Free vibration of a square cylinder at low Reynolds numbers

Free vibrations of a square cylinder at zero incidence are studied. A stabilized space-time finite-element method has been used to discretize the equations of fluid flow in two-dimensions. The cylinder mounted on elastic supports, with low mass ratio (m⁎=10), is allowed to undergo in-line as well as transverse vibrations. The coefficient of structural damping is assigned a zero value. The Reynolds number, Re, is based on the edge length of the cylinder and free-stream speed. The reduced natural frequency is defined as FN=14.39/Re. For a blockage of 0.05, results are presented for 60 ≤ Re ≤ 250 . As the Reynolds number is progressively increased, the regimes of flow observed are ‘primary’ lock-in, desynchronization and ‘secondary’ lock-in. It is found that the width of the ‘primary’ lock-in zone for a square cylinder with sharp corners is very narrow. The variation of transverse response with Re is associated with the initial, lower and ‘secondary’ lower branches. Transition between the initial and lower branches is hysteretic. Hysteresis in the cylinder response as well as aerodynamic coefficients is observed close to the lower-Re extremity of the ‘primary’ lock-in zone. A hysteretic jump at Re=87 marks the onset of lock-in regime. The 2S mode of vortex shedding is observed along the initial branch. Along the lower branch, the modes of vortex shedding are 2S and C(2S). Vortex shedding along the ‘secondary’ lower branch is via the 2S and 2P modes. The free vibration characteristics of a square differ, in several fundamental aspects, from those of a circular cylinder reported by Prasanth and Mittal (2008a,b). Compared to the circle, a much narrower lock-in zone along with smaller (about 50%) transverse amplitude is observed for the square for Re ≤ 150 . The lock-in extremities of a circular cylinder for Re ≤ 150 and B=0.05 are associated with two hysteresis loops as opposed to one for a square cylinder.