# Coupling Motion and Energy Harvesting of Two Side-by-Side Flexible Plates in a 3D Uniform Flow
## 1 Introduction  
Interactions between flexible structures and viscous flow are ubiquitous and familiar in nature and engineering applications. The physical mechanisms underlying these interactions are of interest because they provide us novel insights into the fluid dynamics in biological growth or locomotion. In natural creatures, the flexibility of the structure plays an important role to improve its body stability or propulsion performance in surrounding flow. For instance, through its passive shape reconfiguration in the flow, a tree leaf can achieve substantial and beneficial drag reduction [1; 2]. Similarly, a fish can exploit energy from the surrounding vortex street and achieve cost savings by undulating its body [3; 4; 5]. The famous experiment conducted by [PERSON] _et al._ revealed that a trout can decrease muscle activity by synchronizing its kinematics to incoming vortices appropriately [6]. Further, a dead fish was observed to propel upstream when its flexible body resonated with the wake of a bluff cylinder [7]; this entirely passive locomotion without energy input was also found in a simple articulated fish-like system [8]. Regarding the applications, the underlying hydrodynamic principles also inspire researchers to develop advanced equipment involving the interactions of unsteady flows and deformable bodies. Recently, the robotic models of fish swimming have been of great interest in ocean engineering, and a wide diversity of approaches have been taken for the mechanical design of fish-inspired systems [9; 10; 11],such as a carangiform fish robot [12; 13] and a batoid-inspired robot [14; 15; 16]. Besides the active flow control mechanisms used in bionic propulsion, passive flapping or vibrating dynamics have also been applied in developing renewable energy harvesters, which are usually based on compliant materials, such as elastic-mounted cylinders and piezoelectric membranes [17; 18; 19]. Although these interaction phenomena are common, they still puzzle researchers due to their complicated dynamics. To obtain physical insights into the underlying mechanisms and dynamics of the flexible structure-fluid interactions, flexible filaments or plates are usually adopted due to their simplicities and usefulness [20; 21]. The deformation of a flexible filament or plate in a motion flow is physically determined by the competition between the fluid load on it and internal elasticity. During the past few years, these challenging problems have attracted many researches. [PERSON] first conducted experiments on a flag in a wind tunnel and observed the oscillation modes of the waving motions [20]. The existence of bistability for an isolated filament, including stretched-straight mode and self-sustained flapping mode, was observed in [PERSON] and [PERSON]'s research [22], where they carried out a series of computer simulations by an immersed boundary method. [PERSON] and [PERSON]'s research further revealed that the sustained flapping only occurred when the filament mass was included, and the bistability depended on the filament's length and initial condition [22]. Moreover, three distinct regimes, including the straight, regular flapping and chaotic flapping regimes, have been observed by changing the mass ratio, bending rigidity or incoming velocity [23; 24; 25]. Additionally, flow past multiple filaments or plates arranged side-by-side is also an important topic for understanding the mechanism of fish schooling or birds flocking. Both in-phase, out-of-phase and erratic flapping interaction modes have been found by experiments and numerical simulations [26; 27; 28]. More recently, it was found that schooling fish can improve locomotion efficiency in side-by-side configuration [29; 30]. Although most of the flexible bodies physically exist with a finite aspect ratio, only a few 3D simulations of an isolated flexible plate have been reported [31; 32; 33; 34].