Mass conservation issues of moving interface flows modelled by the interface capturing approach

Many industrial applications of mechanical engineering involve moving interface flows. Numerous examples include actual problems such as coating process, metal forming, sluice gates, tank sloshing and dam break. Numerical simulation of moving interface flows is extremely complex, since it involves the solution of the incompressible Navier-Stokes equations coupled with moving interface tracking. The numerical approach has to identify the unknown interface, to follow its kinematics and to resolve a strong coupling between the interface propagation and dynamics of the continuum. Over the past 30 years, researchers have put a lot of effort into developing various numerical methods to simulate the moving interface flows governed by the Navier-Stokes equations. All numerical methods for modelling of the moving interface flows are based on interface tracking approach or interface capturing approach. In former, the liquid region is subdivided by a mesh, while each cell is deformed according to the movement of the interface and computed velocities. The earliest works [1] were based on the Lagrangian description of motion. However, this approach requires remeshing procedures to avoid of computation failure due to serious distortion of cells or elements [2]. Various interface tracking methods [3, 4] for attaching the interface to a mesh surface were developed during the past decades using the finite element method (FEM). These methods are unable to cope naturally with interface interacting with itself by folding or rupturing. In the interface capturing approach, the mesh remains fixed and moving interface can not be directly defined by the mesh nodes. Therefore, additional technique is necessary to define the areas occupied by fluid or gas on either side of the interface. The marker-and-cell method [5], the volume of fluid method [6] and the level set method [7] are well known methods using the interface capturing approach. These methods require no geometry manipulations after the mesh is generated and can be applied to interfaces of a complex topology [8]. However, the location of the interface is not explicit and, sometimes, the appropriate boundary conditions cannot be prescribed with a required accuracy.