Tyre feathering

"If a new tyre is punished too heavily, too quickly, its critical tread compound temperature may be exceeded and blisters will appear and burst...Heavy cars, unsympathetically-driven cars and badly set-up cars may burn-out their tyres (particularly at the front) in a less abrupt manner. This phenomenon, known as 'graining' or 'feathering', rolls little pieces of rubber off the tread - diminishing its grip. A watchful driver will recognise the condition as one edge of his front tyres - or of the rears seen in his mirrors - will blacken. The car's cornering power will be limited and if he presses on the condition will worsen. A sensitive driver will ease off, roll-away the loose granulation, and after a few laps will see his tyres regain their normal appearance and will find much of their adhesion restored."

(Doug Nye, Motor Racing in Colour, Blandford Press, 1978, p44).

BBC Sport and Potential Flow theory

It is a fact, sadly unacknowledged, that BBC Sport's default background animation represents a doublet from Potential Flow theory.

Potential Flow theory is a branch of aerodynamics in which flows are idealised as being inviscid and irrotational. This means, respectively, that there is no friction resistance to shear between adjacent layers of fluid, and there is no vorticity. Hence, Potential Flow theory does not recognise the existence of boundary layers adjacent to solid objects.

Now, there is, within aerodynamics, a distinction between circulation and rotation, which has the potential (if you'll excuse the term) to confuse. In a flow with circulation, you can integrate the velocity vector around a closed loop and obtain a non-zero value. In a rotational flow, the vorticity field is non-zero.

In a Potential Flow (guaranteed, by definition, to be irrotational), if the region of space occupied by the fluid is simply connected topologically-speaking, (entailing that any loop can be smoothly deformed to a point) then the flow will have zero circulation. However, if the region is not simply connected, then the irrotational flow can possess circulation. The presence of solid objects in a fluid prohibits the region of space occupied by the fluid from being simply connected, (in 2-dimensional terms, at least), hence Potential Flows around solid objects can possess circulation.

This loophole (if you'll excuse the term) within Potential Flow theory enables one to represent the circulatory flow around wing sections. Because the equations of Potential Flow theory are linear, one can superpose several solutions of the theory to obtain other solutions. To represent the flow around a cylinder, for example, one superposes a uniform flow with a so-called doublet. The latter provides the stagnation points to the flow at the leading and trailing points of the cylinder.

To represent the circulatory flow around a wing section one basically just adds a free vortex to the superposition.

Hence, despite the BBC's apparent aversion to covering all the Grands Prix in a Formula One season, aerodynamics is clearly a subject close to their heart.

Descent to bathos

Whilst the odiferous obligations of life suggest a strictly utilitarian approach to bathroom use amongst much of the population, there are those of us who continue to resist the anxious, volatile, blinding spray of the shower-head, and stubbornly insist on prolonged semi-submergence in a liquid heat source as an opportunity for rest, relaxation, and consumption of excellent prose.

This sub-class of the population, however, has to endure a flagrant design flaw which besets many a bathroom, yet receives scant public attention. This, namely, is the tendency of negligent bathwrights to erroneously align the tap-end of the tub towards the centre of the bath-space. This places the reclined reader's head at the opposite end of the bath-axis, where illumination from the central, ceiling-located light source is typically inadequate.

Until bathwrights can be prevailed upon to mend their ways, there is, fortunately, an interim measure which can be undertaken: remove a large mirror from the hallway, and balance it on the junction between the bath and the ceramically tessellated wall; obtain a lofty chair, a long electrical extension cable, and place an anglepoise lamp on the chair, facing the mirror. This cunning configuration will provide the recumbent, heat-absorbent reader with an adequate level of reflected light, from which the world of the imagination can once more be accessed, and the stresses of the modern world temporarily dissolved. There is nothing which could possibly go wrong.

City-Link's delivery standards

Autocourse 2012-13 arrived today. Courtesy of City-Link, it was left unprotected, outside the house, on one of the wettest days of the year. The book is ruined.

I presume the package was left in such a manner by an individual who faces a daily struggle to successfully wield a knife and fork.

I'm certainly looking forward to asking the Managing Director of City-Link, Mr David Smith, how employing such individuals contributes to the company's business plan.

Red Bull's double-DRS

Is there a connection between Red Bull's new double-DRS system, and their unique 'underpass' duct?

To recall, the underpass is a means of separating the 'coke-bottle' flow along the flank of the sidepod, from the exhaust-flow, sweeping down from the top of the sidepod. The coke-bottle flow feeds the starter-motor slot and the top surface of the diffuser's trailing edge, while the exhaust jet partially seals the side of the diffuser and increases the flow over the rear brake-duct assembly.

The underpass is fed by the flow along the flanks of the sidepods, which in turn is fed by the front-wing wake. By connecting the flow along the flanks of the sidepods to the low pressure area under the beam wing, the underpass not only assists with rear downforce, but also pulls the air faster over the front-wing.

Red Bull's double-DRS system purportedly stalls the central section of the beam wing, (although Craig Scarborough suggests that it is the tips of the beam-wing which are being stalled, in order to reduce the wing-tip vortex drag). 

The central part of the beam wing is the section which is pulling the air out of the underpass. Thus, if the beam wing stalls, then the underpass stalls, the flow along the flanks of the sidepods weakens, and front-wing downforce and drag are reduced.

The image above here illustrates how the front-wing streamlines on a generic open-wheeled race-car are the same streamlines which pass along the flanks of the sidepods, and thence between the rear wheels, (although there is no sidepod undercut or beam wing in this illustrative case, courtesy of the 2012 University of Southampton Racecar Aerodynamics MSc Group Design Project).

Red Bull introduced a smaller underpass inlet for the Korean Grand Prix, and if the double-DRS really does stall the centre of the beam wing, it would certainly make sense to change the underpass as well. Autosport's Mark Hughes comments in his Korean Grand Prix report that the changes to the sidepod area gave "more downforce and more diffuser stall." 

One can speculate then, that Red Bull's double-DRS is a system which reduces front-wing drag as well as helping to stall the beam wing and diffuser.

Lasers, plasmas and diffusers

The first laser, invented in 1960 by Theodore Maiman, an engineer-turned-physicist, was a product of the aerospace industry. Maiman was working for Hughes Research Laboratories, and was given nine months and $50,000 to make a laser work by Mr Hughes, (And then there was light, Pauline Rigby, Physics World, May 2010).

Perhaps it's time, then, for the introduction of the laser into Formula One, not merely as a ride-height sensor, but as a flow control device.

Racing cars have, traditionally, used the short-range repulsive forces of solid surfaces to control the flow of air. This, however, is merely one particular, very convenient solution to the engineering problem. Lasers can also control the flow of air, either by directly delivering radiation pressure to a specific region of the airflow, or by creating a plasma whose pressure can instead be used to the same effect.

The question here is merely one of practicality. Do modern lasers combine sufficient power in a lightweight, compact package? Well, probably not quite yet, but there are already some tantalising glimpses of the possibilities.

The Curiosity Rover currently exploring Mars is equipped with a Laser-Induced Breakdown Spectroscopy (LIBS) system. The laser vaporises rocks some distance away, and a separate camera system analyses the light to make inferences about the chemical composition of the rock. Most intriguingly, the weight of this powerful laser system was reduced to 500g.

A LIBS system focuses a laser on an object and ablates the surface layer of the object to create a plasma plume. This high-temperature plume has a momentum flux, and it has long been suggested that this could be used to propel lightweight objects.

In terms of an immediate Formula One application, however, one might install a laser unit low down in the aft region of the sidepods, and train the laser light upon an aluminium foil surface attached to the floor in the region where one currently finds vortex generators, just inside the rear wheel. The resulting plasma plume could be used as a surrogate exhaust jet to seal the diffuser, with perhaps the odd magnet to focus the plasma. Depending upon how the regulations evolve, KERS energy could be used to power the laser.

There are some potential hazards, such as the possibility of vaporising the rear end of the car if the laser is incorrectly focused, but these are small matters in comparison to the potential advantages of a laser-sealed diffuser system.

Race strategy equations



William Mulholland, erstwhile Vehicle Dynamics Engineer for McLaren Inter-Planetary, wrote an interesting introduction to the mathematics of Formula 1 race strategy a few years ago.

Mulholland's account considers only the effect of fuel weight rather than tyre performance, and dates back to the refuelling era, but it's still a decent account of the basic concepts. 

With t₀ denoting the notional lap-time without any fuel weight, W denoting the lap-time deficit per lap of fuel onboard, and l denoting the distance travelled in laps, then the first expression here defines the time which elapses between pitstops for fuel on laps L₁ and L₂.

The solution to this integral is then provided by the expression on the left.

Taking an intrepid approach, we can generalise these equations to include the effect of tyre performance deterioration, and deal with the absence of refuelling.

With t₀ now denoting the lap-time on pristine tyres and empty fuel tanks, L_end denoting the number of laps in the race, and T denoting the lap-time deficit per lap travelled on a set of tyres, we obtain the expression below for the stint-time between laps L₁ and L₂:

The solution to this integral is provided by the following expression:





In the absence of interference from other cars, the optimal number of stops, and the optimal timing of those stops, can be calculated once the time lost during each pit-stop is added to the time which elapses during each stint of the race.

Obviously, these equations are still highly idealised. The actual lap-time deficit T per lap travelled on a set of tyres will be fuel-load and track-condition dependent, hence in reality this will be a function T(l) rather than a constant.

Moreover, the presence of interference from other cars changes the optimal strategy, and introduces uncertainties. Dropping into slower traffic after a pitstop, and being unable to overtake that traffic, prevents a driver exploiting the full performance potential of the car at that point in time. Hence, the optimal number of pitstops in the presence of other cars tends to be less than the optimal number in the absence of other cars.

The unpredictable behaviour of other cars, and the possible occurrence of chance events such as rain and safety cars, entails the introduction of probability distributions over the optimal number and timing of pitstops. Calculating these optima in the presence of other cars becomes an exercise in game theory, where the effect of a decision is influenced by the decisions of other agents, and where the decision of an agent is influenced by that agent's beliefs about the anticipated decisions of the other agents.

Bayesian networks are precisely designed to capture the conditional probabilistic relationships between numerous chance events and unpredictable decisions. Hence Bayesian networks might be very useful for updating the most likely optimal strategy as a race progresses in real-time.

Michael Schumacher's marmite nightmare

Richard Feynman once remarked that experimental particle physics is somewhat akin to smashing watches together, and examining the various gears, cogs and springs which fly out, in order to better understand how such an intricate device is first put together.

Inspired by his academic counterpart, one can only assume that Michael Schumacher has recently developed a deep interest in the transmission internals and rear-wing assembly of a Formula One car, and wishes to better understand how they are put together.

Hurtling down Esplanade Drive at the re-start of last week's Singapore Grand Prix, approaching the ninety-degree right of turn 14, Michael appeared either to mis-judge his braking, or under-anticipate the braking point of those on worn tyres ahead of him. Locking up all four wheels, and slewing marginally sideways, Michael plunged into the rear of Jean-Eric Vergne's Toro Rosso.

On time-scales brief to the human eye, this triggered a complex hierarchy of energy degrading processes over a range of different length-scales. The front-wing of the Mercedes broke off and partially shattered as the nose of the Mercedes was driven into the rear of the Italian car, lifting it into the air. As the Mercedes rode up and over the rear crash structure on the Toro Rosso, the detached front-wing was briefly trapped under the rear of Vergne's car. The front wheels of the Mercedes folded inwards, the rear-wing of the Toro Rosso shattered into a characteristic fragment-size distribution (large numbers of small fragments, small number of large fragments), and a titanium end-plate of the Mercedes front-wing was simultaneously dragged along the road, raising a cascade of sparks, diffraction-spiked in subsequent photos like an astrophysical star cluster. It was a frightening freeze-frame moment of beautiful complexity.

The ultimate cause of the accident, however, may lie with a comment Schumacher made in the October issue of F1 Racing magazine. Here, Michael was quizzed with readers' questions, and at one point was asked if, looking back over his career, he had any regrets. The answer was terse, but revealing:

"Jerez. In 1997."

Now, far from being a confession or admission, this actually constituted an implicit rebuttal. Michael was, in effect, saying that he doesn't regret any of the litany of other transgressions of which he is often accused: the other deliberate accidents; the dragging of wounded cars back onto the track to get races red-flagged; the deliberate blocking of the track in final qualifying, etc etc. None of that is regretted, presumably because Michael was only stripped of points for that one incident in 1997, when, as Nigel Roebuck put it, the FIA came down on him "like a tonne of feathers."

Michael's latest denial was only going to provoke further action from the Karma Police, who've had the former wunderkind in their cross-hairs for some time now. The combination of circumstances on the restart at Singapore was simply their latest move; an elegantly planned and executed slice of 'fate'.

Elsewhere in the same interview, Michael was asked if there was something which irritates him about Formula One today, and his response was somewhat enigmatic: "Black gold." Prompted to elaborate, Michael merely added "Think about it."

Could Michael possibly be expressing concern about the use of petroleum-derived energy sources in Formula One? Surely not; after all, the total greenhouse gas emissions from the sport are an infinitesimal fraction of global emissions, and McLaren Pan-Galactic, for one, announced late last year that they're actually a carbon-neutral organisation. 

Perusing the list of other possible referents for 'black gold', one's eye immediately alights upon that notoriously polarizing condiment, marmite. Joining up the dots to understand Michael's source of vexation, it should be remembered that Lewis Hamilton is Formula One's Mr Marmite: loved by some, loathed by others.

And, as we all found out on Friday this week, Michael's marmite discomfort transpired to be fully justified.

Quasi-particles

A recent BBC Horizon documentary on the smallest things in the universe, claimed to present experimental proof that the electron has been split.

With the tone of indulgent bemusement that characterises contemporary television accounts of modern physics, the viewer was told that, in principle, it is possible to split the fundamental properties of the electron. These properties, we were told, are (intrinsic) spin, (electric) charge and orbital (angular momentum), and in the experiment in question, "when the x-ray beam strikes, the electron split into new quasi-particles, called spinons, orbitons and holons, which carry the properties of the electron, and can travel off in different directions."

In point of fact, the electron hasn't been split at all. Quasi-particles are collective excitations in the state of condensed matter, typically the atomic lattice of a crystal. Some of these collective states possess properties which formally resemble properties possessed by states of the individual electron. In 1996, holons and spinons were produced for the first time in the collective state of a crystal, and now a new collective state, containing orbitons and spinons, has been created.

Quasi-particles, however, are certainly of very deep interest, not least because they reveal that particle-like entities can be created as nothing more than transient disturbances within a substrate. As philosopher of physics David Wallace points out, quasi-particles "can be created and annihilated; they can be scattered off one another; they can be detected (by, for instance, scattering them off 'real' particles like neutrons); sometimes we can even measure their time of flight...We have no more evidence than this that 'real' particles exist...and yet they consist only of a certain pattern within the constituents of the solid-state system in question," (p51, The Emergent Multiverse, OUP, 2012).

Moreover, according to quantum field theory, elementary particles such as electrons are merely excited states of underlying quantum fields. This in itself should undermine confidence in the fundamentality of so-called elementary particles, but unfortunately quantum field theory provides a rather sparse characterisation of what a quantum field actually is, merely specifying it to be a self-adjoint operator-valued field on space-time (technically an operator-valued 'distribution'). The significance of quasi-particles is that they provide a very concrete substrate upon which particle-like entities can be realised, and a discrete substrate at that.

Perhaps, then, there is no lowest level to the structure of the universe; no foundations and no basement level, just an infinite multi-storey subterranean car-park.

Ride-height matters

There was a suggestion on this blog at the beginning of the season that Mercedes's DRS-activated F-duct might be utilising more than one means to reduce straightline drag.

In this context, Adrian Newey recalls that in the early 1990s, when active suspension was permitted, "We realised in the wind-tunnel that if we lowered the rear and raised the front, you could stall the diffuser and that reduced the drag of the car significantly...I can't remember the figure but that would give them something like an extra 10 kph," (p233-234 in Williams, Maurice Hamilton, 2009). 



Stalling the front-wing not only reduces drag, but by virtue of reducing front downforce also increases front ride-height. Hence, using DRS to stall the front-wing could, in principle, also stall the diffuser.

Admittedly, diffuser dimensions were somewhat different in the early 1990s, and there hasn't been any suggestion from rival engineers this year that Mercedes are indeed exploiting this mechanism. It would, for a start, require a car very softly sprung in dive. Nevertheless, it's interesting to look at the ride-height map above, reproduced from Toet, Zhang and Zeridan's 2006 paper, Ground Effect Aerodynamics of Racecars.

The top diagram provides the front downforce contour map, while the image at the bottom offers the rear downforce contours. In each case, front ride-height is on the vertical axis, and rear ride-height is on the horizontal.

It can be seen that front downforce is maximised when the car has maximum rake, with a low front ride-height and a high rear ride-height. This is simply because the front-wing operates in ground-effect. Rear downforce, however, is slightly more subtle because the diffuser downforce is dependent upon (i) the strength of the side-edge vortices, and (ii) the prevention of turbulent ingress from the rotating rear wheels. As the rear ride-height lowers, downforce increases up to a certain point, but if the car goes any lower, the side-edge vortices dilate and weaken, and the diffuser stalls. There is therefore something of an escarpment in the rear downforce contour map.



The other feature of interest in this rear downforce map is the sheer cliff evident when the front ride-height increases to about 30mm. This may be the behaviour Adrian Newey was exploiting in the early 1990s. Sadly, no indication is given of the provenance of these diagrams; they are referred to merely as the downforce contours for a 'generic open wheeled race car'.

One might wish to compare and contrast with some actual ride-height maps for the F1-2000 Ferrari, (Ferrari Formula 1, Peter Wright, David Bull Publishing, p120), reproduced on the left here. One can see the same diagonal pattern to the front downforce contours, and a similar rear downforce plateau. Sadly, these diagrams don't extend up to a front ride-height of 30mm, so it's not possible to discern if the diffuser of an F1 car from the early 2000s could be stalled by raising the front ride-height.

As to what the current state-of-play is, these are questions which can, perhaps, only be answered by a tweet from Lewis Hamilton.