Curved flow and the Arrows A3

After something of a sustained gestation period, the publication of F1 Retro 1980 is imminent, so it's a good opportunity to take a look at one of the more interesting aerodynamic experiments seen that season: the underbody venturi extensions on the Arrows A3 at Brands Hatch. 

This was the latest in a series of attempts to improve upon the original F1 ground-effect concept. In 1979, the Lotus 80 and the Arrows A2 had both attempted to extend the area of the underbody, but both had failed to reap the expected benefits.

The Lotus 80, in its initial configuration, featured skirts under the nose, and separate skirts extending all the way from the leading edge of the sidepods, inside the rear wheels, to the back of the car. The failure of the Lotus 80 is commonly attributed both to an ineffective skirt system, and an insufficiently rigid chassis.  

The Arrows A2 featured an engine and gearbox inclined at 2.5 degrees in an attempt to exploit the full width of the rear underbody. In its original configuration the A2 also dispensed with a conventional rear-wing, replacing it with a flap mounted across the rear-deck. The sidepod skirts were complemented by a parallel pair of skirts running inside the width of the rear wheels to the back of the car. Unfortunately, the higher CoG at the back entailed the car had to be run with a stiff rear anti-roll bar, detracting from the handling, (Tony Southgate - From Drawing Board to Chequered Flag, MRP 2010, p108).

The 1980 Arrows A3 was a more conventional car, with the engine and gearbox returned to a horizontal inclination. However, at Brands Match in 1980, Arrows experimented, like the initial Lotus 80, with skirts under the nose. Developed in the Imperial College wind-tunnel, the Arrows version of the idea had skirts suspended from sponsons attached to the lower edges of the monocoque, running back beneath the lower front wishbones to the leading edge of the sidepods. At the same event, the team also tried extending the rear underbody all the way to the trailing edge of the rear suspension, with bulbous fairings either side of the gearbox fairing. This was done with the avowed intention of sealing the underbody from the detrimental effects of rear wheel turbulence.

Sadly, although the nose-skirts were intended to cure understeer, it was reported that they actually exacerbated the understeer.

Now, many aerodynamic difficulties encountered in this era of Formula One were actually just a manifestation of inadequate stiffness in the chassis or suspension. However, for the sake of argument, let's pursue an aerodynamic hypothesis to explain why the nose-skirts on the A3 worsened its understeer characteristic.

The nose skirts on the Lotus 80 and Arrows A3 would have suffered from the fact that a Formula 1 car has to generate its downforce in a state of yaw. Thus, in a cornering condition, a car is subjected to a curved flow-field. This is difficult to replicate in a wind-tunnel, hence a venturi tunnel design which worked well in a straight-ahead wind-tunnel condition could have failed dramatically under curved flow conditions. To understand this better, a short digression on curved flow and yaw angles is in order.

The first point to note is that a car follows a curved trajectory through a corner, hence if we switch to a reference frame in which the car is fixed but the air is moving, then the air has to follow a curved trajectory. If we freeze the relative motion mid-corner, with the car pointing at a tangent to the curve, then the air at the front of the car will be coming from approximately the direction of the inside front-wheel, while the air at the back of the car will be coming from an outer direction.

That's the simplest way of thinking about it, but there's a further subtlety. The negotiate a corner, a car generates: (i) a lateral force towards the centre of the corner's radius of curvature; and (ii) a yaw moment about its vertical axis.

Imagine the two extremes of motion where only one of these eventualities occur. In the first case, the car would continue pointing straight ahead, but would follow a curved path around the corner, exiting at right-angles to its direction of travel. In the second case, it would spin around its vertical axis while its centre-of-mass continued to travel in a straight line.

In the first case, the lateral component of the car's velocity vector corresponds to a lateral component in the airflow over the car. The angle which the airflow vector subtends to the longitudinal axis of the car, is the same along the length of the vehicle.

In the second case, the spinning motion also induces an additional component to the airflow over the car. It's a solid body spinning about its centre of mass with a fixed angular velocity, and the tangential velocity of that spin induces an additional component to the airflow velocity along the length of the car. However, the further away a point is from the axis of rotation, the greater the tangential velocity; such points have to sweep out circles of greater circumference than points closer to the centre of mass, hence their tangential velocity is greater.

Curved-flow, side-slip and yaw-angle. (From 'Development methodologies for Formula One aerodynamics', Ogawa et al, Honda R&D Technical Review 2009).

Now imagine the two types of motion combined. The result is depicted above, in the left-part of the diagram. The white arrows depict the component of the airflow due to 'side-slip': the car's instantaneous velocity vector subtends a small angle to the direction in which its longitudinal axis is pointing. In the reference frame in which the car is fixed, this corresponds to a lateral component in the direction of the airflow which is constant along on the length of the car.

When the yaw moment of the car is included (indicated by the curved blue arrow about the centre-of-mass), it induces an additional airflow component, indicated by the green arrows. Two things should be noted: (i) the green arrows at the front of the car point in the opposite direction from the green arrows at the rear; and (ii) the magnitude of the green arrows increases with distance from the centre of mass. The front of the car is rotating towards the inside of the corner, while the rear of the car is rotating away, hence the difference in the direction of the green arrows. And, as we explained above, the tangential velocity increases with distance from the axis of rotation, hence the increase in the magnitude of the green arrows.

The net result, indicated by the red arrows, is that the yaw-angle of the airflow has a different sign at the front and rear of the car, and the magnitude of the yaw angle increases with distance from the centre-of-mass. (The red arrows in the diagram are pointing in the direction in which the car is travelling; the airflow direction is obtained by reversing these arrows).

So, to return to 1980, the Arrows A3 design trialed at Brands Hatch moved the mouth of the venturi tunnel forward to the nose of the car. The further forward the mouth, the greater the angle of the curved onset flow to the longitudinal axis of the car, and the further away it is from the straight-ahead condition. Hence, the curved flow might well have separated from the leading edge of the skirt on the side of the car facing the inside of the corner, injecting a turbulent wake directly down the centre of the underbody. In this respect, the conventional location of the venturi inlets on a 1980 F1 car, (i.e., behind the front wheel centreline), would have reduced yaw sensitivity.

Front-wings and rear-wings certainly have to operate in state of yaw, and do so with a relatively high level of success. However, such devices have a larger aspect-ratio than an underbody venturi, which has to keep its boundary layer attached for a much longer distance.

It should also be noted that the flow through the underbody tunnels, like that through any type of duct, suffers from ‘losses’ which induce drag. The energy budget of a flow-field can be partitioned into kinetic energy, pressure-energy, and ‘internal’ heat energy. Viscous friction in the boundary layers, and any turbulence which follows from separation in the duct, creates heat energy, and irreversibly reduces the sum of the mean-flow kinetic energy and the pressure energy.

These energy losses are proportional to the length of the duct, the average flow velocity through the duct, and inversely proportional to the effective cross-sectional diameter of the duct. Due to such losses, it is not possible for full pressure recovery to be attained in the diffuser and its wake, and this will contribute to the total drag of the car. Hence, whilst underbody downforce comes with less of a drag penalty than that associated with inverted wings in freestream flow, it is nevertheless true that the longer the venturi tunnels, and the greater the average velocity of the underbody flow, the greater the drag of the car. 

 

Moreover, the longer the mouth and throat of a venturi tunnel, the thicker the boundary layer at the start of the pressure recovery region, and the more prone it will be to separation in that adverse pressure gradient. All of which mitigates against a quick and easy gain from extending the area of the underbody.

Driverless cars and cities

Driverless cars are somewhat in the news this year, with Ford investing $1bn to meet their objective of launching a fleet of autonomous cars in 2021. Coincidentally, the July 2017 issue of 'Scientific American' features an article extolling the virtues of a driverless future in modern cities. The article is written by Carlo Ratti and Assaf Biderman, who work for something called the 'Senseable Lab' at the Massachusetts Institute of Technology. 

 A number of the claims made in the article are worth reviewing. Let's start with the following:

"On average, cars sit idle 96 percent of the time. That makes them ideal candidates for the sharing economy...The potential to reduce congestion is enormous...'Your' car could give you a lift to work in the morning and then, rather than sitting in a parking lot, give a lift to someone else in your family - or to anyone else in your neighbourhood or social media community...a city might get by with just 20 percent the number of cars now in use...fewer cars might also mean shorter travel times, less congestion and a smaller environmental impact."

A number of thoughts occur in response to these claims:

1) Ride-sharing would reduce the number of cars, not the number of journeys. Every journey which currently takes place would still take place, but in addition would be all the journeys made when a car needs to travel from the point where one passenger disembarks to the point where the next embarks. At present, each journey contains a passenger; with the proposed ride-sharing of driverless cars, there would be additional journeys in which the cars contain no passengers at all. All other things being equal, that would increase congestion and pollution, not reduce it. 

2) The modern technological world, including the GPUs and artificial neural networks which have created the possibility of driverless vehicles, has been built upon the wealth of a capitalist economy. Such an economy is driven by, amongst other things, the incentivization of private ownership. In particular, people like owning their own cars. It's not clear why a technological development alone, such as that of the driverless car, will prompt society to adopt a new model of shared ownership.

3) Not everyone lives in cities. Universities tend to be located in cities, hence many academics fall into the habit of thinking that everyone lives and works in cities. Many people live outside cities, and drive into them to their places of work. They drive into the cities from different places at the same time each morning. For such people, there needs to be a one-to-one correspondence between cars and passengers.

4) People like the convenience and efficacy of having a car parked adjacent to their home or place of work. If you're a parent, and your child falls ill at home, or there's an accident at school, you want to drive there immediately, not wait for a shared resource to arrive.

5) If cars are constantly in use, their components will degrade in a shorter period of time, so maintenance costs will greater, and the environmental impact of manufacturing new tyres, batteries etc. will be greater.

So that's just for starters. What else do our MIT friends have to say? Well, they next claim that "vacant parking lots could be converted to offer shared public amenities such as playgrounds, cafes, fitness trails and bike lanes."

Unfortunately, most car parks are privately owned, either by retail outlets or employers. If they become redundant, then those private companies will either extend their existing office space or floor space, or sell to the highest bidder. Car-parks are unlikely to become playgrounds.

The authors then claim that current traffic-light controlled intersections could be managed in the style of air traffic control systems: 

"On approaching an intersection, a vehicle would automatically contact a traffic-management system to request access. It would then be assigned an individualized time, or 'slot', to pass through the intersection.

"Slot-based intersections could significantly reduce queues and delays...Analyses show that systems assigning slots in real time could allow twice as many vehicles to cross an intersection in the same amount of time as traffic lights usually do...Travel and waiting times would drop; fuel-consumption would go down; and less stop-and-go traffic would mean less air pollution."

Sadly, this is a concept which seems to imagine that cities consist of grids of multi-lane highways. Most cities in the world don't. And in every city, the following 'min-max' principle of road-capacity applies:

For a sequence of interconnected roads, given the capacity (i.e., the maximum flow-rate) in each component, the capacity of the entire sequence is the minimum of those individual capacities. 

Hence, even if the capacity of every multi-lane intersection in a city is doubled, the capacity of a linked sequence is determined by the component with the lowest capacity. In many cities, multi-lane highways taper into single-lane roads, and it is the single-lane roads which limit the overall capacity. Doubling the capacity of intersections would merely change the spatial distribution of the queues.

So, all in all, not a positive advert for driverless cars.