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 compute 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 drops of 1 mm ≤ d < 1.5 mm, an assumption of viscosity ratio μ = 200 (greater than that of water versus air) seems to be necessary to account for the vortex shedding in the unsteady wake and subsequent reduction of the internal circulation intensity. An additional assumption for adjusting the value of Reynolds number Re is needed to match the drag coefficient value consistent with the measured (known) terminal velocity, for 1.5 mm ≤ d ≤ 5 mm. Now the terminal velocity cannot be determined as part of the solution and Re differs from that evaluated from the actual physical properties. But 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 > 5 mm, it seems that the adjustment of viscosity ratio μ in addition to Re enables obtaining the drop shape with axis ratio comparable to experimental data, otherwise the solution would over-estimate the drop deformation.

Research highlights
► We compute 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 drops of 1 mm < d < 1.5 mm, an assumption of viscosity ratio μ = 200 is used to account for the vortex shedding in the unsteady wake and subsequent reduction of the internal circulation intensity.
► For larger drops, reasonable shape can be obtained with additional assumptions for matching the drag coefficient value consistent with the measured (known) terminal velocity, etc.