- [Jeffrey Knutsen] In this screen cast,
we're going to examine the effect
of a heated plate on the velocity profile
and the temperature
of a relatively cool fluid flowing down its length.
Let's begin by defining our coordinate axis,
x will be in the direction of flow,
at x equals 0 we've got the leading edge of the plate,
y will be the vertical direction,
the distance above the plate itself
and z will be a direction orthogonal to both of these.
Now let's make the assumption
that the plate is relatively wide compared
to the flow direction so we don't really need a z direction,
the plate's gonna be so wide in that direction
so ultimately this is going to be a two dimensional problem.
We'll be looking at a number of different fluids
in this screen cast but in all cases
the speed of the fluid far away
from the plate is going to be u infinity,
and we'll set that equal to 0.05 meters per second.
The temperature of this fluid far away from the plate
we'll call t infinity and we'll set that to 0 degrees C.
And we're also going to say the temperature of the plate,
the temperature at the surface is going
to be a hundred degrees C.
We'll call the length of our plate L
and we'll set that equal to one meter.
What we're going to explore are a number of graphs
in which we've got y on the vertical axis,
x on the horizontal axis, I'm showing the plate down here,
this again has a surface temperature we'll set
at a hundred degrees C, at a distance far away from it,
we'll set t infinity is equal to 0 degree C
and the vertical axis are gonna range arbitrarily
from 0 to 0.1 meters,
do keep in mind that, that dimension goes up to infinity.
And on this part of the graph we've got x equals 0
which represents the leading edge of the plate,
and I've got x equals one meter
which represents the trailing edge of the plate.
And when we show these simulations, we'll have a color scale
on the right that goes from the surface temperature
of a hundred degree C down to t infinity, 0 degree C.
Some of the other trends that we're going
to see is the momentum boundary layer
and I'm gonna use a dashed line
for the momentum boundary layer.
It'll be more obvious when I show the graphs
or call from fluid mechanics that the boundary layer,
the definition of the boundary layer is a location
where you the horizontal component
of the fluid velocity is equal to 99% of u infinity,
so in our case, 99% of 0.05 meters per second
is equal to 0.0495 meters per second.
So when you see these dash lines,
just know that the horizontal component
of velocity is equal to 0.0495 meters per second.
The thickness of the momentum boundary layer we'll denote
with the lowercase delta and that's a function of x,
so we'll find that the thickness
of the momentum boundary layer, the leading edge
of the plate is much smaller than the thickness
of the momentum boundary layer downstream.
Another thing you're going to see is a solid line
which we'll call the thermal boundary layer.
The definition of the thermal boundary layer is
that the temperature anywhere on this solid line is equal
to 99% of t infinity plus 1% of the surface temperature.
So in our case, with the surface temperature
of a hundred degree C, t infinity of 0 degree C,
we'll find that the temperature everywhere
on the thermal boundary layer is going
to be equal to 1 degree C.
We're going to define the thickness
of our thermal boundary layer
with the Greek letter again delta
with a subscript t to represent temperature,
and that's also a function of x.
And the thermal boundary layer is thinner
at the leading edge of the plate
and it gets larger as we move downstream.
So to reiterate, the solid line,
everywhere on the solid line,
the temperature is equal to 1 degree C
and everywhere on the dashed line, the speed
of the fluid will be equal to 0.0495 meters per second.
So a couple of parameters that we need to define,
the first of which is the Reynolds number,
which we'll define is rho times u infinity
times some length scale.
And now the length scale I'm gonna use is the x coordinate
so the distance from the leading edge
of the plate divided by the dynamic viscosity.
To figure out what the length scale is,
a method of notation is to use a subscript x just to denote
that x is the length scale.
And by the definition of our kinematic viscosity,
this is just equal to u infinity times x divided
by the kinematic viscosity.
Note that when we define it this way,
the Reynolds number is a function of x,
at the leading edge of the plate
the Reynolds number is going to be equal to 0.
so at this dashed line,
the Reynolds number is equal to 0.
As we move down the length of the plate as x gets larger,
everywhere on this dash line, the Reynolds number
is continuously increasing,
and it's increasing in proportion to the distance
at which we've moved downstream of the plate.
As we continue all the way down, at the end of the plate
we'll be a value of x equals L,
and we're gonna define this one now, the length scale,
we're gonna use Re sub L to define the length scale
and that's going to be equal to u infinity times L divided
by the kinematic viscosity.
So the highest Reynold's number we're going to see
in this scenario will occur at the end of the plate,
so let's back up a second.
One thing that we need to confirm
is that there's no region on the plate
in which the Reynolds number is greater than 500,000
because if that's the case,
if this was at 500,000 for example,
we would find laminar flow upstream
of that dashed line and that laminar flow will turn
into turbulent flow once Re sub x exceeds 500,000.
And this type of flow we can't handle mathematically
the way we're gonna look at these simulations
so we're just gonna confirm that the Reynolds number
Re sub L is less than 500,000,
so that's a major assumption that we're making
is that the flow is laminar.
Another dimensionless quantity we need to define
is known as the prandtl number.
And the prandtl number is the ratio
of the kinematic viscosity divided
by the thermal diffusivity.
So fluids that tend to have a low viscosity
and a high conductivity would tend
to have relatively low prandtl numbers,
something like this might be mercury for example.
On the other end of the spectrum,
fluids with a high kinematic viscosity
and a low thermal diffusivity will tend
to have very large prandtl numbers.
An example on this extreme would be something
like motor oil or olive oil, for example.
What we're going to observe is that the prandtl number
will affect the relative thickness
of a thermal and the momentum boundary layers,
such that a prandtl numbers smaller than one will find
that the momentum boundary layer will be thinner
than the thermal boundary layer,
a fluid that had a prandtl number of one will find
that the two boundary layers are going
to have the same thickness.
And then for prandtl numbers greater than one,
we'll find that the momentum boundary layer is thicker
than the thermal boundary layer.
We'll also find that the thickness
of the momentum boundary layer is a strong function
of the Reynolds number,
such that a very large Reynolds number will result
in a thin momentum boundary layer,
so small value of delta.
A large Reynolds number would be in our case,
will be a fluid with a small viscosity.
These are the four fluids that we'll be looking at,
I've provided prandtl numbers, kinematic viscosity
and the Reynolds number at the trailing edge of the plate.
Note that so we'll be working with air, water,
mercury and oil, we'll have Reynolds numbers that range
from over 400,000 down to about 50 between mercury and oil.
And the prandtl numbers are going to range several orders
of magnitude between something really small for mercury
and something really large for oil,
so do keep in mind how they will impact
the relative thickness of a thermal
in momentum boundary layers.
And because we're holding the speed
of the fluid and the length of the plates constant,
we'll find that the Reynolds number scales
with the kinematic viscosity.
Fluids with a very large kinematic viscosity
will have small res numbers like oil,
conversely, fluids with small kinematic viscosity
will exhibit a large Reynolds number.
So let's take a look at our first simulation, which is air.
We zoom in on it, what's immediately apparent
are the velocity profiles, they start out at u infinity
and right when they hit the momentum boundary layer
which is the dashed line,
the velocity is 99% of that free stream velocity.
And then due to the no slip condition,
the velocity goes to zero at the surface of the plate.
As we move down stream, we'll find that the plate has more
and more influence on the velocity profile,
the thickness of the momentum boundary layer increases.
We're now about halfway down the length of the plate,
as we continue down to the very end of the plate
at one meter, we'll find that the plate
has a very large influence on the velocity profile.
So we'll see in this instance, the velocity is 99%
of the free stream velocity way up at that value of y
and then a more gradual decrease down to a velocity of zero.
And what we also see is its temperature is equal
to a hundred degrees C at the surface of the plate
and the temperature is equal to t infinity far away from it.
At the solid liner recall this is
where the temperature is equal to one degree C.
Because air has a prandtl number that's less
than one, about 0.7, we find that the thickness
of the thermal boundary layer is a little bit larger
than the thickness of the momentum layer
and that's because for air the prandtl number
is equal to about 0.7,
recall its small prandtl numbers means that the thickness
of the thermal boundary layer will be a great deal larger
than the thickness of the momentum boundary layer.
The next simulation we'll look at is water.
And what's immediately apparent
is that the thickness of both the thermal
and the momentum boundary layers are quite a bit thinner.
Compare the thickness of the momentum boundary layer
for water and that for air, much higher Reynolds number,
much thinner momentum boundary layer.
Another thing to observe is that the prandtl number
for water is greater than one,
the prandtl number is seven as opposed to air
which had a prandtl number that was less than one.
Compare the relative thicknesses, here's for water,
the momentum boundary layers thicker
than that of the thermal boundary layer,
and here it is for air,
the momentum boundary layer is thinner
than that of the thermal boundary layer.
The next simulation is that of mercury,
we have a very large Reynolds number
in excess of 400,000, compared the thickness
of the momentum boundary layer
between mercury and water and air,
we'll find that the momentum boundary layer
is very thin for mercury.
Interestingly, what we'll also observe is
that the prandtl number for mercury
is so small, has a very high thermal conductivity
and a relatively small kinematic viscosity.
If we zoom in on the end of the plate, what we find is
that the fluid is being heated up by the plate
but it's velocity profile is not being affected,
its velocity profile is not being affected
until it's below the momentum boundary layer,
so there's a very high diffusion of heat into the fluid
but relatively small impact on the velocity profile.
And let's take a look at our last fluid, oil.
Oil is a very large prandtl number,
a very large kinematic viscosity
and consequently will have a very small Reynolds number.
Immediately apparent is that the thickness
of the momentum boundary layer is off the charts,
it's much larger than 0.1 meters
and we see that within the velocity profiles as well.
There's no region unless we zoomed out
on these charts where we'd ever see the velocity being equal
to the free stream velocity,
oil just has that much viscosity.
Let's zoom in and compare the effects.
Oil again is on the top, mercury is on the bottom.
Again, what we found for mercury is
that the temperature diffused readily
into the mercury itself,
the thermal conductivity of mercury is really high.
It diffused into a region
in which the fluid velocity was not even impacted
so it diffused outside of the momentum boundary layer.
In contrast, oil has a prandtl number on the order
of 10,000 and what we see is a very large impact
on the velocity profile,
and a relatively small difference in temperature
so the thickness of the thermal boundary layer
is small and the thickness
of the momentum boundary layer is gigantic.
In fact, if we scale out quite a bit here,
I've got a vertical axis of 0.5 meters instead of 0.1
which was the original vertical axis,
and you can see just how much larger
the momentum boundary layer is
than the thermal boundary layer.
And finally, here's a hypothetical fluid
in which the prandtl number is equal to one.
And what we observe is that
the thickness of the thermal boundary layer is equal
to the thickness of the momentum boundary layer,
so both found boundary layers fall right
on top of one another.
If we zoom in on the plot, we see that the velocity is equal
to 99% of the free stream velocity at the same location
in which the temperature is equal to one degree C,
and that's true along the entire length of our plate.