**7. Conclusions**

This paper describes a comprehensive approach to identifying suitable mesh properties for CFD simulations of airfoil-shaped ship rudders. The goal is to build up an efficient mesh with a sufficient resolution. The suggested mesh properties are summarized in Table 30 for different research objectives. Important conclusions drawn from this work include:


**Objective Properties Settings** Accurate prediction for both *CL* and *CD* Mesh type H-mesh *Re* Actual Reynolds number whenever possible Domain size As large as possible or at least 30 *c* around and 60 *c* after the profile Chord-wise spacing More than 692 nodes for *c* = 1 m Layer-wise growth rate 1.05 Number of radial nodes 150 nodes Number of wake nodes 300 nodes Number of mesh cells No less than 4.62 × 105 Accurate *CL* only with less mesh number Mesh type C-mesh *Re* <sup>6</sup> × 106 Domain size 30 *c* around and 60 *c* after the profile Chord-wise spacing 670 nodes for *c* = 1 m Layer-wise growth rate 1.10 Number of radial nodes 150 nodes Number of wake nodes 300 nodes Number of mesh cells About 2.51 × <sup>10</sup><sup>5</sup> Acceptable *CL* and *CD* with simple meshing strategy Mesh type Hybrid-mesh *Re* <sup>6</sup> × 106 Domain size 30 *c* around and 60 *c* after the profile Chord-wise spacing 2.23 × <sup>10</sup>−<sup>4</sup> *<sup>c</sup>* Layer-wise growth rate 1.10 Element size 0.125 *c* Number of mesh cells About 7.19 × <sup>10</sup><sup>5</sup>

**Table 30.** Recommended mesh properties for different objective.

We realize that the above conclusions are based on a case study of 2D ship rudders. For unconventional rudder types, such as flap rudders and fishtail rudders, the conclusions should not be applied directly. For instance, the gap of the flap rudder and the concave part of the fishtail rudder may require an additional number of cells. What's more, attention should be paid to mesh elements in span-wise directions for 3D rudders. Considering larger calculation errors with larger angles of attack, extra efforts are also needed in applications of other CFD methods, such as Large-Eddy Simulation (LES) and Direct Numerical Simulation (DNS), which commonly require a much finer mesh than the RANS method. Therefore, further study is required to determine the most suitable mesh for different rudder types and meshing strategies in 3D configurations.

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

**Funding:** This research was funded by National Natural Science Foundation of China (62003250), Research on Intelligent Ship Testing and Verification (2018473), Program of Marine Economy Development Special Fund (Six Marine Industries) under Department of Natural Resources of Guangdong Province (Project No. GDNRC[2021]59).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data supporting reported results can be found in Ladson 1988, Abbott 1959, Jacobs 1937.

**Conflicts of Interest:** The authors declare no conflict of interest.
