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.