Two-phase flows are involved in many industrial and environmental applications. Indeed, fronts or interfaces occur in very different problems of fluid mechanics such as spray formation, wave breaking or oil transportation. Then, understanding the dynamic of this kind of complex flows is of great scientific importance and remains a challenging task. Even if literature is extensive on this topic, accurate simulation of fluid flows with sharp front presents many problems and the difficulty in solving the full Navier–Stokes equations in the presence of a deforming interface is considerable. In past few years, major progress has been achieved in computing multiphase flows.

For stationary and regular grid, an approach to track interfaces consists in capturing directly the presence of the front on the Eulerian grid. The VOF method, where the interface is tracked by a marker function or the Marker and Cell method where particles are used to identify each fluid are the oldest but still popular front-capturing approach. The level-set method where the interface is tracked by a distance function has been developed more recently on this purpose also. The main difficulty of this kind of approaches is to maintain the sharpness of the front during the flow motion and to account for flow structures whose characteristic size is smaller or equal to the mesh cell length.

Another class of method is Lagrangian, where the grid follows the interface. These methods are accurate but involve high complexity grids for three-dimensional flows.

The front-tracking approach is a class of method where a specific mesh of front is used to track explicitly the interface on a Cartesian grid. This approach involves an additional difficulty since two grids have to be taken into account and the application of this method to three-dimensional flows can be sometimes painful. However, the sharpness of the front is always maintained even for under-resolved Eulerian grid making the front-tracking approach one of the most accurate methods to track interfaces.

We describe here a two-dimensional conservative front-tracking method for multi-scale and multiphase flows.

The markers used to represent the interface are connected by linear segments to avoid the complexity of high order interpolation and allow a simple three-dimensional extension. To maintain a good accuracy of the front, a reseeding operation is used to keep an almost constant distance between markers and we ensure at each time step that no markers are deleted in high curvature zone to keep also a good interface shape. The velocity interpolation on the markers node is made by a divergence free parabolic edge reconstruction method. This allows keeping a very good accuracy on the markers velocity. Moreover, we propose two volume conservation algorithms which avoid volume loss while keeping a good accuracy and interface shape. The curvature at each markers node or the mean curvature on the elements front can be calculated efficiently here. A strategy has been developed to take into account the topology changes of fluid interfaces such as drops or bubbles coalescence or films that rupture. This task is achieved by modifying the front connectivity when it is necessary in an appropriate way.

In order to validate the proposed front-tracking method, two analytical test cases are considered for which the interface is widely stretched and deformed to show the ability of the front-tracking method to track interfaces even for complex cases.

A complete convergence study is realized on these test cases and a comparison between different existing approaches and our front-tracking method is achieved.

The conservative front-tracking method computed in this work is shown to be able to describe interfaces with high accuracy even for under-resolved Eulerian grids.