The propulsion of a pitching flexible plate in a uniform flow is investigated numerically. The effects of bending stiffness ( $K$ ), pitching amplitude ( $A_L$ ) and frequency ( $St$ ) on the wake patterns, thrust generations and propulsive performances of the fluid–plate system are analysed. Four typical wake patterns, i.e. von Kármán, reversed von Kármán, deflected and chaotic wakes, emerge from various kinematics, and the $St-A_L$ wake maps are given for various $K$ . The drag-to-thrust transitions (DTT) and the wake transitions (WT) between the von Kármán and reversed von Kármán wakes are examined. Results indicate that the WT and DTT boundaries can be scaled by the chord-averaged distance of travel, $\mathcal {L}$ , which leads to $\mathcal {L}\times St \approx 1$ and $\mathcal {L}\times St \approx 1.2$ , respectively. Further, the resonance mechanism for the performance enhancement is revealed and confirmed in a wide range of parameters. The dimensionless average speed of plate, $\mathcal {U^*}\left (=\mathcal {L}\times St\right )$ , is adopted merely to characterize the propulsive performances. For the first time, the $\mathcal {U^*}$ -based scaling laws for the thrust and power are revealed in pitching rigid and flexible plates for various $A_L$ and $St$ . This study may deepen our understanding of biological swimming and flying, and provide a guide for bionic design.

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Dynamics of two-dimensional flow past a rigid flat plate with a trailing closed flexible filament acting as a deformable afterbody are investigated numerically by an immersed boundary-lattice Boltzmann method for the fluid flow and a finite element method for the filament motion. The effects of Reynolds number ( $Re$ ) and length ratio ( $Lr$ ) on the flow patterns and dynamics of the rigid-flexible coupling system are studied. Based on our numerical results, five typical state modes have been identified in $Lr\unicode{x2013}Re$ plane in terms of the filament shape and corresponding dynamics, i.e. static deformation, micro-vibration, multi-frequency flapping, periodic flapping and chaotic flapping modes, respectively. Benefiting from the passive flow control by using the flexible filament as a deformable afterbody, the coupled system may enjoy a significant drag reduction (up to $22\,\%$ ) compared with bare plate scenarios ( $Lr=1$ ). Maximum drag reduction achieved at $L_{c,{min}} \in [1.8, 2]$ is often accompanied by the onset of the system state transition. The flow characteristic and its relation to the change in hydrodynamic drag are further explored in order to reveal the underlying mechanisms of the counterintuitive dynamical behaviour of the coupled system. The scaling laws for the form drag and the friction drag, which arise from the pressure and viscous effects, respectively, are proposed to estimate the overall drag acting on the system. The results obtained in the present study may shed some light on understanding the dynamical behaviour of rigid-flexible coupling systems.

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