A Higher-Order VOF Interface Reconstruction Scheme for Non-Orthogonal Structured Grids - with Application to Surface Tension Modelling

The volume-of-fluid (VOF) method [24] is widely used to track the interface for the purpose of simulating liquid-gas interfacial flows numerically. The key strength of VOF is its mass conserving property. However, interface reconstruction is required when geometric properties such as curvature need to be accurately computed. For surface tension modelling in particular, computing the interface curvature accurately is crucial to avoiding so-called spurious or parasitic currents. Of the existing VOF-based schemes, the height-function (HF) method [10, 16, 18, 42, 46, 53] allows accurate interface representation on Cartesian grids. No work has hitherto been done to extend the HF philosophy to non-orthogonal structured grids. To this end, this work proposes a higher-order accurate VOF interface reconstruction method for non-orthogonal structured grids. Higher-order in the context of this work denotes up to 4 th-order. The scheme generalises the interface reconstruction component of the HF method. Columns of control volumes that straddle the interface are identified, and piecewise-linear interface constructions (PLIC) are computed in a volume-conservative manner in each column. To ensure efficiency, this procedure is executed by a novel sweep-plane algorithm based on the convex decomposition of the control volumes in each column. The PLIC representation of the interface is then smoothed by iteratively refining the PLIC facet normals. Rapid convergence of the latter is achieved via a novel spring-based acceleration procedure. The interface is then reconstructed by fitting higher-order polynomial curves/surfaces to local stencils of PLIC facets in a least squares manner [29]. Volume conservation is optimised for at the central column. The accuracy of the interface reconstruction procedure is evaluated via grid convergence studies in terms of volume conservation and curvature errors. The scheme is shown to achieve arbitrary-order accuracy on Cartesian grids and up to fourth-order accuracy on non-orthogonal structured grids. The curvature computation scheme is finally applied in a balanced-force continuum-surface-force (CSF) [4] surface tension scheme for variable-density flows on nonorthogonal structured grids in 2D. Up to fourth-order accuracy is demonstrated for the Laplace pressure jump in the simulation of a 2D stationary bubble with a high liquid-gas density ratio. A significant reduction in parasitic currents is demonstrated. Lastly, second-order accuracy is achieved when computing the frequency of a 2D inviscid oscillating droplet in zero gravity. The above tools were implemented and evaluated using the Elemental®multi-physics code and using a vertex-centred finite volume framework. For the purpose of VOF advection the algebraic CICSAM scheme (available in Elemental®) was employed