Welcome back to the video course on fluid
mechanics. The last lecture we were discussing
about the drag and lift forces coming on the
immersed bodies in fluids. So we have seen
the expression for the drag force and lift
force and various situations where the drag
and lift acts on the bodies then how it can
be calculated. Also we have seen that, depending
upon the shape of the body, depending upon
the profile of the body and then how we place
it just like in the case airfoil, we have
seen the drag changes and also lift forces
changes.
So also we have seen mainly, that we have
the pressure drag and the friction drag with
respect to the viscous effects of the fluid.
Now before further discussing on how to find
out the drag coefficient for laminar turbulent,
with respect to the boundary layers, we will
just see initially, we will discuss how this
we can reduce the drag effects on bodies,
the way the moving bodies or stationary bodies
in fluid. So first the topic which we are
going to discuss is reduction measures of
pressure and friction drag.
Here we can see in this slide here, we have
an aircraft and then while it is flying we
can see that various forces acting or the
drag in this direction. Then there will be
a trust in this direction and then the weight
of the aircraft and then lift. So as we have
already discussed the drag is a mechanical
force generated by a solid object moving through
a fluid as in the case of an aircraft. The
main drag effects are here, the pressure drag
and friction drag. So what are the various
possibilities depending upon whether it is
blend body or it is streamline body? Depending
upon the shape of the body and profile the
drag effect will be different. How we can
reduce this pressure and friction drag? That
is what we want to discuss here.
As I mentioned earlier, if you consider an
airfoil as in this slide, airfoil is placed
just within. If the free stream flow is coming
like this and the airfoil is placed in the
direction of the free stream, then we can
see that the drag effect due to the pressure
drag effect will be much lower and only friction
drag will be there, but when we slightly tilt
the airfoil in such a way that for example
the angle of attack is 5% as in this figure.
Then we can see that the drag effect, the
friction drag effect. The drag effect compared
to the pressure drag will be more. So depending
upon the angle attack the second figure again,
the angle of attack is 40 degree.
That we can see again the flow phenomena surrounding
the airfoil is changing. What we can observe
here with respect to this slide is that, depending
upon the placement of the location or how
the profile of the body which is immersed
in the fluid flow. Accordingly, the drag effect
whether it is pressure drag or the friction
drag it changes, its nature changes. The question
here is how we can reduce the pressure and
friction drag depending upon the shape or
whether depending upon the placement of this
say as in the case of this airfoil.
Now the pressure drag we assumed the drag
is concerned either it can be mainly it can
be pressure drag or the friction drag. The
pressure drag can be reduced. Two important
steps which we can use it here, one is making
the body as streamlined. The pressure drag
effect can be reduced by making the body as
streamlined and the second possibilities that
we can reduce the length of the component
wherever applicable.
So example, the wing span of an aircraft,
we can see that here is the aircraft and then
the wing span. Depending upon how you place
the wing and then how it is oriented and if
you reduce the length component then the pressure
drag can be reduced. So like this various
measures we can adopt to reduce the pressure
drag. Second kind of drag is the friction
drag we can reduce the friction drag by making
the surface smooth.
One of the step which we can adopt to reduce
the friction drag is the surface is rough
makes it more smooth and also the flow should
be laminar for the greater portion of the
body. The flow is laminar then the drag friction
drag effect to be much lesser compared to
the turbulent flow effects. Flow should be
laminar for the later portion of the body.
So these are the two measures possible with
respect to the friction drag to reduce the
friction drag.
Now we have seen with respect to the pressure
drag if you make the body streamline, then
the pressure drag is reduced. So that we have
already seen here, make the body streamline.
We will discuss here briefly about the streamline
body. A body is said to be streamlined if
its shape is such that, a separation occurs
towards the rearmost part of the body.
We call a body as streamlined if the shape
of the body is such way that the boundary
layer separation occurs towards the rearmost
part of the body. In that case, we can see
that the wake formation is very small. It
offers least possible resistance to the flow
of air water or other fluids where the body
is immersed. We can see the airfoil which
we have already discussed earlier, airfoil
or the fish. You can see that the shape of
the fish is in such a way that it is we can
say that the body is streamlined, so that
the wake formation is very small and then
the shape is such a way that when the fish
moves the separation occurs towards the rearmost
part of the body. Here we can see that whenever
the airfoil is placed like this, we can see
the flow is coming like this and then the
wake formation is only due to the placement
in such a way that the body is streamlined.
Here the wake is very small and then the drag
effect will be much less especially the pressure
drag effect will be much smaller. Similarly,
the fish in the case of a fish also we have
shape of the fish is in such way that we can
say that the wake formation is very small
and the separation the boundary layer separation
occurs towards the rearmost part of the body
of the fish. In the same airfoil if you place
it in other way as we have seen the airfoil
is concerned, it is how you are placing it
also important, how is the angle of attack
accordingly the drag effect changes? So this
is about the streamline body and then the
next kind of body which we have seen is so
called Bluff body.
In the case Bluff body say as in the case
of a sphere or the case of a circular cylinder
we can see that the body shape is such that
separation occurs much ahead of the rearmost
part of the body. You can see that this is
the sphere here, the separation occurs much
ahead of the rearmost part of the body. That
is why it is called as a bluff body and then
you can see the wake formation is large as
in the case of a sphere or a cylinder and
object as shown in this figure. The body is
as far as with respect to the drag is concerned,
we can say to reduce the drag effect, we consider
that we should have a streamline body. So
either the streamline body or bluff body is
concerned, you can see the wake flow formation
will be much larger and then the pressure
drag effect will be much more, so if you consider
the drag on streamline and bluff body.
So here you can see that, if you consider
circular cylinder like this is the blunt body
or the bluff body. So you can see that the
separated flow is here.Here the pressure drag
in the case of a bluff body, the pressure
drag will be much higher. So here in this
figure the friction drag is shown as red color
and the pressure drag is indicated as the
yellow color.
We can see that in the case of bluff body
like in the case of a circular cylinder the
pressure drag will be much more and then friction
drag or skin friction drag will be much smaller
and also the total drag will be much higher
compare to the streamline body. So the case
of an airfoil which is a streamline body,
we can see that the skin friction drag. Since
here we can see that skin friction drag will
be more compared to the pressure drag. The
pressure drag here is also indicate as yellow
color and the skin friction drag is shown
as red.
So you can see that the relative drag force
will be much smaller for a streamline body
as shown in this figure. So here the friction
drag is lower than the pressure drag in the
case of bluff body. So bluff body, the friction
drag is smaller and pressure drag is more.
Then in the case of streamline body the friction
drag is higher and then the pressure drag
is much smaller. These are the effect with
respect to whether the body is a bluff body
or whether the body is a streamline body.
So accordingly, we can say that the drag effect
is the pressure drag or the friction drag
which one is more in the case whether it is
a bluff body or it is a streamline body. So
the drag effect is concerned, this special
drag and the friction drag are the two most
important effect, as far as the drag is concerned
other than this pressure drag and the friction
drag. There can also be some other kinds of
drag. Depending upon say for example, if you
consider this slide, the movement of a boat,
here a boat is going through the water where
also the effects are there, and then there
are so many other kinds of drags are also
possible. Except pressure and friction drag
there are different kinds of drag. So these
are first one is called induced drag and second
one is called profile drag, and third one
is called heel drag, and fourth one is called
residuary drag, fifth one is called added
wave drag.
If you consider the moment of a boat then
we can see that many of this drag effect will
be coming over the boat, which is moving over
the water like this. But most of the time
the pressure drag and the friction drag are
the most important. The important components
of the drag force but others are also possible
like induced or profile drag or the heel drag
like that. Let us see the definition of this
induced or these various kinds of other drags.
So now the induced drag, actually it is the
drag force induced in the body due to the
lift forces acting on it. We have seen when
body is immersed in the fluid either in the
moving fluid or depending upon whether the
body is moving or the fluid it is moving.
So with respect to the immersed body we have
the drag and as well as lift. So this lift
components also may be induced some kinds
of drag that is called induced drag.
Here we can see that if you consider an airfoil
like in this figure here, then we can see
that the drag force will be in this direction
and then the lift force will be the normal
direction of flow direction. Then we can see
that since the airfoil is placed with respect
to some angle of the free stream flow then
we can see that an induced drag will be coming
here. So this is due to the lift force. If
there is no lift force, when there is no induced
drag so as in the case of an airfoil you can
see that when the angle of attack increases
the induced drag also increases. So actually
this is this induced drag is concerns this
depends upon how we are placing the body or
how it is how its profile is there and how
you say place the body with respect to the
flow direction of the fluid. This is so called
induced drag and then next one is the profile
drag. This profile drag is developed say for
example if you consider the airfoil if it
is infinitely long then the flow it to be
two dimensions, so this is coming from the
profile of the of the body. That is why it
is called profile drag. This depends on the
shape or profile of the shape, if you consider
the airfoil it depends upon the shape and
how you say place it or the profile. So that
is called profile drag and then third kind
of drag here is called heel drag.
This heel drag, this is due to the change
in drag sometimes says negative. This occurs
due to the change in the hull shape of a boat
below the surface. We have seen this boat
moment here, so if the hull shape depending
upon the hull shape of the boat then you can
see that there is possibility of the say there
is just change in the drag effect and its
occurs due to the shape which shape you gives
say in the case of for a boat below the surface
how it is provided and then how it is moving.
So that is called the heel drag and then the
next kind of drag is called residuary drag.
This is actually the combination of the, we
have seen that when the boat is moving, there
is also waves will be there on the water surface
and then say how this waves with respect to
moment of the boat. So this residuary drag
is coming, it is the actually the combination
of the wave making and viscous pressure drag
or the form drag. This is the resistance caused
by the creation of bow and stern waves; we
can see that here in this figure. When this
boat is moving how it behaves and how the
waves are coming so with respect to the drag
formation that drag is called residuary drag
and also then added wave drag this is the
additional drag caused by the oncoming waves.
These are some of the other kinds of drag
than the pressure drag or the friction drag
which be consider, which are the most important
kinds of drag or the fluid immersed in a body
or the body moving through fluid. Now we have
seen various kind of drag here, let us see
how this drag effect will be there with respect
to say a boat moving through say through the
sea or a lake, what kind of say how this various
kinds of drag effect will be coming over the
boat. Here this slide show say with respect
to the speed of the boat, the drag is also
plotted here. So you can see that this figure
shows the contribution of different drag forces.
Here this line shows the total drag, you can
see that this friction drag is which is also
predominant here, this is the friction drag.
And then we have this residuary drag and then
this heel drag is very small and then we have
this blue color here is this color here induced
drag and then added wave drag, so that the
total drag is this line. This line is the
total drag and then also you can see that
this depends upon the speed of the boat with
respect the speed of the boat is increasing
you can see that the drag effect is also increasing
as plotted here.
The various kinds of drag here we have seen
and then say with respect to the movement
of boat how the drag variations for with respect
to various components, that is what is explained
here in this slide. Now we have seen various
kinds of drag and how the drag can be reduced
with respect to shape or with respect to the
placement of the of the body in the fluid.
Now we will analyze how we can calculate,
we can find this drag coefficient or how to
find out the drag and lift with respect to
the fluid flow and then with respect to the
body which we consider. As we have already
defined the drag force say we have already
seen the equation for drag force and lift
force.
If you analyze the say whenever we consider
the a body immersed in fluid, with respect
to the drag effect and lift effect, if you
consider a dimension analysis and say we can
see that with respect to the various parameters
which have to be considered while dealing
with a drag force are the length of the body,
the dimensions, length, breath and other see
the depth or dimensions and then rho is the
density of the fluid and mu is the coefficient
of dynamics viscosity and E is the velocity
of the velocity force or with respect to the
fluid and g is the acceleration due to gravity
and then the free stream velocity u infinity.
So these are the important parameters we have
to consider while considering, while analyzing
the drag force and the lift force. Similarly,
the drag force can be considered as a function
of this length L of the density rho.
And the coefficient viscosity mu and the elasticity
E and access due to gravity and the free stream
velocity. Similarly, the lift force also can
be considered as s function of these parameters
as shown here FL is equal to f2 as a function
of L, rho, mu, E, g and u infinitive. So we
can do and analysis with respect to these
various parameters. Using the dimensional
analysis we can show that especially while
doing experiment if you are doing the dimension
analysis through dimension analysis we can
show that this drag force and then lift force
say if you consider FD the drag force FD is
divide by rho L square u infinitive square
where u infinitive is the free stream velocity,
so that is equal to as a function of say f3
here rho u infinitive L by mu u infinitive
square by L g u infinitive by square root
of E by rho. Similarly, this lift force we
can write as FL divide by rho L square u infinitive
square is equal to as a function of f4 rho
u infinitive L by mu u infinitive square by
L g u infinitive by square root of E by rho.
So we can represent in the dimension analysis
like this with respect to the drag force and
then with respect to the lift force after
writing like this, we can see that, so we
can we have seen that this coefficient of
drag and coefficient of lift we can write
in terms of this. So the coefficient of drag
is equal to the drag force FD divide by half
rho u infinitive square into A where A is
the area of the body which we considered,
so whether where u infinitive is the free
stream velocity. Similarly, coefficient of
lift we can write as CL is equal to the lift
force FL divided by half rho u infinitive
square A. Now if you consider this with respect
to the analysis here the dimension analysis
which is explained here in this slide, we
can write this coefficient of drag it can
be put as a function of the Reynolds number,
Froude number and Mach number.
So you can see this component here rho infinitive
L by mu represent the Reynolds number and
the second term u infinitive square L g representing
the Froude number and this u infinitive by
square root of E by rho represent the Mach
number. So we can represent the coefficient
of drag and the coefficient of lift in terms
of the as a function of the Reynolds number,
Froude number and the Mach number. Generally,
when we consider the drag and lift effect
and then when we are finding the coefficient
of drag and coefficient of lift we can show
that, this Froude number and Mach number that
term or that effect will be much smaller,
so we can neglect it and generally we can
represent this CD coefficient of drag as a
function of Reynolds number and also coefficient
of lift as function of Reynolds number by
neglecting this Froude number and Mach number
which we considered here in this analysis.
This way we can represent the coefficient
of drag and coefficient of lift.
Now we will analyze the drag effect say initially
with respect to the small Reynolds number
and then we will see the laminar flow with
especially with respect to a flat plate, we
see the drag force for the coefficient of
drag and its various parameters, coefficient
of drag and other parameters for flat plate.
Initially let us discuss about the drag at
small Reynolds number, so this strokes conducted
number of experiments with respect to the
drag effect and then say especially in the
case of Reynolds number is small Reynolds
number say especially when the Reynolds number
is less than 1, so this category of flow as
been classified as creeping flow which we
discussed earlier. Here you can see that the
drag is proportional to the, so through experiments
stocks shows that the drag is proportional
to the free stream velocity u infinitive.
If you consider as in this figure say if you
consider a sphere the creeping flow so wherever
the Reynolds number is very small so that
flow which is categorized as creeping flow
so we can see that here the free stream is
coming here and then if you consider sphere
like this the creeping flow over a sphere
say the radius R and if you consider an angle
theta like this, so here the number of experiments
conducted by stocks and then he has shown
that the deformation drag for a creeping flow
wherever the Reynolds number is very small
as we have seen here say Reynolds number less
than 1, there stocks showed that this deformation
drag FD is equal to 3 pi mu u infinitive into
d, where mu is the dynamic coefficient viscosity;
u infinitive is the free stream velocity;
and d is the diameter of the sphere, as shown
here this is the diameter of the sphere; and
u infinitive is the free stream velocity.
So for small Reynolds number considers the
creeping flow stokes showed that this deformation
drag FD is equal to 3 pi mu u infinitive into
d.
So this known as Stokes flow where this equation
is applicable that kind of flow is called
as stokes flow and as far as drag is concerned
stokes shows that generally this after the
total drag two third is the, say in the case
of a creeping flow say if we consider the
creeping flow over a sphere is shown that
two third is friction drag and one third is
generally pressure drag.
So this is the case where the Reynolds number
is small say generally less than 1, where
stokes stored this drag effect is concerned
two third will be generally the friction drag
and one third is the pressure drag and also
say if we consider this stokes flow around
a sphere then the coefficient of drag we can
show that the coefficient of drag is equal
to 24 mu divided by rho u infinitive d, d
is the diameter of sphere, so that is equal
to 24 by Re. This is valid for Reynolds number
less than .1 and when the Reynolds number
is between .1 to 1, we can show that this
coefficient drag will be 24 by Re into one
plus 3 by 16 into Re.
Where Re is the Reynolds number which we considered
and again say large number of experiments
conducted for different Reynolds number and
this coefficient of drag we can show that
it will be CD is equal to 24 by Re into 1
plus 3 by 16 Re to the power 1 by 3 for Reynolds
number up to 100. So this we can see that
the coefficient of drag is varying with respect
to the Reynolds number as shown in this slide.
So this creeping flow which we discussed earlier
also with respect to the stokes flow, these
kinds of analysis very important especially
in civil engineering and sedimentation and
silting problem is there in water and also
the settlement of dust in the atmosphere,
we can utilize this stokes flow and then with
respect to the we can calculate the coefficient
of drag and also the drag force coming on
the particle.
So that is the case where the Reynolds number
is so small where we consider the creeping
flow, we have seen the drag effect in the
coefficient of drag and then the drag force.
Now we will further analyze here the drag
effect coefficient of drag and various other
parameters with respect to the flow over a
flat plate. We have already seen this flow
over a flat plate case, when we consider the
flow over a flat plate that can be three conditions
first one is the boundary layer formation
is that is laminar boundary layer; and then
there is a transition states; and then there
can be turbulent zone. If you consider the
drag over a flat plate we can see that generally
there is no variation pressure in the flow
direction as we have discussed earlier so
that velocity gradient is constant.
So in this case we can see that here we have
the laminar zone and then there is transition
zone and here with respect to boundary layer
formation we have the turbulent zone. Now
with respect to these various cases, we will
determine the drag how to find out the drag
for a laminar case and then turbulent case
and also the transition case.
When we consider as we discussed for drag
over a flat plate boundary layer is fully
laminar. So generally it happens when the
length of the plate is short or initially
as we have seen in the previous slide initially
it can be laminar and then transition and
turbulent or say only laminar boundary layer
you if the length of the plate is very small
and then the velocity is also the free stream
velocity is small and then second case is
when the boundary layer is fully turbulent
so it develops after a certain distance along
the plate as we have seen here. So here finally
here the boundary layer is totally turbulent.
Then the third case is boundary layer in transition
from laminar to turbulent, so this is the
case where the transition zone here we can
see that this is a laminar here it is turbulent
so here is the transition zone. Here we want
to find out the drag force with respect to
say for the flow over a flat plate since this
is one of the generally used to case to analyze
various parameters, so here also to analyze
the drag force the coefficient of drag we
use the flow over a flat plate problem.
Now first, let us consider the laminar boundary
layer formation or for flow over a flat plate
in the flow is fully laminar boundary layer
and then how we can get the drag coefficient
and then the drag force. So with respect to
the flow over a flat plate for fully laminar
boundary layer Prandtl analyze this problem
by using the Prandtl’s boundary layer equations
which we have discussed earlier, so you can
see that here the this is the flat plate here
and then flow free stream velocity is coming
like this and then you have the boundary layer
formation.
So this is the boundary layer thickness and
here the u infinitive is the free stream velocity.
By considering the Prandtl’s boundary layer
equations from the law of similarity, we can
write say the velocity at any location u by
u infinitive is a function of y, x, nu and
u infinitive where nu is the dynamitic viscosity
here, so this is equal to F eta so where eta
we can represent as this y is in this direction
eta is equal to y by delta the location which
we consider where the velocity is u and the
location is at y.
So now with respect to this say from the boundary
layer flow analysis of unsteady motion Prandtl’s
analysis showed that this boundary layer is
proportional to square root of nu into t where
nu is the kinematics viscosity and t is the
time required for a fluid particle to travel
a distance x with velocity u infinitive. Here
you can see this figure earlier, so the flow
is taking place like this and then we have
this the fully laminar boundary layer, delta
the boundary layer thickness is proportional
to square root of nu into t where t is the
time required for a fluid particle to travel
distance x with velocity u infinitive the
free stream velocity.
So this we can write this is proportional
square root of nu X by u infinitive or we
can write delta is equal to X by square root
of Rx where this Rx is the is the Reynolds
number at that particular location where we
considered. So now in the previous slide we
have seen this y is represent as this eta
is equal to y by delta.
This is equal to y into square root of u infinitive
by nu into X, so with respect to this equation
here Prandtl’s analysis shown that eta is
equal to y into square root of u infinitive
by nu into X .Here if you introduce the stream
function which is defined as sie is equal
to integral u dy so that is equal to u infinitive
delta integral F eta d eta, so that is equal
to u infinitive substitute for delta we can
write u infinitive into square root of nu
x by u infinitive into function of F eta d
eta. So that we can write sie is equal to
stream function is equal to square root of
u infinitive into nu into X into f eta where
F eta is equal to integral f eta coming from
here f eta d eta.
Now we know that the velocity can be represent
as u is equal to del sie by del y, so that
we can write u is equal to del sie by del
eta into del eta by del y. This is equal to
u infinitive f dash eta. So velocity is represented
as at any location u is equal to u infinitive
f das eta and then the velocity gradient we
can write as del u by del y is equal to del
u by del eta into del eta by del y, so this
equal to u infinitive square root of u infinitive
by nu x into f double dash eta, so second
derivative. This f dash f eta, so here first
derivative here the second derivative or double
dash eta and then the shear stress on the
surface of the flat plate from the Newton’s
law we can write tow is equal to mu du by
dy at y is equal to 0, so that is equal to
mu u infinitive square root of u infinitive
by nu X f double dash at zero location.
So this is the shear stress on the surface
of the flat plate. And then by using this
Prandtl’s equations Blasius derived this
solution by the discussed steps and then the
Blasius solution for the boundary layer flow
analysis or flat plate at eta is equal to
0, he showed that f double dash 0 is equal
to 0.332, so that we can write say from the
velocity profile obtained by Blasius solution,
the boundary layer thickness delta is equal
to 5 into X by square root of Rx where Rx
is the Reynolds number at that particular
location which we consider at say where u
by u infinitive is equal to .992 with respect
to the definition. So that delta is obtained
as 5 into X by square root of Rx where Rx
is the Reynolds number and also here delta
star the displacement thickness is equal to
1.73 X Blasius showed that delta star is equal
to 1.73 X divided by square root of Rx.
Where Rx is the Reynolds number at that particular
location and the momentum thickness theta
is equal to .664 into X by square root of
Rx where Rx is the Reynolds number at that
location. Now from the equation number 1 here
is substitute in this equation number 1 we
can show that tow0 is equal to .332 mu u infinitive
square root of rho u infinitive by mu X so
that is equal to 0.332 rho u infinitive square
by square root of Rx where Rx is the local
Reynolds number represent as represented as
rho u infinitive into X by mu. So this way
by using the Prandtl’s equations Blasius
derived this expression for delta, delta star,
theta and then also expression for tow0 the
shear stress on the flat plate.
And then now we want to see with respect to
drag effect so the when we consider the local
friction drag coefficient with respect to
this figure here, local means at the particular
location so here Cf is equal to the local
friction drag coefficient Cf is equal to friction
drag divided by dynamic pressure drag so that
we can write Cf is equal to the friction drag
is even a tow0 into A by this dynamic pressure
drag is obtained half rho u infinitive square
into A, this is equal to tow0 by half rho
u infinitive square. So this he sort with
respect to the previous expressions here for
tow0 which is obtained here, so Blasius showed
that Cf is equal to 0.664 by square root of
Rx where Rx is the local Reynolds number.
So this Cf is the local friction drag coefficient
for the laminar boundary layer formation over
a flat plate. Now the friction drag say over
one side of the plate of length l per unit
width we can write as FDf is equal to integral
0 to L tow0 dx, so this is equal to integral
zero to L 0.332 divided by u infinitive square
root of u infinitive X by mu into rho u infinitive
square dx. So this is equal to 0.664 rho u
infinitive square into square root of nu L
by u infinitive, this is the friction drag
and then from this FDf the friction drag over
one side of the plate, so the average coefficient
of friction drag we can get from this expression
this Cf is equal to FDf by half rho u infinitive
square into L, so this is equal to .664 u
infinitive square, square root of nu L by
u infinitive half rho u infinitive square
into L, so that is equal to 1.328 divided
by square root of RL where RL is the Reynolds
number at the trailing edge of the drag plate.
Finally, here we got the coefficient of friction
drag for the plate Cf as Cf is equal to 1.328
divide by square root of RL. In the previous
space here what we considered here is the
local friction drag coefficient which we got
as .664 divide by square root of Rx, so here
we got as Cf is equal to 1.328 divide by square
root of RL where RL is the Reynolds number
at the end of the plate which we consider.
Now like this here and various parameters
have been derived for the coefficient of drag
whether it is local or at the end of the plate,
so with respect to this we have seen say by
starting from the Prandtl’s boundary layer
equations the parameters like delta, delta
star, and then momentum thickness etcetera
are derived by Blasius in the earlier slides
which we have discussed. Now we will discuss
say a small example here, so the example here
is say we have to calculate the friction drag
on a flat plate so here this a flat plate
problem. So calculate the friction drag on
a flat plate of 25 centimeter wide and 60
centimeter long placed longitudinally in a
stream of oil of relative density 0.945 and
kinematic viscosity .8 stokes flowing with
a free stream velocity of 3 meter per second
and also find the thickness of the boundary
layer and shear stress at the trailing edge.
So here the problem is this is the free stream
velocity we have free stream velocity of 3
meter per second and then the rate density
of the fluid is .945 and length of the plate
is 60 centimeter and width is 25 centimeter
and kinematic viscosity is .8 stokes.
This is the section of the plate, so for this
problem we have to find out the thickness
of the boundary layer and the shear stress
at the trailing edge. We have already seen
for these kinds of problem enough for the
we can first find out the at the trailing
edge what will be the Reynolds number, so
that we can see that with respect to the whether
the flow is laminar or not and then we can
use the equations which we have discussed
earlier to get the boundary layer thickness
and the shear stress.
Here for this problem the Reynolds number
at the trailing edge Rel is equal to uL by
nu, so 3 into .6 by .8 into 10 to the power
minus 4, since here .8 stokes is the kinematic
viscosity. So this is equal to 2.25 into 10
to the power 4, so Reynolds number at the
trailing edge of the plate is 2.25 into 10
to the power 4. So here we can see that 1
stoke is equal to 10 to the power minus 4
meter square per second, so here say we have
when we discussed the initially about the
boundary layer we have seen say the especially
for the case of a for flow over a flat plate
then the Reynolds number is less than 5 into
10 to the power 5 we are categorize the boundary
layer as laminar, so here we got 2.25 into
10 to the power 4 as a Reynolds number. So
you can see that the boundary layer is laminar
so we can use the equations which we have
seen earlier since that boundary layer is
laminar in nature, so the Blasius solution
we can use so delta is equal to 5 into x by
square root of Rx, that is the in the boundary
layer thickness equation E 1 by Blasius.
So at the trailing edge means at the end of
the plate, so here at this location, so that
means the boundary layer thickness here so
we can see that L is equal to .6 so delta
L is equal to 5 into .6 divide by and Reynolds
number square root of 2.25 into 10 to the
power 4 which is Reynolds number so we get
the boundary layer thickness at the trailing
edges 0.02 meter and now the second problem
here is the find out the shear stress at the
trailing edge.
So to find out the shear stress we have already
seen this equation here so this equation is
here towL is equal to half rho u infinitive
square Cf L so that is equal to half rho u
infinitive square so here this Cf L is seen
0.664 square root of Rel, so this is equal
to rho is .945 into 1000 into say, free stream
velocity is 3, so 3 square into .664 divided
by 2 into square root of 2.25 into 10 to the
power 4, so this will give the shear stress
at the trailing edge as 18.82 Newton per meter
square. And then the drag on one side of the
plate FD is equal to CDf into L into B into
rho infinitive square by 2 and then the CDf
that means the when we consider say here the
say total drag so we have seen the drag is
concerned the local drag coefficient is concerned
local drag coefficient as well as the full
drag coefficient as we have seen here the
average coefficient of friction drag CDf 1.328
root RL so this equation we can utilize here
so that CDf is equal to 1.328 divided by square
root of Rel so that is equal to 1.328 divide
by square root of 2.25 into 10 to the power
4 so this is CDf is equal to 8.85 into 10
to the power minus 3.
And then the force due to drag we can calculate
FD is equal to CDf into area L into B into
rho u infinitive square by 2, so this is equal
to 8.85 into 10 to the power minus 3 which
is the CDf into .6 L is .6 and B is .25 here
into .945 so here B is 25 centimeter so that
is why .25 into .945 into thousand into 3
square by 2, so this gives 5.695 Newton. So
total drag force on both sides of the plate
is equal to say here this calculation is for
one side both sides to be two times the drag
force calculated. So 2 into 5.695 is equal
to that gives 11.29 Newton. So like this we
can solve the drag force with respect to the
laminar boundary layer formation in the case
of a flat plate and now the next case is drag
over flat plate where the boundary layer is
turbulent.
So we have seen so now the laminar we have
seen now if the boundary layer is fully turbulent
then say the various equations here we consider
for turbulent boundary layer. Now here say
we have seen when we discuss the boundary
layer earlier, we have seen the momentum integral
equation, with respect to this for fully developed
turbulent boundary layer we can see over a
flat plate the momentum integral equation
maybe written in slightly different form as
d by dx of integral 0 to delta u into u infinitive
minus u dy that is equal to tow0 by rho as
in equation number 1.
Where u infinitive is the free stream velocity
and mu is the velocity at any location tow0
is the boundary shear stress and rho is the
density and then Prandtl’s experimentally
showed that for turbulent boundary layer we
can write this u by u infinitive is equal
to y by delta to the power 1 by 7 so this
is coming from the experimental observation
by Prandtl’s, so this is u by u infinitive
is equal to y by delta to the power 1 by 7
as in equation number 2 this is known as Blasius
one seventh power law, so this is through
experimental observations. And then also the
velocity distribution turbulent flow over
a flat plate say when the Reynolds number
is less than 10 to the power 7 through experiments
it was be shown that u by square root of tow0
by rho is equal to 8.74 into square root of
tow0 by rho into y by nu to the power 1 by
7.
And also here this say with respect to the
Blasius analysis and using the momentum integral
equations same with respect to the pipe flow
for turbulent flow in pipes it is shows that
tow0 is equal to 0.034 rho v square vd by
nu to the power minus on1e by 4 as in equation
number 3 and the average velocity for with
respect to pipe flow we can show that v is
equal to .817 the maximum velocity u max,
so here the maximum velocity of the free stream
velocity so that is equal to .817 u infinitive.
So by using this here we can show that tow0
is equal to zero .0233 rho u infinitive square
u infinitive delta by nu to the power minus
1 by 4 as in equation number 4.
So now if you use the previous equation, equation
number 1 and then equation number 2, equation
number 2 here and then equation number 4,
we can show that delta by x is equal to for
turbulent boundary layer delta by x is equal
to .379 divide by Rx to the power 1 by 5 as
in equation number 5. So this is to find out
the boundary layer at any location with respect
to the turbulent boundary layer formation
and now if you use this equation number 5
and 4 together, we can show that tow0 is equal
to .0295 rho u infinitive square Rx to the
power minus 1 by 5 as in equation number 6.
So now for turbulent boundary layer local
friction coefficient we can find out Cf is
equal to tow0 by half rho u infinitive square
as we have discussed earlier so this for turbulent
boundary layer the local friction coefficient
we can show that Cf is equal to .059 divided
by Rx to the power 1 by 5 where Rx is the
Reynolds number local Reynolds number as in
equation number 7.
In the friction drag per unit with for one
side of the plate is we can integrate FDf
is equal to integral zero to L tow0 dx so
this is equal to 0.0638 L rho u infinitive
square RL to the power minus 1 by 5 and then
and hence Cf is equal to FDf by half rho u
infinitive square L. This is we can show that
Cf is equal to 0.074 dived by RL to the power
1 by 5 as given equation number 8. So this
equation number 8 through experiment shows
that this equation number 8 is valid for Reynolds
number in the range of 5 into 10 to the power
5 and between 10 to the power 7 the Reynolds
number 5 into 10 to the power 5 to 10 to the
power 7 and when again Prandtl calculated
and also shown through experiment starts when
the Reynolds number is greater than 10 to
the power 7 he showed that this the friction
coefficient Cf is equal to 0.455 divided by
log 10 RL to the power 2.58.
So this is thoroughly seen by Prandtl’s
and verified through experiment as shown in
this equation 9. So this gives for turbulent
boundary layer this gives the various equation
friction coefficient and the coefficient say
the average friction drag coefficient for
the turbulent boundary layer. Next say we
will be discussing further on the drag force
and the drag coefficient in the transition
case and also further with respect to various
shapes and with respect to various body parts
how the drag coefficient changes will be discussing
in the next lecture.
