Let me welcome you all to this introductory
course on CFD or Computational Fluid Dynamics.
So, in this particular course we will emphasize
on the fundamental principles that govern
the implementation of CFD in practical applications.
Now, the first question should come that,
what is CFD, why are we doing CFD and when
should we do CFD. We will try to answer these
one by one.
So, what is CFD? CFD has a full form of Computational
Fluid Dynamics, but it's scope is not only
limited to fluid dynamics. So, when we say
computational fluid dynamics, we essentially
mean computational transport phenomena, so
which involve computational fluid dynamics,
heat transfer, mass transfer or any process
which involves transport phenomena with it.
We will come into that in more details subsequently.
Now, next we will try to see that, when CFD;
that is, what are the important and challenging
applications that we have for CFD these days.
So let us look into some of the interesting
applications of CFD.
So, first application which has been the most
common one is the aerospace application. So,
if you see the aerospace applications, we
will see that, let us consider an aircraft
as an example. So, if you consider an aircraft,
it has interior and an exterior. So, when
we consider the design of an aircraft, you
have to design the interior that is the air
conditioning and ventilation system, how should
it work, so there is an interior and there
is also an exterior. And when you have an
exterior, you are basically considering an
aero file section and there is a flow past
an aero file section, which you are intending
to model. Also you have combustion chamber
within the aircraft and you need to know about
the transport phenomena within the combustion
chamber. This is an example of a case when
there is some unintentional incidence that
is a fire that is set up in the aircraft,
then what happens. And one can have CFD for
spacecraft applications also. So, that is
like for aerospace applications.
Next automobile applications. So, if you look
into automobile applications, what you see
is that again the basic philosophy is very
similar, you want to see the fluid flow behaviour
inside the car, as an example. The whole idea
is to make a good design, so that the passenger
feels comfortable inside the car, but the
other important consideration is the external
flow, that is how fluid flow takes place across
and outside the car. Because that sort of
determines the drag force on the car and that
is where most of the clever design goes into.
But one should not undermine the importance
of design that goes for designing the ventilation
system inside the car. Then there is the combustion
chamber and combustion chamber design is also
very important from the transport phenomena
view point.
When we say automobiles and maybe aircraft
applications, these are very traditional applications
of CFD, and that is why we have begun with
those applications. But there are many applications
which are not so traditional, but which are
very importantly used in modern day applications,
like the biomedical applications. So, you
can see that you have basically interesting
flow patterns in bronchial tubes and these
are some of the examples which show that how
these flow patterns are there. Like this is
the human heart being modeled as a sort of
a pump in CFD applications. Now, this is flow
through arteries and this is also an interesting
case where fluid flow analysis is very important.
This is an example, where it is not fluid
flow, but heat transfer. So, this example
is like you have a tumor and somebody is trying
to have a laser treatment to destroy that
tumor. So, how the temperature is distributed
across the tumor on the laser treatment, that
visualization is there in this view graph.
So, like for a medical practitioner this is
very important because medical practitioner
can get an idea that what will be the extent
of destruction of the cells with the laser
treatment. So, accordingly the parameters
of the laser treatment may be sort of designed,
so that healthy cells will not be damaged.
Of course, there may be some damage, but it
will not be damaged beyond a critical extent
and only the tumor cells which are designed
to be destroyed will be destroyed by the laser
treatment.
CFD has lots of applications in chemical industries.
And if you see, this is a nice example where
you have mixing of two different chemical
streams and the interface emits some bubbles.
And so it is a very complicated interfacial
phenomenon and CFD sort of tries to capture
these type of phenomena by involved computational
modeling. You can also have like injection
of streams or separation mixing, these are
all important chemical engineering applications
and CFD has a big role to play in many of
these applications.
Now, chemical engineering application is a
sort of a traditional application, but you
can also have not so traditional applications,
just like the biomedical we gave one example,
another example is electronics. So, these
days, like with the advent of new generation
electronic devices, the sizes of electronic
devices are getting more and more reduced.
But if you are still having the same power
rating, then what happens? Then basically
what you have to do is, you have to dissipate
a much greater amount of heat per unit volume
than what you had to dissipate for larger
devices; that means, you must have more efficient
electronics cooling strategies, so like no
matter whether it is your laptop or desktop
or any other electronic device you require
efficient cooling strategies.
So, if you are cooling it by different means
for example, if you have a fan cooling or
like cooling through different other technologies,
we will not go into such technologies, but
there are interesting technologies these days,
one has to assess that what is the fluid flow
and heat transfer around the flow passages
in electronic devices. I mean, it may not
always be a forced convection type of cooling,
you can also have a cooling because of a phase
change in a loop system, which is called as
a heat pipe. So, you can have for example,
a system where you have a hot spot in the
device and there is a fluid which gets evaporated
by taking heat. And then it the fluid gets
transported to a different place which is
colder and then it gets condensed. So, you
have a cyclic evaporation and condensation
process. In this way, the entire energy is
sort of handled in the form of latent heat.
So, there is no sensible change in temperature
and there is a sort of a cyclic phenomenon
that goes on occurring.
There are lots of other energy related applications,
like you can see this is an example of emission
characteristics from a coal fired boiler.
You can have interesting flow in fluidized
based systems and separators and so on, and
like, these are many different energy related
applications, like for example, this is like
a wind turbine. So, you can have very interesting
energy related applications, and fluid flow
and heat transfer and mass transfer in these
many of these applications stands out to be
very critical and CFD plays a big role towards
understanding that.
Now, when you have fluids, many times we forget
that fluids interact with their boundaries
and that give rise to fluid structure interactions.
So, when we have fluid structure interactions,
it is important to model the CFD with a structural
model. So, CFD is basically a fluid dynamics
model, but you should also have a model for
solid mechanics, a structural mechanics, and
there must be an interaction between these
models, which can give rise to interesting
fluid structure interactions.
Look at this example. So, here there is a
balloon which sort of interacts with the surface
and tends to get deformed as it interacts
with the surface. So, there is a model of
the surface because it has its own elasticity
and stiffness. And the fluid filled balloon
has its own modeling, the balloon has a membrane,
so it has also its own flexibility. So all-in-all,
apparently it might appear to be very simple
problem, but it is actually a very complex
problem in fluid mechanics.
In this example what is shown is that there
are dolphins which are sort of interacting
with a pool of fluid. So, dolphins are sort
of models. See, CFD is all about modeling,
it is not that somebody is representing the
real dolphin, but representing it may be like
a flexible membrane. So, you have a flexible
membrane that represents or that models the
dolphin artificially. And that sort of interacts
with the fluid and creates a deformation in
the fluid itself as well as there is a deformation
in the body of the dolphins. So, that their
interactions it is a very interesting problem
and that this type of interesting problems
can also be addressed through CFD.
In general, fluid structure interaction problems
talk about some structures which sort of have
some vibration or motion, and it interacts
with an insufficient fluid which is there
in the surroundings, and how they interact
eventually with each other. So, the fluid
interacts with the structure, the structure
interacts with the fluid. It is not a one-way
coupling, but it is a two-way coupling, that
is very important. Otherwise, one can have
a separate model for fluid and a separate
model for solid, but whatever is the disturbance
that the fluid gives to the structure, the
structure has a deformation on the basis of
that. That deformation is fed back to the
fluid and that modifies the velocity field
within the fluid. So, it is a two-way coupling
and a dynamic coupling. And the great challenge
is to actually take into account all the intricacies
of these dynamics, while modeling fluid structure
interactions.
CFD has lots of applications in larger scales
also, like marine applications. So, you can
have a flow past ships and boats and even
submarines, and though these are of much larger
length scales than many of the examples that
we have seen till now, but like these are
still very important, and huge component of
the marine design now these days relies on
the use of CFD.
CFD has lots of applications in materials
processing or manufacturing. So, like this
is an example where one is trying to simulate
the grain growth in a material, which sort
of the nature of which gives rise to the material
properties. So, this is an outcome of a CFD
simulation of how the grain grows. Of course,
this always need not be a deterministic simulation,
the one can also use stochastic simulations
like Monte Carlo simulations and so on. So,
CFD need not always be a deterministic solution,
it need not always give a deterministic set
of approach, but there are stochastic approach
is also in CFD, like in this example one such
approach is used. This is an example, where
one is interested about the transport phenomena
during a bath-welding process. So like, you
are interested to create a joint by bringing
two materials from the two sides and that
bath-welded joint is sort of like, what is
the transport phenomena there, that is sort
of represented in this view graph.
Here, what was shown in the picture was the
growth of a dendritic crystal. It is a solidification
of a dendritic crystal from a melt, and you
can see interesting patterns as the material
is being processed. And you can also have
interesting applications in mould filling.
So, if you are filling a mould with a molten
metal and then you can see that, like it is
very important to know that how well the mould
will be filled up because of a particular
filling process. Because if the filling process
is not appropriate, it can give rise to casting
defects, and those such kind of defects are
not wanted, and one cannot do lots and lots
of trials and experiments because that will
be hugely expensive. So, if CFD gives a sort
of a guideline, that these are the processing
parameters that are required to have a very
efficient mould filling process, then that
sort of can optimize the process and make
sure that defects are minimized and so on.
So, in processing of materials one faces many
challenges. For example, like in many cases,
we require that the product should be as homogeneous
as possible. Now, what happens, materials
have impurities and when there is a fluid
flow these impurities are getting transported
from one point to another with the flow, and
many of the materials actually have to undergo
these process, because many of the processing
of materials involve the transport of molten
material.
So, if you have a solidification process,
that means prior to the solidification the
material was there in a molten state, just
like a casting process. Now, if you have impurities,
the impurities will be transported by convection
from one point to another point with the molten
material. Now, once it solidifies, whatever
was the impurity at that particular location
it will be frozen at that location and it
will give rise to a distribution of species
across the material. And if this distribution
of species is undesirable, then that will
give rise to undesirable mechanical properties.
So, to know that how to impose a control over
the distribution of species that are being
added to a particular system, one has to sort
of assess the velocity and the temperature
field and the species concentration field
within the domain, which constitutes the domain
on the materials processing application.
Now, if you consider applications in small
scales, there are interesting applications
of CFD in micro-fluidics or micro-scale fluid
flows, I mean these are some interesting examples.
So, here what is the purpose? The purpose
is to mix two different fluids. Now, micro-fluidics
deals with studies on fluid flows at small
scales in channels of micron or sub-micron
dimensions. In such small length scales, usually
it is not possible to take the advantage of
turbulence, because Reynolds number is very
small. So, if you want to mix two streams,
there are different ways in which you can
do that. One is by clever designing of the
channel geometry, that is not shown in this
figure. But what is shown in this figure is
that you can also try to have a good mixing
by using a pulsating flow. And with a pulsetized
flow input the objective is to create a good
mixing between different fluid layers. In
micro-fluidics, one also has important studies
considering or concerning droplet dynamics.
So, this is an example, where in a junction,
this is a t-type of junction because of sheer
droplets are produced. And then droplets are
elongated and sort of compressed and so on.
So, they are deformed and it is very important
to see that how these droplets evolve. Because
many times you put reactants with these droplets
and allow reactions to get performed in small
scales, those are called as micro-deactors.
And therefore, it is very important to understand
that how the droplet dynamics evolves in small
scales.
Now, in micro-fluidics also fluid structure
interactions are important. This is an example,
where you have an oscillating flap located
within a micro channel or a small channel.
So, what this flap tries to do is, it tries
to oscillate with a particular frequency and
amplitude and there is some material which
wants to undergo a reaction at the wall. So,
this flap what it is trying to do, it is trying
to force the material on the wall at certain
locations where the reactants are kept. And
it is not only at one single spot where the
reactants are kept, there are different locations
where these are kept. So, one needs to optimize
the movement of these flaps, so that all the
reactant locations get the sample for reaction.
And these flaps essentially act like a combination
of a mixer and a pump. And because mixing
is not so easy in small scales, this is one
of the strategies by which one can have a
better mixing by using mechanical flaps in
small scales.
Of course, there are many other applications,
but our objective is not to go deep into any
application, but to just appreciate that different
fields have different challenging applications
of CFD, the whole objective is not to go deep
into any particular field at this particular
stage.
Now, just if you come to some fields of entertainment
like or sports may be, there is a huge application
of CFD in sports. So, for example, if there
is somebody who is sort of driving a racing
car, then how the fluid flows past that. It
is very important because again in such applications
it is a sort of a very important idea how
to minimize the drag, and then to do that
one needs to design the car. And to design
the car one needs to taste it. And of course,
if one fabricates it in a real size all the
time and does the testing, then it may get
over expensive. So, before the actual fabrication,
during the design stage CFD is very commonly
used. So, even when human beings are running,
so how the fluid flow takes place around human
beings when they are running around the body
contour. Because that sort of dictates that
what is the total drag force and the running
posture interacts very nicely with the fluid
flow, in a way that one can have a technique
by which one can minimize the drag force under
certain conditions.
The dynamics of sports balls is also very
important, like golf ball, cricket ball all
these things, that how fluid flow takes place
around these. Because it is the extent of
turbulence or the nature of the flow, whether
it is laminar or turbulent, that sort of many
times dictates the movement of the ball. So,
you can have the swing of the ball and you
can have other sorts of, other types of lateral
movement.
So, in sports CFD is very extensively used
for scientific purposes. In other applications
of mechanical engineering, like the one of
the traditional applications is turbo machines.
So, in turbo machines CFD is very commonly
used for design of like blades or flow passages,
so to say in general, those may be blades
or impellers. So, you can see that these are
wheels of Helton turbine or the buckets. So,
you can see that you have a wheel and then
on the top of that, I mean on the periphery
of that, you have buckets and fluid jet is
interacting with the bucket and this entire
process is being simulated by CFD.
So, you can have CFD applications in turbo
machines, where you can sort of access in
a great details of how fluid flow takes place,
for example, across the impeller of a pump.
Now, this is very important because like analytically
it is virtually impossible to analyze such
flows, because the blade passages are very
complicated. It is not as simple as like flow
over a flat plate or flow in a parallel plate
channel or flow in a circular pipe, because
of such complicated geometry across which
the fluid flow is taking place, it is almost
inevitable that one has to go through computational
roots for solving such problems.
So, we have seen some applications of CFD
and we have now developed an idea that in
these applications why CFD is important, but
the bigger issue or the bigger question is,
is CFD inevitable; that means, is it so that
we require CFD in all cases. That is one interesting
question that we would like to answer. Because
many times, because this is a CFD course,
we will be having a bias towards CFD. So,
we will say that yes CFD is necessary. But
to be honest, let us try to answer this question
that, is CFD necessary in all cases. So, that
is the second question that we wanted to answer;
that is, when CFD and why CFD. So, we have
understood by this time that CFD is about
numerical simulation of some governing equations
for transport phenomena; that is, in a summarized
form. But to solve the transport phenomena
problem, you could also have an analytical
approach and you could also have an experimental
approach. So, these are alternative approaches
to CFD.
So, question is that when should we prefer
other approaches or when should we prefer
CFD. Let us try to look into that. So, first
of all experimental investigation, we should
keep in mind that there is no substitute for
experimentation, because seeing is believing.
So, there are many idealizations that we can
sort of do, while we are doing a numerical
modeling or analytical modeling, but when
we do experiment and see something is happening,
that is the reality what is happening.
But having said that, there are certain important
limitations or restrictions that can go with
experimental investigation or experimental
modeling. What are those? So, when we say
modeling it can be both experimental modeling
and numerical modeling. We are now focusing
on experimental modeling. Experimental modeling
in full scale is expensive and often impossible.
So, like, for example, if you are interested
to study fluid flow in a steel making vessel.
So, in a steel plant, in a steel making vessel,
there is a very interesting flow pattern,
because you may have jets of oxygen being
input to the fluid. You can have the jets
coming from both the top and from the bottom.
And so there can be multi-faced flows in the
system. So, it can be a very complicated flow
and eventually the final property of the steel
is a strong function of how the flow is taking
place in the steel making vessel. But imagine
that you have a molten steel at such a high
temperature plus the medium is also not transparent.
So, you cannot have a full scale model with
a real fluid under those applications.
So, it could be possible in this case, it
is in the real case it is not possible to
study the fluid flow exactly in the system.
There are certain examples where it is possible,
but it is very expensive. So in general, full
scale modeling may be expensive, often impossible.
Expensive full scale modeling means you are
designing an aircraft. So, before designing
or before coming up with a final dimensions,
if you want to make a lot of trial by experimenting
with a real sized aircraft, then that is very
expensive, I mean, one should not go for such
a design because it will involve a lot of
expense. Plus just like any sort of any tool
has its own errors, experiment also has important
errors, which are related to measurements.
So, it is not that seeing is, believing, yes
seeing is believing, but measuring is not
always believing. So, whatever you are measuring
may have lot of error associated with that.
Now, many times because of limitations with
the full-scale modeling, one goes for a scaled
model. The scaled model may be smaller than
the actual model, it may also be larger than
the actual model, depending on whatever is
convenient. Now, scaled model should satisfy
certain important objectives. That once you
have a scaled model, you should not scale
it up or scale it down in such a way that
the flow physics has changed. For example,
say you are interested to study flow in a
micro capillary. Now, you say that well it
is difficult for me to do experiments in a
micro capillary, so I will do same experiment
in a pipe, which is sort of geometrically
similar version of the capillary, but hugely
expanded in dimensions.
So, geometrically you can maintain the similarity,
but the physics of the problem might change.
Because in the micro capillary, surface tension
force is very important, but when you go to
a large pipe, surface tension force may not
be that important. So, when you are considering
a scaled model, one has to keep in mind that
at least the physics does not change. But
many times one has to make a compromise, and
once that compromise is made it becomes a
simplified version of the reality, but not
the reality itself.
Sometimes it is very difficult to extrapolate
results from the model to what will happen
for a prototype, because of incomplete similarities.
Because many times, exact all sort of similarities;
that is, the physical similarity, geometrical
similarity, kinematic similarity, dynamic
similarity, all these similarities may not
be properly maintained, and then it may be
difficult to extrapolate the results. Plus
measurement errors are there even in the scaled
model, not that in scaled models measurement
errors are not present. Next, so this is about
experimental investigation. So, experimental
investigation has many advantages, here we
have only itemized the critical concerns,
it is not that we have considered just the
positive points, but we have considered it
very critically, where the constraints are
also taken into account.
Theoretical calculations, if you come to theoretical
calculations, theoretical calculations for
the same problem will have two sorts of approaches.
One is an analytical solution. Analytical
solution, if it exists there is nothing like
it, because it is exact and it gives exact
point to point variation without any approximation.
But the question is, does the analytical solution
exist for the case that you are interested
for? Well, for most of the practical cases
analytical solution does not exist. So, it
exists only for a few cases, those few cases
are very important. See, we will see that
CFD cannot stand alone by itself, it requires
a good help from experimental as well as analytical
solutions, because you require to benchmark
your solutions. Say, you are solving a very
complex problem of fluid flow and you have
written a code or you have developed a code,
how do you validate that? You can validate
that against some problems, which are either
standardized in their solution by analytical
means or there are benchmark experiments with
which you can compare. So, you cannot really
discuss CFD devoid of the considerations of
analytical solutions and experimental solutions,
that is why we are bringing these into perspective.
So, analytical solutions exist only for a
few cases. And in cases where they exist,
sometimes they become very complex.
So, I have seen students by looking into analytical
solutions of that problem so easily, sometimes
they get driven away from the problem that
they never come back for solving the problem.
And the reason is that sometimes the expressions
are so big and so cumbersome, as if like wherever
inside those expressions some physics is hidden,
it is very difficult for beginner to appreciate
that in many cases. But if analytical solution
exist, there lies its own elegance and its
own beauty and it needs to be appreciated
and it can be used as a benchmark for more
complicated numerical solutions.
Numerical solutions exist for almost every
problem. Of course, whenever you gain something,
you lose something also. When you say that
numerical solution exists for almost every
problem, what we mean by numerical solution
is that you are getting solutions at discrete
points in the domain, not continuously at
all locations. So, the continuous nature of
the solution gets sacrificed, but at the expense
of that you at least get solution for even
complex problems.
So, if you compare modeling versus experimentation,
what are the advantages of modeling? Advantages
of modeling are as follows, they are cheaper
than experimentation in many cases. For example,
if you have developed a particular code or
you have got a code which can be used for
a particular problem, then you can simulate
it for different cases and for that expenses
do not mount up. So, it is the same code that
you are running with different parameters.
Of course, you are incurring computational
time. And if it is a highly sensitive code
to computational effort; that is, you are
for example, doing a direct numerical simulation
for a turbulent flow problem or solving a
problem in a highly paralyzed computing environment,
then it involves a lot of cost, it does not
mean that computation is free. But only thing
is that many times it may still be less expensive
than the corresponding trials, which one could
have done with experimental investigation.
With modeling you get more complete information.
Why? Say, with experiment, you do not really
have a scope of getting all the details of
all the variables that you were looking for.
Because for example, you are interested for
some flow measurement, temperature measurement,
velocity measurement, of course, you can do
that within the complete flow field by numerical
simulation, you can also do it also by experimental
simulation. But if you were interested to
do that, it will involve a huge expense. Not
only that, in many cases it is ruled out as
we said. For example, if you interested in
a flow visualization for a molted steel, say,
if you want to get velocity profiles within
that or velocity vectors within that, numerically
you can generate, experimentally it is not
possible. One can handle any degree of complexity,
that is it can be a very complex problem and
one can handle it, but there are important
buts. What are those buts? Like, when you
are handling a very complex problem, many
times what is not known is a correct description
of the boundary condition. So, you see a numerical
solution is as good as the input that is going
with the solution. So, if you are solving
a problem numerically and you are getting
a solution, whether you need to believe the
solution or not, it is up to the level of
the input. So, what we mean by the level of
the input? The input mainly deals with the
input of thermo physical properties in CFD
applications, like for example, input of properties
as a function of temperature.
So, if you are giving the properties as a
function of temperature and composition very
accurately, then that is a very good thing,
but where from you will get that. So, you
must have reliable sources of property data,
that is one thing. The other thing is boundary
conditions. So, many times we give boundary
conditions in numerical simulation which are
not correct representatives of physical reality,
like for example, we say isothermal boundary
conditions. So, in reality, whoever has done
experiment will understand that it is one
of the toughest things to maintain an isothermal
boundary, even in a very simple case.
So, whatever we are sort of planning to be
isothermal, in the real experimental case,
it may not be maintained as isothermal. So,
that is one example. So, the understanding
is that there can be lots and lots of uncertainties
with regard to the boundary conditions. And
one has to consider these uncertainties very
seriously, because sometimes the problem may
be so non-linear that with a slight change
in the initial condition, with a slight change
in boundary condition, the solution may change,
the solution may bifurcate to an entirely
different one from what you are interested
to get.
And that is where it can be very critical.
So, we should not basically be so emphatic
and say it is disadvantage of modeling, but
may be one can say some limitations that modeling
deals with the mathematical description and
not with the reality. So, when you say a boundary
condition, it is a mathematical description,
that mathematical description may correspond
to the reality, may not consider also to the
reality. So, one has to keep in mind that
it is your responsibility as an analyzer to
make sure that it deals with a reality as
much as possible to whatever extent possible.
But mathematical description can be inadequate.
So, how it can be inadequate? So, we have
considered the property data, we have considered
the boundary condition, what about the governing
equations themselves? So, are the governing
equations representing the correct physics
that you are looking for? May be or may not
be. May be the correct physics that you are,
or the physics that you are trying to sort
of capture, cannot be captured by the equations
that you are employing. So, then no matter
whether you are using correct properties or
correct boundary conditions, your solutions
will not be the desired ones, simply because
you are not representing the correct physics.
Sometimes, modeling particularly for non-linear
problems can be tricky, because non-linear
problems may have multiple solutions. And
by modeling, you might get each of these solutions
by having different initial guesses. For example,
if you are using iterative schemes for solutions,
so the question is what would be your correct
solution. Because may be, when you are doing
experiments, you will get only one solution
during one experiment. So, how do you relate
your experiments with the modeling, that is
a very important concern so to say.
So, to summarize this discussion on modeling
versus experimentation, we can say that there
is actually no substitute for experimentation,
but there are limitations associated with
experimentation, maybe because of expenses
or sometimes it is because of the inability
to capture many things during experiments,
like for example, capture the velocity field
in a non-transparent material flow, as an
example. So, because of these limitations
one cannot do experiments on a trial basis.
So, if one is interested to go for design,
then the best route for solving a problem
may be like this, that first make a model
of the problem, and then try to approach the
problem mathematically, come up with some
solutions, either numerically or analytically
depending on whatever is more convenient.
Of course, if analytical solution exists that
will remain to be more convenient, but if
it does not exist, then one has to go for
numerical solutions. And from those solutions,
reduce your number of trials, so that you
come up with only a few sets of data with
which you do final set of limited and very
controlled experiments.
So, study of CFD is not to make us go away
from experiments, but actually to help us
in doing better experiments, so that we can
get a domain where we have very restricted
and very highly refined sets of data towards
a good experiment design. And then on the
basis of that design, if you do experiments,
that will give us a good feel of the reality
what is happening. So, then with that experiment
we can compare some of the CFD output, some
of the CFD output we cannot compare with experiment,
because such details you may not get from
experiment. But if other details are verified,
then of course we can say that the CFD model
is representing the reality in a very nice
way. And then you can do many sets of parametric
studies and come up with nice conclusions
which sort of may not be possible with just
experimentation. So, many times in old days,
experimentation was like a hit and miss type
of trial. So like, it is just like an alchemist,
where one is mixing A with B to see that whether
C is becoming gold. Those approaches towards
experiments were very ancient. Now-a-days
approaches to experimentation is very sophisticated.
And in fact, CFD plays a big role towards
good experimentation. Now, next what we will
do is. So, we have seen that, what is CFD,
and sort of why we are doing CFD, and how
do you compare modeling with experimentation,
and when to do experimentation, and when to
do modeling and so on. Now, next what we will
try to do is, we will try to see that what
are the basic principles that govern the implementation
of CFD. And when we say that what are the
basic principles that govern the implementation
of CFD, we have to keep in mind certain things.
So, what are those certain things. So, those
certain things are fundamental 
principles of conservation. So, fundamental
principles of conservation are the principles,
which govern the basic equations that we commonly
use for CFD. And not only CFD, but even analytical
fluid dynamics. So, to understand that what
are the basic principles that go behind, we
will try to learn these principles of conservation
in a very generic manner to begin with, and
then we will try to apply it for the transport
of mass, momentum, heat like that.
To do that, we will first see that what essentially
does, conservation talk about. We will start
with an example, that example is not falling
in the purview of heat transfer, mass transfer
or momentum transfer, but just to make sure
that you are not very disinterested, we will
consider money transfer. So, let us say that
you have a bank account and you are interested
to transfer some money to the bank account
and withdraw some money from the bank account.
So, let us consider ancient times, not the
present internet banking system. Let us consider
that it is a traditional era, when you are
going to the bank and you are opening a bank
account. So, you are depositing some money
to the bank account and your bank account
is activated. Now, say, every month you get
some salary, scholarship, whatever, and there
is a direct transfer of money to your bank
account. So, your bank account is getting
enriched. Now, suddenly one fine morning you
feel that you need to withdraw a good amount
of money.
Remember we are talking about old age, when
it is not an ATM by which you are operating.
So, you are physically going to the bank and
you are withdrawing some money. After withdrawing
the money, you are checking that what is there
in the bank account. In fact, this same procedure
you can do with the ATM also, it is not that
you cannot do, but just for simplicity we
say that you are checking the bank account
in which some money has gone into it, to begin
with, you have withdrawn some money. So, you
have some money in, you have some money out,
which you have taken out. And you are interested
to see now that what is the balance that you
are having now. So, you can see that whatever
was the money in minus whatever was the money
out is not the balance that you are getting,
you are having the more balance than that,
because the bank is generous to give you some
interest. So, there is some source of your
extra funding and that is by interest provided
by the bank. So, this source we may also call
as generation.
So, we can say that, in minus out plus the
generation equal to the net change of money
in your bank account. So, that is the statement
of conservation of your bank account management
or bank balance. Believe it or not, this is
the same principle which governs the behavior
of conservation of any other physical variable,
maybe mass, momentum, energy or whatever.
So, for many of those, what we do is, we express
this as rate, rate of in minus out plus generation
is equal to rate of change.
And there are principles which sort of try
to execute this conservation principle, I
mean, there are mathematical formalisms which
try to execute this conservation principle
and apply it to the conservation of mass,
momentum, energy and so on. So, this bank
account which we have just considered as a
specific example, if you consider it as a
more generic case, you can consider it to
be a region in space across which different
quantities can flow, the quantity here is
money, but it can also be mass, momentum,
energy like that.
So, this by definition is a control volume.
So, across this control volume there is transport
of some quantity. So, this control volume
what is that? It is a identified region in
space across which matter and energy can flow.
So, when you have a control volume, we can
write a balance for the control volume, true.
Now when we write a balance for the control
volume, where lies the difficulty? The difficulty
lies in using some of the basic equations
in mechanics, which are not inherently developed
for control volume.
For example, Newton's laws of motion, those
are not inherently developed for control volumes,
those are inherently developed for identified
sets of particles. So, those equations are
inherently suitable for a Lagranian description;
that is, you have identified motions of particles
and for each particle you are writing equations
of motions. So, you have a trajectory of each
particle which is governed by the Newton's
laws of motion. On the other hand, this approach
is called as Eulerian approach, where you
have a Eulerian control volume and you are
interested to study the transport behavior
across that.
All of us know that for fluid mechanics, it
is more convenient to use the control volume
or the Eulerian approach. Because fluid is
continuously reforming, if you are identifying
some particle in a fluid, then what will happen?
There will be numerous particles which will
be evolving in very complicated ways, until
and unless the flow is very simple. On the
other hand, if you just focus your attention,
as if you were sitting with a camera and focusing
attention on a specified region, you are not
having to track the particle, but what you
are just seeing is, what is happening across
your focused region.
So, the Eulerian approach is more convenient,
only problem is basic equations of mechanics
and thermodynamics are not originally developed
for Eulerian description. So, we must have
a transformation from Lagranian to Eulerian
description, so that we can use that Eulerian
description, at the same time we can use the
basic laws of mechanics that were originally
developed in the Lagranian frame. And that
description is given by the Reynolds transport
theorem, which we will just revisit.
Let us say that we have a system. System 
is essentially an identified collection of
particles of fixed mass and identity. So,
it is a representative of the Lagranian description.
So, we have a system at time t, we consider
the same system in a different configuration
at time t plus delta t, where delta t is small.
Actually because delta t is small, these two
configurations are almost coincident, but
just for clarity we have shown them distinctly.
Now, this overlapping of the systems at two
different instances gives rise to three different
regions. One is this region 1, then this region
common region 2 and the third one is the region
3. Let us say that we have some property N,
which is extensive property, any extensive
property. So, N at time t is N occupying the
region 1 at time t plus n occupying the region
2 at time t, because at time t it is region
1 plus region 2.
N at time t plus delta t is N at 2 at time
t plus delta t plus N at 3 at time t plus
delta t. We are interested to find out what
is dN dt of the system; that is, the rate
of change of N for the system. So to do that,
we can write dN dt of the system is equal
to limit as delta t tends to 0 N at t plus
delta t for the system minus N at t divided
by delta t.
So, N at t plus delta t, you can see that
it has combination from region 2 and region
3, out of these region 2 is common. So, we
can leave that apart and write it or break
it up into two different terms. One is N at
2 at t plus delta t minus N at 2 at t by delta
t in the limit as delta t tends to 0 plus,
we can write something bit qualitatively,
plus we have this term N at 3 at time t plus
delta t divided by delta t, what is that.
See, at the region 3, whatever is the property,
that property is ready to leave the control
volume, what is the control volume here?
Remember when delta t tends to 0, what is
your identified region in space? When delta
t tends to 0, this 2 is your identified region
in space. So, 2 is your control volume and
when delta t tends to 0, your system and control
volume almost coincide. Because these are
almost merging configurations, but your control
volume is 2. So, when you have 3; that means
what, it is ready to leave the control volume
and when something is there at 1; that means,
it is something ready to enter the control
volume.
So, we can say that this is nothing but, plus
rate of outflow minus inflow of the property
across the control volume. So, we can write
this, rate of outflow minus inflow of the
property across the control volume in a bit
of a mathematical way.
Like, just we can first write this one. This
one is nothing but, partial derivative of
N with respect to time in the control volume.
Because here we are fixing the position and
changing N with respect to time, that is why
it is the partial derivative.
Now, let us look into the last term rate of
outflow minus inflow. So, for that we define
small n as capital N per unit mass. Now, when
there is a surface on the control volume,
let us say that there is a small area dA and
the fluid is having a velocity of V. So, what
is the volume flow rate there? What you have
to do is to just construct a unit normal vector,
let us say eta is a unit normal vector to
the area. So, V dot eta dA is the volume flow
rate. That times the density is the mass flow
rate. And small n is the property per unit
mass. So, n times this, is the total transport
of property. So, if you integrate it over
the control volume, it can give outflow minus
inflow. Because if V dot eta is positive;
that means, outflow if V dot eta is negative;
that means, inflow. So, if you consider it
over the entire control volume, it automatically
takes into account outflow minus inflow.
So, with that consideration, we can finally
write the conservation expression as follows,
dN dt of the system is equal to... One important
thing to mention is that, this V is not the
absolute velocity, this is velocity of the
fluid relative to the control volume, because
that dictates what is the flow rate. So, we
will call it V relative. This expression is
a mathematical representation of a conservation
principle known as Reynolds transport theorem.
We will stop here today and in the next class
we will see that how to apply this theorem
for various applications of conservation,
and which leads to different conservation
equations that we will use for CFD applications.
