  The correspondence of secondary spanwise structures and pressure gradient is
numerically evaluated for a foil, performing heaving and pitching motion, at a
range of phase offsets (90$^\circ$ $\le \phi \le$ 270$^\circ$) and reduced
frequency (0.32 $\le St_c \le$ 0.56). The Reynolds number is $Re =$ 8000. The
wake is shown to be dominated by secondary hairpin-like structures that are
formed due to an elliptic instability prompted by the paired primary and
secondary leading edge vortex ($LEV$). The weaker secondary $LEV$ undergoes a
core deformation, resulting in streamwise vorticity outflux across the span of
the foil, and hence, the growth of hairpin-like structures. Evaluating pressure
gradients on the surface of the foil reveals a unique fundamental measure to
quantitatively characterize the growth of these coherent structures. Their
dominant presence can be directly linked to the growth of the secondary $LEV$
formed due to the large-scale interactions under localized adverse pressure
gradients. These promote a streamwise flow compression in neighboring regions
of the primary $LEV$. This association also presents a vivid consistency across
a range of kinematics. Therefore, this correspondence provides a novel
procedure to investigate the mechanisms involved in the formation of secondary
structures in the wake of an oscillating foil.
