Welcome to the lecture on Introduction to
Turbulence in Fluid Flow. So, we had the discussion
about the governing equations for fluid flow,
and when we talk about fluid flow then in
that case the flow may be laminar or may be
turbulent and we need to know you know things
about turbulence because most of the flows
which we will deal they will be of the turbulent
nature.
So, what are the essential terminologies?
And what turbulence means? What needs to be
modeled while we are solving for the you know
fluid flow problems in the tundish or in any
vessel where the flow is turbulent? So, we
will have some introduction about these terms
and we will know more about these turbulent
flows modeling and their associated terms
you know in our coming lectures.
So, as you know that when we talk about the
flow then we basically decide by the Reynolds
number whether the flow is laminar or turbulent.
So, the Reynolds number value, so that will
be less, in that case it will be laminar and
if it is more, so at low Reynolds number flow
will be laminar and at high Reynolds number
flow is observed to be you know turbulent.
So, your Reynolds number for that the expression
will be re equal to u l upon eta. So, you
will have these characteristic velocity characteristic
length and then the kinematic viscosity of
the fluid in the denominator. So, that way
you get these value of the Reynolds number
and then you differentiate whether the flow
is laminar or the turbulent.
So, certainly, there will be also a situation
when there will be transition from laminar
to turbulence, so that will be you know another
thing which will be there. Now, Reynolds number
of flow gives a measure of relative importance
of inertia forces and viscous forces. So,
you know as we know that Reynold number is
inertia force upon viscous force, so if it
is of smaller value it means viscous forces
are more dominant. And if the Reynolds number
is very high it means inertia forces are dominating,
so that way you know whenever you will have
those reasons, so depending upon the Reynolds
number we can say that which kind of forces
are you know dominating which is more you
know prominent.
Now, what happens that when we talk about
the turbulence? So, basically you will have
fluctuations. Now, the difference between
the laminar and turbulence is that in case
of turbulence what was happening that those
length scales become quite large over which
the mixing takes place. There will be fluctuations
in the variable values like velocity, pressure
or source, so they will be fluctuating over
the mean values. So, in fact, the fluctuation
in the velocities will give rise to another
additional stress which we have seen that
certainly that, in earlier lectures we have
seen that you have stress terms and they are
expressed in terms of you know velocity gradient
term and all that.
Now, in this case your fluctuation terms which
you are encountering because of the turbulence,
now they lead to extra or additional stress
terms. So, they are known Reynolds stress
terms. So, they need to be modeled, they need
to be taken into account. So, so these things
are important when we talk about the Reynolds
number of flow and as you see that you will
have depending upon the number you see that
which force is important, now you know that
you have a critical Reynolds number. So, before
below that it is laminar and above that it
is you know turbulent.
So, when the value of the critical Reynolds
number, I mean value of the Reynolds number
is below the critical Reynolds number then
the flow is laminar and in that you have the
adjacent you know when the flow is smooth
and adjacent layers of fluid slide past each
other. So, that is what the new traits of
the laminar flow is that your flow will be
smooth and your adjacent layers will be sliding
one above other. So, they will not be mixing
you know across you know in the cross devised
manner.
So, they will be going and then there may
be some diffusion of the atoms will be taking
place you know from the bottom layer to the
top layer or top layer to the bottom layer
like that. So, that is what is happening in
the case of the laminar flow. And in that
case if your applied boundary conditions are
not changing with the time then we call it
the flow is steady and the regime will be
the laminar flow.
Now, there is another regime that is your
turbulent regime and that is that you are
encountering when you see that the Reynolds
number value is going you know beyond a certain
value that is critical value and in that basically
there will be complicated series of events
will be taking place. So, you that I mean
there will be radical change of the flow.
So, you will have a lot of you know events
taking place in that and basically what happens
that finally, the behavior of the fluid that
becomes a random and chaotic.
In this it is a in a in a in a in a very defined
manner the flow is flowing one above other
flow is there, but in that case it becomes
very chaotic as well as random, there will
be mixing, there will be eddies, smaller eddies
will be there, larger eddies will be there,
so all the diffusion which is taking place
they will be at higher scales. So, because
the there may be you know diffusion or the
mixing over the larger you know bulk, so maybe
if you have 1, 2, 3 or 4 and 5 layers, so
maybe 1 and 5 layer may mix because of these
turbulence. So, that is the basic difference.
So, your behavior that that is what it there
are radical changes in the flow character
you have your flow becomes random and chaotic
and you know that regime which you get that
is known as the you know turbulent flow.
So, if you try to see the you know the velocity
measurement in case of a turbulent flow then
the velocity measurement basically will be
like you will have the time here and if your
velocity is here; so, what happens that your
flow will be, so suppose this is u, so your
actual flow will be like it will be going
like this. So, it will be going, it will be
fluctuating. So, in normal case you can assume
it to be you know fluctuating over a certain
mean value. So, that is. So, you will have
a steady state mean value that is u. So, you
will your you know this value that is will
be in the case of turbulent flow you will
have a mean value and you have the fluctuating
component.
So, basically at any time u t will be u, so
that will be your mean component and then
you will have a fluctuating component, so
that is u prime t. So, that way the turbulence
you know in case of turbulence the flow is
basically characterized, and you will have
this for all the properties like either as
u or v, t or w or p. So, for all that you
will have a mean value and a fluctuating component
this component basically you know that component
is this component above the mean that is the
fluctuating component that is u prime t. So,
that way your so you will have many statistical
descriptors will be used while we try to know
more about the you know the turbulent flow.
Now, while in normal case when your you will
have two space dimensions for the velocity
of pressure now turbulent fluctuations when
there are they will have the 3-dimensional
character. So, that is the trait about the
turbulence. Now, in this case what happens
that you have the presence of eddies and if
you look at the you know turbulent flow. So,
in that flow structures you will have the
presence of eddies.
So, these eddies may be of you know smaller
dimension or the larger dimension and they
can be or in of size, so they can maybe of
a smaller size, intermediate size or even
the larger size may be to the extent of the
length of the vessel or so. So, that way you
will have the presence of eddies and in that
because of these eddies you know the point
which are largely separated in the vessel
they also come in intimate contact because
of the flow structures in the case of turbulent
flow and there will be very high value of
diffusive transfer as in transfer of mass,
heat or momentum in the case of turbulence.
So, that is the basically trait about the
turbulent flow. So, you will have the production
of or the genesis of very high value of the
coefficient of diffusion when we talk about
the turbulent flows.
Now, the thing is that in this case you have
the characteristic velocity and the characteristic
length, and they are of the larger eddies
are basically same of the for the larger eddies
suppose the characteristic length that is
l will be same as the you know length scale
l of the mean flow. So, similarly the velocity
also characteristic velocity for the larger
eddies will be same as that of the mean flow
velocity u. So, that is normally there in
the case of the turbulent you know flows.
Now, when we talk about the other scales or
the other properties, suppose if you go for
those when the turbulent is very small, suppose
turbulent if you talk about the Reynolds number.
Now, in the Reynolds number you have the ratio
that is inertia force by viscous force. So,
when your inertia force and viscous force
will be somewhat nearly equal in that case
Reynolds number is close to 1, so that tells
that your viscous force is quite high. So,
it is something that that is why it is close
to about 1. Otherwise when you go in the upstream,
so that is your depending upon, so if you
take suppose the u l upon eta, if the u is
1 meter per second, l is 1 meter and your
eta in normal case becomes 10 raised to the
power minus 6 or so, so that in that also
you see that the value of the Reynolds number
becomes 10 raised to the power 6. So, that
is how you know the, so that will be inertia
dominated kind of flow.
So, we will come to this further if you try
to know about the turbulence. So, now, you
will have the transition to turbulence and
in the case of a different kind of flows you
will have a Reynold number which will say
that here the transition takes place and you
know in the case of pipe flow transition you
will have the category of flow without inflexion
point.
So, basically there are two cases there may
be the presence of inflexion point when you
have the transition to turbulence or you may
have the case without the you know the presence
of these inflexion points. So, that way you
will have, so that depends upon the different
situations.
Now, the viscosity of hydrodynamic stability
predicts that these flows are unconditionally
stable to infinitely disturb to disturbances
at all Reynolds number in case of pipe flow.
And in the case of pipe flow, the transition
to turbulence will be taking place between
Reynolds number 2000 and 100000. So, basically
normally what we might have studied about
the transitions in case of these you know
pipe flows.
Before that when we are talking about the
turbulence. So, in that we talked about these
smaller eddies and the larger eddies, and
the you know the larger it is basically they
will be dominated by the inertia effects and
the viscous effects are negligible in that
case. So, you know, so normally this larger
eddies which are there, so these are basically
inviscid, are effectively inviscid.
So, basically if you look at the values u
l upon eta, so in that case if you take u
and l and what we saw, so if you are taking
that length as a characteristic length and
we are taking larger eddies into account in
that case those the Reynolds number becomes
very high, so inertia force is dominating.
So, in that case your the flow is inviscid,
viscous forces are negligible, their effect
is negligible. So, that is why we take it
as the inviscid.
Now, in those cases your angular momentum
is conserved and that is why that leads to
you know I mean during that vortex stretching
this angular momentum is said to be conserved.
So, that will you know lead to the rotation
rate to further you know increase and the
you know radius of the cross section will
decrease.
So, that way you will have that kind of flow
structure which will be you know coming up
in the case of the turbulence and you know
you will have the and also what happens that
you will have the mean flow, you will have
the larger eddies, you have a smaller eddies.
So, they will be deriving the energy from
the mean flow, so that way and also the turbulence
is said to be self dissipating also. So, those
terms we will be discussing you know later.
Now, coming to the you know the turbulence
and the transition to the turbulence if you
look at. So, this we discussed that for the
pipe flow that was the case where and for
different kind of flows you will have the
in different way you will have the you know
transition taking place and we are discussing
that when we talk about the you know velocity
profile where that will be susceptible to
the different kind of flows and creating the
instability.
So, the inflexion point which we were discussing
that you will have you know when you may have
the you know instability and the velocity
profiles may be like if your velocity profile
goes like this, so basically this is the point
of inflexion. So, this is your y and this
is your velocity.
Similarly, you know when you have the, you
can have the transition of the turbulence
also, transition to turbulence even without
the inflexion point, so your situation may
go like this. So, that maybe in this case
you know you have this is your inviscid instability.
So, in that case you know viscous effects
are negligible and this is your viscous instability.
So, this is basically the you know velocity
profile which you see. So, this is with the
presence of inflexion point and this is without
the presence of the inflexion point. So, that
is what is normally happening when we have
the transition to turbulence you know taking
place.
Now, that is basically for the different you
know there may be different kind of flows,
you have jet flow, you have also the, so in
that jet flow and then you have the mixing
or you have the you know many cases, so mixing
or flow behind the wake and all that. So,
in all those cases you will have you must
have the idea about how you know the turbulence
is takes place, how there is you know how
I suppose in jet.
So, it will be moving this way moving fluid
and that will be interacting with the stationary
fluid which is there, so accordingly you will
have a deformation and all that. So, these
needs to be you know we expect that you have
some understanding of these phenomena and
that will help you basically, in understanding
basically the terminologies which will be
coming later related to the turbulence.
Now, coming to some descriptors of the turbulent
flow. So, when we as we discussed that in
the case of turbulent flow you have a mean
component and one is your fluctuating component.
So, you know, so what we do is normally the
mean phi of the flow property, so you will
have the flow property phi and this will be
defined as the mean phi plus phi prime t.
So, that way we define.
So, basically for the for the turbulent flow
you know for any property for turbulent flow
when we talk about any property of phi at
time t, so this phi at you know time t. So,
that will be defined as the mean value. So,
that is phi, so and plus you will have the
fluctuating component of phi dash phi prime
t. So, that way your this is how the property
will be there in the case of the turbulent
flow.
Now, the thing is that for as far as this
fluctuating part is there. So, you know for
this the if you take the time varying component
and if you take its value, so there, so basically
its time mean will be 0 for the fluctuating
component. So, so you will have the other,
there are different ways to express this and
one is the time average or mean. So, for any
property you know phi the mean phi that mean
will be denoted as, so that will be basically,
so that is shown here. So, your this mean
of the flow property is defined as we are
taking 1 by delta t and 0 to delta t, you
have sine of phi t delta t. So, that way we
define the mean of the flow property.
And then you have the you know fluctuating
component also. So, you may you know we should
take that limit towards the infinity you know,
but then you will have, but you have to take
the delta t you know cautiously because you
have certainly a size of there is a limitation
on that limitation is put by the size of the
eddies. So, that you know slowest variation
will be because of the largest eddies. So,
accordingly you will have to take the delta
t.
Now, if you take the time average value of
the fluctuations. So, by definition the time
average value of the fluctuation will be taken
as 0. So, if your the fluctuation which we
have seen in the first case that you have
the mean value and then fluctuation taking
place, so basically it is assumed that the
if you take the time average of this value
it will be coming to 0. So, that is why you
have a mean value and then you have the time
average value. So, we write normally when
we do not take the t into every time for t
notation, so you will be writing phi equal
to mean phi plus, phi prime, so that way we
are writing you know these values.
Now, apart from that you know you have other
properties. So, other properties will be your
now this fluctuation component of the suppose
velocity we are taking into account. So, this
fluctuation component will also be leading
to the energy also. So, the total kinetic
energy that. So, these fluctuating parts which
will that give rise to the kinetic energy
they are the turbulent kinetic energy known
as.
So, you will have the turbulent kinetic energy.
So, this turbulent of kinetic energy then
you know per unit mass. So, this will be by
the respective velocity fluctuations and that
will be equal to half of u prime square plus
v prime square and plus, so you will have
a square and then and then w prime square.
So, that way this term because of this fluctuation
part which is squared. So, that gives you
basically the turbulent kinetic energy you
know per unit mass and this is basically to
be taken into account and we deal with the
turbulent flows modeling in that case this
will be used.
Similarly, you have based on this you define
the turbulence intensity. So, that is basically
you know defined, and turbulence intensity
will be the average you know rms, so it will
be depending upon the you know rms velocity.
Now, before that we need to know what is the
you know variance and rms.
So, you know first of all the we are talking
about the variance, so the variance we will
be we define as 1 by delta t and phi square
phi prime square dt. So, that is how the variance
is calculated. You will have the fluctuation
part, it is it is square, and then we are
dividing by delta t if you are integrating
over delta t.
So, this way we are getting the variance and
the rms value of this fluctuating part, so
that phi rms that will be the square root
of this. So, this is your you know square
root. And basically what happens that before
that we need to understand that the u prime
square or v prime square or w prime square
bar on all these components so, they give
rise to the some of the you know stress terms.
So, you know, so that is why and these are
and these stresses which are because of these
fluctuation components we call it as Reynolds
stresses that is what we discussed. And then
we defined this turbulent kinetic energy that
will be K equal to half of u prime square
plus v prime square by w prime square. And
the turbulence intensity, that is the average
rms velocity divided by the reference mean
flow. So, this will be you know average rms
velocity divided by reference mean flow, so
mean flow velocity.
So, if you take reference mean flow velocity
as we u reference, so what you do is you can
derive it through this K, so that will be
2 by 3 of this K. So, so T i will be 2 by
3 k and its half because you will have a one
velocity component, so you will have 3 by
2. So, 2 by 3 you multiply and, so it will
be velocity component that is your rms velocity
and then it will be divided by the u reference.
So, this way you define these you know turbulence
intensity and this will be one of the you
know parameters which will be used when we
are going to give the boundary conditions
in the case of the flow where the turbulence
will be used.
Apart from that you have even the terms like
the moment of the these different fluctuating
variables, so that will lead to the terms
like u prime, v prime bar or v prime, w prime
bar that leads to these you know stresses.
So, so that those terms those will be the
additional shear stresses. They will be coming
up you know when we deal with turbulence.
So, when we go to deal with the Navier-Stoke
for that Reynold averaging Navier-Stoke equations,
where we take this turbulence flow into account
in that case we will see that these terms
will also come into picture that will be the
extra Reynolds, that is shear stress that
will be you know coming up.
So, accordingly now you know these terms need
to be understood and that we will see now
you know more of these terminologies in our
you know coming lectures.
Thank you very much.
