1. THEME 1. INTRODUCTION
Aerodynamics of an aircraft is the science on the general laws of air motion and
its specific features at flow around an aircraft and its parts, on forces and moments
which are affecting the plane and its parts, about thermal effect of a flow aboard the
plane. It is based on the laws of physics, mechanics and thermodynamics.
The knowledge of aerodynamics of an aircraft is a necessary condition for
consequent study of such course, as «Flight dynamics», «Aircraft structure» and
«Designing of aircraft», «Production technology of aircraft».
The following example shows a role of aerodynamics in aircraft creation. The
fivefold increase of the expenses on aerodynamic researches is profitable, if it results in
increase of lift-to-drag ratio at 1 %.
The large role in research of aerodynamics of an aircraft and its parts is played by
experimental researches (wind-tunnel tests and flight experiment). The especially
important role is played by experiment as a way of checking the theoretical data.
1.1. Sections of aerodynamics.
The division of aerodynamics into sections is made on speeds and altitudes of
flight. Such division is conditional, since the basic criteria are the limitations or
assumptions introduced during the studies or research of the aerodynamic characteristics
of an aircraft.
The division of aerodynamics on speeds is conducted depending on a Mach
number, which is a measure of the compressibility of air flow. The Mach number is
non-dimensional value equal to ratio of body velocity to a local velocity of a sound:
V∞
M∞ = , where V∞ is the speed of motion of an aircraft; a is the speed of a sound.
a
So, the first section of aerodynamics studies motion of bodies at Mach numbers
lying in limits M ∞ ≤ 0 .4 . M ∞ ≤ 0 .4 - but again it is conditional number up to which
liquid (air) is possible to be considered as the incompressible medium. Air behaves just
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2. as a liquid, thus the compressibility of the environment does not influence the
aerodynamic characteristics. The error at M ∞ = 0 .4 can be no more than 2% .
This section is named «Aerodynamics of the incompressible environments».
The second section is named «Subsonic aerodynamics». In this section the
motion of an aircraft is studied at Mach numbers lying in limits 0 .4 ≤ M ∞ ≤ M∗ (from
M ∞ ≥ 0 .4 up to M ∞ ≤ M* ). The number M ∞ (undisturbed subsonic flow), at which
somewhere on a surface of a streamlined body for the first time local velocity of a flow
reaches speed of a sound (V∞ = a ), is named critical and is marked as M* . The critical
Mach number M* depends on the shape of a streamlined body. In an airfoil point where
the gas flow speed is maximum, according to a Bernoulli's relation speed of a sound is
determined as a = a min . Therefore, maximum value of a local Mach number
M = M max is reached there where stream cross-section is the least. To this point there
corresponds also minimum value of a coefficient of pressure C p = C p min . The Mach
number M* is always less than or equal to one ( M* ≤ 1 ). The number M ∞ = M* is the
highest limit of numbers M ∞ , at which the ratios obtained for completely subsonic flow
are fair.
The third section is named «Transonic aerodynamics». At Mach numbers
M ∞ > M* on a streamlined surface both subsonic and supersonic zones of flow take
place. The zones with subsonic speeds do not fade away at once at reaching supersonic
speed of flight. Depending on the shape of an airfoil it occurs at Mach numbers
M ∞ ≈ 1.1 ÷ 1.2 . Such flow mode is also named transonic. Section «Transonic
aerodynamics» studies speed range M* ≤ M ∞ ≤ 1.1 ÷ 1.2 . The upper Mach number
M ∞ ≈ 1.2 also is selected conditionally, and in some books it is possible to meet values
M ∞ ≈ 1.25 .
The fourth section «Supersonic aerodynamics» is limited by range of Mach
numbers 1.1 ÷ 1.2 ≤ M ∞ ≤ 4 ÷ 5 .
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3. The fifth section is named «Hypersonic aerodynamics». It corresponds to Mach
numbers more than M ∞ ≤ 4 ÷ 5 . For example, "Shuttle" or "Buran" enter into
atmosphere with Mach numbers M ∞ ≈ 20 ÷ 25 .
Besides, aerodynamics is divided on altitudes of flight H . The main criterion of
division is the Knudsen number.
λ
Knudsen number k n = , where λ is free length of molecules run, l is reference
l
size of liquid flow.
In standard conditions λ ≈ 10 − 6 m , at t oC = 15 , P ≈ 760 mm . Hg .
Depending on Knudsen number aerodynamics is divided onto the following
sections:
«Aerodynamics of continuum». Values of number k n ≤ 0 .1 and altitude of
flight H ≤ 80 Km correspond to this section.
«Aerodynamics of the strongly rarefied environment». Values of number
k n > 10 and altitude of flight H > 120 Km correspond to this section.
It is possible to consider air as continuum at k n ≤ 0 .01 in the given course. For
modern aircraft which are flying at altitudes up to H < 40 Km , this condition is
performed.
1.2. Aircraft and its main structural members
Let's proceed to consideration of an aircraft and its parts. We will introduce
general concepts and aerodynamic characteristics, we will show on examples a role of
the aerodynamic characteristics in formation of flight properties of an aircraft.
An airplane is an aircraft heavier than air having a power plant for obtaining
thrust and wings for creating lift.
Describing the shape of an aircraft, these concepts: are used a base plane of an
aircraft and base system of coordinates.
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4. The aircraft base plane is the plane, concerning which the majority of structural
members of an aircraft are located symmetrically on the left and on the right. This plane
is often named as a plane of symmetry.
Base system of coordinates is the right rectangular coordinate system
0 R x R y R z R , fixed concerning an aircraft. An origin 0 R is named as an aircraft base
point, Axis 0 R x R - aircraft base axis. The base point is in a base plane of an aircraft.
Its position is determined from task to be solved. The axes 0 R x R and 0 R y R are also in
an aircraft base plane. The first is directed forwards, the second - upwards. The axis
0 R z R is directed along the right half wing.
The main parts of an aircraft (fig. 1.1) are: a wing, fuselage (body), tail unit,
landing gears and power plant.
1 - wing;
2 - fuselage;
3 - power plant;
4 - horizontal tail;
5 - elevator;
6 - stabilizer;
7 - vertical tail;
8 - rudder;
9 - fin;
10 - flaps;
11 - aileron.
Fig. 1.1. Main parts of an aircraft
Wing is the main lifting surface of an aircraft. The wing is designed to create
lifting force necessary for aircraft gravity balance. The wing usually has a plane of
symmetry.
Fuselage is designed for accommodation of the crew, passengers, equipment,
fuel, freights and power plant. Usually fuselage creates small lift and considerable drag.
Power plant consists of engines with devices and systems providing their
operation, air intakes, propellers and nozzles. The power plant is intended for thrust
creation.
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5. Tail unit of an aircraft consists of horizontal tails and vertical tails and is
designed for maintenance of stability and controllability in longitudinal and lateral
motion.
Landing gears consist of a landing gear, high-lift devices, accelerating and
braking devices.
1.3. Coordinate system
While studing aircraft aerodynamics body 0 xyz and wind 0 xa ya za coordinate
systems (Fig. 1.2) are more often used. Both coordinate systems are right rectangular.
The body coordinate system is fixed
relatively to an aircraft and moves together with it.
Its origin 0 is usually placed in a center of mass.
The axes 0x , 0 y , 0 z are named as longitudinal,
normal and transverse axes. The axes 0x and 0 y
are located in a base plane of an aircraft. The axis
Fig. 1.2. Coordinate systems
0x is directed from an aircraft tail section to the
nose part, the axis 0 y is directed towards top part of an aircraft. The axis 0 z goes
perpendicularly to an aircraft base plane and is directed to the right side of an aircraft.
Beginning of wind coordinate system 0 xa ya za usually is also placed in the center
of mass. There distinguish a wind axis 0 xa , lift axis 0 ya and lateral axis 0 za . The wind
axis 0 xa is directed posigrade of an aircraft. The lift axis 0 ya lies in a base plane of an
aircraft (or in a plane parallel it) and is directed to an aircraft top. The lateral axis 0 za
passes so that it has supplemented axes 0 xa and 0 ya up to the right coordinate system.
The wind system is not rigidly connected with an aircraft and can change the orientation
in relation to it during the flight.
The orientation of an aircraft relatively to the velocity vector is determined by
angle of attack α and angle of slip β . An angle of attack α is an angle between a
projection of velocity vector to a vehicle plane of symmetry (base plane of an aircraft)
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6. 0 xy and centerline 0x . A slip angle β is an angle between velocity vector and plane of
symmetry 0 xy .
In some cases normal coordinate system
0 x g y g zg (Fig 1.3) is used. It is the mobile right
system. Its beginning 0 is combined with the
beginning of body coordinate system. The axis
0 y g is directed upwards along a local vertical, and
directions of axes 0 x g and 0 z g are selected
according to the task to be solved. The plane
0 x g zg is always located horizontally in this
Fig. 1.3. Normal Coordinate
system coordinate system.
The angle between the axis 0 x g and
projection of a centerline to a horizontal plane is named as yaw angle and designated as
ψ . The angle between the aircraft centerline 0x and horizontal plane 0 x g zg is named
as the pitch angle and designated as ϑ . The angle between the transverse axis 0 z and
axis 0 z g of normal coordinate system, displaced in the position at which yaw angle is
equal to zero ( ψ = 0 ), is named as the bank angle and designated as γ .
1.4. Aerodynamic forces and moments.
Coefficients of aerodynamic forces and moments.
The main vector of forces system which affect onto a flight vehicle at its motion
from the air, is named as full aerodynamic force and is designated as R A . The concept
of aerodynamic force is usable not only to an aircraft as a whole, but also to its parts: a
wing, a fuselage and so on.
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7. Components of full aerodynamic force
X * , Y , Z along axes of body coordinate
system are determined by making projections
of R A on these axes 0 xyz . The component
X * taken with a converse sign is named as
aerodynamic longitudinal force and
designated as X. Aerodynamic force
Fig. 1.4. Components of aerodynamic components Y, Z are named as
force in body coordinate aerodynamic normal and aerodynamic
transversal forces. Forces X , Y , Z can be both positive and negative depending on
the shape of an aircraft and the mode of flight (Fig. 1.4).
Let's project force R A onto axes of
wind coordinate system 0 xa ya za . Let's
designate its projections as X* , Ya , Z a .
a
Taken with a converse sign the component
X* is named as drag force and designated as
a
X a . The drag force is always positive.
Aerodynamic force components Ya , Z a are
named as aerodynamic lifting force and
aerodynamic lateral force. They can be both
positive, and negative (Fig. 1.5).
Fig. 1.5. Aerodynamic force In aerodynamics it is accepted to work
components in wind coordinate system not with absolute forces values but with
values of their coefficients. Having divided values of the aerodynamic forces on
dynamic pressure q∞ = ρ∞ V∞
2
(where ρ ∞ is the density of an undisturbed air flow,
2
V∞ is undisturbed air flow velocity ran against the plane at versed motion) and on the
reference area S , we get coefficients of aerodynamic forces:
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8. X Y Z
Cx = ; Cy = ; Cz = ; (1.1)
q∞ S q∞ S q∞ S
Xa Y Z
C xa = ; C ya = a ; C za = a . (1.2)
q∞ S q∞ S q∞ S
The coefficients C x , C y , C z , C ya , C za are named as coefficients of
aerodynamic longitudinal, normal, transversal, lifting and lateral force, and C xa is the
drag coefficient.
As the reference area S it can be adopted for definition of coefficients of
aerodynamic forces:
• Gross wing area while aircraft considering;
• Area of wing formed by outer panels while considering a wing separately;
• Mid-section area in case of considering a fuselage, engines, nacelles etc.
Let's proceed to consideration of the aerodynamic moments. Let's put an origin of
a body system in the center of mass and we can assume this point as a point of reduction
of aerodynamic forces. The moment M caused by these forces is named as the
aerodynamic moment. The aerodynamic moment components along axes of body
coordinate system are designated as M x ,
M y , M z and named as aerodynamic roll
moment, aerodynamic yaw moment and
aerodynamic pitch moment (Fig. 1.6).
Let's introduce non-dimensional
Fig. 1.6. Components of the
coefficients of the moments:
aerodynamic moment
Mx My Mz
mx = ; my = ; mz = ,
q∞ S l q∞ S l q∞ S b
(1.3)
where l is the reference length, usually it is a wing span; b is the chord of a wing,
usually it is the length of the mean aerodynamic chord.
In case of the aircraft parts under consideration the reference area and reference
linear dimensions of these parts are used as S , b , l in the reduced formulae.
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9. The coefficients mx , m y , mz are named as coefficients of aerodynamic roll, yaw
and pitch moments.
While considering the aerodynamic forces both moments and their coefficients
the word "aerodynamic" can be omitted if doesn’t cause an error explanation of these
terms.
Till now we spoke only about summarized forces and moments. But in some
cases it is necessary to know local forces which are affecting on unit area of an aircraft
surface or on its separate parts in specified point. Aerodynamic forces caused by
pressure distribution along an aircraft surface are usually determined by overpressures.
An overpressure is usually expressed in shares of undisturbed flow drag, i.e. as non-
dimensional value which is named as coefficient of pressure:
p − p∞
Cp = . (1.4)
q∞
Let's write down also formulae
determining proportions between forces
coefficients in body and wind coordinate
systems. Let's consider flow about the wing
with infinite span by flat flow under some
angle of attack (Fig. 1.7). Let's direct an axis
xa along undisturbed stream velocity, axis
Fig. 1.7. Aerodynamic forces in wind ya - perpendicularly to axis xa to the airfoil
and body coordinate systems top outline. An axis x of body coordinate
system will be directed along chord, axis y - perpendicularly to axis x to the upper
outline. We will place an origin of both systems in a center of pressure. Center of
pressure of an airfoil is the crosspoint of action line of resultant aerodynamic force of
the airfoil with a chord or its prolongation. As it follows from fig. 1.7:
Ya = Y cos α − X sinα ;⎫
⎬ (1.5)
X a = X cos α + Y sinα ⎭
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10. Y = Ya cos α + X a sin α ;⎫
⎬ (1.6)
X = X a cos α − Ya sinα ⎭
Having forces substituted by their expressions under the formulae (1.1), (1.2) and
having reduced the identical coefficients, we will receive:
C y = C y cos α − C x sinα ;⎫ ⎪
a
⎬ (1.7)
C x a = C x cos α + C y sin α ⎪
⎭
C y = C ya cos α + C xa sinα ;⎫
⎪
⎬ (1.8)
C x = C xa cos α − C ya sinα ⎪⎭
At small angles of attack α it is possible to assume cos α ≈ 1 , sinα ≈ α . Besides
it is possible to neglect C xa << C ya and addend C xa sin α . Therefore it is possible to
write down expressions (1.7) and (1.8) at small angles of attack as:
Cy ≈ Cy ; ⎫
⎪
a
⎬ (1.9)
Cxa ≈ Cx + C y α ⎪
⎭
C y ≈ C ya ; ⎫
⎪
⎬ (1.10)
C x ≈ C x a − C ya α ⎪
⎭
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