In this work, we present a computational study of the flow structure and forces on a cylinder vibrating both transversely and in-line to a uniform stream. The in-line frequency is equal to twice the transverse frequency, while the ratio of the in-line to the transverse oscillation amplitude varies from zero (transverse vibration only) to infinity (in-line oscillation only). For all intermediate values, the cylinder thus follows an “eight”-like of trajectory, emulating the motion of real cylindrical structures undergoing vortex-induced vibrations. For a flow from left to right, we distinguish between a “counter-clockwise” mode (if the upper part of the trajectory is traversed counter-clockwise) and a “clockwise” mode (if the upper part of the trajectory is traversed clockwise). Here, we use a spectral element method, and perform simulations for a Reynolds number of 400. We focus on a value of the transverse oscillation frequency equal to half the natural frequency of the Ka´rma´n vortex street (sub-harmonic excitation). Results are compared against cases corresponding to resonant forcing, previously studied by the research team. In all cases, the flow properties are greatly influenced by the direction in which the cylinder is traversed. In particular, the “counter-clockwise” mode is characterized by higher values of the forces acting on the cylinder, as well as by higher values of the power transfer from the flow to the cylinder. The case of sub-harmonic excitation is unique, in that the power transfer remains negative for all values of the non-dimensional excitation amplitude, i.e. corresponds to damping. Flow visualization reveals a variety of vortex patterns in the wake, in particular regular patterns at sub-harmonic excitation, and complex vortex streets at high amplitude resonant forcing.

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