Welcome back to the last lecture of this module
computational fluid dynamics and the last
lecture we ended at a point where we saw the
Reynolds shear stress equation. We discussed
about the different terms in it you do not
need to worry about the mathematical equation
of those terms, but some common terms like
Reynolds shear stress, those things are recommended
that you remember them, but not the complex
terms. So, the up in proceeding in the direction
we are going to study the closure problem.
So, the effect of the Reynolds shear stress
rho in tau i j as you can see on the slide
now the effect of rho in tau i j on the mean
flow is like that of a stress term.
So I am going to write here rho in tau i j
is actually Reynolds shear stress. So it is
like shear stress to obtain u i and pressure.
So, u i is the average flow velocities and
pressure from RANS equation, we need to model
this shear stress rho in tau i j is a function
of the average flow so if you look at the
equations. So take you back to this.
Where we have written the Reynolds average
here you see this particular equation this
is average quantity that we need to find out
this we need to find out these are the independent
variables. D i j we also know because this
is the average value and it comprises of only
this is the only term that has fluctuations
and this increases our complexity. So, we
need to be able to equate tau ij to something
that is known known or some unknown which
we are actually calculating.
So, the best ways to put it in some form of
an average value and to do that, that particular
problem is called a closure problem. So, we
will go back to the closure problem so as
I said, to obtain u i and pressure from Reynolds
equation, we need to model this Reynolds's
shear stress given by rho into tau i j as
a function of average flow, this will remove
the fluctuations and this process is called
the closure problem.
There are elegant ways the first of the methods
is called the k epsilon model. So, this in
this technique the model focuses on the mechanism
that effects the turbulent kinetic energy,
k stands for kinetic energy. So, the instantaneous
kinetic energy k as a function of time k t
of a turbulent flow is the sum of the mean
kinetic energy and the turbulent kinetic energy
k. So, k of t can be written as capital K
+ small k.
So, this is small k 
and capital K is written is simply half u
square + v squared + w squared actually u
bar v bar and w bar and small k is written
is half of u dash squared + v dash squared
+ w dash squared.
The governing equations for this are one we
have continuity equation and the other is
so, the way we write is So, D D t of tau i
j that is what we are modeling is -1 by rho
del p del x i + del x j into 2 nu + nu t into
del i j. How does equations have been derived
we are not going to discuss but these are
the 2 equations that we use for momentum.
See, it looks like a momentum equation itself
so modeling of the Reynolds shear stress in
as the momentum equation.
So, if you see there is a term called nu T.
So, this nu t is the eddy viscosity. So, this
actually is not should better be called as
turbulent eddy viscosity, nu T, it can be
actually dimensionally related to kinetic
energy k kinetic energy dissipation rate epsilon
through nu T is written as c mu into k square
by epsilon. So, nu T we got also able to find
on infinite terms of k and epsilon and we
substitute this into this equation and C mu
is a non-dimensional constant. Which values
we know from experiments for uniform and isotropic
turbulence there is no production or diffusion
of turbulent kinetic energy, therefore, del
k delta t will be equal to - epsilon E.
The turbulent kinetic energy and the energy
dissipation can be determined from the following
equation. So, now, we said that this is the
equation which we are going to use for determining
tau i j and that we can use in our average
equation Reynolds average Navier stokes equation,
we saw that there are some terms one is nu
t that we do not know again others are again
like del p bar, D i j bar. Those are known
means do quantity that are in terms of average
quantities.
So nu T is something that we yet do not know.
And we write it nu to in terms of C mu into
k square by epsilon. So, again, we have 2
things to again find out k and epsilon we
still we do not know so, the next step is
finding out this turbulent kinetic energy
k and epsilon that that is how this equation
gets in the model gets its name. And to be
able to do that we have 2 another equation
1 is in terms of kinetic energy 1 is in terms
of epsilon. And this is production of turbulent
kinetic energy this term here. So, we use
these 2 equation to solve for k and epsilon.
And the values of the constant in above equation
are C mu = 0.09, sigma k = 1 if you see, C
mu was there first, and then there is sigma
k here then there is sigma E here and also
there are some terms there is sigma there
is the C epsilon 1 there is C epsilon 2. So,
those things are already known these are the
values that we use so, what we do is we solve
the k and epsilon equations use those equations
to find out the turbulent kinetic turbulent
eddy viscosity nu T put it into their an Reynolds
shear stress equation.
And use it in the average equation of Reynolds
Navier Stokes equation. So, this for just
writing it down solve for k and epsilon then
nu T is related to k square by epsilon. Nu
T, put this in Reynolds shear stress equation
and use that in Reynolds average Navier stokes
equations. It is a quite a complex way, but
it gives good results.
Most of the turbulence model in the world
follow k and epsilon, there is one other turbulence
model for k omega. So, k is the same kinetic
energy omega is somehow related to dissipation
epsilon, but that is also the outside the
scope, but it is better to remember the name.
So, the other model is k omega model like
epsilon, these are the two world's most widely
used turbulence models, and some of each of
them have their advantages and disadvantages.
So, another method apart from Reynolds average
navier stokes equation for solving the turbulence,
for solving the computational fluid dynamics
navier stokes equation is called direct numerical
simulation. So, in direct numerical simulation
DNS navier stokes equation or simulated numerically
without any turbulence model. So, they solve
for the exact solution when the governing
equations of turbulent flows are discretized
with sufficient spatial resolution and high
order numerical accuracy, it is known as full
turbulence simulation FTS.
So, in this modeling Reynolds number is expressed
as UL by nu, where R e represents the ratio
of the inertial forces to viscous forces or
Reynolds number, use the characteristic velocity
L is the characteristic length or Reynolds
number can also be written as square by nu
divided by L by U. Numerator and denominators
both have dimensions of time so, this L square
by nu is the characteristic timescale for
viscous diffusion whereas L by U is the characteristics
timescale for advection.
The magnitude of inertial terms are much higher
than viscous term in high Reynolds number
flows that is true, because it is the ratio
of viscous diffusion because the ratio of
the inertial forces to viscous forces. So,
can we conclude that viscous effects are unimportant
in turbulent flows no it is very important
because turbulent energy is dissipated in
form of heat by viscous effects? Therefore,
the viscous effects are very important in
high energy turbulent flows.
To sustain the fluctuations the supply of
kinetic energy must be balanced by these dissipation
of turbulent energy this is important thing
supply of kinetic energy must be balanced
by the dissipation of turbulent energy. So,
from the above consideration what we get is
all definitely outside the scope the derivation
of this that L by eta is of the order of Reynolds
number to the power 3 by 4 this this has been
obtained and this eta is a Kolmogorov length
scale. So, it is the length scale at which
the energy is dissipated with viscosity nu
you should remember the order and everything
but the derivation is not required.
So, what can we convert conclusions can be
drawn from the above expect expression this
is important so, L by eta so length of the
flow divided by the Kolmogorov length scale
at which energy is dissipated is the function
of Reynolds number to the power 3 by 4. So,
if this is the equation what are the conclusions
from it? This means that the length scale
at which the dissipation takes place is much
smaller than the characteristic length scale.
L by eta is proportional to R e to the power
so in turbulent flow and Reynolds number is
very high.
So that means L by eta is very high. This
means numerator is much, much larger than
the denominator. That is implying the length
scale at which dissipation takes place is
much eta is much smaller than the characteristic
length scale L. Valid conclusion from the
equation. The ratio of these 2 length scale
is proportional to R e to the power 3 by 4
as we have seen in the last slide.
Second thing is the energy is passed from
large vortices, that means, the flow is of
the scale of length of L and T to smaller
vortices that is, so, because see the dissipation
happens at the Kolmogorov length scale. So,
the eddies become transfer energy to each
other they keep on losing the energy you know
and until they become they come into range
of this Kolmogorov length scale and at this
point this energy is dissipated through the
heat.
So, in order to simulate all the scales in
turbulent flow So, now you see we have a length
of scale from capital L to eta it is Kolmogorov
length scale which is of the order of several
order of magnitude less than millimeters the
computational domain must be sufficiently
large than the characteristic length scale
L . So, to see there will be what is this
which will be of the length L as well so,
the computational domain must be sufficiently
large than the character length L.
But also it is important that the grid size
must be smaller than this may then this then
this it is very small of the order of 10 to
the power -5, 6 meters and length the scale
of the typically of the order of let us say
1 or 2 meters or 10 to the power 1 meter,
let us say so to be able to model all the
type of energies energy in direct numerical
simulation, the two things that are important
is that the total domain means the study area
should be larger than the length L which is
of the order of meters several time 100 of
meters but the grid size must be smaller than
this Kolmogorov length scale which is very,
small.
The implication of this is that for 3 dimensional
simulation of turbulent flow we will require
at least L by eta to the power 3 that means,
Reynolds number 2 the power 9 by 4 grid points
let us say R e is very, not very high Reynolds
number let us see Reynolds number is 10 to
the power 4. So, how many grids is needed?
R e that is 10 to the power 4 raise to the
power 9 by 4. So, we need 10 to the power
nine grids almost so much high more than billions.
You know around billion grids, which currently
looking at the capacity of our computational
facilities it is not possible. So, these things
like R e to the power 9 by 4, R e to the power
3 by 4 those are the term which you must the
values which you must remember for the computational
cost of DNS is very high.
Another such technique is called Large Eddy
simulation. See in the DNS one important thing
to note was that we had the best accuracy
but lot of computational time is required
LES is sort of a tradeoff between the Reynolds
average in Reynolds average we do many approximations
so the results are not that accurate compare
to DNS, but LES is something which is a tradeoff
between DNS and Reynolds average navier stokes
equation. So there is a big difference in
the behaviors of large and small IDs in turbulent
flow field.
We were talking in DNS about the length scales,
we said that there will be vortices or a LES
that are as big as the length of the flow,
there will be LES at the time of dissipation
if heat will be very, very small, let us say
order of 10 to the power -5 -6 which also
means that there is actually big difference
in the behavior of these Eddies large Eddies
will have a different behavior and smaller
these will have a small bit, I mean, a different
behavior.
So what are large Eddies they are large, Eddies
are more, anisotropic. And their behavior
is dictated by the geometry of the problem
domain. And the boundary conditions the larger
Eddies, they also must depend upon the they
will also depend upon the body forces acting
whereas small Eddies they are nearly isotropic
they are very small they have not generally
a universal behavior as demonstrated by Kolmogorov
which we have not read in this course.
But just take it for granted that these have
a universal behavior important thing to remember
is that the large eddies extract energy from
the mean flow, so, larger that is more than
energy will be there, and they take energy
from the mean flow that the flow real flow.
Whereas as small eddies take energy from the
a little bit larger eddies which takes more
energy from the larger Eddies than them. So
energy is in form of a cascade. So, this this
term is called Kolmogorov hypothesis. Again,
I am telling it is outside the scope, but
it is better to mention that this is actually
a major hurdle in the path of research just
trying to develop a universal turbulence model,
which is what is the hurdle the variation
of eddies from large scale to small scale?
And the fact that the largest scale eddies
are not universal in nature only smaller eddies
are a single turbulence model must be able
to describe the collective behavior of all
the eddies. However, in the dependence of
large eddies on various parameter complicates
the problem smaller eddies no problem because
it is universal.
So, in LES the larger eddies are computed
with a time dependent simulation where the
influence of this small eddies are incorporated
through turbulence model. So, we say in LES
2 parts large Eddies and there are small eddies.
So they are solved through time dependent
simulation and we account for everything for
all large eddies but small eddies are so much
the, the effects of the small eddies is taken
through a turbulence model.
(Refer Slide time: 21:48)
So our domain size does not is not mean we
do not need to be, you know, accounting for
the smaller eddies. We make our grid size
that it captures the largest of the eddies.
So, LES uses spatial filtering operation to
separate the large and the small eddies as
I have told you and the filtered navier stokes
equation are used as the governing equation
for large Eddy simulation.
Now, what would be the filter means up till
what time what lens should we cut it the filter
width is set to be close to the size of the
mesh. There are some terms that when we LES
when we use the size of the mesh 1 is the
Grid Scale GS. The scales that are directly
solved for on the grid are called the grid
scales for large eddies they are they are
the grid scales are used for the large eddies
and for the smaller one the Subgrid Scales
SGS.
So, small scales are not captured by those
grid correct because they are larger in size
to capture those eddies. The grid should be
at least more I mean smaller than those so
sub grid scale modeling is used for smaller
eddies.
Now the filtering the filtering operation
that we have talked about is defined by a
filter function g of x, x dash, delta as it
is a very complex but there is a filtering
operation that you must know that in LES we
use a filtering operation to cut off the smaller
eddies and solve in reality for the larger
eddies and then approximate this smaller eddies
is using the turbulent model. So this is a
filtered function delta is a filter with which
is mostly said to the size of the mesh.
And this is the unfiltered function, phi x
dash t. The over bar in phi bar x t denotes
is spatial filtering and not time averaging.
This is very important. So until now, when
it there was bar we used to do average in
time, but this 1 is spatial filtering spatial
filtering. So, average in space.
So examples of filtering functions are there
is a top hat or box filter, where g x see,
you see there is a G here, this is a filtering
function and this therefore this is a filtered
function. So g xx can be given by different
box filter, it is a different study in its
own is a Gaussian filter. So remembering the
name of the filter is good enough you do not
need to remember those equations. So for 3d
computation, the filter with this is important.
The filter of it is given by this you must
remember that delta is given by cube root
off under root x, root y and root z where
delta x, delta y and delta z or the length
width and height of the grid cells respectively.
So, you saw what G is and what delta is this
delta is here.
(Refer Slide time: 25:19)
So, now coming to the governing equations
of LES the filtered momentum equation using
the grid scale variables can be written so,
this is again it is a space averaging. The
influence of sub grid scale eddies introduced
through SGS stress, sub grid scale stress
as SGS and that have written as tau i j you
see here is a term tau i j just similar to
the Navier stokes equation and that is written
as i u j u i u j whole bar - u i bar into
u j bar, these are against space average in
LES.
Where tau i j is L i j + C i j + R i j. The
L i j is the Leonard term given by this there
is a cross term called C i j the SGS Reynolds
Stress R i j.
So, this gives you an overall idea the differences
between the Reynolds shear stress in a little
bit more detail. The DNS the idea of DNS,
why it is computationally so expensive has
been reduced. We went through the term like
R e to the power 3 by 4 number of grid points
are equal to R e to the power 9 by 4. The
basic idea of LES where we studied that we
basically use the filtering function to model
the larger these various sub grid scale modeling
is used to model the smaller Eddies.
So, information like this what are the different
terms Leonard term, Cross term and SGS Reynolds
stresses and things like that have been covered
in the turbulence modeling part. Many complex
equations are not supposed to be remembered
by you, but an overall idea about those equations
or whatever required in this particular module.
And this actually concludes our module on
computational introduction to computational
fluid dynamics.
I hope you enjoyed this particular session,
this particular module of the course and as
always, these are the references of then I
mean the reference books that you can actually
use. And thank you so much for listening and
I will see you next week.
