Dynamics of wall-mounted tandem flexible plates with unequal lengths in a laminar boundary layer

This study presents a numerical investigation of wall-mounted tandem flexible plates with unequal lengths in a laminar boundary layer flow, examining both two-dimensional (2-D) and three-dimensional (3-D) configurations. Key parameters influencing the system include the plate’s bending stiffness ( $K$ ), Reynolds number ( ${Re}$ ) and length ratio ( $L^*$ ). Five motion modes are identified: dual collapse (DC), flapping collapse (FC), dual flapping (DF), static flapping (SF) and dual static (DS). A phase diagram in the ( $K,L^*$ ) space is constructed to illustrate their regimes. We focus on DF and SF modes, which significantly amplify oscillations in the downstream plate – critical for energy harvesting. These amplification mechanisms are classified into externally driven and self-induced modes, with the self-induced mechanism, which maximises the downstream plate’s amplitude, being the main focus of our study. A rigid–flexible (RF) configuration is introduced by setting the upstream plate as rigid, showing enhanced performance at high ${Re}$ , with oscillation amplitudes up to 100 % larger than the isolated flexible (IF) plate configuration. A relation is developed to explain these results, relating oscillation amplitude to trailing-edge velocity, oscillation frequency and chord length. Force analysis reveals that the RF configuration outperforms both IF and flexible–flexible (FF) configurations. Unlike frequency lock-in, the RF configuration exhibits frequency unlocking, following a $-2/3$ scaling law between the Strouhal number ( $St$ ) and ${Re}$ . Results from the 3-D RF configuration confirm that the 2-D model remains applicable, with the self-induced amplification mechanism persisting in 3-D scenarios. These findings enhance understanding of fluid–structure interactions, and offer valuable insights for designing efficient energy harvesting systems.