Wednesday, July 02, 2014

Formula 3 airflow restrictors and black holes

The July issue of RaceTech magazine contains a decent article by Marco de Luca and Angelo Camerini on the principles behind airflow restrictors, such as those used to limit engine power in endurance racing and Formula 3.

The authors explain that as the RPM of the engine increases, the suction of the engine lowers the pressure in the downstream portion of the air intake duct. With an airflow restrictor, this duct consists of an inlet, a converging section, a throat, and (in some cases, but not in F3) a diverging section. As the downstream pressure lowers, the airflow velocity through the throat increases until, eventually, supersonic speeds are reached. In this condition, the flow in the duct is 'choked'.

Under normal conditions, when the RPM of the engine increases the pressure drop is communicated by means of pressure waves travelling upstream to the external atmosphere at the speed of sound. When the flow velocity becomes supersonic in the throat of the intake duct, these pressure waves can no longer breach the throat, and the engine's demand for greater mass-flow is unsatiated.


What the RaceTech authors sadly omit to mention, however, are the similarities between such airflow regimes and the event horizons of black holes. 

Physicists have been aware for some decades that fluid flow can influence the propagation of sound in the same way that black holes can influence the propagation of light. The classic example of such an analogy is perhaps the Laval nozzle. This contains a converging section, which accelerates a subsonic upstream flow so that it reaches the speed of sound in the throat of the nozzle. Then, (unlike the case of the air restrictor), the airflow is maintained in a supersonic condition downstream. 

The subsonic region of the 'acoustic geometry' corresponds to the exterior of the blackhole spacetime geometry; the throat corresponds to the event horizon; and the supersonic region corresponds to the interior of the black hole. 

Sound waves propagating upstream in the subsonic region are doppler shifted to longer wavelengths, just like the light escaping from the clutches of a black hole. Moreover, all sound waves in the supersonic region are swept further downstream, just as light is unable to escape the interior of a black hole. (See this Scientific American article by Theodore A. Jacobson and Renaud Parentani for an accessible introduction to the field of acoustic black holes.)


I look forward to a subsequent article by Marco, which explains the analogies between exhaust systems and white holes.

Saturday, February 15, 2014

Nigel Bennett, Gordon Murray, and vortex generators

Erstwhile Formula 1 and Penske designer Nigel Bennett has published a superb autobiography, Inspired to Design, which provides a reminder that several important aerodynamic concepts, prevalent in Formula 1 to this day, were actually invented in Indycar.

One of the recollections in the book even suggests that the use of vortex generators to enhance underbody downforce, was co-conceived by Bennett and Tony Purnell:

"Tony Purnell and I discussed some research he was doing at Cambridge University regarding laser viewing of vortex sheets, an element of which was trying to measure the low pressure generated at the centre of a vortex. Tony explained that if the vortex was trained to run between two plain surfaces, the low pressure would act on those surfaces.

"So, in our wind-tunnel tests, we set out to see if we could use this phenomenon to create more downforce from the car, and sure enough, it worked in that by creating a vortex at the front of the underbody such that it directed air at the underwing and chassis intersection, we were able to gain some 30-40lb [13-18kg] of downforce (full-size at 150mph) without an increase in drag. We developed a series of triangular sharks' teeth, fitted at an angle to the normal air stream just in front of the lower edge of the radiator intake duct, and the air would spill off these and form the swirling vortex. Later work using flow visualisation techniques showed where this vortex ran, and indeed, other vortices from the outer shelf edge did much the same thing in the outer rear corners," (p97).

It seems, then, that Bennett and Purnell were the first to systematically investigate and apply vortex generators. This work appears to have been undertaken as part of the design for the 1988 Penske PC 17. However, it should be recalled that Gordon Murray (featured in this month's Motorsport magazine podcast) introduced inch-deep vortex generators on the underside of the 1975 Brabham BT44, also with the intention of creating downforce. (Murray explains this in diagrammatic form when interviewed by Steve Rider for Sky Sports' F1 Legends Series).

For those seeking a rigorous insight into vortex generators, Lara Schembri Puglisevich has recently submitted a PhD thesis at the University of Loughborough, reporting the results of Large-Eddy Simulations of vortical flows in ground effect. This work includes a comparison (pictured below) of a vortex generator above: (i) a smooth, stationary ground plane; (ii) a smooth, moving ground plane; and (iii) a rough, stationary ground plane. The images show vorticity isosurfaces, colour-contoured by streamwise velocity. The flow is from left-to-right, with the vortex generator suspended from the floor above.

This is the first attempt to understand the potential interaction between a vortex and the roughness of the ground plane. Unfortunately, it wasn't possible to make the rough ground plane a moving plane, hence the stationary ground plane builds up its own boundary layer, which interacts with the vorticity shed by the vortex generator.

Nevertheless, these LES images vividly demonstrate just how 'messy' real vortices are.



Saturday, January 18, 2014

Red Bull's Y250 and the Batchelor vortex

Armchair aerodynamicists were presented with a rare treat last Autumn when cold, humid, early-morning conditions at Austin vividly revealed the Y250 vortices shed by several Formula 1 cars. Prominent amongst them was Red Bull's stable, gently corkscrewing version, which almost resembled a piece of aerogel taped to the front-wing.
 


In fact, it's worth emphasising that the condensation of water vapour only takes place in the vortex core, where the temperature and pressure is at its lowest, hence the Red Bull Y250 vortex is liable to be larger than these images suggest.


This is nicely exemplified in the diagrams, above and below, of a trailing wing-tip vortex, taken from Doug McLean's excellent book Understanding Aerodynamics (Wiley, 2012), but attributed there to Spalart.

Here and below, we will deal with a cylindrical coordinate system, in which there is an axial coordinate, a radial coordinate, and a circumferential coordinate.

The image above displays the circumferential velocity (the continuous, bold line) as a function of radial distance from the centre of the vortex. The circumferential velocity is the component of velocity around the longitudinal axis of the vortex; we will denote it below as v(r). r1 denotes the radius of the vortex core, while r2 denotes the radius of the vortex as a whole.

The image below displays the pressure as a function of radial distance. Clearly, the pressure only declines significantly within the vortex core.
 

This, however, begs the question: 'What defines the radius of a vortex, and what defines the radius of the vortex core?' To answer this, recall first that a vortex is loosely defined as a region of concentrated vorticity. Now, non-zero vorticity requires the infinitesimal fluid parcels to be rotating about their own axes as they follow their trajectories in the flow field. Merely being entrained in a flow which swirls about a global centre of rotation is insufficient. In fact, a so-called 'free vortex' has no vorticity at all!

A free vortex is defined by a circumferential velocity profile v(r) = r-1. To calculate the vorticity in an axial direction ωz, one can use the following simple formula:
If you insert v(r) = r-1 into this formula, and take the derivative (recalling the Leibniz rule for the derivative of a product), you can verify that the two resulting terms cancel, yielding zero vorticity in an axial direction.

At the opposite extreme to a free vortex is a rigid body vortex, in which there is no shear between concentric rings of fluid, and the vortex rotates like a solid body, with a circumferential velocity profile of v(r) ~ r.

A more realistic vortex model is intermediate between these two extremes: the vortex core resembles a rigid body vortex, whilst outside the core the velocity profile blends into that of a free vortex. The circumferential velocity initially increases with radial distance from the centre, reaches a peak, and then begins to decline. The radial distance at which the circumferential velocity peaks is, by convention, defined as the radius of the vortex core. In the case of a simple vortex model the radial distance at which the velocity blends into the r-1 profile is defined as the radius of the vortex (although many attempts at more precise definitions, applicable to generic vortices, have been proposed).

Perhaps the best starting point for a realistic vortex model is the Batchelor q-vortex. This is still highly idealised because it assumes that there is no stretching of the vortex in an axial direction, that there is no radial velocity component, and that the remaining components of velocity vary only in a radial direction. It is, nevertheless, a good start.

The circumferential velocity of a Batchelor vortex is given by the following function of the radial distance, where the value of q determines the strength of the vortex:


The axial velocity, meanwhile, is given by the following expression, where W0 is the freestream axial velocity.

The choice of plus or minus determines whether the vortex core has an axial velocity deficit or surplus with respect to the freestream. (A deficit will make the vortex susceptible to breakdown, but that's another story).

If we plug the expression for the Batchelor circumferential velocity profile v(r) into the formula for the axial vorticity ωz, we obtain the following expression for the axial vorticity as a function of the radial coordinate:


The Batchelor vortex is often termed a Gaussian vortex, due to the presence of the exp(-r2) term,which gives the axial vorticity the same characteristic 'bell-shaped' profile as a Gaussian probability distribution. This can be seen in the chart below, where the axial vorticity is plotted by the red-coloured line:


The circumferential ('tangential') velocity in the Batchelor vortex is plotted in the chart below, and compared with the profile of a free vortex. One can see that the velocity profile resembles that of a solid body, v(r) ~ r, inside the vortex core, and then eventually blends into the free vortex profile, v(r) = r-1, as advertised.


Whilst the complexity of the real-world quickly overwhelms such analytical mathematical models, vortices like Red Bull's Y250 can be seen as perturbations and variations of the Batchelor vortex, with axial pressure gradients, axial curvature, and so forth.

It's always nice to have a mental model of the simplest version of something.

Sunday, December 29, 2013

Lotus 72 CFD


For those interested in a bit of retrospective CFD, I've written an aerodynamic analysis of the 1970 Lotus 72, as a contribution to Mark Hughes's latest work, F1 Retro 1970.

We were extremely fortunate here in being able to utilise the CFD resources of Sharc, and in particular the patient and responsive cooperation of Richard Bardwell. Many thanks to all concerned!

You can read the full text in the published work, but there is an omission in that, to avoid deterring the non-technical reader, I decided not to list there the full details of the CFD configuration.

For the technically curious, however, I can now put that to rights: We used a k-epsilon turbulence model, and employed a mesh containing approximately 20 million cells, with a boundary layer mesh comprising 3 layers. A y+1 of around 30 was chosen. The vehicle was simulated with rotating wheels on a rolling road at a freestream airspeed of 50m/s. The sidepod radiator and airbox inlets were treated as outlets (if you see what I mean).

Two ride-height combinations were used: 40mm front/60mm rear, and 60mm front/90mm rear. Runs were conducted for the Straight Ahead case, and with 4 degrees of steering angle. The frontal area used in the lift-coefficient calculations was 1.372msq. For each CFD case, the solver was run for 1000 iterations.

Tuesday, November 12, 2013

The legality of Brabham's 1983 World Championship

A couple of recent pieces in the motorsport press have raised separate issues over the legality of the Brabham-BMW which won the 1983 World Drivers' Championship in the hands of Nelson Piquet. Gary Watkins's Autosport article re-considered the exotic, ex-Luftwaffe fuel brew used by BMW in the latter stages of the season, while Mike Doodson's Motorsport article elicited the following admission from then-Chief Mechanic Charlie Whiting that, "All I will say is that we always, um, attempted to make the car as light as possible."

Indeed, and not just in qualifying it seems, for Gordon Murray rather gave the game away earlier this year with the following comment:

"Whenever we planned to stop - we could go without on street circuits - we tended to do 60-70 per cent of the race on the first set of tyres because that meant we could run very close to, or under, the minimum limit before adjusting the weight with the amount of fuel we put in," (Motorsport, May 2013, p86).

That's a pretty unambiguous admission that the Brabhams ran under the legal weight limit in many 1983 Grands Prix, and used fuel-weight as ballast. If you go through the races in 1983 you'll see that Piquet's Brabham was almost always the last of the leading runners to pit for fuel, and that he sometimes pitted 5-10 laps or so after the Ferraris and Renaults, his championship competitors. At Hockenheim, for example, Prost's Renault stopped on lap 20, while Piquet stopped on lap 30.

Why would Brabham want to stop after everyone else? Well, in the era of refuelling, a car on empty tanks and worn tyres was generally faster than a car which was fuelled-up on fresh tyres. Moreover, in 1983 the Ferraris were shod with bias-ply Goodyear tyres, and thereby tended to suffer greater tyre warm-up difficulties than the Michelin-tyred Brabham. Thus, it was in this 5-10 lap window that the Brabham would often make hay.

At Spa, Tambay's Ferrari was running ahead of Piquet until stopping on lap 21; Piquet stopped on lap 24, and jumped ahead of the Ferrari, forcing Tambay to re-pass some laps later. Similarly, at the Osterreichring, leader Arnoux's Ferrari stopped on lap 28, Piquet on lap 31, after which Piquet emerged ahead, forcing Arnoux to re-pass some laps later.

So why, then, wouldn't everyone schedule their pit-stops at 3/4 distance? Well, that extra fuel-weight costs lap-time, and it costs you lap-time on each and every lap that you carry the extra weight around on your back. If you started a race with about 20kg of extra fuel, that would cost you about 0.6seconds of lap-time, which over 30 laps would mount up to a very substantial 18 seconds or so.

However, if your car was 20kg beneath the legal weight limit when drained of fuel, you could start the race at the same weight as your competitors, but with the ability to run 5 laps or so further. You would suffer no disadvantage in the early stages of the race, and you would also be able to jump ahead of your competitors by running longer. Perfect.

In fact, just about the only time Piquet didn't run long was in the final race at Kyalami, when he shot off into the distance on a light fuel load, pitting on lap 28 of 77, able to resume without having lost the lead. It's possible that the Brabham started the race over the legal weight limit, but was running underweight for a significant portion of this stint. Moreover, in the late stages of the race Piquet slowed considerably, sacrificing what appeared to be an easy victory. Contrary to the explanation given on the day, that Piquet was merely trying to ensure the reliability of his car, he might also have been minimising fuel consumption to ensure the car was actually over the legal weight limit at the end of the race!

Monday, October 21, 2013

Ferrari's illegal brake ducts


When Ferrari launched their 1976 car, the 312T2, it appeared with a pair of outrageous front brake-duct appendages. These extended forwards, and curved around the inner front shoulder of the tyres, presumably with the intention of reducing front-wheel drag and turbulence.


These brake-duct extensions appeared only once during the racing season, in modified form, at the French Grand Prix (above). Pete Lyons reported in Autosport/Autocourse that "As a member of the CSI [the sport's governing body], Jabby Crombac pointed out that these appeared to contravene the regulations about 'movable aerodynamic devices' and the first session times for both cars were disallowed."

Note that these brake ducts were declared illegal, not because they intruded into a region from which bodywork was prohibited, but because they constituted movable aerodynamic devices. That is to say, being attached to the wheel uprights, they moved with respect to the sprung mass of the car, the reference frame against which movability is judged in this context.

It is a curiosity, then, that despite this precedent, and despite the fact that section 3.15 of the current Formula One Technical Regulations still requires bodywork influencing the aerodynamics to be "immobile in relation to the sprung part of the car," brake ducts are explicitly exempted. Devices such as those pictured below seem not dissimilar to those on the 312T2 at Paul Ricard in 1976. Perhaps Ferrari should ask for their Friday morning times to be reinstated...

Monday, September 30, 2013

Rush and the 1976 Austrian Grand Prix

Rush is currently No. 1 at the UK box-office, and Ron Howard's populist mix of stereotypical rivalry, heroism, sex and danger, is sure to attract a new generation of fans to the fight between 3-stop strategies and pace-managed 2-stoppers.

For those seeking the real thing from 1976, this was the video of the entire Austrian Grand Prix at the Osterreichring. It has now been stripped from Youtube by the guardians of copyright, who don't appear to be protecting any commercial interest given that there is no alternative outlet in which the entire race can be viewed. 

Which is a pity, because the original Osterreichring was a majestic, Styrian theatre of motorsport. The 1976 race started with localised showers of rain, until, as Pete Lyons reported, "The Sun now broke out on the highest slopes above, kindling the tents there into incandescent yellows and reds. Within minutes the entire landscape was brilliant, rain-washed green." 

Proper drivers, proper cars, and a proper circuit.


Wednesday, September 25, 2013

Chapman, side-thrust, and the America's Cup

In the summer of 1975, Colin Chapman composed a list of requirements for a Future F1 Car.  Reproduced in Karl Ludvigsen's excellent engineering biography (Colin Chapman - Inside the Innovator), many of the points continue to be relevant today. In particular, the list includes the following laudable objectives:

5...We must get maximum cornering force from the tyres. This is maximised by:
(i) The largest possible contact patches.
(ii) With the softest compound.
(iii) Kept in contact with the ground as long as possible.
(iv) With highest possible download.
(v) Spread as evenly as possible over the contact patch.
(vi) And spread as evenly over the four contact patches in proportion to the sideloads they have to carry.

In a more quirky vein, Chapman includes the following speculative thought:

9. Total cornering force can also be increased aerodynamically
Should we try to use vertical lifting surfaces to provide additional side load derived from the speed and yaw angle of the car whilst cornering?

Which is interesting, because apart from the use of fins atop the engine cover, there appears to have been little effort in Formula 1 to generate a direct aerodynamic side-thrust. In contrast, it seems to be an extremely important part of racing yacht design, of which The America's Cup might be held as the foremost example.

If a yacht is your chosen mode of travel, and the wind rather inconveniently happens to be blowing from a direction close to the direction in which you wish to travel, you can still generate a thrust in that direction by means of some aerodynamic and hydrodynamic magic.

Firstly, you use your sail as an aerofoil, and generate low pressure on one side of it, so that an aerodynamic force is produced at right angles to the effective direction in which the wind is travelling. This alone wouldn't get you to where you want to go, but here you can use the fact that there are actually two fluids in play: air and water. Whilst your sail can generate a force from the airflow, the hull of your yacht can also generate a force from the flow of water. With a yaw angle between your direction of travel and the effective wind direction, the water will accelerate around one side of the hull, creating a hydrodynamic side-force which can be used to cancel out the sideways component of the force generated by the sail. What remains of the aerodynamic force is a component pointing in the direction you wish to travel!











As Alfio Quarteroni explains (Mathematical Models in Science and Engineering, from which the diagram above is taken), the presence of two fluids with different densities and viscosities, separated by a free surface endowed with surface tension and variable curvature, adds many interesting dimensions to the fluid mechanical problem. Moreover, the sail itself needs to be treated as an aero-elastic medium, deforming in response to the pressure field upon it, and thereby changing the airflow, in a coupled manner. Seen in this light, it's no surprise that The America's Cup once exerted such a pull over the imagination of Chapman's modern counterpart, Adrian Newey.

Friday, August 16, 2013

What is a quantum field?

Philosopher of Physics Meinard Kuhlmann has a brilliant article, What is real?, in the August 2013 edition of Scientific American.

Kuhlmann provides a clear account of why the particle and field ontologies provide equally inadequte interpretations of quantum field theory. Whilst most physicists tend to resort to the lazy claim that the particle and field concepts are somehow 'complementary', Kuhlmann points out that this doesn't help "because neither of these conceptions works even in those cases where we are supposed to see one or the other aspect in purity."

Kuhlmann's account of how a quantum field is mathematically defined is particularly striking: "A classical field is like a weather map that shows the temperature in various cities. The quantum version is like a weather map that does not show you '40 degrees', but '√'.

The article concludes with a nice explanation of two alternative ontologies: structural realism, and the bundle theory of properties.

If you want an insight into the philosophical problems of modern physics, this is an excellent introduction.

Sunday, July 28, 2013

Front-wing vortex breakdown

With the assistance of Mercedes GP, Jacques Heyder-Bruckner completed a PhD thesis in 2011 which analysed the front wing-wheel interaction on a racing car. Of particular interest in this work is the fact that Detached Eddy Simulation (DES) was used to represent the phenomenon of vortex breakdown, which occurs when the front-wing approaches low ride-heights under conditions of roll and pitch.

Whilst Reynolds Averaged Navier-Stokes (RANS) simulations are the CFD workhorse of modern motorsport, RANS is known to be inadequate for representing separated flows, and in particular for representing large-scale vortical phenomena, such as vortex evolution and breakdown. In contrast, Detached Eddy Simulations use RANS to represent the attached flow, but directly solve the Navier-Stokes equations in the regions of separated flow.



















Q-criterion isosurfaces of the vortex breakdown phenomenon, taken from the instantaneous DES flowfield, are depicted here. Bruckner points out that “the vortex breakdown moves forward as the wing is moved closer to the ground…The large vortex expansion…is composed of a recirculation region enclosed by the spiralling tail shed from the vortex breakdown.This causes high pressure fluctuations on the endplate and flap, resulting in a more unstable wing with variations in downforce and drag three times larger than at higher ride-heights.” (p116).