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.