  In the present study, the axisymmetric natural oscillations of a liquid drop
supported by a flat surface is investigated by direct numerical simulation. The
liquid-gas interface is captured using a geometric volume-of-fluid (VOF)
method. A parametric study is carried out by varying the equilibrium contact
angle and the gravitational Bond number (Bo). Both positive and negative
gravities are considered, and thus the results cover both pendant and sessile
drops. To incorporate the effect of contact line mobility, the two asymptotic
limits, namely the pinned contact line (PCL) and free contact line (FCL)
conditions, are considered and their effects on the drop oscillation features
are characterized. The predicted oscillation frequencies for PCL and FCL serve
as the upper and lower bounds for general situations. The drop oscillation is
initiated by increasing the gravity magnitude for a short time. The first mode
due to the drop centroid translation dominates the excited oscillation. The
oscillation frequency scales with the capillary frequency, and the normalized
frequency monotonically decreases with the equilibrium contact angle. For zero
gravity, the computed frequencies for all contact angles agree remarkably well
with the inviscid theory for both the PCL and FCL conditions. The kinetic
energy correction factor is introduced to account for the additional
contribution of the oscillation-induced internal flow to the overall kinetic
energy of the drop. Both the frequency and the kinetic energy correction factor
increase with Bo, decrease with the contact angle, and increase when the
contact line condition changed from FCL to PCL. The variation of oscillation
frequency due to the change of Bo is particularly significant when the contact
angle is large.
