Is the 'Equal Transit' Theory Really True? A Closer Look at Airplane Flight

[daniel_i_l]

For as long as I can remember the anwer I've heard to this question was the "Equal Transit" theory - that since the top of the wing is longer that the bottom the air has to go more quickly over the top inorder to "keep up" with the bottom causing a lower pressure on the top than the bottom. But this doesn't really make sense - why should the air on top "care" about the air on the bottom?
When I looked on Nasa's website I found a different explanation:
http://www.grc.nasa.gov/WWW/K-12/airplane/right2.html
and a debunking of the "Equal Transit" theory:
http://www.grc.nasa.gov/WWW/K-12/airplane/wrong1.html

So why have I never come across the explanation Nasa gives and everyone gives the "Equal Transit" theory? Which explanation is true? Or is it both?
Thanks.

 

[Doc Al]

The "equal transit" explanation, while popular, is complete nonsense.

 

[daniel_i_l]

Even if the "equal transit" explanation of the velocity difference is silly, the speed of the air over the top of the wing in greater than on the bottom due to the shape. But what lifts the plane - the pressure difference, Nasa's "turning of air" or a combination of the two?
Thanks.

 

[Doc Al]

I didn't look at the NASA links (yet), but here's a decent description of how airplanes fly: http://www.aviation-history.com/theory/lift.htm"

 

[D H]

Ultimately it is the turning of the airflow downward: Newton's third law. However, you can push air down using any old slab of plywood that is angled with respect to the airflow. This will of course generate a lot of drag; it is not a good airfoil. In a good airfoil it is the upper surface of the wing that does the lions share of the turning the airflow rather than the lower surface. To get a good picture of this it is good to look at things from the perspective of Bernoulli's principle.

 

[russ_watters]

That said, the low pressure (plus the coanda effect) above the wing causes the air above the wing to be pushed down. So it's just different ways of saying the same thing. Perhaps it is easier to quantify or explain using Newton's Third, but I don't know. I prefer the pressure/coanda effect explanation because the Newton's Third is a "what", but not a "why".

 

[russ_watters]

The "equal transit" explanation, while popular, is complete nonsense.

Though I realize it is wrong, too-emphatic rejections of it can lead to misnomers as well - they imply that the Bernoulli equation is inapplicable and there is no area of low pressure above the wing. I don't like that NASA link for that reason. The NASA link combines the "longer path" and "equal transit time" explanations into one, and I don't think that that is correct. IE:

Let's use the information we've just learned to evaluate the various parts of the "Equal Transit" Theory.

{Lifting airfoils are designed to have the upper surface longer than the bottom.} This is not always correct. The symmetric airfoil in our experiment generates plenty of lift and its upper surface is the same length as the lower surface.

That's wrong. As can clearly be seen from giving the airfoil some positive angle of attack, the stagnation point moves down on the leading edge of the airfoil, so the air going over the top surface does take a longer path than the air on the bottom surface.

The other two bullet points (including the one addresses Bernoulli's equation) are correct, though one is just pointing out what those two misnomers get right (the way Bernoulli's eq can tell us the pressure gradient).

 

[rcgldr]

A summary of posts I've made in previous threads:

After visiting a large number of web sites, my conclusion is that lift is a combination of Coanda effect, "void effect" and simple deflection, all of which result in the "downwards" acceleration of air. Coanda effect explains how laminar flow follows a convex suface. "Void effect" explains how turbulent flow follows a convex surface. Concave surfaces simply deflect airflow. The curvature of air flow accelerates the air and generates lift. "Void effect" explains how drag is developed "behind" a wing, while direct forward deflection of air accounts for the drag in front of a wing, along with friction along the surface of a wing.

Except for a special class of airfoils, most of the air flow over and under a wing is turbulent, with only a portion of the air flow being laminar near the leading edge. For most wings, the flow transitions from laminar to turbulent flow above and below a wing, detaching during the transition, but reattaching after the transition. This happens in the first 30% of the chord length or sooner on a "normal" airfoil, and between the first 30% to 70% of chord lengh for a "laminar" airfoil (by definition). In some cases, rough surfaces and/or turbalators are used to cause the transition to occur at a specific position on an air foil. In the case of gliders, an "oil flow test" is done to visualise this transition. A bead of oil is placed on the leading edge of the wings, the glider is flown for a while at a fixed speed, then landed and the oil pattern observed. It's common practice to do this in glider magazine reviews.

http://www.standardcirrus.org/Turbulators.html

Oil flow testing is also done in wind tunnels:

http://www.hisacproject.com/news.html

At this web site, pages 4 and 5 discuss how little air flow is laminar over many wings, and how "laminar" air foils increase laminar flow to 30% or more over the chord length of a wing. In the case of gliders, laminar "bubbles" result in either more drag or less lift so the laminar air flow is deliberately broken up sooner than it normally would via rougher surfaces or turbulators (this is mentioned in the article). The laminar section starts mid way down page 4:

http://www.dreesecode.com/primer/airfoil4.html

"All airfoils must have adverse pressure gradients on their aft end. The usual definition of a laminar flow airfoil is that the favorable pressure gradient ends somewhere between 30% and 75% of chord."

http://www.aviation-history.com/theory/lam-flow.htm

The next website does a descent job of explaining lift, but with a bit too much emphasis on Coanda effect, ignoring void effect and turbulent flow, but towards the end of this web page, there's a diagram of a wind blowing over a roof, and although the air downwind of the roof is turbulent, it's also at lower pressure, due to void effect. Since both laminar and tubulent air flows contribute to lift, both cases should have been covered better than it was at this web page:

"The physical cause of low or high pressure is the forced normal (perpendicular) acceleration of streaming air caused by obstacles or curved planes in combination with the Coanda-effect.":

http://user.uni-frankfurt.de/~weltner/Mis6/mis6.html

videos

Assuming this next video isn't GGI, it appears to be a series of pictures of a flame aimed at various angles over an glowing (from the heat) airfoil at a fixed angle of about 45 degrees. As the flame angle is made more horizontal, the effective angle of attack becomes higher. What I call "void" effect is more evident here, as the flame flow is detaches from the aft end of the airfoil at low effective angle of attack. At higher effective angle of attack, the flame flow detaches from the "upper" surface of the airfoil, but it's still accelerated (curved) "downwards", while below the airfoil there is significant direct deflection. About 28 seconds into this video (you can hold it at this position), the downwards curvature of the flame over the "top" of the wing is still evident, in spite of the large amount of apparent detachment.

http://www.youtube.com/watch?v=hkJaTTIiXSc&fmt=18

Next is a link to a small wind tunnel video, considered a "2d" airflow (equivalent to a 3d wing with infinite wingspan). Air speed is slow, chord length is small, so the Reynolds number is quite low, and the air flow is much more laminar and the angle of attack before stall is much higher than it would be if everything were scaled up to a faster speed and a larger size. The smallish wind tunnel also prevents any significant upwards or downwards flow of air, so the air flow is not the same as it would be in an open environment. Wind tunnels that are much larger than the wing or model being tested, such as the one linked to above showing oil flow testing are much closer to "real world" environment. The transition into the stalled condition is very abrupt. In the segment annotated as "stall", there's virtually no lift, but near the end of the video, that starts off "flow attached", then "stall", there's still significant lift although there is a stall.

http://www.youtube.com/watch?v=6UlsArvbTeo&fmt=18

For this model, the stalling angle of attack is fairly small:
http://www.youtube.com/watch?v=5wIq75_BzOQ&fmt=18

Another wind tunnel, slow air speed, short chord, but not as much as the first video. Again the nature of the wing tunnel (proably drawing air inwards from the right), prevents the air flow from remaining deflected, and skews what would happened in an open environment:

http://www.youtube.com/watch?v=TGUSmdFmXDg&fmt=18

Regarding equal transit times, here are a couple of links to pictures of a flat top, curved bottom, pre-shuttle prototype:

M2-F2 glider with F104 chase plane:
m2-f2.jpg

M2-F3 rocket powered model (reached a speed of Mach 1.6) with B52:
m2-f3.jpg[/QUOTE]

 

[JonnyV70]

Would the same physics apply to a paper airplane or does the weight difference in this case change the physics? I always thought paper airplanes flew or glided due to the large surface area to weight ratio.

 

[Renge Ishyo]

So why have I never come across the explanation Nasa gives and everyone gives the "Equal Transit" theory? Which explanation is true? Or is it both?
Thanks.

The equal transit theory is good enough for teaching people beginning physics. Its value lies in its simplicity. Yes, it is a crude approximation to the "truth," but so is most of beginning physics (if not all of it).

The equal transit theory doesn't gives you a *completely* inaccurate view of how things work. Calling it "incorrect" is extremely unfair, it implies that the theory is completely false. This is not so. As NASA mentions, it is still a fact that "flow over the top of a lifting airfoil does travel faster than the flow beneath the airfoil". It is still a fact that "The upper flow is faster and from Bernoulli's equation the pressure is lower. The difference in pressure across the airfoil produces the lift". These are the two main points of the equal transit theory, and they hold up just fine.

It is just that the equal transit theory makes many simplifying approximations about the flow of the air itself that cause it to deviate slightly from the real situation. For one thing the theory *assumes* laminar flow and so *assumes* that the fluid will recombine at both sides of the wing without "breaking" structure. This approximation turns out to be too simple a view (for one thing, we probably can guess that real air does not flow over a wing in straight uniform lines that never cross each other and never break). In fact, the proof that the equal transit theory is just an approximation is that under careful observation, the air over the top of the wing surpasses the flow over the bottom as they emerge from the other side of the wing. This can be seen clearly in this video yanked from the post above (watch from 25s to 50s):

http://www.youtube.com/watch?v=6UlsArvbTeo&fmt=18

The question is, does this small difference justify coming up with a more complicated model of airflow, or does the simplified model come close enough to reality to be of use? It probably depends on what your needs are. If you are a student who is just trying to get the general idea of how wings generate lift the crude approximation is probably just fine (and this is probably the reason this theory predominates in education), but if you are an engineer for NASA building wings for a new airplane you might have to use more complicated airflow models to get more accurate calculations.

 

[rcgldr]

The equal transit theory is good enough for teaching people beginning physics.

Equal transit theory doesn't exist in the real world. There is no tendency for separated air flows to remerge with the same relative postion they had before being separated. These curved bottom flat top lifting bodies have the "hump" on the bottom, it's a much longer path to go around the bottom than the top and yet they glide (M2-F2) or fly (M2-F3) very well:

M2-F2 glider with F104 chase plane:
m2-f2.jpg

M2-F3 rocket powered model (reached a speed of Mach 1.6) with B52:
m2-f3.jpg

As NASA mentions, it is still a fact that "flow over the top of a lifting airfoil does travel faster than the flow beneath the airfoil".

But as shown in the case of the M2-F2 and M2-F3, this has nothing to do with the distance the air has to travel across a wing.

The air over the top of the wing surpasses the flow over the bottom as they emerge from the other side of the wing. This can be seen clearly in this video yanked from the post above (watch from 25s to 50s):

http://www.youtube.com/watch?v=6UlsArvbTeo&fmt=18

Note my comment: Again the nature of the wind tunnel prevents the air flow from remaining deflected, and skews what would happen in an open environment.

Does this small difference justify coming up with a more complicated model of airflow, or does the simplified model come close enough to reality to be of use?

It's not a small difference, it is an incorrect explanation versus a correct explanation. If you want simple, then use the Newton approach: wings produce lift by applying a downwards force to the air, which reacts with an equal and opposite upwards force to the wing, in accordance with Newton's third law. Newton's third law holds true regardless of wing effeciency.

Bernoulli effects are related to the efficiency of an air foil. The acceleration of air results in an increase in kinetic energy of the air. Effecient airfoils obtain most of this increase in kinetic energy of air through a Bernoulli like conversion of pressure energy into kinetic energy. Inefficient airfoils have less of this Bernoulli like conversion of energy and consume more energy in order to produce lift. Since wings aren't 100% efficient, there is always some non-Bernoulli related increase in the total energy of the air while producing lift. Bernoulli doesn't cover this aspect of producing lift, where overall work is done on the air increasing it's total energy.

The downwards force occurs due to the combination of an effective angle of attack and a forwards speed. The downwards force results in a downwards acceleration of air. The lower surface of a wing simply deflects air downwards. The upper surface draws air downwards, to fill in the void left behind by the upper surface of a wing, and because of friction and viscosity, all of which create a Coanada like effect, even in the case of turbulent flow.

Aerodynamic forces are the result of accelerations of the air, not relative velocities. Regardless of the frame of reference, the wing itself or the air itself, most of the acceleration of the air is downwards corresponding to lift, and some of it is forwards corresponding to drag.

The lower pressure area above a wing accelerates air toward that low pressure area in all directions, except the air can't flow through (it can flow around) the wing itself, resulting in a net downwards (and forwards) acceleration of air. The higher pressure below a wing causes air to accelerate away from that high pressure area in all directions, except that air can't flow through the wing, also resulting in a net downwards and forwards acceleration of air.

Things get more complicated because air can flow around a wing. Vortices are created at the wing tips. Air flow near the leading edge of a wing is diverted somewhat over the wing, reducing the overall pressure differential.

questions that I don't have good answers for:

It seems that it should be more efficient to draw air downwards from above a wing than to deflect it from below. The total acceleration of the air results in an average terminal velocity of the air mostly forwards and somewhat downwards, and a large increase in kinetic energy of the air. For efficient airfoils, most of this increase in kinetic energy occurs through a Bernoulli like conversion of pressure energy into kinetic energy, reducing the amount of energy required to produce lift. Above a wing, the pressure is reduced below ambient, and air will accelerate towards this low pressure zone, exchanging pressure energy for kinetic energy as it accelerates towards this moving low pressure zone, with the wing just moving past before the downwards component of air flow "catches up".

I don't fully understand the process of deflection below a wing; mechanical defelection of air would seem to increase both pressure and kinetic energy, consuming a significant amount of energy, but the M2-F2 lifting body glider, a deflection based airfoil, glides reasonably well. Using a wing based reference, the air stream is slowed down and the pressure is increased, a Bernoulli like conversion. However, an air based frame of reference has to work just as well, and in this case the wing accelerates the air downwards (and somewhat forwards), and this acceleration is due to a moving high pressure area under the wing. The work done could be based on the integral of the downwards force across the vertical component of distance that the lower surface of the wing is in "contact" with the air, which is the wing chord distance times the sin of the angle of attack. However this same concept of force times vertical component of distance of the wing in "contact" with the air could also be applied to the upper surface of a wing. The camber in air foils helps in that the net vertical component of distance is reduced (the distance is "up" for the leading portion, then "down" for the trailing portion), and fully cambered airfoils are generally more efficient than partially cambered airfoils.

 

[Renge Ishyo]

It's not a small difference, it is an incorrect explanation versus a correct explanation.

The books I have read that have used this theory have been very careful to stipulate that the model that gives rise to the equal transit theory is based on the assumption that there is laminar flow and that the flow lines never cross or break. If these assumptions are *accepted* (nevermind whether or not they carry over into "actual" reality) then the conclusions of the theory follow. Again I don't like the word "incorrect" versus "correct" explanation because by assuming laminar flow the model never pretended to be anything more than a crude approximation anyway. The kinetic theory of gases is also a crude approximation that is used in education as a teaching tool. So is it wise to substitute that out for the much more difficult models because the assumptions that bring about the kinetic theory are "incorrect"? Crude simplified approximations still have a place in Physics even if you personally get to leave them behind as you become more of an expert in the field.

If you want simple, then use the Newton approach: wings produce lift by applying a downwards force to the air, which reacts with an equal and opposite upwards force to the wing, in accordance with Newton's third law. Newton's third law holds true regardless of wing effeciency.

If you tried that explanation in a beginning physics class the first thing a student would ask in response to "wings produce lift by applying a downwards force to the air" would be "how does it do this"? Then you are still left with the problem of trying to explain how the airflow across the wing generates this force. As a physicist you may want to give the more accurate explanation as above, but as a teacher of beginning physics you probably wouldn't succeed if you did so...

 

[checksix]

An interesting book on this subject is "Stop Abusing Bernoulli! - How Airplanes Really Fly"

https://www.amazon.com/dp/0964680629/?tag=pfamazon01-20

 

[Holocene]

Why does an airplane fly?

Magic.

Seriously, as I pilot, I was actually taught the so-called "equal transit" theory. Later I learned it was false, and I was pissed that not only was I taught nonsene, but that publishers wrere willing to include this garbage in textbooks.

Seriously, why has the bogus explanation persisted?

 

[jtbell]

Why does an airplane fly? Because driving is too slow.

(Sorry, couldn't resist the temptation! )

 

[rcgldr]

If you want simple, then use the Newton approach: wings produce lift by applying a downwards force to the air, which reacts with an equal and opposite upwards force to the wing, in accordance with Newton's third law.

The downwards force occurs due to the combination of an effective angle of attack and a forwards speed.

If you tried that explanation in a beginning physics class the first thing a student would ask in response to "wings produce lift by applying a downwards force to the air" would be "how does it do this"?

The downwards force occurs due to the combination of an effective angle of attack and a forwards speed.This can be demonstrated with a fan blowing air and a flat piece of cardboard, changing the angle on the cardboard changes the amount of lift and drag. Or the students, as passengers in car or a school bus, can stick their arms out the window of the moving car and by angling their hand, make their arms "fly".

The books I have read that have used this theory have been very careful to stipulate that the model that gives rise to the equal transit theory is based on the assumption that there is laminar flow and that the flow lines never cross or break. If these assumptions are *accepted* (nevermind whether or not they carry over into "actual" reality) then the conclusions of the theory follow. Again I don't like the word "incorrect" versus "correct" explanation because by assuming laminar flow the model never pretended to be anything more than a crude approximation anyway.

The issue isn't about an inverse relationship between airspeed^2 (kinetic energy) and pressure (energy), since this relationship does exist in a theortical no work situation and approximated in a real world low work situation.

The "incorrect" part is implying that there is some tendency for air molecules to regroup back to their previous relative positions after separation by an air foil, as if there was some type of memory based attraction between air molecules separated into independent streams by a solid object. An example I saw on one website (will try to find it later) included a picture of a hoop (circular cylinder) cut open with one end facing into the wind and the other end facing away from the wind. The air molecules that flow inside and around this hoop will be displaced far away from the molecules that they were previously adjancent to. As already mentioned, the longer surface on the M2-F2 and M2-F3 is on the bottom, the top is a flat surface, so the distance the air has to travel is much longer under (big curve) these lifting bodies than over (a straight path); if equal transit theory were true, then the M2-F2 and M2-F3 couldn't glide or fly.

The Bernoulli relationship also needs to work in the case air is used as a frame of reference. It's a dynamic situation, the wing passes horizontally through the air (horizontal by definition in this case, with lift perpendicular to the direction of travel, and drag in the direction of travel), with a stream of air flowing downwards and somwhat forwards towards the wing and passing behind the wing as it passes forwards through a volume of air.

 

[rwjefferson]

As fluid air mass stream passes height of contour, fluid mass inertia creates lower pressure over wing. Low pressure over wing accelerates air stream downward as per Bernoulli. In verse (equal and opposite), wing is forced upward.

Peace
rwj

 

[Type_R]

For as long as I can remember the anwer I've heard to this question was the "Equal Transit" theory - that since the top of the wing is longer that the bottom the air has to go more quickly over the top inorder to "keep up" with the bottom causing a lower pressure on the top than the bottom. But this doesn't really make sense - why should the air on top "care" about the air on the bottom?
When I looked on Nasa's website I found a different explanation:
http://www.grc.nasa.gov/WWW/K-12/airplane/right2.html
and a debunking of the "Equal Transit" theory:
http://www.grc.nasa.gov/WWW/K-12/airplane/wrong1.html

So why have I never come across the explanation Nasa gives and everyone gives the "Equal Transit" theory? Which explanation is true? Or is it both?
Thanks.

Easy...stick your hand out the window while riding in a car and make it flat with palm facing down. Now direct it upwards you'll feel a lift. Now direct it downwards... you'll feel a drag.
This is caused by the different pressure of wind underneath your palm versus the pressure of air above your hand.

 

[rcgldr]

Since someone else blew the dust off this thread ... I thought I'd do a bit more cleaning.

As fluid air mass stream passes height of contour, fluid mass inertia creates lower pressure over wing.

Inertia allows the lower pressure zone to exist, but it doesn't create the low pressure areas. Most of the lowering of pressure is due to what I call "void effect", from Wiki on wings: In that case a low pressure region is generated on the upper surface of the wing which draws the air above the wing downwards towards what would otherwise be a void after the wing had passed.

http://en.wikipedia.org/wiki/Wing

I thought "void theory" was more common, but I've only seen it used at a few sites in refernce to lift. I seem to be the main user of the term, except for the wiki article and this one The plate scoops out a void ...

http://www.terrycolon.com/1features/fly.html

The issue is that fluids and gasses prevent the creation of voids, so what would be a void ends up as a low pressure area. A better term would be "void abhorence theory". The upper surface of a wing "attempts" to create a void, but this can't be done in a fluid or gas, and the result is a low pressure zone. Usually "void theory" is used to explain why most of the drag on a land vehicle, like a bus on a highway, occurs at the back of the bus.

Here is a link about propellers, where the amount of work done on air is significant:

http://www.grc.nasa.gov/WWW/K-12/airplane/propanl.html

Note that both a wing and a propeller operate in their own induced wash, and perform work (a non-Bernoulli interaction) on the air, but the amounts are much less in the case of a wing. Since wings are so close to being 100% efficient (some are over 99% efficient), the work related aspects are often ignored (except for drag related effects).

Another tid-bit for thought. The downforce applied by the wing to the air propagates through the air and eventually ends up as downforce applied by the air to the surface of the Earth (coexistant with an upwards force from the surface of the Earth onto the air). This effect is more commonly presented as birds or a model flying inside a closed container, but the Earth and it's atomosphere are part of a closed system.

 

[Bad Monkey]

I don't see how "void abhorance" could contribute to lift. It will imply a horizontal drag on the wing, or a vertical drag contributing to a lower terminal velocity if the wing is falling. If the wing is flying and not stalled, I don't see that lowered pressure from relatively static air helps add a positive vertical force.

 

[rcgldr]

I don't see how "void abhorance" could contribute to lift. It will imply a horizontal drag on the wing.

Note that the uppper surface of a wing is cambered (or angled downwards for a flat wing) at a slight angle of attack. Using the air as a frame of reference, the surface is moving "away" from the air downwards much more slowly than that surface is moving forwards, so the air will take the path of least acceleration, which will be mostly downwards (lift) and only a tiny amount forwards (drag).

This is why streamlining the back end of land speed vehicles with long tapered tails works. It minimizes the amount of forwards acceleration of air (drag) by letting the surrounding air accelerate inwards at a relatively slow rate towards the receding tapered tail as opposed to accelerating forwards at a very fast rate to chase the tail. For example, a tear drop shape is very efficient at reducing drag, and many air foil cross sections are basically thinned and cambered tear drop like shapes.

Getting back to a wing, the top half is similar to the tapered tail, but here the goal is to generate significant downwards acceleration of air, with minimal forward acceleration. This works even with a flat board wing if the angle of attack is reasonable.

In my opinion, what occurs at the bottom of an effcient wing is also tricky, as the air diverted downwards from under an efficient wing also experiences a reduction in pressure. Part of this is because the lower still pressure air that is flowing downwards from above the wing past the aft end of the wing is helping to draw the air from under the wing into that flow.

 

[Steve Stone]

Hi everyone,

Some great reading there!

I have always been amazed how a jumbo jet can get tanks landrovers and men into the air because the wings look tiny compared to the hold.

There have been some interesting points here on how lift is acieved; but all that said how is it possible for a stunt plane to fly upside down?

I thought at first it was an optical illusion in that the plane had forward movement but was in reality falling because wings simply would not work inverted.

But then a few years ago I saw a stunt plane fly over us (the crowd) very low with smoke coming from the tail. It was easy to see the plane was flying in a straight line by lining up the smoke trail with the horizon by eye.

Thanks, Steve

 

[rwjefferson]

Inertia allows the lower pressure zone to exist, but it doesn't create the low pressure areas.

The wing and fluid mass inertia creates the lower pressure zone. Although it is the wing sweeps air out from underneath the air over the wing, if there were no inertia, there would be no lower pressure behind the height of contour.

Peace
rwj

 

[rwjefferson]

…but all that said how is it possible for a stunt plane to fly upside down? Thanks, Steve

You seem to be confused by the dogma that says a wing creates lift because the upper surface is longer than the lower. By reality, any wing that deflects air can create lift.
A wing lifts by forcing air mass downward.

The shape of the wing is based on how to most efficiently accomplish this with minimum drag.
Most stunt planes have symmetrical wings.

Peace
rwj

 

[rcgldr]

How is it possible for a stunt plane to fly upside down?

As a passenger in moving car, stick your hand out the window, straighten it out and angle it up to get lift, and down to get downforce. Although somewhat rare these days, hobby stores have these very cheap small balsa gliders that use flat board wings.

As stated above, the curved surfaces on a wing aren't required in order to produce lift, they just consume less energy in the process. Symmetrical airfoils are used on stunt planes, the curve on top and bottom are identical. Symmetrical airfoils are also commonly used on helicopters, to prevent the downwards pitching torque on the blades that results from using cambered air foils (the rotor blades are designed to flex a bit to optimize peformance over a range of pitch angles, and a pitching torque would interfere with this design strategy).

Regarding inverted flight, perhaps you missed the links to these lifting bodel NASA models, flat tops, very curved bottoms, great real world examples that disprove equal transit theory:

M2-F2 glider (next to F104):
http://www.dfrc.nasa.gov/Gallery/Photo/M2-F2/Medium/EC66-1567.jpg

M2-F3 rocket powered model (top speed was Mach 1.6):
http://www.dfrc.nasa.gov/Gallery/Photo/M2-F3/Medium/EC71-2774.jpg

 

[Steve Stone]

Thanks for that

Steve