# Numerical Simulation of Irregular Breaking Waves Using a Coupled Artificial Compressibility Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Numerical Method

#### 3.1. Spatial Discretization

#### 3.2. Temporal Discretization

#### 3.3. Turbulence Modelling

## 4. Irregular Wave Generation

#### 4.1. Numerical Wave Tank

#### 4.2. Irregular Wave Generation Framework

#### 4.3. Model Validation

## 5. Numerical Results

#### 5.1. Submerged Trapezoid Bar

^{−1}.

#### 5.1.1. Long Waves

#### 5.1.2. Short Waves

#### 5.2. Piecewise Sloped Seabed

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Free surface elevation profiles for grid resolutions $G1,G2,G3$ against the analytical solution from linear wave theory.

**Figure 3.**Free surface elevation profiles for timesteps $dt={T}_{p}/400,{T}_{p}/800,{T}_{p}/1600$ against the analytical solution from linear wave theory.

**Figure 5.**Configuration of the numerical wave tank for the case of irregular wave propagation over a breaker bar.

**Figure 6.**Normalized wave energy density spectra for the case of irregular wave propagation over a breaker bar (${T}_{p}=2.50$ s and ${H}_{s}=0.029$ m).

**Figure 8.**Normalized wave energy density spectra for the case of irregular wave propagation over a breaker bar (${T}_{p}=1.0$ s and ${H}_{s}=0.041$ m).

**Figure 10.**Configuration of the numerical wave tank for the case of irregular wave propagation over a sloped bottom.

**Figure 11.**Reconstructed free-surface elevation profiles for the case of irregular wave propagation over a sloped bottom [40].

**Figure 12.**Normalized wave energy density spectra for the case of irregular wave propagation over a sloped bottom.

**Figure 13.**Snapshots of the eddy viscosity for the case of irregular wave propagation over a sloped bottom.

Nodes | $\mathit{d}\mathit{x}$ (m) | Cells per ${\mathit{H}}_{\mathit{s}}$ | Cells per ${\mathit{\lambda}}_{\mathit{p}}$ | |
---|---|---|---|---|

G1 | 49000 | 0.040 | 7 | 470 |

G2 | 105000 | 0.030 | 12 | 647 |

G3 | 225000 | 0.015 | 18 | 1177 |

Nodes | Cells per ${\mathit{H}}_{\mathit{s}}$ | ${\mathit{d}\mathit{x}}_{1}$ (m) | ${\mathit{d}\mathit{x}}_{2}$ (m) | |
---|---|---|---|---|

G1 | 140000 | 18 | 0.03 | 0.02 |

G2 | 240000 | 28 | 0.02 | 0.01 |

G3 | 356000 | 34 | 0.015 | 0.075 |

**Table 3.**Reduction in ${E}_{rms}$ for the three gauge positions relative to the previous grid/timestep selection (%).

WG1 | WG2 | WG3 | |
---|---|---|---|

G1 | - | - | - |

G2 | 0.085 | 0.387 | 0.931 |

G3 | 0.007 | 0.154 | 1.047 |

${T}_{p}/1000$ | - | - | - |

${T}_{p}/2000$ | 0.405 | 2.449 | 4.129 |

${T}_{p}/4000$ | 0.157 | 0.944 | 3.385 |

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**MDPI and ACS Style**

Dermatis, A.; Ntouras, D.; Papadakis, G.
Numerical Simulation of Irregular Breaking Waves Using a Coupled Artificial Compressibility Method. *Fluids* **2022**, *7*, 235.
https://doi.org/10.3390/fluids7070235

**AMA Style**

Dermatis A, Ntouras D, Papadakis G.
Numerical Simulation of Irregular Breaking Waves Using a Coupled Artificial Compressibility Method. *Fluids*. 2022; 7(7):235.
https://doi.org/10.3390/fluids7070235

**Chicago/Turabian Style**

Dermatis, Athanasios, Dimitrios Ntouras, and George Papadakis.
2022. "Numerical Simulation of Irregular Breaking Waves Using a Coupled Artificial Compressibility Method" *Fluids* 7, no. 7: 235.
https://doi.org/10.3390/fluids7070235