Annals of Mathematical Sciences and Applications

Volume 5 (2020)

Number 1

Mathematical sciences related to theoretical physics, engineering, biology and economics

Guest Editor: Tony Wen-Hann Sheu, National Taiwan University

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

Pages: 75 – 100



Po-Wen Hsieh (Department of Applied Mathematics, National Chung Hsing University, Taichung City, Taiwan)

Suh-Yuh Yang (Department of Mathematics, National Central University, Taoyuan City, Taiwan)

Cheng-Shu You (Department of Applied Mathematics, Feng Chia University, Taichung City, Taiwan)


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.


incompressible Navier–Stokes equations, fluid-solid interaction, free-falling body, sedimentation, immersed boundary method, direct-forcing method, projection scheme

2010 Mathematics Subject Classification

Primary 65M06. Secondary 76M20.

This work was supported by the Ministry of Science and Technology of Taiwan under grants MOST 106-2115-M-005-005-MY2 (Po-Wen Hsieh), MOST 106-2115-M-008-014-MY2 (Suh-Yuh Yang), and MOST 107-2115-M-035-007-MY2 (Cheng-Shu You).

The research of Suh-Yuh Yang was also partially supported by the National Center for Theoretical Sciences, Taiwan.

Received 12 August 2019

Accepted 8 November 2019

Published 27 February 2020