Flow past a freely vibrating elliptic cylinder at 45∘ incidence

Abstract

Undamped transverse-only flow-induced vibrations (FIV) of an elliptic cylinder of mass ratio, $m^{\ast }=10$ at 45° incidence are investigated via two-dimensional computations at Reynolds numbers, $Re=100$ and 200. Using quasi-steady theory, it is illustrated that the asymmetric oscillator does not gallop at $Re=100$ and 200. Resolution of hysteresis-free solutions at $Re=100$ is a novel finding. As compared to $Re=100$, response at $Re=200$ is associated with additional branches: a lower branch, a terminal branch and a third regime of desynchronization. Assuming harmonic lift and response, mathematical expressions are obtained for modified dimensionless circular frequency, ${{{\omega }_N}^{\ast }}^2$ and modified damping. The variation of ${{{\omega }_N}^{\ast }}^2$ with reduced speed, $U^{\ast }$ reveals excellent collapse with predicted dynamic response. For FIV at $Re=200$ and not at $Re=100$, a second regime of significant vibrations develops in the neighbourhood of $U^{\ast }=8$ in addition to the first one around $U^{\ast }=4$. The period doubling bifurcation occurring around $U^{\ast }=8$ is an 1:2 sub-harmonic synchronization; it halves the oscillation frequency that in turn closely approaches reduced natural frequency of the cylinder. In this regime, the wake mode is found to be 2(2S). Leontini et al. (2018) resolved periodic doubling bifurcation for FIV of an inclined elliptic cylinder using a low $m^{\ast }$ of unity. The occurrences of second lock-in and period doubling therefore appear not to be a function of $m^{\ast }$; they are rather Reynolds number phenomena.