Numerical solutions of free-surface incompressible flows

This study is dedicated to the development, implementation, and analysis of a numerical method for computer simulations of multicomponent flows involving capillary free surfaces. For this purpose, a numerical scheme for the incompressible Navier-Stokes equations is presented. The scheme is an implementation of the projection method by the method of Finite Elements. The convection-diffusion sub-step is solved using a conforming linear finite element for the velocity, while the projection sub-step is solved using a nonconforming linear finite element for the velocity and piecewise constant pressure. The end-of-step velocity is locally pointwise divergence-free, which is a desired feature, since it allows for improved mass conservation.;This projection scheme is employed for computer simulations of multicomponent incompressible flows. Discontinuous pressure and low-order velocity approximations provide consistent handling of discontinuities in the solution, as long as computational cells are not intersected by moving interfaces. A robust algorithm for local grid alignment is proposed. A reference grid is maintained and used on every timestep to produce a new computational grid. Few nodes in the reference grid that are close to the interface are projected onto it, so that the computational grid contains no edges intersected by the interface and has the same connectivity as the reference grid. The unchanging connectivity makes parallelization easier and more effective. The interfaces are approximated in the vicinity of each node by a part of a sphere, which is also used for the computation of surface tension.;Both the proposed projection scheme and the local grid alignment are validated on a number of numerical examples.