# A Comparative Study of Computational Methods for Wave-Induced Motions and Loads

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

#### 2.1. Strip Theory Method

#### 2.2. Rankine Source Boundary Element Method

#### 2.3. Field Method

#### 2.4. Green Function Boundary Element Method

## 3. Investigated Ships and Model Tests

## 4. Computational Setup

#### 4.1. Strip Theory Method

#### 4.2. Rankine Source Boundary Element Method

#### 4.3. Field Method

#### 4.4. Green Function Boundary Element Method

## 5. Results

#### 5.1. Regular Waves

#### 5.1.1. Cruise Ship

#### 5.1.2. Containership

#### 5.1.3. LNG Carrier

#### 5.1.4. Chemical Tanker

#### 5.2. Irregular Extreme Waves

#### 5.2.1. Investigated Sea States

#### 5.2.2. Time Histories

#### Cruise Ship

#### LNG Carrier

#### 5.2.3. Short-Term Statistics

#### Cruise Ship

#### Chemical Tanker and LNG Carrier

#### Containership

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | The propeller’s rate of revolution of the physical models at CEHIPAR was PID-controlled to maintain the mean forward speed. It was aimed to bypass the uncertainty of this condition influenced by the specific control mechanism. |

**Figure 2.**Overview about numerical grids for cruise ship (

**top left**), containership (

**top right**), liquid natural gas (LNG) carrier (

**bottom left**), chemical tanker (

**bottom right**).

**Figure 3.**Cruise ship: computed and measured RAOs. (

**a**) Heave motion ζ and (

**b**) pitch motions ϑ. ζ

_{a}is the wave amplitude, k the wave number, λ the wave length and L

_{pp}the ship length.

**Figure 4.**Cruise ship: computed and measured RAOs for midship vertical bending moment ${M}_{y}$. $\rho $ is the water density, g the gravity constant, B the ship’s breadth and ${L}_{pp}$ the ship length.

**Figure 5.**Containership: computed and measured RAOs: (

**a**) heave motion ζ and (

**b**) pitch motion ϑ. ζ

_{a}is the wave amplitude, k the wave number.

**Figure 6.**Containership: midship vertical bending moment ${M}_{y}$. $\rho $ is the water density, g the gravity constant, B the ship’s breadth and ${L}_{pp}$ the ship length.

**Figure 7.**LNG carrier: computed and measured RAOs. (

**a**) Heave motion ζ and (

**b**) pitch motions ϑ. ζ

_{a}is the wave amplitude, k the wave number, λ the wave length and L

_{pp}the ship length.

**Figure 8.**LNG carrier: computed and measured RAOs. (

**a**) Surge motions ζ and (

**b**) midship vertical bending moment M

_{y}(right). ρ is the water density, g the gravity constant, B the ship’s breadth and L

_{pp}the ship length.

**Figure 9.**LNG carrier: with Green functions boundary element method computed effects of water depth on (

**a**) heave and (

**b**) pitch motions.

**Figure 10.**LNG carrier: with Green functions boundary element method computed effects of water depth on (

**a**) surge motion and (

**b**) midship vertical bending moment.

**Figure 11.**Chemical tanker: computed and measured RAOs, (

**a**) heave motion ζ and (

**b**) pitch motion ϑ. ζ

_{a}is the wave amplitude, k the wave number, λ the wave length and L

_{pp}the ship length.

**Figure 12.**Chemical tanker: computed and measured RAOs, (

**a**) surge motion ζ and (

**b**) midship vertical bending moment ${M}_{y}$. $\rho $ is the water density, g the gravity constant, B the ship’s breadth, ${L}_{pp}$ the ship length.

**Figure 13.**Chemical tanker: time histories of field method computed and measured (normalized) midship vertical bending moment ${M}_{y}$. $\rho $ is the water density, g the gravity constant, B the ship’s breadth, ${L}_{pp}$ the ship length and ${\zeta}_{a}$ the free surface elevation.

**Figure 14.**Exemplary field method computed pressure and velocity distribution in the fluid domain surrounding the chemical tanker at the symmetry plane (y = 0). An extreme wave ($H\approx 15$ m and $\lambda \approx 200$ m) impinges the vessel’s bow. The orbital velocity field of the wave crest and wave troughs are cut off from the sea-bed (indicated by solid bottom-line).

**Figure 15.**Cruise ship: comparison of time histories obtained with field method and experiments for (

**a**) the free surface elevation. The lower figure (

**b**) shows the field method and strip method computed vertical bending moment amidships in comparison with model test results.

**Figure 16.**Cruise ship: comparison of time histories obtained with field method, strip method and experiments for (

**a**) heave and (

**b**) pitch motions.

**Figure 17.**LNG carrier: field method computed and measured time histories of (

**a**) free surface elevation and (

**b**) midship vertical bending moment.

**Figure 18.**LNG carrier: field method computed and measured time histories of (

**a**) heave and (

**b**) pitch motions.

**Figure 19.**LNG carrier: pressure sensor locations at the ship’s stern and bow [54].

**Figure 20.**LNG carrier: field method computed time histories of pressures at (

**a**) sensor 10 and (

**b**) sensor 34.

**Figure 21.**LNG carrier: (

**a**) field method computed and measured time histories of free surface elevation above deck and (

**b**) wave gauge arrangement on the physical model [54].

**Figure 22.**Cruise ship: short-term statistics of (

**a**) free surface elevation and (

**b**) midship vertical bending moment.

**Figure 26.**Chemical tanker vs. LNG carrier: short-term statistics of (

**a**) heave and (

**b**) pitch motions, model test results.

Cruise Ship | Containership | LNG Carrier | Chemical Tanker | |
---|---|---|---|---|

Length overall [m] | 238.00 | 349.00 | 197.10 | 170.00 |

Length bet. perpendiculars [m] | 216.80 | 333.44 | 186.90 | 161.00 |

Moulded breadth [m] | 32.20 | 42.80 | 30.38 | 28.00 |

Design draft [m] | 7.20 | 13.1 | 8.40 | 9.00 |

Block coefficient [-] | 0.65 | 0.62 | 0.73 | 0.75 |

Displacement [t] | 34,087 | 125,604 | 35,355 | 30,707 |

Mass moment of inertia (Ixx) [${\mathrm{kgm}}^{2}$] | 5.62 × 10${}^{9}$ | 3.65 × 10${}^{10}$ | 4.90 × 10${}^{9}$ | 2.73 × 10${}^{9}$ |

Mass moment of inertia (Iyy) [${\mathrm{kgm}}^{2}$] | 1.00 × 10${}^{11}$ | 8.59 × 10${}^{11}$ | 5.95 × 10${}^{10}$ | 3.30 × 10${}^{10}$ |

Longitudinal Center of Gravity [m] | 99.60 | 161.94 | 94.88 | 82.51 |

Vertical Center of Gravity [m] | 15.30 | 19.20 | 8.24 | 6.20 |

Grid | ${\mathit{H}}_{\mathit{s}}/\mathbf{\Delta}\mathit{z}$ | $\mathit{\lambda}/\mathbf{\Delta}\mathit{x}$ | ${\mathit{T}}_{\mathit{p}}$/$\mathbf{\Delta}\mathit{t}$ | Number of Cells |
---|---|---|---|---|

Rigid hulls | 10 to 20 | 70 to 160 | 800 to 950 | 600,000–1,800,000 |

Flexible hulls | 15 to 25 | 100 to 200 | 950 to 1260 | 800,000–2,000,000 |

**Table 3.**Parameters for the determination of response amplitude operators (RAOs) and applied methods.

Vessel | $\mathit{\mu}$ [deg] | v [kn] | Response Quantity | Field Method | Rankine Source Method | STRIP Method | Green Function Method | Experiment |
---|---|---|---|---|---|---|---|---|

Cruise Ship | 180 | $6.0$ | ${M}_{y}$, $\vartheta $, $\zeta $ | ✓ | ✓ | ✓ | ✓ | ✓ |

$\xi $ | ✓ | ✓ | ✓ | ✓ | ||||

Containership | 180 | $15.0$ | ${M}_{y}$, $\vartheta $, $\zeta $ | ✓ | ✓ | ✓ | ||

$\xi $ | ✓ | |||||||

LNG carrier | 180 | 0 | ${M}_{y}$, $\vartheta $, $\zeta $ | ✓ | ✓ | ✓ | ✓ | ✓ |

$\xi $ | ✓ | ✓ | ✓ | |||||

Chemical tanker | 180 | 0 | ${M}_{y}$, $\vartheta $, $\zeta $ | ✓ | ✓ | ✓ | ✓ | ✓ |

$\xi $ | ✓ | ✓ | ✓ |

Ship | ${\mathit{H}}_{\mathit{s}}$ [m] | ${\mathit{T}}_{\mathit{p}}$ [s] | $\mathit{\gamma}$ [-] | s [-] | v [kn] | ${\mathit{T}}_{\mathit{d}}$ [s] | Field Method | Rankine Source Method | STRIP Method | Experiment |
---|---|---|---|---|---|---|---|---|---|---|

Cruise ship | 10.5 | 12.22 | 3.3 | 0.075 | 6.0 | 1600 | ✓ | ✓ | ✓ | ✓ |

Containership | 12.5 | 11.80 | 5.0 | 0.089 | 15.0 | 1400 | ✓ | ✓ | ||

LNG carrier | 10.5 | 12.22 | 3.3 | 0.075 | 0 | 2700 | ✓ | ✓ | ✓ | |

Chemical tanker | 10.5 | 12.22 | 3.3 | 0.075 | 0 | 2700 | ✓ | ✓ | ✓ |

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**MDPI and ACS Style**

Ley, J.; el Moctar, O. A Comparative Study of Computational Methods for Wave-Induced Motions and Loads. *J. Mar. Sci. Eng.* **2021**, *9*, 83.
https://doi.org/10.3390/jmse9010083

**AMA Style**

Ley J, el Moctar O. A Comparative Study of Computational Methods for Wave-Induced Motions and Loads. *Journal of Marine Science and Engineering*. 2021; 9(1):83.
https://doi.org/10.3390/jmse9010083

**Chicago/Turabian Style**

Ley, Jens, and Ould el Moctar. 2021. "A Comparative Study of Computational Methods for Wave-Induced Motions and Loads" *Journal of Marine Science and Engineering* 9, no. 1: 83.
https://doi.org/10.3390/jmse9010083