  The short-term transient falling dynamics of a dripping water drop in
quiescent air has been investigated through both simulation and experiment. The
focus is on the short term behavior and the time range considered covers about
eight dominant second-mode oscillations of the drop after it is formed. Due to
the small fluid inertia the growth of the drop is quasi-static and is well
captured by the static pendant drop theory. Nevertheless, the pinching dynamics
and the resulting post-formation state of the drop trigger a nonlinear
oscillation when the drop falls. The initial shape of the drop when it is just
formed is decomposed into spherical harmonic modes. The pinching dynamics such
as interface overturning introduces small-scale variation on the drop contour,
which in turn contributes to the finite amplitudes of the higher-order modes.
Furthermore, the initial kinetic energy when the droplet is just formed is as
important as the initial surface energy contained in the drop shape, and is
found to amplify the initial oscillation amplitude and to induce a phase shift
in the oscillation of all the modes. By incorporating both the initial surface
and kinetic energy, the linear model for a free drop oscillation yields very
good predictions for the second and third modes. The mode amplitude spectra
show both the primary frequencies that are consistent with the Lamb's theory
and the secondary frequencies arising from different modes due to nonlinear
inter-mode coupling. The complex transient flow inside and outside the drop is
induced by the interaction between the falling motion and the nonlinear
oscillation. The streamlines indicate that the internal flow is substantially
different from the Hill vortex for a falling drop without oscillation. The
temporal evolutions of both the internal flow and the wake morphology follow
the dominant second oscillation mode.
