Numerical simulations of free surface flows on adaptive cartesian grids

The paper studies a method for numerical simulation of free surface flows of viscous incompressible fluids. The approach is based on the level set method for capturing free surface evolution and features compact finite difference approximations of fluid and level set equations on locally refined and dynamically adapted octree cartesian grids. We consider in detail an extension of staggered grid stable approximation of the NavierStokes and level-set equations for the case of evolving free boundary and dynamically refined / coarsened octree meshes. The problem of surface reconstruction and preserving distance property of the discrete level set function is addressed. Several re-initialization techniques are introduced, compared, and their impact on the accuracy of flow computation is considered. Numerical examples include analytical tests, the breaking dam, and the oscillating droplet benchmark problems. It is demonstrated that altogether the considered techniques and algorithms result in an accurate and efficient approach to modeling incompressible viscous flows with free surfaces.