Conservative multilevel discrete unified gas kinetic scheme for modeling multiphase flows with large density ratios

A novel multilevel discrete unified gas kinetic scheme (MDUGKS) is proposed to efficiently model multiphase flows with large density ratios. By constructing the kinetic flux with a mutable time interval depending on the local mesh spacing, the MDUGKS overcomes the stability problems encountered by the standard DUGKS when operating with the multilevel mesh. With the interpolation of macroscopic variables and mesoscopic distributions handled separately, the moments of distribution functions are maintained consistent with the conservative flow variables. Two kinetic equations corresponding to the conservative Allen-Cahn equation and the hydrodynamic equation are individually solved by the MDUGKS, and six benchmark problems have been conducted to evaluate its performance. Numerical solutions in steady cases produced by the MDUGKS are in well accordance with the theoretical predictions. A limiting density ratio of 109 is achieved in the quiescent droplet. The dynamic processes in unsteady cases anticipated by the MDUGKS agree well with the reference predictions. Comparative results also demonstrate that the MDUGKS behaves consistently with different types of meshes. With the employment of the adaptive multilevel mesh, 80% improvement in computational efficiency could be achieved compared with the uniform mesh. Considering the kinetic nature and the high efficiency, the MDUGKS offers a powerful tool for presenting meaningful insight into understanding the realistic multiphase systems at the mesoscopic scale.