So how does such a shape create streamwise vorticity? Well, the answer lies in a subfield of fluid mechanics called 'secondary flows', (with thanks to Professor Gary Coleman of Southampton University, for pointing me in the direction of this field). Such flows typically involve a primary flow - with the streamlines oriented in a particular direction, and a vorticity field perpendicular to the primary flow - in which there is also some type of differential convection to the primary flow. ('Convection' here simply means the transport of fluid by bulk motion, sometimes referred to as advection if there is any confusion with thermal convection). This differential convection tilts and stretches the vorticity lines, increasing the magnitude of the vorticity, and re-directing it in a streamwise orientation. The streamlines corresponding to this vorticity constitute the secondary motion, superimposed upon the primary streamlines.

This type of secondary flow is exactly what Red Bull are using to create separated streamwise vortices from the boundary layer on their front-wing. But before proceeding further, let's establish some notation. In what follows, we shall denote the streamwise direction as x, the direction normal to the wing as y, and the spanwise direction as z. We also have three components for the velocity vector field, which will be denoted as U, V and W, respectively. There is also a vorticity vector field, whose components will be denoted as ω

_{x}, ω

_{y}, and ω

_{z}.

On the underside of the front-wing is a boundary layer, and like all boundary layers, there is a velocity gradient ∂U/∂y in a direction normal to the wing, given that the velocity is zero at the solid surface. This entails that the boundary layer possesses vorticity in a spanwise direction ω

_{z}. The vortex lines in this boundary layer are perpendicular to the streamwise direction of flow. The trick is then to convert some of this spanwise vorticity into streamwise vorticity ω

_{x}. It transpires that the way to do this is to create a lateral pressure gradient ∂p/∂z.

Now, the front-wing operates in ground effect, so the pressure in an elevated mini-arch will be less than it is underneath the adjacent portion of the main plane, creating just such a pressure gradient. The crucial point is that this lateral pressure gradient corresponds to the creation of a spanwise-gradient in the streamwise velocity ∂U/∂z > 0. To see why this is crucial, however, we need to look at the Vorticity Transport Equation (VTE) for ω

_{x}, the streamwise component of vorticity. The effect in question can be seen by studying incompressible, inviscid, laminar flow, so we can simplify the VTE by omitting the turbulent and viscous terms to obtain:

Dω

_{x}/Dt = ω

_{x}(∂U/∂x) + ω

_{y}(∂U/∂y) + ω

_{z}(∂U/∂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, we started with ω

_{z}> 0 in the boundary layer, and by virtue of creating a lateral pressure gradient, we also have ∂U/∂z > 0. This means that the third term on the right-hand side in the equation above is positive, which (assuming the other pair of terms are non-negative) entails that Dω

_{x}/Dt > 0.

Thus, the creation of the spanwise-gradient in the streamwise velocity ∂U/∂z, skews the initially spanwise vortex lines ω

_{z}until they possess a significant component ω

_{x}in a streamwise direction. The lateral pressure gradient has created streamwise vorticity.

As Peter Bradshaw writes, "if the lateral deflection that produces longitudinal vorticity extends for only a small

*spanwise*direction, then the longitudinal vorticity becomes concentrated into a vortex," (

*Turbulent secondary flows*, Ann. Rev. Fluid Mechanics 1987, p64). Which is exactly what Red Bull, and for that matter, many other Formula 1 teams, do when they incorporate mini-arches into their front-wings.

As a final aside, note that there is an interesting duality at the heart of fluid mechanics, namely that between a description which uses the velocity and pressure fields, and a description which uses the vorticity field instead. The vorticity has been described as "the sinews and muscles of fluid mechanics," (Kuchemann 1965,

*Report on the IUTAM Symposium on concentrated vortex motions in fluids*, Fluid Mech. 21). P.A. Davidson points out that in the case of incompressible flow, because pressure waves can travel infinitely fast, the velocity vector field is a non-local field; the vorticity field, in contrast, is local. "While linear momentum can be instantaneously redistributed throughout space by the pressure field, vorticity can only spread through a fluid in an incremental fashion, either by diffusion or else by material transport (advection). Without doubt, it is the vorticity field, and not [the velocity field], which is the more fundamental," (

*Turbulence*, 2004, p39).

An aerodynamicist with an especially strong visual imagination, perhaps someone who had been stimulated to develop such mental capabilities to compensate for dyslexia, might be able to develop a better understanding of the fluid flow around a Formula 1 car by thinking in terms of vorticity, or by developing the ability to mentally switch back and forth between the vorticity and velocity representations. Such an individual might even reject tools such as CAD and CFD, preferring instead to work on a drawing board...

## No comments:

Post a Comment