The global nature of aerodynamics

It's been a commonplace observation in motorsport over the years, that simply copying the front or rear-end of a successful car, won't necessarily assist the performance of your own car.

It's a message that Gary Anderson reiterates in his assessment of the Red Bull front-wing: "I guarantee that if you put a Red Bull front wing on any other F1 car in a wind tunnel, it would be worse than that team's current configuration. That's because it's not just the wing that makes the Red Bull the best car aerodynamically, it's the whole aero philosophy of the car, with everything working in sync with each other."

Now, it's certainly easy to understand that the shape of the front-wing will influence the pattern of airflow over the rest of the car. The airflow of a racing car possesses a well-defined directionality which enables one to distinguish between that which is upstream and that which is downstream. And, at first sight, this seems to be an asymmetrical relationship, in the sense that the upstream airflow clearly influences and determines the nature of the airflow downstream. However, aerodynamics, (and fluid mechanics in general), has a slightly spooky aspect to it, in the sense that the downstream airflow also influences and determines the upstream airflow.

To take a simple example, if one creates a blockage downstream of the front-wing, then this will reduce the velocity of the airflow over the front-wing, reducing the downforce it generates. Conversely, if one accelerates the airflow downstream of the front-wing, then this will also accelerate the airflow over the front-wing, increasing the downforce it generates.

There are also more subtle examples of this type of aerodynamic spookiness. Back in 2006, the change to 2.4 litre V8 engines in Formula 1 triggered the proliferation of so-called flow conditioners atop the nose of the chassis. These appendages re-aligned the upper-body airflow in a manner which reduced the rate of airflow over the front-wing, but increased it over the rear-wing. Recall that the downforce generated by a wing is proportional to both the speed of the airflow over it, and the angle-of-attack of the wing. Thus, Formula 1 teams wishing to compensate for the loss of engine power, could reduce the drag generated by the rear wing for a given level of downforce, by reducing the angle-of-attack, but increasing the airflow over it. (See 2006 Formula 1 Review, Mark Hughes, Autosport December 14th/21st).

The fact that information downstream influences and determines the information upstream, and vice versa, is grounded in the mathematics used to represent steady-state, subsonic (i.e., relatively incompressible) airflow. Crucially, the Navier-Stokes equations for such an airflow regime are elliptic. (Diagrams here courtesy of Computational Fluid Dynamics, J.Blazek, Elsevier 2001).

Elliptic equations are typical, mathematically speaking, of equilibrium situations, and in these circumstances, the information at any point is capable of influencing the information at any other point. In the case of steady-state aerodynamics, this means that the airflow information at any point is capable of influencing the airflow information at any other point, upstream or downstream. Hence, the aerodynamicist must seek optimal global solutions, rather than simply working from the front of the car rearwards.

Now, ellipticity is only one of the possible personalities which differential equations can possess. Other equations, for example, are hyperbolic or parabolic. In the case of hyperbolic equations, the information at a point will only influence the part of the domain enclosed within a cone-shaped region emanating from that point. Meanwhile, in the case of a parabolic equation, the information at a point will influence a block-shaped region downstream of that point.

Whilst the airflow over a racing car travelling in a straight line at a constant speed corresponds to a solution of the steady-state, incompressible Navier-Stokes equations, when a car is subject to yaw under cornering conditions, the airflow corresponds to a solution of the unsteady, incompressible Navier-Stokes equations. In this case, the equations are parabolic, and what happens at the front-wing is extra-crucial, for the information at any point upstream influences the entire solution downstream.

As ever, the key is to be a leader rather than a follower.