The flow pattern around a NACA0012 airfoil undergoing harmonic plunging motion corresponding to the deflected wake phenomenon reported by Jones and Platzer (Exp Fluids 46:799-810, 2009) is investigated in detail using direct numerical simulations. An arbitrary Lagrangian-Eulerian formulation based on an unstructured side-centered finite volume method is utilized in order to solve the incompressible unsteady Navier-Stokes equations. The Reynolds number is chosen to be 252, and the reduced frequency of plunging motion (k = 2 pi fc/U (a)) and the plunge amplitude non-dimensionalized with respect to chord are set to be 12.3 and 0.12, respectively, as in the experimental study of Jones and Platzer (2009). The present numerical simulations reveal a highly persistent transient effect, and it takes two orders of magnitude larger duration than the heave period to reach the time-periodic state. In addition, the three-dimensional simulation reveals that the flow field is three-dimensional for the parameters used herein. The calculation reproduces the deflected wake and shows a good agreement with the experimental wake pattern. The instantaneous vorticity contours, finite-time Lyapunov exponent fields and particle traces are presented along with the aerodynamic parameters including the lift and thrust coefficients.