Lift (force)

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For other uses, see Lift (disambiguation).
Boeing 747-8F landing

A fluid flowing past the surface of a body exerts a force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction.[1] It contrasts with the drag force, which is the component of the surface force parallel to the flow direction. If the fluid is air, the force is called an aerodynamic force. In water, it is called a hydrodynamic force.

Lift is the force generated by propellers and wings to propel aircraft and keep them in the air. Birds, bats, insects, fish, flying reptiles, and even falling plant seeds have usefully exploited lift for millions of years.


Forces on an airfoil

Lift is most commonly associated with the wing of a fixed-wing aircraft, although lift is also generated by propellers, kites, helicopter rotors, rudders, sails and keels on sailboats, hydrofoils, wings on auto racing cars, wind turbines and other streamlined objects. Lift is also exploited in the animal world, and even in the plant world by the seeds of certain trees[citation needed]. While the common meaning of the word "lift" assumes that lift opposes weight, lift in its technical sense can be in any direction since it is defined with respect to the direction of flow rather than to the direction of gravity. When an aircraft is flying straight and level (cruise) most of the lift opposes gravity.[2] However, when an aircraft is climbing, descending, or banking in a turn the lift is tilted with respect to the vertical.[3] Lift may also be entirely downwards in some aerobatic manoeuvres, or on the wing on a racing car. In this last case, the term downforce is often used. Lift may also be largely horizontal, for instance on a sail on a sailboat.

An airfoil is a streamlined shape that is capable of generating significantly more lift than drag.[4] Non-streamlined objects such as bluff bodies and flat plates may also generate lift when moving relative to the fluid, but will have a higher drag coefficient dominated by pressure drag.[5]

Description of lift on an airfoil[edit]

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.[6][7][8][9][10] For example, there are explanations based directly on Newton’s laws of motion and explanations based on Bernoulli’s principle. Either can be used to explain lift.[11][12] This article will start with a simple explanation based on Newton's laws; more complicated and alternative explanations will follow.

Newton's laws: lift and the deflection of the flow[edit]


Streamlines around an airfoil in a wind tunnel. Note the curved streamlines above and below the foil, and the overall downward deflection of the air.

Lift is a reaction force—an airfoil deflects the air as it passes the airfoil.[13] Since the foil must exert a force on the air to change its direction, the air must exert a force of equal magnitude but opposite direction on the foil. 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.[15][16][18][19][20][21][22][23]

This follows from the second and third of Newton's laws of motion: The net force on an object is equal to its rate of momentum change, and: To every action there is an equal and opposite reaction.[24]

The air changes direction as it passes the airfoil and follows a path that is curved. Whenever airflow changes direction, a reaction force is generated opposite to the directional change.[25]

Pressure differences[edit]

Lift may also be described in terms of air pressure: pressure is the normal force per unit area. Wherever there is net force there is also a pressure difference, thus deflection/flow turning indicates the presence of a net force and therefore a pressure difference. The direction of the net force implies that the average pressure on the upper surface of the wing is lower than the average pressure on the underside.[26][27][28]

Whenever a fluid follows a curved path, there is a pressure gradient perpendicular to the flow direction. This direct relationship between curved streamlines and pressure differences was derived from Newton's second law by Leonhard Euler in 1754:

\frac{\operatorname{d}p}{\operatorname{d}R}= \rho \frac{v^2}{R}

where R is the radius of curvature, p is the pressure, ρ is the density, and v is the velocity. This formula shows that higher velocities and tighter curvatures create larger pressure differentials and that for straight flow (R → ∞) the pressure difference is zero.[29]

Flow on both sides of the wing[edit]

Variation of the pressure around an airfoil as obtained by a solution of the Euler equations.

In the picture above, observe that the air is turned both above and below the wing so both the upper and lower surface contribute to the flow turning and therefore the lift. In fact, for typical airfoils at subsonic speeds the top surface contributes more flow turning than the bottom surface, and the pressure deviation along the top is significantly larger than along the bottom. A common explanation describes lift as merely the result of the air molecules bouncing off the lower surface of the wing, but since this ignores the airflow around the top of the wing it usually leads to incorrect results. However, at hypersonic speeds, this model becomes applicable.[30][31][32]

Angle of attack[edit]

The arrow is the vector representing the velocity of the air in the free stream around a stationary two-dimensional airfoil. The upper red line is the chord line of the airfoil and the lower red line is parallel to the arrow. The angle α is the angle of attack.

The angle of attack is the angle between an airfoil and the oncoming air. A symmetrical airfoil will generate zero lift at zero angle of attack. But as the angle of attack increases, the air is deflected through a larger angle and the vertical component of the airstream velocity increases, resulting in more lift. For small angles a symmetrical airfoil will generate a lift force roughly proportional to the angle of attack.[33][34]

As the angle of attack grows larger, the lift reaches a maximum at some angle; increasing the angle of attack beyond this critical angle of attack causes the air to become turbulent and separate from the wing; there is less deflection downward so the airfoil generates less lift. The airfoil is said to be stalled.[35]

Cambered airfoils will generate lift at zero angle of attack. When the chordline is horizontal, the trailing edge has a downward direction and since the air follows the trailing edge it is deflected downward.[36] When a cambered airfoil is upside down, the angle of attack can be adjusted so that the lift force is upwards. This explains how a plane can fly upside down.[37][38]

Limitations of deflection/turning[edit]

  • While the theory correctly reasons that deflection implies that there must be a force on the wing, it does not explain why the air is deflected. Intuitively, one can say that the air follows the curve of the foil,[39] but this is not very rigorous or precise.
  • The theory, while correct in as far as it goes, is not sufficiently detailed to support the precise calculations required for engineering.[40][41][42] Thus, textbooks on aerodynamics use more complex models to provide a full description of lift.

Bernoulli's principle: lift, pressure, and speed[edit]

Bernoulli's principle states that within an airflow of constant energy, when the air flows through a region of lower pressure it speeds up and vice versa.[43] Thus, there is a direct mathematical relationship between the pressure and the speed, so if one knows the speed at all points within the airflow one can calculate the pressure, and vice versa. For any airfoil generating lift, there must be a pressure imbalance, i.e. lower average air pressure on the top than on the bottom. Bernoulli's principle states that this pressure difference must be accompanied by a speed difference.

Bernoulli's principle does not explain why the air flows faster over the top of the wing; to explain that requires some other physical reasoning.[44] It is in providing that additional reasoning where some explanations oversimplify things.

Conservation of mass[edit]

Streamlines around an airfoil in a wind tunnel. Note the narrower streamtubes above and the wider streamtubes below the foil.

If one takes the experimentally observed flow around an airfoil as a starting point, then lift can be explained in terms of pressures using Bernoulli's principle and conservation of mass.[14]

Returning to the picture from the previous section, the flow approaching an airfoil can be divided into streamtubes, which are defined based on the area between two streamlines. By definition, fluid never crosses a streamline in a steady flow. Assuming that the air is incompressible, the rate of flow (e.g. liters or gallons per minute) must be constant within each streamtube since matter is not created or destroyed. If a streamtube becomes narrower, the flow speed must increase in the narrower region to maintain the constant flow rate. This idea is called "conservation of mass", and for incompressible flow mass is conserved within each streamtube.[45]

The picture shows that the upper stream tubes constrict as they flow up and around the airfoil. Conservation of mass says that the flow speed must increase as the stream tube area decreases.[14] Similarly, the lower stream tubes expand and the flow slows down.

From Bernoulli's principle, the pressure on the upper surface where the flow is moving faster is lower than the pressure on the lower surface where it is moving slower. The pressure difference thus creates a net aerodynamic force, pointing upward and downstream to the flow direction. The component of the force perpendicular to the free stream is lift; the component parallel to the freestream is drag. In conjunction with this force by the air on the airfoil, the airfoil imparts an equal-and-opposite force on the surrounding air that creates the downwash, in accordance with Newton's third law. Measuring the momentum transferred to the downwash is another way to determine the amount of lift on the airfoil.[46]

Limitations of explanations based on Bernoulli's principle[edit]

  • The explanation above does not explain why the streamtubes change size. To see why the air flows the way it does requires more sophisticated analysis.[47][48][49]
  • Sometimes a geometrical argument is offered to demonstrate why the streamtubes change size: it is asserted that the top "obstructs" or "constricts" the air more than the bottom, hence narrower streamtubes. 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 do not predict experimental results.[50][51][52][53]
  • In deriving Bernoulli's principle, assumptions may be made (such as constant energy or incompressible flow) that are not applicable to real-world airfoils. For instance, a sailboat that is accelerating is removing energy from the flow while an airplane in level flight is adding energy to the flow, so energy is not constant. For high speed aircraft moving at transonic speeds the effects of compressibility can't be neglected.[54][55][56][57]
  • A common explanation using Bernoulli's principle asserts that the air must traverse both the top and bottom in the same amount of time and that this explains the increased speed on the (longer) top side of the wing. But this assertion is false; it is typically the case that the air parcels traveling over the upper surface will reach the trailing edge before those traveling over the bottom.[58]

A more detailed physical description[edit]

Flow around an airfoil: the dots move with the flow. Note that the velocities are much higher at the upper surface than at the lower surface. The black dots are on timelines, which split into two – an upper and lower part – at the leading edge. The part of a timeline below the airfoil does not catch up with the one above. Colors of the dots indicate streamlines. The airfoil is a Kármán–Trefftz airfoil, with parameters μx = –0.08, μy = +0.08 and n = 1.94. The angle of attack is 8°, and the flow is a potential flow.

Lift is generated in accordance with the fundamental principles of physics. The most relevant physics reduce to three principles:[59]

In addition, one needs an expression relating the fluid stresses (consisting of pressure and shear stress components) to the properties of the flow.[60][61] The pressure depends on the other flow properties, such as its mass density, through the (thermodynamic) equation of state, while the shear stresses are related to the flow through the air's viscosity.[60]

Navier–Stokes and Euler equations[edit]

The above conditions can be expressed as partial differential equations that constrain the fluid flow. Unlike the physics of rigid bodies where the equations of motion describe the physical location of the rigid body, fluid dynamics describes the flow as a velocity field, i.e. a solution associates each point in space with a vector that represents the speed and direction of the flow at that point. Once the velocity field is solved for, other quantities of interest may be computed from the velocity field. For instance, Bernoulli's equation can be used to determine the pressure at each point on the airfoil from the speed of the velocity field. The pressures can then be "added up" (integrated) to determine the net aerodynamic force.

Application of the viscous shear stresses to Newton's second law for an airflow results in the Navier–Stokes equations. These equations are notoriously difficult to solve, but in many instances approximations suffice for a good description of lifting airfoils. In large parts of the flow viscosity may be neglected; such an inviscid flow can be described mathematically through the Euler equations, resulting from the Navier-Stokes equations when the viscosity is neglected. Neither the Navier-Stokes equations nor the Euler equations lend themselves to exact analytic solutions; usually engineers have to resort to numerical solutions to solve them, but Euler's equation can be solved by making further simplifying assumptions.[62]

Potential flow, the Kutta condition, and circulation[edit]

Uniform flow plus vortex flow (circulation) gives the total flow below.

One further simplifying approximation is to assume that the flow is irrotational, i.e. that the flow does not rotate around itself. Mathematically, this is expressed by saying that the curl of the flow field is everywhere equal to zero. Irrotational flows have two nice properties: 1) an irrotational flow can be expressed as the gradient of a scalar function called a potential (the flow is then called potential flow) 2) solutions to complicated flows can be expressed as the sum of simpler flow fields. These two properties make the calculations tractable.[63][64]

In particular, solutions can be expressed as the sum of a uniform flow (i.e. a steady flow equal to the free stream velocity) plus a free vortex flow, i.e. a circular flow around the airfoil with the speed inversely proportional to the radius.[65] (Vortex flow is irrotational, except at the center of its circulation; since the center is inside the airfoil, the flow itself remains irrotational.) A vortex flow of any strength may be added to this uniform flow and the equation is solved, thus there are many flows that solve the Euler equations.[66]

In order to arrive at a unique (physical) solution, one can apply the Kutta condition, which says that for steady flow the rear stagnation point is coincident with the trailing edge of the airfoil. The magnitude of the vortex flow is adjusted so that the Kutta condition is met. Another way to say this is that the airflow around the airfoil develops enough vorticity to satisfy the Kutta condition. Vorticity is measured by a number called the circulation, so an airfoil in a steady flow will develop sufficient circulation to satisfy the Kutta condition.

Streamlines and streamtubes around a NACA 0012 airfoil at moderate angle of attack. The stagnation line is the boundary between the two shaded streamtubes.

The image to the right shows the streamlines over a NACA 0012 airfoil computed using potential flow theory and applying the Kutta condition as a boundary condition.[67] The stagnation streamline is the streamline that intersects the airfoil. In the front it typically intersects the airfoil on the lower surface near the leading edge. The stagnation streamline leaves the airfoil at the sharp trailing edge, according to the Kutta condition.

The physical interpretation of circulation is that the vortex flow adds to the uniform flow above the wing and subtracts from it below the wing. Thus, the speed of the air above the wing is higher than below it, as can be seen in the narrower streamtube. By Bernoulli's principle, this implies a pressure difference and therefore lift. The amount of lift is directly proportional to the circulation.[68]

Flowfield formation[edit]

Consider the case of an airfoil accelerating from rest in a viscous flow. When there is no flow, there is no lift and the aerodynamic forces acting on the airfoil are zero. At the instant when the flow is “turned on”, the flow is undeflected downstream of the airfoil and there are two stagnation points on the airfoil (where the flow velocity is zero): one near the leading edge on the bottom surface, and another on the upper surface near the trailing edge. The dividing line between the upper and lower streamtubes intersects the body at the stagnation points. Since the flow speed is zero at these points, by Bernoulli's principle the static pressure at these points is at a maximum. As long as the second stagnation point is at its initial location on the upper surface of the wing, the circulation around the airfoil is zero and, in accordance with the Kutta–Joukowski theorem, there is no lift. The net pressure difference between the upper and lower surfaces is zero.[69]

The effects of viscosity are contained within a thin layer of fluid called the boundary layer, close to the body. As flow over the airfoil commences, the flow along the lower surface turns at the sharp trailing edge and flows along the upper surface towards the upper stagnation point. The flow in the vicinity of the sharp trailing edge is very fast and the resulting viscous forces cause the boundary layer to accumulate into a vortex on the upper side of the airfoil between the trailing edge and the upper stagnation point.[70] This is called the starting vortex. The starting vortex and the bound vortex around the surface of the wing are two halves of a closed loop. As the starting vortex increases in strength the bound vortex also strengthens, causing the flow over the upper surface of the airfoil to accelerate and drive the upper stagnation point towards the sharp trailing edge. As this happens, the starting vortex is shed into the wake,[71] and is a necessary condition to produce lift on an airfoil. If the flow were stopped, there would be a corresponding "stopping vortex".[72] Despite being an idealization of the real world, the “vortex system” set up around a wing is both real and observable; the trailing vortex sheet most noticeably rolls up into wing-tip vortices.

The upper stagnation point continues moving downstream until it is coincident with the sharp trailing edge (as stated by the Kutta condition). The flow downstream of the airfoil is deflected downward from the free-stream direction and, from the reasoning above in the basic explanation, there is now a net pressure difference between the upper and lower surfaces and an aerodynamic force is generated.

Other alternative explanations for the generation of lift[edit]

Many other alternative explanations for the generation of lift by an airfoil have been put forward, a few of which are presented here. Most of them are intended to explain the phenomenon of lift to a general audience. Although the explanations may share features in common with the explanations above, additional assumptions and simplifications may be introduced. This can reduce the validity of an alternative explanation to a limited sub-class of lift generating conditions, or might not allow a quantitative analysis. Several theories introduce assumptions which proved to be wrong, like the equal transit-time theory.

False explanation based on equal transit-time[edit]

An illustration of the (incorrect) equal transit-time explanation of airfoil lift.

Basic or popular sources often describe the "Equal Transit-Time" theory of lift, which incorrectly assumes that the parcels of air that divide at the leading edge of an airfoil must rejoin at the trailing edge, forcing the air traveling along the longer upper surface to go faster. Bernoulli's Principle is then cited to conclude that since the air moves slower along the bottom of the wing, the air pressure must be higher, pushing the wing up.[73]

However, there is no physical principle that requires equal transit time and experimental results show that this assumption is false. In fact, the air moving over the top of an airfoil generating lift moves much faster than the equal transit theory predicts.[74] Further, the theory violates Newton's third law of motion, since it describes a force on the wing with no opposite force.[75][76][77][78][79]

The assertion that the air must arrive simultaneously at the trailing edge is sometimes referred to as the "Equal Transit-Time Fallacy".[80][81][82][83][84]

Coandă effect[edit]

Main article: Coandă effect

In a limited sense, 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. The effect is named for Henri Coandă, the Romanian aerodynamicist who exploited it in many of his patents.

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 to explain why the air flow remains attached to the top side of an airfoil.[85] Jef Raskin,[86] for example, describes a simple demonstration, using a straw to blow over the upper surface of a wing. The wing deflects upwards, thus demonstrating that the Coandă effect creates lift. This demonstration correctly demonstrates the Coandă effect as a fluid jet (the exhaust from a straw) adhering to a curved surface (the wing). However, the upper surface in this flow is a complicated, vortex-laden mixing layer, while on the lower surface the flow is quiescent. The physics of this demonstration are very different from that of the general flow over the wing.[87] The usage in this sense is encountered in some popular references on aerodynamics.[85][86] In the aerodynamics field, the Coandă effect is commonly defined in the more limited sense above[87][88][89] and viscosity is used to explain why the boundary layer attaches to the surface of a wing.[72]

Methods to determine lift on an airfoil[edit]

Lift coefficient[edit]

Main article: Lift coefficient

If the lift coefficient for a wing at a specified angle of attack is known (or estimated using a method such as thin airfoil theory), then the lift produced for specific flow conditions can be determined using the following equation:[90]

L = \tfrac12\rho v^2 A C_L


Kutta–Joukowski theorem[edit]

Lift can be calculated using potential flow theory by imposing a circulation. It is often used by practising aerodynamicists as a convenient quantity in calculations, for example thin-airfoil theory and lifting-line theory.

The circulation \Gamma is the contour integral of the tangential velocity of the air on a closed loop (also called a 'circuit') around the boundary of an airfoil. It can be understood as the total amount of "spinning" (or vorticity) of air around the airfoil. The section lift/span L' can be calculated using the Kutta–Joukowski theorem:[46]

L' =  \rho v \Gamma\,

where  \rho is the air density,  v is the free-stream airspeed. Kelvin's circulation theorem states that circulation is conserved for any circuit moving with the fluid.[92] When an aircraft is at rest, circulation is zero.

The challenge when using the Kutta–Joukowski theorem to determine lift is to determine the appropriate circulation for a particular airfoil. In practice, this is done by applying the Kutta condition, which uniquely prescribes the circulation for a given geometry and free-stream velocity.

A physical understanding of the theorem can be observed in the Magnus effect, which is a lift force generated by a spinning cylinder in a freestream. Here the necessary circulation is induced by the mechanical rotation acting on the boundary layer, causing it to induce a faster flow around one side of the cylinder and a slower flow around the other. The asymmetric distribution of airspeed around the cylinder then produces a circulation in the outer inviscid flow.[93]

Pressure integration[edit]

The force on the wing can be examined in terms of the pressure differences above and below the wing, which can be related to velocity changes by Bernoulli's principle.

The total lift force is the integral of vertical pressure forces over the entire wetted surface area of the wing:[94]

L = \oint p\mathbf{n} \cdot\mathbf{k} \; \mathrm{d}A


  • L is the lift,
  • A is the wing surface area,
  • p is the value of the pressure,
  • n is the normal unit vector pointing into the wing, and
  • k is the vertical unit vector, normal to the freestream direction.

The above lift equation neglects the skin friction forces, which typically have a negligible contribution to the lift compared to the pressure forces. 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 induced drag in a 3D wing). If we use the spanwise vector j, we obtain the side force Y.

  D_p &= \oint p\mathbf{n} \cdot\mathbf{i} \; \mathrm{d}A
  Y   &= \oint p\mathbf{n} \cdot\mathbf{j} \; \mathrm{d}A

One method for calculating the pressure is Bernoulli's equation, which is the mathematical expression of Bernoulli's principle. This method ignores the effects of viscosity, which can be important in the boundary layer and to predict friction drag, which is the other component of the total drag in addition to Dp.

The Bernoulli principle states that the sum total of energy within a parcel of fluid remains constant as long as no energy is added or removed. It is a statement of the principle of the conservation of energy applied to flowing fluids.

A substantial simplification proposes that other forms of energy changes are inconsequential during the flow of air around a wing and that energy transfer in/out of the air is not significant, so the sum of pressure energy and speed energy for any particular parcel of air must be constant. Consequently, an increase in speed must be accompanied by a decrease in pressure and vice-versa. It is named for the Dutch-Swiss mathematician and scientist Daniel Bernoulli, though it was previously understood by Leonhard Euler and others.

Bernoulli's principle provides an explanation of pressure difference in the absence of air density and temperature variation (a common approximation for low-speed aircraft). If the air density and temperature are the same above and below a wing, a naive application of the ideal gas law requires that the pressure also be the same. Bernoulli's principle, by including air velocity, explains this pressure difference. The principle does not, however, specify the air velocity. This must come from another source, e.g., experimental data.

In order to solve for the velocity of inviscid flow around a wing, the Kutta condition must be applied to simulate the effects of viscosity. The Kutta condition allows for the correct choice among an infinite number of flow solutions that otherwise obey the laws of conservation of mass and conservation of momentum.

Lift forces on bluff bodies[edit]

Further information: Vortex shedding and Vortex-induced vibration

The flow around bluff bodies – i.e. without a streamlined shape, or stalling airfoils – may also generate lift, besides a strong drag force. This lift may be steady, or it may oscillate due to vortex shedding. Interaction of the object's flexibility with the vortex shedding may enhance the effects of fluctuating lift and cause vortex-induced vibrations.[95] For instance, the flow around a circular cylinder generates a Kármán vortex street: vortices being shed in an alternating fashion from each side of the cylinder. The oscillatory nature of the flow is reflected in the fluctuating lift force on the cylinder, whereas the mean lift force is negligible. The lift force frequency is characterised by the dimensionless Strouhal number, which depends (among others) on the Reynolds number of the flow.[96][97]

For a flexible structure, this oscillatory lift force may induce vortex-induced vibrations. 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. Such vibrations may pose problems and threaten collapse in tall man-made structures like industrial chimneys.[95]

See also[edit]

References and notes[edit]

  1. ^ "What is Lift?". NASA Glenn Research Center. Retrieved March 4, 2009. 
  2. ^ The amount of lift will be (usually slightly) more or less than gravity depending on the thrust level and vertical alignment of the thrust line. A side thrust line will result in some lift opposing side thrust as well.
  3. ^ Clancy, L.J., Aerodynamics, Section 14.6
  4. ^ Clancy, L.J., Aerodynamics, Section 5.2
  5. ^ Drag of Blunt Bodies and Streamlined Bodies
  6. ^ "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
  7. ^ "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 pg 300
  8. ^ "One explanation of how a wing of an airplane 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
  9. ^ "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 No.January 1, 1987
  10. ^ "The lift on the body is's the re-action 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
  11. ^ "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
  12. ^ Ison, David, "Bernoulli Or Newton: Who's Right About Lift?", Plane & Pilot, retrieved January 14, 2011 
  13. ^ "...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 Edition, John Wiley & Sons, p. 378 
  14. ^ a b c Anderson, John D. (2004), Introduction to Flight (5th ed.), McGraw-Hill, pp. 352–361, §5.19, ISBN 0-07-282569-3 
  15. ^ "The wing deflects the airflow such that the mean velocity vector behind the wing is canted slightly downward (…). Hence, the wing imparts a downward component of momentum to the air; that is, the wing exerts a force on the air, pushing the flow downward. From Newton's third law, the equal and opposite reaction produces a lift."[14]
  16. ^ "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, as shown in Fig. 2. 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 pg 373
  17. ^ Weltner, Klaus; Ingelman-Sundberg, Martin, Physics of Flight – reviewed 
  18. ^ "The cause of the aerodynamic lifting force is the downward acceleration of air by the airfoil... "[17]
  19. ^ "The main fact of all heaver-than-air flight is this: the wing keeps the airplane up by pushing the air down." In: Langewiesche, Wolfgang (1990), Stick and Rudder: An Explanation of the Art of Flying, McGraw-Hill, pp. 6–10, ISBN 0-07-036240-8 
  20. ^ "The lift generated by an airplane is, on account of the principles of action and reaction, necessarily connected with a descending current in all its details." Ludwig Prandtl, as quoted by John D. Anderson in Introduction to Flight pg 332
  21. ^ "Birds and aircraft fly because they are constantly pushing air downwards: L = dp/dt Here L is the lift force and dp/dt is the rate at which downward momentum is imparted to the airflow." Flight without Bernoulli Chris Waltham THE PHYSICS TEACHER Vol. 36, Nov. 1998
  22. ^ "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 
  23. ^ "That's what the wings are for. They divert the air they reach and deflect it downwards." Cliff Swartz Numbers Count The Physics Teacher Vol 34 Dec 1996 pg 536
  24. ^ Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1963), The Feynman Lectures on Physics, Reading, Mass.: Addison-Wesley, ISBN 0-201-02116-1 , Vol. 1, §10–1 and §10–2.
  25. ^ "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. Retrieved July 7, 2009. 
  26. ^ 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 0-521-66396-2 
  27. ^ Robert P. Bauman (February 2007), The Bernoulli Conundrum 
  28. ^ "...if a streamline is curved, there must be a pressure gradient across the streamline..."Babinsky, Holger (November 2003), "How do wings work?", Physics Education 
  29. ^ 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 pg 53
  30. ^ NASA, Glenn Research Center. "Aerodynamic Forces". Retrieved August 15, 2011. 
  31. ^ "...newtonian theory is used frequently to estimate the pressure distribution of a hypersonic body." Fundamentals of Aerodynamics 3rd ed. by John D. Anderson, Jr. McGraw-Hill 2001 ISBN 0-07-237335-0 p. 686.
  32. ^
  33. ^ "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 Department of Physics and Astronomy
  34. ^ "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 pg 52
  35. ^ "The decrease 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 pg 52
  36. ^ "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 pg 52
  37. ^ "...the important thing about an aerofoil (say an aircraft wing) 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
  38. ^ "It requires adjustment of the angle of attack, but as clearly demonstrated in almost every air show, it can be done." hyperphysics Georgia State University Department of Physics and Astronomy
  39. ^ Most students will be happy with the streamline pattern around a lifting wing ... because it intuitively looks right Babinsky, Holger (November 2003), "How do wings work?", Physics Education 
  40. ^ "We have used a very simple physical model relying only on Newton’s second law to reproduce all the salient features of a rigorous fluid dynamical treatment of flight... The model has its limitations; we cannot calculate real performance with it." Waltham, Chris (November 1998), "Flight Without Bernoulli", The Physics Teacher 
  41. ^ "Measuring lift by measuring the increase in downward vertical velocity in the flow coming off the trailing edge of the airfoil is conceptually possible. This downward velocity is definitely there and is known as downwash. I have never heard of anyone actually measuring it with sufficient precision to calculate lift, not because it is physically unsound but because it is not a practical experiment." Charles N. Eastlake An Aerodynamicist’s View of Lift, Bernoulli, and Newton THE PHYSICS TEACHER Vol. 40, March 2002
  42. ^ "Finally we obtain dp/dz = p v^2/R. Curved streamlines within a flow are related to pressure gradients. Unfortunately this equation cannot be integrated directly. The integration requires the knowledge of the total flow field." Physics of Flight - reviewed by Klaus WELTNER
  43. ^ "A complete statement of Bernoulli's Theorem is as follows: "In a flow where no energy is being added or taken away, the sum of its various energies is a constant: consequently where the velocity increasees the pressure decreases and vice versa."" Norman F Smith Bernoulli, Newton and Dynamic Lift Part I School Science and Mathematics Vol 73 Issue 3
  44. ^ "Although the Bernoulli principle can be used to determine the net force and the lifting force upon a solid body once the pressure and velocity distribution around the body have been specified, this flow regime is a consequence of viscosity, and cannot be explained without recourse to the Coanda effect, circulation, and the curvature of the wing. Hence the Bernoulli principle explanation is subservient to an explanation of lift which employs the latter concepts." Gordon McCabe Explanation and discovery in aerodynamics
  45. ^ "The effect of squeezing streamlines together as they divert around the front of an airfoil shape is that the velocity must increase to keep the mass flow constant since the area between the streamlines has become smaller." Charles N. Eastlake An Aerodynamicist’s View of Lift, Bernoulli, and Newton THE PHYSICS TEACHER Vol. 40, March 2002
  46. ^ a b Landau, L. D.; Lifshitz, E. M. (1987), Fluid mechanics, Course of Theoretical Physics 6 (2nd revised ed.), Pergamon Press, ISBN 0-08-033932-8, OCLC 15017127  , pp. 68–69 and pp. 153–155.
  47. ^ "There is no way to predict, from Bernoulli's equation alone, what the pattern of streamlines will be for a particular wing." Halliday and Resnick Fundamentals of Physics 3rd Ed. Extended pg 378
  48. ^ "The generation of lift may be explained by starting from the shape of streamtubes above and below an airfoil. With a constriction above and an expansion below, it is easy to demonstrate lift, again via the Bernoulli equation. However, the reason for the shape of the streamtubes remains obscure..." Jaakko Hoffren Quest for an Improved Explanation of Lift American Institute of Aeronautics and Astronautics 2001 pg 3
  49. ^ "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
  50. ^ "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
  51. ^ "A concept...uses a symmetrical convergent-divergent channel, like a longitudinal section of a Venturi tube, as the starting point. It is widely known that, 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
  52. ^ "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
  53. ^ "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 (Fig. 1). 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 pg 84.
  54. ^ "One of the complicated equations with which we bedevil our students is due to Bernoulli.... Once having derived the equation, however, we like to apply it to all sorts of phenomena that don’t satisfy the constraints. For instance, we ascribe lift on airfoils to the air going over the top of the foil with greater speed and thus less pressure. But air isn’t incompressible and the streamlines are all mixed up and dissipated with turbulence." Misconceptions for Grown-Ups Clifford E. Swartz THE PHYSICS TEACHER Vol. 36, April 1998 pg 200
  55. ^ "Unfortunately, applying Bernoulli's equation and ignoring the problem's complexity ... (gives) the false impression that restrictions used in deriving an equation like Bernoulli's equation can be ignored in the later applications of that equation. Bernoulli's equation is derived assuming incompressible fluids, and therefore can only be applied to air in special circumstances. Our objections to the use of Bernoulli's equation to explain airfoil lift in introductory texts are that it sets a bad example in illustrating to students that it is all right (perhaps even good physics) to apply an equation to a situation in which it appears to violate the conditions assumed in deriving the equation, and that it does not lead to a clear understanding of how airfoils generate lift." Huebner & Jagannathan letter published in American Journal of Physics Vol 56 No 9 1988 pg 855
  56. ^ "Another fundamental problem with this description is that the air’s pressure and speed are not related by the Bernoulli equation for a real wing in flight! The Bernoulli equation is a statement of the conservation of energy. For it to be applied the system must be in equilibrium and no energy added to the system. As you will see in the discussion below, a great deal of energy as added to the air. Before the wing came by the air was standing still. After the passage of the wing there is a great deal of air in motion. A 250-ton jet at cruse speed is doing a lot of work to stay in the air. Much of the fuel that is burned is adding energy to the air to create lift. Thus the Bernoulli equation is not applicable." The Newtonian Description of Lift of a Wing-Revised David F. Anderson & Scott Eberhardt
  57. ^ "Although the Bernoulli equation is very useful for qualatative descriptions of many features of fluid flow, such descriptions are often grossly inaccurate when compared with the quantitative results of experiments. Prominent reasons for the discrepancies are that gasses like air are hardly incompressible, and liquids like water are hardly inviscid, which invalidates the assumptions made in deriving the Bernoulli equation. In addition it is often difficult to maintain steady, streamlined flow without turbulence, and the introduction of turbulence can greatly affect the results" Paul A. Tipler and Gene Mosca Physics for Scientists and Engineers 6th ed. 2008 W. H. Freeman and Company pg 445 isbn 1-4292-0124-X
  58. ^ "...there is nothing in aerodynamics requiring the top and bottom flows having to reach the trailing edge at the same time. This idea is a completely erroneous explanation for lift. The flow on top gets to the trailing edge long before the flow on the bottom because of the circulation flow field." Arvel Gentry Origins of Lift
  59. ^ "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
  60. ^ a b Batchelor, G.K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press, pp. 141–151, ISBN 0-521-66396-2 
  61. ^ "The main relationships comprising the NS equations are the basic conservation laws for mass, momentum, and energy. To have a complete equation set we also need an equation of state relating temperature, pressure, and density, and formulas defining the other required gas properties." Doug McLean Understanding Aerodynamics: Arguing from the Real Physics page 14 Wiley Print ISBN 9781119967514 Online ISBN 9781118454190
  62. ^ Jaakko Hoffren Quest for an Improved Explanation of Lift Section 4.3 American Institute of Aeronautics and Astronautics 2001
  63. ^ "...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
  64. ^ Elements of Potential Flow California State University Los Angeles
  65. ^ ""...we can visualize the true flow over an the superposition of a uniform flow and a circulatory flow..." John D Anderson Introduction to Flight 4th Ed. pg 325
  67. ^ Potential Flow over a NACA Four-Digit Airfoil from the Wolfram Demonstrations Project
  68. ^ "On the top side of the airfoil, the circulation vectors are in the same direction as the free stream direction, therefore causing the flow to speed up. This increase in speed means lower pressures (according to Bernoulli's equation). On the bottom side of the airfoil, the circulation vector is opposite the general flow direction so the fluid tends to be slowed down resulting in increased pressure. The difference in pressure forces between the top and bottom sides of an airfoil are what gives us lift." Arvel Gentry The Origins of Lift January 2006
  69. ^ Karamacheti, Krishnamurty (1980), Principles of Ideal-Fluid Aerodynamics (Reprint ed.), Robert E. Krieger 
  70. ^ Clancy, L.J., Aerodynamics, Figure 4.7
  71. ^ Clancy, L.J., Aerodynamics, Figure 4.8
  72. ^ a b White, Frank M. (2002), Fluid Mechanics (5th ed.), McGraw Hill 
  73. ^ "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, pp. 451
  74. ^ "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 (March 15, 2006). "Incorrect Lift Theory". NASA. Retrieved August 12, 2010. 
  75. ^ "Unfortunately, this explanation falls to earth 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 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, pp. 451
  76. ^ Anderson, David (2001), Understanding Flight, New York: McGraw-Hill, pp. 15–16, ISBN 0-07-136377-7, "The first thing that is wrong is that the principle of equal transit times is not true for a wing with lift." 
  77. ^ Anderson, John (2005). Introduction to Flight. Boston: McGraw-Hill Higher Education. p. 355. ISBN 0072825693. "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" 
  78. ^ Cambridge scientist debunks flying myth UK Telegraph 24 Jan 2012
  79. ^ Flow Visualization. National Committee for Fluid Mechanics Films/Educational Development Center. 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.
  80. ^ A false explanation for lift has been put forward in mainstream books, and even in scientific exhibitions. Known as the "equal transit-time" explanation, it states that the parcels of air which are divided by an airfoil must rejoin again; because of the greater curvature (and hence longer path) of the upper surface of an aerofoil, the air going over the top must go faster in order to "catch up" with the air flowing around the bottom. Therefore, because of its higher speed the pressure of the air above the airfoil must be lower. Despite the fact that this "explanation" is probably the most common of all, it is false. It has recently been dubbed the "Equal transit-time fallacy"."Fixed wing aircraft facts and how aircraft fly". Retrieved July 7, 2009. 
  81. ^ leaves the impression that Professor Bernoulli is somehow to blame for the "equal transit time" fallacy... John S. Denker (1999). "Critique of "How Airplanes Fly"". Retrieved July 7, 2009. 
  82. ^ The fallacy of equal transit time can be deduced from consideration of a flat plate, which will indeed produce lift, as anyone who has handled a sheet of plywood in the wind can testify. Gale M. Craig. "Physical principles of winged flight". Retrieved July 7, 2009. 
  83. ^ Fallacy 1: Air takes the same time to move across the top of an aerofoil as across the bottom. Peter Eastwell (2007), "Bernoulli? Perhaps, but What About Viscosity?", The Science Education Review 6 (1), retrieved July 14, 2009. 
  84. ^ "There is a popular fallacy called the equal transit-time fallacy that claims the two halves rejoin at the trailing edge of the aerofoil." Ethirajan Rathakrishnan Theoretical Aerodynamics John Wiley & sons 2013 section 4.10.1
  85. ^ a b Anderson, David; Eberhart, Scott (1999), How Airplanes Fly: A Physical Description of Lift, retrieved June 4, 2008 
  86. ^ a b Raskin, Jef (1994), Coanda Effect: Understanding Why Wings Work, archived from the original on September 28, 2007 
  87. ^ a b Auerbach, David (2000), "Why Aircraft Fly", Eur. J. Phys. 21 (4): 289–296, Bibcode:2000EJPh...21..289A, doi:10.1088/0143-0807/21/4/302 
  88. ^ Denker, JS, Fallacious Model of Lift Production, retrieved 2008-08-18 
  89. ^ Wille, R; Fernholz, H (1965), "Report on the first European Mechanics Colloquium, on the Coanda effect", J. Fluid Mech. 23 (4): 801–819, Bibcode:1965JFM....23..801W, doi:10.1017/S0022112065001702 
  90. ^ Anderson, John D. (2004), Introduction to Flight (5th ed.), McGraw-Hill, pp. 257–261, ISBN 0-07-282569-3 
  91. ^ Yoon, Joe (2003-12-28), Mach Number & Similarity Parameters,, retrieved 2009-02-11 
  92. ^ Clancy, L.J., Aerodynamics, Section 7.27
  93. ^ Clancy, L.J., Aerodynamics, Sections 4.5 and 4.6
  94. ^ Anderson, John D. (2004), Introduction to Flight, Section 5.7 (5th edition), McGraw-Hill. ISBN 0-07-282569-3
  95. ^ 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 
  96. ^ Sumer, B. Mutlu; Fredsøe, Jørgen (2006), Hydrodynamics around cylindrical structures (revised ed.), World Scientific, pp. 6–13, 42–45 & 50–52, ISBN 981-270-039-0 
  97. ^ Zdravkovich, M.M. (2003), Flow around circular cylinders 2, Oxford University Press, pp. 850–855, ISBN 0-19-856561-5 

Further reading[edit]

  • Introduction to Flight, John D. Anderson, Jr., McGraw-Hill, ISBN 0-07-299071-6 – The author is the Curator of Aerodynamics at the Smithsonian Institution's National Air & Space Museum and Professor Emeritus at the University of Maryland.
  • Understanding Flight, by David Anderson and Scott Eberhardt, McGraw-Hill, ISBN 0-07-136377-7 – The authors are a physicist and an aeronautical engineer. They explain flight in non-technical terms and specifically address the equal-transit-time myth. Turning of the flow around the wing is attributed to the Coanda effect, which is quite controversial.
  • Aerodynamics, Clancy, L.J. (1975), Pitman Publishing Limited, London ISBN 0-273-01120-0.
  • Aerodynamics, Aeronautics, and Flight Mechanics, McCormick, Barnes W., (1979), Chapter 3, John Wiley & Sons, Inc., New York ISBN 0-471-03032-5.
  • Fundamentals of Flight, Richard S. Shevell, Prentice-Hall International Editions, ISBN 0-13-332917-8 – This book is primarily intended as a text for a one semester undergraduate course in mechanical or aeronautical engineering, although its sections on theory of flight are understandable with a passing knowledge of calculus and physics.
  • "Observation of Perfect Potential Flow in Superfluid", Paul P. Craig and John R. Pellam (1957) Physical Review 108(5), pp. 1109–1112, doi:10.1103/PhysRev.108.1109 – Experiments under superfluidity conditions, resulting in the vanishing of lift in inviscid flow since the Kutta condition no longer is satisfied.
  • "Aerodynamics at the Particle Level", Charles A. Crummer (2005, revised 2012) – A treatment of aerodynamics emphasizing the particle nature of air, as opposed to the fluid approximation commonly used.
  • "Flight without Bernoulli" Chris Waltham Vol. 36, Nov. 1998 THE PHYSICS TEACHER – using a physical model relying only on Newton’s second law, the author presents a rigorous fluid dynamical treatment of flight.
  • Bernoulli, Newton, and Dynamic Lift Norman F. Smith School Science and Mathematics vol 73 Part I: Part II

External links[edit]