Design, analysis and verification of a volume-of-fluid model with interface-capturing scheme

The Volume-of-Fluid (VoF) model with interface-capturing scheme, available in many commercial and open-source CFD packages, may be widely used for simulations of free-surface hydrodynamics but is certainly not used without difficulties. While practitioners have come to expect conservation of mass and momentum, unconditional stability, second-order of accuracy and good iterative convergence, none of these qualities are evident when a free surface gets involved. In this paper, we investigate how these qualities are affected by the discontinuous nature of the volume fraction, which requires specialized numerical techniques for advection, for the interpolation of material properties, for the hydrostatic balance and for the pressure-weighted interpolation used in finite-volume methods with co-located variables. We find that conservation properties and good iterative convergence can still be attained, but not second-order of accuracy and unconditional stability. Code verification is presented to substantiate these findings, with cases ranging from one to three dimensions, from uniform to locally refined grids and from the stand-alone volume fraction equation to the complete set of mass, momentum and volume fraction equations. For the latter, a novel manufactured solution based on a sinusoidal wave in deep water is introduced. These tests permit not only to quantitatively assess the results for comparison with theoretical predictions but also ensure that our findings are characteristic of the VoF method and not of a particular implementation.