A direct-forcing immersed boundary projection method for simulating the dynamics of freely falling solid bodies in an incompressible viscous fluid^∗

In this paper, we develop a new direct-forcing immersed boundary approach combined with the Choi-Moin projection scheme for simulating the dynamics of freely falling solid bodies in an incompressible viscous fluid. At first, the solid object region is regarded as made of fluid and we then introduce a virtual force distributed only on that region that enforces it to behave like a real solid body with the solid velocity. The time integration of the momentum equation is performed by using a third-order Runge-Kutta formula for the convection and a second-order Crank-Nicolson formula for the diffusion. Moreover, second-order centered differences over a staggered Cartesian grid are employed for all the spatial discretizations in the projection scheme. We also integrate a collision model into the method for circular particles to mimic the repulsion force arising from body-body or body-wall collisions in the fluid-solid interaction process. The most advantageous feature of the proposed method is that it is conceptually simple and rather easy to implement without involving any discrete Dirac delta functions or post interpolations for accuracy like most immersed boundary methods in the literature. Several numerical experiments are carried out to illustrate the effectiveness of the newly proposed method.