Modes of synchronisation in the wake of a streamwise oscillatory cylinder

Abstract

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 $(f_{d})$ at relatively low amplitudes $(A)$ and a constant Reynolds number of 175 (based on the free-stream velocity). The vortex shedding is investigated, especially when the shedding frequency $(f_{s})$ synchronises with the driving frequency. A series of modes of synchronisation are presented, which are referred to as the $p/q$ modes, where $p$ and $q$ are natural numbers. When a $p/q$ mode occurs, $f_{s}$ is detuned to $(p/q)f_{d}$, representing the shedding of $p$ pairs of vortices over $q$ cycles of cylinder oscillation. The $p/q$ modes are further characterised by the periodicity of the transverse force over every $q$ cycles of oscillation and a spatial–temporal symmetry possessed by the global wake. The synchronisation modes $(p/q)$ 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 $p$ and $q$ smaller than 10), the probability of occurrence of synchronisation modes with $q$ being an even number is much higher than $q$ being an odd number, which is believed to be influenced by the natural even distribution of vortices in the wake of a stationary cylinder.