Welcome back students in this lecture, we
will we are going to continue what we started
in the last lecture with the grid generation.
So, we finished that this slide. Therefore,
we are going to proceed from this point onwards
and see what further analysis can be done.
So, we talked about grids so, there are actually
2 types of grids. One is a structured grid
and the structured grid means the grids are
regular and coherent structure to the mesh
layer. These are the simplest the structured
grid and they are generally uniform rectangular
grid those are called the structured grids.
So, the structured grids look like this one
here. So, you see they have a uniform rectangular
grid. So these all the small shapes are rectangular.
Structured grids are not limited to rectangular
grids only. So, these ones this could be off
actually of any shape here it chose the rectangular
grids only but you see these rectangular grids
are decreasing in size.
The second one are the unstructured grids.
So, the grid cell arrangement is irregular
and has no symmetry pattern if you see in
the last one, there was a symmetry pattern
here. If you consider this one you see symmetry
cause this and here it is completely symmetry
this one in the unstructured grids, the cell
arrangement is irregular and has no symmetric
pattern something like this, so, see the triangles,
these are off, no specific, same type. You
see this the cells here are very small, smallest
smaller than these ones here largest.
So, now after this regeneration we comes to
the solver stage. So, in this stage that the
governing differential equations are solved
by an approximate numerical technique after
specifying the boundary and the initial conditions.
So, the actual real solution of those differential
equations is done at this stage called the
solver stage and in post processing, the extraction
of results and visualization how the results
appear is done. So, when it comes to extraction
of results and visualization flow field variables
are plotted and analyze graphically. So, this
is the most common thing to do in the post
processing.
So, you use you know we mentioned about boundary
and initial conditions and that is what we
are going to discuss next. Boundary condition
the same governing equations are valid for
all compressible Newtonian fluid flow problems.
So, if the same equations are solved for all
types of problems, and how can we achieve
different solutions for different flow situations
involving different geometries that is the
question because which governing equations
are we going to talk we are talking about
the continuity equation, Navier stokes equation.
So, this is common for all the fluid flow
problems? Then how do we know and how do we
achieve different solutions for different
flow situations involving different geometries
and this happens because of something called
boundaries conditions.
So the answer is boundary conditions of the
problem. So, boundary conditions, I will tell
you what that is for example, if the flow
is occurring in a tank itself the velocity
is 1 meters per second, this is one continuously
occurring. So, this is 1 boundary condition,
there could be other where you know the velocities
could be 10 meters per second for example,
so, these are different boundary conditions
that is why we get different results.
Suppose, 1 of the boundary conditions could
be this is closed at this end and the other
one that it could be open. So, this is another
boundary condition so, because of this difference
in boundary conditions we have different solutions
of the problem.
Now, as I said I mentioned about being closed
and you know being open there is a wall at
the boundary these wall and other concepts
you have read in your previous lectures of
hydraulic engineering. So, we are going to
talk about the wall boundary condition, since
fluid cannot pass through a wall the normal
component of the velocity relative to the
wall is set to 0 and this is what is this
called this is called no slip condition. Therefore,
due to the no slip condition the tangential
component of the velocity at a stationary
wall is set to 0.
So, you see there if there is an inlet here,
there is a wall here there is an outlet. So,
the velocity at this point does not matter
what the velocity here is at this particular
point anywhere around across this wall the
velocity will be 0 due to 
no slip condition.
No not talking about the inflow and the outflow
boundary conditions. So, you see this is inlet
this is outlet. So, the here from here the
inflow will be there, and from here out flow
will be there. In means coming in and out
means going out. So, at a velocity inlet or
outlet the velocity of the incoming or outgoing
flow specified along the inlet outlet phase
at pressure inlet and outlet the total pressure
along the inlet and outlet phase specified.
So, the in the inlet and outlet can be the
specification can be in 2 forms, whether if
whether we want to specify the inlet flow
velocities at the inlet or outlet or the pressure.
1 typical example here is the for the pressure
is the pipe flow. Here it is open channel
flow.
So, there is one question which I would want
to know, solve solver discuss. So there is
a CFD code which is used to solve a 2 dimensional
in 2 dimension X and Y incompressible laminar
flow without free surfaces. The fluid is Newtonian.
So appropriate boundary conditions are used.
So the question here is now lists the unknowns
in the problem and list the corresponding
equations to be solved by the computer.
So first important information that we have
is that it is 2 dimensional in nature. So,
the unknowns, you can start guessing the unknowns
first is going to be velocity you which is
along X direction. Secondly, the velocity
V, which is along Y direction. The third one
is going to be since there is no more third
dimension there is not going to be a W direct
W velocity, but there definitely will be p
that is pressure and how are we going to solve
these which equations are we going to solve
for U V and P.
We are going to use 3 we see there are 3 equations.
So, the first equation is very common conservation
equation which is the continuity equation.
The 
second one will be the X momentum equation
so momentum equation in X direction. We will
have a Y momentum equation. So, this is very
quiet simple to you know imagine or guess
that in 2 dimension only 2 dimensional velocities
and the pressure is going to be the unknown
and therefore, to solve we just need the three
equations continuity, x momentum and y momentum
equations.
So, now, we come to the, in the solver stage
we come to we have this way we have mentioned
about the boundary condition and now we come
to discuss about the partial differential
equations. So, partial differential equation
PDE is an equation stating a relationship
between a function of two or more independent
variables and the partial derivatives of this
function with respect to the independent variable.
So, this is the definition of partial differential
equation.
So, it states 
it states the relationship between a function
of two or more independent variables and the
partial derivatives of this function with
respect to the independent variable. For example,
this equation del square f by del x square
+ del square f by del y square = 0. So, these
are independent variables x and y and f is
an independent variable and this equation
is called Laplace equation from your previous
experience you should be knowing this another
equation is del f by del t = alpha del f squared
by del x square.
So, this is a diffusion equation the third
equation which we have written is del square
f by del t square = c square del square f
by del x square and this equation is a wave
equation. So, these are the 3 most common
types of equation in partial differential
equation that we know.
So, this is solution of the partial differential
equation is that particular function f x y
or f x, t, which satisfies the partial differential
equation in the domain of interest domain
is given by D x y or D x, t, f x y is when
the function is based only on x and y, f x
t when if it is dependent on only x and t.
Further f of x, y or f of x, t. These are
2 different type of partial differential equations
f x, y or f x, t they also satisfy the boundary
and our initial conditions specified at the
domain of interest. So, they must be satisfying
they must be satisfied at all places in the
domain or and at all times for which the calculation
is being done.
So, going to take a slight detour, because
it is important that we try to remember what
the classification of partial differential
equations look like. So, they are general
quasilinear second order non-homogeneous partial
differential equation in 2 independent variables
can be written as Af xx + Bf xy + Cf yy +
Df x + Ef y + F into f, this is a general
quasilinear second order we have x x, x y
and y y that is second order non-homogeneous
or partial differential equation.
So, A, B and C may depend on x, y, f x and
f y, D, E and F may depend on x y and f. So,
this is this means quasilinear. G is non-homogeneous
term and may depend on x and y that is why
we read non-homogeneous this is not equal
to 0.
So, the classification depends on the sign
of the discriminant B squared - 4AC as so,
in this particular equation Af xx + Bf xy
+ Cf yy + Df x + Ef y + F f = G, the solution
will depend on the value of B squared - 4AC.
And how does it depend if B squared - 4AC
is less than 0, then it is going to be elliptical
partial differential equation. If B squared
- 4AC is going to be a parabolic partial differential
equation, if B squared - 4AC is greater than
0 then it is going to be a hyperbolic partial
differential equation. You this you can recall
from your mathematics class.
Now going to domain of dependence and range
of influence. So, if you consider a point
P in the solution domain, so, there is a point
in the domain P and let the solution at P
if we assume that the solution at that particular
point P is f x p, y p we assume that then
the domain of dependence of P is the region
of solution domain upon which f x p, y p depends.
So, region of influence of P the region of
solution domain in which the solution of x,
y f of x, y is influenced by the solution
at P which is f x p, y p. So, for an elliptical
partial differential equation the entire solution
domain is both the domain of dependence and
range of influence of every point in the solution
domain. So, there are 2 things domain of dependence
of P and secondly, there is a range of influence
of P which we have defined as the domain of
dependence of phase the region of solution
domain upon which x p f of x p, y p depends
whereas, the range of influence of phase the
range of solution domain in which the solution
of except of x, y is influenced by the solution
at p.
So, applying this to an elliptic partial differential
equation the entire solution domain is both
the domain of dependence and the range of
influence of every point in the solution domain.
So, the horizontal hatching here these ones
shows the domain of dependence whereas the
vertical hatching the shows the range of influence.
Now, for a parabolic PDE this is the domain
of dependence so, horizontal hatching whereas
the vertical hatching like these ones 
these are the range of influences. For a hyperbolic
partial differential equation, the horizontal
hatching shows the domain of dependence this
one whereas the vertical hatching shows the
range of influence. So, for 3 different type
of solutions we said profiles, elliptic PDE,
parabolic PDE and hyperbolic PDE.
For elliptic PDE we have seen that the domain
of dependence is the entire solution and also
the domain of dependence. Whereas, the parabolic
and hyperbolic PDE the horizontal hatching
as shown in the figures here shows the domain
of dependence whereas the vertical hatching
shows the range of influence.
Now the classification of the physical problems
can be classified into equilibrium problems
or propagation problems, the third is Eigen
problems. So, any physical problems can be
classified into 3 different forms.
What are the equilibrium problems are the
steady state problems in closed domain steady
state means, there is no dependence on time
there is an equilibrium. So, example is Laplace
equation of such type of problems. So, here
the solution f of x, y is governed by an electrical
partial differential equation subject to boundary
conditions specified at each point on the
boundary B of the domain.
So, solution of Laplace equation is governed
by an elliptic partial differential equation
is an important thing to remember. And of
course, the solution will depend upon the
different type of boundary conditions that
has been specified at each point on the boundary
B of the domain. So, we have to specify suppose
this is the you know, so, we have to specify
boundary here also at all those points 
something like this.
So, if this is the figure you see you see,
because this is an elliptical PDE the domain
of independent dependence and range of influence
are same all the domain and this is the boundary.
So, we should be able to supply the boundary
conditions at all the points, which is denoted
by B here.
The second type of problems or propagation
problems. So, an example is initial value
problems in open domains. So, open with respect
to one of the independent variables example
time the solution f of x, t in the domain
is marched forward from the initial stage.
So, we know things that time t = 0, then we
go from time t = 0 so let us say t = 2 seconds
then time t = 2 seconds and time t = 3 seconds.
So, the marching of the solution is guided
and modified by the boundary conditions. So,
if we have a different boundary conditions,
we still have to specify the boundary condition
at time t = 0 initial at initial point, and
we will also have to specify the initial problems
at initial values at all the points in the
domain. So boundary conditions and also the
value at all the points in the domain initially
to.
Example 1 here is the diffusion equation,
you see this is an open boundary and we have
to go from it is in x and this is in time
t, this is x direction. So, this are solved
by the parabolic partial differential equation
parabolic PDE diffusion equation, second example
is a wave equation. So, the wave equation
is solved by the hyperbolic PDE hyperbolic
partial differential equation. So, as we have
seen 3 different type of equations have different
partial differential equation profiles 1 was
elliptical, Laplace equation the diffusion
equation has parabolic partial differential
equation and a wave is hyperbolic partial
differential equations.
Now the last one in this set is the Eigen
problems. So, problems where the solution
exists only for special values of parameter
of the problem. So, it the solution will not
be there for all the values of the parameters.
So, and these special values are called the
Eigen values hence, these problems have involved
additional step of determining the Eigen values
is the solution procedure.
So, this is about the 4 type of problems that
we were talking so, now, from this point onward
we will proceed to the discretization technique
that is the in we start with the finite difference
method first. There are significant benefits
in obtaining a theoretical prediction of a
physical phenomenon. So, the phenomenon of
interest here are governed by differential
equations concept that is replacing the continuous
information contained in the exact solution
of the differential equation with discrete
values.
There is something called the Taylor Series
Formulation which we generally use, usually
finite difference equation consists of approximating
the derivatives in the differential equations
via a truncated Taylor series. So, how do
we approximate the derivatives in the differential
equation using a truncated Taylor series,
which looks like this? I am pretty sure you
have read that in your math class.
So, phi 1 is written as phi 2 - delta x del
phi by del x at 2 + half delta x whole squared
del squared phi into del x whole squared at
2. So, at series like this goes on with alternate
- and + signs. So, this is called the truncated
Taylor series or phi 3 can be written as phi
2 + delta x del phi del x + half del x squared
and it can go on like this.
So, truncating the series just after the third
term adding and subtracting the 2 equations
so, you see there were 2 equations phi 1 and
phi 3 both were return in terms of phi 2.
If what we do if we just do you know if we
first stopped the series just after the third
term and add and subtract the 2 equation then
we obtain del phi by del x at .2 will be phi
3 - phi 1 by 2 delta x and also del square
phi by del x square can be written us phi
1 + phi 3 - 2phi 2 by delta x square.
So, what we have seen here, we have used the
truncated Taylor series in general to write
these values phi 1 and phi 3 and phi 2 and
in once we add those and once we subtract
those we get terms like these so, the substitution
of such expression into differential equation
leads to finite difference equation.
Say like this so, analytical solution of the
partial differential equation provide us with
closed form expressions which depicts the
variation of the independent variable in the
domain, so, this is the domain here 
and the numerical solution based on finite
differences provide us with the values at
discrete points in domain which are known
at grid points. So, the difference between
the analytical or the mathematical solution.
What does analytical solution of the partial
differential equation provide us they provide
us with closed form expressions we depict
the variation of the dependent variable in
the entire domain. Whereas, using the numerical
simulation based on finite difference it provides
us with the values at discrete points in the
domain. So, it will provide us the value here.
Here, whereas mathematical solution will give
us closed form expression which is valid everywhere
not only at some points.
So, of course, if we are able to obtain the
analytical solution that is the best case
scenario, but in majority most of the cases
actually majority of the cases we are not
able to do that therefore, we resort to the
technique of finite difference method.
So, I think this is a nice point to stop and
in the next lecture we will start with elementary
finite difference quotients. So, thank you
so much for listening and I will see you in
the next lecture.
