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Mr swordfish (talk | contribs) Latest proposal from Doug McLean - posted here to do a diff Tag: Reverted |
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==Understanding lift as a physical phenomenon== |
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The flow around a lifting wing is a complex fluid-mechanics phenomenon that can be understood on essentially two levels: |
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1) The level of the rigorous science represented by the [[#Mathematical theories of lift|mathematical theories]], which are based on established laws of physics and represent the flow accurately, but which require solving partial differential equations, and |
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2) The level of qualitative physical explanations without math, which are less rigorous. Correctly explaining lift in these qualitative terms is difficult because the cause-and-effect relationships involved are subtle. A [[#A more comprehensive physical explanation|comprehensive explanation]] that captures all of the essential aspects is rather long. There are also many simplified explanations, and most readers will likely already have been exposed to one or more of them. But simplifying the explanation of lift is inherently problematic, and all of the known [[#Simplified physical explanations of lift on an airfoil|simplified explanations]] leave significant parts of the phenomenon unexplained and have other significant flaws<ref>Doug McLean ''Aerodynamic Lift, Part 2: A comprehensive Physical Explanation'' The Physics teacher, November, 2018</ref>. These issues are discussed in connection with each of the explanations presented below. |
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==Simplified physical explanations of lift on an airfoil== |
==Simplified physical explanations of lift on an airfoil== |
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[[Image:Airfoil cross section.jpg|thumb|240px|right|A cross-section of a wing defines an airfoil shape.]] |
[[Image:Airfoil cross section.jpg|thumb|240px|right|A cross-section of a wing defines an airfoil shape.]] |
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An [[airfoil]] is a streamlined shape that is capable of generating significantly more lift than drag.<ref>Clancy, L. J., ''Aerodynamics'', Section 5.2</ref> A flat plate can generate lift, but not as much as a streamlined airfoil, and with somewhat higher drag. |
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Many different simplified explanations have been proposed. Most follow either of two basic approaches, based either on Newton's laws of motion or on Bernoulli's principle. |
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There are several ways to explain how an airfoil generates lift. Some are more complicated or more mathematically rigorous than others; some have been shown to be incorrect.<ref name="nasa_equal_transit">"There are many theories of how lift is generated. Unfortunately, many of the theories found in encyclopedias, on web sites, and even in some textbooks are incorrect, causing unnecessary confusion for students." NASA {{cite web|url=https://www.grc.nasa.gov/WWW/K-12/airplane/wrong1.html|title=Incorrect lift theory #1|date=Aug 16, 2000|access-date=June 27, 2021|archive-url=https://web.archive.org/web/20140427084226/http://www.grc.nasa.gov/WWW/K-12/airplane/wrong1.html|archive-date=April 27, 2014}}</ref><ref>"Most of the texts present the Bernoulli formula without derivation, but also with very little explanation. When applied to the lift of an airfoil, the explanation and diagrams are almost always wrong. At least for an introductory course, lift on an airfoil should be explained simply in terms of Newton’s Third Law, with the thrust up being equal to the time rate of change of momentum of the air downwards." Cliff Swartz et al. ''Quibbles, Misunderstandings, and Egregious Mistakes - Survey of High-School Physics Texts'' THE PHYSICS TEACHER Vol. 37, May 1999 p. 300 [http://aapt.scitation.org/doi/abs/10.1119/1.880274]</ref><ref>"One explanation of how a wing . . gives lift is that as a result of the shape of the airfoil, the air flows faster over the top than it does over the bottom because it has farther to travel. Of course, with our thin-airfoil sails, the distance along the top is the same as along the bottom so this explanation of lift fails." ''The Aerodynamics of Sail Interaction'' by Arvel Gentry Proceedings of the Third AIAA Symposium on the Aero/Hydronautics of Sailing 1971 {{cite web|url=http://www.arvelgentry.com/techs/The%20Aerodynamics%20of%20Sail%20Interaction.pdf|title=Archived copy|access-date=12 July 2011|url-status=dead|archive-url=https://web.archive.org/web/20110707172946/http://www.arvelgentry.com/techs/The%20Aerodynamics%20of%20Sail%20Interaction.pdf|archive-date=July 7, 2011|df=mdy-all}}</ref><ref>"An explanation frequently given is that the path along the upper side of the aerofoil is longer and the air thus has to be faster. This explanation is wrong." ''A comparison of explanations of the aerodynamic lifting force'' Klaus Weltner '' Am. J. Phys. Vol.55 January 1, 1987</ref><ref>"The lift on the body is simple...it's the reaction of the solid body to the turning of a moving fluid...Now why does the fluid turn the way that it does? That's where the complexity enters in because we are dealing with a fluid. ...The cause for the flow turning is the simultaneous conservation of mass, momentum (both linear and angular), and energy by the fluid. And it's confusing for a fluid because the mass can move and redistribute itself (unlike a solid), but can only do so in ways that conserve momentum (mass times velocity) and energy (mass times velocity squared)... A change in velocity in one direction can cause a change in velocity in a perpendicular direction in a fluid, which doesn't occur in solid mechanics... So exactly describing how the flow turns is a complex problem; too complex for most people to visualize. So we make up simplified "models". And when we simplify, we leave something out. So the model is flawed. Most of the arguments about lift generation come down to people finding the flaws in the various models, and so the arguments are usually very legitimate." Tom Benson of NASA's Glenn Research Center in an interview with AlphaTrainer.Com {{cite web|url=http://www.alphatrainer.com/pages/corner.htm|title=Archived copy - Tom Benson Interview|access-date=26 July 2012|url-status=dead|archive-url=https://web.archive.org/web/20120427005906/http://www.alphatrainer.com/pages/corner.htm|archive-date=April 27, 2012}}</ref> Most simplified explanations follow one of two basic approaches, based either on [[Newton's laws of motion]] or on [[Bernoulli's principle]].<ref>{{cite book |last=McLean |first=Doug |date=2012 |title=Understanding Aerodynamics: Arguing from the Real Physics |page=281 |isbn=978-1119967514|quote=Another argument that is often made, as in several successive versions of the Wikipedia article “Aerodynamic Lift,” is that lift can always be explained either in terms of pressure or in terms of momentum and that the two explanations are somehow “equivalent.” This “either/or” approach also misses the mark.}}</ref><ref>"Both approaches are equally valid and equally correct, a concept that is central to the conclusion of this article." Charles N. Eastlake ''An Aerodynamicist’s View of Lift, Bernoulli, and Newton'' THE PHYSICS TEACHER Vol. 40, March 2002 {{cite web|url=http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf|title=Archived copy|access-date=10 September 2009|url-status=dead|archive-url=https://web.archive.org/web/20090411055333/http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf|archive-date=April 11, 2009}}</ref><ref>{{citation|url=http://www.planeandpilotmag.com/component/zine/article/289.html|last=Ison|first=David|title=Bernoulli Or Newton: Who's Right About Lift?|magazine=Plane & Pilot|access-date=January 14, 2011|url-status=dead|archive-url=https://web.archive.org/web/20150924073958/http://www.planeandpilotmag.com/component/zine/article/289.html|archive-date=September 24, 2015}}</ref><ref>Doug McLean ''Aerodynamic Lift, Part 2: A comprehensive Physical Explanation'' The Physics teacher, November, 2018</ref> |
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===Explanation based on flow deflection and Newton's laws=== |
===Explanation based on flow deflection and Newton's laws=== |
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The airflow changes direction as it passes the airfoil and follows a path that is curved downward. According to Newton's second law, this change in flow direction requires a downward force applied to the air by the airfoil. Then Newton's third law requires the air to exert an upward force on the airfoil; thus a reaction force, lift, is generated opposite to the directional change. In the case of an airplane wing, the wing exerts a downward force on the air and the air exerts an upward force on the wing.<ref>"Lift is a force generated by turning a moving fluid... If the body is shaped, moved, or inclined in such a way as to produce a net deflection or turning of the flow, the local velocity is changed in magnitude, direction, or both. Changing the velocity creates a net force on the body." {{cite web|publisher=NASA Glenn Research Center|title=Lift from Flow Turning|url=https://www.grc.nasa.gov/WWW/K-12/airplane/right2.html |date=May 27, 2000 |access-date=June 27, 2021 |archive-url=https://web.archive.org/web/20110705131653/http://www.grc.nasa.gov/WWW/K-12/airplane/right2.html|archive-date=July 5, 2011}}</ref><ref>"Essentially, due to the presence of the wing (its shape and inclination to the incoming flow, the so-called angle of attack), the flow is given a downward deflection. It is Newton’s third law at work here, with the flow then exerting a reaction force on the wing in an upward direction, thus generating lift." Vassilis Spathopoulos - Flight Physics for Beginners: Simple Examples of Applying Newton’s Laws ''The Physics Teacher'' Vol. 49, September 2011 p. 373 [https://archive.today/20130618032326/http://tpt.aapt.org/resource/1/phteah/v49/i6/p373_s1]</ref><ref>"The main fact of all heavier-than-air flight is this: ''the wing keeps the airplane up by pushing the air down.''" In: Langewiesche - ''Stick and Rudder'', p. 6</ref><ref>"Birds and aircraft fly because they are constantly pushing air downwards: L = Δp/Δt where L= lift force, and Δp/Δt is the rate at which downward momentum is imparted to the airflow." ''Flight without Bernoulli'' Chris Waltham ''THE PHYSICS TEACHER'' Vol. 36, Nov. 1998 {{cite web|url=http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/fly_no_bernoulli.pdf|title=Archived copy|access-date=4 August 2011|url-status=live|archive-url=https://web.archive.org/web/20110928200519/http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/fly_no_bernoulli.pdf|archive-date=September 28, 2011}}</ref><ref>Clancy, L. J.; ''Aerodynamics'', Pitman 1975, p. 76: "This lift force has its reaction in the downward momentum which is imparted to the air as it flows over the wing. Thus the lift of the wing is equal to the rate of transport of downward momentum of this air."</ref><ref>"...if the air is to produce an upward force on the wing, the wing must produce a downward force on the air. Because under these circumstances air cannot sustain a force, it is deflected, or accelerated, downward. Newton's second law gives us the means for quantifying the lift force: F<sub>lift</sub> = m∆v/∆t = ∆(mv)/∆t. The lift force is equal to the time rate of change of momentum of the air." {{cite journal|last1=Smith|first1=Norman F.|year=1972|title=Bernoulli and Newton in Fluid Mechanics|journal=The Physics Teacher|volume=10|issue=8|page=451|doi=10.1119/1.2352317|bibcode=1972PhTea..10..451S}}</ref> |
The airflow changes direction as it passes the airfoil and follows a path that is curved downward. According to Newton's second law, this change in flow direction requires a downward force applied to the air by the airfoil. Then Newton's third law requires the air to exert an upward force on the airfoil; thus a reaction force, lift, is generated opposite to the directional change. In the case of an airplane wing, the wing exerts a downward force on the air and the air exerts an upward force on the wing.<ref>"Lift is a force generated by turning a moving fluid... If the body is shaped, moved, or inclined in such a way as to produce a net deflection or turning of the flow, the local velocity is changed in magnitude, direction, or both. Changing the velocity creates a net force on the body." {{cite web|publisher=NASA Glenn Research Center|title=Lift from Flow Turning|url=https://www.grc.nasa.gov/WWW/K-12/airplane/right2.html |date=May 27, 2000 |access-date=June 27, 2021 |archive-url=https://web.archive.org/web/20110705131653/http://www.grc.nasa.gov/WWW/K-12/airplane/right2.html|archive-date=July 5, 2011}}</ref><ref>"Essentially, due to the presence of the wing (its shape and inclination to the incoming flow, the so-called angle of attack), the flow is given a downward deflection. It is Newton’s third law at work here, with the flow then exerting a reaction force on the wing in an upward direction, thus generating lift." Vassilis Spathopoulos - Flight Physics for Beginners: Simple Examples of Applying Newton’s Laws ''The Physics Teacher'' Vol. 49, September 2011 p. 373 [https://archive.today/20130618032326/http://tpt.aapt.org/resource/1/phteah/v49/i6/p373_s1]</ref><ref>"The main fact of all heavier-than-air flight is this: ''the wing keeps the airplane up by pushing the air down.''" In: Langewiesche - ''Stick and Rudder'', p. 6</ref><ref>"Birds and aircraft fly because they are constantly pushing air downwards: L = Δp/Δt where L= lift force, and Δp/Δt is the rate at which downward momentum is imparted to the airflow." ''Flight without Bernoulli'' Chris Waltham ''THE PHYSICS TEACHER'' Vol. 36, Nov. 1998 {{cite web|url=http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/fly_no_bernoulli.pdf|title=Archived copy|access-date=4 August 2011|url-status=live|archive-url=https://web.archive.org/web/20110928200519/http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/fly_no_bernoulli.pdf|archive-date=September 28, 2011}}</ref><ref>Clancy, L. J.; ''Aerodynamics'', Pitman 1975, p. 76: "This lift force has its reaction in the downward momentum which is imparted to the air as it flows over the wing. Thus the lift of the wing is equal to the rate of transport of downward momentum of this air."</ref><ref>"...if the air is to produce an upward force on the wing, the wing must produce a downward force on the air. Because under these circumstances air cannot sustain a force, it is deflected, or accelerated, downward. Newton's second law gives us the means for quantifying the lift force: F<sub>lift</sub> = m∆v/∆t = ∆(mv)/∆t. The lift force is equal to the time rate of change of momentum of the air." {{cite journal|last1=Smith|first1=Norman F.|year=1972|title=Bernoulli and Newton in Fluid Mechanics|journal=The Physics Teacher|volume=10|issue=8|page=451|doi=10.1119/1.2352317|bibcode=1972PhTea..10..451S}}</ref> |
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The downward turning of the flow is not produced solely by the lower surface of the airfoil, and the air flow above the airfoil accounts for much of the downward-turning action (reference?). |
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The downward turning of the flow is not produced solely by the lower surface of the airfoil, and the air flow above the airfoil accounts for much of the downward-turning action.<ref>"...when one considers the downwash produced by a lifting airfoil, the upper surface contributes more flow turning than the lower surface." ''Incorrect Theory #2'' Glenn Research Center NASA https://www.grc.nasa.gov/WWW/K-12/airplane/wrong2.html</ref><ref>" This happens to some extent on both the upper and lower surface of the airfoil, but it is much more pronounced on the forward portion of the upper surface, so the upper surface gets the credit for being the primary lift producer. " Charles N. Eastlake ''An Aerodynamicist’s View of Lift, Bernoulli, and Newton'' ''THE PHYSICS TEACHER'' Vol. 40, March 2002 [http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf PDF] {{webarchive|url=https://web.archive.org/web/20090411055333/http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf|date=April 11, 2009}}</ref><ref>"The pressure reaches its minimum value around 5 to 15% chord after the leading edge. As a result, about half of the lift is generated in the first 1/4 chord region of the airfoil. Looking at all three angles of attack, we observe a similar pressure change after the leading edge. |
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Additionally, in all three cases, the upper surface contributes more lift than the lower surface. As a result, it is critical to maintain a clean and rigid surface on the top of the wing. This is why most airplanes are cleared of any objects on the top of the wing." ''Airfoil Behavior: Pressure Distribution over a Clark Y-14 Wing'' David Guo, College of Engineering, Technology, and Aeronautics (CETA), Southern New Hampshire University https://www.jove.com/v/10453/airfoil-behavior-pressure-distribution-over-a-clark-y-14-wing</ref><ref>"There’s always a tremendous amount of focus on the upper portion of the wing, but the lower surface also contributes to lift." ''Bernoulli Or Newton: Who’s Right About Lift?'' David Ison Plane & Pilot Feb 2016</ref> |
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This explanation is correct |
This explanation is correct as far as it goes but is incomplete. It doesn't explain how the airfoil can impart downward turning to a much deeper swath of the flow than it actually touches. Furthermore, it doesn't mention that the lift force is exerted by [[#Pressure differences|pressure differences]], and doesn't explain how those pressure differences are sustained.<ref>Doug McLean ''Aerodynamic Lift, Part 2: A comprehensive Physical Explanation'' The Physics teacher, November, 2018</ref> |
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====Controversy regarding the Coandă effect==== |
====Controversy regarding the Coandă effect==== |
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More broadly, some consider the effect to include the tendency of any fluid [[boundary layer]] to adhere to a curved surface, not just the boundary layer accompanying a fluid jet. It is in this broader sense that the Coandă effect is used by some popular references to explain why airflow remains attached to the top side of an airfoil.<ref name="scotteberhart"/><ref name="raskin"/> This is a controversial use of the term "Coandă effect"; the flow following the upper surface simply reflects an absence of boundary-layer separation, thus it is not an example of the Coandă effect.<ref>Auerbach (2000)</ref><ref>Denker (1996)</ref><ref>Wille and Fernholz(1965)</ref><ref>{{citation|last=White|first=Frank M.|title=Fluid Mechanics|year=2002|edition=5th|publisher=McGraw Hill}}</ref> |
More broadly, some consider the effect to include the tendency of any fluid [[boundary layer]] to adhere to a curved surface, not just the boundary layer accompanying a fluid jet. It is in this broader sense that the Coandă effect is used by some popular references to explain why airflow remains attached to the top side of an airfoil.<ref name="scotteberhart"/><ref name="raskin"/> This is a controversial use of the term "Coandă effect"; the flow following the upper surface simply reflects an absence of boundary-layer separation, thus it is not an example of the Coandă effect.<ref>Auerbach (2000)</ref><ref>Denker (1996)</ref><ref>Wille and Fernholz(1965)</ref><ref>{{citation|last=White|first=Frank M.|title=Fluid Mechanics|year=2002|edition=5th|publisher=McGraw Hill}}</ref> |
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===Explanations based on an increase in flow speed and Bernoulli's principle=== |
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There are two common versions of this explanation, one based on "equal transit time", and one based on " |
There are two common versions of this explanation, one based on "equal transit time", and one based on "streamtube pinching". |
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====Equal transit time==== |
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[[File:Equal transit-time NASA wrong1 en.svg|thumb|right|586px|An illustration of the incorrect equal transit-time explanation of airfoil lift.<ref name="nasa_equal_transit"></ref>]] |
[[File:Equal transit-time NASA wrong1 en.svg|thumb|right|586px|An illustration of the incorrect equal transit-time explanation of airfoil lift.<ref name="nasa_equal_transit"></ref>]] |
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====Equal transit time <span class="anchor" id="False explanation based on equal transit-time"></span>==== |
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The "equal transit time" explanation starts by arguing that the flow over the upper surface is faster than the flow over the lower surface because the path length over the upper surface is longer and must be traversed in equal transit time |
The "equal transit time" explanation starts by arguing that the flow over the upper surface is faster than the flow over the lower surface because the path length over the upper surface is longer and must be traversed in equal transit time<ref>Burge, Cyril Gordon (1936). Encyclopædia of aviation. London: Pitman. p. 441. "… the fact that the air passing over the hump on the top of the wing will have to speed up more than that flowing beneath the wing, in order to arrive at the trailing edge in the same time."</ref><ref>Illman, Paul (2000). The Pilot's Handbook of Aeronautical Knowledge. New York: McGraw-Hill. pp. 15–16. ISBN 0071345191. When air flows along the upper wing surface, it travels a greater distance in the same period of time as the airflow along the lower wing surface."</ref><ref>Dingle, Lloyd; Tooley, Michael H. (2005). Aircraft engineering principles. Boston: Elsevier Butterworth-Heinemann. p. 548. ISBN 0-7506-5015-X. The air travelling over the cambered top surface of the aerofoil shown in Figure 7.6, which is split as it passes around the aerofoil, will speed up, because it must reach the trailing edge of the aerofoil at the same time as the air that flows underneath the section."</ref>. [[Bernoulli's principle]] states that under certain conditions increased flow speed is associated with reduced pressure. It is concluded that the reduced pressure over the upper surface results in upward lift<ref>"The airfoil of the airplane wing, according to the textbook explanation that is more or less standard in the United States, has a special shape with more curvature on top than on the bottom; consequently, the air must travel farther over the top surface than over the bottom surface. Because the air must make the trip over the top and bottom surfaces in the same elapsed time ..., the velocity over the top surface will be greater than over the bottom. According to Bernoulli's theorem, this velocity difference produces a pressure difference which is lift." ''Bernoulli and Newton in Fluid Mechanics'' Norman F. Smith ''The Physics Teacher'' November 1972 Volume 10, Issue 8, p. 451 [http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PHTEAH000010000008000451000001&idtype=cvips&doi=10.1119/1.2352317&prog=normal] {{dead link|date=January 2018|bot=InternetArchiveBot|fix-attempted=yes}}</ref>. |
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A serious flaw in the equal transit time explanation is that it doesn't correctly explain what causes the flow to speed up |
A serious flaw in the equal transit time explanation is that it doesn't correctly explain what causes the flow to speed up<ref>Doug McLean ''Aerodynamic Lift, Part 2: A comprehensive Physical Explanation'' The Physics teacher, November, 2018</ref>. The longer-path-length explanation is simply wrong. No difference in path length is needed, and even when there is a difference, it's typically much too small to explain the observed speed difference <ref>Craig(1997)</ref>. This is because the assumption of equal transit time is wrong. There is no physical principle that requires equal transit time and experimental results show that this assumption is false.<ref>"Unfortunately, this explanation [fails] on three counts. First, an airfoil need not have more curvature on its top than on its bottom. Airplanes can and do fly with perfectly symmetrical airfoils; that is with airfoils that have the ''same'' curvature top and bottom. Second, even if a humped-up (cambered) shape is used, the claim that the air must traverse the curved top surface in the same time as it does the flat bottom surface...is fictional. We can quote no physical law that tells us this. Third—and this is the most serious—the common textbook explanation, and the diagrams that accompany it, describe a force on the wing with no net disturbance to the airstream. This constitutes a violation of Newton's third law." ''Bernoulli and Newton in Fluid Mechanics'' Norman F. Smith ''The Physics Teacher'' November 1972 Volume 10, Issue 8, p. 451 {{cite web|url=http://tpt.aapt.org/resource/1/phteah/v10/i8|title=Archived copy|access-date=4 August 2011|url-status=dead|archive-url=https://web.archive.org/web/20120317075304/http://tpt.aapt.org/resource/1/phteah/v10/i8|archive-date=March 17, 2012}}</ref><ref> |
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{{Citation|last=Anderson|first=David|title=Understanding Flight|publisher=McGraw-Hill|location=New York|year=2001|isbn=978-0-07-136377-8|quote=The first thing that is wrong is that the principle of equal transit times is not true for a wing with lift.|page=15}}</ref><ref>{{cite book|last=Anderson|first=John|title=Introduction to Flight|publisher=McGraw-Hill Higher Education|location=Boston|year=2005|isbn=978-0072825695|page=355|quote=It is then assumed that these two elements must meet up at the trailing edge, and because the running distance over the top surface of the airfoil is longer than that over the bottom surface, the element over the top surface must move faster. This is simply not true}}</ref><ref>{{cite web|url=https://www.telegraph.co.uk/science/science-news/9035708/Cambridge-scientist-debunks-flying-myth.html|title=Archived copy|access-date=10 June 2012|url-status=dead|archive-url=https://web.archive.org/web/20120630121849/http://www.telegraph.co.uk/science/science-news/9035708/Cambridge-scientist-debunks-flying-myth.html|archive-date=June 30, 2012}} ''Cambridge scientist debunks flying myth'' UK Telegraph 24 January 2012</ref><ref>{{cite AV media|url=http://web.mit.edu/hml/ncfmf.html|title=Flow Visualization|publisher=National Committee for Fluid Mechanics Films/Educational Development Center|access-date=January 21, 2009|url-status=live|archive-url=https://web.archive.org/web/20161021122939/http://web.mit.edu/hml/ncfmf.html|archive-date=October 21, 2016}} A visualization of the typical retarded flow over the lower surface of the wing and the accelerated flow over the upper surface starts at 5:29 in the video.</ref><ref>"...do you remember hearing that troubling business about the particles moving over the curved top surface having to go faster than the particles that went underneath, because they have a longer path to travel but must still get there at the same time? This is simply not true. It does not happen." Charles N. Eastlake ''An Aerodynamicist’s View of Lift, Bernoulli, and Newton'' ''THE PHYSICS TEACHER'' Vol. 40, March 2002 [http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf PDF] {{webarchive|url=https://web.archive.org/web/20090411055333/http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf|date=April 11, 2009}}</ref> In fact, the air moving over the top of an airfoil generating lift moves ''much'' ''faster'' than the equal transit theory predicts.<ref>"The actual velocity over the top of an airfoil is much faster than that predicted by the "Longer Path" theory and particles moving over the top arrive at the trailing edge before particles moving under the airfoil." {{cite web|url=https://www.grc.nasa.gov/WWW/K-12/airplane/wrong1.html |date=Aug 16, 2000 |title=Incorrect Lift Theory #1 |author=Glenn Research Center|publisher=NASA |access-date=June 27, 2021| archive-url=https://web.archive.org/web/20140427084226/http://www.grc.nasa.gov/WWW/K-12/airplane/wrong1.html|archive-date=April 27, 2014}}</ref> The much higher flow speed over the upper surface can be clearly seen in the animated flow visualization adjacent to [[#The wider flow around the airfoil|The wider flow around the airfoil]]. |
{{Citation|last=Anderson|first=David|title=Understanding Flight|publisher=McGraw-Hill|location=New York|year=2001|isbn=978-0-07-136377-8|quote=The first thing that is wrong is that the principle of equal transit times is not true for a wing with lift.|page=15}}</ref><ref>{{cite book|last=Anderson|first=John|title=Introduction to Flight|publisher=McGraw-Hill Higher Education|location=Boston|year=2005|isbn=978-0072825695|page=355|quote=It is then assumed that these two elements must meet up at the trailing edge, and because the running distance over the top surface of the airfoil is longer than that over the bottom surface, the element over the top surface must move faster. This is simply not true}}</ref><ref>{{cite web|url=https://www.telegraph.co.uk/science/science-news/9035708/Cambridge-scientist-debunks-flying-myth.html|title=Archived copy|access-date=10 June 2012|url-status=dead|archive-url=https://web.archive.org/web/20120630121849/http://www.telegraph.co.uk/science/science-news/9035708/Cambridge-scientist-debunks-flying-myth.html|archive-date=June 30, 2012}} ''Cambridge scientist debunks flying myth'' UK Telegraph 24 January 2012</ref><ref>{{cite AV media|url=http://web.mit.edu/hml/ncfmf.html|title=Flow Visualization|publisher=National Committee for Fluid Mechanics Films/Educational Development Center|access-date=January 21, 2009|url-status=live|archive-url=https://web.archive.org/web/20161021122939/http://web.mit.edu/hml/ncfmf.html|archive-date=October 21, 2016}} A visualization of the typical retarded flow over the lower surface of the wing and the accelerated flow over the upper surface starts at 5:29 in the video.</ref><ref>"...do you remember hearing that troubling business about the particles moving over the curved top surface having to go faster than the particles that went underneath, because they have a longer path to travel but must still get there at the same time? This is simply not true. It does not happen." Charles N. Eastlake ''An Aerodynamicist’s View of Lift, Bernoulli, and Newton'' ''THE PHYSICS TEACHER'' Vol. 40, March 2002 [http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf PDF] {{webarchive|url=https://web.archive.org/web/20090411055333/http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf|date=April 11, 2009}}</ref> In fact, the air moving over the top of an airfoil generating lift moves ''much'' ''faster'' than the equal transit theory predicts.<ref>"The actual velocity over the top of an airfoil is much faster than that predicted by the "Longer Path" theory and particles moving over the top arrive at the trailing edge before particles moving under the airfoil." {{cite web|url=https://www.grc.nasa.gov/WWW/K-12/airplane/wrong1.html |date=Aug 16, 2000 |title=Incorrect Lift Theory #1 |author=Glenn Research Center|publisher=NASA |access-date=June 27, 2021| archive-url=https://web.archive.org/web/20140427084226/http://www.grc.nasa.gov/WWW/K-12/airplane/wrong1.html|archive-date=April 27, 2014}}</ref>. The much higher flow speed over the upper surface can be clearly seen in the animated flow visualization adjacent to [[#The wider flow around the airfoil|The wider flow around the airfoil]]. |
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[[File:Streamlines around a NACA 0012.svg|thumb|300px|Streamlines and streamtubes around an airfoil generating lift. Note the narrower streamtubes above and the wider streamtubes below.]] |
[[File:Streamlines around a NACA 0012.svg|thumb|300px|Streamlines and streamtubes around an airfoil generating lift. Note the narrower streamtubes above and the wider streamtubes below.]] |
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Like the equal transit time explanation, the " |
Like the equal transit time explanation, the "streamtube pinching" explanation argues that the flow over the upper surface is faster than the flow over the lower surface, but gives a different reason for the difference in speed. It argues that the curved upper surface acts as more of an obstacle to the flow, forcing the streamlines to pinch closer together, making the streamtubes narrower. When streamtubes become narrower, conservation of mass requires that flow speed must increase<ref>J. D. Anderson (2008), section 5.19</ref>. Reduced upper-surface pressure and upward lift follow from the higher speed by [[Bernoulli's principle]], just as in the equal transit time explanation. |
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One serious flaw in the streamtube pinching explanation is that it doesn't really explain how streamtube pinching comes about at all, let alone why it's greater over the upper surface than the lower surface. Another flaw is that conservation of mass isn't a satisfying physical reason why the flow would speed up. Really explaining why something speeds up requires identifying the force that makes it accelerate<ref>Doug McLean ''Understanding Aerodynamics'', section 7.3.1.5, Wiley, 2012</ref>. |
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====Issues common to both versions of the Bernoulli-based explanation==== |
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A serious flaw common to all the Bernoulli-based explanations is that they imply that a speed difference can arise from causes other than a pressure difference, and that the speed difference then leads to a pressure difference, by Bernoulli’s principle. This implied one-way causation is a misconception. The real relationship between pressure and flow speed is a [[#Mutual interaction of pressure differences and changes in flow velocity|mutual interaction]]<ref>Doug McLean ''Aerodynamic Lift, Part 2: A comprehensive Physical Explanation'' The Physics teacher, November, 2018</ref>. |
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One serious flaw in the obstruction explanation is that it doesn't explain how streamtube pinching comes about, or why it's greater over the upper surface than the lower surface. For conventional wings that are flat on the bottom and curved on top this makes some intuitive sense, but it does not explain how flat plates, symmetric airfoils, sailboat sails, or conventional airfoils flying upside down can generate lift, and attempts to calculate lift based on the amount of constriction or obstruction do not predict experimental results.<ref>"The problem with the 'Venturi' theory is that it attempts to provide us with the velocity based on an incorrect assumption (the constriction of the flow produces the velocity field). We can calculate a velocity based on this assumption, and use Bernoulli's equation to compute the pressure, and perform the pressure-area calculation and the answer we get does not agree with the lift that we measure for a given airfoil." NASA Glenn Research Center {{cite web|url=https://www.grc.nasa.gov/WWW/K-12/airplane/wrong3.html |title=Incorrect lift theory #3|date=Aug 16, 2000 |access-date=27 June 2021 |archive-url=https://web.archive.org/web/20120717222459/http://www.grc.nasa.gov/WWW/k-12/airplane/wrong3.html|archive-date=July 17, 2012}}</ref><ref>"A concept...uses a symmetrical convergent-divergent channel, like a longitudinal section of a Venturi tube, as the starting point . . when such a device is put in a flow, the static pressure in the tube decreases. When the upper half of the tube is removed, a geometry resembling the airfoil is left, and suction is still maintained on top of it. Of course, this explanation is flawed too, because the geometry change affects the whole flowfield and there is no physics involved in the description." Jaakko Hoffren ''Quest for an Improved Explanation of Lift'' Section 4.3 American Institute of Aeronautics and Astronautics 2001 {{cite web|url=http://corsair.flugmodellbau.de/files/area2/LIFT.PDF|title=Archived copy|access-date=26 July 2012|url-status=dead|archive-url=https://web.archive.org/web/20131207102746/http://corsair.flugmodellbau.de/files/area2/LIFT.PDF|archive-date=December 7, 2013}}</ref><ref>"This answers the apparent mystery of how a symmetric airfoil can produce lift. ... This is also true of a flat plate at non-zero angle of attack." Charles N. Eastlake ''An Aerodynamicist’s View of Lift, Bernoulli, and Newton'' {{cite web|url=http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf|title=Archived copy|access-date=10 September 2009|url-status=dead|archive-url=https://web.archive.org/web/20090411055333/http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf|archive-date=April 11, 2009}}</ref><ref>"This classic explanation is based on the difference of streaming velocities caused by the airfoil. There remains, however, a question: How does the airfoil cause the difference in streaming velocities? Some books don't give any answer, while others just stress the picture of the streamlines, saying the airfoil reduces the separations of the streamlines at the upper side. They do not say how the airfoil manages to do this. Thus this is not a sufficient answer." Klaus Weltner ''Bernoulli's Law and Aerodynamic Lifting Force'' The Physics Teacher February 1990 p. 84. [http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PHTEAH000028000002000084000001&idtype=cvips&prog=normal] {{dead link|date=January 2018|bot=InternetArchiveBot|fix-attempted=yes}}</ref> Another flaw is that conservation of mass isn't a satisfying physical reason why the flow would speed up. Really explaining why something speeds up requires identifying the force that makes it accelerate.<ref>Doug McLean ''Understanding Aerodynamics'', section 7.3.1.5, Wiley, 2012</ref> |
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In addition, as explained below under [[#A more comprehensive physical explanation|a more comprehensive physical explanation]], producing a lift force requires maintaining pressure differences in both the vertical and horizontal directions. The Bernoulli-only explanations don't explain how the pressure differences in the vertical direction are sustained. That is, they leave out the flow-deflection part of the interaction<ref>Doug McLean ''Aerodynamic Lift, Part 2: A comprehensive Physical Explanation'' The Physics teacher, November, 2018</ref>. |
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Although the two simple explanations above are incorrect, there is nothing incorrect about Bernoulli's principle or the fact that the air goes faster on the top of the wing |
Although the two simple explanations above are incorrect, there is nothing incorrect about Bernoulli's principle or the fact that the air goes faster on the top of the wing. In airfoil flows, Bernoulli's principle is applicable<ref>Doug McLean ''Aerodynamic Lift, Part 1: The Science'' The Physics teacher, November, 2018</ref> to the flow outside the [[#Boundary layer and profile drag|boundary layer]] and can be used correctly as part of a more complicated explanation of lift.<ref>"There is nothing wrong with the Bernoulli principle, or with the statement that the air goes faster over the top of the wing. But, as the above discussion suggests, our understanding is not complete with this explanation. The problem is that we are missing a vital piece when we apply Bernoulli’s principle. We can calculate the pressures around the wing if we know the speed of the air over and under the wing, but how do we determine the speed?" ''How Airplanes Fly: A Physical Description of Lift'' David Anderson and Scott Eberhardt {{cite web|url=http://www.allstar.fiu.edu/aero/airflylvl3.htm|title=Archived copy|access-date=26 January 2016|url-status=live|archive-url=https://web.archive.org/web/20160126200755/http://www.allstar.fiu.edu/aero/airflylvl3.htm|archive-date=January 26, 2016}}</ref> |
Revision as of 15:15, 23 August 2021
Understanding lift as a physical phenomenon
The flow around a lifting wing is a complex fluid-mechanics phenomenon that can be understood on essentially two levels:
1) The level of the rigorous science represented by the mathematical theories, which are based on established laws of physics and represent the flow accurately, but which require solving partial differential equations, and
2) The level of qualitative physical explanations without math, which are less rigorous. Correctly explaining lift in these qualitative terms is difficult because the cause-and-effect relationships involved are subtle. A comprehensive explanation that captures all of the essential aspects is rather long. There are also many simplified explanations, and most readers will likely already have been exposed to one or more of them. But simplifying the explanation of lift is inherently problematic, and all of the known simplified explanations leave significant parts of the phenomenon unexplained and have other significant flaws[1]. These issues are discussed in connection with each of the explanations presented below.
Simplified physical explanations of lift on an airfoil
Many different simplified explanations have been proposed. Most follow either of two basic approaches, based either on Newton's laws of motion or on Bernoulli's principle.
Explanation based on flow deflection and Newton's laws
An airfoil generates lift by exerting a downward force on the air as it flows past. According to Newton's third law, the air must exert an equal and opposite (upward) force on the airfoil, which is lift.[2][3][4][5]
The airflow changes direction as it passes the airfoil and follows a path that is curved downward. According to Newton's second law, this change in flow direction requires a downward force applied to the air by the airfoil. Then Newton's third law requires the air to exert an upward force on the airfoil; thus a reaction force, lift, is generated opposite to the directional change. In the case of an airplane wing, the wing exerts a downward force on the air and the air exerts an upward force on the wing.[6][7][8][9][10][11]
The downward turning of the flow is not produced solely by the lower surface of the airfoil, and the air flow above the airfoil accounts for much of the downward-turning action (reference?).
This explanation is correct as far as it goes but is incomplete. It doesn't explain how the airfoil can impart downward turning to a much deeper swath of the flow than it actually touches. Furthermore, it doesn't mention that the lift force is exerted by pressure differences, and doesn't explain how those pressure differences are sustained.[12]
Controversy regarding the Coandă effect
Some versions of the flow-deflection explanation of lift cite the Coandă effect as the reason the flow is able to follow the convex upper surface of the airfoil. The conventional definition in the aerodynamics field is that the Coandă effect refers to the tendency of a fluid jet to stay attached to an adjacent surface that curves away from the flow, and the resultant entrainment of ambient air into the flow.[13][14][15]
More broadly, some consider the effect to include the tendency of any fluid boundary layer to adhere to a curved surface, not just the boundary layer accompanying a fluid jet. It is in this broader sense that the Coandă effect is used by some popular references to explain why airflow remains attached to the top side of an airfoil.[16][17] This is a controversial use of the term "Coandă effect"; the flow following the upper surface simply reflects an absence of boundary-layer separation, thus it is not an example of the Coandă effect.[18][19][20][21]
Explanations based on an increase in flow speed and Bernoulli's principle
There are two common versions of this explanation, one based on "equal transit time", and one based on "streamtube pinching".
Equal transit time
The "equal transit time" explanation starts by arguing that the flow over the upper surface is faster than the flow over the lower surface because the path length over the upper surface is longer and must be traversed in equal transit time[23][24][25]. Bernoulli's principle states that under certain conditions increased flow speed is associated with reduced pressure. It is concluded that the reduced pressure over the upper surface results in upward lift[26].
A serious flaw in the equal transit time explanation is that it doesn't correctly explain what causes the flow to speed up[27]. The longer-path-length explanation is simply wrong. No difference in path length is needed, and even when there is a difference, it's typically much too small to explain the observed speed difference [28]. This is because the assumption of equal transit time is wrong. There is no physical principle that requires equal transit time and experimental results show that this assumption is false.[29][30][31][32][33][34] In fact, the air moving over the top of an airfoil generating lift moves much faster than the equal transit theory predicts.[35]. The much higher flow speed over the upper surface can be clearly seen in the animated flow visualization adjacent to The wider flow around the airfoil.
Obstruction of the flow
Like the equal transit time explanation, the "streamtube pinching" explanation argues that the flow over the upper surface is faster than the flow over the lower surface, but gives a different reason for the difference in speed. It argues that the curved upper surface acts as more of an obstacle to the flow, forcing the streamlines to pinch closer together, making the streamtubes narrower. When streamtubes become narrower, conservation of mass requires that flow speed must increase[36]. Reduced upper-surface pressure and upward lift follow from the higher speed by Bernoulli's principle, just as in the equal transit time explanation.
One serious flaw in the streamtube pinching explanation is that it doesn't really explain how streamtube pinching comes about at all, let alone why it's greater over the upper surface than the lower surface. Another flaw is that conservation of mass isn't a satisfying physical reason why the flow would speed up. Really explaining why something speeds up requires identifying the force that makes it accelerate[37].
Issues common to both versions of the Bernoulli-based explanation
A serious flaw common to all the Bernoulli-based explanations is that they imply that a speed difference can arise from causes other than a pressure difference, and that the speed difference then leads to a pressure difference, by Bernoulli’s principle. This implied one-way causation is a misconception. The real relationship between pressure and flow speed is a mutual interaction[38].
In addition, as explained below under a more comprehensive physical explanation, producing a lift force requires maintaining pressure differences in both the vertical and horizontal directions. The Bernoulli-only explanations don't explain how the pressure differences in the vertical direction are sustained. That is, they leave out the flow-deflection part of the interaction[39].
Although the two simple explanations above are incorrect, there is nothing incorrect about Bernoulli's principle or the fact that the air goes faster on the top of the wing. In airfoil flows, Bernoulli's principle is applicable[40] to the flow outside the boundary layer and can be used correctly as part of a more complicated explanation of lift.[41]
- ^ Doug McLean Aerodynamic Lift, Part 2: A comprehensive Physical Explanation The Physics teacher, November, 2018
- ^ "...the effect of the wing is to give the air stream a downward velocity component. The reaction force of the deflected air mass must then act on the wing to give it an equal and opposite upward component." In: Halliday, David; Resnick, Robert, Fundamentals of Physics 3rd Ed., John Wiley & Sons, p. 378
- ^ Anderson and Eberhardt (2001)
- ^ Langewiesche (1944)
- ^ "When air flows over and under an airfoil inclined at a small angle to its direction, the air is turned from its course. Now, when a body is moving in a uniform speed in a straight line, it requires force to alter either its direction or speed. Therefore, the sails exert a force on the wind and, since action and reaction are equal and opposite, the wind exerts a force on the sails." In: Morwood, John, Sailing Aerodynamics, Adlard Coles Limited, p. 17
- ^ "Lift is a force generated by turning a moving fluid... If the body is shaped, moved, or inclined in such a way as to produce a net deflection or turning of the flow, the local velocity is changed in magnitude, direction, or both. Changing the velocity creates a net force on the body." "Lift from Flow Turning". NASA Glenn Research Center. May 27, 2000. Archived from the original on July 5, 2011. Retrieved June 27, 2021.
- ^ "Essentially, due to the presence of the wing (its shape and inclination to the incoming flow, the so-called angle of attack), the flow is given a downward deflection. It is Newton’s third law at work here, with the flow then exerting a reaction force on the wing in an upward direction, thus generating lift." Vassilis Spathopoulos - Flight Physics for Beginners: Simple Examples of Applying Newton’s Laws The Physics Teacher Vol. 49, September 2011 p. 373 [1]
- ^ "The main fact of all heavier-than-air flight is this: the wing keeps the airplane up by pushing the air down." In: Langewiesche - Stick and Rudder, p. 6
- ^ "Birds and aircraft fly because they are constantly pushing air downwards: L = Δp/Δt where L= lift force, and Δp/Δt is the rate at which downward momentum is imparted to the airflow." Flight without Bernoulli Chris Waltham THE PHYSICS TEACHER Vol. 36, Nov. 1998 "Archived copy" (PDF). Archived (PDF) from the original on September 28, 2011. Retrieved 4 August 2011.
{{cite web}}
: CS1 maint: archived copy as title (link) - ^ Clancy, L. J.; Aerodynamics, Pitman 1975, p. 76: "This lift force has its reaction in the downward momentum which is imparted to the air as it flows over the wing. Thus the lift of the wing is equal to the rate of transport of downward momentum of this air."
- ^ "...if the air is to produce an upward force on the wing, the wing must produce a downward force on the air. Because under these circumstances air cannot sustain a force, it is deflected, or accelerated, downward. Newton's second law gives us the means for quantifying the lift force: Flift = m∆v/∆t = ∆(mv)/∆t. The lift force is equal to the time rate of change of momentum of the air." Smith, Norman F. (1972). "Bernoulli and Newton in Fluid Mechanics". The Physics Teacher. 10 (8): 451. Bibcode:1972PhTea..10..451S. doi:10.1119/1.2352317.
- ^ Doug McLean Aerodynamic Lift, Part 2: A comprehensive Physical Explanation The Physics teacher, November, 2018
- ^ Cite error: The named reference
auerbach
was invoked but never defined (see the help page). - ^ Denker, JS, Fallacious Model of Lift Production, archived from the original on March 2, 2009, retrieved 18 August 2008
- ^ Wille, R.; Fernholz, H. (1965), "Report on the first European Mechanics Colloquium, on the Coanda effect", J. Fluid Mech., 23 (4): 801, Bibcode:1965JFM....23..801W, doi:10.1017/S0022112065001702
- ^ Cite error: The named reference
scotteberhart
was invoked but never defined (see the help page). - ^ Cite error: The named reference
raskin
was invoked but never defined (see the help page). - ^ Auerbach (2000)
- ^ Denker (1996)
- ^ Wille and Fernholz(1965)
- ^ White, Frank M. (2002), Fluid Mechanics (5th ed.), McGraw Hill
- ^ Cite error: The named reference
nasa_equal_transit
was invoked but never defined (see the help page). - ^ Burge, Cyril Gordon (1936). Encyclopædia of aviation. London: Pitman. p. 441. "… the fact that the air passing over the hump on the top of the wing will have to speed up more than that flowing beneath the wing, in order to arrive at the trailing edge in the same time."
- ^ Illman, Paul (2000). The Pilot's Handbook of Aeronautical Knowledge. New York: McGraw-Hill. pp. 15–16. ISBN 0071345191. When air flows along the upper wing surface, it travels a greater distance in the same period of time as the airflow along the lower wing surface."
- ^ Dingle, Lloyd; Tooley, Michael H. (2005). Aircraft engineering principles. Boston: Elsevier Butterworth-Heinemann. p. 548. ISBN 0-7506-5015-X. The air travelling over the cambered top surface of the aerofoil shown in Figure 7.6, which is split as it passes around the aerofoil, will speed up, because it must reach the trailing edge of the aerofoil at the same time as the air that flows underneath the section."
- ^ "The airfoil of the airplane wing, according to the textbook explanation that is more or less standard in the United States, has a special shape with more curvature on top than on the bottom; consequently, the air must travel farther over the top surface than over the bottom surface. Because the air must make the trip over the top and bottom surfaces in the same elapsed time ..., the velocity over the top surface will be greater than over the bottom. According to Bernoulli's theorem, this velocity difference produces a pressure difference which is lift." Bernoulli and Newton in Fluid Mechanics Norman F. Smith The Physics Teacher November 1972 Volume 10, Issue 8, p. 451 [2] [permanent dead link]
- ^ Doug McLean Aerodynamic Lift, Part 2: A comprehensive Physical Explanation The Physics teacher, November, 2018
- ^ Craig(1997)
- ^ "Unfortunately, this explanation [fails] on three counts. First, an airfoil need not have more curvature on its top than on its bottom. Airplanes can and do fly with perfectly symmetrical airfoils; that is with airfoils that have the same curvature top and bottom. Second, even if a humped-up (cambered) shape is used, the claim that the air must traverse the curved top surface in the same time as it does the flat bottom surface...is fictional. We can quote no physical law that tells us this. Third—and this is the most serious—the common textbook explanation, and the diagrams that accompany it, describe a force on the wing with no net disturbance to the airstream. This constitutes a violation of Newton's third law." Bernoulli and Newton in Fluid Mechanics Norman F. Smith The Physics Teacher November 1972 Volume 10, Issue 8, p. 451 "Archived copy". Archived from the original on March 17, 2012. Retrieved 4 August 2011.
{{cite web}}
: CS1 maint: archived copy as title (link) - ^
Anderson, David (2001), Understanding Flight, New York: McGraw-Hill, p. 15, ISBN 978-0-07-136377-8,
The first thing that is wrong is that the principle of equal transit times is not true for a wing with lift.
- ^ Anderson, John (2005). Introduction to Flight. Boston: McGraw-Hill Higher Education. p. 355. ISBN 978-0072825695.
It is then assumed that these two elements must meet up at the trailing edge, and because the running distance over the top surface of the airfoil is longer than that over the bottom surface, the element over the top surface must move faster. This is simply not true
- ^ "Archived copy". Archived from the original on June 30, 2012. Retrieved 10 June 2012.
{{cite web}}
: CS1 maint: archived copy as title (link) Cambridge scientist debunks flying myth UK Telegraph 24 January 2012 - ^ Flow Visualization. National Committee for Fluid Mechanics Films/Educational Development Center. Archived from the original on October 21, 2016. Retrieved January 21, 2009. A visualization of the typical retarded flow over the lower surface of the wing and the accelerated flow over the upper surface starts at 5:29 in the video.
- ^ "...do you remember hearing that troubling business about the particles moving over the curved top surface having to go faster than the particles that went underneath, because they have a longer path to travel but must still get there at the same time? This is simply not true. It does not happen." Charles N. Eastlake An Aerodynamicist’s View of Lift, Bernoulli, and Newton THE PHYSICS TEACHER Vol. 40, March 2002 PDF Archived April 11, 2009, at the Wayback Machine
- ^ "The actual velocity over the top of an airfoil is much faster than that predicted by the "Longer Path" theory and particles moving over the top arrive at the trailing edge before particles moving under the airfoil." Glenn Research Center (Aug 16, 2000). "Incorrect Lift Theory #1". NASA. Archived from the original on April 27, 2014. Retrieved June 27, 2021.
- ^ J. D. Anderson (2008), section 5.19
- ^ Doug McLean Understanding Aerodynamics, section 7.3.1.5, Wiley, 2012
- ^ Doug McLean Aerodynamic Lift, Part 2: A comprehensive Physical Explanation The Physics teacher, November, 2018
- ^ Doug McLean Aerodynamic Lift, Part 2: A comprehensive Physical Explanation The Physics teacher, November, 2018
- ^ Doug McLean Aerodynamic Lift, Part 1: The Science The Physics teacher, November, 2018
- ^ "There is nothing wrong with the Bernoulli principle, or with the statement that the air goes faster over the top of the wing. But, as the above discussion suggests, our understanding is not complete with this explanation. The problem is that we are missing a vital piece when we apply Bernoulli’s principle. We can calculate the pressures around the wing if we know the speed of the air over and under the wing, but how do we determine the speed?" How Airplanes Fly: A Physical Description of Lift David Anderson and Scott Eberhardt "Archived copy". Archived from the original on January 26, 2016. Retrieved 26 January 2016.
{{cite web}}
: CS1 maint: archived copy as title (link)