Level set method on adaptive Cartesian grids for multiphase flow computation

The Level Set Method, a powerful numerical approach for analyzing and computing the motion of interfaces, has been studied and extended to 2D adaptive Cartesian grids in order to achieve higher accuracy and better efficiency. First, a fast quadtree-based grid generator is developed to produce 2D adaptive Cartesian mesh using an object-oriented programming model. After that, both finite volume and semi-Lagrangian methods are developed to solve the level set equation. Second-order accuracy in smooth regions, good stability and convergence to viscosity solutions are demonstrated by the finite volume level set method. Versatile motions of interfaces under passive transport velocity field, first and second order geometric velocity fields are simulated with the semi-Lagrangian method to show its ability. A particle correction algorithm is developed to further improve the accuracy. The results are excellent and have a very good mass-conserving property. Finally, the level set solver is coupled with a characteristics upwind finite volume incompressible flow solver to tackle multi-phase flow problems. By using unstructured adaptive Cartesian grids with automatic refinement near interfaces, the solution accuracy and efficiency is dramatically improved. Several representative multi-phase flow problems are tackled to evaluate the effectiveness of the method. The computational results using the adaptive grids are compared with the results using uniform grids to demonstrate the efficiency and accuracy of the method.