Ocean Engineering 209 (2020) 107300 Available online 6 June 2020 0029-8018/© 2020 Elsevier Ltd. All rights reserved. Numerical investigation of streamwise vibration of an elastically mounted circular cylinder in oscillatory fow Erfan Taheri a , Ming Zhao a, * , Helen Wu a , Feifei Tong b a School of Engineering, Western Sydney University, Penrith, 2751, NSW, Australia b Department of Civil, Environmental and Mining Engineering, The University of Western Australia, Crawley, WA, 6009, Australia A R T I C L E INFO Keywords: Oscillatory fow Vortex induced vibration Low Reynolds number ABSTRACT Vibration of an elastically mounted circular cylinder subjected to an oscillatory fow is investigated by two- dimensional numerical simulations. A low Reynolds number of Re ¼ 150 and two Keulegan–Carpenter numbers of 5 and 10 are chosen in this study. A wide range of frequency ratios, which is defned as the ratio of the oscillatory fow frequency to the natural frequency, are studied. The cylinder, with mass ratios of 1, 2 and 3, is free to vibrate at zero damping ratio along the streamwise direction only. The vibration velocity of the cylinder relative to the fuid motion (referred to as relative velocity) is found to vary signifcantly with the frequency ratio. The amplitude of the relative velocity is greater than the amplitude of the oscillatory fow velocity as the frequency ratio is less than a critical value, which is slightly smaller than 1. The amplitude of the relative velocity signifcantly reduces as the frequency ratio is greater than this critical value. The change of the phase difference between the cylinder motion and fuid motion is identifed as the cause for the increase or decrease of the relative velocity. The phase changes from 90 � to 90 � as the frequency ratio exceeds 1. The drag force is found to be zero at a frequency ratio of 1. 1. Introduction As fuid fow comes in contact with an object, it may produce fow induced vibration. A topic of interest in fuid-structure interaction is the study of vortex shedding and its impact on a cylinder. The vortex shedding in the wake of a circular cylinder in a steady fow occurs when the Reynolds number (Re) exceeds approximately 47 (Williamson, 1996). The Re is defned as Re¼UD/ν, where U is the fuid velocity, D is the diameter of the cylinder and ν is the kinematic viscosity of the fuid. Barkley and Henderson (1996) found the wake fow be- comes unstable and three-dimensional as Re exceeds 188:5� 1:0, and becomes fully turbulent when Re > 300. It was demonstrated that for an elastically-mounted cylinder subject to vortex shedding, the amplitude of cylinder vibration is dependent on the cylinder’s mass, natural frequency and damping ratio (Raghavan, 2011). The synchronization between vortex shedding and cylinder vi- bration frequency is found to occur within a range of reduced velocities, which is also referred to as lock-in regime (Govardhan and Williamson, 2000). The reduced velocity is defned as U r ¼ U/(f n D), where U is the incoming fuid velocity and f n is the natural frequency of the cylinder. Modir et al. (2016) and Cen et al. (2016) conducted experiments of vortex-induced vibration (VIV) of a fexibly mounted cylinder at different mass ratios and found the lowest mass ratio produced the largest vibration amplitude. Mass ratio is the ratio of the cylinder mass to the displaced fuid mass. In many numerical studies of VIV of a cir- cular cylinder, the damping is zero in order to achieve the maximum response amplitude (Silva et al., 2016; Chen et al., 2019). In offshore engineering, wave-induced water motion is commonly modelled by oscillatory fow. The interaction between circular cylinders and oscillatory fow is relevant to engineering applications, such as offshore pipelines, riser pipes, mooring cables, etc. In addition to Re, the effects of oscillatory fow on vortex shedding is also dependent on the Keulegan-Carpenter (KC) number. The KC is defned as KC¼U m T/D, where U m and T are the amplitude and period of the oscillatory fow velocity, respectively. The relationship between the KC number and the amplitude of the fuid motion (a) is KC ¼ 2πa/D. The Stokes number (β) is used to describe the ratio between Re and KC, i.e. β ¼ Re/KC ¼ D 2 / (νT). Increasing KC may also induce the fow transition from laminar to turbulent (Sarpkaya, 1986). The generation of vortices in oscillatory fow occurs every half period of oscillatory movement, provided that KC is greater than 1.1 (Sarpkaya, 1986). As the KC is increased (β ¼ 50–800 * Corresponding author. E-mail address: m.zhao@westernsydney.edu.au (M. Zhao). Contents lists available at ScienceDirect Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng https://doi.org/10.1016/j.oceaneng.2020.107300 Received 12 November 2019; Received in revised form 24 March 2020; Accepted 25 March 2020