Effect of mass ratio on free vibrations of a square cylinder at low Reynolds numbers

A stabilized space–time finite-element method is used to study the effect of oscillator mass ratio, m⁎ on in-line and transverse free vibrations of a rigid square cylinder at zero incidence in two-dimensions. The mass ratios considered are 1, 5, 10 and 20. The reduced natural frequency is FN=14.39/Re where Re, the Reynolds number, is based on the edge length of the square cylinder and free-stream speed. The structural damping coefficient is assigned a zero value. Results are presented for 50≤Re≤250. The cylinder may undergo vortex-induced vibrations (VIV) and/or galloping. It is found that the occurrence of galloping is a function of mass ratio. Galloping is not observed for the low mass ratio considered (m⁎=1), but strong galloping effects are realized for m⁎≥5. The absence of galloping for m⁎=1 marks significant difference in frequency, response and force characteristics as compared to the cases of higher mass ratios. The response behaviour of m⁎=1 cylinder is characterized by the initial and lower branches. For m⁎≥5 an additional galloping branch (Sen and Mittal, 2011. Journal of Fluids and Structures 27, 875–884) is observed. The onset of galloping is marked with the occurrence of mismatch of frequency of vortex-shedding and body oscillation. The Reynolds number or reduced speed marking the onset of lock-in increases with increasing m⁎. In contrast, the Re or reduced speed for onset of galloping decreases with increase in m⁎ and varies as m⁎−1.3. The vortex-shedding is characterized by the 2S and C(2S) modes in the VIV regime. It is 2S during galloping for low oscillation amplitude and changes to 2P+2S when the transverse displacement surpasses a threshold value (0.7D, approximately where D is the edge length of square). Weak and strong hystereses at the onset of lock-in and galloping, respectively, are displayed by the m⁎=5 cylinder. No hysteresis is observed for m⁎=1. As m⁎ increases, the primary hysteresis becomes stronger and secondary hysteresis disappears. The Re, at which the phase jump of ≈180° between lift and transverse response occurs, is virtually independent of m⁎. Unlike a freely vibrating circular cylinder where the maximum transverse response increases with decreasing m⁎, the variation of oscillation amplitude with m⁎ for square cylinder is non-monotonic in both the lock-in and galloping zones.