Bernoulli's Principle and Airplane Aerodynamics

A Critical Analysis

The aerodynamic lift on the wing of an airplane (airfoil) is generally explained by the argument that the faster speed of the air along the top of the wing leads to reduced air pressure there and hence produces a lift (Bernoulli's Law). Using this argument, one should also expect a lift for a symmetric wing profile as shown in Fig.1.

Symmetric wing profile   Fig.1

However, if one considers the problem from a microscopic point of view, one comes to a different conclusion: upward and downward forces should exactly cancel for a symmetric wing profile. This is easy to see if one simplifies the situation and replaces the curved wing surface by two plane sections (Fig.2)

Schematic illustration of symmetric wing profile   Fig.2

If the wing is stationary, the pressure on all parts of the wing is identical, i.e. there is no lift. If the wing is moving in the indicated direction and assuming an inviscid gas, the front half of the upper wing surface experiences an increased pressure because of the increased speed and number of air molecules hitting it (due to the orientation of the surface, this creates a downward force). On the other hand, the rear half experiences a reduced pressure because the of the reduced speed and number of air molecules hitting it (creating a lift) (for a more detailed theoretical analysis of this see the page regarding aerodynamic drag and lift). Overall, there is consequently no lift, but only an anti-clockwise torque. It is obvious that an overall lift is only achieved if the rear section of the wing has a larger area than the front section, i.e. one would get the maximum lift for the following profile (Fig.3)

Schematic illustration of wing profile yielding lift   Fig.3

and wing profiles are actually asymmetric in this sense (see for instance
On the other hand, the reverse situation (Fig.4) should lead to a downward force, although Bernoulli's Law would again predict a lift.

Schematic illustration of wing profile yielding downward force   Fig.4

Note: the above arguments assume that the lower surface of the wing is always parallel to the velocity vector, i.e. the pressure acting on it is unchanged; by varying the 'angle of attack' of the wing the amount of lift can of course be changed arbitrarily and one could even generate a lift for the bottom image (Fig.4).

In any case, it is clear that an airflow parallel to a surface can not transfer any momentum to it and therefore not exert any force on it. This invalidates Bernoulli's equation as an explanation for the aerodynamic lift. The enhanced airflow speed around certain sections of the wing is not the cause of the aerodynamic lift, but both the lift and the speed enhancement are separate consequences of the pressure changes at the different wing sections caused by the motion of the wing in the viscous air.
In this way one has also to interpret the frequently given example of blowing over a piece of paper. In fact, if one puts a sheet of paper flat on a table, fixes it to the edge of the table and blows over it from the edge, the paper will not lift by one millimeter, despite the motion of the air which according to Bernoulli's law should cause an underpressure.
The apparent attraction that is observed when blowing between two sheets of paper can be either explained by the fact that the sheets are in fact not exactly parallel to the airflow but bend away from it (hence reducing the pressure on the surface), or by the circumstance that the airflow does not cover the whole width of the paper (which leads to the stationary molecules being pulled into the airstream by means of friction (viscosity), which again reduces the pressure because molecules are removed from between the sheets; one can verify this by just using two narrow (1cm wide) strips of paper; these show no attraction but tend to stay parallel).
Either of these two mechanism should indeed be responsible for many of the phenomena attributed to Bernoulli's Principle.

It should therefore be obvious that Bernoulli's law is only a viable physical explanation in cases where the viscosity of the medium is instrumental for the considered effect. Contrary to some scientific misconceptions, this is neither the case for the aerodynamic lift associated with airplanes nor for the drag of objects moving through a medium (see my separate page regarding aerodynamic drag and lift for a more detailed theoretical analysis of these issues).

Note 1: it is frequently claimed by other critics of Bernoulli's law in this context, that Newton's law of action and reaction in connection with the observed 'downwash' of air near the wing is the explanation for the lift force acting on an airplane (see Ref. 1, Ref. 2). This view has to be rejected as well: this is not a problem of an action at a distance, but the only way a force can be exerted on a wing is by an increase of the number and speed of air molecules hitting the wing surface. Everything else, including the 'downwash', is merely a consequence of this, not the cause (some people argue that this cause/effect issue would be merely a semantic problem, but a) it is worrying if physicists don't care any more about the correct chain of cause and effect and b) the above examples show clearly that this can indeed be of practical relevance).

Note 2: In contrast to the usual aerodynamic lift, the well known Magnus effect due to the rotation of objects does however not exist in an inviscid gas and can only be explained in terms of Bernoulli's principle:
consider a rotating ball that is moving through an inviscid gas (i.e. molecules interacting with the ball but not with each other): if the surface of the ball would be mathematically smooth, then the rotation would actually be without any effect at all because the air molecules would just bounce off like for a non-rotating sphere, but even for a realistic rough surface (obviously a surface can not be smoother than about 1 atomic radius), the overall effect still cancels to zero: the pressure on the side rotating against the airstream is higher at the front but smaller at the back (and the other way around for the co-rotating side) so overall there is no resultant force on the ball but merely a torque that slows down the rotation.
Hydrodynamics arguments (i.e. Bernoulli's principle) are therefore required to explain the Magnus effect but not for the aerodynamic lift.

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Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)
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