A Perturbation Model of Raindrop Oscillation Characteristics with Aerodynamic Effects

Abstract

An asymptotic analysis with the method of multiple-parameter perturbations has been carried out to examine the basic features of drop oscillations in a uniform flow field. The quiescent drop shape has an oblate deformation resulting from a potential-flow pressure distribution. This equilibrium distortion leads to a frequency splitting that eliminates the so-called mode degeneracies found by Rayleigh for small-amplitude oscillations about an undeformed spherical drop. Our results show that the characteristic frequencies for the zonal (axisymmetric) modes increase whereas those of sectoral modes decrease as the velocity of the external uniform flow increases. This trend agrees with the observational data and the results of the spheroidal model presented by Beard. The fine structure in the frequency spectrum of falling water drops may play a role in mode selection as observed in experiments. In addition, the one-to-one correspondence of characteristic frequencies and oscillation modes provides a theoretical basis for extracting more information from the experimental data. The modification of the drop oscillation mode shapes due to the coupling between the external flow field and oscillatory motion in the drop is also derived from the solutions of higher-order perturbations. Corresponding to each characteristic frequency, the oscillation mode shape contains more than one spherical harmonic.