Lift (force)

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This article is about lift force in fluid dynamics. For other uses, see lift.
Airbus A380 taking off during the Paris Air Show in 2007.
Look up lift in Wiktionary, the free dictionary.

A fluid flowing relative to a body exerts a force on it, called a "fluid-dynamic force", or "aerodynamic force" if the fluid is air. Lift is the component of this force which is perpendicular to the oncoming flow direction.[1] It contrasts with the drag force, which is the component of the aerodynamic force parallel to the flow direction. An airfoil is a streamlined body that is capable of generating significantly more lift than drag.[2]

Lift is commonly associated with the wing of a fixed-wing aircraft, although lift is also generated by propellers; helicopter rotors; rudders, sails and keels on sailboats; hydrofoils; wings on auto racing cars; wind turbines and other streamlined objects. While common meanings of the word "lift" suggest that lift opposes gravity, lift can be in any direction. When an aircraft is flying straight and level (cruise) all of the lift opposes gravity. However, when an aircraft is climbing, descending, or banking in a turn, for example, the lift is tilted with respect to the vertical and the lift is greater than, or less than, the weight of the aircraft.[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.

Non-streamlined objects such as bluff bodies and plates (not parallel to the flow) may also generate lift when moving relative to the fluid. 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.[4]

Contents

[edit] Description of lift on an airfoil

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

In the last principle, 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.[5] Application of the viscous shear stresses to Newton's second law for an airflow results in the Navier–Stokes equations. 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.

The Euler equations for a steady and inviscid flow can be integrated along a streamline, resulting in Bernoulli's equation. The particular form of Bernoulli's equation found depends on the equation of state used.[6] At low Mach numbers, compressibility effects may be neglected, resulting in an incompressible flow approximation. In incompressible and inviscid flow the Bernoulli equation is just an integration of Newton's second law—in the form of the description of momentum evolution by the Euler equations—along a streamline.[6]

Explaining lift while considering all of the principles involved is a complex task and is not easily simplified.[7][8] As a result, there are numerous different explanations of lift with different levels of rigour and complexity. For example, there is an explanation based directly on Newton’s laws of motion; and an explanation based on Bernoulli’s principle. Neither of these explanations is incorrect, but each appeals to a different audience.[9]

In order to explain lift as it applies to an airplane wing, consider the incompressible flow around a 2-D, symmetric airfoil at positive angle of attack in a uniform free stream. Instead of considering the case where an airfoil moves through a fluid as seen by a stationary observer, it is equivalent and simpler to consider the picture when the observer follows the airfoil and the fluid moves past it. The explanation presented first is a summary of the most scientifically complete and accepted explanation; alternative and generally more accessible explanations follow below.

[edit] Lift in an established flow

Streamlines around a NACA 0012 airfoil at moderate angle of attack.

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 (which can be derived from Newton's second law) and conservation of mass.[7]

The image to the right shows the streamlines over a NACA 0012 airfoil computed using potential flow theory, a simplified model of the real flow. The flow approaching an airfoil can be divided into two streamtubes, which are defined based on the area between two streamlines. By definition, fluid never crosses a streamline in a steady flow; hence mass is conserved within each streamtube. One streamtube travels over the upper surface, while the other travels over the lower surface; dividing these two tubes is a dividing line (the stagnation streamline) that intersects the airfoil on the lower surface, typically near to the leading edge. The stagnation streamline leaves the airfoil at the sharp trailing edge, a feature of the flow known as the Kutta condition. In calculating the flow shown, the Kutta condition was imposed as an initial assumption; the justification for this assumption is explained below.

The upper stream tube constricts as it flows up and around the airfoil, a part of the so-called upwash. From the conservation of mass, the flow speed must increase as the stream tube area decreases.[7] The area of the lower stream tube increases, causing the flow inside the tube to slow down. 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.

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. The pressure difference thus creates a net aerodynamic force, pointing upward and downstream to the flow direction. The component of the force normal to the free stream is considered to be lift; the component parallel to the free stream is drag. In conjunction with this force by the air on the airfoil, by Newton's third law, the airfoil imparts an equal-and-opposite force on the surrounding air that creates the downwash. Measuring the momentum transferred to the downwash is another way to determine the amount of lift on the airfoil.[10]

[edit] Flowfield formation

In attempting to explain why the flow follows the upper surface of the airfoil, the situation gets considerably more complex. It is here that many simplifications are made in presenting lift to various audiences, some of which are presented below.

Consider the case of an airfoil accelerating from rest in a viscous flow. Lift depends entirely on the nature of viscous flow past certain bodies[11]: in inviscid flow (i.e. assuming that viscous forces are negligible in comparison to inertial forces), there is no lift without imposing a net circulation, the proper amount of which can be determined by applying the Kutta condition. In a viscous flow like in the physical world, however, the lift and other properties arise naturally as described here.

When there is no flow, there is no lift and the 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 mentioned above 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.

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.[12] 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, [13] and is a necessary condition to produce lift on an airfoil. If the flow were stopped, there would be a corresponding "stopping vortex".[14] 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.

[edit] Alternative explanations for the generation of lift

It is amazing that today, almost 100 years after the first flight of the Wright Flyer, groups of engineers, scientists, pilots, and others can gather together and have a spirited debate on how an airplane wing generates lift. Various explanations are put forth, and the debate centers on which explanation is the most fundamental.

– John D. Anderson[7]

Many alternative explanations for the generation of lift by an airfoil have been put forward, of which a few 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 explanation above, additional assumptions and simplifications may be introduced. This reduces the validity of an alternative explanation to a limited sub-class of lift generating conditions, or does not allow a quantitative analysis. Several theories introduce assumptions which proved to be wrong, like the equal transit-time theory.

[edit] Equal transit-time

An illustration of the equal transit-time theory.
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 = –0.1, μy = +0.1 and n = 1.96. The angle of attack is 8°, and the flow is a potential flow.

An explanation of lift frequently encountered in basic or popular sources is the equal transit-time theory. Equal transit-time states that because of the longer path of the upper surface of an airfoil, the air going over the top must go faster in order to catch up with the air flowing around the bottom.[15] Such that the parcels of air that are divided at the leading edge, and travel above and below an airfoil must rejoin when they reach the trailing edge. However, this is not accurate[16] and the fact that this is not generally the case can be readily observed.[17] Although it is true that the air moving over the top of a wing generating lift does move faster, there is no requirement for equal transit time. In fact the air moving over the top of an airfoil generating lift is always moving much faster than the equal transit theory would imply.[18]

Further information: List of works with the equal transit-time fallacy

[edit] Coandă effect

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.

One of the first known uses was in his patent for a high-lift device[19] that used a fan of gas exiting at high speed from an internal compressor. This circular spray was directed radially over the top of a curved surface shaped like a lens to decrease the pressure on that surface. The total lift for the device was caused by the difference between this pressure and that on the bottom of the craft. Two aircrafts, the Antonov An-72 and An-74 "Coaler", use the exhaust from top-mounted jet engines flowing over the wing to enhance lift,[20] as do the Boeing YC-14 and the McDonnell Douglas YC-15.[21] [22] The effect is also used in high-lift devices such as a blown flap.[23]

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 lift.[24] Jef Raskin[25], for example, describes a simple demonstration, using a straw to blow over the upper surface of a wing. The wing deflects upwards, thus supposedly 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 the lower surface the flow is quiescent. The physics of this demonstration are very different from that of the general flow over the wing.[26] The usage in this sense is encountered in some popular references on aerodynamics.[24][25] In the aerodynamics field, the Coandă effect is commonly defined in the more limited sense above[26][27][28] and viscosity is used to explain why the boundary layer attaches to the surface of a wing.[14]

[edit] In terms of a difference in areas

When a fluid flows relative to a solid body, the body obstructs the flow, causing some of the fluid to change its speed and direction in order to flow around the body. The obstructive nature of the solid body causes the streamlines to move closer together in some places, and further apart in others.[7][29][30] When fluid flows past a 2-D cambered airfoil at zero angle of attack, the upper surface has a greater area (that is, the interior area of the airfoil above the chordline) than the lower surface and hence presents a greater obstruction to the fluid than the lower surface.[29] This asymmetry causes the streamlines in the fluid flowing over the upper surface to move closer together than the streamlines over the lower surface. As a consequence of mass conservation, the reduced area between the streamlines over the upper surface results in a higher velocity than that over the lower surface. The upper streamtube is squashed the most in the nose region ahead of the maximum thickness of the airfoil, causing the maximum velocity to occur ahead of the maximum thickness.[7]

In accordance with Bernoulli's principle, where the fluid is moving faster the pressure is lower, and where the fluid is moving slower the pressure is greater. The fluid is moving faster over the upper surface, particularly near the leading edge, than over the lower surface so the pressure on the upper surface is lower than the pressure on the lower surface. The difference in pressure between the upper and lower surfaces results in lift.[7]

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

The direct relationship between the lift force and the downward deflection of the flow (downwash) behind an airfoil helps to obtain an insight into the phenomenon of lift. This explanation relies on 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. [31] For a fluid, the momentum density is equal to mass density multiplied by the flow velocity.[32] Now consider an airfoil in straight flight at constant speed, and observe the airflow from the frame of reference in which the velocity of the airfoil is zero. John D. Anderson explains the relationship between lift and downwash as follows: "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."[7]

This explanation of lift was used in the 1944 book Stick and Rudder by aviation writer Wolfgang Langewiesche: "… the wing keeps the airplane up by pushing the air down."[33] Consequently, the lift is directly related to the deflection of the flow field behind the airfoil.[34][10]

One may be tempted to think that lift can be explained along this line of reasoning, saying that lift on the airfoil is caused by the downward deflection of the airflow. However, this is a misinterpretation of Newton's laws. One can just as well argue that the downwash behind the airfoil is an effect of the lift force: according to Newton's third law the upward lift is accompanied by a downward force on the air. This latter force is pushing the air down, which results in a downward deflection of the velocity vectors behind the airfoil.[7]

Newton's laws of motion are not sufficient to explain lift. Also a force model is needed. To see this, consider for example an object under the influence of gravity: Newton's law of gravity is required in addition to his laws of motion, in order to be able to describe the object's motion.[35] So also for air flows a force model is needed, relating flow properties to resulting stresses. Primarily, in the case of aerodynamics, this is an equation of state describing the air pressure dependence on the other flow characteristics.

In case of the flow around an airfoil at low Mach numbers, the usual force model results from the incompressible flow approximation. Then the mass density is constant and the pressure is such that it constraints the flow to be isochoric — air parcels are of constant volume as they move along with the flow.[36]

[edit] Methods of determining lift on an airfoil

[edit] Lift coefficient

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:[37]

L = \tfrac12\rho v^2 A C_L

where

[edit] Kutta–Joukowski theorem

Main article: Kutta–Joukowski theorem

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

The circulation Γ is the line integral of the velocity of the air, in a closed loop 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:[10]

L' = -\rho V \Gamma\,

where ρ is the air density, V is the free-stream airspeed. Kelvin's circulation theorem states that circulation is conserved.[39] There is conservation of the air's angular momentum. When an aircraft is at rest, there is no circulation.

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 free stream. 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.[40]

[edit] Pressure integration

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:[41]

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

where:

  • 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.

\begin{align} D_p &= \oint p\mathbf{n} \cdot\mathbf{i} \; \mathrm{d}A, \\[1.2ex] Y &= \oint p\mathbf{n} \cdot\mathbf{j} \; \mathrm{d}A. \end{align}

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 of this proposes that as 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, then 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 should be noted that this is not a causational relationship. Rather, it is a coincidental relationship, whatever causes one must also cause the other as energy can neither be created nor destroyed. 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 inertia and 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.

[edit] Lift forces on bluff bodies

Flow separation and a Von Kármán vortex street behind a circular cylinder. The flow is from the right to the left. Part of the cylinder can be seen at the right edge of the image. Two locations of flow separation from the cylinder are clearly visible.
Further information: Vortex shedding and Vortex-induced vibration

The flow around bluff bodies may also generate lift, besides a strong drag force. 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.[42][43]

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, even collapse, in man-made tall structures like for instance industrial chimneys, if not properly taken care of in the design.[4]

[edit] See also

[edit] References and notes

  1. ^ "What is Lift?". NASA Glenn Research Center. http://www.grc.nasa.gov/WWW/K-12/airplane/lift1.html. Retrieved on 2009-03-04. 
  2. ^ Clancy, L.J., Aerodynamics, Section 5.2
  3. ^ Clancy, L.J., Aerodynamics, Section 14.6
  4. ^ a b Williamson, C.H.K.; Govardhan, R. (2004), "Vortex-induced vibrations", Annual Review of Fluid Mechanics 36: 413–455, doi:10.1146/annurev.fluid.36.050802.122128 
  5. ^ a b Batchelor, G.K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. pp. 141–151. ISBN 0521663962. 
  6. ^ a b Batchelor, G.K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. pp. 156–164, 265–266, 382–383, 507–509. ISBN 0521663962. 
  7. ^ a b c d e f g h i Anderson, John D. (2004), Introduction to Flight (5th ed.), McGraw-Hill, pp. 352–361, §5.19, ISBN 0-07-282569-3 
  8. ^ NASA Glenn Research Center, Bernoulli and Newton, http://www.grc.nasa.gov/WWW/K-12/airplane/bernnew.html, retrieved on 2008-04-19 
  9. ^ Ison, David, "Bernoulli Or Newton: Who's Right About Lift?", Plane & Pilot, http://www.planeandpilotmag.com/aircraft/specifications/diamond/2007-diamond-star-da40-xl/289.html, retrieved on 2008-04-21 
  10. ^ a b c Landau, L. D.; Lifshitz, E. M. (1986). 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.
  11. ^ Karamacheti, Krishnamurty (1980), Principles of Ideal-Fluid Aerodynamics (Reprint ed.), Robert E. Krieger 
  12. ^ Clancy, L.J., Aerodynamics, Figure 4.7
  13. ^ Clancy, L.J., Aerodynamics, Figure 4.8
  14. ^ a b White, Frank M. (2002), "Fluid Mechanics" (5th ed.), McGraw Hill 
  15. ^ Anderson, David (2001). Understanding Flight. New York: McGraw-Hill. pp. 15–16. ISBN 0071363777. "The first thing that is wrong is that the principle of equal transit times is not true for a wing with lift." 
  16. ^ “This is simply not true.” Anderson, John D. Jr, Introduction to Flight, p.355 (5th edition), McGraw-Hill ISBN 0-07-282569-3
  17. ^ Flow Visualization. National Committee for Fluid Mechanics Films/Educational Development Center. Retrieved on 2009-01-21. 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.
  18. ^ Glenn Research Center (2006-03-15). "Incorrect Lift Theory". NASA. http://www.grc.nasa.gov/WWW/K-12/airplane/wrong1.html. Retrieved on 2008-03-27. 
  19. ^ US patent 2108652
  20. ^ Antonov, Oleg Konstantinovich (24-May), http://www.aviation.ru/An 
  21. ^ Neely, Mike (2008), http://www.theaviationzone.com/factsheets/amst.asp, retrieved on 21-July-2008 
  22. ^ Pike, John (2008), http://www.globalsecurity.org/military/systems/aircraft/c-14.htm, retrieved on 23-July-2008 
  23. ^ Englar, Robert J. (June 2005), "Overview of Circulation Control Pneumatic Aerodynamics: Blown Force and Moment Augmentation and Modification as Applied Primarily to Fixed-Wing Aircraft", Proceedings of the 2004 NASA/ONR Circulation Control Workshop, Part 1, NASA/ONR, pp. 37–99 
  24. ^ a b Anderson, David; Eberhart, Scott (1999), How Airplanes Fly: A Physical Description of Lift, http://www.allstar.fiu.edu/AERO/airflylvl3.htm, retrieved on 2008-06-04 
  25. ^ a b Raskin, Jef (1994), Coanda Effect: Understanding Why Wings Work, http://jef.raskincenter.org/published/coanda_effect.html 
  26. ^ a b Auerbach, David (2000), "Why Aircraft Fly", Eur. J. Phys. 21: 289–296, doi:10.1088/0143-0807/21/4/302 
  27. ^ Denker, JS, Fallacious Model of Lift Production, http://www.av8n.com/how/htm/spins.html#sec-coanda-fallacy, retrieved on 2008-08-18 
  28. ^ Wille, R; Fernholz, H (1965), "Report on the first European Mechanics Colloquium, on the Coanda effect", J. Fluid Mech. 23: 801–819, doi:10.1017/S0022112065001702, http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=370712 
  29. ^ a b Smith, Hubert (1992), Illustrated Guide to Aerodynamics, McGraw-Hill Professional, pp. 17–23, ISBN 0830639012 
  30. ^ Brandt, Steven A.; Stiles, Randall J. (2004), Introduction to Aeronautics: A Design Perspective, AIAA, p. 74, ISBN 1563477017 
  31. ^ Feynman, RichardP.; 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.
  32. ^ Anderson, David F.; Eberhardt, Scott (2001), Understanding Flight, McGraw-Hill, pp. 24–28, ISBN 0071363777 
  33. ^ "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 0070362408 
  34. ^ More precisely: the lift is directly related to the flux of the vertical component of momentum through a vertical plane behind the airfoil. This vertical plane is perpendicular to the oncoming flow direction, in a frame of reference moving with the airfoil.
  35. ^ 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, §9–4 and §12–1.
  36. ^ Marshall, Jeffrey S. (2001), Inviscid incompressible flow, Wiley-IEEE, pp. 53–54, ISBN 0471375667 
  37. ^ Anderson, John D. (2004), Introduction to Flight (5th ed.), McGraw-Hill, pp. 257–261, ISBN 0-07-282569-3 
  38. ^ Yoon, Joe (2003-12-28), Mach Number & Similarity Parameters, Aerospaceweb.org, http://www.aerospaceweb.org/question/aerodynamics/q0156.shtml, retrieved on 2009-02-11 
  39. ^ Clancy, L.J., Aerodynamics, Section 7.27
  40. ^ Clancy, L.J., Aerodynamics, Sections 4.5 and 4.6
  41. ^ Anderson, John D. (2004), Introduction to Flight, Section 5.7 (5th edition), McGraw-Hill. ISBN 0-07-282569-3
  42. ^ Sumer, B. Mutlu; Fredsøe, Jørgen (2006), Hydrodynamics around cylindrical structures (revised ed.), World Scientific, pp. 6–13, 42–45 & 50–52, ISBN 9812700390 
  43. ^ Zdravkovich, M.M. (2003), Flow around circular cylinders, 2, Oxford University Press, pp. 850–855, ISBN 0198565615 

[edit] Further reading

  • Introduction to Flight, John D. Anderson, Jr., McGraw-Hill, ISBN 0-07-299071-6 — The author is the Curator of Aerodynamics at the National Air & Space Museum Smithsonian Institute 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), Section 4.8, Pitman Publishing Limited, London ISBN 0 273 01120 0.
  • Quest for an improved explanation of lift Jaako Hoffren (Helsinki Univ. of Technology, Espoo, Finland) AIAA-2001-872 Aerospace Sciences Meeting and Exhibit, 39th, Reno, NV, Jan. 8-11, 2001 — This paper focuses on a physics-based explanation of lift. Calculation of lift based on circulation with artificially imposed Kutta condition is interpreted as a mathematical model, having limited "real-world" physics, resulting from the assumption of potential flow. Also the role of viscosity is discussed. Author's claim is that viscosity is not important for lift generation.
  • 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.

[edit] External links

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