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# Lies, More Lies, and Arithmetic

09/30/2006 4:00 PM

We all know how an airplane wing works: the top surface is longer than the bottom, so the air across the top must flow faster than the air across the bottom. Just look in any basic physics text. (Bernoulli, and all that.) The calculation involved is easy. Using Bernoulli's principal, we know that the difference in pressure (top vs bottom) is equal to the difference in the squares of the velocities. [If you look up the formula, you will find one of those funny looking letters, rho, which stands for the mass density of air. In English units, that measurement is in slugs per cubic foot, and at STP, its value is .0024. In doing the math below, you probably won't need that, because it cancels out (given that the top and bottom of the wing are flying on the same day through the same conditions.) But you may find that you want to double check by calculating the actual pressures.]

So here is the challenge, which probably seems a little elementary to many of you (it doesn't require any calculus, and doesn't even require algebra, unless substituting numbers for letters counts.) But even if it seems elementary, go ahead and try it; it won't take long, and you'll win the admiration of your peers if you can get this to work out right. All you need to do, is run the numbers (which you can darn near do in your head) to show that my old Beech Sport would be capable of supporting its own weight at 49 knots, the published stall speed (the speed at which it essentially stops flying and begins to sink like a stone) . Here are some actual measurements. The top surface of the wing is 1.008 times longer than the bottom: but use 1.01 for ease: we're trying to prove the thing can fly, not that it can't. 49 knots = 83 FPS. Consider the velocity across the bottom of the wing to be 83 FPS, and that across the top to be 83.83 (1 percent higher). My plane's weight, with fuel and me (assuming 1G): 2000#. Wing area: 146 square feet. SP = 14.7 psi, and the wing area, in sq inches, for the lazy among you is 21,024. (To those of you who work in metric units, I'd say "Hey, your units already make more sense, and the numbers are simpler, so to avoid giving you an advantage, you have to do the conversions yourself.")

So all you need to do is show that the differential pressure is about 1/10 psi, and you've shown that we can lift 2102 pounds, which is enough above the weight of my plane that I'll feel safe in flying it, knowing the physics work. Post your answers, and a clear (maybe step-by-step) rationale, to impress your peers. If you come up with an answer that doesn't seem to work, just google for Bernoulli's principle to make sure you have the right formula. You might even separately calculate the pressures top and bottom, just to convince yourself that you are doing the math right. Warning: I know some pretty good mathematicians who must have goofed up somehow, because they couldn't get the numbers to work out.

Extra credit: for those of you who were able to get the right answer: My friend has a Pitts Special, which flies ever so much better than my own plane, especially aerobatically. Do the math to show how this is possible with its symmetrical wing profile, in which the top surface and bottom surface are the same length.

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#1

### Re: Lies, More Lies, and Arithmetic

10/01/2006 12:15 AM

I know nothing about lift, but with enough prop power, who needs lift.

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#3

### Re: Lies, More Lies, and Arithmetic

10/01/2006 1:55 AM

People with limited fuel resources! It consumes far less power to use an airfoil than pure propulsion.

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#4

### Re: Lies, More Lies, and Arithmetic

10/01/2006 5:59 AM

Propellers and turbine blades are aerofoils.

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#2

### Re: Lies, More Lies, and Arithmetic

10/01/2006 12:15 AM

Bernoulli doesn't fly the airplane, Newton flies it. The air on the bottom of the wing is trying to get to the top and lifts the wing. The airplane's angle of attack creates a downwash, and the opposite reaction creates lift. If anyone tries to do the math, Bernoulli will contribute less than 10% of the requirement.

I would rather take a ride in your friend's Pitts Special than try to apply Bernoulli's formula to a symmetrical wing to calculate lift!

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#5

### Re: Lies, More Lies, and Arithmetic

10/01/2006 6:34 AM

With 20+ years in aviation I have a firm belief in the following:

1. If you add the Length of the props and get a length longer than the wing, it will fly. If the mounted props are a substantial portion of the visible wing leading edge, it will fly well. If you are going to screw your way through the air, better you be well equipped. I would point to your example of the Pitts and the Cessna. I would also point to two of my favorite flyers the P-3 and the C-130, both with more prop than wing.

2. Jets fly for the same reason you must deal with crying children, or avoid your angry spouse; natural avoidance of noise. If you make enough noise something has to move. Since the earth is fixed, that only leaves the craft to move. Hence two of the noisiest aircraft are two of the best fliers: the F-18 and the F-4. And the F-4 illustrates that aerodynamics has nothing to do with the issue.

3. Helicopters fly because the earth rejects them as unnatural. Also there is a strong component of #2 involved.

4. Sailplanes fly because of natural attraction. Sailplanes are beautiful, flying is beautiful, its like trying to pry magnets apart. God flys a sailplane.

I should point out that I have 20+ years in avionics, building equipment to tell you if you are flying, where you are flying, and if you did fly. And making it possible for said unhappy spouse to phone you while you are flying. And while I may make you NOT fly, I don't make you fly. That is usually done by a disgruntled customer at least two connections away from anywhere you wanted to be.

Emmett

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#8

### Re: Lies, More Lies, and Arithmetic

10/01/2006 12:53 PM

Excellent! Some of the best explanations for flight I've read!

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#9

### Re: Lies, More Lies, and Arithmetic

10/01/2006 1:12 PM

But wait!! My physics teacher taught me, back in the late 60's, long after airplanes were developed, that Bernoulli was the pilot. Here's an explanation from How Things Work, 1988.

It is written.

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#10

### Re: Lies, More Lies, and Arithmetic

10/01/2006 3:02 PM

Hi Ken, it seems from your challenge that you take the air over the low-pressure side of the wing to take as much time to reach the trailing edge as the air 'under' the wing. Is it not a requirement that the transit time of air on the low pressure side must be less than the transit time of air on the high pressure side in order to generate lift?

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#13

### Re: Lies, More Lies, and Arithmetic

10/02/2006 10:49 AM

I'd have to agree with you, but my old physics teacher would not. Even now, many science museums, websites, and elementary texts provide an explanation like the one from How Things Work.

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#6

### Re: Lies, More Lies, and Arithmetic

10/01/2006 10:56 AM

Angle of attack!

Take a bit of thin wood etc. (stiff balsa about 3 ft long by 3 in wide, 1/8th in thick is ideal). Whirl it round, holding it at varying angles of attack to its perceived airflow. Tilt it up - you'll get lift, tilt it down - it'll dive.

(Acknowledgements to Ried, who's comment I didn't read well enough 1st time round).

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#7

### Re: Lies, More Lies, and Arithmetic

10/01/2006 12:45 PM

Another fun thing to do with wood: go to Home Depot, and buy a 1/4" 4' x 8' sheet of plywood. While still in the store, hold it over your head, so the entire sheet is about parallel to the floor. Now lower one side so the sheet is about 10 degrees from horizontal. Walk out into the parking lot like this, on a day with 20 knot winds. Report your findings.

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#11

### Re: Lies, More Lies, and Arithmetic

10/01/2006 3:34 PM

Bin there, done that - it's Fun!

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#12

### Re: Lies, More Lies, and Arithmetic

10/01/2006 11:45 PM

I notice everyone else posting is as lazy as I am and did not tackle the math . . .

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#14

### Re: Lies, More Lies, and Arithmetic

10/02/2006 11:56 AM

The math works out like this:

½ ρ (83.832 - 832) = .0012 x (7027.46 – 6889) = .166 lb/ft2. This, times the wing area (146 square feet) works out to about 46 pounds, or about 2 % of the required lift.

I think it is interesting that the Bernoulli explanation still shows up as much as it does. If the intent was to make things simple, then why not just say that a wing deflects air downward, and as a result the wing is pushed up. Some links to plausible explanations of how things fly:

http://regenpress.com/

http://www.av8n.com/how/htm/airfoils.html

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#17

### Re: Lies, More Lies, and Arithmetic

10/03/2006 2:49 AM

Just to see what would happen I decided to work what the differences between the upper and lower cord lengths would be on my H-206 Hornet glider and here is what I got.

Wing Area = WA = 9.8m2

Maximum Take Off Weight = M = 420Kg

Gravity = 9.8ms-1

Stall Velocity = VS = 42Kts = 21.6ms-1

Upper Cord ÷ Lower Cord = Δ

Density of Air = ρ = 1.29

At the stall speed the velocity over the wing is VS therefore

Upper Velocity = Lower Velocity x Δ or VU = ΔVL

Bernoulli's principal

P1 + ½ ρV12 = P2 + ½ ρV22

P1 – P2 = ½ρ(V22 – V12)

And the lift created is the pressure difference times the surface area so substituting we get

And solving this gives us

and now solving for the know conditions we get

Meaning that the velocity over the upper wing surface is 1.55 times the velocity over the lower wing surface which would indicate that the upper cord was 55% longer than the lower cord.

Now strangely enough if you look at a hemisphere the ratio of the radius to the half circle circumference happens to be 1.57 which is awfully close to the result above. Draw your own conclusions.

This was also an attempt at inserting equations into a CR-4 post. It was fiddly an the results weren't real great but there is a way.

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#20

### Re: Lies, More Lies, and Arithmetic

10/03/2006 10:55 AM

I'm envious of your glider. Although I've had loads of fun in powered aircraft, I've spent little time in gliders, but I've really enjoyed that little bit of time. Gliding seems to be flight at it purest and most efficient.

An acquaintance from my college days is Karl Striedick, who has flown on the order of 1000 nautical miles from a single Jeep tow launch. I think he still lives in State College PA, on top of a ridge, where from his grass strip he can launch and fly close to the entire length of the Appalachian Mountains, which fill most of the east coast of the US. He's flown from central PA to Nashville TN, and back on one launch.

It would be interesting to measure the actual upper surface length vs lower surface length on your glider. You'd need to set the craft at the attitude that corresponds to the stall angle of attack and then make a pencil line where a plumb line touches the leading edge. Measuring back to the trailing edge, along both surfaces, I'd guess the difference top/bottom might be as high as 1.05. (I have a full scale wing rib template in the shop which I just measured. It has unusually high asymmetry, and it came out at 1.04, but your wing might be even more asymmetric.)

I think you meant to say that the ratio of the diameter to the half circle length would be 1.57. (The radius to half circle length would be π) << there's a pretty yucky looking symbol, eh?

While on the subject of glider wings: most modern glider wings have a design lift coefficient of about .6. When they are flying, this CL is achieved at 2o angle of attack. (Therefore, the wing is mounted to the plane with this angle of incidence, so that fuselage drag is minimized.) At 0o angle of attack, these wings produce a CL of .4 (about all that is required for cruise flight in many aircraft). So... it's clearly not just angle of attack, nor just Bernoulli, nor just circulation, nor entirely magic.

But Karl's flights would have to argue in favor of magic, wouldn't they? (Imagine if our cars got similar mileage: 1000 miles on the several teaspoons of gas the Jeep burns per launch.)

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#22

### Re: Lies, More Lies, and Arithmetic

10/03/2006 11:34 AM

Sorry I did mean the diameter compared to the half circle.

As for gliding the world record is held by a New Zealand pilot and is 2,192.9 Km. He started in the middle of the south island flew to the southern end of the south island then headed north across the Cook Straight to the north of the north island then back to his starting point. Most of the way he rode the wave in the atmosphere that is caused by the roaring 40s as they hit New Zealand. Similar to what your friend did.

In Australia we don't have any real big mountains, well in reality we don't have any mountains at all as the maximum peak is only 7,310 feet so we don't get a lot of ridge soaring in. We do however get mind blowing thermals that can give you 1,000 feet per minute and take you up to 20,000 feet on a really good day. I have managed 13,500 feet on a couple of occasions. I only called it off due to lack of oxygen.

My most memorable experience was flying tight formation with a wedge tailed eagle with a 3 m (10 foot) wing span. I was close enough to see the mottling on his feathers. Gliding in my opinion is as close as you can come to soaring like a bird, you can defiantly soar with them.

PS I DO NOT RECOMEND approaching large birds in flight unless you are sure that bird doesn't see you as a threat.

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#23

### Re: Lies, More Lies, and Arithmetic

10/04/2006 9:34 AM

Oh no. Now we've done it. The three ways to derail any discussion is to mention politics, religion, or worst.... flying!

One of my most pleasant flying memories also has to do with flying with the birds. A friend and I delivered a J3 Piper Cub from Pittsburgh, Pennsylvania, to Melbourne Florida over two days, dawn to dusk each day. Nothing for navigation other than a kerosene compass, and a handheld vhf for communications. Due to the low landing speed of the Cub, we could safely fly a low altitude. Trucks on I-95 (a major N-S highway) would routinely pass us. Several times we flew beside eagles, who could almost keep up with us. We really had the feeling of being kindred spirits. The eagles were no doubt thinking: now if that's not the ugliest bird I've ever seen!

Wow, the senery on the New Zealand flight would have been breathtaking!

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#28

### Re: Lies, More Lies, and Arithmetic

05/08/2009 9:33 PM

Hi everyone.

The thing is Lift is not about the upper and lower cords of the wing. It's about the available room for air to flow above and below (imagine vertical cross-sections of the flow, not of the wing). As inside a narrowing pipe, what matters is the cross section variation. And air flows along the wing, above and below, restricted by the wing and the Potential Flow, which are pretty much like a pipe.

Plus, for a correct calculation, you would have to integrate the static pressure on the wing surface.

Maybe my other post will clarify...or not!

See you.

RGO

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#29

### Re: Lies, More Lies, and Arithmetic

05/09/2009 7:19 AM

G'day RGO,

I have read your post in the thread Myths: How does an aeroplane fly? or what is wrong with so called popular science and given you a good answer vote for it.

Ok, I believe I understand what you were trying to explain and yes as far as I know that's what happens.

However, that's not the point of this thread.

The problem is that if you plug real world numbers into the concept of your explanation and Bernoulli's Principle the theoretical amount of lift such an aerofoil creates is dramatically lower than the real world.

In my post #17 I took the real world measurements from a Glasflügel H-206 Hornet glider. I then used the concepts presented in your post and Bernoulli to calculate how much longer the upper path needed to be than the lower path in order to support the aircraft's mass of 420 kg and resultant weight of 4.116 kN.

Now this is where the problem is because it the numbers gave a result that the upper path needs to be 1.55 times longer than the lower path to support the 4.116 kN at the gliders Vs (stall speed) of 42 knots (21.6 ms-1).

I have a considerable amount of time in the H-206 so I can testify that it definitely is capable of flying and has a stall speed of around 40 knots. However I can also categorically state that the upper wing surface is more like 1.05 rather than the calculated 1.55 times as long as the lower wing surface.

Clearly there must be something else at work here and that's what this thread is about.

Regards, masu

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#30

### Re: Lies, More Lies, and Arithmetic

05/11/2009 5:18 PM

Hi masu and thanks for the vote.

Maybe you're right, I didn't quite get the real point.

Anyway, it's being fun to participate.

See you all later.

RGO

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#15

### Re: Lies, More Lies, and Arithmetic

10/02/2006 8:39 PM

Sorry ken. Airfoils really have little to do with lift. It is all about angle of attack.

I spoke to this topic once before, but take a look at a wing of an aerobatic aircraft. You will see that the wing is perfectly symmetrical on top and bottom. The plane will fly just as easily upside down as right side up.

However, the angle of attack of the wing is what generates lift.

I will give you credit for the math problem!

You are right. For whatever reason, there is a lot of misconception about how a wing works.

Now, how many of you think it is the rudder that turns the airplane?

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#16

### Re: Lies, More Lies, and Arithmetic

10/02/2006 10:06 PM

Actually, airfoils have a great deal to do with lift. Just try taking off with only a fuselage. Maybe you meant to say that Bernoulli has little to do with lift. Which is partly true. Actually Bernoulli effects do come into play, but the flow across the top of the wing is much faster than the classic explanation, do to circulation.

Angle of attack has a huge effect, with lift increasing in linear fashion from 0 degrees up to very close to the stall angle of attack. (In fact, a convenient rule of thumb for many wing profiles is that the angle of attack/10 gives you the coefficient of lift.)

But angle of attack alone does not explain how many planes can be flown (at high speed of course) with a wing angle of attack of 0 degrees.

The point of this math exercise is to suggest how easily incorrect explanations of physics, and other things, get into the mainstream. Even FAA texts used the How Things Work explanation when I began flying. It is interesting to see, I think, that the HTW explanation can only account for 1/50 of the lift actually required to fly the plane.

Of the links above, the one that gives the most complete explanation, in my view, is the av8n one, although another one gives an interesting Von Karmann quote on the value of keeping things simple even if misleading: basically he advocates baffling em with BS.

Everyone knows what makes an airplane turn: the copilot. The pilot is usually sleeping.

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#18

### Re: Lies, More Lies, and Arithmetic

10/03/2006 3:19 AM

I agree with Ken's: "Actually, airfoils have a great deal to do with lift."

Airfoils are all about efficiently generating lift. Flat plates can generate lift, but at a huge cost of drag, when compared to airfoils shaped for economical performance for the specific aircraft type. I think fully aerobatic aircraft get away with symmetrical wing profiles because they don't care about efficiency - they do it with brute force.

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#19

### Re: Lies, More Lies, and Arithmetic

10/03/2006 4:05 AM

They also spend quite a lot of time flying upside down!

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#21

### Re: Lies, More Lies, and Arithmetic

10/03/2006 11:07 AM

Well put. In fact, in purpose-designed aerobatic planes, especially those which are built for mere mortals to fly (like the Pitts Special) inefficiency is advantageous. With the drag of wires, struts, fixed gear, double wings, etc. when you point the thing downhill, you are less likely to exceed Vne inadvertently. My own plane, which was certified for aerobatics, (but not purpose-built for it) was much more slippery, which made it a little less safe. Also, the lack of power meant that to do loops you would have to enter at just under Vne. While flying, you'd think you were doing a nice circular loop. But from the ground perspective, you were really writing a lower case cursive "l". Not pretty, but loads of fun, anyway.

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#24

### Re: Lies, More Lies, and Arithmetic

10/04/2006 10:04 AM

Hi Ken, your 'almost Vne for entering a loop' made me recall my old Harvard trainer days. Same thing. Do a wingover for a dive, pull up short of Vne and 'will' it over the top of the loop. Or, much safer, roll it out at the top, if you still have speed!

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#25

### Re: Lies, More Lies, and Arithmetic

10/04/2006 10:57 AM

You hit it on the nose: "'will' it over the top of the loop"! I can't tell you how many times I would roll out of a loop at the top. I'd get back on the ground, and someone would comment on the nice Immelmans. If only they knew!

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#26

### Re: Lies, More Lies, and Arithmetic

10/06/2006 11:04 PM

Flying bricks, baby!

See #5 above: With enough thrust, you can make a brick (F-4) fly!

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#27

### Re: Lies, More Lies, and Arithmetic

07/19/2008 11:18 AM

hope you are ok?

I am really interested in the good and clever way you described this thing on flying. I am going to have a going with the Math.

In the mean time, am I right in thinking that a by=planes wing as you describe, is affectively a wider wring so it gives more lift and controllability? You do not have to answer yet. If you want to wait for some answers to the 'riddle' you pose.

Just one last question...........what math style would you usually use to work this out? I see no problem in changing inches to centimeters at all. But the answer will look different I think?............As there is 6.25² Cm's to a square inch.

babybear

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