Conservative three-dimensional unsplit geometric volume-of-fluid method with piecewise-parabolic interface reconstructions

The volume of fluid (VOF) method is widely used to track the interface between two fluids in numerical simulations of interfacial flows. Accurate interface reconstruction is critical for transporting fluid moments and estimating the interface curvature for surface tension. Existing state-of-the-art techniques use piecewise-linear interfaces, while recent advancements have introduced piecewise parabolic-interfaces. Higher-order surface reconstruction requires solving an optimization problem that is computationally expensive, highly non-convex and nonlinear, and sensitive to initial guesses. With the goal of lowering the computational cost of the piecewise-parabolic reconstruction of the interface, we propose new techniques that do not require the iterative solution of a nonlinear minimization problem while retaining satisfactory accuracy. Specific focus is put on accurately representing the interface in regions of low resolution and minimizing the presence of flotsam and jetsam that tend to pollute simulations, giving rise to unphysical results. The proposed techniques are evaluated on a range of implicit surface reconstruction test cases for studying their convergence behavior. Their performance is also assessed on advection test cases that utilize a conservative three-dimensional unsplit geometric VOF method. Lastly, the proposed VOF framework is combined with a Navier-Stokes solver to simulate a range of canonical flows driven by surface tension.