An Immersed Boundary Method for flows around rectangular objects

A second-order accurate and highly efficient Immersed Boundary Method (IBM) is presented for simulating flows along rectangular non-moving solid objects. In this method a rectangular object is placed on a staggered Cartesian grid such that its boundary coincides with grid points for the boundary-normal velocity component. By imposing forces at the grid points nearest to and on the boundary, the no-slip condition for the boundary-parallel velocity components is satisfied exactly, while the no-penetration condition for the boundary-normal velocity component is satisfied to a very good approximation. The accuracy of the IBM requires a pressure-correction method in which the correction pressure is small, which is accomplished by adding a fraction of the correction pressure to the pressure
after every time step. It is shown that for the currently used second-order Adams-Bashforth scheme this fraction must not exceed one for maintaining stability. Furthermore, a von Neumann stability analysis has been performed, from which it is argued that the forces imposed in the IBM will usually not affect the numerical stability. The method has been successfully applied to both laminar and turbulent flows through a porous medium consisting of a periodic three-dimensional regular array of cubes.