Wednesday, December 23, 2015
Tornados and the Y250 wing-tip vortex
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 Ux, Uy, and Uz, There is also a vorticity vector field, whose components will be denoted as ωx, ωy, and ωz.
The vorticity vector field ω is defined as the curl of the velocity vector field:
ω = (ωx , ωy, ωz)
= (∂Uz/∂y − ∂Uy/∂z , ∂Ux/∂z − ∂Uz/∂x , ∂Uy/∂x − ∂Ux/∂y)
We're also interested here in the Vorticity Transport Equation (VTE) for ωx, the streamwise component of vorticity. In this context we can simplify the VTE by omitting turbulent, viscous and baroclinic terms to obtain:
Dωx/Dt = ωx(∂Ux/∂x) + ωy(∂Ux/∂y) + ωz(∂Ux/∂z)
The left-hand side here, Dωx/Dt, is the material derivative of the x-component of vorticity; it denotes the change of ωx 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 ωy, and a lateral gradient in streamwise velocity, ∂Ux/∂y ≠ 0.
2) A vertical gradient in the lateral component of velocity, ∂Uy/∂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 ∂Ux/∂y ≠ 0. Given that ωy ≠ 0 in the boundary layer of the wing, combining this with ∂Ux/∂y ≠ 0 entails that the second term on the right-hand side in the VTE is non-zero, which entails that Dωx/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 ωx 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 ∂Uy/∂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 ωx:
ωx = ∂Uz/∂y − ∂Uy/∂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.