A freely rotating square cylinder at various rotation centers in a low Reynolds number range

Flow-induced rotation of a rigid square cylinder is a fundamental and scientific problem, commonly observed in nature and engineering, such as the slowly dropping particles in water. Previous studies have reported abundant interesting phenomena and conclusions for a square cylinder rotating around its centroid. However, the rotation center may vary due to the uneven mass distribution, and this situation has not been investigated yet. Therefore, this study presents a numerical analysis of a freely rotating square cylinder through two-dimensional simulations at Reynolds numbers ranging from 10 to 150, considering various rotation centers. For cylinders with axially or rotationally symmetrical rotation centers, the rotation response exhibits a highly similar pattern, regardless of the occurrence of deflection (a phenomenon that the square cylinder rotates from its initial position to a new equilibrium one, presenting a non-zero time-averaged rotation angle), indicating its independence from symmetrical rotation centers. A regime map categorizes the rotation responses into five modes (I–V) based on the Reynolds number and rotation center: (I) no oscillation and no deflection, (II) periodic oscillation without deflection, (III) random oscillation, (IV) no oscillation after deflection, and (V) periodic oscillation after deflection. Deflection occurs when the rotation center deviates from the x axis. The cylinder remains stationary at low Reynolds numbers while transitions to periodic or random oscillations at higher values, where the critical Reynolds number varies with various rotation centers. For each mode, the moment generation mechanism is elucidated by examining instantaneous flow fields and surface pressure and shear stress distributions within one rotation response cycle. The time-averaged rotation angles are independent of the Reynolds number for non-deflecting and π/4-deflecting cases, while they decrease from 0.4 to 0.16 with increasing Reynolds number when the rotation center is off the x axis and the diagonal. Shifting the rotation center forward enhances the rotation response, lowers the critical Reynolds number below 30, and alters the rotation frequency trend from linear decay to logarithmic growth.