The effects of flexibility on the wake structures of a foil under a heaving motion in a viscous uniform flow are numerically studied using an immersed boundary method. An inspection of the phase diagram of the wake structures in a map of the chord-length-based dimensionless heaving amplitude (AL) and Strouhal number (StL) shows that the wake transition boundaries of the rigid foil are well predicted by constant amplitude-based Strouhal number (StA) lines, similar to previous studies. However, the wake transition boundaries of the flexible foil are predictable by constant StA lines only for high StL cases. A large deformation angle of a flexible foil by the amplitude difference and phase difference between the leading and trailing edge cross-stream displacements reduces the effective leading edge velocity, with an accompanying decrease in the leading edge circulation. However, the trailing edge circulation for a flexible foil is increased due to increased trailing edge amplitude. The sum of the leading and trailing edge circulations plays an important role in determining the wake pattern behind a rigid and flexible foil, and wake transitions are observed beyond critical circulations. The decrease in the thrust coefficient for large values of StL and AL is closely associated with the generation of a complex wake pattern behind a foil, and the complex wake is a direct consequence of sufficiently large leading edge circulation. A critical effective phase velocity in a vortex dipole model is proposed to predict the maximum thrust coefficient without a complex wake pattern.
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