Turbulence in Singapore

The fourth Grand Prix of Singapore will take place this weekend, and whilst the city-state forms an impressive backdrop for a race, overtaking has been notoriously difficult in previous years here. As difficult, in fact, as it is at Valencia, another street-circuit situated at sea level.

Which is intriguing, because the atmospheric density is greater at sea level, and this has certain aerodynamic ramifications. Assuming the pressure at Singapore is 101 kPa (the standard sea level pressure), a temperature of 20 degrees Celsius corresponds to an air density of 1.196 kg/m³.

Singapore, however, is also notoriously humid, and because water vapour is lighter than dry air, humid air is less dense than dry air. Assuming a relative humidity of 80%, and a temperature of 20 degrees, the air density at Singapore is about 1.190 kg/m³. (The tables here are taken from Soil mechanics for unsaturated soils, Fredlund and Rahardjo, p23-24).

In contrast, at a circuit such as Spa Francorchamps, which lies at an altitude of about 400m, the standard atmospheric pressure is about 95 kPa, and at a temperature of 20 degrees the air density is only 1.124 kg/m³.

Now, greater air density increases downforce and drag, but it also changes the Reynolds number:

Re = (airspeed x length x air density)/viscosity of air

The viscosity of air increases as a function of temperature, (as tabulated on the left here), but is largely independent of pressure. Hence, the increased air density entails that the Reynolds number of the airflow at Singapore (and Valencia) will be slightly greater than it is at venues such as Spa.

How much greater? Well, by a factor of 1.190/1.124 = 1.058. In other words, the Reynolds number at Singapore is about 5% greater than it is at Spa.

Now, the Reynolds number specifies the ratio of the inertial forces to the viscous forces, and this is important for quantifying the effect of turbulence. The greater the Reynolds number, the more turbulent the flow. In particular, as a rule-of-thumb, viscous dissipation of turbulent energy only kicks-in when the Reynolds number of the turbulent eddies approaches unity. Thus, if the air density at Singapore is 5% greater than that at Spa, the viscous dissipation of turbulence doesn't kick-in until the turbulent eddies reach a size about 5% smaller than at Spa.

If the premises here are correct, and the reasoning is valid, then the cars will have a slightly longer turbulent wake at Singapore (and Valencia), than they have at venues such as Spa. That would help to explain why it's so difficult to overtake at Singapore (and Valencia).

Nevertheless, circuit design is still by far the most important factor. Zandvoort, after all, was situated amongst the sand dunes bordering the North Sea, yet the racing there was amongst the best you could care to see.

The Miles-Phillips Mechanism

Two distinct mechanisms have been proposed to explain the means by which the wind is capable of generating waves and perturbations on the surface of lakes and oceans: Kelvin-Helmholtz instability (KHI), and the Miles-Phillips Mechanism.

Now, KHI reputedly requires a minimum wind speed of 6 m s^-1 to make waves grow against the competing effects of gravity and surface tension. Thus, whilst KHI is relevant to the generation of large wavelength perturbations, it is the Miles-Phillips Mechanism which is relevant to low wind speeds, and short-wavelength perturbations. In particular, the Miles-Phillips Mechanism involves a resonant interaction between the surface of the water and turbulent fluctuations in the air.

So, in the interests of science, I wandered down acorn-strewn paths to my local lake, to see if I could identify the Miles-Phillips Mechanism in action. What I observed over the course of several days, were a complex sequence of meta-stable and transient patterns. All the photos here were taken at the same time of day, around 2pm.

The first couple of pictures are from Friday 16th September. There was a light breeze blowing from left-to-right here, and this appeared to maintain a band of short wavelength perturbations in the middle of the lake. There is a clearly-defined transition, however, towards the margins of the lake, where the ripples were of a visibly longer wavelength. The shorter modes completely disrupt the reflective properties of the lake, but you can still see distorted images of the surrounding trees in the areas with the longer wavelength disturbances.

This is in sharp contrast with the pattern exhibited on two days previously, when a stable pattern of short-wavelength perturbations covered most of the lake. Note the absence of any reflective images at all.

On Sunday 18th September, the breeze was light, but rapidly fluctuating, and bands of short-wavelength perturbations would arise, and then dissipate, over a timescale of just a few minutes. In the first photo here, virtually the entire surface is smooth and reflective...

But within little more than five minutes, a band of short-wavelength ripples had covered the middle of the lake.

Such patterns would rise and fall, and drift back and forth across the lake as the local wind shifted and fluctuated. The wind variation was imperceptible from the viewpoint of the observer, and the patterns became as inexplicable and mesmerising as a mere screen-saver.

The cost of motorsport books

Here's a rather stark illustration of US/UK pricing differentials.

The Autocourse 60 Years of Grand Prix Motor Racing is available on Amazon.co.uk for a whopping £44.96. Exactly the same book is also available on Amazon.com for the rather more affordable sum of $41.97, which equals £26.58 at current exchange rates.

There is, it seems, a somewhat different pricing strategy in different markets...

Formula 1 aerodynamics in the 1970s

For most of the 1970s, there seems to have been a fundamental schism in the front-end aerodynamic concept of Formula 1 cars. Some of the cars, such as the McLarens, Lotuses and Ferraris, continued to run with front wings, but another group appeared to abandon that concept for most of the decade, running instead a front spoiler/airdam/splitter. This latter group included luminaries such as March, Brabham and Tyrrell, with Jackie Stewart winning the 1971 and 1973 World Championships in Tyrrell designs sporting just such a front-end.

So what was the idea? Well, part of the motivation was presumably to reduce the lift, drag and turbulence created by the front wheels. The front spoilers were much wider than front wings, and partially shrouded the front wheels, diverting airflow down the sides of the car.

So that was part of the idea. The other possible motive is perhaps more interesting, because it involves ground-effect. A spoiler/airdam provides a vertical barrier which (i) maximises the high pressure stagnation point at the front of the car, and (ii) accelerates the airflow through the restricted gap between the spoiler/airdam and the ground surface. A horizontal splitter projecting from the bottom of the spoiler/airdam then takes advantage of the high pressure of the stagnation point to generate some extra downforce.

A front airdam/spoiler is partially, then, a ground-effect device, which perhaps explains why cars such as the Brabhams and Tyrrells were still able to win Grands Prix against those utilising conventional front-wing arrangements. The photo here shows a Tyrrell running quite a degree of rake, which would serve to accentuate the ground-effect of the front spoiler.

So, perhaps surprisingly, ground-effect in Formula 1 actually predates the underbody venturi tunnels and skirts used on the Lotus 78/79. And in fact, Gordon Murray began experimenting with ground-effect on the Brabham BT44 back in 1974, arriving at "an inch-deep underbody vee, something like a front airdam, but halfway down the car." (Vacuum Clean-Up, Adam Cooper, Motorsport, May 1998, pp64-69).

The introduction of underbody venturi and skirts presumably spelt the death-knell for front spoilers, as the emphasis then shifted to feeding the underbody with as much airflow as possible. Still, it would be interesting to hear from those involved, what the initial impetus was for adopting those spoilers, and how effective they really were.

Suspension camber in Grand Prix racing

Formula One's latest cause celebre revolves around Red Bull's decision to race at Spa with a greater degree of negative front-wheel camber than recommended by Pirelli.

Negative camber simply means that both wheels are inclined inwards at the top. The benefit of this is that the outer wheel generates greater lateral force on the entry to a corner (so-called camber thrust, similar to the way a motorbike rider generates lateral force by keeling the bike over), but the disadvantage is that the inner shoulders of both front tyres will suffer greater stress when the car runs in a straightline, and at Spa this caused both Red Bull drivers to suffer tyre blisters.

It's interesting to recall, however, that in the pre-war era of Grand Prix racing, the cars were actually set-up with visible levels of positive front-end camber. In other words, the front-wheels were inclined outwards at the top.

So why was this? Well, there seem to be at least two distinct reasons. The first was relevant prior to the mid-1930s, when cars employed what now look like rather primitive beam axle front suspension systems. Under the extra load generated by braking, the front axle would sag, and pull the front wheels inward at the top, as illustrated in this diagram taken from Matt Joseph's excellent 'Collector Car Restoration Bible: Practical Techniques for Professional Results'. Thus, a degree of positive static camber was necessary to offset this effect.

The eventual transition to independent, double-wishbone, ball-joint suspension, meant that wheel camber was no longer affected by the loads generated under straightline braking (or acceleration). However, even after the adoption of more modern suspension in the mid-1930s, the Mercedes and Auto Union Grand Prix cars continued to run with appreciable levels of positive camber. The primary reason for this appears to involve a concept called the scrub radius.

Now, when the front wheels of a car are steered, the wheels pivot around some axis. Originally, this steering axis was implemented with a physical rod called a king-pin, which was attached to each end of the beam axle. With independent, double-wishbone suspension, this king-pin is replaced by the line drawn between the upper and lower ball-joints at the outer end of the wishbones. This axis is also the line along which the weight of the car is projected down to the ground. The distance between the point where this line intersects the ground and the contact patch of the tyre, is called the scrub radius.

As Joseph explains (p261), a non-zero scrub radius causes several problems: it puts large forces into the king-pins; it acts like a lever, thereby putting large shocks into the steering; and it makes it harder to steer a car. Positive camber was the common solution devised for minimising the scrub radius. If the wheels are inclined outwards at the top, then the contact patches will be placed directly under, or at least closer to, the point where the steering axis intersects the road surface.

There's just one more complication to consider. Under the chassis roll generated by cornering, a double-wishbone suspension system will experience a positive camber increment on the more heavily loaded outer wheel, and a negative camber change on the lightly-loaded inner wheel. By setting a car up with a degree of positive static camber, this will result in the outer wheel acquiring an even greater degree of positive camber during cornering, while the inner wheel reaches a more vertical inclination, as nicely demonstrated in the photo of the Mercedes above.

Spot the difference

This is Vittorio Brambilla, otherwise know as the Monza Gorilla, and best remembered for crashing immediately after winning the 1975 Austrian Grand Prix.

Not to be confused with...

Michela Vittoria Brambilla, Italian beauty queen, philosophy-graduate, businesswoman, and erstwhile Minister of Tourism in Silvio Berlusconi's government.