On the Convergence of Normal and Curvature Calculations with the Height Function Method for Two-Phase Flow

The volume-of-fluid (VOF) method is widely used for multiphase flow simulations, where the VOF function implicitly represents the interface through the volume fraction field. The height function (HF) method on a Cartesian grid integrates the volume fractions of a column of cells across the interface. A stencil of three consecutive heights and centered finite differences compute the unit normal n and the curvature κ with second-order convergence with grid refinement. The interface line can cross more than one cell of the column, and the value of the geometrical properties of the interface should be interpolated in the cut cells. We propose a numerical algorithm to interpolate the geometrical data that removes the inconsistency between theoretical and numerical results presented in many papers. A constant approximation in the column of cells provides first-order convergence with grid refinement, while linear and quadratic interpolations indicate second-order convergence. The numerical results obtained with analytical curves agree with the theoretical development presented in this study.