic nozzle flows (except in special, nonequilibrium, hightemperature
flows, which are beyond the scope of this book).
Also from Example 4.5, at the exit, Ve= 1188 m/s and Te= 300 K. Hence,
ae=y'yRTe =y'l.4(287)(300) =347m/s
M = V,, = 1188 ~
e ae 347 L:J
4.10 LOWSPEED SUBSONIC WIND TUNNELS
Throughout the remainder of this book the aerodynamic fundamentals and tools
(equations) developed in previous sections will be applied to specific problems of
interest. The first of these will be a discussion of lowspeed subsonic wind tunnels.
In the first place, what are wind tunnels, any kind of wind tunnels? In the
most basic sense, they are groundbased experimental facilities designed to
produce flows of air (or sometimes other gases) which simulate natural flows
occurring outside the laboratory. For most aerospace engineering applications,
wind tunnels are designed to simulate flows encountered in the flight of airplanes,
missiles, or space vehicles. Since these flows have ranged from the 27 rni/h speed
of the early Wright Flyer to the 25,000 rni/h reentry velocity of the Apollo lunar
spacecraft, obviously many different types of wind tunnels, from low subsonic to
hypersonic, are necessary for the laboratory simulation of actual flight conditions.
However, referring again to Figure 1.27, flow velocities of 300 mijh or less were
the flight regime of interest until about 1940. Hence, during the first four decades
of human flight, airplanes were tested and developed in wind tunnels designed to
simulate lowspeed subsonic flight. Such tunnels are still in use today but are
complemented by transonic, supersonic, and hypersonic wind tunnels as well.
The essence of a typical lowspeed subsonic wind tunnel is sketched in Figure
4.11. The airflow with pressure p
1
enters the nozzle at a low velocity V
1
, where the
area is A
1
• The nozzle converges to a smaller area A
2
at the test section. Since we
are dealing with lowspeed flows, where M is generally less than 0.3, the flow will
be assumed to be incompressible. Hence, Eq. ( 4.3) dictates that the flow velocity
increases as the air flows through the convergent nozzle. The velocity in the test
section is then, from Eq. ( 4.3),
(4.56)
110 INTRODUCTION TO FLIGHT
Settling
chamber
(reservoir)
Nozzle
2 ~
pz, Ai
Test section
Figure 4.11 Simple schematic of a subsonic wind tunnel.
Diffuser
After flowing over an aerodynamic model (which may be a model of a complete
airplane or part of an airplane, such as a wing, tail, or engine nacelle), the air
passes into a diverging duct called a diffuser, where the area increases and
velocity decreases to A
3
and V
3
, respectively. Again, from continuity,
A2
V3 = AV2
The pressure at various locations in th; wind tunnel is related to the\,elocity,
through Bernoulli's equation, Eq. (4.9a), for incompressible flow.
(4.57)
From Eq. (4.57), as V increases, p decreases; hence p
2
< p
1
, that is, the test
section pressure is smaller than the reservoir pressure upstream of the nozzle. In
many subsonic wind tunnels, all or part of the test section is open, or vented, to
the surrounding air in the laboratory. In such cases, the outside air pressure is
communicated directly to the flow in the test section, and p
2
= 1 atm. Down
stream of the test section, in the diverging area diffuser, the pressure increases as
velocity decreases. Hence, p
3
> p
2
. If A
3
= A
1
, then from Eq. (4.56), V
3
= V
1
;
and from Eq. (4.57), p
3
= p
1
• (Note: In actual wind tunnels, the aerodynamic
drag created by the flow over the model in the test section causes a loss of
momentum not included in the derivation of Bernoulli's equation; hence in
reality, p
3
is slightly less than p
1
, because of such losses.)
In practical operation of this type of wind tunnel, the test section velocity is
governed by the pressure difference p
1
 Pi and the area ratio of the noz;:le,
A
2
/ A
1
, as follows. From Eq. (4.57),
2 2 ( ) 2
V2 =  P1  P2 + VI
p
(4.58)
From Eq. (4.56), V
1
= (A
2
/A
1
)V
2
. Substituting this into the righthand side of
Eq. (4.58), we obtain
(4.59)
BASIC AERODYNAMICS 111
Solving Eq. (4.59) for V
2
yields
(4.60)
The area ratio A
2
/ Ai is a fixed quantity for a wind tunnel of given design. Hence,
the "control knob" of the wind tunnel basically controls Pi  p
2
, which allows
the windtunnel operator to control the value of test section velocity V
2
via Eq.
(4.60).
In subsonic wind tunnels, the most convenient method of measuring the
pressure difference p
1
 p
2
, hence of measuring V
2
via Eq. (4.60), is by means of
a manometer. A basic type of manometer is the U tube shown in Figure 4.12.
Here, the left side of the tube is connected to a pressure p
1
, the right side of the
tube is connected to a pressure p
2
, and the difference D..h in the heights of a fluid
in both sides of the U tube is a measurement of the pressure difference p
2
 p
1
.
This can easily be demonstrated by considering the force balance on the liquid in
the tube at the two cross sections cut by plane BB, shown in Figure 4.12. Plane
BB is drawn tangent to the top of the column of fluid on the left. If A is the
crosssectional area of the tube, then p
1
A is the force exerted on the left column
of fluid. The force on the right column at plane BB is the sum of the weight of
the fluid above plane BB and the force due to the pressure P2A. The volume of
the fluid in the right column above BB is A D..h. The specific weight (weight per
unit volume) of the fluid is w = p
1
g, where p
1
is the density of the fluid and g is
the acceleration of gravity. Hence, the total weight of the column of fluid above
BB is the specific weight times the volume, that is, wA D..h. The total force on the
righthand cross section at plane BB is then p
2
A + wA D..h. Since the fluid is
stationary in the tube, the forces on the left and righthand cross sections must
balance, i.e., they are the same. Hence,
PiA = P2A + wAD..h
or
I P1  P2 = w/:::,,h I
(4.61)
Pl
P2
B B
B
Figure 4.12 Force diagram for a manometer.
112 INTRODUCTION TO FLIGHT
If the lefthand side of the U tube manometer were connected to the reservoir in a
subsonic tunnel (point 1 in Figure 4.11) and the righthand side were connected
to the test section (point 2), then !::..h of the U tube would directly measure the
velocity of the airflow in the test section via Eqs. (4.61) and (4.60).
Example 4.8 In a lowspeed subsonic wind tunnel, one side of a mercury manometer is
connected to the settling chamber (reservoir) and the other side is connected to the test section.
The contraction ratio of the nozzle, A
2
/A
1
, equals 1/15. The reservoir pressure and temperature
are p
1
=1.1 atm and T
1
= 300 K, respectively. When the tunnel is running, the height difference
between the two columns of mercury is 10 cm. The density of liquid mercury is 1.36X10
4
kg/m
3
. Calculate the airflow velocity in the test section, V
2
.
SOLUTION
l:lh =10 cm= 0.1 m
w (for mercury)= p
1
g = (1.36X10
4
kg/m
3
)( 9.8 m/s
2
)
w = 1.33x10
5
N/m
3
From Eq. (4.61)
p
1
 p
2
= wl:lh = (1.33 X 10
5
N/m3)(0.l m) = 1.33 x 10
4
N/m2
To find the velocity V
2
, use Eq. (4.60). However, in Eq. (4.60) we need a value of density p. This
can be found from the reservoir conditions by using the equation of state. (Remember: 1
atm = 1.01X10
5
N/m
2
.)
=A = u (1.01x105) = 1.29 k /m3
Pi RT1 287(300) g
Since we are dealing with a lowspeed subsonic flow, assume p
1
= p = const. Hence, from Eq.
(4.60)
Note: This answer corresponds to approximately a Mach number of 0.4 in the test section, one
slightly above the value of 0.3 that bounds incompressible flow. Hence, our assumption of
p = const in this example is inaccurate by about 8 percent.
4.11 MEASUREMENT OF AIRSPEED
In the previous section, we demonstrated that the airflow velocity in the test
section of a lowspeed wind tunnel (assuming incompressible flow) can be
obtained by measuring p
1
 p
2
• However, the previous analysis implicitly as
sumes that the flow properties are reasonably constant over any given cross
section of the flow in the tunnel (socalled quasionedimensional flow). If the
flow is not constant over a given cross section, for example, if the flow velocity in
the middle of the test section is higher than near the walls, then V
2
obtained from
the preceding section is only a mean value of the test section velocity. For this
reason and for many other aerodynamic applications, it is important to obtain a
BASIC AERODYNAMICS 113
point measurement of velocity at a given spatial location in the flow. This
measurement can be made by an instrument called a Pi totstatic tube, as de
scribed below.
First, we must add to our inventory of aerodynamic definitions. We have
been glibly talking about the pressures at points in flows, such as points 1 and 2
in Figure 4.5. However, these pressures are of a special type, called static. Static
pressure at a given point is the pressure we would feel if we were moving along
with the flow at that point. It is the ramification of gas molecules moving about
with random motion and transferring their momentum to or across surfaces, as
discussed in Chap. 2. If we look more closely at the molecules in a flowing gas, we
see that they have a purely random motion superimposed on a directed motion
due to the velocity of the flow. Static pressure is a consequence of just the purely
random motion of the molecules. When an engineer or scientist uses the word
"pressure," it always means static pressure unless otherwise identified, and we
will continue such practice here. In all of our previous discussions so far, the
pressures have been static pressures.
A second type of pressure is commonly utilized in aerodynamics, namely,
total pressure. To define and understand total pressure, consider again a fluid
element moving along a streamline, as shown in Figure 4.4. The pressure of the
gas in this fluid element is the static pressure. However, now imagine that we grab
hold of this fluid element and slow it down to zero velocity. Moreover, imagine
that we do this isentropically. Intuitively, the thermodynamic properties p, T, and
p of the fluid element would change as we bring the element to rest; they would
follow the conservation laws we have discussed previously in this chapter. Indeed,
as the fluid element is isentropically brought to rest, p, T, and p would all
increase above their original values when the element was moving freely along the
streamline. The values of p, T, and p of the fluid element after it has been
brought to rest are called total values, i.e., total pressure p
0
, total temperature T
0
,
etc. Thus, we are led to the following precise definition:
Total pressure at a given point in a flow is that pressure that would exist if the
flow were slowed down isentropically to zero velocity.
There is a perspective to be gained here. Total pressure p
0
is a property of the
gas flow at a given point. It is something that is associated with the flow itself.
The process of isentropically bringing the fluid element to rest is just an
imaginary mental process we use to define the total pressure. It does not mean
that we actually have to do it in practice. In other words, considering again the
flow sketched in Figure 4.5, there are two pressures we can consider at points 1, 2,
etc., associated with each point of the flow: a static pressure p and a total
pressure p
0
, where Po > p.
For the special case of a gas that is not moving, i.e., the fluid element has no
velocity in the first place, then static and total pressures are synonymous: p
0
= p.
This is the case in common situations such as the stagnant air in the room, gas
confined in a cylinder, etc.
114 INTRODUCTION TO FLIGHT
~ Pitottube
~ ~ ~ ~ ~ ~ ~ ~ ~ ~
c
~ ~
Open end
Figure 4.13 Sketch of a Pitot
tube.
There is an aerodynamic instrument that actually measures the total pressure
at a point in the flow, namely, a Pitot tube. A basic sketch of a Pitot tube is
shown in Figure 4.13. It consists of a tube placed parallel to the flow and open to
the flow at one end (point A). The other end of the tube (point B) is closed. Now
imagine that the flow is first started. Gas will pile up inside the tube. After a few
moments, there will be no motion inside the tube because the gas has nowhere to
gothe gas will stagnate once steadystate conditions have been reached. In fact,
the gas will be stagnant everywhere inside the tube, including at point A. As a
result, the flow field sees the open end of the Pitot tube (point A) as an
obstruction, and a fluid element moving along the streamline, labeled C, has no
choice but to stop when it arrives at point A. Since no heat has been exchanged,
and friction is negligible, this process will be isentropic, i.e., a fluid element
moving along streamline C will be isentropically brought to rest at point A by the
very presence of the Pitot tube. Therefore, the pressure at point A is, truly
speaking, the total pressure p
0
• This pressure will be transmitted throughout the
Pitot tube, and if a pressure gauge is placed at point B, it will in actuality
measure the total pressure of the flow. In this fashion, a Pitot tube is an
instrument that measures the total pressure of a flow.
By definition, any point of a flow where V = 0 is called a stagnation point. In
Figure 4.13, point A is a stagnation point.
Consider the arrangement shown in Figure 4.14. Here we have a uniform flow
with velocity V
1
moving over a flat surface parallel to the flow. There is a small
hole in the surface at point A called a static pressure orifice. Since the surface is
parallel to the flow, only the random motion of the gas molecules will be felt by
the surface itself. In other words, the surface pressure is indeed the static pressure
p. This will be the pressure at the orifice at point A. On the other hand, the Pitot
tube at point B in Figure 4.14 will feel the total pressure p
0
, as discussed above.
If the static pressure orifice at point A and the Pitot tube at point B are
BASIC AERODYNAMICS 115
Total pressure measured here
Flow with velocity V1
/Static pressure orifice
Static
pressure
measured here
B
Pitot tube
Differential pressure gauge
Figure 4.14 Schematic of a Pitotstatic measurement.
connected across a pressure gauge, as shown in Figure 4.14, the gauge will
measure the difference between total and static pressure, p
0
 p.
Now we arrive at the main thrust of this section. The pressure difference
Po  p, as measured in Figure 4.14, gives a measure of the flow velocity V
1
. A
combination of a total pressure measurement and a static pressure measurement
allows us to measure the velocity at a given point in a flow. These two
me.asurements can be combined in the same instrument, a Pitotstatic probe, as
illustrated in Figure 4.15. A Pitotstatic probe measures p
0
at the nose of the
probe and p at a point on the probe surface downstream of the nose. The
pressure difference Po  p yields the velocity V
1
, but the quantitative formulation
differs depending on whether the flow is lowspeed (incompressible), highspeed
subsonic, or supersonic.
A Incompressible Flow
Consider again the sketch shown in Figure 4.14. At point A, the pressure is p and
the velocity is V
1
. At point B, the pressure is p
0
and the velocity is zero. Applying
Bernoulli's equation, Eq. (4.9a), at points A and B, we obtain
p + =
Po
Static Dynamic Total
pressure pressure
Total pressure .
felt here Static pressure felt here
1177777777777777
2722772777277272Z
Pitotstatic probe
(4.62)
Figure 4.15 Schematic of a
Pitotstatic probe.
116 INTRODUCTION TO FLIGHT
In Eq. (4.62), for dynamic pressure q we have the definition
q = tpV2 (4.63)
which is frequently employed in aerodynamics; the grouping tpV
2
is termed the
dynamic pressure for flows of all types, incompressible to hypersonic. From Eq.
(4.62),
(4.64)
This relation holds for incompressible flow only. The total pressure equals the sum
of the static plus the dynamic pressure. Also from Eq. ( 4.62),
(4.65)
Equation (4.65) is the desired result; it allows the calculation of flow velocity from
a measurement of p
0
 p, obtained from a Pitotstatic tube. Again, we emphasize
that Eq. (4.65) holds only for incompressible flow.
A Pitot tube can be used to measure the flow velocity at various points in the
test section of a lowspeed wind tunnel, as shown in Figure 4.16. The total
pressure at point B is obtained by the Pitot probe, and the static pressure, also at
point B, is obtained from a static pressure orifice located at point A on the wall
of the closed test section, assuming that the static pressure is constant throughout
the test section. This assumption of constant static pressure is fairly good for
subsonic windtunnel test sections and is commonly made. If the test section is
/Pitot probe
~ B . J < ~
A /Static pressure orifice
~ · (a) Closed test section
~ J
A•
~ P=latm
(b) Open test section
Figure 4.16 Pressure measure
ments in open and closed test
sections of subsonic wind tun
nels.
BASIC AERODYNAMICS 117
open to the room, as also sketched in Figure 4.16, then the static pressure at all
points in the test section is p = 1 atm. In either case, the velocity at point A is
calculated from Eq. ( 4.65). The density p in Eq. ( 4.65) is a constant (incom
pressible flow). Its value can be obtained by measurements of p and T somewhere
in the tunnel, using the equation of state to calculate p = p/RT. These measure
ments are usually made in the reservoir upstream of the nozzle.
Either a Pitot tube or a Pitotstatic tube can be used to measure the airspeed
of airplanes. Such tubes can be seen extending from airplane wingtips, with the
tube oriented in the flight direction, as shown in Figure 4.17. If a Pitot tube is
used, then the ambient static pressure in the atmosphere around the airplane is
obtained from a static pressure orifice placed strategically on the airplane surface.
Its location is placed where the surface pressure is nearly the same as the pressure
of the surrounding atmosphere. Such a location is found by experience. It is
generally on the fuselage somewhere between the nose and the wing. The values
of p
0
obtained from the wingtip Pitot probe and p obtained from the static
pressure orifice on the surface allow the calculation of the airplane's speed
through the air using Eq. ( 4.65), as long as the airplane's velocity is low enough to
justify the assumption of incompressible flow, i.e., for velocities less than 300 ftjs.
In actual practice, the measurements of p
0
and p are joined across a differential
pressure gauge which is calibrated in terms of airspeed, using Eq. ( 4.65). This
airspeed indicator is a dial in the cockpit, with units of velocity, say miles per
hour, on the dial. However, in determining the calibration, i.e., in determining
what values of miles per hour go along with given values of p
0
 p, the engineer
must decide what value of p to use in Eq. ( 4.65). If p is the true value, somehow
measured in the actual air around the airplane, then Eq. ( 4.65) gives the true
airspeed of the airplane:
V. = v 2( Po  P)
true p
(4.66)
However, the measurement of atmospheric air density directly at the airplane's
location is difficult. Therefore, for practical reasons, the airspeed indicators on
lowspeed airplanes are calibrated by using the standard sealevel value of Ps in
Flight
direction
Pitot tube
Wing
Figure 4.17 Sketch of wingmounted Pitot probe.
118 INTRODUCTION TO FLIGHT
Eq. (4.65). This gives a value of velocity called the equivalent airspeed:
V = _ /2( Po  P)
e V Ps
(4.67)
The equivalent airspeed v. differs slightly from V'irue• the difference being the
factor (p/p.)
1
1
2
• At altitudes near sea level, this difference is small.
Example 4.9 The altimeter on a lowspeed Cessna 150 private aircraft reads 5000 ft. By an
independent measurement, the outside air temperature is 505°R If a Pitot tube mounted on the
wingtip measures a pressure of 1818 lb/ft
2
, what is the true velocity of the airplane? What is the
equivalent airspeed?
SOLUTION An altimeter measures the pressure altitude (see discussion in Chap. 3). From the
standard atmosphere table, Appendix B, at 5000 ft, p =1761 lb/ft
2
. Also, the Pitot tube
measures total pressure; hence
Po  p = 18181761 = 57 lb/ft
2
The true airspeed can be obtained from Eq. (4.66); however, we need p, which is obtained from
the equation of state. For the outside, ambient air,
p 1761 3 3
p= RT=
1716
(
505
) =2.03X10 slug/ft
From Eq. (4.66),
2
<
51
> = 1237 ft/s I
2.mx10
3
Note: Since 88 ft/s = 60 mijh, V.rue = 237(60/88) = 162 mijh.
The equivalent airspeed (that which would be read on the airspeed indicator in the cockpit)
is obtained from Eq. (4.67), where Ps = 0.002377 slug/ft
3
(the standard sealevel value). Hence,
from Eq. (4.67),
V = v 2( Po  p) =
e Ps
2(57) = 1219 ftjs I
2.377x10
3
Note that there exists a 7.6 percent difference between V.rue and Ve.
B Subsonic Compressible Flow
The results of the previous section are valid for airflows where M < 0.3, that is,
where the flow can be reasonably assumed to be incompressible. This is the flight
regime of, for example, small, piston engine private aircraft. For higherspeed
flows, but where the Mach number is still less than 1 (highspeed subsonic flows),
other equations must be used. This is the flight regime of commercial jet
transports such as the Boeing 747 and the McDonnelDouglas DC10 and of
many military aircraft. For these cases, compressibility must be taken into
account, as follows.
Consider the definition of enthalpy h = e + pu. Since h = cPT and e = cJ,
then cPT = cvT + RT, or
(4.68)
BASIC AERODYNAMICS 119
Divide Eq. (4.68) by cP:
1
_.!_=yl=R
y Y cP
or (4.69)
Equation (4.69) holds for a perfect gas with constant specific heats. It is a
necessary thermodynamic relation for use in the energy equation, as follows.
Consider again a Pitot tube in a flow, as shown in Figures 4.13 and 4.14.
Assume the flow velocity V
1
is high enough that compressibility must be taken
into account. As usual, the flow is isentropically compressed to zero velocity at
the stagnation point on the nose of the probe. The values of the stagnation, or
total, pressure and temperature at this point are p
0
and T
0
, respectively. From the
energy equation, Eq. ( 4.42), written between a point in the freestream flow where
the temperature and velocity are T
1
and V
1
, respectively, and the stagnation point,
where the velocity is zero and the temperature is T
0
,
or
cPT
1
+ ! V/ = cpTo
To V12
=l+
Tl 2cPT
1
Substitute Eq. (4.69) for cP in Eq. (4.70):
T
0
V/ y  1 V
1
2
=l+ =l+
T1 2[ yR/( y  l)] T
1
2 yRT
1
However, from Eq. (4.54) for the speed of sound,
a ~ yRT
1
Thus, Eq. (4.71) becomes
Since the Mach number M
1
= Vi/a
1
, Eq. (4.72) becomes
(4.70)
(4.71)
(4.72)
(4.73)
Since the gas is isentropically compressed at the nose of the Pitot probe in Figures
4.13 and 4.14, Eq. (4.37) holds between the freestream and the stagnation point.
120 INTRODUCTION TO FLIGHT
That is, Po/P
1
= (p
0
/p
1
)Y = (T
0
/T
1
)Yl<rll_ Therefore, from Eq. (4.73), we
obtain
= 1 +M
2
Po ( y  1 )rl<rll
P1 2 1
(4.74)
= 1 +M
2
Po ( y  1 )l/(rll
P1 2 1
(4.75)
Equations (4.73) to (4.75) are fundamental and important relations for
compressible, isentropic flow. They apply to many other practical problems in
addition to the Pitot tube. It should be noted that Eq. (4.73) holds for adiabatic
flow, whereas Eqs. (4.74) and (4.75) contain the additional assumption of friction
less (hence isentropic) flow. Also, from a slightly different perspective, Eqs. (4.73)
to (4.75) determine the total temperature, density, and pressure, T
0
, p
0
, and p
0
, at
any point in the flow where the static temperature, density, and pressure are T
1
,
p
1
, and p
1
and where the Mach number is M
1
. In other words, reflecting the
earlier discussion of the definition of total conditions, Eqs. (4.73) to (4.75) give
the values of p
0
, T
0
, and p
0
that are associated with a point in the flow where the
pressure, temperature, density, and Mach number are p
1
, T
1
, p
1
, and M
1
. These
equations also demonstrate the powerful influence of Mach number in
aerodynamic flow calculations. It is very important to note that the ratios T
0
/T
1
,
p
0
/p
1
, and p
0
/p
1
are functions of M
1
only (assuming y is known; y = 1.4 for
normal air).
Returning to our objective of measuring airspeed, and solving Eq. (4.74) for
M
1
, we obtain
M   1
2 _ 2 [(Po )<r lJIY ]
1 y  1 P1
(4.76)
Hence, for subsonic compressible flow, the ratio of total to static pressure, p
0
/p
1
,
is a direct measure of Mach number. Thus, individual measurements of p
0
and p
1
in conjunction with Eq. (4.76) can be used to calibrate an instrument in the
cockpit of an airplane called a Mach meter, where the dial reads directly in terms
of the flight Mach number of the airplane.
To obtain the actual flight velocity, recall that M
1
= V
1
/a
1
, hence Eq. (4.76)
becomes
(4.77a)
Equation (4.77) can be rearranged algebraically as
vz = __ 1_ o 1 + l _ l
2a2 [( p _ p )<rlJIY ]
1 y  1 P1
(4.77b)
BASIC AERODYNAMICS 121
Equations (4.77a) and (4.77b) give the true airspeed of the airplane. However,
they require a knowledge of a
1
, hence T
1
. The static temperature in the air
surrounding the airplane is difficult to measure. Hence, all highspeed (but
subsonic) airspeed indicators are calibrated from Eq. (4.77b), assuming that a
1
is
equal to the standard sealevel value as= 340.3 m/s = 1116 ftjs. Moreover, the
airspeed indicator is designed to sense the actual pressure difference p
0
 p
1
, in
Eq. (4.77b), not the pressure ratio p
0
/p
1
, as appears in Eq. (4.77a). Hence, the
form of Eq. (4.77b) is used to define a calibrated airspeed as follows:
v2 = __ s_ 0 1 + 1  1
2a 2 [( p  p )<rlllY ]
cal Y  1 Ps
(4.78)
where as and Ps are the standard sealevel values of the speed of sound and static
pressure, respectively.
Again, emphasis is made that Eqs. ( 4. 76) to ( 4. 78) must be used to measure
airspeed when M
1
> 0.3, that is, when the flow is compressible. Equations based
on Bernoulli's equation, such as Eqs. (4.66) and (4.67), are not valid when
M
1
> 0.3.
Example 4.10 A highspeed subsonic McDonnelDouglas DC10 airliner is flying at a pressure
altitude of 10 km. A Pitot tube on the wingtip measures a pressure of 4.24X10
4
N/m
2
.
Calculate the Mach number at which the airplane is flying. If the ambient air temperature is 230
K, calculate the true airspeed and the calibrated airspeed.
SOLUTION From the standard atmosphere table, Appendix A, at an altitude of 10,000 m,
p = 2.65x10
4
N/m
2
. Hence, from Eq. (4.76),
M 2 = _2 [(Po )<yllh  l] = _2 [( 4.24X10
4
)
0
.
286
 l]
I y1 P1 1.41 2.65X10
4
Thus I M
1
= 0.8481
It is given that T
1
= 230 K, hence
a
1
=JyRTi =Jl.4(287)(230) = 304.0 m/s
From Eq. (4.77)
V/ = 2a1
2
[( Po )<Y  ll/y  l] = 2(304.0)
2
[( 4.24 )
0
·
286
 l]
y1 p
1
1.41 2.65
I vi= 258 m/s \ true airspeed
Note: As a check, from the definition of Mach number,
V
1
= M
1
a
1
= 0.848(304.0) = 258 m/s
122 INTRODUCTION TO FLIGHT
The calibrated airspeed can be obtained from Eq. (4.78).
V2 =s __ O __ I +l l
2a2[(pp )(yllh]
cal Y 1 Ps
= 2(340.3)
2
[( 4.24x10
4
 2.65x10
4
+ 1)
0
·
286
1]
1.41 1.01x10
5
V.:a1 = 157 m/s I
Note that the difference between true and calibrated airspeeds is 39 percent. Note: Just out of
curiosity, let us calculate V
1
the wrong way, i.e., let us apply Eq. (4.66), which was obtained from
Bernoulli's equation for incompressible flow. Equation (4.66) does not apply to the highspeed
case of this problem, but let us see what result we get anyway.
= J!.J_ = 2.65X104 = 0.4 k /m3
P RT
1
287(230) g
From Eq. (4.66)
V. =. / 2(Po  P)
true v p
2( 4.24 2.65) x 10
4
0.4 = 282 m/s (incorrect answer)
Comparing with V
1
= 258 m/s obtained above, an error of 9.3 percent is introduced in the
calculation of true airspeed by using the incorrect assumption of incompressible flow. This error
grows very rapidly as the Mach number approaches unity, as discussed in a subsequent section.
C Supersonic Flow
Airspeed measurements in supersonic flow, that is, M > 1, are qualitatively
different from those for subsonic flow. In supersonic flow, a shock wave will form
ahead of the Pitot tube, as shown in Figure 4.18. Shock waves are very thin
regions of the flow (for example, 10
4
cm), across which some very severe changes
in the flow properties take place. Specifically, as a fluid element flows through a
shock wave:
1. The Mach number decreases.
2. The static pressure increases.
3. The static temperature increases.
Shock wave
Figure 4.18 Pitot tube in supersonic flow.
Shock wave
M
1
> l
P1
T,
v,

Po,
To,
M2 <M1
P2 > P1
T1 > T1
V2 < V1
Po
2
< Po
1
To
2
= To
1
Pi tot
tube
4. The flow velocity decreases.
5. The total pressure p
0
decreases.
BASIC AERODYNAMICS 123
Figure 4.19 Changes across a shock wave in front of
a Pitot tube in supersonic flow.
6. The total temperature T
0
stays the same for a perfect gas.
These changes across a shock wave are shown in Figure 4.19.
How and why does a shock wave form in supersonic flow? There are various
answers with various degrees of sophistication. However, the essence is as follows.
Refer to Figure 4.13, which shows a Pitot tube in subsonic flow. The gas
molecules that collide with the probe set up a disturbance in the flow. This
disturbance is communicated to other regions of the flow, away from the probe,
by means of weak pressure waves (essentially sound waves) propagating at the
local speed of sound. If the flow velocity V
1
is less than the speed of sound, as in
Figure 4.13, then the pressure disturbances (which are traveling at the speed of
sound) will work their way upstream and eventually be felt in all regions of the
flow. On the other hand, refer to Figure 4.18, which shows a Pitot tube in
supersonic flow. Here, V
1
is greater than the speed of sound. Thus, pressure
disturbances which are created at the probe surface and which propagate away at
the speed of sound cannot work their way upstream. Instead, these disturbances
coalesce at a finite distance from the probe and form a natural phenomenon
called a shock wave, as shown in Figures 4.18 and 4.19. The flow upstream of the
shock wave (to the left of the shock) does not feel the pressure disturbance; i.e.,
the presence of the Pitot tube is not communicated to the flow upstream of the
shock. The presence of the Pitot tube is felt only in the regions of flow behind the
shock wave. Thus, the shock wave is a thin boundary in a supersonic flow, across
which major changes in flow properties take place and which divides the region of
undisturbed flow upstream from the region of disturbed flow downstream.
Whenever a solid body is placed in a supersonic stream, shock waves will
occur. An example is shown in Figure 4.20, which shows photographs of the
supersonic flow over several aerodynamic shapes. The shock waves, which are
generally not visible to the naked eye, are made visible in Figure 4.20 by means of
a specially designed optical system called a schlieren system. (An example where
shock waves are sometimes visible to the naked eye is on the wing of a high speed
subsonic transport such as a Boeing 707. As we will discuss shortly, there are
regions of local supersonic flow on the upper surface of the wing, and these
(a)
(h)
(c)
Figure 4.20 (a) Shock waves on a sweptwing airplane (left) and on a straightwing airplane (right).
Schlieren pictures taken in a supersonic wind tunnel at NASA Ames Research Center. ( h) Shock
waves on a blunt body (left) and sharpnosed body (right). ( c) Shock waves on a model of the Gemini
manned space capsule. (Courtesy of NASA Ames Research Center.) Parts hand care shadowgraphs of
the flow.
124
BASIC AERODYNAMICS 125
supersonic regions are generally accompanied by weak shock waves. If the sun is
almost directly overhead and if you look out the window along the span of the
wing, you can sometimes see these waves dancing back and forth on the wing
surface.)
Consider again the measurement of airspeed in a supersonic flow. The
measurement is complicated by the presence of the shock wave in Figure 4.18
because the flow through a shock wave is nonisentropic. Within the thin structure
of a shock wave itself, very large friction and thermal conduction effects are
taking place. Hence, neither adiabatic nor frictionless conditions hold, and hence
the flow is not isentropic. As a result, Eq. (4.74) and hence Eqs. (4.76) and (4.77a)
do not hold across the shock wave. A major consequence is that the total pressure
Po is smaller behind the shock than in front of it. In turn, the total pressure
measured at the nose of the Pitot probe in supersonic flow will not be the same
value as associated with the freestream, i.e., as associated with M
1
• Consequently,
a separate shock wave theory must be applied to relate the Pitot tube measure
ment to the value of M
1
. This theory is beyond the scope of our presentation, but
the resulting formula is given below for the sake of completeness:
Po
2
__ ( ( y + 1)
2
M
1
2
)rl<rll 1  y + 2yM/
P1 4yM/  2( y  1) y + 1
(4.79)
This equation is called the Rayleigh Pitot tube formula. It relates the Pitot tube
measurement of total pressure behind the shock, p
02
, and a measurement of
freestream static pressure (again obtained by a static pressure orifice somewhere
on the surface of the airplane) to the freestream supersonic Mach number M
1
. In
this fashion, measurements of Po, and p
1
, along with Eq. (4.79), allow the
calibration of a Mach meter for supersonic flight.
Example 4.11 An experimental rocketpowered aircraft is flying with a velocity of 3000 mi/hat
an altitude where the ambient pressure and temperature are 151 lb/ft
2
and 390°R, respectively.
A Pitot tube is mounted in the nose of the aircraft. What is the pressure measured by the Pitot
tube?
SOLUTION First, we ask the question, is the flow supersonic or subsonic, i.e., what is M
1
? From
Eq. (4.54)
a
1
= JyRT
1
= /1.4(1716)(390) = 968.0 ft/s
v
1
= 3000(88/60) = 4400 ftjs
V
1
4400
M1 = = 968.0 = 4.55
Hence, M
1
> l; the flow is supersonic. There is a shock wave in front of the Pi tot tube; therefore,
Eq. (4.74) developed for isentropic flow does not hold. Instead, Eq. (4.79) must be used.
Po,=( (y+l)
2
M
1
2
)r/<rllly+2yM
1
2
P1 4yM
1
2
2(yl) y+l
= ( (2.4)
2
(4.54)
2
)
3
\l.4+2(1.4)(4.54)
2
=27
4(1.4)( 4.54)
2
 2(0.4) 2.4
Thus Po,= 27p
1
= 27(151) = 4077 lb/ft
2
126 INTRODUCTION TO FLIGHT
Note: Again, out of curiosity, let us calculate the wrong an'swer. If we did not take into account
the shock wave in front of the Pitot tube at supersonic speeds, then Eq. (4.74) would give
Po ( y1 )r/<rl> ( 0 4 2)
3
·
5
p;_= 1+
2
M
1
2
= 1+2(4.54) =304.2
Thus p
0
= 304.2Pi = 304.2(151) = 45,931 lb/ft
2
incorrect answer
Note that the incorrect answer is off by a factor of more than 10!
D Summary
As a summary on the measurement of airspeed, note that different results apply
to different regimes of flight: lowspeed (incompressible), highspeed subsonic,
and supersonic. These differences are fundamental and serve as excellent exam
ples of the application of the different laws of aerodynamics developed in
previous sections. Moreover, many of the formulas developed in this section apply
to other practical problems, as discussed below.
4.12 SUPERSONIC WIND TUNNELS AND ROCKET ENGINES
For more than a century, projectiles such as bullets and artillery shells have been
fired at supersonic velocities. However, the main aerodynamic interest in super
sonic flows occurred after World War II with the advent of jet aircraft and
rocketpropelled guided missiles. As a result, almost every aerodynamic labora
tory has an inventory of supersonic and hypersonic wind tunnels to simulate
modern highspeed flight. In addition to their practical importance, supersonic
wind tunnels are an excellent example of the application of the fundamental laws
of aerodynamics. The flow through rocket engine nozzles is another example of
the same laws. In fact, the basic aerodynamics of supersonic wind tunnels and
rocket engines are essentially the same, as discussed below.
First, consider isentropic flow in a stream tube, as sketched in Figure 4.1.
From the continuity equation, Eq. ( 4.2),
pAV = const
or In p + In A + In V = In ( const)
Differentiating, we obtain
Recalling the momentum equation, Eq. (4.8) (Euler's), we obtain
dp = pVdV
Hence
 dp
p = VdV
Substitute Eq. (4.81) into (4.80):
_ dpVdV + dA + dV = O
dp A V
(4.80)
(4.81)
(4.82)
BASIC AERODYNAMICS 127
Since the flow is isentropic,
dp 1
dp = dp/dp = (dp_/_d_p_)_isentropic  a 2
1 1
Thus, Eq. (4.82) becomes
Rearranging, we get
or
dA = VdV _ dV = ( V
2
_ l) dV
A a2 V a2 V
(4.83)
Equation (4.83) is called the areavelocity relation, and it contains a wealth of
information about the flow in the stream tube shown in Figure 4.1. First, note the
mathematical convention that an increasing velocity and an increasing area
correspond to positive values of dV and dA, respectively, whereas a decreasing
velocity and a decreasing area correspond to negative values of dV and dA. This
is the normal convention for differentials from differential calculus. With this in
mind, Eq. (4.83) yields the following physical phenomena:
1. If the flow is subsonic (M < 1), for the velocity to increase (dV positive), the
area must decrease (dA negative); i.e., when the flow is subsonic, the area
must converge for the velocity to increase. This is sketched in Figure 4.21a.
This same result was observed in Sec. 4.2 for incompressible flow. Of course,
incompressible flow is, in a sense, a singular case of subsonic flow, where
M+ 0.
2. If the flow is supersonic (M > 1), for the velocity to increase (dV positive),
the area must also increase (dA positive); i.e., when the flow is supersonic, the
area must diverge for the velocity to increase. This is sketched in Figure
4.21b.
3. If the flow is sonic (M = 1), then Eq. (4.83) yields for the velocity
>
• •
(a)
dV 1 dA 1 dA
v
(b)
=
M
2
1 A
0 A
I
I
I
I
I
M= 1
(c)
Figure 4.21 Results from the areavelocity relation.
(4.84)
128 INTRODUCTION TO FLIGHT
which at first glance says that dV/V is infinitely large. However, on a
physical basis, the velocity, and hence the change in velocity dV, must at all
times be finite. This is only common sense. Thus, looking at Eq. (4.84), the
only way for dV/V to be finite is to have dA/A = 0, so
dV 1 dA 0 f" . b
V =
0
A =
0
= mite num er
i.e., in the language of differential calculus, dV / V is an indeterminate form
of zero over zero and hence can have a finite value. In turn, if dA/ A = 0, the
stream tube has a minimum area at M = 1. This minimum area is called a
throat and is sketched in Figure 4.21c.
Therefore, in order to expand a gas to supersonic speeds, starting with a
stagnant gas in a reservoir, the above discussion says that a duct of a sufficiently
convergingdiverging shape must be used. This is sketched in Figure 4.22, where
typical shapes for supersonic windtunnel nozzles and rocket engine nozzles are
shown. In both cases, the flow starts out with a very low velocity, V ::::: 0, in the
reservoir, expands to high subsonic speeds in the convergent section, reaches
Mach 1 at the throat, and then goes supersonic in the divergent section down
stream of the throat. In a supersonic wind tunnel, smooth, uniform flow at the
nozzle exit is usually desired, and therefore a long, gradually converging and
diverging nozzle is employed, as shown at the top of Figure 4.22. For rocket
engines, the flow quality at the exit is not quite as important, but the weight of the
nozzle is a major concern. For the weight to be minimized, the engine's length is
minimized, which gives rise to a rapidly diverging, belllike shape for the
supersonic section, as shown at the bottom of Figure 4.22. A photograph of a
typical rocket engine is shown in Figure 4.23.
iExit

Reservoir
Po
To
v""o
M< I
Subsonic
Throat
t
M= I

M> I
1 ~ Flow
Supersonic
(a) Supersonic windtunnel nozzle
Combustion
chamber
Po M < l M ;"I
To
v"" 0
M>1
Flow
Supersonic
(b) Rocketengine nozzle
Figure 4.22 Supersonic nozzle shapes.
Pe
T,
BASIC AERODYNAMICS 129
The real flow through nozzles such as those sketched in Figure 4.22 is closely
approximated by isentropic flow because little or no heat is added or taken away
through the nozzle walls and a vast core of the flow is virtually frictionless.
Therefore, Eqs. (4.73) to (4.75) apply to nozzle flows. Here, the total pressure and
temperature p
0
and T
0
remain constant throughout the flow, and Eqs. ( 4. 73) to
(4.75) can be interpreted as relating conditions at any point in the flow to the
stagnation conditions in the reservoir. For example, consider Figure 4.22, which
illustrates the reservoir conditions p
0
and T
0
where V :::::: 0. Consider any cross
section downstream of the reservoir. The static temperature, density, and pressure
at this section are T
1
, p
1
, and p
1
, respectively. If the Mach number M
1
is known
at this point, T
1
, p
1
, and p
1
can be found from Eqs. (4.73) to (4.75) as:
T1 = To[1 +HY  1)M/r
1
(4.85)
P1 = Po[l +HY  l)M1
2
]l/(yl)
P1 = Po[l +HY  l)M1
2
]y/(yl)
(4.86)
(4.87)
Again, Eqs. (4.85) to (4.87) demonstrate the power of the Mach number in
making aerodynamic calculations. The variation of Mach number itself through
the nozzle is strictly a function of the ratio of the crosssectional area to the throat
area, A/ Ar This relation can be developed from the aerodynamic fundamentals
already discussed; the resulting form is
(
A )
2
1 [ 2 ( y  1
2
)]<y+ll/(ylJ
 = 1+M
A
1
M
2
y + 1 2
(4.88)
Therefore, the analysis of isentropic flow through a nozzle is relatively
straightforward. The procedure is summarized in Figure 4.24. Consider that the
nozzle shape, hence A/A
1
, is given as shown in Figure 4.24a. Then, from Eq.
Figure 4.23 A typical rocket engine. Shown is a small rocket designed by MesserschmittBolkow
Blohm for European satellite launching.
130 INTRODUCTION TO FLIGHT
( 4.88), the Mach number can be obtained (implicitly). Its variation is sketched in
Figure 4.24b. Since Mis now known through the nozzle, then Eqs. (4.85) to (4.87)
give the variations of T, p, and p, which are sketched in Figure 4.24c to e. The
directions of these variations are important and should be noted. From Figure
4.24, the Mach number continuously increases through the nozzle, going from
near zero in the reservoir to M = 1 at the throat and to supersonic values
downstream of the throat. In turn, p, T, and p begin with their stagnation values
in the reservoir and continuously decrease to low values at the nozzle exit. Hence,
a supersonic nozzle flow is an expansion process, where pressure decreases
through the nozzle. In fact, it is this pressure decrease that provides the mechani
cal force for pushing the flow through the nozzle. If the nozzle shown in Figure
4.24a is simply set out by itself in a laboratory, obviously nothing will happen;
the air will not start to rush through the nozzle by its own accord. Instead, to
establish the flow sketched in Figure 4.24, we must provide a highpressure source
at the inlet, and/or a lowpressure source at the exit, with the pressure ratio just
the right value, as prescribed by Eq. ( 4.87) and as sketched in Figure 4.24c.
(a)
(b)
(c) p
Po
(d) T
(e)
To
p
Po
I
I
I
I
I
I
Flow
l=zt=
I
1.0 :
I
I
x
x
Figure 4.24 Variation of Mach num
ber, pressure, temperature, and den
sity through a supersonic nozzle.
BASIC AERODYNAMICS 131
Example 4.12 You are given the job of designing a supersonic wind tunnel that has a Mach 2
flow at standard sealevel conditions in the test section. What reservoir pressure and temperature
and what area ratio A./A, are required to obtain these conditions?
SoLUTION The static pressure Pe= 1 atm = 1.01x10
5
N/m
2
, and the static temperature T. =
288.16 K, from conditions at standard sea level. These are the desired conditions at the exit of
the nozzle (the entrance to the test section). The necessary reservoir conditions are obtained from
Eqs. (4.85) and (4.87):
To=l+ ylM2=1+1.41(2)2=l8
T, 2 e 2 ·
Thus, T
0
=I.ST.=1.8(288.16) = J 518.7 K \
P
[
Y
1 ]y/(y1)
p: = 1 + 
2
 M/ = (1.8)
3 5
= 7 .82
Thus, Po= 7.82p. = 7.82(1.01 X10
5
) =/ 7.9x10
5
N/m
2
/
The area ratio is obtained from Eq. (4.88):
(
A.)2 1 [ 2 ( y1 2)]<r+IJ/<r1i
 =  1+M
A, M2 y +1 2
=   1 + 2
2
= 2 85
1 [ 2 ( 0.4 )](2.4)/(0.4)
22 2.4 2 .
Hence
B
9 .
t
Example 4.13 The reservoir temperature and pressure of a supersonic wind tunnel are 600°R
and 10 atm, respectively. The Mach number of the flow in the test section is 3. A bluntnosed
model like that shown at the left in Figure 4.20b is inserted in the testsection flow. Calculate the
pressure, temperature, and density at the stagnation point (at the nose of the body).
SOLUTION The flow conditions in the test section are the same as at the nozzle exit. Hence, in
the test section, the exit pressure is obtained from Eq. (4.87), recalling that 1 atm = 2116 lb/ft
2
:
= (1+1( l)M2]r/(rll
Pe Po 2 Y e
= (10)(2116)[ 1 + 0.5(0.4)(3)
2
] 
3
·
5
= 576 lb/ft
2
The pressure at the stagnation point on the model is the total pressure behind a normal wave
because the stagnation streamline has traversed the normal portion of the curved bow shock
wave in Figure 4.20b and then has been isentropically compressed to zero velocity between the
shock and the body. This is the same situation as that existing at the mouth of a Pitot tube in
132 INTRODUCTION TO FLIGHT
supersonic flow, as described in Sec. 4.11 C. Hence, the stagnation pressure is given by Eq.
(4.79):
Po,2 = Pstag = Y + 1 Me 1  Y + 2yMe
[
( )
2 2 ] y /( y l) 2
P1 Pe 4yM'/  2( y  1) y + 1
[
2 2 ] 3.5 2
Pstag 2.4 (3) 1  1.4 + 2(1.4){3 )
 = = 12.06
Pe 4(1.4)(3)
2
 2(0.4) 2.4
Pstag = 12.06pe = 12.06(576) = 6947 lb/ft
2
The total temperature (not the static temperature) at the nozzle exist is the same as the reservoir
temperature
To,e =To
because the flow through the nozzle is isentropic and hence adiabatic. For an adiabatic flow, the
total temperature is constant, as demonstrated by Eq. (4.42), where at two different points in art
adiabatic flow with different velocities if the flow is adiabatically slowed to zero velocity at both
points, we obtain
hence T
0
,
1
= Ta.
2
; that is, the total temperature at the two different points is the same.
Therefore, in the present problem, the total temperature associated with the testsection flow is
equal to the total temperature throughout the nozzle expansion, T
0
e = Ta = 600°R. [Note that
the static temperature of the testsection flow is 214.3°R, obtained' from Eq. (4.85).] Moreover,
in traversing a shock wave (see Figure 4.19), the total temperature is unchanged; i.e., the total
temperature behind the shock wave on the model is also 600°R (although the static temperature
behind the shock is Jess than 600°R). Finally, since the flow is isentropically compressed to zero
velocity at the stagnation point, the stagnationpoint temperature is the total temperature, which
also stays constant through the isentropic compression. Hence, the gas temperature at the
stagnation point is
From the equation of state
Pstag 6947
Pstag = = 1716(600)
stag
0.0067 slug/ft
3
4.13 DISCUSSION ON COMPRESSIBILITY
We have been stating all along that flows where M < 0.3 can be treated at
essentially incompressible and, conversely, flows where M 0.3 should be treated
as compressible. We are now in a position to prove this.
Consider a gas at rest (V = 0) with density p
0
. Now accelerate this gas
isentropically to some velocity V and Mach number M. Obviously, the thermody
namic properties of the gas will change, including the density. In fact, the change
in density will be given by Eq. (4.75):
Po ( y  1 z)l/(rll
= 1 +M
p 2
BASIC AERODYNAMICS 133
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mach number
Figure 4.25 Density variation with Mach number for y = 1.4, showing region where the density
change is less than 5 percent.
For y = 1.4, this variation of p/p
0
is given in Figure 4.25. Note that, for
M < 0.3, the density change in the flow is less than 5 percent; i.e., the density
is essentially constant for M < 0.3, and for all practical purposes the flow is
incompressible. Therefore, we have just demonstrated the validity of the state
ment
For M < 0.3, the flow can be treated as incompressible.
4.14 INTRODUCTION TO VISCOUS FLOW
To this point, we have dealt exclusively with frictionless flows, and we have
adequately treated a number of practical problems with this assumption. How
ever, there are numerous other problems in which the effect of friction is
dominant, indeed, in which the complete nature of the problem is governed by the
presence of friction between the airflow and a solid surface. A classic example is
sketched in Figure 4.26, which shows the lowspeed flow over a sphere. At the left
is sketched the flow field for frictionless flow. The streamlines are symmetrical
and amazingly, there is no aerodynamic force on the sphere. The pressure
distribution over the forward surface exactly balances that over the rearward
134 INTRODUCTION TO FLIGHT
Frictionless flow: no drag
V Separated
Real flow: finite drag
Figure 4.26 Comparison between ideal frictionless flow and real flow with the effects of friction.
surface, and hence there is no drag (no force in the flow direction). However, this
purely theoretical result is contrary to common sense; in real life there is a drag
force on the sphere tending to retard the motion of the sphere. The failure of the
theory to predict drag was bothersome to early nineteenth century aerodynami
cists and was even given a name, d 'Alembert's paradox. The problem is caused
by not including friction in the theory. The real flow over a sphere is sketched on
the right in Figure 4.26. The flow separates on the rearward surface of the sphere,
setting up a complicated flow in the wake and causing the pressure on the
rearward surface to be less than on the forward surface. Hence, there is a drag
force exerted on the sphere, as shown by D in Figure 4.26. The difference between
the two flows in Figure 4.26 is simply friction, but what a difference!
Consider the flow of a gas over a solid surface, such as the airfoil sketched in
Figure 4.27. According to our previous considerations of frictionless flows, we
have considered the flow velocity at the surface as being a finite value, such as V
2
shown in Figure 4.27; i.e., due to the lack of friction, the streamline right at the
surface slips over the surface. In fact, we stated that if the flow is incompressible,
V
2
can be calculated from Bernoulli's equation:
P1 + 1PV/ = P2 + 1PV/
However, in real life, the flow at the surface adheres to the surface because of
friction between the gas and the solid material; i.e., right at the surface, the flow
velocity is zero, and there is a thin region of retarded fl.ow in the vicinity of the
surface, as sketched in Figure 4.28. This region of viscous flow which has been
retarded due to friction at the surface is called a boundary layer. The inner edge of
the boundary layer is the solid surface itself, such as point a in Figure 4.28, where
V = 0. The outer edge of the boundary layer is given by point b, where the flow
velocity is essentially the value given by V
2
in Figure 4.27. That is, point b in
Figure 4.28 is essentially equivalent to point 2 in Figure 4.27. In this fashion, the
flow properties at the outer edge of the boundary layer in Figure 4.28 can be
calculated from a frictionless flow analysis, as pictured in Figure 4.27. This leads
Figure 4.27 Frictionless flow.
The streamline that is right on
the surface slips over the surface.
BASIC AERODYNAMICS 135
Figure 4.28 Flow in real life, with friction. The thickness of the boundary layer is greatly overem
phasized for clarity.
to an important conceptual point in theoretical aerodynamics: a flow field can be
split into two regions, one region where friction is important, namely, in the
boundary layer near the surface, and another region of frictionless flow (some
times called potential flow) outside the boundary layer. This concept was first
introduced by Ludwig Prandtl in 1904, and it revolutionized modern theoretical
aerodynamics.
It can be shown experimentally and theoretically that the pressure through
the boundary layer in a direction perpendicular to the surface is constant. That is,
letting Pa and Pb be the static pressures at points a and b, respectively, in Figure
4.28, then Pa= Pb· This is an important phenomenon. This is why a surface
pressure distribution calculated from frictionless flow (Figure 4.27) many times
gives accurate results for the reallife surface pressures; it is because the friction
less calculations give the correct pressures at the outer edge of the boundary layer
(point b ), and these pressures are impressed without change through the boundary
layer right down to the surface (point a). The above statements are reasonable for
slender aerodynamic shapes such as the airfoil in Figure 4.28; they do not hold
for regions of separated flow over blunt bodies, as previously sketched in Figure
4.26. Such separated flows will be discussed in a subsequent section.
Refer again to Figure 4.28. The boundary layer thickness 8 grows as the flow
moves over the body; i.e., more and more of the flow is affected by friction as the
distance along the surface increases. In addition, the presence of friction creates a
shear stress at the surface, Tw. This shear stress has dimensions of force/area and
acts in a direction tangential to the surface. Both 8 and Tw are important
quantities, and a large part of boundary layer theory is devoted to their calcula
tion. As we will see, Tw gives rise to a drag force called skin friction drag, hence
attesting to its importance. In subsequent sections, equations for the calculation
of 8 and Tw will be given.
Looking more closely at the boundary layer, a velocity profile through the
boundary layer is sketched in Figure 4.29. The velocity starts out at zero at the
surface and increases continuously to its value of V
2
at the outer edge. Let us set
up coordinate axes x and y such that x is parallel to the surface and y is normal
to the surface, as shown in Figure 4.29. By definition, a velocity profile gives the
variation of velocity in the boundary layer as a function of y. In general, the
velocity profiles at different x stations are different.
The slope of the velocity profile at the wall is of particular importance,
because it governs the wall shear stress. Let (dV/dy)y=o be defined as the velocity
136 INTRODUCTION TO FLIGHT
I
•
1
_Velocity profile through
,.. the boundary layer
Figure 4.29 Velocity profile through a boundary layer.
gradient at the wall. Then the shear stress at the wall is given by
(
dV)
'T =µ, 
w dy y0
(4.89)
where µ, is called the absolute viscosity coefficient (or simply the viscosity) of the.
gas. The viscosity coefficient has dimensions of mass/(length)(time), as can be
verified from Eq. (4.89) combined with Newton's second law. It is a physical
property of the fluid; µ, is different for different gases and liquids. Also, µ, varies
with T. For liquids, µ, decreases as T increases (we all know that oil gets
" thinner" when the temperature is increased). But for gases, µ, increases as T
increases (air gets "thicker" when temperature is increased). For air at standard
sealevel temperature,
µ, = 1.7894 x io
5
kg/(m)(s) = 3.7373 x 10
7
slug/(ft)(s)
The variation of µ, with temperature for air is given in Figure 4.30.
2.1
2.0
0
1.9
x
00
1.8
a
""'
1.7
::i
1.6
L5
250 270 290 310 330 350
Temperature. K
Figure 4.30 Variation of viscosity coefficient with temperature.
BASIC AERODYNAMICS 137
Leading
edge 
\ //y
Figure 4.31 Growth of the boundary layer thickness.
In this section, we are simply introducing the fundamental concepts of
boundary layer flows; such concepts are essential to the practical calculation of
aerodynamic drag, as we will soon appreciate. In this spirit, we introduce another
important dimensionless "number," a number of importance and impact on
aerodynamics equal to those of the Mach number discussed earlierthe Reynolds
number. Consider the development of a boundary layer on a surface, such as the
flat plate sketched in Figure 4.31. Let x be measured from the leading edge, i.e.,
the front tip of the plate. Let V
00
be the flow velocity far upstream of the plate.
(The subscript oo is commonly used to denote conditions far upstream of an
aerodynamic body, socalled freestream conditions.) The Reynolds number Rex is
defined as
(4.90)
Note that Rex is dimensionless and that it varies linearly with x. For this reason,
Rex is sometimes called a "local" Reynolds number, because it is based on the
local coordinate x.
To this point in our discussion on aerodynamics, we have always considered
flow streamlines to be smooth and regular curves in space. However, in a viscous
flow, and particularly in boundary layers, life is not quite so simple. There are two
basic types of viscous flow:
1. Laminar flow, where the streamlines are smooth and regular and a fluid
element moves smoothly along a streamline
2. Turbulent flow, where the streamlines break up and a fluid element moves in a
random, irregular, and tortuous fashion
If you observe the smoke rising from a lit cigarette, as sketched in Figure 4.32,
you see first a region of smooth flowlaminar flowand then a transition to
irregular, mixedup flowturbulent flow. The differences between laminar and
turbulent flow are dramatic, and they have major impact on aerodynamics. For
example, consider the velocity profiles through a boundary layer, as sketched in
Figure 4.33. The profiles are different, depending on whether the flow is laminar
or turbulent. The turbulent profile is "fatter," or fuller, than the laminar profile.
For the turbulent profile, from the outer edge to a point near the surface, the
velocity remains reasonably close to the freestream velocity; it then rapidly
138 INTRODUCTION TO FLIGHT
l
_:}:'
Laminar
t
Cigarette
Figure 4.32 Smoke pattern from a cigarette.
decreases to zero at the surface. In contrast, the laminar velocity profile gradually
decreases to zero from the outer edge to the surface. Now, consider the velocity
gradient at the wall, (dV/dy)ro' which is the reciprocal of the slope of the
curves shown in Figure 4.33 evaluated at y = 0. From Figure 4.33, it is clear that
[ ( ~ ; yo for laminar flow] < [ ( ~ ; r o for turbulent flow]
Recalling Eq. (4.89) for rw leads us to the fundamental and highly important fact
y
v ~
/
/
/
.... ""
 
Turbulent/
I
I
I
I
I
I
I
I
I
I
I
I
Figure 4.33 Velocity profiles for
laminar and turbulent boundary
layers. Note that the turbulent
boundary layer thickness is larger
than the laminar boundary layer
thickness.
BASIC AERODYNAMICS 139
0.8
0.6
0.4
r Minimum pressure
Point of maximum thickness
~
NACA 0012
Standard airfoil
Figure 4.34 Comparison of conventional and laminar flow airfoils. The pressure distributions shown
are the theoretical results obtained by the NACA and are for 0° angle of attack. The airfoil shapes are
drawn to scale.
that laminar shear stress is less than turbulent shear stress.
Tw laminar < Tw turbulent
This obviously implies that the skin friction exerted on an airplane wing or body
will depend on whether the boundary layer on the surface is laminar or turbulent,
with laminar flow yielding the smaller skin friction drag.
It appears to be almost universal in nature that systems with the maximum
amount of disorder are favored. For aerodynamics, this means that the vast
majority of practical viscous flows are turbulent. The boundary layers on most
practical airplanes, missiles, ship hulls, etc., are turbulent, with the exception of
small regions near the leading edge, as we will soon see. Consequently, the skin
friction on these surfaces is the higher, turbulent value. For the aerodynamicist,
who is usually striving to reduce drag, this is unfortunate. However, the skin
friction on slender shapes, such as wing cross sections (airfoils), can be reduced
140 INTRODUCTION TO FLIGHT
p p ~
~ 1 . 0
2 Poo oo
0.8
0.6
0.4
0.4
Point of maximum thickness
r Minim um pressure
Laminar flow airfoil
Figure 4.34 (continued)
by designing the shape in such a manner to encourage laminar flow. Figure 4.34
indicates how this can be achieved. Here, two airfoils are shown; the standard
airfoil on the left has a maximum thickness near the leading edge, whereas the
laminar flow airfoil has its maximum thickness near the middle of the airfoil. The
pressure distributions on the top surface of the airfoils are sketched above
the airfoils in Figure 4.34. Note that for the standard airfoil, the minimum
pressure occurs near the leading edge, and there is a long stretch of increasing
pressure from this point to the trailing edge. Turbulent boundary layers are
encouraged by such increasing pressure distributions. Hence, the standard airfoil
is generally bathed in long regions of turbulent flow, with the attendant high skin
friction drag. On the other hand, note that for the laminar flow airfoil, the
minimum pressure occurs near the trailing edge, and there is a long stretch of
decreasing pressure from the leading edge to the point of minimum pressure.
Laminar boundary layers are encouraged by such decreasing pressure distribu
tions. Hence, the laminar flow airfoil is generally bathed in long regions of
laminar flow, thus benefiting from the reduced skin friction drag.
BASIC AERODYNAMICS 141
The North American P51 Mustang, designed at the outset of World War II,
was the first production aircraft to employ a laminar flow airfoil. However,
laminar flow is a sensitive phenomenon; it readily gets unstable and tries to
change into turbulent flow. For example, the slightest roughness of the airfoil
surface caused by such reallife effects as protruding rivets, imperfections in
machining, and bug spots can cause a premature transition to turbulent flow in
advance of the design condition. Therefore, most laminar flow airfoils used on
production aircraft do not yield the extensive regions of laminar flow that are
obtained in controlled laboratory tests using airfoil models with highly polished,
smooth surfaces. From this point of view, the early laminar flow airfoils were not
successful. However, they were successful from an entirely different point of view;
namely, they were found to have excellent highspeed properties, postponing to a
higher flight Mach number the large drag rise due to shock waves and flow
separation encountered near Mach 1. (Such highspeed effects are discussed in
Secs. 5.95.11.) As a result, the early laminar flow airfoils were extensively used
on jetpropelled airplanes during the 1950s and 1960s and are still employed
today on some modern highspeed aircraft.
Given a laminar or turbulent flow over a surface, how do we actually
calculate the skin friction drag? The answer is given in the following two sections.
4.15 RESULTS FOR A LAMINAR BOUNDARY LAYER
Consider again the boundary layer flow over a flat plate as sketched in Figure
4.31. Assume that the flow is laminar. The two physical quantities of interest are
the boundary layer thickness 8 and shear stress Tw at location x. Formulas for
these quantities can be obtained from laminar boundary layer theory, which is
beyond the scope of this book. However, the results, which have been verified by
experiment, are as follows. The laminar boundary layer thickness is
1" 5.2x l .
u = ro;: ammar
yRex
(4.91)
where Rex = p
00
V
00
x/µ,
00
as defined in Eq. (4.90). It is remarkable that a phenom
enon as complex as the development of a boundary layer, which depends at least
on density, velocity, viscosity, and length of the surface, should be described by a
formula as simple as Eq. (4.91). In this vein, Eq. (4.91) demonstrates the powerful
influence of Reynolds number Rex in aerodynamic calculations.
Note from Eq. (4.91) that laminar boundary layer thickness varies inversely
as the square root of the Reynolds number. Also, since Rex = p
00
V
00
x/µ.
00
, then
from Eq. (4.91) 8 a: x
1
1
2
; that is, the laminar boundary layer grows parabolical/y.
The local shear stress Tw is also a function of x, as sketched in Figure 4.35.
Rather than dealing with Tw directly, aerodynamicists find it more convenient to
142 INTRODUCTION TO FLIGHT
Figure 4.35 Variation of shear stress with
x distance along the surface.
define a local skin friction coefficient c
1
x as
C = Tw =:
Ix  lp V 2 q
20000
00
(4.92)
The skin friction coefficient is dimensionless and is defined as the local shear
stress divided by the dynamic pressure at the outer edge of the boundary. From
laminar boundary layer theory,
0.664 1 .
c =  ammar
Ix JRex
(4.93)
where, as usual, Rex= p
00
V
00
x/µ,
00
• Equation (4.93) demonstrates the convenience
of defining a dimensionless skin friction coefficient. On one hand, the dimensional
shear stress Tw (as sketched in Figure 4.35) depends on several quantities such as
p
00
, V
00
, and Rex; on the other hand, from Eq. (4.93), c
1
, is a function of Rex only.
This convenience obtained from using dimensionless coefficients and numbers
reverberates throughout aerodynamics. Relations between dimensionless quanti
ties such as those given in Eq. (4.93) can be substantiated by dimensional analysis,
a formal procedure to be discussed later.
Combining Eqs. (4.92) and (4.93), we can obtain values of Tw from
( )
0.664q
00
7' =/ x =
w JRex
(4.94)
Note from Eqs. (4.93) and (4.94) that both c
1
x and Tw for laminar boundary layers
vary as x
1
1
2
; that is, c
1
x and Tw decrease along the surface in the flow direction,
as sketched in Figure 4.35. The shear stress near the leading edge of a fiat plate is
greater than near the trailing edge. •
The variation of local shear stress Tw along the surface allows us to calculate
the total skin friction drag due to the airflow over an aerodynamic shape. Recall
from Sec. 2.2 that the net aerodynamic force on any body is fundamentally due to
the pressure and shear stress distributions on the surface. In many cases, it is this
total aerodynamic force that is of primary interest. For example, if you mount a
fiat plate parallel to the airstream in a wind tunnel and measure the force exerted
on the plate, by means of a balance of some sort, you are not measuring the local
BASIC AERODYNAMICS 143
Voo
~ ~  ~
J_ .. __ • xL , l_1
Figure 4.36 Total drag is the integral of the local shear stress over the surface.
shear stress Tw; rather, you are measuring the total drag due to skin friction being
exerted over all the surface. This total skin friction drag can be obtained as
follows.
Consider a flat plate of length L and unit width oriented parallel to the flow,
as shown in perspective in Figure 4.36. Consider also an infinitesimally small
surface element of the plate of length dx and width unity, as shown in Figure
4.36. The local shear stress on this element is Tx, a function of x. Hence, the force
on this element due to skin friction is Tw dx (1) = Tw dx. The total skin friction
drag is the sum of the forces on all the infinitesimal elements from the leading to
the trailing edge; i.e., the total skin friction drag D
1
is obtained by integrating ,.x
along the surface, as
Let us define a total skin friction drag coefficient C
1
as
C = DI
1 qooS
(4.95)
(4.96)
(4.97)
where Sis the total area of the plate, S = L(l). Thus, from Eqs. (4.96) and (4.97),
D
1
l .328q
00
L
Cf= qooL(l) ( )1;2
qooL PooVooL/µ.oo
or C =
1
·
328
laminar
f yReL
(4.98)
where the Reynolds number is now based on the total length L, that is,
ReL = p
00
V
00
L/µ.
00
•
Do not confuse Eq. (4.98) with Eq. (4.93); they are different quantities. The
local skin friction coefficient c
1
x in Eq. (4.93) is based on the local Reynolds
number Rex= p
00
V
00
x/µ.
00
and is a function of x. On the other hand, the total
144 INTRODUCTION TO FLIGHT
skin friction coefficient c
1
is based on Reynolds number for the plate length L,
ReL = PooVooL/µoo.
Emphasis is made that Eqs. (4.91), (4.93), and (4.98) apply to laminar
boundary layers only; for turbulent fl.ow the expressions are different. Also, these
equations are exact only for lowspeed (incompressible) fl.ow. However, they have
been shown to be reasonably accurate for highspeed subsonic flows as well. For
supersonic and hypersonic flows, where the velocity gradients within the boundary
layer are so extreme and where the presence of frictional dissipation creates very
large temperatures within the boundary layer, the form of these equations can still
be used for engineering approximations, but p and µ must be evaluated at some
reference conditions germane to the flow inside the boundary layer. Such matters
are beyond the scope of this book.
Example 4.14 Consider the flow of air over a small flat plate which is 5 cm long in the flow
direction and 1 m wide. The freestream conditions correspond to standard sea level, and the flow
velocity is 120 m/s. Assuming laminar flow, calculate:
(a) The boundary layer thickness at the downstream edge (the trailing edge)
(b) The drag force on the plate
SOLUTION
(a) At the trailing edge of the plate, where x = 5 cm= 0.05 m, the Reynolds number is,
from Eq. (4.90),
p
00
V
00
x (1.225 kg/m
3
)(120 m/s)(0.05 m)
Re = = ~ ~ ~ ~ ~
x /loo l.789Xl0
5
kg/(m)(s)
= 4.11 xl0
5
From Eq. (4.91),
8 = ~ = 5.2(0.05) = 14.06 x io4 m I
Re/12 (4.llxl0s)1;2
Note how thin the boundary layer isonly 0.0406 cm at the trailing edge.
(b) To obtain the skin friction drag, Eq. (4.98) gives, with L = 0.05 m,
c = 1.328 = 1.328 = 2.07x103
I ReLl/2 (4.11 XlOs)l/2
The drag can be obtained from the definition of the skin friction drag coefficient, Eq. (4.97), once
q
00
and S are known.
q
00
= tP
00
V
00
2
= Hl.225)(120)
2
= 8820 N/m
2
S = 0.05(1) = 0.05 m
2
Thus, from Eq. (4.97), the drag on one surface of the plate (say the top surface) is
(top) D
1
= q
00
SC
1
=8820(0.05)(2.07X10
3
) = 0.913 N
Since both the top and bottom surfaces are exposed to the flow, the total friction drag will be
double the above result:
Total D
1
=2(0.913)=11.826 NI
4.16 RESULTS FOR A TURBULENT BOUNDARY LAYER
Under the same fl.ow conditions, a turbulent boundary layer will be thicker than a
laminar boundary layer. This comparison is sketched in Figure 4.37. Unlike
BASIC AERODYNAMICS 145
V=
Figure 4.37 Turbulent boundary layers are thicker than laminar boundary layers.
laminar flows, no exact theoretical results can be presented for turbulent boundary
layers. The study of turbulence is a major effort in fluid dynamics today; so far,
turbulence is still an unsolved theoretical problem and is likely to remain so for
an indefinite time. In fact, turbulence is one of the major unsolved problems in
theoretical physics. As a result, our knowledge of 8 and rw for turbulent
boundary layers must rely on experimental results. Such results yield the follow
ing approximate formulas for turbulent flow.
0.37x
8 = 
0

2
turbulent
Rex·
(4.99)
Note from Eq. (4.99) that a turbulent boundary grows approximately as x
4
1
5
•
This is in contrast to the slower x
1
1
2
variation for a laminar boundary layer. As a
result, turbulent boundary layers grow faster and are thicker than laminar
boundary layers.
The total skin friction coefficient is given approximately as
E
0.074 b 1
~ tur uent
(4.100)
Note that for turbulent flow, Cf varies as L 1!
5
; this is in contrast to the L 
1
1
2
variation for laminar flow. Hence Cf is larger for turbulent flow, which precisely
confirms our reasoning at the end of Sec. 4.14, where we noted that rw(laminar)
< rw(turbulent). Also note that Cf in Eq. (4.100) is once again a function of ReL'
Values of Cf for both laminar and turbulent flows are commonly plotted in the
form shown in Figure 4.38. Note the magnitude of the numbers involved in
Figure 4.38. The values of ReL for actual flight situations may vary from 10
5
to
10
8
or higher; the values of Cf are generally much less than unity, on the order of
102.
Example 4.15 Consider the same flow over the same flat plate as in Example 4.14; however,
assume that the boundary layer is now completely turbulent. Calculate the boundary layer
thickness at the trailing edge and the drag force on the plate.
SOLUTION From Example 4.14, ~ = 4.11 x 10
5
. From Eq. (4.99), for turbulent flow,
0.37x 0.37(0.05) I _
3
I
8 =  =
0
= 1.39 X 10 m
Rex0.2 ( 4.11 X 105) 2
146 INTRODUCTION TO FLIGHT
0.010
0.008
0.006
0.004
Ct
0.003
0.002
0.00 I '''''''
Figure 4.38 Variation of skin
friction coefficient with Reynolds
number for lowspeed flow. Com
parison of laminar and turbulent
flow.
10
5
10
6
10
10
Note: Compare this result with the laminar flow result from Example 4.14.
l.39xl03 = 3.42
4.06Xl0
4
Note that the turbulent boundary layer at the trailing edge is 3.42 times thicker than the laminar
boundary a sizable amount! From Eq. (4.100)
C1 = 0.074 = 0.074 = 0.00558
ReL02 (4.11 x10s)o2
On the top surface,
D
1
= q
00
SC
1
= 8820(0.05)(0.00558) = 2.46 N
Considering both top and bottom surface, we have
Total D
1
=2(2.46)=14.92 N I
Note that the turbulent drag is 2.7 times larger than the laminar drag.
4.17 TRANSITION
In Sec. 4.15 we discussed the flow over a flat plate as if it were all laminar.
Similarly, in Sec. 4.16 we assumed all turbulent flow. In reality, the flow always
starts out from the leading edge as laminar. Then, at some point downstream of
the leading edge, the laminar boundary layer becomes unstable and small
"bursts" of turbulent flow begin to grow in the flow. Finally, over a certain region
called the transition region, the boundary layer becomes completely turbulent. For
purposes of analysis, we usually draw the picture shown in Figure 4.39, where a
laminar boundary starts out from the leading edge of a flat plate and grows
parabolically downstream. Then, at the transition point, it becomes a turbulent
boundary layer growing at a faster rate, on the order of x
4
1
5
downstream. The
value of x where transition is said to take place is the critical value xcr· In turn,
xcr allows the definition of a critical Reynolds number for transition as
(4.101)
BASIC AERODYNAMICS 147
·

Transition 
\ /
~ Tttrbulent
Figure 4.39 Transition from laminar to turbulent flow. The boundary layer thickness is exaggerated
for clarity.
Volumes of literature have been written on the phenomenon of transition
from laminar to turbulent flow. Obviously, because Tw is different for the two
flows, knowledge of where on the surface transition takes place is vital to an
accurate prediction of skin friction drag. The location of the transition point (in
reality, a finite region) depends on many quantities, such as the Reynolds number,
Mach number, heat transfer to or from the surface, turbulence in the freestream,
surface roughness, pressure gradient, etc. A comprehensive discussion of transi
tion is beyond the scope of this book. However, if the critical Reynolds number is
given to you (usually from experiments for a given type of flow), then the location
of transition xcr can be obtained directly from the definition, Eq. (4.101).
For example, assume that you have an airfoil of given surface roughness in a
flow at a freestream velocity of 150 m/s and you wish to predict how far from the
leading edge transition will take place. After searching through the literature for
lowspeed flows over such surfaces, you may find that the critical Reynolds
number determined by experience is approximately Rex = 5 x 10
5
. Applying
this "experience" to your problem, using Eq. (4.101) and ~ ~ s u m i n the thermody
namic conditions of the airflow correspond to standard sea level, you find
(1.789 x 10
5
kg/ms)(5 x 10
5
)
'=''' = 0 .047 m
(1.225 kg/m
3
)(150 m/s)
Note that the region of laminar flow in this example is smallonly 4.7 cm
between the leading edge and the transition point. If now you double the
freestream velocity to 300 m/s, the transition point is still governed by the critical
Reynolds number Rex" = 5 X 10
5
. Thus,
x = (1.789 x 105)(5 x 105) = 0.0235 m
er 1.225(300)
Hence, when the velocity is doubled, the transition point moves forward half the
distance to the leading edge.
In summary, once you know the critical Reynolds number, you can find xcr
from Eq. (4.101). However, an accurate value of Rex applicable to your problem
must come from somewhereexperiment, free fli°ght, or some semiempirical
theoryand this may be difficult to obtain. This situation provides a little insight
into why basic studies of transition and turbulence are needed to advance our
148 INTRODUCTION TO FLIGHT
understanding of such flows and to allow us to apply more valid reasoning to the
prediction of transition in practical problems.
Example 4.16 The wingspan of the Wright Flyer I biplane is 40 ft 4 in, and the planform area of
each wing is 255 ft
2
(see Figures 1.1 and 1.2). Assume the wing shape is rectangular (obviously
not quite the case, but not bad), as shown in Figure 4.40. If the Flyer is moving with a velocity of
30 mijh at standard sealevel conditions, calculate the skin friction drag on the wings. Assume
the transition Reynolds number is 6.5x10
5
. The areas of laminar and turbulent flow are
illustrated by areas A and B, respectively, in Figure 4.40.
SOLUTION The general procedure is:
(a) Calculate D
1
for the combined area A+ B assuming the flow is completely turbulent.
(b) Obtain the turbulent D
1
for area B only by calculating the turbulent D
1
for area A and
subtracting this from the result of part (a).
(c) Calculate the laminar D
1
for area A.
(d) Add results from parts (b) and (c) to obtain total drag on the complete surface A+ B.
First, obtain some useful numbers in consistent units: b = 40 ft 4 in= 40.33 ft. Let S = planform
area= A+ B = 255 ft
2
• Hence c = S/b = 255/40.33 = 6.32 ft. At standard sea level, p
00
=
0.002377 slug/ft
3
and µ
00
=3.7373X10
7
slug/(ft)(s). Also V
00
= 30 mijh = 30(88/60) =
44 ftjs.
Thus
Re = p00 V00 c = 0.002377(44)(6.32)
' µ00 3.7373Xl0
7
= 1.769x10
6
This is the Reynolds number at the trailing edge. To find xc,,
= (6.5X10
5
)(3.7373X10
7
) =
2
.
32
ft
0.002377(44)
Figure 4.40 Planform view of surface experiencing transition from laminar to turbulent flow.
BASIC AERODYNAMICS 149
We are now ready to calculate drag. Assume that the wings of the Wright Flyer I are thin enough
that the flat plate formulas apply.
(a) To calculate turbulent drag over the complete surface S =A+ B, use Eq. (4.100):
c
1
= 0.014 = 0.014 =
0
_
00417
Re1,
02
(1.769X10
6
)
02
q
00
= }p
00
V
00
2
= t{0.002377)( 44)
2
= 2.30 lb/ft
2
( D
1
) s = q
00
S0 = 2.30(255) (0.00417) = 2.446 lb
(b) For area A only, assuming turbulent flow,
c = 0.074 = 0.074 = 0.00509
I (RexJ02 (6.5X105)02
(DI) A= q
00
AC1=2.30(2.32X40.33)(0.00509) = 1.095 lb
Hence, the turbulent drag on area B only is
(DI)
8
= (DI), (DI) A= 2.4461.095 =1.351 lb
(c) Considering the drag on area A, which is in reality a laminar drag, we obtain from Eq.
(4.98)
1.328
C=
1 (Re )o.s
x,,
1.328
= 0.00165
(6.5X10
5
)
05
(DJ) A= q
00
AC/ = 2.30(2.32X40.33)(0.00165) = 0.354 lb
(d) The total drag D
1
on the surface is
D
1
= (laminar drag on A)+ (turbulent drag on B)
= 0.354lb+1.351 lb =1.705 lb
This is the drag on one surface. Each wing has a top and bottom surface, and there are two
wings. Hence, the total skin friction drag on the complete biplane wing configuration is
DI= 4(1.705) = 16.820 lb I
4.18 FLOW SEPARATION
We have seen that the presence of friction in the flow causes a shear stress at the
surface of a body, which in tum contributes to the aerodynamic drag of the body:
skin friction drag. However, friction also causes another phenomenon called flow
separation, which in tum creates another source of aerodynamic drag called
pressure drag due to separation. The real flow field about a sphere sketched in
Figure 4.26 is dominated by the separated flow on the rearward surface. Conse
quently, the pressure on the rearward surface is less than the pressure on the
forward surface, and this imbalance of pressure forces causes a drag, hence the
term "pressure drag due to separation." In comparison, the skin friction drag on
the sphere is very small.
Another example of where flow separation is important is the flow over an
airfoil. Consider an airfoil at a low angle of attack (low angle of incidence) to the
150 INTRODUCTION TO FLIGHT
0.2
1.0
NASA LS( 1)  0417 airfoil
Angle of attack= 0°
0.6 x/c
Here, dp/dx is(+); this
is an adverse pressure gradient,
but it is moderate
Figure 4.41 Pressure distribution over the top surface for attached flow over an airfoil. Theoretical
data for a modern NASA lowspeed airfoil, from NASA Conference Publication 2046, Advanced
Technology Airfoil Research, Vol. II, March 1978, p. 11. (After McGhee, Beasley, and Whitcomb.)
flow, as sketched in Figure 4.41. The streamlines move smoothly over the airfoil.
The pressure distribution over the top surface is also shown in Figure 4.41. Note
that the pressure at the leading edge is high; the leading edge is a stagnation
region, and the pressure is essentially stagnation pressure. This is the highest
pressure anywhere on the airfoil. As the flow expands around the top surface of
the airfoil, the surface pressure decreases dramatically, dipping to a minimum
pressure, which is below the freestream static pressure p
00
• Then, as the flow
moves farther downstream, the pressure gradually increases, reaching a value
slightly above freestream pressure at the trailing edge. This region of increasing
pressure is called a region of adverse pressure gradient, defined as a region where
dp / dx is positive. This region is so identified in Figure 4.41. The adverse pressure
gradient is moderate, that is, dp / dx is small, and for all practical purposes the
flow remains attached to the airfoil surface, as sketched in Figure 4.41. The drag
on this airfoil is therefore mainly skin friction drag D
1
.
Now consider the same airfoil at a very high angle of attack, as shown in
Figure 4.42. First, assume that we had some magic fluid that would remain
attached to the surfacepurely an artificial situation. If this were the case, then
the pressure distribution on the top surface would follow the dashed line in
Figure 4.42. The pressure would drop precipitously downstream of the leading
P P ~
1.0
l.p v 2
2 ~ ~
0
1
 2
3
4
5
6
7
8
0.2
NASA LS(!)  0417 airfoil
Angle of attack= 18.4°
0.4 0.6
/
/
0.8 1.0
.;
,,,"' x/c
;< Pressure distribution
I
I
I
I
I
/ with separation
I ~
I
Pressure distribution if there
were no separation; dp/dx
I
I
I
I
I
I
I
I
is ( +) and large
BASIC AERODYNAMICS 151
Figure 4.42 Pressure distribution over the top surface for separated flow over an airfoil. Theoretical
data for a modern NASA lowspeed airfoil, from NASA Conference Publication 2045, Part 1,
Advanced Technology Airfoil Research, vol. I, March 1978, p. 380. (After Zumwalt and Nack.)
edge to a value far below the freestream static pressure Poo Farther downstream
the pressure would rapidly recover to a value above p
00
• However, in this
recovery, the adverse pressure gradient would no longer be moderate, as was the
case in Figure 4.41. Instead, in Figure 4.42, the adverse pressure gradient would
be severe, that is, dp / dx would be large. In such cases, the real flow field tends to
separate from the surface. Therefore, in Figure 4.42, the real flow field is sketched
with a large region of separated flow over the top surface of the airfoil. In this real
separated flow, the actual surface pressure distribution is given by the solid curve.
152 INTRODUCTION TO FLIGHT
In comparison to the dashed curve, note that the actual pressure distribution does
not dip to as low a pressure minimum and that the pressure near the trailing edge
does not recover to a value above p
00
• This has two major consequences, as can be
seen from Figure 4.43. Here, the airfoil at a large angle of attack (thus, with flow
separation) is shown with the real surface pressure distribution symbolized by the
solid arrows. Pressure always acts normal to a surface. Hence the arrows are all
perpendicular to the local surface. The length of the arrows denotes the magni
tude of the pressure. A solid curve is drawn through the base of the arrows to
form an "envelope" to make the pressure distribution easier to visualize. On the
other hand, if the flow were not separated, i.e., if the flow were attached, then the
pressure distribution would be that shown by the dashed arrows (and the dashed
envelope). The solid and dashed arrows in Figure 4.43 qualitatively correspond to
the solid and dashed pressure distribution curves, respectively, in Figure 4.42.
The solid and dashed arrows in Figure 4.43 should be looked at carefully.
They explain the two major consequences brought about by the separated flow
over the airfoil. The first consequence is a loss of lift. The aerodynamic lift (the
vertical force shown in Figure 4.43) is derived from the net component of a
pressure distribution in the vertical direction. High lift is obtained when the
pressure on the bottom surface is large and the pressure on the top surface is
small. Separation does not affect the bottom surface pressure distribution. How
t Lattached flow
I
I
I
I
I L separated flow
I
I
I
I
Note: Length of the arrows denoting
pressure is proportional to p · Pref•
where Pref is an arbitrary reference
pressure slightly less than the minimum
pressure on the airfoil
Attached flow    
Separated flow 
/ D separated
I
Figure 4.43 Qualitative comparison of pressure distribution, lift, and drag for attach
0
d and separated
flow. Note that for separated flow, the lift decreases and the drag increases.
BASIC AERODYNAMICS 153
ever, comparing the solid and dashed arrows on the top surface just downstream
of the leading edge, we find the solid arrows indicate a higher pressure, hence
lower lift, when the flow is separated. In addition, the pressure drag is derived
from the net component of the pressure distribution in the horizontal direction.
This is shown as the horizontal force D in Figure 4.43. If the flow were not
separated, the horizontal force due to pressure (the dashed arrows) would exactly
cancel in the left and right directions, and there would be no drag due to pressure
(again we have d' Alembert's paradox). However, in the real separated flow, the
horizontal components of pressure do not cancel, and for the conditions in Figure
4.43, there is a net pressure force in the drag direction.
Therefore, two major consequences of the flow separating over an airfoil are:
1. A drastic loss of lift (stalling)
2. A major increase in drag, caused by pressure drag due to separation
When the wing of an airplane is pitched to a high angle of attack, the wing can
stall, i.e., there can be a sudden loss of lift. Our discussion above gives the
physical reasons for this stalling phenomenon. Additional ramifications of stalling
will be discussed in Chap. 5.
Before ending this discussion of separated flow, let us ask the question; why
does a flow separate from a surface? The answer is com.Pined in the concept of an
adverse pressure gradient ( dp / dx is positive) and the velocity profile through the
boundary layer, as shown in Figure 4.33. If dp/ dx is positive, then the fluid
elements moving along a streamline have to work their way "uphill" against an
increasing pressure. Consequently, the fluid elements will slow down under the
influence of an adverse pressure gradient. For the fluid elements moving outside
the boundary layer, where the velocity (and hence kinetic energy) is high, this is
not much of a problem. The fluid element keeps moving downstream. However,
consider a fluid element deep inside the boundary layer. Looking at Figure 4.33,
we see its velocity is small. It has been retarded by friction forces. The fluid
element still encounters the same adverse pressure gradient, but its velocity is too
low to negotiate the increasing pressure. As a result, the element comes to a stop
somewhere downstream and then reverses its direction. Such reversed flow causes
the flow field in general to separate from the surface, as shown in Figure 4.42.
This is physically how separated flow develops.
Reflecting once again on Figure 4.33, note that turbulent boundary layers
have fuller velocity profiles. At a given distance from the surface (a given value of
y ), the velocity of a fluid element in a turbulent boundary is higher than in a
laminar boundary layer. Hence, in turbulent boundary layers, there is more flow
kinetic energy nearer the surface. Hence, the flow is less inclined to separate. This
leads to a very fundamental fact. Laminar boundary layers separate more easily
than turbulent boundary layers. Hence, to help prevent flow field separation, we
want a turbulent boundary layer.
154 INTRODUCTION TO FLIGHT
4.19 SUMMARY OF VISCOUS EFFECTS ON DRAG
We have seen that the presence of friction in a flow produces two sources of drag:
1. Skin friction drag DJ due to shear stress at the wall
2. Pressure drag due to flow separation, DP, sometimes identified as form drag
The total drag which is caused by viscous effects is then
D
DJ + DP
(4.102)
Total drag Drag due Drag due to
due to viscous to skin separation
effects friction (pressure drag)
Equation (4.102) contains one of the classic compromises of aerodynamics. In
previous sections, we have pointed out that skin friction drag is reduced by
maintaining a laminar boundary layer over a surface. However, we have also
pointed out at the end of Sec. 4.18 that turbulent boundary layers inhibit flow
separation; hence pressure drag due to separation is reduced by establishing a
turbulent boundary layer on the surface. Therefore, in Eq. (4.102) we have the
following compromise:
D=
+
Less for laminar, More for laminar,
more for turbulent less for turbulent
Consequently, as discussed at the end of Sec. 4.14, it cannot be said in general
that either laminar or turbulent flow is preferable over the other. Any preference
depends on the specific application. For example, for a blunt body such as the
sphere in Figure 4.26, the drag is mainly pressure drag due to separation; hence,
turbulent boundary layers reduce the drag on spheres and are therefore prefer
able. (We will discuss this again in Chap. 5). On the other hand, for a slender
body such as a sharp, slender cone or a thin airfoil at small angles of attack to the
flow, the drag is mainly skin friction drag; hence, laminar boundary layers are
preferable in this case. For inbetween cases, the ingenuity of the designer along
with practical experience helps to determine what compromises are best.
As a final note to this section, the total drag D given by Eq. (4.102) is called
profile drag because both skin friction and pressure drag due to separation are
ramifications of the shape and size of the body, i.e., the "profile" of the body. The
profile drag D is the total drag on an aerodynamic shape due to viscous effects.
However, it is not in general the total aerodynamic drag on the body. There is one
more source of drag, induced drag, which will be discussed in the next chapter.
4.20 HISTORICAL NOTE: BERNOULLI AND EULER
Equation ( 4.9) is one of the oldest and most powerful equations in fluid dynamics.
It is credited to Daniel Bernoulli, who lived during the eighteenth century; little
BASIC AERODYNAMICS 155
did Bernoulli know that his concept would find widespread application in the
aeronautics of the twentieth century. Who was Bernoulli, and how did Bernoulli's
equation come about? Let us briefly look into these questions; the answers will
lead us to a rather unexpected conclusion.
Daniel Bernoulli (17001782) was born in Groningen, Netherlands, on
January 29, 1700. He was a member of a remarkable family. His father, Johann
Bernoulli, was a noted mathematician who made contributions to differential and
integral calculus and who later became a doctor of medicine. Jakob Bernoulli,
who was Johann's brother (Daniel's uncle), was an even more accomplished
mathematician; he made major contributions to the calculus; he coined the term
"integral." Sons of both Jakob and Johann, including Daniel, went on to become
noted mathematicians and physicists. The entire family was Swiss and made their
home in Basel, Switzerland, where they held various professorships at the Univer
sity of Basel. Daniel Bernoulli was born away from Basel only because his father
spent 10 years as professor of mathematics in the Netherlands. With this type of
pedigree, Daniel could hardly avoid making contributions to mathematics and
science himself.
And indeed he did make contributions. For example, he had insight into the
kinetic theory of gases; he theorized that a gas was a collection of individual
particles moving about in an agitated fashion and correctly associated the
increased temperature of a gas with increased energy of the particles. These ideas,
originally published in 1738, were to lead a century later to a mature under
standing of the nature of gases and heat and helped to lay the foundation for the
elegant kinetic theory of gases.
Daniel's thoughts on the kinetic motion of gases were published in his book
Hydrodynamica (1738). However, this book was to etch his name more deeply in
association with fluid mechanics than kinetic theory. The book was started in
1729, when Daniel was a professor of mathematics at Leningrad (then St.
Petersburg) in Russia. By this time he was already well recognized; he had won 10
prizes offered by the Royal Academy of Sciences in Paris for his solution of
various mathematical problems. In his Hydrodynamica (which was written en
tirely in Latin), Bernoulli ranged over such topics as jet propulsion, manometers,
and flow in pipes. He also attempted to obtain a relationship between pressure
and velocity, but his derivation was obscure. In fact, even though Bernoulli's
equation, Eq. (4.9), is usually ascribed to Daniel via his Hydrodynamica, the
precise equation is not to be found in the book! The picture is further com
plicated by his father, Johann, who published a book in 1743 entitled Hydraulica.
It is clear from this latter book that the father understood Bernoulli's theorem
better than the son; Daniel thought of pressure strictly in terms of the height of a
manometer column, whereas Johann had the more fundamental understanding
that pressure was a force acting on the fluid. However, neither of the Bernoullis
understood that pressure is a point property. That was to be left to Leonhard
Euler.
Leonhard Euler (17071783) was also a Swiss mathematician. He was born at
Basel, Switzerland, on April 15, 1707, seven years after the birth of Daniel
156 INTRODUCTION TO FLIGHT
Bernoulli. Euler went on to become one of the mathematical giants of history, but
his contributions to fluid dynamics are of interest here. Euler was a close friend of
the Bernoullis; indeed, he was a student of Johann Bernoulli at the University of
Basel. Later, Euler followed Daniel to St. Petersburg, where he became a
professor of mathematics. It was here that Euler was influenced by the work of
the Bernoullis in hydrodynamics, but more influenced by Johann than by Daniel.
Euler originated the concept of pressure acting at a point in a gas. This quickly
led to his differential equation for a fluid accelerated by gradients in pressure, the
same equation we have derived as Eq. (4.8). In turn, Euler integrated the
differential equation to obtain, for the first time in history, Bernoulli's equation,
just as we have obtained Eq. (4.9). Hence we see that Bernoulli's equation, Eq.
( 4.9), is really a historical misnomer. Credit for Bernoulli's equation is legitimately
shared by Euler.
4.21 HISTORICAL NOTE: THE PITOT TUBE
The use of a Pitot tube to measure airspeed is described in Sec. 4.11; indeed, the
Pi tot tube is today so commonly used in aerodynamic laboratories and on aircraft
that it is almost taken for granted. However, this simple little device has had a
rather interesting and somewhat obscure history, as follows.
The Pitot tube is named after its inventor, Henri Pitot (16951771). Born in
Aramon, France, in 1695, Pitot began his career as an astronomer and mathema
tician. He was accomplished enough to be elected to the Royal Academy of
Sciences, Paris, in 1724. About this time, Pitot became interested in hydraulics
and, in particular, in the flow of water in rivers and canals. However, he was not
satisfied with the existing technique of measuring the flow velocity, which was to
observe the speed of a floating object on the surface of the water. So he devised
an instrument consisting of two tubes; one was simply a straight tube open at one
end which was inserted vertically into the water (to measure static pressure), and
the other was a tube with one end bent at right angles, with the open end facing
directly into the flow (to measure total pressure). In 1732, between two piers of a
bridge over the Seine River in Paris, he used this instrument to measure the flow
velocity of the river. This invention and first use of the Pitot tube was announced
by Pitot to the Academy on November 12, 1732. In his presentation, he also
presented some data of major importance on the variation of water flow velocity
with depth. Contemporary theory, based on experience of some Italian engineers,
held that the flow velocity at a given depth was proportional to the mass above it;
hence the velocity was thought to increase with depth. Pitot reported the stunning
(and correct) results, measured with his instrument, that in reality the flow
velocity decreased as the depth increased. Hence, the Pitot tube was introduced
with style.
Interestingly enough, Pitot's invention soon fell into disfavor with the en
gineering community. A number of investigators attempted to use just the Pitot
tube itself, without a local static pressure measurement. Others, using the device
BASIC AERODYNAMICS 157
under uncontrolled conditions, produced spurious results. Various shapes and
forms other than a simple tube were sometimes used for the mouth of the
instrument. Moreover, there was no agreedupon rational theory of the Pitot tube.
Note that Pitot developed his instrument in 1732, six years before Daniel
Bernoulli's Hydrodynamica and well before Euler had developed the Bernoullis'
concepts into Eq. (4.9), as discussed in Sec. 4.20. Hence, Pitot used intuition, not
theory, to establish that the pressure difference measured by his instrument was
an indication of the square of the local flow velocity. Of course, as described in
Sec. 4.11, we now clearly understand that a Pitotstatic device measures the
difference between total and static pressure and that for incompressible flow, this
difference is related to the velocity squared through Bernoulli's equation; i.e.,
from Eq. (4.62),
However, for more than 150 years after Pitot's introduction of the instrument,
various engineers attempted to interpret readings in terms of
Po  p = 1KpV
2
where K was an empirical constant, generally much different than unity. Con
troversy was still raging as late as 1913, when John Airey, a professor of
mechanical engineering from the University of Michigan, finally performed a
series of wellcontrolled experiments in a water tow tank, using Pitot probes of six
different shapes. These shapes are shown in Figure 4.44, which is taken from
Airey's paper in the April 17, 1913, issue of the Engineering News entitled" Notes
on the Pitot Tube." In this paper, Airey states that all his measurements indicate
that K = 1.0 within 1 percent accuracy, independent of the shape of the tube.
Moreover, he presents a rational theory based on Bernoulli's equation. Further
comments on these results are made in a paper entitled "Origin and Theory of the
Pitot Tube" by A. E. Guy, the chief engineer of a centrifugal pump company in
Pittsburgh, in a later, June 5, 1913, issue of the Engineering News. This paper also
helped to establish the Pitot tube on firmer technical grounds.
It is interesting to note that neither of these papers in 1913 mentioned what
was to become the most prevalent use of the Pitot tube, namely, the measurement
of airspeed for airplanes and wind tunnels. The first practical airspeed indicator, a
Venturi tube, was used on an aircraft by the French Captain A. Eteve in January
1911, more than seven years after the first powered flight. Later in 1911, British
engineers at the Royal Aircraft Establishment (RAE) at Farnborough employed a
Pitot tube on an airplane for the first time. This was to eventually evolve into the
primary instrument for flight speed measurement.
There was still controversy over Pitot tubes, as well as the need for reliable
airspeed measurements, in 1915, when the brandnew National Advisory Com
mittee for Aeronautics (NACA) stated in its First Annual Report that "an
important problem to aviation in general is the devising of accurate, reliable and
durable air speed meters. . . . The Bureau of Standards is now engaged in
158 INTRODUCTION TO FLIGHT
Water surface

 
(a)
Water surface
(d)
into
(b)
(e)

.s
"'"
N
l
I l7
(c)
===  =======
(f)
Figure 4.44 Six forms of Pitot tubes
tested by John Airey. (From En
gineering News, vol. 69, no. 16, p.
783, April 1913.)
investigation of such meters, and attention is invited to the report of Professor
Herschel and Dr. Buckingham of the bureau on Pitot tubes." The aforementioned
report was NACA report no. 2, part 1, "The Pitot Tube and other Anemometers
for Aeroplanes," by W. H. Herschel, and part 2, "The Theory of the Pitot and
Venturi Tubes," by E. Buckingham. Part 2 is of particular interest. In clear terms,
it gives a version of the theory we have developed in Sec. 4.11 for the Pitot tube,
and moreover it develops for the first time the theory for compressible subsonic
flowquite unusual for 1915! Buckingham shows that, to obtain 0.5 percent
accuracy with the incompressible relations, V
00
should not exceed 148 mi/h = 66.1
m/s. However, he goes on to state that "since the accuracy of better than 1.0
percent can hardly be demanded of an airplane speedometer, it is evident that for
all ordinary speeds of flight, no correction for compressibility is needed .... " This
was certainly an appropriate comment for the "ordinary" airplanes of that day;
indeed, it was accurate for most aircraft until the 1930s.
In retrospect, we see that the Pitot tube was invented almost 250 years ago
but that its use was controversial and obscure until the second decade of powered
flight. Then, between 1911 and 1915, one of those "explosions" in technical
advancement occurred. Pitot tubes found a major home on airplanes, and the
appropriate theory for their correct use was finally established. Since then, Pitot
tubes have become commonplace. Indeed, the Pitot tube is usually the first
aerodynamic instrument introduced to students of aerospace engineering in their
laboratory studies.
BASIC AERODYNAMICS 159
4.22 HISTORICAL NOTE: THE FIRST WIND TUNNELS
Aerospace engineering in general, and aerodynamics in particular, is an em
pirically based discipline. Discovery and development by experimental means
have been its lifeblood, extending all the way back to George Cayley (see Chap.
1). In turn, the workhorse for such experiments has been predominantly the wind
tunnel, so much so that today most aerospace industrial, government, and
university laboratories have a complete spectrum of wind tunnels ranging from
low subsonic to hypersonic speeds.
It is interesting to reach back briefly into history and look at the evolution of
wind tunnels. Amazingly enough, this history goes back more than 400 years,
because the cardinal principle of windtunnel testing was stated by Leonardo da
Vinci near the beginning of the sixteenth century as follows:
For since the action of the medium upon the body is the same whether the body moves in a
quiescent medium, or whether the particles of the medium impinge with the same velocity upon
the quiescent body; let us consider the body as if it were quiescent and see with what force it
would be impelled by the moving medium.
This is almost selfevident today, that the lift and drag of an aerodynamic
body are the same whether it moves through the stagnant air at 100 mijh or
whether the air moves over the stationary body at 100 mi/h. This concept is the
very foundation of windtunnel testing.
The first actual wind tunnel in history was designed and built over 100 years
ago by Francis Wenham at Greenwich, England, in 1871. We have met Mr.
Wenham once before, in Sec. 1.4, where his activity in the Aeronautical Society of
Great Britain was noted. Wenham's tunnel was nothing more than a 10ftlong
wooden box with a square cross section, 18 in on a side. A steamdriven fan at the
front end blew air through the duct. There was no contour, hence no aerodynamic
control or enhancement of the flow. Plane aerodynamic surfaces were placed in
the airstream at the end of the box, where Wenham measured the lift and drag on
weighing beams linked to the model.
Thirteen years later, Horatio F. Phillips, also an Englishman, built the second
known wind tunnel in history. Again, the flow duct was a box, but Phillips used
steam ejectors (highspeed steam nozzles) downstream of the test section to suck
air through the tunnel. Phillips went on to conduct some pioneering airfoil testing
in his tunnel, which will be mentioned again in Chap. 5.
Other wind tunnels were built before the turning point in aviation in 1903.
For example, the first wind tunnel in Russia was due to Nikolai Joukowski at the
University of Moscow in 1891 (it had a 2in diameter). A larger, 7 in x 10 in
tunnel was built in Austria in 1893 by Ludwig Mach, son of the famed scientist
and philosopher Ernst Mach, after whom the Mach number is named. The first
tunnel in the United States was built at the Massachusetts Institute of Technology
in 1896 by Alfred J. Wells, who used the machine to measure the drag on a fiat
plate as a check on the whirling arm measurements of Langley (see Sec. 1.8).
160 INTRODUCTION TO FLIGHT
Another tunnel in the United States was built by Dr. A. Heb Zahm at the
Catholic University of America in 1901. In light of these activities, it is obvious
that at the turn of the twentieth century aerodynamic testing in wind tunnels was
poised and ready to burst forth with the same energy that accompanied the
development of the airplane itself.
It is fitting that the same two people responsible for getting the airplane off
the ground should also have been responsible for the first concentrated series of
windtunnel tests. As noted in Sec. 1.9, the Wright brothers in late 1901
concluded that a large part of the existing aerodynamic data was erroneous. This
led to their construction of a 6ftlong 16insquare wind tunnel powered by a
twobladed fan connected to a gasoline engine. An original photograph of the
Wrights' wind tunnel in their Dayton bicycle shop is shown in Figure 4.45. They
designed and built their own balance to measure the ratios of lift to drag. Using
this apparatus, Wilbur and Orville undertook a major program of aeronautical
research between September 1901 and August 1902. During this time, they tested
over 200 different airfoil shapes manufactured out of wax. The results from these
tests constitute the first major impact of windtunnel testing on the development
of a successful airplane. As we quoted in Sec. 1.9, Orville said about their results:
"Our tables of air pressure which we made in our wind tunnel would enable us to
Figure 4.45 An original photograph of the Wright brothers' wind tunnel in their bicycle shop in
Dayton, Ohio, 1901 to 1902.
BASIC AERODYNAMICS 161
calculate in advance the performance of a machine." What a fantastic develop
ment! This was a turning point in the history of windtunnel testing, and it had as
much impact on that discipline as the December 17, 1903, flight had on the
airplane.
The rapid growth in aviation after 1903 was paced by the rapid growth of
wind tunnels, both in numbers and in technology. For example, tunnels were built
at the National Physical Laboratory in London in 1903, in Rome in 1903, in
Moscow in 1905, in Gottingen, Germany (by the famous Dr. Ludwig Prandtl,
originator of the boundary layer concept in fluid dynamics) in 1908, in Paris in
1909 (including two built by Gustave Eiffel, of tower fame), and again at the
National Physical Laboratory in 1910 and 1912.
All of these tunnels, quite :11aturally, were lowspeed facilities, but they were
pioneering for their time. Then, in 1915, with the creation of the NACA (see Sec.
2.6), the foundation was laid for some major spurts in windtunnel design. The
first NACA wind tunnel became operational at the Langley Memorial Aeronauti
cal Laboratory at Hampton, Virginia, in 1920. It had a 5ftdiameter test section
which accommodated models up to 3.5 ft wide. Then in 1923, in order to simulate
the higher Reynolds numbers associated with flight, the NACA built the first
variabledensity wind tunnel, a facility that could be pressurized to 20 atm in the
flow and therefore obtain a 20fold increase in density, hence Re, in the test
section. During the 1930s and 1940s, subsonic wind tunnels grew larger and
larger. In 1931, an NACA wind tunnel with a 30 ft x 60 ft oval test section went
into operation at Langley with a 129mi/h maximum flow velocity. This was the
first milliondollar tunnel in history. Later, in 1944, a 40 ft x 80 ft tunnel with a
flow velocity of 265 mi/h was initiated at Ames Aeronautical Laboratory at
Moffett Field, California. This is still the largest wind tunnel in the world today.
Figure 4.46 shows the magnitude of such tunnels: whole airplanes can be
mounted in the test section!
The tunnels mentioned above were lowspeed, essentially incompressible flow
tunnels. They were the cornerstone of aeronautical testing until the 1930s and
remain an important part of the aerodynamic scene today. However, airplane
speeds were progressively increasing, and new wind tunnels with higher velocity
capability were needed. Indeed, the first requirement for highspeed subsonic
tunnels was established by propellersin the 1920s and 1930s the propeller
diameters and rotational speeds were both increasing so as to encounter com
pressibility problems at the tips. This problem led the NACA to build a
12indiameter highspeed tunnel at Langley in 1927. It could produce a test
section flow of 765 mi/h. In 1936, to keep up with increasing airplane speeds,
Langley built a large 8ft highspeed wind tunnel providing 500 mijh. This was
increased to 760 mi/h in 1945. An important facility was built at Ames in 1941, a
16ft tunnel with an airspeed of 680 mi/h. A photograph of the Ames 16ft tunnel
is shown in Figure 4.47 just to provide a feeling for the massive size involved with
such a facility.
In the early 1940s, the advent of the V2 rocket as well as the jet engine put
supersonic flight in the minds of aeronautical engineers. Suddenly, the require
162 INTRODUCTION TO FLIGHT
Figure 4.46 A subsonic wind tunnel large enough to test a fullsize airplane. The NASA Langley
Research Center 30 ft x 60 ft tunnel.
BASIC AERODYNAMICS 163
Figure 4.47 The Ames 16ft highspeed subsonic wind tunnel, illustrating the massive size that goes
along with such a wind tunnel complex. (Courtesy NASA Ames Research Center.)
ment for supersonic tunnels became a major factor. However, supersonic flows in
the laboratory and in practice date farther back than this. The first supersonic
nozzle was developed by Laval about 1880 for use with steam turbines. This is
why convergentdivergent nozzles are frequently called Laval nozzles. In 1905,
Prandtl built a small Mach 1.5 tunnel at Gottingen to be used to study steam
turbine flows and (of all things) the moving of sawdust around sawmills.
The first practical supersonic wind tunnel for aerodynamic testing was
developed by Dr. A. Busemann at Braunschweig, Germany, in the mid1930s.
Using the "method of characteristics" technique, which he had developed in 1929,
Busemann designed the first smooth supersonic nozzle contour which produced
shockfree isentropic flow. He had a diffuser with a second throat downstream to
decelerate the flow and to obtain efficient operation of the tunnel. A photograph
of Busemann's tunnel is shown in Figure 4.48. All supersonic tunnels today look
essentially the same.
Working from Busemann's example, the Germans built two major supersonic
tunnels at their research complex at Peenemunde during World War II. These
were used for research and development of the V2 rocket. After the war, these
tunnels were moved almost in total to the U.S. Naval Ordnance Laboratory (later,
one was moved to the University of Maryland), where they are still in use today.
However, the first supersonic tunnel built in the United States was designed by
Theodore von Karman and his colleagues at the California Institute of Technol
ogy in 1944 and was built and operated at the Army Ballistics Research
Laboratory at Aberdeen, Maryland, under contract with Cal Tech. Then, the
164 INTRODUCTION TO FLIGHT
J1
Figure 4.48 The first practical supersonic wind tunnel, built by A. Busemann In the mid1930s.
(Courtesy of A. Busemann.)
1950s saw a virtual harvest of supersonic wind tunnels, one of the largest being
the 16 ft x 16 ft continuously operated supersonic tunnel of the Air Force at the
Arnold Engineering Development Center (AEDC) in Tennessee.
About this time, the development of the intercontinental ballistic missile
(ICBM) was on the horizon, soon to be followed by the space program of the
1960s. Flight vehicles were soon to encounter velocities as high as 36,000 ftjs in
the atmo.>phere, hypersonic velocities. In turn, hypersonic wind tunnels ( M > 5)
were suddenly in demand. The first hypersonic wind tunnel was operated by the
NACA at Langley in 1947. It had an 11insquare test section capable of Mach 7.
Three years later, another hypersonic tunnel went into operation at the Naval
Ordnance Laboratory. These tunnels are distinctly different from their supersonic
relatives in that, to obtain hypersonic speeds, the flow has to be expanded so far
that the temperature decreases to the point of liquifying the air. To prevent this,
all hypersonic tunnels, both old and new, have to have the reservoir gas heated to
temperatures far above room temperature before its expansion through the
nozzle. Heat transfer is a problem to highspeed flight vehicles, and such heating
problems feed right down to the groundtesting facilities for such vehicles.
In summary, modern windtunnel facilities slash across the whole spectrum of
flight velocities, from low subsonic to hypersonic speeds. These facilities are part
of the everyday life of aerospace engineering; hopefully, this brief historical
sketch has provided some insight into their tradition and development.
BASIC AERODYNAMIC'S 165
4.23 HISTORICAL NOTE: OSBORNE REYNOLDS
AND HIS NUMBER
In Secs. 4.14 to 4.17, we observed that the Reynolds number. defined in Eq. (4.90)
as Re = p
00
V
00
x/µ
00
, was the governing parameter for viscous flow. Boundary
layer thickness, skin friction drag, transition to turbulent flow, and many other
characteristics of viscous flow depend explicitly on the Reynolds number. Indeed.
we can readily show that the Reynolds number itself has physical meaningit is
proportional to the ratio of inertia forces to viscous forces in a fluid flow. Clearly.
the Reynolds number is an extremely important dimensionless parameter in fluid
dynamics. Where did the Reynolds number come from? When was it first
introduced, and under what circumstances? The Reynolds number is named after
a manOsborne Reynolds. Who was Reynolds? The purpose of this section is to
address these questions.
First, let us look at Osborne Reynolds, the man. He was born on October 23.
1842, in Belfast, Ireland. He was raised in an intellectual family atmosphere; his
father had been a fellow of Queens College, Cambridge, a principal of Belfast
Collegiate School, headmaster of Dedham Grammar School in Essex, and finally
rector at DebachwithBoulge in Suffolk. Indeed, Anglican clerics were a tradition
in the Reynolds family; in addition to his father, his grandfather and great
grandfather had also been rectors at Debach. Against this background, Osborne
Reynolds's early education was carried out by his father at Dedham. In his teens,
he already showed an intense interest in the study of mechanics, for which he had
a natural aptitude. At the age of 19, he served a short apprenticeship in
mechanical engineering before attending Cambridge University a year later.
Reynolds was a highly successful student at Cambridge, graduating with the
highest honors in mathematics. In 1867, he was elected a fellow of Queens
College, Cambridge (an honor earlier bestowed upon his father). He went on to
serve one year as a practicing civil engineer in the office of John Lawson in
London. However, in 1868, Owens College in Manchester (later to become the
University of Manchester) established its chair of engineeringthe second of its
kind in any English university (the first was the chair of civil engineering
established at the University College, London, in 1865). Reynolds applied for this
chair, writing in his application:
From my earliest recollection I have had an irresistible liking for mechanics and the physical
laws on which mechanics as a science are based. In my boyhood I had the advantage of the
constant guidance of my father, also a lover of mechanics and a man of no mean attainment in
mathematics and their application to physics.
Despite his youth and relative lack of experience, Reynolds was appointed to the
chair at Manchester. For the next 37 years he would serve as a professor at
Manchester until his retirement in 1905.
During those 37 years, Reynolds distinguished himself as one of history's
leading practitioners of classical mechanics. For his first years at Manchester, he
166 INTRODUCTION TO FLIGHT
worked on problems involving electricity, magnetism, and the electromagnetic
properties of solar and cometary phenomena. After 1873, he focused on fluid
mechanicsthe area in which he made his lasting contributions. For example., he
(1) developed Reynolds's analogy in 1874, a relation between heat transfer and
frictional shear stress in a fluid, (2) measured the average specific heat of water
between freezing and boiling, which ranks among the classic determinations of
physical constants, (3) studied water currents and waves in estuaries, (4) devel
oped turbines and pumps, and (5) studied the propagation of sound waves in
fluids. However, his most important work, and the one which gave birth to the
concept of the Reynolds number, was reported in 1883 in a paper entitled "An
Experimental Investigation of the Circumstances which Determine whether the
Motion of Water in Parallel Channels Shall Be Direct or Sinuous, and of the Law
of Resistance in Parallel Channels." Published in the Proceedings of the Royal
Society, this paper was the first to demonstrate the transition from laminar to
turbulent flow and relate this transition to a critical value of a dimensionlless
parameterlater to become known as the Reynolds number. Reynolds studied
this phenomenon in the water flow through pipes. His experimental apparatus is
illustrated in Figure 4.49, taken from his original 1883 paper. (Note that before
the day of modern photographic techniques, some technical papers contained
Figure 4.49 Osborne Reynolds's apparatus for his famous pipeflow experiments. This figure is from
his original paper, referenced in the text.
BASIC AERODYNAMICS 167
rather elegant hand sketches of experimental apparatus, of which Figure 4.49 is
an example.) Reynolds filled a large reservoir with water, which fed into a glass
pipe through a larger bellmouth entrance. As the water flowed through the pipe,
Reynolds introduced dye into the middle of the stream, at the entrance of the bell
mouth. What happened to this thin filament of dye as it flowed through the pipe
is illustrated in Figure 4.50, also from Reynolds's original paper. The flow is from
right to left. If the flow velocity was small, the thin dye filament would travel
downstream in a smooth, neat, orderly fashion, with a clear demarcation between
the dye and the rest of the water, as illustrated in Figure 4.50a. However, if the
flow velocity was increased beyond a certain value, the dye filament would
suddenly become unstable and would fill the entire pipe with color, as shown in
Figure 4.50b. Reynolds clearly pointed out that the smooth dye filament in Figure
4.50a corresponded to laminar flow in the pipe, whereas the agitated and totally
diffused dye filament in Figure 4.50b was due to turbulent flow in the pipe.
Furthermore, Reynolds studied the details of this turbulent flow by visually
observing the pipe flow illuminated by a momentary electric spark, much as we
would use a strobe light today. He saw that the turbulent flow consisted of a large
number of distinct eddies, as sketched in Figure 4.50c. The transition from
laminar to turbulent flow occurred when the parameter defined by pVD /µ,
exceeded a certain critical value, where p was the density of the water, V was the
mean flow velocity, µ,was the viscosity coefficient, and D was the diameter of the
pipe. This dimensionless parameter, first introduced by Reynolds, later became
known as the Reynolds number. Reynolds measured the critical value of this
number, above which turbulent flow occurred, as 2300. This original work of
(]
:
(a)
ace:
(b)
(j
UC:
(c)
Figure 4.50 Development of turbulent flow in pipes. as observed and sketched by Reynolds. This
figure is from his original paper, referenced in the text.
168 INTRODUCTION TO FLIGHT
Reynolds initiated the study of transition from laminar to turbulent flow as a new
field of research in fluid dynamicsa field which is still today one of the most
important and insufficiently understood areas of aerodynamics.
Reynolds was a scholarly man with high standards. Engineering education
was new to English universities at that time, and Reynolds had definite ideas
about its proper form. He felt that all engineering students, no matter what their
specialty, should have a common background based on mathematics, physics, and
in particular the fundamentals of classical mechanics. At Manchester, he organized
a systematic engineering curriculum covering the basics of civil and mechanical
engineering. Ironically, despite his intense interest in education, as a lecturer in
the classroom Reynolds left something to be desired. His lectures were hard to
follow, and his topics frequently wandered with little or no connection. He was
known to have new ideas during the course of a lecture and to spend the
remainder of the lecture working out these ideas on the board, seemingly
oblivious to the students in the classroom. That is, he did not "spoonfeed" his
students, and many of the poorer students did not pass his courses. In contrast,
the best students enjoyed his lectures and found them stimulating. Many of
Reynolds's successful students went on to become distinguished engineers and
scientists, the most notable being Sir J. J. Thomson, later the Cavendish Professor
of Physics at Cambridge; Thomson is famous for first demonstrating the existence
of the electron in 1897, for which he received the Nobel prize in 1906.
In regard to Reynolds's interesting research approach, his student, colleague,
and friend, Professor A. H. Gibson, had this to say in his biography of Reynolds,
written for the British Council in 1946:
Reynolds' approach to a problem was essentially individualistic. He never began by reading what
others thought about the matter, but first thought this out for himself. The novelty of his
approach to some problems made some of his papers difficult to follow, especially those written
during his later years. His more descriptive physical papers, however, make fascinating reading,
and when addressing a popular audience, his talks were models of clear exposition.
At the turn of the century, Reynolds's health began to fail, and he subse
quently had to retire in 1905. The last years of his life were ones of considerably
diminished physical and mental capabilities, a particularly sad state for such a
brilliant and successful scholar. He died at Somerset, England, in 1912. Sir
Horace Lamb, one of history's most famous fluid dynamicists and a longtime
colleague of Reynolds, wrote after Reynolds's death:
The character of Reynolds was, like his writings, strongly individual. He was conscious of the
value of his work, but was content to leave it to the mature judgement of the scientific world. For
advertisement he had no taste, and undue pretensions on the part of others only elicited a
tolerant smile. To his pupils he was most generous in the opportunities for valuable work which
he put in their way, and in the share of cooperation. Somewhat reserved in serious or personal
matters and occasionally combative and tenacious in debate, he was in the ordinary relations of
life the most kindly and genial of companions. He had a keen sense of humor and delighted in
startling paradoxes, which he would maintain, half seriously and half playfully, with astonishing
ingenuity and resource. The illness which at length compelled his retirement was felt as a
grievous calamity by his pupils, his colleagues and other friends throughout the country.
BASIC AERODYNAMICS 169
The purpose of this section has been to relate the historical beginnings of the
Reynolds number in fluid mechanics. From now on, when you use the Reynolds
number, hopefully you will view it not only as a powerful dimensionless parame
ter governing viscous flow, but also as a testimonial to its originatorone of the
famous fluid dynamicists of the nineteenth century.
4.24 HISTORICAL NOTE:
PRANDTL AND THE DEVELOPMENT OF THE
BOUNDARY LAYER CONCEPT
The modern science of aerodynamics has its roots as far back as Isaac Newton,
who devoted the entire second book of his Principia (1687) to fluid
dynamicsespecially to the formulation of "laws of resistance" (drag). He noted
that drag is a function of fluid density, velocity, and shape of the body in motion.
However, Newton was unable to formulate the correct equation for drag. Indeed,
he derived a formula which gave the drag on an inclined object as proportional to
the sine squared of the angle of attack. Later, Newton's sinesquared law was used
to demonstrate the "impossibility of heavierthanair flight" and served to hinder
the intellectual advancement of flight in the nineteenth century. Ironically, the
physical assumptions used by Newton in deriving his sinesquared law approxi
mately reflect the conditions of hypersonic flight, and the "newtonian law" has
been used since 1950 in the design of high Mach number vehicles. However,
Newton correctly reasoned the mechanism of shear stress in a fluid. In section 9
of book 2 of the Principia, Newton states the following hypothesis: "The
resistance arising from want of lubricity in the parts of a fluid is ... proportional
to the velocity with which the parts of the fluid are separated from each other."
This is the first statement in history of the friction law for laminar flow; it is
embodied in Eq. (4.89), which describes a socalled "newtonian fluid."
Further attempts to understand fluid dynamic drag were made by the French
mathematician Jean le Rond d' Alembert, who is noted for developing the calculus
of partial differences (leading to the mathematics of partial differential equations).
In 1768, d' Alembert applied the equations of motion for an incompressible,
inviscid (frictionless) flow about a twodimensional body in a moving fluid and
found that no drag is obtained. He wrote: "I do not see then, I admit, how one
can explain the resistance of fluids by the theory in a satisfactory manner. It
seems to me on the contrary that this theory, dealt with and studied with
profound attention gives, at least in most cases, resistance absolutely zero: a
singular paradox which I leave to geometricians to explain." That this theoretical
result of zero drag is truly a paradox was clearly recognized by d' Alembert, who
also conducted experimental research on drag and who was among the first to
discover that drag was proportional to the square of the velocity, as derived in
Sec. 5.3 and given in Eq. (5.18).
170 INTRODUCTION TO FLIGHT
D' Alembert's paradox arose due to the neglect of friction in the classical
theory. It was not until a century later that the effect of friction was properly
incorporated in the classical equations of motion by the work of M. Navier
(17851836) and Sir George Stokes (18191903). The socalled NavierStokes
equations stand today as the classical formulation of fluid dynamics. However, in
general they are nonlinear equations and are extremely difficult to solve; indeed,
only with the numerical power of modern highspeed digital computers are
"exact" solutions of the NavierStokes equations finally being obtained for
general flow fields. Also in the nineteenth century, the first experiments on
transition from laminar to turbulent flow were carried out by Osborne Reynolds
(18421912), as related in Sec. 4.23. In his classic paper of 1883 entitled "An
Experimental Investigation of the Circumstances which Determine whether the
Motion of Water in Parallel Channels Shall Be Direct or Sinuous, and of the Law
of Resistance in Parallel Channels," Reynolds observed a filament of colored dye
in a pipe flow and noted that transition from laminar to turbulent flow always
corresponded to approximately the same value of a dimensionless number,
pVD/µ., where D was the diameter of the pipe. This was the origin of the
Reynolds number, defined in Sec. 4.14, and discussed at length in Sec. 4.23.
Therefore, at the beginning of the twentieth century, when the Wright
brothers were deeply involved in the development of the first successful airplane,
the development of theoretical fluid dynamics still had not led to practical results
for aerodynamic drag. It was this environment into which Ludwig Prandtl was
born on February 4, 1875, at Freising, in Bavaria, Germany. Prandtl was a genius
who had the talent of cutting through a maze of complex physical phenomena to
extract the most salient points and put them in simple mathematical form.
Educated as a physicist, Prandtl was appointed in 1904 as professor of applied
mechanics at Gottingen University in Germany, a post he occupied until his
death in 1953.
In the period 19021904, Prandtl made one of the most important contribu
tions to fluid dynamics. Thinking about the viscous flow over a body, he reasoned
that the flow velocity right at the surface was zero and that if the Reynolds
number was high enough, the influence of friction was limited to a thin layer
(Prandtl first called it a transition layer) near the surface. Therefore, the analysis
of the flow field could be divided into two distinct regionsone close to the
surface, which included friction, and the other farther away, in which friction
could be neglected. In one of the most important fluid dynamics papers in history,
entitled "Uber Flussigkeitsbewegung bei sehr kleiner Reibung," Prandtl reported
his thoughts to the Third International Mathematical Congress at Heidelberg in
1904. In this paper, Prandtl observed:
A very satisfactory explanation of the physical process in the boundary layer (Grenzschicht)
between a fluid and a solid body could be obtained by the hypothesis of an adhesion of the fluid
to the walls, that is, by the hypothesis of a zero relative velocity between fluid and wall. If the
viscosity is very small and the fluid path along the wall not too long, the fluid velocity ought to
BASIC AERODYNAMICS 171
resume its normal value at a very short distance from the wall. In the thin transition layer
however, the sharp changes of velocity, even with small coefficient of friction, produce marked
results.
In the same paper, Prandtl's theory is applied to the prediction of fl.ow
separation:
In given cases, in certain points fully determined by external conditions, the fluid flow ought to
separate from the wall. That is, there ought to be a layer of fluid which, having been set in
rotation by the friction on the wall, insinuates itself into the free fluid, transforming completely
the motion of the latter ....
Prandtl's boundary layer hypothesis allows the NavierStokes equations to be
reduced to a simpler form; by 1908, Prandtl and one of his students, H. Blasius,
had solved these simpler boundary layer equations for laminar fl.ow over a fiat
plate, yielding the equations for boundary layer thickness and skin friction drag
given by Eqs. (4.91) and (4.93). Finally, after centuries of effort, the first rational
resistance laws describing fluid dynamic drag due to friction had been obtained.
Prandtl's work was a stroke of genius, and it revolutionized theoretical
aerodynamics. However, possibly due to the language barrier, it only slowly
diffused through the worldwide technical community. Serious work on boundary
layer theory did not emerge in England and the United States until the 1920s. By
that time, Prandtl and his students at Gottingen had applied it to various
aerodynamic shapes and were including the effects of turbulence.
Prandtl has been called the "father of aerodynamics," and rightly so. His
contributions extend far beyond boundary layer theory; e.g., he pioneered the
development of wing lift and drag theory, as will be seen in the next chapter.
Moreover, he was interested in more fields than just fluid dynamicshe made
several important contributions to structural mechanics as well.
As a note on Prandtl's personal life, he had the singleness of purpose which
seems to drive many giants of humanity. However, his almost complete preoc
cupation with his work led to a somewhat naive outlook on life. Theodore von
Karman, one of Prandtl's most illustrious students, relates that Prandtl would
rather find fancy in the examination of children's toys than participate in social
gatherings. When Prandtl was near 40, he suddenly decided that it was time to get
married, and he wrote to a friend for the hand of one of his two
daughtersPrandtl did not care which one! During the 1930s and early 1940s,
Prandtl had mixed emotions about the political problems of the day. He con
tinued his research work at Gottingen under Hitler's Nazi regime but became
continually confused about the course of events. Von Karman writes about
Prandtl in his autobiography:
I saw Prandtl once again for the last time right after the Nazi surrender. He was a sad figure. The
roof of his house in Gottingen, he mourned, had been destroyed by an American bomb. He
couldn't understand why this had been done to him! He was also deeply shaken by the collapse
of Germany. He lived only a few years after that, and though he did engage in some research
work in meteorology, he died, I believe, a broken man, still puzzled by the ways of mankind.
172 INTRODUCTION TO FLIGHT
Prandtl died in Gottingen on August 15, 1953. Of any fluid dynamicist or
aerodynamicist in history, Prandtl came closest to deserving a Nobel prize. Why
he never received one is an unanswered question. However, as Jong as there are
flight vehicles, and as long as students study the discipline of fluid dynamics, the
name of Ludwig Prandtl will be enshrined for posterity.
4.25 CHAPTER SUMMARY
A few of the important concepts from this chapter are summarized as follows:
1. The basic equations of aerodynamics, in the form derived here, are:
Continuity:
Momentum: dp = pVdV
Energy:
(4.2)
(4.8)
(4.42)
These equations hold for a compressible flow. For an incompressible flow, we
have:
Continuity: (4.3)
Momentum: (4.9a)
Equation (4.9a) is called Bernoulli's equation.
2. The change in pressure, density, and temperature between two points in an
isentropic process is given by
3. The speed of sound is given by
a=
( : ) isentropic
(4.48)
For a perfect gas, this becomes
a= /yRT (4.54)
4. The speed of a gas flow can be measured by a Pitot tube, which senses the
total pressure p
0
. For incompressible flow,
(4.66)
BASIC AERODYNAMICS 173
For subsonic compressible flow,
v/ = ~ f l [ ( ;: r1)/y 1] (4.77a)
For supersonic flow, a shock wave exists in front of the Pitot tube, and Eq.
(4.79) must be used in lieu of Eq. (4.77a) in order to find the Mach number of
the flow.
5. The areavelocity relation for isentropic flow is
dA = ( M z  1) dV
A V
(4.83)
From this relation, we observe that (1) for a subsonic flow, the velocity
increases in a convergent duct and decreases in a divergent duct, (2) for a
supersonic flow, the velocity increases in a divergent duct and decreases in a
convergent duct, and (3) the flow is sonic only at the minimum area.
6. The isentropic flow of a gas is governed by
(4.74)
To=l+ ylM2
T1 2 i
(4.73)
p ( y  1 )l/(y1)
_Q= 1 +
2
M/
P1
(4.75)
Here T
0
, p
0
, and p
0
are the total temperature, pressure, and density,
respectively. For an isentropic flow, p
0
= const throughout the flow. Simi
larly, p
0
= const and T
0
= const throughout the flow.
7. Viscous effects create a boundary layer along a solid surface in a flow. In this
boundary layer, the flow moves slowly and the velocity goes to zero right at
the surface. The shear stress at the wall is given by
T = JL( dV) (4.89)
w dy y ~ O
The shear stress is larger for a turbulent boundary layer than for a laminar
boundary layer.
8. For a laminar boundary layer, on a fiat plate,
(4.91)
and (4.98)
where l) is the boundary layer thickness, cf is the total skin friction drag
174 INTRODUCTION TO FLIGHT
coefficient, and Re is the Reynolds number;
Re = Poc,Voox
x /Loo
(local Reynolds number)
(plate Reynolds number)
Here, x is the running length along the plate, and L is the total length of the
plate.
9. For a turbulent boundary layer on a fiat plate,
8 = 0.37x (4.99)
Rexo.2
c = 0.074
f ReLo.2
(4.100)
10. Any real flow along a surface first starts out as laminar but then changes into
a turbulent flow. The point where this transition effectively occurs (in reality,
transition occurs over a finite length) is designated xcr· In turn, the critical
Reynolds number for transition is defined as
(4.101)
11. Whenever a boundary layer encounters an adverse pressure gradient (a region
of increasing pressure in the flow direction), it can readily separate from the
surface'. On an airfoil or wing, such flow separation decreases the lift and
increases the drag.
BIBLIOGRAPHY
Airey, J., "Notes on the Pitot Tube," Engineering News, vol. 69, no. 16, April 17, 1913, pp. 782783.
Goin, K. L., "The History, Evolution, and Use of Wind Tunnels," AIAA Student Journal, February
1971, pp. 313.
Guy, A. E., "Origin and Theory of the Pitot Tube," Engineering News, vol. 69, no. 23, June 5, 1913,
pp. 11721175.
Kuethe, A. M., and Schetzer, J. D., Foundations of Aerodynamics, Wiley, New York, 1959.
Pope, A. Aerodynamics of Supersonic Flight, Pitman, New York, 1958.
von Karman, T., Aerodynamics, McGrawHill, New York, 1963.
PROBLEMS
4.1 Consider the incompressible flow of water through a divergent duct. The inlet velocity and area
are 5 ft/s and 10 ft
2
, respectively. If the exit area is four times the inlet area, calculate the water flow
velocity at the exit.
BASIC AERODYNAMICS 175
4.2 In the above problem, calculate the pressure difference between the exit and the inlet. The density
of water is 62.4 lbm/ft
3
.
4.3 Consider an airplane flying with a velocity of 60 m/s at a standard altitude of 3 km. At a point
on the wing, the airflow velocity is 70 m/s. Calculate the pressure at this point. Assume incom
pressible flow.
4.4 An instrument used to measure the airspeed on many early lowspeed airplanes, principally
during 19191930, was the venturi tube. This simple device is a convergentdivergent duct. (The front
section's crosssectional area A decreases in the flow direction, and the back section's crosssectional
area increases in the flow direction. Somewhere in between the inlet and exit of the duct, there is a
minimum area, called the throat.) Let A
1
and A
2
denote the inlet and throat areas, respectively. Let
p
1
and p
2
be the pressures at the inlet and throat, respectively. The venturi tube is mounted at a
specific location on the airplane (generally on the wing or near the front of the fuselage), where the
inlet velocity V
1
is essentially the same as the freestream velocity, i.e., the velocity of the airplane
through the air. With a knowledge of the area ratio A
2
/ A
1
(a fixed design feature) and a measurement
of the pressure difference p
1
 p
2
, the airplane's velocity can be determined. For example, assume
A
2
/ A
1
= 1/4, and Pi  p
2
= 80 lb/ft
2
. If the airplane is flying at standard sea level, what is its
velocity?
4.5 Consider the flow of air through a convergentdivergent duct, such as the venturi described in
Prob. 4.4. The inlet, throat, and exit areas are 3, 1.5, and 2 m
2
, respectively. The inlet and exit
pressures are 1.02 X 10
5
and 1.00 X 10
5
N/m
2
, respectively. Calculate the flow velocity at the throat.
Assume incompressible flow with standard sealevel density.
4.6 An airplane is flying at a velocity of 130 mijh at a standard altitude of 5000 ft. At a point on the
wing, the pressure is 1750.0 lb/ft
2
• Calculate the velocity at that point, assuming incompressible flow.
4.7 Imagine that you have designed a lowspeed airplane with a maximum velocity at sea level of 90
m/s. For your airspeed instrument, you plan to use a venturi tube with a 1.3: 1 area ratio. Inside the
cockpit is an airspeed indicatora dial that is connected to a pressure gauge sensing the venturi tube
pressure difference p
1
 p
2
and properly calibrated in terms of velocity. What is the maximum
pressure difference you would expect the gauge to experience?
4.8 A supersonic nozzle is also a convergentdivergent duct, which is fed by a large reservoir at the
inlet to the nozzle. In the reservoir of the nozzle, the pressure and temperature are 10 atm and 300 K,
respectively. At the nozzle exit, the pressure is 1 atm. Calculate the temperature and density of the
flow at the exit. Assume the flow is isentropic and, of course, compressible.
4.9 Derive an expression for the exit velocity of a supersonic nozzle in terms of the pressure ratio
between the reservoir and exit, p
0
/p,, and the reservoir temperature T
0
.
4.10 Consider an airplane flying at a standard altitude of 5 km with a velocity of 270 m/s. At a point
on the wing of the airplane, the velocity is 330 m/s. Calculate the pressure at this point.
4.11 The mass flow of air through a supersonic nozzle is 1.5 lbm/s. The exit velocity is 1500 ftjs, and
the reservoir temperature and pressure are 1000°R and 7 atm, respectively. Calculate the area of the
nozzle exit. For air, cP = 6000 ft· lb/(slug)(
0
R).
4.12 A supersonic transport is flying at a velocity of 1500 mi/hat a standard altitude of 50,000 ft.
The temperature at a point in the flow over the wing is 793.32°R. Calculate the flow velocity at that
point.
4.13 For the airplane in Prob. 4.12, the total crosssectional area of the inlet to the jet engines is 20
ft
2
. Assume that the flow properties of the air entering the inlet are those of the freestream al!ead of
the airplane. Fuel is injected inside the engine at a rate of 0.05 lb of fuel for every pound of air
flowing through the engine (i.e., the fuelair ratio by mass is 0.05). Calculate the mass flow (in slugs/s)
that comes out the exit of the engine.
4.14 Calculate the Mach number at the exit of the nozzle in Prob. 4.11.
4.15 A Boeing 747 is cruising at a velocity of 250 m/s at a standard altitude of 13 km. What is its
Mach number?
176 INTRODUCTION TO FLIGHT
4.16 A highspeed missile is traveling at Mach 3 at standard sea level. What is its velocity in miles per
hour?
4.17 Calculate the flight Mach number for the supersonic transport in Prob. 4.12.
4.18 Consider a lowspeed subsonic wind tunnel with a nozzle contraction ratio of 1: 20. One side of
a mercury manometer is connected to the settling chamber, and the other side to the test section. The
pressure and temperature in the test section are 1 atm and 300 K, respectively. What is the height
difference between the two columns of mercury when the test section velocity is 80 m/s?
4.19 We wish to operate a lowspeed subsonic wind tunnel so that the flow in the test section has a
velocity of 200 mi/h at standard sealevel conditions. Consider two different types of wind tunnels:
(a) a nozzle and a constantarea test section, where the flow at the exit of the test section simply
dumps out to the surrounding atmosphere; i.e., there is no diffuser, and (b) a conventional
arrangement of nozzle, test section, and diffuser, where the flow at the exit of the diffuser dumps out
to the surrounding atmosphere. For both wind tunnels (a) and (b) calculate the pressure differences
across the entire wind tunnel required to operate them so as to have the given flow conditions in the
test section. For tunnel (a) the crosssectional area of the entrance is 20 ft
2
, and the crosssectional
area of the test section is 4 ft
2
• For tunnel (b) a diffuser is added to (a) with a diffuser area of 18 ft
2
.
After completing your calculations, examine and compare your answers for tunnels (a) and (b).
Which requires the smaller overall pressure difference? What does this say about the value of a
diffuser on a subsonic wind tunnel?
4.20 A Pitot tube is mounted in the test section of a lowspeed subsonic wind tunnel. The flow in
the test section has a velocity, static pressure, and temperature of 150 mijh, 1 atm, and 70°F,
respectively. Calculate the pressure measured by the Pitot tube.
4.21 The altimeter on a lowspeed Piper Aztec reads 8000 ft. A Pitot tube mounted on the wingtip
measures a presure of 1650 lb/ft
2
. If the outside air temperature is 500°R, what is the true velocity of
the airplane? What is the equivalent airspeed?
4.22 The altimeter on a lowspeed airplane reads 2 km. The airspeed indicator reads 50 m/s. If the
outside air temperature is 280 K, what is the true velocity of the airplane?
4.23 A Pitot tube is mounted in the test section of a highspeed subsonic wind tunnel. The pressure
and temperature of the airflow are 1 atm and 270 K, respectively. If the flow velocity is 250 m/s, what
is the pressure measured by the Pitot tube?
4.24 A highspeed subsonic Boeing 707 airliner is flying at a pressure altitude of 12 km. A Pitot tube
on the vertical tail measures a pressure of 2.96 X 10
4
N/m
2
. At what Mach number is the airplane
flying?
4.25 A highspeed subsonic airplane is flying at Mach 0.65. A Pitot tube on the wingtip measures a
pressure of 2339 lb/ft
2
. What is the altitude reading on the altimeter.
4.26 A highperformance F16 fighter is flying at Mach 0.96 at sea level. What is the air temperature
at the stagnation point at the leading edge of the wing?
4.27 An airplane is flying at a pressure altitude of 10 km with a velocity of 596 m/s. The outside air
temperature is 220 K. What is the pressure measured by a Pitot tube mounted on the nose of the
airplane?
4.28 The dynamic pressure is defined as q = 0.5pV
2
• For highspeed flows, where Mach number is
used frequently, it is convenient to express q in terms of pressure p and Mach number M rather than
p and V. Derive an equation for q = q(p, M).
4.29 After completing its mission in orbit around the earth, the space shuttle enters the earth's
atmosphere at very high Mach number and, under the influence of aerodynamic drag, slows as it
penetrates deeper into the atmosphere. (These matters are discussed in Chap. 8.) During its
atmospheric entry, assume that the shuttle is flying at the Mach number M corresponding to the
altitudes h :
h, (km) 60 50 40 30 20
M 17 9.5 5.5 3
BASIC AERODYNAMICS 177
Calculate the corresponding values of the freestream dynamic pressure at each one of these flight path
points. Suggestion: Use the result from Prob. 4.28. Examine and comment on the variation of q"' as
the shuttle enters the atmosphere.
4.30 Consider a Mach 2 airstream at standard sealevel conditions. Calculate the total pressure of this
flow. Compare this result with (a) the stagnation pressure that would exist at the nose of a blunt body
in the flow and (b) the erroneous result given by Bernoulli's equation, which of course does not apply
here.
4.31 Consider the flow of air through a supersonic nozzle. The reservoir pressure and temperature are
5 atm and 500 K, respectively. If the Mach number at the nozzle exit is 3, calculate the exit pressure,
temperature, and density.
4.32 Consider a supersonic nozzle across which the pressure ratio is Pe/Po = 0.2. Calculate the ratio
of exit area to throat area.
4.33 Consider the expansion of air through a convergentdivergent supersonic nozzle. The Mach
number varies from essentially zero in the reservoir to Mach 2.0 at the exit. Plot on graph paper the
variation of the ratio of dynamic pressure to total pressure as a function of Mach number, i.e., plot
q/p
0
versus M from M = 0 to M = 2.0.
4.34 The wing of the Fairchild Republic AlOA twinjet closesupport airplane is approximately a
rectangular shape with a wingspan (the length perpendicular to the flow direction) of 17.5 m and a
chord (the length parallel to the flow direction) of 3 m. The airplane is flying at standard sea level
with a velocity of 200 m/s. If the flow is considered to be completely laminar, calculate the boundary
layer thickness at the trailing edge and the total skin friction drag. Assume the wing is approximated
by a flat plate.
4.35 In Prob. 4.34, assume the flow is completely turbulent. Calculate the boundary layer thickness at
the trailing edge and the total skin friction drag. Compare these turbulent results with the above
laminar results.
4.36 If the critical Reynolds number for transition is 10
6
, calculate the skin friction drag for the wing
in Prob. 4.34.
CHAPTER
FIVE
AIRFOILS, WINGS, AND OTHER
AERODYNAMIC SHAPES
There can be no doubt that the inclined plane is the true principle of aerial navigation by mechanical
means.
Sir George Cayley, 1843
5.1 INTRODUCTION
It is remarkable that the modern airplane as we know it today, with its fixed wing
and vertical and horizontal tail surfaces, was first conceived by George Cayley in
1799, more than 175 years ago. He inscribed his first concept on a silver disc
(presumably for permanence) shown in Figure 1.5. It is also remarkable that
Cayley recognized that a curved surface (as shown on the silver disc) creates more
lift than a flat surface. Cayley's fixedwing concept was a true revolution in the
development of heavierthanair flight machines. Prior to his time, aviation
enthusiasts had been doing their best to imitate mechanically the natural flight of
birds, which led to a series of humanpowered flappingwing designs ( ornithopters),
which never had any real possibility of working. In fact, even Leonardo da Vinci
devoted a considerable effort to the design of many types of ornithopters in the
late fifteenth century, of course to no avail. In such ornithopter designs, the
flapping of the wings was supposed to provide simultaneously both lift (to sustain
the machine in the air) and propulsion (to push it along in flight). Cayley is
responsible for directing people's minds away from imitating bird flight and for
separating the two principles of lift and propulsion. He proposed and demon
strated that lift can be obtained from a fixed, straight wing inclined to the
airstream, while propulsion can be provided by some independent mechanism
178
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 179
such as paddles or airscrews. For this concept and for his many other thoughts
and inventions in aeronautics, Sir George Cayley is truly the parent of modern
aviation. A more detailed discussion of Cayley's contributions is given in Chap. 1.
However, emphasis is made that much of the technology discussed in the present
chapter had its origins at the beginning of the nineteenth centurytechnology
that came to fruition on December 17, 1903, near Kitty Hawk, North Carolina.
The following sections develop some of the terminology and basic
aerodynamic fundamentals of airfoils and wings. These concepts form the heart
of airplane flight and they represent a major excursion into aeronautical engineer
ing.
5.2 AIRFOIL NOMENCLATURE
Consider the wing of an airplane, as sketched in Figure 5.1. The crosssectional
shape obtained by the intersection of the wing with the perpendicular plane
shown in Figure 5.1 is called an airfoil. Such an airfoil is sketched in Figure 5.2,
which defines some basic terminology. The major design feature of an airfoil is
the mean camber line, which is the locus of points halfway between the upper and
lower surfaces as measured perpendicular to the mean camber line itself. The
most forward and rearward points of the mean camber line are the leading and
trailing edges, respectively. The straight line connecting the leading and trailing
edges is the chord line of the airfoil, and the precise distance from the leading to
the trailing edge measured along the chord line is simply designated the chord of
the airfoil, given by the symbol c. The camber is the maximum distance between
the mean camber line and the chord line, measured perpendicular to the chord
line. The camber, the shape of the mean camber line, and to a lesser extent, the
thickness distribution of the airfoil essentially control the lift and moment
characteristics of the airfoil.
Figure 5.1 Sketch of a wing and
airfoil.
180 INTRODUCTION TO FLIGHT
.Thickness
Mean camber line
Leading edge
__ J ___ _
Trailing edge
Camber I
Chordci•
Figure 5.2 Airfoil nomenclature. The shape shown here is a NACA 4415 airfoil.
More definitions are illustrated in Figure 5.3, which shows an airfoil inclined
to a stream of air. The freestream velocity V
00
is the velocity of the air far
upstream of the airfoil. The direction of V
00
is defined as the relative wind. The
angle between the relative wind and the chord line is the angle of attack a of the
airfoil. As described in Chaps. 2 and 4, there is an aerodynamic force created by
the pressure and shear stress distributions over the wing surface. This resultant
force is shown by the vector R in Figure 5.3. In tum, the aerodynamic force R
can be resolved into two forces, parallel and perpendicular to the relative wind.
The drag D is always defined as the component of the aerodynamic force parallel
to the relative wind. The lift L is always defined as the component of the
aerodynamic force perpendicular to the relative wind.
In addition to lift and drag, the surface pressure and shear stress distributions
also create a moment M which tends to rotate the wing. To see more clearly how
this moment is created, consider the surface pressure distribution over an airfoil
Cl
(
L,_R
;I
I I
I I
I I
I I
/ I
I I
I I
I I
I
~
ID
e ~
Relative wind
Figure 5.3 Sketch showing the definitions of lift, drag, moments, angle of attack, and relative wind.
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 181
as sketched in Figure 5.4 (we will ignore the shear stress for this discussion).
Consider just the pressure on the top surface of the airfoil. This pressure gives rise
to a net force F
1
in the general downward direction. Moreover, F
1
acts through a
given point on the chord line, point 1, which can be found by integrating the
pressure times distance over the surface (analogous to finding the centroid or
center of pressure from integral calculus). Now consider just the pressure on the
bottom surface of the airfoil. This pressure gives rise to a net force F
2
in the
general upward direction, acting through point 2. The total aerodynamic force on
the airfoil is the summation of F
1
and F
2
, and lift is obtained when F
2
> Fi.
However, note from Figure 5.4 that F
1
and F
2
will create a moment which will
tend to rotate the airfoil. Moreover, the value of this aerodynamically induced
moment depends on the point about which we chose to take moments. For
example, if we take moments about the leading edge, the aerodynamic moment is
designated M LE· It is more common in the case of subsonic airfoils to take
moments about a point on the chord at a distance c/4 from the leading edge, the
quarterchord point, as illustrated in Figure 5.3. This moment about the quarter
chord is designated Mc
14
. In general, MLE * Mc
14
. Intuition will tell you that
lift, drag, and moments on a wing will change as the angle of attack a changes. In
fact, the variations of these aerodynamic quantities with a represent some of the
most important information that an airplane designer needs to know. We will
address this matter in the following sections. However, it should be pointed out
that, although M LE and Mc
14
are both functions of a, there exists a certain point
on the airfoil about which moments essentially do not vary with a. This point is
defined as the aerodynamic center, and the moment about the aerodynamic center
is designated Mac· By definition,
Mac= const
F
2
Note: Length of the arrow denoting pressure
is proportional top  pref, where Pref is an
arbitrary reference pressure slightly less than
the minimum pressure on the airfoil.
Figure 5.4 The physical origin of moments on an airfoil.
182 INTRODUCTION TO FLIGHT
independent of angle of attack. The location of the aerodynamic center for real
aerodynamic shapes can be found from experiment. For lowspeed subsonic
airfoils, the aerodynamic center is generally very close to the quarterchord point.
5.3 LIFT, DRAG, AND MOMENT COEFFICIENTS
Again appealing to intuition, we note that it makes sense that for an airplane in
flight, the actual magnitudes of L, D, and M depend not only on a, but on
velocity and altitude as well. In fact, we can expect that the variations of L, D,
and M depend at least on:
1. Freestream velocity Vw
2. Freestream density p
00
, that is, on altitude.
3. Size of the aerodynamic surface. For airplanes, we will use the wing area S to
indicate size.
4. Angle of attack a.
5. Shape of the airfoil.
6. Viscosity coefficient µ
00
(because the aerodynamic forces are generated in
part from skin friction distributions).
7. Compressibility of the airflow. In Chap. 4 we demonstrated that compressibil
ity effects are governed by the value of the freestream Mach number,
M
00
= V
00
/a
00
• Since V
00
is already listed above, we can therefore designate
a
00
as our index for compressibility.
Hence, we can write that, for a given shape airfoil at a given angle of attack,
(5.1)
and D and M are similar functions.
In principle, for a given airfoil at a given angle of attack, we could find the
variation of L by performing a myriad of windtunnel experiments wherein V
00
,
Pw S, µw and a
00
are individually varied and then try to make sense out of the
resulting huge collection of data. This is the hard way. Instead, we could ask the
question, are there groupings of the quantities Vw p
00
, S, µ
00
, a
00
, and L such
that Eq. (5.1) can be written in terms of fewer parameters? The answer is yes. In
the process of developing this answer, we will gain some insight into the beauty of
nature as applied to aerodynamics.
The technique we will apply is a simple example of a more general theoretical
approach called dimensional analysis. Let us assume that Eq. (5.1) is of the
functional form
(5.2)
where Z, a, b, d, e, and fare dimensionless constants. However, no matter what
the values of these constants may be, it is a physical fact that the dimensions of
the left and righthand sides of Eq. (5.2) must match; i.e., if L is a force (say in
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 183
newtons), then the net result of all the exponents and multiplication on the
righthand side must also produce a result with dimensions of a force. This
constraint will ultimately give us information on the values of a, b, etc. If we
designate the basic dimensions of mass, length, and time by m, I, and t,
respectively, then the dimensions of various physical quantities are as given
below.
Physical Quantity Dimensions
ml/t
2
(from Newton's second law)
l/t
m/1
3
,2
l/t
m/ft
Thus, equating the dimensions of the left and righthand sides of Eq. (5.2), we
obtain
(5.3)
Consider mass m. The exponent of m on the lefthand side is 1. Thus, the
exponents of m on the right must add up to 1. Hence
l=b+f (5.4)
Similarly, for time t we have
2 = a  e  f (5.5)
and for length /,
1 = a  3b + 2d + e  f (5.6)
Solving Eqs. (5.4) to (5.6) for a, b, and d in terms of e and f yields
b = 1  f (5.7)
a = 2  e  f (5.8)
d=1L (5.9)
2
Substituting Eqs. (5. 7) to (5.9) into (5.2) gives
L = Z(V )2e/ lfs11;2 e I
oo Pao aoo µoo
(5.10)
Rearranging Eq. (5.10), we find
L = ZpOOVOO 2s( ~ ) e ( µ00 1/2) I
00 PooVooS
(5.11)
Note that a
00
/V
00
= l/M
00
, where M
00
is the freestream Mach number. Also
184 INTRODUCTION TO FLIGHT
note that the dimensions of S are 1
2
; hence the dimension of s
1
1
2
is /, purely a
length. Let us choose this length to be the chord c by convention. Hence,
µ
00
/ p
00
V
00
S
112
can be replaced in our consideration by the equivalent quantity
µoo/PooVooc
However, µ
00
/p
00
V
00
c = l/Re, where Re is based on the chord length c. Hence,
Eq. (5.11) becomes
L = Zp00V00
2
s( ~ o o ) e ~ e r
(5.12)
We now define a new quantity called the lift coefficient c
1
as
c,/2 = z( ~ 0 0 ) e ~ e r
( 5 .13)
Then, Eq. (5.12) becomes
L = 1PooVoo
2
Sc, (5.14)
Recalling from Chap. 4 that the dynamic pressure is q
00
= ip
00
V
00
2
, we transform
Eq. (5.14) into
L =
qoo
x s x c, ( 5 .15)
i i i i
Lift Dynamic Wing Lift
pressure area
coefficient
Look what has happened! Equation (5.1), written from intuition, but not very
useful, has cascaded to the simple, direct form of Eq. (5.15), which contains a
tremendous amount of information. In fact, Eq. (5.15) is one of the most
important relations in applied aerodynamics. It says that the lift is directly
proportional to the dynamic pressure (hence to the square of the velocity). It is
also directly proportional to the wing area Sand to the lift coefficient c
1
• In fact,
Eq. (5.15) can be turned around and used as a definition for the lift coefficient:
(5.16)
That is, the lift coefficient is always defined as the aerodynamic lift divided by the
dynamic pressure and some reference area (for wings, the convenient reference
area S, as we have been using).
The lift coefficient is a function of M
00
and Re as reflected in Eq. (5.13).
Moreover, since M
00
and Re are dimensionless and since Z was assumed initially
as a dimensionless constant, from Eq. (5.13) c
1
is dimensionless. This is also
consistent with Eqs. (5.15) and (5.16). Also, recall that the above derivation was
carried out for an airfoil of given shape and at a given angle of attack a. If a were
to vary, then c
1
would also vary. Hence, for a given airfoil,
(5.17)
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 185
This relation is important. Fix in your mind that lift coefficient is a function of
angle of attack, Mach number, and Reynolds number.
Performing a similar dimensional analysis on drag and moments, beginning
with relations analogous to Eq. (5.1), we find that
(5.18)
where cd is a dimensionless drag coefficient, and
I M = q00 Sccm I
(5.19)
where cm is a dimensionless moment coefficient. Note that Eq. (5.19) differs
slightly from Eqs. (5.15) and (5.18) by the inclusion of the chord length c. This is
because L and D have dimensions of a force, whereas M has dimensions of a
forcelength product.
The importance of Eqs. (5.15) to (5.19) cannot be emphasized too much. They
are fundamental to all of applied aerodynamics. They are readily obtained from
dimensional analysis, which essentially takes us from loosely defined functional
relationships [such as Eq. (5.1)] to welldefined relations between dimensionless
quantities [Eqs. (5.15) to (5.19)]. In summary, for an airfoil of given shape, the
dimensionless lift, drag, and moment coefficients have been defined as
(5.20)
where
(5.21)
Reflecting for an instant, we find there may appear to be a conflict in our
aerodynamic philosophy. On the one hand, Chaps. 2 and 4 emphasized that lift,
drag, and moments on an aerodynamic shape stem from the detailed pressure and
shear stress distributions on the surface and that measurements and/or calcula
tions of these distributions, especially for complex configurations, are not trivial
undertakings. On the other hand, Eqs. (5.20) indicate that lift, drag, and moments
can be quickly obtained from simple formulas. The bridge between these two
outlooks is, of course, the lift, drag, and moment coefficients. All the physical
complexity of the flow field around an aerodynamic body is implicitly buried in
c
1
, cd, and cm. Before the simple Eqs. (5.20) can be used to calculate lift, drag,
and moments for an airfoil, wing, or body, the appropriate aerodynamic coeffi
cients must be known. From this point of view, the simplicity of Eqs. (5.20) is a
bit deceptive. These equations simply shift the forces of aerodynamic rigor from
the forces and moments themselves to the appropriate coefficients instead. So we
are now led to the questions, how do we obtain values of c
1
, c d• and cm for given
186 INTRODUCTION TO FLIGHT
configurations, and how do they vary with a::, Mw and Re? The answers are
introduced in the following sections.
5.4 AIRFOIL DA TA
A goal of theoretical aerodynamics is to predict values of ct, cd, and cm from the
basic equations and concepts of physical science, some of which were discussed in
previous chapters. However, simplifying assumptions are usually necessary to
make the mathematics tractable. Therefore, when theoretical results are obtained,
they are generally not "exact." The use of highspeed digital computers to solve
the governing flow equations is now bringing us much closer to the accurate
calculation of aerodynamic characteristics; however, there are still limitations
imposed by the numerical methods themselves, and the storage and speed
capacity of current computers is still not sufficient to solve many complex
aerodynamic flows. As a result, the practical aerodynamicist has to rely upon
direct experimental measurements of ct, cd, and cm for specific bodies of interest.
A large bulk of experimental airfoil data was compiled over the years by the
National Advisory Committee for Aeronautics (NACA), which was absorbed in
the creation of the National Aeronautics and Space Administration (NASA) in
1958. Lift, drag, and moment coefficients were systematically measured for many
airfoil shapes in lowspeed subsonic wind tunnels. These measurements were
carried out on straight, constantchord wings which completely spanned the
tunnel test section from one side wall to the other. In this fashion, the flow
essentially "saw" a wing with no wingtips, and the experimental airfoil data were
thus obtained for "infinite wings." (The distinction between infinite and finite
wings will be made in subsequent sections.) Some results of these airfoil measure
ments are given in Appendix D. The first page of Appendix D gives data for ct
and cm,c/
4
versus angle of attack for the NACA 1408 airfoil. The second page
gives cd and cm,ac versus ct for the same airfoil. Since Ct is known as a function
of a from the first page, then the data from both pages can be crossplotted to
obtain the variation of c d and cm ac versus a::. The remaining pages of Appendix
D give the same type of data for different standard NACA airfoil shapes.
Let us examine the variation of ct with a more closely. This variation is
sketched in Figure 5.5. The experimental data indicate that ct varies linearly with
a over a large range of angle of attack. Thinairfoil theory, which is the subject of
more advanced books on aerodynamics, also predicts the same type of linear
variation. The slope of the linear portion of the lift curve is designated as
a
0
= dc/da = lift slope. Note that in Figure 5.5, when a = 0, there is still a
positive value of c
1
, that is, there is still some lift even when the airfoil is at zero
angle of attack to the flow. This is due to the positive camber of the airfoil. All
airfoils with such camber have to be pitched to some negative angle of attack
before zero lift is obtained. The value of a when lift is zero is defined as the zero
lift angle of attack a L ~ o and is illustrated in Figure 5.5. This effect is further
demonstrated in Figure 5.6, where the lift curve for a cambered airfoil is
Ct, max
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 187
maximum cf,
stalling angle
of attack
Stall due to
flow separation
°'
Figure 5.5 Sketch of a typical
lift curve.
compared with that for a symmetric (no camber) airfoil. Note that the lift curve
for a symmetric airfoil goes through the origin. Refer again to Figure 5.5, at the
other extreme: for large values of a, the linearity of the lift curve breaks down. As
a is increased beyond a certain value, c
1
peaks at some maximum value, c
1
max•
and then drops precipitously as a is further increased. In this situation, w h r ~ the
lift is rapidly decreasing at high a, the airfoil is stalled.
The phenomenon of airfoil stall is of critical importance in airplane design. It
is caused by flow separation on the upper surface of the airfoil. This is illustrated
in Figure 5.7, which again shows the variation of c
1
versus a for an airfoil. At
point 1 on the linear portion of the lift curve, the flow field over the airfoil is
attached to the surface, as pictured in Figure 5.7. However, as discussed in Chap.
4, the effect of friction is to slow the airflow near the surface; in the presence of
Cambered airfoil Symmetric airfoil
°'
DI[,= 0
Figure 5.6 Comparison of lift curves for cambered and symmetric airfoils.
188 INTRODUCTION TO FLIGHT
Separated flow
Attached flow
Figure 5.7 Flow mechanism associated with stalling.
an adverse pressure gradient, there will be a tendency for the boundary layer to
separate from the surface. As the angle of attack is increased, the adverse pressure
gradient on the top surface of the airfoil will become stronger, and at some value
of athe stalling angle of attackthe flow becomes separated from the top
surface. When separation occurs, the lift decreases drastically and the drag
increases suddenly. This is the picture associated with point 2 in Figure 5.7. (It
would be well for the reader to review at this stage the discussion on flow
separation and its effect on pressure distribution, lift, and drag ih Sec. 4.18.)
The nature of the flow field over the wing of an airplane that is below, just
beyond, and way beyond the stall is shown in Figures 5.8a, b, and c, respectively.
These figures are photographs of a windtunnel model with a wingspan of 6 ft.
The entire model has been painted with a mixture of mineral oil and a fluorescent
powder, which glows under ultraviolet light. After the wind tunnel is turned on,
the fluorescent oil indicates the streamline pattern on the surface of the model. In
Figure 5.8a, the angle of attack is below the stall; the flow is fully attached, as
evidenced by the fact that the high surface shear stress has scrubbed most of the
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 189
oil from the surface. In Figure 5.8b, the angle of attack is slightly beyond the stall.
A large, mushroomshaped, separated flow pattern has developed over the wing,
with attendant highly threedimensional, lowenergy recirculating flow. In Figure
5.8c, the angle of attack is far beyond the stall. The flow over almost the entire
wing has separated. These photographs are striking examples of different types of
flow that can occur over an airplane wing at different angles of attack, and they
graphically show the extent of the flow field separation that cart occur.
The lift curves sketched in Figures 5.5 to 5.7 illustrate the type of variation
observed experimentally in the data of Appendix D. Returning to Appendix D,
note that the lift curves are all virtually linear up to the stall. Singling out a given
airfoil, say the NACA 2412 airfoil, also note that ct versus a is given for three
different values of the Reynolds number from 3.1 x 10
6
to 8.9 x 10
6
. The lift
curves for all three values of Re fall on top of each other in the linear region, i.e.,
Re has little influence on ct when the flow is attached. However, flow separation is
a viscous effect, and as discussed in Chap. 4, Re is a governing parameter for
viscous flow. Therefore, it is not surprising that the experimental data for Ct max in
the stalling region are affected by Re, as can be seen by the slightly different
variations of ct at high a for different values of Re. In fact, these lift curves at
different Re answer part of the question posed in Eq. (5.17): the data represent
ct= /(Re). Again, Re exerts little or no effect on ct except in the stalling region.
On the same page as c
1
versus a, the variation of cm, c/
4
versus a is also
given. It has only a slight variation with a and is almost completely unaffected by
Re. Also note that the values of cm, c/
4
are slightly negative. By convention, a
positive moment is in a clockwise direction; it pitches the airfoil towards larger
angles of attack, as shown in Figure 5.3. Therefore, for the NACA 2412 airfoil,
with cm,c/
4
negative, the moments are counterclockwise, and the airfoil tends to
pitch downward. This is characteristic of all airfoils with positive camber.
On the page following Ct and cm,c/
4
, the variation of cd and cm,ac is given
versus ct. Because c
1
varies linearly with a, the reader can visualize these curves of
cd and cm,ac as being plotted versus a as well; the shapes will be the same. Note
that the drag curves have a "bucket" type of shape, with minimum drag occurring
at small values of c
1
(hence there are small angles of attack). As a goes to large
negative or positive values, cd increases. Also note that cd is strongly affected by
Re, there being a distinct drag curve for each Re. This is to be expected because
the drag for a slender aerodynamic shape is mainly skin friction drag, and from
Chap. 4 we have seen that Re strongly governs skin friction. With regard to cm,ac•
the definition of the aerodynamic center is clearly evident; cm,ac is constant with
respect to a. Also, it is insensitive to Re and has a small negative value.
Refer to Eqs. (5.21); the airfoil data in Appendix D experimentally provide
the variation of c
1
, cd, and cm with a and Re. The effect of M
00
on the airfoil
coefficients will be discussed later. However, emphasis is made that the data in
Appendix D were measured in lowspeed subsonic wind tunnels. Hence, the flow
was essentially incompressible. Thus, ct, cm,c/
4
, cd, and cm,ac given in Appendix
D are incompressible flow values. It is important to keep this in mind during our
subsequent discussions.
190 INTRODUCTION TO FLIGHT
(a)
(h)
Figure 5.8 Surface oilflow patterns on a windtunnel model of a Grumman American Yankee taken
by Dr. Allen Winkelmann in the Glenn L. Martin Wind Tunnel <it the University of Maryland. The
mixture is mineral oil and a fluorescent powder, and the photographs were taken under ultraviolet
light. (a) Below the stall. The wing is at a = 4 °, where the flow is attached. ( b) Very near the stall.
The wing is at a = 11 °, where the highly threedimensional separated flow is developing in a
mushroom cell pattern. ( c) Far above the stall. The wing is at a = 24 °, where the flow over almost the
entire wing has separated.
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 191
( c)
Example 5.1 A model wing of constant chord length is placed in a lowspeed subsonic wind
tunnel, spanning the test section. The wing has a NACA 2412 airfoil and a chord length of 1.3 m.
The flow in the test section is at a velocity of 50 m/s at standard sealevel conditions. If the wing
is at a 4° angle of attack, calculate (a) c
1
, cd, and cm,c/
4
and (b) the lift, drag, and moments
about the quarter chord, per unit span.
SOLUTION
(a) From Appendix D, for a NACA 2412 airfoil at a 4° angle of attack,
c, = 0.63
cm,c/4 = 0.035
To obtain cd, we must first check the value of the Reynolds number:
p
00
V
00
c (1.225 kg/m3)(50 m/s)(l.3 m)
Re = = "='''''''
µoo l.789xlo
5
kg/(m){s)
= 4.45x10
6
For this value of Re and for c
1
= 0.63, from Appendix D,
I Cd= 0.0071
(b) Since the chord is 1.3 m and we want the aerodynamic forces and moments per unit
span (a unit length along the wing, perpendicular to the flow), then S = c(l) = 1.3(1) = 1.3 m
2
•
Also
192 INTRODUCTION TO FLIGHT
From Eq. (5.20),
L = %0Sc1=1531(1.3)(0.63) = j 1254 NI
Since 1 N = 0.2248 lb, then also
L = (1254 N)(0.2248 lb/N) = 281.9 lb
D = q
00
Scd = 1531(1.3)(0.007) = j u.9 NI
=13.9(0.2248) = 3.13 lb
Note: The ratio of lift to drag, which is an important aerodynamic quantity, is
.!:_ = .!i = 1254 = 90 2
D Cd 13.9 .
M,;4=q
00
Scm.c/4c=1531(1.3)( 0.035)(1.3)
I M,
14
= 90.6 (N)(m) I
Example 5.2 The same wing in the same flow as in Example 5.1 is pitched to an angle of attack
such that the lift per unit span is 700 N (157 lb).
(a) What is the angle of attack?
(b) To what angle of attack must the wing be pitched to obtain zero lift?
SOLUTION
Thus
(a) From the previous example,
q
00
=1531 N/m
2
S =1.3 m
2
c
1
= _L_ =
7
00 = 0.352
q
00
S 1531(1.3)
From Appendix D for the NACA 2412 airfoil, the angle of attack corresponding to c
1
= 0.352 is
(b) Also from Appendix D, for zero lift, that is, c
1
= 0,
I ~ o =  2.20 I
5.5 INFINITE VERSUS FINITE WINGS
As stated in Sec. 5.4, the airfoil data in Appendix D were measured in lowspeed
subsonic wind tunnels where the model wing spanned the test section from one
side wall to the other. In this fashion, the ft.ow sees essentially a wing with no
wingtips, i.e., the wing could in principle be stretching from plus infinity to minus
infinity in the spanwise direction. Such an infinite wing is sketched in Figure 5.9,
where the wing stretches to ± oo in the z direction. The ft.ow about this wing
varies only in the x and y directions; for this reason the flow is called twodimen
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 193
+oo
Figure 5.9 Infinite (twodimensional) wing.
sional. Thus, the airfoil data in Appendix D apply only to such infinite (or
twodimensional) wings. This is an important point to keep in mind.
On the other hand, all real airplane wings are obviously finite, as sketched in
Figure 5.10. Here, the top view (planform view) of a finite wing is shown, where
the distance between the two wingtips is defined as the wingspan b. The area of
the wing in this planform view is designated, as before, by S. This leads to an
important definition which pervades all aerodynamic wing considerations, namely,
the aspect ratio AR.
b2
Aspect ratio = AR = S
The importance of AR will come to light in subsequent sections.
Rightwing tip
b ; wingspan
Leftwing tip Figure 5.10 Finite wing; plan view (top).
(5.22)
194 INTRODUCTION TO FLIGHT
The flow field about a finite wing is threedimensional and is therefore
inherently different from the twodimensional flow about an infinite wing. As a
result, the lift, drag, and moment coefficients for a finite wing with a given airfoil
shape at a given a are different from the lift, drag, and moment coefficients for an
infinite wing with the same airfoil shape at the same a. For this reason, the
aerodynamic coefficients for a finite wing are designated by capital letters, Cv Cv,
CM; this is in contrast to those for an infinite wing, which we have been
designating as c
1
, cd, and cm. Note that the data in Appendix D are for infinite
(twodimensional) wings, i.e., the data are for c
1
, cd, and cm. In a subsequent
section, we will show how to obtain the finite wing aerodynamic coefficients from
the infinite wing data in Appendix D. Our purpose in this section is simply to
underscore that there is a difference.
5.6 PRESSURE COEFFICIENT
We continue with our parade of aerodynamic definitions. Consider the pressure
distribution over the top surface of an airfoil. Instead of plotting the actual
pressure (say in units of N/m
2
), let us define a new dimensionless quantity called
the pressure coefficient cp as
C = P  Poo _ P  Poo
p qoo 1 V 2
IPoo oo
(5.23)
The pressure distribution is sketched in terms of CP in Figure 5.11. This figure is
worth close attention, because pressure distributions found in the aerodynamic
literature are usually given in terms of the dimensionless pressure coefficient. Note
from Figure 5.11 that CP at the leading edge is positive because p > pOO" However,
as the flow expands around the top surface of the airfoil, p decreases rapidly, and
1.5
0.5
0.5
1.0
1.0
Lower surface
Figure 5.11 Distribution of pressure
coefficient over the top and bottom
surfaces of a NACA 0012 airfoil at
3.93° angle of attack. M
00
= 0.345,
Re = 3.245 X 10
6
• Experimental data
x/c from Ohio State University, in NACA
Conference Publication 2045, part I,
Advanced Technology Airfoil Research,
vol. I, p. 1590. (After Freuler and
Gregorek.)
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 195
CP goes negative in those regions where p < p
00
• By convention, plots of CP for
airfoils are usually shown with negative values above the abscissa, as shown in
Figure 5.11.
The pressure coefficient is an important quantity; for example, the distribu
tion of CP over the airfoil surface leads directly to the value of c
1
, as will be
discussed in a subsequent section. Moreover, considerations of CP lead directly to
the calculation of the effect of Mach number M
00
on the lift coefficient. To set the
stage for this calculation, consider CP at a given point on an airfoil surface. The
airfoil is a given shape at a fixed angle of attack. The value of CP can be measured
by testing the airfoil in a wind tunnel. Assume that, at first, V
00
in the tunnel test
section is low, say, M
00
< 0.3, such that the flow is essentially incompressible. The
measured value of CP at the point on the airfoil will therefore be a lowspeed
value. Let us designate the lowspeed (incompressible) value of cp by cp,O· If voo
is increased but M
00
is still less than 0.3, CP will not change, that is, CP is
essentially constant with velocity at low speeds. However, if we now increase V
00
such that M
00
> 0.3, then compressibility becomes a factor, and the effect of
compressibility is to increase the absolute magnitude of CP as M
00
increases. This
variation of CP with M
00
is shown in Figure 5.12. Note that, at M
00
::::: 0,
CP = Cp,o· As M
00
increases to M
00
::::: 0.3, CP is essentially constant. However, as
M
00
is increased beyond 0.3, CP increases dramatically. (That is, the absolute
magnitude increases; if cp,0 is negative, then cp will become an increasingly more
negative number as Moo increases, whereas if cp,O is positive, then cp will become
an increasingly more positive number as M
00
increases.) The variation of CP with
1.6
"
0.
"'
"O
"
1.2
><
t:::
'
0
~
·=
"'
<=: ..><
"' u
0.8
<=:"'
0 t::
~ '
<=: '
· 0
8. .£
"' ""
<=:
~ '
"' "O 0.4
l;,,.)1:),.§
Figure 5.12 Plot of the PrandtlGlauert rule for Cp.o = 0.5.
196 INTRODUCTION TO FLIGHT
M
00
for high subsonic Mach numbers was a major focus of aerodynamic research
after World War II. An approximate theoretical analysis yields the result
(5.24)
Equation (5.24) is called the PrandtlG/auert rule. It is reasonably accurate for
0.3 < M
00
< 0.7. For M
00
> 0.7, its accuracy rapidly diminishes; indeed, Eq.
(5.24) predicts that CP becomes infinite as M
00
goes to unityan impossible
physical situation. (It is well to note that nature abhors infinities as well as
discontinuities that are sometimes predicted by mathematical, but approximate,
theories in physical science.) There are more accurate, but more complicated,
formulas than Eq. (5.24) for nearsonic Mach numbers. However, Eq. (5.24) will
be sufficient for our purposes.
Formulas such as Eq. (5.24), which attempt to predict the effect of M
00
on CP
for subsonic speeds, are called compressibility corrections, i.e., they modify (cor
rect) the lowspeed pressure coefficient Cp.o to take into account the effects of
compressibility which are so important at high subsonic Mach numbers.
Example 5.3 The pressure at a point on the wing of an airplane is 7.58x10
4
N /m
2
. The airplane
is flying with a velocity of 70 m/s at conditions associated with a standard altitude of 2000 m.
Calculate the pressure coefficient at this point on ihe wing.
SOLUTION For a standard altitude of 2000 m,
p
00
=7.95Xl0
4
N/m
2
POX>= 1.0066 kg/m3
Thus, q
00
= &p
00
V
00
2
= t<J.0066)(70)
2
= 2466 N/m
2
. From Eq. (5.23),
p Poo (7.587.95)X10
4
cp = q:; = 2466
Example 5.4 Consider an airfoil mounted in a lowspeed subsonic wind tunnel. The flow velocity
in the test section is 100 ftjs, and the conditions are standard sea level. If the pressure at a point
on the airfoil is 2102 lb/ft
2
, what is the pressure coefficient?
SOLUTION
From Eq. (5.23),
%o = i p
00
V
00
2
= H 0.002377 slug/ft
3
) (100 ftjs }2
=ll.89 lb/ft
2
C = p Poo = 21022116 ~
p q
00
11.89 L::_j
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 197
Example 5.5 In Example 5.4, if the flow velocity is increased such that the freestream Mach
number is 0.6, what is the pressure coefficient at the same point on the airfoil?
SOLUTION First of all, what is the Mach number of the flow in Example 5.4? At standard sea
level,
T, = 518.69 R
Hence, a
00
= yRT
00
=yll.4(1716)(518.69) =1116 ft/s
Thus, in Example 5.4, M
00
= V
00
/ a
00
= 100/1116 = 0.09a very low value. Hence, the flow in
Example 5.4 is essentially incompressible, and the pressure coefficient is a lowspeed value, that
is, Cp.o = 1.18. Thus, if the flow Mach number is increased to 0.6, from the PrandtlGlauert
rule, Eq. (5.24),
c
C = p,O
p ( 2)1/2
1 M
00
1.18
5.7 OBTAINING LIFT COEFFICIENT FROM Cp
If you are given the distribution of pressure coefficient over the top and bottom
surfaces of an airfoil, you can calculate c
1
in a straightforward manner. Consider
a segment of an infinite wing, as shown in Figure 5.13. Assume the segment has
unit span and chord c. Let the x direction correspond to the direction of V
00
, that
is, the relative wind, and let c be aligned parallel to V
00
for the time being; i.e., let
a = 0. Also, let s be the distance measured along the surface from the leading
edge. Also, the angle between the normal to the airfoil surface and a line
dscoslJ=dx
IJ dx
LE (leading edge)
TE (trailing edge)
Figure 5.13 Sketch showing how the pressure distribution can be integrated to obtain lift per unit
span.
198 INTRODUCTION TO FLIGHT
perpendicular to the freestream (but in the plane of the airfoil section) is given by
0. Also consider an infinitesimal strip of the surface area with length ds and unit
width, as shown by the dotted lines in Figure 5.13. The aerodynamic force due to
pressure on this strip is p ds (1), which acts normal to the surface. Its component
in the lift direction (perpendicular to the relative wind) is ( p cos 0) ds. Adding a
subscript u to designate the pressure on the upper surface of the airfoil, as well as
a negative sign to indicate the force is directed downward (we use the convention
that a positive force is directed upward), the contribution to lift of the pressure on
the infinitesimal strip is  Pu cos() ds. If the contributions from all the strips on
the upper surface are added from the leading to the trailing edges, we obtain, by
letting ds approach zero, the integral
J
TE
 PucosOds
LE
This is the force in the lift direction due to the pressure distribution on the upper
surface. A similar term is obtained for the lower surface of the airfoil, and hence
the total lift acting on an airfoil of unit span is given by
J
TE JTE
L = PtCosOds  pucosOds
LE LE
(5.25)
where pt denotes the pressure on the lower surface. From the small triangle in the
corner of Figure 5.13, we see the geometric relationship dscos() = dx. Thus, Eq.
(5.25) becomes
L = { ~ t x  {Pudx
0 0
(5.26)
Adding and subtracting p
00
, we find Eq. (5.26) becomes
L =[(Pt  p
00
) dx  [(Pu  p
00
) dx
0 0
(5.27)
From the definition of lift coefficient, Eq. (5.16),
L L L
Ct= qooS = qooc(l) = qooc
(5.28)
Combining Eqs. (5.27) and (5.28) yields
1 ic Pt  Poo 1 ic Pu  Poo
ct=  dx   dx
C 0 qoo C 0 qoo
(5.29)
Note that
Pt  Poo = CP t = pressure coefficient on lower surface
qoo '
Pu ;OOPoo = cp, u = pressure coefficient on upper surface
Hence, Eq. (5.29) becomes
(5 .30)
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 199
Equation (5.30) is a useful relationship; it demonstrates that the lift coefficient
can be obtained by integrating CP over the airfoil surface. If you have plots of
pressure coefficient data on the upper and lower surface vs. distance x, then the
lift coefficient is equal to the net area between the curves, divided by the chord
length.
Recall that the derivation of Eq. (5.30) assumed that c, x, and V
00
were all in
the same direction. Hence Eq. (5.30) is valid only for a = 0, and it is a good
approximation for a small a. For a more detailed and accurate derivation which
is valid for any angle of attack, see Chap. 1 of Anderson, Fundamentals of
Aerodynamics (McGrawHill, 1984).
Example 5.6 Consider an airfoil with chord length c and the running distance x measured along
the chord. The leading edge is located at x/c = 0 and the trailing edge at x/c = 1. The
pressurecoefficient variations over the upper and lower surfaces are given respectively as
c = 1  3 0 0 ~ )
2
p. u c
x
cp "= 2.2211 + 2.2111
. c
x
C / = 1  0.95
p, c
Calculate the lift coefficient.
SOLUTION From Eq. (5.30)
x
forO s  s 0.1
c
x
forO.l s  s 1.0
c
x
forO s  s 1.0
c
x 11 (x )21
1
x 10.1 (x )3IO.l x
1
1.0 (x )z,l.O
c, =  0.475    + 100  + 2.2277  1.1388 
c 0 c 0 c 0 c 0 c 0.1 c 0.1
c, = 1  0.475  0.1 + 0.1 + 2.2277  0.22277  1.1388 + 0.011388 = 11.40 I
It is interesting to note that, when Cp,
1
and Cp," are plotted on the same graph versus x/ c,
represents the area between the two curves, as shown in Figure 5.14. Also note that the CP
variations given analytically in this problem are only crude representations of a realistic case and
should not be taken too seriously, since the purpose of this example is simply to illustrate the use
of Eq. (5.30).
200 INTRODUCTION TO FLIGHT
1.0
v"
"
"
'()
s
"
8 0
!:!
x/c
J:
1.0
Figure 5.14 Sketch of the pressure coefficient over the upper and lower surfaces of an airfoil showing
that the area between the two curves is the lift coefficient.
5.8 COMPRESSIBILITY CORRECTION FOR
LIFT COEFFICIENT
The pressure coefficients in Eq. (5.30) can be replaced by the compressibility
correction given in Eq. (5.24), as follows
1 ic ( cp,1 cp,Jo 1 l lc
C1 =  v dx = v  ( CP I  CP J
0
dx
c lM
2
lM
2
co' '
0 00 00
(5.31)
where again the subscript 0 denotes lowspeed incompressible flow values. How
ever, referring to the form of Eq. (5.30), we see that
1 c
fo ( Cp,/  Cp,u}
0
dx := C1,0
where c
1
,
0
is the lowspeed value of lift coefficient. Thus, Eq. (5.31) becomes
( 5 .32)
Equation (5.32) gives the compressibility correction for lift coefficient. It is subject
to the same approximations and accuracy restrictions as the PrandtlGlauert rule,
Eq. (5.24). Also note that the airfoil data in Appendix D were obtained at low
speeds, hence the values of lift coefficient obtained from Appendix D are c
1
,
0
•
Finally, in reference to Eq. (5.17), we now have a reasonable answer to how c
1
varies with Mach number. For subsonic speeds, except near Mach 1, the lift
coefficient varies inversely as (1  M
00
2
)
1
1
2
•
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 201
Example 5.7 Consider a NACA 4412 airfoil at an angle of attack of 4°. If the freestream Mach
number is 0.7, what is the lift coefficient?
SOLUTION From Appendix D, for o: = 4°, ct= 0.83. However, the data in Appendix D were
obtained at low speeds, hence the lift coefficient value obtained above, namely 0.83, is really ct.o:
l't,O = 0.83
For high Mach numbers, this must be corrected according to Eq. (5.32):
Ct Q 0.83
Ct= , =
( 1  Moo 2 )112 (1  0. 72) 1;2
I Ct= 1.16 at M00 = 0.71
5.9 CRITICAL MACH NUMBER AND CRITICAL
PRESSURE COEFFICIENT
Consider the flow of air over an airfoil. We know that, as the gas expands around
the top surface near the leading edge, the velocity and hence the Mach number
will increase rapidly. Indeed, there are regions on the airfoil surface where the
local Mach number is greater than M
00
• Imagine that we put a given airfoil in a
wind tunnel where M
00
= 0.3 and that we observe the peak local Mach number
on the top surface of the airfoil to be 0.435. This is sketched in Figure 5.I5a.
Imagine that we now increase M
00
to 0.5; the peak local Mach number will
correspondingly increase to 0.772, as shown in Figure 5.I5b. If we further
increase M
00
to a value of 0.61, we observe that the peak local Mach number is
Mpeak = 0.435
(a)
M== 0.5
(b)
Mpeak = 1.0, sonic flow first encountered on airfoil
Critical Mach number
for the airfoil
(c)
Figure 5.15 Illustration of critical Mach number.
202 INTRODUCTION TO FLIGHT
H
Cp,cr
(+)
pressure point on
the airfoil
Figure 5.16 Illustration of critical pressure coefficient.
1.0, locally sonic flow on the surface of the airfoil. This is sketched in Figure
5.15c. Note that the flow over an airfoil can locally be sonic (or higher), even
though the freestream Mach number is subsonic. By definition, that freestream
Mach number at which sonic flow is first obtained somewhere on the airfoil
surface is called the critical Mach number of the airfoil. In the above example, the
critical Mach number Mer for the airfoil is 0.61. As we will see later, Mer is an
important quantity, because at some freestream Mach number above Mer the
airfoil will experience a dramatic increase in drag.
Returning to Figure 5.15, the point on the airfoil where the local Mis a peak
value is also the point of minimum surface pressure. From the definition of
pressure coefficient, Eq. (5.23), CP will correspondingly have its most negative
value at this point. Moreover, according to the PrandtlGlauert rule, Eq. (5.24),
as M
00
is increased from 0.3 to 0.61, the value of CP at this point will become
increasingly negative. This is sketched in Figure 5.16. The specific value of CP
that corresponds to sonic flow is defined as the critical pressure coefficient Cp,er·
In Figures 5.15a and 5.15b CP at the minimum pressure point on the airfoil is less
negative than Cp,er; however, in Figure 5.15c, CP = Cp,er (by definition).
Consider now three different airfoils ranging from thin to thick, as shown in
Figure 5.17. Concentrate first on the thin airfoil. Because of the thin, streamlined
profile, the flow over the thin airfoil is only slightly perturbed from its freestream
values. The expansion over the top surface is mild, the velocity increases only
slightly, the pressure decreases only a relative small amount, and hence the
magnitude of CP at the minimum pressure point is small. Thus, the variation of
CP with M
00
is shown as the bottom curve in Figure 5.17. For the thin airfoil,
cp,O is small in magnitude, and the rate of increase of cp as Moo increases is also
relatively small. In fact, because the flow expansion over the thin airfoil surface is
mild, M
00
can be increased to a large subsonic value before sonic flow is
encountered on the airfoil surfc.ce. The point corresponding to sonic flow
conditions on the thin airfoil is labeled point a in Figure 5.17. The values of CP
and M
00
at point a are Cp,er and Men respectively, for the thin airfoil, by
definition. Now consider the airfoil of medium thickness. The flow expansion
over the leading edge for this medium airfoil will be stronger, the velocity will
"
0
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 203
Thin C
1.5,__ __ _
Thin airfoil
0.5
Mer Mer
(thick) (thin)
MCI
(medium)
1.0
Figure 5.17 Critical pressure coefficient and critical Mach numbers for airfoils of different thick
nesses.
increase to larger values, the pressure will decrease to lower values, and the
absolute magnitude of CP is larger. Thus, the pressure coefficient curve for the
medium thickness airfoil will lie above that for a thin airfoil, as demonstrated in
Figure 5.17. Moreover, because the flow expansion is stronger, sonic conditions
will be obtained sooner (at a lower M
00
). Sonic conditions for the medium airfoil
are labeled as point b in Figure 5.17. Note that point b is to the left of point a,
that is, the critical Mach number for the mediumthickness airfoil is less than Mer
for the thin airfoil. The same logic holds for the pressure coefficient curve for the
thick airfoil, where Cp,cr and Mer are given by point c. Emphasis is made that the
thinner airfoils have higher values of Mer· As we will see, this is desirable, and
hence all airfoils on modern, highspeed airplanes are thin.
The pressure coefficient curves in Figure 5 .17 are shown as solid curves. On
these curves, only points a, b, and c are critical pressure coefficients, by
definition. However, these critical points by themselves form a locus represented
204 INTRODUCTION TO FLIGHT
by the dotted curve in Figure 5.17; i.e., the critical pressure coefficients them
selves are given by a curve of Cp.cr = f(M
00
), as labeled in Figure 5.17. Let us
proceed to derive this function. It is an important result, and it also represents an
interesting application of our aerodynamic relationships developed in Chap. 4.
First, consider the definition of CP from Eq. (5.23):
c = p  Poo = Poo (L  1) (5.33)
p qoo qoo Poo
From the definition of dynamic pressure,
_ 1 2 1 Poo ( ) 2 _ 1 Voo
2
( )
qoo = IPooVoo = 2 YPoo YPoo voo  2 YPoo/Poo YPoo
However, from Eq. (4.53), a
00
2
= yp
00
/p
00
• Thus
1 voo 2 y 2
qoo = l 2 YPoo = 2 PooMoo
aoo
(5.34)
We will return to Eq. (5.34) in a moment. Now, recall Eq. (4.74) for isentropic
flow,
Po ( Y  1 2) rl<rll
= l+M
p 2
This relates the total pressure p
0
at a point in the flow to the static pressure p and
local Mach number M at the same point. Also, from the same relation,
Po ( Y  1 2)r/<rll
= l+M
Poo 2 oo
This relates the total pressure p
0
in the freestream to the freestream static
pressure p
00
and Mach number M
00
• For an isentropic flow, which is a close
approximation to the actual, reallife, subsonic flow over an airfoil, the total
pressure remains constant throughout. (We refer to more advanced books in
aerodynamics for a proof of this fact.) Thus, if the two previous equations are
divided, Po will cancel, yielding
_L  ( 1 + HY  l)Moo2) y/(y1)
 (5.35)
Poo 1 +Hy  l)M
2
Substitute Eqs. (5.34) and (5.35) into (5.33):
Poo P Poo 1 + 2 Y  1 Moo
[(
1
( ) 2)r/<rll l
cp = qoo (Poo 
1
) = hPooMoo
2
1 +HY  l)M
2
 l
C = _2_[( 1 +Hy l)Moo2)r/(yl) 1]
p yMoo
2
1 +HY  l)M
2
(5.36)
For a given freestream Mach number M
00
, Eq. (5.36) relates the local value of CP
to the local M at any given point in the flow field, hence at any given point on the
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 205
B
Cp,o .i..::..
Figure 5,18 Determmation of criti
cal Mach number,
airfoil surface. Let us pick that particular point on the surface where M = 1.
Then, by definition, CP = Cp.cr· Putting M = 1 into Eq. (5.36), we obtain
C = _2_[(2 +(y  l)M"/)rl<r1i 1]
p,cr M 2 Y + 1
y 00
(5.37)
Equation (5.37) gives the desired relation Cp,cr = f( M
00
). When numbers are fed
into Eq. (5.37), the dotted curve in Figure 5.17 results. Note that, as M
00
increases, cp,cr decreases.
Example 5.8 Given a specific airfoil, how can you estimate its critical Mach number?
SOLUTION There are several steps to this process, as follows.
(a) Obtain a plot of Cp,cr versus M from Eq. (5.37). This is illustrated by curve A in Figure
5.18. This curve is a fixed "universal" curve, which you can keep for all such problems.
(b) From measurement or theory for lowspeed flow, obtain the minimum pressure
coefficient on the top surface of the airfoil. This is Cp,o shown as point B in Figure 5.18.
(c) Using Eq. (5.24), plot the variation of this minimum pressure coefficient versus M
00
•
This is illustrated by curve C.
(d) When curve C intersects curve A, then the minimum pressure coefficient on the top
surface of the airfoil is equal to the critical pressure coefficient, and the corresponding M
00
is the
critical Mach number. Hence, point D is the solution for Mer
5.10 DRAGDIVERGENCE MACH NUMBER
We now turn our attention to the airfoil drag coefficient cd. Figure 5.19 sketches
the variation of c d with M
00
• At low Mach numbers, less than Mer' cd is virtually
constant and is equal to its lowspeed value given in Appendix D. The flow field
206 INTRODUCTION TO FLIGHT
c
a
Mer Mdrag 1.0
divergence
Figure 5.19 Variation of drag coefficient with Mach number.
about the airfoil for this condition (say, point a in Figure 5.19) is noted in Figure
5.20a, where M < 1 everywhere in the flow. If M
00
is increased slightly above
Mc,, a "bubble" of supersonic flow will occur surrounding the minimum pressure
point, as shown in Figure 5.20b. Correspondingly, cd will still remain reasonably
low, as indicated by point b in Figure 5.19. However, if M
00
is still further
increased, a very sudden and dramatic rise in the drag coefficient will be
observed, as noted by point c in Figure 5.19. Here, shock waves suddenly appear
Mer < Moo < Mctrag
divergence
Moo> Mctrag
divergence
M< I
c
(a)
(b)
M <
1
 Shock wave
//
//M> I M< I
Separated flow
( c)
Figure 5.20 Physical mechanism of
drag divergence. (a) Flow field
associated with point a in Figure
5.19. (b) Flow field associated with
point b in Figure 5.19. ( c) Flow
field associated with point c in
Figure 5.19.
s
0.8
0.7
0.6
0.5
;;:
0.4
" ·n
e
0.3
" 0
u
.::::
0.2
;::i
0.1
0
0.1
0.2
'
I
I
1
!
' r1\
, ....c.
I
4
::::::= [ ,J
: 3 
' (
2 '\
' !,  (
+
1 , o \V
!'"':I '\
I , 1, ,V r
I •
. l\'.()>  v
' ' deg
' J
l l .l.LJ _l j_J__,
t_j.L
0 0.1 0.2 0.3 0.4 0.5 0.6 0. 7 0.8 0.9 1.0
Mach number
(a)
0.18
0.17
0.16
0.15
0.14
0.13
0.12
<.'i' 0.11
;;:
.!l
0.10 u
E
"
0
0.09 u
Oil
"'
0 0.08
0.07
0.06
0.05
0.04
0.03
O.Q2
O.QJ
I
'
,_
.......
'
'
'
'
J
 /1
:; /
...._ I
._ J II/.
/1 71
I
 5 [//'/1 ll'.o,
1_.,____,_____.__4 ID deg _
 3 o
1  _,, ,,,,
I I I l I I I I I I
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mach number M
(b)
Figure 5.21 Variation of (a) lift coefficient and ( b) drag coefficient versus Mach number with angle of attack as a parameter
for an NACA 2315 airfoil. (Windtunnel measurements at the NACA Langley Memorial Laboratory.)
208 INTRODUCTION TO FLIGHT
c _____ ====
H
Figure 5.22 Shape of a typical supercritical airfoil and its pressure coefficient distribution over the top
surface.
in the flow, as sketched in Figure 5.20c. The shock waves themselves are
dissipative phenomena, which result in an increase in drag on the airfoil. But in
addition, the sharp pressure increase across the shock waves creates a strong
adverse pressure gradient, hence causing the flow to separate from the surface. As
discussed in Sec. 4.18, such flow separation can create substantial increases in
drag. Thus, the sharp increase in cd shown in Figure 5.19 is a combined effect of
shock waves and flow separation. The freestream Mach number at which cd
begins to increase rapidly is defined at the dragdivergence Mach number and is
noted in Figure 5.19. Note that
Mer < Mdragdivergence < 1.0
The shock pattern sketched in Figure 5.20c is characteristic of a flight regime
called transonic. When 0.8 :;;; M
00
:;;; 1.2, the flow is generally designated as
transonic flow and it is characterized by some very complex effects only hinted
at in Figure 5.20c. To reinforce these comments, Figure 5.21 shows the variation
of both c
1
and cd as a function of Mach number with angle of attack as a
parameter. The airfoil is a standard NACA 2315 airfoil. Figure 5.21, which shows
actual windtunnel data, illustrates the massive transonicflow effects on both lift
and drag coefficients. The analysis of transonic flows has been one of the major
challenges in modern aerodynamics. Only in recent years, since about 1970, have
computer solutions for transonic flows over airfoils come into practical use; these
numerical solutions are still in a state of development and improvement. Tran
sonic flow has been a hard nut to crack.
The airfoils on modern subsonic jet aircraft, such as the McDonnellDouglas
DC10 widebody transport, are relatively thin profiles designed to increase the
dragdivergence Mach number. In fact, NASA (mainly under the direction of
Richard Whitcomb at the Langley Research Center) has developed a new
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 209
supercritical airfoil, designed to place the dragdivergence Mach number ex
tremely close to 1.0. Supercritical airfoils are of the general shape shown in
Figure 5.22. Here, the maximum thickness is designed close to the trailing edge.
The flow over the airfoil is largely supersonic, and the airfoil shape is designed to
discourage the formation of shock waves.
These considerations of Mer and dragdivergence Mach number are im
portant to subsonic transport design. The cruising speeds of such airplanes can be
increased by incorporating airfoils with high values of Mdrag divergence· Alterna
tively, in the spirit of energy conservation, supercritical airfoils should allow less
drag, hence less fuel consumption, at the same cruising speed. In either case, the
advantages are obvious, and this helps to explain why there has been a modern
resurgence of interest in highspeed airfoil design.
As a final note in this section, the next time you have an opportunity to fly in
a jet airliner and the sun is directly overhead (around noon and early afternoon),
look out along the span of the wing. Due to the refraction of light waves through
shock waves, you can sometimes see with the naked eye the transonic shock waves
dancing about on the wing surface.
5.11 WAVE DRAG (AT SUPERSONIC SPEEDS)
To this point, we have discussed airfoil properties at subsonic speeds. When M
00
is supersonic, a major new physical phenomenon is introduced: shock waves. We
have previously alluded to shock waves in Sec. 4.11 C in conjunction with the
Pitottube measurement of supersonic airspeeds. With respect to airfoils (as well
as all other aerodynamic bodies) shock waves in supersonic flow create a new
source of drag called wave drag. In this section, we will simply highlight some of
the ideas involving shock waves and the consequent wave drag; a detailed study
of shock wave phenomena is left to more advanced texts in aerodynamics.
To obtain a feel for how a shock is produced, imagine that we have a small
source of sound waves, a tiny "beeper" (something like a tuning fork). At time
t = 0, assume the beeper is at point P in Figure 5.23. At this point, let the beeper
emit a sound wave, which will propagate in all directions at the speed of sound a.
Also, let the beeper move with velocity V, where V is less than the speed of
sound. At time t, the sound wave will have moved outward by a distance at, as
shown in Figure 5.23. At the same time t, the beeper will have moved a distance
Vt, to point Q. Since V < a, the beeper will always stay inside the sound wave. If
the beeper is constantly emitting sound waves as it moves along, these waves will
constantly move outward, ahead of the beeper. As long as V < a, the beeper will
always be inside the envelope formed by the sound waves.
On the other hand, assume the beeper is moving at supersonic speed, that is,
V > a. At time t = 0, assume the beeper is at point R in Figure 5.24. At this
point, let the beeper emit a sound wave, which, as before, will propagate in all
directions at the speed of sound a. At time t the sound wave will have moved
outward by a distance at, as shown in Figure 5.24. At the same time t, the beeper
will have moved a distance Vt, to point S. However, since V > a, the beeper will
210 INTRODUCTION TO FLIGHT
Location of sound
wave at time t
Location of
beeper at ~
time t
I Vt
Ql p
Location of beeper
at time = 0; it gives
off a sound wave
at t = 0
Beeper stays inside Figure S.23 Beeper moving less than the speed
the sound wave of sound.
now be outside the sound wave. If the beeper is constantly emitting sound waves
as it moves along, these waves will now pile up inside an envelope formed by a
line from point S tangent to the circle formed by the first sound wave, centered at
point R. This tangent line, the line where the pressure disturbances are piling up,
is called a Mach wave. The vertex of the wave is fixed to the moving beeper at
point S. In supersonic flight, the air ahead of the beeper in Figure 5.24 has no
warning of the approach of the beeper. Only the air behind the Mach wave has
felt the presence of the beeper, and this presence is communicated by pressure
Location of
beeper at
time t
Location of beeper at
time t = 0; it gives off
a sound wave at t = 0
Location of sound
wave at time t
Beeper stays outside
the sound wave
Figure S.24 The origin of Mach waves and shock waves. Beeper is moving faster than the speed of
sound.
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 211
Figure 5.25 Mach waves on a needlelike body.
(sound) waves confined inside the conical region bounded by the Mach wave. In
contrast, in subsonic flight, the air ahead of the beeper in Figure 5.23 is
forewarned about the oncoming beeper by the sound waves. In this case, there is
no piling up of pressure waves; there is no Mach wave.
Hence, we can begin to feel that the coalescing, or piling up, of pressure
waves in supersonic flight can create sharply defined waves of some sort. In
Figure 5.24, the Mach wave that is formed makes an angleµ with the direction of
movement of the beeper. This angle, defined as the Mach angle, is easily obtained
from the geometry of Figure 5.24, as follows:
. at a 1
smµ =Vt= V = M
Hence Mach angle = µ = arcsin ! (5.38)
In real life, a very thin object (such as a thin needle) moving at M
00
> 1
creates a very weak disturbance in the flow, limited to a Mach wave. This is
sketched in Figure 5.25. On the other hand, an object with some reasonable
thickness, such as the wedge shown in Figure 5.26, moving at supersonic speeds
will create a strong disturbance called a shock wave. The shock wave will be
inclined at an oblique angle /3, where /3 > µ, as shown in Figure 5.26. As the flow
Figure 5.26 Oblique shock waves on a
wedgetype body.
212 INTRODUCTION TO FLIGHT
Poo
Wave drag
net drag due to
higher pressure
behind the
shock wave
Figure 5.27 Pressure distribution on a wedge at supersonic speeds; origin of wave drag.
moves across the oblique shock wave, the pressure, temperature, and density
increase, and the velocity and Mach number decrease.
Consider now the pressure on the surface of the wedge, as sketched in Figure
5.27. Since p increases across the oblique shock wave, then at the wedge surface,
p > p
00
• Since the pressure acts normal to the surface and the surface itself is
inclined to the relative wind, there will be a net drag produced on the wedge, as
seen by simple inspection of Figure 5.27. This drag is called wave drag, because it
is inherently due to the pressure increase across the shock wave.
In order to minimize the strength of the shock wave, all supersonic airfoil
profiles are thin, with relatively sharp leading edges. (The leading edge of the
Lockheed F104 supersonic fighter is almost razor thin.) Let us approximate a
thin supersonic airfoil by the flat plate illustrated in Figure 5.28. The flat plate is
inclined at a small angle of attack a to the supersonic freestream. On the top
surface of the plate, the flow field is turned away from the freestream through an
Poo
Pressure
distribution
Expansion
Shock
wave
Flow field
Expansion
wave
'L
Dwave
Figure 5.28 Flow field and pressure distribution for a flat plate at angle of attack in supersonic flow.
There is a net lift and drag due to the pressure distribution set up by the shock and expansion waves.
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 213
expansion wave at the leading edge; an expansion wave is a fanshaped region
through which the pressure decreases. At the trailing edge on the top side, the
flow is turned back into the freestream direction through an oblique shock wave.
On the bottom surface of the plate, the flow is turned into the freestream, causing
an oblique shock wave with an increase in pressure. At the trailing edge, the flow
is turned back to the freestream direction through an expansion wave. (Details
and theory for expansion waves, as well as shock waves, are beyond the scope of
this bookyou will have to simply accept on faith the flow field sketched in
Figure 5.28 until your study of aerodynamics becomes more advanced.) The
expansion and shock waves at the leading edge result in a surface pressure
distribution where the pressure on the top surface is less than Pw whereas the
pressure on the bottom surface is greater than p
00
• The net effect is an aerodynamic
force normal to the plate. The components of this force perpendicular and
parallel to the relative wind are the lift and supersonic wave drag, respectively.
Approximate relations for the lift and drag coefficients are
4a
c, = (5.39)
(Moo21)1;2
4a
2
and (5.40)
(Moo21)1;2
A subscript w has been added to the drag coefficient to emphasize that it is the
wave drag coefficient. Equations (5.39) and (5.40) are approximate expressions,
useful for thin airfoils at small to moderate angles of attack in supersonic flow.
Note that as M
00
increases, both c
1
and cd decrease. This is not to say that the lift
and drag forces themselves decrease with Moo Quite the contrary, for any flight
regime, as the flight velocity increases, L and D usually increase because the
dynamic pressure, q
00
= tpV
00
2
, increases. In the supersonic regime, L and D
increase with velocity, even though c
1
and cd,w decrease with M
00
according to
Eqs. (5.39) and (5.40).
Example 5.9 Consider a thin supersonic airfoil with chord length c = 5 ft in a Mach 3
freestream at a standard altitude of 20,000 ft. The airfoil is at an angle of attack of 5°.
(a) Calculate the lift and wave drag coefficients and the lift and wave drag per unit span.
(b) Compare these results with the same airfoil at the same conditions, except at Mach 2.
SOLUTION (a) In Eqs. (5.39) and (5.40) the angle of attack a must be in radians. Hence
5
Also
a = 5° =  rad = 0.0873 rad
57.3
VM
00
2
 1 = ~ = 2 8 2 8
4a 4(0.0873) ~
c, = = 0.123
VM
00
2 _ 1 2.828
4a
2
Cd, w = V;=M=oo=2=_=1
4(0.0873)
2
8
 = 0.0108
2.828
214 INTRODUCTION TO FLIGHT
At 20,000 ft, p
00
= 1.2673 X 10
3
slug/ft
3
and T = 447.43°R. Hence
(b)
a
00
= /yRT
00
= /1.4(1716)(447.43) = 1037 ft/s
V
00
= M
00
a
00
= 3(1037) = 3111 ft/s
q
00
= }p
00
V
00
2
= }(1.2673 x 10
3
)(3111)
2
= 6133 lb/ft
2
L (per unit span) = q
00
cc
1
= 6133(5)(0.123) = 13772 lb I
Dw (per unit span) = q
00
ccd. w = 6133(5)(0.0108) = 1331.2 lb I
VM
00
2
 1 = = 1.732
4a = 4(0.0873) = I
0
.
2011
c, = V M,,,2  1 1.732
4a
2
4(0.0873)
2
B
c  ==== = = 0.0176
d.w  ./ 2 1.732
yM
00
 1
Note: At Mach 2, c
1
and cd w are higher than at Mach 3. This is a general result; both c
1
and
cd w decrease with increasing Mach number, as clearly seen from Eqs. (5.39) and (5.40). Does
this mean that L and Dw also decrease with increasing Mach number? Intuitively this does not
seem correct. Let us find out.
voo = aOOMOO = 1037(2) = 2074 ft/s
q
00
= }p
00
V
00
2
= }(1.2673 X 10
3
)(2074)
2
= 2726 Jb/ft
2
L (per unit span) = q
00
cc
1
= 2726(5)(0.207) = 12821 lb I
Dw (per unit span) = q
00
ccd, w = 2726(5)(0.0176) = 1240 lb I
Hence, there is no conflict with our intuition. As the supersonic Mach numbers increase, L and
Dw also increase although the lift and drag coefficients decrease.
5.12 SUMMARY OF AIRFOIL DRAG
Amplifying on Eq. (4.102), we can write the total drag of an airfoil as the sum of
three contributions:
D =DI+ DP+ Dw
where D = total drag on the airfoil
D
1
= skin friction drag
DP= pressure drag due to flow separation
Dw = wave drag (present only at transonic and supersonic speeds; zero for
subsonic speeds below the dragdivergence Mach number)
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 215
In terms of the drag coefficients, we can write:
where cd, cd.f' cd,p' and cd, w are the total drag, skin friction drag, pressure drag,
and wave drag coefficients, respectively. The sum cd.f + cd,p is called the profile
drag coefjicient; this is the quantity that is given by the data in Appendix D. The
profile drag coefficient is relatively constant with M
00
at subsonic speeds.
The variation of cd with M
00
from incompressible to supersonic speeds is
sketched in Figure 5.29. It is important to note the qualitative variation of this
curve. For M
00
ranging from zero to drag divergence, cd is relatively constant; it
consists entirely of profile drag. For M
00
from drag divergence to slightly above 1,
the value of cd skyrockets; indeed, the peak value of cd around M
00
=1 can be an
order of magnitude larger than the profile drag itself. This large increase in c d is
due to wave drag associated with the presence of shock waves. For supersonic
Mach numbers, cd decreases approximately as (M
00
2
1)
1
1
2
.
The large increase in the drag coefficient near Mach 1 gave rise to the term
"sound barrier" in the 1940s. There was a camp of professionals who at that time
felt that the sound barrier could not be pierced, that we could not fly faster than
the speed of sound. Certainly, a glance at Eq. (5.24) for the pressure coefficient in
subsonic flow, as well as Eq. (5.40) for wave drag in supersonic flow, would hint
that the drag coefficient might become infinitely large as M
00
approaches 1 from
either the subsonic or supersonic sides. However, such reasoning is an example of
a common pitfall in science and engineering, namely, the application of equations
outside their ranges of validity. Neither Eq. (5.24) nor (5.40) is valid in the
transonic range near M
00
=1. Moreover, remember that nature abhors infinities.
In real life, c d does not become infinitely large. To get past the sound barrier, all
that is needed (in principle) is an engine with enough thrust to overcome the high
(but finite) drag.
Profile drag
Mdrag 1.0
divergence
__ ]_
VM'!,,, I
Figure S.29 Variation of drag coefficient with Mach number for subsonic and supersonic speeds.
216 INTRODUCTION TO FLIGHT
5.13 FINITE WINGS
We now return to the discussion initiated in Sec. 5.5. Our considerations so far
have dealt mainly with airfoils, where the aerodynamic properties are directly
applicable to infinite wings. However, all real wings are finite, and for practical
reasons we must translate our knowledge about airfoils to the case where the wing
has wingtips. This is the purpose of the next two sections.
The fundamental difference between flows over finite wings as opposed to
infinite wings can be seen as follows. Consider the front view of a finite wing as
sketched in Figure 5.30a. If the wing has lift, then obviously the average pressure
over the bottom surface is greater than that over the top surface. Consequently,
there is some tendency for the air to "leak," or flow, around the wingtips from
the high to the lowpressure sides, as shown in Figure 5.30a. This flow estab
lishes a circulatory motion which trails downstream of the wing. The trailing
circular motion is called a vortex. There is a major trailing vortex from each
wingtip, as sketched in Figure 5.30b and as shown in the photograph in Figure
5.31.
These wingtip vortices downstream of the wing induce a small downward
component of air velocity in the neighborhood of the wing itself. This can be seen
intuitively from Figure 5.30b; the two wingtip vortices tend to drag the surround
ing air around with them, and this secondary movement induces a small velocity
component in the downward direction at the wing. This downward component is
called downwash, given by the symbol w.
An effect of downwash can be seen in Figure 5.32. As usual, V
00
designates
the relative wind. However, in the immediate vicinity of the wing, V
00
and w add
Vortex
(,_===========:=Low===pressure ~
\__: High pressure J _)
Front view of wing
(a)
(b)
Wingtip
vortices
Figure 5.30 Origin of wingtip vortices on a finite wing.
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 217
Figure 5.31 Wingtip vortices made visible by smoke ejected at the wing tips of a Boeing 727 test
airplane. (NASA.)
vectorally to produce a "local" relative wind which is canted downward from the
original direction of V
00
• This has several consequences:
1. The angle of attack of the airfoil sections of the wing is effectively reduced in
comparison to the angle of attack of the wing referenced to V
00
•
2. There is an increase in the drag. The increase is called induced drag, which
has at least three physical interpretations. First, the wingtip vortices simply
Relative wind
(free stream)
Tip vortex
~ ~
Local flow in The wingtip .
vicinity of wing vortex which trails
downstream causes
downwash, w
Figure 5.32 The origin of downwash.
218 INTRODUCTION TO FLIGHT
alter the flow field about the wing in such a fashion as to change the surface
pressure distributions in the direction of increased drag. An alternate ex
planation is that, because the local relative wind is canted downward (see
Figure 5.32), the lift vector itself is "tilted back," hence it contributes a
certain component of force parallel to V
00
, that is, a drag force. A third
physical explanation of the source of induced drag is that the wingtip vortices
contain a certain amount of rotational kinetic energy. This energy has to
come from somewhere; indeed, it is supplied by the aircraft propulsion
system, where extra power has to be added to overcome the extra increment
in drag due to induced drag. All three of these outlooks of the physical
mechanism of induced drag are synonymous.
5.14 CALCULATION OF INDUCED DRAG
Consider a section of a finite wing as shown in Figure 5.33. The angle of attack
defined between the mean chord of the wing and the direction of V
00
(the relative
wind) is called the geometric angle of attack a. However, in the vicinity of the
wing the local flow is (on the average) deflected downward by the angle a;
because of downwash. This angle a;, defined as the induced angle of attack, is the
difference between the local flow direction and the freestream direction. Hence,
although the naked eye sees the wing at an angle of attack a, the airfoil section
itself is seeing an effective angle of attack which is smaller than a. Letting aeff
denote the effective angle of attack, we see from Figure 5.33 that aeff = a  a;.
Let us now adopt the point of view that, because the local flow direction in
the vicinity of the wing is inclined downward with respect to the freestream, the
lift vector remains perpendicular to the local relative wind and is therefore tilted
back through the angle a;. This is shown in Figure 5.33. However, still consid
ering drag to be parallel to the freestream, we see that the tiltedlift vector
Geometric angle of attack
~
Relative wind
Local flow direction
Figure 5.33 The origin of induced drag.
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 219
contributes a certain component of drag. This drag is the induced drag D;. From
Figure 5.33,
D; = L sin a;
Values of a; are generally small, hence sina; :::::: a;. Thus
D; = La; (5.41)
Note that in Eq. (5.41) a; must be in radians. Hence, D; can be calculated from
Eq. (5.41) once a; is obtained.
The calculation of a; is beyond the scope of this book. However, it can be
shown that the value of a; for a given section of a finite wing depends on the
distribution of downwash along the span of the wing. In turn, the downwash
distribution is governed by the distribution of lift over the span of the wing. To
see this more clearly, consider Figure 5.34, which shows the front view of a finite
wing. The lift per unit span may vary as a function of distance along the wing
because:
1. The chord may vary in length along the wing.
2. The wing may be twisted such that each airfoil section of the wing is at a
different geometric angle of attack.
3. The shape of the airfoil section may change along the span.
Shown in Figure 5.34 is the case of an elliptical lift distribution (the lift per unit
span varies elliptically along the span), which in turn produces a uniform
downwash distribution. For this case, incompressible flow theory predicts that
a;=
~ ~ (5.42)
where CL is the lift coefficient of the finite wing and AR= b
2
/S is the aspect
ratio defined in Eq. (5.22). Substituting Eq. (5.42) into (5.41) yields
CL
D; =La;= L
77
AR
However, L = q
00
SCL; hence from Eq. (5.43),
CL2
D; = qooS 7TAR
or
Front view of wing
c2
L
7rAR
/Lift per unit span as a function of
/ distance along the spanthis is the
' lift distribution
' '
"'
w, the down wash distribution which
results from the given lift distribution
Figure 5.34 Lift distribution and downwash distribution.
(5.43)
(5.44)
220 INTRODUCTION TO FLIGHT
Defining the induced drag coefficient as CD,;= D/q
00
S, we can write Eq. (5.44) as
c2
L
CD,i = 7TAR
( 5 .45)
This result holds for an elliptical lift distribution, as sketched in Figure 5.34. For
a wing with the same airfoil shape across the span and with no twist, an elliptical
lift distribution is characteristic of an elliptical wing planform. (The famous
British Spitfire of World War II was one of the few aircraft in history designed
with an elliptical wing planform. Wings with straight leading and trailing edges
are more economical to manufacture.)
For all wings in general, a span efficiency factor e can be defined such that
c 2
C =L
D,i 7TeAR
( 5 .46)
For elliptical planforms, e = l; for all other planforms, e < 1. Thus, CD,i and
hence induced drag is a minimum for an elliptical planform. For typical subsonic
aircraft, e ranges from 0.85 to 0.95. Equation (5.46) is an important relation. It
demonstrates that induced drag varies as the square of the lift coefficient; at high
lift, such as near CL,mw the induced drag can be a substantial portion of the total
drag. Equation (5.46) also demonstrates that as AR is increased, induced drag is
decreased. Hence, subsonic airplanes designed to minimize induced drag have
highaspect ratio wings (such as the long, narrow wings of the Lockheed U2
highaltitude reconnaissance aircraft).
It is clear from Eq. (5.46) that induced drag is intimately related to lift. In
fact, another expression for induced drag is drag due to lift. In a fundamental
sense, that power provided by the engines of the airplane to overcome induced
drag is the power required to sustain a heavierthanair vehicle in the air, the
power necessary to produce lift equal to the weight of the airplane in flight.
In light of Eq. (5.46), we can now write the total drag coefficient for a finite
wing at subsonic speeds as
Total Profile
drag drag
Induced
drag
( 5 .47)
Keep in mind that profile drag is composed of two parts; drag due to skin
friction, cd,f' and pressure drag due to separation, cd,p; that is, cd = cd.f + cd,p·
Also keep in mind that cd can be obtained from the data in Appendix D. The
quadratic variation of CD with CL given in Eq. (5.47), when plotted on a graph,
leads to a curve as shown in Figure 5.35. Such a plot of CD versus CL is called a
drag polar. Much of the basic aerodynamics of an airplane is reflected in the drag
polar, and such curves are essential to the design of airplanes. You should become
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 221
familiar with the concept of drag polar. Note that the drag data in Appendix D
are given in terms of drag polars for infinite wings, that is, cd is plotted versus c
1
.
However, induced drag is not included in Appendix D because cD.i for an
infinite wing (infinite aspect ratio) is zero.
Example 5.10 Consider the Northrop F5 fighter airplane, which has a wing area of 170 ft
2
. The
wing is generating 18,000 lb of lift. For a flight velocity of 250 mi/h at standard sea level,
calculate the lift coefficient.
SOLUTION The velocity in consistent units is
88
V
00
= 250
60
= 366.7 ftjs
q
00
= tp
00
V
00
2
= }(0.002377)(366.7)
2
= 159.8 lb/ft
2
Hence c =  = = 06626
L 18,000 B
L q
00
S (159.8)(170) .
Example S.11 The wingspan of the Northrop F5 is 25.25 ft. Calculate the induced drag
coefficient and the induced drag itself for the conditions of Example 5.10. Assume e = 0.8.
SOLUTION The aspect ratio is AR= b
2
/S = (25.25)
2
/170 = 3.75. Since CL = 0.6626, from
Example 5.10, then from Eq. (5.46),
c/ (0.6626)
2
EJ
c    = 0.0466
D.i  '1TeAR  '11"(0.8)(3.75)
From Example 5.10, q
00
= 159.8 lb/ft
2
. Hence
D; = q
00
SCD,i = {159.8)(170)(0.0466) = 11266 lb I
Example S.12 Consider a flying wing (such as the Northrop YB49 of the early 1950s) with a
wing area of 206 m2, an aspect ratio of 10, a span effectiveness factor of 0.95, and a NACA 4412
airfoil. The weight of the airplane is 7.5 X 10
5
N. If the density altitude is 3 km and the flight
velocity is 100 m/s, calculate the total drag on the aircraft.
Cv
Figure S.35 Sketch of a drag polar, i.e., a
CL plot of drag coefficient vs. lift coefficient.
222 INTRODUCTION TO FLIGHT
SOLUTION First, obtain the lift coefficient. At a density altitude of 3 km= 3000 m, p
00
= 0.909
kg/m
3
(from Appendix A).
q
00
= !p
00
V
00
2
= !{0.909)(100/ = 4545 N/m
2
L= W= 7.5x10
5
N
L 7.5x10
5
c == =08
L q
00
S 4545(206) .
Note: This is a rather high lift coefficient, but the velocity is lownear the landing speedhence
the airplane is pitched to a rather high angle of attack to generate enough lift to keep the airplane
flying.
Next, obtain the induced drag coefficient:
CL
2
0.8
2
CD,;= weAR = w(0.95)(10) =
0
·
021
The profile drag coefficient must be estimated from the aerodynamic data in Appendix D.
Assume that cd is given by the highest Reynolds number data shown for the NACA 4412 air
foil in Appendix D, and furthermore, assume that it is in the "drag bucket." Hence, from
Appendix D,
Cd"" 0.006
Thus, from Eq. (5.47), the total drag coefficient is
= 0.006 + 0.021 = 0.027
Note that the induced drag is about 3.5 times larger than profile drag for this case, thus
underscoring the importance of induced drag.
Therefore, the total drag is
D = q
00
SCD = 4545(206)(0.027) = 12.53X10
4
NI
5.15 CHANGE IN THE LIFT SLOPE
The aerodynamic properties of a finite wing differ in two major respects from the
data of Appendix D, which apply to infinite wings. The first difference has
already been discussed, namely, the addition of induced drag for a finite wing.
The second difference is that the lift curve for a finite wing has a smaller slope
than the corresponding lift curve for an infinite wing with the same airfoil cross
section. This change in the lift slope can be examined as follows. Recall that
because of the presence of downwash, )Vhich is induced by the trailing wingtip
vortices, the flow in the local vicinity of the wing is canted downward with respect
to the freestream relative wind. As a result, the angle of attack which the airfoil
section effectively sees, called the effective angle of attack aeff• is less than the
geometric angle of attack a. This situation is sketched in Figure 5.36. The
difference between a and aeff is the induced angle of attack a;, first introduced in
Sec. 5.14, where a; = a  aeff· Moreover, for an elliptical lift distribution, Eq.
(5.42) gives values for the induced angle of attack a;= CJ?TAR. Extending Eq.
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 223
 direction

Figure 5.36 Relation between the geometric, effective, and induced angles of attack.
(5.42) to wings of any general planform, a new span effectiveness factor e
1
can be
defined such that
( 5 .48)
where e
1
and e [defined for induced drag in Eq. (5.46)] are theoretically different
but are in practice approximately the same value for a given wing. Note that Eq.
(5.48) gives a; in radians. For a; in degrees,
(5.48')
Emphasis is made that the flow over a finite wing at an angle of attack a is
essentially the same as the flow over an infinite wing at an angle of attack aerr·
Keeping this in mind, assume that we plot the lift coefficient for the finite wing
CL versus the effective angle of attack, aeff = a  <X;, as shown in Figure 5.37a.
Because we are using aeff• the lift curve should correspond to that for an infinite
wing; hence the lift curve slope in Figure 5.37a is a
0
, obtained from Appendix D
for the given airfoil. However, in real life our naked eyes cannot see aeff; instead,
what we actually observe is a finite wing at the geometric angle of attack a (the
actual angle between the freestream relative wind and the mean chord line).
Hence, for a finite wing, it makes much more sense to plot CL versus a, as shown
in Figure 5.37b, rather than versus aeff• as shown in Figure 5.37a. For example,
CL versus a would be the result most directly obtained from testing a finite wing
in a wind tunnel, because a (and not aerr) can be measured directly. Hence, the
lift curve slope for a finite wing is defined as a= dCJda, where a* a
0
. Noting
that a > aeff from Figure 5.36, the abscissa of Figure 5.37b is stretched out more
than the abscissa of Figure 5.37a; hence the lift curve of Figure 5.37b is less
inclined, that is, a < a
0
• The effect of a finite wing is to reduce the lift curve slope.
However, when the lift is zero, CL= 0, and from Eq. (5.48), a;= 0. Thus, at zero
lift, a = aerr· In terms of Figures 5.37a and 5.37b, this means that the angle of
attack for zero lift, aL=O• is the same for the finite and infinite wings. Thus, for
finite wings, aLo can be obtained directly from Appendix D.
Question: if we know a
0
(say, from Appendix D), how do we find a for a
finite wing with a given aspect ratio? The answer can be obtained by examining
224 INTRODUCTION TO FLIGHT
a
0
, lift curve slope for an infinite wing
o:  o:;, effective angle of attack
(a)
a, lift curve slope for the finite wing
°'L =O
o:, geometric angle of attack
(b)
Figure 5.37 Distinction between the lift curve slopes for infinite and finite wings.
Figure 5.37. From Figure 5.37a,
d(aa;)
=Go
Integrating, we find
(5.49)
Substituting Eq. (5.48') into Eq. (5.49), we obtain
(5.50)
Solving Eq. (5.50) for CL yields
a
0
a const
c  + 
L  1 + 57.3a
0
/?Te
1
AR 1 + 57.3a
0
/?Te
1
AR
(5.51)
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 225
Differentiating Eq. (5.51) with respect to a, we get
1 + 51.3a
0
/'7Te
1
AR
(5.52)
However, from Figure 5.37b, by definition, dCdda =a. Hence, from Eq. (5.52)
(5.53)
Equation (5.53) gives the desired lift slope for a finite wing of given aspect ratio
AR when we know the corresponding slope a
0
for an infinite wing. Remember:
a
0
is obtained from airfoil data such as in Appendix D. Also note that Eq. (5.53)
verifies our previous qualitative statement that a < a
0
.
In summary, a finite wing introduces two major changes to the airfoil data in
Appendix D:
1. Induced drag must be added to the finite wing:
c2
CD +
L
=
Cd
Total Profile
?Te AR
drag drag
Induced
drag
2. The slope of the lift curve for a finite wing is less than that for an infinite
wing; a< a
0
.
Example 5.13 Consider a wing with an aspect ratio of 10 and a NACA 23012 airfoil section.
Assume Re = 5 X 10
6
. The span efficiency factorise= e
1
= 0.95. If the wing is at a 4° angle of
attack, calculate CL and Cv.
SOLUTION Since we are dealing with a finite wing but have airfoil data (Appendix D) for infinite
wings only, the first job is to obtain the slope of this lift curve for the finite wing, modifying the
data from Appendix D.
The infinite wing lift slope can be obtained from any two points on the linear curve. For the
NACA 23012 airfoil, for example (from Appendix D),
c
1
=1.2 at aeff = 10°
c
1
= 0.14 at aeff = 0°
Hence
dc1 1.20.14_1.06_
0106
d
ao = da = 100  10  . per egree
Also from Appendix D,
and
The lift slope for the finite wing can now be obtained from Eq. (5.53).
a= ao =
0
·
106
= 0.088 er de ree
1+57.3a
0
/we
1
AR 1+57.3(0.106)/w(0.95)(10) p g
226 INTRODUCTION TO FLIGHT
Ata=4°,
CL=a(aaLo)
= 0.088[4° (1.5)]
= 0.088(5.5)
I C1, = 0 4 ~
The total drag coefficient is given by Eq. (5.47):
c = c + _!:l_ = 0.006 + 0.4
842
0
J l'TeAR 1'1'(0.95)(10)
= 0.006 + 0.0078 =I 0.01381
5.16 SWEPT WINGS
Almost all modern highspeed aircraft have sweptback wings, such as shown in
Figure 5.38b. Why? We are now in a position to answer this question.
We will first consider subsonic flight. Consider the planview of a straight
wing, as sketched in Figure 5.38a. Assume this wing has an airfoil cross section
with a critical Mach number Mer = 0.7. (Remember from Sec. 5.10 that for M
00
slightly above Mw there is a large increase in drag. Hence, it is desirable to
increase Mer as much as possible in highspeed subsonic airplane design.) Now
assume that we sweep the wing back through an angle of, say, 30°, as shown in
Figure 5.38b. The airfoil, which still has a value of Mer = 0.7, now "sees"
essentially only the component of the flow normal to the leading edge of the
wing; i.e., the aerodynamic properties of the local section of the swept wing are
governed mainly by the flow normal to the leading edge. Hence, if M
00
is the
freestream Mach number, the airfoil in Figure 5.38b is seeing effectively a smaller
Mach number, M
00
cos 30°. As a result, the actual freestream Mach number can
be increased above 0.7 before critical phenomena on the airfoil are encountered.
In fact, we could expect that the critical Mach number for the swept wing itself
would be as high as 0.7 /cos 30° = 0.808, as shown in Figure 5.38b. This means
that the large increase in drag (as sketched in Figure 5.19) would be delayed to
M
00
much larger than Mer for the airfoilin terms of Figure 5.38, something
much larger than 0.7, and maybe even as high as 0.808. Therefore, we see the
main function of a swept wing. By sweeping the wings of subsonic aircraft, drag
divergence is delayed to higher Mach numbers.
In real life, the flow over the swept wing sketched in Figure 5.38b is a fairly
complex threedimensional flow, and to say that the airfoil sees only the compo
nent normal to the leading edge is a sweeping simplification. However, it leads to
a good rule of thumb. If !J is the sweep angle, as shown in Figure 5.38b, the
actual critical Mach number for the swept wing is bracketed by:
M f
. f .
1
actual Mer
or arr 01 < . <
er for swept wmg
Mer for airfoil
cos!J
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 227
Assume Mer for
wing= 0.7
fl A;rloH "'Hoo
LJ with Mer 0.7
Now sweep the same wing by 30°
(a)
(b)
Airfoil section
with Mer= 0.7
Figure 5.38 Effect of a swept wing on critical Mach number.
For supersonic flight, swept wings are also advantageous, but not quite from
the same point of view as described above for subsonic fl.ow. Consider the two
swept wings sketched in Figure 5.39. For a given M
00
> 1, there is a Mach cone
with vertex angle µ, equal to the Mach angle [recall Eq. (5.38)]. If the leading
edge of a swept wing is outside the Mach cone, as shown in Figure 5.39a, the
component of the Mach number normal to the leading edge is supersonic. As a
I \
I \
I \
I \
(a) (b)
Figure 5.39 Swept wings for supersonic flow. (a) Wing swept outside the Mach cone. ( b) Wing swept
inside the Mach cone.
228 INTRODUCTION TO FLIGHT
result, a fairly strong oblique shock wave will be created by the wing itself, with
an attendant large wave drag. On the other hand, if the leading edge of the swept
wing is inside the Mach cone, as shown in Figure 5.39b, the component of the
Mach number normal to the leading edge is subsonic. As a result, the wave drag
produced by the wing is less. Therefore, the advantage of sweeping the wings for
supersonic flight is in general to obtain a decrease in wave drag, and if the wing is
swept inside the Mach cone, a considerable decrease can be obtained.
The quantitative effects of maximum thickness and wing sweep on wavedrag
coefficient are shown in Figure 5.40a and b, respectively. For all cases, the wing
aspect ratio is 3.5 and the taper ratio (tip to root chord) is 0.2. Clearly, thin wings
with large angles of sweepback have the smallest wave drag. The effect of wing
sweep on the lifttodrag ratio is shown in Figure 5.41. Note that for a subsonic
airplane, sweeping the wing reduces CL more than CD and hence gives a lower
L/ D ratio. In contrast, for a supersonic airplane at Mach 2.2, sweeping the wing
reduces both CL and Cv correspondingly and has virtually no effect on L/D.
As a final note, we might observe that not all supersonic aircraft have swept
wings. In supersonic flight, wave drag can also be reduced by using stubby, very
low aspect ratio, straight wings with thin, sharp airfoils (such as on the Lockheed
F104). In supersonic flight, wave drag is generally much larger, hence much more
important, than induced drag. Thus, lowaspect ratio wings for supersonic
aircraft reduce wave drag much more than they increase induced drag. This
results in a net decrease in total drag. Thus, lowaspect ratio wings are efficient
for supersonic flight, whereas highaspect ratio wings are efficient for subsonic
flight.
~ 0.03
v
~
e 0.02
8
00
!':
'O
00
0.0 I
=
'i<
E
.11=11°
Sweep angle measured at quarterchord line
n = 35°
LJ
A = 3.5: A= 0.2
t/c, %
6
r4
E o c._ _ __._ __ _,__ __ .L._ __,
n = 47°
fl
.11, deg
r11
r35
r47
·c o.6 0.8 1.0 1.2 14 0.6 0.8 1.0 1.2 1.4
:1!
Mach number M Mach number M
(a) (b)
Figure 5.40 Sketch of the variation of minimum wingdrag coefficient versus Mach number with (a)
wing thickness as a parameter (i::l = 47°) and (b) wing sweepback angle as a parameter (t/c = 4
percent). (From L. Loftin, Quest/or Performance, NASA SP 468, 1985.)
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 229
20
16
'' 12
0
.,,
::
&
'i'
.::::
;J
4
0 20 40 60 80 100 120
Wing sweepback angle, deg
Figure 5.41 Variation of lifttodrag ratio with wing sweep. Windtunnel measurements at the NASA
Langley Research Center. (From Loftin, NASA SP 468, 1985.)
5.17 FLAPSA MECHANISM FOR HIGH LIFT
An airplane normally encounters its lowest flight velocities at takeoff or landing,
two periods that are most critical for aircraft safety. The slowest speed at which
an airplane can fly in straight and level flight is defined as the stalling speed V,,
1
•
11
•
Hence, the calculation of V,,
1
a1
1
, as well as aerodynamic methods of making V,,
1
an
as small as possible, are of vital importance.
The stalling velocity is readily obtained in terms of the maximum lift
coefficient, as follows. From the definition of C v
L = q
00
SCL =
Thus (5.54)
In steady, level flight, the lift is just sufficient to support the weight W of the
aircraft, that is, L = W. Thus
(5.55)
Examining Eq. (5.55), for an airplane of given weight and size at a given altitude,
we find the only recourse to minimize V
00
is to maximize CL. Hence, stalling
230 INTRODUCTION TO FLIGHT
voo
c
Figure 5.42 Illustration of a single
flap.
speed corresponds to the angle of attack that produces CL.max:
V.tall = / p s
~
oo L,max
(5.56)
In order to decrease V.iall• CL max must be increased. However, for a wing
with a given airfoil shape, CL.max i ~ fixed by nature; i.e., the lift properties of an
With flaps
10 20
°'L=O
with flaps
°'L=O
no flaps
°'(degrees)
Figure 5.43 Illustration of the effect of flaps on the lift curve. The numbers shown are typical of a
modem mediumrange jet transport.
e
"
0
u
E
" E
·;;
"' E
"
·E
"'
4.0
3.6
3.2
2.8
2.4
2.0
l.6
1.2
0.8
0.4
0
.(i)
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 231
Figure 5.44 Typical values of
airfoil maximum lift coefficient for
various types of highlift devices:
(1) airfoil only, (2) plain flap, (3)
split flap, (4) leadingedge slat, (5)
singleslotted flap, (6) double
slotted flap, (7) doubleslotted flap
in combination with a leadingedge
slat, (8) addition of boundarylayer
suction at the top of the airfoil.
(From Loftin, NASA SP 468,
1985.)
airfoil, including maximum lift, depend on the physics of the flow over the airfoil.
In order to assist nature, the lifting properties of a given airfoil can be greatly
enhanced by the use of "artificial" highlift devices. The most common of these
devices is the simple flap at the trailing edge of the wing, as sketched in Figure
5.42. When the flap is rotated downward, the camber of the airfoil is effectively
increased, with a consequent increase in CL. max for the wing. At the same time,
the zerolift angle of attack is shifted to a more negative value. These trends are
illustrated in Figure 5.43, which compares the variation of CL with a for a wing
with and without flaps. Also, for some of the airfoils given in Appendix D, lift
curves are shown with the effect of flap deflection included.
The increase in CL max due to flaps can be dramatic. If the flap is designed
not only to rotate but also to translate rearward so as to increase the
effective wing area, CL, max can be increased by approximately a factor of 2. If
additional highlift devices are used, such as slats at the leading edge, slots in the
surface, or mechanical means of boundary layer control, then CL, max can some
times be increased by a factor of 3 or more, as shown in Figure 5.44. For an
interesting and more detailed discussion of various highlift devices, the reader is
referred to the books by McCormick and Shevell (see the Bibliography at the end
of this chapter).
5.18 THE AERODYNAMICS OF CYLINDERS AND SPHERES
Consider the lowspeed subsonic fl.ow over a sphere or an infinite cylinder with its
axis normal to the flow. If the fl.ow were inviscid (frictionless), the theoretical flow
pattern would look qualitatively as sketched in Figure 5.45a. The streamlines
232 INTRODUCTION TO FLIGHT
would form a symmetrical pattern; hence the pressure distributions over the front
and rear surfaces would also be symmetrical, as sketched in Figure 5.45b. This
symmetry creates a momentous phenomenon, namely, that there is no pressure
drag on the sphere if the flow is frictionless. This can be seen by simple inspection
of Figure 5.45b: the pressure distribution on the front face (  90° ::;; 0 ::;; 90°)
creates a force in the drag direction, but the pressure distribution on the rear face
(90° ::;; 0 ::;; 270°), which is identical to that on the front face, creates an equal
and opposite force. Thus, we obtain the curious theoretical result that there is no
drag on the body, quite contrary to everyday experience. This conflict between
theory and experiment was well known at the end of the nineteenth century and
is called d 'Alembert's paradox.
The actual flow over a sphere or cylinder is sketched in Figure 4.26; as
discussed in Sec. 4.18, the presence of friction leads to separated flows in regions
of adverse pressure gradients. Examining the theoretical inviscid pressure distri
bution shown in Figure 5.45b, we find on the rear surface (90° ::;; 0 ::;; 270°) the
pressure increases in the flow direction; i.e., an adverse pressure gradient exists.
Thus, it is entirely reasonable that the reallife flow over a sphere or cylinder
would be separated from the rear surface. This is indeed the case, as first shown
in Figure 4.26 and as sketched again in Figure 5.46a. The real pressure distribu
tion that corresponds to this separated flow is shown as the solid curve in Figure
5.46b. Note that the average pressure is much higher on the front face (  90° < 0
< 90°) than on the rear face (90° < 0 < 270°). As a result, there is a net drag
force exerted on the body. Hence, natural and experience are again reconciled,
and d'Alembert's paradox is removed by a proper account of the presence of
friction.
c
p
(+)
(a)
(b)
Figure 5.45 Ideal frictionless flow over a sphere.
(a) Flow field. ( b) Pressure coefficient distribu
tion.
c p
(+)
() 0
,_,
I
I
I
I
(a)
,\
I \
I \
I \
I \
180°
(b)
\
\
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 233
Figure 5.46 Real separated flow over a sphere;
separation is due to friction. (a) Flow field. ( b)
Pressurecoefficient distribution.
It is emphasized that the flow over a sphere or cylinder, and therefore the
drag, is dominated by flow separation on the rear face. This leads to an
interesting variation of CD with the Reynolds number. Let the Reynolds number
be defined in terms of the sphere diameter D: Re = p
00
V
00
D / µ
00
• If a sphere is
mounted in a lowspeed subsonic wind tunnel and the freestream velocity is
varied such that Re increases from 10
5
to 10
6
, a curious, almost discontinuous
drop in CD is observed at about Re= 3 X 10
5
• This behavior is sketched in
Figure 5.47. What causes this precipitous decrease in drag? The answer lies in the
different effects of laminar and turbulent boundary layers on flow separation. At
the end of Sec. 4.18 it was noted that laminar boundary layers separate much
more readily than turbulent boundary layers. In the flow over a sphere at
0.4
0.09
Re
Figure 5.47 Variation of drag coeffi
cient with Reynolds number for a
sphere in lowspeed flow.
234 INTRODUCTION TO FLIGHT
Re < 3 x 10
5
, the boundary layer is laminar. Hence, the flow is totally separated
from the rear face, and the wake behind the body is large, as sketched in Figure
5.48a. In turn, the value of CD is large, as noted at the left of Figure 5.47 for
Re < 3 < 10
5
• On the other hand, as Re is increased above 3 x 10
5
, transition
takes place on the front face, the boundary layer becomes turbulent, and the
separation point moves rearward. (Turbulent boundary layers remain attached
for longer distances in the face of adverse pressure gradients.) In this case, the
wake behind the body is much smaller, as sketched in Figure 5.48b. In turn, the
pressure drag is less and CD decreases, as noted as the right of Figure 5.47.
Therefore, to decrease the drag on a sphere or cylinder, a turbulent boundary
layer must be obtained on the front surface. This can be made to occur naturally
by increasing Re until transition occurs on the front face. It can also be forced
artificially at lower values of Re by using a rough surface to encourage early
transition or by wrapping wire or other protuberances around the surface to
create turbulence. (The use of such artificial devices is sometimes called "tripping
the boundary layer.")
It is interesting to note that the dimples on the surface of a golf ball are
designed to promote turbulence and hence reduce the drag on the ball in flight.
Indeed, some very recent research has shown that polygonal dimples result in less
Laminar boundary layer
(a)
Laminar boundary layer
(b)
Figure 5.48 Laminar and turbulent flow over a sphere.
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 235
drag than the conventional circular dimples on golf balls; but a dimple of any
shape leads to less drag than a smooth surface (table tennis balls have more drag
than golf balls).
5.19 HOW LIFT IS PRODUCEDSOME ALTERNATE
EXPLANATIONS
As an epilogue to our discussions of airfoils and wings, it is fitting to examine
once again the source of aerodynamic lift. As emphasized in the previous
chapters, the fundamental source of lift is the pressure distribution over the wing
surface. This is sketched in Figure 5.49, which illustrates .the low pressure on the
top surface, the high pressure on the bottom surface, and the resulting net lift.
An alternate explanation is sometimes given: The wing deflects the airflow
such that the mean velocity vector behind the wing is canted slightly downward,
as sketched in Figure 5.50. Hence, the wing imparts a downward component of
momentum to the air; i.e., the wing exerts a force on the air, pushing the flow
downward. From Newton's third law, the equal and opposite reaction produces a
lift. However, this explanation really involves the effect of lift, and not the cause.
In reality, the air pressure on the surface (see Figure 5.49) is pushing on the
surface, hence creating lift. As a result of the equalandopposite principle, the
airfoil surface pushes on the air, imparting a force on the airflow which deflects
the velocity downward. Hence, the net rate of change of downward momentum
created in the airflow because of the presence of the wing can be thought of as an
~
L




Note: The length of the arrows denoting pressure is
proportional to p  pref. where pref is an arbitrary
reference pressure slightly less than the minimum
pressure on the airfoil.
Figure 5.49 Pressure distribution is the source of lift.


236 INTRODUCTION TO FLIGHT
Voo
L = time rate of change of
momentum of airflow in
the downward direction
Figure 5.50 Relationship of lift to the time rate of change of momentum of the airflow.
effect due to the surface pressure distribution; the pressure distribution by itself is
the fund&mental cause of lift.
A third argument, called the circulation theory of lift, is sometimes given for
the source of lift. However, this turns out to be not so much an explanation of lift
per se, but rather more of a mathematical formulation for the calculation of lift
for an airfoil of given shape. Moreover, it is mainly applicable to incompressible
flow. The circulation theory of lift is elegant and welldeveloped; it is also beyond
the scope of this book. However, some of its flavor is given as follows.
Consider the flow over a given airfoil, as shown in Figure 5.51. Imagine a
closed curve C drawn around the airfoil. At a point on this curve, the flow
velocity is V and the angle between V and a tangent to the curve is 8. Let ds be
an incremental distance along C. A quantity called the circulation T is defined as
I
I
\
/
I
\
' ......
..........


f = Vcos8ds
c
Figure 5.51 Diagram for the circulation theory of lift.
(5.57)
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 237
that is, r is the line integral of the component of flow velocity along the closed
curve C. After a value of f is obtained, the lift per unit span can be calculated
from
(5.58)
Equation (5.57) is the KuttaJoukowsky theorem; it is a pivotal relation in the
circulation theory of lift. The object of the theory is to (somehow) calculate r for
a given V
00
and airfoil shape. Then Eq. (5.58) yields the lift. A major thrust of
ideal incompressible flow theory, many times called potential flow theory, is to
calculate r. Such matters are discussed in moreadvanced aerodynamic texts (see,
for example, Anderson, Fundamentals of Aerodynamics, McGrawHill, 1984).
The circulation theory of lift is compatible with the true physical nature of
the flow over an airfoil, as any successful mathematical theory must be. In the
simplest sense, we can visualize the true flow over an airfoil, shown at the right of
Figure 5.52, as the superposition of a uniform flow and a circulatory flow, shown
at the left of Figure 5.52. The circulatory flow is clockwise, which when added to
the uniform flow, yields a higher velocity above the airfoil and a lower velocity
below the airfoil. From Bernoulli's equation, this implies a lower pressure on the
top surface of the airfoil and a higher pressure on the bottom surface, hence
generating lift in the upward direction. The strength of the circulatory contribu
tion, defined by Eq. (5.57), is just the precise value such that when it is added to
the uniform flow contribution, the actual flow over the airfoil leaves the trailing
edge smoothly, as sketched at the right of Figure 5.52. This is called the Kutta
condition and is one of the major facets of the circulation theory of lift.
Again, keep in mind that the actual mechanism that nature has of communi
cating a lift to the airfoil is the pressure distribution over the surface of the
airfoil, as sketched 1n Figure 5.49. In turn, this pressure distribution ultimately
causes a time rate of change of momentum of the airflow, as shown in Figure
L
~
Unifonn flow Pure circulation Incompressible flow over an airfoil
Figure 5.52 Addition of two elementary flows to synthesize a more complex flow. If one or more of
the elementary flows has circulation, then the synthesized flow also has the same circulation. The lift
is directly proportional to the circulation.
238 INTRODUCTION TO FLIGHT
5.50, a principle which can be used as an alternate way of visualizing the
generation of lift. Finally, even the circulation theory of lift stems from the
pressure distribution over the surface of the airfoil, because the derivation of the
KuttaJoukowsky theorem, Eq. (5.58), involves the surface pressure distribution.
Again, for more details, consult Anderson, Fundamentals of Aerodynamics
(McGrawHill, 1984).
5.20 HISTORICAL NOTE: AIRFOILS AND WINGS
We know that George Cayley introduced the concept of a fixedwing aircraft in
1799; this has been discussed at length in Secs. 1.3 and 5.1. Moreover, Cayley
appreciated the fact that lift is produced by a region of low pressure on the top
surface of the wing and high pressure on the bottom surface and that a cambered
shape produces more lift than a flat surface. Indeed, Figure 1.5 shows that Cayley
was thinking of a curved surface for a wing, although the curvature was due to the
wind billowing against a loosely fitting fabric surface. However, neither Cayley
nor any of his immediate followers performed work even closely resembling
airfoil research or development.
It was not until 1884 that the first serious airfoil developments were made. In
this year Horatio F. Phillips, an Englishman, was granted a patent for a series of
doublesurface, cambered airfoils. Figure 5.53 shows Phillips's patent drawings
for his airfoil section. Phillips was an important figure in late nineteenth century
aeronautical engineering; we have met him before, in Sec. 4.22, in conjunction
with his ejectordriven wind tunnel. In fact, the airfoil shapes in Figure 5.53 were

,
_,,,

Figure 5.53 Doublesurface airfoil sections by Phillips. The six upper shapes were patented by
Phillips in 1884; the lower airfoil was patented in 1891.
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 239
the result of numerous windtunnel experiments in which Phillips examined
curved wings of "every conceivable form and combination of forms." Phillips
widely published his results, which had a major impact on the aeronautics
community. Continuing with his work, Phillips patented more airfoil shapes in
1891. Then, moving into airplane design in 1893, he built and tested a large
multiplane model, consisting of a large number of wings, each with a 19ft span
and a chord of only 1 in, which looked like a venetian blind! The airplane was
powered by a steam engine with a 6.5ft propeller. The vehicle ran on a circular
track and actually lifted a few feet off the ground, momentarily. After this
demonstration, Phillips gave up until 1907, when he made the first tentative hop
flight in England in a similar, but gasolinepowered, machine, staying airborne for
about 500 ft. This was his last contribution to aeronautics. However, his pioneer
ing work during the 1880s and 1890s clearly designates Phillips as the grandparent
of the modern airfoil.
After Phillips, the work on airfoils shifted to a search for the most efficient
shapes. Indeed, work is still being done today on this very problem, although
much progress has been made. This progress covers several historical periods, as
described below.
A The Wright Brothers
As noted in Secs. 1.8 and 4.22, Wilbur and Orville Wright, after their early
experience with gliders, concluded in 1901 that many of the existing "air
pressure" data on airfoil sections were inadequate and frequently incorrect. In
order to rectify these deficiencies, they constructed their own wind tunnel (see
Figure 4.45), in which they tested several hundred different airfoil shapes between
September 1901 and August 1902. From their experimental results, the Wright
brothers chose an airfoil with a 1/20 maximum cambertochord ratio for their
successful Wright Flyer I in 1903. These windtunnel tests by the Wright brothers
constituted a major advance in airfoil technology at the turn of the century.
B British and United States Airfoils (1910 to 1920)
In the early days of powered flight, airfoil design was basically customized and
personalized; very little concerted effort was made to find a standardized, efficient
section. However, some early work was performed by the British Government at
the National Physical Laboratory (NPL), leading to a series of Royal Aircraft
Factory (RAF) airfoils used on World War I airplanes. Figure 5.54 illustrates the
shape of the RAF 6 airfoil. In the United States, most aircraft until 1915 used
either an RAF section or a shape designed by the Frenchman Alexandre Gustave
Eiffel. This tenuous status of airfoils led the NACA, in its First Annual Report in
1915, to emphasize the need for "the evolution of more efficient wing sections of
practical form, embodying suitable dimensions for an economical structure, with
moderate travel of the center of pressure and still affording a large angle of attack
240 INTRODUCTION TO FLIGHT
RAF6
USA 6 Figure 5.54 Typical airfoils in 1917.
combined with efficient action." To this day, more than 70 years later, NASA is
still pursuing such work.
The first NACA work on airfoils was reported in NACA report no. 18,
"Aerofoils and Aerofoil Structural Combinations," by Lt. Col. Edgar S. Gorrell
and Major H. S. Martin, prepared at the Massachusetts Institute of Technology
(M.l.T.) in 1917. Gorrell and Martin summarized the contemporary airfoil status
as follows:
Mathematical theory has not, as yet, been applied to the discontinuous motion past a cambered
surface. For this reason, we are able to design aerofoils only by consideration of those forms
which have been successful, by applying general rules learned by experience, and by then testing
the aerofoils in a reliable wind tunnel.
In NACA report No. 18, Gorrell and Martin disclosed a series of tests on the
largest single group of airfoils to that date, except for the work done at NPL and
by Eiffel. They introduced the U.S.A. airfoil series and reported windtunnel data
for the U.S.A. 1 through U.S.A. 6 sections. Figure 5.54 illustrates the shape of the
U.S.A. 6 airfoil. The airfoil models were made of brass and were finite wings with
a span of 18 in and a chord of 3 in, that is, AR = 6. Lift and drag coefficients
were measured at a velocity of 30 mi/h in the M.l.T. wind tunnel. These airfoils
represented the first systematic series originated and studied by the NACA.
C 1920 to 1930
Based on their windtunnel observations in 1917, Gorrell and Martin stated that
slight variations in airfoil design make large differences in the aerodynamic
performance. This is the underlying problem of airfoil research, and it led in the
1920s to a proliferation of airfoil shapes. In fact, as late as 1929, F. A. Louden, in
his NACA report no. 331, entitled "Collection of Wind Tunnel Data on Com
monly Used Wing Sections," stated that "the wing sections most commonly used
in this country are the Clark Y, Clark Y15, Gottingen G387, G398, G436,
N.A.C.A. M12, Navy N9, N10, N22, R.A.F.15, Sloane, U.S.A.27, U.S.A.35A,
U.S.A.35B." However, help was on its way. As noted in Sec. 4.22, the NACA
built a variabledensity wind tunnel at Langley Aeronautical Laboratory in 1923,
a wind tunnel that was to become a workhorse in future airfoil research, as
emphasized below.
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 241
D The Early NACA FourDigit Airfoils
In a classic work in 1933, order and logic were finally brought to airfoil design in
the United States. This was reported in NACA report no. 460, "The Characteris
tics of 78 Related Airfoil Sections from Tests in the VariableDensity Wind
Tunnel," by Eastman N. Jacobs, Kenneth E. Ward, and Robert M. Pinkerton.
Their philosophy on airfoil design was as follows:
Airfoil profiles may be considered as made up of certain profilethickness forms disposed about
certain mean lines. The major shape variables then become two, the thickness form and the
meanline form. The thickness form is of particular importance from a structural standpoint. On
the other hand, the form of the mean line determines almost independently some of the most
important aerodynamic properties of the airfoil section, e.g., the angle of zero lift and the
pitchingmoment characteristics. The related airfoil profiles for this investigation were derived by
changing systematically these shape variables.
They then proceeded to define and study for the first time in history the
famous NACA fourdigit airfoil series, some of which are given in Appendix D of
this book. For example, the NACA 2412 is defined as a shape that has a
maximum camber of 2 percent of the chord (the first digit); the maximum camber
occurs at a position of 0.4 chord from the leading edge (the second digit), and the
maximum thickness is 12 percent (the last two digits). Jacobs and his colleagues
tested these airfoils in the NACA variabledensity tunnel using a 5 in x 30 in
finite wing (again, an aspect ratio of 6). In NACA report no. 460, they gave curves
of Cv Cv, and L/ D for the finite wing. Moreover, using the same formulas
developed in Sec. 5.15, they corrected their data to give results for the infinite
wing case, also. After this work was published, the standard NACA fourdigit
airfoils were widely used. Indeed, even today the NACA 2412 is used on several
light aircraft.
E Later NACA Airfoils
In the late 1930s, the NACA developed a new camber line family to increase
maximum lift, with the 230 camber line being the most popular. Combining with
the standard NACA thickness distribution, this gave rise to the NACA fivedigit
airfoil series, such as the 23012, some of which are still flying today, for example,
on the Cessna Citation and the Beech King Air. This work was followed by
families of highspeed airfoils and laminar flow airfoils in the 1940s.
To reinforce its airfoil development, in 1939 the NACA constructed a new
lowturbulence twodimensional wind tunnel at Langley exclusively for airfoil
testing. This tunnel has a rectangular test section 3 ft wide and 71' ft high and can
be pressurized up to 10 atm for highReynolds number testing. Most im
portantly, this tunnel allows airfoil models to span the test section completely,
thus directly providing infinite wing data. This is in contrast to the earlier tests
described above, which used a finite wing of AR = 6 and then corrected the data
to correspond to infinite wing conditions. Such corrections are always compro
242 INTRODUCTION TO FLIGHT
mised by tip effects. (For example, what is the precise span efficiency factor for a
given wing?) With the new twodimensional tunnel, vast numbers of tests were
performed in the early 1940s on both old and new airfoil shapes over a Reynolds
number range from 3 to 9 million and at Mach numbers less than 0.17 (incom
pressible flow). The airfoil models generally had a 2ft chord and completely
spanned the 3ft width of the test section. It is interesting to note that the lift and
drag are not obtained on a force balance. Rather, the lift is calculated by
integrating the measured pressure distribution, and the drag is calculated from
Pitot pressure measurements made in the wake downstream of the trailing edge.
However, the pitching moments are measured directly on a balance. A vast
amount of airfoil data obtained in this fashion from the twodimensional tunnel
at Langley was compiled and published in a book entitled Theory of Wing
Sections Including a Summary of Airfoil Data, by Abbott and von Doenhoff, in
1949 (see Bibliography at end of this chapter). It is important to note that all the
airfoil data in Appendix D are obtained from this reference, i.e., all the data in
Appendix D are direct measurements for an infinite wing at essentially incom
pressible flow conditions.
F Modern Airfoil Work
Priorities for supersonic and hypersonic aerodynamics put a stop to the NACA
airfoil development in 1950. Over the next 15 years specialized equipment for
airfoil testing was dismantled. Virtually no systematic airfoil research was done in
the United States during this period.
However, in 1965 Richard T. Whitcomb made a breakthrough with the
NASA supercritical airfoil. This revolutionary development, which allowed the
design of wings with high critical Mach numbers (see Sec. 5.10), served to
reactivate the interest in airfoils within NASA. Since that time, a healthy program
in modern airfoil development has been reestablished. The lowturbulence, pres
surized, twodimensional wind tunnel at Langley is back in operation. Moreover,
a new dimension has been added to airfoil research: the highspeed digital
computer. In fact, computer programs for calculating the flow field around
airfoils at subsonic speeds are so reliable that they shoulder some of the routine
testing duties heretofore exclusively carried by wind tunnels. The same cannot yet
be said about transonic cases, but current research is focusing on this problem. A
recent and interesting survey of modern airfoil activity within NASA is given by
Pierpont in Astronautics and Aeronautics (see Bibliography).
Of special note is the modern lowspeed airfoil series, designated LS(l),
developed by NASA for use by general aviation on light airplanes. The shape of a
typical LS(l) airfoil is contrasted with a "conventional" airfoil in Figure 5.55. Its
lifting characteristics, illustrated in Figure 5.56, are clearly superior and should
allow smaller wing areas, hence less drag, for airplanes of the type shown in
Figure 5.55.
In summary, airfoil development over the past 100 years has moved from an
ad hoc individual process to a very systematic and logical engineering process. It
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 243
.......... Advanced technology airfoil [LS( l )041 7]
··Conventional airfoil (65 series)
Figure 5.55 Shape comparison between the modem LS(l)0417 and a conventional airfoil. The higher
lift obtained with the LS(l)0417 allows a smaller wing area and hence lower drag. (NASA.)
is alive and well today, with the promise of major advancements in the future
using both wind tunnels and computers.
G Finite Wings
Some historical comments on the finite wing are in order. Francis Wenham (see
Chap. 1), in his classic paper entitled Aerial Locomotion given to the Aeronautical
2.0
g
1.8
(f
"
"'
·v
8
1.6
"' 0
u
.;:
·E 1.4
u
2
" 2
·;;;
1.2
" ::E
LS(l)0417
10 15
Reynolds number, R X 10
6
Low speed, M "" 0.2
20 Figure 5.56 Comparison of maximum lift
coefficients between the LS(l)0417 and con
ventional airfoils. (NASA.)
244 INTRODUCTION TO FLIGHT
Society of Great Britain on June 27, 1866, theorized (correctly) that most of the
lift of a wing occurs from the portion near the leading edge and hence a long,
narrow wing would be most efficient. In this fashion, he was the first person in
history to appreciate the value of highaspect ratio wings for subsonic flight.
Moreover, he suggested stacking a number of long, thin wings above each other to
generate the required lift, and hence he became an advocate of the multiplane
concept. In turn, he built two fullsize gliders in 1858, both with five wings each,
and successfully demonstrated the validity of his ideas.
However, the true aerodynamic theory and understanding of finite wings did
not come until 1907, when the Englishman Frederick W. Lanchester published his
book Aerodynamics. In it, he outlined the circulation theory of lift, developed
independently about the same time by Kutta in Germany and Joukowsky in
Russia. More importantly, Lanchester discussed for the first time the effect of
wingtip vortices on finite wing aerodynamics. Unfortunately, Lanchester was not
a clear writer; his ideas were extremely difficult to understand, and hence they did
not find application in the aeronautical community.
In 1908, Lanchester visited Gottingen, Germany, and fully discussed his wing
theory with Ludwig Prandtl and his student, Theodore von Karman. Prandtl
spoke no English, Lanchester spoke no German, and in light of Lanchester's
unclear ways of explaining his ideas, there appeared to be little chance of
understanding between the two parties. However, in 1914, Prandtl set forth a
simple, clear, and correct theory for calculating the effect of tip vortices on the
aerodynamic characteristics of finite wings. It is virtually impossible to assess how
much Prandtl was influenced by Lanchester, but to Prandtl must go the credit of
first establishing a practical finite wing theory, a theory which is fundamental to
our discussion of finite wings in Secs. 5.13 to 5.15. Indeed, Prandtl's first
published words on the subject were:
The lift generated by the airplane is, on account of the principle of action and reaction,
necessarily connected with a descending current in all its details. It appears that the descending
current is formed by a pair of vortices, the vortex filaments of which start from the airplane
wingtips. The distance of the two vortices is equal to the span of the airplane, their strength is
equal to the circulation of the current around the airplane, and the current in the vicinity of the
airplane is fully given by the superposition of the uniform current with that of a vortex consisting
of three rectilinear sections.
Prandtl's pioneering work on finite wing theory, along with his ingenious
concept of the boundary layer, has earned him the title "parent of aerodynamics."
In the four years following 1914, he went on to show that an elliptical lift
distribution results in the minimum induced drag. Indeed, the first coining of the
terms "induced drag" and "profile drag" was made in 1918 by Max Munk in a
note entitled "Contribution to the Aerodynamics of the Lifting Organs of the
Airplane." Munk was a colleague of Prandtl's, and the note was one of several
classified German wartime reports on airplane aerodynamics.
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 245
5.21 HISTORICAL NOTE: ERNST MACH AND HIS NUMBER
Airplanes that fly at Mach 2 are commonplace today. Indeed, highperformance
military aircraft such as the Lockheed SR71 Blackbird can exceed Mach 3. As a
result, the term "Mach number" has become part of our general language the
average person in the street understands that Mach 2 means twice the speed of
sound. On a more technical basis, the dimensional analysis described in Sec. 5.3
demonstrated that aerodynamic lift, drag, and moments depend on two important
dimensionless productsthe Reynolds number and the Mach number. Indeed, in
a more general treatment of fluid dynamics, the Reynolds number and Mach
number can be shown as the major governing parameters for any realistic flow
field; they are among a series of governing dimensionless parameters called
similarity parameters. We have already examined the historical source of the
Reynolds number in Sec. 4.23; let us do the same for the Mach number in the
present section.
The Mach number is named after Ernst Mach, a famous nineteenth century
physicist and philosopher. Mach was an illustrious figure with widely varying
interests. He was the first person in history to observe supersonic flow and to
understand its basic nature. Let us take a quick look at this man and his
contributions to supersonic aerodynamics.
Ernst Mach was born at Turas, Moravia, in Austria, on February 18, 1838.
Mach's father and mother were both extremely private and introspective intellec
tuals. His father was a student of philosophy and classical literature; his mother
was a poet and musician. The family was voluntarily isolated on a farm, where
Mach's father pursued an interest of raising silkwormsindeed pioneering the
beginning of silkworm culture in Europe. At an early age, Mach was not a
particularly successful student. Later, Mach described himself as a "weak pitiful
child who developed very slowly." Through extensive tutoring by his father at
home, Mach learned Latin, Greek, history, algebra, and geometry. After marginal
performances in grade school and high school (due not to any lack of intellectual
ability, but rather to a lack of interest in the material usually taught by rote),
Mach entered the University of Vienna. There, he blossomed, spurred by interest
in mathematics, physics, philosophy, and history. In 1860, he received a Ph.D. in
physics, writing a thesis entitled "On Electrical Discharge and Induction." By
1864, he was a professor of physics at the University of Graz. (The variety and
depth of his intellectual interests at this time are attested by the fact that he
turned down the position of a chair in surgery at the University of Salzburg to go
to Graz.) In 1867, Mach became a professor of experimental physics at the
University of Praguea position he would occupy for the next 28 years.
In today's modern technological world, where engineers and scientists are
virtually forced, out of necessity, to peak their knowledge in very narrow areas of
extreme specialization, it is interesting to reflect on the personality of Mach, who
was the supreme generalist. Here is only a partial list of Mach's contributions, as
demonstrated in his writings: physical optics, history of science, mechanics,
246 INTRODUCTION TO FLIGHT
philosophy, origins of relativity theory, supersonic flow, physiology, thermody
namics, sugar cycle in grapes, physics of music, and classical literature. He even
wrote on world affairs. (One of Mach's papers commented on the "absurdity
committed by the statesman who regards the individual as existing solely for the
sake of the state"; for this, Mach was severely criticized at that time by Lenin.)
We can only sit back with awe and envy for Mach, whoin the words of the
American philosopher William Jamesknew "everything about everything."
Mach's contributions to supersonic aerodynamics were highlighted in a paper
entitled "Photographische Fixierung der <lurch Projektile in der Luft eingeleiten
Vorgange," given to the Academy of Sciences in Vienna in 1887. Here, for the
first time in history, Mach showed a photograph of a shock wave in front of a
bullet moving at supersonic speeds. This historic photograph, taken from Mach's
original paper, is shown in Figure 5.57. Also visible are weaker waves at the rear
of the projectile and the structure of the turbulent wake downstream of the base
region. The two vertical lines are trip wires designed to time the photographic
light source (or spark) with the passing of the projectile. Mach was a precise and
careful experimentalist; the quality of the picture shown in Figure 5.57, along
with the very fact that he was able to make the shock waves visible in the first
place (he used an innovative technique called the shadowgram), attest to his
exceptional experimental abilities. Note that Mach was able to carry out such
experiments involving splitsecond timing without the benefit of electronics
indeed, the vacuum tube had not yet been invented.
Figure 5.57 Photograph of a bullet in supersonic flight, published by Ernst Mach in 1887.
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 247
Figure 5.58 Ernst Mach (18381916).
Mach was the first to understand the basic characteristics of supersonic flow.
He was the first to point out the importance of the flow velocity V relative to the
speed of sound a and to note the discontinuous and marked changes in a flow
field as the ratio V /a changes from below 1 to above 1. He did not, however, call
this ratio the Mach number. The term "Mach number" was not coined until 1929,
when the wellknown Swiss engineer Jacob Ackeret introduced this terminology,
in honor of Mach, at a lecture at the Eidgenossiche Technische Hochschule in
Zurich. Hence, the term "Mach number" is of fairly recent usage, not introduced
into the English literature until 1932.
Mach was an active thinker, lecturer, and writer up to the time of his death
on February 19, 1916, near Munich, one day after his seventyeighth birthday.
His contributions to human thought were many, and his general philosophy on
epistemologya study of knowledge itselfis still discussed in college classes in
philosophy today. Aeronautical engineers know him as the originator of super
sonic aerodynamics; the rest of the world knows him as a man who originated the
following philosophy, as paraphrased by Richard von Mises, himself a wellknown
mathematician and aerodynamicist of the early twentieth century:
Mach does not start out to analyze statements, systems of sentences, or theories, but rather the
world of phenomena itself. His elements are not the simplest sentences, and hence the building
stones of theories, but ratherat least according to his way of speakingthe simplest facts,
248 INTRODUCTION TO FLIGHT
phenomena, and events of which the world in which we live and which we know is composed.
The world open to our observation and experience consists of "colors, sounds, warmths,
pressures, spaces, times, etc." and their components in greater and smaller complexes. All we
make statements or assertions about, or formulate questions and answers to, are the relations in
which these elements stand to each other. That is Mach's point of view.*
We end this section with a photograph of Mach, taken about 1910, shown in
Figure 5.58. It is a picture of a thoughtful, sensitive man; no wonder that his
philosophy of life emphasized observation through the senses, as discussed by von
Mises above. In honor of his memory, an entire research institute, the Ernst Mach
Institute in West Germany, is named. This institute deals with research in
experimental gas dynamics, ballistics, highspeed photography, and cinematogra
phy. Indeed, for a much more extensive review of the technical accomplishments
of Mach, see the recent paper authored by a member of the Ernst Mach Institute,
H. Reichenbach, entitled "Contributions of Ernst Mach to Fluid Mechanics," in
Annual Reviews of Fluid Mechanics, vol. 15, 1983, pp. 128 (published by Annual
Reviews, Inc., Palo Alto, California).
5.22 HISTORICAL NOTE: THE FIRST MANNED
SUPERSONIC FLIGHT
On October 14, 1947, a human being flew faster than the speed of sound for the
first time in history. Imagine the magnitude of this accomplishmentjust 60
years after Ernst Mach observed shock waves on supersonic projectiles (see Sec.
5.21) and a scant 44 years after the Wright brothers achieved their first successful
powered flight (see Secs. 1.1 and 1.8). It is almost certain that Mach was not
thinking at all about heavierthanair manned flight of any kind, which in his day
was still considered to be virtually impossible and the essence of foolish dreams.
It is also almost certain that the Wright brothers had not the remotest idea that
their fledgling 30mi/h flight on December 17, 1903, would ultimately lead to a
manned supersonic flight in Orville's lifetime (although Wilbur died in 1912,
Orville lived an active life until his death in 1948). Compared to the total
spectrum of manned flight reaching all the way back to the ideas of Leonardo da
Vinci in the fifteenth century (see Sec. 1.2), this rapid advancement into the realm
of supersonic flight is truly phenomenal. How did this advancement occur?. What
are the circumstances surrounding the first supersonic flight? Why was it so
important? The purpose of this section is to address these questions.
Supersonic flight did not happen by chance; rather, it was an inevitable result
of the progressive advancement of aeronautical technology over the years. On one
hand we have the evolution of highspeed aerodynamic theory, starting with the
pioneering work of Mach, as described in Sec. 5.21. This was followed by the
*From Richard von Mises, Positivism, A Study in Human Understanding, Braziller, New York,
1956.
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 249
development of supersonic nozzles by two European engineers, Carl G. P. de
Laval in Sweden and A. B. Stodola in Switzerland. In 1887 de Laval used a
convergentdivergent supersonic nozzle to produce a highvelocity flow of steam
to drive a turbine. In 1903 Stodola was the first person in history to definitely
prove (by means of a series of laboratory experiments) that such convergent
divergent nozzles did indeed produce supersonic flow. From 1905 to 1908, Prandtl
in Germany took pictures of Mach waves inside supersonic nozzles and developed
the first rational theory for oblique shock waves and expansion waves. After
World War I, Prandtl studied compressibility effects in highspeed subsonic flow.
This work, in conjunction with independent studies by the English aerodynami
cist Herman Glauert, led to the publishing of the PrandtlGlauert rule in the late
1920s (see Sec. 5.6 for a discussion of the PrandtlGlauert rule and its use as a
compressibility correction). These milestones, among others, established a core of
aerodynamic theory for highspeed flowa core that was wellestablished at least
20 years before the first supersonic flight. (For more historical details concerning
the evolution of aerodynamic theory pertaining to supersonic flight, see Anderson,
Modern Compressible Flow: With Historical Perspective, McGrawHill, 1982.)
On the other hand, we also have the evolution of hardware necessary for
supersonic flight. The development of highspeed wind tunnels, starting with the
small 12inchdiameter highspeed subsonic tunnel at the NACA Langley Memo
rial Aeronautical Laboratory in 1927 and continuing with the first practical
supersonic wind tunnels developed by Adolf Busemann in Germany in the early
1930s, is described in Sec. 4.22. The exciting developments leading to the first
successful rocket engines in the late 1930s are discussed in Sec. 9.12. The
concurrent invention and development of the jet engine, which would ultimately
provide the thrust necessary for everyday supersonic flight, are related in Sec.
9.11. Hence, on the basis of the theory and hardware which existed at that time,
the advent of manned supersonic flight in 1947 was a natural progression in the
advancement of aeronautics.
However, in 1947 there was one missing link in both the theory and the
hardware the transonic regime, near Mach 1. The governing equations for
transonic flow are highly nonlinear and hence are difficult to solve. No practical
solution of these equations existed in 1947. This theoretical gap was compounded
by a similar gap in wind tunnels. The sensitivity of a flow near Mach 1 makes the
design of a proper transonic tunnel difficult. In 1947, no reliable transonic wind
tunnel data were available. This gap of knowledge was of great concern to the
aeronautical engineers who designed the first supersonic airplane; and it was the
single most important reason for the excitement, apprehension, uncertainty, and
outright bravery that surrounded the first supersonic flight.
The unanswered questions about transonic flow did nothing to dispel the
myth of the "sound barrier" that arose in the 1930s and 1940s. As discussed in
Sec. 5.12, the very rapid increase in drag coefficient beyond the dragdivergence
Mach number led some people to believe that humans would never fly faster than
sound. Grist was lent to their arguments when, on September 27, 1946, Geoffrey
deHavilland, son of the famous British airplane designer, took the D.H. 108
250 INTRODUCTION TO FLIGHT
Swallow up for an attack on the world's speed record. The Swallow was an
experimental jetpropelled aircraft, with swept wings and no horizontal tail.
Attempting to exceed 615 mi/h on its first highspeed, lowaltitude run, the
Swallow encountered major compressibility problems and broke up in the air.
DeHavilland was killed instantly. The sound barrier had taken its toll. It was
against this background that the first supersonic flight was attempted in 1947.
During the late 1930s, and all through World War II, some visionaries clearly
saw the need for an experimental airplane designed to probe the mysteries of
supersonic flight. Finally, in 1944, their efforts prevailed; the Army Air Force, in
conjunction with the NACA, awarded a contract to the Bell Aircraft Corporation
for the design, construction, and preliminary testing of a manned supersonic
airplane. Designated the XS1 (Experimental SonicI), this design had a fuselage
shaped like a 50caliber bullet, mated to a pair of very thin (thicknesstochord
ratio of 0.06), lowaspect ratio, straight wings, as shown in Figure 5.59. The
aircraft was powered by a fourchamber liquidpropellant rocket engine mounted
in the tail. This engine, made by Reaction Motors and designated the XLRll,
produced 6000 lb of thrust by burning a mixture of liquid oxygen and diluted
alcohol.
The Bell XS1 was designed to be carried aloft by a parent airplane, such as
the giant Boeing B29, and then launched at altitude; in this fashion, the extra
weight of fuel that would have been necessary for takeoff and climb to altitude
was saved, allowing the designers to concentrate on one performance
aspectspeed. Three XSls were ultimately built, the first one being completed
just after Christmas 1945. There followed a year and a half of gliding and then
powered tests, wherein the XS1 was cautiously nudged toward the speed of
sound.
Figure 5.59 The Bell XS1, the first supersonic airplane, 1947. (National Air and Space Museum.)
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 251
Muroc Dry Lake is a large expanse of fiat, hard lake bed in the Mojave
Desert in California. Site of a U.S. Army highspeed flight test center during
World War II, Muroc was later to become known as Edwards Air Force Base,
now the site of the Air Force Test Pilots School and the home of all experimental
highspeed flight testing for both the Air Force and NASA. On Tuesday, October
14, 1947, the Bell XS1, nestled under the fuselage of a B29, was waiting on the
flight line at Muroc. After intensive preparations by a swarm of technicians, the
B29, with its cargo took off at 10 A.M. On board the XS1 was Captain Charles E.
(Chuck) Yeager. That morning, Yeager was in excruciating pain from two broken
ribs fractured during a horsebackriding accident earlier that weekend; however,
he told virtually no one. At 10:26 A.M. at an altitude of 20,000 ft, the Bell XS1,
with Yeager as its pilot, was dropped from the B29. What happened next is one
of the major milestones in aviation history. Let us see how Yeager himself recalled
events, as stated in his written flight report:
Date:
Pilot:
Time:
14 October 1947
Capt. Charles E. Yeager
14 Minutes
9th Powered Flight
1. After normal pilot entry and the subsequent climb. the XS1 was dropped from the B29 at
20,000' and at 250 MPH IAS. This was slower than desired.
2. Immediately after drop, all four cylinders were turned on in rapid sequence, their operation
stabilizing at the chamber and line pressures reported in the last flight. The ensuing climb was
made at .85.88 Mach, and, as usual it was necessary to change the stabilizer setting to 2 degrees
nose down from its predrop setting of 1 degree nose down. Two cylinders were turned off
between 35,000' and 40,000', but speed had increased to .92 Mach as the airplane was leveled off
at 42,000'. Incidentally, during the slight pushover at this altitude, the lox line pressure dropped
perhaps 40 psi and the resultant rich mixture caused the chamber pressures to decrease slightly.
The effect was only momentary, occurring at .6 G's, and all pressures returned to normal at 1 G.
3. In anticipation of the decrease in elevator effectiveness at all speeds above .93 Mach,
longitudinal control by means of the stabilizer was tried during the climb at .83, .88, and .92
Mach. The stabilizer was moved in increments of !t degree and proved to be very effective;
also, no change in effectiveness was noticed at the different speeds.
4. At 42,000' in approximately level flight, a third cylinder was turned on. Acceleration was
rapid and speed increased to .98 Mach. The needle of the machmeter fluctuated at this reading
momentarily, then passed off the scale. Assuming that the offscale reading remained linear, it is
estimated that 1.05 Mach was attained at this time. Approximately 30% of fuel and lox remained
when this speed was reached and the motor was turned off.
5. While the usual lift buffet and instability characteristics were encountered in the .88.90 Mach
range and elevator effectiveness was very greatly decreased at .94 Mach, stability about all three
axes was good as speed increased and elevator effectiveness was regained above .97 Mach. As
speed decreased after turning off the motor, the various phenomena occurred in reverse sequence
at the usual speeds, and in addition, a slight longitudinal porpoising was noticed from .98.96
Mach which was controllable by elevators alone. Incidentally, the stabilizer setting was not
changed from its 2 degrees nose down position after trial at .92 Mach.
6. After jettisoning the remaining fuel and lox at 1 G stall was performed at 45,000'. The flight
was concluded by the subsequent glide and a normal landing on the lakebed.
CHARLES E. YEAGER
Capt. Air Corps
252 INTRODUCTION TO FLIGHT
In reality, the Bell SX1 had reached M
00
= 1.06, as determined from official
NACA tracking data. The duration of its supersonic flight was 20.5 s, almost
twice as long as the Wright brothers' entire first flight just 44 years earlier. On
that day, Chuck Yeager became the first person to fly faster than the speed of
sound. It is a fitting testimonial to the aeronautical engineers at that time that the
flight was smooth and without unexpected consequences. An aircraft had finally
been properly designed to probe the "sound barrier," which it penetrated with
relative ease. Less than a month later, Yeager reached Mach 1.35 in the same
airplane. The sound barrier had not only been penetrated, it had been virtually
destroyed as the myth it really was.
As a final note, the whole story of the human and engineering challenges that
revolved about the quest for and eventual achievement of supersonic flight is
fascinating, and it is a living testimonial to the glory of aeronautical engineering.
The story is brilliantly spelled out by Dr. Richard Hallion, earlier a curator at the
Air and Space Museum of the Smithsonian Institution and now chief historian at
Edwards Air Force Base, in his book Supersonic Flight (see Bibliography at the
end of this chapter). The reader should make every effort to study Hallion's story
of the events leading to and following Yeager's flight in 1947.
5.23 HISTORICAL NOTE:
THE X15FIRST MANNED HYPERSONIC AIRPLANE
AND STEPPINGSTONE TO THE SPACE SHUTTLE
Faster and higherfor all practical purposes, this has been the driving potential
behind the development of aviation since the Wrights' first successful flight in
1903. (See Sec. 1.11 and Figures 1.27 and 1.28.) This credo was never more true
than during the 15 years following Chuck Yeager's first supersonic flight in the
Bell XS1, described in Sec. 5.22. Once the "sound barrier" was broken, it was left
far behind in the dust. The next goal became manned hypersonic flightMach 5
and beyond.
To accomplish this goal, the NACA initiated a series of preliminary studies in
the early 1950s for an aircraft to fly beyond Mach 5, the definition of the
hypersonic flight regime. This definition is essentially a rule of thumb; unlike the
severe and radical flow field changes which take place when an aircraft flies
through Mach 1, nothing dramatic happens when Mach 5 is exceeded. Rather, the
hypersonic regime is simply a very high Mach number regime, where shock waves
are particularly strong and the gas temperatures behind these shock waves are
high. For example, consider Eq. (4.73), which gives the total temperature T
0
, that
is, the temperature of a gas which was initially at a Mach number M
1
and which
has been adiabatically slowed to zero velocity. This is essentially the temperature
at the stagnation point on a body. If M
1
= 7, Eq. (4.73) shows that (for y = 1.4),
T
0
/T
1
= 10.8. If the flight altitude is, say, 100,000 ft where T
1
= 419°R, then
T
0
= 4525°R = 4065°Ffar above the melting point of stainless steel. There
fore, as flight velocities increase far above the speed of sound, they gradually
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 253
approach a "thermal barrier," i.e., those velocities beyond which the skin temper
atures become too high and structural failure can occur. As in the case of the
sound barrier, the thermal barrier is only a figure of speechit is not an inherent
limitation on flight speed. With proper design to overcome the high rates of
aerodynamic heating, vehicles have today flown at Mach numbers as high as 36
(the Apollo lunar return capsule, for example). (For more details on highspeed
reentry aerodynamic heating, see Sec. 8.11.)
Nevertheless, in the early 1950s, manned hypersonic flight was a goal to be
achievedan untried and questionable regime characterized by high tempera
tures and strong shock waves. The basic NACA studies fed into an industrywide
design competition for a hypersonic airplane. In 1955, North American Aircraft
Corporation was awarded a joint NACAAir ForceNavy contract to design and
construct three prototypes of a manned hypersonic research airplane capable of
Mach 7 and a maximum altitude of 264,000 ft. This airplane was designated the
X15 and is shown in Figure 5.60. The first two aircraft were powered by
Reaction Motors LRll rocket engines, with 8000 lb of thrust (essentially the same
as the engine used for the Bell XS1). Along with the third prototype, the two
aircraft were later reengined with a more powerful rocket motor, the Reaction
Motors XLR99, capable of 57,000 lb of thrust. The basic internal structure of the
airplane was made from titanium and stainless steel, but the airplane skin was
Figure 5.60 The North American X15, the first manned hypersonic airplane. (North American/
Rockwell Corporation.)
254 INTRODUCTION TO FLIGHT
Inconel Xa nickel alloy steel capable of withstanding temperatures up to
1200°F. (Although the theoretical stagnation temperature at Mach 7 is 4065°F, as
discussed above, the actual skin temperature is cooler because of heat sink and
heat dissipation effects.) The wings had a low aspect ratio of 2.5 and a thickness
tochord ratio of 0.05both intended to reduce supersonic wave drag.
The first X15 was rolled out of the North American factory at Los Angeles
on October 15, 1958. Then Vice President Richard M. Nixon was the guest of
honor at the rollout ceremonies. The X15 had become a political, as well as a
technical, accomplishment, because the United States was attempting to heal its
wounded pride after the Russians had launched the first successful unmanned
satellite, Sputnik I, just a year earlier (see Sec. 8.15). The next day, the X15 was
transported by truck to the nearby Edwards Air Force Base (the site at Muroc
which saw the first supersonic flights of the Bell XS1 ).
As in the case of the XS1, the X15 was designed to be carried aloft by a
parent airplane, this time a Boeing B52 jet bomber. The first free flight, without
power, was made by Scott Crossfield on June 8, 1959. This was soon followed by
the first powered flight on September 17, 1959, when the X15 reached Mach 2.1
in a shallow climb to 52,341 ft. Powered with the smaller LRl 1 rocket engines,
the X15 set a speed record of Mach 3.31 on August 4, 1960, and an altitude
record of 136,500 ft just eight days later. However, these records were just
transitory. After November 1960, the X15 received the more powerful XLR99
engine. Indeed, the first flight with this rocket was made on November 15, 1960;
on this flight, with power adjusted to its lowest level and with the air brakes fully
extended, the X15 still hit 2000 mi/h. Finally, on June 23, 1961, hypersonic
flight was fully achieved when U.S. Air Force test pilot Major Robert White flew
the X15 at Mach 5.3 and in so doing accomplished the first "milepersecond"
flight in an airplane, reaching a maximum velocity of 3603 mi/h. This began an
illustrious series of hypersonic flight tests, peaked by a flight at Mach 6.72 on
October 3, 1967, with Air Force Major Pete Knight at the controls.
Experimental aircraft are just thatvehicles designed for specific experimen
tal purposes, which, after they are achieved, lead to the end of the program. This
happened to the X15 when on October 24, 1968, the last flight was carried
outthe 199th of the entire program. A 200th flight was planned, partly for
reasons of nostalgia; however, technical problems delayed this planned flight until
December 20, when the X15 was ready to go, attached to its B52 parent plane as
usual. However, of all things, a highly unusual snow squall suddenly hit Edwards,
and the flight was cancelled. The X15 never flew again. In 1969, the first X15
was given to the National Air and Space Museum of the Smithsonian, where it
now hangs with distinction in the Milestones of Flight Gallery, along with the
Bell XS1.
The X15 opened the world of manned hypersonic flight. The next hypersonic
airplane was the space shuttle. The vast bulk of aerodynamic and flight dynamic
data generated during the X15 program carried over to the space shuttle design.
The pilots' experience with lowspeed flights in a highspeed aircraft with low
lifttodrag ratio set the stage for flight preparations with the space shuttle. In
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 255
these respects, the X15 was clearly the major steppingstone to the space shuttle
of the 1980s.
5.24 CHAPTER SUMMARY
Some of the aspects of this chapter are highlighted below.
1. For an airfoil, the lift, drag, and moment coefficients are defined as
D
c =
d qooS
where L, D, and M are the lift, drag, and moments per unit span and
S = c(l).
For a finite wing, the lift, drag, and moment coefficients are defined as
where L, D, and M are the lift, drag, and moments for the complete wing,
and S is the wing planform area.
For a given shape, these coefficients are a function of angle of attack,
Mach number, and Reynolds number.
2. The pressure coefficient is defined as
C = P  Poo
p lp v 2
2 00 00
(5.23)
3. The PrandtlGlauert rule is a compressibility correction for subsonic flow:
(5.24)
where Cp,O and CP are the incompressible and compressible pressure coeffi
cients, respectively. The same rule holds for the lift and moment coefficients,
e.g.,
(5.32)
4. The critical Mach number is that freestream Mach number at which sonic
flow is first achieved at some point on a body. The dragdivergence Mach
number is that freestream Mach number at which the drag coefficient begins
to rapidly increase due to the occurrence of transonic shock waves. For a
given body, the dragdivergence Mach number is slightly higher than the
critical Mach number.
256 INTRODUCTION TO FLIGHT
5. The Mach angle is defined as
. 1
µ = arcsm M
6. The total drag coefficient for a finite wing is equal to
c2
CD= cd + T e ~ R
(5.38)
(5.47)
where cd is the profile drag coefficient and CL
2
/( ?TeAR) is the induceddrag
coefficient.
7. The lift slope for a finite wing, a, is given by
(5.53)
where a
0
is the lift slope for the corresponding infinite wing.
BIBLIOGRAPHY
Abbott, I. H., and von Doenhoff, A. E., Theory of Wing Sections, McGrawHill, New York, 1949 (also
Dover, New York, 1959).
Anderson, John D., Jr., Fundamentals of Aerodynamics, McGrawHill, New York, 1984.
Dommasch, D. 0., Sherbey, S.S., and Connolly, T. F., Airplane Aerodynamics, 4th ed., Pitman, New
York, 1968.
Hallion, R., Supersonic Flight (The Story of the Bell X1 and Douglas D558), Macmillan, New York,
1972.
McCormick, B. W., Aerodynamics, Aeronautics and Flight Mechanics, Wiley, New York, 1979.
Pierpont, P. K., "Bringing Wings of Change," Astronautics and Aeronautics, vol. 13, no. 10, October
1975, pp. 2027.
Shapiro, A.H., Shape and Flow: The Fluid Dynamics of Drag, Anchor, Garden City, New York, 1961.
Shevell, R. S., Fundamentals of Flight, PrenticeHall, Englewood Cliffs, NJ, 1983.
von Karman, T. (with Lee Edson), The Wind and Beyond, Little, Brown, Boston, 1967 (an autobio
graphy).
PROBLEMS
5.1 By the method of dimensional analysis, derive the expression M = q
00
Sccm for the aerodynamic
moment on an airfoil, where c is the chord and cm is the moment coefficient.
5.2 Consider an infinite wing with a NACA 1412 airfoil section and a chord length of 3 ft. The wing
is at an angle of attack of 5° in an airflow velocity of 100 ftjs at standard sealevel conditions.
Calculate the lift, drag, and moment about the quarter chord per unit span.
5.3 Consider a rectangular wing mounted in a lowspeed subsonic wing tunnel. The wing model
completely spans the test section so that the flow "sees" essentially an infinite wing. If the wing has a
NACA 23012 airfoil section and a chord of 0.3 m, calculate the lift, drag, and moment about the
quarter chord per unit span when the airflow pressure, temperature, and velocity are 1 atm, 303 K,
and 42 m/s, respectively. The angle of attack is 8°.
AIRFOILS, WINGS, AND OTHER AERODYNAMIC SHAPES 257
5.4 The wing model in Prob. 5.3 is pitched to a new angle of attack, where the lift on the entire wing
is measured as 200 N by the windtunnel force balance. If the wingspan is 2 m, what is the angle of
attack?
5.5 Consider a rectangular wing with a NACA 0009 airfoil section spanning the test section of a wind
tunnel. The test section airflow conditions are standard sea level with a velocity of 120 mi/h. The wing
is at an angle of attack of 4°, and the windtunnel force balance measures a lift of 29.5 lb. What is the
area of the wing?
5.6 The ratio of lift to drag, L/D, for a wing or airfoil is an important aerodynamic parameter.
Indeed, it is a direct measure of the aerodynamic efficiency of the wing. If a wing is pitched through a
range of angle of attack, the L/ D first increases, then goes through a maximum, and then decreases.
Consider an infinite wing with a NACA 2412 airfoil. Estimate the maximum value of L/D. Assume
the Reynolds number is 9 X 10
6
.
5.7 Consider an airfoil in a freestream with a velocity of 50 m/s at standard sealevel conditions. At a
point on the airfoil, the pressure is 9.5 X 10
4
N/m
2
. What is the pressure coefficient at this point?
5.8 Consider a lowspeed airplane flying at a velocity of 55 m/s. If the velocity at a point on the
fuselage is 62 m/s, what is the pressure coefficient at this point?
5.9 Consider a wing mounted in the test section of a subsonic wind tunnel. The velocity of the airflow
is 160 ftjs. If the velocity at a point on the wing is 195 ftjs, what is the pressure coefficient at this
point?
5.10 Consider the same wing in the same wind tunnel as in Prob. 5.9. If the test section air
temperature is 510°R, and if the flow velocity is increased to 700 ftjs, what is the pressure coefficient
at the same point?
5.11 Consider a wing in a highspeed wing tunnel. At a point on the wing, the velocity is 850 ftjs. If
the test section flow is at a velocity of 780 ft/s, with a pressure and temperature of 1 atm and 505°R,
respectively, calculate the pressure coefficient at the point.
5.12 If the test section flow velocity in Prob. 5.11 is reduced to 100 ftjs, what will the pressure
coefficient become at the same point on the wing?
5.13 Consider a NACA 1412 airfoil at an angle of attack of 4°. If the freestream Mach number is 0.8,
calculate the lift coefficient.
5.14 A NACA 4415 airfoil is mounted in a highspeed subsonic wind tunnel. The lift coefficient is
measured as 0.85. If the test section Mach number is 0.7, at what angle of attack is the airfoil?
5.15 Consider an airfoil at a given angle of attack, say a
1
• At low speeds, the minimum pressure
coefficient on the top surface of the airfoil is 0.90. What is the critical Mach number of the airfoil?
5.16 Consider the airfoil in Prob. 5.15 at a smaller angle of attack, say a
2
. At low speeds, the
minimum pressure coefficient is  0.65 at this lower angle of attack. What is the critical Mach number
of the airfoil?
5.17 Consider a uniform flow with a Mach number of 2. What angle does a Mach wave make with
respect to the flow direction?
5.18 Consider a supersonic missile flying at Mach 2.5 at an altitude of 10 km. Assume the angle of the
shock wave from the nose is approximated by the Mach angle (i.e., a very weak shock). How far
behind the nose of the vehicle will the shock wave impinge upon the ground? (Ignore the fact that the
speed of sound, hence the Mach angle, changes with altitude.)
5.19 The wing area of the Lockheed F104 straightwing supersonic fighter is approximately 210 ft
2
. If
the airplane weighs 16,000 lb and is flying in level flight at Mach 2.2 at a standard altitude of 36,000 ft,
estimate the wave drag on the wings.
5.20 Consider a flat plate at ~ angle of attack of 2° in a Mach 2.2 airflow. (Mach 2.2 is the cruising
Mach number of the Concorde supersonic transport, for example.) The length of the plate in the flow
direction is 202 ft, which is the length of the Concorde. Assume that the freestream conditions
correspond to a standard altitude of 50,000 ft. The total drag on this plate is the sum of wave drag
and skinfriction drag. Assume that a turbulent boundary layer exists over the entire plate. The
258 INTRODUCTION TO FLIGHT
results given in Chap. 4 for skinfriction coefficients hold for incompressible flow only; there is a
compressibility effect on Cf such that its value decreases with increasing Mach number. Specifically,
at Mach 2.2 assume that the Cf given in Chap. 4 is reduced by 20 percent.
(a) Given all the above information, calculate the total drag coefficient for the plate.
(b) If the angle of attack is increased to 5 °, assuming Cf stays the same, calculate the total drag
coefficient.
(c) For these cases, what can you conclude about the relative influence of wave drag and
skinfriction drag?
5.21 The Cessna Cardinal, a singleengine light plane, has a wing with an area of 16.2 m2 and an
aspect ratio of 7.31. Assume the span efficiency factor is 0.62. If the airplane is flying at standard
sealevel conditions with a velocity of 251 km/h, what is the induced drag when the total weight is
9800 N?
5.22 For the Cessna Cardinal in Prob. 5.21, calculate the induced drag when the velocity is 85.5
km/h (stalling speed at sea level with flaps down).
5.23 Consider a finite wing with an area and aspect ratio of 21.5 m
2
and 5, respectively (this is
comparable to the wing on a Gates Learjet, a twinjet executive transport). Assume the wing has a
NACA 65210 airfoil, a span efficiency factor of 0.9, and a profile drag coefficient of 0.004. If the wing
is at a 6° angle of attack, calculate CL and Cv.
5.24 During the 1920s and e . .1rly 1930s, the NACA obtained windtunnel data on different airfoils by
testing finite wings with an aspect ratio of 6. These data were than "corrected" to obtain infinite wing
airfoil characteristics. Consider such a finite wing with an area and aspect ratio of 1.5 ft
2
and 6,
respectively, mounted in a wind tunnel where the test section flow velocity is 100 ft/s at standard
sealevel conditions. When the wing is pitched to a= 2°, no lift is measured. When the wing is
pitched to a = 10°, a lift of 17.9 lb is measured. Calculate the lift slope for the airfoil (the infinite
wing) if the span effectiveness factor is 0.95.
5.25 A finite wing of area 1.5 ft
2
and aspect ratio of 6 is tested in a subsonic wind tunnel at a velocity
of 130 ft/s at standard sealevel conditions. At an angle of attack of 1°, the measured lift and drag
are 0 and 0.181 lb, respectively. At an angle of attack of 2°, the lift and drag are measured as 5.0 lb
and 0.23 lb, respectively. Calculate the span efficiency factor and the infinite wing lift slope.
5.26 Consider a light, singleengine airplane such as the Piper Super Cub. If the maximum gross
weight of the airplane is 7780 N, the wing area is 16.6 m
2
, and the maximum lift coefficient is 2.1 with
flaps down, calculate the stalling speed at sea level.
CHAPTER
SIX
ELEMENTS OF
AIRPLANE PERFORMANCE
First Europe, and then the globe, will he linked hy flight, and nations so knit together that thev will
grow to he nextdoor neighbors. This conquest of the air will proue, ultimatelv, to he man's greatest
and most glorious triumph. What railways have done for nations, airways will do for the world.
6.1 INTRODUCTION
Claude GrahameWhite
British Aviator, 1914
Henson's aerial steam carriage of the midnineteenth century (see Figure 1.11) was
pictured by contemporary artists as flying to all corners of the world. Of course,
questions about how it would fly to such distant locations were not considered by
the designers. As with most early aeronautical engineers of that time, their main
concern was simply to lift or otherwise propel the airplane from the ground; what
happened once the vehicle was airborne was viewed with secondary importance.
However, with the success of the Wright brothers in 1903, and with the subse
quent rapid development of aviation during the preWorld War I era, the
airborne performance of the airplane suddenly became of primary importance.
Some obvious questions were, and still are, asked about a given design. What is
the maximum speed of the airplane? How fast can it climb to a given altitude?
How far can it fly on a given tank of fuel? How long can it stay in the air?
Answers to these and similar questions constitute the study of airplane perform
ance, which is the subject of this chapter.
In previous chapters, the physical phenomena producing lift, drag, and
moments of an airplane were introduced. Emphasis was made that the
aerodynamic forces and moments exerted on a body moving through a fluid stem
259
260 INTRODUCTION TO FLIGHT
from two sources:
1. The pressure distribution
2. The shear stress distribution
both acting over the body surface. The physical laws governing such phenomena
were examined, with various applications to aerodynamic flows.
In the present chapter, we begin a new phase of study. The airplane will be
considered a rigid body on which is exerted four natural forces: lift, drag,
propulsive thrust, and weight. Concern will be focused on the movement of the
airplane as it responds to these forces. Such considerations form the core of flight
dynamics, an important discipline of aerospace engineering. Studies of airplane
performance (this chapter) and stability and control (Chap. 7) both fall under the
heading of flight dynamics.
In these studies, we will no longer be concerned with aerodynamic details;
rather, we will generally assume that the aerodynamicists have done their work
and provided us with the pertinent aerodynamic data for a given airplane. These
data are usually packaged in the form of a drag polar for the complete airplane,
given as
c2
CD= CD e + Al,R
. 7Te
(6.la)
Equation (6.la) is an extension of Eq. (5.47) to include the whole airplane. Here,
CD is the drag coefficient for the complete airplane; CL is the total lift coefficient,
including the small contributions from the horizontal tail and fuselage; and CD. e
is defined as the parasite drag coefficient, which contains not only the profile drag
of the wing [cd in Eq. (5.47)] but also the friction and pressure drag of the tail
surfaces, fuselage, engine nacelles, landing gear, and any other component of the
airplane which is exposed to the airflow. At transoni and supersonic speeds, CD, e
also contains wave drag. Because of changes in the flow field around the airplane
especially changes in the amount of separated flow over parts of the airplaneas
the angle of attack is varied, CD, e will change with angle of attack; that is, CD, e is
itself a function of lift coefficient. A reasonable approximation for this function is
CD,e = CD,O + rCL2
where r is an empirically determined constant. Hence, Eq. (6.la) can be written
as
CD= cD,O +(r +
(6.lb)
In Eqs. (6.la) and (6.lb), e is the familiar span efficiency factor, which takes into
account the nonelliptical lift distribution on wings of general shape (see Sec.
5.14). Let us now redefine e such that it also includes the effect of the variation of
parasite drag with lift; i.e., let us write Eq. (6.lb) in the form
c2
L
CD= CD 0 + AR
. 7Te
(6.lc)
ELEMENTS OF AIRPLANE PERFORMANCE 261
where C D.o is the parasite drag coefficient at zero lift and the term CL
2
/7TeAR
includes both induced drag and the contribution to parasite drag due to lift. In
Eq. (6.lc), our redefined e, which now includes the effect of r from Eq. (6.lb). is
called the Oswald efficiency factor (named after W. Bailey Oswald, who first
established this terminology in NACA report no. 408 in 1933). In this chapter, the
basic aerodynamic properties of the airplane will be described by Eq. (6.lc), and
we will consider both CD.o and e as known aerodynamic quantities, obtained
from the aerodynamicist. We will continue to designate C//7TeAR by CD.i•
where CD
1
now has the expanded interpretation as the coefficient of drag due to
lift, including both the contributions due to induced drag and the increment in
parasite drag due to angle of attack different than aL = 0. For compactness, we
will designate CD.o as simply the parasite drag coefficient, although we recognize it
to be more precisely the parasite drag coefficient at zero lift, i.e., the value of CD. e
when a= a ~ o ·
To study the performance of an airplane, the fundamental equations which
govern its translational motion through the air must first be established, as
follows.
6.2 EQUATIONS OF MOTION
Consider an airplane in flight, as sketched in Figure 6.1. The flight path (direction
of motion of the airplane) is inclined at an angle () with respect to the horizontal.
In terms of the definitions in Chap. 5, the flight path direction and the relative
wind are along the same line. The mean chord line is at a geometric angle of
attack a with respect to the flight path direction. There are four physical forces
L
I
I
I
I e
'.. v
w
Figure 6.1 Force diagram for an airplane in night.
e
262 INTRODUCTION TO FLIGHT
acting on the airplane:
1. Lift L, which is perpendicular to the flight path direction
2. Drag D, which is parallel to the flight path direction
3. Weight W, which acts vertically toward the center of the earth (and hence is
inclined at the angle 8 with respect to the lift direction)
4. Thrust T, which in general is inclined at the angle aT with respect to the flight
path direction
'
The force diagram shown in Figure 6.1 is important. Study it carefully until you
feel comfortable with it.
The flight path shown in Figure 6.1 is drawn as a straight line. This is the
picture we see by focusing locally on the airplane itself. However, if we stand
back and take a wider view of the space in which the airplane is traveling, the
flight path is generally curved. This is obviously true if the airplane is maneuver
ing; even if the airplane is flying "straight and level" with respect to the ground, it
is still executing a curved flight path with a radius of curvature equal to the
absolute altitude ha (as defined in Sec. 3.1).
When an object moves along a curved path, the motion is called curvilinear,
as opposed to motion along a straight line, which is rectilinear. Newton's second
law, which is a physical statement that force = mass x acceleration, holds in
either case. Consider a curvilinear path. At a given point on the path, set up two
mutually perpendicular axes, one along the direction of the flight path and the
other normal to the flight path. Applying Newton's law along the flight path,
dV
L:F
11
=ma= mdt (6.2)
where L:F
11
is the summation of all forces parallel to the flight path, a = dV / dt is
the acceleration along the flight path, and V is the instantaneous value of the
airplane's flight velocity. (Velocity V is always along the flight path direction, by
definition.) Now, applying Newton's law perpendicular to the flight path, we have
v2
LFl. = m (6.3)
re
where L:F .i. is the summation of all forces perpendicular to the flight path and
V
2
/re is the acceleration normal to a curved path with radius of curvature re. This
normal acceleration V
2
/r, should be familiar from basic physics. The righthand
side of Eq. (6.3) is nothing other than the "centrifugal force."
Examining Figure 6.1, we see that the forces parallel to the flight path
(positive to the right, negative to the left) are
L:F
11
= TcosaT  D  Wsin8 (6.4)
and the forces perpendicular to the flight path (positive upward and negative
downward) are
L:Fl. = L + TsinaT  Wcos8 (6.5)
ELEMENTS OF AIRPLANE PERFORMANCE 263
Combining Eq. (6.2) with (6.4), and (6.3) with (6.5) yields
I Tcorn, D  WsinO m'!f I
v2
L + Tsinar  Wcos8 = m
rc
(6.6)
(6.7)
Equations (6.6) and (6.7) are the equations of motion for an airplane in transla
tional flight. (Note that an airplane can also rotate about its axes; this will be
discussed in Chap. 7. Also note that we are not considering the possible sidewise
motion of the airplane perpendicular to the page of Figure 6.1.)
Equations (6.6) and (6.7) describe the general twodimensional translational
motion of an airplane in accelerated flight. However, in the first part of this
chapter we are interested in a specialized application of these equations, namely,
the case where the acceleration is zero. The performance of an airplane for such
unaccelerated flight conditions is called static performance. This may, at first
thought, seem unduly restrictive; however, static performance analyses lead to
reasonable calculations of maximum velocity, maximum rate of climb, maximum
range, etc.parameters of vital interest in airplane design and operation.
With this in mind, consider level, unaccelerated flight. Referring to Figure
6.1, level flight means that the flight path is along the horizontal; that is, 8 = 0.
Unaccelerated flight means that the righthand sides of Eqs. (6.6) and (6.7) are
zero. Therefore, these equations reduce to
Tcosar = D
L + Tsinar = W
(6.8)
(6.9)
Furthermore, for most conventional airplanes, aT is small enough that cos aT:::: 1
and sina:T:::: 0. Thus, from Eqs. (6.8) and (6.9),
(6.10)
(6.11)
Equations (6.10) and (6.11) are the equations <1f motion for level, unaccelerated
flight. They can also be obtained directly from Figure 6.1, by inspection. In level,
unaccelerated flight, the aerodynamic drag is balanced by the thrust of the engine,
and the aerodynamic lift is balanced by the weight of the airplanealmost trivial,
but very useful, results.
Let us now apply these results to the static performance analysis of an
airplane. The following sections constitute the building blocks for such an
analysis, which ultimately yields answers to such questions as how fast, how far,
how long, and how high a given airplane can fly. Also, the discussion in these
sections relies heavily on a graphical approach to the calculation of airplane
264 INTRODUCTION TO FLIGHT
performance. In modern aerospace engineering, such calculations are made
directly on highspeed digital computers. However, the graphical illustrations in
the following sections are essential to the programming and understanding of
such computer solutions; moreover, they help to clarify and explain the concepts
being presented.
6.3 THRUST REQUIRED FOR LEVEL, UNACCELERATED FLIGHT
Consider an airplane in steady level flight at a given altitude and a given velocity.
For flight at this velocity, the airplane's power plant (e.g., turbojet engine or
reciprocating enginepropeller combination) must produce a net thrust which is
equal to the drag. The thrust required to obtain a certain steady velocity is easily
calculated as follows. From Eqs. (6.10) and (5.18)
and from Eqs. (6.11) and (5.15)
L = W = q
00
SCL
Dividing Eq. (6.12) by (6.13) yields
T CD
w CL
(6.12)
(6.13)
(6.14)
Thus, from Eq. (6.14), the thrust required for an airplane to fly at a given velocity
in level, unaccelerated flight is
w
L/D
(6.15)
(Note that a subscript R has been added to thrust to emphasize that it is thrust
required.)
Thrust required TR for a given airplane at a given altitude, varies with
velocity Voc,. The thrustrequired curve is a plot of this variation and has the
general shape illustrated in Figure 6.2. To calculate a point on this curve, proceed
as follows:
1. Choose a value of V
00
•
2. For this V
00
, calculate the lift coefficient from Eq. (6.13):
(6.16)
Note that p
00
is known from the given altitude and S is known from the
given airplane. The CL calculated from Eq. (6.16) is that value necessary for
the lift to balance the known weight W of the airplane.
1000
800
:2
. 600
h"'
400
200
0 JOO
(L/Dlmax
200
v ~ ft/s
300
ELEMENTS OF AIRPLANE PERFORMANCE 265
400
Figure 6.2 Thrustrequired curve. The results on this and subsequent figures correspond to answers
for some of the sample problems in Chap. 6.
3. Calculate CD from the known drag polar for the airplane:
c2
CD= CD 0 + ALR
. 7Te
where CL is the value obtained from Eq. (6.16).
4. Form the ratio CL/CD.
5. Calculate thrust required from Eq. (6.15).
The value of TR obtained from step 5 is that thrust required to fly at the
specific velocity chosen in step 1. In tum, the curve in Figure 6.2 is the locus of all
such points taken for all velocities in the flight range of the airplane. The reader
should study Example 6.1 at the end of this section in order to become familiar
with the above steps.
Note from Eq. (6.15) that TR varies inversely as L/D. Hence, minimum
thrust required will be obtained when the airplane is flying at a velocity where
L/ D is maximum. This condition is shown in Figure 6.2.
The lifttodrag ratio L/ D is a measure of the aerodynamic efficiency of an
airplane; it only makes sense that maximum aerodynamic efficiency should lead
to minimum thrust required. Consequently, lifttodrag ratio is an important
aerodynamic consideration in airplane design. Also note that L/ D is a function
of angle of attack, as sketched in Figure 6.3. For most conventional subsonic
airplanes, L/ D reaches a maximum at some specific value of a, usually on the
order of 2 to 5 °. Hence, when an airplane is flying at the velocity for minimum TR
as shown in Figure 6.2, it is simultaneously flying at the angle of attack for
maximum L/ D as shown in Figure 6.3.
266 INTRODUCTION TO FLIGHT
(L/D)max
Figure 6.3 Lifttodrag ratio versus angle of
attack.
As a corollary to this discussion, note that different points on the thrust
required curve correspond to different angles of attack. This is emphasized in
Figure 6.4, which shows that, as we move from right to left on the thrustrequired
curve, the airplane angle of attack increases. This also helps to explain physically
why TR goes through a minimum. Recall that L = W = q
00
SCL. At high veloci
ties (point a in Figure 6.4), most of the required lift is obtained from high
dynamic pressure q
00
; hence CL and therefore a are small. Also, under the same
conditions, drag (D = q
00
SCv) is relatively large because q
00
is large. As we move
to the left on the thrustrequired curve, q
00
decreases; hence CL and therefore a
must increase to support the given airplane weight. Because q
00
decreases, D and
hence TR initially decrease. However, recall that drag due to lift is a component
of total drag and that Cv,; varies as Cl. At low velocities, such as at point b in
Figure 6.4, qoo is low, and hence CL is large. At these conditions, cD,i increases
rapidly, more rapidly than q
00
decreases, and D, hence TR, increases. This is why,
a
Increasing angle of attack
Increasing velocity
Figure 6.4 Thrustrequired curve with associated angleofattack variation.
ELEMENTS OF AIRPLANE PERFORMANCE 1.67
starting at point a, TR first decreases as voe decreases, then goes through a
minimum and starts to increase, as shown at point b.
Recall from Eq. (6.lc) that the total drag of the airplane is the sum of
parasite drag and drag due to lift. The corresponding drag coefficients are CD.o
and CD;= CL
2
/'1TeAR, respectively. At the condition for minimum TR, there
exists ~ interesting relation between cD.O and cD.i' as follows. From Eq. (6.10)
TR= D = qooSCD = qooS( cD.O + CD_;)
(6.17)
Parasite TR Induced TR
Note that, as identified in Eq. (6.17), the thrust required can be considered the
sum of parasite thrust required (thrust required to balance parasite drag) and
induced thrust required (thrust required to balance drag due to lift). Examining
Figure 6.5, we find induced TR decreases but parasite TR increases as the velocity
is increased. (Why?)
Recall that CL= W/q
00
S. From Eq. (6.17),
(6.18)
Also
dTR dV
00

dVOO dq
00
( 6.19)
From calculus, we find that the point of minimum TR in Figure 6.2 corresponds
to dTR/dV
00
= 0. Hence, from Eq. (6.19), minimum TR also corresponds to
/
/
/
/
I
I
I
/
,/,I' Parasite TR
.,,,.,,''
' I 
..... ....__I_.,,,...
* Induced TR
__ ,....... I  ________ _.. I
Figure 6.5 Comparison of induced and parasite thrust required.
268 INTRODUCTION TO FLIGHT
dTR/dq
00
= 0. Differentiating Eq. (6.18) with respect to qoc and setting the
derivative equal to zero, we have
Thus
CD,O =
w2
However
qoo 2s2
w2
w2
=0
q
00
2
S?TeAR
q
00
2
S
2
?TeAR
__!!::__  c 2
( r
qooS  L
Hence, Eq. (6.20) becomes
c 2
C = __ L_ = C
D,o ?TeAR D,;
Parasite drag = drag due to lift
(6,20)
(6.21)
Equation (6.21) yields the interesting aerodynamic result that at minimum thrust
required, parasite drag equals drag due to lift. Hence, the curves for parasite and
induced TR intersect at the velocity for minimum TR (i.e., for maximum L/ D), as
shown in Figure 6.5. We will return to this result in Sec. 6.13.
Example 6.1 For all the examples given in this chapter, two types of airplanes will be considered:
(a) A light, singleengine, propellerdriven, private airplane, approximately modeled after
the Cessna Skylane shown in Figure 6,6. For convenience, we will designate our hypothetical
Figure 6.6 The hypothetical CP1 studied in Chap. 6 sample problems is modeled after the Cessna
Skylane shown here. (Cessna Aircraft Corporation.)
ELEMENTS OF AIRPLANE PERFORMANCE 269
airplane as the CP1, having the following characteristics:
Wingspan= 35.8 ft
Wing area= 174 ft
2
Normal gross weight = 2950 lb
Normal capacity: 65 gallons of aviation gasoline
Power plant: onepiston engine of 230 hp at sea level
Specific fuel consumption= 0.45 lb/(hp)(h)
Parasite drag coefficient= Cv.o = 0.025
Oswald efficiency factor= e = 0.8
Propeller efficiency= 0.8
(b) A jetpowered executive aircraft, approximately modeled after the Cessna Citation 3
shown in Figure 6.7. For convenience, we will designate our hypothetical jet as the CJ1, having
the following characteristics:
Wingspan= 53.3 ft
Wing area= 318 ft
2
Normal gross weight= 19,815 lb
Fuel capacity: 1119 gallons of kerosene
Power plant: two turbofan engines of 3650 lb of thrust each at sea level
Specific fuel consumption= 0.6 lb of fuelj(lb thrust)(h)
Parasite drag coefficient= C
0
.
0
= 0.02
Oswald efficiency factor= e = 0.81
By the end of this chapter, all the examples taken together will represent a basic
performance analysis of these two aircraft.
Figure 6.7 The hypothetical CJ1 studied in Chap. 6 sample problems is modeled after the Cessna
Citation 3 shown here. (Cessna Aircraft Corp.)
270 INTRODUCTION TO FLIGHT
In this example, only the thrust required is considered. Calculate the TR curves at sea level
for both the CP1 and the CJ1.
SOLUTION
(a) For the CP1 assume V
00
= 200 ftjs = 136.4 mijh. From Eq. (6.16),
cl = w 2950 = o.357
·
2
s H0.002377)(200)
2
(174)
The aspect ratio is
AR= b2 = (35.8)2 = 7 37
s 174 .
Thus, from Eq. (6.lc),
CL
2
(0.357)
2
CD= cD.O + '!TeAR = 0.025 + ( )( )
'TT 0.8 7.37
= 0.0319
Hence
L CL 0.357
D =CD= 0.0319 =ll.
2
Finally, from Eq. (6.15),
w 2950
TR= L/D = 11.2
To obtain the thrustrequired curve, the above calculation is repeated for many different values
of V
00
• Some sample results are tabulated below.
v""' ft/s CL CD L/D TR.lb
100 1.43 0.135 10.6 279
150 0.634 0.047 13.6 217
250 0.228 0.028 8.21 359
300 0.159 0.026 6.01 491
350 0.116 0.026 4.53 652
The above tabulation is given so that the reader can try such calculations and compare the
results. Such tabulations will be given throughout this chapter. They are taken from a computer
calculation where 100 different velocities are used to generate the data. The TR curve obtained
from these calculations is given in Figure 6.2.
(b) For the CJ1 assume V
00
= 500 ft/s = 341 mijh. From Eq. (6.16),
CL= w 19,815
tPooVoo
2
S (0.002377)(500)\318)
= 0.210
The aspect ratio is
AR= b2 = (53.3)2 =8 93
s 318 .
Thus, Eq. (6.lc).
CL
2
(0.21)
2
CD= CD.O + = 0.02+
'!TeAR "'(0.81 )(8.93)
= 0.022
Hence
L CL 0.21
D = CD = 0.022 =
9
·
55
ELEMENTS OF AIRPLANE PERFORMANCE 271
10.0
9.0
8.0
7.0
~
6.0
'
0
x
5.0
:e
._,<>:
4.0
V max= 975 ft/s
I
3.0
I
2.0
11 ____
I
I
I
I
1.0
I V1 I
I
I
I
I
0 200 400 600 800 1000 1200
V ft/s
Figure 6.8 Thrustrequired curve for the CJ1.
Finally, from Eq. (6.15)
r: = ~ = 19.815=12075 lb I
R L/D 9.55
A tabulation for a few different velocities follows.
V
00
, ft/s C1_ Co L/D TR.lb
300 0.583 0.035 16.7 1188
600 0.146 0.021 6.96 2848
700 0.107 0.021 5.23 3797
850 0.073 0.020 3.59 5525
1000 0.052 0.020 2.61 7605
The thrustrequired curve is given in Figure 6.8.
6.4 THRUST AVAILABLE AND MAXIMUM VELOCITY
Thrust required TR, described in the previous section, is dictated by the
aerodynamics and weight of the airplane itself; it is an airframeassociated
272 INTRODUCTION TO FLIGHT

] Reciprocating
enginepropeller
·a combination
0
"'
(a)
Turbojet
engine
(b)
Figure 6.9 Thrustavailable curves for piston enginepropeller combination and for a turbojet engine.
phenomenon. In contrast, the thrust available TA is strictly associated with the
engine of the airplane; it is the propulsive thrust provided by an enginepropeller
combination, a turbojet, a rocket, etc. Propulsion is the subject of Chap. 9. Suffice
it to say here that reciprocating piston engines with propellers exhibit a variation
of thrust with velocity, as sketched in Figure 6.9a. Thrust at zero velocity (static
thrust) is a maximum and decreases with forward velocity. At nearsonic flight
speeds, the tips of the propeller blades encounter the same compressibility
problems discussed in Chap. 5, and the thrust available rapidly deteriorates. On
the other hand, the thrust of a turbojet engine is relatively constant with velocity,
as sketched in Figure 6.9b. These two power plants are quite common in aviation
today; reciprocating enginepropeller combinations power the average light,
general aviation aircraft, whereas the jet engine is used by almost all large
commercial transports and military combat aircraft. For these reasons, the
performance analyses of the present chapter will consider only the above two
propulsive mechanisms.
Consider a jet airplane flying in level, unaccelerated flight at a given altitude
and with the velocity V
1
, as shown in Figure 6.8. Point 1 on the thrustrequired
curve gives the value of TR for the airplane to fly at velocity V
1
. The pilot has
adjusted the throttle such that the jet engine provides thrust available just equal
to the thrust required at this point; TA = TR. This partialthrottle TA is illustrated
by the dashed curve in Figure 6.8. If the pilot now pushes the throttle forward
and increases the engine thrust to a higher value of TA, the airplane will accelerate
to a higher velocity. If the throttle is increased to full position, maximum TA will
ELEMENTS OF AIRPLANE PERFORMANCE 273
be produced by the jet engine. In this case, the speed of the airplane will further
increase until the thrust required equals the maximum TA (point 2 in Figure 6.8).
It is now impossible for the airplane to fly any faster thah the velocity at point 2;
otherwise, the thrust required would exceed the maximum thrust available from
the power plant. Hence, the intersection of the TR curve (dependent on the
airframe) and the maximum TA curve (dependent on the engine) defines the
maximum velocity Vmax of the airplane at the given altitude, as shown in Figure 6.8.
Calculating the maximum velocity is an important aspect of the airplane design
process.
Conventional jet engines are rated in terms of thrust (usually in pounds).
Hence, the thrust curves in Figure 6.8 are useful for the performance analysis of a
jetpowered aircraft. On the other hand, piston engines are rated in terms of
power (usually horsepower). Hence, the concepts of TA and TR are inconvenient
for propellerdriven aircraft. In this case, power required and power available are
the more relevant quantities. Moreover, considerations of power lead to results
such as rate of climb and maximum altitude for both jet and propellerdriven
airplanes. Therefore, for the remainder of this chapter, emphasis will be placed on
power rather than thrust, as introduced in the next section.
Example 6.2 Calculate the maximum velocity of the CJ1 at sea level (see Example 6.1).
SOLUTION The information given in Example 6.1 states that the power plant for the CJ1
consists of two turbofan engines of 3650 lb of thrust each at sea level. Hence,
TA = 2(3650) = 7300 lb
Examining the results of Example 6.1, we see TR= TA= 7300 lb occurs when V
00
= 975 ft/s (see
Figure 6.8). Hence
I vmax = 975 ft/s = 665 mijh I
It is interesting to note that, since the sealevel speed of sound is 1117 ft/s. the maximum
sealevel Mach number is
vmax 975
Mmax=a= 1117 =0.87
In the present examples, Cv.o is assumed constant; hence, the drag polar does not include
dragdivergence effects, as discussed in Chap. 5. Because the dragdivergence Mach number for
this type of airplane is normally on the order of 0.82 to 0.85, the above calculation indicates that
Mmax is above drag divergence, and our assumption of constant Cv.o becomes inaccurate at this
high a Mach number.
6.5 POWER REQUIRED FOR LEVEL, UNACCELERATED FLIGHT
Power is a precisely defined mechanical term; it is energy per unit time. The
power associated with a moving object can be illustrated by a block moving at
constant velocity V under the influence of the constant force F, as shown in
Figure 6.10. The block moves from left to right through the distance d in a time
274 INTRODUCTION TO FLIGHT
Time= t1 Time= t
2
Figure 6.10 Force, velocity, and power of a moving body.
interval t
2
 t
1
. (We assume that an opposing equal force not shown in Figure
6.10, say due to friction, keeps the block from accelerating.) Work is another
precisely defined mechanical term; it is force multiplied by the distance through
which the force moves. Moreover, work is energy, having the same units as
energy. Hence
P
energy force X distance f distance
ower = . = . = orce x . 
time ttme time
Applied to the moving block in Figure 6.10, this becomes
Power = F(d) = FV
t2  11
(6.22)
where d/(t
2
 t
1
) is the velocity V of the object. Thus, Eq. (6.22) demonstrates
that the power associated with a force exerted on a moving object is force x
velocity, an important result.
Consider an airplane in level, unaccelerated flight at a given altitude and with
velocity Voo The thrust required is TR. From Eq. (6.22), the power required PR is
therefore
I PR= TRVoo I (6.23)
The effect of the airplane aerodynamics (CL and Cn) on PR is readily
obtained by combining Eqs. (6.15) and (6.23):
w
PR= TRVoo = CL/Cn V00 (6.24)
From Eq. (6.11),
Hence (6.25)
2W
3
Cn
2
1
 cx:
PooSCL3 CL3/2/Cn
(6.26)
4.0
3.0
I
0
x
"'
;e 2.0
.:::
"
o...""
1.0
0 100 200
v ~ . ft/s
ELEMENTS OF AIRPLANE PERFORMANCE 275
300 400
Figure 6.11 Powerrequired curve for the CP1 at sea level.
In contrast to thrust required, which varies inversely as CL/CD [see Eq. (6.15)],
power required varies inversely as CL
3
1
2
/CD.
The powerrequired curve is defined as a plot of PR versus V
00
, as sketched in
Figure 6.11; note that it qualitatively resembles the thrustrequired curve of
Figure 6.2. As the airplane velocity increases, PR first decreases, goes through a
minimum, and then increases. At the velocity for minimum power required, the
airplane is flying at the angle of attack which corresponds to a maximum
CL3/2/CD.
In Sec. 6.3, we demonstrated that minimum TR aerodynamically corresponds
to equal parasite and induced drag. An analogous but different relation holds at
minimum PR. From Eqs. (6.10) and (6.23),
PR= TRVoo = DV()() = q()()s( cD,O + T T ~ l ~ ) v()()
c 2
PR= qooSCD oVoo + qooSVoo ALR
, 'TTe
Parasite power
required
Induced power
required
(6.27)
Therefore, as in the earlier case of TR, the power required can be split into the
respective contributions needed to overcome parasite drag and drag due to lift.
These contributions are sketched in Figure 6.12. Also as before, the aerodynamic
conditions associated with minimum PR can be obtained from Eq. (6.27) by
setting dPR/dV
00
= 0. To do this, first obtain Eq. (6.27) explicitly in terms of V
00
,
276 INTRODUCTION TO FLIGHT
Figure 6.12 Comparison of induced, parasite, and net power required.
(6.28)
For minimum power required, dPR/dV
00
= 0. Differentiating Eq. (6.28) yields
W
2
/tp
00
V
00
2
S
?Te AR
for minimum PR
Hence, the aerodynamic condition that holds at minimum power required is
I Cv.o = }Cv,; I (6.29)
The fact that parasite drag is onethird the drag due to lift at minimum PR is
reinforced by examination of Figure 6.12. Also note that point 1 in Figure 6.12
corresponds to Cv.o = CD,i• that is, minimum TR; hence, V
00
for minimum PR is
less than that for minimum TR.
The point on the powerrequired curve that corresponds to minimum TR is
easily obtained by drawing a line through the origin and tangent to the PR curve,
as shown in Figure 6.13. The point of tangency corresponds to minimum TR
(hence maximum L/D). To prove this, consider any line through the origin and
intersecting the PR curve, such as the dashed line in Figure 6.13. The slope of this
ELEMENTS OF AIRPLANE PERFORMANCE 277
Figure 6.13 The tangent to the powerrequired curve locates the point of minimum thrust required
(and hence the point of maximum L/D).
line is PR/V
00
• As we move to the right along the PR curve, the slope of an
intersecting line will first decrease, then reach a minimum (at the tangent point),
and then again increase. This is clearly seen simply by inspection of the geometry
of Figure 6.13. Thus, the point of tangency corresponds to a minimum slope,
hence a minimum value of PR/V
00
• In turn, from calculus this corresponds to
d(PR/V
00
) d(TRV
00
/V
00
) dTR
''"=' = =  = 0
dV
00
dV
00
dV
00
The above result yields dTR/ dV
00
= 0 at the tangent point, which is precisely the
mathematical criterion for minimum TR. Correspondingly, L/ D is maximum at
the tangent point.
Example 6.3 Calculate the powerrequired curves for (a) the CP1 at sea level and for (b) the
CJ1 at an altitude of 22,000 ft.
SOLUTION
(a) For the CP1 the values of TR at sea level have already been tabulated and graphed in
Example 6.1. Hence, from Eq. (6.23),
we obtain the following tabulation:
v,
ftjs
100
150
250
300
350
279
217
359
491
652
PR,
ft· lb/s
27,860
32,580
89,860
147,200
228,100
The powerrequired curve is given in Figure 6.11.
278 INTRODUCTION TO FLIGHT
0.4
CJI
0.3
22,000 ft
I
s
x
;s
0.2
it:
o...""
0.1
0 200 400 800 1000
Figure 6.14 Powerrequired curve for the CJ1 at 22,000 ft.
(b) For the CJ1 at 22,000 ft, p
00
= 0.001184 slug/ft
3
. The calculation of Tn is done with
the same method as given in Example 6.1, and PR is obtained from Eq. (6.23). Some results are
tabulated below.
Voo, TR, PR.
ft/s C1. Co L/D lb ft · lb/s
300 1.17 0.081 14.6 1358 0.041 x 10
7
500 0.421 0.028 15.2 1308 0.065 x 10
7
600 0.292 0.024 12.3 1610 0.097 x 10
7
800 0.165 0.021 7.76 2553 0.204 x 10
7
1000 0.105 0.020 5.14 3857 0.386 x 10
7
The reader should attempt to reproduce these results.
The powerrequired curve is given in Figure 6.14.
6.6 POWER AVAILABLE AND MAXIMUM VELOCITY
Note again that PR is a characteristic of the aerodynamic design and weight of
the aircraft itself. In contrast, the power available PA is a characteristic of the
power plant. A detailed discussion on propulsion is deferred until Chap. 9;
however, the following comments are made to expedite our performance analyses.
ELEMENTS OF AIRPLANE PERFORMANCE 279
A Reciprocating EnginePropeller Combination
A piston engine generates power by burning fuel in confined cylinders and using
this energy to move pistons, which in turn deliver power to the rotating crankshaft,
as schematically shown in Figure 6.15. The power delivered to the propeller by
the crankshaft is defined as the shaft brake power P (the word "brake" stems
from a method of laboratory testing which measures the power of an engine by
loading it with a calibrated brake mechanism). However, not all of P is available
to drive the airplane; some of it is dissipated by inefficiencies of the propeller
itself (to be discussed in Chap. 9). Hence, the power available to propel the
airplane, PA, is given by
I PA = 11P I
(6.30)
where 1J is the propeller efficiency, 1J < 1. Propeller efficiency is an important
quantity and is a direct product of the aerodynamics of the propeller. It is always
less than unity. For our discussions here, both 1J and P are assumed to be known
quantities for a given airplane.
A remark on units is necessary. In the engineering system, power is in
ft · lb/s; in the SI system, power is in watts [which are equivalent to N · m/s].
However, the historical evolution of engineering has left us with a horrendously
inconsistent (but very convenient) unit of power which is widely used, namely,
horsepower. All reciprocating engines are rated in terms of horsepower, and it is
important to note that
1 horsepower = 550 ft · lb/s
= 746W
Therefore, it is common to use shaft brake horsepower bhp in place of P, and
horsepower available hpA in place of PA" Equation (6.30) still holds in the form
hp A = ( 1} )(bhp) (6.30')
However, be cautious. As always in dealing with fundamental physical relations,
Reciprocating
engine
p
Shaft
Propeller
Figure 6.15 Relation between shaft
brake power and power available.
280 INTRODUCTION TO FLIGHT
PA
C". propeller combination
(a) (b)
Figure 6.16 Power available for piston enginepropeller combination and for the jet engine.
units must be consistent; therefore, a good habit is to immediately convert hp to
ft · lb/s or to watts before starting an analysis. This approach will be used here.
The poweravailable curve for a typical piston enginepropeller combination
is sketched in Figure 6.16a.
B Jet Engine
The jet engine (see Chap. 9) derives its thrust by combustion heating an incoming
stream of air and then exhausting this hot air at high velocities through a nozzle.
The power available from a jet engine is obtained from Eq. (6.22) as
( 6.31)
Recall from Figure 6.9b that TA for a jet engine is reasonably constant with
velocity. Thus, the poweravailable curve varies essentially linearly with V
00
, as
sketched in Figure 6.16b.
For both the propeller and jetpowered aircraft, the maximum flight velocity
is determined by the highspeed intersection of the maximum PA and the PR
curves. This is illustrated in Figure 6.17. Because of their utility in determining
other performance characteristics of an airplane, these power curves are essential
to any performance analysis.
Example 6.4 Calculate the maximum velocity for (a) the CP1 at sea level and (b) the CJ1 at
22,000 ft.
SOLUTION
(a) For the CP1 the information in Example 6.1 gave the horsepower rating of the power
plant at sea level as 230 hp. Hence,
hp A= ( 1) )(bhp) = 0.80(230) = 184 hp
The results of Example 6.3 for power required are replotted in Figure 6.l 7a in terms of
horsepower. The horsepower available is also shown, and Vmax is determined by the intersection
700
600
500
...
"
"
400
0
0.
"
0
:r: 300
200
100
0
10
9
8
7
M
I
0
6
x
...
"
"
0
0.
"
0
4
:r:
3
2
0 200
JOO
CP1
sea level
200
v_, ft/s
(a)
CJI
22,000 ft
400
ELEMENTS OF AIRPLANE PERFORMANCE 281
I
I
: V max = 265 ft/s
I
300
600 800
v_, ft/s
(b)
400
V max = 965 ft/s
1000 1200
Figure 6.17 Poweravailable and powerrequired curves, and the determination of maximum velocity.
(a) Propellerdriven airplane. ( b) Jetpropelled airplane.
282 INTRODUCTION TO FLIGHT
of the curves as
\ vm .. = 265ft/s=181 mi/h \
(b) For the CJ1, again from the information given in Example 6.1, the sealevel static
thrust for each engine is 3650 lb. There are two engines, hence TA = 2(3650) = 7300 lb. From
Eq. (6.31), PA = TAVoo, and in terms of horsepower, where is in pounds and v"' in ftjs,
Let hPA.o be the horsepower at sea level. As we will see in Chap. 9, the thrust of ajet engine is, to
a first approximation, proportional to the air density. If we make this approximation here, the
thrust at altitude becomes
T _..f!_T
A.alt  Po A.O
and hence
h _..f!_h
PA.alt  Po PA.O
For the CJ1 at 22,000 ft, where p = 0.001184 slug/ft
3
,
The results of Example 6.3 for power required are replotted in Figure 6.l 7b in terms of
horsepower. The horsepower available, obtained from the above equation, is also shown, and
Vmax is determined by the intersection of the curves as
I vm .. = 965 ft/s = 658 mijh \
6.7 ALTITUDE EFFECTS ON POWER REQUIRED AND
AVAILABLE
With regard to PR, curves at altitude could be generated by repeating the
calculations of the previous sections, with p
00
appropriate to the given altitude.
However, once the sealevel PR curve is calculated by means of this process, the
curves at altitude can be more quickly obtained by simple ratios, as follows. Let
the subscript "O" designate sealevel conditions. From Eqs. (6.25) and (6.26),
(6.32)
(6.33)
ELEMENTS OF AIRPLANE PERFORMANCE 283
where V
0
, PR.o, and p
0
are velocity, power, and density at sea level. At altitude,
where the density is p, these relations are
(6.34)
(6.35)
Now, strictly for the purposes of calculation, Let CL remain fixed between sea \
level and altitude. Hence, because CD = CD
0
+ CL
2
/7TeAR, CD also remains
fixed. Dividing Eq. (6.34) by (6.32), and (6.35>' by (6.33), we obtain
(
Po )
1
/
2
V..1t = Vo p
(6.36)
and
p P __Q
(
p )1/2
R,alt  R,0 p
( 6.37)
Geometrically, these equations allow us to plot a point on the PR curve at altitude
from a given point on the sealevel curve. For example, consider point 1 on the
sealevel PR curve sketched in Figure 6.18. By multiplying both the velocity and
power at point 1 by (p
0
/p)
1
1
2
, a new point is obtained, point 2 in Figure 6.18.
Point 2 is guaranteed to fall on the curve at altitude because of our analysis
above. In this fashion, the complete PR curve at altitude can be readily obtained
from the sealevel curve. The results are qualitatively given in Figure 6.19, where
the altitude curves tend to experience an upward and rightward translation as
well as a slight clockwise rotation.
2
Point corresponding to a given CL
PR, alt
Point corresponding
to same CL, but at
PR,
0

1
.. altitude
I I
I I
I I
I I
V
0
v.Jt
Figure 6.18 Correspondence of points on sealevel and altitude powerrequired curves.
284 INTRODUCTION TO FLIGHT
Figure 6.19 Effect of altitude on power required.
With regard to PA, the lower air density at altitude invariably causes a
reduction in power for both the reciprocating and jet engines. In this book we will
assume PA and TA to be proportional to ambient density, as in Example 6.4.
Reasons for this will be made clear in Chap. 9. For the reciprocating engine, the
loss in power can be delayed by using a supercharger. Nevertheless, the impact on
airplane performance due to altitude effects is illustrated in Figures 6.20a and
6.20b for the propeller and jetpowered airplanes, respectively. Both PR and
maximum PA are shown; the solid curves correspond to sea level, and the dashed
curves to altitude. From these curves, note that Vmax varies with altitude. Also
note that at high enough altitude, the lowspeed limit, which is usually dictated by
~ t a l l may instead be determined by maximum PA. This effect is emphasized in
Figure 6.21, where maximum PA has been reduced to the extent that, at velocities
just above stalling, PR exceeds PA For this case, we make the interesting
conclusion that stalling speed cannot be reached in level, steady flight.
To this point in our discussion, only the horizontal velocity perform
anceboth maximum and minimum speeds in steady, level flighthas been
emphasized. We have seen that maximum velocity of an airplane is determined by
the highspeed intersection of the PA and PR curves and that the minimum
velocity is determined either by stalling or by the lowspeed intersection of the
power curves. These velocity considerations are an important part of airplane
performance; indeed, for some airplanes, such as many military fighter planes,
squeezing the maximum velocity out of the aircraft is the pivotal design feature.
However, this is just the beginning of the performance story; we will proceed to
examine other important characteristics in the remaining sections of this chapter.
Example 6.5 Using the method of this section, from the CJ1 powerrequired curve at 22,000 ft
in Example 6.4, obtain the CJ1 powerrequired curve at sea level. Compare the maximum
velocities at both altitudes.
p
ELEMENTS OF AIRPLANE PERFORMANCE 285
I
I
I
I
I
I
I /&
I
I
/.:::
I
: I !>
I/'<.<>;
l
11
I I
I I
/ I
/ I
ta\t1W I
PA:.,_ /I I
/ I " I 
 / 1o
/ IB l..!l
/ 1·z I"'
_ _,,/ l';;j li;l
I • I •
11 1J
I I
(a)
p
(b)
Figure 6.20 Effect of altitude on maximum velocity. (a) Propellerdriven airplane. ( b) Jetpropelled
airplane.
286 INTRODUCTION TO FLIGHT
p
I
I I
I I
I Vmin
I
v,,.11
Voo
Figure 6.21 Situation when
minimum velocity at altitude is
greater than stalling velocity.
SOLUTION From Eqs. (6.36) and (6.37), corresponding points on the powerrequired curves for
sea level and altitude are
( )
1/2
Vo= V.1t :o
and
( )
1/2
hpR.O = hpR.alt :o
We are given V.1t and hpR.alt for 22,000 ft from the CJ1 curve in Example 6.4. Using the above
formulas, we can generate V
0
and hPR.o as in the following table, noting that
(
.!!_)1/2  ( 0.001184 )1/2 
Po  0.002377  0.706
Given Points Generated Points
V.1t•
( :Jl/2
Vo,
ftjs
hPR.alt
ft/s hpR.O
200 889 0.706 141 628
300 741
l
212 523
500 1190 353 840
800 3713 565 2621
1,000 7012 706 4950
These results, along with the hpA curves for sea level and 22,000 ft, are plotted in Figure 6.22.
Looking closely at Figure 6.22, note that point 1 on the hpR curve at 22,000 ft is used to generate
point 2 on the hpR curve at sea level. This illustrates the idea of this section. Also, note that Vmax
at sea level is 975 ftjs = 665 mijh. This is slightly larger than Vmax at 22,000 ft, which is 965
ftjs = 658 mi/h.
ELEMENTS OF AIRPLANE PERFORMANCE '1J.r7
6.8 RA TE OF CLIMB
Visualize a McDonnellDouglas DC9 transport powering itself to takeoff speed
on an airport runway. It gently lifts off at about 160 rni/h, the nose rotates
upward, and the airplane rapidly climbs out of sight. In a matter of minutes, it is
cruising at 30,000 ft. This picture prompts the following questions: How fast can
the airplane climb? How long does it take to reach a certain altitude? The next
two sections provide some answers.
Consider an airplane in steady, unaccelerated, climbing flight, as shown in
Figure 6.23. The velocity along the flight path is V
00
, and the flight path itself is
14
13
12
CJI
11
10
"' I
9
s
x
....
22,000 ft
" 8
0
..
"
0
7
:i::
6
5
4
3
~ 1 1 ~
2
~ I I ~
~ 11 b;
~ 1 1 ~
::..
5
1 I g
11::..
0 200 400 600 800 1000 1200
v ~ , ft/s
Figure 6.22 Altitude effects on Vmax for the CJ1.
288 INTRODUCTION TO FLIGHT
'l>.:s. /
'1,'1' /
//"\
fl
Horizontal l

\
\
\
\
eA
w
Figure 6.23 Airplane in climbing flight.
\
\
\
R/C = V oo sin fl
inclined to the horizontal at the angle 0. As always, lift and drag are perpendicu
lar and parallel to V
00
, and the weight is perpendicular to the horizontal. Thrust T
is assumed to be aligned with the flight path. Here, the physical difference from
our previous discussion on level flight is that T is not only working to overcome
the drag, but for climbing flight it is also supporting a component of weight.
Summing forces parallel to the flight path, we get
T= D + WsinO
and perpendicular to the flight path, we have
L = Wcos()
(6.38)
(6.39)
Note from Eq. (6.39) that the lift is now smaller than the weight. Equations (6.38)
and (6.39) represent the equations of motion for steady, climbing flight and are
analogous to Eqs. (6.10) and (6.11) obtained earlier for steady, horizontal flight.
Multiply Eq. (6.38) by V
00
:
TV DV
00 00
= V sin()
w 00
(6.40)
Examine Eq. (6.40) closely. The righthand side, V
00
sin(), is the airplane's vertical
velocity, as illustrated in Figure 6.23. This vertical velocity is called the rate of
climb R/C:
J R/C = V00 sin() I (6.41)
ELEMENTS OF AIRPLANE PERFORMANCE 289
On the lefthand side of Eq. (6.40), TV
00
is the power available, from Eq. (6.31),
and is represented by the PA curves in Figure 6.16. The second term on the
lefthand side of Eq. (6.40) is DV
00
, which for level flight is the power required, as
represented by the PR curve in Figure 6.11. For climbing flight, however, DV
00
is
no longer precisely the power required, because power must be applied to
overcome a component of weight as well as drag. Nevertheless, for small angles of
climb, say () < 20°, it is reasonable to neglect this fact and to assume that the
DV
00
term in Eq. (6.40) is given by the levelflight PR curve in Figure 6.11. With
this,
/ TV
00
 D V
00
= excess power I ( 6.42)
where the excess power is the difference between power available and power
required, as shown in Figures 6.24a and 6.24b, for propellerdriven and jet
powered aircraft, respectively. Combining Eqs. (6.40) to (6.42), we obtain
R/C = excess power
w
where the excess power is clearly illustrated in Figure 6.24.
( 6.43)
Again, emphasis is made that the PR curves in Figures 6.24a and 6.24b are
taken, for convenience, as those already calculated for level flight. Hence, in
conjunction with these curves, Eq. (6.43) is an approximation to the rate of climb,
good only for small 0. To be more specific, a plot of DV
00
versus V
00
for climbing
flight [which is exactly called for in Eq. (6.40)] is different from a plot of DV
00
versus V
00
for level flight [which is the curve assumed in Figure 6.24 and used in
Eq. (6.43)] simply because Dis smaller for climbing than for level flight at the same
V
00
• To see this more clearly, consider an airplane with W = 5000 lb, S = 100 ft
2
,
CD,o = 0.015, e = 0.6, and AR= 6. If the velocity is V
00
= 500 ftjs at sea level,
and if the airplane is in level flight, then CL = L/ q
00
S = W / ~ p
0
V
0 0
2
S = 0.168.
(a) (b)
Figure 6.24 Illustration of excess power. (a) Propellerdriven airplane. ( b) Jetpropelled airplane.
290 INTRODUCTION TO FLIGHT
In turn
c2
L
CD= CD 0 + AR = 0.015 + 0.0025 = 0.0175
, TTe
Now, consider the same airplane in a 30° climb at sea level, with the same
velocity V
00
= 500 ft/s. Here the lift is smaller than the weight, L = Wcos 8,
and therefore CL = W cos 30° / tp
00
V
00
2
S = 0.145. In turn CD = CD,o +
Cf/TTeAR = 0.015 + 0.0019 = 0.0169. This should be compared with the higher
value of 0.0175 obtained above for level flight. As seen in this example, for steady
climbing flight, L (hence CL) is smaller, and thus induced drag is smaller.
Consequently, total drag for climbing flight is smaller than that for level flight at
the same velocity.
Return again to Figure 6.24, which corresponds to a given altitude. Note that
the excess power is different at different values of V
00
• Indeed, for both the
propeller and jetpowered aircraft there is some V
00
at which the excess power is
maximum. At this point, from Eq. (6.43), R/C will be maximum.
R/C maximum excess power
max = W
(6.44)
This situation is sketched in Figure 6.25a, where the power available is that at full
throttle, i.e., maximum PA. The maximum excess power, shown in Figure 6.25a,
via Eq. (6.44) yields the maximum rate of climb that can be generated by the
airplane at the given altitude. A convenient graphical method of determining
maximum R/C is to plot R/C versus V
00
, as shown in Figure 6.25b. A horizontal
tangent defines the point of maximum R/C. Another useful construction is the
hodograph diagram, which is a plot of the airplane's vertical velocity Vv versus its
horizontal velocity Vh. Such a hodograph is sketched in Figure 6.26. Remember
that R/C is defined as the vertical velocity, R/C = Vv; hence a horizontal
tangent to the hodograph defines the point of maximum R/C (point 1 in Figure
6.26). Also, any line through the origin and intersecting the hodograph (say, at
point 2) has the slope Vv/Vh; hence, from the geometry of the velocity compo
nents, such a line makes the climb angle (} with respect to the horizontal axis, as
shown in Figure 6.26. Moreover, the length of the line is equal to V
00
• As this line
is rotated counterclockwise, R/C first increases, then goes through its maximum,
and then decreases. Finally, the line becomes tangent to the hodograph at point 3.
This tangent line gives the maximum climb angle for which the airplane can
maintain steady flight, shown as (}max in Figure 6.26. It is interesting that
maximum R/C does not occur at maximum climb angle.
The large excess power and high thrust available in modern aircraft allow
climbing flight at virtually any angle. Indeed, modern highperformance military
aircraft (such as the F14, F15, and F16) can accelerate to supersonic speeds
flying straight up! For such large climb angles, the previous analysis is not valid.
Instead, to deal with large 8, the original equations of motion [Eqs. (6.38) and
(6.39)] must be solved algebraically, leading to an exact solution valid for any
value of 0. The details of this approach can be found in the books by Dommasch,
Maximum
excess
power
Vmax R/C
Maximum PA
(a)
Maximum R/C
(b)
ELEMENTS OF AIRPLANE PEllFORMANCE 291
Figure 6.25 Determination of maximum
rate of climb for a given altitude.
Sherbey, and Connolly and by Perkins and Hage (see Bibliography at the end of
this chapter).
Returning briefly to Figures 6.24a and 6.24b for the propellerdriven and
jetpowered aircraft, respectively, an important difference in the lowspeed rate
ofclimb performance can be seen between the two types. Due to the poweravail
able characteristics of a piston enginepropeller combination, large excess powers
are available at low values of V
00
, just above the stall. For an airplane on its
Vv
I
I
I
I
I
/ Maximum R/C
7==
/,
Ve, max V max R/C
Figure 6.26 Hodograph for climb performance at a given altitude.
292 INTRODUCTION TO FLIGHT
landing approach, this gives a comfortable margin of safety in case of a sudden
waveoff (particularly important for landings on aircraft carriers). In contrast, the
excess power available to jet aircraft at low V
00
is small, with a correspondingly
reduced rateofclimb capability.
Figures 6.25b and 6.26 give R/C at a given altitude. In the next section we
will ask how R/C varies with altitude. In pursuit of an answer, we will also find
the answer to another question, namely, how high can the airplane fly.
Example 6.6 Calculate the rate of climb vs. velocity at sea level for (a) the CP1 and (b) the CJ1.
SOLUTION
(a) For the CP1, from Eq. (6.43)
R/C =excess power= PA  PR
w w
With power in footpounds per second and W in pounds, for the CP1, this equation becomes
From Example 6.3, at V
00
=150 ftjs, PR= 0.326x10
5
ft· lb/s. From Example 6.4, PA=
550(hpA) = 550(184) = 1.012x10
5
ft· lb/s. Hence,
R/c
 (1.0120.326)x10
5
_ f
 2950  23.3 t/s
In terms of feet per minute,
R/C = 23.3(60) = 11395 ft/min I at V
00
=150 ftjs
This calculation can be repeated at different velocities, with the following results:
Voo, R/C,
ft/s ft/min
100 1492
130 1472
180 1189
220 729
260 32.6
These results are plotted in Figure 6.27.
(b) For the CJ1, from Eq. (6.43),
PA  PR 550(hpA  hpR)
R/C=w= 19,815
1600
1400
1200
1000
.E
_§_
<!::
800
~
I>::
600
400
200
0 50
CP1
sea level
100 150
V . ft/s
Figure 6.27 Sealevel rate of climb for the CP1.
10,000
8,000
:::
6,000
i
<!::
u
~
4,000
2,000
0 200 400
CJI
600
v ~ ft/s
Figure 6.28 Sealevel rate of climb for the CJ1.
200 250 300
l:iOO 1000
293
294 INTRODUCTION TO FLIGHT
From the results and curves of Example 6.5, at V
00
= 500 ftjs, hpR = 1884, and hp A = 6636.
Hence
66361884
R/C = 550
19 815
=132 ftjs
or R/C = 132( 60) = 7914 ft/min at V
00
= 500 ft/s
Again, a short tabulation for other velocities is given below for the reader to check.
Voo, R/C,
ftjs ft/min
200 3546
400 7031
600 8088
800 5792
950 1230
These results are plotted in Figure 6.28.
6.9 GLIDING FLIGHT
Consider an airplane in a poweroff glide, as sketched in Figure 6.29. The forces
acting on this aircraft are lift, drag, and weight; the thrust is zero because the
power is off. The glide flight path makes an angle () below the horizontal. For an
equilibrium unaccelerated glide the sum of the forces must be zero. Summing
Horizontal
T
w
Figure 6.29 Airplane in poweroff gliding flight.
_ /
ELEMENTS OF AIRPLANE PERFORMANCE 295
forces along the flight path, we have
D = WsinO
and perpendicular to the flight path
(6.45)
L = WcosO (6.46)
The equilibrium glide angle can be calculated by dividing Eq. (6.45) by (6.46),
yielding
sinO D
=
cos 0 L
or
D
(6.47)
Clearly, the glide angle is strictly a function of the lifttodrag ratio; the higher
the L/ D, the shallower the glide angle. From this, the smallest equilibrium glide
angle occurs at (L/D)max' which corresponds to the maximum range for the
glide.
Example 6.7 The maximum lifttodrag ratio for the CP1 is 13.6. Calculate the minimum glide
angle and the maximum range measured along the ground covered by the CP1 in a poweroff
glide that starts at an altitude of 10,000 ft.
SOLUTION The minimum glide angle is obtained from Eq. (6.47) as
1 1
tan8min = (L/D)max 13.6
I nmin = 4.2° I
The distance covered along the ground is R, as shown in Figure 6.30. If h is the altitude at the
Figure 6.30 Range covered in an equilibrium glide.
296 INTRODUCTION TO FLIGHT
start of the glide, then
h L
R=  =h
tanO D
Hence, Rmax = h(!:_) = 1000(13.6)
D max
I Rmax = 136,000 ft = 25.6 mi I
Example 6.8 Repeat Example 6.7 for the CJ1, for which the value of (L/D)max is 16.9.
SOLUTION
1
tanOmin = 
( L/D)max
I (}min= 3.39° I
1
16.9
Rmax = h(!:_) = 10,000(16.9)
D max
I Rmax = 169,000 ft = 32 mi I
Note the obvious fact that the CJ1, with its higher value of (L/D)max• is capable of a larger
glide range than the CP1.
Example 6.9 For the CP1, calculate the equilibrium glide velocities at altitudes of 10,000 and
2000 ft, each corresponding to the minimum glide angle.
SOLUTION
Combining this with Eq. (6.46) gives
or
WcosO = }pooVoo
2
SCL
v: =
00
2cos0 W
where W /Sis defined as the wing loading, more fully described in Sec. 6.17. From this equation
we see that the higher the wing loading, the higher the glide velocity. This makes sense: a heavier
airplane with a smaller wing area is going to glide to the earth's surface at a greater velocity.
Note, however, that the glide angle, and hence range, depends not on the weight of the airplane
or its wing loading but exclusively on the value of (L/D)m.., which is an aerodynamic property
of the airframe design. A higher wing loading simply means that the airplane will have a faster
glide and will reach the earth's surface sooner. From Example 6.1 we have for the CP1,
w 2950
 =  = 16.95 lb/ft
2
s 174
Also from the tabulation in Example 6.1, we see that (L/D)max = 13.6 corresponds to a lift
coefficient CL= 0.634. (Note that both L/D and CL are functions of angle of attack of the
ELEMENTS OF AIRPLANE PERFORMANCE 297
airplane; these are aerodynamic data associated with the airframe and are not influenced by the
flight conditions. Hence, CL= 0.634 at maximum L/D, no matter whether the airplane is in
level flight, climbing, or in a glide.) Therefore, at 10,000 ft, where p
00
= 0.0017556 slug/ft
3
, we
have
v. =
00
(2 cos 4.2°)(16.95)
0.0017556(0.634)
I voo = 174.3 ftjs I
at h = 10,000 ft
At 2000 ft, p
00
= 0.0022409 slug/ft
3
. Hence
v. =
00
(2 cos 4.2°)(16.95)
0.0022409(0.634)
I voo = 154.3 ftjs I
at h = 2000 ft
Note that the equilibrium glide velocity decreases as altitude decreases.
6.10 ABSOLUTE AND SERVICE CEILINGS
The effects of altitude on PA and PR were discussed in Sec. 6.7 and illustrated in
Figures 6.20a and 6.20b. For the sake of discussion, consider a propellerdriven
airplane: the results of this section will be qualitatively the same for a jet. As
altitude increases, the maximum excess power decreases, as shown in Figure 6.31.
In turn, maximum R/C decreases. This is illustrated by Figure 6.32, which is a
plot of maximum R/C versus altitude, but with R/C as the abscissa.
There is some altitude high enough at which the PA curve becomes tangent to
the PR curve (point 1 in Figure 6.33). The velocity at this point is the only value at
which steady, level flight is possible; moreover, there is zero excess power, hence
+
Maximum
/
/
;:
/
..,,. .... _ ....
Maximum excess
power at altitude
Figure 6.31 Variation of excess power with altitude.
Voo
298 INTRODUCTION TO FLIGHT
32
28
Absolute ceiling
24
"' I
20
s
x
.:::
,; 16
'O
·E
<
12
8 CP1
4
0 0.4 0.8 1.2 1.6
Maximum R/C, ft/min X I Q3
Figure 6.32 Determination of absolute and service ceilings for the CP1.
zero maximum rate of climb, at this point. The altitude at which maximum
R/C = 0 is defined as the absolute ceiling of the airplane. A more useful quantity
is the service ceiling, defined as that altitude where the maximum R/C = 100
ftjmin. The service ceiling represents the practical upper limit of steady, level
flight.
The absolute and service ceilings can be determined as follows:
1. Using the technique of Sec. 6.8, calculate values of maximum R/C for a
number of different altitudes.
2. Plot maximum rate of climb vs. altitude, as shown in Figure 6.32.
3. Extrapolate the curve to 100 ft/min and 0 ftjmin to find the service and
absolute ceilings, respectively, as also shown in Figure 6.32.
Figure 6.33 Powerrequired and power
V oo available curves at the absolute ceiling.
ELEMENTS OF AIRPLANE PERFORMANCE 299
Example 6.10 Calculate the absolute and service ceilings for (a) the CP1 and (b) the CJ1.
SOLUTION
(a) For the CP1, as stated in Example 6.1, all the results presented in all the examples of
this chapter are taken from a computer program which deals with 100 different velocities, each
at different altitudes, beginning at sea level and increasing in 2000ft increments. In modem
engineering, using the computer to take the drudgery out of extensive and repeated calculations
is an everyday practice. For example, note from Example 6.6 that the maximum rate of climb at
sea level for the CP1 is 1500 ft/min. In essence, this result is the product of all the work
performed in Examples 6.1 to 6.6. Now, to obtain the absolute and service ceilings, these
calculations must be repeated at several different altitudes in order to find where R/C = 0 and
100 ft/min, respectively. Some results are tabulated and plotted below; the reader should take
the time to check a few of the numbers.
Altitude, Maximum R/C,
ft ft/min
0 1500
4,000 1234
8,000 987
12,000 755
16,000 537
20,000 331
24,000 135
26,000 40
These results are plotted in Figure 6.32. From these numbers, we find:
Absolute ceiling (R/C = 0) is 127,000 ft I
Service ceiling (R/C = 100 ft/min) is j 25,000 ft I
(b) For the CJ1, utilizing the results from Examples 6.1 to 6.6 and making similar
calculations at various altitudes, we tabulate the following results:
Altitude, Maximum R/C,
ft ft/min
0 8118
6,000 6699
12,000 5448
18,000 4344
24,000 3369
30,000 2502
36,000 1718
These results are plotted in Figure 6.34.
300 INTRODUCTION TO FLIGHT
50
45
40
35
CJI
I
:2
30
x
¢::
,,; 25
'O
.s
:;;:
20
15
10
5
0 2 4 6 8 10
Maximum R/C, ft/min X JQ3
Figure 6.34 Determination of absolute and service ceilings for the CJ1.
From these results, we find:
Absolute ceiling (R/C = 0) is 149,000 ft I
Service ceiling (R/C = 100 ftjmin) is 148,000 ft I
6.11 TIME TO CLIMB
To carry out its defensive role adequately, a fighter airplane must be able to climb
from sea level to the altitude of advancing enemy aircraft in as short a time as
possible. In another case, a commercial aircraft must be able to rapidly climb to
high altitudes to minimize the discomfort and risks of inclement weather and to
minimize air traffic problems. As a result, the time for an airplane to climb to a
given altitude can become an important design consideration. The calculation of
the time to climb follows directly from our previous discussions, as described
below.
The rate of climb was defined in Sec. 6.8 as the vertical velocity of the
airplane. Velocity is simply the time rate of change of distance, the distance here
ELEMENTS OF AIRPLANE PERFORMANCE 301
being the altitude h. Hence, R/C = dh/ dt. Therefore,
dh
dt = R/C
( 6.48)
In Eq. (6.48), dt is the small increment in time required to climb a small
increment dh in altitude. Therefore, from calculus, the time to climb from one
altitude, h
1
, to another, h
2
, is obtained by integrating Eq. (6.48):
1h2 __!!!!_
t R/C
h,
Normally, time to climb is considered from sea level, where h
1
= 0. Hence, the
time to climb to any given altitude h
2
is
t = 1h2 _:!!!____
o R/C (
6
.4
9
)
To calculate t graphically, first plot (R/q
1
versus h, as shown in Figure
6.35. The area under the curve from h = 0 to h = h
2
is the time to climb to
altitude h
2
•
0
x
5::_
s
" E
:§
E
I
G
Example 6.11 Calculate and compare the time required for (a) the CP1 and (b) the CJ1 to
climb to 20,000 ft.
SOLUTION
(a) For the CP1, from Eq. (6.49), the time to climb is equal to the shaded area under the
curve shown in Figure 6.35. The resulting area gives time to climb as \27.0min.\
(b) For the CJ1, Eq. (6.49) is plotted in Figure 6.36. The resulting area gives time to climb
as\ 3.5min. \
3.2
2.8
2.4
2.0
1.6
1.2
0.8
0.4
Altitude h, ft X I 03
Figure 6.35 Determination of time to climb
for the CP1.
302 INTRODUCTION TO FLIGHT
0
x
:::,
i
" E
·=
·a
I
c
0.4
0.2
0 4
CJI
8 16
Altitude h, ft X J03
20 24
Figure 6.36 Determination of time to climb
for the CJ1.
Note that the CJ1 climbs to 20,000 ft in oneeighth of the time required by the CP1: this is
to be expected for a highperformance executive jet transport in comparison to its propellerdriven
piston engine counterpart.
6.12 RANGE AND ENDURANCEPROPELLERDRIVEN
AIRPLANE
When Charles Lindbergh made his spectacular solo flight across the Atlantic
Ocean on May 2021, 1927, he could not have cared less about maximum
velocity, rate of climb, or time to climb. Uppermost in his mind was the
maximum distance he could fly on the fuel supply carried by the Spirit of St.
Louis. Therefore, range was the allpervasive consideration during the design and
construction of Lindbergh's airplane. Indeed, throughout all of twentiethcentury
aviation, range has been an important design feature, especially for transcon
tinental and transoceanic transports and for strategic bombers for the military.
Range is technically defined as the total distance (measured with respect to
the ground) traversed by the airplane on a tank of fuel. A related quantity is
endurance, which is defined as the total time that an airplane stays in the air on a
tank of fuel. In different applications, it may be desirable to maximize one or the
other of these characteristics. The parameters which maximize range are different
from those which maximize endurance; they are also different for propeller and
jetpowered aircraft. The purpose of this section is to discuss these variations for
the case of a propellerdriven airplane; jet airplanes will be considered in the next
section.
A Physical Considerations
One of the critical factors influencing range and endurance is the specific fuel
consumption, a characteristic of the engine. For a reciprocating engine, specific
fuel consumption (commonly abbreviated as SFC) is defined as the weight of fuel
ELEMENTS OF AIRPLANE PERFORMANCE 303
consumed per unit power per unit time. As mentioned earlier, reciprocating engines
are rated in terms of horsepower, and the common units (although inconsistent)
of specific fuel consumption are
SFC = lb of fuel
(bhp )(h)
where bhp signifies shaft brake horsepower, discussed in Sec. 6.6.
First, consider endurance. On a qualitative basis, in order to stay in the air
for the longest period of time, common sense says that we must use the minimum
number of pounds of fuel per hour. On a dimensional basis this quantity 1s
proportional to the horsepower required by the airplane and to the SFC:
Therefore, minimum pounds of fuel per hour is obtained with minimum hpR.
Since minimum pounds of fuel per hour gives maximum endurance, we quickly
conclude that
Maximum endurance for a propellerdriven airplane occurs when the airplane is
flying at minimum power required.
This condition is sketched in Figure 6.37. Furthermore, in Sec. 6.5 we have
already proven that minimum power required corresponds to a maximum value
of C]/
2
/Cv [see Eq. (6.26)]. Thus
Maximum endurance for a propellerdriven airplane occurs when the airplane is
flying at a velocity such that CL
3
1
2
/C
0
is maximum .
. : : : : . . 9 ~
V for
maximum
endurance
I
I
I
I
I
I
V for
maximum
range
v 00
Figure 6.37 Points of maximum
range and endurance on the power
required curve for a propellerdriven
airplane.
304 INTRODUCTION TO FLIGHT
Now, consider range. In order to cover the longest distance (say, in miles),
common sense says that we must use the minimum number of pounds of fuel per
mile. On a dimensional basis, we can state the proportionality
lb of fuel (SFC)(hpR)
 a: '"'''
mi V
00
(Check the units yourself, assuming V
00
is in miles per hour.) As a result,
minimum pounds of fuel per mile is obtained with a minimum hpR/V
00
• This
minimum value of hpR/V
00
precisely corresponds to the tangent point in Figure
6.13, which also corresponds to maximum L/D, as proved in Sec. 6.5. Thus
Maximum range for a propellerdriven airplane occurs when the airplane is
flying at a velocity such that CL/Cn is maximum.
This condition is also sketched in Figure 6.37.
B Quantitative Formulation
The important conclusions written above in italics were obtained from purely
physical reasoning. We will develop quantitative formulas which substantiate
these conclusions and which allow the direct calculation of range and endurance
for given conditions.
In this development, the specific fuel consumption is couched in units that are
consistent, i.e.,
lb of fuel
(ft· lb/s)(s)
or
N of fuel
(J/s)(s)
For convenience and clarification, c will designate the specific fuel consumption
with consistent units.
Consider the product cP dt, where P is engine power and dt is a small
increment in time. The units of this product are (in the English engineering
system)
lb of fuel ft · lb
cPdt = (f )() (s) =lb of fuel
t · lb/s s s
Therefore, cP dt represents the differential change in the weight of the fuel due to
consumption over the short time period dt. The total weight of the airplane, W, is
the sum of the fixed structural and payload weights, along with the changing fuel
weight. Hence, any change in W is assumed to be due to the change in fuel
weight. Recall that W denotes the weight of the airplane at any instant. Also, let
W
0
= gross weight of the airplane (weight with full fuel and payload), W( = weight
of the fuel load, and W
1
= weight of the airplane without fuel. With these
ELEMENTS OF AIRPLANE PERFORMANCE 305
considerations, we have
and
or
W1=WoUJ
dUf = dW =  cP dt
dW
dt= 
cP
(6.50)
The minus sign in Eq. (6.50) is necessary because dt is physically positive (time
cannot move backward, except in science fiction novels), while at the same
time Wis decreasing (hence dW is negative). Integrating Eq. (6.50) between time
t = 0, where W = W
0
(fuel tanks full), and time t = E, where W = W
1
(fuel
tanks empty), we find
(6.51)
In Eq. (6.51), E is the endurance in seconds.
To obtain an analogous expression for range, multiply Eq. (6.50) by V
00
:
V
00
dW
v dt =   (6.52)
00
cP
In Eq. (6.52), V
00
dt is the incremental distance ds covered in time dt.
V
00
dW
ds= 
cP
(6.53)
The total distance covered throughout the flight is equal to the integral of Eq.
(6.53) from s = 0, where W = W
0
(full fuel tank), to s = R, where W = W
1
(empty fuel tank):
or (6.54)
In Eq. (6.54), R is the range in consistent units, such as feet or meters.
Equations (6.51) and (6.54) can be evaluated graphically, as shown in Figure
6.38a and 6.38b for range and endurance, respectively. Range can be calculated
accurately by plotting V
00
/ cP versus W and taking the area under the curve from
W
1
to W
0
, as shown in Figure 6.38a. Analogously, the endurance can be
calculated accurately by plotting ( cP) 
1
versus W and taking the area under the
curve from W
1
to W
0
, as shown in Figure 6.38b.
306 INTRODUCTION TO FLIGHT
v_
cP
(a)
w
I
cP
Figure 6.38 Determination of range and endurance.
w
(b)
Equations (6.51) and (6.54) are accurate formulations for endurance and
range. In principle, they can include the entire flighttakeoff, climb, cruise, and
landingas long as the instantaneous values of W, V
00
, c, and P are known at
each point along the flight path. However, Eqs. (6.51) and (6.54), although
accurate, are also long and tedious to evaluate by the method discussed above.
Therefore, simpler but approximate analytic expressions for R and E are useful.
Such formulas are developed below.
C Breguet Formulas (PropellerDriven Airplane)
For level, unaccelerated flight, we demonstrated in Sec. 6.5 that PR = DV
00
• Also,
to maintain steady conditions, the pilot has adjusted the throttle such that power
available from the enginepropeller combination is just equal to the power
required: PA = PR= DV
00
• In Eq. (6.50), P is the brake power output of the
engine itself. Recall from Eq. (6.30) that PA = 71P, where 71 is the propeller
efficiency. Thus
PA DVOO
P==
1/ 1/
(6.55}
Substitute Eq. (6.55) into (6.54):
R = lwo V
00
dW = 1% V
00
71 dW = 1Wo71 dW (
6
.
56
)
cP cDV
00
cD
W
1
W
1
W
1
Multiplying Eq. (6.56) by W/W and noting that for steady, level flight, W = L,
we obtain
(6.57}
Unlike Eq. (6.54), which is exact, Eq. (6.57) now contains the direct assumption
ELEMENTS OF AIRPLANE PERFORMANCE 307
of level, unaccelerated flight. However, for practical use, it will be further
simplified by assuming that 'IJ, L/D = CJCn, and c are constant throughout
the flight. This is a reasonable approximation for cruising flight conditions. Thus,
Eq. (6.57) becomes
(6.58)
Equation (6.58) is a classic formula in aeronautical engineering; it is called the
Breguet range formula, and it gives a quick, practical estimate for range which is
generally accurate to within 10 to 20 percent. Also, keep in mind that, as with all
proper physical derivations, Eq. (6.58) deals with consistent units. Hence, R is in
feet or meters when c is in consumption of fuel in lb/(ft · lb/s)(s) or N/(J/s)(s),
respectively, as discussed in part B above. If c is given in terms of brake
horsepower and if R is desired in miles, the proper conversions to consistent
units should be made before using Eq. (6.58).
Look at Eq. (6.58). It says all the things that common sense would expect,
namely, to maximize range for a reciprocatingengine, propellerdriven airplane,
we want:
1. The largest possible propeller efficiency 'IJ.
2. The lowest possible specific fuel consumption c.
3. The highest ratio of W
0
/W
1
, which is obtained with the largest fuel weight
WF
4. Most importantly, flight at maximum L/ D. This confirms our above argu
ment in part A that, for maximum range, we must fly at maximum L/ D.
Indeed, the Breguet range formula shows that range is directly proportional
to L/ D. This clearly explains why high values of L/ D (high aerodynamic
efficiency) have always been of importance in the design of airplanes. This
importance was underscored in the 1970s due to the increasing awareness of
the need to conserve energy (hence fuel).
A similar formula can be obtained for endurance. Recalling that P = DV
00
/11
and that W = L, Eq. (6.51) becomes
E = Jwo dW = 1Wo !l. dW = 1Wo !l. ___!:::_ dW
cP c DV
00
c DV
00
W
W1 W
1
W
1
Since L = W= ip
00
V
00
2
SCv then V
00
= ,,j2W/p
00
SCL.
E = 1Wo21_ CL/ p00 SCL dW
c cn 2 w312
W1
Thus
308 .INTRODUCTION TO FLIGHT
Assuming that Cv CD, TJ, c, and p
00
(constant altitude) are all constant, this
equation becomes
E= 2!1 CL3/2(pooS)1;2[w112]wo
c CD 2 W1
or
c 3/2
E = '!!._ _L_(
2
S)1;2(W 1;2 _ W, 1;2)
c C Poo 1 o
D
(6.59)
Equation (6.59) is the Breguet endurance formula, where E is in seconds (con
sistent units).
Look at Eq. (6.59). It says that to maximize endurance for a reciprocating
engine, propellerdriven airplane, we want:
1. The highest propeller efficiency TJ.
2. The lowest specific fuel consumption c.
3. The highest fuel weight Uf, where W
0
= W
1
+ Uf·
4. Flight at maximum CL
312
/CD. This confirms our above argument in part A
that, for maximum endurance, we must fly at maximum CL
312
/CD.
5. Flight at sea level, because E a: p
00
112
, and p
00
is largest at sea level.
It is interesting to note that, subject to our approximations, endurance depends
on altitude, whereas range [see Eq. (6.59)) is independent of altitude.
Remember that the discussion in this section pertains only to a combination
of piston engine and propeller. For a jetpowered airplane, the picture changes,
as will be discussed in the next section.
Example 6.12 Estimate the maximum range and maximum endurance for the CP1.
SOLUTION The Breguet range formula is given by Eq. (6.58) for a propellerdriven airplane. This
equation is
T/ CL Wo
R=ln
c Cv W1
with the specific fuel consumption c in consistent units, say (lb fuel)/(ft · lb/s)(s) or simply per
foot. However, in Example 6.1, the SFC is given as 0.45 lb of fuelj(hp)(h). This can be changed
to consistent units, as
lb lhp lh
c = 0.45 
(hp}(h) 550 ft· lb/s 3600 s
= 2.27 X 10
7
ftI
In Example 6.1, the variation of CL/Cv = L/D was calculated versus velocity. The variation of
CL
312
/Cv can be obtained in the same fashion. The results are plotted in Figure 6.39.
From these curves,
max ( ~ : = 13.62
(
c 3/2)
max ~ D = 12.81
These are results pertaining to the aerodynamics of the airplane; even though the above plots
14
12
10
8
6
4
2
0
CJI
100
ELEMENTS OF AIRPLANE PERFORMANCE 309
     Maximum CL /CD
    Maximum CL
312
/CD
200 300
V
00
, ft/s
400
Figure 6.39 Aerodynamic ratios for the CP1
at sea level.
were calculated at sea level (from Example 6.1), the maximum values of CL/CD and C1.
3
1
2
/CD
are independent of altitude, velocity, etc. They depend only on the aerodynamic design of the
aircraft.
The gross weight of the CP1 is W
0
= 2950 lb. The fuel capacity given in Example 6.1 is 65
gallons of aviation gasoline, which weighs 5.64 lb/gallon. Hence, the weight of the fuel,
~ = 65(5.64) = 367 lb. Thus, the empty weight W
1
= 2950 367 = 2583 lb.
Returning to Eq. (6.58)
R = .!!_ CL In W
0
= 0.8 (l3.
62
) In 2950
c CD W1 2.21x10
1
2583
IR= 6.38Xl0
6
ft I
Since 1 mi = 5280 ft, then
R =
6
·
3
:
2
~ ~
6
= J 1207 mi I
The endurance is given by Eq. (6.59):
c 3/2
E= .!!_ _L_(2 S)1;2( W, 1/2  w,1/2)
C CD Poo 1 0
Because of the explicit appearance of p
00
in the endurance equation, maximum endurance will
310 INTRODUCTION TO FLIGHT
occur at sea level, p
00
= 0.002377 slug/ft
3
. Hence,
E = 0.
8
(12.81 )[2(0.002377)(174) ]
112
(
1
  
1
)
2.27x10
7
2583
1
1
2
2950
1
1
2
I £=5.19Xl0
4
sl
Since 3600 s = 1 h,
E
= 5.19xl04 = ~ h
3600 ~
6.13 RANGE AND ENDURANCEJET AIRPLANE
For a jet airplane, the specific fuel consumption is defined as the weight of fuel
consumed per unit thrust per unit time. Note that thrust is used here, in contradis
tinction to power, as in the previous case for a reciprocatingenginepropeller
combination. The fuel consumption of a jet engine physically depends on the
thrust produced by the engine, whereas the fuel consumption of a reciprocating
engine physically depends on the brake power produced. It is this simple
difference which leads to different range and endurance formulas for a jet
airplane. In the literature, thrustspecific fuel consumption (TSFC) for jet engines
is commonly given as
TSFC = lb of fuel
(lb of thrust)(h)
(Note the inconsistent unit of time.)
A Physical Considerations
The maximum endurance of a jet airplane occurs for minimum pounds of fuel per
hour, the same as for propellerdriven aircraft. However, for a jet,
lb o ~ fuel = (TSFC) TA
where TA is the thrust available produced by the jet engine. Recall that in steady,
level, unaccelerated flight, the pilot has adjusted the throttle such that thrust
available TA just equals the thrust required TR: TA = TR. Therefore, minimum
pounds of fuel per hour corresponds to minimum thrust required. Hence, we
conclude that
Maximum endurance for a jet airplane occurs when the airplane is flying at
minimum thrust required.
This condition is sketched in Figure 6.40. Furthermore, in Sec. 6.3, minimum
V for
maximum
endurance
V for
maximum
range
ELEMENTS OF AIRPLANE PERFORMANCE 311
Figure 6.40 Points of maximum range and endurance on the thrustrequired curve.
thrust required was shown to correspond to maximum L/ D. Thus
Maximum endurance for a jet airplane occurs when the airplane is flying at a
velocity such that CL/Cv is maximum.
Now, consider range. As before, maximum range occurs for a minimum
pounds of fuel per mile. For a jet, on a dimensional basis,
lb of fuel (TSFC) TA
mi
Recalling that for steady, level flight, TA = TR, we note that minimum pounds of
fuel per mile corresponds to a minimum TR/V
00
• In turn, TR/V
00
is the slope of a
line through the origin and intersecting the thrustrequired curve; its minimum
value occurs when the line becomes tangent to the thrustrequired curve, as
sketched in Figure 6.40. The aerodynamic condition holding at this tangent point
is obtained as follows. Recalling that for steady, level flight, TR = D, then
1
C i;2/C
L D
Hence, minimum TR/V
00
corresponds to maximum CzY
1
/Cv. In turn, we con
clude that
Maximum range for a jet airplane occurs when the airplane is flying at a velocity
such that CL
1
1
1
/Cv is maximum.
312 INTRODUCTION TO FLIGHT
B Quantitative Formulation
Let c
1
be the thrustspecific fuel consumption in consistent units, e.g.,
lb of fuel N of fuel
(lb of thrust) ( s)
or
(N of thrust)(s)
Let dW be the elemental change in weight of the airplane due to fuel consump
tion over a time increment dt. Then
or
(6.60)
Integrating Eq. (6.60) between t = 0, where W = W
0
, and t = E, where W = W
1
,
we obtain
Recalling that TA = TR = D and W = L, we have
E = 1Wo 2_ L dW
c
1
D W
W1
(6.61)
(6.62)
With the assumption of constant c
1
and CJCn = L/D, Eq. (6.62) becomes
(6.63)
Note from Eq. (6.63) that for maximum endurance for a jet airplane, we want:
1. Minimum thrustspecific fuel consumption c
1
•
2. Maximum fuel weight Uf·
3. Flight at maximum L/ D. This confirms our above argument in part A that,
for maximum endurance for a jet, we must fly such that L/ D is maximum.
Note that, subject to our assumptions, E for a jet does not depend on pOO' that is,
it is independent of altitude.
Now consider range. Returning to Eq. (6.60) and multiplying by VOO' we get
V
00
dW
ds = V dt =   (6.64)
oo ct TA
where ds is the increment in distance traversed by the jet over the time increment
dt. Integrating Eq. (6.64) from s = 0, where W = W
0
, to s = R, where W = W
1
,
ELEMENTS OF AIRPLANE PERFORMANCE 313
we have
R = [R ds = 1w1 Voo dW
c1TA
0 Wo
(6.65)
However, again noting that for steady, level flight, the engine throttle has been
adjusted such that TA= TR and recalling from Eq. (6.15) that TR= W/(CL/CD),
we rewrite Eq. (6.65) as
(6.66)
Since V
00
= J2W/p
00
SCL, Eq. (6.66) becomes
R = 1Wof£ CLl/2/CD dW
P
s c w112
JfJ. oo I
(6.67)
Again, assuming constant cl' Cv CD, and p
00
(constant altitude), we rewrite Eq.
(6.67) as
R = [£ CLl/2 2._ f Wo dW
P
s c c w
1
1
2
00 D I W1
R = 2  _L_ ( Wol/2  wl 1/2)
[£
1c1/2 .
c1 CD
(6.68)
Note from Eq. (6.68) that to obtain maximum range for a jet airplane, we want:
1 Minimum thrustspecific fuel consumption c
1
•
2. Maximum fuel weight
3. Flight at maximum CL
1
/CD. This confirms our above argument in part A
that for maximum range, a jet must fly at a velocity such that CL
112
/CD is
maximum.
4. Flight at high altitudes, i.e., low Pw Of course, Eq. (6.68) says that R
becomes infinite as p
00
decreases to zero, i.e., as we approach outer space.
This is physically ridiculous, however, because an airplane requires the
atmosphere to generate lift and thrust. Long before outer space is reached,
the assumptions behind Eq. (6.68) break down. Moreover, at extremely high
altitudes, ordinary turbojet performance deteriorates and c
1
begins to in
crease. All we can conclude from Eq. (6.68) is that range for a jet is poorest
at sea level and increases with altitude, up to a point. Typical cruising
altitudes for subsonic commercial jet transports are from 30,000 to 40,000 ft;
for supersonic transports they are from 50,000 to 60,000 ft.
314 INTRODUCTION TO FLIGHT
16
4
0
Example 6.13 Estimate the maximum range and endurance for the CJ1.
SOLUTION From the calculations of Example 6.1, the variation of CL/C
0
and CL
1
/
2
/C
0
can
be plotted vs. velocity, as given in Figure 6.41. From these curves, for the CJ1.
In Example 6.1, the specific fuel consumption is given as TSFC = 0.6 (lb fuel)/(lb
thrust)(h). In consistent units,
lb 1 h
c, =
0
·
6
(lb )(h) 3600 s
Also, the gross weight is 19,815 lb. The fuel capacity is 1119 gallons of kerosene, where 1 gallon
of kerosene weighs 6.67 lb. Thus, HJ= 1119(6.67) = 7463 lb. Hence, the empty weight is
W
1
= W
0
 HJ =19,8157463 =12,352 lb.
The range of a jet depends on altitude, as shown by Eq. (6.68). Assume the cruising altitude
is 22,000 ft, where p
00
= 0.00184 slug/ft
3
. From Eq. (6.68), using information from Example 6.1,
200 400 600
V
00
, ft/s
800
Figure 6.41 Aerodynamic ratios for the CJ1 at sea
level.
ELEMENTS OF AIRPLANE PERFORMANCE 315
we obtain
=2
2
(
1
) (23 4)(19 815
1
1
2
12 352
1
1
2
)
(0.001184)(318) 1.667x10
4
. • •
IR =19.2X10
6
ft I
In miles
R = 19.2x10
6
~
5280 ~
The endurance can be found from Eq. (6.63):
or in hours
(16 9)ln 19,815
1.667x10
4
. 12,352
I£= 4.79Xl0
4
s I
E= 4.79X10
4
~
3600 ~
6.14 RELATIONS BETWEEN CD
0
AND CD;
, ,
In the previous sections, we have observed that various aspects of the perform
ance of different types of airplanes depend on the aerodynamic ratios CL
1
1
2
;cD,
CL/CD, or CL
3
1
2
/CD. Moreover, in Sec. 6.3, we proved that at minimum TR, drag
due to lift equals parasite drag, that is, CD
0
= CD;· Analogously, for minimum
PR, we proved in Sec. 6.5 that CD,o = tcD,';. In this section, such results will be
obtained strictly from aerodynamic considerations. The relations between CD
0
and CD.i depend purely on the conditions for maximum CL
1
1
2
/CD, CdCD, ~ r
CL
3
1
2
/CD; their derivations do not have to be associated with minimum TR or PR,
as they were in Secs. 6.3 and 6.5.
For example, consider maximum L/D. Recalling that CD= CD.o +
CL
2
/?TeAR, we can write
2
CD,o + CL j?TeAR
(6.69)
For maximum DdCD, differentiate Eq. (6.69) with respect to CL and set the
316 INTRODUCTION TO FLIGHT
result equal to zero:
Thus,
or
d( CL/CD) CD,o + CL
2
/7TeAR  CL(2CLf7TeAR)
= =0
dCL ( cD,O + CL2/7TeAR)2
c2
C = __ L_
D,o 7TeAR
I CD,O = CD,i I for ( L
D max
(6.70)
Hence, Eq. (6.70), which is identical to Eq. (6.21), simply stems from the fact that
L/ D is maximum. The fact that it also corresponds to minimum TR is only
because TR happens to be minimum when L/D is maximum.
Now consider maximum CL
312
/Cv. By setting d(CL
312
)/dC
1
= 0, a deriva
tion similar to that above yields
J Cv,o = }Cv,; J
(
CL
3
/
2
)
for 
C
D max
(6.71)
Again, Eq. (6.71), which is identical to Eq. (6.29), simply stems from the fact that
CL
312
/CD is maximum. The fact that it also corresponds to minimum PR is only
because PR happens to be minimum when CL
312
/CD is maximum.
Similarly, when CL
112
/CD is maximum, setting d(CL
112
)/dCL = 0 yields
(
CL
1
/
2
)
for 
C
D max
I Cv,o = 3Cv,; I
(6.72)
You should not take Eqs. (6.71) and (6.72) for granted; derive them yourself.
We stated in Example 6.12 that the maximum values of CL
112
/CD, CLfCD,
and CL
312
/CD are independent of altitude, velocity, etc.; rather, they depend
only on the aerodynamic design of the aircraft. The results of this section allow
us to prove this statement, as follows.
First, consider again the case of maximum CLfCD. From Eq. (6.70),
Thus,
CL2
CD 0 =CD i = AR
, , 'TTe
Substituting Eqs. (6.73) and (6.74) into Eq. (6.69), we obtain
CL 7TeAR 7TeAR
  o   ~ =  ~ = ;=======
2CL
2
/7TeAR 2CL 2J7TeARCv,o
(6.73)
(6.74)
(6.75)
ELEMENTS OF AIRPLANE PERFORMANCE 317
Hence, the value of the maximum C£1Cv is obtained from Eq. (6.75) as
( Cv,
0
'1teAR)
112
2Cv,o
(6.76)
Note from Eq. (6.76) that (C£1Cv)max depends only one, AR, and Cv
0
, which
are aerodynamic design parameters of the airplane. In particular, (
does not depend on altitude. However, note from Figures 6.39 and 6.41 that
maximum C£1Cv occurs at a certain velocity, and the velocity at which
(C£1Cv)max is obtained does change with altitude.
and
In the same vein, it is easily shown that
(
D max
(
ci1
2
)
D max
( tcv,
0
7TeAR)1
14
1Cv,o
( 3Cv,
0
7TeAR)
314
4Cv,o
(6.77)
(6.78)
Prove this yourself.
Example 6.14 From the equations given in this section, directly calculate ( CL/Cv)max and
(CL
312
/Cv)max for the CP1.
SOLUTION From Eq. (6.76),
From Eq. (6.78),
(
ci1
2
)
D max
( Cv.oweAR)1;2
2Cv,o
[0.025w(0.8)(7.37)]1
12
_
(2)(0.025) 
(3Cv,
0
weAR)
314
4Cn,o
[(3)(0.025)w(0.8)(7.37)]
314
 = 12 8
(4)(0.025) .
Return to Example 6.12, where the values of (CdCv)max and (Ccl
2
/Cv)max were obtained
graphically, i.e., by plotting CLfC
0
and CL
312
/C
0
and finding their peak values. Note that the
results obtained from Eqs. (6.76) and (6.78) agree with the graphical values obtained in Example
6.12 (as they should); however, the use of Eqs. (6.76) and (6.78) is much easier and quicker than
plotting a series of numbers and finding the maximum.
Example 6.15 From the equations given in this section, directly calculate (CL
1
/
2
/Cv)max and
(CLfCv)max for the CJ1.
318 INTRODUCTION TO FLIGHT
SOLUTION From Eq. (6.77),
(
c ~ 1 1 2
D max
From Eq. (6.76)
(tcv.
0
weARf
14
jCD.O
[l(0.02)w(0.81)(8.93)]1
14
EJ
. = 23 4
Ho.02) .
( weARCv,
0
)
112
2Cv.o
= [ w(0.81)(8.93)(0.02)]
112
= ~
(2)(0.02) ~
These values agree with the graphically obtained maximums in Example 6.13.
6.15 TAKEOFF PERFORMANCE
To this point in our discussion of airplane performance, we have assumed that all
accelerations are zero, i.e., we have dealt with aspects of static performance as
defined in Sec. 6.2. For the remainder of this chapter, we will relax this restriction
and consider several aspects of airplane performance that involve finite accelera
tion, such as takeoff and landing runs, turning flight, and accelerated rate of
climb.
To begin with, we ask the question, what is the running length along the
ground required by an airplane, starting from zero velocity, to gain flight speed
and lift from the ground? This length is defined as the ground roll, or liftoff
distance, sLo·
To address this question, let us first consider the accelerated rectilinear
motion of a body of mass m experiencing a constant force F, as sketched in
Figure 6.42. From Newton's second law,
or
Body at time t = 0
,,
I I F =constant
I V= 0 I
L __ _J
s=O
dV
F= ma= m
dt
F
dV= dt
m
Body at time t
Cffe"""'
I
(6.79)
Figure 6.42 Sketch of a body moving under the influence of a constant force F, starting from rest
(V = 0) at s = 0 and accelerating to velocity Vat distance s.
ELEMENTS OF AIRPLANE PERFORMANCE 319
Assume that the body starts from rest ( V = 0) at location s = 0 at time t = 0 and
is accelerated to velocity V over the distance s at time t. Integrating Eq. (6.79)
between these two points, and remembering that both F and m are constant, we
have
or
Solving for t,
i
v Fit
dV =  dt
o m o
F
V= t
m
Vm
t=
F
(6.80)
(6.81)
Considering an instant when the velocity if V, the incremental distance ds
covered during an incremental time dt is ds = V dt. From Eq. (6.80), we have
F
ds = V dt = tdt (6.82)
m
Integrating Eq. (6.82),
i
s Fit
ds =  tdt
o m o
F t
2
or s= 
m 2
(6.83)
Substituting Eq. (6.81) into (6.83), we obtain
(6.84)
Equation (6.84) gives the distance required for a body of mass m to accelerate to
velocity V under the action of a constant force F.
Now consider the force diagram for an airplane during its ground roll, as
illustrated in Figure 6.43. In addition to the familiar forces of lift, drag, thrust,
L
T
w
Figure 6.43 Forces acting on an airplane during takeoff and landing.
320 INTRODUCTION TO FLIGHT
and weight, the airplane also experiences a resistance force R due to rolling
friction between the tires and the ground. This resistance force is given by
R=µ,(WL) (6.85)
where W  L is the net normal force exerted between the tires and the ground
and µ, is the coefficient of rolling friction. Summing forces parallel to the
ground, and employing Newton's second law, we have
dV
F = T D  R = T D  µ,(w L) = mdt (6.86)
Let us examine Eq. (6.86) more closely. It gives the local instantaneous
acceleration of the airplane, dV/dt, as a function of T, D, W, and L. For
takeoff, over most of the ground roll, T is reasonably constant (this is particu
larly true for a jetpowered airplane). Also, Wis constant. However, both L and
D vary with velocity, since
(6.87)
and
2
( C
2
L )
D = }poc,Voo S CD O + cp
' '1TeAR
(6.88)
The quantity of cf> in Eq. (6.88) requires some explanation. When an airplane is
flying close to the ground, the strength of the wingtip vortices is somewhat
diminished due to interaction with the ground. Since these tip vortices induce
downwash at the wing (see Sec. 5.13), which in turn generates induced drag (see
Sec. 5.14), the downwash and hence induced drag are reduced when the airplane
is flying close to the ground. This phenomenon is called ground effect and is the
cause of the tendency for an airplane to flare, or "float," above the ground near
the instant of landing. The reduced drag in the presence of ground effect is
accounted for by cf> in Eq. (6.88), where cf> :=; 1. An approximate expression for cf>,
based on aerodynamic theory, is given by McCormick (see Bibliography at the
end of this chapter) as
(16 h/b )
2
cf> = 1 + (16 h/b )
2
(6.89)
where h is the height of the wing above the ground and b is the wingspan.
In light of the above, to accurately calculate the variation of velocity with
time during the ground roll, and ultimately the distance required for liftoff, Eq.
(6.86) must be integrated numerically, taking into account the proper velocity
variations of L and D from Eqs. (6.87) and (6.88), respectively, as well as any
velocity effect on T. A typical variation of these forces with distance along the
ground during takeoff is sketched in Figure 6.44. Note from Eq. (6.84) that s is
proportional to V
2
, and hence the horizontal axis in Figure 6.44 could just as well
be V
2
• Since both D and L are proportional to the dynamic pressure, q
00
= }p
00
V
00
2
, they appear as linear variations in Figure 6.44. Also, Figure 6.44 is
drawn for a jetpropelled airplane; hence T is relatively constant.
ELEMENTS OF AIRPLANE PERFORMANCE 321
At the point of takeoff, L = W

T (for a jet)
T[D+µ,(WL)]
D ;µ,(WL!._ ____ J_ __ _
;
0 Distance along ground, s
Figure 6.44 Schematic of a typical variation of forces acting on an airplane during takeoff.
A simple but approximate expression for the liftoff distance sw can be
obtained as follows. Assume that T is constant. Also, assume an average value for
the sum of drag and resistance force, [D + µ,(W  L)lave' such that this average
value, taken as a constant force, produces the proper liftoff distance sLo· Then,
we consider an effective constant force acting on the airplane during its takeoff
ground roll as
Feff = T [D + µ,(W  L)lave =canst (6.90)
These assumptions are fairly reasonable, as seen from Figure 6.44. Note that
the sum of D + µ,(W  L) versus distance (or V
2
) is reasonably constant, as
shown by the dashed line in Figure 6.44. Hence, the accelerating force, T 
[D + µ,(W  L)], which is illustrated by the difference between the thrust curve
and the dashed line in Figure 6.44, is also reasonably constant. Now return to Eq.
(6.84). Considering F given by Eq. (6.90), V = Vw (the liftoff velocity), and
m = W/g, where g is the acceleration of gravity, Eq. (6.84) yields
(6.91)
In order to ensure a margin of safety during takeoff, the liftoff velocity is
322 INTRODUCTION TO FLIGHT
typically 20 percent higher than stalling velocity. Hence, from Eq. (5.56), we have
2W
Vw = 1.2V.1a11 = 1.2 (6.92)
Substituting Eq. (6.92) into (6.91), we obtain
1.44W
2
SLo=
gp""SCL,max{T [D + µ,(W L)Jave}
(6.93)
In order to make a calculation using Eq. (6.93), Shevell (see Bibliography at the
end of this chapter) suggests that the average force in Eq. (6.93) be set equal to its
instantaneous value at a velocity equal to 0.7 Vw, that is,
[D + µ,(W L)Jave = [D + µ,(W L)]o.1 Vw
Also, experience has shown that the coefficient of rolling friction, µ,, in Eq. (6.93)
varies from 0.02 for a relatively smooth paved surface to 0.10 for a grass field.
A further simplification can be obtained by assuming that thrust is much
larger than either D or R during takeoff. Referring to the case shown in Figure
6.44, this simplification is not unreasonable. Hence, ignoring D and R compared
to T, Eq. (6.93) becomes simply
1.44W
2
SLO =
gp""SCL,max T
(6.94)
Equation (6.94) illustrates some important physical trends, as follows:
1. Liftoff distance is very sensitive to the weight of the airplane, varying directly
as W
2
• If the weight is doubled, the ground roll of the airplane is quadrupled.
2. Liftoff distance is dependent on the ambient density pO()' If we assume that
thrust is directly proportional to p
00
, as stated in Sec. 6.7, that is, Tex p
00
,
then Eq. (6.94) demonstrates that
This is why on hot summer days, when the air density is less than on cooler
days, a given airplane requires a longer ground roll to get off the ground.
Also, longer liftoff distances are required at airports which are located at
higher altitudes (such as at Denver, Colorado, a mile above sea level).
3. The liftoff distance can be decreased by increasing the wing area, increasing
CL.max• and increasing the thrust, all of which simply make common sense.
The total takeoff distance, as defined in the Federal Aviation Requirements
(FAR), is the sum of the ground roll distance sw and the distance (measured
along the ground) to clear a 35ft height (for jetpowered civilian transports) or a
50ft height (for all other airplanes). A discussion of these requirements, as well as
ELEMENTS OF AIRPLANE PERFORMANCE 323
more details regarding the total takeoff distance, is beyond the scope of this book.
See the books by Shevell and McCormick listed in the Bibliography at the end of
this chapter for more information on this topic.
Example 6.16 Estimate the liftoff distance for the CJ1 at sea level. Assume a paved runway,
hence µ, = 0.02. Also, during the ground roll, the angle of attack of the airplane is restricted by
the requirement that the tail not drag the ground, and therefore assume that CL, max during
ground roll is limited to 1.0. Also, when the airplane is on the ground, the wings are 6 ft above
the ground.
SoLuTION Use Eq. (6.93). In order to evaluate the average force in Eq. (6.93), first obtain the
ground effect factor from Eq. (6.89), where h/b = 6/53.3 = 0.113.
(16 h/b )
2
</> = = 0.764
1 + (16 h/b )
2
Also, from Eq. (6.92),
2W
Vw = 1.2 V.1a11 = 1.2
= 1.2
2(19,815)
= 230 ftjs
0.002377(318) (1.0)
Hence, 0.7 Vw = 160.3 ft/s. The average force in Eq. (6.93) should be evaluated at a velocity of
. 160.3 ft/s. To do this, from Eq. (6.87) we get
= = 9712 lb
Equation (6.88) yields
D = rp00 V00
2
s( Cv,o + </>
= Ho.002377)(160.3)
2
(318)[ 0.02 + o.764
= 520.7 lb
Finally, from Eq. (6.93),
l.44W
2
1.44(19,815)
2
32.2(0.002377)(318)(1.0){7300  [520.7 + (0.02)(19,815  9712)]}
= 13532 ft I
Note that [ D + µ,( W  L )lave = 722.8 lb, which is about 10 percent of the thrust. Hence, the
assumption leading to Eq. (6.94) is fairly reasonable, i.e., that D and R can sometimes be
. ignored compared with T.
324 INTRODUCTION TO FLIGHT
6.16 LANDING PERFORMANCE
Consider an airplane during landing. After the airplane has touched the ground,
the force diagram during the ground roll is exactly the same as that given in
Figure 6.43, and the instantaneous acceleration (negative in this case) is given by
Eq. (6.86). However, we assume that in order to minimize the distance required to
come to a complete stop, the pilot has decreased the thrust to zero at touchdown,
and therefore the equation of motion for the landing ground roll is obtained from
Eq. (6.86) with T = 0.
dV
D µ,(w L) = mdt (6.95)
A typical variation of the forces on the airplane during landing is sketched in
Figure 6.45. Designate the ground roll distance between touchdown at velocity VT
and a complete stop by sL. An accurate calculation of sL can be obtained by
numerically integrating Eq. (6.95) along with Eqs. (6.87) and (6.88).
However, let us develop an approximate expression for sL which parallels the
philosophy used in Sec. 6.15. Assume an average constant value for D + µ,(W 
L) which effectively yields the correct ground roll distance at landing, s L· Once
again, we can assume that [D + µ,(W  L )lave is equal to its instantaneous
value evaluated at 0.7 VT.
F= [D+µ,(WL)Lve= [D+µ,(WL)]o.1vr (6.96)
L
D+µ,(WL'!___



Distance along ground, s s=O
Figure 6.45 Schematic of a typical variation of forces acting on an airplane during landing.
ELEMENTS Of AIRPLANE PERFORMANCE 325
(Note from Figure 6.45 that the net decelerating force, D + µ.,(W  L), can vary
considerably with distance, as shown by the dashed line. Hence, our assumption
here for landing is more tenuous than for takeoff.) Returning to Eq. (6.82),
integrate between the touchdown point, where s = s L and t = 0, and the point
where the airplane's motion stops, where s = 0 and time equals t.
f
0
ds = ; j
1
tdt
SL 0
F t
2
or
s = 
L m 2
(6.97)
Note that, from Eq. (6.96), F is a negative value; hence sL in Eq. (6.97) is
positive.
Combining Eqs. (6.81) and (6.97), we obtain
V
2
m
s = 
L 2F
(6.98)
Equation (6.98) gives the distance required to decelerate from an initial velocity
V to zero velocity under the action of a constant force F. In Eq. (6.98), F is
given by Eq. (6.96), and V is VT. Thus, Eq. (6.98) becomes
V/(W/g)
sL = 2(D + µ.,(W L)]o.7 vT
(6.99)
In order to maintain a factor of safety,
VT = 1.3V.tall = 1.3 (6.100)
Substituting Eq. (6.100) into (6.99), we obtain
1.69W
2
(6.101)
During the landing ground roll, the pilot is applying brakes; hence, in Eq. (6.101)
the coefficient of rolling friction is that during braking, which is approximately
µ. r = 0.4 for a paved surface.
Modern jet transports utilize thrust reversal during the landing ground roll.
Thrust reversal is created by ducting air from the jet engines and blowing it in the
upstream direction, opposite to the usual downstream direction when normal
thrust is produced. As a result, with thrust reversal, the thrust vector in Figure
6.43 is reversed and points in the drag direction, thus aiding the deceleration and
shortening the ground roll. Designating the reversed thrust by TR, Eq. (6.95)
becomes
dV
T  D  11. (W  L) = m
R rr dt
( 6.102)
326 INTRODUCTION TO FLIGHT
Assuming that TR is constant, Eq. (6.101) becomes
1.69W
2
SL= gp
00
SCL,max{TR+ [D+µ,(WL)]o.?vr}
(6.103)
Another ploy to shorten the ground roll is to decrease the lift to near zero,
hence impose the full weight of the airplane between the tires and the ground and
increase the resistance force due to friction. The lift on an airplane wing can be
destroyed by spoilers, which are simply long, narrow surfaces along the span of
the wing, deflected directly into the flow, thus causing massive flow separation
and a striking decrease in lift.
The total landing distance, as defined in the FAR, is the sum of the ground
roll distance plus the distance (measured along the ground) to achieve touchdown
in a glide from a 50ft height. Such details are beyond the scope of this book; see
the books by Shevell and McCormick (listed in the Bibliography at the end of this
chapter) for more information.
Example 6.17 Estimate the landing ground roll distance at sea level for the CJ1. No thrust
reversal is used; however, spoilers are employed such that L = 0. The spoilers increase the
parasite drag coefficient by 10 percent. The fuel tanks are essentially empty, so neglect the
weight of any fuel carried by the airplane. The maximum lift coefficient, with flaps fully
employed at touchdown, is 2.5.
SOLUTION The empty weight of the CJ1 is 12,352 lb. Hence,
Vr=l.3V,,au=l.3f S2CW
V Poo L.max
=1.3
2(12,352) 6 f
0.002377(318)(2.5) =
148
· tjs
Thus, 0.7Vr = 104 ftjs. Also, Cv.o = 0.02 + 0.1(0.02) = 0.022. From Eq. (6.88), with CL = 0
(remember, spoilers are employed, destroying the lift),
D = iP
0
,,V
00
2
SCv.o = }(0.002377)(104)2(318)(0.022) = 89.9 lb
From Eq. (6.101), with L = 0,
l.69W
2
SL=
gp
00
SCL.max ( D + µ,W)o 7V,
1.69(12,352)
2
32.2(0.002377)(318) (2.5) [89 .9 + (0.4) (12352) l
= 1842 ft I
6.17 TURNING FLIGHT AND THE Vn DIAGRAM
To this point in our discussion of airplane performance, we have considered
rectilinear motion. Our static performance analyses dealt with zero acceleration
leading to constant velocity along straightline paths. Our discussion of takeoff
ELEMENTS OF AIRPLANE PERFORMANCE 327
and landing performance involved rectilinear acceleration, also leading to motion
along a straightline path. Let us now consider some cases involving radial
acceleration, which leads to curved flight paths; i.e., let us consider the turning
flight of an airplane. In particular, we will examine three specialized cases: (1) a
level turn, (2) a pullup, and (3) a pulldown. A study of the generalized motion of
an airplane along a threedimensional flight path is beyond the scope of this book.
A level turn is illustrated in Figure 6.46. Here, the wings of the airplane are
banked through the angle </>; hence the lift vector is inclined at the angle </> to the
vertical. The bank angle </> and the lift L are such that the component of the lift
in the vertical direction exactly equals the weight:
Lcoscp = W
Horizontal plane
w
Top view of
horizontal plane
Figure 6.46 An airplane in a level tum.
R
e
R
Front view
328 INTRODUCTION TO FLIGHT
and therefore the airplane maintains a constant altitude, moving in the same
horizontal plane. However, the resultant of L and W leads to a resultant force F,,
which acts in the horizontal plane. This resultant force is perpendicular to the
flight path, causing the airplane to turn in a circular path with a radius of
curvature equal to R. We wish to study this turn radius R, as well as the turn rate
dO / dt.
From the force diagram in Figure 6.46, the magnitude of the resultant force is
F,. = /L2  w2
We introduce a new term, the load factor n, defined as
L
n=
W
(6.104)
(6.105)
The load factor is usually quoted in terms of "g's"; for example, an airplane with
lift equal to five times the weight is said to be experiencing a load factor of 5 g's.
Hence, Eq. (6.104) can be written as
F,. = w ~ (6.106)
The airplane is moving in a circular path at the velocity Voo Therefore, the radial
acceleration is given by V
00
2
/R. From Newton's second law
v
2
w v
2
F = m ~ = ~
' R g R
Combining Eqs. (6.106) and (6.107) and solving for R, we have
v2
R = oo
g ~
(6.107)
(6.108)
The angular velocity, denoted by w = dO/dt, is called the turn rate and is given
by V
00
/ R. Thus, from Eq. (6.108), we have
( 6.109)
For the maneuvering performance of an airplane, both military and civil, it is
frequently advantageous to have the smallest possible R and the largest possible
w. Equations (6.108) and (6.109) show that to obtain both a small turn radius and
a large turn rate, we want:
1. The highest possible load factor (i.e., the highest possible L/W)
2. The lowest possible velocity
Consider another case of turning flight, where an airplane initially in straight
level flight (where L = W) suddenly experiences an increase in lift. Since L > W,
ELEMENTS OF AIRPLANE PERFORMANCE 329
e
R
L
w
Figure 6.47 The pullup maneuver.
the airplane will begin to tum upward, as sketched in Figure 6.47. For this pullup
maneuver, the flight path becomes curved in the vertical plane, with a tum rate
w = dO/dt. From the force diagram in Figure 6.47, the resultant force F,. is
vertical and is given by
F,. = L  W = W(n  1)
From Newton's second law
v
2
w v
2
F = m ~ = ~
' R g R
Combining Eqs. (6.110) and (6.111) and solving for R,
v 2
R = oo
g(n  1)
and since w = V
00
/R,
(6.110)
(6.111)
(6.112)
(6.113)
A related case is the pulldown maneuver, illustrated in Figure 6.48. Here, an
airplane is initially level flight suddenly rolls to an inverted position, such that
both L and W are pointing downward. The airplane will begin to tum downward
330 INTRODUCTION TO FLIGHT
w
R
L
e
Figure 6.48 The pulldown maneuver.
in a circular flight path with a turn radius R and turn rate w = dO / dt. By an
analysis similar to those above, the following results are easily obtained:
v2
R = oo
g(n + 1)
(6.114)
g(n + 1)
w=  (6.115)
Prove this to yourself.
Considerations of turn radius and turn rate are particularly important to
military fighter aircraft; everything else being equal, those airplanes with the
smallest R and largest w will have definite advantages in air combat. Highper
formance fighter aircraft are designed to operate at high load factors, typically
from 3 to 10. When n is large, then n + 1 = n and n  1 = n; for such cases,
Eqs. (6.108), (6.109), and (6.112) to (6.115) reduce to
v2
R ~
gn
and
gn
w=
voo
Let us work with these equations further. Since
L = tPooVoo
2
SCL
2
2L
v =
00 PooSCL
then
(6.116)
(6.117)
(6.118)
ELEMENTS OF AIRPLANE PERFORMANCE 331
Substituting Eqs. (6.118) and (6.105) into Eqs. (6.116) and (6.117), we obtain
2L 2 W
R= =
p00SCLg(L/W) p
00
CLg S
(6.119)
and
gn
w = ;=====
J2L/p
00
SCL
( 6.120)
Note that in Eqs. ( 6.119) and ( 6.120) the factor W / S appears. This factor occurs
frequently in airplane performance analyses and is labeled as
w
 = wing loading
s
Equations (6.119) and (6.120) clearly show that airplanes with lower wing
loadings will have smaller turn radii ano larger turn rates, everything else being
equal. However, the design wing loading of an airplane is usually determined by
factors other than maneuvering, such as payload, range, and maximum velocity.
As a result, wing loadings for light, general aviation aircraft are relatively low,
but those for highperformance military aircraft are relatively large. Wing load
ings for some typical airplanes are listed below.
Airplane
Wright Flyer (1903)
Beechcraft Bonanza
McDonnell Douglas F15
General Dynamics F16
W/S, lb/ft
2
1.2
18.8
66
74
From the above, we conclude that a small, light aircraft such as the Beechcraft
Bonanza can outmaneuver a larger, heavier aircraft such as the F16 because of
smaller turn radius and larger turn rate. However, this is really comparing apples
and oranges. Instead, let us examine Eqs. (6.119) and (6.120) for a given airplane
with a given wing loading and ask the question, for this specific airplane, under
what conditions will R be minimum and w maximum? From these equations,
clearly R will be minimum and w will be maximum when both CL and n are
maximum. That is,
2 w
Rmin = 
PoogCL,ma:x S
PooCL,ma:x nma:x
2(W/S)
(6.121)
(6.122)
332 INTRODUCTION TO FLIGHT
Also note from Eqs. (6.121) and (6.122) that best perform_ance will occur at sea
level, where p
00
is maximum.
There are some practical constraints on the above considerations. First, at
low speeds, nmax is a function of CL.max itself, because
L tPooVoo
2
SCL
n =  =
w w
c
n = .1 V 2 L,max
max 2Poo oo W/S
and hence (6.123)
At higher speeds, n max is limited by the structural design of the airplane. These
considerations are best understood by examining Figure 6.49, which is a diagram
showing load factor vs. velocity for a given airplanethe Vn diagram. Here,
curve AB is given by Eq. (6.123). Consider an airplane flying at velocity V
1
,
where V
1
is shown in Figure 6.49. Assume that the airplane is at an angle of
attack such that CL < CL.max· This flight condition is represented by point 1 in
Figure 6.49. Now assume that the angle of attack is increased to that for
obtaining CL,max' keeping the velocity constant at V
1
. The lift increases to its
maximum value for the given V
1
, and hence the load factor, n = L/W, reaches
its maximum value, nmax' for the given V
1
. This value of nmax is given by Eq.
(6.123), and the corresponding flight condition is given by point 2 in Figure 6.49.
If the angle of attack is increased further, the wing stalls, and the load factor
drops. Therefore, point 3 in Figure 6.49 is unobtainable in flight. Point 3 is in the
"stall region" of the Vn diagram. Consequently, point 2 represents the highest
possible load factor that can be obtained at the given velocity Vi Now, as V
1
is
increased, say to a value of V
4
, then the maximum possible load factor nmax also
increases, as given by point 4 in Figure 6.49 and as calculated from Eq. (6.123).
However, n max cannot be allowed to increase indefinitely. Beyond a certain value
of load factor, defined as the positive limit load factor and shown as the
horizontal line BC in Figure 6.49, structural damage may occur to the aircraft.
The velocity corresponding to point B is designated as V*. At velocities higher
than V*, say V
5
, the airplane must fly at values of CL less than CL,max so that the
positive limit load factor is not exceeded. If flight at CL max is obtained at
velocity Vs, corresponding to point 5 in Figure 6.49, then damage will
occur. The righthand side of the Vn diagram, line CD, is a highspeed limit. At
velocities greater than this, the dynamic pressure becomes so large that again
structural damage may occur to the airplane. (This maximum velocity limit is, by
design, much larger than the levelflight Vmax calculated in Secs. 6.4 to 6.6. In fact,
the structural design of most airplanes is such that the maximum velocity allowed
by the Vn diagram is sufficiently greater than maximum diving velocity for the
airplane.) Finally, the bottom part of the Vn diagram, given by curves AE and
ED in Figure 6.49, corresponds to negative absolute angles of attack, i.e.,
negative load factors. Curve AE defines the stall limit. (At absolute angles of
attack less than zero, the lift is negative and acts in the downward direction. If
the wing is pitched downward to a large enough negative angle of attack, the flow
ELEMENTS OF AIRPLANE PERFORMANCE 333
will separate from the bottom surface of the wing and the downwardacting lift
will decrease in magnitude; i.e., the wing "stalls.") Line ED gives the negative
limit load factor, beyond which structural damage will occur.
As a final note concerning the Vn diagram, consider point B in Figure 6.49.
This point is called the maneuver point. At this point, both CL and n are
simultaneously at their highest possible values that can be obtained anywhere
throughout the allowable flight envelope of the aircraft. Consequently, from Eqs.
(6.121) and (6.122), this point corresponds simultaneously to the smallest possible
turn radius and the largest possible turn rate for the airplane. The velocity
corresponding to point B is called the corner velocity and is designated by V* in
Figure 6.49. The corner velocity can be obtained by solving Eq. (6.123) for
velocity, yielding
V* = (6.124)
In Eq. (6.124), the value of nmax corresponds to that at point B in Figure 6.49.
The corner velocity is an interesting dividing line. At flight velocities less than V*,
it is not possible to structurally damage the airplane due to the generation of too
much lift. In contrast, at velocities greater than V*, lift can be obtained that can
structurally damage the aircraft (for example, point 5 in Figure 6.49), and the
pilot must make certain to avoid such a case.
12
10
8
6
"
....
~
PooWw limU 1'"'1 '"'" '
E 4
u
Stall area
~
'O
2
"' 0
,_l
A
0
2
Stall area
4
6
Figure 6.49 Vn diagram for a typical jet trainer aircraft. (U.S. Air Force Academy.)
334 INTRODUCTION TO FLIGHT
Figure 6.50 General Dynamics F16 in 90° vertical accelerated climb. (U.S. Air Force.)
6.18 ACCELERATED RATE OF CLIMB (ENERGY METHOD)*
Modern highperformance airplanes, such as the supersonic General Dynamics
F16 shown in Figure 6.50, are capable of highly accelerated rates of climb.
Therefore, the performance analysis of such airplanes requires methods that go
beyond the static rate of climb considerations given in Secs. 6.8 to 6.11. The
purpose of the present section is to introduce one such method, namely a method
dealing with the energy of an airplane. This is in contrast to our previous
discussions that have dealt explicitly with forces on the airplane.
*This section is based in part on material presented by the faculty of the department of
aeronautics at the U.S. Air Force Academy at its annual aerodynamics workshop, held each July at
Colorado Springs. This author has had the distinct privilege to participate in this workshop since its
inception in 1979. Special thanks for this material go to Col. James D. Lang, Major Thomas Parrot,
and Col. Daniel Daley.
ELEMENTS OF AIRPLANE PERFORMANCE 335
Consider an airplane of mass m in flight at some altitude h and with some
velocity V. Due to its altitude, the airplane has potential energy PE equal to mgh.
Due to its velocity, the airplane has kinetic energy KE equal to tm V
2
• The total
energy of the airplane is the sum of these energies; i.e.,
Total aircraft energy= PE+ KE= mgh + tmV
2
(6.125)
The energy per unit weight of the airplane is obtained by dividing Eq. (6.125) by
W = mg. This yields the specific energy, denoted by H., as
or
H=
e
PE+ KE
w
G

g
mgh + tmV
2
mg
(6.126)
The specific energy He has units of height and is therefore also called the energy
height of the aircraft. Thus, let us become accustomed to quoting the energy of an
airplane in terms of the energy height He, which is simply the sum of the
potential and kinetic energies of the airplane per unit weight. Contours of
constant He are illustrated in Figure 6.51, which is an "altitudeMach number
map." Here, the ordinate and abscissa are altitude h and Mach number M,
respectively, and the dashed curves are lines of constant energy height.
100
..._

90 .......
....... 't
....._ loo
80
......_ ·Oa


....... ....._O I{
b 70

'  80
'
........... 000
'
x
'....:r
' 60
...
'
......
.......
'
""'

..._Go
'
._.D
50 ....._ ·Oa
'
B '·.!!/t
'
\
E 40
c
' '
\
<!'. ...
....... '
'\
\
'
'\
\
30
"'o
'
\
\
......_ ·Oa
'
\
20
' 0/) '\
\
\

't
\
\
..... <o
'
\
...... ,.oa
'
..._B
\
\
\ ,o,,
\
,r
'
I I
0 0.5 1.0 1.5 2.0 2.5
Mach number M
Figure 6.51 AltitudeMach number map showing curves of constantenergy height. These are uni
versal curves that represent the variation of kinetic and potential energies per unit weight. They do
not depend on the specific design factors of a given airplane.
336 INTRODUCTION TO FLIGHT
To obtain a feeling for the significance of Figure 6.51, consider two airplanes,
one flying at an altitude of 30,000 ft at Mach 0.81 (point A in Figure 6.51) and
the other flying at an altitude of 10,000 ft at Mach 1.3 (point B). Both airplanes
have the same energy height, 40,000 ft (check this yourself by calculation).
However, airplane A has more potential energy and less kinetic energy (per unit
weight) than airplane B. If both airplanes maintain their same states of total
energy, then both are capable of "zooining" to an altitude of 40,000 ft at zero
velocity (point C) simply by trading all their kinetic energy for potential energy.
Consider another airplane, flying at an altitude of 50,000 ft at Mach 1.85,
denoted by point D in Figure 6.51. This airplane will have an energy height of
100,000 ft and is indeed capable of zooining to an actual altitude of 100,000 ft by
trading all of its kinetic energy for potential energy. Airplane D is in a much
higher energy state (He= 100,000 ft) than airplanes A and B (which have
He = 40,000 ft). Therefore, airplane D has a much greater capability for speed
and altitude performance than airplanes A and B. In air combat, everything else
being equal, it is advantageous to be in a higher energy state (have a larger HJ
than your adversary.
How does an airplane change its energy state; e.g., in Figure 6.51, how could
airplanes A and B increase their energy height to equal that of D? To answer
this question, return to the force diagram in Figure 6.1 and the resulting equation
of motion along the flight path, given by Eq. (6.6). Assuining that ar is small, Eq.
(6.6) becomes
dV
TD WsinO = m
dt
Recalling that m = W/g, Eq. (6.127) can be rearranged as
T  D = w(sin 0 + dV)
g dt
Multiplying by V/W, we obtain
TV DV V dV
 = VsinO + 
W g dt
(6.127)
(6.128)
Exainining Eq. (6.128) and recalling some of the definitions from Sec. 6.8, we
observe that VsinO = R/C = dh/dt and that
TV  D V excess power
W W =Ps
where the excess power per unit weight is defined as the specific excess power and
is denoted by Ps. Hence, Eq. (6.128) can be written as
dh V dV
p =  + 
s dt g dt
(6.129)
Equation (6.129) states that an airplane with excess power can use this excess for
rate of climb (dh/dt) or to accelerate along its flight path (dV/dt) or for a
ELEMENTS OF AIRPLANE PERFORMANCE 337
combination of both. For example, consider an airplane in level flight at a
velocity of 800 ft/s. Assume that when the pilot pushes the throttle all the way
forward, an excess power is generated in the amount Ps = 300 ftjs. Equation
(6.129) shows that the pilot can choose to use all this excess power to obtain a
maximum unaccelerated rate of climb of 300 ftjs ( dV / dt = 0, hence Ps = dh/ dt
= R/C). In this case, the velocity along the flight path stays constant at 800 ftjs.
Alternatively, the pilot may choose to maintain level flight ( dh/ dt = 0) and to
use all this excess power to accelerate at the rate of dV / dt = gPs/ V =
32.2(300)/800 = 12.1 ft/s
2
• On the other hand, some combination could be
achieved, such as a rate of climb dh/dt = 100 ft/s along with an acceleration
along the flight path of dV / dt = 32.2(200)/800 = 8.1 ft/s
2
. [Note that Eqs.
(6.128) and (6.129) are generalizations of Eq. (6.43). In Sec. 6.8, we assumed that
dV / dt = 0, which resulted in Eq. (6.43) for a steady climb. In the present section,
we are treating the more general case of climb with a finite acceleration.] Now
return to Eq. (6.126) for the energy height. Differentiating with respect to time,
we have
dHe dh V dV
=+
dt dt g dt
( 6.130)
The righthand sides of Eqs. (6.129) and (6.130) are identical, hence we see that
[!] (6.131)
That is, the time rate of change of energy height is equal to the specific excess
power. This is the answer to the question at the beginning of this paragraph. An
airplane can increase its energy state simply by the application of excess power.
In Figure 6.51, airplanes A and B can reach the energy state of airplane D if they
have enough excess power to do so.
This immediately leads to the next question, namely, how can we ascertain
whether or not a given airplane has enough Ps to reach a certain energy height?
To address this question, recall the definition of excess power as illustrated in
Figure 6.24, i.e., the difference between power available and power required. For
a given altitude, say h, the excess power (hence Ps) can be plotted vs. velocity (or
Mach number). For a subsonic airplane below the dragdivergence Mach num
ber, the resulting curve will resemble the sketch sown in Figure 6.52a. At a given
altitude h
1
, Ps will be an inverted, Ushaped curve. (This is essentially the same
type of plot as shown in Figures 6.27 and 6.28.) For progressively higher
altitudes, such as h
2
and h
3
, Ps becomes smaller, as also shown in Figure 6.52a.
Hence, Figure 6.52a is simply a plot of Ps versus Mach number with altitude as
a parameter. These results can be crossplotted on an altitudeMach number
map using Ps as a parameter, as illustrated in Figure 6.52b. For example,
consider all the points on Figure 6.52a where Ps = O; these correspond to points
along a horizontal axis through Ps = 0, such as points a, b, c, d, e, and f in
Figure 6.52a. Now replot these points on the altitudeMach number map in
Figure 6.52b. Here, points a, b, c, d, e, and f form a bellshaped curve, along
338 INTRODUCTION TO FLIGHT
400
o ~ 200
A
0.4
Mach number M
B
0.8
0.8
(a)
(b)
Figure 6.52 Construction of the
specific excess power contours in
the altitudeMach number map
for a subsonic airplane below the
dragdivergence Mach number.
These contours are constructed for
a fixed load factor; if the load
factor is changed, the Ps contours
will shift.
which Ps = 0. This curve is called the Ps contour for Ps = 0. Similarly, all points
with Ps = 200 ftjs are on the horizontal line AB in Figure 6.52a, and these
points can be crossplotted to generate the Ps = 200 ftjs contour in Figure 6.52b.
In this fashion, an entire series of Ps contours can be generated in the
altitudeMach number map. For a supersonic airplane, the Ps versus Mach
number curves at different altitudes will appear as sketched in Figure 6.53a. The
"dent" in the Ushaped curves around Mach 1 is due to the large drag increase in
the transonic flight regime (see Sec. 5.10). In turn, these curves can be crossplotted
on the altitudeMach number map, producing the Ps contours as illustrated in
Figure 6.53b. Due to the doublehumped shape of the Ps curves in Figure 6.53a,
the Ps contours in Figure 6.53b have different shapes in the subsonic and
supersonic regions. The shape of the Ps contours shown in Figure 6.53b is
characteristic of most supersonic aircraft. Now, we are close to the answer to our
question at the beginning of this paragraph. Let us overlay the Ps contours, say
from Figure 6.53b, and the energy states illustrated in Figure 6.51all on an
altitudeMach number map. We obtain a diagram, as illustrated in Figure 6.54.
600
"'
.::::
0..,
400
200
..,
h3
"
""
E
hz
<
h1
h1 <hz <h3
0 1.0
(a)
0 1.0
Mach number M
(b)
2.0
2.0
ELEMENTS OF AIRPLANE PERFORMANCE 339
M
Figure 6.53 Specific excess power
contours for a supersonic airplane.
In this figure, note that the Ps contours always correspond to a given airplane at a
given load factor, whereas the He lines are universal fundamental physical curves
that have nothing to do with any given airplane. The usefulness of Figure 6.54 is
that it clearly establishes what energy states are obtainable by a given airplane.
The regime of sustained flight for the airplane lies inside the envelope formed by
the Ps = 0 contour. Hence, all values of He inside this envelope are obtainable by
the airplane. A comparison of figures like Figure 6.54 for different airplanes will
clearly show in what regions of altitude and Mach number an airplane has
maneuver advantages over another.
Figure 6.54 is also useful for representing the proper flight path to achieve
minimum time to climb. For example, consider two energy heights, He. i and
He,
2
, where He,
2
> He,i· The time to move between these energy states can be
obtained from Eq. (6.131), written as
dHe
dt=
ps
340 INTRODUCTION TO FLIGHT
JOO 
90
80

 ...... 80,000 ft
......
.......
'....
......
"' I
0
70
......
......
I 

x

30
20
IO
0 0.5 1.0 1.5 2.0 2.5
Mach number M
Figure 6.54 Overlay of Ps contours and specific energy states on an altitudeMach number map. The
Ps values shown here approximately correspond to a Lockheed F104G supersonic fighter. Load
factor n = 1. W = 18,000 lb. Airplane is at maximum thrust. The path given by points A through I is
the flight path for minimum time to climb.
Integrating between He,l and He,
2
, we have
(6.132)
From Eq. (6.132), the time to climb will be a minimum when Ps is a maximum.
Looking at Figure 6.54, for each He curve, there is a point where Ps is a
maximum. Indeed, at this point, the Ps curve is tangent to the He curve. Such
points are illustrated by points A to I in Figure 6.54. The heavy curve through
these points illustrates the variation of altitude and Mach number along the flight
path for minimum time to climb. The segment of the flight path between D and
D' represents a constant energy dive to accelerate through the dragdivergence
region near Mach 1.
As a final note, analyses of modem highperformance airplanes make exten
sive use of energy concepts such as those described above. Indeed, military pilots
fly with Ps diagrams in the cockpit. Our purpose here has been to simply
introduce some of the definitions and basic ideas in involving these concepts. A
more extensive treatment is beyond the scope of this book.
ELEMENTS OF AIRPLANE PERFORMANCE 341
6.19 A COMMENT
We end the technical portion of this chapter by noting that detailed computer
programs now exist within NASA and the aerospace industry for the accurate
estimation of airplane performance. These programs are usually geared to specific
types of airplanes, e.g., general aviation aircraft (light single or twinengine
private airplanes), military fighter aircraft, and commercial transports. Such
considerations are beyond the scope of this book. However, the principles
developed in this chapter are steppingstones to moreadvanced studies of air
plane performance; the Bibliography at the end of this chapter provides some
suggestions for such studies.
6.20 HISTORICAL NOTE:
DRAG REDUCTIONTHE NACA COWLING AND
THE FILLET
The radial piston engine came into wide use in aviation during and after World
War I. As described in Chap. 9, a radial engine has its pistons arranged in a
circular fashion about the crankshaft, and the cylinders themselves are cooled by
airflow over the outer finned surfaces. Until 1927, these cylinders were generally
directly exposed to the main airstream of the airplane, as sketched in Figure 6.55.
As a result, the drag on the enginefuselage combination was inordinately high.
The problem was severe enough that a group of aircraft manufacturers met at
Langley Field on May 24, 1927, to urge NACA to undertake an investigation of
means to reduce this drag. Subsequently, under the direction of Fred E. Weick, an
extensive series of tests was conducted in the Langley 20ft propeller research
tunnel using a Wright Whirlwind J5 radial engine mounted to a conventional
fuselage. In these tests, various types of aerodynamic surfaces, called cowlings,
were used to cover, partly or completely, the engine cylinders, directly guiding
part of the airflow over these cylinders for cooling purposes but at the same time
not interfering with the smooth primary aerodynamic flow over the fuselage. The
Figure 6.55 Engine mounted with no cowling.
342 INTRODUCTION TO FLIGHT
Figure 6.56 Engine mounted with full cowling.
best cowling, illustrated in Figure 6.56, completely covered the engine. The results
were dramatic! Compared with the uncowled fuselage, a full cowling reduced the
drag by a stunning 60 percent! This is illustrated in Figure 6.57, taken directly
from Weick's report entitled "Drag and Cooling with Various Forms of Cowling
for a Whirlwind Radial AirCooled Engine," NACA technical report no. 313,
published in 1928. After this work, virtually all radial engineequipped airplanes
since 1928 have been designed with a full NACA cowling. The development of
this cowling was one of the most important aerodynamic advancements of the
1920s; it led the way to a major increase in aircraft speed and efficiency.
A few years later, a second major advancement was made, but by a com
pletely different group and on a completely different part of the airplane. In the
early 1930s, the California Institute of Technology at Pasadena, California,
established a program in aeronautics under the direction of Theodore von
Karman. Von Karman, a student of Ludwig Prandtl, became probably the
leading aerodynamicist of the 19201960 time period. At Caltech, von Karman
established an aeronautical laboratory of high quality, which included a large
200
160
120
:2
oii
i1
Q
80
40
0 8 16 24
Dynamic pressure, lb/ft
2
Figure 6.57 Reduction in drag due to a full
cowling.
Fillet
Figure 6.58 Illustration of the wing fillet.
ELEMENTS OF AIRPLANE PERFORMANCE 343
No fillet
(sharp corner)
Fillet
344 INTRODUCTION TO FLIGHT
subsonic wind tunnel funded by a grant from the Guggenheim Foundation. The
first major experimental program in this tunnel was a commercial project for the
Douglas Aircraft Company. Douglas was designing the DC1, the forerunner of a
series of highly successful transports (including the famous DC3, which revo
lutionized commercial aviation in the 1930s). The DC1 was plagued by unusual
buffeting in the region where the wing joined the fuselage. The sharp comer at the
juncture caused severe flow field separation, which resulted in high drag as well as
shed vortices which buffeted the tail. The Caltech solution, which was new and
pioneering, was to fair the trailing edge of the wing smoothly into the fuselage.
These fairings, called fillets, were empirically designed and were modeled in clay
on the DC1 windtunnel models. The best shape was found by trial and error.
The addition of a fillet (see Figure 6.58) solved the buffeting problem by
smoothing out the separated flow and hence also reduced the interference drag.
Since that time, fillets have become a standard airplane design feature. Moreover,
the fillet is an excellent example of how university laboratory research in the
1930s contributed directly to the advancement of practical airplane design.
6.21 HISTORICAL NOTE:
EARLY PREDICTIONS OF
AIRPLANE PERFORMANCE
The airplane of today is a modem work of art and engineering. In tum, the
prediction of airplane performance as described in this chapter is sometimes
viewed as a relatively modem discipline. However, contrary to intuition, some of
the basic concepts have roots deep in history; indeed, some of the very techniques
detailed in previous sections were being used in practice only a few years after the
Wright brothers' successful first flight in 1903. This section traces a few historic
paths for some of the basic ideas of airplane performance, as follows:
1. Some understanding of the power required PR for an airplane was held by
George Cayley. He understood that the rate of energy lost by an airplane in a
steady glide under gravitational attraction must be essentially the power that
must be supplied by an engine to maintain steady, level flight. In 1853,
Cayley wrote:
The whole apparatus when loaded by a weight equal to that of the man intended ultimately to try
the experiment, and with the horizontal rudder [the elevator] described on the essay before sent,
adjusted so as to regulate the oblique descent from some elevated point, to its proper pitch, it
may be expected to skim down, with no force but its own gravitation, in an angle of about 11
degrees with the horizon; or possibly, if well executed, as to direct resistance something less, at a
speed of about 36 feet per second, if loaded 1 pound to each square foot of surface. This having
by repeated experiments, in perfectly calm weather, been ascertained, for both the safety of the
man, and the datum required, let the wings be plied with the man's utmost strength; and let the
angle measured by the greater extent of horizontal range of flight be noted; when this point, by
repeated experiments, has been accurately found, we shall have ascertained a sound practical
ELEMENTS OF AIRPLANE PERFORMANCE 345
basis for calculating what engine power is necessary under the same circumstances as to weight
and surface to produce horizontal flight. ...
2. The drag polar, a concept introduced in Sec. 5.14, sketched in Figure 5.35,
and embodied in Eq. (6.la), represents simply a plot of CD versus Cv
illustrating that CD varies as the square of CL. A knowledge of the drag polar
is essential to the calculation of airplane performance. It is interesting that
the concept of the drag polar was first introduced by the Frenchman M. Eiffel
about 1890. Eiffel was interested in determining the laws of resistance (drag)
on bodies of various shapes, and he conducted such drag measurements by
dropping bodies from the Eiffel Tower and measuring their terminal velocity.
3. Some understanding of the requirements for rate of climb existed as far back
as 1913, when in an address by Granville E. Bradshaw before the Scottish
Aeronautical Society in Glasgow in December, the following comment was
made: "Among the essential features of all successful aeroplanes [is that] it
shall climb very quickly. This depends almost entirely on the weight efficiency
of the engine. The rate of climb varies directly as the power developed and
indirectly as the weight to be lifted." This is essentially a partial statement of
Eq. (6.43).
4. No general understanding of the prediction of airplane performance existed
before the twentieth century. The excellent summary of aeronautics written
by Octave Chanute in 1894, Progress in Flying Machines, does not contain
any calculational technique even remotely resembling the procedures set forth
in this chapter. At best, it was understood by that time that lift and drag
varied as the first power of the area and as the second power of velocity, but
this does not constitute a performance calculation. However, this picture
radically changed in 1911. In that year, the Frenchman Duchene received the
Monthyon Prize from the Paris Academy of Sciences for his book entitled
The Mechanics of the Airplane: A Study of the Principles of Flight. Captain
Duchene was a French engineering officer, born in Paris on December 27,
1869, educated at the famous Ecole Polytechnique, and later assigned to the
fortress at Toul, one of the centers of "aerostation" in France. It was in this
capacity that Captain Duchene wrote his book during 19101911. In this
book, the basic elements of airplane performance, as discussed in this
chapter, are put forth for the first time. Duchene gives curves of power
required and power available, as we illustrated in Figure 6.l 7a; he discusses
airplane maximum velocity; he also gives the same relation as Eq. (6.43) for
rate of climb. Thus, some of our current concepts for the calculation of
airplane performance date back as far as 19101911four years before the
beginning of World War I, and only seven years after the Wright brothers'
first flight in 1903. Later, in 1917, Duchene's book was translated into English
by John Ledeboer and T. O'B. Hubbard (see Bibliography at the end of this
chapter). Finally, during 19181920, three additional books on airplane
346 INTRODUCTION TO FLIGHT
performance were written (again, see Bibliography), the most famous being
the authoritative Applied Aerodynamics by Leonard Bairstow. By this time the
foundations discussed in this chapter had been well set.
6.22 HISTORICAL NOTE:
BREGUET AND THE RANGE FORMULA
LouisCharles Breguet was a famous French aviator, airplane designer, and
industrialist. Born in Paris on January 2, 1880, he was educated in electrical
engineering at the Lycee Condorcet, the Lycee Carnot, and the Ecole Superieure
d'Electricite. After graduation, he joined the electrical engineering firm of his
father, Maison Breguet. However, in 1909 Breguet built his first airplane and then
plunged his life completely into aviation. During World War I, his airplanes were
massproduced for the French air force. In 1919, he founded a commercial airline
company which later grew into Air France. His airplanes set several longrange
records during the 1920s and 1930s. Indeed, Breguet was active in his own aircraft
company until his death on May 4, 1955, in Paris. His name is associated with a
substantial part of French aviation history.
The formula for range of a propellerdriven airplane given by Eq. (6.58) has
also become associated with Breguet's name; indeed, it is commonly called the
Breguet range equation. However, the reason for this association is historically
obscure. In fact, the historical research of the present author can find no
substance to Breguet's association with Eq. (6.58). On one hand, we find
absolutely no reference to airplane range or endurance in any of the airplane
performance literature before 1919, least of all a reference to Breguet. The
authoritative books by Cowley and Levy (1918), Judge (1919), and Bairstow
(1920) (see Bibliography at the end of this chapter) amazingly enough do not
discuss this subject. On the other hand, in 1919, NACA report no. 69, entitled "A
Study of Airplane Ranges and Useful Loads," by J. G. Coffin, gives a complete
derivation of the formulas for range, Eq. (6.58), and endurance, Eq. (6.59).
However, Coffin, who was director of research for Curtiss Engineering Corpora
tion at that time, gives absolutely no references to anybody. Coffin's work appears
to be original and clearly seems to be the first presentation of the range and
endurance formulas in the literature. However, to confuse matters, we find a few
years later, in NACA report no. 173, entitled "Reliable Formulae for Estimating
Airplane Performance and the Effects of Changes in Weight, Wing Area or
Power," by Walter S. Diehl (we have met Diehl before, in Sec. 3.6), the following
statement: "The common formula for range, usually credited to Breguet, is easily
derived." Diehl's report then goes on to use Eq. (6.58), with no further reference
to Breguet. This report was published in 1923, four years after Coffin's work.
Consequently, to say the least, the proprietorship of Eq. (6.58) is not clear. It
appears to this author that, in the United States at least, there is plenty of
documentation to justify calling Eq. (6.58) the CoffinBreguet range equation.
ELEMENTS OF AIRPLANE PERFORMANCE 347
However, it has come down to us through the ages simply as Breguet's equation,
apparently without documented substance.
6.23 HISTORICAL NOTE:
AIRCRAFT DESIGNEVOLUTION AND REVOLUTION
Sit back for a moment and think about the evolution of the airplane, beginning
with Sir George Cayley's 1804 handlaunched glider. Indeed, Figure 1.8 (Cayley's
own sketch of this aircraft) shows the first airplane with a modern configuration.
Now jump ahead a century in the design of the airplane to Figure 1.2, the Wright
brothers' historic photograph of their first successful flight in 1903; this is the
true beginning of the practical airplane. Finally, jump another 80 years to Figure
6.7, which shows a modern jet aircraft. Put these three aircraft side by side in
your mind: Cayley's glider, the Wright Flyer, and the Cessna Citation 3. What a
testimonial to the evolution of airplane design! Each machine is totally different,
each being the product of three different worlds of scientific and engineering
understanding and practice. One must marvel at the rapid technical progress,
especially in the twentieth century, that brings us to the present status of airplane
design represented by the modern, fast, highflying jet aircraft shown in Figure
6.7. What were the major technical milestones in this progress? What were the
evolutionary (and sometimes revolutionary) developments that swept us from
Cayley's seminal concepts to the modern airplane? The eyeopening and exciting
answers to these questions would require a separate book to relate, but in this
section we highlight a few aspects of the technical progression of airplane design,
using some of the technology we have covered in the present chapter on airplane
performance.
In order to provide a technical focus for our discussion we chose two
aerodynamic parameters as figures of merit to compare and evaluate different
airplane designs. The first is the zerolift drag coefficient CD.o' an important
characteristic of any airplane because it has a strong effect on the maximum flight
speed. Recall that at Vmax for an airplane since the angle of attack (and hence the
induced drag) is small, the total drag given by the drag polar in Eq. (6.lc) is
dominated by CD
0
at high speeds. Everything else being equal, the lower the
CD
0
, the faster the airplane. The other aerodynamic figure of merit highlighted
here is the lifttodrag ratio and especially its maximum value (L/D)max· As we
have already seen, L/ D is a measure of the aerodynamic efficiency of an
airplane, and it affects such flight characteristics as endurance and range. We will
use both CD
0
and (L/D)max to illustrate the historical progress in airplane
design. '
We start with the airplanes of Cayley early in the nineteenth century because
they were the first designs to exemplify the fixedwing heavierthanair aircraft we
know today. Return again to Figure 1.8, showing the first airplane with a modern
configuration, with a fixed wing for lift, a tail for stability, and a fuselage
348 INTRODUCTION TO FLIGHT
connecting the two. The mechanism of propulsion (in this case a hand launch) is
separate from the mechanism of lift. The amount of technical knowledge Cayley
was able to incorporate in his design is best reflected in his famous "triple paper"
of 18091810 (see Sec. 1.3). The technical concepts of CD
0
and L/D did not
exist in Cayley's day, but he reflects a basic intuition about these quantities in his
triple paper. For example, Cayley used a method called newtonian theory (which
will be derived in Chap. 10) to estimate the aerodynamic force on an inclined
plane (the wing). This theory takes into account only the pressure acting on the
surface; surface shear stress (and hence friction drag) was not fully appreciated in
Cayley's time; furthermore there were no methods for its prediction. The newton
ian theory predicts a net force perpendicular to the inclined plane and therefore
contains a component of drag. Cayley makes reference to this "retarding force"
due to the component of the aerodynamic pressure force acting along the flow
direction. In modern terms, we call this component of drag the "drag due to lift"
or more specifically (since only pressure is considered) the "induced drag."
Cayley goes on to say (in discussing the flight of birds), "In addition to the
retarding force thus received is the direct resistance, which the bulk of the bird
opposes to the current. This is a matter to be entered into separately from the
principle now under consideration." Here, Cayley is discussing what we would
today call the parasite drag (sum of pressure drag due to separation and
skinfriction drag) due primarily to the body of the bird. Although Cayley was on
the right track conceptually, he had no method of calculating the parasite drag,
and measurements (made with a whirling arm such as sketched in Figure 1.7)
were wholly unreliable.Therefore, we have no value of CD,o for Cayley's 1804
glider in Figure 1.8.
Although Cayley did not identify and use the concept of L/ D directly, in his
triple paper he refers to his glider sailing "majestically" from the top of a hill,
descending at an angle of about 18° with the horizon. Using the results of Sec.
6.9 dealing with a poweroff glide, we can today quickly calculate that the L/D
ratio for the glider was 3.08, not a very impressible value. Typical values of L/ D
for modern airplanes are 15 to 20 and for modern gliders above 40. Cayley did
not have an efficient airplane nor did he know about aspectratio effects. Today
we know that lowaspect ratio wings such as used by Cayley (aspect ratio about
1) are very inefficient because they produce large amounts of induced drag.
The technical evolution of airplane design after Cayley was gradual and
evolutionary during the remainder of the nineteenth century. The change that
occurred with the Wright Flyer (Figures 1.1 and 1.2) was revolutionary (1)
because the Wrights ultimately relied on virtually no previous data, doing
everything themselves (see Sec. 1.8) and (2) because it was the first successful
flying machine. The aerodynamic quality of the Wright Flyer is discussed by
Culick and Jex, who report modern calculations and measurements of the drag
polar for the Wright Flyer (Figure 6.59). The experimental data were obtained on
a model of the Wright Flyer mounted in a wind tunnel at the California Institute
of Technology. The theoretical data are supplied by a modern vortexlattice
computer program for calculating lowspeed incompressible inviscid flow. (Since
ELEMENTS OF AIRPLANE PERFORMANCE 349
1.2
1.0
0.8
Lift
coefficient
c,
CD = 0.117 Flight trim
c, = 0.62 (Model) lift coefficient
1.2
1.0
0.6 V//ffe//ffeffehti
0.4 •
•
0.2
•
0.04
0.2
•
• 0.2
Drag coefficient CD
Theory
4 8 12 16
a, deg
Figure 6.59 Drag polar and lift curve for the 1903 Wright Flyer. Experimental data are from modem
experiments using models of the Wright Flyer in modern wind tunnels. The vortexlattice theory is a
modem computer calculation. The values of CL, Cv, and a correspond to equilibrium trimmedflight
conditions (see Chap. 7), highlighted by the horizontal bar across the figure. (From Culick and Jex.)
these methods do not include the effects of friction, they cannot be used to
predict flow separation.) The data in Figure 6.59 show that Cv
0
is about 0.12
and the maximum lift coefficient nearly 1.1. Moreover, by drawin'g a straight line
from the origin tangent to the drag polar curve we see that the value of (L/D)max
is about 5.7. By present standards, the Wright Flyer was not an aerodynamic
masterpiece, but in 1903 it was the only successful flying machine in existence.
Moreover, compared with Cayley's airplanes, the Wright Flyer was a revolu
tionary advancement in design.
After the Wright Flyer, advances in airplane design grew almost exponen
tially in the last half of the twentieth century. Using our two figures of merit,
Cv,o and (L/ D)max• we can identify three general periods of progress in airplane
design during the twentieth century, as shown in Figures 6.60 and 6.61. Values of
Cv,o (Figure 6.60) and (L/D)max (Figure 6.61) for representative airplanes are
shown versus time in years. These data are obtained from Loftin, an authoritative
publication which the interested reader is encouraged to examine; it contains
detailed case studies of the technical design of many famous aircraft. The data
for Cv
0
in Figure 6.60 suggest that airplane design has gone through three major
evolutionary periods, distinguished from each other by a dramatic change. For
example, the period of strutandwire biplanes (such as the SPAD XIII shown in
Figure 6.62) extends from the Wright Flyer to the middle or end of the 1920s.
Here, values of Cv
0
are typically on the order of 0.4, a high value due to the
large form drag (pressure drag due to flow separation) associated with the bracing
struts and wires between the two wings of a biplane. In the late 1920s a
revolution in design came with the adoption of the monoplane configuration
coupled with the NACA cowl (see Sec. 6.20). The resulting second period of
350 INTRODUCTION TO FLIGHT
0.6
0.5
0
\..)c:;
0.4
·a
8
0.3
..
'O
0.2
" N
O.l
• Period of
3
strutandwire biplanes
!
2•
Je
4•
Period of mature propellerdriven
monoplanes with NACA cowling
7•
S •6 Be
•
13
• 12 •15
Year
Period of modern
jet airplanes
Figure 6.60 Use of zerolift drag coefficient to illustrate three general periods of twentiethcentury
airplane design. The numbered data points correspond to the following aircraft: (1) SPAD XIII, (2)
Fokker DVII, (3) Curtiss JN4H Jenny, (4) Ryan NYP (Spirit of St. Louis), (5) Lockheed Vega, (6)
Douglas DC3, (7) Boeing B17, (8) Boeing B29, (9) North American P51, (10) Lockheed P80, (11)
North American F86, (12) Lockheed F104, (13) McDonnell F4E, (14) Boeing B52, (15) General
Dynamics FlllD.
22
20
18
16
14
g 12
§:
10
8
6
4
0
1910
Period of
strutandwire
biplanes
1920
10 . . d
• Penod ot mo ern
8
• jet airplanes 15
6. 9. •11
7.
;;r;;;aMej
• 4 propellerdriven
1930
monoplanes with
NACA cowling
1940
Year
1950 1960 1990
Figure 6.61 Use of lifttodrag ratio to illustrate three general periods of twentiethcentury airplane
design.
ELEMENTS OF AIRPLANE PERFORMANCE 351
Figure 6.62 The French SPAD XIII, an example of the strutandwire biplane period. Captain Eddie
Rickenbacker is shown at the front of the airplane. (U.S. Air Force.)
design evolution (exemplified by the DC3 shown in Figure 6.63) is characterized
by Cv
0
values on the order of 0.27. In the mid1940s, the major design rev
o l u t o ~ was the advent of the jetpropelled airplane. This period, which we are
still in today (reflected in the famous F86 of the Korean War era, shown in
Figure 6.64) is represented by Cv,o values on the order of 0.15.
As a note of caution, we add that Cv.o must not be taken as an unfailing
indication of drag performance for the comparison of different airplanes. Since
Figure 6.63 The Douglas DC3, an example of the period of mature propellerdriven monoplanes
with the NACA cowling and wing fillets. (Douglas Aircraft Company.)
352 INTRODUCTION TO FLIGHT
Figure 6.64 The North American F86, one of the most successful modern jet airplanes from the early
1950s. (North American/ Rockwell.)
Cv
0
is obtained by dividing the zerolift drag by wing area (and of course by
qJ, an airplane with an exceptionally high wing area (hence low wing loading
W /S) will result in a low Cv
0
even though the aircraft is an intrinsically
highdrag design. For example, from Figure 6.59, the Wright Flyer has a Cv
0
value of 0.12, which in Figure 6.60 would place it in the same Cv,o range as jet
airplanes. In conjunction with other parameters, however, relative values of Cv,o
allow some judgment of the progress of airplane design over the years.
The use of (L/D)max as an aerodynamic figure of merit is less ambiguous.
As seen in Figure 6.61, where ( L/ D) max is plotted versus years, the data points
for the same airplanes as in Figure 6.60 group themselves in the same three
design periods as deduced from Figure 6.60. Note that compared with the value
of 5.7 for the Wright Flyer, the average value of (L/D)max for World War I
airplanes was about 8not a great improvement. After the introduction of the
monoplane with the NACA cowling, typical (L/D)max values averaged substan
tially higher, on the order of 12 or sometimes considerably above. [The Boeing
B29 bomber of World War II fame had a (L/D)max value of nearly 17, the
highest for this period. This was in part due to the exceptionally large wing
aspect ratio of 11.5 in a period when wing aspect ratios were averaging on the
order of 6 to 8.] Today, (L/D)max values for modern aircraft range over the
whole scale, from 12 or 13 for highperformance military jet fighters to nearly 20
and above for large jet bombers and civilian transports such as the Boeing 747.
This section has given you the chance to think about the progress in aircraft
design in terms of some of the aerodynamic performance parameters discussed in
the present chapter.
ELEMENTS OF AIRPLANE PERFORMANCE 353
6.24 CHAPTER SUMMARY
A few of the important aspects of this chapter are listed below.
1. For a complete airplane, the drag polar is given as
CL2
Cv = Cv.o + 'lTeAR
(6.lc)
where Cv.o is the parasite drag coefficient at zero lift and the term CL
2
/'1TeAR
includes both induced drag and the contribution to parasite drag due to lift.
2. Thrust required for level, unaccelerated flight is
w
TR= L/D (6.15)
Thrust required is a minimum when L/ D is maximum.
3. Power required for level, unaccelerated flight is
2W3Cv2
PooSCL3
Power required is a minimum when CL
3
1
2
/Cv is a maximum.
4. The rate of climb, R/C = dh/dt, is given by
dh TV DV V dV

dt w g dt
(6.26)
(6.129)
where (TV  DV)/W = P
5
, the specific excess power. For an unaccelerated
climb, dV / dt = 0, and hence
R/C = dh = TV  DV
dt w
5. In a poweroff glide, the glide angle is given by
1
tanO = L/D
( 6.43)
(6.47)
6. The absolute ceiling is defined as that altitude where maximum R/C = 0.
The service ceiling is that altitude where maximum R/C = 100 ftjmin.
7. For a propellerdriven airplane, range R and endurance E are given by
T/ CL Wo
R=ln (6.58)
c Cv W
1
and
c 3/2
E = '!!_ _L_(
2
s)112(w 112 _ w, 1;2)
c C Poo i o
D
(6.59)
Maximum range occurs at maximum CL/Cv. Maximum endurance occurs at sea
level with maximum CL
312
/Cv.
354 INTRODUCTION TO FLIGHT
8. For a jetpropelled airplane, range and endurance are given by
{£
1 c112
R = 2  _L_( Wol/2  wll/2)
ct CD
and
9. At maximum CL
312
/CD, cD,0 = tcD, ;· For this case,
(
ci1
2
)
(3CD,
0
?TeAR)
314
D max
At maximum CJCD, CD,o = CD,;· For this case,
( )
1/2
CD,o?TeAR
At maximum CL
112
/CD, CD,o = 3CD,i· For this case
(
c ~ 1 1 2
D max
10. Takeoff ground roll is given by
11. The landing ground roll is
( fCD,o?TeAR)I/4
fCD,o
1.44W
2
1.69W
2
SL= [ ( ]
gpooSCL,max D + µ, W L) ave
12. The load factor is defined as
n = L/W
13. In level turning flight, the turn radius is
v 2
R = oo
g ~
and the turn rate is
w=
(6.68)
(6.63)
(6.78)
(6.76)
(6.77)
(6.93)
(6.101)
(6.105)
( 6.108)
(6.109)
14. The Vn diagram is illustrated in Figure 6.49. It is a diagram showing load
factor vs. velocity for a given airplane, along with the constraints on both n
and V due to structural limitations. The Vn diagram illustrates some par
ticularly important aspects of overall airplane performance.
ELEMENTS OF AIRPLANE PERFORMANCE 355
15. The energy height (specific energy) of an airplane is given by
v2
H =h+
e 2g
This, in combination with the specific excess power,
TVDV
P=
s
w
(6.126)
leads to the analysis of acceleratedclimb performance using energy consider
ations only.
BIBLIOGRAPHY
Bairstow, L., Applied Aerodynamics, Longmans, London, 1920.
Cowley, W. L., and Levy, H., Aeronautics in Theory and Experiment, E. Arnold, London, 1918.
Culick, F. E. C., and Jex, "Aerodynamics, Stability and Control of the 1903 Wright Flyer," pp. 1943
in Howard Wolko (ed.), The Wright Flyer; An Engineering Perspective, Smithsonian Press,
Washington, 1987.
Dommasch, D. 0., Sherbey, S.S., and Connolly, T. F., Airplane Aerodvnamics, 3rd ed., Pitman, New
York, 1961.
Duchene, Captain, The Mechanics of the Airplane: A Study of the Principles of Flight (transl. by J. H.
Ledeboer and T. O'B. Hubbard), Longmans, London, 1917.
Hale, F. J., Introduction to Aircraft Performance, Selection and Design, Wiley, New York, 1984.
Judge, A. W., Handbook of Modern Aeronautics, Appleton, London, 1919.
Loftin, Lawrence, Quest for Performance: The Evolution of Modern Aircraft, NASA SP468, 1985.
McCormick, B. W., Aerodynamics, Aeronautics and Flight Mechanics, Wiley, New York, 1979.
Perkins, C. D., and Hage, R. E., Airplane Performance, Stability and Control, Wiley, New York, 1949.
Shevell, R. S., Fundamentals of Flight, PrenticeHall, Englewood Cliffs, NJ, 1983.
PROBLEMS
6.1 Consider an airplane patterned after the twinengine Beechcraft Queen Air executive transport.
The airplane weight is 38,220 N, wing area is 27.3 m2, aspect ratio is 7.5, Oswald efficiency factor is
0.9, and parasite drag coefficient Cv.o = 0.03. Calculate the thrust required to fly at a velocity of 350
km/h at (a) standard sea level and (b) an altitude of 4.5 km.
6.2 An airplane weighing 5000 lb is flying at standard sea level with a velocity of 200 mijh. At this
velocity, the L/ D ratio is a maximum. The wing area and aspect ratio are 200 ft
2
and 8.5, re
spectively. The Oswald efficiency factor is 0.93. Calculate the total drag on the airplane.
6.3 Consider an airplane patterned after the Fairchild Republic A10, a twinjet attack aircraft. The
airplane has the following characteristics: wing area= 47 m
2
, aspect ratio= 6.5, Oswald efficiency
factor = 0.87, weight = 103,047 N, and parasite drag coefficient = 0.032. The airplane is equipped
with two jet engines with 40,298 N of static thrusts each at sea level.
(a) Calculate and plot the powerrequired curve at sea level.
(b) Calculate the maximum velocity at sea level.
(c) Calculate and plot the powerrequired curve at 5 km altitude.
(d) Calculate the maximum velocity at 5 km altitude. (Assume the engine thrust varies directly
with freestream density.)
6.4 Consider an airplane patterned after the Beechcraft Bonanza Vtailed, singleengine light private
airplane. The characteristics of the airplane are as follows: aspect ratio = 6.2, wing area = 181 ft
2
,
356 INTRODUCTION TO FLIGHT
Oswald efficiency factor = 0.91, weight = 3000 lb, and parasite drag coefficient = 0.027. The airplane
is powered by a single piston engine of 345 hp maximum at sea level. Assume the power of the engine
is proportional to freestream density. The twobladed propeller has an efficiency of 0.83.
(a) Calculate the power required at sea level.
(b) Calculate the maximum velocity at sea level.
(c) Calculate the power required at 12,000 ft altitude.
(d) Calculate the maximum velocity at 12,000 ft altitude.
6.5 From the information generated in Prob. 6.3, calculate the maximum rate of climb for the
twinjet aircraft at sea level and at an altitude of 5 km.
6.6 From the information generated in Prob. 6.4, calculate the maximum rate of climb for the
singleengine light plane at sea level and at 12,000 ft altitude.
6.7 From the rate of climb information for the twinjet aircraft in Prob. 6.5, estimate the absolute
ceiling of the airplane. (Note: Assume maximum R/C varies linearly with altitudenot a precise
assumption, but not bad, either.)
6.8 From the rate of climb information for the singleengine light plane in Prob. 6.6, estimate the
absolute ceiling of the airplane. (Again, make the linear assumption described in Prob. 6.7.)
6.9 The maximum lifttodrag ratio of the World War I Sopwith Camel was 7.7. If the aircraft is in
flight at 5000 ft when the engine fails, how far can it glide in terms of distance measured along the
ground?
6.10 For the Sopwith Camel in Prob. 6.9 calculate the equilibrium glide velocity at 3000 ft,
corresponding to the minimum glide angle. The aspect ratio of the airplane is 4.11, the Oswald
efficiency factor is 0.7, the weight is 1400 lb, and the wing area is 231 ft
2
.
6.11 Consider an airplane with a parasite drag coefficient of 0.025, an aspect ratio of 6.72, and an
Oswald efficiency factor of 0.9. Calculate the value of (L/D)max·
6.12 Consider the singleengine light plane described in Prob. 6.4. If the specific fuel consumption is
0.42 lb of fuel per horsepower per hour, the fuel capacity is 44 gallons, and the maximum gross weight
is 3400 lb, calculate the range and endurarice at standard sea level.
6.13 Consider the twinjet airplane described in Prob. 6.3. The thrustspecific fuel consumption is
1.0 N of fuel per Newton of thrust per hour, the fuel capacity is 1900 gallons, and the maximum gross
weight is 136,960 N. Calculate the range and endurance at a standard altitude of 8 km.
6.14 Derive Eqs. (6.71) and (6.72).
6.15 Derive Eqs. (6.77) and (6.78).
6.16 Estimate the sealevel liftoff distance for the airplane in Prob. 6.3. Assume a paved runway.
Also, during the ground roll, the angle of attack is restricted by the requirement that the tail not drag
the ground. Hence, assume CL max during the grouhd roll is limited to 0.8. Also, when the airplane is
on the ground, the wings are 5 ft above the ground.
6.17 Estimate the sealevel liftoff distance for the airplane in Prob. 6.4. Assume a paved runway, and
CL, max = 1.1 during the ground roll. When the airplane is on the ground, the wings are 4 ft above the
ground.
6.18 Estimate the sealevel landing ground roll distance for the airplane in Prob. 6.3. Assume the
airplane is landing at full gross weight. The maximum lift coefficient with flaps fully employed at
touchdown is 2.8. After touchdown, assume zero lift.
6.19 Estimate the sealevel landing ground roll distance for the airplane in Prob. 6.4. Assume the
airplane is landing with a weight of 2900 lb. The maximum lift coefficient with flaps at touchdown is
1.8. After touchdown, assume zero lift.
6.20 For the airplane in Prob. 6.3, the sealevel comer velocity is 250 mijh, and the maximum lift
coefficient with no flap deflection is 1.2. Calculate the minimum tum radius and maximum turn rate
at sea level.
6.21 The airplane in Prob. 6.3 is flying at 15,000 ft with a velocity of 375 mijh. Calculate its specific
energy at this condition.
CHAPTER
SEVEN
PRINCIPLES OF STABILITY
AND CONTROL
An important problem to aviation is ... improvement in the form of the aeroplane leading toward
natural inherent stability to such a degree as to relieve largely the attention of the pilot while still
retaining sufficient flexibility and control to maintain any desired path, without seriously impairing
the efficiency of the design.
From the First Annual Report of the NA CA, 1915
7.1 INTRODUCTION
The scene: A French army drill field at IssylesMoulineauxjust outside Paris. The
time: The morning of January 13, 1908. The character: Henri Farman, a bearded,
Englishborn but Frenchspeaking aviator, who had flown for his first time just
four months earlier. The action: A delicately constructed VoisinFarman Ibis
biplane (see Figure 7.1) is poised, ready for the takeoff in the brisk Parisian wind,
with Farman seated squarely in front of the 50hp Antoinette engine. The winds
ripple the fabric on the Voisin's box kiteshaped tail as Farman powers to a
bumpy liftoff. Fighting against a head wind, he manipulates his aircraft to a
marker 1000 m from his takeoff point. In a struggling circular turn, Farman
deflects the rudder and mushes the biplane around the marker, the wings
remaining essentially level to the ground. Continuing in its rather wide and
tenuous circular arc, the airplane heads back. Finally, Farman lands at his
original takeoff point, amid cheers from the crowd that had gathered for the
occasion. Farman has been in the air for 1 min and 28 sthe longest flight in
Eurppe to that dateand has just performed the first circular flight of 1km
extent. For this, he is awarded the Grand Prix d'Aviation. (Coincidentally, in the
crowd is a young Hungarian engineer, Theodore von Karman, who is present only
due to the insistence of his female companipnwaking at 5 A.M. in order to see
history be made. However, von Karman is mesmerized by the flight, and his
357
358 INTRODUCTION TO FLIGHT
Figure 7.1 The VoisinFarman Ibis plane. (National Air and Space Museum.)
interest in aeronautical science is catalyzed. Von Karman will go on to become a
leading aerodynamic genius of the first half century of powered flight.)
The scene shifts to a small race track near Le Mans, France. The time: Just
seven months later, August 8, 1908. The character: Wilbur Wright, intense,
reserved, and fully confident. The action: A new Wright type A biplane (see
Figure 1.22), shipped to France in crates and assembled in a friend's factory near
Le Mans, is ready for flight. A crowd is present, enticed to the field by much
advance publicity and an intense curiosity to see if the "rumors" about the
Wright brother<>' reported success were really true. Wilbur takes off. Using the
Wrights' patented concept of twisting the wingtips ("wing warping"), Wilbur is
able to bank and turn at will. He makes two graceful circles and then effortlessly
lands after 1 min and 45 s of flight. The crowds cheer. The French press is almost
speechless but then heralds the flight as epochmaking. The European aviators
who witness this demonstration gaze in amazement and then quickly admit that
the Wrights' airplane is far advanced over the best European machines of that
day. Wilbur goes on to make 104 flights in France before the end of the year and
in the process transforms the direction of aviation in Europe.
The distinction between the two scenes above, and the reason for Wilbur's
mastery of the air in comparison to Farman's struggling circular flight, involves
stability and control. The Voisin aircraft of Farman, which represented the best
European state of the art, had only rudder control and could make only a
laborious, flat turn by simply swinging the tail around. In contrast, the Wright
airplane's wingtwisting mechanism provided control of roll, which when com
bined with rudder control, allowed effortless turning and banking flight, figure
eights, etc. Indeed, the Wright brothers were "airmen" (see Chap. 1) who
concentrated on designing total control into their aircraft before adding an engine
for powered flight. Since those early days, airplane stability and control has been
a dominant aspect of airplane design. This is the subject of the present chapter.
PRINCIPLES OF STABILITY AND CONTROL 359
Roll
x
u
~ L', p
~
Pitch
M,Q
y
Yaw
z
Figure 7.2 Definition of the airplane's axes along with the translational and rotational motion along
and about these axes.
Airplane performance, as discussed in Chap. 6, is governed by forces (along and
perpendicular to the flight path), with the translational motion of the airplane as a
response to these forces. In contrast, airplane stability and control, discussed in the
present chapter, are governed by moments about the center of gravity, with the rota
tional motion of the airplane as a response to these moments. Therefore, moments
and rotational motion are the main focus of this chapter.
Consider an airplane in flight, as sketched in Figure 7.2. The center of gravity
(the point through which the weight of the complete airplane effectively acts) is
denoted as cg. The xyz orthogonal axis system is fixed relative to the airplane; the
x axis is along the fuselage, the y axis is along the wingspan perpendicular to the
x axis, and the z axis is directed downward, perpendicular to the xy plane.
The origin is at the center of gravity. The translational motion of the airplane is
given by the velocity components U, V, and W along the x, y, and z directions,
respectively. (Note that the resultant freestream velocity V
00
is the vector sum of
U, V, and W.) The rotational motion is given by the angular velocity components
P, Q, and R about the x, y, z axes, respectively. These rotational velocities are
due to the moments L', M, and N about the x, y, and z axes, respectively. (The
prime is put over the symbol "L" so that the reader avoids confusing it with lift.)
Rotational motion about the x axis is called roll; L' and P are the rolling
360 INTRODUCTION TO FLIGHT
moment and veloeity, respectively. Rotational motion about the y axis is called
pitch; M and Q are the pitching moment and velocity, respectively. Rotational
motion about the z axis is called yaw; N and R are the yawing moment and
velocity, respectively.
There are three basic controls on an airplanethe ailerons, elevator, and
rudderwhich are designed to change and control the moments about the x, y,
and z axes. These control surfaces are shown in Figure 7.3; they are flaplike
surfaces that can be deflected back and forth at the command of the pilot. The
ailerons are mounted at the trailing edge of the wing, near the wingtips. The
elevators are located on the horizontal stabilizer. In some modern aircraft, the
complete horizontal stabilizer is rotated instead of just the elevator (socalled
flying tails). The rudder is located on the vertical stabilizer, at the trailing edge.
Just as in the case of wing flaps discussed in Sec. 5.17, a downward deflection of
the control surface will increase the lift of the wing or tail. In turn, the moments
will be changed, as sketched in Figure 7.4. Consider Figure 7.4a. One aileron is
deflected up, the other down, creating a differential lifting force on the wings, thus
contributing to the rolling moment L'. In Figure 7.4b, the elevator is deflected
Aileron
Wing
Elevators
Horizontal stabilizer
Aileron
Vertical stabilizer
Figure 7.3 Some airplane nomenclature.
PRINCIPLES OF STABILITY AND CONTROL 361
Aileron up
Aileron down
(a)
Elevator up
(b)
(c)
Figure 7.4 Effect of control deflections on roll, pitch, and yaw. (a) Effect of aileron deflection; lateral
control. ( b) Effect of elevator deflection; longitudinal control. ( c) Effect of rudder deflection;
directional control.
upward, creating a negative lift at the tail, thus contributing to the pitching
moment M. In Figure 7.4c, the rudder is deflected to the right, creating a leftward
aerodynamic force on the tail, thus contributing to the yawing moment N.
Rolling (about the x axis) is also called lateral motion. Referring to Figure
1.4a, we see that ailerons control roll; hence they are known as lateral controls.
Pitching (about the y axis) is also called longitudinal motion. In Figure 1.4b, we
see that elevators control pitch; hence they are known as longitudinal controls.
Yawing (about the z axis) is also called directional motion. Figure 7.4c shows that
the rudder controls yaw; hence it is known as the directional control.
All the above definitions and concepts are part of the basic language of
airplane stability and control; they should be studied carefully. Also, in the
362 INTRODUCTION TO FLIGHT
process, the following question becomes apparent: what is meant by the words
"stability and control" themselves? This question is answered in the next section.
7.2 DEFINITION OF STABILITY AND CONTROL
There are two types of stability: static and dynamic. They can be visualized as
follows.
A Static Stability
Consider a marble on a curved surface, such as a bowl. Imagine that the bowl is
upright and the marble is resting inside, as shown in Figure 7.5a. The marble is
stationary; it is in a state of equilibrium, which means that the moments acting on
the marble are zero. If the marble is now disturbed (moved to one side, as shown
by the dotted circle in Figure 7.5a) and then released, it will roll back toward the
bottom of the bowl, i.e., toward its original equilibrium position. Such a system is
statically stable. In general, we can state that
If the forces and moments on the body caused by a disturbance tend initially to
return the body toward its equilibrium position, the body is statically stable. The
body has positive static stability.
Now, imagine the bowl is upsidedown, with the marble at the crest, as shown
in Figure 7.5b. If the marble is placed precisely at the crest, the moments will be
zero and the marble will be in equilibrium. However, if the marble is now
(a)
(b)
777 777777777
(c)
Figure 7.5 Illustration of static stability. (a) Statically stable system. ( h) Statically unstable system.
( c) Statically neutral system.
PRINCIPLES OF STABILITY AND CONTROi, 363
disturbed (as shown by the dotted circle in Figure 7.5b), it will tend to roll down
the side, away from its equilibrium position. Such a system is statically unstable.
In general, we can state that
If the forces and moments are such that the body continues to move away from
its equilibrium position after being disturbed, the body is statically unstable.
The body has negative static stability.
Finally, imagine the marble on a flat horizontal surface, as shown in Figure
7.5c. Its moments are zero; it is in equilibrium. If the marble is now disturbed to
another location, the moments will still be zero, and it will still be in equilibrium.
Such a system is neutrally stable. This situation is rare in flight vehicles, and we
will not be concerned with it here.
Emphasis is made that static stability (or the lack of it) deals with the initial
tendency of a vehicle to return to equilibrium (or to diverge from equilibrium)
after being disturbed. It says nothing about whether it ever reaches its equilibrium
position, nor how it gets there. Such matters are the realm of dynamic stability, as
follows.
B Dynamic Stability
Dynamic stability deals with the time history of the vehicle's motion after it
initially responds to its static stability. For example, consider an airplane flying at
an angle of attack ae such that its moments about the center of gravity are zero.
The airplane is therefore in equilibrium at ae; in this situation, it is trimmed, and
ae is called the trim angle of attack. Now assume that the airplane is disturbed
(say, by encountering a wind gust) to a new angle of attack a as shown in Figure
7.6. The airplane has been pitched through a displacement a  ae. Now, let us
observe the subsequent pitching motion after the airplane has been disturbed by
the gust. We can describe this motion by plotting the instantaneous displacement
vs. time, as shown in Figure 7.7. Here a  ae is given as a function of time t. At
t = 0, the displacement is equal to that produced by the gust. If the airplane is
Figure 7.6 Disturbance from the equilibrium angle of attack.
364 INTRODUCTION TO FLIGHT
Initial
disturbance
(a) (b)
Figure 7.7 Examples of dynamic stability. (a) Aperiodic; (b) Damped oscillations.
statically stable, it will initially tend to move back toward its equilibrium position,
that is, a  ae will initially decrease. Over a lapse of time, the vehicle may
monotonically "homein" to its equilibrium position, as shown in Figure 7.7a.
Such motion is called aperiodic. Alternately, it may first overshoot the equi
librium position and approach ae after a series of oscillations with decreasing
amplitude, as shown in Figure 7.7b. Such motion is described as damped
oscillations. In both situations, Figures 7.7a and 7.7b, the airplane eventually
returns to its equilibrium position after some interval of time. These two situa
tions are examples of dynamic stability in an airplane. Thus, we can state that
A body is dynamically stable if, out of its own accord, it eventually returns to
and remains at its equilibrium position over a period of time.
On the other hand, after initially responding to its static stability, the airplane
may oscillate with increasing amplitude, as shown in Figure 7.8. Here, the
equilibrium position is never maintained for any period of time; the airplane in
this case is dynamically unstable (even though it is statically stable). Also, it is
theoretically possible for the airplane to pitch back and forth with constant
Increasing oscillations
Figure 7.8 An example of dynamic instability.
PRINCIPLES OF STABILITY AND CONTROL 365
amplitude oscillations. This is an example of a dynamically neutral body; such a
case is of little practical interest here.
It is important to observe from the above examples that a dynamically stable
airplane must always be statically stable. On the other hand, static stability is not
sufficient to ensure dynamic stability. Nevertheless, static stability is usually the
first stability characteristic to be designed into an airplane. (There are some
exceptions, to be discussed later.) Such considerations are of paramount impor
tance in conventional airplanes, and therefore most of the present chapter will
deal with static stability and control. A study of dynamic stability, although of
great importance, requires rather advanced analytical techniques beyond the
scope of this book.
C Control
The conventional control surfaces (elevators, ailerons, and rudder) on an airplane
were discussed in Sec. 7 .1 and sketched in Figures 7 .3 and 7.4. Their function is
usually (1) to change the airplane from one equilibrium position to another and
(2) to produce nonequilibrium accelerated motions such as maneuvers. The study
of the deflections of the ailerons, elevators, and rudder necessary to make the
airplane do what we want and of the amount of force that must be exerted by the
pilot (or the hydraulic boost system) to deflect these controls is part of a discipline
called "airplane control," to be discussed later in this chapter.
D The Partial Derivative
Some physical definitions associated with stability and control have been given
above. In addition, a mathematical definition, namely, that of the partial deriva
tive, will be useful in the equations developed later, not only in this chapter but in
our discussion of astronautics (Chap. 8) as well. For those readers with only a
nodding acquaintance of calculus, hopefully this section will be selfexplanatory;
for those with a deeper calculus background, this should serve as a brief review.
Consider a function, say /(x), of a single variable x. The derivative of /(x) is
defined from elementary calculus as
df = lim (f(x + ~ x  /(x))
dx ~ x   o ~ x
Physically, this limit represents the instantaneous rate of change of f(x) with
respect to x.
Now consider a function which depends on more than one variable, say, for
example, the function g(x, y, z), which depends on the three independent vari
ables x, y, and z. Let x vary while y and z are held constant. Then, the
instantaneous rate of change of g with respect to x is given by
a g = lim ( g ( x + ~ x , y, z)  g ( x' y' z) )
ax ~ x   o ~ x
366 INTRODUCTION TO FLIGHT
Here, ag/ ax is the partial derivative of g with respect to x. Now, let y vary
while holding x and z constant. Then the instantaneous rate of change of g with
respect to y is given by
ag = lim
ay ily+O
Here, ag/ a y is the partial derivative of g with respect to y. An analogous
definition holds for the partial derivative with respect to z, ag;az.
In this book, we will use the concept of the partial derivative as a definition
only. The calculus of partial derivatives is essential to the advanced study of
virtually any field of engineering, but such considerations are beyond the scope of
this book.
Example 7.1 If g = x
2
+ y
2
+ z
2
, calculate Bg/ az.
SOLUTION From the definition given above, the partial derivative is taken with respect to z
holding x and y constant.
ag a(x
2
+y
2
+z
2
) ax
2
ay
2
az
2
az = iiz =;;;+;;;+a;
= 0+0+2z = 2z
7.3 MOMENTS ON THE AIRPLANE
A study of stability and control is focused on moments: moments on the airplane
and moments on the control surfaces. At this stage, it would be well for the reader
to review the discussion of aerodynamically produced moments in Sec. 5.2. Recall
that the pressure and shear stress distributions over a wing produce a pitching
moment. This moment can be taken about any arbitrary point (the leading edge,
the trailing edge, the quarter chord, etc.). However, there exists a particular point
about which the moments are independent of angle of attack. This point is
defined as the aerodynamic center for the wing. The moment and its coefficient
about the aerodynamic center are denoted by Mac and CM,ac• respectively, where
CM,ac = Mac/qooSc.
Reflecting again on Sec. 5.2, consider the force diagram of Figure 5.4.
Assume the wing is flying at zero lift; hence F
1
and F
2
are equal and opposite
forces. Thus, the moment established by these forces is a pure couple, which we
know from elementary physics can be translated anywhere on the body at
constant value. Therefore, at zero lift, Mac = Mc
14
= Many point· In tum
CM,ac = ( CM,c/4) L=O = ( CM,anypoint) L=O
This says that the value of CM ac (which is constant for angles of attack) can be
obtained from the value of the coefficient about any point when the wing
is at the zerolift angle of attack aL=o· For this reason, Mac is sometimes called
the zerolift moment.
PRINCIPLES OF STABILITY AND CONTROL 367
Lwing
T cg
w
Figure 7.9 Contributions to the moment about the center of gravity of the airplane.
The aerodynamic center is a useful concept for the study of stability and
control. In fact, the force and moment system on a wing can be completely
specified by the lift and drag acting through the aerodynamic center, plus the
moment about the aerodynamic center, as sketched in Figure 7.9. We will adopt
this convention for the remainder of the present chapter.
Now consider the complete airplane, as sketched in Figure 7.9. Here, we are
most concerned with the pitching moment about the center of gravity of the
airplane, Meg· Clearly, by examination of Figure 7.9, Meg is created by (1) L, D,
and Mac of the wing, (2) lift of the tail, (3) thrust, and (4) aerodynamic forces and
moments on other parts of the airplane, such as the fuselage and engine nacelles.
(Note that weight does not contribute, since it acts through the center of gravity.)
These contributions to Meg will be treated in detail later. The purpose of Figure
7.9 is simply to illustrate the important conclusion that a moment does exist
about the center of gravity of an airplane, and it is this moment which is
fundamental to the stability and control of the airplane.
The moment coefficient about the center of gravity is defined as
(7 .1)
Combining the above concept with the discussion of Sec. 7.2, we find an airplane
is in equilibrium (in pitch) when the moment about the center of gravity is zero;
i.e., when Meg= CM.cg= 0, the airplane is said to be trimmed.
7.4 ABSOLUTE ANGLE OF ATTACK
Continuing with our collection of tools to analyze stability and control, consider a
wing at an angle of attack such that lift is zero; i.e., the wing is at the zerolift
angle of attack a ~ o • as shown in Figure 7.lOa. With the wing in this orientation,
draw a line through the trailing edge parallel to the relative wind V
00
• This line is
368 INTRODUCTION TO FLIGHT
(a)
C<a
J_
(b)
Figure 7.10 Illustration of the zerolift line and absolute angle of attack. (a) No lift; ( b) with lift.
defined as the zerolift line for the airfoil. It is a fixed line; visualize it frozen into
the geometry of the airfoil, as sketched in Figure 7.lOa. As discussed in Chap. 5,
conventional cambered airfoils have slightly negative zerolift angles; therefore
the zerolift line lies slightly above the chord line, as shown (with overemphasis)
in Figure 7.lOa.
Now consider the wing pitched to the geometric angle of attack a such that
lift is generated, as shown in Figure 7.lOb. (Recall from Chap. 5 that the
geometric angle of attack is the angle between the freestream relative wind and
the chord line.) In the same configuration, Figure 7.lOb demonstrates that the
angle between the zerolift line and the relative wind is equal to the sum of a plus
the absolute value of aL=o· This angle is defined as the absolute angle of attack
aa. From Figure 7.lOb, aa =a+ aL=o (using aL=O in an absolute sense). The
geometry of Figures 7.lOa and 7.lOb should be studied carefully until the concept
of aa is clearly understood.
The definition of the absolute angle of attack has a major advantage. When
aa = 0, then L = 0, no matter what the camber of the airfoil. To further
illustrate, consider the lift curves sketched in Figure 7.11. The conventional plot
(as discussed in detail in Chap. 5), CL versus a, is shown in Figure 7.lla. Here,
the lift curve does not go through the origin, and of course aL=O is different for
different airfoils. In contrast, when CL is plotted versus aa, as sketched in Figure
7.llb, the curve always goes through the origin (by definition of aa). The curve in
Figure 7.llb is identical to that in Figure 7.lla; only the abscissa has been
translated by the value aL=O·
The use of aa in lieu of a is common in studies of stability and control. We
will adopt this convention for the remainder of this chapter.
PRINCIPLES OF STABILITY AND CONTROL 369
CL
(a) (b)
Figure 7.11 Lift coefficient vs. (a) geometric angle of attack and (b) absolute angle of attack.
7.5 CRITERIA FOR LONGITUDINAL STATIC STABILITY
Static stability and control about all three axes shown in Figure 7.2 is usually a
necessity in the design of conventional airplanes. However, a complete description
of all three typeslateral, longitudinal, and directional static stability and
control (see Figure 7.4)is beyond the scope of this book. Rather, the intent here
is to provide only the flavor of stability and control concepts, and to this end only
the airplane's longitudinal motion (pitching motion about the y axis) will be
considered. This pitching motion is illustrated in Figure 7.4b. It takes place in the
plane of symmetry of the airplane. Longitudinal stability is also the most
important static stability mode; in airplane design, windtunnel testing, and flight
research, it usually earns more attention than lateral or directional stability.
Consider a rigid airplane with fixed controls, e.g., the elevator in some fixed
position. Assume the airplane has been tested in a wind tunnel or free flight and
that its variation of Meg with angle of attack has been measured. This variation is
illustrated in Figure 7.12, where CM.cg is sketched versus aa. For many conven
tional airplanes, the curve is nearly linear, as shown in Figure 7.12. The value of
CM.cg at zero lift (where <Xa = 0) is denoted by cM,O· The value of CY.a where
(+)
()
Figure 7.12 Moment coefficient curve with a negative slope.
370 INTRODUCTION TO FLIGHT
CM, cg is negative
CM, cg= 0
(a)
(c)
Figure 7. 13 Illustration of static stability: (a) Equilibrium position (trimmed). ( h) Pitched upward by
disturbance. ( c) Pitched downward by disturbance. In both h and c, the airplane has the initial
tendency to return to its equilibrium position.
Meg = 0 is denoted by ae; as stated in Sec. 7.3, this is the equilibrium, or trim,
angle of attack.
Consider the airplane in steady, equilibrium flight at its trim angle of attack
ae as shown in Figure 7.13a. Suddenly, the airplane is disturbed by hitting a wind
gust, and the angle of attack is momentarily changed. There are two possibilities:
an increase or a decrease in aa. If the airplane is pitched upward, as shown in
Figure 7.13b, then aa > ae. From Figure 7.12, if aa > ae, the moment about the
center of gravity is negative. As discussed in Sec. 5.4, a negative moment (by
convention) is counterclockwise, tending to pitch the nose downward. Hence, in
Figure 7.13b, the airplane will initially tend to move back toward its equilibrium
position after being disturbed. On the other hand, if the plane is pitched
downward by the gust, as shown in Figure 7.13c, then aa < ae. From Figure 7.12,
the resulting moment about the center of gravity will be positive (clockwise) and
will tend to pitch the nose upward. Thus, again we have the situation where the
airplane will initially tend to move back toward its equilibrium position after
being disturbed. From Sec. 7.2, this is precisely the definition of static stability.
Therefore, we conclude that an airplane which has a CM,cg versus aa variation
like that shown in Figure 7.12 is statically stable. Note from Figure 7.12 that cM,O
is positive and that the slope of the curve, ac M,cg/ aaa, is negative. Here, the
partial derivative, defined in Sec. 7.2 D, is used for the slope of the moment
coefficient curve. This is because (as we shall see) CM,cg depends on a number of
PRINCIPLES OF STABILITY AND CONTROL 371
Figure 7.14 Moment coefficient curve with
a positive slope.
other variables in addition to a.a, and therefore it is mathematically proper to use
acM.cg/aaa rather than dCM.cg/daa to represent the slope of the line in Figure
7.12. As defined in Sec. 7.2 D, acM.cJ aaa symbolizes the instantaneous rate of
change of CM,cg with respect to a.a, with all other variables held constant.
Consider now a different airplane, with a measured CM.cg variation as shown
in Figure 7.14. Imagine the airplane is flying at its trim angle of attack ae as
shown in Figure 7.15a. If it is disturbed by a gust, pitching the nose upward, as
shown in Figure 7.15b, then a.a > ae. From Figure 7.14 this results in a positive
(clockwise) moment, which tends to pitch the nose even further away from its
CM, cg is positive
(a)
(/J)
CM, cg is negative
(c)
Figure 7.15 Illustration of static instability. (a) Equilibrium position (trimmed). (h) Pitched upward
by disturbance. ( c) Pitched downward by disturbance. In both h and c, the airplane has the initial
tendency to diverge farther away from its equilibrium position.
372 INTRODUCTION TO FLIGHT
equilibrium position. Similarly, if the gust pitches the nose downward (Figure
7.15c), a negative (counterclockwise) moment results, which also tends to pitch
the nose further away from its equilibrium position. Therefore, because the
airplane always tends to diverge from equilibrium when disturbed, it is statically
unstable. Note from Figure 7.14 that cM,O is negative and acM,cg/aaa is positive
for this airplane.
For both airplanes, Figures 7.12 and 7.14 show a positive value of ae. Recall
from Figure 6.4 that an airplane moves through a range of angle of attack as it
flies through its velocity range from V.1a11 (where aa is the largest) to vmax (where
aa is the smallest). The value of ae must fall within this flight range of angle of
attack, or else the airplane cannot be trimmed for steady flight. (Remember that
we are assuming a fixed elevator position: we are discussing "stickfixed" stabil
ity.) When ae does fall within this range, the airplane is longitudinally balanced.
From the above considerations, we conclude the following.
The necessary criteria for longitudinal balance and static stability are:
1. CM,O must be positive.
2. BCM,cg/Baa must be negative.
That is, the CM.cg curve must look like Figure 7.12.
Of course, implicit in the above criteria is that ae must also fall within the flight
range of angle of attack for the airplane.
We are now in a position to explain why a conventional airplane has a
horizontal tail (the horizontal stabilizer shown in Figure 7.3). First, consider an
ordinary wing (by itself) with a conventional airfoil, say an NACA 2412 section.
Note from the airfoil data in Appendix D that the moment coefficient about
aerodynamic center is negative. This is characteristic of all airfoils with positive
camber. Now, assume that the wing is at zero lift. In this case, the only moment
on the wing is a pure couple, as explained in Sec. 7.3; hence, at zero lift the
moment about one point is equal to the moment about any other point. In
particular,
for zero lift (wing only) (7.2)
On the other hand, examination of Figure 7.12 shows that CM,o is by definition,
the moment coefficient about the center of gravity at zero lift (when aa = 0).
Hence, from Eq. (7.2)
(wing only) (7 .3)
Equation (7.3) demonstrates that, for a wing with positive camber ( CM,ac nega
tive) CM,o is also negative. Hence, such a wing by itself is unbalanced. To rectify
this situation, a horizontal tail must be added to the airplane, as shown in Figures
7.16a and 7.16b. If the tail is mounted behind the wing, as shown in Figure 7.16a,
and if it is inclined downward to produce a negative tail lift as shown, then a
PRINCIPLES OF STABILITY AND CONTROL J73
Center of gravity of airplane
 . ~
Positively cambered wing at Ci = 0 Tail with negative lift
(a)
Tail with positive lift Positively cambered wing at Ci = 0
(b)
Figure 7.16 (a) Conventional wingtail combination. The tail is set at such an angle as to produce
negative lift, thus providing a positive CM.o· (h) Canard wingtail combination. The tail is set at such
an angle as to produce positive lift, thus providing a positive CM.O·
clockwise moment about the center of gravity will be created. If this clockwise
moment is strong enough, it will overcome the negative CM.ac• and CM,o for the
wingtail combination will become positive. The airplane will then be balanced.
The arrangement shown in Figure 7.16a is characteristic of most conven
tional airplanes. However, the tail can also be placed ahead of the wing, as shown
in Figure 7.16b; this is called a canard configuration. For a canard, the tail is
inclined upward to produce a positive lift, hence creating a clockwise moment
about the center of gravity. If this moment is strong enough, CM,o for the
wingtail combination will become positive and again the airplane will be bal
anced. Unfortunately, the forwardlocated tail of a canard interferes with the
smooth aerodynamic flow over the wing. For this and other reasons, canard
configurations have not been popular. Of course, a notable exception were the
Wright Flyers, which were canards. In fact, it was not until 1910 that the Wright
brothers went to a conventional arrangement. Using the word "rudder" to mean
elevator, Orville wrote to Wilbur in 1909 that "the difficulty in handling our
machine is due to the rudder being in front, which makes it hard to keep on a
level course .... I do not think it. is necessary to lengthen the machine, but to
simply put the rudder behind instead of before." Originally, the Wrights thought
the forwardlocated elevator would help to protect them from the type of fatal
crash encountered by Lilienthal. This rationale persisted until the design of their
model Bin 1910. Finally, a modern example of a canard is the North American
XB70, an experimental supersonic bomber developed for the Air Force in the
1960s. The canard surfaces ahead of the wing are clearly evident in the photo
graph shown in Figure 7.17.
374 INTRODUCTION TO FLIGHT
Figure 7.17 The North American XB70. Note the canard surfaces immediately behind the cockpit.
(Rockwell International Corp.)
In retrospect, using essentially qualitative arguments based on physical
reasoning and without resort to complicated mathematical formulas, we have
developed some fundamental results for longitudinal static stability. Indeed, it is
somewhat amazing how far our discussion has progressed on such a qualitative
basis. However, we now turn to some quantitative questions. For a given airplane,
how far should the wing and tail be separated in order to obtain stability? How
large should the tail be made? How do we design for a desired trim angle ae?
These and other such questions are addressed in the remainder of this chapter.
7.6 QUANTITATIVE DISCUSSION: CONTRIBUTION OF
THE WING TO Meg
The calculation of moments about the center of gravity of the airplane, Meg' is
critical to a study of longitudinal static stability. The previous sections have
already underscored this fact. Therefore, we now proceed to consider individually
the contributions of the wing, fuselage, and tail to moments about the center of
gravity of the airplane, in the end combining them to obtain the total Meg·
Consider the forces and moments on the wing only, as shown in Figure 7.18.
Here, the zerolift line is drawn horizontally for convenience; hence, the relative
wind is inclined at the angle aw with respect to the zerolift line, where aw is the
absolute angle of attack of the wing. Let c denote the mean zerolift chord of
the wing (the chord measured along the zerolift line). The difference between
PRINCIPLES OF STABILITY AND CONTROL 375
zc
Figure 7.18 Airfoil nomenclature and geometry.
the zerolift chord and the geometric chord (as defined in Chap. 5) is usually
insignificant and will be ignored here. The center of gravity for the airplane is
located a distance he behind the leading edge, and zc above the zerolift line, as
shown. Hence, h and z are coordinates of the center of gravity in fractions of
chord length. The aerodynamic center is a distance h ac c from the leading edge.
The moment of the wing about the aerodynamic center ~ the wing is denoted by
Mac , and the wing lift and drag are Lw and Dw, respectively, as shown. As usual,
Lw ind Dw are perpendicular and parallel, respectively, to the relative wind.
We wish to take moments about the center of gravity with pitchup moments
positive as usual. Clearly, from Figure 7.18, Lw, Dw, and Mac . all contribute to
moments about the center of gravity. For convenience, split· Lw and Dw into
components perpendicular and parallel to the chord. Then, referring to Figure
7.18, we find the moments about the center of gravity of the airplane due to the
wing are
Mcgw = Macw + Lwcosaw(hc  hacwc) + Dwsinaw(hc  hac/)
+Lwsinawzc  Dwcosawzc (7.4)
[Study Eq. (7.4) and Figure 7.18 carefully, and make certain that you understand
each term before progressing further.] For the normalflight range of a conven
tional airplane, aw is small; hence, the approximation is made that cos aw :::::: 1
and sinaw :::::: aw (where aw is in radians). Then Eq. (7.4) becomes
Mcgw = Macw +(Lw + Dwaw)(h  hacJc +(Lwaw  Dw)zc (7.5)
Dividing Eq. (7.5) by q
00
Sc and recalling that CM= M/q
00
Sc, we obtain the
moment coefficient about the center of gravity as
CM.cgw = CM,ac. +(CL.w + CD,waw)(h  hacJ
(7.6)
For most airplanes, the center of gravity is located close to the zerolift line;
376 INTRODUCTION TO FLIGHT
hence z is usually small ( z ::::: 0) and will be neglected. Furthermore, aw (in
radians) is usually much less than unity, and Cv,w is usually less than CL.w;
hence, the product Cv.waw is small in comparison to CL,w· With these assump
tions, Eq. (7.6) simplifies to
I c = c + c (h  h ) I M,cgw M,acw L,w acw
(7 .7)
Referring to Figure 7.llb, we find CL,w = (dCL,w/da)aw = awaw, where aw is
the lift slope of the wing. Thus, (Eq. (7.7) can be written as
(7.8)
Equations (7.7) and (7.8) give the contribution of the wing to moments about the
center of gravity of the airplane, subject of course to the above assumptions.
Closely examine Eqs. (7.7) and (7.8) along with Figure 7.18. On a physical basis,
they state that the wing's contribution to Meg is essentially due to two factors: the
moment about the aerodynamic center, Mac , and the lift acting through the
moment arm (h  hac )c. w
The above results"' are slightly modified if a fuselage is added to the wing.
Consider a cigarshaped body at an angle of attack to an airstream. This
fuselagetype body experiences a moment about its aerodynamic center, plus
some lift and drag due to the airflow around it. Now consider the fuselage and
wing joined together: a wingbody combination. The airflow about this wingbody
combination is different from that over the wing and body separately; aerodynamic
interference occurs where the flow over the wing affects the fuselage flow, and vice
versa. Due to this interference, the moment due to the wingbody combination is
not simply the sum of the separate wing and fuselage moments. Similarly, the lift
and drag of the wingbody combination are affected by aerodynamic interference.
Such interference effects are extremely difficult to predict theoretically. Conse
quently, the lift, drag, and moments of a wingbody combination are usually
obtained from windtunnel measurements. Let CL and CM ac be the lift
wb ' wb
coefficient and moment coefficient about the aerodynamic center, respectively, for
the wingbody combination. Analogous to Eqs. (7.7) and (7.8) for the wing only,
the contribution of the wingbody combination to Meg is
I c = c + c (h  h ) I
M, cgwb M, acwb Lwb acwb
(7.9)
IC = C +a a (h  h ) I
M,cgwb M,acwb wb wb acwb
(7 .10)
where awb and awb are the slope of the lift curve and absolute angle of attack,
respectively, for the wingbody combination. In general, adding a fuselage to a
wing shifts the aerodynamic center forward, increases the lift curve slope, and
contributes a negative increment to the moment about the aerodynamic center.
PRINCIPLES OF STABILITY AND CONTROL 377
Emphasis is again made that the aerodynamic coefficients in Eqs. (7.9) and (7.10)
are almost always obtained from windtunnel data.
Example 7.2 For a given wingbody combination, the aerodynamic center lies 0.05 chord length
ahead of the center of gravity. The moment coefficient about the aerodynamic center is 0.016.
If the lift coefficient is 0.45, calculate the moment coefficient about the center of gravity.
SOLUTION From Eq. (7.9),
C =C +C (hh )
M.c&wb M,acwb Lwb acwb
where
CM.acwb = 0.016
Thus cM.cgwb = 0.016+0.45(0.05) =I 0.00651
Example 7.3 A wingbody model is tested in a subsonic wind tunnel. The lift is found to be zero
at a geometric angle of attack a= 1.5 °. At a= 5 °, the lift coefficient is measured as 0.52.
Also, at a= 1.0 °and 7.88 °, the moment coefficients about the center of gravity are measured as
 0.01 and 0.05, respectively. The center of gravity is located at 0.35c. Calculate the location of
the aerodynamic center and the value of CM.•cwb·
SOLUTION First, calculate the lift slope:
dCL 0.520 0.52
awb = da = 5(1.5) =Ts= 0.08 per degree
Write Eq. (7.10),
C =CM +a a b(hh )
M,cgwb ,acwb wb w acwb
evaluated at a= 1.0 °[remember that a is the geometric angle of attack, whereas in Eq. (7.10)
awb is the absolute angle of attack]:
0.0l=CMac +0.08(1+1.5)(hhac )
• wb wb
Then evaluate it at a= 7.88 °:
0.05=CMac +0.08(7.88+1.5)(hhac )
• wb wb
The above two equations have two unknowns, CM.acwb and h  h•cwb· They can be solved
simultaneously.
Subtracting the second equation from the first, we get
0.06 = 00.55( h  hacwJ
0.06
h  hacwb = 0.55 = 0.11
The value of h is given: h = 0.35. Thus
hac = 0.350.11 =I 0.241
wb
378 INTRODUCTION TO FLIGHT
In turn 0.01 = CM,acwb +0.08(11.5)(0.11)
CM.acwb=J0.0321
7.7 CONTRIBUTION OF THE TAIL TO Meg
An analysis of moments due to an isolated tail taken independently of the
airplane would be the same as that given for the isolated wing above. However, in
real life the tail is obviously connected to the airplane itself; it is not isolated.
Moreover, the tail is generally mounted behind the wing; hence it feels the wake
of the airflow over the wing. As a result, there are two interference effects that
influence the tail aerodynamics:
1. The airflow at the tail is deflected downward by the downwash due to the
finite wing (see Secs. 5.13 and 5.14); i.e., the relative wind seen by the tail is
not in the same direction as the relative wind V
00
seen by the wing.
2. Due to the retarding force of skin friction and pressure drag over the wing,
the airflow reaching the tail has been slowed. Therefore, the velocity of the
relative wind seen by the tail is less than V
00
• In turn, the dynamic pressure
seen by the tail is less than q
00
•
These effects are illustrated in Figure 7.19. Here V
00
is the relative wind as
seen by the wing, and V' is the relative wind at the tail, inclined below V
00
by the
downwash angle e. The tail lift and drag, L
1
and DI' are (by defimtion)
perpendicular and parallel, respectively, to V'. In contrast, the lift and drag of the
complete airplane are always (by definition) perpendicular and parallel, respec
tively, to V
00
• Therefore, considering components of L
1
and D
1
perpendicular to
V
00
, we demonstrate in Figure 7.19 that the tail contribution to the total airplane
lift is L
1
cos e  D, sin e. In many cases, e is very small, and thus L, cos e 
Figure 7.19 Flow and force diagram in the vicinity of the tail.
PRINCIPLES OF STABILITY AND CONTROL 379
Dt sin e ::::: Lt. Hence, for all practical purposes, it is sufficient to add the tail lift
directly to the wingbody lift to obtain the lift of the complete airplane.
Consider the tail in relation to the wingbody zerolift line, as illustrated in
Figure 7.20. It is useful to pause and study this figure. The wingbody combina
tion is at an absolute angle of attack awb· The tail is twisted downward in order to
provide a positive CM.o' as discussed at the end of Sec. 7.5. Thus the zerolift line
of the tail is intentionally inclined to the zerolift line of the wingbody combina
tion at the tailsetting angle it· (The airfoil section of the tail is generally
symmetric, for which the tail zerolift line and the tail chord line are the same.)
The absolute angle of attack of the tail, at is measured between the local relative
wind V' and the tail zerolift line. The tail has an aerodynamic center, about
which there is a moment Mac and through which Lt and Dt act perpendicular
and parallel, respectively, to 'V'. As before, V' is inclined below V
00
by the
downwash angle e; hence Lt makes an angle awb  e with the vertical. The tail
aerodynamic center is located a distance It behind and zt below the center of
gravity of the airplane. Make certain to carefully study the geometry shown in
Figure 7.20; it is fundamental to the derivation which follows.
Split Lt and Dt into their vertical components, Lt cos ( awb  e) and
Dt sin ( awb  e ), and their horizontal components, Lt sin ( awb  e) and
Dt cos ( awb  e ). By inspection of Figure 7 .20, the sum of moments about the
center of gravity due to LP DP and Mac of the tail is
t
(7 .11)
Here, Meg, denotes the contribution to moments about the airplane's center of
gravity due to the horizontal tail.
In Eq. (7.11), the first term on the righthand side, ltLtcos(awb  e), is by
far the largest in magnitude. In fact, for conventional airplanes, the following
simplifications are reasonable:
1. zt <<It
2. Dt «Lt
3. The angle awb  e is small, hence sin ( awb  e) ::::: 0 and cos ( awb  e) ::::: 1.
4. Mac is small in magnitude.
t
With the above approximations, which are based on experience, Eq. (7.11) is
dramatically simplified to
(7 .12)
Define the tail lift coefficient, based on freestream dynamic pressure q
00
= !P
0
,,V
00
2
and the tail planform area St, as
(7.13)
!
1· I,
Center of gravity EB,,
Zero lift line of wing body
Ciwb
y
Figure 7.20 Geometry of wingtail combination.
z,
V d > ~
~
PRINCIPLES OF STABILITY AND CONTROL 381
Combining Eqs. (7.12) and (7.13), we obtain
(7 .14)
Dividing Eq. (7.14) by q
00
Sc, where c is the wing chord and S is the wing
planform area,
Meg, = C   l 1S1 C
qooSc  M,cg,  cS L,t
(7.15)
Examining the righthand side of Eq. (7.15), note that /
1
S
1
is a volume characteris
tic of the size and location of the tail and that cS is a volume characteristic of the
size of the wing. The ratio of these two volumes is called the tail volume ratio VH,
where
(7 .16)
Thus, Eq. (7.15) becomes
I CM.cg,=  VHCL.t I
(7.17)
The simple relation in Eq. (7.17) gives the total contribution of the tail to
moments about the airplane's center of gravity. With the simplifications above
and referring to Figure 7.20, Eqs. (7.12) and (7.17) say that the moment is equal
to tail lift operating through the moment arm /
1
•
It will be useful to couch Eq. (7.17) in terms of angle of attack, as was done in
Eq. (7.10) for the wingbody combination. Keep in mind that the stability
criterion in Figure 7.12 involves acM.cg/ aaa; hence equations in terms of aa are
directly useful. Specifically, referring to the geometry of Figure 7.20, we see that
the angle of attack of the tail is
(7 .18)
Let a
1
denote the lift slope of the tail. Thus, from Eq. (7.18),
(7.19)
The downwash angle e is difficult to predict theoretically and is usually obtained
from experiment. It can be written as
E = fo + ( ;: ) awb
(7 .20)
where e
0
is the downwash angle when the wingbody combination is at zero lift.
Both e
0
and ae/ aa are usually obtained from windtunnel data. Thus, combining
Eqs. (7.19) and (7.20) yields
(7.21)
382 INTRODUCTION TO FLIGHT
Substituting Eq. (7.21) into (7.17), we obtain
(7.22)
Equation (7.22), although lengthier than pq. (7.17), contains the explicit depen
dence on angle of attack and will be usefµl for our subsequent discussions.
7.8 TOTAL PITCHING MOMENT ABOUT THE
CENTER OF GRAVITY
Consider the airplane as a whole. The total Meg is due to the contribution of the
wingbody combination, plus that of the tail:
(7.23)
Here, CM cg is the total moment coefficient about the center of gravity for the
complete airplane. Substituting Eqs. (7.9) and (7.17) into (7.23),
I c = c + c (h  h )  v c I
M,cg M.acwb Lwb acwh H L,t
(7.24)
In terms of angle of attack, an alternate expression can be obtained by substitut
ing Eqs. (7.10) and (7.22) into (7.23):
(7 .25)
The angle of attack needs further clarification. Referring again to Figure 7.12,
we find the moment coefficient curve is usually obtained from windtunnel data,
preferably on a model of the complete airplane. Hence, aa in Figure 7.12 should
be interpreted as the absolute angle of attack referenced to the zerolift line of the
complete airplane, which is not necessarily the same as the zerolift line for the
wingbody combination. This comparison is sketched in Figure 7.21. However, for
many conventional aircraft, the difference is small. Therefore, in the remainder of
this chapter, we will assume the two zerolift lines in Figure 7.21 to be the same.
Thus, awb becomes the angle of attack of the complete airplane, aa. Consistent
with this assumption, the total lift of the airplane is due to the wingbody
combination, with the tail lift neglected. Hence, CLwb = CL and the lift slope
awb = a, where CL and a are for the complete airplane. With these interpreta
tions, Eq. (7.25) can be rewritten as
(7.26)
PRINCIPLES OF STABILITY AND CONTROL 383
CY.wb
(
Figure 7.21 Zerolift line of the wingbody compared with that of the comp:ete airplane.
Equation (7.26) is the same as Eq. (7.25), except that the subscript "wb" on some
terms has been dropped in deference to properties for the whole airplane.
Example 7.4 Consider the wingbody model in Example 7.3. The area and chord of the wing are
0.1 m
2
and 0.1 m, respectively. Now assume that a horizontal tail is added to this model. The
distance from the airplane's center of gravity to the tail's aerodynamic center is 0.17 m, the tail
area is 0.02 m
2
, the tailsetting angle is 2.7 °, the tail lift slope is 0.1 per degree, and from
experimental measurement, Eo = 0 and aE/ aa = 0.35. If a= 7.88 °, calculate CM.cg for the
airplane model.
SOLUTION From Eq. (7.26)
where CM.acwb = 0.032 (from Example 7.3)
a= 0.08 (from Example 7.3)
aa = 7.88+1.5 = 9.38° (from Example 7.3)
(from Example 7.3)
V = l,S, = 0.17(0.02) =
0
_
34
H cS 0.1(0.1)
a,= 0.1 per degree
~ = 3 5
aa .
Eo = 0
384 INTRODUCTION TO FLIGHT
Thus CM.cg= 0.032 + (0.08)(9.38) [ 0.110.34 i
0
1
8
(10.35)] +0.34(0.l )(2.7 +O)
=0.0320.125+0.092=10.0651
7.9 EQUATIONS FOR LONGITUDINAL STATIC STABILITY
The criteria necessary for longitudinal balance and static stability were developed
in Sec. 7.5. They are (1) CM,o must be positive and (2) acM,cg/aaa must be
negative, both conditions with the implicit assumption that ae falls within the
practical flight range of angle of attack; i.e., the moment coefficient curve must be
similar to that sketched in Figure 7.12. In turn, the ensuing sections developed a
quantitative formalism for static stability culminating in Eq. (7.26) for CM.cg· The
purpose of the present section is to combine the above results in order to obtain
formulas for the direct calculation of CM,O and acM,cg/ aaa. In this manner, we
will be able to make a quantitative assessment of the longitudinal static stability
of a given airplane, as well as point out some basic philosophy of airplane design.
Recall that, by definition, c M,O is the value of c M,cg when aa = 0, that is,
when the lift is zero. Substituting aa = 0 into Eq. (7.26), we directly obtain
I cM,0 = (CM.cg) L0 = cM,acwb + VHat(it + eo) I (7.27)
Examine Eq. (7.27). We know that CM,o must be positive in order to balance
the airplane. However, the previous sections have pointed out that CM ac is
• wb
negative for conventional airplanes. Therefore, VHa
1
(i
1
+ e
0
) must be positive
and large enough to more than counterbalance the negative CM,ac· Both VH and
a
1
are positive quantities, and e
0
is usually so small that it exerts only a minor
effect. Thus, i
1
must be a positive quantity. This verifies our previous physical
arguments that the tail must be set at an angle relative to the wing in the manner
shown in Figures 7.16a and 7.20. This allows the tail to generate enough negative
lift to produce a positive CM,o·
Consider now the slope of the moment coefficient curve. Differentiating Eq.
(7.26) with respect to aa, we obtain
acM,cg = a[(h  h ) V ar(1 ~ ]
aaa acwb H a aa
(7.28)
This equation clearly shows the powerful influence of the location h of the center
of gravity and the tail volume ratio VH in determining longitudinal static stability.
Equations (7.27) and (7.28) allow us to check the static stability of a given
airplane, assuming we have some windtunnel data for a, al' CM ac , e
0
, and
• wb
ae/ aa. They also establish a certain philosophy in the design of an airplane. For
example, consider an airplane where the location h of the center of gravity is
essentially dictated by payload or other mission requirements. Then, the desired
PRINCIPLES OF STABILITY AND CONTROL 385
amount of static stability can be obtained simply by designing VH large enough,
via Eq. (7.28). Once VH is fixed in this manner, then the desired CM
0
(or the
desired ae) can be obtained by designing i
1
appropriately, via Eq. (7.27). Thus,
the values of CM,O and acM,cg/aaa basically dictate the design values of i, and
VH, respectively (for a fixed centerofgravity location).
Example 7.5 Consider the wingbodytail windtunnel model of Example 7.4. Does this model
have longitudinal static stability and balance?
SOLUTION From Eq. (7.28)
aCM,cg=a[(hh )V
aaa acwb H a aa
where, from Examples 7.3 and 7.4,
a= 0.08
VH = 0.34
a
1
= 0.1 per degree
aa .
Thus
aCM,cg [ 0.1 ]
a;:= 0.08 0.110.34 0.08 (10.35)
= 10.01331
The slope of the moment coefficient curve is negative, hence the airplane model is statically
stable.
However, is the model longitudinally balanced? To answer this, we must find CM,O• which
in combination with the above result for acM,cg / aa, will yield the equilibrium angle of attack
ae. From Eq. (7.27),
where from Examples 7.3 and 7.4,
Thus CM,O = 0.032+0.34(0.1)(2.7)
= 0.06
From Figure 7.12, the equilibrium angle of attack is obtained from
0 = 0.060.0133ae
Thus ae = 4.5 °
Clearly, this angle of attack falls within the reasonable flight range. Therefore, the airplane is
longitudinally balanced as well as statically stable.
386 INTRODUCTION TO FLIGHT
7.10 THE NEUTRAL POINT
Consider the situation where the location h of the center of gravity is allowed to
move, with everything else remaining fixed. In fact, Eq. (7.28) indicates that static
stability is a strong function of h. Indeed, the value of acM.cg/ aaa can always be
made negative by properly locating the center of gravity. In the same vein, there is
one specific location of the center of gravity such that acM,cg/ aaa = 0. The value
of h when this condition holds is defined as the neutral point, denoted by hn.
When h = h n' the slope of the moment coefficient curve is zero, as illustrated in
Figure 7.22.
The location of the neutral point is readily obtained from Eq. (7.28) by
Setting h = hn and JCM,cg/Jaa = 0, as follows.
0 = a[h  h  V a
1
(1  ~ ] (7.29)
n acwb Ha Ja
Solving Eq. (7.29) for hn,
(7.30)
Examine Eq. (7.30). The quantities on the righthand side are, for all practical
purposes, established by the design configuration of the airplane. Thus, for a
given airplane design, the neutral point is a fixed quantity, i.e., a point that is
frozen somewhere on the airplane. It is quite independent of the actual location h
of the center of gravity.
The concept of the neutral point is introduced as an alternative stability
criterion. For example, inspection of Eqs. (7.28) and (7.30) show that JCM,cg/Jaa
is negative, zero, or positive depending on whether h is less than, equal to, or
greater than hn. These situations are sketched in Figure 7.22. Remember that h is
measured from the leading edge of the wing, as shown in Figure 7.18. Hence,
a
0
Figure 7.22 Effect of the location
of the center of gravity, relative to
the neutral point, on static stabil
ity.
PRINCIPLES OF STABILITY AND CONTROL 387
h < h n means that the center of gravity location is forward of the neutral point.
Thus, an alternative stability criterion is:
For longitudinal static stability, the position of the center of gravity must always
be forward of the neutral point.
Recall that the definition of the aerodynamic center for a wing is that point
about which moments are independent of angle of attack. This concept can now
be extrapolated to the whole airplane by considering again Figure 7.22. Clearly,
when h = h n, C M,cg is independent of angle of attack. Therefore, the neutral
point might be considered the aerodynamic center of the complete airplane.
Again examining Eq. (7.30), we see that the tail strongly influences the
location of the neutral point. By proper selection of the tail parameters, principal
ly Vil, hn can be located at will by the designer.
Example 7.6 For the windtunnel model of Examples 7.3 to 7.5, calculate the neutral point
location.
SOLUTION From Eq. (7.30)
a, ( aE)
h,,=hac +Vlf la
wb a a
where hacwb = 0.24 (from Example 7.3).
Thus
0.1
h,, = 0.24+0.34 0.08 (10.35)
lh,,=0.5161
Note from Example 7.3 that h = 0.35. Compare this center of gravity location with the neutral
point location of 0.516. The center of gravity is comfortably forward of the neutral point; this
again confirms the results of Example 7.5 that the airplane is statjcally stable.
7.11 THE STATIC MARGIN
A corollary to the above discussion can be obtained, as follows. Solve Eq. (7.30)
for hacwb·
(7 .31)
Note that in Eqs. (7.29) to (7.31), the value of VH is not precisely the same
number as in Eq. (7.28). Indeed, in Eq. (7.28), VH is based on the moment arm /
1
measured from the center of gravity location, as shown in Figure 7.20. In
contrast, in Eq. (7.29), the center of gravity location has been moved to the
neutral point, and VH is therefore based on the moment arm measured from the
neutral point location. However, the difference is usually small, and this effect will
be ignored here. Therefore, substituting Eq. (7.31) into Eq. (7.28) and cancelling
388 INTRODUCTION TO FLIGHT
Static margin
(ac)wb
ich
ichn +l
Figure 7.23 Illustration of the static margin.
the terms involving VH, we obtain
ac
~ = a h  h )
aaa n
(7.32)
The distance, h n  h, is defined as the static margin and is illustrated in Figure
7.23. Thus, from Eq. (7.32),
 a ( h n  h) =  a X static margin (7.33)
Equation (7.33) shows that the static margin is a direct measure of longitudi
nal static stability. For static stability, the static margin must be positive.
Moreover, the larger the static margin, the more stable is the airplane.
Example 7.7 For the windtunnel model of the previous examples, calculate the static margin.
SOLUTidN From Example 7.6, hn = 0.516 and h = 0.35. Thus; by definition,
static margin= h.,  h = 0.5160.35 =I 0.1661
For a check on the consistency of our calculations, consider Eq. (7.33).
acM.cg . .
a  =  a x stahc margm
aa
= 0.08(0.166)
=  0.0133 per degree
This is the same value calculated in Example 7.5; our calculations are indeed consistent.
7.12 THE CONCEPT OF STATIC LONGITUDINAL CONTROL
A study of stability and control is doublebarrelled. The first aspect that of
stability itselfhas been the subject of the preceding sections. However, for the
remainder of this chapter, the focus will tum to the second aspectcontrol.
PRINCIPLES OF STABILITY AND CONTROL 389
Consider a statically stable airplane in trimmed (equilibrium) flight. Recalling
Figure 7.12, the airplane must therefore be flying at the trim angle of attack ae. In
turn, this value of ae corresponds to a definite value of lift coefficient, namely, the
trim lift coefficient CL .. For steady, level flight, this corresponds to a definite
velocity, which from (6.25) is
(7.34)
Now assume that the pilot wishes to fly at a lower velocity V
00
< At a lower
velocity, the lift coefficient, hence the angle of attack, must be increased in order
to offset the decrease in dynamic pressure (remember from Chap. 6 that the lift
must always balance the weight for steady, level flight). However, from Figure
7.12, if a is increased, CM.cg becomes negative (i.e., the moment about the center
of gravity is np longer zero), and the airplane is no longer trimmed. Consequently,
if nothing else is changed about the airplane, it cannot achieve steady, level,
equilibrium flight at any other velocity than or at any other angle of attack
than ae.
Obviously, this is an intolerable situationan airplane must be able to
change its velocity at the will of the pilot and still remain balanced. The only way
to accomplish this is to effectively change the moment coefficient curve for the
airplane. Say that the pilot wishes to fly at a faster velocity but still remain in
steady, level, balanced flight. The lift coefficient must decrease, hence a new angle
of attack an must be obtained, where an < ae. At the same time, the moment
coefficient curve must be changed such that CM.cg = 0 at an. Figures 7.24 and
7.25 demonstrate two methods of achieving this change. In Figure 7.24, the slope
is made more negative, such that CM.cg goes through zero at an. From Eq. (7.28)
or (7.32), the slope can be changed by shifting the center of gravity. In our
example, the center of gravity must be shifted forward. Otto Lilienthal (see Sec.
1.5) used this method in his gliding flights. Figure 1.15 shows Lilienthal hanging
loosely below his glider; by simply swinging his hips, he was able to shift the
center of gravity and change the stability of the aircraft. This principle is carried
over today to the modern hang gliders for sports use.
Figure 7.24 Change in trim angle of attack due to
change in slope of moment coefficient curve.
390 INTRODUCTION TO FLIGHT
Figure 7.25 Change in trim angle of attack due to
change in CM.o·
However, for a conventional airplane, shifting the center of gravity is highly
impractical. Therefore, another method for changing the moment curve is em
ployed, as shown in Figure 7.25. Here, the slope remains the same, but CM.o is
changed such that CM.cg = 0 at an. This is accomplished by deflecting the
elevator on the horizontal tail. Hence, we have arrived at a major concept of
static, longitudinal control, namely, that the elevator deflection can be used to
control the trim angle of attack, hence to control the equilibrium velocity of the
airplane.
Consider Figure 7.25. We stated above, without proof, that a translation of
the moment curve without a change in slope can be obtained simply by deflecting
the elevator. But how and to what extent does the elevator deflection change
CM.cg? To provide some answers, first consider the horizontal tail with the elevator
fixed in the neutral position, i.e., no elevator deflection, as shown in Figure 7.26.
The absolute angle of attack of the tail is a,, as defined earlier. The variation of
tail lift coefficient with a, is also sketched in Figure 7.26; note that it has the same
general shape as the airfoil and wing lift curves discussed in Chap. 5. Now,
E
;I<
V'
Figure 7.26 Taillift coefficient curve
ex, with no elevator deflection.
PRINCIPLES OF STABILITY AND CONTROL 391
c
Figure 7.27 Taillift coefficient with elevator deflection.
assume that the elevator is deflected downward through the angle Se, as shown in
Figure 7.27. This is the same picture as a wing with a deflected flap, as was
discussed in Sec. 5.17. Consequently, just as in the case of a deflected flap, the
deflected elevator causes the tail lift coefficient curve to shift to the left, as shown
in Figure 7.27. By convention (and for convenience later on), a downward
elevator deflection is positive. Therefore, if the elevator is deflected by an angle of,
say, 5° and then held fixed as the complete tail is pitched through a range of at,
the tail lift curve is translated to the left. If the elevator is then deflected farther,
say to 10°, the lift curve is shifted even farther to the left. This behavior is clearly
illustrated in Figure 7.27. Note that for all the lift curves, the slope, acL.t;aaf' is
the same.
With the above in mind, now consider the tail at a fixed angle of attack, say
(ath· If the elevator is deflected from, say, 0 to 15°, CL.i will increase along the
vertical dashed line in Figure 7 .27. This variation can be crossplotted as CL, t
versus Se, as shown in Figure 7.28. For most conventional airplanes, the curve in
Figure 7.28 is essentially linear, and its slope, acL. ti ase, is called the elevator
control effectiveness. This quantity is a direct measure of the "strength" of the
elevator as a control; because Se has been defined as positive for downward
deflections, the acL, i/ ase is always positive.
Consequently, the tail lift coefficient is a function of both a
1
and Se (hence,
the partial derivative notation is used, as discussed earlier). Keep in mind that,
physically, acL,i/aat is the rate of change Of CL,I with respect to at, keeping Se
constant; similarly, acL,i1ase is the rate of change of cL,t with respect to Se,
392 INTRODUCTION TO FLIGHT
I Constant °'r I
Figure 7.28 Taillift coefficient vs. elevator deflection at constant angle of attack; a cross plot of
Figure 7.27.
keeping a, constant. Hence, on a physical basis,
acL I acL I
CL,t = aa; at+ ~ e
(7 .35)
Recalling that the tail lift slope is at= acL,,;aat, Eq. (7.35) can be written as
acL I
CL,t = at<X.t + JSse
e
(7 .36)
Substituting Eq. (7.36) into (7.24), we have for the pitching moment about the
center of gravity,
(7 .37)
Equation (7.37) gives explicitly the effect of elevator deflection on moments about
the center of gravity of the airplane.
The rate of change of CM.cg due only to elevator deflection is, by definition,
acM,cg/aSe. This partial derivative can be found by differentiating Eq. (7.37) with
respect to Se, keeping everything else constant.
acM,cg acL,t
as =  V Ha8
e e
(7.38)
Note that, from Figure 7.28, acL,,;ase is constant; moreover, VH is a specific
value for the given airplane. Thus, the righthand side of Eq. (7.38) is a constant.
Therefore, on a physical basis, the increment in CM.cg due only to a given elevator
deflection Se is
(7.39)
PRINCIPLES OF STABILITY AND CONTROL 393
Original cM,O
New cM,O
Original trim angle of attack
New trim angle of attack
Figure 7.29 Effect of elevator deflection on moment coefficient.
Equation (7.39) now answers the questions asked earlier: how and to what
extent the elevator deflection changes CM.cg· Consider the moment curve labeled
l)e = 0 in Figure 7.29. This is the curve with the elevator fixed in the neutral
position; it is the curve we originally introduced in Figure 7.12. If the elevator is
now deflected through a positive angle (downward), Eq. (7.39) states that ail
points on this curve will now be shifted down by the constant amount ~ C M c g ·
Hence, the slope of the moment curve is preserved; only the value of C M,o is
changed by elevator deflection. This now proves our earlier statement made in
conjunction with Figure 7.25.
For emphasis, we repeat the main thrust of this section. The elevator can be
used to change and control the trim of the airplane. In essence, this controls the
equilibrium velocity of the airplane. For example, by a downward deflection of
the elevator, a new trim angle an, smaller than the original trim angle ae, can be
obtained. (This is illustrated in Figure 7.29.) This corresponds to an increase in
velocity of the airplane.
As another example, consider the two velocity extremesstalling velocity
and maximum velocity. Figure 7.30 illustrates the elevator deflection necessary to
trim the airplane at these two extremes. First consider Figure 7.30a, which
corresponds to an airplane flying at V
00
:::::: V:ian· This would be the situation
on a landing approach, for example. The airplane is flying at CL , hence the
angle of attack is large. Therefore, from our previous discussion,mathe airplane
must be trimmed by an upelevator position, i.e., by a negative l>e. On the
other hand, consider Figure 7.30b, which corresponds to an airplane flying at
V
00
:::::: Vmax (near full throttle). Because q
00
is large, the airplane requires only
a small CL to generate the required lift force; hence the angle of attack is
small. Thus, the airplane must be trimmed by a downelevator position, i.e., by a
positive l>e.
394 INTRODUCTION TO FLIGHT
v o o ~ v m x
(a)
(b)
oe
positive
Figure 7.30 Elevator deflection required for trim at (a) low flight velocity. and ( b) high flight
velocity.
7.13 CALCULATION OF ELEVATOR ANGLE TO TRIM
The concepts and relations developed in Sec. 7.12 allow us now to calculate the
precise elevator deflection necessary to trim the airplane at a given angle of
attack. Consider an airplane with its moment coefficient curve given as in Figure
7.31. The equilibrium angle of attack with no elevator deflection is ae. We wish to
trim the airplane at a new angle of attack an. What value of l>e is required for this
purpose?
To answer this question, first write the equation for the moment curve with
l>e = 0 (the solid line in Figure 7.31). This is a straight line with a constant slope
equal to ac M,cg/ aaa and intercepting the ordinate at c M,O· Hence, from analytic
geometry, the equation of this line is
(7.40)
Figure 7.31 Given the equilibrium angle of attack at zero elevator deflection, what elevator deflection
is necessary to establish a given new equilibrium angle of attack?
PRINCIPLES OF STABILITY AND CONTROL 395
Now assume the elevator is deflected through an angle Se. The value of CM,cg will
now change by the increment ~ C M , c g and the moment equation given by Eq.
(7.40) is now modified as
(7.41)
The value of 11CM,cg was obtained earlier as Eq. (7.39). Hence, substituting Eq.
(7.39) into (7.41), we obtain
(7.42)
Equation (7.42) allows us to calculate CM,cg for any arbitrary angle of attack
aa and any arbitrary elevator deflection Se. However, we are interested in the
specific situation where CM,cg = 0 at aa = an and where the value of Se necessary
to obtain this condition is Se = Strim· That is, we want to find the value of Se
which gives the dashed line in Figure 7.31. Substituting the above values into Eq.
(7.42),
and solving for strim' we obtain
CM.O + ( acM,cg/ aau} <Xn
strim= VH( acL,i1ase)
(7.43)
Equation (7.43) is the desired result. It gives the elevator deflection necessary to
trim the airplane at a given angle of attack an. In Eq. (7.43), VH is a known value
from the airplane design, and CM,0' acM,cg/aaa, and acL,/aSe are known
values usually obtained from windtunnel or freeflight data.
Example 7.8 Consider a fullsize airplane with the same aerodynamic and design characteristics
as the windtunnel model of Examples 7.3 to 7.7. The airplane has a wing area of 19 m
2
, a weight
of 2.27x10
4
N, and an elevator control effectiveness of 0.04. Calculate the elevator deflection
angle necessary to trim the airplane at a velocity of 61 m/s at sea level.
SOLUTION First, we must calculate the angle of attack for the airplane at V
00
= 61 m/s. Recall
that
2W 2(2.27Xl0
4
)
C
1
== =0.52
• PooV00
2
S 1.225(61)
2
(19)
From Example 7.3, the lift slope is a= 0.08 per degree. Hence, the absolute angle of attack of
the airplane is
= CL = 0.52 = 6 50
aa a 0.08 ·
From Eq. (7.43), the elevator deflection angle required to trim the airplane at this angle of
396 INTRODUCTION TO FLIGHT
attack is
where
Thus, from Eq. (7.43),
{i. = CM,O +( 8CM,cg/8aa)an
tnm VH(acL.1/Blie)
CM,O = 0.06
CM.cg= 0.0133
aaa
a,,= 6.5°
VH = 0.34
acL.1=o04
a lie .
(from Example 7.5)
(from Example 7.5)
(this is the aa calculated above)
(from Example 7.4)
(given above)
O· 194
•  0.06+(0.0133)(6.5) ~
tnm 0.34(0.04) .
Recall that positive li is downward. Hence, to trim the airplane at an angle of attack of 6.5°, the
elevator must be deflected upward by 1.94°.
7.14 STICKFIXED VERSUS STICKFREE STATIC STABILITY
The second paragraph of Sec. 7.5 initiated our study of a rigid airplane with fixed
controls, e.g., the elevator fixed at a given deflection angle. The ensuing sections
developed the static stability for such a case, always assuming that the elevator
can be deflected to a desired angle 8, but held fixed at that angle. This is the
situation when the pilot (human or automatic) moves the control stick to a given
position and then rigidly holds it there. Consequently, the static stability that we
have discussed to this point is called stickfixed static stability. Modern highper
formance airplanes designed to fly near or beyond the speed of sound have
hydraulically assisted power controls; therefore a stickfixed static stability analy
sis is appropriate for such airplanes.
On the other hand, consider a control stick connected to the elevator via wire
cables without a power boost of any sort. This was characteristic of most early
airplanes until the 1940s and is representative of many light, general aviation,
private aircraft of today. In this case, in order to hold the stick fixed in a given
position, the pilot must continually exert a manual force. This is uncomfortable
and impractical. Consequently, in steady, level flight, the control stick is left
essentially free; in turn, the elevator is left free to float under the influence of the
natural aerodynamic forces and moments at the tail. The static stability of such
an airplane is therefore called stickfree static stability. This will be the subject of
the remainder of this chapter.
PRINCIPLES OF STABILITY AND CONTROL 397
7.15 ELEVATOR HINGE MOMENT
Consider a horizontal tail with an elevator which rotates about a hinge axis, as
shown in Figure 7.32. Assume the airfoil section of the tail is symmetrical, which
is almost always the case for both the horizontal and vertical tail. First, consider
the tail at zero angle of attack, as shown in Figure 7.32a. The aerodynamic
pressure distribution on the top and bottom surfaces of the elevator will be the
same, i.e., symmetrical about the chord. Hence, there will be no moment exerted
on the elevator about the hinge line. Now assume that the tail is pitched to tpe
angle of attack a,, but the elevator is not deflected, that is, 8e = 0. This is
illustrated in Figure 7.32b. As discussed in Chap. 5, there will be a low pressure
on the top surface of the airfoil and a high pressure on the bottom surface.
V'
E
pressure
. _ ___.,,:_ d1stnbuttons

(a)
V'
(b)
V'
(c)
Figure 7.32 Illustration of the aerodynamic generation of elevator hinge moment. (a) No hinge
moment; (b) hinge moment due to angle of attack; (c) hinge moment due to elevator deflection.
398 INTRODUCTION TO FLIGHT
Figure 7.33 Nomenclature and geometry for hinge moment coefficient.
Consequently, the aerodynamic force on the elevator will not be balanced, and
there will be a moment about the hinge axis tending to deflect the elevator
upward. Finally, consider the horizontal tail at zero angle of attack but with the
elevator deflected downward and held fixed at the angle 8e, as shown in Figure
7.32c. Recall from Sec. 5.17 that a flap deflection effectively changes the camber
of the airfoil and alters the pressure distribution. Therefore, in Figure 7.32c, there
will be low and high pressures on the top and bottom elevator surfaces, respec
tively. As a result, a moment will again be exerted about the hinge line, tending to
rotate the elevator upward. Thus, we see that both the tail angle of attack a, and
the elevator deflection 8e result in a moment about the elevator hinge linesuch
a moment is defined as the elevator hinge moment. It is the governing factor in
stickfree static stability, as discussed in the next section.
Let He denote the elevator hinge moment. Also, referring to Figure 7.33, the
chord of the tail is c ,, the distance from the leading edge of the elevator to the
hinge line is cb, the distance from the hinge line to the trailing edge is ce, and that
portion of the elevator planform area that lies behind (aft of) the hinge line is Se.
The elevator hinge moment coefficient, Ch; is then defined as
He
ch,= 1 v 2s
2Poo oo ece
where V
00
is the freestream velocity of the airplane.
{7.44)
Recall that the elevator hinge moment is due to the tail angle of attack and
the elevator deflection. Hence, Ch is a function of both a, and 8e. Moreover,
experience has shown that at both subsonic and supersonic speeds, Ch is
approximately a linear function of a.
1
and 8e. Thus, recalling the definition or' the
partial derivative in Sec. 7.2 D, the hinge moment coefficient can be written as
{7.45)
where ach I aat and ach I a8e are approximately constant. However, the actual
magnitude; of these o n s t ~ n t values depend in a complicated way on ce/c
1
, ch/ce,
PRINCIPLES OF STABILITY AND CONTROL 399
the elevator nose shape, gap, trailing edge angle, and planform. Moreover, He is
very sensitive to local boundary layer separation. As a result, the values of the
partial derivatives in Eq. (7.45) must almost always be obtained empirically, such
as from windtunnel tests, for a given design.
Consistent with the convention that downward elevator deflections are posi
tive, hinge moments which tend to deflect the elevator downward are also defined
as positive. Note from Figure 7.32b that a positive a
1
physically tends to produce
a negative hinge moment (tending to deflect the elevator upward). Hence ach I a at
is usually negative. (However, if the hinge axis is placed very far back, ne;r the
trailing edge, the sense of He may become positive. This is usually not done for
conventional airplanes.) Also, note from Figure 7.32c that a positive Se usually
produces a negative He, hence aCh/a8e is also negative.
7.16 STICKFREE LONGITUDINAL STATIC STABILITY
Let us return to the concept of stickfree static stability introduced in Sec. 7.14. If
the elevator is left free to float, it will always seek some equilibrium deflection
angle such that the hinge moment is zero, that is, He = 0. This is obvious,
because as long as there is a moment on the free elevator, it will always rotate. It
will come to rest (equilibrium) only for that position where the moment is zero.
Recall our qualitative discussion of longitudinal static stability in Sec. 7.5.
Imagine that an airplane is flying in steady, level flight at the equilibrium angle of
attack. Now assume the airplane is disturbed by a wind gust and is momentarily
pitched to another angle of attack, as was sketched in Figure 7.13. If the airplane
is statically stable, it will initially tend to return toward its equilibrium position.
In subsequent sections, we saw that the design of the horizontal tail was a
powerful mechanism governing this static stability. However, until now, the
elevator was always considered fixed. On the other hand, if the elevator is allowed
to float freely when the airplane is pitched by some disturbance, the elevator will
seek some momentary equilibrium position different from its position before the
disturbance. This deflection of the free elevator will change the static stability
characteristics of the airplane. In fact, such stickfree stability is usually less than
stickfixed stability. For this reason, it is usually desirable to design an airplane
such that the difference between stickfree and stickfixed longitudinal stability is
small.
With the above in mind, consider the equilibrium deflection angle of a free
elevator. Denote this angle by 8rrw as sketched in Figure 7.34. At this angle,
He = 0. Thus, from Eq. (7.45),
(7.46)
decreases the static stab1hty of the airplane.
Example 7.9 Consider the airplane of Example 7.8. Its elevator hinge moment derivatives are
ach /aa, = 0.008 and ach /afie = 0.013. Assess the stickfree static stability of this
airplane. '
SOLUTION First, obtain the free elevator factor F defined from Eq. (7.48),
aeronauuca1 engineers, espouseo me concept or mnerem staouny lStaucauy stao1e
400 INTRODUCTION TO FLIGHT
I ~ O I
V'
Figure 7.34 Illustration of free elevator deflection.
402 INTRODUCTION TO FLIGHT
where a, =0.1
()CL.r = 0 04
ase .
(from Example 7.4)
(from Example 7.8)
1  0.008
F=lQ.l(0.04) _0.0l3 =0.754
The stickfree static stability characteristics are given by Eqs. (7.51) to (7.53). First, from Eq.
(7.51),
where
Thus,
v
11
= o.34
Eo = 0
(from Example 7.3)
(from Example 7 .4)
(from Example 7.4)
(from Example 7.4)
Cf.t.o = 0.032 + 0.754(0.34)(0.l )(2.7)
\ Cf.t.o = 0.0371
This is to be compared with CM.a= 0.06 obtained for stickfixed conditions in Example 7.5.
From Eq. (7.52),
where (from Example 7.3)
404 INTRODUCTION TO FLIGHT
After Penaud's work, the attainment of "inherent" (static) stability became
a dominant feature in aeronautical design. Lilienthal, Pilcher, Chanute, and
Langley all strived for it. However, static stability has one disadvantage: the more
stable the airplane, the harder it is to maneuver. An airplane that is highly stable
is also sluggish in the air; its natural tendency to return to equilibrium somewhat
defeats the purpose of the pilot to change its direction by means of control
deflections. The Wright brothers recognized this problem in 1900. Since Wilbur
and Orville were "airmen" in the strictest meaning of the word, they aspired for
quick and easy maneuverability. Therefore, they discarded the idea of inherent
stability that was entrenched by Cayley and Penaud. Wilbur wrote that "we ...
resolved to try a fundamentally different principle. We would arrange the
machine so that it would not tend to right itself." The Wright brothers designed
their aircraft to be statically unstable! This feature, along with their development
of lateral control through wing warping, is primarily responsible for the fantastic
aerial performance of all their airplanes from 1903 to 1912 (when Wilbur died).
Of course, this design feature heavily taxed the pilot, who had to keep the
airplane under control at every instant, continuously operating the controls to
compensate for the unstable characteristics of the airplane. Thus, the Wright
airplanes were difficult to fly, and long periods were required to train pilots for
these aircraft. In the same vein, such unstable aircraft were more dangerous.
These undesirable characteristics were soon to be compelling. After Wilbur's
dramatic public demonstrations in France in 1908 (see Sec. 1.8), the European
designers quickly adopted the Wrights' patented concept of combined lateral and
PRINCIPLES OF STABILITY AND CONTROL 401
stability when the elevator is free. The magnitude of this reduction is developed
below.
Consider now the moment about the center of gravity of the airplane. For a
fixed elevator, the moment coefficient is given by Eq. (7.24),
c = c + c (h  h )  v c
M ,cg M,acwb Lwb acwb H L, t
For a free elevator, the tail lift coefficient is now changed to C{,
1
• Hence, the
moment coefficient for a free elevator, C/.J,cg' is
C' = C + C (h  h )  V C'
M,cg M,acwb Lwb acwb H L,t
(7.49)
Substituting Eq. (7.48) into (7.49),
(7.50)
Equation (7.50) gives the final form of the moment coefficient about the
center of gravity of the airplane with a free elevator.
Using Eq. (7.50), the same analyses as given in Sec. 7.9 can be used to obtain
equations for stickfree longitudinal static stability. The results are:
(7.51)
(7.52)
aC'
~ a(h' h)
aa n
(7.53)
Equations (7.51), (7.52), and (7.53) apply for stickfree conditions, denoted by the
prime notation. They should be compared with Eqs. (7.27), (7.30), and (7.33),
respectively, for stickfixed stability. Note that h ~  h is the stickfree static
margin; because F < 1.0, this is smaller than the stickfixed static margin.
Hence, it is clear from Eqs. (7.51) to (7.53) that a free elevator usually
decreases the static stability of the airplane.
Example 7.9 Consider the airplane of Example 7.8. Its elevator hinge moment derivatives are
ach /aa, = 0.008 and ach /a8e = 0.013. Assess the stickfree static stability of this
airplane. '
SOLUTION First, obtain the free elevator factor F defined from Eq. (7.48),
402 INTRODUCTION TO FLIGHT
where a,= 0.1
acL.r = o 04
ase .
{from Example 7.4)
(from Example 7.8)
1 0.008
F=lQ.1{0.04) _
0
.
013
=0.754
The stickfree static stability characteristics are given by Eqs. (7.51) to (7.53). First, from Eq.
(7.51),
where
Thus,
CM.acwb = 0.032
V
11
= 0.34
Ea= 0
(from Example 7.3)
(from Example 7.4)
{from Example 7.4)
{from Example 7.4)
Cf.t.o = 0.032+0.754(0.34)(0.1)(2.7)
j c,;.,. 0 = o.037 J
This is to be compared with CM.a= 0.06 obtained for stickfixed conditions in Example 7.5.
From Eq. (7.52),
where hacwb = 0.24
aa .
a= 0.08
(from Example 7.3)
(from Example 7.4)
(from Example 7 .4)
= 0.24 + 0.754(0.34)(
{10.35)
I h;, = 0.4481
This is to be compared with hn = 0.516 obtained for stickfixed conditions in Example 7.6. Note
that the neutral point has moved forward for stickfree conditions, hence decreasing the stability.
In fact, the stickfree static margin is
h;,  h = 0.4480.35 = 0.098
This is a 41 percent decrease, in comparison with the stickfixed static margin from Example 7.7.
Finally, from Eq. (7.53),
acM.cg (, )
a;;=  a h,,  h = 0.08(0.098) = C5
Thus, as expected, the slope of the stickfree moment coefficient curve, although still negative, is
small in absolute value.
In conclusion, this problem indicates that stickfree conditions cut the static stability of our
hypothetical airplane by nearly onehalf. This helps to dramatize the differences between
stickfixed and stickfree considerations.
PRINCIPLES OF STABILITY AND CONTROL 403
7.17 A COMMENT
This brings to a close our technical discussion of stability and control. The
preceding sections constitute an introduction to the subject; however, we have
just scratched the surface. There are many other considerations: control forces,
dynamic stability, lateral and directional stability, etc. Such matters are the
subject of more advanced studies of stability and control and are beyond the
scope of this book. However, this subject is one of the fundamental pillars of
aeronautical engineering, and the interested reader can find extensive presenta
tions in books such as those of Perkins and Hage, and Etkin (see Bibliography at
the end of this chapter).
7.18 HISTORICAL NOTE: THE WRIGHT BROTHERS
VERSUS THE EUROPEAN PHILOSOPHY
ON STABILITY AND CONTROL
The two contrasting scenes depicted in Sec. 7.1the lumbering, belabored flight
of Farman vs. the relatively effortless maneuvering of Wilbur Wrightunder
score two different schools of aeronautical thought during the first decade of
powered flight. One school, consisting of virtually all early European and U.S.
aeronautical engineers, espoused the concept of inherent stability (statically stable
aircraft); the other, consisting solely of Wilbur and Orville Wright, practiced the
design of statically unstable aircraft that had to be controlled every instant by the
pilot. Both philosophies have their advantages and disadvantages, and because
they have an impact on modern airplane design, let us examine their background
more closely.
The basic principles of airplane stability and control began to evolve at the
time of George Cayley. His glider of 1804, sketched in Figure 1.8, incorporated a
vertical and horizontal tail that could be adjusted up and down. In this fashion,
the complete tail unit acted as an elevator.
The next major advance in airplane stability was made by Alphonse Penaud,
a brilliant French aeronautical engineer who committed suicide in 1880 at the age
of 30. Penaud built small model airplanes powered by twisted rubber bands, a
precursor of the flying balsaandtissue paper models of today. Penaud's design
had a fixed wing and tail, like Cayley's even though at the time Penaud was not
aware of Cayley's work. Of particular note was Penaud's horizontal tail design,
which was set at a negative 8 °with respect to the wing chord line. Here we find
the first true understanding of the role of the tailsetting angle i
1
(see Secs. 7 .5
and 7.7) on the static stability of an airplane. Penaud flew his model in the
Tuileries Gardens in Paris on August 18, 1871, before members of the Societe de
Navigation Aerienne. The aircraft flew for 11 s, covering 131 ft. This event, along
with Penaud's theory for stability, remained branded on future aeronautical
designs right down to the present day.
404 INTRODUCTION TO FLIGHT
After Penaud's work, the attainment of "inherent" (static) stability became
a dominant feature in aeronautical design. Lilienthal, Pilcher, Chanute, and
Langley all strived for it. However, static stability has one disadvantage: the more
stable the airplane, the harder it is to maneuver. An airplane that is highly stable
is also sluggish in the air; its natural tendency to return to equilibrium somewhat
defeats the purpose of the pilot to change its direction by means of control
deflections. The Wright brothers recognized this problem in 1900. Since Wilbur
and Orville were "airmen" in the strictest meaning of the word, they aspired for
quick and easy maneuverability. Therefore, they discarded the idea of inherent
stability that was entrenched by Cayley and Penaud. Wilbur wrote that "we ...
resolved to try a fundamentally different principle. We would arrange the
machine so that it would not tend to right itself." The Wright brothers designed
their aircraft to be statically unstable! This feature, along with their development
of lateral control through wing warping, is primarily responsible for the fantastic
aerial performance of all their airplanes from 1903 to 1912 (when Wilbur died).
Of course, this design feature heavily taxed the pilot, who had to keep the
airplane under control at every instant, continuously operating the controls to
compensate for the unstable characteristics of the airplane. Thus, the Wright
airplanes were difficult to fly, and long periods were required to train pilots for
these aircraft. In the same vein, such unstable aircraft were more dangerous.
These undesirable characteristics were soon to be compelling. After Wilbur's
dramatic public demonstrations in France in 1908 (see Sec. 1.8), the European
designers quickly adopted the Wrights' patented concept of combined lateral and
directional control by coordinated wing warping (or by ailerons) and rudder
deflection. But they rejected the Wrights' philosophy of static instability. By 1910,
the Europeans were designing and flying aircraft that properly mated the Wrights'
control ideas with the longestablished static stability principles. On the other
hand, the Wrights stubbornly clung to their basic unstable design. As a result, by
1910 the European designs began to surpass the Wrights' machines, and the lead
in aeronautical engineering established in America in 1903 now swung to France,
England, and Germany, where it remained for almost 20 years. In the process,
static stability became an unquestioned design feature in all successful aircraft up
to the 1970s.
It is interesting that very modern airplane design has returned full circle to
the Wright brothers' original philosophy, at least in some cases. Recent lightweight
military fighter designs, such as the F16, F17, and F18, are statically unstable
in order to obtain dramatic increases in maneuverability. At the same time, the
airplane is instantaneously kept under control by computercalculated and electri
cally adjusted positions of the control surfacesthe "fly by wire" concept. In this
fashion, the maneuverability advantages of static instability can be realized
without heavily taxing the pilot: the work is done by electronics! Even when
maneuverability is not a prime feature, such as in civil transport airplanes, static
instability has some advantages. For example, the tail surfaces for an unstable
airplane can be smaller, with a subsequent savings in structural weight and
reductions in aerodynamic drag. Hence, with the advent of the "fly by wire"
PRINCIPLES OF STABILITY AND CONTROL 405
system, the cardinal airplane design principle of static stability may be somewhat
relaxed in the future. The Wright brothers may indeed ride again!
7.19 HISTORICAL NOTE: THE DEVELOPMENT
OF FLIGHT CONTROLS
Figure 7.3 illustrates the basic aerodynamic control surfaces on an airplane, the
ailerons, elevator, and rudder. They have been an integral part of airplane designs
for most of the twentieth century, and we take them almost for granted. But
where are their origins? When did such controls first come into practical use?
Who had the first inspirations for such controls?
In the previous section, we have already mentioned that by 1809 George
Cayley employed a movable tail in his designs, the first effort at some type of
longitudinal control. Cayley's idea of moving the complete horizontal tail to
obtain such control persisted through the first decade of the twentieth century.
Henson, Stringfellow, Penaud, Lilienthal, the Wright brothers all envisioned or
utilized movement of the complete horizontal tail surface for longitudinal control.
It was not until 19081909 that the first "modern" tail control configuration was
put into practice. This was achieved by the French designer Levavasseur on his
famous Antoinette airplanes, which had fixed vertical and horizontal tail surfaces
with movable, flaplike rudder and elevator surfaces at the trailing edges. So the
configuration for elevators and rudders as shown in Figure 7.3 dates back to 1908,
five years after the dawn of powered flight.
The origin of ailerons (a French word for the extremity of a bird's wing) is
steeped in more history and controversy. It is known that the Englishman M. P.
W. Boulton patented a concept for lateral control by ailerons in 1868. Of course,
at that time no practical aircraft existed, so the concept could not be demon
strated and verified, and Boulton's invention quickly retreated to the background
and was forgotten. Ideas of warping the wings or inserting vertical surfaces
(spoilers) at the wingtips cropped up several times in Europe during the late
nineteenth century and into the first decade of the twentieth century, but always
in the context of a braking surface which would slow one wing down and pivot
the airplane about a vertical axis. The true function of ailerons or wing warping,
that for lateral control for banking and consequently turning an airplane, was not
fully appreciated until Orville and Wilbur incorporated wing warping on their
Flyers (see Chap. 1). The Wright brothers' claim that they were the first to invent
wing warping may not be historically precise, but clearly they were the first to
demonstrate its function and to obtain a legally enforced patent on its use
(combined with simultaneous rudder action for total control in banking). The
early European airplane designers did not appreciate the need for lateral control
until Wilbur's dramatic public flights in France in 1908. This is in spite of the fact
that Wilbur had fully described their wing warping concept in a paper at Chicago
on September 1, 1901, and again on June 24, 1903; indeed, Octave Chanute
clearly described the Wrights' concept in a lecture to the Aero Club de France in
406 INTRODUCTION TO FLIGHT
Paris in April 1903. Other aeronautical engineers at that time, if they listened, did
not pay much heed. As a result, European aircraft before 1908, even though they
were making some sustained flights, were awkward to control.
However, the picture changed after 1908, when in the face of the indisputable
superiority of the Wrights' control system, virtually everybody turned to some
type of lateral control. Wing warping was quickly copied and was employed on
numerous different designs. Moreover, the idea was refined to include movable
surfaces near the wingtips. These were first separate "winglets" mounted either
above, below, or between the wings. But, in 1909, Henri Farman (see Sec. 7.1)
designed a biplane named the Henri Farman III, which included a flaplike aileron
at the trailing edge of all four wingtips; this was the true ancestor of the
conventional modernday aileron, as sketched in Figure 7.3. Farman's design was
soon adopted by most designers, and wing warping quickly became passe. Only
the Wright brothers clung to their old concept; a Wright airplane did not
incorporate ailerons until 1915, six years after Farman's development.
7.20 HISTORICAL NOTE: THE "TUCKUNDER" PROBLEM
A quick examination of Figure 7.20, and the resulting stability equations such as
Eqs. (7.26), (7.27), and (7.28), clearly underscores the importance of the down
wash angle e in determining longitudinal static stability. Downwash is a rather
skittish aerodynamic phenomenon, very difficult to calculate accurately for real
airplanes and therefore usually measured in windtunnel tests or in free flight. A
classic example of the stability problems that can be caused by downwash, and
how windtunnel testing can help, occurred during World War II, as described
below.
In numerous flights during 1941 and 1942, the Lockheed P38, a twinengine,
twinboomed, highperformance fighter plane (see Figure 7.35), went into sudden
dives from which recovery was exceptionally difficult. Indeed, several pilots were
killed in this fashion. The problem occurred at high subsonic speeds, usually in a
dive, where the airplane had a tendency to nose over, putting the plane in yet a
steeper dive. Occasionally, the airplane would become locked in this position, and
even with maximum elevator deflection, a pullout could not be achieved. This
"tuckunder" tendency could not be tolerated in a fighter aircraft which was
earmarked for a major combat role.
Therefore, with great urgency, NACA was asked to investigate the problem.
Since the effect occurred only at high speeds, usually above Mach 0.6, com
pressibility appeared to be the culprit. Tests in the Langley 30 ft by 60 ft
lowspeed tunnel and in the 8ft highspeed tunnel (see Sec. 4.22) correlated the
tuckunder tendency with the simultaneous formation of shock waves on the wing
surface. Such compressibility effects were discussed in Secs. 5.9 and 5.10, where it
was pointed out that, beyond the critical Mach number for the wing, shock waves
will form on the upper surface, encouraging flow separation far upstream of the
trailing edge. The P38 was apparently the first operational airplane to encounter
PRINCIPLES OF STABILITY AND CONTROL 407
fu
11.
' I
. t:i 
/
Figure 7.35 The Lockheed P38 of World War II fame.
this problem. The test engineers at Langley made several suggestions to rectify the
situation, but they all involved major modifications of the airplane. For a model
already in production, a quicker fix was needed.
Next, the 16ft highspeed wind tunnel at the NACA Ames Aeronautical
Laboratory in California (see again Sec. 4.22) was pressed into service on the P38
problem. Here, further tests indicated that the shockinduced separated flow over
the wing was drastically reducing the lift. In turn, because the downwash is
directly related to lift, as discussed in Secs. 5.13 and 5.14, the downwash angle e
was greatly reduced. Consequently (see Figure 7.20), the tail angle of attack a,
was markedly increased. This caused a sharp increase in the positive lift on the
tail, creating a strong pitching moment, nosing the airplane into a steeper dive.
After the series of Ames tests in April 1943, Al Erickson of NACA suggested the
408 INTRODUCTION TO FLIGHT
addition of flaps on the lower surface of the wing at the 0.33c point in order to
increase the lift, hence increase the downwash. This was the quick fix that
Lockheed was looking for, and it worked.
7.21 CHAPTER SUMMARY
Some of the important points of this chapter are given below.
1. If the forces and moments on a body caused by a disturbance tend initially to
return the body toward its equilibrium position, the body is statically stable.
In contrast, if these forces and moments tend initially to move the body away
from its equilibrium position, the body is statically unstable.
2. The necessary criteria for longitudinal balance and static stability are (a) CM
0
must be positive, (b) acM,cJaaa must be negative, and (c) the trim angle
attack ae must fall within the flight range of angle of attack for the airplane.
These criteria may be evaluated quantitatively for a given airplane from
and
acM,cg = a[(h  h )  V at(l 
aaa acwb H a aa
where the tail volume ratio is given by
list
(7.27)
(7.28)
3. The neutral point is that location of the center of gravity where ac M.cgl aaa
= 0. It can be calculated from
(7.30)
4. The static margin is defined as hn  h. For static stability, the location of the
center of gravity must be ahead of the neutral point; i.e., the static margin
must be positive.
5. The effect of elevator deflection 8e on the pitching moment about the center
of gravity is given by
CM.cg= cM,acwb + cL,wb(h  hac)  vH( a tat+ 8e) (7.37)
6. The elevator deflection necessary to trim an airplane at a given angle of
attack an is
CM,O +(8CM,cg/aaJan
strim= VH( acL,t;ase)
(7.43)
PRINCIPLES OF STABILITY AND CONTROL 409
BIBLIOGRAPHY
Etkin, B., Dynamics of Flight, Wiley, New York, 1959.
GibbsSmith, C. H., Aviation: An Historical Survey from its Origins to the End of World War Tl, Her
Majesty's Stationery Office, London, 1970.
Perkins, C. D., and Hage, R. E., Airplane Performance, Stability, and Control, Wiley, New York,
1949.
PROBLEMS
7.1 For a given wingbody combination, the aerodynamic center lies 0.03 chord length ahead of the
center of gravity. The moment coefficient about the center of gravity is 0.0050, and the lift coefficient
is 0.50. Calculate the moment coefficient about the aerodynamic center.
7.2 Consider a model of a wingbody shape mounted in a wind tunnel. The flow conditions in the test
section are standard sealevel properties with a velocity of 100 m/s. The wing area and chord are 1.5
m2 and 0.45 m, respectively. Using the windtunnel force and momentmeasuring balance, the
moment about the center of gravity when the lift is zero is found to be 12.4 N · m. When the model
is pitched to another angle of attack, the lift and moment about the center of gravity are measured to
be 3675 N and 20.67 N · m, respectively. Calculate the value of the moment coefficient about the
aerodynamic center and the location of the aerodynamic center.
7.3 Consider the model in Prob. 7.2. If a mass of lead is added to the rear of the model such that the
center of gravity is shifted rearward by a length equal to 20 percent of the chord, calculate the moment
about the center of gravity when the lift is 4000 N.
7.4 Consider the wingbody model in Prob. 7.2. Assume that a horizontal tail with no elevator is
added to this model. The distance from the airplane's center of gravity to the tail's aerodynamic center
is 1.0 m. The area of the tail is 0.4 m
2
, and the tailsetting angle is 2.0°. The lift slope of the tail is 0.12
per degree. From experimental measurement, e
0
= 0 and Be/ Ba = 0.42. If the absolute angle of
attack of the model is 5° and the lift at this angle of attack is 4134 N, calculate the moment about the
center of gravity.
7.5 Consider the wingbodytail model of Prob. 7.4. Does this model have longitudinal static stability
and balance?
7.6 For the configuration of Prob. 7.4, calculate the neutral point and static margin. h = 0.26.
7.7 Assume that an elevator is added to the horizontal tail of the configuration given in Prob. 7.4. The
elevator control effectiveness is 0.04. Calculate the elevator deflection angle necessary to trim the
configuration at an angle of attack of 8 °.
7.8 Consider the configuration of Prob. 7.7. The elevator hinge moment derivatives are BCh /Ba,=
0.007 and BChj BB, =  0.012. Assess the stickfree static stability of this configuration. '
CHAPTER
EIGHT
ASTRONAUTICS
It is difficult to say what is impossible, for the dream of yesterday is the hope of today and the reality
of tomorrow.
Robert H. Goddard, at his high school graduation, 1904
Houston, Tranquillity Base here. The Eagle has landed.
8.1 INTRODUCTION
Neil Armstrong, in a radio transmission to Mission
Control, at the instant of the first manned landing
on the moon, u ~ y 20, 1969
Spacethat last frontier, that limitless expanse which far outdistances the reach
of our strongest telescopes, that region which may harbor other intelligent
civilizations on countless planets; spacewhose unknown secrets have attracted
the imagination of humanity for centuries and whose technical conquest has
labeled the latter half of the twentieth century as the "space age"; spaceis the
subject of this chapter.
To this point in our introduction of flight, emphasis has been placed on
aeronautics, the science and engineering of vehicles which are designed to move
within the atmosphere and which depend on the atmosphere for their lift and
propulsion. However, as presented in Sec. 1.8, the driving force behind the
advancement of aviation has always been the desire to fly higher and faster. The
ultimate, of course, is to fly so high and so fast that you find yourself in outer
space, beyond the limits of the sensible atmosphere. Here, motion of the vehicle
takes place only under the influence of gravity and possibly some type of
propulsive force; however, the mode of propulsion must be entirely independent
of the air for its thrust. Therefore, the physical fundamentals and engineering
principles associated with space vehicles are somewhat different from those
410
ASTRONAUTICS 411
Figure 8.1 Earth orbit.
Figure 8.2 The Skylaban earth satellite. (NASA.)
412 INTRODUCTION TO FLIGHT
associated with airplanes. The purpose of this chapter is to introduce some of the
basic concepts of space flight, i.e., to introduce the discipline of astronautics. In
particular, the early sections of this chapter will emphasize the calculation and
analysis of orbits and trajectories of space vehicles operating under the influence
of gravitational forces only (such as in the vacuum of free space). In the later
sections, we will consider several aspects of the reentry of a space vehicle into the
earth's atmosphere, especially the reentry trajectory and aerodynamic heating of
the vehicle.
The space age formally began on October 4, 1957, when the Soviet Union
launched Sputnik /, the first artificial satellite to go into orbit about the earth.
Figure 8.3 Earthmoon mission (not to scale).
ASTRONAUTICS 413
Unlike the first flight of the Wright brothers in 1903, which took years to have
any impact on society, the effect of Sputnik I on the world was immediate. Within
12 years, people had walked on the moon, and after another 7 years, unmanned
probes were resting on the surfaces of Venus and Mars. A variety of different
space vehicles designed for different missions have been launched since 1957.
Most of these vehicles fall into three main categories, as follows:
I. Earth satellites, launched with enough velocity to go into orbit about the
earth, as sketched in Figure 8.1. As we will show later, velocities on the order
of 26,000 ftjs (7.9 km/s) are necessary to place a vehicle in orbit about the
earth, and these orbits are generally elliptical. Figure 8.2 shows a photograph
of an artificial earth satellite.
2. Lunar and interplanetary vehicles, launched with enough velocity to over
come the gravitational attraction of the earth and travel into deep space.
Velocities of 36,000 ftjs (approximately 11 km/s) or larger are necessary for
this purpose. Such trajectories are parabolic or hyperbolic. A typical path
from the earth to the moon is sketched in Figure 8.3; here, the space vehicle is
first placed in earth orbit, from which it is subsequently boosted by onboard
rockets to an orbit about the moon, from which it finally makes a landing on
the moon's surface. This is the mode employed by all the Apollo manned
Figure 8.4 The Apollo spacecraft. (Smithsonian National Air and Space Museum.)
414 INTRODUCTION TO FLIGHT
Figure 8.5 Earth orbit with lifting reentry.
lunar missions, beginning with the historic first moon landing on July 20,
1969. A photograph of the Apollo spacecraft is shown in Figure 8.4.
3. Space shuttles, designed to take off from the earth's surface, perform a
mission in space, and then return and land on the earth's surface, all
selfcontained in the same vehicle. These are lifting reentry vehicles, designed
with a reasonable L/ D ratio to allow the pilot to land the craft just like an
airplane. Earth orbit with a lifting reentry path is sketched in Figure 8.5. The
first successful flight of a space shuttle into space, with a subsequent lifting
reentry and landing, was carried out by NASA's Columbia during the period
April 1214, 1981. A photograph of the space shuttle is given in Figure 8.6.
Finally, a discussion of astronautics, even an introductory one, requires
slightly more mathematics depth than just basic differential and integral calculus.
Therefore, this chapter will incorporate more mathematical rigor than other parts
of this book. In particular, some concepts from differential equations must be
employed. However, it will be assumed that the reader has not had exposure to
such mathematics, and therefore the necessary ideas will be introduced in a
selfcontained fashion.
ASTRONAUTICS 415
Figure 8.6 The space shuttle. (NASA.)
8.2 DIFFERENTIAL EQUATIONS
Consider a dependent variable r which depends on an independent variable t.
Thus, r = f(t). The concept of the derivative of r with respect tot, dr/dt, has
been used frequently in this book. The physical interpretation of dr / dt is simply
the rate of change of r with respect to t. If r is a distance and t is time, then
dr / dt is the rate of change of distance with respect to time, i.e., velocity. The
second derivative of r with respect to t is simply
d(dr/dt} = d
2
r
dt  dt
2
This is the rate of change of the derivative itself with respect to t. If r and t are
distance and time, respectively, then d
2
r/dt
2
is the rate of change of velocity
with respect to time, i.e., acceleration.
A differential equation is simply an equation which has derivatives in some of
its terms. For example
d
2
r dr
3
 + r  2t = 2
dt2 dt
(8.1)
is a differential equation; it contains derivatives along with the variables r and t
416 INTRODUCTION TO FLIGHT
themselves. By comparison, the equation
t2
r+= 0
r
is an algebraic equation; it contains only r and t without any derivatives.
To find a solution of the differential equation, Eq. (8.1), means to find a
functional relation r = f(t) which satisfies the equation. For example, assume
that r = t
2
• Then, dr/dt = 2t and d
2
r/dt
2
= 2. Substitute into Eq. (8.1):
2 + t
2
2t  2t
3
= 2
2 + 2t
3
 2t
3
= 2
2=2
Hence, r = t
2
does indeed satisfy the differential equation, Eq. (8.1). Thus, r = t
2
is called a solution of that equation.
Calculations of space vehicle trajectories involve distance r and time t. Some
of the fundamental equations involve first and second derivatives of r with
respect tot. To simplify the notation in these equations, we now introduce the dot
notation for time derivatives, i.e.,
r = dr
dt
d
2
r
r=
dt2
A single dot over the variable means the first time derivative of that variable; a
double dot means the second derivative. For example, the differential equation,
Eq. (8.1), can be written as
;: + rr  2t
3
= 2 (8.2)
Equations (8.1) and (8.2) are identical; only the notation is different. The dot
notation for time derivatives is common in physical science; you will encounter it
frequently in moreadvanced studies of science and engineering.
8.3 LAGRANGE'S EQUATION
In physical science, a study of the forces and motion of bodies is called
mechanics. If the body is motionless, this study is further identified as statics; if
the body is moving, the study is one of dynamics. In this chapter, we are
concerned with the dynamics of space vehicles.
Problems in dynamics usually involve the use of Newton's second law,
F = ma, where Fis force, m is mass, and a is acceleration. Perhaps the reader is
familiar with various applications of F = ma from basic physics; indeed, we
applied this law in Chap. 4 to obtain the momentum equation in aerodynamics
and again in Chap. 6 to obtain the equations of motion for an airplane. However,
in this section we introduce an equation, Lagrange's equation, which is essentially
ASTRONAUTICS 417
x
W=mg
Figure 8.7 Falling body.
a corollary to Newton's second law. The use of Lagrange's equation represents an
alternative approach to the solution of dynamics problems in lieu of F = ma; in
the study of space vehicle orbits and trajectories, Lagrange's equation greatly
simplifies the analysis. We will not derive Lagrange's equation; rather, we will
simply introduce it by way of an example and then, in the next section, apply it to
obtain the orbit equation. A rigorous derivation of Lagrange's equation is left to
moreadvanced studies of mechanics.
Consider the following example. A body of mass m is falling freely in the
earth's gravitational field, as sketched in Figure 8.7. Let x be the vertical distance
of the body from the ground. If we ignore drag, the only force on the body is its
weight w directed downward. By definition, the weight of a body is equal to its
mass m times the acceleration of gravity, g; w = mg. From Newton's second law,
F= ma (8.3)
The force is weight, directed downward. Since the direction of positive x is
upward, then a downwardacting force is negative. Hence,
F= w =mg
From the discussion in Sec. 8.2, the acceleration can be written as
d
2
x
a==x
dt
2
Substituting Eqs. (8.4) and (8.5) into (8.3) yields
mg= mx
(8.4)
(8.5)
(8.6)
Equation (8.6) is the equation of motion for the body in our example. It is a
differential equation whose solution will yield x = f(t). Moreover, Eq. (8.6) was
obtained by the application of Newton's second law.
Now consider an alternative formulation of this example, using Lagrange's
equation. This will serve as an introduction to Lagrange's equation. Let T denote
the kinetic energy of the body, where by definition,
T = 1mV
2
= 1m(.x)
2
(8.7)
Let <I> denote the potential energy of the body. By definition, potential energy of a
418 INTRODUCTION TO FLIGHT
body referenced to the earth's surface is the weight of the body times the distance
above the surface:
<I> = wx = mgx (8.8)
Now define the lagrangian function B as the difference between kinetic and
potential energy:
B = T  <I> (8.9)
For our example, combining Eqs. (8.7) to (8.9), we get
B = }m(x)
2
 mgx (8.10)
We now write down Lagrange's equation, which will have to be accepted without
proof; it is simply a corollary to Newton's second law:
!!...(BB) BB= O (
8
.ll)
dt a.x ax
In Lagrange's equation above, recall the definition of the partial derivative given
in Sec. 7.2 D. For example, BB/Bx means the derivative of B with respect to x,
holding everything else constant. Hence, from Eq. (8.10),
= mx (8.12)
and
BB
ax= mg
Substituting Eqs. (8.12) and (8.13) into Eq. (8.11), we have
or, because m is a constant,
!!... ( mx)  (  mg) = 0
dt
d .
m  ( x)  (  mg) = 0
dt
mx +mg= 0
(8.13)
(8.14)
Compare Eqs. (8.14) and (8.6); they are identical equations of motion.
Therefore, we induce that Lagrange's equation and Newton's second law are
equivalent mechanical relations and lead to the same equations of motion for a
mechanical system. In the above example, the use of Lagrange's equation resulted
in a slightly more complicated formulation than the direct use of F = ma.
However, in the analysis of space vehicle orbits and trajectories, Lagrange's
equation is the most expedient formulation, as will be detailed in the next section.
With the above example in mind, a more general formulation of Lagrange's
equation can be given. Again, no direct proof is given; the reader must be content
with the "cookbook" recipe given below, using the above example as a basis for
induction. Consider a body moving in threedimensional space, described by
ASTRONAUTICS 419
some generalized spatial coordinates q
1
, q
2
, and q
3
. (These may be r, 8, and </>
for a spherical coordinate system; x, y, and z for rectangular coordinate system;
etc.) Set up the expression for the kinetic energy of the body, which may depend
on the coordinates q
1
, q
2
, and q
3
themselves as well as the velocities iJi, q
2
, and
q3:
(8.15)
Then, set up the expression for the potential energy of the body, which depends
only on spatial location:
(8.16)
Now form the lagrangian function:
B=Tcl> (8.17)
Finally, obtain three equations of motion (one along each coordinate direction)
by writing Lagrange's equation for each coordinate:
. d ( aB) aB
ql coordmate: d a  a= 0
t ql ql
. d ( aB) aB
q2 coordmate: dt aq2  aq2 = 0
(8.18)
. d ( aB) aB
q
3
coordmate: d y  a = 0
t q3 q3
Let us now apply this formalism to obtain the orbit or trajectory equations
for a space vehicle.
8.4 THE ORBIT EQUATION
Space vehicles are launched from a planet's surface by rocket boosters. The rocket
engines driving these boosters are discussed in Chap. 9. Here, we are concerned
with the motion of the vehicle after all stages of the booster have burned out and
the satellite, interplanetary probe, etc., is smoothly moving through space under
the influence of gravitational forces. At the instant the last booster stage bums
out, the space vehicle is at a given distance from the center of the planet, moving
in a specific direction at a specific velocity. Obviously, nature prescribes a specific
path (a specific orbit about the planet or possibly a specific trajectory away from
the planet) for these given conditions at burnout. The purpose of this section is to
derive the equation which describes this path.
420 INTRODUCTION TO FLIGHT
M
A Force and Energy
Figure 8.8 Movement of a small mass in the gravitational
field of a large mass.
Consider a vehicle of mass m moving with velocity V in the vicinity of a planet of
large mass M, as sketched in Figure 8.8. The distance between the centers of the
two masses is r. In a stroke of genius during the last quarter of the seventeenth
century, Isaac Newton uncovered the law of universal gravitation, which states
that the gravitational force between two masses varies inversely as the square of
the distance between their centers. In particular, this force is given by
F = GmM (8.19)
,2
where G is the universal gravitational constant, G = 6.67 x 1 ~
1 1
m
3
/(kg)(s)
2
.
Lagrange's equation deals with energy, both potential and kinetic. First
consider the potential energy of the system shown in Figure 8.8. Potential energy
is always based on some reference point, and for gravitational problems in
astronautics, it is conventional to establish the potential energy as zero at r equal
to infinity. Hence, the potential energy at a distance r is defined as the work done
in moving the mass m from infinity to the location r. Let <I> be the potential
energy. If the distance between M and m is changed by a small increment dr,
then the work done in producing this change is F dr. This is also the change in
potential energy, d<I>. Using Eq. (8.19), we obtain
GmM
d<I> = Fdr = dr
,2
Integrating from r equals infinity, where <I> by definition is 0, tor= r, where the
potential energy is <I> = <I>, we get
f
<I> d<I> = f r GmM dr
0 ,2
00
or I W G,mM I (8.20)
Equation (8.20) gives the potential energy of small mass m in the gravitational
field of large mass M at the distance r. The potential energy at r is a negative
value due to our choice of <I> = 0 at r going to infinity. However, if the idea of a
negative energy is foreign to you, do not be concerned. In mechanical systems, we
ASTRONAUTICS 421
Figure 8.9 Polar coordinate system.
are usually concerned with changes in energy, and such changes are independent
of our choice of reference for potential energy.
Now consider the kinetic energy. Here we need more precisely to establish
our coordinate system. In moreadvanced studies of mechanics, it can be proven
that the motion of a body in a central force field (such as we are dealing with
here) takes place in a plane. Hence, we need only two coordinates to designate the
location of mass m. Polar coordinates are particularly useful in this case, as
shown in Figure 8.9. Here, the origin is at the center of mass M, r is the distance
between m and M, and () is the angular orientation of r. The velocity of the
vehicle of mass m is V. The velocity component parallel to r is V,. = dr / dt = t.
The velocity component perpendicular to r is equal to the radius vector r times
the time rate of change of 8, that is, times the angular velocity; V
8
= r(d8/dt) =
rB. Therefore, the kinetic energy of the vehicle is
(8.21)
B The Equation of Motion
From Eqs. (8.17), (8.20), and (8.21), the lagrangian function is
B = T  <I>= 1m [t
2
+ (r0)
2
] + GmM (8.22)
r
In orbital analysis, it is common to denote the product GM by k
2
. If we are
dealing with the earth, where M = 5.98 X 10
24
kg, then
k
2
=GM= 3.986 x 10
14
m
3
/s
2
Equation (8.22) then becomes
[
2 ( · 2] mk
2
B = 1m t + r8) + 
r
(8.23)
Now invoke Lagrange's equation, Eq. (8.18), where q
1
= () and q
2
= r. First, the
() equation is
!!__ ~ _ aB =
0
dt a8 a8
(8.24)
422 INTRODUCTION TO FLIGHT
From Eq. (8.23)
aB
2
•
. = mr ()
ao
and
aB = O
ao
Substituting Eqs. (8.25) and (8.26) into (8.24) , we obtain
!!_mr
2
il = 0
dt
(8.25)
(8.26)
(8.27)
Equation (8.27) is the equation of motion of the space vehicle in the () direction.
It can be immediately integrated as
I mr
2
0 = const = c
1
I
(8.28)
From elementary physics, linear momentum is defined as mass times velocity.
Analogously, for angular motion, angular momentum is defined as Iii, where I is
the moment of inertia and iJ is the angular velocity. For a point mass m,
I = mr
2
• Hence, the product mr
2
il is the angular momentum of the space vehicle,
and from Eq. (8.28),
I mr
2
0 = angular momentum= const I
For a central force field, Eq. (8.28) demonstrates that the angular momentum is
constant.
Now consider the r equation. From Eq. (8.18), where q
2
= r,
From Eq. (8.23)
!!_ aB _ aB =
0
dt a; ar
aB .
ar= mr
aB
ar
mk2 ·2
  + mrO
r2
Substituting Eqs. (8.30) and (8.31) into (8.29), we get
or
d mk
2
 mt +   mrlJ
2
= 0
dt r2
mk
2
mr  mrlJ
2
+  = 0
r2
(8.29)
(8.30)
(8.31)
(8.32)
(8.33)
Equation (8.28) demonstrated that, since m is constant, r
2
0 is constant. Denote
this quantity by h.
r
2
1J = h = angular momentum per unit mass
ASTRONAUTICS 423
Multiplying and dividing the second term of Eq. (8.33) by r
3
and canceling m
yields
or
r
4
iJ2 mk
2
m'f  m +  = 0
r3 r2
h2 k2
'f+=0
r3 r2
(8.34)
Equation (8.34) is the equation of motion for the space vehicle in the r direction.
Note that both h
2
and k
2
are constants. Recalling our discussion in Sec. 8.2, we
see that Eq. (8.34) is a differential equation. Its solution will provide a relation for
r as the function of time, that is, r = f(t).
However, examine Figure 8.9. The equation of the path of the vehicle in
space should be geometrically given by r = /(0), not r = f(t). We are interested
in this path, i.e., we want the equation of the space vehicle motion in terms of its
geometric coordinates r and 0. Therefore, Eq. (8.34) must be reworked, as
follows.
Let us transform Eq. (8.34) to a new dependent variable u, where
Then
Hence
1
r = 
u
. _ dr d(l/u)
r = dt = dt =
1 du dO
u2 dO dt
1 du
u2 dt
iJ du
u2 dO
hdu
dO
Differentiating Eq. (8.37) with respect to t, we get
'f =  h :i =  h ( :o )
h(d
2
u)d0 = hd
2
uiJ
d0
2
dt d0
2
But from Eq. (8.36), iJ = u
2
h. Substituting into Eq. (8.38), we obtain
'f = h2u2 d2u
d0
2
Substituting Eqs. (8.39) and (8.35) into Eq. (8.34) yields
d
2
u
h
2
u
2
  h
2
u
3
+ k
2
u
2
= 0
d0
2
(8.35)
(8.36)
(8.37)
(8.38)
(8.39)
424 INTRODUCTION TO FLIGHT
or, dividing by h
2
u
2
,
d
2
u k
2
+u=0
d()2 h2
(8.40)
Equation (8.40) is just as valid an equation of motion as is the original Eq. (8.34).
Equation (8.40) is a differential equation, and its solution gives u = f( 8). Specifi
cally, a solution of Eq. (8.40) is
k2
u = J;i. + A cos( ()  C) ( 8 .41)
where A and C are constants (essentially, constants of integration). You should
satisfy yourself that Eq. (8.41) is indeed a solution of Eq. (8.40) by substitution of
(8.41) into (8.40).
Return to the original transformation, Eq. (8.35). Substituting u = l/r into
Eq. (8.41) yields
1
r = 
k2/h2 +A cos(()  C)
(8.42)
Multiply and divide Eq. (8.42) by h
2
/k
2
:
h2/k2
r =
1 + A(h
2
/k
2
) cos(8  C)
(8.43)
Equation (8.43) is the desired equation of the path (the orbit, or trajectory) of
the space vehicle. It is an algebraic equation for r = f( 8); it gives the geometric
coordinates r and () for a given path. The specific path is dictated by the values of
the constants h
2
, A, and C in Eq. (8.43). In turn, refer to Figure 8.10: these
constants are fixed by conditions at the instant of burnout of the rocket booster.
At burnout, the vehicle is at distance rb from the center of the earth, and its
velocity has a magnitude Vb and is in a direction /3b with respect to a perpendicu
lar to r. These burnout conditions completely specify the vehicle's path; i.e., they
determine the values of h
2
, A, and C for Eq. (8.43).
Equation (8.43) is sometimes generically called the "orbit equation." How
ever, it applies to the trajectory of a space vehicle escaping from the gravitational
eb
_____ J_ _____ _
M
m
\
\
Figure 8.10 Conditions at the instant of
burnout.
ASTRONAUTICS 425
field of the earth as well as to an artificial satellite in orbit about the earth. In fact,
what kind of orbit or trajectory is described by Eq. (8.43)? What type of
mathematical curve is it? What physical conditions are necessary for a body to go
into orbit or to escape from the earth? The answers can be found by further
examination of Eq. (8.43), as discussed below.
8.5 SPACE VEHICLE TRAJECTORIES
SOME BASIC ASPECTS
Examine Eq. (8.43) closely. It has the general form
p
r= 
1 + e cos ( ()  C )
(8.44)
where p = h
2
/k
2
, e = A(h
2
/k
2
), and C is simply a phase angle. From analytic
geometry, Eq. (8.44) is recognized as the standard form of a conic section in polar
coordinates; i.e., Eq. (8.44) is the equation of a circle, ellipse, parabola, or
hyperbola, depending on the value of e, where e is the eccentricity of the conic
section. Specifically:
If e = 0, the path is a circle.
If e < 1, the path is an ellipse.
If e = 1, the path is a parabola.
If e > 1, the path is a hyperbola.
These possibilities are sketched in Figure 8.11. Note that point b on these
sketches denotes the point of burnout and that () is referenced to the dashed line
through b, that is, () is arbitrarily chosen as zero at burnout. Then C is simply a
phase angle which orients the x and y axes with respect to the burnout point,
where the x axis is a line of symmetry for the conic section. From inspection of
Figure 8.11, circular and elliptical paths result in an orbit about the large mass M
(the earth), whereas parabolic and hyperbolic paths result in escape from the
earth.
On a physical basis, the eccentricity, hence the type of path for the space
vehicle, is governed by the difference between the kinetic and potential energies of
the vehicle. To prove this, consider first the kinetic energy, T = ~ m V
2
• From Eq.
(8.21),
T= ~ m [ r
+(rb)
2
]
Differentiate Eq. (8.44) with respect to t:
dr [ re sin ( ()  C)] ti
 = f = 
dt 1 + e cos ( ()  C )
(8.45)
426 INTRODUCTION TO FLIGHT
y
v
y
v
Circle e = 0 Ellipse e < l
y
M
Parabola e = l Hyperbola e > I
Figure 8.11 The four types of orbits and trajectories, illustrating the relation of the burnout point and
phase angle with the axes of symmetry.
Substitute Eq. (8.45) into (8.21):
T = m + ,202
1 ( [r
2
e
2
sin
2
(8 c)]tF . )
2 [ 1 + e cos ( 8  C)]
2
(8.46)
Recall that r
2
0 = h, hence 0
2
= h
2
/r
4
. Thus, Eq. (8.46) becomes
1 ( h
2
e
2
sin
2
(8C) h
2
)
T= 2m r
2
[1 + ecos(O C)]
2
+ 72
(8.47)
ASTRONAUTICS 427
Putting the righthand side of Eq .. (8.47) under the same common denominator
and remembering from Eq. (8.44) that
(
h2 )
2
r
2
[ 1 + e cos ( {}  C)]
2
= k
2
we transform Eq. (8.47) into
1 k
4
T = 2 m ,;·d 1 + 2 e cos ( (J  C) + e
2
] (8.48)
The reader should put in the few missing algebraic steps to obtain Eq. (8.48).
Consider now the absolute value of the potential energy, denoted as l<l>I.
From Eq. (8 .. 20),
l<l>I = GMm = k
2
m
r r
Substitute Eq .. (8.44) into (8.49):
k
4
m
I <I> I = h2 [ 1 + e cos ( o  c)]
(8.49)
(8 .. 50)
The difference between the kinetic and potential energies is obtained by
subtracting Eq .. (8.50) from (8.48):
1 k
4
k
4
m
T  I <I> I = 2 m ,;d 1 + 2 e cos ( lJ  C) + e
2
]  J;2 [ 1 + e cos ( {}  C)]
Let H denote T  l<l>I .. Then, Eq .. (8 .. 51) becomes
1 k
4
H = T  l<l>I =  m(1  e
2
)
2 h2
Solving Eq .. (8.52) for e, we get
e=
(8 .. 51)
(8 .. 52)
(8.53)
Equation (8 .. 53) is the desired result, giving the eccentricity e in terms of the
difference between kinetic and potential energies, H ..
Examine Eq .. (8 .. 53) .. If the kinetic energy is smaller than the potential energy,
H will be negative, and hence e < 1.. If the kinetic and potential energies are
equal, H = 0 and e = 1.. Similarly, if the kinetic energy is larger than the
potential energy, H is postive and e > 1.. Referring again to Figure 8 .. 11, we can
428 INTRODUCTION TO FLIGHT
make the following tabulation:
Type of Trajectory e Energy Relation
Ellipse < 1
'mv2 < GMm
2 r
Parabola = 1
'mv2 = GMm
2 r
Hyperbola > 1
'mv2 > GMm
2 r
From this we make the important conclusion that a vehicle intended to escape the
earth and travel into deep space (a parabolic or hyperbolic trajectory) must be
launched such that its kinetic energy at burnout is equal to or greater than its
potential energy, a conclusion that makes intuitive sense even without the above
derivation.
Equation (8.53) tells us more. For example, what velocity is required for a
circular orbit? To answer this question, recall that a circle has zero eccentricity.
Putting e = 0 in Eq. (8.53), we get
0=
2h
2
H
1+
mk
4
H=
mk
4
(8.54) or

2h
2
Recall that H = T  lcf>I = tmV
2
 GMm/r. Hence, Eq. (8.54) becomes
mk
4
GMm
tmV
2
=   +
2h2 r
(8.55)
From Eq. (8.44), with e = 0,
h2
r=
k2
(8.56)
Substitute Eq. (8.56) into (8.55) and solve for V:
m k
2
k
2
m k
2
m
tmV
2
=   +
2 r r 2r
Thus
V=lf
circular velocity (8.57)
Equation (8.57) gives the velocity required to obtain a circular orbit. Recall from
Sec. 8.4 B that k
2
= GM= 3.956 X 10
14
m
3
/s
2
. Assume that r = 6.4 X 10
6
m,
essentially the radius of the earth. Then
V=
3.986 x 10
14
= 7.9 x 10
3
m/s
6.4 x 10
6
ASTRONAUTICS 429
This is a convenient number to remember; circular, or orbital, velocity is 7.9 km/s,
or approximately 26,000 ft/s.
The velocity required to escape the earth can be obtained in much the same
fashion. We have demonstrated above that a vehicle will escape if it has a
parabolic ( e = 1) or a hyperbolic ( e > 1) trajectory. Consider a parabolic trajec
tory. For this, we know that the kinetic and potential energies are equal; T = lcl>I.
Hence,
Solving for V, we get
GMm
!mV
2
=
r r
V ~
parabolic velocity (8.58)
Equation (8.58) gives the velocity required to obtain a parabolic trajectory. This is
called the escape velocity; note by comparing Eqs. (8.57) and (8.58) that escape
velocity is larger than orbital velocity by a factor of .fi. Again assuming r is the
radius of the earth, r = 6.4 X 10
6
m, then escape velocity is 11.2 km/s, or
approximately 36,000 ft/s. Return to Figure 8.10; if at burnout V b ~ 11.2 km/s,
then the vehicle will escape the earth, independent of the direction of motion, f3b.
Example 8.1 At the end of a rocket launch of a space vehicle, the burnout velocity is 9 km/sin a
direction due north and 3° above the local horizontal. The altitude above sea level is 500 mi. The
burnout point is located at the 27th parallel (27°) above the equator. Calculate and plot the
trajectory of the space vehicle.
SOLUTION The burnout conditions are sketched in Figure 8.12. The altitude above sea level is
hG = 500 mi= 0.805x10
6
m
The distance from the center of the earth to the burnout point is (where the earth's radius is
r, = 6.4 X 10
6
m)
rh = r, + hG = 6.4X 10
6
+0.805X10
6
=7.2X10
6
m
As given in Sec. 8.4 B,
Also, as defined earlier,
h = r
2
1J = r(rlJ) = rVo
430 INTRODUCTION TO FLIGHT
Equator
Figure 8.12 Burnout conditions for Example 8.1.
where Vo is the velocity component perpendicular to the radius vector r. Thus,
h = rVfi = rhVcos{Jh = (7.2 X 10
6
)(9 X 10
3
) cos3°
=6.47x10
10
m
2
/s
h
2
=4.188x10
21
m
4
/s
2
Hence
= ':}.,, = 4.188x1021 = 1.0506x107
P k
2
3.986Xl0
14
m
The trajectory equation is given by Eq. (8.44), where the above value of p is the numerator of the
righthand side. To proceed further, we need the eccentricity e. This can be obtained from Eq.
(8.53)
e=
where H/m = (T lcf>l)/m.
Hence
Thus
T V
2
(9X10
3
)
2
== =405x10
7
m
2
/s
2
m 2 2 ·
/
~ = G M = ~ = 3.986x1014 = 5.536x107 m1/s2
m rh rh 7 .2 x 10
6
!!_ = (4.055.536) x10
7
=1.486x10
7
m
2
/s
2
m
= (l+ 2(4.188X10
21
)(1.486X10
7
) )l/Z
(3.986X10
14
)
2
= /0.2166 = 0.4654
Immediately we recognize that the trajectory is an elliptical orbit, because e < 1 and also because
ASTRONAUTICS 431
T < l<l>I. From Eq. (8.44),
p
r=
1 + ecos(O  C)
1.0506 x 10
7
1 + 0.4654cos(O  C)
To find the phase angle C, simply substitute the burnout location ( rh = 7.2X10
6
m and (i = 0°)
into the above equation. (Note that (i = 0° at burnout, and hence (i is measured relative to the
radius vector at burnout, with increasing (i taken in the direction of motion: this is sketched