An innately mass conserving interface capturing method for the modelling of interface advection on unstructured grids

Abstract

In solving two-phase flows, the location of the interface between the phases is necessary to handle interface jump conditions when solving the Navier-Stokes equation. Current interface capturing and advection methods, however, suffer from various issues. The level set method uses the signed distance to the interface and the interface being the zero level set of this function allows the evolution of the level set field to be described with a simple advection equation. This means that no additional steps are required, but solving the advection equation generally does not conserve mass. On the other hand, Volume of Fluid methods utilise the local fluid fractions to represent the phase interface. For these methods, a mass conserving advection algorithm exists, but the absence of an explicit interface requires expensive reconstruction methods to be used instead. Additionally, this Volume of Fluid advection method is subject to a restrictive CFL condition on the time-step and, being dimensionally split, requires a structured grid to be used. Other Volume of Fluid or Moment of Fluid advection methods that do not have these conditions are not mass conserving. Dual interface methods that combine information from the level set and volume fractions are able to achieve higher accuracy, but these method still use Volume of Fluid advection to remain mass conserving and are thus subject to the same conditions. These methods use the level set field to create a cheaper and better justified reconstruction method, which negates one drawback of VoF methods. In this thesis, a method is formulated to allow interface advection on unstructured grids without such a strict CFL condition compared to the MCLS method. To do this, the finite volume level set advection method of the MCLS method is replaced with a nodal-modal discontinuous Galerkin method. The DG method is analogous to the Galerkin finite element methods, but the basis functions are now only valid on one element, and the solution can be multiply defined on the cell boundaries. This allows for level set advection on polygonal cells, and has the added benefit that since cells are semi-independent, level set correction now only has to be applied locally. This, like the finite volume level set advection method, is not necessarily mass conserving though, so the volume of fluid will need to solve this issue. The difference with MCLS is that no volume of fluid advection method is used, since these methods are the parts that introduce the aforementioned problems that are being avoided. Instead, a minimisation method is done on the VOF field in order to obtain a correction which can be applied to the level set and keep it mass conserving. For this minimisation method, a flux condition is used to impose additional constraints on the solution. This condition effectively attempts to match the interface intersections on a cell edge for both adjacent cells, for all cell edges that have a flux going through. The resulting method is tested for four test cases, with solid body translation and rotation, the corner flow test, and the vortex deformation test. In the translation test, there is only vertical velocity, so the flux condition does not take vertical edges into account. To stabilise this test case, level set reinitialisation is applied, which is a technique used to maintain the signed distance property of the level set function to use in the Navier-Stokes evolution method. This keeps the solution in the translation test somewhat representative of the exact interface, but makes the obtained results no longer representative of the actual method. For the other test cases, the method appears to converge when the grid is refined, except for the solid body rotation test using Euler Forward, as this result becomes unstable. However, no apparent order of convergence is present, and the effect of a more accurate time integration method is very inconsistent. To solve this, the method is altered to allow multiple level set advection steps for every optimisation routine. This makes the method better follow the order of convergence of the discontinuous Galerkin method, although this order is slightly lower for the vortex deformation test. The obtained results show that in it's current form, this optimisation based method is likely not usable for general applications, but that with additional work, this kind of method may prove to be more applicable than MCLS.