Towards a high-order two-dimensional volume-of-fluid method for gas-liquid flow simulations

It is well known that current geometric volume-of-fluid (VOF) methods are limited to second-order accuracy due to the piecewise linear nature of the cell pre-images and interface approximations used to compute advective fluxes. We present recent work towards the extension of this framework to high-order accuracy in two dimensions. To this end, we provide closed-form expressions for the area of a Jordan curve clipped by a parabola. The Jordan curves we consider are reconstructed as a composite of quadratic Bézier curves and approximate the pre-image of a computational cell for transporting volume fractions in a semi-Lagrangian fashion. The expressions for computing the clipped area are derived from applying the divergence theorem across parameterized intersections between the Jordan curve cell and the parabola. The resulting analytical expressions have been verified using several carefully engineered test configurations and Monte Carlo integration. We also present means to reconstruct cellwise pre-images as composite quadratic Bézier curves that guarantee exact volume conservation. Ultimately, the proposed framework is used to conservatively advect the VOF indicator function with high-order accuracy. It is tested over a range of canonical advection test-cases.