# Numerical Study of a Model and Full-Scale Container Ship Sailing in Regular Head Waves

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## Abstract

**:**

## 1. Introduction

^{9}, and it was applied to the Athena R/V for resistance, propulsion, seakeeping, and maneuvering simulations. The wall function model showed good agreement with the near-wall turbulence model and experimental data for resistance, propulsion, and boundary layer profiles. The study proved that there is a viable alternative to the near-wall turbulence model for ship hydrodynamics applications, especially for full-scale simulations where the near-wall grid resolution is prohibitive.

_{2}) emissions due to operation in waves. Their study contributes to the literature by providing a comprehensive and detailed analysis of the scale effects in ship hydrodynamics using a state-of-the-art CFD method.

## 2. Numerical Approach

## 3. Computational Strategy

#### 3.1. Ship Geometry

#### 3.2. Test Cases

#### 3.3. Computational Domain

#### 3.4. Computational Grid and Grid Convergence Test

#### 3.5. Simulation Setup

## 4. Results and Discussions

#### 4.1. Verification and Validation

#### 4.1.1. Calm Water Conditions

#### 4.1.2. Head Wave Conditions

#### 4.2. Scale Effect

_{AW}, and nondimensional heave and pitch amplitudes, as previously explained. The schematic diagram for the added wave resistance coefficient is plotted in Figure 9 for model- and full-scale ships as a function of the nondimensionalized wavelength at different wave heights.

## 5. Conclusions

- On the validation side, the CFD results were in good agreement with EFD data, especially in terms of total resistance coefficients, with a relative difference 1.22%; however, for the vertical motions, the values were slightly significant, at 6.88% for sinkage and 3.96% for trim. This indicates that the simulation is accurate, and the result deviation is acceptable in calm water condition.
- As far as the verification study is concerned, the simulations showed a monotonic convergence for all the test cases performed in this study. The numerical errors are more affected by the grid compared to the time step. The time step convergence showed that 600 iterations per wave period is sufficient for effective prediction of the simulation parameters. Further refinement of the time step had no impact on the results. On the other hand, the grid refinement was shown to have a significant effect on the results; the finer the grid, the better the accuracy. This complies with the theory and common practice in CFD simulations.
- Result validation in waves had a significant relative difference compared to that in calm water. Total resistance coefficient difference had an average value of 6.4%, while the average values for heave and pitch nondimensionalized parameters were 15.2% and 20.7%, respectively. The maximum difference for resistance was less than 10%, for heave it was within 29%, and it was within 34% for pitch motion. Though the difference between EFD and CFD was significant for ship sailing in waves, taking into consideration the fact that the study proposes a comparison between the same ship at different scales, the results were sufficient for comparative purposes, especially for the main target of the study, which is the added resistance in waves.
- Free-surface predictions showed that there was a slight difference observed between the model and full scale, especially at the stern zone and downstream; this was apparently related to the difference in form between model and full scale at the stern shoulders. Further investigation in that scope will be carried in separate research.
- The flow in the propeller plane was also investigated, revealing an obvious difference between model and full scale comparing the velocity contours and vortex formation in the propeller plane. The vortical formations are more significant at the model scale, while the contours are finely distributed, with fewer vortices at the full scale.
- A boundary layer fluctuation was observed as the ship sails in waves for both model and full scale. The effect was more accentuated at the model scale. Results showed that the change is significant, and may have an impact on the propulsion performance in waves. This is a subject of concern for the authors and shall be investigated in the future.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Comparison between CFD and EFD curves for (

**a**) total resistance coefficient in waves, (

**b**) heave amplitude, and (

**c**) pitch amplitude.

**Figure 5.**Comparison between CFD and EFD curves for (

**a**) total resistance coefficient in wave amplitude, (

**b**) heave amplitude, and (

**c**) pitch amplitude for $\lambda /{L}_{pp}=0.651$ and ${H}_{w}/{L}_{pp}=0.010$.

**Figure 6.**Comparison between CFD and EFD curves for (

**a**) total resistance coefficient in wave amplitude, (

**b**) heave amplitude, and (

**c**) pitch amplitude for $\lambda /{L}_{pp}=0.851$ and ${H}_{w}/{L}_{pp}=0.013$.

**Figure 7.**Comparison between CFD and EFD curves for (

**a**) average total resistance coefficient in waves, (

**b**) heave amplitude, and (

**c**) pitch amplitude for $\lambda /{L}_{pp}=1.150$ and ${H}_{w}/{L}_{pp}=0.020$.

**Figure 8.**Comparison between CFD and EFD curves for (

**a**) total resistance coefficient in wave amplitude, (

**b**) heave amplitude, and (

**c**) pitch amplitude for $\lambda /{L}_{pp}=1.951$ and ${H}_{w}/{L}_{pp}=0.032$.

**Figure 9.**Comparison between model-scale and full-scale curves for wave added resistance coefficient in the case of (

**a**) ${H}_{w}/{L}_{pp}=0.010$, (

**b**) ${H}_{w}/{L}_{pp}=0.020$, and (

**c**) ${H}_{w}/{L}_{pp}=0.032$.

**Figure 10.**Comparison between model-scale and full-scale curves for heave amplitude in the case of (

**a**) ${H}_{w}/{L}_{pp}=0.010$, (

**b**) ${H}_{w}/{L}_{pp}=0.020$, and (

**c**) ${H}_{w}/{L}_{pp}=0.032$.

**Figure 11.**Comparison between model-scale and full-scale curves for pitch amplitude in the case of (

**a**)${H}_{w}/{L}_{pp}=0.010$, (

**b**) ${H}_{w}/{L}_{pp}=0.020$, and (

**c**) ${H}_{w}/{L}_{pp}=0.032$.

**Figure 13.**Comparison between model scale and full scale for wave elevation in regular head waves for $\lambda /{L}_{pp}=1.951$ and ${H}_{w}/{L}_{pp}=0.032$.

**Figure 14.**Comparison between model scale and full scale for mass fraction distribution in regular head waves for $\lambda /{L}_{pp}=1.951$ and ${H}_{w}/{L}_{pp}=0.032$.

**Figure 15.**Comparison between model scale and full scale for axial velocity in the propeller in regular head waves for $\lambda /{L}_{pp}=1.951$ and ${H}_{w}/{L}_{pp}=0.032$. Section at propeller plane.

Main Particulars | Symbol | Model Scale | Full Scale |
---|---|---|---|

Length between perpendiculars | ${L}_{PP}\left[\mathrm{m}\right]$ | 6.0702 | 230.0 |

Length of waterline | ${L}_{WL}\left[\mathrm{m}\right]$ | 6.1357 | 232.5 |

Maximum beam of waterline | ${B}_{WL}\left[\mathrm{m}\right]$ | 0.8498 | 32.3 |

Draft | $T\left[\mathrm{m}\right]$ | 0.2850 | 10.8 |

Displacement volume | $\nabla \left[{\mathrm{m}}^{3}\right]$ | 0.9571 | 53,030 |

Block coefficient | ${C}_{B}\left[-\right]$ | 0.6505 | 0.6505 |

Wetted surface area | ${S}_{W}\left[{\mathrm{m}}^{2}\right]$ | 6.6978 | 9539 |

$\mathit{F}\mathit{r}$ | $\mathit{\lambda}/{\mathit{L}}_{\mathit{p}\mathit{p}}$ | ${\mathit{\lambda}}_{\mathit{m}\mathit{o}\mathit{d}\mathit{e}\mathit{l}\mathit{s}\mathit{c}\mathit{a}\mathit{l}\mathit{e}}\left[\mathbf{m}\right]$ | ${\mathit{\lambda}}_{\mathit{f}\mathit{u}\mathit{l}\mathit{l}-\mathit{s}\mathit{c}\mathit{a}\mathit{l}\mathit{e}}\left[\mathbf{m}\right]$ | ${\mathit{H}}_{\mathit{w}}/{\mathit{L}}_{\mathit{p}\mathit{p}}$ | ${\mathit{H}}_{\mathit{w}\mathit{m}\mathit{o}\mathit{d}\mathit{e}\mathit{l}\mathit{s}\mathit{c}\mathit{a}\mathit{l}\mathit{e}}\left[\mathbf{m}\right]$ | ${\mathit{H}}_{\mathit{w}\mathit{f}\mathit{u}\mathit{l}\mathit{l}-\mathit{s}\mathit{c}\mathit{a}\mathit{l}\mathit{e}}\left[\mathbf{m}\right]$ |
---|---|---|---|---|---|---|

0.261 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

0.651 | 3.949 | 149.628 | 0.010 | 0.063 | 2.349 | |

0.020 | 0.123 | 4.660 | ||||

0.032 | 0.196 | 7.426 | ||||

0.851 | 5.164 | 195.664 | 0.010 | 0.063 | 2.349 | |

0.013 | 0.078 | 2.955 | ||||

0.020 | 0.123 | 4.660 | ||||

0.032 | 0.196 | 7.426 | ||||

1.150 | 6.979 | 264.434 | 0.010 | 0.063 | 2.349 | |

0.020 | 0.123 | 4.660 | ||||

0.032 | 0.196 | 7.426 | ||||

1.951 | 11.840 | 448.618 | 0.010 | 0.063 | 2.349 | |

0.020 | 0.123 | 4.660 | ||||

0.032 | 0.196 | 7.426 |

Scale | Parameter | ${\mathit{\epsilon}}_{21}$ | ${\mathit{\epsilon}}_{32}$ | ${\mathit{R}}_{\mathit{G}}$ | ${\mathit{\delta}}_{\mathit{G}}$ | ${\mathit{U}}_{\mathit{G}}$ |
---|---|---|---|---|---|---|

MODEL-SCALE | ${R}_{{T}_{W}}$ | 6.394 | 11.023 | 0.580 | 8.832 | 8.832 |

${R}_{{T}_{W}}Amp.$ | 6.118 | 9.966 | 0.614 | 9.727 | 9.727 | |

$zAmp.$ | −0.008 | −0.012 | 0.689 | −0.018 | 0.018 | |

$\theta Amp.$ | −0.002 | −0.003 | 0.593 | −0.003 | 0.003 | |

FULL-SCALE | ${R}_{{T}_{W}}$ | 4.529 | 7.113 | 0.637 | 7.935 | 7.935 |

${R}_{{T}_{W}}Amp.$ | 7.779 | 17.827 | 0.525 | 8.584 | 8.584 | |

$zAmp.$ | −0.008 | −0.017 | 0.458 | −0.007 | 0.009 | |

$\theta Amp.$ | −0.002 | −0.008 | 0.227 | −0.001 | 0.003 |

$\mathit{F}\mathit{r}$ | $\mathit{v}\left[\mathbf{m}/\mathbf{s}\right]$ | ${\mathit{C}}_{{\mathit{T}}_{\mathit{C}}}$ | $\mathit{z}$ | $\mathit{\theta}$ |
---|---|---|---|---|

0.261 | 2.017 | −1.22% | −6.88% | 3.96% |

$\mathit{\lambda}/{\mathit{L}}_{\mathit{p}\mathit{p}}$ | ${\mathit{H}}_{\mathit{w}}/{\mathit{L}}_{\mathit{p}\mathit{p}}$ | ${\mathit{C}}_{{\mathit{T}}_{\mathit{W}}}$ | $\mathit{z}/{\mathit{\zeta}}_{\mathit{s}}$ | $\mathit{\theta}/\mathit{k}{\mathit{\zeta}}_{\mathit{s}}$ |
---|---|---|---|---|

0.651 | 0.010 | −1.87% | 14.42% | 34.51% |

0.851 | 0.013 | −4.56% | −29.11% | 33.81% |

1.150 | 0.020 | 9.95% | 14.37% | −3.61% |

1.951 | 0.032 | 9.32% | 2.74% | 10.88% |

$\mathit{\lambda}/{\mathit{L}}_{\mathit{p}\mathit{p}}$ | ${\mathit{H}}_{\mathit{w}}/{\mathit{L}}_{\mathit{p}\mathit{p}}$ | ${\mathit{C}}_{\mathit{A}\mathit{W}}$ | $\mathit{z}/{\mathit{\zeta}}_{\mathit{s}}$ | $\mathit{\theta}/\mathit{k}{\mathit{\zeta}}_{\mathit{s}}$ |
---|---|---|---|---|

0.651 | 0.010 | 3.37% | −1.09% | −10.59% |

0.020 | −6.70% | 1.24% | 5.57% | |

0.032 | −9.29% | 1.34% | −2.10% | |

0.851 | 0.010 | 4.17% | 1.89% | −1.73% |

0.020 | −5.38% | −4.14% | −3.24% | |

0.032 | −2.66% | −3.11% | −5.92% | |

1.150 | 0.010 | −7.81% | −3.08% | −6.83% |

0.020 | −9.07% | 2.32% | −5.69% | |

0.032 | −10.61% | −2.66% | −8.42% | |

1.951 | 0.010 | 11.20% | −0.23% | −0.20% |

0.020 | 2.78% | 0.84% | 0.98% | |

0.032 | 4.36% | 2.74% | 2.79% |

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## Share and Cite

**MDPI and ACS Style**

Mandru, A.; Rusu, L.; Bekhit, A.; Pacuraru, F.
Numerical Study of a Model and Full-Scale Container Ship Sailing in Regular Head Waves. *Inventions* **2024**, *9*, 22.
https://doi.org/10.3390/inventions9010022

**AMA Style**

Mandru A, Rusu L, Bekhit A, Pacuraru F.
Numerical Study of a Model and Full-Scale Container Ship Sailing in Regular Head Waves. *Inventions*. 2024; 9(1):22.
https://doi.org/10.3390/inventions9010022

**Chicago/Turabian Style**

Mandru, Andreea, Liliana Rusu, Adham Bekhit, and Florin Pacuraru.
2024. "Numerical Study of a Model and Full-Scale Container Ship Sailing in Regular Head Waves" *Inventions* 9, no. 1: 22.
https://doi.org/10.3390/inventions9010022