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...