**4. Conclusions**

In this paper, we extended our work of single-phase IBM for RANS simulations of fixed and moving objects in air–water two-phase turbulent flows. The VoF method with a compression term was used to simulate the air–water interface. Compared with the simulations of single-phase flows, the momentum equation for two-phase flows has the gradient of density term because of buoyancy. Due to the large ratio of density between air and water, a correct density distribution near the IB surface is necessary to represent the homogeneous Neumann wall condition of density on body-fitted boundaries. This is satisfied indirectly by setting the homogeneous Neumann boundary condition of the volume fraction *α*, and it corresponds to a neutral contact angle boundary condition *θ* = *π*/2. In the present work, the volume fraction *α* is extended into the solid domain by a field extension operation. The process involves interpolation of *α* one layer a time into the solid domain using the inverse square distance method. For turbulence modeling, the same methodology in our previous work is adopted, i.e., the Spalart–Allmaras turbulence model with an IB wall function.

A series of test cases are presented to evaluate the accuracy of the IBM. The cases consider both fixed and moving IB. Various aspects are carefully investigated, including the deformation of the air–water interface, the flow fields and the forces on the IB surfaces. Comparisons between the present results and other resourses demonstrate that the proposed IBM is capable of simulating the interaction between the air–water interface and the solid boundaries. Finally, we investigate the simulation of a KCS ship model advancing with a rotating rudder. The accuracy and flexibility of the IBM is fully demonstrated via the combined usage of the IB surfaces with unstructured body-fitted approach.

**Author Contributions:** Conceptualization, H.Y. and K.M.; methodology, H.Y.; software, H.Y. and Y.C.; validation, H.Y., Y.C. and K.M.; formal analysis, H.Y.; investigation, H.Y., Y.C., and K.M.; resources, H.Y. and K.M.; data curation, H.Y.; writing—original draft preparation, H.Y.; writing—review and editing, Y.C. and K.M.; visualization, H.Y.; supervision, K.M.; project administration, K.M.; funding acquisition, K.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Ford Motor Company and the United States Office of Naval Research.

**Acknowledgments:** The authors gratefully acknowledge the financial support of Ford Motor Company and the United States Office of Naval Research.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
