An immersed boundary method for non-uniform Cartesian grids

Many kinds of immersed boundary method have been developed, but most of them have been used in uniform grids with discrete Dirac delta functions. Therefore, the distribution of Lagrangian points over the immersed surface is usually made uniformly. However, when any immersed boundary method is to be applied to non-uniform grids, uniform distribution might not be optimum for good performance. Recently, Akiki and Balachandar (2016) proposed a method to distribute the Lagrangian points nonuniformly over the surface of a sphere near the wall, but it cannot not be extended to more general shape of immersed surface. We propose a method that is capable for properly distributing the Lagrangian points over any kind of surface by considering the size of nearby Eulerian grids. Present method first finds intersection points between immersed surface and nonuniform Cartesian grids. Then, the centroid of the intersection points is projected on the immersed surface to be designated by Lagrangian point. This procedure guarantees one Lagrangian point per the Eulerian grid cell. This method is validated for various problems such as flows around a settling sphere, a moving sphere in the near-wall region and a tilted ellipsoid near the wall.