Welcome all of you for this third lectures
on fluid mechanics. So today we will cover
on flow analysis of very complex flow processes,
how you can solve very complex flow processes.
So that is what I will discuss today.
Considering that I will cover today systems
and control volumes; what is the difference
between system, what is difference between
control volume. Then I will give a very interesting
examples of a bird under the wind flow conditions.
Then we will talk about what type of flow
analysis techniques are available and how
we solve very complex flow problems using
these analysis techniques and then I will
talk about velocity field, the pressure field,
density field.
And the after that we will talk about very
interesting part is the streamline, the path
line and the streak lines which is very much
required for analyzing or visualizing a fluid
flow problems. Then again, I will talk about
these virtual fluid ball concept what we introduced
in the first class. So same things how we
can use it for very complex flow problems.
Considering that let me look at what so far
we have discussed.
One is very basic equations of Newton's laws
of viscosity which is establish a relationship
between the shear stress and the shear strain
rate and also we discussed about how these
the the mu the dynamic of viscosities depends
upon the pressure and the temperatures and
that relationships are different for the fluid
and for the gas part and after that we just
know some of the basic definitions about density,
specific gravity, specific weight and the
surface tension.
With this knowledge, let us go to the next
level that if I have a very complex problem,
fluid flow problems, how do you solve that
ones.
First let us talk what is the system, what
is the control volume. The system is a quantity
of matter or the region in a space chosen
for the study. For example I have considered
a 2 kg of gas which is having 1 meter cube
volumes. And if I heat this gas, if I give
a temperature to this gas, then what will
happen? This gas will be expanded.
So this is a system, that means we have a
fixed amount of the mass of gas we consider
it is a system and this system has a boundary
and the surroundings. So the boundary in this
case is the the surface where the heat flux
is coming into the gas, gas gets expanded.
Because of that the the boundary at this stop
is a moving boundary conditions whereas other
directions at the fixed boundary conditions.
So the basically when you talk about the system
we have the boundary we have a surroundings.
Mostly when you talk about the systems we
consider a fixed mass of the fluid. And how
it interacts with the boundary with respect
to the heat, mass, and momentum exchange through
these boundaries. That is what is called the
systems.
Here very clearly say that it consider a quantity
of matters or the the mass components as a
system. But many of the times we cannot solve
the problems within system approach which
in generally follow in thermodynamics. But
in case of the fluid flow problems, we go
for a space defined by a particular volume
okay. Like for example, I have a this problem.
If you look it this is what my control volume.
This is the space what I have considered as
a control volume and the fluid is coming from
this sites and this piston is moving in these
conditions. So this is what the control volume
and there is the the surface confined to this
control volume is called the control surface.
That is what the the control surface. The
control surface can be a fixed surface or
can be a movable boundary conditions or these
control surface here the mass or momentum
flux entering to this control volume.
So we have a control volume the fixed regions
or the movable regions on the space that what
we consider it and it is confined by the surface
we call the control surface and this control
surface through these control surface the
fluid mass momentum exchange mass or energy
mass comes into the control volumes. But if
you look it another case like you have the
nozzles, you have the flow is coming from
the left to the right and it has consider
this control volume like this okay.
So in this case, this is the inlet path and
this is a outlet path. The flow is coming
from from the left side and it is going out
the outsides. We have considered the boundary.
One is the real boundary conditions another
the imaginary boundary conditions. So you
have considered a fixed the space in the fluid
regions. Within that there may be a system
of or the a fixed mass of the fluid is make
enter to this control volumes.
After certain times may again go out from
these control surface. So we have a two approach.
One is a system approach another is a control
volume approach. The mostly in the fluid mechanics
problems what we will solve it we will follow
the control volume approach. That means we
will define a regions defined by the the surface
that is your control surface.
Through this control surface the fluid mass,
the fluid momentum flux or the energy flux
will come into this control volumes. Also
will be go out across this control surface
as as is the outlet conditions. So the mostly
in fluid mechanics we follow the control volume
approach, which is easy to solve as compared
to the system approach where you have to consider
a fixed mass of the fluid and you track over
that which is very difficult when you have
a very complex problems.
Whereas if you consider a control volume that
means you consider a fixed regions of the
space maybe in a fixed conditions or the movable
conditions. That not a big problems. Here
the easy things is that there will be a very
defined surface, control surface in which
the momentum flux, the energy flux will come
into the control volumes. Also there will
be a defined surface.
On that defined surface the mass flux, momentum
flux, energy flux will go out. So the reasons
we define it is is the control volumes and
the surface what you define is it the control
surface. So it is easy to solve the fluid
mechanics problem using the control volume
approach.
So most of my lectures I will cover through
these the control volume approach as compared
to the system system approach which mostly
follow in the problems like what I am talking
about heat flow and moving boundary conditions
like a piston conditions which is most of
the times in thermodynamics it is followed
it but in case of the fluid flow, we consider
the control volume approach to solve the problems.
Now, if you look it the next very interesting
problems what I have to give a illustrations
to you that if you look at this very beautiful
bird sitting on a branch. If there is a wind
movement is coming from this and this wind
movements consider let me the this speed is
increasing from 10 km/hr to 50 km/hr okay.
The speed of the wind is increasing from 10
km to 50 km per hour.
The question which comes it at which speed
this bird cannot hold this branch. That means
after that critical speed this bird has to
fly from this place okay that is very interesting
topic what you can understand it. That means
what we are looking it that there is a fluid
flow is coming from these sides is having
a speed let be the V the speed is what is
coming upon that.
Because of that here you are going to have
a two force components okay. One will be the
drag force and other will be the lift force
and will have a result and ports what is occurring,
because this fluid where it is passing through
that that what will create a pro structure
such a way that there will be a drag force
there will be the lift force.
And you will have a resultant force what is
occurring because this fluid which is passing
through that, that what will create a flow
structure such a way that there will be a
drag force, there will be the lift force.
Also the resultant force will react like that.
So these resultant force when you cross it
the strength of the the holding of the bird
on this at that time, bird has to fly from
these places.
So we are looking it that at which speed the
force will be coming such a way that, that
amount of the force this bird cannot withstand
with this holding. So this if you look at
that is very interesting problems. But it
is also the complex problem in the safe. If
you look at this, the beautiful bird shape
okay. It is very interesting, the geometry
what you have.
So also, as the wind flow is happening it
that bird how it is responding which is a
totally different aspect, we are not going
to that details. So we are trying to look
at how this force is going to happen it. So
we can conduct the experimental way to compute
it what will be the drag force, what will
be the lift force, what will be the resultant
forces or we can follow a analytical methods.
That means we can follow laws of conservations
like mass conservation, momentum conservation,
energy conservation, then we take a appropriate
control volumes. Then you try to find out
what could be the approximate the drag force
and the lift force on this case. Or we go
for very much the computational methods. The
recently people has has been using this the
computational methods to solve these type
of problems to find out what will be the drag
and the lift forces.
So we have three ways to solve this problems.
The experimental ways, analytical ways and
also the computational ways. That means with
help of the computers, by solving a set of
nonlinear partial differential equations,
we can find out what could be the pressure
field, what could be the velocity field that
what I will introduce you that. Based on that
we can compute it, what will be the drag force,
the lift force, and what will be the resultant
force.
And at which force component this bird can
withstand and beyond that it cannot. So that
is what is the very interesting study can
be looked upon.
So similar way, that is what I am telling
it, there are three ways to do the solve the
any flow complex flow problems, like you can
have experimental methods. That means you
can have a prototype and make a scaled models.
That means you can reduce the the flow and
the geometry of the problem in such a way
that you do a scaled model.
Then you do a wind tunnel or the flume test
to measure the velocity, pressure and the
density. So once you measure the velocity
and pressure and density that means you solve
the problems. You know, how the flow is happening
in terms of velocity, in terms of pressure,
and same way the density. The second approach
is analytical approach which mostly in the
fluid mechanics books, we will cover with
the analytical approach.
In that what do we do it we have a problems.
We try to use a control volumes, is a bigger
control volume we try to use it. And we try
to understand it where the mass flux is coming,
the momentum flux is coming and which are
the boundary there is no flux of mass, momentum,
energy is passing through that. Equating applying
this all mass, momentum, energy conservation
with a integral approach we can find out the
gross velocity, pressure and density.
Still I will talk about the gross. It is not
the very detail distributions of the velocity
the distribution on the pressure or the density
we will get it. In a very average type of
conditions what we can predict it as a gross
characteristics what will go through this
analytical methods. One is experimental methods.
The second is the analytical methods. Third
is which is the last one of two decades is
very famous is the computational fluid dynamics.
In which what we do it any of the fluid problems
okay, we define through these mass conservations
and momentum, energy conservation into a set
of partial differential equations. This most
often is a nonlinear partial differential
equations and both equations we try to solve
the numerically. As we solve this numerically,
we get the velocity pressure and the density
distribution, but these are all approximate
solution.
It is not the true solutions, because they
are the numerical solution is approximation
solutions to this numerical partial differential
equation. So we get approximate solutions
of the velocity field, the pressure field
and the density field. So that in summary,
I can say that we have three ways to solve
the problems. One is conducting experiments
to find out the velocity, pressure, and density
field.
Second one is that we use appropriate control
volume, apply basic conservation equations
like mass conservations, momentum conservations,
and energy conservation equation. Then we
try to get it what is the gross velocity distributions,
the pressure distributions and the density
distribution. These are gross level. It is
not gives a very detailed like the we get
it the wind tunnel or the numerical methods.
In a computational methods as you know it
now we have a lot of supercomputers, we can
solve many of the complex problems, fluid
flow problems, that is what we do it any of
the fluid flow problems we convert to a set
of partial differential equations by applying
same basic principle of mass, momentum and
energy. But here these the control volumes
are the unit what we consider that is what
is infinitely small.
And then we try to solve that equation using
a numerical methods. And then we get this
approximate values of velocity, pressure,
and the density field. So we have a three
ways to solve any complex fluid flow problems
experimentally, analytically and computational
way.
Now if you let me summarize that way, there
are three basic ways to solve the problems.
One is a bigger control volumes where we have
an integral analysis or the analytical methods
ways. You take a smaller control volume, which
is very close to infinitely small. Then you
get a set of a differential equation problem.
So that is the reason we call differential
analysis.
Then the experimental study, as I said that
we need to have a scaled models. So we will
discuss more in the latter half of my lecture
series that there is a techniques to how to
scale the models from the prototype to scaled
models. How to get different geometry scale,
flow scale all the similarity concept what
we will discuss in later on. So that way you
have control volume, infinite small systems.
That means analytical and you have a computational
methods techniques, then experimental methods.
And as I already said that, these three basic
equations always need to be applied for any
fluid flow problems. They are conservations
of mass, the linear momentums or the angular
momentums, the first law of the thermodynamics
that is conservations of energy.
But apart from these equations we adopt a
state relationship. That means a relationship
between a density and the pressure and temperature,
the ideal gas laws. So these type of the relationship
between one variable to other the independent
variable like pressure and temperatures, we
can establish that relationship and that relationship
is called the state relationship.
And at the last one what I can say that not
only know this what type of flow problems
also we should have a very good understanding
how to define the boundary conditions. That
means, you have to give a appropriate boundary
conditions to solve the problems.
So the flow analysis techniques what is available
to us with a very basic conservation equations
as we apply for solid mechanics here also
we follow conservation of mass, momentum,
linear momentum equation, energy conservation
equations. Apart from that, we look for appropriate
boundary conditions and the state relationship
to solve this problems.
So this is what the basic strategy to solve
any fluid flow problems and a fluid specialist
has to have a confidence or knowledge on how
to define the boundary conditions, what type
of equations to apply it and which condition
he has to go for a control volume approach
or the differential analysis or the experimental
analysis that all it depends upon a knowledge
on the fluid mechanics.
So that is my idea is that we should have
a very indepth knowledge of the fluid mechanics
then we can analyze a complex fluid flow problems
taking appropriate analysis techniques. Now
let us come to a very interesting examples
here, we have given it here.
That they let be there is a weather radar
is there. You know it nowadays weather radars
are there to measure the rainfall, the wind
velocities and all. That is what is a on the
hilltop and it has a stand of 10 meters and
the weather radar tower is about 10 meters.
And the wind is moving at a speed of 150 km/hr
and the velocity is 0 at this point.
If you have that conditions the questions
is coming to design this radar systems we
need to find out what could be the maximum
of split force and the drag force of this
radar systems when you the wind speed is close
to 150 km/hr of reaction of reaction. If that
is the problems, for these type of complex
problems we do not have any analytical solutions.
So what we go for? We go for a scaled models
in a wind tunnel, okay?
So that means in a wind tunnel, we set up
the scaled models. As I say the scaled models
means we reduce the dimension, we reduce the
flow velocities or the densities. How to do
that we will discuss that in a dimensional
analysis chapters, but lets us you understand
it that these are the real problems.
And and we reduce to a scale models which
is if you look it here is 100 meters here
is 10 centimeters, the 1 centimeters, 1 centimeter.
So we scale the models okay. We reduce the
dimensions. Similar way we have reduced the
velocity and this is the facilities is there
in the wind tunnels and we have the velocity
flow is going like this and at each point
of this grid we will measure the pressure,
velocity components okay
So if you look it that, so we have the wind
tunnels and we will fix up these the scale
models of the hills, the radar systems as
equivalent to a spherical body and connecting
part and we generate the flow systems which
is you will have the flow distributions like
this and with having a velocity 0.52 meter
per second and each grid point will measure
it what is the pressure, what is the velocity,
what is the density.
For example, if you look it that point, like
the wind tunnel facilities what you have at
the IIT Guwahati in the Department of Mechanical
Energy. See if you can look at this wind tunnel
the setups okay there is a inflow, this will
be the outflow and this is what the test sections
what is there and these are all the recording
systems for velocity measurement and the pressure
measurements.
So if you can look at these type of wind tunnels
they are most of the advanced fluid mechanics
labs this type of wind tunnel systems are
there where we generate the wind movement
through a test sections like this. If you
look it this is the or the test sections okay.
In the test section we generate the wind flow.
To measure the velocity we have a Pitot static
tubes. I will discuss more detail what is
it a Pitot static tube in the later on.
So you can now know it there are the instruments
which can measure the velocity, not only this
one direction, it can measure the three dimensional
velocity component. That means it can measure
the u, v, and the w components, the three
velocity distributions component can measure
it. And similar way we can have a nanometer.
I will discuss what is a nanometer and all
the things later on.
That the instrument can use to measure the
pressure. So we can measure the velocity,
we can measure the pressure and this is the
wind tunnels where you can the variable the
speed, the speed of the winds, which will
pass through these the test sections. Then
we can measure the velocity and the pressure.
Similar we can have also measure the density.
So if you look it this way, that means what
I am talking about that I have a scale models
put into the wind tunnel facilities. Then
at each point we are measuring the pressure
and the velocity and the density. It is the
measurement, the physical measurement of pressure
and velocity and the density.
Now if you look it for example, we got for
each grid point this type of velocity factors.
It has the directions, it has a magnitude.
So we get the velocity at each points. So
interestingly you can see that there will
be the high velocity zones, there will be
a low velocity zones and there could be a
vertex formation. But that is not our interest
now. We just measure the at each grid point
the velocity factors, velocity magnitudes.
So that means we know u, v, w component, the
scalar component of velocity we know it. That
is what is called the velocity field. Now
you can understand it if I take a more number
of grid points the the velocity field will
have a more accuracy. So if I take a less
number of the grid point for the measurements,
I will have a less accuracy in defining the
velocity field because the sampling points
are less.
So that depending upon the flow problems,
you can decide that how much the spacing,
how much locations you should collect the
velocity or the pressure or the density content
and once you know these pressure components
that means I know it approximately how the
pressure is varying with the space and the
time. I know it how it varies because these
are the measurement values.
The similar way, if I have vectors, this is
what I measure it the vector which is the
sample point vectors, which have x and the
t. So this is what called the pressure field
and the velocity field. Now if you look it
that my prime objective as a engineer is that
I have to design what could be the lift force
and the drag force because of 150 km/hr wind
is blowing over these tower.
What we can do it one we can measure the pressure.
That means you knew the pressure point P 1,
P 2, P 4, P 5 like this. You just integrate
that places you can find out what will be
the drag force and the lift force components.
So this drag and lift force can be used to
design this the civil engineering design the
structural engineering designs of this tower.
this the mounting staff or the foundations
will design such a way that whenever when
you have a 150 km/hr the winds pass over this
these tower system, these structures would
be in safe, okay. So that is what to make
it that we should know exactly what is the
drag force and the lift force is happening
it.
So with this example, if you can know it,
that with measurements with conducting the
experiments, we can compute it the velocity
field and the pressure field and knowing this
pressure field and velocity field can compute
it, what will be the gross or split force
is going to act it and the drag force and
these two force can be used to design these
structures for wind speed of 150 km/hr.
So this is what is experimental way we can
conduct it. But you can always think it that
the conducting this type of experiment is
always expensive and for that we need to have
a wind tunnel facilities to solve these problems.
Today's you know it there are big wind tunnel
facilities are there. They full prototype
of truck they simulate it.
So that is what am I pointing that the the
facilities nowadays the wind tunnel facility
what is there in all over the world, the automobile
engineers they use the full truck not the
scaled models, they conduct the wind tunnel
test to know it what will be the drag force,
the lift force for different flow conditions
like different velocity, different fix and
drag confusions they try to do solve that
type of problems the very complex fluid flow
problems with full scale wind tunnel lab facility.
So it is possible today's it is not that difficult
that to conduct the wind tunnel test, but
it is expensive as compared to the other methods
what we are going to discuss it.
Second one you know it we are looking for
a analytical solutions. That means the fluid
flow problems we are more or less very simplified
cases like for examples, we have a flow that
means flow velocity is coming from this striking
on the horizontal floor here. The flow jet
is coming and striking over a horizontal floor.
And our interest to know it what will be the
pressure distributions, what will be the velocity
distributions.
These problems we can identify it is a two
dimensional problems we can make it because
jet direction it does not have a much component.
So incompressible flow and we can simplify
to a steady flow. So we have simplified the
flow now okay as a two dimensional incompressible
and steady flow. Because of that, we can use
this the mass conservations and linear momentum
equations.
Okay, I am not going more detailed. Anyway,
we will discuss that how to apply the mass
conservations and linear momentum equations
here. Considering that if I do that, I will
get a solutions of this. These are unintegral
solutions of the scalar velocity component
of u, the v as a function of ax and ay and
the pressure will be function of x and y;
a is constant and P naught is a pressure at
this point at the where we have the regions.
The pressure at that point which called the
stagnation pressure. So if I know these equations,
which satisfy this conservation equations
and the linear momentum equations and the
boundary conditions at this point as well
as this boundary conditions here, then this
is what my solutions of the equations which
is the analytical solutions of equations.
That means, if I take any point here, if I
have a x 1 and y 1 is the coordinate I just
substituting this x and y 1 I can get the
u 1 component v 1 component and the P 1 component.
Here the u 2 component v 2 component and P
2component. So here what we have drawn it
as the water jet is coming it and tracking
on the horizontal floors. As you know it that
this flow is a symmetric problems.
So exactly at the center the flow velocity
will be zero. That is what you can say that
if you substitute x and y equal to zero v
and v component will be zero. So that condition
is satisfied and the pressure if you look
it that it will be varied with a x square
plus y square term is a is a equations of
a circles okay it is more or less the equations
of a circle.
That is the reason you will have a unit for
the pressure, the constant pressures line
like this, the concentric circles like that,
where it will be half concentric circles.
So this is what called isobar. The line of
equal pressures. So you will get the equal
pressure line. And if you know the u, v and
that you can draw the flow stream lines. That
means flow will come like this okay the pressure
diagrams like that.
So for a very simple case, we can get a analytical
solutions like u and v and w and the pressure
and that analytical solutions can help us
to know the velocity, the pressure distribution
of these problems and it satisfy conservation
equations, mass conservation equations, linear
momentum equations. Also it satisfy the boundary
conditions at the floor also flow inject what
is coming it.
So that is what I say that there is a experimental
way to do that thing. Very very simple case,
we can simplify it and we can like it a two
dimensional incompressible steady flow, this
is what the total simplification of problems
are. These assumptions are hold good for these
type of problems. Then you apply the mass
conservation linear momentum equations. Then
you get these solutions.
How to get this u and v equations and the
pressure that what we will discuss later,
but at present you know that we can get it
the functional relationship of u, v with respect
to a Cartesian coordinate of x, y, z and these
problems becomes a steady problems. So there
is no time components. So we will have the
solution of u, v, w and the pressure component.
Okay, so from that two examples, one the wind
flow over a weather radar setup, second is
a simple flow jet impacting on the floor.
We tried to understand it, the pressure field
and the velocity field.
So now I am just defining them the velocity
field, when we are talking about we are talking
this velocity as a vector quantity, which
vary in a space in case of the Cartesian coordinate
system of x, y, z and the time. But most often
for easy point of view, we resolve this velocity
factor component into a scalar component in
Cartesian coordinate systems like the i and
j and k.
As you know from vector rotations of x and
y, j these are the unit vectors. So you will
have the u velocity components along this
x directions. The v velocity components, small
v velocity component in y direction and w
is a velocity component z. All will have a
scalar component having a functions with the
space x, y, z and the time; v will be x, y,
z and the time and w is that. So we define
a velocity field.
That means getting a velocity field either
from experimental study or analytical study.
Similar way we can get it from computational
methods, which later on I will introduce to
more detail to you. So that way we will get
a velocity field. Similar way if I know the
velocity field, we can compute it what will
be the the acceleration, the rate of change
of the velocity gradient, velocity, you know
it the accelerations that the component you
will get it.
About these derivations we will come it when
we have fluid kinematics. We will talk about
that, but you can see that any of the flow
field conditions we can have a velocity field.
That means we know the velocity distributions
with respect to space and the time. Similar
way if I know the velocity distributions,
we can compute the acceleration distribution
over that the fluid flow problems.
Now as already I discussed that we talk about
the pressures which is very dynamic variables,
the pressure distribution play the major roles
because as you know it the flow is come from
high energy to the low energy. Many of the
time this pressure the gradients indicates
for us which directions flow will be there.
So that is the reasons we always look at the
gradient of the force which drive the flow.
Mostly this energy drive the flow, but many
of the cases the other component whenever
is less, the pressure gradient itself will
indicate it which direction the flow is going
on. So the the computing the pressure and
the pressure gradient that what is major component
in the fluid flow problems. Some of the case
studies like a is there we also very worried
about the places that when it goes below a
particular pressure this vapor pressure then
the water converts from the liquid to the
vapor.
So that is what creates the problems of gravitations.
That what I will just discuss in a example
problems, how the gravitation processes occurs
it. So we try to look it which are the reasons
we have low flow speed flow zone or the high
speed flow zones, how the pressure distributions
are changes it. All we can try to look it
in a fluid flow problems, if I have either
pressure variations if I know which varies
with respect to the positions and also the
time. That means is a scalar components.
It varies with a space to space, the locations
to locations. Also it varies with time. So
in case of steady problems, we the time component
goes out. So we have the pressures which varies
with space only. So if you know the velocity
field that means velocity variations with
space and time, the pressure variations with
space and time then more or less you have
solved that fluid flow problems.
But some of the cases like when you have the
heat exchange is going on drastically in a
fluid flow where there is a lot of temperature
gradients are there, then we apply the first
law of thermodynamics to get it the temperature
field. So there are the, the problems here
is not only the fluid flow problems there
will be the heat transfer problems.
We there is a lot of gradient of temperatures
are there, the heat transfers are there. The
same way for that fluid flow problems, we
can have the temperatures is a function of
the space and the time. So to solve this,
we have to follow first law of thermodynamics,
the heat transfer equations to solve this
problem. Mostly in these fluid mechanics course
I will not go more detail about these thermal
flow, the heat transfer problems more, the
flow due to the temperature gradients that
part.
And this is as you know is very basic things
that how the temperatures related to the at
different centigrade to the Kelvin scale.
That is what the point.
And second point what I am to discuss is the
density of the flow. You know it this mass
per unit volume or it indicates the mass of
the fluid mass of the fluid and that what
per unit volume we quantified it. So if you
multiply the volume, you know this what is
the amount of the mass is there and based
on that we can find out which is a heavier
mass or the lighter mass, the fluid as they
have a heavier and lighter mass.
The energy, the kinetic energy, potential
energy and how that is what varying it that
is what is related to the mass properties.
So that way the density plays a major role
for us. Some of the cases when you have the
flow is compressible, your density is also
the significantly varies with the positions
and the time.
So we can solve the problems to get these
density variations with positions and the
time and that is is a variable for the gases,
but for the density in the liquids nearly
constant as most of the examples what I have
I will discuss it. We will talk about the
liquid flows not the gas flows. So that is
the case if the density of the liquids will
be really constant.
Even if in case of the gas flows also as discussed
earlier, when you have a Mach number less
than 0.3 also we can consider as incompressible
flow, incompressible flow. So density does
not vary significant. So that way, if you
look at that, as a fluid mechanics specialist,
we will try to solve only two fields, pressure
field and the velocity field.
Because most of the fluid flow problems what
we have considered where the Mach number is
less than 0.3 unless there is supersonic flow
and all which is for the rocket flow and all
the concept. So we need not to have a density
field that for this problems, what we consider
it, but the major things what we look at how
the pressure field and the velocity field.
How does it varies with the space and the
time.
These two things are more important for us,
the pressure field and velocity field. Since
we consider the problems which the problem,
the flow is a Mach number is less than 0.3.
So flow is incompressible in nature. So we
can just need to know it the pressure field
and the velocity field.
If it is not, then you have to go for that
means if your Mach number is more than 0.3
conditions in your flow field conditions then
we have to consider the density variations
which will be the varies with the space and
the time component. So in that case, we will
have a the pressure, the velocity, and the
density variations. So three fields we need
to know, define the flow when you have a flow
incompressible.
But when you have the Mach number greater
than 0.3 the flow is compressible. Then you
need to have a density field, the pressure
field, and the velocity field. These three
field we need to define it when the flow is
compressible. If flow is incomprehensible,
that what we can get it when the Mach number
is less than 0.3. In that conditions only
we need a pressure field and the velocity
filed.
So what I am to tell you that fluid flow problems
are very interesting problems. It is a solvable
problems at present. Only we need to know
it how the pressure varies it, how the velocity
varies it. But as I showed it two examples,
one is the flow around a bird which is very
complex geometry and another is the flow load
on a weather towers which is looks like a
spherical ball.
So if you look it that as we go for a complex
problems, so getting these pressure filed,
velocity field, it is not that easy. So that
is the reasons we follow experimental methods,
analytical methods, and numerical methods
to solve the problems. In fluid mechanics
problems, the a specialist who can visualize
the flow better and then he can simplify the
flow problems and then he can solve the problem.
The flow visualization is a major issue and
how to visualize the flow. That means how
to determine that how what could be a tentative
flow patterns or the flow patterns are obtaining
from either experimental results or analytical
methods or the computational fluid dynamics
methods we should try to understand what we
are getting the flow patterns.
Are they correct or a fluid mechanics specialist
he can looking these problems he can visualize
it that this could be expected flow pattern
could be there. So define these flow patterns
we define technically three lines.
One is the streamline, pathline, and the streakline.
Let us look at the definitions. The the streamline,
a line everywhere it is a tangent to the velocity
vector at given instant. That means time equal
to zero or we take a snapshot okay. And at
each point, if I draw this the tangent that
tangent should give a directions of the velocity
vector okay. So if you look at this that as
I draw this, these tangent line, this line
tangent should match with the directions of
the velocity.
So if that is the conditions and join that
line is called the streamlines. So if you
look at that one case, the streamline goes
like this, one case streamlines goes like
this, like this. like this. Other way round,
if you have that streamlines at a point if
you draw the tangent and that the tangent
is a direction of the velocity. So we can
find out the directions of velocity if I know
the streamlines.
Or if I know this direction of the velocity
by measuring any experimentally work and connecting
that lines such a way that it will have a
tangent to that velocity. Tangent and the
velocity direction matches each other. That
what will give a streamline. So we will have
the streamline patterns like that.
So if you look it that the flow is going this
direction, this direction, and this direction,
but one of the easy things here in the streamline
if you look it if this is the velocity, the
direction of the velocity, velocity that means
there is no component is working on this time.
There is no velocity component. The normal
components is zero. The flow is going tangential
to that. There is no component on this.
That means, there is no flow goes through
this ones. That is the reasons of what we
define it if I have draw the streamlines,
and it is occupied a certain space like this
the steam cube. In that case is a very simplified
case is now the only the flow will be enter
from the side and go out from this. Since
the streamlines does not allow any flow cross
through that as the definitions indicates
for us.
So there will be no flow will go through these
ones. So we consider a streamline, a stream
tube which is composition of streamline such
a way that there will be inflow and outflow
and across the stream tubes, there will be
no flow component. So this is what the imaginary
the stream tube will generate it to solve
the problems because it gives us that there
is no velocity, no mass flux, no momentum
flux comes into that.
So some of the times we consider the stream
tube as a control volume. We solve the problems
as a stream tube as a control volume, then
we solve it because we get imaginary the boundary
where there is no mass flux, no momentum flux.
And it is easy for us to solve the problems
because it is only having inflow and the outflow.
So we use the stream tube concept and the
streamline concept.
So again I am to repeat it the streamline
is a line everywhere the tangent through the
velocity vector at given instant of time,
as a snapshot. You take a snapshot, at that
snapshot, if you draw a line the each tangent
of the line will indicate us the directions
of the velocity. So that is the basic point.
That is what we call the streamline. Second
is the pathline.
If you can understand it that the actually
path traversed by a given fluid particle.
You consider a fluid particle and at different
time interval you find out where that fluid
particles.
Like the next examples, like I have a fluid
particles at these points at the t 1 time
and t 2 time, t 3 time, t 4 time, t 5 time,
t 6 time. It is a different time. How from
this position to this position, this positions
like this. Then if I draw this line which
is called the pathlines. That means we define
the path of the fluid particles, but it is
traced by the maybe last few minutes, last
few seconds drawing that will draw the pathline.
So it is a time informations are there the
how the fluid particles are passing through
at different time. So we are tracking over
fluid particles, we are talking about that.
So how the at the different time it is moving
it. This is called the pathline. Similar way
if you look at this, another point is the
the streakline. What it says that the locus
of the particles that have earlier passed
through a prescribed point.
That means you have defined a point, at that
point the fluid particles are passing through
that. So that means the position is fixed.
At that point, which are the fluid particles
have already passed through that and those
if you color it or those you mark it that
the lines will indicate as the streakline.
So if you look at these problems that let
I have the flow like, this is coming from
these.
We have a channels. I have at this point I
have a color dye putting into there. So that
means which are the fluid particles are coming
to where I am making them to either a red
color or the blue colors okay, any of the
colors okay you can use as a color dye. So
put in a color. So as after few minute if
you look it these particles will move it the
second particles will move it and this color
dye pattern what will get it after t 1 time
that what will give us the streaklines.
So try to understand there is the streamline,
pathline, and the streaklines. The streamlines
talks about the how presentations of the velocity
vectors at a particular instant of the time.
Whereas the pathlines which talk about us
the path traversed by a fluid particles of
a durations of t. What are the path, what
is the different positions it should path
it.
The streaklines what is talk about that it
is a locus of the particles which have passed
through a fixed points. So if you look at
this way, we use a pathline for a some problems
to solve it. The streaklines to solve the
some problems, to visualize the problems.
Similar way we use the streamlines more upon
we use the steamlines which as I try to explaining
is that if you know the streamlines you know
these the velocity the directions of the velocity
vectors, which gives us that there is no flow
cross through that and we can compose a stream
tube concept as a control volume and you can
solve the many problems.
So mostly in case of the analytical methods
we follow the stream tube or streamlines methods
more accurately, but the experimental technique
when you visit either we follow for a techniques
like a pathlines like we track a fluid particles
at different instant of time then trace on
that or we do a dye a color a series of the
fluid particles then find out the color dye
pattern. So that patterns will be like a streakline.
So these two most upon use for an experimental
works to know it how the flow patterns or
flow visualizations will do it. Then later
on with wind lab experiment I will demonstrate
to you how interesting flow patents we get
it for different conditions. So but very interestingly
that you see that if you have a steady flow
okay if the flow parameters characteristics
the velocity, pressure they do not change
with the time then all the steamlines, streakline,
pathlines are the same.
So definitely they are not will be the different
the same things will happen it if you have
the steady problems. That means your pressure,
the velocity that they do not depend upon
the time variabilities. With this let us come
back to that just to have a if you look at
any fluid flow problems which are very complex
and most often this fluid flow in a natural
systems is much more complex as compared to
manmade systems like as I given a example
of bird and the weather towers.
Similar way you can imagine very complex problems
would it happens is that. So as I told you
that we need to visualize the flow. If you
visualize the flow you solve the problems.
The for example, if I take these problems
that flow past a cylinder. That means I have
a cylinder, a velocity of v uniform velocity
of v is passing through this okay? As I told
it earlier I will have a virtual fluid balls
okay it is not real fluid balls these are
virtual fluid balls. I have a ball 1, 2, and
3 and this is the fixed. And these balls are
having a velocity v as equal to the velocity
of uniform flow.
And as you can see it very clearly that the
ball 2 at t 1, t 2, t 3, t 4, t 5 will move
like this, which will be defined as a streamlines
in case of the steady flow it will define
a streamline or pathline or streaklines. Similar
way if I take a fluid flow ball is third one
that what will we define it the another streamlines,
pathline, streaklines like this. So we can
with help of this conceptualization we can
draw it what could be the approachment the
streamline conditions when flow pass a cylinder.
But what it happens to the 1 which the ball
is goes after certain time dash over this.
As you know it at this point the velocity
will be zero.
So these concept what we have said it this
is what approximately can draw the flow field
the streamline patterns. But when you close
to the cases when you have a velocity reduces
and we have a hypothesis now, that it may
degenerated it make it bigger ball to smaller
ball and we can find out these are what zone
of influence. So if you have a art, how to
draw a streamlines of a flow conditions that
means you have solved many problems.
See here I am with a example of a virtual
fluid ball concept, you just think it balls
are rolling it and dashing over a cylinder
is there. So because of the dashing there
will be a regions which will have a effect,
there will be regions will not have a effect.
The balls of these ones will move like this.
So if I can draw that, I can draw the streamlines.
I can draw the streamlines.
So that means I am just hypothetically I am
considering the balls are rolling with the
velocity v and there is a zone of influences
and those the regions the balls will be disintegrated
into smaller part beyond that, they may not
have a effect and that what we can move it.
So just to visualize the fluid flow we have
brought this concept of virtual fluid balls.
That means if you want to have a very complex
problems you can change you can make a more
number balls are moving it and interacting
with the structures and how the zone of influence
how the streamline patterns will be there,
that what we can generate it and we will try
to use this concept to define the laminar
flow and the turbulent flow which is will
be more interesting to you to visualize the
flow.
With this let me summarize today's lectures
that we define what is the systems and the
control volumes and the fluid mechanics problems.
There are three tools are available to us,
the experimental, analytical and computational
approach. The last two decades people have
been using the computational methods more
extensively solve very, very complex fluid
flow problems.
Because of that, we have fuel efficient aircraft,
the fuel efficient spacecraft. So all these
are possible because of the use of the computational
fluid dynamics. The use of the computational
fluid dynamics help us to predict the weather
which is also a fluid flow and heat transfer
problems.
Also the parallely as I say that there are
a lot of experimental facility has developed
in the world that can not not necessarily
will we do in a scaled models can do a full
prototype of models, which is used in automobile
industries or the aerospace industry. They
try to use the full scale models and try to
look into what is space technology centers
they use the full scale models to test it.
So these are not is possible. But before that
as you know these analytical methods is give
us a basic knowledge how the fluid flow problems
happens it with the help of a control volumes
with the basic energy conservation equations
and mass conservation movement. And these
are the three approaches. Next is integral,
differential and the dimensional analysis.
As we have talked about that radar problems
and the bird problems. Then very simple way
I just define it the again the virtual fluid
balls concept what we should try to understand
it. Then we can visualize the flow and once
you visualize the flow then you solve the
problems very systematic way. At the end we
have learned also the three lines. One is
streamline, pathline, and streakline.
The streamline the line everywhere tangent
to velocity vector gives at a particular instant.
That means it talks about the tangents is
a parallel to the the directions of velocity
vectors. And the actual path traversed by
the fluid particle is pathline. And the locus
of the particles that have the passed through
the prescribed point is called the streak
plane. With these definitions let me conclude
this class. Thank you.
