A general implicit direct forcing immersed boundary method for rigid particles

This paper presents a new immersed boundary method for rigid particles of arbitrary shape and arbitrary density, which can be exactly zero. Especially in the latter case, the coupling of the fluid and the solid part requires special numerical techniques to obtain stability. Exploiting the direct forcing approach an algorithm with strong coupling between fluid and particles is developed, which is exempt from any global iteration between the fluid part and the solid part within a single time step. Starting point is a previously proposed method restricted to spherical particles [37]. It is proved that the coupling concept can be generalized to rigid particles with complex shapes, which is not obvious a priori. The extension requires various non-trivial methodological extensions, especially with respect to the angular motion of the particle. Using analytical techniques it is demonstrated that the implicit treatment of the coupling force in the equations of motion results in additional terms related to a surrounding numerical fluid layer. As a main improvement over other non-iterative methods the proposed scheme is unconditionally stable for the entire range of density ratios and particle shapes allowing large time steps, with Courant numbers around unity. To date, no other non-iterative coupling approach offers such a generality regarding fluid-particle interactions. In addition to the detailed description of the underlying methodology and its differences from other methods, the paper provides all modifications required to improve immersed boundary methods for particulate flows from weak to strong coupling. Furthermore, an extensive validation of the scheme for particles of different density ratios and shapes is presented. The accuracy of the method as well as the convergence behavior are assessed by systematic studies.