Sunday, July 14, 2019

The problem of refuelling

FIA President Jean Todt has floated the idea of re-introducing refuelling to Formula 1, largely it seems to reduce the running-weight of the cars. According to Andrew Benson's BBC report, "Todt said he had been warned that the reintroduction of refuelling would likely lead to teams' race strategies being too similar to each other but countered that that was a product of there being too much simulation in F1."

Quite. So let's have a quick look at why refuelling doesn't necessarily make race-strategy more interesting. Suppose we have the following fairly typical parameter values:

Tyre-deg: 0.05 sec/lap
Fuel-effect: 0.033 sec/kg
Fuel-consumption: 1.5 kg/lap

Suppose that the deterministically optimal first pit-stop would be after 20 laps. That requires a fuel-load of 30kg. Suppose that the second stint would also require a fuel-load of 30kg. The lap-time penalty for 30kg of fuel would be 30*0.033 = 1 sec.

After 20 laps, the cumulative tyre degradation would be 20*0.05 = 1 sec.

Suppose two cars with identical zero-fuel-load lap-times are racing each other. As we approach the first pit-stop window, there would be no benefit of the car behind trying to pit first to undercut the car ahead: the penalty of taking on 30kg of fuel cancels out the advantage of switching to a fresh set of tyres. The conventional undercut logic, which permits overtaking between equally matched cars, would be lost.

With this set of parameter values, there would also be no benefit to the car behind running longer: the cumulative tyre deg cancels out the benefit of lapping on almost empty fuel tanks. However, one might suspect that this result only follows from choosing a special set of parameter values, so let's assume that the tyre-deg is lower, at 0.03 sec/lap. Surely this would tip the balance in favour of running longer?

Well, suppose the car behind is planning to run 3 laps further in the race, to lap 23. That requires a starting fuel-load of 23*1.5 = 34.5kg. That's an extra 4.5kg which the car behind needs to carry around for the first 20 laps of the race. With a fuel-effect of 0.033 sec/kg, that's a lap-time penalty of 0.033*4.5 = 0.15 secs on every lap of the first 20 laps. So, if we assume both cars are running in free air, the car behind would have lost 3 secs of cumulative time after 20 laps (assuming the extra weight didn't also increase the tyre-deg).

With the assumed tyre-deg of 0.03 secs/lap, the cumulative deg after 20 laps would be 0.6 secs. Which is less than the 1 sec penalty for taking on 30kg of fuel. Hence, the car running to lap 23 would make up 0.4 secs/lap on the car which has pitted on lap 20. Over 3 laps, that would be 1.2 secs.

Would the car behind be able to overcut the car ahead? Unfortunately not. That extra fuel-weight has already cost it 3 secs over the first 20 laps of the race. The 1.2 secs regained still leaves a net loss of 1.8 secs when it finally pits on lap 23.

Obviously, if both cars were running in traffic, and the car ahead was unable to exploit its superior potential lap-time over the first 20 laps, then the overcut might still work.

In summary, however, we can see why refuelling pushes strategies towards the deterministic optima: if you try to overcut an opponent, the greater fuel-weight necessary for that is counter-productive; conversely, if there's any benefit to be had from undercutting an opponent, that benefit would be even greater in the absence of refuelling. 

Saturday, June 29, 2019

Flocculation and the Payne effect

With the quality of racing in contemporary Formula One reaching something of a nadir, some parties have sought a quick-fix by proposing that Pirelli revert to their thicker-gauge 2018 tread design. With tyres back on the agenda, then, perhaps it's a good moment to look a little bit deeper at the composition of a racing tyre tread. 

A modern pneumatic tyre-tread contains rubber. Rubber itself consists of long chains of polymer molecules. The chains are mutually entangled, and in its raw form it is a highly-viscous liquid. It is not, however, elastic. It only becomes a viscoelastic solid when it undergoes 'vulcanization', whereby sulphur crosslinks are created between the molecular chains. This transforms the already entangled collection of polymer chains into a 3-dimensional network. 

So far, so familiar. However, a modern tyre-tread is a rubber composite. In addition to the network of vulcanized rubber, it contains a network of 'filler' particles. These filler particles are not just dispersed as isolated particles in the rubber matrix; rather, they agglomerate into their own 3-dimensional network. (The term for this agglomeration is 'flocculation').

The rubber network and filler network interpenetrate each other. Hence, the elasticity, viscosity, and ultimately the frictional grip of a tyre-tread is attributable to three sources: (i) the cross-links and friction between rubber polymer molecules; (ii) the bonds between filler particles; and (iii) the bonds between filler particles and the rubber molecules.

'High-performance' racing tyres, of course, are something of a world of their own, and tend to use carbon-black as a filler in high concentrations because it increases hysteresis (i.e., viscous dissipation) and grip. One can find statements in the academic literature such as the following:

"For a typical rubber compound, roughly half of the energy dissipation during cyclic deformation can be ascribed to the agglomerated filler, the rest coming from [rubber polymer] chain ends and internal friction [of polymer network chains]," (Ulmer, Hergenrother and Lawson, 1988, 'Hysteresis Contributions in Carbon Black-filled rubbers containing conventional and tin end-modified polymers').

Given the higher concentration of filler in a racing tyre, one might expect more than half of the energy dissipation, and therefore the frictional grip, to come from the agglomerated filler.

And now comes the interesting bit. Filled rubber compounds suffer from the 'Payne effect'. This is typically defined by the variation in both the storage modulus, and the loss modulus (or tan-delta) of the tyre when it is subjected to a strain-sweep under cyclic loading conditions. (The storage modulus is related to the elasticity or stiffness of the material, and the loss modulus is related to the viscous dissipation).

Typical graphs, such as that above, show that the storage modulus decreases as the amplitude of the strain is increased, whilst the loss-modulus or tan-delta reaches a peak at strains of 5-10%.

The Payne effect is typically attributed to the breaking of bonds between filler particles, as Pirelli World Superbike engineer Fabio Meni attests:

'Riders constantly talk about how their tires "take a step down" after a few laps, so I asked Meni what physical process in the rubber is responsible for this perceived drop in properties. "This has a name", he began. "It is called the Payne effect."

"In the compound," Meni continued, "the carbon-black particles are not present as separate entities but exist as aggregates - clusters of particles. As the tire is put into service, the high strains to which it is subjected have the effect of breaking up these aggregates over time, and this alters the rubber's properties."

Meni went on to say that it is not so much that the tire loses grip as it feels different to the rider. This is 'the step' that the rider feels after a few laps, after which the tire's properties may change little through the rest of the race.

In fact, I would quibble with this slightly: in the world of Formula One tyres, the way in which a tyre is treated at the beginning of a stint will often determine the subsequent degradation slope. If you abuse a tyre, it remembers it, and punishes you. Damage to the filler network appears to reduce grip, not merely soften a tyre.

One intriguing twist to the Payne effect is that there may be circumstances in which it is possible for a tyre to recover from damage to the filler network: "Much of the softening remains when the amplitude [of the strain in a cyclic strain-sweep] is reduced back to small values and the original modulus is recovered only after a period of heating at temperatures of the order of 100 degrees C or higher," (A.N.Gent, 'Engineering with Rubber', 2012, p115).

So heat is capable of annealing a damaged filler network, restoring the bonds between filler particles. The anneal temperature quoted here by Gent is not dissimilar to the maximum tyre-blanket temperatures currently permitted by Pirelli in Formula One...

As a final flourish on this subject, for those who like a bit of scanning electron microscopy, images of filler-reinforced rubber which has been in a state of slip across a rough surface, reveal that there is a modified 'dead' surface layer, about a micron-thick, in which the carbon-black filler particles are absent, (image below from work conducted by Marc Masen of Imperial College).

To paraphrase Homer Simpson, "Here's to tyres: the cause of, and solution to, all of Formula One's problems."

Wednesday, February 06, 2019

Assessing the nuclear winter hypothesis

After falling into disrepute for some years, the nuclear winter hypothesis has enjoyed something of a renaissance over the past decade. In the January 2010 edition of Scientific American, two of the principal proponents of the hypothesis, Alan Robock and Owen Brian Toon, published an article summarising recent work. This article focused on the hypothetical case of a regional nuclear war between India and Pakistan, in which each side dropped 50 nuclear warheads, with a yield of 15-kilotons each, on the highest population density targets in the opponent's territory.

Robock and his colleagues assumed that this would result in at least 5 teragrams of sooty smoke reaching the upper troposphere over India and Pakistan. A climate model was developed to calculate the effects, as Robock and Toon report:

"The model calculated how winds would blow the smoke around the world and how the smoke particles would settle out from the atmosphere. The smoke covered all the continents within two weeks. The black, sooty smoke absorbed sunlight, warmed and rose into the stratosphere. Rain never falls there, so the air is never cleansed by precipitation; particles very slowly settle out by falling, with air resisting them...

"The climatic response to the smoke was surprising. Sunlight was immediately reduced, cooling the planet to temperatures lower than any experienced for the past 1,000 years. The global average cooling, of about 1.25 degrees Celsius (2.3 degrees Fahrenheit), lasted for several years, and even after 10 years the temperature was still 0.5 degree C colder than normal. The models also showed a 10 percent reduction in precipitation worldwide...Less sunlight and precipitation, cold spells, shorter growing seasons and more ultraviolet radiation would all reduce or eliminate agricultural production.," (Scientific American, January 2010, p78-79).

These claims seem to have been widely believed within the scientific community. For example, in 2017 NewScientist magazine wrote a Leader article on the North Korean nuclear problem, which asserted that: "those who study nuclear war scenarios say millions of tonnes of smoke would gush into the stratosphere, resulting in a nuclear winter that would lower global temperatures for years. The ensuing global crisis in agriculture – dubbed a “nuclear famine” – would be devastating," (NewScientist, 22nd April 2017).

But is there any way of empirically testing the predictions made by Robock and his colleagues? Well, perhaps there is. In 1945, the Americans inflicted an incendiary bombing campaign on Japan prior to the use of nuclear weapons. Between March and June of 1945, Japan's six largest industrial centres, Tokyo, Nagoya, Kobe, Osaka, Yokohama and Kawasaki, were devastated. As military historian John Keegan wrote, “Japan's flimsy wood-and-paper cities burned far more easily than European stone and mid-June...260,000 people had been killed, 2 million buildings destroyed and between 9 and 13 million people made July 60 per cent of the ground area of the country's sixty larger cities and towns had been burnt out,” (The Second World War, 1989, p481).

This devastation created a huge amount of smoke, so what effect did it have on the world's climate? Well Robock and Brian Zambri have recently published a paper, 'Did smoke from city fires in World War II cause global cooling?', (Journal of Geophysical Research: Atmospheres, 2018, 123), which addresses this very question.

Robock and Zambri  use the following equation to estimate the total mass of soot $M$ injected into the lower stratosphere:
M = A\cdot F\cdot E\cdot R \cdot L \;.
$$ $A$ is the total area burned, $F$ is the mass of fuel per unit area, $E$ is the percentage of fuel emitted as soot into the upper troposphere, $R$ is the fraction that is not rained out, and $L$ is the fraction lofted from the upper troposphere into the lower stratosphere. Robock and Zambri then make the following statements:

"Because the city fires were at nighttime and did not always persist until daylight, and because some of the city fires were in the spring, with less intense sunlight, we estimate that L is about 0.5, so based on the values above, M for Japan for the summer of 1945 was about 0.5 Tg of soot. However, this estimate is extremely uncertain."

But then something strange happens at this point, because the authors make no attempt to quantify the uncertainty, or to place confidence intervals around their estimate of 0.5 teragrams.

I'll come back to this shortly, but for the moment simply note that 0.5 teragrams is one-tenth of the amount of soot which is assumed to result from a nuclear exchange between India and Pakistan, a quantity of soot which Robock and his colleagues claim is sufficient to cause a worldwide nuclear winter.

Having obtained their estimate that 0.5 teragrams of soot reached the lower stratosphere in 1945, Robock and Zambri examine the climate record to see if there was any evidence of global cooling. What they find is a reduction in temperatures at the beginning of 1945, before the bombing of Japan, but no evidence of cooling thereafter: "The injection of 0.5–1 Tg of soot into the upper troposphere from city fires during World War II would be expected to produce 0.1–0.2 K global average cooling...when examining the observed signal further and comparing them to natural variability, it is not possible to detect a statistically significant signal."

Despite this negative result, Robock and Zambri defiantly conclude that "Nevertheless, these results do not provide observational support to counter nuclear winter theory." However, the proponents of the nuclear winter hypothesis now seem to have put themselves in the position of making the following joint claim:

'5 teragrams of soot would cause a global nuclear winter, but the 0.5 teragrams injected into the atmosphere in 1945 didn't make a mark in the climatological record.'

Unfortunately, their analysis doesn't even entitle them to make this assertion, precisely because they failed to quantity the uncertainty in that estimate of 0.5 teragrams. The omission rather stands out like a sore thumb, because there are well-known, routine methods for calculating such uncertainties.

Let's go through these methods, starting with the formula $M = A\cdot F\cdot E\cdot R \cdot L \;.$ The uncertainty in the input variables here propagates through to the uncertainty in the output variable, the mass $M$. It seems reasonable to assume that the input variables here are mutually independent, so the uncertainty $U_M$ in the output variable can be inferred by a simple formula from the uncertainties attached to each of the input variables:
U_M = \sqrt{(U_A^2 + U_F^2+U_E^2+U_R^2+U_L^2)} \;.
$$ $U_A$ is the uncertainty in the total area burned, $U_F$ is the uncertainty in the mass of fuel per unit area, $U_E$ is the uncertainty in the percentage of fuel emitted as soot into the upper troposphere, $U_R$ is the uncertainty in the fraction that is not rained out, and $U_L$ is the uncertainty in the fraction lofted from the upper troposphere into the lower stratosphere.

Next, to infer confidence intervals, we can follow the prescriptions of the IPCC, the Intergovernmental Panel on Climate Change. The 2010 Scientific American article boasts that Robock is a participant in the IPCC, so he will surely be familiar with this methodology.

First we note that because $M$ is the product of several variables, its distribution will tend towards a lognormal distribution, or at least a positively skewed distribution resembling the lognormal. The IPCC figure below depicts how the upper and lower 95% confidence limits can be inferred from the uncertainty in a lognormally distributed quantity. The uncertainty $U_M$ corresponds to the 'uncertainty half-range' in IPCC terms.  

The IPCC figure "illustrates the sensitivity of the lower and upper bounds of the 95 percent probability range, which are the 2.5th and 97.5th percentiles, respectively, calculated assuming a lognormal distribution based upon an estimated uncertainty half-range from an error propagation approach. The uncertainty range is approximately symmetric relative to the mean up to an uncertainty half-range of approximately 10 to 20 percent. As the uncertainty half-range, U, becomes large, the 95 percent uncertainty range shown [in the Figure above] becomes large and asymmetric, "(IPCC Guidelines for National Greenhouse Gas Inventories - Uncertainties, 3.62).

So, for example, given the large uncertainties in the input variables, the uncertainty half-range $U_M$ for the soot injected into the lower stratosphere in 1945 might well reach 200% or more. In this event, the upper limit of the 95% confidence interval would be of the order of +300%. That's +300% relative to the best estimate of 0.5 Tg. Hence, at the 95% confidence level, the upper range might well extend to the same order of magnitude as the hypothetical quantity of soot injected into the stratosphere by a nuclear exchange between India and Pakistan. 

Thus, the research conducted by Robock and Zambri fails to exclude the possibility that the empirical data from 1945 falsifies the nuclear winter hypothesis for the case of a regional nuclear exchange. 

In a sense, then, it's clear when Robock and Zambri refrained from including confidence limits in their paper. What's more perplexing is how and why this got past the referees at the Journal of Geophysical Research...

Sunday, January 13, 2019

Thruxton British F3 1989

The thickness of the atmospheric thermal boundary layer falls to a global minimum over Thruxton. Hence Thruxton is very cold. So much so, in fact, that the British Antarctic Survey have a station there, built into the noise-attenuation banking at the exit of The Complex, (much like a Hobbit-hole), where the younger scientists train to work in a frozen environment before travelling to the Halley Research Station on the Brunt ice shelf.

The late Paul Warwick in the Intersport Reynard. Puzzlingly, in the background there appear to be no takers for the shelter provided by the parasols.
In 1869, John Tyndall discovered why the sky is blue. If he'd lived in Thruxton, the question wouldn't even have occurred to him. Note the characteristic Wiltshire combination of distant mist, a stand of lifefless trees, and flat wind-swept expanses.
Marshals assist a driver who has entered a turnip field. The Wiltshire economy is entirely dependent upon (i) the annual turnip yield, and (ii) government subsidies into the thousand-year consultation process for a Stonehenge bypass/tunnel. The buildings in the background are what people from Wiltshire refer to as a 'collection of modern luxury flats and town-houses.' 
One of the drivers is distracted by an ancient ley line running tangential to the Brooklands kink.

Friday, January 11, 2019

Silverstone Tyre Test 1990

Generally speaking, it was impossible to see a car with the naked eye at Silverstone. However, with the assistance of the world's best astronomical optics, I was occasionally able to pluck an image out of the infinitesimally small strip separating the cold, grey sky from the wooden fence posts and metal railings.

Satoru Nakajima in the pioneering raised-nose Tyrrell 019. This image was obtained with the Wide Field and Planetary Camera on the Hubble Space Telescope.
Alessandro Nannini in the Benetton. This shot was taken with the 100-inch reflector on Mount Wilson.
Nigel Mansell, lighting up the front brake discs as he prepares for the turn-in to Copse. Nigel set the fastest lap on the day I attended the test; a mid-season pattern of performance which led Ferrari to reward Nigel by handing his chassis over to Prost.
Ayrton Senna in characteristic pose, head dipped forward and tilted towards the rapidly approaching apex of Copse corner .

Thursday, January 03, 2019

Formula One and Electro-Aerodynamics

Most travelling Formula One engineers probably think that an 'ionic wind' is the result of over-indulgence at the end-of-season curry night. On the contrary, in late 2018 a group of researchers from MIT published a paper in Nature detailing how an ionic wind was used for the first-ever flight of a heavier-than-air, self-propelled device with no mechanical moving parts.

The ionic wind was created by generating an ionic cascade between the paired elements in an array of high-voltage electrodes. Each positive electrode was a 0.2mm stainless-steel wire supported in front of a wing-section. The corresponding negative electrode was a thin layer of aluminium foil on the downstream wing-section. The ions are accelerated in the electric field, and impart some of their momentum to the ambient air-flow, thereby generating a forward thrust, (and in this case, presumably, some lift).

Ultra-light power-sources were used: a custom-made 600W battery, and a custom-made High-Voltage Power Converter (HVPC), yielding a DC voltage of ~40kV. The battery weighed only 230g, and the HVPC weighed 510g.

The thrust generated by the experimental device was ~3N, from a wing-span of 5.14m, so the thrust itself isn't about to grab the attention of the Formula One community. However, one might instead be tempted to re-task such ionic wind devices with accelerating the boundary layer flow in certain areas, enabling one to avoid separation at moments of extremis.
Ionic wind accelerating the flow at the bottom of the boundary layer. (From 'Ionic winds for locally enhanced cooling', Go, Garimella, Fisher & Mongia, Journal of Applied Physics 102, 2007). This retards separation by delaying the point at which the slope of the velocity profile, at the wall, becomes zero.
Plasma-actuators for boundary layer control have been under aeronautical development for some years, and unfortunately their use in Formula One seems to have already been proscribed. The Technical Working Group notes for December 2006 contain a request for clarification on the issue from James Allison, and in response Charlie Whiting declares that he had "already given a negative opinion, based on moving parts influencing the car's aerodynamics."

This is a slightly puzzling response, because the whole point about plasma actuators and ionic winds is that they involve no moving mechanical parts. The objects in motion are electrical currents, and the ambient airflow itself, both of which are considered to be consistent with the regulations, and indeed necessary for the function of a Formula One car.

So perhaps there's a future here for electro-aerodynamics in Formula One. It would be an exciting line of research, and one which might also be considered beneficial to Formula One's environmental credentials.  

Monday, December 31, 2018

Andrew Ridgeley crashes at Brands

It's the 1986 Cellnet Superprix, the rain is relentless, and it's dark, very dark. Too dark, in fact, for those amateur photographers foolish enough to have arrived with nothing other than 100 ASA film...

The field is strong, containing future luminaries such as Martin Donnelly, Perry McCarthy, Damon Hill, Gary Brabham, Andy Wallace, David Leslie and Julian Bailey. Also entered is Andrew Ridgeley, George Michael's partner in contemporary pop-duo Wham!

Failing to compensate for the reduction in the effective power-spectrum of the road-surface, Ridgeley Go-Goes into the gravel at Druids...
His momentum is somewhat checked by the Cellnet cars of Ross Cheever and David Hunt, which have already found the tyre-wall.
Nothing for it but to head back to the Kentagon for a warming cup of tea.

Friday, December 28, 2018

Brands Hatch F3000 1987

Second installment of the McCabism photographic archive. The place once again is Brands Hatch, this time for a round of the European F3000 championship. By 1987 the Grand Prix had moved to the numbing wind-swept expanses of Silverstone, so this was very much Brands' biggest single-seater race of the year.

F3000 cars of the era tended to be sans engine cover. This is the Madgwick Motorsport Lola of Andy Wallace.
The absent engine cover facilitated tall intake trumpets, which maximise torque at lower revs, F3000 engines being restricted to 9,000rpm. (When the intake valve opens, it creates a rarefaction wave, which propagates to the top of the open-ended trumpet, and reflects as a compression wave. The lower the revs, the longer the necessary trumpet length so that the compression wave returns just as the intake valve is closing, pushing more mass into the combustion chamber). This is the Pavesi Racing Ralt of Pierluigi Martini.
This is a motor racing circuit. It possesses gradient and contour, exists in a natural setting, and is distinguishable from a go-kart track.
First corner of the race, and Julian Bailey has immediately overtaken the front-row sitters, Gugelmin and Moreno in their Ralts.
Andy Wallace moves past Moreno into 3rd, but Bailey looks comfortable in the lead.
Stefano Modena ahead of Yannick Dalmas, the latter displaying a black tyre-mark on his nose-cone as evidence of prior contact with Modena. Dalmas had tapped Modena into a spin at Druids, but courteously waited for Modena to resume ahead of him.
Michel Trolle has a territorial dispute at Druids with A Yorkshireman. This inevitably results in a large accident, Trolle's car flipping upside down onto the top of the tyre-wall.
A slightly shocked Trolle stumbles to his feet after being hauled from the upturned car.
The race is red-flagged and Bailey awarded victory.

Sunday, December 23, 2018

Brands Hatch Tyre Test 1986

Time to dip into the McCabism personal photographic archive. The occasion here is the 1986 pre-Grand Prix Tyre Test at Brands Hatch.

Proper drivers, proper cars, and a proper circuit.

Nigel Mansell zipping down the pit-straight in a car weighing less than 700kg.
Stefan Johansson's Ferrari, in the days before people from Northern Europe and South Africa introduced the Scuderia to the exciting world of stable balanced downforce.
Nelson Piquet, plotting and scheming as he brakes for the hairpin at Druids.

Ayrton Senna negotiating the curved flow at Hawthorns.
Poking my camera over the bridge parapet to get a shot of Keke Rosberg accelerating down the hill to Clearways.
Nigel Mansell in the pits, complaining of an excessive surface/bulk tyre temperature delta, and some inconsistent behaviour from the Y250 vortex.
Ayrton Senna cresting the rise after the sylvan delights of Dingle Dell, and preparing to turn into Dingle Dell corner.
Keke Rosberg in the McLaren, somehow living to tell the tale without a halo for protection.

Thursday, July 19, 2018

Understanding tyre compound deltas

Pirelli revealed at the beginning of the 2018 F1 season that it was using new software to help it choose the three tyre compounds available at each race. Pirelli's Racing Manager Mario Isola commented:

"It's important that we collect the delta lap times between compounds to decide the selection. If we confirm the numbers that we have seen in Abu Dhabi [testing in November] - between soft and supersoft we had 0.6s, and supersoft to ultrasoft was 0.4s - depending on that, we can fine tune the selection and try to choose the best combination."

Getting the tyre compound deltas correct is indeed a crucial part of F1 race strategy, so let's review some of the fundamental facts about these numbers. The first point to note is that tyres are a performance multiplier, rather than a performance additive

To understand this in the simplest possible terms, consider the following equation:
$$F_y = \mu F_z $$ This states that the lateral force $F_y$ generated by a tyre is a product of the coefficient of friction $\mu$, and the vertical load $F_z$. All other things being equal, the greater the lateral force generated by a car in the corners, the faster the laptime. (Note, however, that in many circumstances one would wish to work with lateral acceleration rather than lateral force, given the influence of car-mass on lateral acceleration).

Now, suppose we have a base compound. Let's call it the Prime, and let's denote its coefficient of friction as $\mu_P$. Let's consider a fixed car running the Prime tyre with: (i) a light fuel-load, and (ii) a heavy fuel-load. 

Let's really simplify things by supposing that the performance of the car, and its laptime, can be reduced to a single vertical load due to downforce alone, and a single lateral force number. When the car is running a heavy fuel load, it will generate a downforce $F_z$, but when it's running a light fuel load it will be cornering faster, so the vertical load due to downforce will be greater, $F_z + \delta F_z$. (Recall that the contribution of greater fuel weight to vertical load results in a net loss of lateral acceleration due to weight transfer). The lateral forces will be as follows:

Prime tyre. High fuel-load

$\mu_P  F_z $

Prime tyre. Low fuel-load

$\mu_P (F_z + \delta F_z) = \mu_P F_z + \mu_P\delta F_z$

Now, let's suppose that there is a softer tyre compound available. Call it the Option. Its coefficient of friction $\mu_O$ will be greater than that of the Prime, $\mu_O = \mu_P + \delta \mu$. 

Consider the performance of the same car on the softer compound, again running a light fuel-load and a heavy fuel-load:

Option tyre. High fuel-load

$\mu_O  F_z = ( \mu_P +\delta \mu )  F_z $

Option tyre. Low fuel-load

$\mu_O (F_z + \delta F_z) = ( \mu_P +\delta \mu )(F_z + \delta F_z) $

So far, so good. Now let's consider the performance deltas between the Option and the Prime, once again using lateral force as our proxy for laptime. 

High-fuel Option-Prime delta

$( \mu_P +\delta \mu )  F_z-\mu_P  F_z = \delta \mu F_z$

Low-fuel Option-Prime delta

$( \mu_P +\delta \mu )(F_z + \delta F_z)-\mu_P (F_z + \delta F_z)=\delta \mu (F_z + \delta F_z)$

Notice that sneaky extra term, $\delta \mu \delta F_z$, in the expression for the low-fuel compound delta? As a consequence of that extra term, the Option-Prime delta is greater on a low fuel load than a heavy fuel-load. As promised, tyre-grip is a performance multiplier.

If you scrutinise the compound deltas in each FP2 session, you'll see that the low-fuel compound deltas from the beginning of the session are indeed greater than those from the high-fuel running later in the session. 

Given that the compound deltas input into race-strategy software are generally high-fuel deltas, one could make quite a mistake by using those low-fuel deltas. In fact, parties using low-fuel deltas might be surprised to see more 1-stop races than they were expecting.

There is another important consequence of the fact that tyres are performance multipliers: the pace gap between faster cars and slower cars increases when softer tyres are supplied. The faster cars have more downforce, and therefore more vertical load $F_z$ than the slower cars, at any equivalent fuel-weight. The delta in vertical load is multiplied by the delta in the coefficient of friction, and all things being equal, the faster cars duly benefit from that extra $\delta \mu \delta F_z$.  

Of course, that qualification about 'all things being equal', hides some complex issues. For example, softer tyres have a lower 'cornering stiffness', (i.e., the gradient of lateral force against slip-angle). A softer tyre therefore generates peak grip at a higher slip-angle than a harder tyre. If the aerodynamics of a car are particularly susceptible to the steering angle of the front wheels, then such a car might struggle to gain proportionately from the greater grip theoretically afforded by a softer tyre. Such a car would also appear to gain, relative to its opposition, towards the end of a stint, when the tyres are worn and their cornering stiffness increases.

Notwithstanding such qualifications, the following problem presents itself: the softer the tyres supplied to the teams in an attempt to enhance the level of strategic variety, the greater the pace-gaps become, and the less effect that strategic variety has...

Tuesday, May 15, 2018

Front-wing in yaw

Armchair aerodynamicists might be interested in a 2015 paper, 'Aerodynamic characteristics of a wing-and-flap in ground effect and yaw'. 

The quartet of authors from Cranfield University analyse a simple raised-nose and front-wing assembly, consisting of a main-plane and a pair of flaps, equipped with rectangular endplates. On each side of the wing, three vortices are created: an upper endplate vortex, a lower endplate vortex, and a vortex at the inboard edge of the flap. (The latter is essentially a weaker version of the Y250 which plays such an important role in contemporary F1 aerodynamics). 

The authors assess their front-wing in yaw, using both CFD and the wind-tunnel, and make the following observations:

1) In yaw, vortices generated by a lateral movement of air in the same direction as the free-stream, increase in strength, whereas those which form due to air moving in the opposite direction are weakened.

2) The leeward side of the wing generates more downforce than the windward side. This is due to an increase in pressure on the leeward pressure surface and a decrease in suction on the windward suction surface. The stagnation pressure is increased on the inner side of the leeward endplate, and the windward endplate partially blocks the flow from entering the region below the wing.

3) A region of flow separation occurs on the windward flap suction surface.

4) Trailing edge separation occurs in the central region of the wing. This is explained by the following: (i) The aluminium wing surface was milled in the longitudinal direction, hence there is increased surface roughness, due to the material grain, for air flowing spanwise across the surface; (ii) There is a reduction in the mass flow-rate underneath the wing; (iii) The effective chord-length has increased in yaw.

5) The vortices follow the free-stream direction. Hence, for example, the windward flap-edge vortex is drawn further towards the centreline when the wing is in yaw.

One comment of my own concerns the following statement:

"The yaw rate for a racing car can be high, up to 50°/sec, but is only significant aerodynamically during quick change of direction events, such as initial turn-in to the corner. The yaw angle, however, is felt throughout the corner and is usually in the vicinity of 3-5°. Although the yaw angle changes throughout the corner the yaw rate is not sufficiently high, other than for the initial turn-in event, to warrant any more than quasi-static analysis."

This is true, but it's vital to point out that the stability of a car in the dynamic corner-entry condition determines how much speed a driver can take into a corner. If the car is unstable at the point of corner-entry, the downforce available in a quasi-static state of yaw will be not consistently accessible. 

Aerodynamicists have an understandable tendency to weight conditions by their 'residency time'. i.e., the fraction of the grip-limited portion of a lap occupied by that condition. The fact that the high yaw-rate corner-entry condition lasts for only a fraction of a second is deceptive. Minimum corner speed depends not only on the downforce available in a quasi-static state of yaw, but whether the driver can control the transition from the straight-ahead condition to the quasi-static state of yaw.