An optimal approach for velocity interpolation in multilevel VOF method

The use of mesh refinement techniques is becoming more and more popular in computational fluid dynamics, from multilevel approaches to adaptive mesh refinement. In particular, direct numerical simulations of two-phase flow have seen recently the introduction of a finer grid for interface representation. In general, the mesh used for surface tension calculations requires higher resolution than the grid where the conservation equations are discretized and the introduction of these auxiliary grids overcomes some well-known limitations of Volume-of-Fluid (VOF) and Level Set methods. In particular for VOF methods, the interface discontinuity at the cell boundary can cause the deformation of the interface shape at low resolution even for simple translations and solid-body rotations. In fact different interface lines interact with each other in the reconstruction step at a distance of a couple of grid spacings, causing distortion and eventually numerical break-up of the interface. The methodology we present provides a smoother description of the interface and its geometrical properties which are necessary for an accurate capillary force evaluation. The use of different grids for Navier-Stokes equations and interface representation requires the projection of the velocity field from the coarse to the higher resolution grid. This procedure is not trivial, especially for incompressible flows where the divergence-free constraint must be satisfied on both grids. In this paper we present a new method to estimate the velocity field on the fine mesh based on an optimal approach. This algorithm allows us to satisfy the divergence-free constraint in every cell of the fine grid in two- and three-dimensional geometry. This is achieved by a constrained minimization of an objective functional. The proposed functional is the integral of the square of the error between the computed and the linear interpolation velocity field. The minimization is subject to the divergence-free constraint which therefore must be always satisfied. In this way, the projected velocity field is divergence-free and the interface advection does not generate unrealistic situations that could lead to an unphysical behavior. In the last part of the paper we present some results obtained with the above method in two-phase flow simulations in order to assess its reliability and robustness.