An Unstructured Immersed Boundary Method for Simulation of Flows over Bottom Topography

An immersed boundary method (IBM) for unstructured grids is presented for the simulation of flows over bottom topography. We implement the method in a nonhydrostatic, unstructured-grid code that solves the Navier-Stokes equations under the Boussinesq approximation. The equations are discretized on a staggered Cartesian z-level grid in the vertical plane and an unstructured triangular grid in the horizontal plane, and the solution to the hydrodynamics is based on a three dimensional semi-implicit free-surface approach together with the pressure-correction method for the non-hydrostatic pressure. In the simulation domain, the cells which contain the physical boundaries are treated as the ghost cells. Thus, each ghost cell itself has an immersed boundary, which is used to represent the presence of the physical boundaries based on the assumption that boundaries are piecewise linear within each cell. In the ghost cell, rather than directly imposing the desired boundary condition on the cell edge, which is the case for the z-level grids in the absence of an immersed boundary, the ghost-cell quantities are imposed at those edges to satisfy the desired boundary conditions at the piecewise-linear immersed boundaries. The ghost-cell quantities are obtained using linear extrapolations from the known interior points. Before solving the discrete governing equations, the spatial coefficients for the extrapolation are obtained based on geometries. Unlike previous studies on IBM, which impose the forcing term to make velocities to satisfy the desired boundary conditions, this study focuses on applying the immersed boundary method for solving the pressure field. The potential flow over a Gaussian hump is simulated using the proposed immersed boundary method and the results are compared with the simulation without IBM.