Abstract. The conventional or standard explanation of aerodynamic lift states the higher streaming velocity at the upper side of the airfoil as cause of the lower pressure, due to Bernoulli’s law. But a higher streaming velocity is the effect of a lower pressure and never its cause. The cause of the aerodynamic lifting force is the downward acceleration of air by the airfoil - which depends on the angle of attack and its velocity.
In relation to the airfoil the normal acceleration of the air in case of curved streamlines must be regarded which results in pressure gradients perpendicular to the streamlines and reaction forces acting perpendicular on the deflecting surfaces.
1. Introduction
The explanation of the aerodynamic lift has a long history but, there is controversy regarding the fundamental physics and their relation to Newton’s mechanics to date [1,2,3,4,]. This topic could be one of the most interesting and motivating in physics education. But the physics of flight nearly disappeared from the curriculum in schools and basic physics courses in most European countries. One reason, why this topic is neglected by teachers despite of students interest might be the fact that the conventional explanation of the aerodynamic lift is not convincing enough and not related to fundamental mechanics.
Niermann [5] analized all German language textbooks published within the the last one hundred years investigating the explanations of aerodynamic lift. The main result is that explanations based on Bernoulli’s law are dominating since the 1920s. He found the same to be true for a survey of contemporary American and English textbooks. A prominent excemption is the textbook of Halliday and Resnick [6].
2. Analysis of explanations based on Bernoulli’s law
The explanation of the aerodynamic lift in textbooks is based on Bernoulli’s law and the velocity distribution of the streaming air passing an airfoil.
Firstly Bernoulli’s law is stated. It says that there is an reverse relation between flow velocity and static pressure: If the flow velocity goes up the static pressure goes down and vice versa.
Secondly, streamlines of air passing an airfoil are demonstrated and analysed. Figure 1. It is stressed that the air at the upper side flows faster than the air at the lower side. Finally Bernoulli’s law is applyed to the different flow velocities at the upper and lower side of the wing resulting in a lower pressure at the upper surface and a higher pressure at the lower surface. These pressure differences produce a force on the airfoil - the lift.
This explanation, however, is not complete. If different flow
velocities are to cause different pressures a physical reason
for the fact that the air flows faster at the upper side given
too. Three types of reasons can be found.
Figure 1: Air flow around an airfoil.
i Analysis of path lengths: Referring to figure 1 it is argued [6] ‘The profile of a wing is such that the air has further to travel over the upper surface and hence has to flow faster to maintain streamline flow’.
In a wellknown German university textbooks it reads [7]:
‘If air is streaming from the left against the airfoil or if the airfoil moves in air at rest to the left the air will be separated. Since the path length along the upper side of the airfoil exceeds the path length along the lower side the velocity of the air at the upper side must exceed the velocity at the lower side.’
These arguments are based on a tacit hypothesis: Air which is adjacent before the separation by the airfoil has to meet again behind the airfoil.
In 1972 Smith [1] pointed out that there is no law or rule in physics to require adjacent air before the separation to meet behind the airfoil again after separation. Adjacent air before the separation indeed does not meet behind airfoil. Figure 2 shows experimental evidence of air streaming along an airfoil. Smoke tracer mark the streamlines.
The figure shows at the leading edge adjacent air before the separation. The figure shows at the trailing edge that the air which passed along the upper surface travelled even further creating a shift between the upper and lower air. This disapproves the arguments based on differences in path length.
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ii Using the concept of Circulation: In this type of reasoning the velocity distribution is explained using the concept of circulation. It starts from the description of a potential flow around the airfoil. Next, a circulation flow is superimposed in such a way that the stationary flow around the airfoil is obtained. The circulation must be such that the flow is even at the trailing edge. By this procedure a flow of the air is achieved which can be used to calculate the velocity distribution and hence the pressure distribution.
The concept of circulation is a sophisticated mathematical discription of the velocity distribution but not the cause of the latter.
iii Some textbooks give reasons like [9]:
‘Because of the shape of the wing the air travels faster over the curved upper surface than it does over the flatter lower surface’
This type of reasoning makes the geometry of the profile responsible for the behaviour of the streamlines without giving a physical cause why and how the streaming velocities are influenced by the airfoil.
This leaves the fundamental question open: How to generate the velocity distribution around an airfoil.
3. A general view - Aerodynamic lifting force as reaction force while air is accelerated downwards
For simplicity reasons we first discuss the lift by the rotor
of a helicopter or the propulsion force of a propeller or a jet.
In both cases an air stream is generated and air is accelerated.
To accelerate air a force must be excerted on the air and the
reaction force acts on the rotor or the propeller. Quantitatively
this force equals the change of momentum of the air stream: 
.
Basically, the same physics apply to the airfoil. The airfoil acts as a slightly curved plane moved horizontally with a small angle of attack. It accelerates air initially at rest downwards.
This vertical acceleration of air can be demonstrated by a simple experiment: A piece of cotton wool or tissue paper is freely suspended to indicate the movement of the air. According to figure 3a, an airfoil is moved horizontally below the cotton wool. It swings downward indicating a downward motion of the air. Figure 3b. If the airfoil is moved above the cotton wool the same downward movement can be observed. Also, this experiment can be used to demonstrate qualitatively the impact of the angle of attack and of the velocity on the downward motion and consequently on the lifting force.
Viewed from the aircraft the airfoil deflects the horizontal flow of the air downwards. This vertical motion is called downwash and can be demonstrated as well. A thread of wool must be glued to the trailing edge of an airfoil. If positioned in the air stream of a blower or fan the direction of the thread follows the direction of the trailing edge indicating the direction of the air stream behind the airfoil. If the angle of attack is varied the direction of the thread varys too. This demonstrates that the direction of the air flow near the airfoil can be manipulated by the position of the airfoil itself. An airfoil may be produced by cardboard glued together.
Figure 3: A cotton wool as indicator of air movements.
Some important relations can easily be derived from the vertical flow of momentum caused by the airfoil.
Lift and angle of attack: The air flow near the airfoil follows the geometrical shape of the latter’s surface. The air stream is deflected downwards approximately proportional to the angle of attack. Consequently the lift is approximately proportional to the angle of attack too. This holds for angles of attack between -10° and 15°. With an exceeding angle of attack the air stream ceases to follow the surface homogenously creating turbulence. In aviation this process is called ‘stall’.
It should be noted that the trailing edge of a strongly curved profile - used for aircrafts with low velocities - points downward. Even if the angle of attack is zero the air behind the airfoil has a vertical velocity component. Consequently the airofoil produces lift.
Velocity and lift: The geometry of the streaming flow remains the same if the velocity is doubled. Two factors double:
Combining the effects the lift is to increase four times if the streaming velocity is doubled.
Lift and air density: The reaction forces are proportional to the accelerated mass and therefore proportional to the density of the air. At an altitude of 12.000 m density and air pressure are approximately a quarter of their standards at sea level. Consequently the lift is reduced to a quarter as well. This loss can be compensated by doubling the velocity.
4. The one-dimensional Euler-Equation and the generation of pressure
To understand the origin of the pressure distribution along the surface of an airfoil in detail we must refer to the Euler equations. Those describe the relation between pressure gradients and acceleration of incompressible fluids without friction. Euler applied Newton’s laws to the motion of fluids. We refer to the most simple form - the one-dimensional Euler equation - which holds for stationary flow confined by streamlines.[10,11] Gravitational effects are excluded. We assume a cubical volume. Figure 4. For the mass D m confined in the volume the basic equation is:
We analyse separately tangential acceleration, figure 4, and normal acceleration, figure 5.
4.1 A tangential acceleration in s-direction is the result of a force. A force in s-direction occurs when the pressure acting at the faces A at the back is higher than the pressure in front: It must be stressed that an acceleration in s-direction is caused by a decrease of pressure in s-direction.
Inserting the mass 
and 
we
get
This equation is transformed to
Solving the definite integral we arrive at the Bernoulli equation:
4.2 Normal acceleration. See figure 5
A normal acceleration within curved streamlines needs a higher pressure at the outer lateral face than at the inner lateral face.
According to figure 5:
Inserting the mass of the volume 
we arrive at
The acceleration in direction of the center of curvature is well known. It is the centripetal acceleration of a circular motion.
(R = radius of curvature, v = streaming velocity)
Finally we obtain
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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.
Nonetheless, the analysis of normal acceleration of air serves as an explanation for the generation of regions with lower or higher pressure for the flow around an airfoil.
We refer to the stationary flow near an airfoil. The streaming air passing the airfoil cannot penetrate the surface and is forced to move on streamlines which surround the airfoil and follow its geometrical shape. Close to the airfoil the flow is forced to approximate the latter’s geometry. This is due to the Coanda effect.[12] The motion near the airfoil is a forced motion determined by the shape of the airfoil and the latter’s position in relation to the direction of the flow (angle of attack).
At the upper surface of the airfoil the acceleration is directed to the center of curvature, i.e. mainly downwards. The necessary pressure gradient is created by a slight ‘removal’ of the air from the surface reducing the pressure and creating a pressure gradient in the vertical direction. Thus a pressure gradient is established which ensures that the flow follows the shape of the surface. Colloquially one might say that the pressure gradient its created by the centrifugal force of the air flowing around the surface. From the curvature of the streamlines the pressure gradients and consequently the distribution of pressure of the surface of an airfoil may be derived.
At the upper surface the pressure going outward must grow. Since we have normal pressure in a greater distance we have lower pressure at the surface.
Furthermore a consequence of the lower pressure at the upper surface is the positive tangential acceleration of the incoming air. The problem of how to explain the faster motion of the air at the upper surface is now solved. It is the lower pressure that makes the air accelerate and flow faster.
5. Notes on the origin of the conventional explanation
More than a hundred years ago Otto Lilienthal [13] explained the aerodynamical lift correctly and clearly. He compared a curved plane with a flat plane. He referred to figure 6 and wrote ‘The air passing the planes is accelerated downwards in both cases. The air below has to go down and the air passing the upper side has to fill the space above. The deflection of the air stream downwards happens abruptly at the front edge of an even plane. This gives rise to turbulence and vortices. It is different with the curved plane. The air flow passing the front edge will be deflected gradually from its horizontal direction and led downward. The flow gains a horizontal velocity component without any sudden impact. It is clear that only the curved plane - provided its direction at the front edge parallels the original direction of the flow - will divert the air stream downwards with less turbulence in a direction which is given by the tangent at the trailing edge of the plane. The vertical momentum of the air stream makes for the upward force acting on the airfoil.’
The explanation based on the relation between aerodynamic lift and the acceleration of a downward air flow prevailed in textbooks in this simple form until 1920 without having been elaborated further. By approximately the year 1920, when aviation gained much interest in science and public, the explanation based on Bernoulli’s law appeared and displaced the explanation based on reaction forces.
In any case it was necessary that the explanation of lift using Bernoulli’s law had to be complemented by giving a cause for the higher streaming velocity of the wing’s upper surface.
Thus the origin of the path length reasoning may be found in a diagram given by Prandtl in 1922 [14] figure 7.
Dotted lines connect air volumes originally adjacent. With this diagram Prandtl tried to show that air at the inner layers stick to the surface. In respect to one point this diagram is not correct. It indicates that the volumes remain at the same vertical position and it indicates that originally adjacent air meets again at the end of the airfoil. This diagram miss the phase shift shown in figure 2, which seems not to have been observed by Prandtl at that time.
Diagrams of this type might have misled scolars to the hypothesis that adjacent air has to meet again after passing the airfoil.
6. The flow and the system of vortices
Regarding the total flow of a streaming around an airfoil we have to add details. If the airfoil generates low pressure at its upper side and high pressure at its lower side this causes lateral movements rotating to the ends of the wing. Below the wing air moves outwards and above the wing air moves inwards. Beyond the ends of the airfoil air moves even upwards. Thus a system of vortices is generated figure 10.
The vortices behind the wing are directed clockwise at one side and counterclockwise at the other side. The system of vortices is of a remarkable stability and moves downward as a whole.
7. Summary
The conventional explanation of aerodynamical lift based on Bernoulli’s law and velocity differences mixes up cause and effect. The faster flow at the upper side of the wing is the consequence of low pressure and not its cause.
The generation of lift by an airfoil can be explained correctly and simply taking the downward acceleration of air into consideration. This approach allows to derive the dependency of lift from angle of attack, flow velocity and the air’s density in a streightforward and coherent way.
A detailed explanation of the generation of pressure differences is possible if the normal acceleration of streaming air is taken into consideration.
8. Literature
[1] Smith, N.F.: Bernoulli and Newton and Fluidmechanics, in: Physics Teacher, 1972, 10, S. 451-455
[2] Fletcher, N.H: Mechanics of Flight, in: Physics Education , 1975, S. 385-389
[3] Weltner, K.: A comparison of explanations of aerodynamical lifting force, in: American Journal of Physics, 1987, Vol. 55, No. 1, pp. 50-54
[4] Baumann, R.; Schwaneberg, R.: Interpretation of Bernoulli’s Equation, The Physics Teacher, Vol. 32, Nov. 1994, pp. 478 - 488
[5] Niermann, K.: Darstellung der Aerodynamik in Schulphysikbüchern, Alsbach/Bergstraße, 1989
[6] Halliday-Resnick: Physics, New York, 1976
[7] Mansfield, M.; O’Sullivan, C.: Understanding Physics, Chichester/ New York 1998
[8] Bergmann/Schäfer: "Lehrbuch der Experimentalphysik" Band 1, Mechanik, Akustik, Wärme, 10. Auflage, Berlin 1990
[9] Cutnell, Johnson: "Physics", New York, Chichester, 1998
[10] Tuckenbrodt, E.; Schlichting, H.: "Aerodynamik des Flugzeuges", Grundlagen as der Strömungsmechanik Aerodynamik des Tragflügels (Teil I), Berlin/Heidelberg/New York, 1967
[11] Weltner, K.: Aerodynamic Lifting Force, In: The Physics Teacher, New York, 1990, No. 2, Vol. 28, pp. 78-82
[12] Weltner, K.: Misinterpretation of Bernoulli’s law, submitted to European Journal of Physics.
[13] Lilienthal, O.: "Der Vogelflug als Grundlage der Fliegekunst. Ein Beitrag zur Systematik der Flugtechnik", Berlin, 1889.
[14] Prandtl, L.: Applications of modern Hydrodynamics to Aeronautics, in: NACA Report, 1921, 116, pp. 161-182
