Iterative Local Bézier Reconstruction Algorithm of Smooth Droplet Surface for the Immersed Boundary Method

Abstract

Empirical formulae of the terminal velocity and shape of a water droplet in microphysics parametrizations are derived from experiments or theoretical works and they are only verified under room temperature and standard atmospheric pressure. A two-phase direct numerical simulation model can be a strong tool to improve those empirical formulae under general conditions. Recently, the global B-spline fitting has been applied and its smooth one dimensional (1D) surface reconstruction of water droplets has enabled stable simulations of falling two dimensional (2D) droplets by the immersed boundary method (IBM). However, an extension of the global reconstruction from 1D to 2D is highly complex and is almost impossible to use in a model. To overcome this limitation, an iterative algorithm is proposed for a local smooth surface reconstruction in this work. One significant advantage is its straightforward extension to 2D surfaces. To test the new method, simulations of an axisymmetric free-oscillating water droplet are compared between the global and local surface reconstructions. A further simulation of a rising air bubble is performed to examine the robustness of the new algorithm for the highly distorted interface. This new method opens a pathway to three dimensional (3D) water droplet simulations by the IBM.