Aerodynamics demystified

I know: you don’t believe me; but you haven’t read Henk Tennekes’s book, The Simple Science of Flight; From Insects to Jumbo Jets. This is a book that should be taught in high school, not to get people into the aerospace industry, but to demonstrate how science works in the real world. It is probably the best example I’ve come across, ever.

I admit, I have an advantage with this book, because I have an engineering background, but the truth is that anyone, with a rudimentary high school education in mathematics, should be able to follow this book. By rudimentary, I mean you don’t need to know calculus, just how to manipulate basic equations. Aerodynamics is one of the most esoteric subjects on the planet – all the more reason that Tennekes’s book should be part of a high school curriculum. It demonstrates the availability of science to the layperson better than any book I’ve read on a single subject.

Firstly, you must appreciate that mathematics is about the relationship between numbers rather than the numbers themselves. This is why an equation can be written without any numbers at all, but with symbols (letters of the alphabet) representing numbers. The numbers can have any value as long as the relationship between them is dictated by the equation. So, for an equation containing 3 symbols, if you know 2 of the values, you can work out the third. Elementary, really. To give an example from Tennekes’s book:

W/S=0.38 V²

Where W is the weight of the flying object (in Newtons), S is the area of the wing (square metres) and V is cruising speed (metres per second). 0.38 is a factor dependent on the angle of attack of the wing (average 6 degrees) and the density of the medium (0.3125 kg/m3; air at sea level). What Tennekes reveals graphically is that you can apply this equation to everything from a fruit fly (Drosophila melanogaster) to an Airbus A380 on what he calls The Great Flight Diagram (see bottom of post). (Mind you, his graph is logarithmic along both axes, but that’s being academic, quite literally.)

I’ve used a small sleight-of-hand here, because the equation for the graph is actually:

W/S = c x W^1/3

W/S (weight divided by wing area which gives pressure) is called ‘wing loading’ and is proportional to the cubed root of the Weight, which is a direct consequence of the first equation (that I haven’t explained, though Tennekes does). Tennekes’s ‘Great Flight Diagram’ employs the second equation, but gives V (flight cruise speed) as one of the axes (horizontal) against Weight (vertical axis); both logarithmic as I said. At the risk of confusing you, the second equation graphs better (it gives a straight line on a logarithmic scale) but the relationships of both equations are effectively entailed in the one graph, because W, W/S and V can all be read from it.

I was amazed that one equation could virtually cover the entire range of flight dynamics for winged objects on the planet. The equations also effectively explain the range of flight dynamics that nature allows to take place. The heavier something is, the faster it has to fly to stay in the air, which is why 747s consistently fly 200 times faster than fruit flies. The equation shows that there is a relationship between weight, wing area and air speed at all scales, and while that relationship can be stretched it has limits. Flyers (both natural and artificial) left of the curve are slow for their size and ones to the right are fast for their size – they represent the effective limits. (A line on a graph is called a ‘curve’, even if it’s straight, to distinguish it from a grid-line.) So a side-benefit of the book is that it provides a demonstration of how mathematics is not only a tool of analysis, but how it reveals nature’s limits within a specific medium – in this case, air in earth’s gravitational field. It reminded me of why I fell in love with physics when I was in high school – nature’s secrets revealed through mathematics.

The iconic Supermarine Spitfire is one of the few that is right on the curve, but, as Tennekes points out, it was built to climb fast as an interceptor, not for outright speed.

Now, for those who know more about this subject than I do, they may ask: what about Reynolds numbers? Well, I know Reynolds numbers are used by aeronautical engineers to scale up aerodynamic phenomena from small scale models they use in wind tunnels to full scale aeroplanes. Tennekes conveniently leaves this out, but then he’s not explaining how we use models to provide data for their full scale equivalents – he’s explaining what happens at full scale no matter what the scale is. So speed increases with weight and therefore scale – we are not looking for a conversion factor to take us from one scale to another, which is what Reynolds numbers do. (Actually, there’s a lot more to Reynolds numbers than that, but it’s beyond my intellectual ken.) I’m not an aeronautical engineer, though I did work in a very minor role on the design of a wind tunnel once. By minor role, I took minutes of the meetings held by the real experts.

When I was in high school, I was told that winged flight was all explained by the Bernoulli effect, which Tennekes describes as a ‘polite fiction’. So, that little gem alone, makes Tennekes’s book a worthwhile addition to any school’s science library.

But the real value in this book comes when he starts to talk about migrating birds and the relationship between energy and flight. Not only does he compare aeroplanes with other forms of transport, thus explaining why flight is the most economical means of travel over long distances, as nature has already proven with birds, but he analyses what it takes for the longest flying birds to achieve their goals, and how they live at the limit of what nature allows them to do. Again, he uses mathematics, that the reader can work out for themselves, to convert calories from food into muscle power into flight speed and distance, to verify that the very best traveled migratory birds don’t cheat nature, but live at its limits.

The most extraordinary example being bar-tailed godwits that fly across the entire Pacific Ocean from Alaska to New Zealand and to Australia’s Northern Territory – a total of 11,000 km non-stop (7,000 miles). It’s such a feat that Tennekes claims it requires a rethink on the metabolic efficiency of the muscles of these birds, and he provides the numbers to support his argument. He also explains how birds can convert fat directly into energy for muscles, something we can’t do (we have to convert it into sugar first). He also explains how some migratory birds even start to atrophy their wing muscles and heart muscles to extend their trip – they literally burn up their own muscles for fuel.

So he combines physics with biology with zoology with mathematics, all in one chapter, on one specific subject: bird migration.

He uses another equation, along with a graphic display of vectors, that explains how flapping wings work on exactly the same principle as ice skating in humans. What’s more, he doesn’t even tell the reader that he’s working with vectors, or use trigonometry to explain it, yet anyone would be able to understand the connection. That’s just brilliant exposition.

In a nutshell (without the diagrams) power equals force times speed: P=FV. For the same amount of Power, you can have a large Force and small Velocity or the converse.

In other words, a large force times a small velocity can be transformed into a small force with a large velocity, with very little energy loss if friction is minimalised. This applies to both skaters and birds. The large force, in skating, is your leg pushing sideways against your skate, with a small sideways velocity, resulting in a large velocity forwards, from a small force on the skate backwards. Because the skate is at a slight angle, the force sideways (from your leg) is much greater than the force backwards, but it translates into a high velocity forwards.

The same applies to birds on their downstroke: a large force vertically, at a slight forward angle, gives a higher velocity forward. Tennekes says that the ratio of wing tip velocity to forward velocity for birds is typically 1 to 3, though varies between 2 and 4. If a bird wants to fly faster, they don’t flap quicker, they increase the amplitude, which, at the same frequency, increases wing tip speed, which increases forward flight speed. Simple, isn’t it? The sound you hear when pigeons or doves take off vertically is there wing tips actually touching (on both strokes). Actually, what you hear is the whistle of air escaping the closed gap, as a continuous chirp, which is their flapping frequency. So when they take off, they don’t double their wing flapping frequency, they double their wing flapping amplitude, which doubles their wing tip speed at the same frequency: the wing tip has to travel double the distance in the same time.

One possible point of confusion is a term Tennekes uses called ‘specific energy consumption’, which is a ratio, not an amount of energy as its description implies. It is used to compare energy consumption or energy efficiency between different birds (or planes), irrespective of what units of energy one uses. The inversion of the ratio gives the glide ratio (for both birds and planes) or what the French call ‘Finesse’ – a term that has special appeal to Tennekes. So a lower energy consumption gives a longer guide ratio, or vice versa, as one would expect.

Tennekes finally gets into esoteric territory when he discusses drag and vortices, but he’s clever enough to perform an integral without introducing his readers to calculus. He’s even more clever when he derives an equation based on vortices and links it back to the original equation that I referenced at the beginning of this post. Again, he’s demonstrating how mathematics keeps us honest. To give another, completely unrelated example: if Einstein’s general theory of relativity couldn’t be linked to Newton’s general equation of gravity, then Einstein would have had to abandon it. Tennekes does exactly the same thing for exactly the same reason: to show that his new equation agrees with what has already been demonstrated empirically. Although it’s not his equation, but Ludwig Prandtl’s, whom he calls the ‘German grandfather of aerodynamics’.

Prandtl based his equation on an analogy with electromagnetic induction, which Tennekes explains in some detail. They both deal with an induced phenomenon that occurs in a circular loop perpendicular to the core axis. Vortices create drag, but in aerodynamics it actually goes down with speed, which is highly counterintuitive, but explains why flight is so economical compared to other forms of travel, both for birds and for planes. The drag from vortices is called ‘induced’ drag, not to be confused with ‘frictional’ drag that does increase with air speed, so at some point there is an optimal speed, and, logically, Tennekes provides the equation that gives us that as well. He also explains how it’s the vortices from wing tips that cause many long distance flyers, like geese and swans, to fly in V formation. The vortex supplies an updraft just aft and adjacent to the wingtip that the following bird takes advantage of.

Tennekes uses his equations to explain why human-powered flight is the reserve of professional cyclists, and not a recreational sport like hang-gliding or conventional gliding. Americans apparently use the term, sailplane, instead of glider, and Tennekes uses both without explaining he’s referring to the same thing.

Tennekes reveals that his doctoral thesis (in 1964) critiqued the Concorde (still on the drawing board back then) as ‘a step backward in the history of aviation.’ This was considered heretical at the time, but not now, as history has demonstrated to his credit.

The Concorde is now given as an example, in psychology, of how humans are the only species that don’t know when to give up (called the ‘Concorde effect’). Unlike other species, humans evaluate the effort they’ve put into an endeavour, and sometimes, the more effort they invest, the more determined they become to succeed. Whether this is a good or bad trait is purely subjective, but it can evolve into a combination of pride, egotism and even denial. In the case of the Concorde, Tennekes likens it to a manifestation of ‘megalomania’, comparable to Howard Hughes’ infamous Spruce Goose.

Tennekes’s favourite plane is the Boeing 747, which is the complete antithesis to the Concorde, in evolutionary terms, and developed at the same time; apparently so it could be converted to a freight plane when supersonic flight became the expected norm. So, in some respects, the 747, and its successors, were an ironic by-product of the Concorde-inspired thinking of the time.

My only criticism of Tennekes is that he persistently refers to a budgerigar as a parakeet. This is parochialism on my part: in Australia, where they are native, we call them budgies.