Boeing 247-D (1934)

Heavier than air machines don't fly.

Odd how even intelligent people can make statements which fly in the face of their experience, maybe because of the compartmentalisation of knowledge. We all do it; it is one of the problems that science has with scientists, and technology with technologists. The sub-head statement was the opinion voiced when the Wright brothers and Bleriot and anyone else who was trying to build aircraft were apparently trying to fly  in the face of physical law. But heavier than air machines had been around for aeons, flying around with apparently no concern for human prejudice. Birds. Evidently heavier than air, since they could be, and had been for the same number of aeons, shot down. I would prefer the term denser than air, but I am accused too often of pedantry so will not air this further.

The basis of flight, or at least the description of it which we call 'physical law', had also been around for a long time before Wright, Bleriot et al. The description is due to Daniel Bernoulli.

The Bernouilli family.

The Bernoulli family was a family of mathematical prodigies. The three best known, though by no means the only, members are Jacques (27th December 1654 - 10th August 1705), Jean (7th August 1667 - 1st January 1748),  and Daniel, Jean's son (8th February 1700 - 17th March 1782). Jean and Jacques were born in Basle (Basel), and all three died there; Daniel was born in Groningen in the Netherlands. The Bernoullis were from a wealthy Antwerpen family which had moved to Switzerland from Holland in the 16th century.

All three were characteristic of their time in that they were natural philosophers. The specialisation which we regard as more or less normal nowadays was not then in evidence. Thus although Jacques was essentially a mathematician, both his brother and his nephew began their careers with the study of medicine. Daniel received his doctorate for a thesis on the action of the lungs (a topic addressed by Robert Hooke - an Old Westminster - somewhat earlier) but was Professor of Mathematics at St Petersburg in 1725. In 1733 he returned to Basle, being successively Professor of Anatomy and Botany, and then of Natural Philosophy until 1777. He worked on probability, partial differential equations, trigonometrical functions, and a number of other topics. The work on fluid dynamics which is relevant to flying was published in Hydrodynamica (1738).

Aerodynamics.

There are lots of books on aerodynamics, and lots of books on physics which deal with aerofoils; the problem is that you have to know whether what you're reading is worth the effort. (You could say the same about this - you could even question my presumption as a Chemist in writing about non-Chemical things. Just think of me as a natural philosopher. Or a plagiarist.) There is a great deal of misinformation about flight - like 'the aerofoil has to be curved' (not true) or the notion that molecules of air that separate at the leading edge of the wing arrive together at the trailing edge - also not true. How would they know? Who would reprimand them were they to behave differently as, indeed, they do. Why not regard the (particulate) stream of air as a stream of bullets hitting the underside of the wing and producing lift? Because it gives incorrect predictions and ignores the critical role of the top surface of the wing, that's why. And if aerofoil curvature is so critical, how come that aircraft can fly upside down? I wondered if a 747 can, even if the airline would lose a few passengers. I asked Boeing via their website;

                                         Me: Can a 747 fly upside-down?
                                         Boeing: Not with me in it!

So I think the answer is probably not, though I am told that one once did a roll as part of a display when the aircraft was first marketed. If you know the answer, please tell. Boeing make many other aircraft which can.

Bernoulli's equation and aerofoils.

Bernoulli's equation describes the flow of an incompressible, non-viscous liquid through a pipe whose bore varies; it is a particular example of the Law of Conservation of Energy. You may be perplexed by the relationship of this to flying; an aerofoil isn't a pipe, and air is both viscous and compressible.

Indeed this has led to plenty of writers to disclaim the relevance of Bernoulli to the problem of 'explaining' flight. This is usually because

• they have not attempted to see whether the deviations from ideal behaviour make any difference
• they have not considered that many physical phenomena can be approached in several ways concerning the physical laws that describe them
• there is misunderstanding about what is meant by a 'Physical Law' in the first place.

        The Bernoulli equation is a special case of the law of conservation of energy. Consider a pipe, carrying an incompressible liquid, which has a constriction in it:

Before you go any further, answer this question: is the pressure in the constriction higher or lower than the pressure outside it? The answers which intuition gives are not always the right ones!

Let the liquid have a density r. The total energy of the liquid is the sum of its potential and kinetic energies, and it is not hard to show that this is given by  p  +  r v². If the total energy of the system is indeed constant, this leads to

p₁  +  r v₁²  =    p₂  +  r v₂²

It is also fairly evident that since the liquid is incompressible, that moving through the smaller bore is travelling faster than that moving through the larger; v₂ > v₁. The result is that the pressure p₂ is lower than the pressure p₁. This may not be the answer you gave earlier, but it is true.

The aerofoil.

The air flowing over a wing is not moving through a pipe in the true sense above, nor is it incompressible. The experimental fact is that it behaves in both respects almost as if it is. The effect on the airflow across the wing is to reduce the pressure above the wing relative to that below it, albeit in a fairly complex manner.

This pressure differential, which is combined also with a downthrust on the air and with a circulatory movement about the wing axis (which gives the wake vortex - see below), appears because the air travelling above the wing is accelerated compared with that below the wing, which is somewhat retarded. This is shown in the figure. I am indebted to 'See how they fly' for this treatment, and I recommend that you look at this excellent and large work - there's a link in the bibliography at the end of this page.

The coloured lines define equal times. The aerofoil has the effect it does only if it is at an angle to the airflow - an aircraft rudder is an aerofoil too, but there's no net force to one side or the other unless it is at an angle to the airstream. Its curvature is not relevant.

As soon as the air approaches the leading edge of the wing, it will go either above or below it. If it is forced above the wing, it effectively enters a small-bore pipe, travels faster, and the pressure falls. The air above the wing travels faster than that below it, as shown be the green lines and the orange lines. Further, the air coming off the trailing edge has a downwards component to its velocity, called the downwash.

The air travelling below the wing has a downwards component imparted to it because of the angle of attack. Contrary to what is supposed, the curvature of the wing is not necessary to produce lift; a flat wing would, but wouldn't be as good because the airflow would be more turbulent.

Bernoulli, then, predicts that there will be a pressure differential between the underside and topside of the wing, giving rise to lift. Some people like to see the lift as a sort of 'bullet' effect, with air hitting the underside of the angled wing forcing it up. This might be fine were it not for the fact that the shape of the top surface does affect the lift - indeed the top surface is often modified with flaps rising on landing.

Circulation.

Bernoulli has been invoked to 'explain' lift, and Newton's Laws have also been used implicitly by suggesting that an upwards force is exerted on the wing by the downwards compnent of the air's velocity. But there's another factor to consider. The acceleration above the wing and the retardation below it can be regarded as a combination of two factors; the flow of air past the wing combined

with a circulatory motion shown in red. The two add above the wing, making the air velocity greater, and subtract below it making it less. So is this circulation real or is it simply a mathematical trick? Ask a pilot of a relatively small aeroplane following a slow-moving large one - like a 747!

The vortex wake from an aircraft, the magnitude of which will affect the intervals between aircraft on approach or departure, can be very serious. It can turn a small aircraft over, and it is one of the factors that has to be borne in mind by Air Traffic Control. The amount of circulation depends on the size of the aircraft and the speed at which it's flying; large, slow aircraft have more circulation than small fast ones.

Circulation can be shown by flying an ordinary business card; hold it with its long edge horizontal, and give it a backwards horizontal twirl. It flies quite well.

There is much more that you can find out - indeed my precis might cause gritted teeth amongst experts, so perhaps you'd better! There is the whole problem of the relationship between lift, thrust, weight and drag, the so-called four forces. This is intended as a taster, not a text. See the bibliography.

The nature of physical law.

A friend of mine, who's a Classicist, once said that he hated the word 'Law' in the way in which it is used by scientists. He didn't see it as something that was 'obeyed by choice' by the Universe - and in any case was often subject to change.

Essentially subject to change, indeed. The nature of physical law is that it is always approximate and subject to experimental refutation; and that it should enable predictions to be made concerning physical systems other than that being investigated.

Aircraft don't obey physical laws; they just fly. Physical laws can be used to describe flight, and it can be used to improve flight, but because natural philosophers like Bernoulli and Newton can write down mathematical equations doesn't mean that the aircraft obeys these laws. A mathematician friend of mine said recently about flight that 'I think you'll find it's a solution to problems in the complex plane...'; no pun was intended. At least I don't think it was! A 747 isn't thinking all the time about how to find the square root of -1. Only people can do that.

This is a complex subject, and involves the odd relationship of mathematics, which is a human acitivity, to the Universe. It's a question which you should pursue if you are interested in science.

Bibliography.

There's a huge number of books on flight and physics; I suggest the following as a starter pack:

• Resnick R & Halliday D, 'Physics', part 1. John Wiley & Sons.  This gives the derivation of the Bernoulli Equation - so do dozens of other texts.
• Tennekes, Henk: 'The Simple Science of Flight'. MIT Press, 1996.  Tennekes uses the original fliers - birds - to illustrate aircraft flight. He's a big 747 fan, too.
• Feynman, Richard P: 'The Nature of Physical Law'. MIT Press.
• And, on the net, chapter 3 (and the rest) of  'See How it Flies'. A long text on flight, which I have used with pleasure, and which is splendidly clear. A must-read if you have found this page amusing.

As a tail-ender, you might speculate whether paint has much effect on flying characteristics....

747 appreciation    747 history    747 statistics    The Jumbo and the Crane Fly    Home Page

Rod Beavon  17 Dean's Yard  London  SW1P  3PB

e-mail:  rod.beavon@westminster.org.uk