In order to simulate numerically the physical behaviour of a free surface between two fluids, we have developped a compressible model, able to correctly reproduce sloshing at high Bond number (capillary forces neglectible). We will first present the model. It is composed of four equations in two space dimensions : two for each fluid mass conservation, and two for momemtum conservation. The gas and liquid phases are supposed to be compressible fluids following an isothermal linearized equation of state. A mechanical equilibrium relation closes the model (the equilibrium model). Relaxing this relation, we obtain a new model (the relaxation model) that is more adapted for numerical treatment. The second part of the paper is dedicated to the presentation of the numerical method that we used. We explain why it does not need a scheme for the volume fraction. In particular, there is no specific algorithm to localize the interface, as in VOF, Level-Set or Front Tracking methods. The last part of the paper is devoted to some validation results, compared to analytical solution or experimental data.
Formation of singularities in compressible fluids in two-space dimensions
