Modes of synchronisation in the wake of a streamwise oscillatory cylinder

A numerical analysis of flow around a circular cylinder oscillating in-line with a steady flow is carried out over a range of driving frequencies at relatively low amplitudes and a constant Reynolds number of 175 (based on the free-stream velocity). The vortex shedding is investigated, especially when the shedding frequency synchronises with the driving frequency. A series of modes of synchronisation are presented, which are referred to as the modes, where and are natural numbers. When a mode occurs, is detuned to , representing the shedding of pairs of vortices over cycles of cylinder oscillation. The modes are further characterised by the periodicity of the transverse force over every cycles of oscillation and a spatial-temporal symmetry possessed by the global wake. The synchronisation modes with relatively small natural numbers are less sensitive to the change of external control parameters than those with large natural numbers, while the latter is featured with a narrow space of occurrence. Although the mode of synchronisation can be almost any rational ratio (as shown for and smaller than 10), the probability of occurrence of synchronisation modes with being an even number is much higher than being an odd number, which is believed to be influenced by the natural even distribution of vortices in the wake of a stationary cylinder.