An anti-singularity Mie-Grüneisen mixture model based on isentropic and Hugoniot curves

The Mie-Grüneisen mixture model is conveniently used in the multi-component problem with Mie-Grüneisen EOS (equation of states). In the Mie-Grüneisen EOS, the isentropic and Hugoniot curves are two typical reference states curves. However, the curves of these two reference states contain singularity points and cause difficulty when the interface is treated by volume fraction, which is accustomed to being used as a color function in traditional models. The difficulty lies in that the volume fraction model produces fragments of fluid volumes near the interface due to its diffused style. Meanwhile, these volume fragments may encounter the singularity points and make the sound velocity abnormally high at the interface in some isentropic reference curves. On the other side, the singularity points may cause the sound velocity to be negative for some Hugoniot reference states and interrupt the calculation. To avoid volume fragments near the interface area, the volume fraction is replaced by mass fraction, in which the relative volume is defined by the reciprocal of proportional density of fluid component. Based on this new definition, the reconstructed relative volume is no less than that of the fluids mixture. Consequently, the sound velocity forms a trough shape at the interface and does not cause high peak value. Moreover, some equations in Mie-Grüneisen mixture model contain the derivative terms of reference states parameters. When these items are defined as weighted average mixture at the interface, they often become negative if weighted average of mass fraction is directly used. To prevent the negative value at the interface, the reference states are optimized at the interface. Numerical examples show that although the mass fraction has negligible effect on the improvement of the accuracy of results, it makes the sound velocity steady under the isentropic reference states of medium, and therefore, less time steps are needed than those used for the volume fraction model. Furthermore, the mass fraction can be used to correct the negative sound velocity in Hugoniot reference states to ensure the smooth and accurate calculations.