Tornados and the Y250 wing-tip vortex

Streamwise vortices occur when fluid spirals around an axis which points in the same direction as the overall direction of fluid flow. In particular, streamwise vortices are generated by aircraft wing-tips, and by the front-wing of a Formula 1 car at the inboard transition between the neutral central section and the inner tip of the main-plane and flap(s). The latter is the so-called Y250 vortex. Surprisingly, the method by which such streamwise vorticity is generated also plays a crucial role in the generation of atmospheric tornados.

Let's begin with the meteorology. A tornado is a funnel of concentrated vertical vorticity in the atmosphere. Most tornados are generated within supercell thunderstorms when the updraft of the storm combines with the horizontal vorticity generated by vertical wind shear. The updraft tilts the horizontal vorticity into vertical vorticity, generating a rotating updraft.

However, there are two distinct types of vertical wind shear: Unidirectional and directional. The former generates crosswise vorticity, whilst the latter generates streamwise vorticity.

When the wind shear associated with a storm is unidirectional, the updraft acquires no net rotation. The updraft raises the crosswise vorticity into a hairpin shape, with one cyclonically rotating leg, on the right as one looks downstream, and an anticyclonic leg on the left. Updrafts only acquire net cyclonic rotation when the horizontal vorticity has a streamwise component. (Diagrams above and below from St Andrews University Climate and Weather Systems website).

Specifically, cyclonic tornado formation requires that the wind veers with vertical height, (meaning that its direction rotates in a clockwise sense).

In effect, the flow of air through the updraft becomes analagous to flow over a hill (personal communication with Robert Davies-Jones): the flow into the updraft has cyclonic vorticity, and the flow velocity there reinforces the vertical velocity of the updraft; the downward flow on the other side, where the anticyclonic vorticity exists, partially cancels the vertical velocity of the updraft. Hence, the cyclonic part of the updraft becomes dominant.

Before we turn to consider wing-tip vortices, we need to recall the mathematical definition of vorticity, and the vorticity transport equation.

Let's start with some notation. In what follows, we shall denote the streamwise direction as x, the lateral (aka 'spanwise' or 'crosswise') direction as y, and the vertical direction as z. The velocity vector field U has components in these directions, denoted respectively as U_x, U_y, and U_z, There is also a vorticity vector field, whose components will be denoted as ω_x, ω_y, and ω_z.

The vorticity vector field ω is defined as the curl of the velocity vector field:

ω = (ω_x , ω_y, ω_z)

= (∂U_z/∂y − ∂U_y/∂z , ∂U_x/∂z − ∂U_z/∂x , ∂U_y/∂x − ∂U_x/∂y)

We're also interested here in the Vorticity Transport Equation (VTE) for ω_x, the streamwise component of vorticity. In this context we can simplify the VTE by omitting turbulent, viscous and baroclinic terms to obtain:

Dω_x/Dt = ω_x(∂U_x/∂x) + ω_y(∂U_x/∂y) + ω_z(∂U_x/∂z)

The left-hand side here, Dω_x/Dt, is the material derivative of the x-component of vorticity; it denotes the change of ω_x in material fluid elements convected downstream by the flow.

Now, for a racecar, streamwise vorticity can be created by at least two distinct front-wing mechanisms:

1) A combination of initial lateral vorticity ω_y, and a lateral gradient in streamwise velocity, ∂U_x/∂y ≠ 0.

2) A vertical gradient in the lateral component of velocity, ∂U_y/∂z ≠ 0, (corresponding to directional vertical wind shear in meteorology).

In the case of the first mechanism, one can vary the chord, camber, or angle of attack possessed by sections of the wing to create a lateral gradient in the streamwise velocity ∂U_x/∂y ≠ 0. Given that ω_y ≠ 0 in the boundary layer of the wing, combining this with ∂U_x/∂y ≠ 0 entails that the second term on the right-hand side in the VTE is non-zero, which entails that Dω_x/Dt ≠ 0. Thus, the creation of the spanwise-gradient in the streamwise velocity skews the initially spanwise vortex lines until they possess a significant component ω_x in a streamwise direction.

However, it is perhaps the second mechanism which provides the best insight into the formation of wing-tip vortices. As the diagram above illustrates for the case of an aircraft wing (G.A.Tokaty, A History and Philosophy of Fluid Mechanics), the spanwise component of the flow varies above and below the wing. This corresponds to a non-zero value of ∂U_y/∂z, and such a non-zero value plugs straight into the definition of the curl of the velocity vector field, yielding a non-zero value for the streamwise vorticity ω_x:

ω_x = ∂U_z/∂y − ∂U_y/∂z

Putting this in meteorological terms, looking from the front of a Formula 1 car (with inverted wing-sections, remember), the left-hand-side of the front-wing has a veering flow-field at the junction between the flap/main-plane and the neutral section. The streamlines are, in meteorological terms, South-Easterlies under the wing, veering to South-Westerlies above. This produces streamwise vorticity of positive sign.

On the right-hand side, the flow-field is backing with increasing vertical height z. The streamlines are South-Westerlies under the wing, backing to South-Easterlies above. This produces streamwise vorticity with a negative sign.

Thus, we have demonstrated that the generation of the Y250 vortex employs the same mechanism for streamwise vorticity formation as that required for tornadogenesis.

The open-tailed box effect

The modern understanding of racecar aerodynamics holds that copious amounts of downforce can be produced by accelerating the airflow under the car, in effect turning the region between the underbody and ground plane into a mobile nozzle.

The Lotus 78 of 1977 famously introduced venturi profiles beneath the car, and sliding skirts to seal the low pressure area thereby created. However, it is less well-known that underbody skirts had fitfully appeared on various cars earlier in the decade. Moreover, it is slightly disconcerting to hear the explanations proffered by several F1 designers from the middle 1970s for the function of these devices.

Gordon Murray introduced inch-deep skirts on the underside of the 1975 Brabham BT44 in conjunction with an overall 'upturned saucer' design, and explains his thinking as follows:

"With any moving form you have a stagnation point where air meets it and decides how much is going to flow over, below or around it...I decided, instead of presenting some sort of parabolic-shaped bluff body to the air, I wouldn't give the air a chance." He sketches a triangular shape. "That way the stagnation point was there," he says, pointing to the leading edge of the triangle's base, which is very low to the ground. "So all the air had to go over the top and you had the minimum coming under the car," (F1 Magazine, May 2001, p140-141).

Gordon Coppuck, however, had already experimented with skirts on the McLaren M23:

"In 1974 at Dijon-Prenois, vertical plastic skirts around the under-periphery of the car were tried, but they quickly wore away on contact with the track. The idea was to exclude air from underneath the car and so minimise lift," (p49, McLaren M23, Ian Wagstaff, Haynes 2013). The skirts were fitted again to the M23 at some races in early 1976, this time provoking complaints from competitors such as Colin Chapman (!) and Ken Tyrrell.

Talk of minimising lift by forcing air over the top of the car seems misguided because the upper surface of a racecar is generally convex, and the air will tend to be accelerated by a convex surface, producing low pressure on the upper surfaces, somewhat counter to the overall objective.

Nevertheless, it seems that there actually was a beneficial effect to be had from partially excluding air from the underbody, and this is clearly explained by Ian Bamsey in his fantastic book The Anatomy and Development of the Sports Prototype Racing Car (Haynes, 1991):

"The [Shadow] DN8 had conventional wings and a flat bottom and, following the fashion of 1976, it was fitted with skirts along the side of its monocoque, these joined in a vee under the nose. Under certain conditions the skirts rubbed on the track and their general effect was to sweep the air aside, in snowplough fashion. Thus, the overall effect was not one of spatial acceleration of the underbody air, it was one of exclusion. The flow blockage allowed the forward migration of the naturally low pressure air at the back of the car into the skirt's exclusion zone. This was the principle of the so-called open tailed box. A box with the road forming its bottom and only its tail open will experience a pressure reduction within as it progresses along the track," (p59).

So, although the effect may be quite weak, it is possible to generate downforce by excluding air from the underbody. 

The L'Oreal Women in Science initiative

"Much remains to be done with regard to gender balance in science. Most tellingly, women account for only 30% of the world’s researchers. There are still great barriers that discourage women from entering the profession and obstacles continue to block progress for those already in the field."

So complains the L'Oreal-UNESCO 'For Women in Science' initiative. Since 1998 this programme has "strived to support and recognize accomplished women researchers, to encourage more young women to enter the profession and to assist them once their careers are in progress," by means of awards, fellowships, and advertising campaigns declaring that 'Science Needs Women'.

In comparison, the plight of men employed in the nursing profession has received little attention. To place this in the type of quantitative context which should appeal to 'Women in Science', the UK Office for National Statistics compiles an Annual Survey of Hours and Earnings (ASHE), based upon a sample taken from HM Revenue and Customs' Pay As You Earn (PAYE) records. Amongst other information, this reveals the number of men and women employed in different professions. The 2015 results estimate that the number of men and women employed in nursing are as follows:

Women in nursing: 673,000

Men in nursing: 109,000

Hence, only 14% of nurses in the UK are men, a figure somewhat lower than the 30% of 'Women in Science' worldwide. This shocking gender imbalance suggests that men are systematically discouraged from entering the nursing profession, are discriminated against within the profession, and have their progress blocked within the field. 

Now, some people might argue that this is only natural because men have a tendency to be more aggressive and competitive than women, a characteristic which makes women rather more suited to the caring professions.

This, however, is merely one of the phony arguments used by the nursing matriarchy to preserve the pre-eminent status of women within the profession. Men have evolved by sexual selection to be more aggressive and competitive in order to make themselves more attractive to women, and thereby enhance their prospects of being chosen for mating. It is therefore women and their mating criteria which are ultimately responsible for the aggressive and competitive nature of men.

Hence, it is about time that L'Oreal expanded its concerns over professional gender imbalance, and initiated a range of awards and fellowships to assist the cause of Men in Nursing (MIN). If possible, the assistance of the BBC should be sought to promulgate a range of positive Male Nursing stereotypes within its programming; for example, all hospital scenes should feature male nurses in prominent roles, leading and directing their female colleagues.

But hold on: what's this on the L'Oreal website?

"More women scientists should also be able to obtain positions of responsibility, just like their male counterparts, so that future generations will have role models to inspire them. The current situation, however, indicates that, well into the third millennium, a considerable discrepancy exists between what society professes to believe and what we actually do."

The third millennium? The third millennium since what exactly? The 'Women in Science' will be able to tell you that Homo Sapiens have been around for approximately 1.8 million years, so that's about one thousand eight hundred millennia. Not three.

Perhaps we should only consider the period of time which has elapsed since Homo Sapiens made the transition from the hunter-gatherer lifestyle to agriculture and settlement. But that would still be about 12,000 years, four times the number of millennia that L'Oreal are willing to acknowledge.

It's the type of error one would expect of a cosmetics-oriented organisation, rather than a scientifically-oriented one. Perhaps, then, we shall have to cast our net more widely to find a suitable sponsor for MIN...