**Haixuan Ye \*, Yang Chen and Kevin Maki**

Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USA; chanyang@umich.edu (Y.C.); kjmaki@umich.edu (K.M.)

**\*** Correspondence: hxye@umich.edu

Received: 24 September 2020; Accepted: 13 October 2020; Published: 19 October 2020

**Abstract:** For numerical simulations of ship and offshore hydrodynamic problems, it is challenging to model the interaction between the free surface and moving complex geometries. This paper proposes a discrete-forcing immersed boundary method (IBM) to efficiently simulate moving solid boundaries in incompressible air–water two-phase flows. In the present work, the air–water two-phase flows are modeled using the Volume-of-Fluid (VoF) method. The present IBM is suitable for unstructured meshes. It can be used combined with body-fitted wall boundaries to model the relative motions between solid walls, which makes it flexible to use in practical applications. A field extension method is used to model the interaction between the air–water interface and the immersed boundaries. The accuracy of the method is demonstrated through validation cases, including the three-dimensional dam-break problem with an obstacle, the water exit of a circular cylinder, and a ship model advancing with a rotating semi-balanced rudder. The flow field, free-surface profile and force on the immersed boundaries (IBs) are in good agreement with experimental data and other numerical results.

**Keywords:** immersed boundary method; air–water two-phase flows; VoF method; OpenFOAM; finite volume method
