• The result is that when the air is viewed as a continuous material, it is seen to be unable to slide along the surface, and the air’s velocity relative to the airfoil decreases
  to nearly zero at the surface (i.e., the air molecules “stick” to the surface instead of sliding along it), something known as the no-slip condition.

• This requires maintaining pressure differences in both the vertical and horizontal directions, and thus requires both downward turning of the flow and changes in flow speed
  according to Bernoulli’s principle.

• [1] Lift is always accompanied by a drag force, which is the component of the surface force parallel to the flow direction.

• Thus the non-uniform pressure is also the cause of the changes in flow speed visible in the flow animation.

• [90] How simpler explanations fall short [edit] Producing a lift force requires both downward turning of the flow and changes in flow speed consistent with Bernoulli’s principle.

• The relationship is thus a mutual, or reciprocal, interaction: Air flow changes speed or direction in response to pressure differences, and the pressure differences are sustained
  by the air’s resistance to changing speed or direction.

• Thus the vertical arrows in the accompanying pressure field diagram indicate that air above and below the airfoil is accelerated, or turned downward, and that the non-uniform
  pressure is thus the cause of the downward deflection of the flow visible in the flow animation.

• Like the equal transit time explanation, the “obstruction” or “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.

• A more comprehensive explanation involves both downward deflection and pressure differences (including changes in flow speed associated with the pressure differences), and
  requires looking at the flow in more detail.

• [83] The pressure difference which results in lift acts directly on the airfoil surfaces; however, understanding how the pressure difference is produced requires understanding
  what the flow does over a wider area.

• According to Newton’s second law, this change in flow direction requires a downward force applied to the air by the airfoil.

• [52] The much higher flow speed over the upper surface can be clearly seen in this animated flow visualization.

• The arrows show the pressure differential from high (red) to low (blue) and hence also the net force which causes the air to accelerate in that direction.

• [60] Basic attributes of lift Lift is a result of pressure differences and depends on angle of attack, airfoil shape, air density, and airspeed.

• 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.

• [7] False explanation based on equal transit-time [edit] 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.

• [81] A more comprehensive physical explanation As described above under “Simplified physical explanations of lift on an airfoil”, there are two main popular explanations:
  one based on downward deflection of the flow (Newton’s laws), and one based on pressure differences accompanied by changes in flow speed (Bernoulli’s principle).

• The oscillatory nature of the flow produces a fluctuating lift force on the cylinder, even though the net (mean) force is negligible.

• [20][21][22] 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.

• The pressure difference that acts on the surface is just part of this pressure field.

• Under certain conditions – for instance resonance or strong spanwise correlation of the lift force – the resulting motion of the structure due to the lift fluctuations may
  be strongly enhanced.

• Sustaining the pressure difference that exerts the lift force on the airfoil surfaces requires sustaining a pattern of non-uniform pressure in a wide area around the airfoil.

• Thus changes in flow direction and speed are directly caused by the non-uniform pressure.

• Pressure differences [edit] Pressure is the normal force per unit area exerted by the air on itself and on surfaces that it touches.

• These differences in the direction and speed of the flow are greatest close to the airfoil and decrease gradually far above and below.

• Note that the downward turning of the flow over the upper surface is the result of the air being pushed downward by higher pressure above it than below it.

• [33][34] 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.

• [64][65] As the angle of attack increases, the lift reaches a maximum at some angle; increasing the angle of attack beyond this critical angle of attack causes the upper-surface
  flow to separate from the wing; there is less deflection downward so the airfoil generates less lift.

• Compared to the predictions of inviscid flow theory, in which there is no boundary layer, the attached boundary layer reduces the lift by a modest amount and modifies the
  pressure distribution somewhat, which results in a viscosity-related pressure drag over and above the skin friction drag.

• [74][75] Under usual flight conditions, the boundary layer remains attached to both the upper and lower surfaces all the way to the trailing edge, and its effect on the rest
  of the flow is modest.

• The changes in flow speed are consistent with Bernoulli’s principle, which states that in a steady flow without viscosity, lower pressure means higher speed, and higher pressure
  means lower speed.

• The asymmetric separation changes the effective shape of the cylinder as far as the flow is concerned such that the cylinder acts like a lifting airfoil with circulation in
  the outer flow.

• The pressure differences follow naturally from Newton’s second law and from the fact that flow along the surface follows the predominantly downward-sloping contours of the
  airfoil.

• [87] Mutual interaction of pressure differences and changes in flow velocity [edit] Pressure field around an airfoil.

• No difference in path length is needed, and even when there is a difference, it is typically much too small to explain the observed speed difference.

• [5] As explained below under a more comprehensive physical explanation, producing a lift force requires maintaining pressure differences in both the vertical and horizontal
  directions.

• 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.

• [41][42][43] Bernoulli’s principle states that under certain conditions increased flow speed is associated with reduced pressure.

• Explanation based on flow deflection and Newton’s laws [edit] When a wing generates lift, it deflects air downward, and to do this it must exert a downward force on the air.

• [85][86] The pressure is also affected over a wide area, in a pattern of non-uniform pressure called a pressure field.

• [46][47][48] In fact, the air moving past the top of an airfoil generating lift moves much faster than equal transit time predicts.

• The pressure differences and the changes in flow direction and speed sustain each other in a mutual interaction.

• The downward deflection and the changes in flow speed are pronounced and extend over a wide area, as can be seen in the flow animation on the right.

• This formula shows that higher velocities and tighter curvatures create larger pressure differentials and that for straight flow (R → ∞), the pressure difference is zero.

• Each of the simplified explanations given above in Simplified physical explanations of lift on an airfoil falls short by trying to explain lift in terms of only one or the
  other, thus explaining only part of the phenomenon and leaving other parts unexplained.

• When a fluid follows a curved path, there is a pressure gradient perpendicular to the flow direction with higher pressure on the outside of the curve and lower pressure on
  the inside.

• 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.

• Overview A fluid flowing around the surface of a solid object applies a force on it.

• [53] Reduced upper-surface pressure and upward lift follow from the higher speed by Bernoulli’s principle, just as in the equal transit time explanation.

• An airfoil affects the speed and direction of the flow over a wide area, producing a pattern called a velocity field.

• [75][76] Stalling [edit] Main article: Stall (fluid dynamics) Airflow separating from a wing at a high angle of attack An airfoil’s maximum lift at a given airspeed is limited
  by boundary-layer separation.

• [2] While the common meaning of the word “lift” assumes that lift opposes weight, lift can be in any direction with respect to gravity, since it is defined with respect to
  the direction of flow rather than to the direction of gravity.

• The direction of the net force implies that the average pressure on the upper surface of the airfoil is lower than the average pressure on the underside.

• The pressure on the lower surface pushes up harder than the reduced pressure on the upper surface pushes down, and the net result is upward lift.

• 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.

• [18] The net force exerted by the air occurs as a pressure difference over the airfoil’s surfaces.

• Aerostatic lift or buoyancy, in which an internal fluid is lighter than the surrounding fluid, does not require movement and is used by balloons, blimps, dirigibles, boats,
  and submarines.

• [55][56][57]] Another flaw is that conservation of mass is not a satisfying physical reason why the flow would speed up.

• The flowing air reacts to the presence of the wing by reducing the pressure on the wing’s upper surface and increasing the pressure on the lower surface.

• Lift also depends on the size of the wing, being generally proportional to the wing’s area projected in the lift direction.

• The direction of the force is different at different locations around the airfoil, as indicated by the block arrows in the pressure field around an airfoil figure.

• [5] Controversy regarding the Coandă effect [edit] Main article: 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.

• If the value of for a wing at a specified angle of attack is given, then the lift produced for specific flow conditions can be determined:[93] where • is the lift force •
  is the air density • is the velocity or true airspeed • is the planform (projected) wing area • is the lift coefficient at the desired angle of attack, Mach number, and Reynolds number[94] Mathematical theories of lift Mathematical theories
  of lift are based on continuum fluid mechanics, assuming that air flows as a continuous fluid.

• Because an airfoil affects the flow in a wide area around it, the conservation laws of mechanics are embodied in the form of partial differential equations combined with a
  set of boundary condition requirements which the flow has to satisfy at the airfoil surface and far away from the airfoil.

• The solution to this problem is to introduce a branch cut, a curve or line from some point on the airfoil surface out to infinite distance, and to allow a jump in the value
  of the potential across the cut.

• The airfoil is assumed to exert a downward force −L’ per unit span on the air, and the proportions in which that force is manifested as momentum fluxes and pressure differences
  at the outer boundary are indicated for each different shape of control volume.

• At this outer boundary distant from the airfoil, the velocity and pressure are well represented by the velocity and pressure associated with a uniform flow plus a vortex,
  and viscous stress is negligible, so that the only force that must be integrated over the outer boundary is the pressure.

• The momentum theorem states that the integrated force exerted at the boundaries of the control volume (a surface integral), is equal to the integrated time rate of change
  (material derivative) of the momentum of fluid parcels passing through the interior of the control volume.

• Comparison of a non-lifting flow pattern around an airfoil; and a lifting flow pattern consistent with the Kutta condition in which the flow leaves the trailing edge smoothly
  Applying potential-flow theory to a lifting flow requires special treatment and an additional assumption.

• Either Euler or potential-flow calculations predict the pressure distribution on the airfoil surfaces roughly correctly for angles of attack below stall, where they might
  miss the total lift by as much as 10–20%.

• [121] There is more downward turning of the flow than there would be in a two-dimensional flow with the same airfoil shape and sectional lift, and a higher sectional angle
  of attack is required to achieve the same lift compared to a two-dimensional flow.

• Determining the net aerodynamic force from a CFD solution requires “adding up” (integrating) the forces due to pressure and shear determined by the CFD over every surface
  element of the airfoil as described under “pressure integration”.

• [131] The lifting flow around a 2D airfoil is usually analyzed in a control volume that completely surrounds the airfoil, so that the inner boundary of the control volume
  is the airfoil surface, where the downward force per unit span is exerted on the fluid by the airfoil.

• [130] Manifestations of lift in the farfield Integrated force/momentum balance in lifting flows [edit] Control volumes of different shapes that have been used in analyzing
  the momentum balance in the 2D flow around a lifting airfoil.

• The problem arises because lift on an airfoil in inviscid flow requires circulation in the flow around the airfoil (See “Circulation and the Kutta–Joukowski theorem” below),
  but a single potential function that is continuous throughout the domain around the airfoil cannot represent a flow with nonzero circulation.

• Quantifying lift Pressure integration [edit] When the pressure distribution on the airfoil surface is known, determining the total lift requires adding up the contributions
  to the pressure force from local elements of the surface, each with its own local value of pressure.

• For a wing of low aspect ratio, such as a typical delta wing, two-dimensional theories may provide a poor model and three-dimensional flow effects can dominate.

• The flow around a lifting airfoil must satisfy Newton’s second law regarding conservation of momentum, both locally at every point in the flow field, and in an integrated
  sense over any extended region of the flow.

• Three-dimensional flow The flow around a three-dimensional wing involves significant additional issues, especially relating to the wing tips.

• By using the streamwise vector i parallel to the freestream in place of k in the integral, we obtain an expression for the pressure drag Dp (which includes the pressure portion
  of the profile drag and, if the wing is three-dimensional, the induced drag).

• For the free-air case (no ground plane), the force exerted by the airfoil on the fluid is manifested partly as momentum fluxes and partly as pressure differences at the outer
  boundary, in proportions that depend on the shape of the outer boundary, as shown in the diagram at right.

• [119] Even for wings of high aspect ratio, the three-dimensional effects associated with finite span can affect the whole span, not just close to the tips.

• The net force due to the lift, acting on the atmosphere as a whole, is therefore zero, and thus there is no integrated accumulation of vertical momentum in the atmosphere,
  as was noted by Lanchester early in the development of modern aerodynamics.

• [100][101] In principle, the NS equations, combined with boundary conditions of no through-flow and no slip at the airfoil surface, could be used to predict lift in any situation
  in ordinary atmospheric flight with high accuracy.

• [127][128][129] The velocity perturbations in the flow around a wing are in fact produced by the pressure field.

• For a flat horizontal rectangle that is much longer than it is tall, the fluxes of vertical momentum through the front and back are negligible, and the lift is accounted for
  entirely by the integrated pressure differences on the top and bottom.

• It is often convenient to quantify the lift of a given airfoil by its lift coefficient , which defines its overall lift in terms of a unit area of the wing.

• [85][113][114] It is a key element in an explanation of lift that follows the development of the flow around an airfoil as the airfoil starts its motion from rest and a starting
  vortex is formed and left behind, leading to the formation of circulation around the airfoil.

• The streamline sketches illustrate one flow pattern with zero lift, in which the flow goes around the trailing edge and leaves the upper surface ahead of the trailing edge,
  and another flow pattern with positive lift, in which the flow leaves smoothly at the trailing edge in accordance with the Kutta condition.

• In the early 20th century, before computers were available, conformal mapping was used to generate solutions to the incompressible potential-flow equation for a class of idealized
  airfoil shapes, providing some of the first practical theoretical predictions of the pressure distribution on a lifting airfoil.

• [132][133][134] For a vertical rectangle that is much taller than it is wide, the unbalanced pressure forces on the top and bottom are negligible, and lift is accounted for
  entirely by momentum fluxes, with a flux of upward momentum that enters the control volume through the front accounting for half the lift, and a flux of downward momentum that exits the control volume through the back accounting for the other
  half.

• [122] The wing is effectively flying in a downdraft of its own making, as if the freestream flow were tilted downward, with the result that the total aerodynamic force vector
  is tilted backward slightly compared to what it would be in two dimensions.

• After the flow leaves the trailing edge, this difference in velocity takes place across a relatively thin shear layer called a vortex sheet.

• The Navier–Stokes equations (NS) provide the potentially most accurate theory of lift, but in practice, capturing the effects of turbulence in the boundary layer on the airfoil
  surface requires sacrificing some accuracy, and requires use of the Reynolds-averaged Navier–Stokes equations (RANS).

• For an extended region, Newton’s second law takes the form of the momentum theorem for a control volume, where a control volume can be any region of the flow chosen for analysis.

• [123] Given the distribution of bound vorticity and the vorticity in the wake, the Biot–Savart law (a vector-calculus relation) can be used to calculate the velocity perturbation
  anywhere in the field, caused by the lift on the wing.

• However, airflows in practical situations always involve turbulence in the boundary layer next to the airfoil surface, at least over the aft portion of the airfoil.

• [112] The linearized theory predicts the general character of the airfoil pressure distribution and how it is influenced by airfoil shape and angle of attack, but is not accurate
  enough for design work.

• However, the potential jump is a free parameter that is not determined by the potential equation or the other boundary conditions, and the solution is thus indeterminate.

• [133][134] Lift reacted by overpressure on the ground under an airplane [edit] Illustration of the distribution of higher-than-ambient pressure on the ground under an airplane
  in subsonic flight An airfoil produces a pressure field in the surrounding air, as explained under “The wider flow around the airfoil” above.

• For steady, level flight, the integrated force due to the pressure differences is equal to the total aerodynamic lift of the airplane and to the airplane’s weight.

• For a steady flow, this can be expressed in the form of the net surface integral of the flux of momentum through the boundary.

• [132][133][134] The free-stream velocity is usually assumed to be horizontal, with lift vertically upward, so that the vertical momentum is the component of interest.

• [92] where: • S is the projected (planform) area of the airfoil, measured normal to the mean airflow; • n is the normal unit vector pointing into the wing; • k is the vertical
  unit vector, normal to the freestream direction.

• Circulation and the Kutta–Joukowski theorem [edit] Circulation component of the flow around an airfoil When an airfoil generates lift, several components of the overall velocity
  field contribute to a net circulation of air around it: the upward flow ahead of the airfoil, the accelerated flow above, the decelerated flow below, and the downward flow behind.

• The lift tends to decrease in the spanwise direction from root to tip, and the pressure distributions around the airfoil sections change accordingly in the spanwise direction.

• One way to resolve this indeterminacy is to impose the Kutta condition,[110][111] which is that, of all the possible solutions, the physically reasonable solution is the one
  in which the flow leaves the trailing edge smoothly.

• [95][96][97] Lift is generated in accordance with the fundamental principles of physics, the most relevant being the following three principles:[98] • Conservation of momentum,
  which is a consequence of Newton’s laws of motion, especially Newton’s second law which relates the net force on an element of air to its rate of momentum change, • Conservation of mass, including the assumption that the airfoil’s surface
  is impermeable for the air flowing around, and • Conservation of energy, which says that energy is neither created nor destroyed.

• [99] To predict lift requires solving the equations for a particular airfoil shape and flow condition, which generally requires calculations that are so voluminous that they
  are practical only on a computer, through the methods of computational fluid dynamics (CFD).

 

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• ^ Kulfan (2010)
• ^
The amount of aerodynamic lift is (usually slightly) more or less than gravity depending on the thrust level and vertical alignment of the thrust line. A side thrust line results in some lift opposing side thrust as well.
• ^ Clancy, L. J., Aerodynamics,
Section 14.6
• ^ Jump up to:a b c d e f g 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
• ^
Jump up to:a b “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 “Incorrect lift
theory #1”. August 16, 2000. Archived from the original on April 27, 2014. Retrieved June 27, 2021.
• ^ “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 [1] Archived August 25, 2019, at the Wayback Machine
• ^ Arvel Gentry Proceedings
of the Third AIAA Symposium on the Aero/Hydronautics of Sailing 1971. “The Aerodynamics of Sail Interaction” (PDF). Archived from the original (PDF) on July 7, 2011. Retrieved July 12, 2011. 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.
• ^ “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
• ^ “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 “Archived
copy – Tom Benson Interview”. Archived from the original on April 27, 2012. Retrieved July 26, 2012.
• ^ Clancy, L. J., Aerodynamics, Section 5.2
• ^ McLean, Doug (2012). Understanding Aerodynamics: Arguing from the Real Physics. p. 281. ISBN
978-1119967514. 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.
• ^ “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 “Archived copy” (PDF). Archived from the original (PDF) on April 11, 2009. Retrieved September 10, 2009.
• ^ Ison, David, “Bernoulli Or Newton: Who’s Right About Lift?”, Plane
& Pilot, archived from the original on September 24, 2015, retrieved January 14, 2011
• ^ “…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)
• ^ Jump up to:a b 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 [2]
• ^ “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 August 4, 2011.
• ^ 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.
• ^ “…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://www1.grc.nasa.gov/beginners-guide-to-aeronautics/foilw2/ Archived February 9, 2023, at the Wayback Machine
• ^
” 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 PDF Archived April 11, 2009, at the Wayback Machine
• ^ “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. 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 Archived August 5, 2021, at the Wayback Machine
• ^
“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
• ^ Auerbach, David (2000), “Why Aircraft
Fly”, Eur. J. Phys., 21 (4): 289, Bibcode:2000EJPh…21..289A, doi:10.1088/0143-0807/21/4/302, S2CID 250821727
• ^ Denker, JS, Fallacious Model of Lift Production, archived from the original on March 2, 2009, retrieved August 18, 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, S2CID 121981660
• ^ Anderson, David; Eberhart, Scott (1999),
How Airplanes Fly: A Physical Description of Lift, archived from the original on January 26, 2016, retrieved June 4, 2008
• ^ Raskin, Jef (1994), Coanda Effect: Understanding Why Wings Work, archived from the original on September 28, 2007
• ^
Auerbach (2000)
• ^ Denker (1996)
• ^ Wille and Fernholz(1965)
• ^ White, Frank M. (2002), Fluid Mechanics (5th ed.), McGraw Hill
• ^ McLean, D. (2012), Section 7.3.2
• ^ McLean, D. (2012), Section 7.3.1.7
• ^ 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 has 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 [3] [permanent dead link]
• ^ Craig G.M. (1997), Stop Abusing Bernoulli
• ^ “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 “Browse – the Physics Teacher”. Archived from the original on March 17, 2012. Retrieved August 4, 2011.
• ^ 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
• ^ “Cambridge
scientist debunks flying myth – Telegraph”. Archived from the original on June 30, 2012. Retrieved June 10, 2012. 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 (August
16, 2000). “Incorrect Lift Theory #1”. NASA. Archived from the original on April 27, 2014. Retrieved June 27, 2021.
• ^ “As stream tube A flows toward the airfoil, it senses the upper portion of the airfoil as an obstruction, and stream tube A must
move out of the way of this obstruction. In so doing, stream tube A is squashed to a smaller cross-sectional area as it flows over the nose of the airfoil. In turn, because of mass continuity (ρ AV = constant), the velocity of the flow in the stream
tube must increase in the region where the stream tube is being squashed.” J. D. Anderson (2008), Introduction to Flight (6th edition), section 5.19
• ^ “The theory is based on the idea that the airfoil upper surface is shaped to act as a nozzle
which accelerates the flow. Such a nozzle configuration is called a Venturi nozzle and it can be analyzed classically. Considering the conservation of mass, the mass flowing past any point in the nozzle is a constant; the mass flow rate of a Venturi
nozzle is a constant… For a constant density, decreasing the area increases the velocity.” Incorrect Theory #3 Glenn Research Center NASA https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/venturi-theory/ Archived February 9, 2023, at the
Wayback Machine
• ^ “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 “Incorrect lift theory
#3”. August 16, 2000. Archived from the original on July 17, 2012. Retrieved June 27, 2021.
• ^ “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 “Archived copy” (PDF). Archived
from the original (PDF) on December 7, 2013. Retrieved July 26, 2012.
• ^ “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 “Archived copy” (PDF). Archived from the original (PDF) on April 11, 2009. Retrieved September 10, 2009.
• ^ “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. [4] [permanent dead
link]
• ^ Doug McLean Understanding Aerodynamics, section 7.3.1.5, Wiley, 2012
• ^ “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 “How Airplanes Fly”. Archived from the original on January 26, 2016. Retrieved January 26, 2016.
• ^ A uniform pressure surrounding
a body does not create a net force. (See buoyancy). Therefore pressure differences are needed to exert a force on a body immersed in a fluid. For example, see: Batchelor, G.K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press,
pp. 14–15, ISBN 978-0-521-66396-0
• ^ “…if a streamline is curved, there must be a pressure gradient across the streamline…”Babinsky, Holger (November 2003), “How do wings work?”, Physics Education, 38 (6): 497, Bibcode:2003PhyEd..38..497B,
doi:10.1088/0031-9120/38/6/001, S2CID 1657792
• ^ Thus a distribution of the pressure is created which is given in Euler’s equation. The physical reason is the aerofoil which forces the streamline to follow its curved surface. The low pressure
at the upper side of the aerofoil is a consequence of the curved surface.” A comparison of explanations of the aerodynamic lifting force Klaus Weltner Am. J. Phys. Vol.55 No.January 1, 1987, p. 53 [5] Archived April 28, 2021, at the Wayback Machine
• ^
“You can argue that the main lift comes from the fact that the wing is angled slightly upward so that air striking the underside of the wing is forced downward. The Newton’s 3rd law reaction force upward on the wing provides the lift. Increasing
the angle of attack can increase the lift, but it also increases drag so that you have to provide more thrust with the aircraft engines” Hyperphysics Georgia State University Dept. of Physics and Astronomy “Angle of Attack for Airfoil”. Archived from
the original on October 14, 2012. Retrieved July 26, 2012.
• ^ “If we enlarge the angle of attack we enlarge the deflection of the airstream by the airfoil. This results in the enlargement of the vertical component of the velocity of the airstream…
we may expect that the lifting force depends linearly on the angle of attack. This dependency is in complete agreement with the results of experiments…” Klaus Weltner A comparison of explanations of the aerodynamic lifting force Am. J. Phys. 55(1),
January 1987 p. 52
• ^ “The decrease[d lift] of angles exceeding 25° is plausible. For large angles of attack we get turbulence and thus less deflection downward.” Klaus Weltner A comparison of explanations of the aerodynamic lifting force Am. J.
Phys. 55(1), January 1987 p. 52
• ^ Clancy (1975), Section 5.2
• ^ Abbott, and von Doenhoff (1958), Section 4.2
• ^ “With an angle of attack of 0°, we can explain why we already have a lifting force. The air stream behind the aerofoil follows
the trailing edge. The trailing edge already has a downward direction, if the chord to the middle line of the profile is horizontal.” Klaus Weltner A comparison of explanations of the aerodynamic lifting force Am. J. Phys. 55(1), January 1987 p. 52
• ^
“…the important thing about an aerofoil . . is not so much that its upper surface is humped and its lower surface is nearly flat, but simply that it moves through the air at an angle. This also avoids the otherwise difficult paradox that an aircraft
can fly upside down!” N. H. Fletcher Mechanics of Flight Physics Education July 1975 [6]
• ^ “It requires adjustment of the angle of attack, but as clearly demonstrated in almost every air show, it can be done.” Hyperphysics GSU Dept. of Physics
and Astronomy [7] Archived July 8, 2012, at the Wayback Machine
• ^ White (1991), Section 1-4
• ^ White (1991), Section 1-2
• ^ Jump up to:a b Anderson (1991), Chapter 17
• ^ Jump up to:a b Abbott and von Doenhoff (1958), Chapter 5
• ^ Schlichting
(1979), Chapter XXIV
• ^ Abbott and Doenhoff (1958), Chapter 8
• ^ Jump up to:a b Williamson, C. H. K.; Govardhan, R. (2004), “Vortex-induced vibrations”, Annual Review of Fluid Mechanics, 36: 413–455, Bibcode:2004AnRFM..36..413W, doi:10.1146/annurev.fluid.36.050802.122128,
S2CID 58937745
• ^ Sumer, B. Mutlu; Fredsøe, Jørgen (2006), Hydrodynamics around cylindrical structures (revised ed.), World Scientific, pp. 6–13, 42–45 & 50–52, ISBN 978-981-270-039-1
• ^ Zdravkovich, M.M. (2003), Flow around circular cylinders,
vol. 2, Oxford University Press, pp. 850–855, ISBN 978-0-19-856561-1
• ^ Clancy, L. J., Aerodynamics, Sections 4.5, 4.6
• ^ McLean (2012), Section 7.3.3
• ^ Jump up to:a b Milne-Thomson (1966), Section 1.41
• ^ Jeans (1967), Section 33.
• ^
Jump up to:a b Clancy (1975), Section 4.5
• ^ Milne-Thomson (1966.), Section 5.31
• ^ McLean 2012, Section 7.3.3.7
• ^ McLean (2012), Section 3.5
• ^ McLean 2012, Section 7.3.3.9″
• ^ McLean 2012, Section 7.3.3.9
• ^ McLean, Doug (2012).
“7.3.3.12”. Understanding Aerodynamics: Arguing from the Real Physics. ISBN 978-1119967514. Doug McLean, Common Misconceptions in Aerodynamics on YouTube
• ^ Anderson (2008), Section 5.7
• ^ Anderson, John D. (2004), Introduction to Flight (5th
ed.), McGraw-Hill, p. 257, ISBN 978-0-07-282569-5
• ^ Yoon, Joe (December 28, 2003), Mach Number & Similarity Parameters, Aerospaceweb.org, archived from the original on February 24, 2021, retrieved February 11, 2009
• ^ Batchelor (1967), Section
1.2
• ^ Thwaites (1958), Section I.2
• ^ von Mises (1959), Section I.1
• ^ “Analysis of fluid flow is typically presented to engineering students in terms of three fundamental principles: conservation of mass, conservation of momentum, and conservation
of energy.” Charles N. Eastlake An Aerodynamicist’s View of Lift, Bernoulli, and Newton The Physics Teacher Vol. 40, March 2002 “Archived copy” (PDF). Archived from the original (PDF) on April 11, 2009. Retrieved September 10, 2009.
• ^ White (1991),
Chapter 1
• ^ Batchelor (1967), Chapter 3
• ^ Aris (1989)
• ^ Jump up to:a b Spalart, Philippe R. (2000) Amsterdam, the Netherlands. Elsevier Science Publishers.
• ^ White (1991), Section 6-2
• ^ Schlichting(1979), Chapter XVIII
• ^ Anderson
(1995)
• ^ “…whenever the velocity field is irrotational, it can be expressed as the gradient of a scalar function we call a velocity potential φ: V = ∇φ. The existence of a velocity potential can greatly simplify the analysis of inviscid flows
by way of potential-flow theory…” Doug McLean Understanding Aerodynamics: Arguing from the Real Physics p. 26 Wiley “Continuum Fluid Mechanics and the Navier–Stokes Equations”. Understanding Aerodynamics. 2012. p. 13. doi:10.1002/9781118454190.ch3.
ISBN 9781118454190.
• ^ Elements of Potential Flow California State University Los Angeles “Faculty Web Directory”. Archived from the original on November 11, 2012. Retrieved July 26, 2012.
• ^ Batchelor(1967), Section 2.7
• ^ Milne-Thomson(1966),
Section 3.31
• ^ Clancy (1975), Section 4.8
• ^ Anderson(1991), Section 4.5
• ^ Clancy(1975), Sections 8.1–8
• ^ von Mises (1959), Section VIII.2
• ^ Anderson(1991), Section 3.15
• ^ Prandtl and Tietjens (1934)
• ^ Batchelor (1967),
Section 6.7
• ^ Gentry (2006)
• ^ McLean (2012), Section 7.2.1
• ^ Milne-Thomson (1966), Section 12.3
• ^ McLean (2012), Section 8.1.3
• ^ McLean (2012), Section 8.1.1
• ^ Hurt, H. H. (1965) Aerodynamics for Naval Aviators, Figure 1.30,
NAVWEPS 00-80T-80
• ^ Lanchester (1907)
• ^ Milne-Thomson (1966), Section 10.1
• ^ Clancy (1975), Section 8.9
• ^ Anderson (1991), Section 5.2
• ^ Batchelor (1967), Section 2.4
• ^ Milne-Thomson (1966), Section 9.3
• ^ Durand (1932),
Section III.2
• ^ McLean (2012), Section 8.1
• ^ Shapiro (1953), Section 1.5, equation 1.15
• ^ Jump up to:a b c d Lissaman (1996), “Lift in thin slices: the two dimensional case”
• ^ Jump up to:a b c Durand (1932), Sections B.V.6, B.V.7
• ^
Jump up to:a b c Batchelor (1967), Section 6.4, p. 407
• ^ Prandtl and Tietjens (1934), Figure 150
• ^ Lanchester (1907), Sections 5 and 112

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