Today, we will start with Boundary Layer Theory.
Boundary Layer Theory is one of the outstanding revolutionary theories that has come in the
history of fluid mechanics and we need to understand the perspective of this, before
coming to a conclusion how important or critical this theory has been. What is the motivation
of learning this theory? Well, starting from designing aircraft wings to understanding
how a cricket ball swings; how a bird flies, all these things can be well addressed if
we know what is boundary layer theory and how to solve the boundary layer equations.
So, that is the vastness of boundary layer theory and it is a subject which can if it
is properly used, it can give rise to such a critical physical cum mathematical understanding
of aerospace science and engineering which cannot be really addressed by simply looking
into the full form of the Navier Stokes equations. Now, before getting into what is boundary
layer theory, we have to understand that what is boundary layer and then the theory aspect
comes. So, to understand what is boundary layer;
let us say that there is a flat plate; infinitely long and we have just considered a finite
length of it. It can be whatever length let us say L and fluid is coming from far stream
with a velocity u infinity; uniform velocity u infinity. Now, let us consider what happens
when the fluid interacts with the plate. Let us consider a section like this.
So, let us say that at the wall there is no slip between the fluid and the solid boundary.
This is the point which is questionable because under certain circumstances, it may become
possible that fluid slips relative to the solid boundary. So, it need not be taken as
a ritual, it is just a very common situation encountered in engineering. So, when you have
this velocity as 0 at here, come to a point which is little bit you know away from this.
So, the velocity here is non-zero, but it is not also u infinity because the fluid here
responds completely to the momentum disturbance imposed by the solid boundary and is arrested
to rest. However, as you go further away and away, the fluid is not directly in contact
with the solid boundary. But the fluid understands that there is a solid boundary. How does it
understand? There is a messenger of momentum disturbance called as viscosity. Through that
messenger the fluid understands that there is a momentum disturbance. In this way the
velocity increases till it comes to u infinity and then, the velocity doesn’t change any
further. So, this is u infinity; all these are u infinity. Come to another section.
So, here at the same layer whatever is this velocity, this velocity will be even less.
Why, because more and more fluid is now in contact with the solid boundary. Then, the
velocity increases and the velocity reaches u infinity not at this height, but let us
say some height which is here. So, or maybe here.
This is u infinity and then, it remains constant when we say it reaches u infinity, remember
it is from a practical engineering consideration because theoretically it reaches u infinity
only at infinity. Practically, it may reach u infinity within a finite distance and if
it does that; then it simplifies our situation considerable. We will see that later on, but
for the time being we are constrained to understand the basics of what is boundary layer.
So, this circled location is the penetration depth up to which the effect of the plate
is felt. Here, this is the penetration depth up to which the effect of plate is felt outside
because of uniform velocity the fluid really does not understand that there is a plate.
So, now, if you draw the locus of these circled points, then you come up with two regions.
This region is a region where the effect of viscosity is important in terms of creating
a velocity gradient. So, this is called as boundary layer and this is called as edge
of the boundary layer. So, in short we will write B.L for boundary layer. This is the
edge of the boundary layer. Let us say that at a at some distance x, this
delta we can clearly see the that this delta is a function of x. So, the next obvious question
will be that how large or how small this delta is? Now, there is a possibility that delta
is very large. Can you tell when delta is very large? Delta is very large when the fluid
is highly viscous; then, a large distance from the wall up to a large distance from
the wall the effect of viscosity, the effect of momentum disturbance of the wall will be
felt. However, if the fluid is less viscous and this is what we are saying qualitatively,
quantitatively we will identify some quantitative parameters because less or more does not have
any you know quantitative meaning. This is just a qualitative way in which we can bring
out the physics. So, if it is very less viscous, then it may
so happen that this layer is very thin and how thin? Maybe delta at a given length L
is much much less than L. If that be the case, then we can develop a nice theory where we
can solve simplified versions of the Navier Stokes equation within this layer and outside,
we can use simple potential flow equations or inviscid irrotational flow equations which
for constant density become Bernoulli’s equation.
So, the entire domain where ideally Navier Stokes equation should have been used, gets
reduced to a very narrow domain within which viscous flow equations. But Navier Stokes
equation, but in a simplified form need to be solved and outside that equation need not
be referred to at all and inviscid equation can be used. In the modern era of CFD, you
people may argue that you know why it is necessary to get so, much restricted that is have a
part of the domain in which viscous flow equation is solved; a major part of the domain were
viscous flow equation is not solved. We can solve blindly the viscous flow equation
full Navier Stokes equation for the entire domain, we have CFD tools, we have nice CFD
software. Now, you have to imagine the era in which this theory was developed; that era
was a time when modern day high performance computing was not available. So, when modern
day high performance computing was not available, in those days because machine was not beating
human beings; human intellect was manifested at its best.
So, human intellect always tried to make an attempt to reduce the computational task by
using judicious combination of physics and mathematics, and that was first attempted
by the genius of a famous engineer known as Prandtl and he came up with this revolutionary
theory. I would say it is revolutionary because prior to Prandtl’s era, fluid mechanics
was governed primarily by mathematicians and it was understood that there are certain classes
of problems were exact solution of Navier Stokes equation exists that is fine.
If it does not exist you know people were debating about you know what could be possible
solutions and all those things, computational tools were not available. But those cases
could not be addressed to come up with solutions which engineers could use for designing of
devices. For example, designing of aircrafts or designing of automobiles these things were
not possible until and unless this beautiful theory which reduces the computational task
to a large extent, not only that it gives a nice physical insight to the problem that
how large the domain may be. Under certain cases, we will identify which
cases they are there is a small part of the domain where viscous effects are important
and the major large part of the domain, where you can use the dynamics of inviscid flow
equations. The interesting thing is that although the fluid still have viscosity there, it does
not have a velocity gradient and that makes the shear stresses vanish. So, that brings
us to the perspective of studying the boundary layer theory. Now, you people may have another
argument, well in Prandtl era may be CFD or computational fluid dynamics was not that
developed. In the modern era, why do you required to
study boundary layered theory because we can run CFD codes. Question is, let us say that
you have a solid boundary may not be as simply as a flat plate. Now, when you have a solid
boundary to understand the velocity gradients at the solid boundary you have to use very
fine messing or very fine grid points close to the solid boundary, but in the outer part,
outside the boundary layer such fine messing may not be required. It will unnecessarily
add to the computational cost. So, there is always a question that how much
distance close to the solid boundary, you may have to use a fine grid and how much;
and beyond what distance that fine gridding is not necessary; how will you know that?
You will know that only when you can make an assessment of how thick it is for a given
physical problem Then only you will cluster fine meshes within that and outside you will
not unnecessarily put huge computational burden by putting large number of grids.
So, the moral of the story is that even in the modern era of CFD, the physical basis
of the boundary layer theory remains as important as it was when it was first introduced. So,
with this little bit of background, what we will do is we will try to write the Navier
Stokes equation appropriate to this physical consideration. So, what are the considerations
that we will keep in mind. So, our assumptions will be steady flow then,
constant properties that we will include homogenous isotropic
all these things just for the sake of understanding, then Newtonian fluid; so, that by a single
viscosity we can describe the constitutive behavior. Then, we will also assume that if
it is constant property, then density is also constant. So, total derivative of density
is 0; that means, it will be incompressible flow. So, and we will assume the two-dimensional
scenario where x coordinate so, if you have a plate like this, x will be along the plate
and y will be normal to the plate. Now, you may argue that you know if you have
such a case; then, what will be your x and y, still you have a global x and global y.
No, in that case we will have a curve fitted x and y that means x and y locally here. Here
x and y like this. So, in this way we will have a x y coordinate system where x and y
are relatively orthogonal, but their orientations are continuously changing as you are moving
along the solid boundary. So, with this, but for a flat plate it remains global x and global
y. So, we are writing first, the conservation of mass or incompressibility condition in
this case ok. So, we will make an order of magnitude analysis.
So, physically what we are considering? We have a boundary layer like this at L we have
a delta, free stream velocity is u infinity. So, we will make an order of magnitude analysis
of this equation. So, in this course so far we have discussed about what is order of magnitude
analysis and we will apply that here. So, this is of the order of u infinity by L. This
is of the order of let us say at the edge of the boundary layer v is v infinity. Remember
the transition from the engineering theory to the mathematical theory has to be.
. So, we call it just v infinity just give a
name. So, we have a nice transition from a mathematical based theory to a engineering
based theory. When we say mathematical based theory this delta is technically infinity.
Engineering based theory is that if the u there becomes 99 percent of u infinity, for
all practical purpose we can assume that that is where the free stream condition is reached.
So, you have a finite boundary layer thickness instead of a mathematically imposed infinite
thickness. So, this is of the order of v infinity by delta; v infinity is v at the far stream.
So, which is v at delta ok. So, now, because these two terms are of the same order, these
two terms must cancel each other to make it 0, they must be of the same order. So, you can write u infinity by L is of the
order of v infinity by delta; that means, v infinity is of the order of u infinity delta
by L. So, one important thing that we can understand is that v infinity is much much
less than u infinity, if delta is much much less than L. But v infinity is never 0, no
matter how much less delta is as compared to L. So, this is a big difference distinction
between this and the fully developed flow. Fully developed flow, you have v equal to
0 here; v may be small as compared to u, but v is not identically 0.
Some students have a misunderstanding that v is 0, because when we draw the velocity
profile in the boundary layer, we only draw the x component of velocity profile. We do
not draw the y component, but y component is very much there. If delta is comparable
with the L, v infinity and u infinity may be comparable. But in boundary layered theory
we are looking for only those conditions for which delta is much much less than L.
So, boundary layer will exist for all viscous flow problems. It may be as small to as large
as you know almost the entire up to infinity, but boundary layer theory is a theory which
captures only those problems where delta is much much less than L. The other part where
boundary layer may be there, but boundary layer theory will not work. So, v infinity
is much much less than u infinity, if delta is much much less than L. Now, we will consider
the x momentum and the y momentum. Because it is a steady flow and all I am just
writing the equation straight away. By this time, you know how to write the Navier Stokes
equation for all. Let us assume that there is no body force. I am deliberately keeping
a space here because we will make some order of magnitude analysis for which I am keeping
some space. So, let us find out the orders of magnitudes of these terms. So, this is
of the order of u infinity into u infinity by L; u infinity square by L. This is of the
order of v infinity into u infinity by delta, but from here you can see that v infinity
is of the order of u infinity delta by L. So, this becomes of the order of again u infinity
square by L. See this is again something which is not intuitive
the reason is before that I think the pressure gradient
term, we have missed. So, let us just write this to complete the equation.
So, coming back to this; so, this term and this term they are of the same order as we
are seeing from here. Natural intuition is that because here is v and v is much much
less than u if delta is much much less than L, this term is much much less than these,
that is not true. Why that is not true is because these gradient is much sharper than
these gradient. So, this term is exactly as important as this term. If you cannot ignore
this term, you cannot ignore this term. Out of these term these two terms, so this
is of the order of and this is of the order of if delta is much much less than L then,
this you can neglect clearly. Pressure gradient, we are not committing at this moment how big
or how small it is. So, here also in the same way, we will write the order of magnitude.
This is u infinity v infinity by L using the continuity equation, this will also become
of the order of u infinity v infinity by L. This is of the order of nu v infinity by L
square, this is of the order of nu v infinity by delta square. Again, this can be neglected.
All these are based on the premises that delta is much much less than L, the entire understanding
is based on that. So, now, let us write the inertia term. Inertia
term is the left hand side in y momentum by inertia term in x momentum. This is of the
order of u infinity v infinity by L by u infinity square by L, right. So, this is of the order
of so, u infinity u infinity gets cancelled; v infinity by u infinity that is of the order
of delta by L. Similarly, viscous term, so this is much much less than 1, if delta much
much less than L. Similarly, viscous term in y momentum by viscous term in x momentum
is of the order of v infinity by u infinity. You can clearly see this term and this term.
So, that is also same conclusion can be drawn from that.
So, if these term is at least one order less than these term and this term is at least
one order less oh sorry if this term is at least one order less than this and this is
at least one order less than this, I mean because only these terms remain this is at
least one order less than this. So, the conclusion from here is a very important
conclusion, this is the conclusion from the analysis of the y momentum equation; that
being the case you can write p as a function of x only. So, this will approximately become
d p d x. So, this leads us to the boundary layer equations. From the Navier Stokes equation
to a simplified Navier Stokes equation within the boundary layer which are called as boundary
layered equations ok. So, now when you have these boundary layered
equations, question is when are these equations valid. To understand this, we will take the
example of flow over a flat plate. Of course, we have taken the consideration that delta
is much much less than L, but a physical problem where some external condition is imposed that
does not understand what is delta, right. So, we need to express this in terms of externally
controllable physical parameters. So example, flow over a flat plate, because
there is no pressure gradient in the boundary layer, you can write d p d x is equal to d
p infinity d x, right. The pressure gradient along x is whatever is imposed from outside
the boundary layer and outside the boundary layer the beauty is that you can use this
Bernoulli’s equation if rho is constant. The reason is that the uniform flow is irrotational
outside the boundary layer it is inviscid. Inviscid flow means irrotational flow will
remain irrotational forever. So, you can use Bernoulli’s equation with an absolute global
constant. So, if that be the case you can then differentiate this with respect to x,
because u infinity does not change with x. If u infinity does not change with x and that
is what is the case for flow over the flat plate, you have d p infinity d x is equal
to 0. So, if d p infinity d x equal to 0, then you
have this one. So, when you have this one, this is of the order of u infinity square
by L and this is of the order of nu u infinity by delta square ok. So, these two terms are
of the same order. So, u infinity square by L is of the order of nu infinity by delta
square. So, delta by L is of the order of u infinity L by nu to the power minus half,
right. So, this is; this means delta by L is of the
order of Reynolds number to the power minus half. So, from here you have a clue now this
is an externally controllable parameter. So, you are able to say that delta is much much
less than L only when Reynolds number is large. So, boundary layer theory is applicable to
large Reynolds number problem because when you have Reynolds number large, how large?
These Reynolds numbers such that delta is at least one order less than L that is delta
by L at least 0.1 or less. So, then you can find out a Reynolds number and for that Reynolds
number or Reynolds number beyond that you can use the boundary layered theory. So, delta
much much less than L will boil down to Reynolds number very large. So, boundary layer theory is applicable, when?
So, boundary layer theory is applicable when delta by L is much much less than 1, which
is equivalent to Reynolds number large and see another case which for flow over a flat
plate is not encountered but for other cases, you have this dp dx and this dp dx can play
a very critical role. For example, if the dp dx is such that there is an adverse pressure
gradient, adverse pressure gradient means pressure is increasing along x. If pressure
increases along x, then there will be a force which is opposite to the motion of the fluid.
The fluid want to move along positive x, but the pressure gradient it is called as adverse
because it is trying to oppose the movement. Along with that the viscous flow also tries
to slow the fluid down. So, the acceleration or the inertia of the fluid may not be sufficient
to overcome this two forces and if it is insufficient, the fluid instead of moving forward may start
to move backwards along the negative x direction and this is called as boundary layer separation.
So, that is possible only if the pressure gradient is adverse. If the pressure gradient
is favorable, then the pressure gradient will drive the flow along positive x and boundary
layer separation may not be possible. But boundary layer separation is possible, if
there is adverse pressure gradient. If there is adverse pressure gradient, then there is
a chance that boundary layer separation occurs and if boundary layer separation occurs, then
there is no more monotonic growth of delta as a function of x and then, we say that boundary
layered theory does not work. So, boundary layered theory is applicable
when delta by L is much much less than 1 and there is no boundary layer separation. This
is not very commonly discussed, but this is very very important. For flow over a flat
plate this does not matter because there is no question of adverse pressure gradient the
pressure gradient is 0; so, there is no question of boundary layer separation.
So, today we have learned that what is boundary layer theory; what are boundary layer equations;
why it is important and what are the assumptions behind the boundary layer theory. We will
take it forward from this in the next lecture. Thank you.
