Falling dynamics of flexible plates with concentrated weight

We investigate the free fall of a flexible plate with a concentrated central weight using numerical simulations to explore the coupling effects of structural flexibility and concentrated loading. Two-dimensional and three-dimensional (3-D) simulations, combining lattice Boltzmann and immersed boundary methods, identify four distinct falling modes: swing, stable falling, fluttering and tumbling. The swing and stable falling modes are reported for the first time in this context. By analysing the balance between bending stiffness

and concentrated weight

upper G G $G$

, we introduce a new dimensionless effective stiffness

italic Kg equals upper K divided by upper G Kg = K / G $ \textit{Kg} = K/G$

, which governs mode transitions and successfully collapses the kinematic data. A bistable transition zone between fluttering and tumbling is observed, with the mode selection dependent on the initial inclination angle, which scales as a power law of

italic Kg Kg $ \textit{Kg}$

. Additionally, within the swing mode, we observe a transition in wake topology from a 2P to a 2S vortex-shedding pattern, modulated by the mass ratio. In 3-D simulations, the finite aspect ratio

upper A Subscript r A r $A_r$

significantly influences mode selection due to pressure leakage at the side edges. While the

italic Kg Kg $ \textit{Kg}$

scaling remains valid for the global dynamics, 3-D effects suppress tumbling in narrow plates and induce spanwise oscillations in the stable falling mode at higher aspect ratios.