The connection between Formula 1 and Dark Energy

Perhaps to the surprise of many aerodynamicists, one of the concepts deeply familiar to them, may actually explain the existence of dark energy.

This ominous form of energy corresponds to an extra source term in the cosmological version of the Einstein field equations, and is commonly held to be responsible for the accelerating expansion of the universe.

The connection with aerodynamics lies with a concept used to represent turbulent flow, called Reynolds Averaging. Despite the supercomputer power available in Formula 1, it remains impractical to solve the exact Navier-Stokes equations for a turbulent flow. Instead, the velocity and pressure fields are split into the sum of a mean field and a fluctuating component, and the Reynolds-Averaged Navier-Stokes equations are solved to calculate the mean fields.

However, the process of taking the average of the Navier-Stokes equations generates an extra term, which is interpreted as an extra source of stress. One thinks of the mean field as having to do work against the extra stresses created by turbulent eddies, as well as the viscous stresses already inherent in the fluid.

Thus, whilst the Navier-Stokes equations for a steady flow are as follows:







The Reynolds-Averaged equations pick up an extra term on the right-hand side:







Now, as cosmologist George Ellis and his colleagues explain, whilst the exact Einstein equations hold at the local scale:



Once one averages the matter-energy and geometry over galactic scales, the Einstein equations pick up an extra term:



Sound familiar? As Ellis et al comment, this "is the effective matter source term representing the effect of averaging out smaller scale structures."

Whether this type of effect can explain the quantitative cosmological data remains to be seen, but perhaps aerodynamicists have something to offer the cosmological community here.

Wastegate-blown diffusers

Whilst the FIA have made extensive efforts to prohibit the use of exhaust-blown diffusers in Formula 1 from 2012 onwards, there appears to be no such proscription on the use of compressed air from the inlet manifold of an engine. Formula 1's engine formula changes to a 1.6 litre turbo-charged V6 in 2014, and the turbine in such an engine is constantly generating compressed air. The inlet manifold of a turbo engine has a blow-off valve, specifically designed to release pressure when the driver lifts off the throttle or the throttle is closed. Thus, the blow-off valve could be vented down to the sides of the diffuser, providing vital extra downforce when a driver comes off the throttle turning into a corner.

In terms of the legality of such a system, there is already a clear precedent. As pointed out in Giorgio Piola's Formula 1 Technical Analysis 2010/2011, the very first exhaust-blown diffuser, the Renault RE40 of 1983, had several outlets blowing each lateral channel of the diffuser; in Piola's diagram (p15), three came from the exhaust, but one came from the turbo wastegate, (in fact, in pictures featured on Craig Scarborough's site, there are only two outlets from the exhaust, and one from the wastegate).

Piola points out that the legality of the system was protested by Brabham at the 1983 Detroit Grand Prix, but the protest was rejected both in situ, and at a subsequent tribunal on 26th July. Hence, whilst the FIA might be able to regulate the position of the exhaust outlets, and the off-throttle pumping effect of the engine, the turbo wastegate could conceivably be used as an independent blowing device in 2014.

McLaren's floor tiles

The BBC recently broadcast a documentary on McLaren's new supercar, the MP4-12C. In the opening minutes of the programme, we follow Ron Dennis around the immaculate McLaren Technology Centre. As Ron strides across the lakefront atrium, he points out a broken floor tile, and expresses his annoyance: "The reality is, when it's changed, it'll be imperfection, because the colour won't match. Tiles come in batches. You can see this one here's been changed, [pointing to a darker tile.] Doesn't that bug you? It bugs me. Big time."

In fact, rather than tiles from different batches being of a different hue, in this case the reason a new tile stands out is probably that the old ones have been subjected to the daylight for a period of time, and due to the slow photodissociation of the pigments inside, will gradually be growing lighter in colour. Hence, a new tile looks darker simply because it's been exposed to daylight for a shorter period of time.

There is, then, a possible means by which McLaren can mitigate this chromo-tessellatory problem: Purchase a sufficient number of spare tiles at the outset, and then allocate a backroom at the MTC for the purpose of exposing these backup tiles to the correct diurnally-averaged spectrum and intensity of artificial sunlight. When a frontline ('customer-facing') tile suffers a fracture, the replacement will have endured the same amount of photodissociation, and will be indistinguishable in colour from its two-dimensional siblings. Problem solved.

Autocourse 2011-2012

This year's Autocourse is printed on golf-leaf infused paper, using inks derived from the pituitary gland of the Himalayan gazelle.

At least, that's the only justification one can imagine for a price-tag of £50.

There was a time when one could look forward to the stunning photography in Autocourse each year, but those days are long gone. This year's edition contains only one memorable image, a two-page spread of Lewis Hamilton and Sebastien Vettel, wheel-to-wheel into Turn 3 at the Hungaroring, Hamilton's outside wheels skirting the grass-verge. Unfortunately, most people will have already seen this image, and the photography elsewhere never rises above the mediocre.

In fact, the quality of the photographic reproduction in Autocourse has become remarkably dark, virtually every image dominated by the sheer quantity of black ink. As an indication of this, it's almost impossible to find an image of a car, taken from the front, in which the undernose splitter can actually be discerned. By way of contrast (excuse the pun), if you happen to have the £4.40 Autosport Formula 1 review at hand, compare the picture on p32-33, taken from Ste Devote at Monaco, with the picture on p153 of Autocourse, taken from exactly the same vantage point. The difference is almost literally night and day.

It's difficult to know whether this is determined by the combination of inks and glossy paper used by Autocourse, or whether there's some artistic motivation behind it. There's almost a photophobic, crepuscular mood running through the annual: an article on Pirelli opens with a two-page spread of Hamilton and Webber in the gloaming at Korea; the team-by-team review begins with a two-page spread of the F1 paddock in semi-darkness; the race reports are prefaced with a two-page spread of a Williams passing through a silhouetted Eau Rouge; there's a two-page spread of Lewis Hamilton beneath leaden skies at the Nurburgring; there's a two-page spread of Singapore, in the darkness; and there's a two-page spread of Jenson celebrating his Japanese victory...in the darkness. It's like a book directed by David Fincher.

Formula 1 should be bright and colourful. Autocourse makes it look like an activity which takes place at 7pm on a damp October day in Macclesfield.

So do you get anything for your £50? Well, yes, you get Mark Hughes's team-by-team analysis, which is reliably superb. Mark, of course, also does something similar in Autosport's Formula 1 review, but the Autocourse version is more detailed in places, and contains extended explanation from each team's technical director. Paddy Lowe and Pat Fry, in particular, are fascinating this year as they explain where things went awry.

So there's something good here, but not £50-worth.

Red Bull and Immersed Boundary Methods

Autosport's recent 2011 Formula 1 review pointed out that whilst Red Bull were the first team to appear with exhausts blowing the outer extremities of the diffuser, "others, notably Renault and Ferrari, had tried the layout in their tunnels before the Red Bull appeared and couldn't make it work, and Newey later confirmed that it actually took months of simulation work to maximise."

So what is it that Red Bull were able to do that other teams weren't? Was it mere persistence in the wind-tunnel with a flow regime that transpired to be extremely sensitive to the exact position and geometry of the exhaust outlet? Or were Red Bull able to apply some form of computer simulation not currently utilised by other teams?

Perhaps the former is the most likely answer, but let's pursue the alternative explanation, and see if we can join up the dots. And let's start with the fact that Red Bull use Ansys Fluent as their CFD package. In this promotional video from late 2010, it's acknowledged that Red Bull use Fluent to model their exhaust flow, (although this obviously doesn't entail that it's their only simulation tool for doing so).

Speaking recently about the High Performance Computing solutions provided by Ansys, Nathan Sykes, CFD Team Leader at Red Bull Racing, pointed out that "To retain freedom to innovate and adapt the car quickly, we rely on a robust modeling process. This puts new designs on the track quickly. To accomplish our goal, we continually need to leverage technologies that help us introduce and evaluate new ideas. With a significant reduction in process times over the last three years, ANSYS HPC solutions have continued to be the tool of choice for us."

Now, the normal aerodynamic optimisation cycle involves shaping a part in CAD, importing it into CFD, meshing it in CFD, running the CFD solver, and then post-processing the results. Meshing, in particular, can be very time-consuming. There is, however, a means of short-circuiting the cycle, called the Immersed Boundary Method, and in an environment such as Formula 1, where aerodynamic turnover is paramount, any team able to successfully implement this method could gain a significant advantage.

Immersed Boundary Methods provide a means for dealing with geometries which may be complex, or in a state of motion. They enable one to mimic the effect that an appendage has on the fluid flow in terms of something called a 'body force'. For example, if a fixed solid object is introduced into a region previously occupied by fluid flow, then the no-slip boundary condition must be imposed on the new surface, (i.e., the velocity there must be zero). In effect, this requires the application of a force which reduces the pre-existing velocity to zero. To calculate the necessary body force, one could in principle insert the necessary acceleration into the (Reynolds-Averaged) Navier-Stokes equations, as below, with u_desired=0 in this case:



Coincidentally, Ansys Fluent 12.0, released in 2009, has an Immersed Boundary module, developed with Cascade Technologies Inc. This is what Ansys said at the time:

A conventional fluid dynamics simulation starts with the transfer of CAD data to a grid-generation package, in which a surface mesh and then a volume mesh are generated before the simulation can be set up and the solution run. The effort and time required for such pre-processing tasks can be significant. For example, in cases with complex or dirty geometry that require CAD cleanup, this part of the process may take 50 percent to 90 percent of the total time required for the simulation. The Immersed Boundary module addresses such issues by providing a rapid, automated, preliminary design approach.

Fluid flow simulations using the Immersed Boundary module for ANSYS FLUENT 12.0 software start with the surface data of the simulation geometry in the STL file format, which is commonly used in rapid prototyping and computer-aided manufacturing. This CAD geometry does not need to be clean, does not require smooth surfaces or geometry connectivity, and may contain overlapping surfaces, small holes and even missing parts. The simulation geometry is meshed automatically. Mesh refinement also is carried out automatically after specifying the desired resolution on the boundaries, ensuring the accuracy required for preliminary design evaluation. Using the immersed boundary meshing technique greatly reduces the amount of time spent preparing the geometry for meshing and creating the mesh.

At first sight, Immersed Boundary Methods do not appear to be available in Star-CCM+, one of Fluent's main competitors. Star-CCM+ does, however, provide a Surface Wrapper, a type of shrink-wrapper, which fixes gaps and overlaps in complex CAD geometries. Nevertheless, in Star-CCM+ it appears to be necessary to create a body-fitted mesh: a surface mesh must be created on the surface imported from CAD, and then a volume mesh is grown outwards from the surface mesh.

Immersed Boundary Methods have become increasingly popular over the past decade, and knowledge of such techniques will have been carried into the world of Formula 1 by many recent PhDs. Nevertheless, it's interesting to speculate whether Red Bull have stolen another march on the opposition here...

Unleashing radiation in a wind-tunnel

There are currently two primary methods of wind-tunnel flow visualisation: Particle Image Velocimetry (PIV) and Laser Doppler Anemometry (LDA). Both techniques seed the airflow with tracer particles, and use lasers and optical detectors and cameras to provoke and record a pattern of scattered light. This poses a problem, in that the wheels, wings, and diffusers of interest to the aerodynamicist, are normally opaque to the passage of optical radiation. Hence, PIV and LDA experiments typically require the construction of transparent wings and aerodynamic appendages.

There is, however, a possible solution to this problem: Why not use radioactive isotopes to obtain quantitative flow data from wind-tunnel testing? One could inject a harmless radioactive tracer into the flow, such as one of those used in the medical imaging industry; technetium-99m-labelled DTPA (diethylene triamine pentaacetic acid) would be an obvious candidate here. One could then use gamma (ray) cameras to image the flow in a similar way that optical cameras are currently used in PIV and LDA.

There would, of course, be the need for some additional precautions. However, an isotope such as technetium-99 is considered sufficiently harmless to be injected into medical patients, and has a half-life of only 6 hours, so a wind-tunnel would not need to be decontaminated by the Nuclear Decommissioning Authority!

In fact, taking a closer look reveals that there are already significant areas of shared technology between wind-tunnel flow visualisation and lung scintigraphy, the use of gamma cameras to record 2-dimensional images formed by the emission of gamma rays from inhaled radioisotopes:

“99mTc labelled aerosols, 0.5-3 [microns] in size, are used routinely in lung ventilation studies. Radiolabelled aerosols are produced by nebulizing 99mTc-DTPA (or other appropriate 99mTc-products) in commerically available nebulizers,” (p276, Fundamentals of Nuclear Pharmacy, 2010, Saha).

When such aerosols are inhaled for lung scintigraphy, droplet sizes must be small enough to permit diffusion deep into the lungs; specifically, diameters smaller than 2 microns are preferred. In the case of wind-tunnel flow visualisation, the tracer particles must follow the flow. Given that the ratio of the tracer particle density to the flow density is typically of the order 10³ in gas flows, it is necessary to use tracer particles of diameter between 0.5 and 5 [microns], (p288, Springer Handbook of Experimental Fluid Mechanics, Tropea, Yarin and Foss, 2007). The method by which such tracer particles are injected into the airflow suggest close reciprocities with lung scintigraphy:


"By far the most common method of seeding gas flows is through liquid atomization. Of the many atomizer types available the common nebulizer used in inhalation devices is the most suitable...The droplet size depends primarily on the atomizing airflow rate and on the liquid used. Typical mean particle sizes range from 0.2 [microns] using DEHS...to 4-5 [microns] with water...For many applications, the common inhalation or medication nebulizer offers an economical solution and can be obtained through medical suppliers," (ibid., p293).

Thus, the medical and wind-tunnel industries already use the same nebulizing technology, and comparable droplet diameters. In particular, technetium-labelled DTPA has a comparable density, in solution, to the DEHS (di-ethyl-hexyl-sebacat) widely used for seeding airflows in PIV experiments.

One potential limiting factor, however, may be the current resolution of gamma-ray cameras. A gamma camera consists of a scintillation crystal, which converts gamma rays into optical-wavelength light, detected by photomultiplier tubes behind the crystal. However, despite a recent breakthrough which demonstrates that gamma rays can be focused, there is currently no equivalent of an optical lens. Instead, a collimator, consisting of an array of tiny pin-holes, is used. The collimator absorbs some of the radiation, limiting the sensitivity of a gamma camera, and also places a limit on the spatial resolution. Typical current resolution is 7-12mm at a distance of 10cm, (p96, Nuclear Medicine Instrumentation, Prekeges, 2009).

Despite such problems, the possibilities for development abound.

How Red Bull create streamwise vorticity

Red Bull arrived in Singapore this year with interesting little mini-arches in their front-wing, where the inner end of each main plane meets the 50cm-wide neutral central section. Craig Scarborough suggested at the time that "this shape is to create a vortex along the Y250 axis." As Craig explains elsewhere, "flow structures along this axis [250mm from the centreline] drive airflow under the floor towards the diffuser and around the sidepod undercuts."

So how does such a shape create streamwise vorticity? Well, the answer lies in a subfield of fluid mechanics called 'secondary flows', (with thanks to Professor Gary Coleman of Southampton University, for pointing me in the direction of this field). Such flows typically involve a primary flow - with the streamlines oriented in a particular direction, and a vorticity field perpendicular to the primary flow - in which there is also some type of differential convection to the primary flow. ('Convection' here simply means the transport of fluid by bulk motion, sometimes referred to as advection if there is any confusion with thermal convection). This differential convection tilts and stretches the vorticity lines, increasing the magnitude of the vorticity, and re-directing it in a streamwise orientation. The streamlines corresponding to this vorticity constitute the secondary motion, superimposed upon the primary streamlines.

This type of secondary flow is exactly what Red Bull are using to create separated streamwise vortices from the boundary layer on their front-wing. But before proceeding further, let's establish some notation. In what follows, we shall denote the streamwise direction as x, the direction normal to the wing as y, and the spanwise direction as z. We also have three components for the velocity vector field, which will be denoted as U, V and W, respectively. There is also a vorticity vector field, whose components will be denoted as ω_x, ω_y, and ω_z.

On the underside of the front-wing is a boundary layer, and like all boundary layers, there is a velocity gradient ∂U/∂y in a direction normal to the wing, given that the velocity is zero at the solid surface. This entails that the boundary layer possesses vorticity in a spanwise direction ω_z. The vortex lines in this boundary layer are perpendicular to the streamwise direction of flow. The trick is then to convert some of this spanwise vorticity into streamwise vorticity ω_x. It transpires that the way to do this is to create a lateral pressure gradient ∂p/∂z.

Now, the front-wing operates in ground effect, so the pressure in an elevated mini-arch will be less than it is underneath the adjacent portion of the main plane, creating just such a pressure gradient. The crucial point is that this lateral pressure gradient corresponds to the creation of a spanwise-gradient in the streamwise velocity ∂U/∂z > 0. To see why this is crucial, however, we need to look at the Vorticity Transport Equation (VTE) for ω_x, the streamwise component of vorticity. The effect in question can be seen by studying incompressible, inviscid, laminar flow, so we can simplify the VTE by omitting the turbulent and viscous terms to obtain:

Dω_x/Dt = ω_x(∂U/∂x) + ω_y(∂U/∂y) + ω_z(∂U/∂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, we started with ω_z > 0 in the boundary layer, and by virtue of creating a lateral pressure gradient, we also have ∂U/∂z > 0. This means that the third term on the right-hand side in the equation above is positive, which (assuming the other pair of terms are non-negative) entails that Dω_x/Dt > 0.

Thus, the creation of the spanwise-gradient in the streamwise velocity ∂U/∂z, skews the initially spanwise vortex lines ω_z until they possess a significant component ω_x in a streamwise direction. The lateral pressure gradient has created streamwise vorticity.

As Peter Bradshaw writes, "if the lateral deflection that produces longitudinal vorticity extends for only a small spanwise direction, then the longitudinal vorticity becomes concentrated into a vortex," (Turbulent secondary flows, Ann. Rev. Fluid Mechanics 1987, p64). Which is exactly what Red Bull, and for that matter, many other Formula 1 teams, do when they incorporate mini-arches into their front-wings.

As a final aside, note that there is an interesting duality at the heart of fluid mechanics, namely that between a description which uses the velocity and pressure fields, and a description which uses the vorticity field instead. The vorticity has been described as "the sinews and muscles of fluid mechanics," (Kuchemann 1965, Report on the IUTAM Symposium on concentrated vortex motions in fluids, Fluid Mech. 21). P.A. Davidson points out that in the case of incompressible flow, because pressure waves can travel infinitely fast, the velocity vector field is a non-local field; the vorticity field, in contrast, is local. "While linear momentum can be instantaneously redistributed throughout space by the pressure field, vorticity can only spread through a fluid in an incremental fashion, either by diffusion or else by material transport (advection). Without doubt, it is the vorticity field, and not [the velocity field], which is the more fundamental," (Turbulence, 2004, p39).

An aerodynamicist with an especially strong visual imagination, perhaps someone who had been stimulated to develop such mental capabilities to compensate for dyslexia, might be able to develop a better understanding of the fluid flow around a Formula 1 car by thinking in terms of vorticity, or by developing the ability to mentally switch back and forth between the vorticity and velocity representations. Such an individual might even reject tools such as CAD and CFD, preferring instead to work on a drawing board...

Linking Red Bull's fuel loads to McLaren's rear-wing

One of the oddities of the 2011 Formula 1 season has been the contrast between the alacrity with which McLaren responded to the failure of their experimental 'bagpipe' exhaust system in pre-season testing, and their belated, late-season introduction of a Red-Bull style rear-wing, featuring a more powerful DRS effect.

Whilst a new and highly effective exhaust-blown diffuser was available on the McLaren from the first race, the new rear-wing combination, with smaller flap and larger main plane, only began to make sporadic appearances in practice from the middle of the season, and the system was only definitively installed for the Japanese Grand Prix at Suzuka.

Now, it's well-understood that a major aerodynamic component cannot be changed independently of the other primary aerodynamic components on a car, so McLaren presumably needed to make changes to the airflow feeding the rear-wing, or deal with the consequent change to the centre-of-pressure, before they could introduce the smaller flap design. Nevertheless, given the clear benefits of a powerful DRS system, as demonstrated by Red Bull from day one, McLaren do seem to have been a little tardy in this respect.

There is, however, a possible exculpatory explanation. Mark Hughes has recently drawn attention to the fact that during Friday practice this year, Red Bull have apparently used a second stint fuel load in the long run phase of these sessions, whilst McLaren have used a first stint fuel load. Conversely, Red Bull have tended to fuel more heavily on the short-run practice laps. This fuel-load combination has disguised Red Bull's real qualifying pace, relative to McLaren, but exaggerated their prospective race pace.

Perhaps, then, in the early stages of the season, McLaren came away from the races believing that their potential qualifying performance was stronger than their potential race performance, relative to Red Bull, and they therefore needed to continue optimising their car for race performance. This, in turn, meant retaining a larger rear-wing flap with a less powerful DRS stall.

Airbox spillage and fluidics

A couple of weeks ago, the FIA issued a Technical Directive to the Formula One teams, announcing that off-throttle blowing of the exhausts will be severely curtailed in 2012 by engine mapping restrictions.

In combination with stringent requirements on the position and angle of the exhaust exits, this is intended to minimise the exploitation of exhaust flow for aerodynamic purposes. It will, however, have a secondary consequence. As Gary Anderson recently explained, off-throttle exhaust flow also serves to reduce spillage from the airbox:

"In the past when the driver closed the throttle to slow for a corner, the airbox spillage became a lot worse. If the airflow attachment on the sides of the engine cover was not good, the performance of the rear wing would be compromised – not something the driver wants under braking or on corner entry.

"Step forward the blown diffuser. Hot or cold blowing allows the engine to work like an air pump, moving this airflow through and out of the exhausts. This reduces the potential turbulent airflow creating negative performance on the rear wing.

If off-throttle blowing of the exhausts is genuinely to be prohibited next year by means of engine mapping restrictions, this will presumably re-create the problem of airflow spilling out of the airbox when the driver lifts off the throttle on turn-in to a corner.

So here's an idea: Why not introduce a fluidic switch which, under certain circumstances, re-routes the airbox airflow through the chassis to the lower leading edge of the sidepods? This could have the joint benefit of boosting the velocity of the underbody flow, and improving airflow to the rear wing, just at the time when the driver most needs it, when the car is in pitch under braking and turn-in.

Exhaust-blown diffusers in 2012?

The 2012 Formula One regulations are intended to prohibit the use of exhaust-blown diffusers: stringent requirements have been placed on the location of the exhaust exit, and a recent announcement from the FIA suggests that engine mapping restrictions will be imposed to eliminate off-throttle pumping of the exhaust jet.

Craig Scarborough has produced a fantastic analysis of the exact restrictions to be placed on the location and orientation of the exhaust exit. In short, these move the exhaust exit to at least 500mm in front of the rear axle line, and 250mm above the reference plane underneath the car. The exhaust exit must also be angled upwards by at least 10 degrees. Hence, it will no longer be possible to blow the exhaust directly between the outer edge of the diffuser and inner face of the rotating rear wheel. Moreover, it will be illegal to place any sprung bodywork in a cone-shaped region, aligned with the exhaust exit, diverging at 3 degrees, and terminating at the rear axle line.

So will this be sufficient to eliminate exhaust-blown diffusers? Well, the first thing to note is that whilst it will be impossible to point the exhaust exit down at the diffuser, this won't necessarily prevent the exhaust jet itself from playing in that direction. When an exhaust jet exits into a cross-stream, the jet almost behaves like a deformable solid, as emphasised by F.L.Parra and K.Kontis in their 2006 paper, Aerodynamic effectiveness of the flow of exhaust gases in a generic formula one car configuration, from which the illustration here is taken.

If the exhaust exit is placed flush in the rearward face of sidepods sweeping downwards at a fairly steep angle, then the freestream airflow could deflect the exhaust jet towards the diffuser. The degree to which the jet is deflected is determined by the ratio between the velocity of the jet and the velocity of the cross-stream flow. The smaller the ratio, the more the jet is deflected.

Hence, there is something of a trade-off necessary here. To allow the exhaust jet to be deflected down towards the diffuser requires a lower exhaust jet velocity, yet for the exhaust jet to be effective in that region, requires higher jet velocities. There may be a compromise solution available here, an optimum exhaust velocity, which permits the jet to be directed towards the outer edge of the diffuser with sufficient velocity to have an effect, but that's something which only CFD and wind-tunnel experimentation will be able to determine...

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.