The clockwork of vortex shedding

A cylindrical object of diameter D immersed into a flow of sufficiently high velocity U sheds vortices at the Strouhal frequency f = SU/D. It is shown here that S depends on the drag coefficient C-D as well as on the Reynolds number. We first prove that the oscillation in the wake is a simple harmonic motion process; then we derive its frequency f = zeta/2 pi 2(1/2) from the momentum equation in transverse direction. The average near wake vorticity zeta is then calculated from the Navier-Stokes equation as a function of the drag coefficient C-D, and the normalized vortex radius alpha = R/D1, so that the Strouhal number S = fD/U can be linked to alpha and C-D. For Reynolds numbers above approximate to 10(3) we find S = S-infinity = 2(-7/4) pi(-1) alpha(-1) (1 + C-D)(1/2). For Re below approximate to 10(3) we obtain S(Re) = S-infinity[(1 + x(2))(1/2)-x], where x = Re-star/Re. The experimental data for circular cylinders are well described by this functional form, yielding by curve fit the empirical constants Re-star = 28, alpha = 0.62, and S-infinity = 0.22.