# Numerical Simulation of a Sandy Seabed Response to Water Surface Waves Propagating on Current

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Fluid Sub-Model

^{3}); ${g}_{i}$ is the gravitational acceleration (m/s

^{2}) and $\mu $ is the molecular viscosity (Pa·s).

^{2}/s

^{2}), and ${\delta}_{ij}$ is the Kronecker delta. The dissipation rate of TKE ($\epsilon $) is defined as:

^{−1}) at a given distance (${l}_{k}$, m) from the starting side of the wave-absorbing layer toward the open boundary, and $\langle {u}_{fi}\rangle {}_{str}$ is the background stream velocity (m/s) that is exempted from damping. The coefficient ${k}_{d}$ is estimated using:

_{0}and k

_{1}(${k}_{1}\ge {k}_{0}$) are the values of ${k}_{d}$ at the starting side of the sponge layer and the open boundary, respectively. The distance ${l}_{k}$ is a variable measured from the starting side of the wave-absorbing layer towards the open boundary. Finally, d is the length of the sponge layer (m). In the present study, k

_{0}= 0, k

_{1}= 1, and d = 2L

_{w}, where ${L}_{w}$ is the incident wavelength.

#### 2.2. Seabed Sub-Model

_{s}is the pore pressure in seabed, ${\gamma}_{w}$ is the unit weight of water, ${n}_{s}$ is the soil porosity and ${k}_{s}$ is the seabed permeability. For a plane strain problem, the volume strain (${\epsilon}_{s}$) and the compressibility of pore fluid (${\beta}_{s}$) are, respectively, defined as follows:

^{9}Pa in the present study), ${P}_{wo}$ is the absolute water pressure, and ${S}_{r}$ is the seabed degree of saturation.

#### 2.3. Boundary Treatment

#### 2.4. Numerical Scheme

^{−4}m/s. The shear modulus ($G$), Poisson’s ratio ($\nu $) and porosity (${n}_{s}$) are set as 1.0 × 10

^{7}N/m

^{2}, 0.333 and 0.3, respectively. As an elastic seabed, the Young’s modulus ($E$) is calculated by:

_{w}) to diminish the influence of fixed boundary, as suggested by Ye and Jeng [13]. Hence, in the present model, the seabed length is set as 3L

_{w}= 213 m along with a seabed thickness of 30 m. Correspondingly, the wave model length is set as 5L

_{w}in which the downstream 2L

_{w}long region is set as a sponge layer to minimize wave reflection.

## 3. Results

#### 3.1. Model Validation

#### 3.2. Hydrodynamics of WCSI

#### 3.3. Seabed Response

#### 3.4. Seabed Liquefaction

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Validation of the fluid sub-model against the experiment data of Umeyama [41].

**Figure 4.**Time series of (

**a**) free surface elevation and (

**b**) wave pressure at location O for various current velocities.

**Figure 5.**Seabed response to combined wave and current loading when current velocity v

_{c}= 0.5 m/s.

**Figure 6.**Seabed response to combined wave and current loading when ${v}_{c}=1\mathrm{m}/\mathrm{s}$ in Ye and Jeng [13].

**Figure 7.**Vertical distributions of minimum wave-induced pore pressure with various current velocities.

Module | Parameter | Notation | Magnitude | Unit |
---|---|---|---|---|

Wave | Water Depth | ${h}_{w}$ | 10 | m |

Wave Height | H | 3 | m | |

Wave Period | T | 8 | s | |

Wavelength | ${L}_{w}$ | 71 | m | |

Current | Velocity | v_{c} | 0, 0.25, 0.5, 0.75, 1 | m/s |

Seabed | Permeability | k_{s} | 1.0 × 10^{−4} | m/s |

Degree of Saturation | S_{r} | 0.985 | - | |

Shear Modulus | G | 1.0 × 10^{7} | N/m^{2} | |

Poisson’s Ratio | $\nu $ | 0.333 | - | |

Porosity | n_{s} | 0.3 | - |

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**MDPI and ACS Style**

Tong, D.; Liao, C.; Chen, J.; Zhang, Q.
Numerical Simulation of a Sandy Seabed Response to Water Surface Waves Propagating on Current. *J. Mar. Sci. Eng.* **2018**, *6*, 88.
https://doi.org/10.3390/jmse6030088

**AMA Style**

Tong D, Liao C, Chen J, Zhang Q.
Numerical Simulation of a Sandy Seabed Response to Water Surface Waves Propagating on Current. *Journal of Marine Science and Engineering*. 2018; 6(3):88.
https://doi.org/10.3390/jmse6030088

**Chicago/Turabian Style**

Tong, Dagui, Chencong Liao, Jinjian Chen, and Qi Zhang.
2018. "Numerical Simulation of a Sandy Seabed Response to Water Surface Waves Propagating on Current" *Journal of Marine Science and Engineering* 6, no. 3: 88.
https://doi.org/10.3390/jmse6030088