# CFD Study on the Influence of Exostructure Elements on the Resistance of a Submarine

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Analysis of Submarine Resistance

#### 2.1. Numerical Model

- Discontinuity of the density field $\rho $:$$\left[\rho \right]={\rho}_{a}-{\rho}_{w},$$
- Continuity of the pressure field p, arising from the dynamic free surface boundary condition:$$\left[p\right]=0,$$
- Continuity of the pressure gradient normalised by density, which is a result of applying the kinematic free surface boundary condition to the momentum equation [11]:$$\left[\frac{\nabla p}{\rho}\right]=0.$$Applying the above equation to various discretisation schemes yields the interface-sensitised schemes for the pressure field. These eliminate the “spurious air velocity” problem, which is manifested in the phase-averaged approach that is more commonly used in numerical hydrodynamics. The reader is directed to [11] for more details.

#### 2.2. Description of the Submarine

#### 2.3. Simulated Conditions

- Geometry 1: Smooth cylindrical hull, representing a simplified geometry of the pressure hull, Figure 2;
- Geometry 2: Cylindrical pressure hull with ring protrusions placed at 2.4 m intervals, with thickness of 240 mm and 60 mm height, Figure 3;
- Geometry 3: Full geometry of the pressure hull and exostructure, Figure 4.

#### 2.4. Description of Simulation Set Up

- Geometry 1: 2,837,383 cells;
- Geometry 2: 2,853,694 cells;
- Geometry 3: 11,091,951 cells.

#### 2.5. Numerical Results

- ${F}_{t}$: Total resistance;
- ${F}_{p}$: Pressure resistance;
- ${F}_{v}$: Viscous resistance;
- ${C}_{t}=\frac{{F}_{t}}{1/2\rho S{V}^{2}}$: Total resistance coefficient;
- ${C}_{p}=\frac{{F}_{p}}{1/2\rho S{V}^{2}}$: Pressure resistance coefficient;
- ${C}_{v}=\frac{{F}_{v}}{1/2\rho S{V}^{2}}$: Viscous resistance coefficient.

## 3. Discussion

#### 3.1. Comparison of Resistance between Geometries

#### 3.2. Free Surface Effects

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

RANSE | Reyndols Averaged Navier–Stokes Equations |

GFM | Ghost Fluid method |

LS | Level Set |

ITTC | International Towing Tank Conference |

## References

- Jiao, H.; Fu, W.; Zhang, L.; Zhao, C.; Zheng, Z. Simulation research and optimization design on towed system of manned submersible. In Proceedings of the 2018 IEEE 8th International Conference on Underwater System Technology: Theory and Applications (USYS), Wuhan, China, 1–3 December 2018. [Google Scholar]
- Jiang, Z.; Lu, B.; Wang, B.; Cui, W.; Zhang, J.; Luo, R.; Luo, G.; Zhang, S.; Mao, Z. A Prototype Design and Sea Trials of an 11,000 m Autonomous and Remotely-Operated Vehicle Dream Chaser. J. Mar. Sci. Eng.
**2022**, 10, 812. [Google Scholar] [CrossRef] - Wei, Z.F.; Wang, M.Y.; Yu, Q.; Yang, S.L. A Design of Resistance Optimization System for Unmanned Submersible Vehicle Based on Response Surface Method. In Proceedings of the 2014 International Conference on Mechanics and Civil Engineering; Chen, W., Wu, X., Xu, J., Eds.; AER-Advances in Engineering Research; Atlantis Press: Amsterdam, The Netherlands, 2014; Volume 7, pp. 128–133. [Google Scholar]
- Kotb, M.A.; Banawan, A.; Ahmed, Y.M. Flow field characteristics past a slow speed tourist submarine and their environmental impacts. In Proceedings of the 7th International Conference on Role of Engineering towards a Better Environment, Alexandria, Egypt, 20–22 December 2008. [Google Scholar]
- Phillips, A.; Furlong, M.; Turnock, S.R. The use of computational fluid dynamics to assess the hull resistance of concept autonomous underwater vehicles. In Proceedings of the OCEANS 2007—Europe, Aberdeen, UK, 18–21 June 2007. [Google Scholar] [CrossRef] [Green Version]
- Khan, S.A.; Fatepurwala, M.A.; Pathan, K.N.; Dabeer, P.S.; Baig, M.A.A. CFD analysis of human powered submarine to minimize drag. Int. J. Mech. Prod. Eng. Res. Dev.
**2018**, 8, 1057–1066. [Google Scholar] [CrossRef] - Karim, M.M.; Rahman, M.M.; Alim, M.A. Performance of SST k-ω turbulence model for computation of viscous drag of axisymmetric underwater bodies. Int. J. Eng. Trans. B Appl.
**2011**, 24, 139–146. [Google Scholar] - Chen, J.; Lv, B.; Peng, L.; Huang, B. Study on resistance characteristics of submarine near water surface. MATEC Web Conf.
**2022**, 355, 01002. [Google Scholar] [CrossRef] - Ćorak, M.; Šperanda, Z.; Čokić, J.; Parunov, J. Structural analysis of tourist submarine with acrylic hull. In Sustainable Development and Innovations in Marine Technologies; Guedes Soares, S., Ergin, S., Eds.; CRC Presss: London, UK, 2022; pp. 381–387. [Google Scholar]
- Vukčević, V.; Jasak, H. A Conservative Level Set Method for Interface Capturing in Two-Phase flows. In Proceedings of the 11th World Congress on Computational Mechanics (WCCM XI), Barcelona, Spain, 20–25 July 2014; pp. 1082–1095. [Google Scholar]
- Vukčević, V. Numerical Modelling of Coupled Potential and Viscous Flow for Marine Applications. Ph.D. Thesis, Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia, 2016. [Google Scholar]
- Vukčević, V.; Jasak, H.; Gatin, I. Implementation of the Ghost Fluid Method for free surface flows in polyhedral Finite Volume framework. Comput. Fluids
**2017**, 153, 1–19. [Google Scholar] [CrossRef] - Menter, F.R. Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications. AIAA J.
**1994**, 32, 1598–1605. [Google Scholar] [CrossRef]

**Figure 2.**Geometry 1: Smooth cylindrical hull, representing a simplified geometry of the pressure hull.

**Figure 3.**Geometry 2: Cylindrical hull with ring protrusions, representing the geometry of the pressure hull with rings.

**Figure 4.**Geometry 3: Geometry of the pressure hull and exostructure used in the CFD simulation. The geometry is divided into a number of different patches denoted on the figure.

**Figure 5.**Geometry 3: Geometry of the pressure hull and exostructure used in the CFD simulation, bottom view.

**Figure 10.**Resistance of individual patches of Geometry 3 when sailing at the free surface (0 m depth).

**Figure 14.**Pressure distribution and free surface geometry at 1.5 and 6 knots for Geometry 3 sailing at the free surface.

**Figure 16.**Pressure distribution for Geometry 1, 2, and 3, from left to bottom right. Pressure is shown at 4.5 knots for Geometry 1 and 2, and at 3.0 knots for Geometry 3.

Length over all | ${L}_{OA}$ | 25.115 m |

Maximal breadth | B | 4.75 m |

Depth | D | 4.74 m |

Draft while surfaced | T | 3.42 m |

Dry weight | m | 142 tons |

Geometry No. | 1 | 2 | 3 |
---|---|---|---|

Speeds in knots | 1.5, 3, 4.5 | 1.5, 3, 4.5 | 1.5, 3, 6 |

Froude numbers | 0.05, 0.1, 0.15 | 0.05, 0.1, 0.15 | 0.05, 0.1, 0.2 |

Reynolds numbers | $1.94\times {10}^{7}$, $3.88\times {10}^{7}$, $5.81\times {10}^{7}$ | $1.94\times {10}^{7}$, $3.88\times {10}^{7}$, $5.81\times {10}^{7}$ | $1.94\times {10}^{7}$, $3.88\times {10}^{7}$, $7.75\times {10}^{7}$ |

Sailing depths | 6 m, 40 m | 0 m, 6 m, 40 m | 0 m, 6 m, 40 m |

Speed, kn | 1.5 | 3 | 6 | ||||||

Fr | 0.05 | 0.10 | 0.20 | ||||||

Depth, m | 0 | 6 | 40 | 0 | 6 | 40 | 0 | 6 | 40 |

${F}_{t},N$ | 3374.8 | 4440.62 | 4407.86 | 12,432.37 | 17,036.13 | 16,650.25 | 51,241.67 | 66,722.5 | 64,331.01 |

${F}_{p},N$ | 3200.03 | 4221.6 | 4189.86 | 11,845.72 | 16,232.87 | 15,887.88 | 49,182.31 | 63,729.13 | 61,480.15 |

${F}_{v},N$ | 174.77 | 219.02 | 218 | 586.65 | 803.26 | 762.36 | 2059.37 | 2993.38 | 2850.87 |

${C}_{t}\times {10}^{3},-$ | 19.64 | 19.11 | 18.97 | 18.03 | 18.33 | 17.92 | 18.35 | 17.95 | 17.31 |

${C}_{p}\times {10}^{3},-$ | 18.62 | 18.17 | 18.04 | 17.17 | 17.47 | 17.1 | 17.62 | 17.15 | 16.54 |

${C}_{v}\times {10}^{3},-$ | 1.02 | 0.94 | 0.94 | 0.85 | 0.86 | 0.82 | 0.74 | 0.81 | 0.77 |

$S,{m}^{2}$ | 563.12 | 761.38 | 761.38 | 565.11 | 761.38 | 761.38 | 571.9 | 761.38 | 761.38 |

Speed, kn | 1.5 | 3 | 4.5 | ||||||

Fr | 0.05 | 0.10 | 0.15 | ||||||

Depth, m | 0 | 6 | 40 | 0 | 6 | 40 | 0 | 6 | 40 |

${F}_{t},N$ | 774.92 | 705.25 | 674.41 | 3102.88 | 2780.04 | 2660.81 | 7348.8 | 6149.9 | 5879.6 |

${F}_{p},N$ | 693.07 | 600.71 | 572.78 | 2820.89 | 2409.24 | 2296.34 | 6787.78 | 5379.74 | 5127.37 |

${F}_{v},N$ | 81.85 | 104.54 | 101.63 | 281.99 | 370.8 | 364.47 | 561.01 | 770.17 | 752.23 |

${C}_{t}\times {10}^{3},-$ | 14.59 | 13.14 | 12.56 | 15.81 | 12.95 | 12.39 | 16.19 | 12.73 | 12.17 |

${C}_{p}\times {10}^{3},-$ | 13.05 | 11.19 | 10.67 | 14.38 | 11.22 | 10.7 | 14.95 | 11.14 | 10.61 |

${C}_{v}\times {10}^{3},-$ | 1.54 | 1.95 | 1.89 | 1.44 | 1.73 | 1.7 | 1.24 | 1.59 | 1.56 |

$S,{m}^{2}$ | 174.1 | 175.91 | 175.91 | 160.78 | 175.91 | 175.91 | 165.31 | 175.91 | 175.91 |

Speed, kn | 1.5 | 3 | 4.5 | |||

Fr | 0.05 | 0.10 | 0.15 | |||

Depth, m | 6 | 40 | 6 | 40 | 6 | 40 |

${F}_{t},N$ | 296.86 | 292.09 | 1153.25 | 1091.43 | 2388.76 | 2331.96 |

${F}_{p},N$ | 144.05 | 140.35 | 604.53 | 545.42 | 1222.67 | 1175.99 |

${F}_{v},N$ | 152.82 | 151.74 | 548.72 | 546.01 | 1166.09 | 1155.97 |

${C}_{t}\times {10}^{3},-$ | 5.6 | 5.51 | 5.44 | 5.15 | 5.01 | 4.89 |

${C}_{p}\times {10}^{3},-$ | 2.72 | 2.65 | 2.85 | 2.57 | 2.56 | 2.47 |

${C}_{v}\times {10}^{3},-$ | 2.88 | 2.86 | 2.59 | 2.58 | 2.45 | 2.42 |

$S,{m}^{2}$ | 173.6 | 173.6 | 173.6 | 173.6 | 173.6 | 173.6 |

Speed, kn | 1.5 | 3 | 4.5 | 6 |

${C}_{ITTC}\times {10}^{3},-$ | 2.79 | 2.42 | 2.28 | 2.23 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gatin, I.; Čokić, J.; Romić, D.; Parunov, J.
CFD Study on the Influence of Exostructure Elements on the Resistance of a Submarine. *J. Mar. Sci. Eng.* **2022**, *10*, 1542.
https://doi.org/10.3390/jmse10101542

**AMA Style**

Gatin I, Čokić J, Romić D, Parunov J.
CFD Study on the Influence of Exostructure Elements on the Resistance of a Submarine. *Journal of Marine Science and Engineering*. 2022; 10(10):1542.
https://doi.org/10.3390/jmse10101542

**Chicago/Turabian Style**

Gatin, Inno, Juvel Čokić, Darjan Romić, and Joško Parunov.
2022. "CFD Study on the Influence of Exostructure Elements on the Resistance of a Submarine" *Journal of Marine Science and Engineering* 10, no. 10: 1542.
https://doi.org/10.3390/jmse10101542