Formula 1 turbines and enthalpy

A couple of interesting developments occurred around the exhaust systems on both the Ferrari and Mercedes-engined Formula 1 cars in 2014: the Ferrari-engined vehicles acquired insulation around the exhaust-pipes, and the Mercedes-equipped cars appeared with a so-called log-type exhaust.

The purpose of the insulation was to increase the temperature of the exhaust gases entering the turbine. Similarly, increasing the exhaust gas temperature was a purported beneficial side-effect of the log-type exhaust on the Mercedes.

A couple of general points about the physics of turbines might provide some useful context here. First, the work done by the exhaust gases on the turbine comes from the total enthalpy (aka stagnation enthalpy) of the exhaust gas flow.

This is perhaps a subtle concept. The total energy E in the fluid-flow through any type of turbine consists of:

E = kinetic energy + potential (gravitational) energy + internal energy

However, to understand the change of fluid-energy between the inlet and outlet of a turbine, it is necessary to introduce the enthalpy h, the sum of the internal energy e and the so-called flow-work pv:

h = e + pv ,

where p is the pressure, and v is the specific volume, (the volume occupied by a unit mass of fluid).

One way of looking at the flow-work is that it is part of the energy expended by the fluid maintaining the flow; the fluid performs work upon itself, (in addition to the external work it performs exerting a torque on the turbine), and this work can be divided into that performed by the pressure gradient and the work done in compression/expansion.

Another way of looking at it is that the energy released into the fluid from a combustion process may have been released at a constant pressure as the fluid performed work expanding against its environment. The internal energy e doesn't take that into account, but the enthalpy h = e + pv does. As the diagram above from Daniel Schroeder's Thermal Physics suggests, the enthalpy counts not only the current internal energy of a system, but the internal energy which would be expended creating the volume which the system occupies.

For a system which is flowing, it possesses energy of motion (kinetic energy) in addition to enthalpy. The so-called total enthalpy h_T is simply the sum of the enthalpy and kinetic energy:

 h_T= e + pv + 1/2 ρ v² ,

where ρ is the mass density and v is the fluid-flow velocity.

This quantity is also called the stagnation enthalpy because if you brought a fluid parcel to a stagnation point, at zero velocity, without allowing any heat transfer to take place to adjacent fluid or solid walls, the kinetic energy component of the total energy in that parcel would be transformed into enthalpy.

In the case of a Formula 1 turbine, there is no difference in the potential energy of the exhaust gas at the inlet and outlet, so this term can be omitted from the expression for the change in energy. What remains entails that the rate at which a turbine develops power is determined by subtracting the enthalpy-flow rate at the outlet from the enthalpy-flow-rate at the inlet. The greater the decrease in total enthalpy, the greater the power generated by the turbine.

As the exhaust gases pass through the turbine, they lose both kinetic energy and static pressure, but gain some internal energy due to friction. As a consequence, the entropy of the exhaust gas increases, and the enthalpy reduction is not quite as large as it would otherwise be (see diagram above from Fluid Mechanics, J.F.Douglas, J.M.Gasiorek and J.A.Swaffield).

However, (and here is the crux of the matter), for a given pressure difference between the turbine inlet and outlet, the reduction in total enthalpy increases with increasing temperature at the inlet. In other words, this is another expression of the fact that the thermal efficiency of a turbine is greater at higher temperatures (a fact which also dominates the design of nuclear reactors).

So, all other things being equal, increasing exhaust gas temperature with insulation or a log-type exhaust geometry will increase the loss of total enthalpy between the inlet and outlet of the turbine, increasing the power generated by the turbine.

However, there is another side to this coin: the required pressure drop between the turbine inlet and outlet for a desired enthalpy-reduction, decreases as the inlet temperature increases. Hence, if there is a required turbine power-level, it can be achieved with a lower pressure drop if the exhaust gases are hotter. This could be important, because the lower the pressure at the inlet side of the turbine, the lower the back-pressure which otherwise potentially inhibits the power generated by the internal combustion engine upstream. So increasing exhaust gas temperatures might be about getting the same turbine power with less detrimental back-pressure on the engine.

Coral reefs and vortices

It seems that counter-rotating vortices are everywhere. The September 2014 edition of the Proceedings of the (US) National Academy of Sciences has published a fascinating study which reveals that coral reefs actively create quasi-steady arrays of counter-rotating vortices.

Corals exist in a symbiotic relationship with algae, which live within the tissue of the coral, and photosynthesise the organic carbon used by the corals to build their calcium-carbonate skeletons. In return, the corals have to provide nutrients for the algae, and remove the excess oxygen produced by photosynthesis.

Until now, it's been assumed that corals were dependent upon molecular diffusion alone to achieve the necessary mass transport. A concentration boundary layer exists at the surface of the coral: the concentration of a molecular species produced by the coral (such as molecular oxygen, O₂) is highest at the surface of the coral, and a concentration gradient exists in the direction normal to the surface of the coral until the edge of the boundary layer is reached, where the concentration matches the ambient level. This concentration gradient drives outward molecular diffusion.

In the presence of an ambient flow, the boundary layer becomes thinner, increasing the steepness of the concentration gradient, and thereby enhancing the mass transfer rate. However, many parts of many coral reefs often experience periods of very low ambient flow, and there was evidence to believe that mass transfer rates were actually higher than could be explained by the ambient flow conditions. (Here there is a similarity with heat transfer within a bundle of nuclear fuel rods, where the rate of thermal mixing was higher than could be explained by turbulent diffusion and thermal conduction alone).

The research just published has revealed that the cilia (tiny hairlike entities) on the surface of the coral polyps are able to create a pattern of counter-rotating vortices which enhance mass transfer rates even in conditions of stagnant ambient flow (see image below). The counter-rotating vortices seem to be produced by the coordinated sweeping motion of the cilia, with one group of cilia sweeping in direction, and another group sweeping in the opposite direction.

The research revealed that the vortices are able to transport dissolved molecules by ~1mm in ~1sec, under conditions which would otherwise require ~1000secs to traverse the same distance by molecular diffusion alone.

It was also found that the location and shape of one such vortex was stable over the 90min period under which the concentration levels of oxygen were measured. The latter produced the image below, showing that one side of the vortex, flowing towards the surface of the coral, had ambient levels of oxygen, whilst the other side of the same vortex transports the oxygenated water away.

CFD lessons from nuclear reactors

The fissile fuel in a commercial nuclear reactor is typically packaged into rods, which are collected together in arrays and placed within vertical cylindrical channels (as seen below for the case of the UK's Advanced Gas-Cooled reactor design). The coolant flows through the vertical channels, and the heat generated by fission is transferred from the surface of the fuel rods to the coolant. The efficiency and safety of the reactor therefore depends upon the efficiency with which the heat is transferred from the surface of the solid elements to the fluid flow. It is well-known that turbulent mixing enhances the efficiency of the heat transfer, and this is duly utilised within reactor design.

One of the requirements of reactor design is to homogenise the cross-channel temperature distribution, from one fuel rod to another, and it was noted in the 1960s that there was a greater degree of cross-channel heat transfer within a bundle of fuel rods than could be accounted for by turbulent diffusion alone.

The geometry created by the bundle of rods is rather differerent from a simple channel-flow problem. Taking a cross-section through a vertical channel, one has a collection of solid discs, each of which is separated from its nearest neighbour by a specified gap. The packing of adjacent cylindrical fuel elements creates a network of sub-channels, joined together by the gaps (see diagram below from A Keshmiri, Three-dimensional simulation of a simplified Advanced Gas-Cooled reactor fuel element, 2011). The coolant naturally flows in an axial direction through both the gaps and the sub-channels.

Experimental work noted that there was cross-channel heat transfer taking place through the gaps between sub-channels. For more than 20 years, it was thought that this heat transfer could be explained by 'secondary flow'. In a turbulent channel flow, the anisotropy of the turbulent stresses induce a component to the mean velocity flow-field which lies in a plane normal to the primary streamwise flow. Unfortunately, the magnitude of this secondary flow was way too small to explain the magnitude of the observed cross-channel mixing.

Only in recent decades has it been realised that the cross-channel mixing is due to a train of periodic vortices created in the sub-channels. The continual passage of these vortices creates a quasi-periodic cross-channel flow pulsation at particular stations along the bundle of fuel-rods. Steady-state CFD studies revealed nothing more than a turbulent channel flow pattern, and completely failed to represent the mixing of the coolant between adjacent sub-channels.

The cross-channel mixing was caused by an unsteady flow pattern which was smeared away in steady-state CFD, yet the coherent vortical structures make a contribution to the thermal mixing which has the same order of magnitude as that from the turbulent diffusion.

The exact mechanism responsible for the creation of this vortex train is not yet fully understood. The basic idea, however, is that the fluid flow is slower in the gaps between the fuel rods than it is in the larger sub-channels, and this creates a shear layer. The shear layer is intrinsically unstable, and breaks up into a train of vortices, in a manner possibly similar to Kelvin-Helmholtz instability. Adjacent sub-channels inherit counter-rotating vortices, so the patterns are not dissimilar to those of a von Karman vortex street shed behind a bluff body (see diagram below from Turbulent vortex trains in narrow square arrayed rod bundles of a dual-cooled nuclear reactor, Taehwan et al).

Note, however, that the vortex train in the bundle of fuel rods is not created by separation, as such. Rather, it is the result of the instability of the shear layers within the interior of the fluid. It is ultimately the geometrical configuration of the fuel rods which creates the unsteady flow pattern, and indeed the cross-channel pulsations are seen to vary as the gap between the fuel elements, and the diameter of the fuel elements, are varied.

The message is clear: even in the absence of separation, be very wary of steady-state CFD...

Adrian Newey and unsteady CFD

The September 2014 issue of Motorsport Magazine contains an interesting article in which Adrian Newey discusses his favourite F1 cars. For disciples of modern F1 aero design, however, two statements catch the attention.

With respect to the 2009 Red Bull RB5, Adrian remarks that "we had a really great design group. We did some good research, understood the flow physics and the packaging." Then, recalling the research conducted for the exhaust-blown area around the spat on the 2011 RB7, Newey states that "it was very clear that the area around the rear tyres was critical...Then the whole research started developing...from steady-state CFD to tyre-dependent CFD and we worked with Renault to understand how the pulsing and acoustics of the exhaust worked."

This suggests that the recent aerodynamic success of the Red Bull has been based upon using unsteady CFD to understand the flow physics in that complex area around the spat. When the car pitches and rolls, not only does the rear ride-height change, but the rear tyre sidewall deforms, and given the sensitivity of the flow in the spat area, this sidewall deflection can crucially affect the performance of the diffuser.

The phrase 'tyre-dependent CFD' could, in isolation, merely imply that a set of steady CFD simulations were conducted, each representing a different degree of roll. However, by placing this phrase in opposition to 'steady-state CFD', it implies that Red Bull conducted unsteady CFD simulations which represented the roll of the car, including the time-evolution of the tyre sidewall profile.

Having said that, even if the solid geometry remains fixed, there is ample reason to believe that unsteady CFD simulations are indispensable for understanding the flow physics of a Formula 1 car.  

Steady-state CFD generates time-averaged images of the flow, and these can be misleading, both because they smear away time-dependent fluctuations in the flow, but also because the time-averaging procedure sometimes generates fictional flow structures which don't actually exist in the any of the instantaneous flow fields.

The image on the left, taken from Jacques Heyder-Bruckner's PhD research on wing-wheel interaction, vividly illustrates how the time-averaged image (top) smears away much of the structure associated with the breakdown of a front-wing endplate vortex (bottom).

The fictional potential of steady-state CFD is exemplified by the common wisdom used to explain the function of a Gurney flap. This claims that there is a stable, counter-rotating vortex pair formed behind the Gurney. As a case in point, the All-American Racers website proffers the following explanation:

"At the trailing edge, the airflow immediately beneath the wing rolls into a small anti-clockwise vortex behind the Gurney. Immediately above this, a second small vortex, rotating in the opposite direction, is formed by the airflow traveling above the wing as it passes over the gurney's lip. together these two vortices form a small separation bubble - a rotating mass of air removed from the main flow - which is somewhat taller overall than the gurney itself.

In clearing this separation bubble, the airflow's vertical deflection is increased and hence downforce increases. Additionally, separation of airflow from the wing's lower surface is postponed, allowing a higher angle of attack to be used before stall, which further enhances the wing's effectiveness."

In reality, there is no such stable vortex pair. Research conducted by David Jeffrey and David Hurst at the turn of the century established that the flow behind a Gurney is intrinsically unsteady, consisting of the continual alternate shedding of discrete vortices, which convect downstream (see the PIV images below, obtained by Jonathan Zerihan, which depict the vorticity contours associated with a Gurney flap in ground-effect at four different ride-heights). The process is not dissimilar to that associated with the von Karman vortex street behind a bluff body:

"The first stage in this shedding cycle begins as the separating shear layer on one side of the body rolls up to form a vortex. As it does so, it draws the separating shear layer over from the other side of the body. This second shear layer contains vorticity of opposing sign, and as it crosses the wake centerline it cuts off the supply of vorticity to the shear layer that is rolling up. At this point, the vortex is shed and moves downstream, while the shear layer on the opposite side starts to roll up, repeating the process.
 
With the Gurney flap the offsurface edge provides a fixed separation point for the pressure-surface shear layer, and this interacts with that separating from the suction surface to form a vortex street, in a manner similar to other bluff bodies."

To understand the flow physics in such circumstances, it necessary to compile a sequence of instantaneous flow images, (a storyboard, if you will). Studying the frozen and often fictional images generated by steady-state CFD simply doesn't cut the mustard.

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.

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.

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 ω_z, 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 ω_z, 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.