Three-dimensional transition of the flow past an elastically mounted circular cylinder

Abstract

Floquet stability analysis and direct simulations of a circular cylinder undergoing vortex-induced vibration (VIV) are presented. Simulation predictions are examined for the reduced velocity range over which there is a strong and periodic resonant response: $U_r \in [4.0, 8.0]$ , focusing on a mass ratio of $m^* = 2.546$ matching a number of previous investigations. Over most of this range, the dominant wake modes present are analogous to modes A, B and QP (quasi-periodic) observed in a stationary circular cylinder wake. However, at $U_r = 4.5$ , the dominant modes are B, QP and a subharmonic mode (SH), whereas at $U_r = 4.0$ , the two-dimensional base state switches to a $P+S$ wake. The critical Reynolds number for two- to three-dimensional transition is observed to decrease with an increase of $U_r$ , in line with a decreasing response amplitude. Over this range, the minimum ${\textit {Re}}$ for which the wake remains two-dimensional is 202, which occurs at $U_r = 7.5$ , but this increases to ${\textit {Re}}_{cr} = 300$ at $U_r = 4.5$ , noting the critical Reynolds number for a stationary circular cylinder is ${\textit {Re}}_{cr}=189$ . The corresponding critical spanwise wavelengths for $U_r = 4.5$ and 8 are $1.4D$ (mode B) and $4.0D$ (mode A), respectively. Simulations indicate that even at $Re=300$ , flow three-dimensionality increases the amplitude of the lower branch considerably. The investigation establishes the role of oscillation amplitude and reduced velocity in three-dimensional transition for elastically mounted systems.