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In this study, the Lagrangian coherent structures (LCSs) of the flow around two-dimensional airfoil with a plunging motion are numerically investigated, in order to reveal the physics of the unsteady aerodynamics in the flapping wings. The present study, in which only low Reynolds ( Re ) flows are considered, focuses on two typical unsteady flows at zero angle of attack, respectively induced by slow and fast plunging motions. To simulate such unsteady flows, the Characteristic Based Split (CBS) scheme combined with Arbitrary Lagrangian–Eulerian (ALE) framework are applied. After that, the LCSs are introduced to study the dynamic properties in the unsteady flow and by using the finite-time Lyapunov exponent (FTLE), the evolution of the vortex structures in the two distinct flow patterns are determined. By presenting the dynamic evolution of the Lagrangian behaviors, such as the formation and transport of the flow patterns, as well as the flow separation, it proves that LCSs can describe the Lagrangian dynamics properly. Moreover, during investigating the formation of leading-edge vortex (LEV), the time-dependent separation profile is found to play a major role. To be specific, within the slow plunging motion, the flow is rolled up to form a LEV downstream the separation point where the unstable manifold attaches. In contrast, within the fast plunging motion, a transfer barrier is generated upstream the leading edge, causing the LEV to form upstream the separation point. It is also found that by studying the dynamic behavior of the intersection between the unstable manifold and the stable manifold, the evolvement of the vortex structures, including the formation and the shedding of them, is more clear. Compared to the traditional visualization techniques, the Lagrangian analysis based on LCSs can provide a deeper insight into the dynamics of the vortex, which plays an important role in understanding the high performance of the unsteady flow induced by flapping wings. [ABSTRACT FROM AUTHOR]