# A Coupled Artificial Compressibility Method for Free Surface Flows

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Numerical Framework

#### 3.1. Spatial Discretization

#### 3.2. Temporal Discretization

#### 3.3. Turbulence Modeling

#### 3.4. Wave Generation and Absorption

#### 3.5. Volume Fraction Boundedness

## 4. Numerical Results

#### 4.1. Numerical Wave Tank

#### 4.1.1. Wave Generation and Absorption

#### 4.1.2. Grid and Timestep Independence

#### 4.1.3. Influence of the Artificial Compressibility $\beta $ Parameter

#### 4.2. Wave Interaction with Variable Bathymetry

#### 4.3. Moonpool-Type Floater

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Discretization of the Hyperbolic System of Equations

## References

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**Figure 3.**Influence of parameters $\alpha $ and n in the generation of a wave. Wave characteristics: $T=5$ s, $H=0.05$ m, $d=0.5$ m, $\lambda =11.08$ m. (

**a**) Wave elevation inside the generation zone after 25 wave periods; (

**b**) Wave elevation at a station after the generation zone.

**Figure 4.**Influence of parameters $\alpha $ and n in the generation of a wave. Wave characteristics: $T=5$ s, $H=0.05$ m, $d=0.5$ m, $\lambda =11.08$ m. (

**a**) Wave elevation inside the damping zone after 25 wave periods; (

**b**) Wave elevation at a station before the damping zone; (

**c**) Amplitude of the 1st harmonic of the wave at various grid locations.

**Figure 5.**Grid and Timestep independence study. Wave characteristics: $T=5$ s, $H=0.05$ m, $d=0.5$, $\lambda =11.08$ m. (

**a**) Grid Independence: Wave elevation 5 wave lengths after the wave generation zone; (

**b**) Grid Independence: Wave elevation at a station 5 wave lengths after the wave generation zone. The snapshot is after 25 wave periods; (

**c**) Grid Independence: Amplitude of the 1st harmonic of the wave at various grid locations; (

**d**) Timestep Independence: Amplitude of the 1st harmonic of the wave at various grid locations.

**Figure 6.**Influence of the AC $\beta $ parameter: Wave characteristics: $T=5$ s, $H=0.05$ m, $d=0.5$ m, $\lambda =11.08$ m. (

**a**) Wave elevation 5 wave lengths after the wave generation zone. The snapshot is after 25 wave periods; (

**b**) Amplitude of the 1st harmonic of the wave at various grid locations; (

**c**) Amplitude of the 2nd harmonic of the wave at various grid locations; (

**d**) mplitude of the 3rd harmonic of the wave at various grid locations.

**Figure 8.**Free surface elevations at the wave probes. Comparison of the numerical results and experimental data (probes 1–6).

**Figure 10.**RAOs of the three motion and the space averaged free surface elevation inside the moonpool for wave steepness $H/\lambda =1/30$. ${\zeta}_{a}$ is the amplitude of the incoming wave.

**Figure 11.**Vorticity contours multiplied by the volume fraction near the moonpool-type structure. The wave steepness is $H/\lambda =1/30$ and the period is $T=0.95$ s.

**Figure 12.**Pressure contours upstream of the moonpool-type structure. The wave steepness is $H/\lambda =1/30$ and the period is $T=0.95$ s.

# Case | Forcing $\mathit{\alpha}$ | Exponent n |
---|---|---|

1 | 60 | 3.5 |

2 | 120 | 3.5 |

3 | 300 | 3.5 |

4 | 60 | 2 |

5 | 60 | 5 |

6 | 600 | 2 |

# Case | Harmonics | |||
---|---|---|---|---|

1st | 2nd | 3rd | 4th | |

1 | 0.40 | 0.45 | 0.88 | 0.87 |

2 | 0.40 | 0.42 | 0.77 | 0.91 |

3 | 0.37 | 0.39 | 0.63 | 1.08 |

4 | 0.44 | 0.38 | 0.69 | 1.29 |

5 | 0.40 | 0.49 | 1.10 | 0.87 |

6 | 0.42 | 0.37 | 0.31 | 1.72 |

# Case | Harmonics | |||
---|---|---|---|---|

1st | 2nd | 3rd | 4th | |

1 | 0.12 | 0.06 | 0.78 | 1.60 |

2 | 0.20 | 0.09 | 0.58 | 1.41 |

3 | 0.30 | 0.27 | 0.33 | 1.15 |

4 | 0.31 | 0.33 | 0.21 | 1.04 |

5 | 0.03 | 0.32 | 1.11 | 1.90 |

6 | 0.42 | 0.71 | 0.32 | 0.35 |

# Case | Forcing $\mathit{\alpha}$ | Exponent n | Length [L] |
---|---|---|---|

1 | 60 | 3.5 | 3 |

2 | 120 | 3.5 | 3 |

3 | 120 | 3.5 | 1.5 |

4 | 120 | 3.5 | 6 |

5 | 120 | 3.5 | 3 * |

6 | 250 | 3.5 | 3 * |

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**MDPI and ACS Style**

Ntouras, D.; Papadakis, G.
A Coupled Artificial Compressibility Method for Free Surface Flows. *J. Mar. Sci. Eng.* **2020**, *8*, 590.
https://doi.org/10.3390/jmse8080590

**AMA Style**

Ntouras D, Papadakis G.
A Coupled Artificial Compressibility Method for Free Surface Flows. *Journal of Marine Science and Engineering*. 2020; 8(8):590.
https://doi.org/10.3390/jmse8080590

**Chicago/Turabian Style**

Ntouras, Dimitris, and George Papadakis.
2020. "A Coupled Artificial Compressibility Method for Free Surface Flows" *Journal of Marine Science and Engineering* 8, no. 8: 590.
https://doi.org/10.3390/jmse8080590