Effect of solid body degrees of freedom on the path instabilities of freely falling or rising flat cylinders
Résumé
In 2007 Fernandes et al. reported a surprising delay of the onset of instabilities for freely rising cylinders of aspect ratio (χ ¼ diameter/thickness of the cylinder) 10 and more. At Reynolds numbers at which instabilities develop in the wake of the same body kept fixed, the path was still found to be vertical and the wake axisymmetric. In this paper, we explain this delay by investigating numerically the transition scenario of the solid–fluid system represented by the freely moving body interacting with the ambient fluid by hydrodynamic forces. We show that the free body degrees of freedom can have a stabilizing effect on the onset of the primary bifurcation. This effect explains, however, only partly the experimental observation. We show that the primary bifurcation is followed by a sequence of weakly oscillating, virtually unobservable, bifurcating states before the observable path oscillations set in. For aspect ratio smaller than 8, the free body degrees of freedom destabilize the system in agreement with expectations. The primary bifurcation is, however, a Hopf bifurcation instead of regular one in the wake of a fixed body. In our study we focus to the intermediate interval of aspect ratio between 8 and 10. We show that, for χ 4 8, the primary Hopf bifurcation is replaced by a new one with much lower frequency and leading to weakly oscillating periodic oscillations, later (for χ 4 9) the Hopf bifurcation is replaced by a regular one disappearing again for very thin cylinders (χ 4 10).