# Comparison between the Lagrangian and Eulerian Approach for Simulating Regular and Solitary Waves Propagation, Breaking and Run-Up

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{t}. The eddy viscosity can be determined from a turbulent time-scale (or velocity scale) and a turbulent length-scale using the turbulent kinetic energy (k) and the turbulence dissipation rate (ε).

_{t}for a wide range of flows, is the two-equation k-ε model [50]. The two-equation k-ε model has been utilized in Shao [51,52] and De Padova et al. [53], showing a good comparison between the numerical and experimental turbulent quantities.

## 2. Lagrangian and Eulerian Methods

#### 2.1. SPH Formulation

**f**represents accelerations due to external forces, such as gravity. In the SPH formalism, the discrete form of the continuity equation at a point i can be written as follows [72]:

_{i}is the numerical speed of sound, V

_{j}is the associated volume of the j-th particle and ${\psi}_{ij}$ is the artificial dissipation term; in the present paper, the artificial dissipation term proposed by Molteni and Colagrossi [74] was chosen.

_{i}is obtained coupling the viscous dissipation in the laminar regime, as approximated by Lo and Shao [68], with a sub-particle scale (SPS) model [71]. The former can be expressed with the following formula:

_{0}is the kinematic viscosity and $\eta $ is a parameter that guarantees a non-singular operator. In the DualSPHysics solver, $\eta $ is equal to 0.001h

^{2}, with $\left\{\eta \in \mathbb{R};{x}_{i}-{x}_{j}\eta \right\}$.

#### 2.2. Non-Hydrostatic Discontinuous/Continuous Galerkin Model

## 3. Applications

#### 3.1. Experimental Set Up

#### 3.1.1. Regular Breaking Waves on a Plane Beach

_{0}, the wave period T, the offshore wavelength L

_{0}, the Irribarren number ξ

_{0}, which has been estimated in section 76, located where the bottom is flat with water depth d equal to 0.70 m. Based on the Irribarren number ξ

_{0}, the two regular tested waves were characterized by a spilling/plunging and plunging breakers, respectively.

#### 3.1.2. Solitary Waves

#### 3.2. Numerical Parameters

#### 3.2.1. SPH Parameters

#### 3.2.2. Eulerian Model Parameters

## 4. Results and Performance Analysis

#### 4.1. Spilling/Plunging and Plunging Breaking Waves on a Plane Beach

_{c}and X

_{m}are the modeled and measured values, respectively, while the bar denotes the average of the modeled and measured values. It takes into account the relative error among experimental and output values, and it will exhibit values closer to one for higher levels of accordance.

#### 4.2. Solitary Waves Propagation

#### 4.3. Performance Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Sketch of the channel with indications of the investigated sections used to calibrate the numerical model.

**Figure 4.**DualSPHysics model: Snapshots of free surface and vorticity field for (

**a**) spilling/plunging breaking wave (T1) and (

**b**) plunging breaking wave (T2).

**Figure 5.**Computed and measured surface elevation for (

**a**) spilling/plunging breaking wave (T1) and (

**b**) plunging breaking wave (T2).

**Figure 6.**Comparison of experimental and numerical skewness of surface wave elevation for: (

**a**) spilling/plunging breaking wave (T1) and (

**b**) plunging breaking wave (T2).

**Figure 7.**Comparison of experimental and numerical surface profiles of a solitary wave over a fringing reef (T3).

**Figure 8.**Comparison of experimental and numerical surface profiles of a solitary wave run-up on a plane beach (T4).

Test | H_{0} (cm) | T (s) | L_{0} (m) | d (m) | ξ_{0} | Breaking Type |
---|---|---|---|---|---|---|

T1 | 11 | 2 | 4.62 | 0.70 | 0.37 | Spilling/plunging |

T2 | 6.5 | 4 | 10.12 | 0.70 | 0.74 | plunging |

Test | Time Simulation (s) | Δx(m) | h/Δx | N_{particles} (−) |
---|---|---|---|---|

SPH_T1 | 100 | 0.01 | 1.5 | 62,892 |

SPH_T2 | 100 | 0.01 | 1.5 | 62.892 |

SPH_T3 | 13 | 0.02 | 1.5 | 50,242 |

SPH_T4 | 45 | 0.025 | 1.5 | 369,885 |

TEST | Time Simulation (s) | Δx, Δy (m) | Wave Breaking Criterion | N_{nodes}(−) |
---|---|---|---|---|

D/C Galerkin_T1 | 60 | 0.04 | $\frac{\frac{\partial \xi}{\partial t}}{\sqrt{g\left|h\right|}}>0.2$ | 5607 |

D/C Galerkin_T2 | 60 | 0.04 | $\frac{\frac{\partial \xi}{\partial t}}{\sqrt{g\left|h\right|}}>0.3$ | 5607 |

D/C Galerkin_T3 | 25 | 0.05 | $\frac{\xi}{h}>0.8$ | 4505 |

D/C Galerkin_T4 | 40 | 0.4 | $\frac{\xi}{h}>0.8$ | 3857 |

I_{W} | |||
---|---|---|---|

Sect. | 49 | 48 | 45 |

SPH_T1 | 0.92 | 0.90 | 0.89 |

D/C Galerkin_T1 | 0.82 | 0.78 | 0.72 |

SPH_T2 | 0.95 | 0.89 | 0.90 |

D/C Galerkin_T2 | 0.91 | 0.85 | 0.75 |

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**MDPI and ACS Style**

De Padova, D.; Calvo, L.; Carbone, P.M.; Maraglino, D.; Mossa, M.
Comparison between the Lagrangian and Eulerian Approach for Simulating Regular and Solitary Waves Propagation, Breaking and Run-Up. *Appl. Sci.* **2021**, *11*, 9421.
https://doi.org/10.3390/app11209421

**AMA Style**

De Padova D, Calvo L, Carbone PM, Maraglino D, Mossa M.
Comparison between the Lagrangian and Eulerian Approach for Simulating Regular and Solitary Waves Propagation, Breaking and Run-Up. *Applied Sciences*. 2021; 11(20):9421.
https://doi.org/10.3390/app11209421

**Chicago/Turabian Style**

De Padova, Diana, Lucas Calvo, Paolo Michele Carbone, Domenico Maraglino, and Michele Mossa.
2021. "Comparison between the Lagrangian and Eulerian Approach for Simulating Regular and Solitary Waves Propagation, Breaking and Run-Up" *Applied Sciences* 11, no. 20: 9421.
https://doi.org/10.3390/app11209421