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.

9 comments:

Peter B said...

Excellent post Professor McCabe.

My questions are as follows:

In the 2013 Autosport Review issue Mark Hughes said that the Y250 vortex on the Red Bull car was fed into the tunnels at the back of the car and emitted over the top of the diffuser preventing it from stalling.
If the low pressure is confined to small radial area in the vortex core how can it really influence the diffuser base pressure ? How does the flow over the top of diffuser help extract the diffuser exit flow ? How does this mingling of flows work ? Does "entrainment" means one flow regime "sucks" another into itself ?

Gordon McCabe said...

Thanks again.

I'm dubious about whether the Y250 itself reaches the tunnels at the back of the Red Bull. The Y250 can influence the diffuser performance, because all the teams use it to push the turbulent front-wheel-wake outboard. But perhaps Red Bull also use the Y250 to inject higher-energy air into those tunnels. I don't know.

The flow over the top of the diffuser transfers momentum to the flow exiting the diffuser by shear stress (friction between adjacent layers of fluid again). By accelerating the air at the diffuser exit that reduces the adverse pressure gradient in the diffuser.

Peter B said...

But how does the adverse pressure gradient in the diffuser get reduced ? Doesn't the diffuser outlet pressure "track" the inlet pressure ?
I mean doesn't lower base pressure
mean lower diffuser inlet depression ?

Gordon McCabe said...

I don't think so, although you shouldn't treat my judgements as gospel! A good comparison might be a multi-element wing, where the acceleration around the leading edge of the flap reduces the adverse pressure gradient the main-plane experiences.

jeanmarc pierre said...

Very good post again.
I don't have time now to understand in the detail but it is a notion that was puzzling me. i will understand

THis vortex visualisation made me better understand what's going on in the area around the bottom of the sidepod, well it maybe wrong, tell me what you think. The big probleme is the turbulent front wheel wake as you write but i think the shield to prevent this wake to feed the diffuser both inside and on top of it is the accelerated air coming from under the nose and precisely restricted to a narrower section as it goes toward the side pod bottom, hence the solution from lotus some years ago with the front exhaust exit which i didn't quite understood. In fact it was directed where i think the undernose stream is coming forming a V shape shield which we can see sometimes. In doing so, a nice gap appears behind (if air no longer go for a gap) that sucks or lets say allow the y250 vortex to flow there and nowhere else. It is similar to what is seen on the F18 fighter for exemple where the Lerx create a vortex upstream and the extension downstream of this edge prevent other flow than the one coming from this vortex to feed this area. My explanation are pretty basics and my english pretty poor sorry but i understand myself...
There is a flow substitution in a way, a magic trick, that's why we see the vortex coming up over the river coming from under the nose then it plunges into the cock bottle shape area feeding it with an air with more dynamic pressure.
It is more to reduce drag at top speed i think than feeding the top of the diffuser which is the role of the clean air coming from the top of the sidepod the roof of an f1 car in a way.

Gordon McCabe said...

That's excellent jeanmarc, I understand what you're saying.

Peter B said...

Hey Professor,

I'm sure you've seen the unsightly noses on this year's F1 car.

Ian Bamsey has written in F1 Race Technology magazine that "in simple
terms the higher or more arched the nose of a contemporary F1 car, the higher the rear downforce potential at the expense of front downforce".

He did not elaborate but in some early 1990's issues of Race Car Engineering he wrote several times
that a high nose doesn't necessarily mean more air squeezes
itself under the car thereby moving faster than with low nose.
He said the air has to be pulled through by the diffuser ?

Why will more mass flow to diffuser
increase downforce ?

Is it possible to use the simplified Bernoulli equation
show how nozzle ahead of diffuser
designed to "match" the nozzle
parameters might lead to more
downforce from the diffuser ?

Gordon McCabe said...

I can't get into specifics these days, but the general idea is that the mass-flow is proportional to the ratio between the underbody outlet area and the underbody inlet area. With relatively incompressible air, and fixed cross-sectional area, greater mass-flow entails greater velocity, which entails lower pressure. And of course you want to feed the underbody with air which has the minimum level of turbulent intensity.

Peter B said...

Heads up !!

http://fluidsengineering.asmedigitalcollection.asme.org/article.aspx?articleid=1767140

Brand new Automotive vortex aero
paper by Josef Katz

Includes new data on vortex wakes and the effect of rake angle of the underbody !!! Fascinating reading !!

Kindest regards,

Peter