Boundary Treatments for Free Boundary Problems in Complex Geometries

We present a straightforward and efficient method for solving the Poisson equation with mixed Dirichlet and Neumann boundary conditions on irregular domains using a Heaviside formulation that can be applied uniformly on the entire domain. This method produces a symmetric positive definite linear system and second-order accurate solutions in both the L 1 and Linfinity norms. We first present an analysis of the accuracy of the solution and its gradient for the Poisson equation with Dirichlet boundary conditions on irregular domains with different treatments at the interface using the Ghost Fluid Method. We demonstrate that a quadratic extrapolation for defining the ghost values and a quadratic interpolation for finding the interface location are necessary to obtain second-order accurate gradients, with the disadvantage being a non-symmetric discretization matrix. Secondly, we applied this method for imposing Dirichlet boundary conditions as part of our mixed boundary conditions scheme. We show that our Heaviside formulation automatically handles the proper application of Neumann boundary conditions with a good approximation of the Heaviside function. We apply several different Heaviside approximations to our scheme in two spatial dimensions, and demonstrate the second-order accuracy of our method in two and three spatial dimensions. Finally, we consider the numerical approximation of the Navier-Stokes equations on irregular domains and propose a novel approach for solving the Hodge projection step. This method is simple, robust, and leads to a symmetric positive definite linear system for both the projection step and for the implicit treatment of the viscosity. We demonstrate the accuracy of our method in the L1 and Linfinity norms. We apply this method to the simulation of a flow past a cylinder in two spatial dimensions and show that our method can reproduce the known stable and unstable regimes as well as correct lift and drag forces. We also apply this method to the simulation of a flow past a sphere in three spatial dimensions at low and moderate Reynolds number to reproduce the known steady axisymmetric and nonaxisymmetric flow regimes. We further apply this algorithm to the coupling of flows with moving rigid bodies.