Principles_of_Flight_JAA_ATPL_theory.pdf

081 PRINCIPLES OF FLIGHT
© G LONGHURST 1999 All Rights Reserved Worldwide

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EDITION 2.00.00 2001
This is the second edition of this manual, and incorporates all amendments to previous editions, in
whatever form they were issued, prior to July 1999.
EDITION 2.00.00 © 1999,2000,2001 G LONGHURST
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TABLE OF CONTENTS
Aerodynamic Principles
Lift
Drag
Stalling
Lift Augmentation
Control
Forces in Flight
Stability
High Speed Flight
Limitations

TABLE OF CONTENTS
Special Circumstances
Propellers

081 Principles of Flight
Units
Systems of Units
Newton's Laws of Motion
The Equation of Impulse
Basic Gas Laws
Airspeed Measurement
Shape of an Aerofoil
The Equation of Continuity
Bernoulli’s Theorem

Chapter 1 Page 1 © G LONGHURST 1999 All Rights Reserved Worldwide
1Aerodynamic Principles
Units
1. In order to define the magnitude of a particular body in terms of mass, length, time,
acceleration etc., it is necessary to measure it against a system of arbitrary units. For example, one
pound (lb) is a unit of mass, so the mass of a particular body may be described as being a multiple
(say 10 lb), or sub-multiple (say ½ lb) of this unit. Alternatively the mass of the body could have
been measured in kilograms, since the kilogram (kg) is another arbitrary unit of mass.
Systems of Units
2. There are a number of systems of units in existence and it is essential when making
calculations to maintain consistency by using only one system. Three well-known consistent systems
of units are the British, the c.g.s. and the S.I. (Systeme Internationale). These are illustrated in
Figure 1-1 below:

FIGURE 1-1
Units of
Measurement
3. The S.I. system of units is the one most commonly used. In this system, one Newton is the
force that produces an acceleration of 1 M/s² when acting upon a mass of 1 kg.
Newton's Laws of Motion
4. The motion of bodies is usually quite complicated, involving several forces acting at the same
time as well as inertia and momentum. Before considering the Laws of Motion, as described by Sir
Isaac Newton, it is necessary to define force, inertia and momentum.
BRITISH C.G.S. S. I.
SYSTEM
LENGTH Foot Centimetre Metre (m)
TIME Second Second Second (s)
ACCELERATION Ft/s² C/s² M/s²
MASS Pound Gram Kilogram (kg)
FORCE Poundal Dyne Newton (N)

5. Force is that which changes a body's state of rest or of uniform motion in a straight line. The
most familiar forces are those which push or pull. These may or may not produce a change of
motion, depending upon what other forces are present. Pressure acting upon the surface area of a
piston exerts a force that causes the piston to move along its cylinder. If we push against the wall of
a building a force is exerted but the wall does not move, this is because an equal and opposite force is
exerted by the wall. Similarly, if a weight of 1 kilogram is resting upon a table there is a force
(gravitational pull) acting upon the weight but, because an equal and opposite force is exerted by the
table, there is no resultant motion.
Force Can Be Quantified
6. Where motion results from an applied force, the force exerted is the product of mass and
acceleration, or:
F= ma
Where: F = Force m = mass and a = acceleration
7. Inertia is the tendency of a body to remain at rest or, if moving, to continue its motion in a
straight line. Newton's first law of motion, often referred to as the law of inertia, states that every
body remains in a state of rest or uniform motion in a straight line unless it is compelled to change
that state by an applied force.

Momentum
8. The product of mass and velocity is called momentum. Momentum is a vector quantity, in
other words it involves motion, with direction being that of the velocity. The unit of momentum has
no name, it is given in kilogram metres per second (kg m/s). Newton's second law of motion states
that the rate of change of momentum of a body is proportional to the applied force and takes place in
the direction in which the force acts.
9. Newton's third law of motion states that to every action there is an equal and opposite
reaction. This describes the situation when a weight is resting upon a table. For a freely falling body
the force of gravity (gravitational pull), measured in Newtons , acting upon it is governed by:
F = mg
where g is acceleration due to gravity 9.81M/s², and m is the
mass of the body in kilograms.
10. If the same body is at rest upon a table it follows that, since there is no motion, there must be
an equal and opposite force exerted by the table.
Motion with Constant Acceleration
11. When acceleration is uniform, that is to say velocity is increasing at a constant rate, the
relationship between acceleration and velocity can be expressed by simple formulae known as the
equations of motion with constant acceleration. Under these circumstances velocity increases by the
same number of units each second, so the increase of velocity is the product of acceleration (a) and
time (t). If the velocity at the beginning of the time interval, (the initial velocity), is given the symbol
(u) and the velocity at the end of the time interval, (the final velocity), is given the symbol (v) then the
velocity increase for a given period of time can be expressed by the equation:

v = u + a.t
12. If it is required to calculate the distance travelled (s) during a period of motion with constant
acceleration, this can be done using the equation:
13. By substitution, using the above two equations, it is possible to develop two more equations:
And:
These are the equations of motion with constant acceleration.
The Equation of Impulse
14. Given that the momentum of a body is the product of its mass and its velocity it follows that,
providing mass and velocity remain constant, momentum will remain constant. A change of velocity
will occur if a force acts upon the body because:
s
1
2
--
- u v
+
( )t
=
s ut
1
2
--
-at2
+
=
v2 u2 2as
+
=
F ma
=

And therefore
15. If the force acts in the direction of motion of the body for a period of time (t), the resultant
acceleration will cause a velocity increase from (u) to (v). This must also cause an increase in
momentum from (mu) to (mv). Combining the equations F = ma and v = u+at gives:
Which transposes to:
16. The change in momentum (final momentum minus initial momentum) due to a force acting
on a body is the product of that force and the time for which it acts. This change in momentum
called the impulse of the force and is usually identified by the symbol J. Hence:
Or:
a
F
m
---
-
=
v u t
F
m
---
-

+
=
Ft mv mu
–
=
J Ft
=
J mv mu
–
=

17. This is the equation of impulse. The S.I. unit of impulse, being the product of force and time,
is the Newton second (Ns). NOT, it should be noted, Newton per second (N/s).
Basic Gas Laws
18. The Gas Laws deal with the relationships between pressure, volume and temperature of a gas.
They are based upon three separate experiments carried out at widely differing times in history.
These experiments investigated:
(a) The relation between volume (V) and pressure (P) at constant temperature (Boyle's
Law).
(b) The relation between volume (V) and temperature (T) at constant pressure (Charles'
Law).
(c) The relation between pressure (P) and temperature (T) at constant volume (Pressure
Law)
Boyle's Law
19. Boyle's Law states that the volume of a fixed mass of gas is inversely proportional to the
pressure, provided that the temperature remains constant. In other words, if the volume of a given
mass of gas is halved its pressure will be doubled or, if its pressure is halved its volume will be
doubled, providing its temperature does not change.
20. This may be expressed mathematically as:
P1V1 P2V2 or PV cons t
tan
= =

Charles' Law
21. Charles Law states that the volume of a fixed mass of gas at constant pressure expands by 1/
273 of its volume at 0°C for every 1°C rise in temperature. In other words, the volume of a given
mass of gas is directly proportional to its (absolute) temperature, providing its pressure does not
change.
Pressure Law
23. The pressure law is the result of experimentation during the nineteenth century by a professor
called Jolly and states that the pressure of a fixed mass of gas at constant volume increases by 1/273
of its pressure at 0°C for every 1°C rise in temperature. In other words, the pressure of a given mass
of gas is directly proportional to its temperature, providing its volume does not change.
V1
T1
------
V2
T2
------ or
V
T
---
- cons t
tan
=

=
P1
T1
-----
-
P2
T2
-----
- or
P
T
--
- cons t
tan
=

=

The Ideal Gas Equation
25. The three equations expressing the Gas Laws can be combined into a single or Ideal Gas
Equation which may be expressed mathematically as:
Static Pressure
26. The static pressure of the atmosphere at any given altitude is the pressure resulting from the
mass of an imaginary column of air above that altitude. In the International Standard Atmosphere
(ISA) at mean sea level the static pressure of the atmospheric air is 1013.25 millibars (mb), which
equates to 14.7 pounds per square inch (psi) or 29.92 inches of mercury (in. Hg). ISA mean sea level
conditions also assume an air density of 1.225 kilograms per cubic metre (kg/m³) and a temperature
of +15°C (288°A). The standard notation for static pressure at any altitude is (P).
Dynamic Pressure
27. Air has density (mass per unit volume) and consequently air in motion has energy and must
exert pressure upon a body in its path. Similarly, a body moving in air will have a pressure exerted
upon it that is proportional to its rate of movement, or velocity (V). This pressure due to motion is
known as dynamic pressure and is given the notation (q).
28. Energy due to motion is kinetic energy (K.E.) and in the S.I. system of units is measured in
joules (j). From Bernoulli’s equation for incompressible flow the kinetic energy due to air movement
may be calculated using the formula:
P1V1
T1
------------
-
P2V2
T2
------------
- or
PV
T
-------
- cons t
tan
=

=

29. To calculate kinetic energy in joules, density (ρ) must be in kilograms per cubic metre (kg/m³)
and velocity (V) in metres per second (m/s). One joule is the work done when a force of 1 newton
moves the point of application of the force 1 metre in the direction of the force.
30. If a volume of moving air is brought to rest, as in an open-ended tube facing into the
airstream, the kinetic energy is converted into pressure energy with negligible losses. Hence, dynamic
pressure:
31. It should be noted that dynamic pressure cannot be measured in isolation, since ambient
atmospheric pressure (static pressure) is always present also. The sum of the two, (q+P), is known
variously as total pressure, stagnation pressure or pitot pressure and is given the notation (H or Ps).
Therefore, dynamic pressure:
KE
1
2
--
-ρV2
=
q
1
2
--
-ρV2
=
q q P
+
( ) P
–
=

Viscosity
32. Viscosity is a measure of the internal friction of a liquid or gas and determines its fluidity, or
ability to flow. The more viscous a fluid, the less readily it will flow. Unlike liquids, which become
less viscous with increasing temperature, air becomes more viscous as its temperature is increased.
The viscosity of air is of significance when considering scale effects in wind tunnel experiments and
in terms of friction effects as it flows over a surface. Changes of density do not affect the air
viscosity.
Density
33. Density (ρ) is defined as mass per unit volume. The density of air varies inversely with
temperature and directly with pressure. When air is compressed, a greater mass can occupy a given
volume or the same mass can be contained in a smaller volume. Its mass per unit volume has
increased so, by definition, its density has increased.
34. When the temperature of a given mass of air is increased it will expand, thus occupying a
greater volume. Assuming that the pressure remains constant the density will decrease because the
mass per unit volume has decreased.
35. Both the above statements assume that the air is perfectly dry. When air is humid, that is it
contains a proportion of water vapour, it becomes less dense. This is because water vapour weighs
less than air and so a given volume of air weighs less if it contains water vapour than if it were dry.
Its mass per unit volume is less.

Airspeed Measurement
Indicated Airspeed (IAS)
36. The speed displayed on the airspeed indicator (ASI) is known as indicated airspeed. It does
not include corrections for instrument errors and static pressure measurement errors (pressure error),
both of which are very small. The indicated airspeed will differ progressively from actual flight speed
as altitude increases and, consequently, density (ρ) decreases (q = ½ρV²). The notation for IAS is
(VI).
Calibrated Airspeed (CAS)
37. Also known as Rectified Airspeed (RAS), this is the speed obtained by applying the
appropriate instrument error and pressure error corrections to the ASI reading. The notation for
CAS is (Vc).
Equivalent Airspeed (EAS)
38. The equation for IAS (dynamic pressure) is derived from Bernoulli’s equation, which assumes
air to be incompressible. Below about 300 knots the compression that occurs when the airflow is
brought to rest (as in the pitot tube) is negligible for most practical purposes, becoming increasingly
significant above that speed. EAS is obtained by applying the compressibility correction to CAS.
The notation for EAS is (Ve).

True Airspeed (TAS)
39. The true airspeed is the actual flight speed relative to the surrounding atmosphere, regardless
of altitude. It must, therefore, take account of air density and is obtained by applying the formula:
where
40. TAS is given the notation (V). At 40,000 ft, where standard density is one-quarter sea level
density, TAS will be twice EAS (√0.25 = 0.5). British ASI’s, in common with most others, are
calibrated for ISA mean sea level density (ρ0), where EAS = TAS. At all greater altitudes TAS will be
greater than EAS by a proportional amount.
Shape of an Aerofoil
41. The terminology for the dimensions that determine the shape of an aerofoil section is shown
in Figure 1-2 below.
TAS
EAS
σ
-----------
-
=
σ relative air density
ρ
ρ0
-----
= =

FIGURE 1-2
Aerofoil Section
Chord Line
42. A straight line joining the leading edge to the trailing edge of the aerofoil.
Chord (c)
43. The distance between leading and trailing edge measured along the chord line.
Thickness/Chord Ratio
44. The maximum thickness of the aerofoil section, expressed as a percentage of chord length. A
typical figure is about 12 per cent. The distance of the point of maximum thickness from the leading
edge, on the chord line, may also be given as a percentage of chord length. Typically it is about 30
percent.

Mean Camber Line
45. A line joining the leading and trailing edges which is equidistant form the upper and lower
surfaces along its entire length.
Camber
46. The displacement of the mean camber line from the chord line. The point of maximum
camber is expressed as a percentage and is the ratio of the maximum distance between mean camber
line and chord line to chord length. The amount of camber and its distribution along the chord
depends largely upon the operating requirements of the aircraft. Generally speaking, the higher the
operating speed of the aircraft the less the camber (i.e. the thinner the wing).
Nose Radius
47. The nose or leading edge radius is the radius of a circle joining the upper and lower surface
curvatures and centred on a line tangential to the curve of the leading edge.
Angle of Attack (α)
48. The angle between the chord line and the relative airflow (RAF). This may also be referred to
as incidence, but must not be confused with the angle of incidence. Furthermore, it is essential to
differentiate between the angle of attack and pitch angle, or attitude, of the aircraft. The latter is, of
course, measured relative to the horizontal plane.

Angle of Incidence
49. The angle between the aircraft wing chord line and the longitudinal centreline of the aircraft
fuselage.
The Wing Shape
50. The shape of an aircraft wing in planform has a great influence on its aerodynamic
characteristics and will be discussed in depth in later chapters. The terminology describing the
dimensions that determine wing shape is listed below.
Wing Span
51. The straight-line distance measured from tip to tip. See Figure 1-3.

FIGURE 1-3
Wing Span
Wing Area
52. The plan surface area of the wing. In a wing of rectangular planform it is the product of span
x chord.

Aspect Ratio
53. The ratio of wing span to mean chord or to wing area.
Wind Loading
54. The weight per unit wing area.
Root Chord
55. The chord length at the centreline of the wing (the mid-point along the span).
Tip Chord
56. The chord length at the wing tip.
Tapered Wing
57. A wing in which the root chord is greater than the tip chord.
Taper Ratio
58. The ratio of tip chord to root chord usually expressed as a percentage.
Quarter Chord Line
59. A line joining the points of quarter chord along the length of the wing.
span
2

Swept Wing
60. A wing in which the quarter chord line is not parallel with the lateral axis of the aircraft. See
Figure 1-4.
Sweep Angle
61. The angle between the quarter chord line and the lateral axis of the aircraft. See Figure 1-4.

FIGURE 1-4
Sweep Angle
Mean Aerodynamic Chord
62. The chord line passing through the geometric centre of the plan area of the wing (ie. the
centroid). See Figure 1-5.

FIGURE 1-5
Mean
Aerodynamic
Chord
Dihedral
63. The upward inclination of the wing to the plane through the lateral axis. See Figure 1-6.

FIGURE 1-6
Dihedral
Anhedral
64. The downward inclination of the wing to the plane through the lateral axis. See Figure 1-7.
FIGURE 1-7
Anhedral
The Equation of Continuity
65. The equation of continuity states that mass cannot be either created or destroyed. Air mass
flow is a constant.

66. Figure 1-8 illustrates the streamline flow of air through a cylinder of uniform diameter. The
air mass flow is the product of the density of the air (ρ), the cross-sectional area of the cylinder (A)
and the flow velocity (V). At any point along the cylinder:
FIGURE 1-8
Streamline Flow
67. Mass flow = ρAV = constant is the general equation of continuity, which applies to both
compressible and incompressible fluids. In compressible flow theory it is convenient to assume that
changes in density can be ignored at speeds below about 0.4 Mach and a simplified equation of
continuity:
Airmass flow ρAV cons t
tan
= =
AV cons t
tan
=

68. Consider now the streamline airflow through a venturi tube as illustrated at Figure 1-9.
Given that mass flow is constant at any point and is the product of AV then at point Y, since the
cross-sectional area (A) is reduced, velocity (V) must increase in order to maintain the equation of
continuity.
FIGURE 1-9
69. In other words, a reduction in cross-sectional area (as in a venturi tube) produces an increase
in velocity and vice-versa.
Bernoulli’s Theorem
70. A gas in motion possesses four types of energy. Potential energy (due to height), heat energy,
pressure energy and kinetic energy (due to motion). Bernoulli demonstrated that in an ideal gas in
steady streamline flow the sum of the energies remains constant. At low subsonic (less than 0.4
mach) flow air can be conveniently regarded as an ideal gas (incompressible and inviscid). In these
circumstances Bernoulli's Theorem can be further simplified by assuming there is no transfer of heat
or work in or out of the gas and by ignoring the insignificant changes in potential energy and heat
energy.
71. For practical purposes then, in streamline flow of air around an aircraft wing at low subsonic
speed:

pressure energy + kinetic energy = constant
72. This can be expressed as:
73. Where P = static pressure. In other words, static pressure + dynamic pressur = constant.
74. From this simplified Bernoulli's Theorem it is evident that an increase in velocity of gas flow
results in a decrease in static pressure, and vice versa. Hence at point Y in Figure 1-9 the increase in
velocity of airflow will produce a decrease in pressure.
P
1
2
--
-ρV2 cons t
tan
=
+

Lift
Airflow Round an Aerofoil
Two-dimensional Flow
Three Dimensional Flow
Wake Turbulence
Lift

Lift
2Lift
Airflow Round an Aerofoil
1. As stated in the previous chapter the relationship between pressure and velocity in the airflow
patterns around an object is that defined by Bernoulli. The airflow impacting on the object at a point
near its leading edge will be brought to rest, or stagnate. At the stagnation point the velocity is zero
and the pressure equal to the total pressure of the air stream, that is to say ambient atmospheric
pressure plus dynamic pressure. As the airflow divides and passes around the object the increases of
local velocity, characterised by closely spaced streamlines, produce decreases of local static pressure.
Pressures in excess of ambient atmospheric pressure are conventionally referred to as positive (+) and
pressures below ambient atmospheric pressure as negative (-). The type of airflow around the body
will be either Steady Streamline Flow or Unsteady Flow.
Steady Streamline Flow
2. In this type of airflow the flow pattern can be represented by streamlines. Where the
streamlines appear close together high local velocities, greater than the free stream velocity, exist.
Where the streamlines are widely separated velocity is lower than free stream velocity. Steady
streamline flow can be divided into two types.
(a) Classical Linear Flow. In this flow pattern the streamlines basically follow the
contours of the body, with no separation of the airflow from the surface. Figure 2-1
illustrates classical linear flow around an aerofoil.

Lift
FIGURE 2-1
(b) Controlled Separated Flow or Leading Edge Vortex Flow. It is possible to design an
aerofoil such that the airflow close to the surface separates at the leading edge and
forms a controlled vortex. The main streamline flow is then around this vortex. This
has advantages with swept-wing and delta wing planforms, as will be shown in later
chapters. This is shown at Figure 2-2.
FIGURE 2-2
Leading Edge
Vortex Flow

Lift
Unsteady Flow
3. Unsteady Flow occurs when the airflow separates from the surface of the body and the flow
parameters (eg. speed, direction, pressure), at any point, vary with time. The flow thus cannot be
represented by streamlines.
Two-dimensional Flow
4. An aerofoil section has only two dimensions, from leading edge to trailing edge and from
upper surface to lower surface, known as chord and its thickness. Hence, when considering airflow
around it, the consideration is limited to flow in two dimensions only.
Aerodynamic Forces on Surfaces
5. Associated with the velocity changes as air flows around an aerofoil there will, as has already
been explained, also be pressure changes. If the aerofoil is inclined to the airflow as shown in
Figure 2-3, it will be seen from the streamlines that the velocity over the upper surface is greater than
that over the lower surface. According to Bernoulli, the greater the velocity the lower the local
pressure, so there is a pressure difference between the upper and lower surfaces such that a force will
be acting upwards.

Lift
FIGURE 2-3
Airflow Around an
Aerofoil
6. The angle at which the aerofoil is inclined to the airflow is called the angle of attack (α). The
greater the angle of attack, the greater the pressure difference and therefore the greater the upward
force produced. This is true up to the point at which the airflow separates from the upper surface,
known as the point of stall. As the airflow approaches the leading edge of the aerofoil it is turned
towards the lower pressure on the upper surface. This effect is known as upwash. As it leaves the
trailing edge it returns to its original, free stream location and this is termed downwash. The upward
force produced as air flows over the aerofoil is the source of lift.
7. Figure 2-4 illustrates the aerodynamic forces acting upon an aerofoil inclined at an angle of
attack (α) to the relative airflow.

Lift
FIGURE 2-4
Aerodynamic
Forces Acting on
an Aerofoil
8. The resultant of all the aerodynamic forces acting on the wing is also referred to as the Total
Reaction. The lift force is the component of the resultant force acting perpendicular to the relative
air flow. The component of the resultant force acting parallel to the relative airflow is known as
drag.
Streamline Pattern and Pressure Distribution
9. Let us now consider the pressure distribution around a symmetrical aerofoil. A symmetrical
aerofoil is one in which the chord line and the mean chord line are co-incident. Figure 2-5a shows
the streamline pattern around a symmetrical aerofoil at zero degrees angle of attack and Figure 2-5b
shows the pressure distribution for the same situation.

Lift
FIGURE 2-5
Symmetrical
Aerofoil at Zero
Lift Angle of
Attack
(a) Streamline
Flow
(b) Pressure
Distribution
10. Notice from Figure 2-5(b) how the pressure distribution around the aerofoil can be
conveniently represented in vector form. The pressure at any point on the upper and lower surfaces
of the aerofoil is represented by a vector at right angles to the surface and whose length is
proportional to the difference between absolute pressure at that point and free stream static pressure.
Conventionally, pressures higher than ambient, ie. positive, are represented by a vector plotted
towards the surface and for negative values, the vector is plotted away from the surface.

Lift
11. At the leading edge of the aerofoil, where the streamlines diverge, positive pressure exists.
Where the airflow is forced to divide and flow around the aerofoil the streamlines are close together
and high local velocities and negative static pressure exists. The negative pressures are the same
above and below the aerofoil, so with no pressure difference between upper and lower surfaces no
lift is generated. The angle of attack at which this occurs is referred to as the zero lift angle of attack,
for that particular aerofoil. It should be noted that a stagnation point also occurs at the trailing edge,
where the flow velocity decreases to free stream velocity.
12. Consider now the symmetrical aerofoil at a positive angle of attack as shown in Figure 2-6.
The greatest local velocities occur where the streamlines are forced into the greatest curvature as
shown at Figure 2-6a. Consequently the highest velocities occur over the forward part of the upper
surface. Upwash is generated ahead of the aerofoil, moving the forward stagnation point under the
leading edge and creating an area of decreased local velocity below the forward part of the lower
surface. Behind the aerofoil downwash is generated.

Lift
FIGURE 2-6
Symmetrical
Aerofoil at Positive
Angle of Attack (a)
Streamline Flow
(b) Pressure
Distribution
13. Figure 2-6(b) illustrates the pressure distribution from which it can be seen that there is a
marked pressure differential between upper and lower surfaces, creating positive lift.

Lift
14. A practical aircraft wing would not normally be of symmetrical aerofoil section, but would
have some positive camber since such a section is capable of producing lift even at very low angles of
attack.
15. Figure 2-7 illustrates the pressure distribution around a conventional cambered aerofoil
inclined at a small positive angle of attack.
FIGURE 2-7
Pressure
Distribution
Around a
Cambered
Aerofoil at a Small
Positive Angle of
Attack

Lift
16. At point A at Figure 2-7, total pressure (pitot pressure, stagnation pressure) prevails and this
is the forward stagnation point. As the air passes over the upper surface towards point B, it is
moving into an area of reducing pressure and point B is where minimum pressure exists on the upper
surface Beyond B, pressure is increasing until total pressure is recovered at the rear stagnation point
C and thus the air travelling from B to C is moving against an adverse pressure gradient. This is
most significant as the only way the air can travel against this adverse pressure gradient is by virtue
of its kinetic energy and should this be insufficient the air flow will break away or separate from the
wing. This concept is fully covered in Chapter 4 under Stalling.
17. Furthermore, if points A and C are stagnation points and there is negative pressure on both
upper and lower surfaces, then points X on the upper surface and Y on the lower surfaces are points
of static pressure. To reduce the effect of the pressure reduction on the lower surface the curvature of
the lower wing surface is kept to a minimum.
Effect of Angle of Attack on Pressure Distribution
18. Figure 2-8 illustrates the pressure distribution around a conventional cambered aerofoil
through the working range of angles of attack. Such an aerofoil produces lift at zero degrees angle of
attack because the aerofoil over the upper surface, with its greater curvature is accelerated more than
over the lower surface creating a pressure differential and thus positive lift. It follows, therefore, that
the zero lift angle of attack is a negative value (for this particular aerofoil it is -4°) when the
decreased pressure above and below the aerofoil is equal and hence no lift is generated.

Lift
19. As the angle of attack is progressively increased, the negative pressure above the upper surface
steadily increases, whilst that below the lower surface decreases. Beyond about +8°the pressure
below the lower surface becomes positive. Thus, it can be seen that with increasing angle of attack,
the pressure differential between upper and lower aerofoil surface increases. However, at the lower
angles of attack, it is the pressure reduction on the upper surface which is largely responsible for the
lift generated whereas at the higher incidence, it is both the reduced pressure on the upper surface
and the increased pressure on the lower surface which contribute to lift generation.

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FIGURE 2-8
Pressure
Distribution
Around an
Aerofoil

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Flow Separation at High Angles of Attack
20. Beyond about +14° in this typical aerofoil the low-pressure area on the upper surface
suddenly reduces as the airflow separates from the surface, becoming unstable and turbulent instead
of streamline, significantly reducing the total lift. The contribution to the total lift produced by the
increased pressure on the lower surface however, remains relatively unchanged. This occurs at the
critical or stalling angle of attack. At angles of attack beyond the stall the aerofoil may be regarded
as a flat plate inclined to the airflow, as shown in Figure 2-9 and the lift produced is as a result of the
stagnation pressure and flow deflection below the plate, but this is more than offset by the high drag
force due to the plate’s resistance to the airflow. Stalling is fully covered in Chapter 4.

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FIGURE 2-9
Flat Plate Effect
Centre of Pressure
21. The pressure differential between the upper and lower surfaces can be conveniently
represented by a single aerodynamic force acting at a particular point on the chord line. This point is
known as the centre of pressure (CP). Both the resultant aerodynamic force and hence lift, and the
point through which it acts (CP) vary with angle of attack.

Lift
22. From Figure 2-8 it can be seen that as the angle of attack increases, the magnitude of the force
increases and the centre of pressure gradually moves forwards towards the leading edge until the
point of stall when, with the aerofoil past its stalling angle of attack (or critical angle), the force
reduces and the CP moves rapidly rearwards. With a cambered aerofoil, the centre of pressure
movement over the normal operating range of angles of attack is no further forward than
approximately 25 - 30% chord, measured from the leading edge. With a symmetrical aerofoil
section there is virtually no movement of the CP over the working range of angles of attack, in
subsonic flight.
23. The movement of the CP with angle of attack is shown for a cambered aerofoil at
Figure 2-10.

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FIGURE 2-10
Centre of
Pressure
Movement

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24. Let us now briefly consider moments, couples and coefficients.
Moment
25. The moment of a force about any point is the product of the force and the perpendicular
distance from the line of action of the force to that point. See Figure 2-11.
FIGURE 2-11
Moment of a
Force

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Couple
26. Two equal forces acting parallel but in opposite directions are called a couple. The moment
of a couple is the product of one of the forces and the perpendicular distance between them. See
Figure 2-12.
FIGURE 2-12
Moment of a
Couple

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Coefficients
27. When considering lift, drag and pitching moments, it is much more convenient to use their
respective co-efficients, CL,CD,CM. These co-efficients are non-dimensional and independent of
density, scale of the aerofoil and velocity prevailing at the time such studies are carried out. They
depend on the shape of the aerofoil and vary with angle of attack.
Aerodynamic Centre
28. An aircraft pitches about the lateral axis which passes through the centre of gravity. The
wing pitching moment is therefore the product of lift and the distance between CG and CP of the
wing. But, as we know, the position of the CP is not fixed and moves with changes in angle of attack
and therefore, calculation of the pitching moment is quite involved and complicated.
29. The pitching moment and hence its coefficient (CM) depends not only on the lift force and the
position of the CP, both of which change with change in angle of attack, but also the point about
which the moment is considered.
30. For example, if we take a point of reference arbitrarily towards the leading edge then the nose
down pitching moment about this point (B), increases with increasing angle of attack because,
although the centre of pressure movement is forward, its effect is less than that of the increasing lift
force, as shown in Figure 2-13.

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FIGURE 2-13
31. Now, about a point towards the trailing edge (A) the nose up pitching moment increases
progressively with increasing incidence as shown at Figure 2-14.

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FIGURE 2-14
Pitching Moment
Change About
Point A
32. It follows, therefore, that if about the leading edge, the nose-down pitching moment
progressively increases and about the trailing edge, the nose-up pitching moment progressively
increases, then there must be a point somewhere on the chord line between points A and B about
which there is no change in pitching moment with changes in angles of attack. This point is the
wing aerodynamic centre, shown at Figure 2-15 and is, at subsonic speeds, approximately at quarter
chord (ie. 25% chord from the leading edge).

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FIGURE 2-15
Pitching Moment
About Wing
Aerodynamic
Centre
33. This can be represented graphically at Figure 2-16 which shows curves of CM plotted against
CL where, conventionally, nose-up pitching moments are referred to as positive, and nose-down,
negative.

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FIGURE 2-16
Cmagainst CL

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34. It can be seen from Figure 2-16 that at zero lift there is a residual pitching moment present.
It is negative, and, by definition, remains constant about the aerodynamic centre up to the stall (ie
CLmax). By reference to Figure 2-17 which shows the pressure distribution around our cambered
aerofoil section at its zero lift angle of attack, it can be seen that the resultant forces due to the
pressure reduction torwards the trailing edge on the upper surface and towards the leading edge on
the lower surface produce a nose-down (negative) pitching moment.
35. The pitching moment coefficient CM at the zero lift angle of attack is referred to as CMo as
shown in Figure 2-16.

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FIGURE 2-17
Pressure
Distribution at
Zero Lift Angle of
Attack

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Three Dimensional Flow
36. When considering airflow around an aircraft wing the flow becomes three-dimensional. This
is because there is an element of spanwise flow above and below the wing in addition to the
chordwise flow already discussed.
Spanwise Flow
37. When an aircraft wing is producing lift the local static pressure on the upper surface is lower
than that on the lower surface. Air will flow from an area of higher pressure to one of lower
pressure. Since a wing is of finite length, this means that air will flow from the under surface, around
the wingtip, to the upper surface. Consequently, a spanwise flow of air occurs from the root
outwards towards the tip on the under surface, around the tip, and from the tip inwards towards the
root on the upper surface. The effect is illustrated at Figure 2-18.
FIGURE 2-18
Spanwise Flow
38. The flow at any point on the trailing edge leaving the upper surface will, therefore, be moving
in a different direction from that leaving the lower surface as shown at Figure 2-19.

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FIGURE 2-19
Upper and Lower
Surface Flow
39. Thus the flow at the trailing edge of the wing where the upper and lower surface flows meet is
of a vortex nature and these vortices are continuously shed all along the trailing edge, shown at
Figure 2-20 in which the trailing edge is viewed from behind.
FIGURE 2-20
Trailing Edge
Vortices

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Wing Tip Vortices
40. For a rectangular planform wing, the spanwise flow at the wing tip is strong, decreasing
inwards from the tip until it is zero at the root. Consequently, the vortices along the trailing edge
tend to roll-up into a concentrated larger vortex towards each tip as shown in Figure 2-21. These
vortices are known as wing-top vortices and, when viewed from behind, rotate in a clockwise
direction on the port wing and anticlockwise on the starboard.

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FIGURE 2-21
Wing Tip Vortices

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Induced Downwash
41. The effect of these trailing vortices is to produce a downward airflow, or downwash which
influences the whole flow over the wing with two important consequences:
(a) The effective angle of attack is reduced by the modified relative airflow and thus the
lift generated is also reduced. This is shown at Figure 2-22. Furthermore, the drag
characteristics of the wing are adversely affected and this induced or vortex drag will
be covered in detail in Chapter 3.
(b) The flow over the tailplane in a conventional aircraft design will be affected by the
downwash such that its effective incidence is also reduced with important
consequences in respect of longitudinal stability, which is covered fully in Chapter 8.
FIGURE 2-22
Downwash Effect
on Angle of Attack

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42. The magnitude of the downwash is determined by the vortex formation which in turn is a
consequence of spanwise flow. Spanwise flow results from wing tip spillage, the magnitude of which
is determined by the pressure differential between upper and lower surfaces. In other words, any
parameter which increases the pressure differential will also increase downwash and its effects.
Spanwise Lift Distribution
43. The distribution of lift along the span of the wing depends upon a number of variables, one of
which is the variation of chord length along the span (in other words, the wing planform). A
rectangular planform (i.e. constant chord throughout the span) wing creates most of its trailing
vortices at the tips, consequently downwash is greatest at the tips. A tapered wing, with the chord
progressively narrowing toward the tip, produces a greater proportion of lift at the centre and the
trailing vortices are greatest towards the wing root.
44. Theoretically a constant downwash condition along the span can be achieved if the lift
increases from zero at the tip to a maximum at the root in an elliptical fashion as shown in
Figure 2-23. Such a condition is highly desirable for the reduction of induced drag, (explained fully
in Chapter 3), and one way of achieving it is with a planform in which the chord increases elliptically
from tip to root. There are, however, manufacturing and structural difficulties with such a planform
and it has been found that a close approximation to elliptical spanwise lift distribution is possible
using a tapered wing with varying aerofoil section.

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FIGURE 2-23
Elliptical Lift
Distribution
45. A convenient way to consider lift distribution is to use the ratio of the lift coefficient at any
given point on the wing span (the local lift coefficient) (Cl), to the overall wing lift coefficient (CL),
and plot this against the semi-span distance. When this is done for an elliptical planform wing a
constant value is obtained from root to tip since:
46. Figure 2-24 shows spanwise lift distribution in this format for a number of wing planforms.
The elliptical planform (A) has only been used in a few cases, most notable being the Spitfire. The
rectangular planform (B) is often used for light private aircraft and trainers, because of its favourable
stall characteristics. Larger aircraft invariably use tapered wings, in order to limit structural weight
and maintain stiffness, with a taper ratio of between 20% and 45% (C) and (D).
Cl
CL
------ 1.0
=

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FIGURE 2-24
Spanwise Lift
Distribution

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Wake Turbulence
47. The trailing vortices produced by a wing when it is creating lift extend for a considerable
distance behind the aircraft. The greater the lift being produced, the stronger these vortices will be.
With a large aircraft these vortices create significant turbulence in the wake of the aircraft and take
several minutes to dissipate. This wake turbulence is sufficient to seriously affect the controllability
of other aircraft entering it and pilots are strongly advised to maintain a specified separation when
following a large aircraft, especially in close proximity to the ground (i.e. during the take-off and
landing phase). The separation required will depend upon the relative sizes of the aircraft and may
be several miles if the following aircraft is much smaller than the leading one.
48. In addition, the strength of the vortices is universally proportional to aircraft speed and
aspect ratio. The increased angle of attack, for a given weight, associated with low speed and
stronger vortices from a low aspect ratio wing will result in increased wake turbulence.
Furtheremore, when trailing edge flaps are extended, extra vortices are shed from the flap tips which
tend to weaken the tip vortices and hasten vortex breakdown.
49. The vortex strength will, therefore, be greatest with increased aircraft weight, reduced speed
and clean configuration ie. shortly after take-off.
50. This wake turbulence is influenced by proximity of the aircraft to the ground and also wind
conditions. The vortices slowly descend under downwash influence to approximately 1000ft below
the aircraft until when, in ground effect, they drift outwards from the generating aircraft’s track.
Any prevailing cross-wind, between 5-10kts, will retain the upwind vortex on the generating
aircraft’s track ie. on the runway after take-off.

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Lift
51. Lift is defined as that component of the total aerodynamic force which is acting perpendicular
to the direction of flight. The magnitude of the total aerodynamic force, and therefore the lift
generated, is dependent upon a number of variables of which the following are the most important:
(a) Free stream velocity (V)
(b) Air density ( )
(c) Wing area (S)
(d) Angle of attack (α)
(e) Wing planform and aerofoil section
(f) Surface condition (rough or smooth)
(g) Air viscosity (µ)
(h) Compressibility of the air.
52. The last two variables, viscosity and compressibility of the air, and their effect on lift will be
discussed in subsequent chapters.
53. However, the major factors are dynamic pressure (½ρV²), wing surface area (S) and the
relative pressure distribution existing on the surface, ie. the coefficient of lift (CL).
ρ

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Coefficient of Lift (CL)
54. The simplified equation for calculating aerodynamic force is ½ρV²S multiplied by a
coefficient proportional to the change in force that occurs when angle of attack is changed. This
coefficient is the lift coefficient (CL) and the equation for lift is:
L = CL ½ V² S
CL for a given aerofoil section and planform allows for varying angle of attack and other variables
not included in the equation. By transposition of formula it can be seen that:
55. The coefficient of lift is the ratio of lift pressure to dynamic pressure.
Effect of Angle of Attack
56. It is convenient to represent lift in coefficient form (CL) and then consider the factors affecting
lift in terms of CL which can then be depicted graphically. It is possible from experimentation to
obtain values of CL and plot them against angle of attack for a given wing at constant airspeed and
air density.
ρ
CL
lift
1
2
--
-ρV2S
----------------
=

Lift
57. Figure 2-25 shows a graph of CL against angle of attack (α) for a moderately cambered
aerofoil section. It will be seen that the curve is linear for the greater part, with the coefficient of lift
beginning to fall off at about +14°. The lift coefficient reaches a maximum value at about +15° (CL
max) as the section reaches stalling angle (αstall), otherwise known as the critical angle (αcrit).
Therefore any further increase in angle of attack results in marked reduction in CL.

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FIGURE 2-25
Lift Curve

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Effect of Camber
58. The slope of the CL/α curve is constant regardless of the camber of the aerofoil, but values of
CL are greater for any given angle of attack in sections of increased camber. This is illustrated at
Figure 2-26.
FIGURE 2-26
Effect of Camber
on the Lift Curve

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59. Note that the curve passing through the origin at Figure 2-26 is representative of a
symmetrical wing section. At zero angle of attack such a section produces no lift since the pressure
distribution on the upper and lower surfaces is identical and there is thus no pressure differential. As
we know from our pressure distribution studies earlier in this chapter, the angle of attack at which
the CL is zero is known as the zero-lift angle of attack. For a symmetrical aerofoil it is 0° and for a
cambered section, typically between -2° and -4°.
Effect of Leading Edge Radius
60. The shape of the leading edge largely determines the stall characteristics of a wing. A bulbous
leading edge with a corresponding large radius results in a well-rounded peak to the CL curve
whereas a small leading edge radius will encourage a leading edge stall as the airflow will be less able
to negotiate the sharper corner at large angles of attack. The peak to the CL curve is much more
pronounced and the small radius produces a correspondingly more abrupt stalling characteristic.

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FIGURE 2-27
Effect of Leading
Edge Radius on
the Lift Curve

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Effect of Aspect Ratio
61. The effective angle of attack is reduced by induced downwash, ie. the downward component
of airflow at the rear of the wind caused by trailing edge vortices. A wing of infinite span has no tip
vortices, no induced downwash and therefore no reduction in the angle of attack and it is a wing of
high aspect ratio which approaches this condition of infinite span. Conversely, a wing of low aspect
ratio, having greater trailing edge vorticity will have a greater reduction in the effective angle of
attack and thus produce less lift than a wing of high aspect ratio, with the same wing area.
Figure 2-28 shows the effect of aspect ratio on the CL curve.

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FIGURE 2-28
Effect of Aspect
Ratio on the Lift
Curve

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Effect of Sweepback
62. If an aircrafts wings are swept back and the wing area remains the same, the aspect ratio
( /area) must be less than its equivalent straight wing. Therefore, the effect on the CL curve for
a swept wing compared to a straight wing is similar to that for a low aspect ratio wing when
compared to a high aspect ratio wing . This effect is shown at Figure 2-29.
FIGURE 2-29
Effect of
Sweepback on the
Lift Curve
span
2

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63. Nevertheless, this does not account for the distinct reduction in CLMAX for highly swept
wings, however this is fully explained in Chapter 4, ‘Stalling’.
Effect of Surface Condition
64. Roughness of the wings surface, especially at or near the leading edge has a considerable
effect particularly on CLMAX. Figure 2-33 shows the reduction in CLMAX for a roughened leading
edge when compared to a relatively smooth surface. Any roughness of the wing surface beyond 25%
has little effect on CLMAX or the curve gradient.

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FIGURE 2-30
Effect of Leading
Edge Roughness
on the Lift Curve

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Effect of Ice and Frost
65. Ice or frost deposits on the aircraft surface will invariably have a detrimental effect upon the
performance of the aeroplane. In either case the aerodynamic shape will be changed and the
boundary layer performance will be altered such that turbulence and separation occur more readily
than with a clean aircraft. Since the wing is responsible for the vast majority of the lift generated the
formation of ice or frost on its surface may cause considerable changes to the aerodynamic
characteristics of the aircraft.
Ice at the Stagnation Point
66. There are essentially two effects of large ice formations on the leading edge of the wing. In
the first place the contour of the aerofoil section may be considerably changed, as shown at
Figure 2-31.
FIGURE 2-31
Effect of Ice on
Leading Edge
67. This will almost certainly induce severe local pressure gradients, reducing boundary layer
velocity locally and possibly causing leading edge separation, with consequent loss of lift. Secondly,
some forms of ice have great surface roughness that significantly increases surface friction. This
reduces the boundary layer energy and increases drag. The overall effect is a decrease in the
maximum lift coefficient and an increase in drag.

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68. The effects in practical terms are that the aircraft will require more power to maintain a given
airspeed, the stalling speed will be higher and the stalling angle of attack will be lower.
69. Leading edge icing is most likely to occur during flight in icing conditions and should be
prevented at the onset by the correct use of anti-icing procedures. Its effect will be most noticeable at
the low flight speeds associated with approach and landing, where the higher stalling speed will
require a higher landing speed.
Surface Ice and Frost
70. A thin layer of ice or frost on the upper surface of the wing may not significantly change the
aerodynamic contour of the aerofoil section. However, the surface roughness, especially of hard
frost, can increase surface friction and reduce boundary layer energy sufficiently to promote a loss of
lift by as much as 25%. There will also be an increase of drag due to the increased skin friction.
71. The loss of boundary layer energy will lead to separation and a reduction in the stalling angle
of attack. The maximum lift coefficient will be reduced and stalling speed will be increased.
72. Surface coatings of frost can occur in flight, but are more commonly associated with ground
formation. Application of adequate ground de-icing and anti-icing procedures in conditions where
ice or frost may form on the upper surfaces of a parked aircraft are essential prior to flight. The loss
of lift and increased drag due to such coatings will seriously reduce take-off performance, to the
extent that the aircraft may have difficulty in becoming airborne. Even if it does, its climb
performance may be degraded to the point where obstacles cannot be cleared.

Lift
73. In addition to the foregoing, there is always a weight penalty associated with ice accretions on
the aircraft. The added weight means that airspeed must be higher or the angle of attack must be
increased to produce the extra lift required. The latter choice is a dangerous one, bearing in mind
that the ice will almost certainly have reduced the stalling angle of attack. It should also be borne in
mind that leading edge icing will probably have rendered the angle of attack indicator inoperative
and that the stall warning system does not compensate for the reduced critical angle.
74. Figure 2-32 shows the effects of ice and frost formation on the wing on the CL - α curve.
FIGURE 2-32
Effect of Ice and
Frost on CL

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Lift Coefficient and Speed for Constant Lift
75. In straight and level flight the lift is equal to the weight and:
76. Wing area (S) does not change and. at constant altitude, density (ρ) remains essentially
constant. In order to maintain constant lift both the lift coefficient (CL) and speed (V) must be kept
constant or, if one increases the other must decrease proportionately. Since varying angle of attack
varies (CL), and the optimum angle of attack has been shown to be about +4°, maintaining constant
lift is best achieved by adjusting airspeed. The relation between the lift coefficient and speed, for
constant lift, is shown in the graph at Figure 2-33.
Lift CL
1
2
--
-ρV2S
=

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FIGURE 2-33
Lift Coefficient and
Airspeed
Relationship for
Constant Lift
77. During flight the weight will progressively decrease as fuel is used and the lift must decrease
accordingly, or the aircraft will climb. The ideal aerodynamic solution to this is to reduce airspeed
progressively, but in commercial operations, for ease of flight planning and other reasons, it is
normal to fly at constant speed and trim the aircraft to reduce angle of attack (incidence)
progressively.

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Self Assessed Exercise No. 1
QUESTIONS:
QUESTION 1.
What is the SI unit of force?
QUESTION 2.
What is momentum?
QUESTION 3.
State Newton's second law of motion.
QUESTION 4.
How is force quantified?
QUESTION 5.
State the formula for dynamic pressure.
QUESTION 6.
What is the name of the line on an aerofoil which is equidistant from the upper and lower surfaces?
QUESTION 7.
What is aspect ratio?

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QUESTION 8.
What do you understand by the Equation of Continuity?
QUESTION 9.
State Bernoulli's Theorem.
QUESTION 10.
What is the pressure of air at a forward stagnation point?
QUESTION 11.
If a wing chord measures 2m and the maximum thickness is 25cm, what is the thickness/chord ratio
as a percentage?
QUESTION 12.
What can you say about an aerofoil where the chordline and mean camber line are coincident?
QUESTION 13.
What is the name given to the angle between a wing chordline and the longitudinal centreline datum
of an aeroplane?
QUESTION 14.
For a cambered aerofoil, a typical angle of attack which will give zero lift is approximately what
angle?

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QUESTION 15.
What do you understand by the centre of pressure?
QUESTION 16.
What happens to the centre of pressure as the angle of attack of a cambered aerofoil increases?
QUESTION 17.
What do you understand by the aerodynamic centre?
QUESTION 18.
For a cambered aerofoil, what pitching moment is produced at the zero lift angle of attack?
QUESTION 19.
In which directions does air tend to flow over the lower and upper wing surfaces?
QUESTION 20.
In general terms, what will be the stalling angle of attack of a low aspect ratio wing compared with a
high aspect ratio one of similar section?

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ANSWERS:
ANSWER 1.
Chap 1 page 1
ANSWER 2.
Chap 1 page 2
ANSWER 3.
Chap 1 page 2
ANSWER 4.
Chap 1 page 2
ANSWER 5.
Chap 1 page 6
ANSWER 6.
Chap 1 page 8
ANSWER 7.
Chap 1 page 10

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ANSWER 8.
Chap 1 page 12
ANSWER 9.
Chap 1 page 14
ANSWER 10.
Chap 2 page 6
ANSWER 11.
12.5% Chap 1 page 9
ANSWER 12.
symmetrical Chap 1 page 9
ANSWER 13.
Chap 1 page 9
ANSWER 14.
Chap 2 page 6
ANSWER 15.
Chap 2 page 8

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ANSWER 16.
Chap 2 page 9
ANSWER 17.
Chap 2 page 12
ANSWER 18.
Chap 2 page 12 13
ANSWER 19.
Chap 2 page 14
ANSWER 20.
Chap 2 page 22

Drag
Zero Lift Drag
Lift Dependent Drag
Total Drag
Speed Stability

Drag
3Drag
1. The total drag force acting upon an aircraft in flight is the sum of all the components of the
total aerodynamic force that are acting parallel and opposite to the direction of flight. Total drag is
made up of those factors arising from the generation of lift (Lift Dependent Drag) and those which
are present when no lift is being generated (Zero Lift Drag).
Zero Lift Drag
2. When an aircraft in flight is not generating any lift there is no component of the total
aerodynamic force acting perpendicular to the flight path. Consequently all of the aerodynamic
force must be acting parallel and opposite to the direction of flight. This force is known as zero lift,
or parasite, drag and comprises surface friction drag, form drag and interference drag.
Profile Drag
3. Otherwise known as boundary layer drag, profile drag is the term used to describe the
combined effects of boundary layer normal pressure drag (form drag) and surface friction drag.

Drag
Form Drag
4. We have already seen that when moving air is either totally or partially brought to rest on the
surface of an object, a pressure greater than static pressure, that is to say total, or stagnation
pressure, is acting on the surface of the body. The velocity differences between leading and trailing
edge mean that there are also pressure differences, pressure at the low velocity trailing edge being
greater than at the relative high velocity leading edge. This adverse pressure gradient opposes the
airflow across the surface, creating pressure drag. Form drag is otherwise known as boundary layer
normal pressure drag and can form a significant portion of the total drag force acting on the aircraft.
5. Consider a circular flat plate, which is placed in a wind tunnel so that the flat surface of the
plate is at right angles to the flow of air. If, over the entire surface area, the air was brought
completely to rest, a pressure equal to the dynamic pressure would be felt at all points. The force
thus created would be equal to the dynamic pressure multiplied by the surface area of the plate, or
½ρV²S, where S is the surface area.
6. The situation is complicated somewhat since the air is not brought totally to rest over the
whole surface. Some of the air flows around the edges, resulting in the formation of a low pressure
area behind the back of the plate. This effectively creates a suction, which tends to retard the airflow
passing the plate (or the passage of the plate through the air).
7. A turbulent wake will form behind the plate, in the case of a flat plate the amount of
turbulence will be considerable and the drag factor therefore will be extremely high. With a
streamlined wing the amount of turbulence will be much lower and therefore the drag factor will be
considerably reduced. We can therefore see that the shape as well as the frontal area will affect the
amount of drag produced. It is the shape, which gives us the co-efficient of drag (CD), and the total
form drag formula now reads:

Drag
8. It is clear that a flat plate is a most inefficient shape to try and move through the air. If we
now consider a sphere of the same diameter as a flat plate placed in the airflow, it is not hard to
visualise that this shape will produce considerably less drag than the flat plate. The nature of the
airflow around these two objects is illustrated at Figure 3-1.
FIGURE 3-1
Airflow Around a
Flat Plate and
Sphere
9. It is seen at Figure 3-1 that the very large turbulent wake created by the flat plate has now
been replaced by a much smaller one as the air flows more smoothly around the surface of the sphere
and the suction drag created behind the sphere is thus reduced. The total amount of form drag
created by the sphere is calculated in exactly the same way , however the co-efficient of
drag for the sphere is very much smaller than that for the flat plate.
Form Drag CD
1
2
--
-ρV
2
S
=
CD
(
1
2
--
-ρV
2
S)

Drag
10. Now consider a streamlined aerofoil (ignoring any lift which it may generate), as illustrated at
Figure 3-2. This shape has a very low co-efficient of drag, since the air can follow the surface of the
shape almost to the trailing edge before it separates from the surface of the aerofoil and becomes
turbulent. The turbulent wake produced by a streamlined shape is therefore very small. It is not
possible to entirely eliminate the turbulent wake, but within limits the streamlined body can be
extended with a consequent reduction in the co-efficient of drag value.
FIGURE 3-2
Airflow Around a
Streamlined
Aerofoil
11. Clearly it would be impractical to extend the length of the aerofoil beyond certain sensible
limits since the increased weight would outweigh the improvement in the co-efficient of drag. In any
event, beyond a certain length, the effect of friction between the air and the aerofoil surface prevents
any further reduction in the drag factor. The amount of streamlining of a body is expressed as a
fineness ratio, which is its length divided by its maximum thickness, as shown at Figure 3-3.

Drag
FIGURE 3-3
Fineness Ratio
Coefficient of Drag (CD)
12. As with lift, drag is an aerodynamic force and may be considered as a coefficient and it is this
coefficient which is the major factor in this drag formula CD½ρV²S. By transposition it can be seen
that:-
13. The coefficient of drag is the ratio of drag pressure to dynamic pressure.
14. Figure 3-4 shows a graph of CD against angle of attack (α). From this it can be seen that at
low angles of attack the drag coefficient is low and it changes only slightly with small changes of
angle of attack. As angle of attack increases however, drag increases and at the upper end of the α
range even small changes in angle of attack produce a significant increase in drag. At the stall a large
increase in drag occurs.
CD
drag
1
2
--
-ρV2S
-----------------
=

Drag
FIGURE 3-4
Drag Curve

Drag
FIGURE 3-5
Lift Curve

Drag
The Lift/Drag Ratio
15. Clearly an aerofoil is at its most efficient when it is generating the greatest possible lift for the
least possible drag. From Figure 3-5 it will be seen that maximum lift is generated at an angle of
attack of about 15°,ie. the stalling angle. From Figure 3-4 it is seen that the least drag occurs at an
angle of attack of about -2°. Neither of these angles is practical for normal flight and neither is
satisfactory, as the ratio of lift to drag is low in each case. What is needed is an operating angle of
attack at which the lift force is high for a low drag force. In other words, a high ratio of lift to drag.
16. By combining Figure 3-4 and Figure 3-5 values of CL and CD for each angle of attack can be
obtained and the ratio CL/CD calculated for each angle. A graph of CL/CD ratio against angle of
attack can then be plotted, from which the best lift/drag ratio angle of attack is evident. Such a
graph is illustrated at Figure 3-6, from which it can be seen that the best lift/drag ratio occurs
typically at about +4°. At this angle the ratio of lift to drag is likely to be between 12:1 and 25:1,
depending upon the aerofoil section used and is referred to as the optimum angle of attack.

Drag
FIGURE 3-6
Lift/Drag Curve

Drag
17. At the zero lift angle of attack the lift/drag ratio is zero, but increases significantly for small
increases in angle up to the optimum angle of attack of about +4°. Beyond this the lift/drag ratio
decreases steadily since, although lift is increasing, drag is increasing at a greater rate. At the stall the
lift/drag ratio falls off markedly.
Surface Friction Drag
18. Modern aircraft have skin surfaces, which appear to be very smooth and polished. A closer
inspection under a magnifying glass would reveal an irregular pitted surface, with the irregularities
having massive dimensions when compared to the individual molecules of air flowing over the
surface. It is not surprising then that the air immediately in contact with the aircraft surface is
brought virtually to rest. This impedes the flow of air layer by layer until the point is reached where
the air is flowing freely at free-stream speed. The total depth of air, which is flowing at less than
99% of free-stream velocity, is known as the boundary layer. The force required to overcome the
shearing friction within the boundary layer is known as surface friction drag and is determined by
the surface area of the aircraft, the viscosity of the air and the rate of change of velocity through the
boundary layer, as illustrated at Figure 3-7.

Drag
FIGURE 3-7
Boundary Layer
19. Within the boundary layer, and certainly within the free-stream air, it is hoped that the
airflow will be laminar from the leading edge almost to the trailing edge. At a point known as the
transition point, the smooth flow breaks down into a turbulent flow, which creates a much thicker
boundary layer, see Figure 3-8.

Drag
FIGURE 3-8
Transition Point
20. The effect of surface friction is, not surprisingly, far more marked in the turbulent part of the
boundary layer than in the laminar part. It is therefore desirable to hang on to the laminar flow for
a long as possible.
21. One of the main factors affecting the position of the transition point is the pressure
distribution of the upper surface of the wing. Transition tends to occur at the minimum pressure
point on the top surface, and this tends to occur at the point of maximum thickness of the wing itself.
Therefore, by designing a wing where the maximum thickness occurs well back from the leading edge
the laminar flow is increased, and the surface friction consequently reduced. A conventional wing
and a laminar flow wing are shown at Figure 3-9.

Drag
FIGURE 3-9
Aerofoil Sections
Interference Drag
22. Where two boundary layers meet, as at the junction between wing and fuselage, turbulent
flow will ensue, leading to an increased pressure difference between leading and trailing surface areas
and resulting pressure drag. The effect can be largely reduced in subsonic flight by adequate fairing
at the junctions.
Profile Drag and Airspeed
23. Any form of profile drag will increase with increasing airspeed, as shown in the graph at
Figure 3-10 and is in fact, proportional to the square of the speed.

Drag
FIGURE 3-10
Profile Drag -
Speed Curve
Lift Dependent Drag
24. When an aircraft is generating lift additional drag is produced. This comprises induced
(vortex) drag plus increases in the components that make up zero lift drag.

Drag
Induced Drag
25. As has been stated, the tip and trailing edge vortices create a downwash that angles the
relative airflow to the direction of flight. The effect of the vortices is to direct the airflow downwards
from the trailing edge. The angle between the airflow as it would be without induced drag, and the
actual airflow, is termed the downwash angle. See Figure 3-11.
FIGURE 3-11
Downwashed
Airflow
26. This induced downwash flow aft of the trailing edge influences the flow over the whole wing,
(see Chapter 2-41), such that the effective angle of attack is reduced. As a result, the lift generated is
also reduced and can only be restored by increasing the angle of attack. This increase in angle of
attack will tilt rearwards the total reaction vector and thus the component parallel to the direction of
flight is increased. This increase in drag, due to the wing vortices is induced (vortex) drag and is
shown in Figure 3-12.

Drag
FIGURE 3-12
Induced (Vortex)
Drag

Drag
27. Figure 3-12(a) represents a section of a two dimensional wing, ie. one of infinite span which is
producing lift, but has no trailing edge vortices.
28. Figure 3-12(b) represents the same section but whose wing is of a finite span and thus having
trailing edge vortices and hence induced downwash flow. The reduced effective angle of attack
results in a reduction in the lift generated as shown.
29. In order to restore the lift to its two dimensional value (ie. value without downwash as at
Figure 3-12(a), the angle of attack must be increased and the subsequent inclination of the total
reaction vector causes an increase in the component parallel to the direction flight (ie. induced drag).
Effect of Airspeed
30. The lift generated by a wing can be increased by increasing the angle of attack for a given
airspeed or by increasing the airspeed for a given angle of attack. The faster you fly, the lower the
angle of attack necessary for the lift required and, the lower the angle of attack the less the
downwash, thus reducing the induced drag, see Figure 3-13.

Drag
FIGURE 3-13
Effect of Speed on
Induced Drag

Drag
31. Suppose, whilst maintaining level flight, the airspeed were to be doubled. The dynamic
pressure producing lift (½ρV²) would be quadrupled. In order to maintain level flight the angle of
attack would have to be reduced, thereby inclining the lift vector forward and reducing induced drag.
Hence, induced drag decreases with increased airspeed being inversely proportional to the square of
the speed, as shown in the graph at Figure 3-14.
FIGURE 3-14
Induced Drag -
Speed Curve

Drag
Effect of Aspect Ratio
32. Aspect ratio is defined as the ratio of the overall wing span to the mean chord. Since the
magnitude of the induced drag of a wing depends upon the magnitude of the tip vortices, anything
that can be done to reduce these vortices must reduce induced drag. The longer and narrower a wing
the less the proportion of airflow around the tips to that over the remainder of the wing and
therefore the less the influence of the tip vortices. In other words, less downwash so less induced
drag than for a wing of the same area, but lower aspect ratio. This is illustrated at Figure 3-15.
FIGURE 3-15
Effect of Aspect
Ratio on Induced
Drag

Drag
33. The greater the aspect ratio, then, the lower the induced drag of the wing. Taken to its
ultimate conclusion, a wing of infinite span would have no induced drag at all. Clearly this is not
feasible in the practical sense, although it can be proved in a wind tunnel experiment with a wing
extending the full width of the tunnel. For aircraft in which low drag at moderate airspeed is a
fundamental requirement, high aspect ratio wings are essential. Examples are sailplanes, long range
patrol aircraft and medium speed transports.
Effect of Planform
34. Since induced drag decreases with increasing airspeed the need for high aspect ratio wings is
less important for aircraft designed to operate at high subsonic or supersonic speeds. Indeed, for
these a low aspect ratio is important because thin aerofoil sections are necessary, demanding a short
wingspan for structural reasons. Concorde, for example, has an aspect ratio of less than 1:1,
whereas a high performance sailplane may have an aspect ratio of 45:1 or greater. Clearly, with
aircraft in which normal operations demand a low aspect ratio, high induced drag at the low speeds
of take-off and landing has to be accepted. Similarly, training aircraft that benefit from the
favourable stall characteristics of a rectangular planform wing suffer from greater induced drag than
would be the case with a tapered wing. An elliptical planform gives the least induced drag for any
given aspect ratio. Sweep back has a similar effect on lift distribution to decreasing taper ratio. A
large sweep back angle tends to increase induced drag.
Effect of Lift and Weight
35. The downwash angle, and therefore the induced drag, depends upon the pressure differential
between upper and lower wing surfaces. Consequently, an increase in lift coefficient (CL), whether
due to increased weight or manoeuvre, must result in greater induced drag at a given speed. Induced
drag varies as CL², and therefore as W² at any given speed.

Drag
Induced Drag Coefficient (CDi)
36. For an elliptical wing planform the coefficient of induced drag is given as:
37. For any other planform a correction factor k is necessary, although a straight tapered wing
with a taper ratio of 2:1 approximates very closely to an elliptical wing. For planforms whose lift
loading is not elliptical k 1. For conventional low speed wings the value of k is usually about 1.1 to
1.3.
38. The above equation for induced drag coefficient shows that:
(a) There is no induced drag on a wing of infinite span (two-dimensional flow only).
(b) The greater the aspect ratio, the less the induced drag.
(c) There is no induced drag on any wing at zero lift.
39. Total drag is the sum of profile drag and induced drag and if the coefficients of total, profile
and induced drag can be represented as CD, CDP=and CDi=respectively, then
CD = CDP=+=CDi
from Paragraph 3-23 above , for a wing with elliptical loading
CDi
CL
2
( )
πA
-----------
=

Drag
However, for any other load distribution
where k is the induced drag factor.
Therefore
and now can, for a given wing, be replaced by a constant K, so that,
CD = CDp=+=K CL²
40. Assuming that CDp=is constant, which is valid over the normal operating range of angles of
attack, a graph of CD against CL² as at Figure 3-16, enables the factor K to be found and hence, for
a given wing, the induced drag factor k can be determined.
CDi
CL
2
πA
--------
-
=
CDi
k CL
πA
-----------
2
=
CD CDp
k CL
πA
-----------
2
+
=
k
πA
-------

Drag
FIGURE 3-16
CD - CL² Curve
Induced Angle of Attack
41. The effect of downwash is to reduce the effective angle of attack. From the induced drag
coefficient equation it is possible to derive an equation for the induced angle of attack (αI):
αi 18.24
CL
A
------
-

=

Drag
42. From the foregoing, if the influence of induced drag upon the graph of lift coefficient (CL)
against angle of attack (α) is considered, it will be seen from Figure 3-17 that the greater the induced
drag the shallower the CL - α curve. Curve A in Figure 3-17 represents a wing of infinite span, in
which there is no downwash (no induced drag) and therefore no reduction of the effective angle of
attack. Curve B represents a wing of finite span in which there is downwash (induced drag) and a
consequent reduction of the effective angle of attack. Curve C also represents a finite wing having
greater induced drag than wing B. As angle of attack increases, downwash increases and the effective
angle of attack progressively decreases. Hence the lift generated is less, at a given actual angle of
attack, the greater the induced drag of the wing.
43. The implication of this is that a wing of low aspect ratio, and therefore higher induced drag,
will have a higher stalling angle of attack than one of high aspect ratio and low induced drag. The
theoretical effect illustrated at Figure 3-17 is not as pronounced in practice.

Drag
FIGURE 3-17
Effect of Aspect
Ratio on Stalling
Angle of Attack
44. When considering the influence of induced drag upon the relationship between lift and drag
coefficients, the lift/drag ratio, it is seen from the graph of CL against CD at Figure 3-18 that as
induced drag increases, the slope of the CL – CD curve decreases, indicating reduced lift/drag ratio.

Drag
FIGURE 3-18
Effect of Aspect
Ratio on CL - CD
(Aeroplane Polar)
#

Drag
Ground Effect
45. During flight in close proximity to the ground the three-dimensional airflow pattern is
changed, since the vertical component of airflow is restricted, or eliminated altogether. This modifies
the upwash, downwash and tip vortices and therefore has a significant effect upon lift and drag.
46. Assuming the same lift coefficient is maintained, as a wing enters ground effect the upwash,
downwash and tip vortices are all reduced, as shown in Figure 3-19. The wing is now behaving as
though its aspect ratio had been increased and, as a consequence, its induced drag coefficient (CDi)
and induced angle of attack (αI) are both less for the same lift coefficient (CL).

Drag
FIGURE 3-19
Ground Effect

Drag
47. For ground effect to be significant the wing must be at a height considerably less than the
span of the wing. When height is equal to half-span the reduction in induced drag coefficient is less
than 10%, but at a height equal to quarter-span this value rises to almost 25%. This is illustrated at
Figure 3-20.
FIGURE 3-20
Effect of Height on
Induced Drag
Coefficient

Drag
48. Because the tip, or trailing vortices are reduced when a wing is in ground effect the spanwise
distribution of lift is altered and the induced angle of attack (αI) is reduced. Consequently a lower
angle of attack (incidence) is necessary to produce the same lift coefficient. In the graph of CL
against α at Figure 3-21 it will be seen that the slope of the curve is increased when the wing is in
ground effect. A lower angle of attack is needed for a given lift coefficient, or greater lift is generated
for a given angle of attack. It should be noted that, when in ground effect, the stalling angle of attack
is reduced.

Drag
FIGURE 3-21
Influence of
Ground Effect on
the Lift Curve
49. It will be appreciated that, for normal flight operations, ground effect is only of significance
during landing and take-off. The general effects are an increase in lift if a constant pitch attitude is
maintained and an increase in speed if the same power is maintained.

Drag
50. Furthermore, the reduced downwash affecting the tailplane will result in a greater
contribution to longitudinal stability and, as the download of the tailplane reduces, there will be a
nose down pitching moment change. However, if the aircraft is of a high tail design, then any
changes in downwash will not affect the tailplane and hence have no effect on longitudinal stability
or trim.
51. Pressure changes at the pressure source associated with changes in upwash and downwash
due to ground effect, will usually cause an increase in static pressure sensed and therefore a reduction
in indicated airspeed and altitude.
Effect on Landing
52. During the final stages of the landing approach there will be a tendency for the aircraft to
float beyond the intended touchdown point if it is brought into ground effect at a constant angle of
attack. The float distance may be considerable if speed is at all excessive.
Effect on Take-Off
53. The main danger from ground effect during take-off arises from the fact that, if constant
angle of attack is maintained, a loss of lift may be experienced as the aircraft leaves ground effect. It
is essential to delay rotation until the recommended airspeed is attained. Otherwise, in marginal
conditions such as high all-up weight, high ambient air temperature and/or low ambient density, it
may prove impossible to climb out of ground effect.

Drag
Wing Tip Design
54. Various design devices are used in an attempt to reduce drag due to tip vortices. Perhaps the
simplest is that employed on some rectangular planform wings on light aircraft, where the wing tip is
deliberately cut off or given an upward or downward bend, as shown in Figure 3-22. The object in
both cases is to encourage separation of the tip airflow, thereby reducing the tip vortex. A reduction
in drag has resulted from these devices.

Drag
FIGURE 3-22
Effect of Wing Tip
Design

Drag
End Plates and Tip Tanks
55. It was thought that the installation of end plates on the wing tips might prevent the formation
of trailing vortices, but it was found in practice that plates large enough to achieve this usually had
detrimental effects upon the handling characteristics of the aircraft. Whilst end plates do not destroy
the trailing vortices they do modify them in a beneficial way. External fuel tanks, or other stores,
mounted on the wing tips can have a small end-plate effect, although their primary advantage is the
relief of wing bending stresses in flight.
Winglets
56. At the wing tip there is a significant sidewash in the form of outwash below the lower surface
and inwash over the upper surface. By mounting a winglet vertically at the wing tip, advantage can
be taken of this sidewash in that the winglet will generate a lifting force, a component of which acts
in the direction of flight, opposing drag. The main tip vortex now forms at the tip of the winglet,
above the main wing, so that downwash from the main wing is reduced. Thus, induced drag is
reduced due to the reduction of the trailing vortices. The effect is illustrated at Figure 3-23. Winglets
can be mounted either above or below the wing tip, or both. However, winglets below the tip are
unusual because by mounting them upwards takes full advantage of upwash.

Drag
FIGURE 3-23
Effect of Winglets

Drag
Wing Span Loading
57. With a tapered wing most of the lift is being produced at the root, and therefore the
downwash at the inboard end is significantly greater than at the outboard end of the wing. This
means that the effective angle of attack is greater at the tip than at the root, as illustrated at Figure 3-
24. This is undesirable for a number of reasons, not least of which is the fact that it will cause the tip
to stall before the root. Additionally, it means that the local wing loading (lift per unit area) tends to
be greater toward the tip. This can be compensated for by mechanically adjusting the effective angle
of attack where necessary, either by twisting the wing or by altering the camber (aerofoil section)
toward the tip.
FIGURE 3-24
Effective Angle of
Attack Spanwise

Drag
Wash-Out
58. The wing is attached to the fuselage of the aeroplane at a particular angle of incidence. If the
angle is maintained throughout the span of a tapered wing, the situation described in the preceding
paragraph will exist. If, however, the angle of incidence is progressively decreased toward the tip a
constant effective angle of attack can be maintained. Decreasing the angle of incidence from root to
tip is known as wash-out. The reverse is known as wash-in.
Change of Camber
59. The same effect as that produced by wash-out can be achieved by a progressive reduction of
camber from root to tip.
Total Drag
60. Of the two types of drag to which an aircraft is subjected, profile drag increases with
increasing airspeed, whilst induced drag decreases with increasing airspeed. By plotting both on a
graph, it is possible to establish a speed at which the sum of the two is a minimum. This is the speed
for minimum drag .See Figure 3-25.
VMD

Drag
FIGURE 3-25
Effect of Speed on
Drag
Drag and Pressure Altitude
61. It should be appreciated that for a given EAS, the dynamic pressure will be the same at all
altitudes and therefore a plot of drag versus EAS will apply at all altitudes. See Figure 3-26.

Drag
FIGURE 3-26
Effect of Altitude
on Drag - EAS
Curve
62. However, for a given EAS, the TAS with increasing altitude increases by the factor , where
is the relative density, and therefore, if drag is now plotted against TAS, the curve is mirrored
progessively to the right with increasing altitude. See Figure 3-27.
1
σ
-------
σ

Drag
FIGURE 3-27
Effet of Altitude
on Drag - TAS
Curve
Speed Stability
63. Stability, which is fully explained in Chapter 8, is the study of the aircrafts response following
a disturbance and its tendency thereafter to return, or otherwise, to its pre-disturbed condition.
When considering speed stability, refer.ence is made to the drag curve shown at Figure 3-27.

Drag
Effect of Speed
Changes on Drag
64. Consider an aircraft flying at speed A. In straight and level flight thrust equals drag and so
the thrust required can be indicated by the horizontal line T1. If now there was an un-demanded
speed increase from A to B, with the thrust remaining as it was, the drag exceeds the thrust and so
the speed will reduce to A. Conversely, if the speed reduces to C, with the thrust remaining constant,
the thrust exceeds the drag and so the speed will increase.
65. This then is a speed stable situation, which will always exist on the front side of the drag
curve, ie. at speeds in excess of VMD.

Drag
66. Now consider the same situation on the back side of the drag curve, in other words where the
aircraft is flying at a speed which is lower than VMD. Initially the speed is D and the thrust T2.
67. If the speed increases to E the drag decreases, the thrust exceeds the drag and so the speed
increases further. This is bad enough, but now consider a drop in speed from D to F. The drag has
increased, the thrust is constant and so the speed decreases further; the drag is increased, the thrust is
insufficient and so on. This then is definitely a speed unstable situation.
68. The effect of increasing profile drag, such as the use of undercarriage or speed brake, will not
alter the induced drag curve, but will steepen the profile drag curve. Thus the point of intersection,
and hence VMD, reduces (although the total drag increases). Thus the speed range where the aircraft
is speed stable will be greater.
69. The speed-instability situation below VMD is particularly important with modern jet
transport aircraft. This is because of the high weights involved, the high angles of attack at the low
airspeeds involved in the approach phase and the slow reaction times of jet engines, when compared
with piston engines.
70. When dealing with induced drag it was seen that an increase in weight resulted in an increase
in induced drag. Since the intersection of the profile and induced drag lines gives VMD we can
therefore see, at Figure 3-28, that VMD increases as the weight increases and speed stability
correspondingly reduces.

Drag
FIGURE 3-28
Effect of Weight
on Speed Stability
71. Using the same sort of logic it can be seen that the higher the aspect ratio the lower the VMD
and speed stability therefore increases.

Stalling
Stalling Speed
Initial Stall in the Spanwise Direction
Stall Warning

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