Three-dimensional flutter dynamics of a rectangular plate: Exploring critical conditions and post-critical regimes

Self-sustained, flow-induced oscillations of a flexible plate have been identified as a viable mechanism for energy harvesting. However, the efficiency of such devices strongly depends on the onset of aeroelastic instabilities. In this work, we investigate numerically the flutter of a rectangular, three-dimensional, inextensible plate invested by a viscous laminar flow. Through a systematic approach, an unprecedentedly broad range of the parametric space spanned by the mass ratio upper M Subscript s M s Mₛ and bending rigidity upper K Subscript upper B K B KB is explored. Specifically, three distinct regimes of response can be identified: (i) fixed-point stability, (ii) limit-cycle oscillations and (iii) irregular flutter. We present a high-resolution map of the flapping states in the upper M Subscript s M s Mₛ – upper K Subscript upper B K B KB plane, where the flapping mode of each case is also determined. The instability onset is rationalised through linear stability analysis, accounting for the bending rigidity and the finite aspect ratio of the plate. Across the parameter space investigated, the flapping frequency exhibits only a weak dependence on flapping amplitude and bending rigidity, except within a lock-in regime near the instability threshold at high Reynolds numbers, where it scales with the plate’s natural frequency. In contrast, the flapping frequency is primarily governed by the mass ratio and the flutter mode, with an additional modulation by the Reynolds number. This Reynolds-number dependence is most pronounced at low mass ratios, where different flutter modes emerge as the Reynolds number varies, and progressively weakens with increasing mass ratio. While low-to-moderate mass ratios exhibit predominantly two-dimensional flutter, spanwise modes arise for mass ratios above approximately 2, with amplitudes that increase with both Reynolds number and mass ratio. The optimal flow configurations for energy harvesting are identified in all flow regimes and rationalised by monitoring the effect of upper M Subscript s M s Mₛ and upper K Subscript upper B K B KB on the corresponding flapping dynamics.