On the numerical oscillation of the direct-forcing immersed-boundary method for moving boundaries

A family of sharp-interface immersed boundary methods using the so-called “direct forcing” approach have been developed for a range of applications because of their simple formulation and relative ease in handling complex geometries. Numerical interpolations are typically adopted in these methods to approximate the flow variables at the grid points near the solid surface, which provides an implicit force in the flow so that the no-slip and no-penetration conditions are satisfied. When dealing with moving boundaries, however, these methods could be prone to numerical oscillations because the nodal points where the direct forcing is applied may change from one time step to next. Noticing that the oscillations are caused by the instantaneous change of the numerical description at the direct-forcing points, we propose a formulation that allows for a smooth transition of the numerical description at these points. This new formulation preserves the spatial accuracy of the original immersed-boundary formulation and can effectively suppress the force oscillations. In this paper, we present a specific example of such formulation in both two- and three-dimensions and validate the implementation for both fixed and moving boundaries. Finally, a full-body simulation of flapping flight is demonstrated using the proposed method.