Section: New Results

Development of a sharp cartesian method for the simulation of flows with high density ratios

We have developed a sharp cartesian method for the simulation of incompressible flows with high density and viscosity ratios, like air-water interfaces. This method is inspired from a second-order cartesian method for elliptic problems with immersed interfaces (Cisternino-Weynans 2012). A classical predictor-corrector algorithm is used to solve the fluid equations, in a non-incremental version, which means that the guess value for the pressure is zero. This choice avoids instability issues due to the discontinuous pressure values when the interface moves. We take into account the viscous forces by regularizing the density and viscosity values. This approach allows for a more straightforward and robust treatment, and has been proven to provide satisfactory accuracies for high Reynolds numbers. To compute the pressure, it is necessary to solve an elliptic problem. This elliptic problem with discontinuous values across an interface is solved with the second order method cited above. The originality of this method lies in the use of unknowns located at the interface. These interface unknowns are used to discretize the flux jump conditions and the elliptic operator accurately enough to get a second order convergence in maximum norm.

Figure 13. Horizontal et vertical velocity profile of an air bubble rising into water.