Free vibrations of an elliptic cylinder at low Reynolds numbers

Vortex-induced vibrations of an elliptic cylinder are investigated in the laminar flow regime, 60≤Re≤140. The cylinder, in its mean position, is oriented such that either the major axis or the minor axis is aligned parallel to the free-stream flow. The aspect ratio of the cylinder, based on the ratio between the diameter transverse to the free-stream direction and its conjugate, is varied between 0.7≤AR≤1.43. A stabilized finite-element method is used to solve the incompressible flow equations and the cylinder motion in two dimensions. The non-dimensional mass of the cylinder is m⁎=10.0. The blockage, based on the transverse diameter of the cylinder and lateral width of the domain, is B = 5%. On the basis of vortex shedding patterns and time histories of cylinder oscillation and aerodynamic coefficients, different flow-regimes/branches of the cylinder response are identified. The classically known initial and lower branches are further split into quasi-periodic and periodic parts. Hysteresis, in the response of the cylinder, is studied near the low- (primary hysteresis) and the high- (secondary hysteresis) Re end of lock-in. The width of primary hysteresis, in general, decreases with increase in AR, and disappears completely for AR = 1.43. The width of secondary hysteresis first increases with increase in AR for 0.7≤AR≤1.14, and then decreases with further increase in AR. The phase difference between the lift force and transverse displacement shows a jump of approximately 180° at the Re where the ratio of the frequency of the time variation of transverse response to the natural frequency attains a value of unity. The mode of vortex shedding associated with all the flow regimes/branches is either 2S or its variant. In the periodic part of the initial branch, the vortex that is about to be shed from the cylinder coalesces with vortices of the same sign in the near wake. This results in stronger vortices and, therefore, relatively larger fluctuations in the lift force. This mode is named as CNW(2S). The peak amplitude of cylinder oscillation increases with increase in aspect ratio. An elliptic cylinder whose minor axis is aligned parallel to the free-stream flow experiences larger amplitude of oscillation compared to the case when the major-axis is aligned parallel to the flow.