Flying Supersonic

Last time (OK, quite a while ago actually), I explained the basic principle (from the Newtonian end of things; we can explain it using pressure, but that’s more complicated) of how wings generate lift when travelling at subsonic speeds, arguably the most important principle of physics affecting our modern world. However, as the second World War came to an end and aircraft started to get faster and faster, problems started to appear.

The first aircraft to approach the speed of sound (Mach 1, or around 700-odd miles an hour depending on air pressure) were WWII fighter aircraft; most only had top speeds of around 400-500mph or so whilst cruising, but could approach the magic number when going into a steep dive. When they did so, they found their aircraft began suffering from severe control issues and would shake violently; there are stories of Japanese Mitsubishi Zeroes that would plough into the ground at full speed, unable to pull out of a deathly transonic dive. Subsequent aerodynamic analyses of these aircraft suggest that if any of them had  in fact broken the sound barrier, their aircraft would most likely have been shaken to pieces. For this reason, the concept of ‘the sound barrier’ developed.

The problem arises from the Doppler effect (which is also, incidentally, responsible for the stellar red-shift that tells us our universe is expanding), and the fact that as an aircraft moves it emits pressure waves, carried through the air by molecules bumping into one another. Since this exactly the same method by which sound propagates in air, these pressure waves move at the speed of sound, and travel outwards from the aircraft in all directions. If the aircraft is travelling forwards, then each time it emits a pressure wave it will be a bit further forward than the centre of the pressure wave it emitted last, causing each wave in front of the aircraft to get closer together and waves behind it to spread out. This is the Doppler Effect.

Now, when the aircraft starts travelling very quickly, this effect becomes especially pronounced, wave fronts becoming compressed very close to one another. When the aircraft is at the speed of sound, the same speed at which the waves propagate, it catches up with the wave fronts themselves and all wave fronts are in the same place just in front of the aircraft. This causes them to build up on top of one another into a band of high-pressure air, which is experienced as a shockwave; the pressure drop behind this shockwave can cause water to condense out of the air and is responsible for pictures such as these.

But the shockwave does not just occur at Mach 1; we must remember that the shape of an aerofoil is such to cause air to travel faster over the top of the wing than it does normally. This means parts of the wing reach supersonic speeds, effectively, before the rest of the aircraft, causing shockwaves to form over the wings at a lower speed. The speed at which this first occurs is known as the critical Mach number. Since these shockwaves are at a high-pressure, then Bernoulli’s principle tells us they cause air to slow down dramatically; this contributes heavily to aerodynamic drag, and is part of the reason why such shockwaves can cause major control issues. Importantly, we must note that shockwaves always cause air to slow down to subsonic speeds, since the shockwave is generated at the point of buildup of all the pressure waves so acts as a barrier between the super- and sub-sonic portions of the airflow. However, there is another problem with this slowing of the airflow; it causes the air to have a higher pressure than the supersonic air in front of the shockwave. Since there is always a force from high pressure to low pressure, this can cause (at speeds sufficiently higher above the critical Mach number) parts of the airflow close to the wing (the boundary layer, which also experience surface friction from the wing) to change direction and start travelling forwards. This causes the boundary layer to recirculate, forming a turbulent portion of air that generates very little lift and quite a lot of drag, and for the rest of the airflow to separate from the wing surface; an effect known as boundary layer separation, (or Mach stall, since it causes similar problems to a regular stall) responsible for even more problems.

The practical upshot of all of this is that flying at transonic speeds (close to and around the speed of sound) is problematic and inefficient; but once we push past Mach 1 and start flying at supersonic speeds, things change somewhat. The shockwave over the wing moves to its trailing edge, as all of the air flowing over it is now travelling at supersonic speeds, and ceases to pose problems, but now we face the issues posed by a bow wave. At subsonic speeds, the pressure waves being emitted by the aircraft help to push air out of the way and mean it is generally deflected around the wing rather than just hitting it and slowing down dramatically; but at subsonic speeds, we leave those pressure waves behind us and we don’t have this advantage. This means supersonic air hits the front of the air and is slowed down or even stopped, creating a portion of subsonic air in front of the wing and (you guessed it) another shockwave between this and the supersonic air in front. This is known as a bow wave, and once again generates a ton of drag.

We can combat the formation of the wing by using a supersonic aerofoil; these are diamond-shaped, rather than the cambered subsonic aerofoils we are more used to, and generate lift in a different way (the ‘skipping stone’ theory is actually rather a good approximation here, except we use the force generated by the shockwaves above and below an angled wing to generate lift). The sharp leading edge of these wings prevents bow waves from forming and such aerofoils are commonly used on missiles, but they are inefficient at subsonic speeds and make takeoff and landing nigh-on impossible.

The other way to get round the problem is somewhat neater; as this graphic shows, when we go past the speed of sound the shockwave created by the aeroplane is not flat any more, but forms an angled cone shape- the faster we go, the steeper the cone angle (the ‘Mach angle’ is given by the formula sin(a)=v/c, for those who are interested). Now, if we remember that shockwaves cause the air behind them to slow down to subsonic speeds, it follows that if our wings lie just behind the shockwave, the air passing over them at right angles to the shockwave will be travelling at subsonic speeds, and the wing can generate lift perfectly normally. This is why the wings on military and other high-speed aircraft (such as Concorde) are ‘swept back’ at an angle; it allows them to generate lift much more easily when travelling at high speeds. Some modern aircraft even have variable-sweep wings (or ‘swing wings’), which can be pointed out flat when flying subsonically (which is more efficient) before being tucked back into a swept position for supersonic flight.

Aerodynamics is complicated.

There is an art, or rather, a knack, to flying…

The aerofoil is one of the greatest inventions mankind has come up with in the last 150 years; in the late 19th century, aristocratic Yorkshireman (as well as inventor, philanthropist, engineer and generally quite cool dude) George Cayley identified the way bird wings generated lift merely by moving through the air (rather than just by flapping), and set about trying to replicate this lift force. To this end, he built a ‘whirling arm’ to test wings and measure the upwards lift force they generated, and found that a cambered wing shape (as in modern aerofoils) similar to that of birds was more efficient at generating lift than one with flat surfaces. This was enough for him to engineer the first manned, sustained flight, sending his coachman across Brompton Dale in 1863 in a homemade glider (the coachman reportedly handed in his notice upon landing with the immortal line “I was hired to drive, not fly”), but he still didn’t really have a proper understanding of how his wing worked.

Nowadays, lift is understood better by both science and the general population; but many people who think they know how a wing works don’t quite understand the full principle. There are two incomplete/incorrect theories that people commonly believe in; the ‘skipping stone’ theory and the ‘equal transit time’ theory.

The ‘equal transit time’ theory is popular because it sounds very sciency and realistic; because a wing is a cambered shape, the tip-tail distance following the wing shape is longer over the top of the wing than it is when following the bottom surface. Therefore, air travelling over the top of the wing has to travel further than the air going underneath. Now, since the aircraft is travelling at a constant speed, all the air must surely be travelling past the aircraft at the same rate; so, regardless of what path the air takes, it must take the same time to travel the same lateral distance. Since speed=distance/time, and air going over the top of the wing has to cover a greater distance, it will be travelling faster than the air going underneath the wing. Bernoulli’s principle tells us that if air travels faster, the air pressure is lower; this means the air on top of the wing is at a lower pressure than the air underneath it, and this difference in pressure generates an upwards force. This force is lift.

The key flaw in this theory is the completely wrong assumption that the air over the top and bottom of the wing must take the same time to travel across it. If we analyse the airspeed at various points over a wing we find that air going over the top does, in fact, travel faster than air going underneath it (the reason for this comes from Euler’s fluid dynamics equations, which can be used to derive the Navier-Stokes equations for aerofoil behaviour. Please don’t ask me to explain them). However, this doesn’t mean that the two airflows necessarily coincide at the same point when we reach the trailing edge of the wing, so the theory doesn’t correctly calculate the amount of lift generated by the wing. This is compounded by the theory not explaining any of the lift generated from the bottom face of the wing, or why the angle wing  is set at (the angle of attack) affects the lift it generates, or how one is able to generate some lift from just a flat sheet set at an angle (or any other symmetrical wing profile), or how aircraft fly upside-down.

Then we have the (somewhat simpler) ‘skipping stone’ theory, which attempts to explain the lift generated from the bottom surface of the wing. Its basic postulate concerns the angle of attack; with an angled wing, the bottom face of the wing strikes some of the incoming air, causing air molecules to bounce off it. This is like the bottom of the wing being continually struck by lots of tiny ball bearings, sort of the same thing that happens when a skimming stone bounces off the surface of the water, and it generates a net force; lift. Not only that, but this theory claims to explain the lower pressure found on top of the wing; since air is blocked by the tilted wing, not so much gets to the area immediately above/behind it. This means there are less air molecules in a given space, giving rise to a lower pressure; another way of explaining the lift generated.

There isn’t much fundamentally wrong with this theory, but once again the mathematics don’t check out; it also does not accurately predict the amount of lift generated by a wing. It also fails to explain why a cambered wing set at a zero angle of attack is still able to generate lift; but actually it provides a surprisingly good model when we consider supersonic flight.

Lift can be explained as a combination of these two effects, but to do so is complex and unnecessary  we can find a far better explanation just by considering the shape the airflow makes when travelling over the wing. Air when passing over an aerofoil tends to follow the shape of its surface (Euler again), meaning it deviates from its initially straight path to follow a curved trajectory. This curve-shaped motion means the direction of the airflow must be changing; and since velocity is a vector quantity, any change in the direction of the air’s movement represents a change in its overall velocity, regardless of any change in airspeed (which contributes separately). Any change in velocity corresponds to the air being accelerated, and since Force = mass x acceleration this acceleration generates a net force; this force is what corresponds to lift. This ‘turning’ theory not only describes lift generation on both the top and bottom wing surfaces, since air is turned upon meeting both, but also why changing the angle off attack affects lift; a steeper angle means the air has to turn more when following the wing’s shape, meaning more lift is generated. Go too steep however, and the airflow breaks away from the wing and undergoes a process called flow separation… but I’m getting ahead of myself.

This explanation works fine so long as our aircraft is travelling at less than the speed of sound. However, as we approach Mach 1, strange things start to happen, as we shall find out next time…

The Conquest of Air

Everybody in the USA, and in fact just about everyone across the world, has heard of Orville and Wilbur Wright. Two of the pioneers of aviation, when their experimental biplane Flyer achieved the first ever manned, powered, heavier-than-air flight on the morning of December 17, 1903, they had finally achieved one of man’s long-held dreams; control and mastery of air travel.

However, what is often puzzling when considering the Wright brothers’ story is the number of misconceptions surrounding them. Many, for instance, are under the impression that they were the first people to fly at all, inventing all the various technicalities of lift, aerofoil structures and control that are now commonplace in today’s aircraft. In fact, the story of flight, perhaps the oldest and maddest of human ambitions, an idea inspired by every time someone has looked up in wonder at the graceful flight of a bird, is a good deal older than either of them.

Our story begins, as does nearly all technological innovation, in imperial China, around 300 BC (the Greek scholar Archytas had admittedly made a model wooden pigeon ‘fly’ some 100 years previously, but nobody is sure exactly how he managed it). The Chinese’s first contribution was the invention of the kite, an innovation that would be insignificant if it wasn’t for whichever nutter decided to build one big enough to fly in. However, being strapped inside a giant kite and sent hurtling skywards not only took some balls, but was heavily dependent on wind conditions, heinously dangerous and dubiously useful, so in the end the Chinese gave up on manned flight and turned instead to unmanned ballooning, which they used for both military signalling and ceremonial purposes. It isn’t actually known if they ever successfully put a man into the air using a kite, but they almost certainly gave it a go. The Chinese did have one further attempt, this time at inventing the rocket engine, some years later, in which a young and presumably mental man theorised that if you strapped enough fireworks to a chair then they would send the chair and its occupants hurtling into the night sky. His prototype (predictably) exploded, and it wasn’t for two millennia, after the passage of classical civilisation, the Dark Ages and the Renaissance, that anyone tried flight again.

That is not to say that the idea didn’t stick around. The science was, admittedly beyond most people, but as early as 1500 Leonardo da Vinci, after close examination of bird wings, had successfully deduced the principle of lift and made several sketches showing designs for a manned glider. The design was never tested, and not fully rediscovered for many hundreds of years after his death (Da Vinci was not only a controversial figure and far ahead of his time, but wrote his notebooks in a code that it took centuries to decipher), but modern-day experiments have shown that his design would probably have worked. Da Vinci also put forward the popular idea of ornithopters, aircraft powered by flapping motion as in bird wings, and many subsequent attempts at flight attempted to emulate this method of motion. Needless to say, these all failed (not least because very few of the inventors concerned actually understood aerodynamics).

In fact, it wasn’t until the late 18th century that anyone started to really make any headway in the pursuit of flight. In 1783, a Parisian physics professor, Jacques Charles, built on the work of several Englishmen concerning the newly discovered hydrogen gas and the properties and behaviour of gases themselves. Theorising that, since hydrogen was less dense than air, it should follow Archimedes’ principle of buoyancy and rise, thus enabling it to lift a balloon, he launched the world’s first hydrogen balloon from the Champs du Mars on August 27th. The balloon was only small, and there were significant difficulties encountered in building it, but in the design process Charles, aided by his engineers the Roberts brothers, invented a method of treating silk to make it airtight, spelling the way for future pioneers of aviation. Whilst Charles made some significant headway in the launch of ever-larger hydrogen balloons, he was beaten to the next significant milestones by the Montgolfier brothers, Joseph-Michel and Jacques-Etienne. In that same year, their far simpler hot-air balloon designs not only put the first living things (a sheep, rooster and duck) into the atmosphere, but, just a month later, a human too- Jacques-Etienne was the first European, and probably the first human, ever to fly.

After that, balloon technology took off rapidly (no pun intended). The French rapidly became masters of the air, being the first to cross the English Channel and creators of the first steerable and powered balloon flights. Finally settling on Charles’ hydrogen balloons as a preferable method of flight, blimps and airships began, over the next century or so, to become an accepted method of travel, and would remain so right up until the Hindenburg disaster of 1937, which rather put people off the idea. For some scientists and engineers, humankind had made it- we could now fly, could control where we were going at least partially independent of the elements, and any attempt to do so with a heavier-than-air machine was both a waste of time and money, the preserve of dreamers. Nonetheless, to change the world, you sometimes have to dream big, and that was where Sir George Cayley came in.

Cayley was an aristocratic Yorkshireman, a skilled engineer and inventor, and a magnanimous, generous man- he offered all of his inventions for the public good and expected no payment for them. He dabbled in a number of fields, including seatbelts, lifeboats, caterpillar tracks, prosthetics, ballistics and railway signalling. In his development of flight, he even reinvented the wheel- he developed the idea of holding a wheel in place using thin metal spokes under tension rather than solid ones under compression, in an effort to make the wheels lighter, and is thus responsible for making all modern bicycles practical to use. However, he is most famous for being the first man ever, in 1853, to put somebody into the air using a heavier-than-air glider (although Cayley may have put a ten-year old in a biplane four years earlier).

The man in question was Cayley’s chauffeur (or butler- historical sources differ widely), who was (perhaps understandably) so hesitant to go in his boss’ mental contraption that he handed in his notice upon landing after his flight across Brompton Dale, stating  as his reason that ‘I was hired to drive, not fly’. Nonetheless, Cayley had shown that the impossible could be done- man could fly using just wings and wheels. He had also designed the aerofoil from scratch, identified the forces of thrust, lift, weight and drag that control an aircraft’s movements, and paved the way for the true pioneer of ‘heavy’ flight- Otto Lilienthal.

Lilienthal (aka ‘The Glider King’) was another engineer, making 25 patents in his life, including a revolutionary new engine design. But his fame comes from a world without engines- the world of the sky, with which he was obsessed. He was just a boy when he first strapped wings to his arms in an effort to fly (which obviously failed completely), and later published works detailing the physics of bird flight. It wasn’t until 1891, aged 43, once his career and financial position was stable and he had finished fighting in the Franco-Prussian War, that he began to fly in earnest, building around 12 gliders over a 5-year period (of which 6 still survive). It might have taken him a while, but once he started there was no stopping him, as he made over 2000 flights in just 5 years (averaging more than one every day). During this time he was only able to rack up 5 hours of flight time (meaning his average flight time was just 9 seconds), but his contribution to his field was enormous. He was the first to be able to control and manoeuvre his machines by varying his position and weight distribution, a factor whose importance he realised was absolutely paramount, and also recognised that a proper understanding of how to achieve powered flight (a pursuit that had been proceeding largely unsuccessfully for the past 50 years) could not be achieved without a basis in unpowered glider flight, in recognising that one must work in harmony with aerodynamic forces. Tragically, one of Lilienthal’s gliders crashed in 1896, and he died after two days in hospital. But his work lived on, and the story of his exploits and his death reached across the world, including to a pair of brothers living in Dayton, Ohio, USA, by the name of Wright. Together, the Wright brothers made huge innovations- they redesigned the aerofoil more efficiently, revolutionised aircraft control using wing warping technology (another idea possibly invented by da Vinci), conducted hours of testing in their own wind tunnel, built dozens of test gliders and brought together the work of Cayley, Lilienthal, da Vinci and a host of other, mostly sadly dead, pioneers of the air.  The Wright brothers are undoubtedly the conquerors of the air, being the first to show that man need not be constrained by either gravity or wind, but can use the air as a medium of travel unlike any other. But the credit is not theirs- it is a credit shared between all those who have lived and died in pursuit of the dream of fling like birds. To quote Lilienthal’s dying words, as he lay crippled by mortal injuries from his crash, ‘Sacrifices must be made’.