Symmetry-breaking bifurcations and sub-harmonic lock-in of a flexible splitter plate in cylinder wake flow

Abstract

This paper investigates the flow past a flexible splitter plate attached to the rear of a fixed circular cylinder at a low Reynolds number of 150. A systematic exploration of the plate length ($L/D$), flexibility coefficient ($S^{*}$), and mass ratio ($m^{*}$) reveals new laws and phenomena. The large-amplitude vibration of the structure is attributed to a resonance phenomenon induced by fluid-structure interaction. The modal decomposition indicates that resonance arises from the coupling between the first and second structural modes, where the excitation of the second structural mode plays a critical role. Due to the combined effects of added mass and periodic stiffness variations, the two modes become synchronized, oscillating at the same frequency while maintaining a fixed phase difference of $π/2$. This further results in the resonant frequency being locked at half of the second natural frequency, which is approximately three times the first natural frequency. A reduction in plate length and an increase in mass ratio are both associated with a narrower resonant locking range, while a higher mass ratio also shifts this range toward lower frequencies. A symmetry-breaking bifurcation is observed for cases with $L/D\leq3.5$, whereas for $L/D=4.0$, the flow remains in a steady state with a stationary splitter plate prior to the onset of resonance. For cases with a short flexible plate and a high mass ratio, the shortened resonance interval causes the plate to return to the symmetry-breaking stage after resonance, gradually approaching an equilibrium position determined by the flow field characteristics at high flexibility coefficients.

Figures (20)

• Figure 1: Schematic view of the configuration: the flexible plate attached to circular cylinder immersed in the laminar flow.
• Figure 2: Variations of the amplitude of vertical displacement at the plate-tip against the normalized bending stiffness $S^{*}$ for the case with $L/D=3.5$.
• Figure 3: Oscillation characteristics of the flexible plate: variation of non-dimensional ($a$) the time-averaged position $Y_{tip}$ and ($b$) amplitude $A_{tip}$ of vertical displacement at the plate tip, with increasing flexibility coefficient $S^{*}$ for the cases with different plate length at $m^{*}=10$.
• Figure 4: Partition of the different vibration modes in the $L/D-S^{*}$ diagram. The region enclosed between the two red lines represents the parameter range in which large-amplitude responses occur. The lower-right corner of the parameter map shows the reappearance of the symmetry-breaking phenomenon at high flexibility parameter values.
• Figure 5: Time histories of the vertical displacement at the plate tip at different vibration modes: ($a$) symmetry-I stage for $S^{*}=5$ and $L/D=3$, ($b$) bifurcation-I stage for $S^{*}=7$ and $L/D=3$, ($c$) symmetry-II stage for $S^{*}=18$ and $L/D=3$, ($d$) symmetry-III stage for $S^{*}=24$ and $L/D=4$.