Tuesday, May 05, 2015

A history of porpoising

Skirted ground-effect Formula 1 cars of the late 1970s and early 1980s were occasionally afflicted by a type of instability referred to as 'porpoising'. Many cars suffered, but the phenomenon is nicely described in Peter Wright in relation to the development of the Lotus T80:

"The car was so sensitive that, above a certain critical speed, it became aerodynamically unstable in pitch. One test day at Silverstone, Mario Andretti coined the term 'porpoising' to describe the phenomenon when he observed daylight under the front wheels while at speed on the straight.

"Since 1977 I had been working with David Williams, Head of the Flight Instrumentation Department at the Cranfield College of Aeronautics. He had designed and built a digital data system for use on the T78 when it had become apparent that it would be absolutely essential to gather data from the chassis in order to progress with the development of ground effect. When the T80 porpoising started, I discussed the phenomenon with him, and he offered to model it and validate the results with the data we had. He established that it was an aero-elasticity problem, akin to flutter in an aircraft wing. The changing aerodynamic loads, as the car bounced and pitched, excited the pitch and heave modes of the sprung mass on its springs and tires." (Formula 1 Technology, p36 and p308.)

However, pace Wright, the same phenomenon had already been identified and named at least as early as the 1940s, albeit in the field of seaplane hydrodynamics; specifically, during the take-off and landing of such craft. A Wartime Report issued by NACA in June 1943, begins:

"Porpoising is a self-sustaining oscillatory motion in the vertical longitudinal plane...Observations of porpoising show that there are two principal oscillatory motions (1) a vertical oscillation of the center of gravity and (2) an angular oscillation about the center of gravity. These two motions are seen to have the same period but to differ in phase." (Some systematic model experiments on the porpoising characteristics of flying-boat hulls, Kenneth S.M. Davidson and F.W.S. Locke Jr, p7).

The British were also heavily involved in the early study of porpoising, an Aeronautical Research Council report in 1954 defining the phenomenon as follows:

"Porpoising, basically, consists of a combination of oscillations in pitch and heave. It includes both stable and unstable oscillations, a stable oscillation being one which damps out. (A review of porpoising instability of seaplanes, A.G.Smith and H.G.White, p5).

All of which is an important reminder that ground-effect was of crucial importance to hydroplanes long before Formula 1 happened upon the phenomenon.

Saturday, April 04, 2015

Optimal control theory and Ferrari's turbo-electric hybrid

The Department of Engineering Science at the University of Oxford, published an interesting paper in 2014 which appears to shed some light on the deployment of energy-recovery systems in contemporary Formula One.

Entitled Optimal control of Formula One car energy recovery systems, (a free version can be downloaded here), the paper considers the most efficient use of the kinetic motor-generator unit (ERS-K), and the thermal motor-generator unit (ERS-H), to minimise lap-time, given the various regulatory constraints. (Recall that the primary constraints are: 100kg fuel capacity, 100kg/hr maximum fuel flow, 4MJ Energy Store capacity, 2MJ per lap maximum energy flow from ERS-K to the Energy Store, and 4MJ per lap maximum energy flow from the Energy Store to the ERS-K). The paper outlines a mathematical approach to this Optimal Control problem, and concludes with results obtained for the Barcelona track.

In the course of the paper, a number of specific figures are quoted for engine power. For example, the power of the internal combustion (IC) engine under the maximum fuel-flow rate, with the turbo wastegate closed, is quoted as 440kW (590bhp); it is claimed that by having the turbo wastegate open, the power of the IC engine can be boosted by 20kW (~27bhp), but in the process the ERS-H has to use 60kW of power from the Energy Store to power the compressor; and with the wastegate closed, the 20kW reduction in IC power is compensated by the 40kW generated by the ERS-H. (Opening the wastegate boosts IC power because the back-pressure in the exhaust system is reduced).

Running with the wastegate closed is therefore considered to be the most efficient solution for racing conditions. However, the paper also considers qualifying conditions, where the Energy Store can be depleted over the course of a lap without any detrimental consequences:

"In its qualifying configuration the engine is run with the waste gate open for sustained periods of time when maximum engine power is needed. During these periods of time the energy store will be supplying both the MGU-K and the MGU-H, with the latter used to drive the engine boost compressor...In contrast to the racing lap, the waste gate is typically open when the engine is being fully fuelled. On the entry to turns 1, 4, 7 and 10 the waste gate is being closed a little before simultaneously cutting the fuel and the MGU-K."

Professor of Control Engineering David Limebeer delivered a presentation of the work at a Matlab conference the same year (video here). Another version of the work, Faster, Higher and Greener, featuring Spa rather than Barcelona, was published in the April 2015 edition of the IEEE Control Systems Magazine. In his Matlab presentation, Professor Limebeer also credits Peter Fussey, Mehdi Masouleh, Matteo Massaro, Giacomo Perantoni, Mark Pullin, and Ingrid Salisbury.

After reading their work, I e-mailed Professor Limebeer, and asked if he'd considered collaborating with a Formula One team. I received a slightly odd response. After a further internet search, I found out why. In the November 2014 issue of Vehicle Electronics, David reports "We have done this work with one of the Formula One teams, but we can’t tell you which one."

Which is totally understandable. University departments have to protect the confidentiality of their work with Formula One teams. Unfortunately, however, the University of Oxford, Department of Engineering Science Newsletter 2013-2014, proudly reveals:

The Ferrari F1 Connection. 
Mr Stefano Domenicali, Scuderia Ferrari Team Principal, visited the Department in May 2013 to deliver the annual Maurice Lubbock Memorial Lecture. During this lecture he announced the evolving research partnership between the University and Ferrari.

DPhil Engineering Science students Chris Lim, Giacomo Perantoni and Ingrid Salisbury are working with Ferrari on novel ways to improve Formula One performance. Chris Lim said: “I’m very excited that I’ll be the first student working with Ferrari in the Department’s Southwell Laboratory, under the supervision of Professor Peter Ireland, the Department’s Professor of Turbomachinery. It’s a privilege to work with a prestigious manufacturer such as Ferrari in an industry like Formula One where the application of thermo-fluids has such a large impact”.

Pictured from left to right are: Chris Lim (postgraduate), Ingrid Salisbury (postgraduate), Mr Stefano Domenicali, Giacomo Perantoni (postgraduate) and Professor David Limebeer (supervisor to Giacomo Perantoni and Ingrid Salisbury). 

In light of this, then, the figures quoted in these papers can be interpreted as pertaining to Ferrari's turbo-electric hybrid. The first paper was submitted for publication in late 2013, and the assumptions used there are the same as those used in the 2015 paper, so it appears that Ferrari development data from no later than 2013 was used throughout.

Monday, March 09, 2015

Adrian Newey and the bar-headed goose

The April edition of Motorsport Magazine contains a fabulous F1 season preview from Mark Hughes, which includes the news that Adrian Newey has recently been taking a break in the Himalayas.

Now, whilst it's likely that the principal purpose of this expedition was to enlighten the Dalai Lama on the importance of using large-eddy simulation to understand the interaction of brake-duct winglets with the spat vortex, it's also possible that Adrian was drawn by the legendary bi-annual migration of the bar-headed goose.


These birds are amongst the highest-flying in the world, and travel across the Himalayas in a single day. William Bryant Logan claims in Air: Restless Shaper of the World (2012), that "the bar-headed goose has been recorded at altitudes of over thirty-three thousand feet. This is the altitude where your pilot remarks that the outside temperature is 40 degrees below zero, where the great fast-flowing rivers of the jet streams set weather systems spinning. The air here contains only one-fifth of the oxygen near sea-level, where the goose winters in lowland India wetlands and marshes. Yet in the space of a few hours the bird can fly from the wetlands to the top of the high peaks and then out onto the world's largest high plateau. There are lower passes through the mountains, but the goose does not take them. It may even preferentially go higher."

However, it seems that some of the claims made for the bar-headed goose lack empirical support. Research led by Bangor University tracked the bar-headed geese with GPS as they migrated over the Himalayas, and reached the following conclusion in 2011:

"Data reveal that they do not normally fly higher than 6,300 m elevation, flying through the Himalayan passes rather than over the peaks of the mountains...It has also been long believed that bar-headed geese use jet stream tail winds to facilitate their flight across the Himalaya. Surprisingly, latest research has shown that despite the prevalence of predictable tail winds that blow up the Himalayas (in the same direction of travel as the geese), bar-headed geese spurn the winds, waiting for them to die down overnight, when they then undertake the greatest rates of climbing flight ever recorded for a bird, and sustain these climbs rates for hours on end."

A more recent iteration of the research, The roller-coaster flight strategy of bar-headed geese conserves energy during Himalayan migration, (Science, 2015), suggests that the geese "opt repeatedly to shed hard-won altitude only subsequently to regain height later in the same flight. An example of this tactic can be seen in a 15.2-hour section of a 17-hour flight in which, after an initial climb to 3200 m, the goose followed an undulating profile involving a total ascent of 6340 m with a total descent of 4950 m for a net altitude gain of only 1390 m. Revealingly, calculations show that steadily ascending in a straight line would have increased the journey cost by around 8%. As even horizontal flapping flight is relatively expensive, the increase in energy consumption due to occasional climbs is not as important as the effect of reducing the general costs of flying by seeking higher-density air at lower altitudes.

"When traversing mountainous areas, a terrain tracking strategy or flying in the cool of the night can reduce the cost of flight in bar-headed geese through exposure to higher air density. Ground-hugging flight may also confer additional advantages including maximizing the potential of any available updrafts of air, reduced exposure to crosswinds and headwinds, greater safety through improved ground visibility, and increased landing opportunities. The atmospheric challenges encountered at the very highest altitudes, coupled with the need for near-maximal physical performance in such conditions, likely explains why bar-headed geese rarely fly close to their altitude ceiling, typically remaining below 6000 m."

Tuesday, March 03, 2015

Driver core-skin temperature gradients and blackouts

Whilst it is highly beneficial to reduce the surface-to-bulk temperature gradient of a racing-tyre, the same cannot be said for the cognitive organisms controlling the slip-angles and slip-ratios of those tyres.

A 2014 paper in the Journal of Thermal Biology, Physiological strain of stock car drivers during competitive racing, revealed that not only does the core body temperature increase during a motor-race, (if we do indeed count a stock-car race as such), but the skin temperature can also rise to such a degree that the core-to-skin temperature delta decreases from ~2 degrees to ~1.3 degrees.

The authors suggest that a reduced core-to-skin temperature gradient increases the cardiovascular stress "by reducing central blood volume." Citing a 1972 study of military pilots, they also suggest that when such conditions are combined with G-forces, the grayout (sic) threshold is reduced.

Intriguingly, in the wake of the Fernando Alonso's alien abduction incident at Barcelona last week, they also assert that "A consequence of this combination may possibly result in a lower blackout tolerance."

Monday, March 02, 2015

McLaren front-wing vortices, circa 2003

Academic dissertations conducted in association with Formula 1 teams tend to be subject to multi-year embargoes. Hence, Jonathan Pegrum's 2006 work, Experimental Study of the Vortex System Generated by a Formula 1 Front Wing, is somewhat outdated, but might still be of some interest to budding aerodynamicists.

Currently an Aerodynamics Team Leader at McLaren, Pegrum's study concentrated on a front-wing configuration not dissimilar from that on an MP4-18/19 (2003-2004).

A constellation of four co-rotating vortices were created: (i) a main bottom edge vortex, generated by the pressure difference across the endplate due to the low pressure under the wing; (ii) a top edge vortex, generated by the pressure difference across the endplate due to the high pressure above the wing; (iii) a canard vortex, a leading edge vortex generated by the semi-delta wing ('canard') attached to the outer surface of the endplate; and (iv) a footplate vortex, generated by the pressure-difference across the footplate operating in ground-effect. 

Pegrum shows (in the absence of a wheel, below), that the strongest vortices are the bottom-edge and top-edge vortices, but all four mutually interact in the manner of unequal, co-rotating vortices, undergoing the early stages of a merger.

Now, whilst co-rotating vortices have a tendency to merge, counter-rotating vortices have a tendency to repel. Pegrum highlights the 1971 work of Harvey and Perry, Flowfield Produced by Trailing Vortices in the Vicinity of the Ground, which demonstrated that when a vortex spinning around an axis in the direction of the freestream passes close to a solid surface, it tends to pull a counter-rotating vortex off the boundary layer of the solid surface, (as illustrated below by Puel and de Saint Victor, Interaction of Wake Vortices with the Ground, 2000). 

The interaction between these counter-rotating vortices is such that the primary vortex is repelled away from the solid surface. This phenomenon, of course, is still very much of interest when it comes to the Y250 vortex and its cousins.

Thursday, February 19, 2015

Proof that Formula 1 was better in the past

If you're a long-time Formula 1 fan, then the chances are that you believe the sport was better in the past. However, the chances are that you will have also read arguments from younger journalists and fans, to the effect that Formula 1 in the modern era is better than it was in the past.

Fortunately, there is an objective means to resolve this dispute: churn.

In sport, churn provides a straightforward measure of the uncertainty of outcome. Churn is simply the average difference between the relative rankings of the competitors at two different measurement points. One can measure the churn at an individual race by comparing finishing positions to grid positions; one can measure the churn from one race to another within a season by comparing the finishing positions in each race; and one can measure the inter-seasonal churn by comparing the championship positions from one year to another.

The latter measure provides an objective means of tracking the level of seasonal uncertainty in Formula 1, and F1 Data Junkie Tony Hirst has recently compiled precisely these statistics, for both the drivers' championship and the constructors' championship, (see figures below). In each case, Hirst compiled the churn and the 'adjusted churn'. The latter is the better measure because it normalises the statistics using the maximum possible value of the churn in each year. The maximum can change as the number of competitors changes.

The results for the drivers' championship indicates that churn peaked in 1980. Given that the interest of many, if not most spectators, is dominated by the outcome of the drivers championship, this suggests that Formula 1 peaked circa 1980.

The results for the manufacturers' championship are slightly different, suggesting that uncertainty peaked in the late 1960s, (although the best-fit line peaks in the middle 1970s).

One could, of course, make the alternative proposal that the churn within individual races is more important to spectators' interest, but at the very least we now have an objective statistical measure which provides good reason for believing that Formula 1 was better in the 1970s and early 1980s.

Monday, February 16, 2015

Lovelock and emergentism

In James Lovelock's 2006 work, The Revenge of Gaia, he concludes the chapter entitled What is Gaia? with a description of the regulator in James Watt's steam engine, and the following argument:

"Simple working regulators, the physiological systems in our bodies that regulate our temperature, blood pressure and chemical composition...are all outside the sharply-defined boundary of Cartesian cause-and-effect thinking. Whenever an engineer like Watt 'closes the loop' linking the parts of his regulator and sets the engine running, there is no linear way to explain its working. The logic becomes circular; more importantly, the whole thing has become more than the sum of its parts. From the collection of elements now in operation, a new property, self-regulation, emerges - a property shared by all living things, mechanisms like thermostats, automatic pilots, and the Earth itself.

"The philosopher Mary Midgley in her pellucid writing reminds us that the twentieth century was the time when Cartesian science triumphed...Life, the universe, consciousness, and even simpler things like riding a bicycle, are inexplicable in words. We are only just beginning to tackle these emergent phenomena, and in Gaia they are as difficult as the near magic of the quantum physics of entanglement."

Now Lovelock is an elegant and fascinating author, but here his thought is lazy, sloganistic and poorly-informed. There are multiple confusions here, and such confusions are endemic amongst a number of writers and journalists who take an interest in science, so let's try and clear them up.

Firstly, we encounter the slogan that a system can be 'more than the sum of its parts'. Unfortunately, the authors who make this statement never seem to conjoin the assertion with a definition of what they mean by the phrase 'sum of its parts'. Most scientists would say that the sum of the parts of a system comprises the parts of the system, their properties, and all the relationships and interactions between the parts. If you think that there is more to a whole system than its parts, their properties and the relationships between the parts, then that amounts to a modern form of vitalism and/or dualism, the notion that living things and/or conscious things depend upon non-physical elements. Calling it 'emergentism' is simply a way of trying to dress up a disreputable idea in different language, rather in the manner than creationism was re-marketed as 'intelligent design'.

Assertions that a system can be more than the sum of its parts are frequently combined with attacks on so-called 'reductionistic' science. Anti-reductionistic authors can often be found pointing out that whole systems possess properties which are not possessed by any of the parts of which that system is composed. However, if such authors think this is somehow anti-reductionistic, then they have profoundly mis-understood what reductionistic science does. Scientists understand that whole systems possess properties which are not possessed by any of the parts; that's precisely because the parts engage in various relationships and interactions. A primary objective of reductionistic science is to try and understand the properties of a whole system in terms of its parts, and the relationships between the parts: diamond and graphite, for example, are both composed of the same parts, (carbon atoms), but what gives diamond and graphite their different properties are the different arrangements of the carbon atoms. Explaining the different properties of carbon and diamond in terms of the different relationships between the parts of which they are composed is a triumph of so-called 'reductionistic' science.

The next confusion we find in Lovelock's argument is the notion that twentieth-century science was somehow linear, or Cartesian, and non-linear systems with feedback somehow lie outside the domain of this world-view. Given the huge body of twentieth-century science devoted to non-linear systems, this will come as something of surprise to many scientists. For example, in General Relativity, (that exemplar of twentieth-century science), the field equations are non-linear. Lovelock might even have heard the phrase 'matter tells space how to curve, and space tells matter how to move'; a feedback cycle, in other words! Yet General Relativity is also a prime exemplar of determinism: the state of the universe at one moment in time uniquely determines its state at all other moments in time. There is clearly no reason to accept the implication that cause-and-effect must be confined to linear chains; non-linear systems with feedback are causal systems just as much as linear systems.

It is amusing the note that Lovelock concludes his attack on so-called 'Cartesian' science with an allusion to quantum entanglement. Clearly, quantum entanglement is a product of quantum physics, that other exemplar of twentieth century physics. So, in one and same breath, twentieth century science is accused of being incapable of dealing with emergentism, yet also somehow yields the primary example of emergentism!

Authors such as Lovelock, Midgley, and their journalistic brethren, are culpable here of insufficient curiosity and insufficient understanding. The arguments they raise against twentieth-century science merely indicate that they have failed to fully understand twentieth-century science and physics.

Tuesday, December 23, 2014

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 hT is simply the sum of the enthalpy and kinetic energy:

 hT= 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.

Saturday, September 20, 2014

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.

Sunday, August 31, 2014

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...