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.

Hitting the hay

OK, so it was history last time, so I’m feeling like a bit of science today. So, here is your random question for today; are the ‘leaps of faith’ in the Assassin’s Creed games survivable?

Between them, the characters of Altair, Ezio and Connor* jump off a wide variety of famous buildings and monuments across the five current games, but the jump that springs most readily to mind is Ezio’s leap from the Campanile di San Marco, in St Mark’s Square, Venice, at the end of Assassin’s Creed II. It’s not the highest jump made, but it is one of the most interesting and it occurs as part of the main story campaign, meaning everyone who’s played the game through will have made the jump and it has some significance attached to it. It’s also a well-known building with plenty of information on it.

[*Interesting fact; apparently, both Altair and Ezio translate as ‘Eagle’ in some form in English, as does Connor’s Mohawk name (Ratonhnhaké;ton, according to Wikipedia) and the name of his ship, the Aquila. Connor itself translates as ‘lover of wolves’ from the original Gaelic]

The Campanile as it stands today is not the same one as in Ezio’s day; in 1902 the original building collapsed and took ten years to rebuild. However, the new Campanile was made to be cosmetically (if not quite structurally) identical to the original, so current data should still be accurate. Wikipedia again tells me the brick shaft making up the bulk of the structure accounts for (apparently only) 50m of the tower’s 98.6m total height, with Ezio’s leap (made from the belfry just above) coming in at around 55m. With this information we can calculate Ezio’s total gravitational potential energy lost during his fall; GPE lost = mgΔh, and presuming a 70kg bloke this comes to GPE lost= 33730J (Δ is, by the way, the mathematical way of expressing a change in something- in this case, Δh represents a change in height). If his fall were made with no air resistance, then all this GPE would be converted to kinetic energy, where KE = mv²/2. Solving to make v (his velocity upon hitting the ground) the subject gives v = sqrt(2*KE/m), and replacing KE with our value of the GPE lost, we get v = 31.04m/s. This tells us two things; firstly that the fall should take Ezio at least three seconds, and secondly that, without air resistance, he’d be in rather a lot of trouble.

But, we must of course factor air resistance into our calculations, but to do so to begin with we must make another assumption; that Ezio reaches terminal velocity before reaching the ground. Whether this statement is valid or not we will find out later. The terminal velocity is just a rearranged form of the drag equation: Vt=sqrt(2mg/pACd), where m= Ezio’s mass (70kg, as presumed earlier), g= gravitational field strength (on Earth, 9.8m/s²), p= air density (on a warm Venetian evening at around 15 degrees Celcius, this comes out as 1.225kg/m3), A= the cross-sectional area of Ezio’s falling body (call it 0.85m², presuming he’s around the same size as me) and Cd= his body’s drag coefficient (a number evaluating how well the air flows around his body and clothing, for which I shall pick 1 at complete random). Plugging these numbers into the equation gives a terminal velocity of 36.30m/s, which is an annoying number; because it’s larger than our previous velocity value, calculated without air resistance, of 31.04m/s, this means that Ezio definitely won’t have reached terminal velocity by the time he reaches the bottom of the Campanile, so we’re going to have to look elsewhere for our numbers. Interestingly, the terminal velocity for a falling skydiver, without parachute, is apparently around 54m/s, suggesting that I’ve got numbers that are in roughly the correct ballpark but that could do with some improvement (this is probably thanks to my chosen Cd value; 1 is a very high value, selected to give Ezio the best possible chance of survival, but ho hum)

Here, I could attempt to derive an equation for how velocity varies with distance travelled, but such things are complicated, time consuming and do not translate well into being typed out. Instead, I am going to take on blind faith a statement attached to my ‘falling skydiver’ number quoted above; that it takes about 3 seconds to achieve half the skydiver’s terminal velocity. We said that Ezio’s fall from the Campanile would take him at least three seconds (just trust me on that one), and in fact it would probably be closer to four, but no matter; let’s just presume he has jumped off some unidentified building such that it takes him precisely three seconds to hit the ground, at which point his velocity will be taken as 27m/s.

Except he won’t hit the ground; assuming he hits his target anyway. The Assassin’s Creed universe is literally littered with indiscriminate piles/carts of hay and flower petals that have been conveniently left around for no obvious reason, and when performing a leap of faith our protagonist’s always aim for them (the AC wiki tells me that these were in fact programmed into the memories that the games consist of in order to aid navigation, but this doesn’t matter). Let us presume that the hay is 1m deep where Ezio lands, and that the whole hay-and-cart structure is entirely successful in its task, in that it manages to reduce Ezio’s velocity from 27m/s to nought across this 1m distance, without any energy being lost through the hard floor (highly unlikely, but let’s be generous). At 27m/s, the 70kg Ezio has a momentum of 1890kgm/s, all of which must be dissipated through the hay across this 1m distance. This means an impulse of 1890Ns, and thus a force, will act upon him; Impulse=Force x ΔTime. This force will cause him to decelerate. If this deceleration is uniform (it wouldn’t be in real life, but modelling this is tricky business and it will do as an approximation), then his average velocity during his ‘slowing’ period will come to be 13.5m/s, and that this deceleration will take 0.074s. Given that we now know the impulse acting on Ezio and the time for which it acts, we can now work out the force upon him; 1890 / 0.074 = 1890 x 13.5 = 26460N. This corresponds to 364.5m/s² deceleration, or around 37g’s to put it in G-force terms. Given that 5g’s has been known to break bones in stunt aircraft, I think it’s safe to say that quite a lot more hay, Ezio’s not getting up any time soon. So remember; next time you’re thinking of jumping off a tall building, I would recommend a parachute over a haystack.

N.B.: The resulting deceleration calculated in the last bit seems a bit massive, suggesting I may have gone wrong somewhere, so if anyone has any better ideas of numbers/equations then feel free to leave them below. I feel here is also an appropriate place to mention a story I once heard concerning an air hostess whose plane blew up. She was thrown free, landed in a tree on the way down… and survived.

EDIT: Since writing this post, this has come into existence, more accurately calculating the drag and final velocity acting on the falling Assassin. They’re more advanced than me, but their conclusion is the same; I like being proved right :).

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’.