Hysteresis in vortex-induced vibrations: critical blockage and effect of m^∗

Abstract

The hysteretic behaviour of a freely vibrating cylinder, near the low-Reynolds-number end of synchronization/lock-in, in the laminar regime is investigated. Computations are carried out using a stabilized finite-element method. The flow remains two-dimensional in this Reynolds number regime. This is verified via comparison of two- and three-dimensional computations. The cylinder is free to undergo crossflow as well as in-line vibrations. The combined effect of mass ratio (1 ≤ m^∗ ≤ 100) and blockage (0.25% ≤ B ≤ 12.5%) is studied in detail. The existence of a critical mass ratio (m^∗_cr = 10.11), below which hysteresis disappears for an unbounded flow situation, is identified. For higher mass ratio the hysteretic behaviour is observed for all blockage. However, the hysteresis loop width is found to vary with B; its variation with m^∗ and B is studied. The concept of critical blockage B_cr is introduced. For B ≤ B_cr the response of the cylinder is virtually the same as that in an unbounded flow domain. The variation of B_cr with m^∗ is investigated. Furthermore, B_cr is found to vary non-monotonically with m^∗ for m^∗ ≤ m^∗_cr and is almost constant for m^∗ ≥ 20. The effect of damping, as well as restricting the cylinder to undergo transverse vibrations only, on the hysteresis behaviour is studied. The transverse-only motion leads to a larger hysteresis loop width compared with the transverse and the in-line motion of the cylinder. An attempt is made to explain this by comparing the results from forced vibrations.