## 1 Introduction  
Flow-induced vibration (FIV) is a common source of vibration in nature. Since its discovery by [PERSON], this phenomenon--which is widely observed in flow environments--has attracted significant attention from researchers. FIV occurs when periodic vortices are generated as fluid flows past bluff bodies, creating alternating low-pressure regions behind them and resulting in vibrations. Numerous studies have investigated this phenomenon [1, 2, 3]. The main forms of FIV include vortex-induced vibration (VIV), galloping, and flutter. Flutter, as a self-excited vibration characterized by large amplitude oscillations [4, 5], occurs when the flow velocity surpasses a critical threshold and has been extensively studied due to its significant impact on the stability of aircraft wings and blades [6, 7]. In recent decades, driven by the pursuit of clean energy, researchers have investigated utilizing specialized devices to convert the mechanical energy from flow-induced vibrations into electrical energy, thereby exploring novel energy harvesting methods [8, 9, 10]. The common methods for converting vibration energy into electrical energy include piezoelectric energy harvesters [11; 12], electromagnetic energy harvesters [13; 14], electrostatic energy harvesters [15; 16], and triboelectric energy harvesters [17; 18]. Among these, piezoelectric harvesters have emerged as a prominent research focus due to their simplicity and high-power output [19; 20]. Given that flutter possesses substantial energy potential, it has been investigated for possible applications in energy harvesting [21; 22]. To understand the mechanisms of flutter-based piezoelectric energy harvester systems, researchers have employed theoretical analyses, experiments, and numerical simulations to explore the impact of various physical parameters on harvester performance. [PERSON] [23] utilized the lattice method and beam theory to establish a fluid-solid coupling model, examining the influence of the airfoil-based's aspect ratio on harvester performance and validating the findings through experiments. Their results indicated that a wider airfoil-based harvester accelerates the onset of limit cycle oscillations (LCOs). [PERSON] [24] compared the effects of steady aerodynamics, unsteady formulations, and unsteady-state aerodynamics on computational accuracy, demonstrating that unsteady-state aerodynamics yield the highest accuracy. [PERSON] [25] compared the accuracy of quasi-steady representations, unsteady formulations, and dynamic stall models, concluding that unsteady formulations offer superior accuracy. Furthermore, [PERSON] studied the impact of elastic coefficients on the performance of airfoil-based flutter energy harvester systems, revealing that lower pitch elastic coefficients and moderate plunge elastic coefficients enhance system performance. To further enhance the performance of airfoil-based piezoelectric energy harvester systems, researchers have explored innovative approaches by incorporating strategic accessories to optimize system performance. [PERSON] et al. [26] developed a novel configuration by integrating a cantilever beam and a cylindrical bluff body into an airfoil-based flutter energy harvester system. The researchers strategically placed a piezoelectric ceramic transducer (PZT) on the cantilever beam to expand the system's operational capabilities. Their proposed design demonstrates remarkable versatility: at velocities below the critical threshold, the cylindrical bluff body effectively induces VIV; while at velocities exceeding the critical point, it triggers limit cycle oscillations (LCOs) in the flutter energy harvester system. This dual functionality significantly improves output power generation during both VIV and LCO conditions. Following this approach, [PERSON] et al. [27] developed an innovative piezoelectric energy harvester system featuring a centrally positioned cylindrical bluff body and symmetrically arranged airfoil-based structures.