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.