Short-term oscillation and falling dynamics for a water drop dripping in quiescent air

The short-term transient falling dynamics of a dripping water drop in quiescent air has been investigated through both simulation and experiment. A representative case with a low inflow rate in the dripping regime is considered. 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 at the inlet, the growth of the drop is quasistatic and is well captured by the static pendant drop theory. Nevertheless, it is demonstrated that the pinching dynamics and the resulting postformation 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 initial mode amplitudes, characterized by the Fourier-Legendre coefficients, are found to be finite for up to the tenth mode. 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 intermode coupling. Moreover, it is worth to noting that the nonlinear effect is most profound for the fourth mode owing to its resonant coupling with the dominant second mode. 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. In the oblate-to-prolate deformation, the internal flow goes against the external flow. As a result, a saddle point arises in the drop, which gives rise to two counterrotating vortices. The vortex dynamics are visualized by the swirling-strength vortex identification criterion and the vorticity. Whereas the potential flow changes direction during a second-mode oscillation cycle, the rotating directions of the vortices remain the same.