So, let us continue our discussion on airfoils;
the general shape and various nomenclatures
about an airfoil.
Airfoils are the cross section of the aircraft
wings, and the general shape is at the nose
or the front part is basically rounded. At
least for subsonic application, the front
part is rounded and then its thickness keeps
on increasing up to certain distance reaches
a maximum, and then again decreases to zero
or nearly zero at the trailing edge.
So, the rounded nose or the leading edge is
rounded and the trailing edge may be either
sharp or also may be rounded, but if it is
rounded, then also its thickness is very small
compared to the front or the nose. Now, we
mentioned that if we join the leading edge
and trailing edge by a straight line; that
line is called the chord of the airfoil.
Now; obviously, question comes that when there
is a sharp trailing edge; that is, a trailing
edge is point, then we can say that this is
a trailing edge, but the leading edge which
is always rounded, which is actually the leading
edge, which point, which point should we connect
to get the chord.
And same question also arise if the trailing
edge is also rounded. You know because of
manufacturing difficulty, many a time the
trailing edge is not exactly a point, but
also little rounded. And in such a case, how
do you look at these trailing edge and leading
edge; the points. This entire region we can
call it the leading edge region, but which
is that point?
Now, in that situation where there is rounded
nose; obviously, the nose is always rounded,
the point of maximum curvature or the center
of the maximum curvature; that is located
and similarly, the trailing edge is also rounded,
then the center of maximum curvature at the
trailing edge; that is also located, and these
two points are joined and then extended to
intersect their airfoils. And wherever it
intersects, that will be taken as the leading
edge point.
So, we need to locate the point of maximum
or center of maximum curvature at the front
as well as at the end, and then join the line.
And that is what is the chord. Let us say
then, that sorry we will have one more…
The chord is usually denoted by c, and it
is a very important reference line because
the incidence or the angle at which it makes
with the flight direction or the relative
wind is called the angle of incidence or angle
of attack. And again that is usually denoted
by alpha. Let us say this is the relative
wind direction, then this angle is called
angle of attack or incidence and it is usually
denoted by alpha. And the maximum distance
of this line from the chord; you will see
this line is not at equidistance from the
chord. Its distance from the chord changes
from point to point.
So, the maximum distance of this line from
the chord expressed as a percentage of chord
is called its camber; the camber of the airfoil.
So, if we say that an airfoil has 3 percent
camber; meaning that the maximum distance
of the camber line from the chord is 0.3 c.
So, the thickness and this camber are always
expressed in terms of percentage chord. And
the airfoils are usually characterized by
this maximum thickness and maximum camber
expressed in terms of percentage chord. So,
if we mention that the airfoil is 12 percent
thick; that means, the maximum thickness of
the airfoils is 12 percent or 0.12 c.
Now, this is the wing cross section. Now let
us look to little bit of wings how the wing
looks like, though of course, it is not the
details of it is not related to aerodynamics
and we will be hardly interested later on
of all these details, but still this is something
all of you must know. The wing cross section
is airfoil, but how is the wing? The wing
is usually hollow, it is not solid.
Besides providing the most important aerodynamic
forces and moments, and also providing support
to the engines, lending gears, and many other
things, wing also serve as the main fuel tank
of the aircraft. That the fuel that the aircraft
needs for its operation is mostly stored within
the wing; that means, that the wing is not
a solid structure, it is hollow. Only the
shape is given by what is called the skin
which is just a thin plate, very thin plate.
It is you can imagine that you have a seat
metal turned in the shape of the wing which
is quite thin. Of course, if it is just like
that, then you can understand that because
of the huge air load that the wing supports,
it would not be able to sustain if it is just
only a seat metal; just only the skin. So,
there are certain internal structures which
give the wing its strength so that it can
maintain its shape because its shape is very
important. These aerodynamic courses and the
moments that the wing provides is just because
of its shape.
So, the shape must be maintained, but just
if you have only a thin metal, sheet metal
turned like that, then of course, the sufficient
strength will not be there. And not only that
you know in early days of aviation, the wing
skin used to be even of cloths; fabric. And
even now, small aircrafts; they have fabric
wing. Now that of course, cannot support these
air pressure or huge air load that the air
craft supports. So, there are internal structures,
but the internal structures are arranged in
such a way that the major part of the wing
is hollow which can be used as fuel tank.
So, let us look to these about these internal
wing structures. Not that they are directly
related to our course of study, to give it
sufficient strength and also to provide connection
with the , there are some number of beams.
Depending upon the aircraft, it might be one
beam or might be two beam or three beam which
are usually having eye cross section and extended
from wing route to wing tip right and extended
throughout. These beam type of structure are
called spars and they provide the most of
the strength that is necessary to support
the loads. These are called spars. Of course,
details of these you will be studying in your
structures course. So, they are called spar.
Also, there are longitudinal members to stiffen
the scheme. So, even if you have that two
or three such spars, then also that the intermediate
space between these two spars or ahead or
behind these spars; the skin there may buckle
or miss sub depressed because of the air load.
The skin is quite thin.
So, to provide support to the skin, there
are stiffners. Again say very small beam,
but they are not connected from upper surface
to lower surface, they are attached to either
upper surface or the lower surface. And they
may of z section or c section or hot hat section,
and they are also extended from route to tip.
They are called stringers or stiffeners. There
will be many such stringers or stiffeners
to stiffen the skin; stringers or stiffeners.
They are called ribs. All of them are of airfoil
shapes. empty spaces or hollow spaces in between.
So, that is what the wing structure is.
In this context, we will also mention that
the wing; it looks to be a one component,
but actually this wing cross section most
often is made up of or the airfoil is not
made up of single component. It is made up
of number of components; that means, in a
practical case, the airfoil can be something
like this. Some cases, this also can be even
two or three. Here I have shown that this
airfoil is in three parts, but these two parts
which are now separated from this main part;
they can fit with it so that you will just
see only a line. When they are fitted together,
you will see just a line without any gap.
However, they can be deflected with respect
to or relative to this main component. Then
these components are in generally called flaps.
In generally, these components are called
flaps, and then they are further named from
their; following the purpose that they are
serving like this; leading edge flap; this
small parts are the leading edge is usually
called a leading edge slat, leading edge slat.
Flap is a general name and there are specific
name based on what purpose they are serving.
One say this flat fitted at the trailing edge
of the wing can be used for two purposes;
one purpose is just to provide little higher
lift during landing and takeoff. The aircraft
when it lands or takes off needs little more
lift than while it flies or rather lift coefficient;
little more lift coefficient. Will say what
those lift coefficient are. And these flaps,
deflecting these flaps that had amount of
lift can be obtained. Same thing; this slat,
actually all these flaps, that is, the general
purpose; that they usually provide when deflected
a little change in lift depending upon in
which direction it is deflected, it may increase
the lift or it may decrease the lift.
Usually if the airfoil has a positive camber
and then the flap is deflected downward, then
the lift is increased. If the flap is deflected
upward, lift is decreased. So, now, imagine
that on two wings, we have two flaps. If both
of them are deflected downward, they will
increase the lift which is required during
landing and takeoff. So, during landing and
takeoff, these flaps are used only for that
purpose to get enhanced lift and then they
are called high lift devices.
Now, think that in one wing, it is deflected
downward, in other wing, it is deflected upward.
What will happen then? In one wing, the lift
will increase, on the other wing, the lift
will decrease. And consequently then the aircraft
will experience a movement about the longitudinal
or what you call the x axis; meaning a rolling
movement. And in such a situation, those flaps
will be called eulerian. When the flaps are
used to provide this rolling movement, they
are called eulerian.
Now, in many aircraft, these purposes are
separated. See the flap is made in two part;
part which is nearer to the which is called
the inner part. So, if this is route part,
so, up to certain distance, up to this part,
it is one and this part is again its separated.
So, this near part, this near part on both
the wing will always be deflected symmetrically;
that means, both will be deflected downward
together and will serve as high lift devices.
You require it during landing and takeoff.
While this outer part; though will be not
used for that purpose, they can be used only
for providing the rolling movement. And since
you know rolling movement is movement about
the longitudinal axis, if these eulerians
are towards the tip, that is beneficial because
in that case, the moment arm becomes longer.
Even with the small difference in force, you
can get larger moment. So, they are more useful
if they are at towards the tip.
So, at this stage, that is all about wing
we wanted to say because these rolling movement
and all those. Only our role here is to find
the pressure distribution, then how this rolling
movements are coming, and how the rolling
movement is going to effect the aircraft motion
is of course the subject matter of flap mechanics,
not aerodynamics
So, this we will not discuss further, but
we should know what those movements are and
what their purpose is, how they can be obtained
or how they can be modified because as an
engineer, that is also one of your job. Not
just to know that what the lift is for a given
wing, but also to know how to change that
lift or how to get more lift by changing the
shape anyway.
The tail plane; as we mentioned that there
are two tail planes and both are basically
small wings. So, their structures and almost
everything are nearly same; however, none
of them are usually having leading edge slat,
usually no horizontal or vertical tail uses
leading edge slat, but they do use that tail
edge flap, but, not for providing high lift.
I mean that is not the main purpose. In their
case, the main purpose of those reading is
flaps for the horizontal tail, the leading
is flap is called elevator. The leading is
flap for the horizontal tail is called elevator.
And once again if a change in lift occurs
there, it again produces a movement, but this
time you see the movement is about the
span wise axis; meaning, it will change the
incidence. Either it will take the nose up
or down; aircraft nose up or down, and that
leading edge flap of the horizontal tail plane
is called elevator. And it provides the pitching
moment necessary to change the incidence of
the aircraft.
If we do not need to change the incidence,
we do not need to use the elevators; that
means, we do not need to deflect the elevators,
they will remain fitted with the horizontal
tail as it is. Horizontal tail; also called
as a stabilizer, horizontal stabilizer. Similarly,
vertical tail also called vertical stabilizer
because the horizontal tail and vertical tail;
they provide the necessary stability to the
aircraft. That is their main purpose, like
the main purpose of the wing is to provide
this aerodynamic force required to fly. The
horizontal tails and vertical tails; the main
purpose of them is to provide the necessary
stability.
See these are the main function. It does not
mean that they do not do anything else. This
horizontal tail of course, always produces
certain amount of lift, irrespective of whatever
small it is, it always produces certain amount
of lift, certain amount of drag, but that
is not their main purpose for which they are
used. Their main purpose is to provide stability
and to hold those elevator and radar.
The flap of the vertical tail is called radar.
It provides the directional control like it
provides a movement about the vertical z axis,
vertical axis and swings the nose to the left
or right; that means, it has sudden control
over the direction of the flight, in which
direction the flighty should go, and it is
called radar.
The difference another difference is there
that in most cases, the airfoils used for
horizontal and vertical tail are symmetric,
are usually of course, there is no hard and
fast rule that they have to be symmetric,
but most cases, they are made symmetric and
in wing, they are always camber Even though
the wing is not there within the , but for
this case, we consider the wing is extended
through the and then the top view of it. If
we look from the top, whatever we will see;
that is what is the plan form and this area
is area of that plan form.
So, as you mentioned the if the wing plan
form is trapezoidal, then basically it is
an area of that trapezoid. As an example,
if you have a cylinder, what is its plan form?
A rectangle, yes.
So, the plan form area of that cylinder is
simply the area of that rectangle. That is
what we are… Similarly, all other forces
are normalized using the same parameter; half
rho u infinity square s. The drag coefficient;
the other force is drag coefficient, sorry
drag force is denoted by D and this is the
drag coefficient.
Similarly, the side force coefficient; let
us say c y also can be written as… The moment
coefficient let us say the pitching moment
coefficient; this is pitching moment coefficient
c m, pitching moment is usually denoted by
m. m by…
Now this is of course, not moment; half rho
infinity square is not a moment. We need another
length to make it moment, and the length is
usually the aerodynamic chord, mean aerodynamic
chord. The yawing moment is usually defined
denoted by n…, not c but b. For the yawing
moment, that is, the moment about the vertical
axis, the appropriate length parameter is
the span; length in that reaction, not the
chord. So, here it is b.
The rolling moment; here we need to be little
careful, rolling moment let us for the time
being because we would not need it, we will
be using let us r; of course, it is not the
usual notation. For pitching moment and yawing
moment, m and n is usual notation. For yawing
moment, the usual notation is l, but we are
not going to use that because we have already
used it for lift.
So, let us write it r and again it is… There
is a little bit of conflict of notation in
fad mechanics and aerodynamics. In fad mechanics,
the notations are much more straight forward.
The three forces are denoted as x y z. The
drag is x, side force is y, the lift is z,
and these three moments; rolling moment is
l, pitching moment is m, yawing moment is
n, but in aerodynamics, lift is always used
as l and hence we have a little problem while
writing rolling moment. So, lets for the time
being we write it r.
Now, when aircraft flies straight and level,
it is going straight. So; obviously, there
is no side force acting. If there is a side
force acting, then of course, it cannot go
straight, and also since it is flying straight
and level; obviously, no moment is acting
on it or all moments are zero.
So, in that situation, all three moments are
zero and side force is zero. Only two nonzero
are lift force and drag force. So, this lift
and drag force are perhaps the most important
of the aerodynamic forces, and as we have
already mentioned, that this lift and drag
forces; and why only lift and drag, all these
forces come because of the pressure and stress
distribution, viscous stress distribution
on the surface of the aircraft and mostly
on the wing.
And once we get the pressure and sheer stress
distribution on the airfoil surface or the
wing surface or to be more accurate, over
the entire aircraft surface, we can find all
these forces and moments; just a matter of
integration. Once we know the pressure and
the stress, we can simply integrate it over
the aircraft surface and get all these forces
and moments. In that process, one may be zero.
And as we mentioned in the beginning that
the role of aerodynamics is to try to find
out that pressure and stress distribution,
to find that pressure and stress distribution
and not only finding the pressure and stress
distribution, also using that knowledge, to
change the pressure distribution or stress
distribution to our advantage, which is the
design problem. The design problem is as you
have that it gives you the task that this
is what we need, how to get it.
So, the question may be like this, that this
is what the pressure distribution we should
have, how to get that pressure distribution;
that becomes a design problem, and what type
of pressure distribution we should have? That
of course, comes with little bit of experience
and little study; different type of analysis
that what type of pressure distribution gives
us the best result. So, we should try to get
that type of pressure distribution. And then
we should look to the problem that if we want
to have this pressure distribution, how we
are going to have the wing, what type of wing
it should be?
So, the aerodynamics and try to answer these
two questions, that if this is the wing or
this is the aircraft, what are the forces
it will have, and if we want to have a little
different type of pressure distribution or
stress distribution, how we will change our
aircraft or we will have a newer aircraft
with different type of these aerodynamics.
And that is what we will try to do over a
number of aerodynamics courses and which we
will try to introduce here.
Now, the pressure distribution and stress
distribution will definitely come from if
we analyze the flow; if we analyze the flow
that occurs over a body. So, here, now the
problem completely changes. Instead of analyzing
the motion of the aircraft, we are now trying
to analyze the motion; the flow motion, the
flow. So, we should have certain as you know
that to analyze any problem, the first thing
that we need is to set up some physical laws
in the mathematical form which gives the answer
to this or the mathematical model, mathematical
model of the problem.
So, the first thing that we should do is model
this flow problem mathematically and then
we will try to solve this problem. The modeling
of the mathematical problem is basically nothing
but expressing the laws of nature or the physical
laws in mathematical form. And you are familiar
with many such physical laws. I think all
of you are familiar with these physical laws;
conservation of mass; that is very a important
physical law, conservation of momentum is
another very important physical law and so
on. There are many such physical laws. So,
if we can express them mathematically for
our problem, we will be able to set the problem
in mathematical form.
So, that will be our first task. And since
our system is basically fluid, so, we have
to set up this system for a fluid system.
You have already set up these systems for
rigid body, you have set up this systems for
some other elastic body or deformable body;
mostly solid. Now same thing we will be doing
now for fluid.
And one more thing, at this stage, perhaps
we should mention, let us say that we know
the pressure and stress distribution on the
body. We have already said that now to get
the forces, it is simply a matter if integration.
So, let us look to that integration at this
stage before we start our modeling the fluid
dynamical or aero dynamical problem.
Let us consider for simplicity on only a cross
section of any arbitrary body to which the
flow is coming with the undisturbed stream
speed of u infinity. Let us consider a small
element of the surface and let us denote this
length by say delta s. This small arc length
is denoted by delta s. You know on this small
element, we have pressure and stress acting.
Pressure; you know always acts normally to
any surface. So, the pressure is normal to
the surface, and the stress is tangential,
we will let us call it tau. Then how much
will be the force? We can have one set up
x and y axis or x and z axis. The other direction
is the y direction. Can we get the force?
The force along this direction will be the
drag force; force along this is direction
will be the lift force because this is same
as the x.
We can even sketch the flow by something like
this, Let us say pressure makes an angle theta,
pressure makes an angle theta with the vertical
axis. Pressure makes an angle theta with the
vertical axis; sorry pressure makes an angle
theta with the horizontal axis. Pressure makes
an angle theta with the horizontal axis or
the x axis. p makes p makes an angle theta
with the x axis. x axis is along flow, along
free stream. Free stream is a stream which
is not disturbed by the body; that means,
if there 
are 
this body were absent, what about the stream
would have been; that you can see that it
is just parallel lines represented by parallel
lines with velocity u infinity at everywhere,
because of this body, that will be disturbed,
and that is called the disturbed flow.
So, our x axis is along this free stream or
along the direction of u infinity. And then
how much will be the lift force? Integrated
over the entire surface into minus p sin theta
plus tau cos theta into the area element.
Area element; how we will you get? We will
consider a unit length along y axis and we
will assume that there is no change along
that for the time being.
Of course, if there is change in that, then
we will consider into delta y or otherwise
now let us consider delta s into 1. So, this
is per unit span. It is called per unit span;
lift force per unit span; that means, we are
considering the span wise length of unity
only. And similarly, the drag force can be
written as sorry this is integrated over the
entire curve… Again this is drag force per
unit span.
Now, at this stage, we may further mention
that at least for the lift, the major contribution
comes from pressure. The contribution from
this stress or sheer stress tau is much smaller
compared to 
the contribution from the pressure or rather
its almost negligibly small compared to the
contribution from pressure. So, amongst the
two, as far as lift is concerned, pressure
is the most important. And at least in this
course and also perhaps next one or two courses
on aerodynamics, most of the time, we 
will be discussing only about the process
where we can consider only pressure, not 
the sheer stress; that is, most often we 
will deal with the situation where there is
no 
sheer stress. The reason is that the contribution
of sheer stress to the lift force is negligibly
small compared to the contribution from pressure.
Of course, that is not true for the drag.
So, in our next lecture now, we will move
on to our proper aerodynamics with this brief
introduction that what aerodynamics intends
to do or what is the subject matter of aerodynamics,
why is in required to do, and then we will
move on to aerodynamics from next lecture
onwards.
