On the Convergence of the 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 mesh integrates the volume fractions of a column of cells across the interface. A stencil of three consecutive heights and centered finite differences computes 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. The continuous height function is used to show that a constant approximation of the two geometrical properties across the column provides first-order convergence, while linear or quadratic interpolations provide second-order convergence. The numerical results agree with the theoretical development presented in this study.