__The
effect of cylinder rotation on vortex shedding__

** Simon J. Tavener** and K. Andrew Cliffe

Colorado State University and Serco
Assurance, U.K.

The loss of stability of the steady, symmetric flow past a
circular cylinder with increasing flow rate, and the periodic shedding of
vortices downstream of the cylinder that results, is a classical and extremely well-studied
example of hydrodynamic instability. Less well known is the experimental
observation that cylinder rotation delays the onset of the vortex street and
that sufficiently large rotation rates can eliminate the vortex shedding
completely. The instability is believed to arise at an Hopf bifurcation point
and the question naturally arises as to whether this surprising
re-stabilization can be explained in terms of the effect of cylinder rotation
on the Hopf bifurcation. Established numerical techniques for locating Hopf
bifurcation points can provide part of the answer, but in an effort to more
closely compare computational results with existing experimental data, a
combined finite-element and spectral technique was developed for computing
periodic solutions of partial differential equations and of the Navier-Stokes equations
in particular. Unlike straightforward simulation, this method has the advantage
of converging to unstable periodic orbits and does not suffer from extremely
slow rates of convergence near Hopf bifurcation points. It allows the energy
contained within each harmonic of the fundamental frequency to be easily
determined, thereby providing a means for deciding whether a sufficient number
of harmonics are present. Fortunately this number is found to be small for the
flows considered here.