Fundamental
dimensions and units
NOTE:-according to this blog we are not getting exponential terms so consider our error and plz understand if it is m2=m square,m3=m cube.
1.1 The greek alphabets:- The Greek alphabet is used extensively in
Europe and the
United States to denote
engineering
quantities (see Table 1.1). Each
letter can have
various meanings, depending on
the context in
which it is used.
( Table is provided)
1.2 Conversions
Units often need
to be converted. The least confusing way to do this is by expressing equality:
For example, to
convert 600 lb thrust to kilograms (kg)
Using 1
kg = 2.205 lb
Hence 600 lb = 272.1 kg
Setting out
calculations in this way can help avoid confusion, particularly when they
involve large numbers and/or several sequential stages
of conversion.
1.3 The Physical Properties of Air:- Air is a compressible fluid. It can
therefore, flow or change its shape when subjected to very small outside
forces because there is
little cohesion of the molecules. If there was no cohesion between the
molecules
and therefore no
internal friction then it would be an ‘ideal’ fluid, but unfortunately such is
not the case.
1.3.1 Fluid Pressure
The pressure in a fluid
at any point is the same in all directions. Any body, irrespective of shape or
position, when immersed
in a stationary fluid is subject to the fluid pressure applied at right angles
to the
surface of that body at
that point.
1.3.2 Static Pressure
The pressure of a
stationary column of air at a particular altitude is that which results from
the mass of
air in the column above
that altitude and acts in all directions at that point. The static pressure
decreases
with increased altitude
as shown in Table 1.3. The abbreviation for the static pressure at any altitude
is P.
1.3.3 Dynamic Pressure
Air in motion has energy
because it possesses density (mass per unit volume) that exerts pressure on any
object in its path.
This is dynamic pressure, which is signified by the notation (q) and is
proportional
to the air density and
the square of the speed of the air. A body moving through the air has a similar
force exerted on it
that is proportional to the rate of movement of the body. The energy due to
this
movement is kinetic
energy (KE), which is equal to half the product of the mass and the square of
the speed.
Bernoulli’s equation
for incompressible airflow states that the kinetic energy of one cubic metre
of air travelling at a
given speed can be calculated from the following formula:
KE = 1/2ρV2
joules
Where ρ is
the air density in kg per m3 and V is the airspeed in metres per second.
Note: 1. A joule is the
work done when the point of application of a force of one Newton is displaced
by one metre in the
direction of the force.
Note: 2. A Newton is
that force that when applied to a mass of one kg produces an acceleration of
1 metre per second per
second.
If a volume of air is
trapped and brought to rest in an open ended tube the total energy remains
constant. If such is
the case then KE becomes pressure energy (PE), which for practical purposes is
equal
to 1/2ρ(Vsquare) Newtons per (msqr). If the area of the tube is
S square metres then:
S (dynamic + static pressure) = 1/2ρV2 Newtons
1.4 Viscosity
Viscosity is a measure
of the degree to which a fluid resists flow under an applied force. A fluid or
liquid that is highly viscous flows less readily than a fluid or liquid that
has low viscosity. The internal friction of the gas or liquid determines its
ability to flow or its fluidity. Viscosity for air is the resistance of one
layer of air to the movement over a neighbouring layer .Dynamic viscosity (μ) is measured in the
SI system, in N s/m2 or pascal seconds (Pa s). Kinematic viscosity (v) is a function of dynamic viscosity.
Kinematic viscosity = dynamic viscosity/density.
1.5 The Equation of Continuity
Mass cannot be created or destroyed. Air mass flow is steady and continual. The equation related will be given later
1.6 Reynolds Number
Reynolds Number (Re) = Inertial Forces/Viscous
Forces
ie.,Re= ρVL/μ
where ,
ρ is density in kg per m3;
V is the velocity in metres per second; L is
the chord length and
μ is the viscosity of the fluid.
A small Reynolds
number is one in
which the viscous force is predominant and indicates a steady
flow and smooth
fluid motion(laminar
flow). A large Reynolds
number is one in
which the inertial force is paramount
and indicates random
eddies and turbulent flow.
1.7 Perfect gas
A perfect (or
‘ideal’) gas is one which follows Boyle’s/Charles’ law
pv =
RT
where:
p =
pressure of the gas
v =
specific volume
T =
absolute temperature
R =
the universal gas constant
1.8 Compressibility
The extent to
which a fluid can be compressed in
volume is
expressed using the compressibility
coefficient B =dv/v =1
dp
k
where dv
= change in
volume
v =
initial volume
dp
= change in
pressure
K =
bulk modulus
SECTION 2
Aeronautical definitions
2.1 Forces and moments
Forces and
moments play an important part in
the science of
aeronautics. The basic definitions are:
Weight force (W):
Weight of
aircraft acting vertically downwards.
Aerodynamic
force:
Force exerted (on an aircraft) by
virtue of
the diversion of
an airstream from its original
path. It is
divided into three components:
lift, drag and
lateral.
Lift force (L)
Force component
perpendicularly ‘upwards’ to the flight direction.
Drag force (D)
Force component
in the opposite direction to flight. Total drag
is subdivided into pressure drag and surface
friction drag.
2.2 Aerofoil Profile
The definitions used
with reference to an aerofoil’s shape are shown in Figure and are as follows:
a)Camber.
The curvature of the
profile view of an aerofoil is its camber. The amount of camber and
its distribution along
the chord length is dependent on the performance requirements of the aerofoil.
Generally, low-speed
aerofoils require a greater amount of camber than high-speed aerofoils. See
Figure
b) Chordline.
The straight line
joining the centre of curvature of the leading-edge radius and the trailing
edge of an aerofoil is
the chordline. See Figure 2.1(a).
c)Chord.
The distance between
the leading edge and the trailing edge of an aerofoil measured along
the chordline is the
chord.
d) Fineness
Ratio. The
fineness ratio is the ratio of the length of a streamlined body to its maximum
width or diameter. A
low fineness ratio has a short and fat shape, whereas a high fineness ratio
describes a long thin object.
e)Leading
Edge Radius. The radius
of a circle, centred on a line tangential to the curve of the leading
edge of an aerofoil,
and joining the curvatures of the upper and lower surfaces of the aerofoil is
the leading-edge radius or nose radius of an aerofoil.
f) Maximum
Thickness. The maximum
depth between the upper and lower surfaces of an aerofoil is its maximum
thickness.
g) Mean
Aerodynamic Chord. The
chordline that passes through the geometric centre of the plan area (the
centroid) of an aerofoil is the Mean Aerodynamic Chord (MAC) and for any
planform is equal to the chord of a rectangular wing having the same wing span
and the same pitching moment and lift.
It is usually located
at approximately 33% of the semi-span from the wing root. The MAC is often used
to reference the location of the centre of gravity particularly with
American-built aeroplanes. It is the primary reference for longitudinal
stability.
h) Mean
Camber Line. The line
joining points that are equidistant from the upper and lower surfaces of an
aerofoil is the mean camber line. Maximum camber occurs at the point where
there is the greatest difference between the mean camber line and the
chordline. When the mean camber line is
above the chordline the
aerofoil has positive camber.
i)Mean
Geometric Chord. This
is the average chord length of an aerofoil is the mean geometric chord or
alternatively it is defined as the wing area divided by the wingspan.
j) Quarter
Chordline. A line
joining the quarter chord points along the length of a wing is the quarter
chordline, i.e. a line joining all of the points 25% of the chord measured from
the leading edge of the aerofoil.
k) Root
Chord. The line
joining the leading edge and trailing edge of a wing at the centreline of the
wing is the root chord
l)Thickness/Chord
Ratio. The ratio
of the maximum thickness of an aerofoil to the chord length expressed as a
percentage is the thickness/chord ratio. This is usually between 10% and 12%.
m) Washout.
A reduction in the
angle of incidence of an aerofoil from the wing root to the wing tip is the
washout. This is also known as the geometric twist of an aerofoil.
2.3 Aerofoil Attitude
The definitions used
with reference to an aerofoil’s attitude shown in Figure 2.2 are as follows:
a. Angle
of Attack. The angle
subtended between the chordline of an aerofoil and the oncoming airflow is the
angle of attack.
b. Angle
of Incidence. The angle
subtended between the chordline and the longitudinal axis is the angle of
incidence. It is usual for the chordline of a commercial transport aeroplane to
be at an angle of incidence of 4◦ when compared with the longitudinal axis because in level flight,
at relatively high speeds, it produces the greatest amount of
lift for the smallest drag penalty. If such is the case, the angle of attack
and the pitch angle will be 4◦ apart,
no matter what the attitude of the aeroplane.
c. Critical
Angle of Attack. The
angle of attack at which the maximum lift is produced is the critical angle of
attack often referred to as the stalling angle of attack. A fixed-wing
aeroplane is stalled at or above the critical angle of attack.
d. Climb
or Descent Angle. The
angle subtended between the flight path of an aeroplane and the horizontal
plane is the climb angle or descent angle.
e. Longitudinal
Dihedral. The
difference between the angle of incidence of the mainplane and the tailplane is
the longitudinal dihedral. See Figure 2.2(b).
f. Pitch
Angle. The angle
subtended between the longitudinal axis and the horizontal plane is the pitch
angle.
g. Pitching
Moment. Certain
combinations of the forces acting on an aeroplane cause it to change its pitch
angle. The length of the arm multiplied by the pitching force is referred to as
a pitching moment and is counteracted by the use of the tailplane. A pitching
moment is positive if it moves the aircraft nose-upward in flight and negative
if it moves it downward.
2.4 Wing Shape
The definitions used
with reference to the shape of an aeroplane’s wing are shown in Figure 2.3 and
are as follows:
a. Aspect
Ratio. The ratio
of the span of an aerofoil to the mean geometric chord is the aspect ratio,
which is sometimes
expressed as the square of the span divided by the wing area.
Aspect Ratio = span
chord
= span2
wing
area
b. Mean
Aerodynamic Chord. The
chordline that passes through the geometric centre of the plan area (the
centroid) of an aerofoil is the Mean Aerodynamic Chord (MAC).
c. Quarter
Chordline. The line
joining all points at 25% of the chord length from the leading edge of an
aerofoil is the quarter chordline.
d. Root
Chord. The
straight line joining the centres of curvature of the leading and trailing
edges of an aerofoil at the root is the root chord.
e. Sweep
Angle. The angle subtended
between the leading edge of an aerofoil and the lateral axis of an aeroplane is
the sweep angle.
f. Taper
Ratio. The ratio
of the length of the tip chord expressed as a percentage of the length of the
root chord is the taper ratio.
g. Tapered
Wing. Any wing on
which the root chord is longer than the tip chord is a tapered wing.
h. Tip
Chord. The length
of the wing chord at the wing tip is the tip chord.
i. Wing
Area. The surface
area of the planform of a wing is the wing area.
j. Swept
Wing. Any wing of
which the quarter chordline is not parallel to the lateral axis is a swept
wing.
k. Wing
Centroid. The
geometric centre on a wing plan area is the wing centroid.
l. Mean
Geometric Chord. The
average chord length of an aerofoil is the mean geometric chord, or
alternatively it is defined as the wing area divided by the wingspan.
m. Wingspan.
The shortest distance
measured between the wing tips of an aeroplane is its wingspan.
2.5 WING LOADING
The angle of
inclination subtended between the front view of a wing 25% chordline (the wing
plane) and the lateral horizontal axis is referred to as:
a. dihedral
if the inclination is
upward.
b. anhedral
if the inclination is
downward.
2.5.1 Wing Loading
The forces acting on a
wing are defined below:
a. Centre
of Pressure (CP). The
point on the chordline through which the resultant of all of the aerodynamic
forces acting on the wing is considered to act is the centre of pressure.
b. Coefficient.
A numerical measure of
a physical property that is constant for a system under specified conditions is
a coefficient, e.g. Cl
is the coefficient of
lift.
c. Load
Factor (n). The total
lift of an aerofoil divided by the total mass is the load factor. i.e.
n = total lift
mass
d. Wing
Loading. The mass
per unit area of a wing is the wing loading. i.e.
Wing Loading = mass
wing area
= M Kg
2.6 Types of flow;-
1)Steady and
unsteady flow:
We will consider the
flow around a body such as a sphere, a cylinder, or an aerofoil. In a steady
flow, there are
no fluctuations and the properties in a point fixed with respect to the body do
not change with time. when the flow properties
in a point moving along
with the body do change in time, we have an unsteady flow.
2)compressible flow:
In a flow around any
orbitary body the property density of the fluid is considered and is no longer
a constant is called a compressible flow
ρ is not constant
3)incompressible flow:
In a flow around any
body if density of the fluid is constant the flow is said to be incompressible
flow.
d ρ=0
,
ρ=constant
4)Inviscid
flow
Some theoretical
approaches of aerodynamics ignore the viscosity of air, which gives an inviscid
or ideal
flow. In such a
flow model there is no internal friction and inviscid flow is also
characterized as frictionless flow.
5)Streamlines
We will examine the two-dimensional
flow around an
aircraft component, in this case an aerofoil section, that is, a cross section in a plane
parallel
to the flow direction
(Figure 3.1). The path of an air element in the flow is called a streamline.1 The flow element has a velocity V at the point A far upstream, where the other
flow properties are p∞, ρ∞ and T∞.When the
element approaches the aerofoil these properties change so that, for instance,
at point B the velocity is greater than V and has a different direction. At each point of the element’s
path, the velocity vector is tangential to the streamline and it follows that
through every point in the flow field there passes one and only one –
streamline. There is one exceptional case: streamlines may be intersecting at a
point in the flow where the local velocity is zero.
Such a point is called a stagnation
point.
2.6 f :Streamlines over an aerofoil
2.7 Equations for steady
flow
2.7.1
Continuity equation
Since there is no mass
transport through the wall of a stream tube, the air mass per unit of time
entering a stream tube or a stream filament is equal to the air mass exiting in
the same period of time, provided the flow is steady.
This
continuity law is based on the principle of mass conservation.Figure 2.7(a)shows
a stream filament with cross sectional areas A1 and
A2 normal to the local direction of flow. Through A1 a volume A1 v1 dt enters each
unit of time dt ,
with a mass equal to dm = ρ1(A1 v1 dt) = (ρ2A2v2 )dt
or ρAv = constant.
This (algebraic) continuity
equation only holds
for steady flows and makes up a simple relation connecting the local density
and velocity to the cross section of a stream filament. For incompressible
flows, the
density is constant and the equation simplifies to
A1 v1 = A2 v2 or Av = constant.
2.7.2 Euler’s
equation
In an incompressible
flow The resultant of all external forces on a body is equal to the product of
the body mass and its acceleration.
dp + ρvdv =0 or
dp/dv = −ρv .
This differential
equation holds for both compressible and incompressible inviscid
flows and is
known as
Euler’s equation
2.7.3
Bernoulli’s equation
P1+ ½ ρv12+
ρgh1 = p2 + ½ ρv22+ ρgh2.
Because P1
and P2 were chosen arbitrarily we can say that along a streamline
p + ½ ρv2 +
ρgh is constant.
This theorem is known as Bernoulli’s
equation,When
we write this equation
per unit of mass,
p/ρ+ ½ v2+ gh = constant.
In the flow around an
aircraft the vertical displacement of elements is mostly small enough to
neglect the change in potential energy relative to the other terms. In this
case, Bernoulli’s equation simplifies to
p + ½ ρv2 = constant. (as.,h1
=h2)
The term ½ ρv2 is called the dynamic pressure q. This is equivalent to the kinetic energy
imparted per second to a unit volume of air of density ρ when it is accelerated to velocity v. In an inviscid incompressible flow, the sum
of
the static and dynamic
pressure is a constant referred to as the total pressure,
pt=p + q =
p + ½ ρv2.
A special case is a stagnation
point, where the
velocity and the dynamic pressure are zero. The stagnation pressure ps is therefore equal to the total
pressure of the
undisturbed flow,
Momentum equation
Newton’s second law of
motion for a solid body with a given mass can be
written as
F = m (dV)
dt
= d(mV )
dt
where mV is the body’s momentum.
Fig1:Element in streamtube
fig2:Control volume over an
aerofoil
ref:-aeronautical engineers data book. by Clifford Matthews BSc, CEng, MBA, FIMechE.
FAA pilots handbook,by FAA usa.
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