# Regular Wave Seakeeping Analysis of a Planing Hull by Smoothed Particle Hydrodynamics: A Comprehensive Validation

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## Abstract

**:**

## 1. Introduction

## 2. The SPH Method

#### 2.1. Mathematical Fundamentals

#### 2.2. Governing Equations

#### 2.3. Rigid Body Dynamics and SPH

#### 2.4. Boundary Conditions

## 3. Experimental Layout

## 4. Numerical Implementation

#### 4.1. Derivation of the Flow Field

#### 4.2. DualSPHysics Framework

## 5. Results

#### 5.1. Roll Decay Test

- Single-DOF numerical roll tests show faster decay terms with respect to multiple-DOF results [67], as the complete reproduction of the complex and non-linear phenomenon is impossible to achieve through strict motion assumptions;
- The SPH resolution strongly influences the results as the motion amplitude decreases, causing a noticeable increase in the numerical damping when the arc covered by the hull approaches the dp value [42].

#### 5.2. Waves over Steady Current

#### 5.3. Hull Motions

#### 5.4. Overall Performance

- (a)
- The vessel approaches the incoming wave and the lift forces increase—the keel is almost completely wetted;
- (b)
- The hull’s bow raises after the large hydrodynamic forces toward the maximum trim angle, and the wetted surface tends to its minimum;
- (c)
- The wave trough approaches as the vessel bow is completely lifted, and the water–keel interface is minimized;
- (d)
- The gravitational force guides the hull downward right before the next wave crest, causing its slamming and a consequential sudden increase in pressure on the keel panels, particularly in the chine regions.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

2D | Two-Dimensional |

3D | Three-Dimensional |

CFD | Computational Fluid Dynamics |

DBC | Dynamic Boundary Condition |

DOF | Degree of Freedom |

FR | Frame of Reference |

$F{r}_{B}$ | Beam Froude Number |

ITTC | International Towing Tank Conference |

mDBC | Modified Dynamic Boundary Condition |

NSS | Naples Systematic Series |

RANS | Reynolds-Averaged Navier–Stokes |

RAO | Response Amplitude Operator |

$Re$ | Reynolds Number |

RZ | Relaxation Zone |

SPH | Smoothed Particle Hydrodynamics |

STL | Standard Triangle Language |

WCSPH | Weakly Compressible SPH |

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**Figure 2.**Absolute and relative reference systems to describe wave propagation over a steady current.

**Figure 4.**Free roll decay test performed in DualSPHysics: sketch of the numerical modeling (

**a**) and comparison between experimental and numerical roll amplitudes with different SPH resolutions (

**b**).

**Figure 6.**Experimental and numerical hull motions (heave and pitch) for 3 of the evaluated wave patterns.

**Figure 7.**Response Amplitude Operator (RAO) for heave (

**a**) and pitch (

**b**) motion for the investigated cases.

**Figure 8.**Velocity field of the incoming waves and current and pressure distribution under the hull’s keel in different phases of the hull motion in wave for Case 4 of Table 2.

Dimensions | Unit | C2s Hull | |
---|---|---|---|

Length overall | ${L}_{OA}$ | [m] | 1.567 |

Length waterline | ${L}_{WL}$ | [m] | 1.440 |

Beam | B | [m] | 0.435 |

Beam waterline | ${B}_{WL}$ | [m] | 0.396 |

Longitudinal CG position | ${L}_{CG}/{L}_{WL}$ | [-] | 0.394 |

Vertical CG position | ${V}_{CG}/{B}_{WL}$ | [-] | 0.501 |

Mass | m | [kg] | 20.91 |

Roll gyradius | ${k}_{xx}$ | [m] | 0.52 |

Pitch gyradius | ${k}_{yy}$ | [m] | 2.79 |

**Table 2.**Simulated wave conditions (deep water waves) as reproduced in the experimental facility; the colored dot represents the cases shown in the following charts.

Case | Angular Frequency ${\mathit{\omega}}_{\mathbf{GEN}}$ [rad/s] | Wavenumber $\mathit{\kappa}$ [1/m] | Relative Wavelength $\mathit{\lambda}/{\mathit{L}}_{\mathbf{WL}}$ [-] | Amplitude H [m] | Wave Steepness $\mathit{\epsilon}=\mathit{\kappa}(\mathit{H}/2)$ [-] | Water Depth d [m] |
---|---|---|---|---|---|---|

1 | 3.456 | 1.217 | 3.585 | 0.1035 | 0.063 | 4.25 |

2 | 3.770 | 1.449 | 3.011 | 0.0440 | 0.032 | |

3 | 4.084 | 1.700 | 2.567 | 0.0741 | 0.063 | |

4 | 5.026 | 2.575 | 1.694 | 0.0489 | 0.052 | |

5 | 6.283 | 4.024 | 1.084 | 0.0313 | 0.063 |

Experimental | ⇒ | Numerical | ||
---|---|---|---|---|

Towing velocity | ${U}_{t}$ | → | Steady current | ${U}_{0}\equiv {U}_{t}$ |

Generated frequency | ${f}_{GEN}$ | → | Encountered frequency | ${f}_{ENC}={f}_{GEN}+{U}_{0}/\lambda $ |

**Table 4.**Response Amplitude Operator (RAO) percentage errors as reported in Figure 7.

$\mathit{\lambda}/{\mathit{L}}_{\mathbf{WL}}$ | Exp $\mathbf{\Delta}\mathit{z}/(\mathit{H}/2)$ | RAO Heave STAR-CCM+ ${\mathit{e}}_{\mathbf{STAR}}$ [%] | SPH ${\mathit{e}}_{\mathbf{SPH}}$ [%] | Exp $\mathit{\theta}/\mathit{\u03f5}$ | RAO Pitch STAR-CCM+ ${\mathit{e}}_{\mathbf{STAR}}$ [%] | SPH ${\mathit{e}}_{\mathbf{SPH}}$ [%] |
---|---|---|---|---|---|---|

3.58 | 1.18 | <1.0 | $-2.5$ | 1.15 | <1.0 | 5.2 |

3.01 | 1.25 | $-6.4$ | $-12.8$ | 1.17 | $-11.1$ | $-11.1$ |

2.57 | 1.16 | 13.8 | $-14.6$ | 1.10 | 15.4 | $-16.4$ |

1.69 | 0.50 | 30.0 | 8.0 | 0.49 | 26.5 | 6.1 |

1.08 | 0.18 | $-11.1$ | $-11.1$ | 0.12 | <1.0 | <1.0 |

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**MDPI and ACS Style**

Capasso, S.; Tagliafierro, B.; Mancini, S.; Martínez-Estévez, I.; Altomare, C.; Domínguez, J.M.; Viccione, G.
Regular Wave Seakeeping Analysis of a Planing Hull by Smoothed Particle Hydrodynamics: A Comprehensive Validation. *J. Mar. Sci. Eng.* **2023**, *11*, 700.
https://doi.org/10.3390/jmse11040700

**AMA Style**

Capasso S, Tagliafierro B, Mancini S, Martínez-Estévez I, Altomare C, Domínguez JM, Viccione G.
Regular Wave Seakeeping Analysis of a Planing Hull by Smoothed Particle Hydrodynamics: A Comprehensive Validation. *Journal of Marine Science and Engineering*. 2023; 11(4):700.
https://doi.org/10.3390/jmse11040700

**Chicago/Turabian Style**

Capasso, Salvatore, Bonaventura Tagliafierro, Simone Mancini, Iván Martínez-Estévez, Corrado Altomare, José M. Domínguez, and Giacomo Viccione.
2023. "Regular Wave Seakeeping Analysis of a Planing Hull by Smoothed Particle Hydrodynamics: A Comprehensive Validation" *Journal of Marine Science and Engineering* 11, no. 4: 700.
https://doi.org/10.3390/jmse11040700