Raindrop shape determined by computing steady axisymmetric solutions for Navier-Stokes equations

To improve the understanding of various physical mechanisms for shaping a raindrop, we present steady axisymmetric solutions of Navier-Stokes equations that also include the free surface deformations. Using a Galerkin finite-element computational method, we are able to obtain solutions capable of describing the raindrop shape along with the associated flow field self-consistently. For drops with diameter d < 1 mm, the drop shape and flow field can be rigorously solved by computing solutions with all the parameters evaluated from the standard known physical properties. For larger drops an assumption of viscosity greater than that of water versus air was necessary to account for the reduction of the internal circulation intensity caused by vortex shedding in the unsteady wake. The value of Reynolds number was also adjusted to match the drag coefficient value consistent with the measured terminal velocity, for d = 1.5 mm to 5 mm. Although the terminal velocity cannot be determined as part of the solution, the flow field might reasonably represent the time-smoothed result of the transient oscillatory flow field that consists of the eddy viscosity. For drops of d > 5mm, the viscosity ratio of air to water was also adjusted to obtain drop shapes with axis ratio comparable to experimental data. By computing solutions at a value of Re consistent with the specified values of Weber number and drag coefficient according to the given drop size d and corresponding terminal velocities, we are able to predict raindrop shape for all drop sizes that are of practical interests without the need of further assumptions.