Wake transition of an unconstrained self-propelled flexible flapping plate

This paper numerically investigates the wake transition of an unconstrained self-propelled flexible flapping plate, which exhibits the ability to move freely both longitudinally and laterally, at a low Reynolds number of 200. By examining crucial parameters, including pitching amplitude ${\theta }_0$, bending stiffness $K$, and mass ratio of the plate to the fluid $M$, the research identifies three distinct wake patterns: symmetric, deflected, and chaotic. The lateral drift speed of the plate ($V$) is used as a quantitative indicator to differentiate between symmetric and asymmetric wake patterns, where $V$ is approximately zero and nonzero, respectively. The paper indicates that the transition from symmetric to asymmetric wakes occurs when the cruising Reynolds number (${\mathrm{Re}}_c$) reaches the critical value (${\mathrm{Re}}_c^{\mathrm{cr}}$), which follows a simple scaling law vs $M$, i.e., ${\mathrm{Re}}_c^{\mathrm{cr}}\sim M^{-1/2}$. The critical dimensionless translational kinetic energy ${\hat{E}}_k$ of the plate remains constant for various $M$ when the wake transition occurs. Analyzing translational kinetic energy not only proves the ${\mathrm{Re}}_c^{\mathrm{cr}}$ scaling, but also offers a unique energy perspective on the phenomenon. In addition, the critical flapping Reynolds number (${\mathrm{Re}}_f^{\mathrm{cr}}$) is found to also satisfy a simple scaling similar to ${\mathrm{Re}}_c^{\mathrm{cr}}$, i.e., ${\mathrm{Re}}_f^{\mathrm{cr}}\sim M^{-1/3}$. Finally, it is revealed that passive lateral oscillation and bending deformation of the plate are two key mechanisms affecting wake symmetry properties. This paper provides insights into flapping-based locomotion.