- [Jeffrey] What you're looking at is a flat surface
and there's some fluid on top of the surface.
The fluid flows along the length of the surface
in this direction.
We're going to assume that the width of the surface
in this direction is infinite,
and we're also going to assume that the height
of the fluid above the surface is also infinite.
Let's establish a coordinate access
in which X represents the distance
from the leading edge of the plate,
Y represents the distance above the plate
and Z represents the distance along the width of the plate.
And again, we're going to assume that the width
of the plate or that Z coordinate is infinite,
and we're also going to assume
that Y is infinite above the surface of the plate.
If we look at the axis from the side,
we've got the X coordinate and the Y coordinate shown here,
where the surface of the plate is shown
at a value of Y equals zero.
Ultimately, what we're trying to figure out
is what the velocity profile looks like as a function of X.
We know that far away from the plate,
the fluid flows with some speed, we'll call it U infinity.
And we also know that at the surface of the plate
and by way of the no slip boundary condition,
the velocity has to be equal to zero.
And these represent two boundary conditions for this flow.
So U is equal to U infinity at Y approaching infinity,
and U is equal to zero at the surface of the plate
or Y equals zero.
Let's consider a location along the length of the plate
that's at, oh, I don't know, 0.2 meters,
something like that.
We're trying to figure out what U is as a function of Y.
So if we drew the velocity profile,
we know that far away from the plate,
U is equal to U infinity and the flow
is uniform far away from the plate.
At the surface of the plate,
we know that the velocity is equal to zero
and the goal here is to try to connect the dots
between Y approaching infinity and Y equal zero.
So drawing the velocity profile,
we see that it is uniform far away from it and somehow,
we have to figure out what the velocity profile looks like
as we get closer to the surface of the plate.
This is what the velocity profile looks like
for air going over the surface of the plate.
And air has a kinematic viscosity of 1.5 times 10
to the minus fifth meter squared per second.
And we're going to assume that the air flows over
the surface of the plate at a speed
of 0.05 meters per second.
So far away from the plate,
we'll find that the speed here is equal
to 0.05 meters per second
and this is at Y approaching infinity,
so far away from the plate.
So if we examine the velocity profile at this location,
we find again that U is equal to zero
at the surface of the plate, and then far away from it,
U is some uniform value of 0.05 meters per second.
Let's start by answering the question,
is the flow laminar or turbulent
over the surface of the plate?
What we'll need to do is calculate the Reynodls number
which is equal to row times U infinity,
multiplied by the coordinate position X,
divided by the dynamic viscosity.
We can also write this as U infinity X divided
by the kinematic viscosity.
Note that the length scale that we're using
for the Reynolds number is the coordinate position.
And to define that, we're going to use the subscript X
for the Reynolds number.
So the Reynolds number is equal to zero
at the leading edge of the plate, where X is equal to zero
and the Reynolds number increases
as you move down the length of the plate.
So for example, the Reynolds number at a distance
of 0.2 meters from the leading edge of the plate
is twice as big as the Reynolds number
at a distance of 0.1 meters from the length of the plate.
To determine whether the flow is laminar or turbulent
at a given location, we calculate the Reynolds number,
and if the Reynolds number is less than 500,000,
the flow will be laminar.
So here, let's calculate the largest value
for the Reynolds number, which is going to occur
at the trailing edge of the plate.
So in this example, I've got a plate that's 0.5 meters long,
in making this calculation,
U infinity is 0.05 meters per second,
the distance L is 0.5 meters and we're gonna divide that
by the kinematic viscosity for air, which is 1.5 times 10
to the minus fifth meters squared per second.
When we make those calculations,
we find that the largest value of the Reynolds number
is equal to 1700, and that's quite a bit smaller
than 500,000.
So we can definitively say that the flow is indeed laminar
along the entire length of the plate.
The next question we need to answer is,
where is the shear stress really small
and where is it really large?
Well, to calculate this, let's look at the shear stress
for a Newtonian fluid.
It's equal to mu times DU DY plus DV DX,
so it's mu times the shear rate.
One thing to consider is that U,
the horizontal component of velocity
is much larger than the vertical component of velocity.
Consequently, this term will be negligibly small compared
to the first one.
So let's just take a look at DU DY.
Let's take a look at DU DY in this region for example,
what we see is a value of U at this location
that's very close to U infinity.
In fact, it's about equal to 0.05 meters per second.
The same thing is true if we move up to this location.
So moving from this value, the smaller value of Y
to the larger value of Y, there was very little difference
in the velocity of the fluid itself.
So at that location within this region, tau is very small,
the sheer stress is very small in that region.
In contrast, if we move from the surface of the plate
to some larger value of Y, directly above it,
we see a very large increase in the velocity.
So in this location, we would find
that DU DY is relatively large,
and within here we see a large value for the shear stress.
We should also consider the question,
where is the speed of the fluid equal
to 99% of U infinity.
So for example, far away from the plate,
U is essentially equal to U infinity at that location,
U is equal to zero at the surface of the plate,
and the value of U equal to 99% of U infinity
would probably be somewhere around this value of Y.
So let's look at it again,
here, U is probably 99% of U infinity.
We could draw another dot at this location,
another dot at that location,
somewhere further away, we'd probably see U infinity.
We would probably find that transition to occur there
and we generally connect the dots.
What we find is a region in which U is smaller
than 99% of U infinity, and up above that region,
U is greater than 99% of U infinity.
If we look at this more quantitatively,
the shaded region in blue represents all locations
in which U is less than 99% of U infinity.
And it's this region
that we're going to call the boundary layer.
And not only is the fluid flowing greater
than 99% of U infinity outside of the boundary layer,
we also see that shear stresses outside
of the boundary layer are relatively small.
And we'll say shear stresses
within the boundary layer are large.
Let's take a look at this a little bit more closely.
What I've got is a simulation now,
and let's look at a differential element at that location.
If we animate this, what we see is that the fluid translates
from left to right and we're going to look
just outside of the boundary layer now.
Locations outside of the boundary layer,
what we see is that the fluid moves in the direction
of the vectors at a speed that's equal to something close
to U infinity outside of the boundary layer.
Within the boundary layer, however,
the shear stresses are quite a bit larger,
and within the boundary layer, the fluid not only flows,
but it's also being deformed.
At the surface of the boundary layer,
this deformation begins to look negligibly small.
So we see the square translating
and it looks just roughly trapezoidal
after we animate it at the surface of the boundary layer.