# Numerical Simulation of 2-D Solitary Wave Run-Up over Various Slopes Using a Particle-Based Method

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

#### 2.1. Governing Equations

#### 2.2. Kernel Function

#### 2.3. Gradient Model

#### 2.4. Laplacian Model

#### 2.5. Incompressibility Model with a Staggered Divergence-Free Model

#### 2.6. Boundary Conditions

#### 2.6.1. Free-Surface Boundary Condition

#### 2.6.2. Wall Boundary Condition

#### 2.7. Turbulence Model

## 3. Numerical Simulations and Discussion

#### 3.1. Hydrostatic Pressure Test

^{2}, and water density is 1000 kg/m

^{3}. Viscous effects are included.

#### 3.2. Solitary Wave Run-Up (Constant Slope Angles)

#### 3.3. Solitary Wave Run-Up (Varying Slope Angle)

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${A}_{1}$, ${A}_{2}$, ${A}_{3}$, ${A}_{4}$ | coefficients for making target wave (-) |

$C$ | wave-velocity (ms^{−1}) |

${C}_{lm}$ | corrective matrix for gradient model (-) |

${C}_{s}$ | Smagorinsky constant (-) |

$d$ | number of space dimensions (-) |

$\mathit{F}$ | external force for unit mass (ms^{−2}) |

${}_{E}$, ${}_{W}$, ${}_{N}$, ${}_{S}$ | centers of the east, west, north, and south control surfaces of particle $l$ (-) |

$g$ | gravitational acceleration (ms^{−2}) |

$h$ | water depth (m) |

$H$ | maximum wave height (m) |

${}_{i}$ | $i$-direction component in the Cartesian coordination system ${x}_{i}$ (-) |

$k$ | turbulent kinetic energy (m^{2}s^{−2}) |

${k}_{0}$ | wave-number (m^{−1}) |

${}_{l}$ or ${}_{m}$ | particle index (-) |

${l}_{0}$ | particle size (m) |

${}^{n}$ | time step index (-) |

${n}^{0}$ | fixed particle number density for the incompressibility (-) |

${n}_{l}$ | particle number density of particle $l$ (-) |

${N}_{0}$ | maximum number of neighboring particles in the initial distribution (-) |

${N}_{l}$ | number of neighboring particles of particle $l$ (-) |

$p$ | pressure (Pa) |

$\mathit{r}$ | position vector between two particles (m) |

${r}_{e}$ | effective radius for particle interactions (m) |

${\mathit{r}}_{l}$, ${\mathit{r}}_{m}$ | position vector of particle $l$ and $m$ (m) |

${\mathit{r}}_{lm}$ | relative position vector between particle $l$ and $m$ (m) |

${r}_{lm}$ | distance between particle $l$ and $m$ $\left[={\left({x}_{lm}^{2}+{y}_{lm}^{2}\right)}^{1/2}=|{\mathit{r}}_{m}-{\mathit{r}}_{l}|\right]$ (m) |

$R$ | wave run-up (m) |

${S}_{ij}$ | strain rate (-) |

${S}_{p}$ | stroke of wavemaker’s paddle (cm) |

$t$ | time (s) |

$\mathit{u}$ | velocity vector (ms^{−1}) |

${\mathit{u}}_{l}^{*}$ | temporary velocity components of particle $l$ (ms^{−1}) |

${\mathit{u}}_{l}^{\prime}$ | velocity correction of particle $l$ (ms^{−1}) |

$\overline{{u}_{i}^{\prime}{u}_{j}^{\prime}}$ | Reynolds stress term for turbulent flows (m^{2}s^{−2}) |

${V}_{lm}$ | statistical volume of particle $l$ (-) |

${U}_{p}$ | speed of wavemaker’s paddle (cms^{−1}) |

$w(\text{\hspace{0.17em}})$ | kernel function (-) |

${w}_{lm}$ | kernel function between particle $l$ and $m$ $\left[=w\left(|{\mathit{r}}_{m}-{\mathit{r}}_{l}|\right)\right]$ (-) |

${x}_{lm}$ | distance between particle $l$ and $m$ in the x-direction (m) |

$X(t)$ | position of wave-maker at time $t$ (m) |

${y}_{lm}$ | distance between particle $l$ and $m$ in the y-direction (m) |

$Z$ | height of wavemaker’s paddle (m) |

${\beta}_{1}$, ${\beta}_{2}$ | parameters for detecting free surface (-) |

${\delta}_{ij}$ | Kronecker’s delta (-) |

$\Delta $ | particle-to-particle spacing (m) |

${\varphi}_{lm}$ | difference of scalar quantities $\varphi $ between particle $l$ and $m$ (m) |

${\varphi}_{l}$, ${\varphi}_{m}$ | scalar quantities $\varphi $ at particles $l$ and $m$ (-) |

$\gamma $ | blending parameter for incompressibility model (-) |

$\lambda $ | parameter that makes the increase in variance equal to that of the analytical solution (-) |

$\theta $ | slope angle of bottom (°) |

$\rho $ | density (kgm^{−3}) |

$\nu $ | kinematic viscosity (m^{2}s^{−1}) |

${\nu}_{t}$ | eddy viscosity (m^{2}/s^{−1}) |

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**Figure 4.**Comparison of the pressure field simulated by the (

**a**) Pusan-National-University-modified moving particle simulation (PNU-MPS) and (

**b**) present refined PNU-MPS adopting the staggered divergence-free model and the moving-particle wall boundary treatment after 3 s of simulations.

**Figure 5.**Time-histories of hydrostatic pressure proved at P1, which are simulated by the (

**a**) PNU-MPS and (

**b**) present refined PNU-MPS adopting the staggered divergence-free model and the moving-particle wall boundary treatment

**Figure 8.**Simulated solitary wave profile compared with analytic solution and distribution of dynamic pressure for wave height ratio H/h = 0.3 (constant slope angle = 90°).

**Figure 12.**Comparison of wave run-up profile over a slope of 26° in case of H/h = 0.35, simulated by (

**a**) fixed- and (

**b**) moving-particle wall treatments.

**Figure 13.**Comparison of time-sequential wave profiles estimated by present simulation and measured by experiment over a slope of 26° in case of H/h = 0.163 at (

**a**) t = 2.68 s, (

**b**) t = 2.88 s, (

**c**) t = 3.28 s, and (

**d**) t = 3.08 s.

**Figure 15.**Time history of wave height for case 1 with particle sizes of (

**a**) 0.005 m and (

**b**) 0.0025 m (solid lines: numerical simulated results, symbols: hydraulic experiments).

**Figure 16.**Time-sequence of turbulent intensity field during wave run-up process including wave breaking and impinging over a vertical slope for case 2.

**Figure 17.**Time history of wave height for case 2 with particle sizes of 0.005 m (solid lines: numerical simulated results, symbols: hydraulic experiments).

L (m) | Z (m) | h (m) | θ (°) | l_{0} (m) | H/h (-) |
---|---|---|---|---|---|

1.5 | 0.3 | 0.1 | 90 | 0.00125, 0.0025, 0.005, 0.010 | 0.1, 0.2, 0.3, 0.4, 0.5,0.6 |

45 | 0.0025 | 0.05, 0.1, 0.2, 0.3, 0.4, 0.5 | |||

26 | 0.15, 0.163, 0.2, 0.25, 0.3, 0.35 |

Case | Breaking | Target Wave Height (-) | A_{1} (-) | A_{2} (-) | A_{3} (-) | A_{4} (-) |
---|---|---|---|---|---|---|

1 | ▲ | 0.3 | 12.85 | 2.93 | 9.42 | −12.85 |

2 | O | 0.7 | 19.63 | 4.368 | 12.25 | −19.63 |

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**MDPI and ACS Style**

Jeong, S.-M.; Park, J.-I.; Park, J.-C.
Numerical Simulation of 2-D Solitary Wave Run-Up over Various Slopes Using a Particle-Based Method. *Water* **2019**, *11*, 462.
https://doi.org/10.3390/w11030462

**AMA Style**

Jeong S-M, Park J-I, Park J-C.
Numerical Simulation of 2-D Solitary Wave Run-Up over Various Slopes Using a Particle-Based Method. *Water*. 2019; 11(3):462.
https://doi.org/10.3390/w11030462

**Chicago/Turabian Style**

Jeong, Se-Min, Ji-In Park, and Jong-Chun Park.
2019. "Numerical Simulation of 2-D Solitary Wave Run-Up over Various Slopes Using a Particle-Based Method" *Water* 11, no. 3: 462.
https://doi.org/10.3390/w11030462