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Welcome back to the video course on fluid mechanics. The last lecture we were discussing
about the boundary layer and its formation and various theories to analyse the boundary
layer for various parameters like velocity, variation shear stress and pressure variation.
Starting from the Prandtl's boundary layer equations we have seen the various solutions
with respect to the blasius solution, then the moment integral equations and then we
have derived various equations which we can utilize to analyse the boundary layer especially
for flow over immersed bodies, flow over a flat plate and also flow through pipe with
respect to the boundary layer formation for the external flows as well as internal flows.
Now, as far as boundary layer is concerned one of the important aspects is the separation
of the boundary layer. We have seen that the boundary layer when it forms with respect
to the flow over the external bodies depending upon the shape of the body, depending upon
the fluid properties the boundary layer thickness can vary; boundary layer thickness can change
with respect to the various parameters. Most of the blend bodies like a circular cylinder
or depending upon the sphere or depending upon the body shape the boundary layer can
separate.
Today, we will discuss the separation of boundary layer so depending upon the case the boundary
layer separate depending upon the shape of the body, depending upon the fluid flow parameters
the boundary layer can separate at various locations with respect to the flow. Here,
we can see the flow over a cylinder is animated; we can see that the flow is coming here and
then we can see that it is separated at some locations. Like this depending upon the problem
here I will show another figure. This video is taken from the book of fluid mechanics
by Munson.
We can see similar beam, the flow over a cube or flow over a rectangular block. You can
see that it is coming here and then separating the boundary layer after sometimes. Depending
upon the case, the flow separation takes place and then the flow at some particular location
the proportion takes place and then we have to find out in the boundary layer and shear
the flow separation starts and what are the various parameters as far as various properties
with respect to the boundary layer separations. When we consider the separation boundary layer
if flow over a boundary occurs there are typically three conditions.
First condition is case A, where pressure decreases in the direction of flow; the fluid
will accelerate and the boundary layer will become thinner, this is called the convergent
type of flow. We can see that here the flow comes and then there is a convergent as far
as the flow is concerned. Then, from here onwards the dp by dx is less than 0 so that
pressure decreases and the fluid will accelerate and the boundary layer will become thinner
in this case. The second case is case B, where if you consider the flow over a flat plate
when we can see that here there is no variation in pressure in the flow direction. Example,
flow along a flat plate velocity gradient is constant. You can see that here the free
stream velocity coming and then the boundary layer is developed and initially laminar and
transition turbulent boundary layer.
Here is the second case; there is no pressure variation in the flow direction. The third
case is here the pressure increases in the direction of flow, that means, here you can
see that here the flow diverges; the dp by dx is greater than 0. You can see that positive
pressure gradient exists and this kind of flow is called divergent flow. So depending
upon case A or case B or case C whether convergent flow or flow over a flat plate which we consider
or the divergent flow, you can see that separation of the boundary layer properties changes.
Especially, in the third case, the last case where we consider the divergent flow you can
see that reverse flow layer tends to push itself between the wall and boundary layer
and in this way separates it from the wall, this phenomenon which separates the boundary
layer from the wall or the body is called separation. So these three cases, I have presented
here to show how the variation takes place and which the most obvious case is where the
separation is immediately visible. In the case c for the divergent flow you can see
that a reverse flow layer tends to push itself between the wall and boundary layer and then
separation starts. This process where the boundary layer separates from the wall or
the body that phenomenon is called boundary layer separation. Generally, this separation,
the location of the boundary layer separation we can find out the velocity gradient del
u by del y in the y direction is equal to 0. If you equate this to 0 then we can find
out the location of the separation points
You can see here in the case of flat plate what is happening and then divergent flows
what is happening we can see. With respect to which we have already analysed with respect
to case C, the divergent flow is very easily visible; how the separation of the phenomena
separation takes place and then we can find out by equating del u by del y is equal to
0 the location of the separation point. Then, we can draw the separation stream line and
then a wake formation also that means due to the pressure difference here the pressure
changes with respect to boundary layer formation and then a wake is formed. This separation
streamlines and wakeup are two important things in the boundary layer separation which we
have to consider while analyzing the boundary layer separation we have to find out the separation
streamline and then we have to see that whether any now pressure or the wake formation takes
place behind the body or after this separation takes place.
This separation and wake formation here this slide shows how the separation point where
two starts and then how the process taking place. The line of zero velocity divide the
forward flow and backward flow leaves the boundary surface at the separation point and
is known as separation streamline. We can just find out the line of zero velocity which
divide the forward flow and backward flow which leaves the boundary surface at the separation
point and we can find out the streamline. The region between the separation, streamline
and the boundary surface is known as wake. Here, we can see that before any wherever
the boundary layer formation takes place and then in the separation starts, if you plot
the velocity it will be in this form and wherever the separation points starts we can see that
it will be going to this level. You can see that after the separation streamline forms.
You can see that the direction of the velocity reverse and you can see that it will be of
this from so that we can find out the separation streamline; we can plot the separation streamline.
Separation streamlines and wake formation, we have to analyse when we analyse the boundary
layer separation we have to discuss with respect to the separation streamlines and the wake
formation.
If you consider a cylinder like this we can see that here the flow past a spherical or
cylindrical object in 2D. We can see that free stream is coming like this and then here
there will be location where the flow velocity is 0 so that is called stagnation points;
here P is the stagnation point. Then you can see that the separation flip flops, the boundary
layer we can see that the about this location at Q the boundary layer separation starts;
we can say that this is the separation point. We can see that here between the two separating
between the separate streamline we can see that wake will be formed. Here, in this figure,
P is the stagnation point, Q is the separation point and then S is where it is located in
the wake as far as the flow over a spherical or cylindrical object in two dimensions is
concerned.
Now, as for as separation wake formation is concerned again we can see depending upon
various parameter whether the cylinder is smooth or whether the cylinder is rough; we
can see that the boundary layer formation as well as the boundary layer separation changes.
So, with respect to the cylindrical flow over a sphere or cylinder in two dimensions which
we have seen in the previous slide, at R separation already took place.
As I mentioned earlier Q is the separation point and then where onwards the separation
already took place and then we can see that the wake is formula and this shows the wake.
We can see that after this within this wake there will be clockwise and anticlockwise
vertices will be formed and then this region is called the wake and this process is called
wake formation as shown in this figure. Here, the free stream is coming like this and then
here is a stagnation point and Q is a separation point and at R the separation already takes
place. Then, we can see that the separating streamline is like this and then in this region
a wake is formed and then S is this point; S is in the zone of wake formation. Depending
upon the case, depending upon whether we are considering circular cylinder or whether depending
upon the smoothness of the cylinder and also the fluid properties we can see that clockwise
and anticlockwise vertices may be formed. Depending upon the case you can see that this
is clockwise and anticlockwise vertices are formed within this way. So this is what we
can generally observe. We can easily visualize experimenting how this separation and then
the wake formation take place. Now, as I mentioned after the separations starts and the separating
streamline also identified then we can see that some distance downstream of the body
we can visualize a regular pattern of two rows of vertices.
With respect to this slide it is very much clear; a sphere is there and then flow takes
place. Then, we can see that at some distance downstream of the body we can visualize a
regular pattern of two rows of vertices and these two rows of vertices move alternative
clockwise and anticlockwise. This is called Karman vortex streamline. We can see that
this is two dimensions Karman vortex. So, depending upon this it is plotted for this
sphere or we can also plot for the circular. Depending upon the various flow parameters
as well as the nature of the sphere or the cylinder integral smoothness then we can see
that a Karman vortex will be formed.
We can see that in regular pattern two rows of vertices which is as we can see that here
it starts and then Karman vortex will be formed with respect to the sphere or cylinder which
we consider. So, this is the after effect of the boundary layer separation and then
within the wake, this Karman vortex is formed. Like this boundary layer separation is very
important in the design of many of the automobiles like car, truck, bus and also this very important
in the case of aero plane, missile, space shuttle or the rocket designer depending upon
the shape. The separation may start or earlier or separation will be at the end. Accordingly,
various fluid flow parameters the velocity, the pressure etc., will be changing. It is
very important to analyse where the boundary layer separation starts and how the wake formation
takes place and its properties. Now, we will just discuss some of the examples.
Boundary layer separation: First case is divergent duct or diffuser. As we have seen earlier,
the flow is coming like this and then there is a divergent duct; it is divergent like
this or as in the case of diffuser you can see that here the boundary layer is formed
and then the separation starts; then, as the angle increases the probability of boundary
layer separation also increases depending upon the case. First example is here: you
can see the divergent duct or diffuser; the boundary layer depending upon the angle, if
the angle is more then we can see the probability of boundary separation also increases. This
is the divergent duct or diffuser where boundary layer separation takes place.
The Second case is a tee junction, this is another example. You can see that flow come
this direction and then it will be going through the direction of B as well as in the direction
of C. Here, the pressures at B and C are higher than at A depending upon the case and there
exist two adverse pressure gradients with respect to A. You can see that in this case
there is a possibility of two sounds of separation as far as the boundary layer is concerned.
You can see that this boundary layer will be formed here and then the separation and
here the boundary layer formation and the separation. There are possibilities of two
sounds of separation so this is the case of a tee junction.
As we have already seen in the third the case the flow is over a cylinder or a sphere. We
consider a ball. The free stream flow over a ball you can see that depending upon the
property this smoothness of the ball we can have a thick wake or a thin wake. In the case
of smooth ball, we can see that this separation starts much earlier. You can see the free
stream is coming like this and then here the laminar boundary layer will be somewhere and
then transition. Then, we can see that the boundary layer separation starts much earlier.
So that at this location the boundary layer separation starts and the separating streamline
is this one and then we can see there will be a thick wake. The reason is that here the
ball which we consider is smooth. We will be having a thick wake as drawn in this slide.
As I mentioned this depends upon the separation as well as the wake formation depends upon
the smoothness of this surface, also over which the flow takes place the same ball which
we consider in the last slide. If the ball is rough or the sphere is rough then we can
see that the boundary layer separation starts later stage and then there will be a thin
wake. So, this shows here the free stream velocity and then laminar boundary layer and
then transition. You can see that there will be a turbulent boundary layer and here due
to the roughness of the ball I for example, in the case of a golf ball or various other
kinds of ball where the surface is rough you can see that separation is much after. Here,
in this location the separation starts and then we can see that this is the separating
streamline. In this case, the separation is much. Here, in this smooth ball case where
the separation is much earlier and here the boundary layer separation is much lighter
so that we will be having a thin wake.
That is why I mentioned, we have to see that not only the flow properties and the fluid
properties but depending upon the case whether the surface is smooth or surface is rough
again the boundary layer characteristics will be different. Then this separation also starts
earlier or later. Also the wake formation will can be thin or thick depending upon the
case. So this is the fourth case where the flow over a rough ball and then the separation
with respect to the rough ball.
Then, the fifth case is the flow past airfoil. This airfoil we can use to see how the separations
take place by placing different angles. Here, we can see that in this slide the flow past
an airfoil. If you place it, the free stream is coming like this and if this is the airfoil
and then you can see that the angle of attack is 0. You can see that if the airfoil is perfectly
made in such that we can have a structure where there is no separation takes place.
You can see that the flow is going like this so there is no boundary layer separation.
This is the case for zero angle of attack but the same airfoil if you just slightly
tilt so that some angle is formed. Then, we can see that for example, 5 degree of angle
of attack, then you can see that the boundary layer will be totally of different nature
and then the separation flow separation also takes place.
Here, in the second figure we can see that these figures are taken from fluid mechanics
by Munson. You can see that in second figure this separation point is here and then you
can see that even there is a possibility of small make formation even for this airfoil
equation. So, the same airfoil if you put it with zero angle of attack then we can see
that no separation take place but if the angle of attack is 5 degree then they will be separation
and so depending upon the angle of attack again the location of separation changes and
also the boundary layer characteristics change. So, this case shows how the flow separation
and then this separation of stream line behave with respect to the placement of the body.
So, this is the fifth case.
Now, we have seen that the flow separation, the boundary layer separation takes place
and it is very important to analyse from where the separation starts and what are the various
characteristics. With respect to the fluid flow parameters it is very important in the
analysis and the design of various bodies wherever this boundary layer is forming and
then separation can take place. So, before discussing further the various applications
of the boundary layer separation. Here, we can see that a figure with respect to for
which here I have shown the variation of the velocity and the pressure. This figure is
taken from the book of fluid mechanics by A.K. Jain. Let us consider the free stream
flow over a curved surface like this. So, this is the curved surface and the free stream
is coming like this. You can see that with respect to the surface a boundary layer will
be formed. Initially, at this location approximately near to this point d consider if you plot
the velocity will be going like this and then we can see that this region of the where the
boundary layer is formed; a special distribution, if you plot you we see that dp by dx will
be negative of dp by dx will be less than 0.
At this location, if you plot the velocity radiant del u by del y at y is equal to 0
we can see that radiant will be greater than 0 and if you take the second derivative of
the velocity you can see that del square u by del y square will be less than 0 at this
location. Then, you can see that due to the curvature of the surface here a separation
occur at this location S. So before that for example at A we can identify a section where
the pressure will be minimum, appear the pressure p is minimum. Then, you can see that at this
location A if you take the velocity gradient you can see that the velocity will be deforming
like this which is keep on reducing like this case at A. At this location, you can see that
the gradient of the velocity del u by del y will be greater than 0 and del square u
by del y square will be almost 0 since this pressure is p minimum. Here, at this location
it will be del square u by del y square will be 0. So, that is at this section at A.
Now, if you consider a section at S where the separation already starts so you can see
that the separation stream line is here; the velocity plot is here. The velocity will be
changing like this and if you plot the velocity gradient you can see that del u by del y at
y is equal to 0 will be 0 near A. So this location near A or at S will be the velocity
gradient which will be zero at this location and del square u by del y square will be also
0 at this location. Then the flow separation takes place and you can see that the pressure
dp by dx is from here onwards after this stream minimum the pressure minimum, the pressure
gradient increases dp by dx will be greater than 0 like this. Then, near E wherever after
the separation takes place we can see that del u by del y at y is equal to 0 will be
less than 0 and del square u by del y square will be greater than 0.
You can see that the velocity if you plot you can see that reversal velocity takes place
after the separation of the stream line or the boundary layer separation we can see that
the velocity will be of the state. Then, here a wake will be formed. So, here with respect
to this figure we have analysed with respect to the flow over a curved surface how the
velocity changes with respect to the boundary layer separation; from starting and after
the separation takes place. Initially, with respect to the boundary layer the velocity
pattern is different and where the stream separation starts, the boundary layer separation
start the velocity parameter the velocity behavior is different with respect to section.
After the separation, you can see that reversal of the flow direction will change in the direction
of the velocity and then that is why we say wake is formed. Then we have also seen how
the pressure is changing. Initially, the pressure gradient is less than 0 and then there is
a location where the pressure is minimum and then the pressure gradient is dp by dx is
greater than 0 after the separations starts. So, this slide shows how the boundary layer
separations with respect to pressure distribution how the parameters like velocity in the gradient
of velocity and the second derivative of the velocity del square u by del y square is changing.
So this way we can analyse various structures of various bran bodies surfaces, how the boundary
layer separation takes place and how the pressure changes and how the velocity gradient changes.
Now, before going to the other topics like drag let us see what the application of these
separations.
We have seen with respect to the shape of the body, with respect to whether it is blend
body or with respect to surplus cylinder or whether it is a sphere or a curve surface
the boundary layer can separate at various locations and then wake formation takes place.
So, I already mentioned there is large number of applications of this boundary layer separation
especially, in the design of aero planes and then space shuttle and also the design of
automobiles.
So, some of the applications of separation are listed here. The potential flow regions
with respect to the boundary layer separation you can see that the potential flow region
does not recombine at the rear end of the body. As we have already seen various cases,
this is because of the existence of the vortex field. We have seen that after the separation
takes place there is possibility within the wake, there is a possibility of vortex field
so that the potential flow which does not recombine at the rear end of the body. Hence,
pressure at the rear end of the body is considerably lower than that is in the free flow region.
The pressure change at the rear end of the body is considerably lower than that is in
the free flow region. After the boundary layer separation and then is the wake formation
or due to the existence of the vortex field what happens after the separation take place.
This principle we can utilize in the design of automobiles and also in the design of various
flying machines like aero planes and other equipments. This is some of the important
applications with respect to the separation of the boundary layer. Here, now in this slide
as I mentioned there are number of applications of this boundary layer formation and its separations.
So you can see that.
The boundary layer separation after its identification of its boundary layer formation separation
we can see that this is very much used especially in the design of vehicles in automobile engineering.
Hence, see that the shape of the cars also change in last few decades since we can see
that wherever the separation for example, in this case here this car is concerned; the
shape of car flow with respect to the boundary layer formation and the separation is plotted
here. So, you can see that at this location the separation starts but here you can see
this car is considered we can see at the rear end separation starts. As we have already
seen in the case of airfoil, if the airfoil is placed in such that angle of attack is
0 so that there can be a condition where there is no separation so that the flow is much
smoother.
Like this in the automobile industry also we can utilize this principle. So that there
will be less resistance by driving the car or while driving the vehicle there will be
of course, wind effect will be there on the vehicle. So, the design of the surface of
the vehicle is such that the separation is taking at the rear end and then the flow resistance
will be much lesser so that we can design, we can see that the shape of the car, how
the shape of the cars are evolved say, starting earlier we can see that in 1980’s and 1950’s
the design was like this and now you can see the latest design of the vehicles you can
see here like this. There will be especially for the case of sports car you can see that
the shape is designed such a way that the separation takes place at the rear end. So
that there will be very less resistance to the moment with respect to the wind effect
on the vehicle.
Similarly, the design of buses, trucks and all these automobiles we can do in such a
way that in the resistance with respect to the wind effect is much lesser on the vehicle
which we consider and then separation we can; if it is at the rear end it will be much better
compared to the case if you say with respect to the vehicle. Here, the study of the boundary
layer separation and its analysis helps in the automobile designing the shape so that
there will be lesser assistance with respect to the wind effect on the automobile.
Like this in a very similar way, when we design the shape of the aero plane its wings and
its various components we can utilize this boundary layer separation theories so that
we can design the aero plane wings and other component such a way that there will be minimum
resistance with respect to the wing on the aero plane. Like this, there are number of
applications we can find out this boundary layer formation and the separation of the
boundary layer. So, further we have seen how the boundary layer forms and then what are
the govern equations and then we have also seen the boundary layer separation and its
applications
Now, another important thing with respect to the external flows are concerned, say flow
over a body or flow over a surface two important parameters are drag and lift. So, next topic
which we will discuss is drag and lift effect. You can see that if you consider the flow
over an airfoil like this we were discussing earlier. You can see the wind of the free
stream velocity is coming and then you can see that there will be two important effects
with respect to the flow over the body which is immersed in a fluid flow or the body which
is moving through a fluid. There are mainly two effects: one is called drag effect and
second one is called lift effect.
With respect to this figure, if we can see that drag is in this direction of the flow
and lift is normal to that. So, drag is the force exerted by the fluid on a body placed
within the stream along the parallel direction of the flow. So, drag is defined as the force
exacted as I mentioned here we are consigned with the external flows where the body is
immersed and fluid flow or the body is moving within the fluid. So, drag is defined as the
force exacted by the fluid on a body placed within the stream along the parallel direction
of the flow; the flow is taking place this direction so the body is the moving body or
the body placed stationery, the body is with respect to the direction of flow what is the
force acting on the body? So that force is called the drag.
This is very important aspect when we consider many design of aero plane, design of some
submarines and many other types of equipment. The force exerted along the direction normal
to the flow is known as lift. So, here the lift is indicated like this. What is the force
acting along the direction normal to the flow is known as the lift. Two important aspects:
one is the drag and the second one is the lift. As far as the external flow circlets
or flow over a surface is concerned or the moment of a body within a fluid is concerned,
two important aspects which we have to consider are drag and lift. So it is all defined here
the drag and lift with respect to the slide here
We will be discussing in details about this drag and lift effect. Now, some of the examples
as I mentioned in this case the submarines are moving through the sea. We can see that
submarines are moving then it is moving through the sea water. You can see there will be of
course with respect to movement of the submarines there will be drag effect which we have already
defined and also there will be lift effect. A sea bird moving through a fluid you can
see that how it is already plot the stream line and how the moving boundary layer we
can see how it forms and then there will be definitely a drag effect on the body surface
of the bird; also, there will be a lift effect. We can visualize this person swimming in the
swimming pool you can see that with respect to the direction of the swimming there will
be drag force and then also and there will be lift force. We can have a number of examples
with respect to the force exerted by wind on a flying aircraft. As I mentioned there
will be the drag and lift are to important force which we have to consider and we have
to critically analyse with respect to the flying aircraft. How much drag will be there?
How much will be the lift? These are two very important parameters which we have to consider
and also in the design of submarine and also many other also the force exerted on a free
falling body like in the sedimentation process in server we have to consider the drag and
lift effect. Some of the examples are shown here with respect to this slide.
Now, further to explain this drag and lift, let us consider an aircraft like this. We
can see that here in this slide this aircraft is moving this direction. So you can see there
will be drag and lift effect on this aircraft; so horizontal plane is like this and vertical
plane is here. With the direction of the aircraft, the aircraft is moving this direction so with
respect to that wind will be coming on the draft. Then, you can see their drag will be
in this direction and then lift will be normal to that; lift will be in this direction and
then of course the weight of the aircraft will also be there which we have to consider.
So, weight of the aircraft w and then drag force d and then due to force in this direction.
So, these are the forces which we have to consider with respect to the flying of this
aircraft. It is the drag and lift effects are marked here in this figure. Now, we have
seen the definition of drag and lift and then we have seen various examples. We have to
critically analyse this drag effect and lift effect of the bodies so especially when we
consider the external flows. We have already defined the drag in the direction of the fluid
flow or the direction of the moment of the body or the vehicle or the body which we consider
lift is normal to that. With respect to this slide here we can identify the drag and the
forces.
You can see that generally the drag and lift forces mainly due to the pressure, the normal
stress variation of the pressure variation and then with respect to the shear stress
acting on the surface. If you consider a small element within a body affected by drag and
lift force as shown in this slide, we can see that this is the body which we consider
and then the pressure is p and if you consider an element variant da then you can see that
the pressure will be p into da and with respect to this angle is theta. We can see that this
direction will be the pressure force p into da sin theta and normal direction it will
be p into da cos theta Similarly if you consider as a shear stress,
with respect to viscous fluid a shear stress will be there on the body. If you consider
the shear stress, you can see with respect to this angle theta this direction shear force
will be tau0 dA cos theta and normal to that a flow direction the sheer force will be tau0
dA sin theta, where tau0 is the shear stress acting on this elemental area. With respect
to this the pressure or the normal stress and the shear stress we can derive the expression
for the drag and lift.
With respect to this figure here we define on that element area which we consider the
lift force is equal to d FL is equal to tau0 dA sin theta minus pdA cos theta so this pdA
cos theta in this direction and tau0 dA sin theta in the opposite direction. That is why
we are considering with negative sign here. The lift force on the element layer is equal
to tau0 dA sin theta minus pdA cos theta is written in this equation number 1. And then
the next the drag force is concerned. You can see that both this force the pdA sin theta
and the with respect to the shear stress both are in the same direction. We can write dFD-
the drag force on the elemental area, dFD is equal to pdA sin theta plus tau0 dA cos
theta as in equation number 2. So these are the equations which we consider for the lift
force and the drag force for which we consider an elementary area
Then, we consider the normal stress or the pressure force and then the shear stress or
the sheer force. Then, we have the form with respect to the direction. With respect to
the angle theta, we go to equations for the lift force on the elemental area and for the
drag force on the element area as shown in equation number 1 and 2. To find out the total
drag force or the area which we consider, we can just integrate with respect to the
area for the two dimension case which we consider. Hence, the drag force is consigned. We can
just integrate. So, this equation number 2 we can integrate. In this case, drag force
is fD is equal to integral pdA sin theta plus integral tau0 dA cos theta with respect to
area A. So, this expression gives the equation for the drag force for the body which we consider.
Similar way the lift force is consigned; we can now say the element area dFL we have seen
with forces on the element area and equation number 1.
We can integrate this expression equation number 1. So that we get a lift forces FL
is equal to integral over the area A tau0 dA sin theta minus integral pdA cos theta
over the area A as in equation number 4. This equation gives the lift force for the body
which we consider. Equation number 3 and 4 gives the equations for the drag force and
then life force for the case which we consider. Now, regarding the life force is concerned
you can see that generally for viscous fluid flow shear stress component of the lift force
maybe neglected and as it is small compared to the pressure force depending upon the case
the exact equation is given by equation number 3 and 4 for drag force and lift force. Depending
up on the case, say, maybe in some cases, the shear stress component we can neglect
and some cases we can neglect the pressure force depending upon the problem. Generally,
if you neglect the shear stress component as for lift force is consigned then we get
this lift force FL is equal to integral f pdA cos theta is minus since we consider this
equation here so this is the lift force.
As I mentioned how we place the body this drag force and lift force effect can be different.
For example, if I consider this as a body, let me consider the plate like this. Now,
I am placing this plate here so this is the plate and then you can see that the fluid
flow is coming like this. Then, we can see as far as the drag force is concerned you
can see that the shear stress effect will be much higher than the pressure force.
In this case, if the board is placed like this we have to consider the shear stress
component will be much larger. So, the drag force effect on this will be mainly with respect
to the shearing effect. If I place the same body in the other direction we can see that
there will be a pressure difference between this and other side and then you can see that
we have to consider the major component in this will be the pressure force. So, accordingly
the equation which we consider, we have to see that whether the pressure drag is much
higher than the drag to the shear effect is much higher.
So this drag condition with respect to the placement of the body with respect to the
body shape is some of the effects are plotted here. If you consider this flat plate which
we consider like this if you put in this location with respect to free stream then you can see
that large vehicle will be formed and here the pressure effect will be much higher. This
is in the case of a cut plate and then as far as sphere is concerned here the boundary
layer formation and wake formation are also shown and then correspondingly the drag effect
will be vary.
So if the sales sphere is put in with fairing like this so that we can see that union boundary
layer separation changes. Then, instead of the pressure effect it will be more, the shear
effect will be much higher and then we have to consider the equation correspondingly and
the same sphere is inside housing like this we can see that here then we can see the boundary
layer expression is at the end. The drag effect will be mainly with respect to the shearing
effect. Like this the drag effect we have to depend upon how you orient the body or
how the body is moving. With respect to that we have to consider whether the pressure effect
in this equation number 3 whether it is major component this pressure effect of the sheer
force or shear stress component accordingly. Similarly, in the case due to force also we
have to consider. Accordingly, now this drag is concerned. We have already seen the major
components are due to the pressure and the pressure variation and then with respect to
shear stress or the viscous effect.
Accordingly, we can classify the drag: first one is the pressure drag. The pressure drag
is caused by the pressure difference between the front and rear of a body. As I mentioned
if the body is placed across the free stream velocity then a pressure drag is created and
the pressure drag is caused by the pressure difference between the front and rear of the
body. Here, you can see a pressure drag is at the rear end is created. In equation 3,
accordingly, the first integral term is called the pressure drag. So, FDP is equal to integral
p dA sin theta. This depends on the form or shape of the object as we are already seen
here. Hence, sometimes we call this form drag. Depending
upon the shape of the body the drag effect is changing. According to the shape this can
be varying. So, that is what is obvious, according the shape the drag effect will be defined.
This is some literature; this special drag is also called as form drag; this dominates
in case of flow over the bluff body. As we can see here this is the bluff body; pressure
drag is dominating. In the case of this here this and here also the pressure drag will
be much but when it is putting this way pressure drag reduces and the shear effect will be
much larger. So, the pressure drag dominates in case of flow over a bluff body.
Another example is placed here. A person is in the swimming pool. Pressure difference
between the front and rear sides, therefore pressure drag exists. So between this side
and the rear side, a pressure drag we have to consider. Then, the second one is the friction
drag. As I mentioned the drag can be mainly due to pressure drag and friction effect or
the viscous effect. So, this friction drag exists due to the viscous effects of the fluid
and second integral term is called the friction drag or the shear drag.
So, FDF is equal to integral tau0 dA cos theta over the area A. This depends on the extent
and character of the boundary layer as we have seen here in the last figure. Here, the
friction drag is much higher; it depends on the extent and character of the boundary layer.
It is sometimes called as viscous drag. The entire friction drag is created within the
boundary layer. Like this we can have the pressure drag and also we can have the friction
drag.
So, here this is an example. For flow past an airfoil, the shear stress acting along
the line of the body causes the friction drag. This shows the skin friction drag. Here, the
friction effect will be or the viscous effect will be much larger and then we have the skin
friction drag. Like this, depending upon the cases we have to consider the pressure drag
or the friction drag. The next lecture, we will be discussing more about the drag and
its effects and then with respect to the laminar boundary layer or with respect to the turbulent
case, how the drag will be? How we will calculate the drag and how various parameters we have
to consider with respect to the drag on the boundary will be discussed in the next lecture.