# Shock-Capturing Boussinesq Modelling of Broken Wave Characteristics Near a Vertical Seawall

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

**M**is the horizontal volume flux expressed as

**u**

_{α}represents the horizontal velocity at a reference elevation, where ${z}_{\alpha}=\zeta h+\beta \eta $ with $\zeta $ = −0.53 and $\beta $ = 0.47 [39].

**u**

_{2}is the depth-dependent correction of velocity at o(μ

^{2}) (μ is the ratio of water depth to wave length) and written as

**V**

_{1}and

**V**

_{2}in Equation (2) are dispersive Boussinesq terms which are expressed as

**V**

_{3}represents the vertical vorticity at o(μ

^{2}) and is written as

**R**stands for diffusive and dissipate terms which include subgrid lateral turbulent mixing and bottom friction. The bottom friction in this study is calculated by a quadratic friction law incorporating a Manning coefficient:

## 3. Laboratory Experiments

_{i}varying from 4.2 cm to 5.5 cm, wave periods T

_{i}from 1.2 s to 1.6 s and water depths at the seawall h

_{s}from 3 cm to 4.5 cm (water depth over flat bottom h from 30 cm to 31.5 cm). Incident wave conditions and water levels were designed to form broken waves with the presence of the seawall in which case waves start to break about 3/2 or one wave length from the seawall. For one wave case with H

_{i}= 4.7 cm, T

_{i}= 1.2 s, h

_{s}= 3 cm, progressive waves with the same incident wave conditions were also performed without the presence of the seawall for comparison. Figure 3 shows the time and spatial variation of root mean square of the surface water fluctuations, η

_{rms}, near the seawall for the above-mentioned case and progressive wave case. η

_{rms}was computed based on the extracted surface water level data of each single wave period so that time-variation of its values can be observed. The origin of the horizontal axis was set at the location of the seawall and the vertical axis is time. The gray color in Figure 3 represents the steel frame of wave flume, behind which no data was captured by the camera. As indicated by the red color in Figure 3a, antinodes of standing wave features can be successfully captured by the image-based measuring system. The positions of antinodes remain nearly the same, which implies steady partial standing waves form in front of the vertical seawall. It was also observed during the experiments that waves started to break around the antinode which is 3/2 wave length (i.e., around x = −140 cm) from the vertical wall. Compared to the progressive waves which start to break around x = −100 cm as indicated in Figure 3b, it can be seen that the formation of antinodes made the breaking points further from the vertical wall than the progressive wave case. More details about the image-based measuring system and data reports can be referred to Liu and Tajima [32].

## 4. Model-Data Comparison

_{i}= 4.7 cm, T

_{i}= 1.2 s, h

_{s}= 3 cm and H

_{i}= 5.5 cm, T

_{i}= 1.2 s, h

_{s}= 4.5 cm are presented here. Figure 4 and Figure 5 demonstrate the predicted and measured spatial distribution of mean water level and η

_{rms}near the seawall for these two wave cases. Mean water level and η

_{rms}were computed from the measured and predicted time series of surface fluctuations of ten wave cycles when waves reached a periodic equilibrium state. The origin of the horizontal axis was also set at the location of seawall. To quantify the performance of the model, the model skill is calculated as [44]:

_{rms}. The coarse grid size tends to underpredict the peak of antinodes since wave breaking mainly happens around the antinodes under broken waves near a vertical seawall and coarse grid size leads to more numerical dissipation similar to the underprediction of peak wave height of progressive wave. In addition, it is noted in Figure 4 that the antinode of mean water level which is located at around x = −50 cm is obviously underestimated by the present model for the case with H

_{i}= 5.5 cm, T

_{i}= 1.2 s, h

_{s}= 4.5 cm. Figure 6 shows the typical snapshots of the collision of incident broken wave and reflected wave at this location (around x = −50 cm) for the wave case with H

_{i}= 4.7 cm, T

_{i}= 1.2 s, h

_{s}= 3 cm and wave case with H

_{i}= 5.5 cm, T

_{i}= 1.2 s, h

_{s}= 4.5 cm. As shown in Figure 6, the collision of broken incident waves and reflected waves in the case with H

_{i}= 5.5 cm, T

_{i}= 1.2 s, h

_{s}= 4.5 cm, which has larger incident wave height and increasing nonlinearity, becomes more violent at this location so that more water splashes out. The splashing water could be captured by the camera during the experiments, while it could not be simulated by the present model, resulting in an underestimation of mean water level by the model. Moreover, the splashing water released from the breaking antinodes also make the surface water fluctuations more complicated so that the present model still cannot reproduce the surface water profiles very accurately. Figure 7 shows the predicted spatial distribution of root mean square of depth-averaged horizontal velocity, U

_{rms}(U = P/D, the vector

**M**in Equation (1) is defined as (P, Q) in the numerical scheme of the present model, in which case P and Q is the component of

**M**along x direction and y direction), near the seawall. As seen in this figure, U

_{rms}is relatively smaller at the positions of antinodes, which confirms that the present model surely captures the key characteristics of partial standing wave near the seawall.

_{i}= 4.7 cm, T

_{i}= 1.2 s, h

_{s}= 3 cm. The notable model of Madsen et al. [25] has been successfully applied to model cross-shore motions of regular waves including various types of breaking on plane beaches and over submerged bars. It is also reported the surface roller type breaking model demonstrates the most appropriate predictive skills of time-varying profiles of surface water compared to the other empirical breaking models [28]. As seen in Figure 8, for the present study, mean water level predicted by the surface-roller breaking model has an obvious setup after wave breaking while η

_{rms}has an obvious decreasing trend. It is also noted that the profile of η

_{rms}predicted by the surface-roller model becomes smoother than the ones of FUNWAVE-TVD and experimental data, especially at the left part of the figure (i.e., −180 < x < −100 cm). As revealed in Liu and Tajima [32], empirical breaking models including surface-roller model and eddy-viscosity model developed based on the progressive wave assumptions take the whole partial standing wave field as pure progressive waves with their breaking criteria and simulated energy dissipation happens both in incident waves and reflected waves while experimental data indicates that dissipation of reflected waves is insignificant actually. Therefore, excess dissipation computed by these breaking models leads to an obvious setup and decreasing of η

_{rms}and smaller reflected waves lead to a smoother profile of η

_{rms}. To the authors’ delight, the present model using shock-capturing method improves these features. The results presented here indicate the shock-capturing method using the ratio of wave height to the local depth as the breaking criteria is more appropriate for modeling the broken waves near a vertical seawall by eliminating the excess dissipation in reflected waves.

## 5. Numerical Experiment Results

_{i}, wave depth at the vertical seawall h

_{s}, and bed slope tanθ, on the wave kinematics (i.e., the root mean square of the surface fluctuations and depth-averaged horizontal velocity) under broken waves near vertical seawalls. Numerical experiment setup is shown in Figure 9. The internal wave maker is 20 m from the seawall and the computational domain with a total length of 26 m is applied with a 3 m sponge layer at the offshore side. The numerical experimental wave conditions for all model runs are listed in Table 3. The tested range of parameters in this numerical study is designed based on the Froude similarity with a geometric scale factor of 1:100. Therefore, at the prototype scale, incident wave heights are from 4 m to 10 m, representing the typical wave heights during typhoon and super typhoon and wave depths at the vertical seawall are from 2 m to 6 m, representing the typical water depths near the vertical seawall subjected to broken waves. For each group in Table 3, only one parameter was changed while keeping other parameters the same. Grid size dx = 0.02 m, Manning coefficient n = 0.012 were used in all numerical experiments as validated above.

_{rms}and U

_{rms}near the seawall for Cases 1–3 of Group 1 whose incident wave heights are 0.04 m, 0.07 m and 0.1 m respectively. U

_{rms}can be somehow regarded as a physically relevant indicator of the potential surrounding scour and wave impact on the seawall. As seen in Figure 10a, the positions of antinodes are nearly the same for these wave cases since they have the same dispersion relationship. It is noted both Case 1 and Case 2 start to break at the antinode around x = −0.5 m, which is 1/2 wave length from the seawall since the elevations of antinodes forming near the seawall continues to increase. For these two wave cases, larger incident wave height induces higher η

_{rms}near the seawall and higher runup on the seawall consequently. On the other hand, waves of Case 3 with the largest incident wave height, 0.1 m, in this group break further from the seawall at the antinode around x = −1.75 m, which is 3/2 wave length from the seawall. However, η

_{rms}of Case 2 and Case 3 become nearly the same between the antinode at around x = −0.5 m and the seawall and the consequent runups on the seawall of these two cases are also close. As seen in Figure 10b, similar to η

_{rms}, for Case 1 and Case 2, which break at the same location, incident wave with larger incident wave height induces higher U

_{rms}near the seawall and for Case 2 and Case 3 U

_{rms}are nearly the same near the seawall.

_{rms}and U

_{rms}near the seawall for Cases 4–6 in Group 2 whose water levels at the seawall, h

_{s}, are 6 cm, 4 cm and 2 cm respectively. h

_{s}is adjusted by changing the deep-water depth over flat bottom in this study. As seen in Figure 11a, the positions of antinodes are different for these wave cases due to different dispersion relationships. Locations of breaking points move further quickly from the seawall as h

_{s}decreases, in which case the η

_{rms}near the seawall and runups on the seawall also decrease significantly. As seen in Figure 11b, shallower h

_{s}induces smaller U

_{rms}near the seawall as well.

_{rms}and U

_{rms}near the seawall for Cases 7–10 in Group 3 whose bed slope angles, tanθ, are 1:15, 1:20, 1:25 and 1:30 respectively. As seen in Figure 12a, the positions of antinodes are slightly different for these wave cases and the locations of breaking points are all at the antinodes which are 1/2 wave length from the seawall. Spatial distributions of the η

_{rms}do not vary much as the seabed slope changes and the runups on the seawall get a little smaller when the bed slope becomes milder. As seen in Figure 12b, U

_{rms}also seem insensitive to the seabed slope.

## 6. Discussions and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Three wave types in front of vertical seawalls [1]. (

**a**) non-breaking standing waves; (

**b**) breaking waves; (

**c**) broken waves.

**Figure 3.**Time and spatial variation of the root mean square of the surface fluctuations near the seawall for (

**a**) broken waves and (

**b**) progressive waves.

**Figure 4.**Comparisons of predicted and measured spatial distribution of (

**a**) mean water level and (

**b**) η

_{rms}for the wave case with H

_{i}= 4.7 cm, T

_{i}= 1.2 s, h

_{s}= 3 cm.

**Figure 5.**Comparisons of predicted and measured spatial distribution of (

**a**) mean water level and (

**b**) η

_{rms}for the wave case with H

_{i}= 5.5 cm, T

_{i}= 1.2 s, h

_{s}= 4.5 cm.

**Figure 6.**Typical snapshots of the collision of incident broken wave and reflected wave for (

**a**) the wave case with H

_{i}= 4.7 cm, T

_{i}= 1.2 s, h

_{s}= 3 cm and (

**b**) wave case with H

_{i}= 5.5 cm, T

_{i}= 1.2 s, h

_{s}= 4.5 cm.

**Figure 7.**Predicted spatial distribution of root mean square of depth-averaged horizontal velocity near the seawall.

**Figure 8.**Comparisons of predicted spatial distribution of (

**a**) mean water level and (

**b**) η

_{rms}between Madsen et al. [25] and FUNWAVE-TVD for the wave case with H

_{i}= 4.8 cm, T

_{i}= 1.2 s, h

_{s}= 3 cm.

**Figure 10.**Computed spatial distributions of (

**a**) η

_{rms}and (

**b**) U

_{rms}near the seawall for Cases 1–3.

**Figure 11.**Computed spatial distributions of (

**a**) η

_{rms}and (

**b**) U

_{rms}near the seawall for Cases 4–6.

**Figure 12.**Computed spatial distributions of (

**a**) η

_{rms}and (

**b**) U

_{rms}near the seawall for Cases 7–10.

The Size of Foundation-Bed | Forming Conditions | Wave Type |
---|---|---|

h_{1}/h_{s} ≥ 2/3 | h_{s} ≥ 2H | Non-breaking standing waves |

h_{s} < 2H, i ≤ 1/10 | Broken waves | |

2/3 ≥ h_{1}/h_{s} > 1/3 | h_{1} ≥ 1.8H | Non-breaking standing waves |

h_{1} < 1.8H | Breaking waves | |

h_{1}/h_{s} ≤ 1/3 | h_{1} ≥ 1.5H | Non-breaking standing waves |

h_{1} < 1.5H | Breaking waves |

_{s}is the water depth at a vertical seawall, h

_{1}is the water depth above the foundation-bed, i is the seabed slope.

Category No. | Governing Equations | Scope of Application | Deficiency |
---|---|---|---|

1 | Energy-balanced equations | Phase-averaged nearshore current field | Limited to the phenomena suitable for linear wave assumptions |

2 | Navier–Stokes equations | Velocity filed and wave impact associated with wave-structure interactions | Limited to the small-scale phenomena due to the high cost of computation |

3 | Boussinesq equations with an ad-hoc dissipation term | Near shore wave processes, including shoaling, dissipation, diffraction, refraction and reflection | Limited to the progressive waves due to the applicability of empirical breaking models |

H_{i} (m) | h_{s} (cm) | tanθ | T_{i} (s) | ||
---|---|---|---|---|---|

Group 1 | Case 1 | 0.04 | 6.0 | 1:30 | 1.2 |

Case 2 | 0.07 | 6.0 | 1:30 | 1.2 | |

Case 3 | 0.1 | 6.0 | 1:30 | 1.2 | |

Group 2 | Case 4 | 0.07 | 6.0 | 1:30 | 1.2 |

Case 5 | 0.07 | 4.0 | 1:30 | 1.2 | |

Case 6 | 0.07 | 2.0 | 1:30 | 1.2 | |

Group 3 | Case 7 | 0.07 | 6.0 | 1:15 | 1.2 |

Case 8 | 0.07 | 6.0 | 1:20 | 1.2 | |

Case 9 | 0.07 | 6.0 | 1:25 | 1.2 | |

Case 10 | 0.07 | 6.0 | 1:30 | 1.2 |

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**MDPI and ACS Style**

Liu, W.; Ning, Y.; Zhang, Y.; Zhang, J.
Shock-Capturing Boussinesq Modelling of Broken Wave Characteristics Near a Vertical Seawall. *Water* **2018**, *10*, 1876.
https://doi.org/10.3390/w10121876

**AMA Style**

Liu W, Ning Y, Zhang Y, Zhang J.
Shock-Capturing Boussinesq Modelling of Broken Wave Characteristics Near a Vertical Seawall. *Water*. 2018; 10(12):1876.
https://doi.org/10.3390/w10121876

**Chicago/Turabian Style**

Liu, Weijie, Yue Ning, Yao Zhang, and Jiandong Zhang.
2018. "Shock-Capturing Boussinesq Modelling of Broken Wave Characteristics Near a Vertical Seawall" *Water* 10, no. 12: 1876.
https://doi.org/10.3390/w10121876