>> Okay. Just a quick summary of where we left off a week ago today. We were
looking at external flow, and we started off with the simplest geometry in
external flow, which is a flat plate. And for that situation, we, I'll draw the
picture over here. It's a laminar from the leading edge, x equals 0. It's
laminar if the Reynolds number is less than or equal to 500,000. So, if that's
satisfied, then if this plate at end of x equal l, the Reynolds number is
500,000 or less, we have all laminar flow. If we have all laminar flow, here are
the equations we looked at last time. First, of all, the boundary layer of
thickness anywhere; now, don't forget what that is. The boundary layer of
thickness is this value right here. It's the height of the boundary layer; we
call it the thickness of the boundary layer at a particular x value. Okay. So,
there it is right there. We then talked about finding the shear stress at the
wall. The shear stress at the wall is the value at y equals 0, and again, y is
measured vertically up from the plate surface. So, y equal 0; that's the shear
stress. Tau w; w means at the wall. Tau, at the wall. Here's the value. We use
the definition of the wall shear stress in terms of a dimensionless term here, C
sub f; if you recall last Monday, C sub f was called the skin friction
coefficient. So, if you work backwards, these two guys are equal. Tau w equal
the same. Solve for C sub f by equating these two values right here, and you get
this value right here, okay. Double check your notes and make sure there's not a
u in front of that square root. I think somebody last time after the lecture
said there was a u out here. It shouldn't be. The u is on the end, u to the
three has power. So, if there's a u after that 332, get rid of it in you notes.
Okay. Back over here, okay. Now, we're interested in several things on flow over
flap, plate and laminar flow. We're interested in finding the bombulator
thickness. Okay, there it is. We're interested in finding the shear stress at
the plate surface as a function of x. There it is. We're interested in finding
the drag force on the plate. Okay, the drag force. The drag force, we integrate
the skin friction times the differential area along the plate surface, da,
because the stress times an area is a force. So, integrate stress times da. That
gives us differential force. Integrate from x equals 0, for instance, to the end
of the plate. Let x equal l, okay; there's the equation when you do that. We put
that down last time. Again, double check, make sure it's u to the three halves.
There's no u in front of the radical sign. And we also define something called
the drag coefficient, capital C skin friction coefficient; lower case c, f means
friction, d means drag force. Our textbook uses capital D for the drag force; I
prefer to use f sub d, I mean, most people think d is diameter. So, I don't want
to confuse anything; so, I'm going to use f sub d as a drag force. The drag
coefficient C sub d times the area of the plate. The plate is l meters long, b
meters in width. That means all the blackboard. Row [phonetic] u score divided
by 2. Since these two guys are equal, you solve out for C sub d the drag
coefficient. The drag coefficient then is right here. So, that's the important
equations that come from laminar theory, and they're derived analytically, okay.
Our next step is to go to turbulent flow, okay. So, now, we have turbulent flow.
We're going to assume to start with that there's turbulent flow all along the
plate, okay. The turbulent flow starts at the leading edge, at 0. It's turbulent
from that point because somebody caused it to be turbulent. I think we may have
mentioned that you can artificially roughen the surface here with sand particles
or sandpaper or very thin piano wire at this point, x equals 0. When you do
that, the boundary layer trips immediately to turbulent flow. So, in the world
of the textbook, it might say, assume turbulent flow from the leading edge.
Okay. That's this picture right here. Assume turbulent flow from the leading
edge. If that's the case, now, you really, the mathematical analysis is very
different from laminar. We don't do it totally mathematically like we did for
laminar. There's no exact solution.
We, numerical methods, computer solutions are used --
We use a one-seventh power law velocity profile with the integral approach, and
that particular velocity profile would look something like this in equation
form.
That's valid for Reynolds number less than or equal to 10 to the fifth, oh, 10
to the seventh, I think it is. Let's double check and see what he puts down
here, if he did. Yeah, 10 to the seventh, I think it is. So, if you do that, and
it's a couple of pages in the textbook, they walk you through it. We're not
going to walk through it, okay. We're going to use the results of that. But if
you assume a power law velocity profile in the boundary layer, turbulent
boundary layer, and you go through the integral approach, what you end up with
is this.
Here's delta as a function of x for a laminar flow. Here's delta as a function
of x for turbulent flow. And tau wall --
And skin friction coefficient. If you do the same thing we did over there, back
it out of this equation. Skin friction coefficient --
Just so you know, this is equation 744 in the white textbook. This is equation
743 --
It was a laminar flow, here they are, tau wall, laminate flow, C sub f, laminate
flow; turbulent flow, tau wall, C sub f here. If you want to find the drag force
over this plate and turbulent flow,
That's equal to our C sub d times the area row u squared over 2 where the drag
coefficient C sub d --
And that goes with this down here. The drag equation for laminar flow or C sub d
is this number. Now, C sub d is this number for turbulent flow. That was now
considered turbulent flow. Now, consider mixed flow.
As the name implies, mixed flow is part laminar and part turbulent boundary
layer.
Flow starts out laminar. It gets to a location called x sub c. The c stands for
critical. So, x sub c is a location where the boundary layer transits from
laminar to turbulent. So, this is laminar; this is turbulent. We're going to
assume that it transitions to turbulent at 5 times 10 to the fifth, okay. So,
the Reynolds number then, 5 times 10 to the fifth equals u x sub c over new
[phonetic]. So, solve that for x sub c. X sub c is equal to 5 times 10 to the
fifth times new over u. For this case then --
F sub b, drag coefficient, okay, times our area, b times l. L is a plate link; b
is the width, times row u squared divided by 2, where c sub d now is this --
Because there's part laminar, part turbulence , so you have to kind of integrate
this in pieces. You integrate the laminar part and the turbulent part. That's
why there's two terms in here with the minus sign. The textbook goes through all
those details, takes about half a page to a page. So, we're going to use it, so
that's what we're doing here. So, there's three cases to worry about. I know
people this morning that had my heat transfer class at 7 or 8 a.m. and 9:15.
This looks very similar to that lecture; of course, it does because we were
doing heat transfer with thermal boundary layers. So, when you get into this
class at this time of day, you got to kind of shift gears. Okay. Now, we're
looking at velocity boundary layers only. Okay. But it's very similar, very
similar. Three possible cases. Case 1, end of the plate, the Reynolds number is
less than 5 times 10 to the fifth. The flow along the plate is all laminar. Here
are the equations. If somehow, the boundary layer has been tripped artificially,
like if it flows over this table here again, and the flow hits this table, so
the sharp edges, the sharp edge, the flow fans over there, well, boundary later
builds up, okay. It might be laminar and after a certain distance x sub c, it
transits to turbulent, but if I put a fine piano wire and glue it on here, or if
I put a glue stick down here and spread sand particles on that glue strip, now
as soon as that velocity hits that particular x equals 0 location with either
sand on there or a piano wire, that's called surface roughness, and the bond
layer trips immediately to turbulent. There is no laminar part, okay. That's
this guy; no laminar part. If the textbook says in a problem, assume the flow
has been artificially cooked at x equals 0, to turbulent flow, this is your
picture. If the textbook problem says nothing at all about depth, then this one,
or this one is your picture, okay. Check the Reynolds number at the end. If it's
less than 5 times 10 to the fifth, use these. If it is greater than 5 times 10
to the fifth, use this. If you want to find the shear stress in this part, use
that, laminar part. Okay. So, there are the equations on the board for the three
possibilities, all laminar flow, all turbulent flow are mixed flow. Okay.
There's also a graph that I'll put on the board here. Let's, okay, I really
don't need that, so I'll put it there, I think. This is Figure 76 from the
textbook. Now, you don't need to copy it down. I've made copies for you, so I'll
give you these copies. I'll pull aboard exactly what's on this page though.
Okay. So, here's what the figure is. On the x-axis, the Reynolds number, based
on l. It's logarithmic 10 to the fifth, 10 to the sixth, 10 to the seventh, and
so on. The y-axis, it's linear.
That's c sub d. Five times 10 to the fifth is right here. Now, you know that --
You know that's a trick question. Yeah, somebody; that's a log axis. Where do
you think 5 times 10 to the fifth is? And the guy says, "Well, I think it should
be right in the middle. No, no, you're not thinking logarithmically. You're
mind's not thinking logarithmically, you know. Where is that? Well, it's 70% of
the way over of the total distance. If that's 10 inches, that's 7 inches over."
So, no. There's where 5 times 10 to the fifth is here. So, some people have
difficulty reading a log scale because they think linearly, like rectangular log
scale. Okay. So, anyway, there's 5 times 10 to the fifth.
Right there. Okay, let's make a dash line there.
Now, you can see here laminar, he says, equation 727; 727 right there. Starts up
here at around 04, goes down here to 5 times 10 to the fifth; stop. Okay. He
marks it laminar equation, I'll put it here, but in the textbook, he says that's
laminar flow, and that's equation 727. At that time, it's going to transist
[phonetic] to turbulent flow. By the way, let me just mention it so you don't
think this is true. This x sub c, does the flow immediately transist from
laminar turbulent at that particular location? And the answer is, no, it
doesn't. There's a grey region here. We don't know where it might transist in
that region. It takes a little distance for it to transist from laminar
turbulent, but for the sake of a textbook world and our homework world and exam
world, we use a magic number. We assume the flow transists exactly at the point
5 times 10 to the fifth; in reality, that's not true. There's a region where it
transists from the laminar to turbulent. Okay. Back over here; all right; magic
number 5 times 10 to the fifth; other people use different numbers. If the free
stream velocity has some turbulence in it, a little bit of turbulence, it's
going to transist to a lower one. If you take really, really, really, really
good care to make this start out to where it's laminar, and incoming flow is
perfectly parallel velocity vectors and let's say the air has been sitting in a
tank for days, there's no motion in the tank over the molecules; they're
perfectly still; then, maybe it won't transist until 7 times 10 to the fifth.
So, it depends on many factors, one being free stream turbulence. But again, in
our case, it's going to be 5 times 10 to the fifth, okay. You can see at that
point, now, let's to go the turbulent one; turbulent smooth starts out here at
about 6 and goes down below 2. So, here's turbulence smooth; it keeps going down
like this. So, this line is turbulent smooth. That's equation, right here, he
tells us to put in [phonetic] 745. This is equation turbulent, that's not
turbulent, here, okay; equation 745. So, plop that guy over here. Now, if it
transists from laminar to turbulent, at what magic number, 5 times 10 to the
fifth; that's right here. You can see what happens. This curve goes up here and
goes like that. And then, he says, this is equation 749, and here it is down
here, equation 749.
So, I erased most of it, but, oh, I didn't; here, it's over here. Okay. If we
know the Reynolds number is less than 5 times 10 to the fifth, on this graph,
what line do we follow? That line right there until you get down to where my
finger is, that line right there. Okay. If the flow is all turbulent from the
leading edge, it's been artificially tripped to terminal for leading edge, which
line do you follow? This line. This line. All turbulent all the way down to
there. Okay. Let's go to where it's mixed flow; starts out laminar and goes to
turbulent. Now, you follow this line on your roadmap. You go down this highway,
stop; then, you go on this shortcut; stop; then, you go down turbulent flow. You
don't have that guy right there. Okay. That's next flow. You want to see sub d
next flow. There it is for you. Read the graph. Not very accurately, of course.
But you can read the graph, or you can use the equations, you know. The choice
is yours. Oh, by the way, there's other curves wicking away bottom, there's
curves that go out like this. These are for all different values of roughness, l
over e --
I didn't mention before, but all these equations are for smooth plates, smooth
plates. If they're rough plates, there's equations in the textbook. We're not
going to worry about rough plates right now. Or you can go on these lines right
here for the relative roughness. Does it matter in laminar flow? No, it doesn't
matter. We engineers love graphs like that. Or you can use the equations. Of
course, you have to if you're doing a code, a computer code, you've got to use
the equations. But the graphs, people are always interested in graphs. They say,
what's going on there? What's caused that to do that like that? And they think
the same; they think the same. Okay. You can guess what I'm going to draw here.
Dimensionless, dimensionless, dimensionless, okay, here we go; one more time,
Emmy 311, dimensionless, dimensionless, dimensionless. Oh yeah, there it is.
Here we go, one more time, is there an equation for laminar flow? Oh, yes, there
is. Laminar flow. Laminar flow. Is there an equation for smooth pipe? Oh, yes,
there is. Smooth pipe. Yeah, there it is. Is there an equation for relative
roughness? Oh yes, there it is. Are they similar? My gosh, are they similar?
Geez! See we engineers think like that a lot, and if that helps you to see
things in the real world, in dream world, fine. That's the whole point of
graphical presentation because they're beautiful. This was named of somebody,
Woody Chart. This has no name. This is Mr. No Name, okay, unfortunately. But
what do they do? This is for flow over a flat plate. This is for flow inside of
a tube. And I'll tell you something, you know, you can have hf --
Oh, my gosh! We engineers think alike. Look over here. Where is he? Okay.
Reynolds number, okay. There he is. Uh-huh. Row b squared squared divided by 2.
Guess what? Flow in the pipe; row b squared divided by 2. Guess what? Geometry.
Guess what? Geometry. Guess what? Dimensionless friction factor. Where'd that
come from? Salina [phonetic] Park with a bunch of equations. Oh, no, no, no, no,
no; we invented that. So, you use this graph and plug it in here. We invented
that, drag coefficient. Boy, are they similar? You'd better believe they're
similar. Oh yeah! That's the beauty of the graphical part. He said, well, why do
you want other -- just use the equations. I know you can. But you don't get a
full understanding of the whole picture until you look at those graphs. I'll
tell you something else, this is a log law graph. This is a straight line. When
I plot this guy, okay, on the log law graph, this is called a power law, if he's
constant, power law. If he [inaudible] our log law graph, you'd probably get a
straight line here too in the laminar region, like a straight line here. Watch
this guy here.
Right there. What's the Reynolds number power? Minus 1. What's the Reynolds
number power? Right there, minus 17. Mm-hmm. So, yeah, they're very similar. So,
you know, this is on your data package for your exam. So, that does, it's hard
to read though, so I would recommend you use the equations. But if you want to
see where you're going or where you're at, look at this thing. You can say, am I
on this curve for all turbulent? Am I on this curve for all laminar? Or am I on
this one, this one, and this one for mixed flow? Like that. Okay. So, let's do
an example. I might want to keep these, so let's leave that go, and I'll start
over here. So, we take an example here, and the one we're going to do is air.
So, example --
Air. Okay. So, here's like 20 degrees C.
So, we'll take u 2 1/2 meters per second. Did I call that b? I think I did. I
got b. B is the width, 3 meters. The plate length is 1/2 a meter. Okay. Got it.
Free stream velocity, da, da, da, da, 20 degrees C air, got it. Okay. Properties
of air at 20 degrees C. Table A2, back of the book. Density, 1.2; kinematic
viscosity, 1.5 times 10 to the fifth minus 5 that is. Kilograms, cubic meter,
meter squared per second. I think that's all we need. Okay. First thing,
Reynolds number, Fl.
If I do that, I get 8.33 times 10 to the fourth.
Which is less than 5 times 10 to the fifth. So, it's all laminar.
Go it. I'm just going to find a lot of stuff about this, so I'm just going to
tell you where to find them. Let's find the boundary layer of thickness at the
end of the plate. Let x equal, l is a half, point 5. Okay. It's all laminar; go
over there. Delta, 5 x over square root Reynolds number at x.
I've got x; I've got Reynolds number; put them in there, 0.0087.
Almost 9 millimeters, a little less than 1 centimeter, 1 centimeter. The plate
length is 1/2 a meter. The plate length is 50 centimeters. Delta, at the end of
the plate, is 1 centimeter, a little less than 1/2 an inch. So, it's very, very
thin with air blowing over the plate at that velocity, okay, very thin. The
boundary layer is usually very, very thin. Okay. Let's go ahead and get c sub f.
So, the skin friction coefficient, c sub f. There it is, .664 square root
Reynolds number.
End of the plate, x equal l. Put l there. Put l there. We're at x equal l. You
can put x or l; it doesn't matter, but we'll just try and -- the actual value of
x is the plate length, l. Put that Reynolds number in here; it's easy. You get
0.00514. Dimensionless, skin friction coefficient. So, the wall shear stress
equals c sub f times row u squared over 2. So, we put c sub f in there, .00514.
The density, 1.2. Velocity, 2.5 squared divided by 2. The wall shear stress at
the end of the plate --
Okay. There it is. Okay. Let's go ahead and get c sub d. It's laminar over the
whole plate. There it is, 1.328 over Reynolds number L to the 1/2. Okay. c sub
d, yeah. We know the Reynolds number 8.33 times 10 to the fourth. Drag
coefficient comes out to be .00460. Drag force equal to drag coefficient times
the area times row u squared divided by 2.
The area of the plate, the width was 3; the length is 1/2; 3 times 1/2.
Density, 1.2; u squared, 2.5 divided by 2. It's really small. For the laminar
flow, you don't expect, and this is really low velocity, so that is 0.026
Newtons. Just so you can see, next, 5.6 miles per hour, so it's really, really
slow velocity, you know, 5.6 miles an hour. That's why the, this is the drag
force and it's really low, and it's laminar flow, which makes it even lower. Big
drag force, turbulent flow, low drag force, laminar flow. All right. Part B.
This is Part A. Part B. Assume boundary layer is trip to turbulent --
At x equals 0, at the leading edge. At the leading edge, the problem it said has
been artificially tripped to 0. Okay. There's the equation, right there. There
are the turbulent flow equations across the whole boundary layer, turbulent
flow. Reynolds number, it's the same, okay. Everything's the same, so put that
guy in there, and you get 0.01585 meters. You can find tau wall anywhere. Give
me an x; I'll find you a tau wall. There's the two equations, not a big deal.
Give me an x; I'll plug it in there, and I'll find tau wall. If I want to, I'll
find the skin friction coefficient. But I'm going, most of the time, we're
interested in the drag force on the object, on a plate; so, I'm going down the
drag force now. Find the drag force on the plate if the flow is turbulent. First
of all, get c sub d. C sub d, .031 divided by Reynolds number, 8.33 times 10 to
the fourth --
Okay. Put him in there; get c sub d, 0.00614. Then, get the drag force, drag
force; okay, 0.00614 times the area of the plate, the top surface of the plate,
1/2 by 3 times the density, 1.2, times u, 2.5 squared divided by 2. The direct
force on the flight, 0.035 Newtons.
Okay. Direct force laminar, .026; direct force turbulent, oh, it's bigger, of
course, .035. That's about a 33% increase over laminar. So, yeah, the turbulent
flow over the flat pipe gives a bigger drag force. Okay. Change your problem
now. Go back here. You say keep that at 1/2. No, I'm sorry; change it to 5. Make
it longer, okay, 5 meters. Ebb flow rate's way too low. I want that to be 56
miles an hour. Now, the airflow is 56 miles an hour. The plate's 5 meters long.
Okay. L went up by a factor of 10. Velocity went up by a factor of 10. They both
went up by a factor of 10. Bigger by a factor of 10; bigger by a factor of 10;
stays the same. Uh-oh, nope, now the Reynolds number at the end of the plate is
definitely greater than 5 times 10 to the fifth. So, now, I say, so, it's a
mixed flow.
Okay. Now, my picture looks like that. Okay. I am going to find the drag force
on this picture now, the drag force. By the way, if I want to find where it
transitions -- I'll do that first. Find x sub c. So, we had it from before, 5
times 10 to the fifth equals the velocity, 25 times x sub c, the critical x
distance divided by the kinematic viscosity. So, x sub c is equal to 0.30
meters. Go back over here. What's l now? L is 5. Where is l x sub c? Point 3,
okay. One, two, three, four, five, close; there's one. Where's .3? Right here.
Okay. Draw a new picture. Oh, there's just this little piece right here,
laminar; then, right here it goes to turbulent. There's a new picture. Okay. So,
back over here. Got it. Now, I want to find c sub d. The drag coefficient, c sub
d. Okay. Mixed flow with c sub d right there. There; it's right there, .031
divided by Reynolds number to the one-seventh.
Minus the 1440 divided by the Reynolds number.
Put the numbers in the Reynolds numbers right up here, 8.33 times 10 to the
sixth. Put that in there. This guy is 0.0046. Drag force then is c sub d times
the area of the plate. The width is 3. Three times 5 times row, v squared
divided by 2. So, the drag force in this case for the mixed flow situation is
16.9 Newtons.
Okay. So, we've got that. Let's see, this was trip to turbulent at x equals 0.
Okay. And that value, mm-hmm. Okay, we don't need this anymore. It's the 10%
rule; if x sub c is less than or equal to 10% of l, assume boundary layer is all
turbulent because the contribution of the laminar parts can be negligible. So,
you might as well assume it's all turbulent. Look over here. If I assume it's
all turbulent, I'll bet mine's going to be pretty close to the real situation,
mixed flow, because that little piece right there is the only laminar part of
the whole boundary layer. So, in our case over here, x sub c was .3 meters. Ten
percent of l; l was 5 meters. Ten percent, 0.5. So, x sub c, 0.3; 0 less than or
equal to; let's find out; 0.50. The answer is yes. You can treat that as if it
were all turbulent. So, that's the rule we'll follow for the class. If x sub c
is less than or equal to 10% of the length of the plate from mixed flow, assume
that it's all turbulent. Okay. If we assume it's all turbulent --
Then we get c sub d, if it's all turbulent, over Reynolds number l to the
one-seventh power, c sub d, 00614, and we get the drag force, c sub d times the
area row u squared over 2 --
Zero point zero three five, 0.035; let's see, where is our mixed flow, the drag
force -- oops, that's the wrong one. Excuse me. Here it is.c sub d is, let me
get this right here now, that was turbulent. C sub d is 0.00312; drag force
17.61 Newtons -- If we do it the right way and assume it's mixed flow, we get
the drag force, 16.7. Because of the 10% rule, if we assume it's all turbulent,
we get 17.61, off by 4%.
Is that okay? Yeah, it's okay. I mean, 4% is not that far off. So, you can use
that rule. So, that's how you use the 10% rule, okay. Okay. Let's erase this
now. Point 0046, .0046, Reynolds number, 8.3 times 10 to the sixth; 8.3 times 10
to the sixth; 8.3 times 10 to the sixth.
That's our Reynolds number. Go up here, go across here, if I had drawn like,
which I don't. Okay, you can look at our picture here. Take 8.33 times 10 to the
sixth. There it is. And we go up there and we're, okay, we're at the lead mixed
flow. Yeah, we're close. We're higher than that, but yeah, that looks right. We
are down here. Let me just make sure it's right, that mixed flow one, c sub b. C
sub d, okay, da, da, da, da, da. I thought so. The answer is correct down here,
but I was looking at the turbulent flow, 0.0030. That's what it is. So, go up
here, go across here; what do I read? Oh, I can read close to .003. So, the
point is you can do the equation. You get .003. You can go to the graph at the
right Reynolds number; you can get close to .003. It might be .0031; I can't
tell, or .0029; I can't tell, but I'm really close with my eyes reading the
graph. You want to go to all turbulent. Okay, let's try that. All turbulent, the
Reynolds number; it's the same, the same. Now, I know how it's drawn; I'll show
you how it's really drawn. That's how it's really drawn if you check your graph.
Okay. Where's all turbulent? This one. Where's the mixed flow? This one. Where's
the .30? Mixed flow. Okay. Where's the all turbulent one? That big black dot. I
read it 0031 with my eyes. What's the equation one? Oh, am I good? I'm off by
six points, six out of 100. That's really good. So, what I'm saying is don't
downplay the graphic; oh, the graph's not accurate? Oh yes, it is. It's as
accurate as you take the time to read it. Just like the movie chart; say, oh, I
use the equations all the time. That's okay, but I guarantee you, I'll be really
close to you with the equations if I read the Moody chart with my eyes because I
know how to read those graphs, okay. That's the whole point. So, whether you
want to read the graph, .30, .31, 30, 312, okay; that's okay. I'm not going to
argue about that difference like that. So, the choice is yours. I think you
should use the equations, but the graph just helps give you a feel. This is a
road map. I have mixed flow. I start here; I go here; make a left turn, go on
here, merge down here. Now, I'm turbulent flow. That's my roadmap. That helps me
visualize what's going on in the problem. Okay. We took that problem a long way.
S, if we're going to stop for the day, we'll pick it up next time. We'll see you
then on Wednesday. Look, don't go anywhere. We're going to give you the exams
now.
