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Airfoil Lifting Force Misconception
Widespread in K-6 Textbooks

William Beaty 1996
º Articles Here
º GIF Diagrams
º Links
º Miscon-ceptions
º Refs



New: common lift miscons, Cambridge U., also Yale, also Vtsium and XKCD comic

Beat around in the underbrush of aerodynamics and you'll encounter an interesting question:
     HOW DO AIRPLANE WINGS *REALLY* WORK?!
Amazingly enough, this question is still argued in many places, from K-6 grade classrooms all the way up to major pilot schools, and even in the engineering departments of major aircraft companies. This is unexpected, since we would assume that aircraft physics was completely explored early this century. Obviously the answers must be spelled out in detail in numerous old dusty aerodynamics texts. However, this is not quite the case. Those old texts contain the details of the math, but it's the *interpretation* of the math that causes the controversy. There is an ongoing Religious War over both the way we should understand the functioning of wings, and over the way we should explain them in children's textbooks. It's even erupted into the news, see news links below. The two sides of the controversy are as follows:

  • The physics explanation, NEWTONIAN or ATTACK ANGLE: wings are forced upwards because they are tilted and they deflect air. The air behind the wing is flowing downwards, while the air far ahead of the wing is not. A wing's trailing edge must be sharp, and it must be aimed diagonally downward if it's to create lift. Both the upper and lower surfaces of the wing act to deflect the air. The upper surface deflects air downwards because the airflow "sticks" to the wing surface and follows the tilted wing (this phenomena is called "Coanda effect" or "Flow Attachment.") Air's inertia is critical: after the wing has passed by, air must remain flowing downwards ...and the lifting force does not arise in extremely viscous fluids. Airplanes fly because of Newton's 3rd law (action/reaction forces,) the law of Conservation of Momentum, and the Coanda effect.
  • The popular explanation, PATH-LENGTH or AIRFOIL-SHAPE: wings do not deflect air: the air far behind the wing is flowing the same as the air far ahead. Instead, wings are essentially "sucked upwards" because the airfoil shape has a longer surface on top. Airfoils are curved on top and flat below, and therefore the air follows a longer path above than below. Hunks of air which get divided at the leading edge of a wing must join each other again at the trailing edge. Since the upper surface of the wing is longer, it causes the upper air to flow faster than the lower, which (by Bernoulli's principle) creates lower pressure above. Because lift is caused by the shape of the wing, wings can create lift at zero attack angle. They can create lift simply from path-length difference which leads to pressure difference, and no air needs to be deflected. After a wing has passed by, the air does not remain moving downwards. (This explanation is seriously flawed.)
This webpage is biased towards the ATTACK ANGLE view. See just about any K-6 science book for the PATH LENGTH argument. (Note that the controversy extends far beyond grade school, and even some pilot training manuals still contain the discredited "path length" explanation.)

Also there are three other explanations of lift: the circulation-based explanation, the flow-turning or streamline-curvature explanation, and the 3D vortex-shedding explanation. These three appear in advanced textbooks, where they form the basis of the mathematics used by aircraft designers. They rely on Bernoulli's equation. The misleading "popular" or "airfoil-shape" explanation commonly appears in children's science books, magazine articles, and in pilot's textbooks. On the other hand, the public rarely if ever encounters explanations based upon circulation, upon vortex shedding, or upon Newton's Laws.

Thin green laser-scanned dye cloud, when 777 model flys through it, it suddenly moves down and develops an opposite-rotating vortex pair.

A possible solution to the controversy:
Billb's balloon analogy to aircraft: vortex shedding
amasci.com/wing/rotbal.html
Note well: Newton and Bernoulli do not contradict each other. Explanations which are based on Newton's and on Bernoulli's principles are completely compatible. Air-deflection and Newton's Laws explain 100% of the lifting force. Air velocity and Bernoulli's equation also explains 100% of the lift. There is no 60% of one and 40% of the other. One of them looks at pressure forces, the other looks at F=mA accelerated mass. For the most part they're just two different ways of simplifying a single complicated subject. Much of the controversy arises because one side or the other insists that only *their* view is correct. They insist that only a *single* explanation is possible, and the opposing view is therefore wrong. In other words... which is the One True Way to crack an egg? This is a war between the Big-endians and Little-endians from "Gulliver's Travels." They simply refuse to acknowledge that there are several valid yet independent approaches to solving the problem. They insist that their version must be the Single Right Answer, the "One True Path," and anyone who disagrees is a dangerous heretic infidel who must be attacked and silenced.

Social psychology aside, there are also several serious mistakes usually associated with the "popular" explanation described above. Those who believe the "popular" explanation are wrongly insisting that any parcels of air divided by the wing's leading edge must meet again at the trailing edge. This is incorrect. Actually it doesn't even occur: experiments easily show that the air above a wing far outraces the air below, and parcels never meet again. (In fact, if a wing is adjusted so the parcels really do merge, this is always the zero-lift configuration!) The same people also believe that wings fly only because of pressure, and that wings don't need to deflect the oncoming air downwards. Also incorrect. These and several other mistakes commonly appear in elementary science texts, as well as in popular articles about aircraft physics. These mistakes change the popular "airfoil-shape" explanation into a system of misconceptions. I explore these below.

Also, those who firmly adhere to the popular explanation have been successful in convincing many authors that there can only be a single best method for explaining aerodynamic lift, and that the "Airfoil-shape" method is far better than the "Attack-angle" method. I strongly disagree with this, and believe that the correct versions of both explanations should be in constant use. Since the Newton method gives a better intuitive grasp of the issues, that method is more appropriate for elementary explanations aimed at the public and for introductory material for science students and pilots. On the other hand, the "Airfoil Shape" or circulation-based explanation is less intuitive, yet it dovetails very well with lifting force calculations, so it is very useful in mathematical modeling, for physics students, for aircraft design, fluid flow simulation software, etc.


The Truth shall set you free... But first it will piss you off!
-Anon

FREQUENTLY ASKED QUESTIONS

1. Your personal theory is wrong, and nobody should listen to you.

Answer: Ha, if it was just me saying it, you'd be wise to be suspicious. On the other hand, Science is based on the questioning of authority. Sometimes the combined voices of famous and important unquestionable authorities are nothing when compared to a single quiet voice who says "and yet it moves." But fortunately where airfoils are concerned, we're way past that part. I'm no Galileo, and it's not just me saying all this stuff.

The latest version of this controversy was started in 1990 by the aerodynamicist Dr. Klaus Weltner with his paper in American Journal of Physics which calls airfoil explanations into question. There were earlier incidents, such as the popular pilot's training book "Stick and Rudder." I first posted my own amateur articles here in 1995 when the web became available. Since then Gale Craig published How Airplanes Really Fly , Jan-Olov Newborg started a campaign to correct many sources, the late Jef Raskin published his article in Quantum, Dr. John Denker of Bell Labs put it in his online pilot's textbook, the NASA Glenn Research Center included the controversy in their public education program, and aerodynamicists Anderson and Eberhardt published a textbook based on those ideas: Understanding Flight. The controversy recently made it into the New York Times as well as several magazines and aerodynamics websites (see links below.) So... if you want to be suspicious, be suspicious of anyone who tries to pretend that no problems exists, or that this controversy is really just some individual's little pet theory.


2. How could so many scientists, engineers, and authors be so wrong?

- First and foremost, the airfoil lifting-force is an example of propulsion, where the airfoil injects energy and momentum into the air. This is forbidden in the Bernoulli world, so Bernoulli's equations cannot explain propellers, jet engines, helicopters, sails, or airplane wings. Bernoulli only works if we transform the airplane into a venturi by making the wingspan infinitely wide. (Imagine an infinitely-wide helicopter prop, or bird wing!) An infinite wing injects zero energy and momentum into the air, instead it's an example of ground-effect flight. It only produces forces between two surfaces.

- Second, the airfoil math is correct, so wing-designs work regardless of the designers' belief system. Airfoils work fine for ground-effect WIG craft or for normal wings. Incorrect beliefs about wing-functions have little impact, as long as engineers don't use the beliefs to alter the fluid simulations.

- Errors can infect grade-school textbooks and spread widely to many books. All aerodynamics people once were kids, so they can pick up a misconception which they never question, and which persists into their adult careers as aircraft experts. Even more advanced textbooks can give explanations which contain errors at much higher level, e.g. the 2D-centric explanations which imply that infinite wings are normal and acceptable, while finite 3D wings are a bizarre special case. That's backwards. It's the 3D wing which is actually real, while the infinitely-wide wings of the 2D world are the unphysical odditiy, and an example of ground-effect flight.

- Henri Coanda's old experimental work on boundary-layer attachment was marginalized, even ridiculed, rather than merged with the rest of aerodynamics or included in college textbooks. Air is nonlinear, with no simple math solutions which simply explain either flow-attachment or turbulence. As a result, a big piece of aerodynamics concepts is missing. After he dies, Nobelist W. Lamb supposedly was hoping to "ask God" how turbulence works. He could have instead asked God an equivalent unsolved question: how do wings really work? Wings work by creating vorticity from nothing, which is also the signature of turbulence. The lifting force is inescapably a product of turbulence, of nonlinear vortex-shedding, so many experts turn away.

- Cambered wings at high Reynolds number have a positive effective attack angle even when the geometrical attack angle is zero. This confuses everyone, even some experts. They see only the zero geometrical angle and believe that the cambered wing cannot deflect air. They don't realize that the down-tilted trailing edge of a cambered wing has far more effect upon an air parcel than does the rest of the entire un-tilted wing. In other words, the sloping rear half of an un-tilted cambered wing is strongly interacting with air because of air's inertia. A cambered wing flings air downwards as if the wing were tilted. A cambered wing can have a large AOA and a zero AOA ...both at the same time. Very confusing to the uninitiated.

- A two-dimensional diagram (also called the 'infinite wing diagram,') is misleading. It depicts ground-effect flight where altitude above a surface is always much less than one wingspan. Any explanation based on this type of diagram does not apply to the vortex-shedding flight of 3-dimensional aircraft when they're far above the ground. These Two-dimensional diagrams are not just simplified, they're genuinely wrong, since typically they neglect to show the floor and ceiling of the 2D wind tunnel which receives the weight of the wing as an instantaneous contact-force. In 2D diagrams the floor and ceiling are an essential part of the system, and their effects do not diminish as they are removed to infinite distance. In other words, two-dimensional airfoil diagrams depict an odd type of "venturi flight" situation, where the wing is trapped in "ground-effect mode," while genuine aircraft fly far from the ground and have no instantaneous weight applied to the Earth's surface. (Then, the ground is illegally erased from the 2D diagram!) To explain lift in high-flying aircraft, we absolutely require a 3D diagram with its vortex downwash wake. Real wings fly because of vortex-shedding, and they're lifted upwards as they fling a mass-bearing vortex-pair downwards. Yet introductory textbooks always use the misleading two-dimensional diagrams which depict only the regime of ground-effect flight: "venturi-flight" of infinitely-wide wings.

- The presence of multiple possible explanations can trigger religious wars, "Swiftian Battles" between adherents to one side and the other. Sometimes one side wins; swaying the audience and stomping out the other explanation... even though parts of both explanations are valid, and even though both explanations are required for complete understanding. We cannot really grasp wing operation unless we know several different ways to explain them. In the same way, toolkits need both hammers AND screwdrivers... and anyone who searches for a "One True Tool," while angrily emptying out the rest of their toolbox, is severely limiting their own expertise.

"I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic."
    - physicist H. Lamb, 1932 address to the British AAS
3. Why are you prejudiced against the Bernoulli-based theory? Bernoulli's equation is perfectly correct.
Huh? Read my stuff again. Please tell me where I attack Bernoulli. Instead I only attack the "popular theory," also called the Equal Transit-Time explanation. By the way, the correct version of the Bernoulli explanation is called Circulation Theory. Another version is called Flow-turning Theory. Anyone who claims to support the Bernoulli side of the controversy, yet isn't familiar with Circulation as explained in intro texts, is laboring in ignorance. Go see John Denker's page for plenty of info and illustrations about circulation-based explanation. On the other hand, yes, Bernoulli can't be used, since real wings function by injecting energy and momentum into the air. Bernoulli doesn't cover that. Instead we need Euler's equations, of which Bernoulli is a subset. We also need fluid simulation, since most instances of Euler (e.g. vortex-shedding) will have only numeric (computer) solutions.























Bill B.'s "Newtonian" Articles & Controversy


The "Airfoil Mistake" in the news

       

Discussions, message threads

Relevant forums

BOOKS

Other websites



"Newtonian" Lifting Force References

Cliff Swartz, "Numbers Count", editorial in THE PHYSICS TEACHER, p536, Vol34, Dec 1996

Gale Craig, NEWTONIAN AERODYNAMICS FUNDAMENTALS, 1995, Regenerative Press, Anderson Indiana 46011, ISBN: 0964680602

Prof. Klaus Weltner, AERODYNAMIC LIFTING FORCE, The Physics Teacher (magazine), Feb 1990, pp78-82

K. Weltner, BERNOULLI'S LAW AND AERODYNAMIC LIFTING FORCE, The Physics Teacher, Feb 1990, pp84-86

K. Weltner, A COMPARISON OF EXPLANATIONS OF THE AERODYNAMIC LIFTING FORCE, Am. J. of Physics, 55 (1) Jan. 1987 pp50-54

Langewiesche, Wolfgang, STICK AND RUDDER, 1975 Tab Books, ISBN: 0070362408

N.H. Fletcher, MECHANICS OF FLIGHT, Physics Education, Wiley, NY 11975, pp385-389

HOW AIRPLANES FLY


THE TWO COMPETING EXPLANATIONS FOUND IN K-6 BOOKS:

Here is the typical "Airfoil shape" or "Popular" explanation of airfoil lift which commonly appears in childrens' science books:

As air approaches a wing, it is divided into two parts, the part which flows above the wing, and the part which flows below. In order to create a lifting force, the upper surface of the wing must be longer and more curved than the lower surface. Because the air flowing above and below the wing must recombine at the trailing edge of the wing, and because the path along the upper surface is longer, the air on the upper surface must flow faster than the air below if both parts are to reach the trailing edge at the same time. The "Bernoulli Principle" says that the total energy contained in each part of the air is constant, and when air gains kinetic energy (speed) it must lose potential energy (pressure,) and so high-speed air has a lower pressure than low-speed air. Therefore, because the air flows faster on the top of the wing than below, the pressure above is lower than the pressure below the wing, and the wing driven upwards by the higher pressure below. In modern wings the low pressure above the wing creates most of the lifting force, so it isn't far from wrong to say that the wing is essentially 'sucked' upwards. (Note however that "suction" doesn't exist, because air molecules can only push upon a surface, and they never can pull.)

MY NOTES:       (1996)

Uh oh, wind tunnel photographs of lift-generating wings reveal a serious problem with the above description! They show that the divided parcels do not recombine at the trailing edge. Whenever an airfoil is adjusted to give lift, then the parcels of air above the wing move far faster than those below, and the lower parcels lag far behind. After the wing has passed by, the parcels remain forever divided. This has nothing to do with the wing's path lengths. This even applies to thin flat wings such as a "flying barn door." The wind tunnel experiments show that the "wing-shape" argument regarding difference in path-length is simply wrong.

Also, real-world aircraft demonstrate another fallacy. In order to create lift, must a wing have greater path length on the upper surface than on the lower? No. Thin cambered (curved) wings such as those on hang gliders and on rubberband-powered balsa gliders, have equal path length above and below, yet they generate lift. Still the air does flow faster above these wings than below. However, since there is no difference in path length, we cannot refer to path length to explain the difference in air speed above and below the thin wing. The typical "airfoil shape" explanation cannot tell us why a paper airplane can fly, because it does not tell us why the air above the paper wing moves faster.

It is also a fallacy that in order to create lift, a wing *must* be more curved on top. In fact, wings which are designed for high speed and aerobatics are symmetrical streamlined shapes, with equal curvature above and below. Some exotic airfoil shapes are even flat on top and more curved on the bottom! (NASA's "supercritical" wing designs, for example.)

If the typical "popular" or "airfoil-shape" explanation is correct, then how can symmetrical wings and thin cambered wings work at all? How can rubberband balsa gliders work? Those who support the "path length" explanation will sometimes suggest that some other method must be used to explain these particular wings. But if so, why then do so many books put forth only the above "popular" explanation as the single explanation of aerodynamic lift? Why do they avoid detailing or even mentioning any other important explanations of lifting force?

The cloth aircraft of old had single-layer wings having identical path length above and below. If the "Wing-shape" or "popular" explanation is correct and path-length is very important, how can the Wrights' flyer have worked at all? Conversely, we do find that thin airfoils such as the Wrights' have faster flow on the upper surface than the lower surface. Since the path lengths are identical, how can we explain this?

The above "path length" viewpoint would predict that the addition of a lump to the top of a wing should always increase the lift (since it increases the upper surface path length.) In fact, the addition of a lump does not increase lift. This suggests that there is a problem with the "airfoil shape" explanation of lift.

Forces on sailboat sails are explained using the typical "pathlength/wingshape" explanation above. But sailboat sails are thin cloth membranes with identical path-lengths on either side. Why should air on either side of a sail have different velocities if the path length is the same?

Children have experience with rubberband-powered balsa wood aircraft having wings composed of a single flat layer of very thin wood. Paper airplanes usually have flat thin wings. These aircraft cannot fly? How can the "path length" version explain their successful operation?

Regardless of the angle of attack, if a wing does not deflect air downwards, it creates no lift at all. To say otherwise would go against the law of Conservation of Momentum. Yet those who believe in the "airfoil-shape" explanation commonly state that wings operate only by pressure, and Newton's laws are unimportant. This is a direct violation of basic physics principles. Bernoulli's equation incorporates basic physics, and anyone who depart from Newton must automatically depart from Bernoulli as well. Besides buoyancy and helium balloons, the only way to remain aloft is to take some matter and accelerate it downwards. The downward force applied to the matter is equal to the upward force applied by the matter against the craft. Rockets work like this, as do ship propellers, jet engines, helicopters, ...and wings!

Some people argue that the "path length" explanation must be right, since some wings generate lift even at zero angle of attack. However, Attack-angle is determined geometrically, by drawing a line between the tip of the leading and trailing edge. This geometrically-determined attack angle can be misleading:

[GIF: add an off-center bump, then tilt the wing to

Small bumps on the leading edge of a blunt-nosed wing have a large effect on where the line is drawn. These bumps strongly affect the determination of "attack angle, yet these bumps may have little if any effect on the lifting forces being generated. Also, once the "zero AOA" geometry has convinced us to tilt the trailing edge downwards, inertial effects will cause the airfoils to deflect air downwards from its trailing edge more than it deflects air upwards at its leading edge. The downward tilt of the trailing edge generates significant lift even when the wing as a whole is lift even angle of attack. This type of wing may APPEAR at zero attack angle. The inertia of air causes the air to flow straight from the trailing edge of the airfoil. Because of inertia, the trailing edge of a cambered airfoil itself behaves as a tilted plane, and therefore the airfoil effectively has a positive angle which causes air to be deflected. Other cambered wings are similar; they still have a positive "effective" attack angle even when their geometrical attack angle is zero. The trailing edge, not the airfoil itself, determines lift.

Some people argue that flat wings, symmetrical aerobatic wings, Supercritical wings, and thin cloth wings do not employ the Bernoulli Effect, and these wings must instead be explained by Newton and attack angle. But as mentioned above, if jet fighters and the Wright Flyer use Attack Angle rather than Bernoulli Effect, why do the books teach only Bernoulli Effect? At the very least, these books are ignoring an entire class of aircraft by never mentioning Attack Angle. However, even these thin wings and symmetrical wings exhibit the full-blown Bernoulli principle! There is a large difference in speed between the upper and lower air streams along flat wings. If a flat sheet of plywood is tilted into the air stream, the air flows faster above the sheet than below, the divided parcels never rejoin, and lift is generated by the pressure difference. But the flat sheet also deflects the air, and just as much lift is generated by deflection of air. In fact, 100% of aerodynamic lift can be explained by pressure forces and the Bernoulli principle. And 100% of lift can be explained by F=mA and Newton's third law. They are two different ways of explaining a single event. However, any appeals to differences in path length are simply wrong, and any book which uses that explanation is acting to spread science misconceptions.


An alternate explanation of lift: "ATTACK ANGLE"

As air flows over a wing, the flow adheres to the surfaces of the wing. This is called flow-attachment, also the "Coanda effect." Because the wing is tilted, the air is deflected downwards as it moves over the wing's surfaces. Air which flows below the wing is pushed downwards by the wing surface, and because the wing pushes down on the air, the air must push upwards on the wing, creating a lifting force. Air which flows over the upper surface of the wing is adhering to the surface also. The wing "pulls downwards" on the air as it flows over the tilted wing and off the trailing edge, and so the air pulls upwards on the wing, creating more lifting force. (Actually the air follows the wing because of reduced pressure, the "pull" is not really an attraction.) The lifting force is created by Newton's Third Law and by conservation of momentum, as the flowing air which has mass is deflected downward as the wing moves forward. Because of Coanda Effect, the upper surface of the wing actually deflects more air than does the lower surface.

My notes on "attack angle":

If you understand the "attack angle" explanation, then the causes of other aircraft phenomena such as wingtip vortex will suddenly become clear. The air at the trailing edge of the wing is streaming downwards into the surrounding still air. The edge of this mass of air curls up as the air moves downwards, creating the "wingtip vortex." A similar effect can be seen when a drop of dye falls into clear water: the edge of the mass of dye curls up as the dye forces itself downwards into the water, resulting in a ring vortex which moves downwards.

There is one major error associated with the "attack angle" explanation. This is the idea that only the LOWER surface of the wing can generate a lifting force. Some people imagine that air bounces off the bottom of the tilted wing, and they come to the mistaken belief that this is the main source of the lifting force. Even Newton himself apparantly made this mistake, and so overestimated the necessary size of man-lifting craft. In reality, air is deflected by both the upper and the lower surfaces of the wing, with the major part being deflected by the upper surface.

Because a large, heavy aircraft must deflect an enormous amount of air downwards, people standing under a low-flying aircraft are, after a short delay, subjected to a huge downblast of air. They are essentially feeling a portion of the pressure which supports the plane. Imagine standing below a helicopter that hovers a few tens of yards above the ground. Enormous downwash? Now imagine that helicopter flying along at 150mph, or imagine the blades detaching and flying away perpendicular to travel, like wings, and you end up with the usual physics of fixed-wing aircraft. All aircraft wings are essentially sucking in air from all directions and flinging it downwards. This fact gets lost when the aircraft moves horizontally much faster than its downwash moves vertically. Some people even come to believe that wings don't deflect air at all, or leave air moving downwards after the aircraft has passed by.

The downwash can be useful: when a cropduster flies low over a field, the spray is injected into the airflow coming from the wings. Rather than trailing straight back behind the craft, the spray is sent downwards by the downwash from the wings. Also, during takeoff the downwash interacts with the ground and causes lift to greatly increase. Pilots often use this effect to gain a large airspeed just after takeoff. Because of downwash "ground effect," their engine needs to do much less work in keeping their aircraft aloft, therefore the extra power available can be used to speed up the plane.

To create adequate lift at extremely low speeds, an airfoil must be operated at a large angle of attack, and this leads to airflow detachment from wing's the upper surface (stall.) To prevent this, the airfoil must be carefully shaped. A good low- speed airfoil is much more curved on the top, since lift can be created only if the wing surface carefully deflects air downwards by adhesion. Thus one origin of the misconception involving "more curved upper surface." The surface must be curved to prevent stall, not to create lift but to avoid losing lift. The situation with the lower surface is different, since the lower surface can deflect the air by collision. Even so, it makes sense to have the lower surface be somewhat concave, so that the air is slowly deflected as it proceeds along, and so the upwards pressure is distributed uniformly over the lower surface.

Why does flowing air adhere to the upper surface of the wing? This is called flow-attachment, also "the Coanda effect." Apparently Dr. Bernoulli has a better PR department than Dr. Coanda, (grin!), since everyone has heard of Bernoulli, while Coanda is rarely mentioned in textbooks.

The only correct part of the "wingshape/pathlength" explanation of lift is the description of the Bernoulli effect itself. But the "Bernoulli Effect" can also be interpreted thus: because the wing is tilted, it creates a pocket of reduced pressure behind its upper surface. Air must rush into this pocket. And at the tilted lower surface, air collides with the surface and creates a region of increased pressure. Any air which approaches the high pressure region is slowed down. Therefore, the pressure is the cause of the air velocity, not vice-versa as in the "airfoil-shape" explanation above. Also, it is wrong to imagine that the low pressure above the wing is caused by the "Bernoulli effect" while the high pressure below the wings is not. Both pressure variations have similar origin, but opposite values.

The "airfoil shape" explanation could be very useful in calculating the lifting force of an airfoil. Knowing the fluid velocity at all points on the airfoil surface, the pressure may be calculated via Bernoulli's equation at all points, and if the pressure at each point is vector summed, the total lifting force upon the wing will be obtained. The trick then is knowing how to obtain the fluid velocities. Appeals to differences in pathlength do not work, so other methods (circulation and Kutta condition) must be used.






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