Formula One and relativity

It would be not inaccurate to say that relativity theory has something of a low profile in Formula One. The recent announcement that gravitational waves have been detected for the first time aroused little more than a grudging blip of interest within the region of the autistic spectrum occupied by F1 vehicle dynamicists, strategists, and aerodynamicists.

It's worth noting, however, that modern F1 operations are heavily dependent upon relativity theory. F1 utilises GPS for its timing systems, and almost all teams use GPS for their trajectory analysis; and GPS, of course, is crucially dependent upon relativity theory.

To accurately establish the position of a car on the surface of the Earth, a GPS receiver must compare the time-stamps on signals it receives from multiple satellites, each one of which is orbiting the Earth at 14,000km/hr. To maintain the desired positional accuracy, the time on each such satellite must be known to within an accuracy of 20-30 nanoseconds.

However, there are two famous relativistic effects which have to be compensated for to maintain such accuracy: (i) special relativistic time dilation; (ii) general relativistic time dilation inside a gravitational well.

Because the satellites are in motion at high speed relative to the reference frame of a car on the surface of the Earth, their clock-ticks are slower by a rate of about 7 microseconds per day. Conversely, because a car lies deeper inside a gravitational well than the satellites, its clock-ticks will slow down by about 45 microseconds per day. The net effect is that the clocks on-board the satellites tick faster than those on-board an Earth-bound GPS receiver by about 35 microseconds per day.

As Richard W. Pogge points outs, "This sounds small, but the high-precision required of the GPS system requires nanosecond accuracy, and 38 microseconds is 38,000 nanoseconds. If these effects were not properly taken into account, a navigational fix based on the GPS constellation would be false after only 2 minutes, and errors in global positions would continue to accumulate at a rate of about 10 kilometers each day! The whole system would be utterly worthless for navigation in a very short time. This kind of accumulated error is akin to measuring my location while standing on my front porch in Columbus, Ohio one day, and then making the same measurement a week later and having my GPS receiver tell me that my porch and I are currently somewhere in the air kilometers away."

Which is worth recalling the next time GPS reveals that Dudley Duoflush is repeatedly missing the apex in Turn 4, or overtook under yellow-flag conditions between Turns 7 and 8.

Red Bull's T-tray wing

Red Bull appeared at the first pre-season Formula 1 test this week with an interesting wing perched atop the T-tray splitter beneath the chassis. As Craig Scarborough points out on Autosport.com, the tips of this wing act as vortex generators. Craig also points out that the idea has been tried before, on the Brawn 001 in 2009.

The interesting thing about such a device is that it's profiled in the manner of an aircraft wing, generating low-pressure above and high pressure below. The consequence of this is that it generates vortices rotating in the same sense as the Y250 vortex on each side of the chassis.

So, for example, looking from a perspective in front of the car, and focusing on the right-hand-side of the chassis, both the Y250 vortex and the T-tray wing vortex rotate in an anticlockwise direction. On the left-hand-side, they both rotate in a clockwise direction.

Now, this is in contrast with the influence provided by a J-vane vortex. As alluded to in Jonathan Pegrum's 2006 academic work, when a vortex spinning around an axis pointing in the direction of the freestream flow passes close to a solid surface, it tends to pull a counter-rotating vortex off the boundary layer of that surface. Hence, when the Y250 vortex passes the J-vanes hanging from the underside of the raised nose on a Formula 1 car, it creates a pair of counter-rotating vortices on each side of the chassis.

For vortices sharing approximately the same rotation axis, it is a general rule that counter-rotating vortices tend to repel each other, whereas co-rotating vortices tend to attract each other. In fact, for a time, co-rotating vortices will orbit a common center of vorticity. This situation will persist so long as they are separated by a distance large compared to their vortex-core radii. Eventually, however, viscous diffusion will enlarge their respective cores, and they will begin to deform each other, eject arms of vorticity, and finally merge into a single, larger vortex.

Because the J-vane vortex rotates in the opposite sense to the Y250, it tends to repel it. Hence, the J-vane can be used to push the Y250 into the optimal position to fulfil its ultimate purpose, which is to push the front-wheel wake further outboard. 

However, the J-vane vortex can only push the Y250. Fitting a T-tray wing, which presumably generates vortices with the same sense of rotation as the Y250 itself, conceivably provides Red Bull with the ability to push and pull the position of the Y250, from two different downstream locations. That possibly improves their ability to fine-tune the position of the Y250 in both a vertical and lateral direction. Alternatively, of course, it may just be designed to interact with the vorticity generated by the bargeboards et al.

Whilst Brawn tried the same concept in 2009, note that the Brawn wasn't fitted with J-vanes, and the presence of a double-diffuser might have reduced the sensitivity of the diffuser to ingress of the front-wheel wake anyway.

Britain braced for -10C winter blast

The Daily Telegraph website has an article about the UK weather-forecast for this weekend. To enhance the restrained and informative nature of the article, which avoids lazy journalistic cliché, I've added my own parenthetical comments in red below:

Britain braced for -10C winter blast. (It's a maritime climate, and it's winter).

Sleet and snow is forecast as far south as Wales and the Midlands by the weekend with wintry showers across the rest of the country. (So not as far South as, say, The South. Not the Isle of Wight, not Bournemouth, but as far South as the Midlands. In a country with a North and a South, the bit roughly halfway between the two is The Midlands. So, the bit which isn't in the South is as far South as the sleet and snow is forecast to reach.)

Storm-hit Britain could be hit with an arctic blast bringing more than three inches of snow towards the end of the week. (It's a maritime climate, and it's winter).

A twist in the jet stream means bitter winds will bring freezing fog and widespread frosts. (A twist? Like when you hold one end of the jet-stream in an aerodynamic clamp, and rotate the other end around its axis? I guess the meanders in a river are colloquially said to be twists, but isn't it really a bend or a kink in the jet-stream?) 

Sleet and snow is forecast as far south as Wales and the Midlands by the weekend while wintry showers are possible across the country. (The headline said the forecast was for wintry showers across the rest of the country, now you're saying they're merely 'possible'. Are they likely or just possible? If you can't tell me, then you've failed in the primary journalistic task of disseminating useful information).

Gareth Harvey, a forecaster with MeteoGroup, said today would be “fairly chilly” and added there would be “widespread frost on Wednesday night, with temperatures between 0C and -3C, which could happen anywhere.” (It's a sad reflection on the modern world if literally 'anywhere' could suffer a frost. In winter, in a maritime climate).

He added: "In the northern part of Scotland, people will wake up to a covering of snow on Thursday morning with accumulations of up to several centimetres. A band of rain and snow will slowly move its way southwards but it will peter out as it reaches central parts." (Not just Scotland, but the 'northern part of Scotland', will wake up to a covering of snow. It's as if the chance of snow in winter increases at higher latitudes).

But the real cold snap will begin on Friday when experts say temperatures could plunge to -10C. (Experts. I didn't realise there were experts involved. These are people who know what they're talking about)

And James Madden, forecaster for Exacta Weather, said cold weather could hold out through the rest of this month. ('Could' or 'likely to'?)

He said: The colder and wintry theme will begin to take more of a stronghold into the second week of February as the UK becomes locked in an icy and wintry grip." ('Stronghold', 'locked', 'grip'. Sounds like some panic-buying in the supermarkets is in order).

The chilly outlooks comes as Britain recovers from the effects of Storm Imogen, which struck on Monday.

Which brings us to the naming of storms. Below the main story we find the following, under the heading 'A-Z of UK storms':

Why do we need to name them? Using a "single authoritative system" helps the media communicate what's happening more effectively, says the Met Office, which in turn increases public awareness.

In what sense, exactly, does the naming of transient patterns in atmospheric airflow constitute an 'authoritative system'. Do anonymous patterns of airflow lack presence in some way? Do anticyclones suffer from poor self-confidence? And how does giving a storm a name help the media to communicate what's happening more effectively? Should we also give economic recessions avuncular names, so that the media can explain more effectively why living standards are falling?  

Chemical adhesion and Formula One tyres

In the early 1980s, John Watson enjoyed what can only be described as a 'spree' of remarkable Grand Prix victories, achieved by overtaking numerous cars from mediocre or lowly grid positions.

Watson's success at Zolder and Detroit in '82, and Long Beach in '83, is commonly ascribed to using a harder compound of tyre, but John himself has commented that "it wasn't so straightforward [as a harder compound] because in those days there were extremely subtle differences between grades, compounds and construction of tyres and Michelin operated with great secrecy anyway," (1982, Christopher Hilton, p126). In particular, John mentions that the tyre he took on the left-hand side at Zolder in '82 was recommended by Michelin's Pierre Dupasquier on the basis of its performance on Bruno Giacomelli's Alfa Romeo at Las Vegas in 1981.

One can hypothesize that John achieved those stunning victories using a Michelin compound which was not only harder, but which generated an unusually high proportion of its grip from chemical adhesion.

In this context, recall that there are two distinct but related mechanisms by which a rubber tyre generates grip: (i) the viscoelastic deformation of the tyre by the 'asperities' in the road surface, ultimately leading to the viscous dissipation of kinetic energy into heat energy; and (ii) chemical adhesion at the interface between the tyre and the road surface.

The viscoelastic mechanism is often dubbed the 'hysteretic friction'. This is not our main concern here, but the interested reader is referred to Tyre friction and self-affine surfaces for an introduction to the representation and role of asperities.

Chemical adhesion is maximised by higher temperatures and higher contact areas. When a tyre gets hotter, it gets softer, and this allows it to deform further into the crenellations in the road surface, increasing the contact area. Hence, adhesion is maximised on smooth, hot surfaces.

Now, let's hypothesise that Las Vegas, Zolder, Detroit and Long Beach shared the following combination of characteristics: the asphalt was very smooth, and, (with the exception of Detroit), somewhere between fairly warm and very hot.

Certainly, Dupasquier has attested to the fact that Long Beach was a low 'severity' surface, (Alpine and Renault, Roy Smith, p148), and it seems likely that Detroit, as another street circuit, would have possessed similar characteristics. Las Vegas was basically just the car-park to a casino, so the same presumably applied there.

Whilst Detroit in '82 was slightly overcast, it was warmer than anticipated at Zolder, and the races at Las Vegas in '81 and Long Beach in '83 were run in high temperatures. On balance, then, Watson's amazing victories were mostly achieved on hot, smooth circuits, and the best tyre on a hot, smooth surface is one which generates a larger proportion of its grip from adhesive friction than hysteretic friction. 

A useful graph in this respect can be found in the latest paper co-authored by rubber-friction expert, B.N.J. Persson, concerning the dependency of rubber friction on normal load, (hereafter referred to as Fortunato et al). The graph, reproduced above, plots viscoelastic friction and adhesive friction as a function of the sliding velocity of a tyre.

The latter concept requires a brief digression: When a tyre is turned at an angle to the direction in which the car is moving, (the so-called slip-angle θ), the contact patch is deformed at a velocity which has a component parallel to the direction in which the tyre is rolling, and a transverse component, perpendicular to the rolling direction. The latter component is the sliding velocity which generates a cornering force. In the figure above, this sliding velocity is plotted in logarithmic form on the horizontal scale. In other words, it expresses the sliding velocity as a power of 10.

If the car-velocity is v_c, and the slip-angle is θ, then the transverse slip-velocity is v_y = v_c Sin θ. Hence, approximately the same slip velocity can be generated by a large slip-angle in a slow-speed corner, and a smaller slip-angle in a high-speed corner. The actual slip velocities seen by an F1 contact patch, of the order ~1 m/s, correspond to a value of 0 on the log scale in the figure above.

Now, the friction coefficient generated by a tyre is actually a function of at least two principal variables: (i) the 'bulk' temperature of the tyre tread, and (ii) the sliding velocity. Hence, the coefficient of friction (mu) should always be imagined as a 2-dimensional surface.

If one represents bulk temperature along the x-axis, sliding velocity along the y-axis, and the friction coefficient as a vertical function mu = f(x,y), then peak adhesive and hysteretic friction can each be pictured as diagonal escarpments running from the bottom-left to the top-right of the horizontal plane. At a fixed sliding velocity, one can plot the mu as a function of bulk temperature; and at a fixed bulk temperature, one can plot the mu as a function of the sliding velocity. The figure above from Fortunato et al represents only a slice of the latter type.
 
As a tyre ages and wears, it loses the ability to generate and retain heat, and its temperature begins to fall. If a driver continued inducing the same slip-velocities as the tyre temperature dropped, then the mu would follow a track parallel with the x-axis, and the drop in grip would be quite precipitous. It's more likely that as a tyre ages, either the cornering speed will reduce, or the driver will fractionally reduce the slip-angles, thereby reducing the slip velocities, and the grip will follow more of a diagonal path, down the ridge of the escarpment towards the bottom left of the mu surface.

Fortunato et al make the crucial point that "at room temperature the maximum in the adhesive contribution is located below the typical slip velocities in tire [sic] applications (1 - 10 m/s), while the maximum in the viscoelastic contribution may be located above typical sliding speeds...Increasing the temperature shifts both [the adhesive and hysteretic mu] towards higher sliding speeds, and also increases the area of real contact A, making the adhesive contribution more important. Depending on the relative importance of the adhesive and viscoelastic contribution to the friction, the friction coefficient may increase or decrease with increasing temperatures."

John Watson's victories in the early 80s were achieved on tyres which took some laps to 'come in', hence this all adds up to a tyre which generated a higher proportion of its grip from adhesion, and only generated peak mu when it had been strained sufficiently to reach a higher temperature. In this case, the greater adhesion at high temperatures more than offset the loss of hysteretic friction.

During the Michelin and Bridgestone tyre war of the early 2000s, Formula One tyres continued to generate a significant proportion of their grip from chemical adhesion, hence a driver was able to 'push' on consecutive laps, without losing grip. The gain in chemical adhesion would offset the loss of hysteretic friction.

In contrast, if we consider the hypothetical case of a tyre which generated only a small proportion of its grip from chemical adhesion, then even before the effects of wear kick-in, a racing driver would find such tyres to be constantly balanced on a knife-edge of hysteretic grip. Push too hard for several laps, and as the tyre gets hotter, it would lose hysteretic grip without a compensating gain in adhesion...

Formula 1 strategy and Nash equilibrium

At first sight, Formula 1 race strategy seems to be an ideal domain for the application of game-theory. There are a collection of non-cooperative agents, each seeking to anticipate the decisions of their competitors, and choose a strategy which maximizes their pay-off. The immediate pay-off at each Grand Prix is championship points.

However, there's a subtlety of game-theory which needs to be appreciated before its most famous concept, that of Nash equilibrium, can be applied.

Let's begin with the game-theory. John Nash demonstrated that a non-cooperative n-player game, in which each player has a finite set of possible strategies, must have at least one point of equilibrium.

This equilibrium is a state in which each player's choice of strategy cannot be improved, given every other player's choice of strategy. In game-theoretic language, each player's pay-off is maximized, given every other player's choice of strategy.

In formal terms, there must be an n-tuple of strategies σ = (σ₁,...,σ_n) in which the pay-off for each player, v_i, is maximized:

v_i(σ) = max v_i (σ₁,...,σ_n),   for i = 1,...,n

where the maximum is taken over all the player-i strategies, σ_i.

The set of strategies adopted by the teams at each Grand Prix should possess at least one such state of Nash equilibrium, (irrespective of whether the competitors are capable of finding that optimal state). However, it's possible to define a simple and realistic scenario which, at first sight, undermines Nash equilibrium.

Suppose that a Ferrari is ahead of a Mercedes in the early laps of a race, but the Mercedes has a pace advantage. Suppose, however, that the pace delta between the cars is less than the minimum threshold for a non-zero probability of the Mercedes overtaking the Ferrari.

Now, for the sake of argument, suppose that due to aerodynamic interference from the wake of the car ahead, the Mercedes cannot follow closer than 1.5 seconds behind the Ferrari, and suppose that tyre degradation is sufficiently low that new tyres provide a 1-lap undercut worth less than 1 second. Even if Mercedes pit first, Ferrari can respond the next lap, and (assuming an error-free stop) will emerge still in the lead.

Clearly, if Mercedes are to beat Ferrari they will need to use a different strategy. Let's make this interesting by postulating that whilst a 1-stop strategy is the fastest 'deterministic' race, a 2-stop strategy is only a second or so slower.

Now, if the Mercedes switches to a 2-stop strategy, it will be out of sync with the Ferrari, will be able to circulate at its true pace, and will be able to beat the Ferrari if the Scuderia remain on a 1-stop. (For the sake of argument, we assume that there are traffic-free gaps into which the Ferrari can pit, without being delayed by other competitors).

However, if Ferrari anticipates this, and plan a 2-stop strategy, it will still win the race. If both cars are on the 2-stop strategy, Mercedes cannot utilise its superior pace.

However, however, if Mercedes anticipates that, it can win the race by sticking to the original 1-stop strategy...which Ferrari, again, can head off by sticking with the 1-stop. And so on, ad infinitum.

Clearly, there is no Nash equilibrium here. Each possible combination of strategies is such that at least one competitor can improve their pay-off by changing strategy, if the other competitor's strategy remains fixed. This structure is depicted graphically above. Each of the four cells represents a possible combination of 1-stop and 2-stop strategies. The pair of numbers in each cell represents the pay-off, in championship points, for Ferrari and Mercedes, respectively.

The coloured arrows indicate how one competitor can always improve their pay-off. For example, the top-left cell represents the case in which Ferrari and Mercedes both pursue a 1-stop strategy. The blue arrow reaching across to the top-right cell indicates that Mercedes can improve their pay-off by switching to a 2-stop strategy, if Ferrari remain wedded to the 1-stop. However, the downward red arrow in the top-right cell indicates that Ferrari can improve their pay-off by switching to a 2-stop if Mercedes remain committed to a 2-stop.

The problem here is that the strategies considered are termed 'pure' strategies in game-theoretic terms. Nash's theorem pertains not to pure strategies, but to probabilistic combinations of pure strategies, called 'mixed' strategies. If there are two possible pure strategies, A and B, a mixed strategy is one in which, for example, you resolve to follow strategy-A 30% of the time, and strategy-B 70% of the time. You must also use a random number generator to enforce the probabilistic split.

A mixed strategy, then, is a rather abstract thing, and not necessarily something which represents human strategic thinking. People often have contingency plans, alternative strategies that they will adopt if certain events occur, but they rarely frame their original strategy in terms of probabilistic mixtures.

In terms of the Formula 1 strategy scenario defined above, there is a state of Nash equilibrium: if Ferrari and Mercedes both adopt the mixed strategy of pursuing a 1-stop with 50% probability and a 2-stop with 50% probability, then neither competitor has a mixed strategy which offers an improvement in terms of their average pay-off.

However, Formula 1 teams are unlikely to adopt such a coin-tossing approach to strategy, so a Grand Prix potentially offers an interesting case study of a non-cooperative n-player game far from Nash equilibrium.

Pantheism and religion

Sandwiched between articles on human flatulence and the hazard posed by pigeon-droppings to electricity pylons, the 2015 Christmas/New Year edition of New Scientist contained an article by theologian Mary-Jane Rubenstein. The main thrust of the article attempts to draw parallels between some ancient philosophies and modern multiverse proposals in cosmology.

Specifically, Mary-Jane argues that the atomists were proposing a type of spatial multiverse, whilst the stoics were advocating a temporal one. Although it's stretching the point somewhat, the majority of the article is quite interesting.

However, as we reach the final paragraphs, Mary-Jane can be found citing a type of pantheism advocated by Nicholas of Cusa:

"Traditionally, Christian doctrine has taught that humans are made in the image of God. Cusa disrupted this idea by saying that the universe, not man, bears the image of God. And if humans are not particularly godlike, then God is not particularly humanoid. God doesn't look like a patriarch in the sky: he looks like the universe."

Now, pantheism is a rather strange notion. It's as if one has responded to the question 'Do unicorns exist as well as horses?' by replying 'Yes, they do, but they don't have horns, and can be identified with, or considered to resemble horses.'

But that's not the main problem with the article. The main problem comes in the final paragraph, where Mary-Jane concludes that because pantheisms "change what it means to be God...we don't need to chose between God and the multiverse...Is it possible that modern cosmology is asking us, not to abandon religion, but to think differently about what it is that gives life, what it is that's sacred, where it is we come from - and where we'll go?"

Whoa! Hold on a cotton-picking minute there, Mary-Jane. Perhaps there were some readers whose blood-flow was devoted more towards the stomach than the brain over the Christmas period, and under such conditions it might be possible to miss the sleight-of-hand here. Under most other conditions it's not too difficult to spot the sudden jump from the abstract metaphysical concept of pantheism to the introduction of religion.

The term 'religion' doesn't just entail a bundle of metaphysical concepts: it means a human institution; it means scripture, liturgy, a priesthood, a dogmatic moral code, the indoctrination of children, and the amplification of tribal behaviour.

That's rather more than pantheism suggests, I fear, and certainly not the answer to any of the questions posed by multiverse cosmology.

Newspaper journalists and the Met Office

It's been a relatively mild winter in Britain this year, and this has deprived newspaper journalists of their normal opportunity for hysterical exaggeration and over-reaction to wintry weather. However, temperatures have fallen this weekend, and, taking a cue from the Met Office's ludicrously patronising weather-warning system, the hyperbole has been flowing:

"Snow and ice sweep across Britain," yells The Guardian, claiming that "A 100-mile wide corridor of snow stretched from north-west Scotland to south-east England overnight." 

"Treacherous driving conditions as snow and ice alert covers more of Britain," shouts The Telegraph headline, "Drivers warned of hazardous conditions after mercury falls to -10C amid 100-mile snow corridor."

The Telegraph article, however, begins to equivocate its message after no more than a couple of sentences, alluding to "many Britons waking up to frosty conditions on Sunday."

Frosty conditions, eh? There's quite a difference between waking up to snowy conditions and waking up to frosty conditions. For a start, whilst people sometimes have to dig their car out of a snow-drift, it's somewhat rarer to dig your car out of a frost-drift.

Scanning further down the page, we find that the Met Office had previously said 'snow had been "expected to fall along a relatively narrow corridor, perhaps only 100 miles wide" and forecaster Sophie Yeomans said "that band of sleet and snow is staying over the country, but it is dying out".'

This reveals that the newspaper journalists have misunderstood the dimensions of the purported 'snow corridor'. The Met Office are using 100 miles as a diminutive term, not as an expansive term. 100 miles is quite a short distance in meteorological terms. The purported 'corridor of snow' is "only 100 miles wide." What's more, it is the width of the corridor which spans 100 miles, not its length.

If we actually scrutinise the shape of the snow corridor in the graphic supplied by the Met Office (below), we can see that its length is much greater than its width. It is much longer than 100 miles. If the journalists were seeking an impressive-sounding length scale to exaggerate the severity of the wintry conditions, they should have quoted its length, not its width. But that would have required additional effort. The Met Office have quoted 100 miles, and it's a nice round number, so that's the length-scale the newspapers are going to quote.

But just look at the length of that snow corridor. That's a lot of snow isn't it? We can tell it's a region of snowfall because there's a snowflake icon, and a sliding-car icon adjoined to the top of the yellow band.

Oh, but hold on, there's also a legend down below which tells us what the yellow shading actually means. It turns out that yellow means 'Be Aware'. Which is a useful piece of advice. Thanks for that. But what exactly does 'Be Aware' mean in this context?

Following the links on the Met Office website to their Weather Warning page, we discover the following definition:

Yellow: Be aware. Severe weather is possible over the next few days and could affect you. Yellow means that you should plan ahead thinking about possible travel delays, or the disruption of your day to day activities. The Met Office is monitoring the developing weather situation and Yellow means keep an eye on the latest forecast and be aware that the weather may change or worsen, leading to disruption of your plans in the next few days. 

So that's not a corridor of predicted snowfall; that's merely a corridor in which the Met Office recommend people should "plan ahead" and "keep an eye on the latest forecast."

In effect, then, The Met Office is saying the following: 'Things are possible, and they may affect you! Don't treat us as an occasional source of information; be dependent upon us; raise your anxiety levels when we tell you to. We are monitoring the developing situation; you need us.'

Hence, whilst The Telegraph article reports that "Parts of the North West were hit by snow overnight, ranging from a light dusting in Manchester to heavier snowfalls on the Pennines and rural Cheshire and Cumbria," we're subsequently informed "that band of sleet and snow is staying over the country, but it is dying out...There is a lot of rain in that but in parts of London there will be sleet falling as well."

So in other words, the story should really be:

'A band of precipitation will cause localised snowfall in the North-West, with sleet turning to rain in other regions.' 

But remember, keep watching and reading those weather forecasts!

Lotus or Shadow?

The February 2016 edition of RaceTech magazine has an interesting article on wind-tunnels by their F1 insider, 'Expert Witness'. However, there may be an error in the caption to one of the photos which accompanies the article, (below).

The caption claims that the photo depicts a Lotus wind-tunnel test from 1972. Which would be surprising, because one would expect a 1972 car to sport a much taller airbox. In fact, not only is the airbox wrong, but the nose doesn't look like a Lotus nose at all.

If I were pressed to identify the model, I would suggest that it is actually a version of Tony Southgate's Shadow DN8 (below), probably from 1976-1977.

 

Certainly, if it does transpire to be a Lotus wind-tunnel test from 1972, it opens a whole new time-travelling window on F1 espionage in the 1970s...

The tortuosity of modern F1 circuits

In recent decades, Formula 1 circuits have tended to lose their character; flowing tracks, sculpted by the contours of the land, have been supplanted by clinical autodromes designed to maximise sponsorship exposure and minimise running costs. 

Perhaps surprisingly, it is soil physics which offers a means of quantifying this loss of flow. Specifically, we need to adapt a quantity called the tortuosity factor, which is used to analyse the permeability of soil to the flow of water.

As Daniel Hillel explains, "The actual length of the path traversed by an average parcel of liquid is greater than the soil column length L, owing to the labyrinthine, or tortuous, nature of the pore passages...Tortuosity can be defined as the average ratio of the actual roundabout path to the apparent, or straight, flow path," (p177, Fundamentals of Soil Physics, Academic Press, 1980).

So, let's calculate and compare the tortuosity of a corner complex on a traditional F1 circuit to that on a modern Hermann Tilke designed circuit. In particular, let's compare the Becketts sequence at Silverstone with the Turn 1/2/3 complex at Shanghai.

Courtesy of Google Maps, the distance between the entry and exit of the Becketts sequence, as the crow flies, is about 517m. The distance along the path of the track is about 600m. Hence the tortuosity of Becketts is T = 600/517 = 1.2

The distance between the entry to Turn 1, and Turn 4 at Shanghai, as the crow flies, is about 145m. The distance along the path of the track is about 650m. Hence, the tortuosity of the opening corner sequence at Shanghai is T = 650/145 = 4.5

So, the tortuosity factor of a modern F1 corner complex can be as much as 3.75 times greater than that of a more traditional sequence. Which is a way of placing a number on how much F1 has lost its soul.

Tornados and the Y250 wing-tip vortex

Streamwise vortices occur when fluid spirals around an axis which points in the same direction as the overall direction of fluid flow. In particular, streamwise vortices are generated by aircraft wing-tips, and by the front-wing of a Formula 1 car at the inboard transition between the neutral central section and the inner tip of the main-plane and flap(s). The latter is the so-called Y250 vortex. Surprisingly, the method by which such streamwise vorticity is generated also plays a crucial role in the generation of atmospheric tornados.

Let's begin with the meteorology. A tornado is a funnel of concentrated vertical vorticity in the atmosphere. Most tornados are generated within supercell thunderstorms when the updraft of the storm combines with the horizontal vorticity generated by vertical wind shear. The updraft tilts the horizontal vorticity into vertical vorticity, generating a rotating updraft.

However, there are two distinct types of vertical wind shear: Unidirectional and directional. The former generates crosswise vorticity, whilst the latter generates streamwise vorticity.

When the wind shear associated with a storm is unidirectional, the updraft acquires no net rotation. The updraft raises the crosswise vorticity into a hairpin shape, with one cyclonically rotating leg, on the right as one looks downstream, and an anticyclonic leg on the left. Updrafts only acquire net cyclonic rotation when the horizontal vorticity has a streamwise component. (Diagrams above and below from St Andrews University Climate and Weather Systems website).

Specifically, cyclonic tornado formation requires that the wind veers with vertical height, (meaning that its direction rotates in a clockwise sense).

In effect, the flow of air through the updraft becomes analagous to flow over a hill (personal communication with Robert Davies-Jones): the flow into the updraft has cyclonic vorticity, and the flow velocity there reinforces the vertical velocity of the updraft; the downward flow on the other side, where the anticyclonic vorticity exists, partially cancels the vertical velocity of the updraft. Hence, the cyclonic part of the updraft becomes dominant.

Before we turn to consider wing-tip vortices, we need to recall the mathematical definition of vorticity, and the vorticity transport equation.

Let's start with some notation. In what follows, we shall denote the streamwise direction as x, the lateral (aka 'spanwise' or 'crosswise') direction as y, and the vertical direction as z. The velocity vector field U has components in these directions, denoted respectively as U_x, U_y, and U_z, There is also a vorticity vector field, whose components will be denoted as ω_x, ω_y, and ω_z.

The vorticity vector field ω is defined as the curl of the velocity vector field:

ω = (ω_x , ω_y, ω_z)

= (∂U_z/∂y − ∂U_y/∂z , ∂U_x/∂z − ∂U_z/∂x , ∂U_y/∂x − ∂U_x/∂y)

We're also interested here in the Vorticity Transport Equation (VTE) for ω_x, the streamwise component of vorticity. In this context we can simplify the VTE by omitting turbulent, viscous and baroclinic terms to obtain:

Dω_x/Dt = ω_x(∂U_x/∂x) + ω_y(∂U_x/∂y) + ω_z(∂U_x/∂z)

The left-hand side here, Dω_x/Dt, is the material derivative of the x-component of vorticity; it denotes the change of ω_x in material fluid elements convected downstream by the flow.

Now, for a racecar, streamwise vorticity can be created by at least two distinct front-wing mechanisms:

1) A combination of initial lateral vorticity ω_y, and a lateral gradient in streamwise velocity, ∂U_x/∂y ≠ 0.

2) A vertical gradient in the lateral component of velocity, ∂U_y/∂z ≠ 0, (corresponding to directional vertical wind shear in meteorology).

In the case of the first mechanism, one can vary the chord, camber, or angle of attack possessed by sections of the wing to create a lateral gradient in the streamwise velocity ∂U_x/∂y ≠ 0. Given that ω_y ≠ 0 in the boundary layer of the wing, combining this with ∂U_x/∂y ≠ 0 entails that the second term on the right-hand side in the VTE is non-zero, which entails that Dω_x/Dt ≠ 0. Thus, the creation of the spanwise-gradient in the streamwise velocity skews the initially spanwise vortex lines until they possess a significant component ω_x in a streamwise direction.

However, it is perhaps the second mechanism which provides the best insight into the formation of wing-tip vortices. As the diagram above illustrates for the case of an aircraft wing (G.A.Tokaty, A History and Philosophy of Fluid Mechanics), the spanwise component of the flow varies above and below the wing. This corresponds to a non-zero value of ∂U_y/∂z, and such a non-zero value plugs straight into the definition of the curl of the velocity vector field, yielding a non-zero value for the streamwise vorticity ω_x:

ω_x = ∂U_z/∂y − ∂U_y/∂z

Putting this in meteorological terms, looking from the front of a Formula 1 car (with inverted wing-sections, remember), the left-hand-side of the front-wing has a veering flow-field at the junction between the flap/main-plane and the neutral section. The streamlines are, in meteorological terms, South-Easterlies under the wing, veering to South-Westerlies above. This produces streamwise vorticity of positive sign.

On the right-hand side, the flow-field is backing with increasing vertical height z. The streamlines are South-Westerlies under the wing, backing to South-Easterlies above. This produces streamwise vorticity with a negative sign.

Thus, we have demonstrated that the generation of the Y250 vortex employs the same mechanism for streamwise vorticity formation as that required for tornadogenesis.

The open-tailed box effect

The modern understanding of racecar aerodynamics holds that copious amounts of downforce can be produced by accelerating the airflow under the car, in effect turning the region between the underbody and ground plane into a mobile nozzle.

The Lotus 78 of 1977 famously introduced venturi profiles beneath the car, and sliding skirts to seal the low pressure area thereby created. However, it is less well-known that underbody skirts had fitfully appeared on various cars earlier in the decade. Moreover, it is slightly disconcerting to hear the explanations proffered by several F1 designers from the middle 1970s for the function of these devices.

Gordon Murray introduced inch-deep skirts on the underside of the 1975 Brabham BT44 in conjunction with an overall 'upturned saucer' design, and explains his thinking as follows:

"With any moving form you have a stagnation point where air meets it and decides how much is going to flow over, below or around it...I decided, instead of presenting some sort of parabolic-shaped bluff body to the air, I wouldn't give the air a chance." He sketches a triangular shape. "That way the stagnation point was there," he says, pointing to the leading edge of the triangle's base, which is very low to the ground. "So all the air had to go over the top and you had the minimum coming under the car," (F1 Magazine, May 2001, p140-141).

Gordon Coppuck, however, had already experimented with skirts on the McLaren M23:

"In 1974 at Dijon-Prenois, vertical plastic skirts around the under-periphery of the car were tried, but they quickly wore away on contact with the track. The idea was to exclude air from underneath the car and so minimise lift," (p49, McLaren M23, Ian Wagstaff, Haynes 2013). The skirts were fitted again to the M23 at some races in early 1976, this time provoking complaints from competitors such as Colin Chapman (!) and Ken Tyrrell.

Talk of minimising lift by forcing air over the top of the car seems misguided because the upper surface of a racecar is generally convex, and the air will tend to be accelerated by a convex surface, producing low pressure on the upper surfaces, somewhat counter to the overall objective.

Nevertheless, it seems that there actually was a beneficial effect to be had from partially excluding air from the underbody, and this is clearly explained by Ian Bamsey in his fantastic book The Anatomy and Development of the Sports Prototype Racing Car (Haynes, 1991):

"The [Shadow] DN8 had conventional wings and a flat bottom and, following the fashion of 1976, it was fitted with skirts along the side of its monocoque, these joined in a vee under the nose. Under certain conditions the skirts rubbed on the track and their general effect was to sweep the air aside, in snowplough fashion. Thus, the overall effect was not one of spatial acceleration of the underbody air, it was one of exclusion. The flow blockage allowed the forward migration of the naturally low pressure air at the back of the car into the skirt's exclusion zone. This was the principle of the so-called open tailed box. A box with the road forming its bottom and only its tail open will experience a pressure reduction within as it progresses along the track," (p59).

So, although the effect may be quite weak, it is possible to generate downforce by excluding air from the underbody.