## Monday, February 14, 2011

### Ducks, geese, and Karman vortex streets

An expert on neutrino dynamics has drawn my attention to a remarkable paper, 'Kinematics of Ducklings Swimming in Formation', written by a certain Frank E.Fish, (who I believe also had a cameo role in Finding Nemo).

Fish explains that the ducks to the rear of a formation utilise the reduced drag afforded by a Karman vortex street, created by the ducks to the front. The ducks at the rear maintain the same velocity as the ducks at the front, but do so with a reduced energy expenditure.

Now, Karman vortex streets are staggered rows of counter-rotating vortices, created by any bluff cylindrical body in relative motion with a Reynolds number greater than 90. (The upper limit at which these vortices are generated is dependent upon the exact shape of the object in question, but in the case of a cylinder, figures such as 107 are quoted). The axis of rotation of these vortices will be in a plane orthogonal to the direction of motion, as seen in the topmost diagram here (taken from Joe Katz's book, Introductory Fluid Mechanics). In the region between the two vortex rows, there is a forward component to the fluid velocity, and in the case of our trailing ducks, it is this which reduces the energy expenditure required to maintain the same speed as the leading ducks. In particular, it seems that the ducks at the rear save energy by reducing the amplitude, rather than the frequency, of their power-strokes.

In contrast, the trailing vortices shed by a wing will have axes of rotation parallel to the direction of motion. Geese, for example, appear to fly in V-formations, not because of Karman vortices, but because they use the trailing wing-tip vortices to gain lift. The outer side of each trailing vortex of the bird in front has an upward component, hence less energy is expended by the bird behind
to maintain the same level of lift.

For a Formula One car, the Reynolds number is of the order of 106, which is just about the upper limit for any sort of vortex street pattern to be distinguishable from random turbulence.

If Karman vortices are generated in the wake of a Formula One car, one presumes that they also interact in a complex manner with the trailing vortices generated by the wings et al...