>> Okay, so picking up where we were last time. We were looking at all the
equations for a flat plate boundary layer where the plates aligned with the
flow. That means the velocity vectors are parallel to the surface of the plate.
We had a set of equations. One set was for a laminar flow. Another set of
equations for turbulent flow from the leading edge. And then we also had an
equation for the [inaudible] flow condition where the flow starts out laminar
from the leading edge and then at a certain distance, x of C down, it transits
to a turbulent boundary layer. And we worked an example problem. And we found
many, many things from that example problem. Okay, so that pretty much ended
that topic. Now we kind of go to the next phase which is here's the flat plate.
Here's the flow. There's a fan in back of me, floor fan. It's blowing air over
the top surface of that plate. Okay, boundary layer builds up. There's drag
force. What causes the drag force? Skin friction. So we have our skin friction
tau sub s. Let's just show that. And we integrated the skin friction over the
length of the plate. And that gave us the drag force. Tau sub S times WDX where
w is the plate width. Now we take that same plate. Now we turn it 90 degrees
like that, very thin plate, okay? This is a sharp-edge plate. This is a very
thing plate. There's a velocity. The velocity vectors now are normal to the flow
field. The plate is normal to the flow field, okay? There's going to create a
drag. I let go of this piece of cardboard with the fan back of me. That
cardboard hits the wall over there probably. I put this like this with the fan
in back of me. Do you think that the plate's, the cardboard's going to go as far
as that carboard piece? No, it's going to float down and go down. This guy gets
blown back here. Take your hand, carefully do this. Maybe there's a passenger in
the car. I don't want to see you driving with just one hand on the wheel. So
you're the passenger in the car. Roll the window down, hold your hand out like
this. Okay, now rotate your hand 90 degrees. Whoa! I feel a much bigger force on
that now. The drag force on that normal orientation is much greater than the
drag force on the tangential orientation. Now this guy here, is there any shear
stress in the x direction? Because the drag force is in the x direction. It's
not going to go up or down. No, it doesn't do that. The drag force is in the x
direction. Is there any shear stress? No. Why not? Because there's no, there's
no plate in the x direction. This is a very thin plate. What creates a drag if
there's no friction, this one, the skin friction causes a drag force. Here it's
not, it's the pressure force. So over here we have a high pressure region. And
in back of it, we have a low pressure region. And that difference in pressure
acting over the plate area creates the drag force on that. Okay? And this, this
drag force is due to pressure differences. So take a flat plate. In this
situation, aligned with the flow. In this situation, normal to the flow. And
things happen. Is there any difference in pressure here and here? No, we assume
the pressure here is the same as the pressure here. The pressure outside the
boundary layer we assume to be constant. No difference in pressure. There is no
pressure drag over here. So here the drag force. Is caused by skin friction.
Skin friction. And we can call this. Friction drag. Over here, this picture
caused by a pressure difference. So drag force. By a pressure difference. Okay
so we've got those two guys on here. This is pressure drag is caught, pressure
or form drag. So. Pressure drag, plate normal. Friction drag, plate tangential.
Okay? If you want to look at, this is the boundary layer. Okay, I'll just sketch
this again. Okay, there's a velocity profile. From x, from y equals zero up.
Don't forget, we measure y vertically up from the plate. Over here when the
velocity vectors hit this plate in the middle stagnation. These velocity vectors
end up looking like this. When they hit that point right there, it's called the
separation point. The flow separates from the plate. The flow physically
separates from the plate. Behind that, there's these swirling eddies. So this is
called the wake region. Where there are eddy flows. It's not a turbulent region.
It's a turbulent region, but it's not turbulent as the way we define turbulence
in a flat plate. It is just mixed up region. If you've ever been out and you've
seen a stream or you're fishing and you see a stream. And there's a boulder in
the middle of the stream, you'll see the water go around the boulder. And, and
behind the boulder you'll see maybe leaves making little circles or bugs making
little circles. They're in the wake region behind, behind the rock in the
stream. They're in here. The leaves and bugs do that. They don't go shooting
down that way. No, no, no. They go around like that. Okay? So, it's a low
pressure region. You've got a van with a back window. And if you've had that
situation, you wonder, you say, "Boy, the back window of my van always gets
dirty. I'm always cleaning the back window of my van. I wonder why that
happens?" Well, all the dust just circulates very slowly when you're driving and
it settles on the back. The front windshield, the dust is blowing away by the
velocity, but the back of the van is in a wake region. That's why they aren't so
efficient, streamlined, obviously. So it looks like this behind your, your van.
Or a bus. Or whatever it is. A tractor trailer truck, same thing. There's a
region back here. Some people and add things back there to, to make the wake
region smaller. By adding things back there to the back of your car, okay? Now,
the real world. Everything is not a flat plate this. Everything is not a flat
plate like this. Typically, obviously you can tell most, most of things we look
at are a combination of both. So. Most things. It's some of each, some of each
typically. And that's the way most things in the real world are. I'm going to
contrast these. One more thing I want to do on here. C sub D is small. I'll, I
think I'll put it right here for here. C sub D is large. Okay. Here was our
graph. Laminar flow, mixed flow, turbulent flow. Run those numbers. C sub D.
Typically on the order of, I think what I read from that was on the order
of--yeah, there it is. Figure 7-6, typical 004. C sub D. Over here, C sub D is
large. Table 7-3. We'll go through it in just a minute, but for right now, table
7-3, C sub D is on the order of 1.22. Wow. Is it bigger than this? It's, it's a
lot bigger than this: .004, same size plate. Same fluid air. Turn it this way. C
sub D: 1.22. Put your hand outside the window the car. You'll tell the
difference. It's big-time difference. Hand like this, hand like this, 60 miles
an hour? Yeah. You can also talk about Reynolds number dependence. Reynolds
number dependence. Over here. Reynolds number dependence. Is small. Effective
plate roughness. Over here. No effective plate roughness. So yeah, they're,
they're greatly different, obviously, from looking at that. It pretty much, I
don't care what the Reynolds number is. It's pretty much in that same ball park.
Over here, boy does it depend on the Reynolds number. Turbulent flow, it goes
down to .0015. Yeah, it's big dependence on the Reynolds number. Not so over
here. Okay? It's almost no effect on Reynolds number. Why? Because pretty much
it always separates at the same point. It always separates there. You don't
expect the flow to say, I'm making you turn. No, it doesn't do stuff like that.
No, uh-uh. It hits the end of that plate, and off it goes, into the wake region.
Okay, so that gets us to besides a flat plate arranged like that, what else
could there be in the world of engineering. Well, we know there can be pipes and
tubes. They're extremely popular thing for engineers. Tons of them out there.
Chemical plant, petroleum refinery, power plants, solar field. You name it,
circular tubes. We engineers have tons of those. So, okay, get rid of these guys
now and look at circular tubes. Okay, so it's not going to be friction drag
totally. It's not going to be [inaudible] drag totally. It's going to be a combo
of both. Alright, so this is circular tubes. And we've got the tube that will
have a few drawings here. Let's take this one first. We'll call this case A.
This is for low Reynolds number. Stagnation position at the front of it with the
velocity vector. And then the velocity vectors go around the back side and
recombine on the back side like this. These are the streamlines. Reynolds number
on the order of one based on diameter. Really, really, really low Reynolds
number, approaching what we call creeping flow. Very slow-moving flow. Then we
go to case B. Stagnation streamline. Flow goes around the cylinder and breaks
off from the cylinder at this point. This is a laminar boundary layer. This on
this side, same thing, breaks off here. The flow goes around like this. This
point's called the separation point. It's the point where DU DY, at that black
dot is zero. At the surface. And if you want to blow it up, it looks something
like this. This is U. DU DY is zero right there. We're measuring Y outward from
the cylinder wall. That's called a separation point. What it implies is that the
boundary layer separates from the surface. The boundary layer separates from the
surface. Behind that region, here it is, there's a wake region. Turbulent
eddies. They call these guys turbulent eddies. Wake region. That's situation B.
Okay, let's take KC. KC stagnation streamline. Laminar boundary layer breaks off
here. Abbreviate it SP separation point. Okay, on the front side of the
cylinder, it passed the 90-degree mark going forward. Now it's on the front side
of the cylinder. The separation point moves forward as the Reynolds number goes
up. Wake region gets bigger. The wake region gets bigger. Separation points move
forward in front of the 90-degree mark towards the front of the cylinder
surface. KC, by the way. This Reynolds number is roughly somewhere from 20-- to
50,000. This guy is from 50-- to 200,000. Fifty thousand, 200,000. Case D.
Stagnation streamline. Now the Reynolds number is so large, the boundary layer
which was laminar, transists to turbulent, and the turbulent boundary layer
sticks to the walls of the cylinder better. So now the wake region behind the
cylinder becomes smaller. This is a laminar boundary layer. This is a turbulent
boundary layer. This is now the separation point. So because the flow transisted
to turbulence, it sticks to the surface better. Why is that? Well. Here's a. A
laminar velocity profile. Here's a typical turbulent velocity profile. What's
the difference in them? The difference is the turbulent velocity profile, the
velocity is bigger near the boundary than the laminar profile. Bigger velocity
vectors. The bigger velocity means they have more momentum, M times V. They
stick to the surface longer because of that momentum. And so they don't separate
until they get around the back side of the cylinder. Eventually they do, but it
takes longer for them to do that. So by transisting from laminar to turbulent,
you reduce the wake region. So the wake region has been reduced. And now--I'm
going to put this right here, okay. This was the reason why the turbulent
boundary layer sticks to the surface better. It has more momentum near the
surface. Okay. Back to here. What are we trying to find? Of course we're trying
to find the drag on the surface. That's the object. Flat plate, find the drag
force. A cylinder, find the drag force. A sphere, find the drag force. For
sphere, it looks very similar. Of course it's three-dimensional. So cylinder's
two dimensional so it's easier to show like this so this is a cylinder. But the
same kinds of things happen for a sphere in three dimensions. Okay, how do we
find the, how do we find the drag force. Okay, go back over here again. Here it
is. Drag force. Okay, kinetic energy times an area times C sub D. There it is
right there. Gotta find C sub D. Similar to someone asks you to find delta P in
a pipe. Find the pressure drop in the pipe. What do you do? You find the
Reynolds number first. Then what so you do? You go to the Moody diagram with a
curve fit equation on it, and you find F. F, L over D, V squared over two G.
Same thing here. What do you do? Get the Reynolds number. Okay? Where do you go?
To a graph or an equation. A graph. Where do you put it then? Over there. And
that equation there. Same procedure. Whether it's flow inside of a pipe. Or flow
outside of a pipe. We think the same way. Okay, this is very complicated stuff.
This is really complicated stuff. Okay? There are no equations that are in our
textbook that talk about C sub D. No. You've gotta--a plot in the book though.
You've got a plot in the textbook. It's C sub D versus the Reynolds numbers.
Let's see which one it is here. Oh, there it is. I think it's. Let's see, nope.
Okay, here we go. It's kind of hard to read because it's so small. It's figure
7-16. So Figure 7-16. So. I made a copy from a different textbook. This is from
the textbook Potter and Wiggert which was our old textbook. But it's got a real
nice big graph. Okay, so there is our picture I'll put on the board, too. But
just so you have one in your hands. And we'll look at--. Let's take, we can only
consider smooth for the picture on the board here. C sub D plotted here.
Reynolds number plotted here. It's a log-log graph. What's the Moody chart? It's
a log-log graph. What's the flat plate skin friction? Logarithmic. What's this
guy? Log-log graph. Okay, well let's put down some numbers here. I'm going to
start at about ten to the third. Fourth, fifth, sixth, seventh. Okay, okay, so
I'm just going to start here at ten to the third, roughly. Kind of flat for a
smooth. Let's, let's do the circular cylinder. It's out there by one. So it's
pretty much just above one until it gets out to about one and a half times ten
to the fifth. One and a half times ten to the fifth out to here. So it stays
about here or a little bit of variation, but we're not going to worry about
that. Okay. One and a half time ten to the fifth. This is a smooth. Oh, I'm
looking at the smooth, yeah, cylinder. Okay. Drops down to about .3 at ten to
the sixth. Ten to the sixth is here, so it drops down here on this graph about
like this. To about .3 and then it recovers and goes back up slowly like that.
That is for a smooth circular cylinder. The graph has a rough cylinder, too, but
we're not going to worry about that right now. For right, we'll do that on
Monday. Alright, now there's also a sphere on there. But I'm not going to draw
that on here right now. It's very similar. This guy goes up here roughly to .75.
About like this, okay, like that. On the [inaudible]. It's still hard to read
the graph, but that's okay. Best we can do. Okay, so. Alright, I'm going to have
to go back down here to. Ten squared. Way back. Ten to the first power, one.
One, this is ten. This is a hundred. Keep it like that, okay. One-tenth. Okay,
that's how it looks on your, on your graph. Alright correlate that C sub D
graph. With these pictures. Okay, let's start over here. Reynolds number about
like one to maybe ten. Okay. Looks like that. One to ten. There's how the
picture matches with the C sub D graph. Situation B, 20 to five times ten to the
fourth. Oh. That's right here, B. From here down to there, then it flattens off
and goes to there. C, okay I had my little erasure there. On the order of ten to
the sixth we had so C is right in here. The wake region is moving forward now,
okay. Roughly 100,000. One times ten to the fifth. Right there. The last one I
had Reynolds number greater than two times ten to the fifth. Ten to the fifth,
ten to the sixth, two times ten to the fifth. Right there. Boy, that C sub D
dropped dramatically. At around ten to the fifth. Boy, it really dropped down.
From on to .3, oh my gosh, a 70% drop in the drag coefficient. Seventy percent
drop. Not five, not ten, 70% drop in the drag coefficient. What caused that?
Well, here it is right here. If you're up here, you're sitting up here.
Separation point, forward of the 90% mark on the cylinder. Suddenly, the laminar
boundary layer. This was a laminar boundary layer. Suddenly it transisted to
turbulent. And kept attached to the cylinder all the way to the back side now.
Which cut the wake region down from a humongous area to a very small area. Boom.
Down goes the drag force. Because you cut down on the wake region. What's a
rak--wake region do? Oh, it's causing a lot of drag. It's causing a lot of drag.
You think a van has good gas mileage, think again. It's got lousy gas mileage
most of the time. Why? Well, okay. I'm sorry, it's right there. What's the back
of the van look like? This. What's the back of a [inaudible] look like? This or
better than that. Oh wow. Wow, wow, wow. So what do you want to do to get better
gas mileage? Reduce that wake region in the back of it. This is the cylinder, so
what, the car's very similar. Similar geometries. Okay? So that's a very
interesting point, how that thing drops off like that. Now, that, your diagram
also has a sphere. So I'll put the smooth sphere down here. It comes down again,
down to around a smooth sphere right at .4. So that was one, that was .3 . So
here's .4. So it comes down to where it's relatively flat at .4 at about ten to
the fourth: .4 is relatively flat. It goes back up to about .5 out to where it
goes down: .5, .4. And when it gets to that point there, you see a two times ten
to the fifth. Boy, the bottom drops out. Not gradually. I mean the bottom drops
out. Down to essentially .2. To there. When it gets down to there, then it
recovers a little bit and goes out like that at .2. There's .3. So this is, this
is for a smooth. Sphere. Okay, so that graph and the graph you have in your
textbook is for the smooth sphere and the smooth--it's also a rough picture on
there. But I'm not going to show both of those. We don't need both of those.
Okay so what do we do? Okay, you know the ru--you know the game. If you've got a
smooth cylinder or a smooth sphere, you find the Reynolds number. Once you find
the Reynolds number, you go to the graph. You read the graph, and from the graph
you put the C sub D in here. Multiply it by the area: row v squared divided by
two. That's the drag force. But area. Okay. Where the area is equal to the
frontal area. For stubby bodies. Such as things like spheres, cylinders, cars,
trucks, missiles. Torpedoes, et cetera. You look at the front, frontal view,
okay? Here's the smooth cylinder. What do I see? I see the diameter times the
length. Diameter times the length. If it's a smooth cylinder, for the area you
put the diameter times the length. If it's a smooth sphere, I've got a baseball
in front of me. What do I see? A circle. Pi r squared. You got it. If it's a
sphere, a is pi r squared. If it's a car, if I face the car in the front, it's
what I see. The area of the front of the car. The area of the front of the bus.
The area of the front of the Metro link train. The area on the front of the
truck. It's the frontal area. Motorcycle rider. It's the front of the area,
including the guy on the motorcycle. Okay? But you've got to, it shifts gears
here now. Or A can be what they call planform area. And that's the area as seen
from above. Not the front view, the area as seen from above looking down on it.
Such as wings, aerospace engineering. Wings. Hydrofoils. Et cetera. Anything
with a wing like structure. The area is defined to be from above looking down
the planform area. Okay, so let's use all this now to work an example. We don't
need this anymore, so I'll put this over here. [Inaudible] tells you for a
sphere, for a sphere 24 over Reynolds number C sub D. So for a sphere, this part
of the curve right here, C sub D equal 25 over Reynolds number. But that's only
for really, really low Reynolds numbers. Don't even use it. Use the graph. But
I'm just saying that because. We know that guy right there. Yep. Laminar flow in
a tube and guess what? F is--does it look similar? Of course it does. I'm trying
to show you the similarity between pipe flow and flow outside the pipe over the
pipe. If the Reynolds number's low enough, you have laminar flow and the
equation's easy. F is 60--if it's not, okay. Now you've got a, a smooth, smooth
pipe. Now you've got a rough pipe and a rougher pipe and a really rough pipe.
Over here you've got a smooth cylinder, a smooth sphere, and then you have a
rough cylinder and a rough sphere on here. But if the Reynolds number's low
enough, you can use that equation over here. If it's less than 2300, you can use
that equation. They're very similar. We engineers think in these veins. The
Moody chart is a lot similar to this chart for C sub D. Okay, so now back over
here. Our example. We're going to look at a sphere, attached to a post. On the
ground. And we have. A free stream approaching this, capital U. And the fluid is
air. And the temperature is 60degrees Fahrenheit. It's going to be in English.
We'll work in English one. Okay, diameter of the sphere is two feet. The
diameter of the post, one foot. And the height of the post is equal to five
feet. Length of the post, five feet. Okay. Find the moment at the base of the
post. Okay, so we have, we have to find the force of course. If you're going to
find the force. You've got to find over there, right-hand side of the board.
Find the drag coefficient. Got it. Drag coefficient, where do you find it? On
the graph right there. Figure 7-16. Got it. Okay, now go through all the
calculations. Step number one, I need the Reynolds number. Okay. I need, I need
the properties. Okay. Air, table's in the back of the book. You've got it. So
get the Reynolds number of the sphere first. And then we'll, we'll go ahead and
get the--let's, let's put that here. I think I'll put that, I'll put the post
here. Okay. The velocity. Let's see what the velocity was. Sixty miles an hour.
Eighty-eight feet per second. So a 60 mile and hour wind, okay. Alright, back
over here. Eighty-eight. Sphere diameter, two feet. Divided 1.8 ten to minus
five. Reynolds number, okay, one times ten to the sixth. Over here, 88 times
one, 1.8 times ten to the minus five. Half of that guy over there. Five times
ten to the fifth, got it. Go to the graph. Alright, the sphere, smooth sphere we
assume. One times ten to the sixth. Right here. Smooth sphere. Oh, it's a case
D. The boundary layer start-off laminar. It transisted the turbulent and it
stuck to the sphere longer. And the wake region is smaller. Okay. C sub D, .2.
Got it. For the post, okay. Post is a cylinder. Five times ten to the fifth.
Okay, right here. Five times ten to the fifth. We go up here. That's the
cylinder, that's right here. That big, black dot, we're right there. And if we
go across here at that point, we find out it's really .3 if I draw, if I have my
drawing drawn correctly. It's at .3. So I'm not going to--I'm going to leave it
like it is right there. Zero point-three. So it's just, it's just turned
turbulent. And dropped down dramatically C sub D, 0.3. Okay, so get the drag
force on the sphere. Alright, now we've got for that, that would be C sub D.
This is for the, this is for the sphere. That area of the sphere is pi over four
D of the sphere. To the fourth. Don't forget for a sphere, the correct area is
the frontal area. You see a baseball, what do you see? I see a circle. What's
the area of a circle? Pi D squared divided by four. Pi D squared. Got it. Put
all those guys in there. You know U 88, you know rho the density. The back of
the book, we've got all those values back there. The density is .00238. Slugs
per cubic foot. Okay. And that gives you the drag force on the sphere. Five
point-eight pounds. Okay. Do the same thing for the post. Drag is equal to C sub
D for the post. This was C sub D for the sphere, area of the sphere. This is C
sub D of the post, the area of the post projected. Rho U squared over two. Area
of the post is D pi D, the circumference, or pi, not pi D. D times L, the length
of it. What do I see as a projected area of that post? Well, it's right here.
Here's my area right here. It's a rectangle. What's the base, the diameter?
What's the height, L? Here's a sphere. What do I see? A circle. What's the area?
Pi D squared divided by four. It's the frontal area though. Got it. Let's see.
Okay, got it. This was C sub D post. This was C sub D sphere. Just so we keep
them separate. This guy here comes out for the post, the drag force on the post:
13.82 pounds. So now I've got them both. So now the moment. Moment is equal to
drag force on the sphere times the moment arm for the sphere. Plus the drag
force on the post. Times the moment arm on the post equals drag force on the
sphere right here. Five point eight. The distance is five plus the radius. Two
feet, radius one foot. Five plus one, six. Times six plus the drag force on the
post: 13.82. Times the moment arm for the post. Halfway up, five over two, two
and a half. Sixty-nine point three pound feet. Sixty-nine point three pound
feet. That's the moment at the base of the post. Now, is everything correct?
Yeah, but there's a lot of assumptions built in. Does the velocity really look
like that? No, no. Down at the ground we know what the velocity is on the
ground: zero. I don't know, does it go up like this real fast? I don't know. I'm
going to make the assumption it's the same from top to bottom. That's a first
cut to the problem. We can change it later. But that's the first cut. So this,
this solution assumes incorrectly of course that the velocity from the ground to
the top is the same. Yeah, okay. So we know that's true. Is the velocity
perfectly like this? Well maybe not. Maybe it's like this. You know, maybe this
is on a hill. And the velocity comes and hits it like that. Okay, I know that's
not going to be right because it's not normal to the, to the axis then. So yeah,
there's some things built into there, but essentially, this is the first cut to
find the moment at the base of that post with a sphere on top. This could be a
water tank in Topeka, Kansas. The thing is what? Sixty feet in diameter? How
high, high is it? A hundred feet above the ground? Here comes a tornado or
something? Oh, yeah. Are those guys worried about this thing staying in the
ground? Oh yeah. Yeah, they are. And who designs that? Guess who designs that?
We do, thank you very much, yeah. We'll tell you how big the bolt should be or
concrete foundation should be for that. That's our job. But somebody's got to go
through this calculation to figure out under very high velocity winds, is it
going to stay attached to the ground? Okay, that's all design part though. Okay,
so that's how we use this. Now, let's take another one. Any questions on that
guy before I erase it? Alright, example. Now we're going to do this. We're going
to take a sphere again. This is going to be a sphere, and this is 30 centimeter,
diameter smooth sphere. Thirty centimeter diameter, okay. Let's do it this way.
Yeah, that's all right. Thirty centimeter diameter smooth sphere. Okay. This
particular problem tells us S is equal to 1.02. Okay. we know from first fluids,
S is equal to rho divided by rho of water in standard conditions. Okay? Specific
gravity. Some people say SG. This author uses just S for that. The ration of the
density of the fluid divided by the density of water at standard conditions.
Sixty Fahrenheit, 20 degrees C. Okay. We're going to--. Drop that in water. Drop
it in water. Will it, will it sink? Well, yeah because its density is bigger
than water. It's going to go down. Specific gravity is greater than one. It's
going to sink. But problem says find the terminal velocity. What does that mean?
That means the velocity after it reaches a steady speed going down. Terminal the
fin--terminal, the final. The final velocity. Oh, it'll accelerate when you let
it go. But eventually it's going to assume a constant velocity as it drops down,
assuming that water stays the same, goes down. Okay. Alright so, let's put that
here then. First thing we'll do. [Inaudible] diagram of the sphere. We know the
sphere has a weight. We know as it's going down, it has a drag force. There it
is over there. What does a drag force try and do? Slow it down. Which way does
it act? Vertically up. We know that there could be, could be a buoyant force.
There will be a buoyant force. What does a buoyant force try to do? Push it back
to the surface. That's why it's called buoyant force. Push it back to the
surface. Which way does a buoyant force act? Vertically up. If it reaches
terminal velocity, what's the equation? The sum of those three forces must
balance out. Okay. So we have the weight acting down is equal to the drag force
plus the buoyant force. Okay, let's do them one at a time. The weight, gamma of
the sphere times the volume of the sphere. Volume with a bar over it so it's not
like velocity. The drag force. C sub D times the correct area. Times rho V
squared divided by two. The buoyant force, the buoyant force is gamma of water.
Times the volume. The volume what? The volume being displaced by the sphere.
Notice that the buoyant force and the weight look very similar. Gamma times the
volume. But they're different. For the weight, what's gamma of the sphere? Well,
the buoyant forces, what's gamma? Ask yourself, what fluid was pushed out of the
way? Answer, water. Okay gamma of water. The sphere pushed that volume of fluid
out of the way. Creates the buoyant force. What am I solving for? Well, of
course V. That's the object. So now it becomes pretty straight forward. We know
what all these guys are. This is, comes out to be S minus one gamma of water
times the volume, four-thirds pi r cubed. Okay, and that's equal to C sub D
times rho V squared. Over two times pi r squared. This is the frontal area.
Don't forget for a sphere, the area. It's a circle. Pi r squared. Where'd that
one come from? Over here. What's gamma of water? S is one for water. What's
gamma of the sphere? Capital S times gamma water. So this term here is this term
minus that term. Okay. Minus sign. We have to get C sub D. So okay. Let's see,
alright. I think we use for C sub D yeah, .5. It's a sphere. Okay. You have to
make a guess for C sub D first. You, you can't find C sub D. Because you can't
find the Reynolds number. Because the Reynolds number has a velocity in it. So
you can't find the Reynolds number. You have to guess C sub D. Let's go back to
fluids one. You've got a flow of water in a pipe, okay. I don't know the
Reynolds number. Okay, what do I do? Okay, let's say, let's say, well, that's
right. This is fine. Do I assume it's laminar? No, probably not. Water's rarely
laminar. Okay. Let's say I have a roughness E over D, .001. Let's just say that.
So here's the curve for .001. I'll show it like this. This is what your
instructor probably told you at ME, or Fluids One. I, I don't know the friction
factor. I've got to guess that friction factor. So you know the story. You guess
the friction factor where it's fully turbulent, where it's flat. That's your
first guess for the friction factor. Where it's flat, what's it's flat, where
it's flat. What if it's a smooth pipe? Down here. What do you guess? I don't
know. Take this number over here, .004. Take this number here, .002. I'm going
to guess between 004 and 0002 I'm going to guess .003. Okay. You guess. But if
you got a flat part of the curve, your best guess is to assume the friction
factor's going to lie out here. The odds are good it will. Over here, don't pick
it over here where there's a low Reynolds level, No. Do the same thinking over
here. Make your first guess where the curve is flat. Oh, look up here. Cylinder,
wow. From 1000, from 1000 to a hundred thousand. It didn't change. My first
guess is going to be right there. Or maybe I'll guess it out there, but I don't
think I will. I'll guess there first. What do I do then? I calculate the
velocity. I get the rentals' number. I go back up here and see if with my
Reynolds number, if it agreed with my guess. If it didn't agree with my guess, I
used new F value, go back, get new velocity until these converge, which they
should. Just like you do for pipe friction. Same procedure. So I, I take my
first guess. It could be .4 or .5. You can take one of those two. So I made my
first guess at .5. Okay, point five. So that would be from here to here. Okay.
Okay, put it, put that guy into there. Solve for V. Get V equal 0.40 meters per
second. Get the Reynolds number. VD over new. Put that v in for there. We know
D. Reynolds number 1.2 times ten to the fifth. Check C sub D graph. Go over
here, 1.2 times ten to the fifth. Right here. One point two times ten to the
fifth. Go up vertically. See what I get. Oh, I get point-five. I'm a good
guesser. Okay. At Reynolds number equal one point two times ten to the fifth. C
sub D is equal to 1.5, checks with my guess. So the velocity, the terminal speed
I got it right there: .4 meters per second. And if you, if you didn't guess
right, you repeat it over again until the Fs converge. Until the Fs converge.
Okay, so good stopping point. We'll stop for today. And then we'll come back on
Monday. And we'll finish up drag force on spheres and cylinders.
