# Modeling of Breaching Due to Overtopping Flow and Waves Based on Coupled Flow and Sediment Transport

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations of Flow and Sediment Transport in Breaching Process

_{t}is the non-equilibrium adaptation length or saturation length of sediment transport [15,16,17]. The total-load sediment transport capacity and the adaptation length are determined by the same method proposed by Wu [15].

## 3. Numerical Methods

#### 3.1. Model Discretization

**S**

_{i, j}represents the source terms evaluated at the cell center.

#### 3.2. Nonnegative Reconstruction of Riemann State

_{x})

_{i}

_{, j}, (h

_{x})

_{i}

_{, j}, ((hu)

_{x})

_{i, j}, and ((hv)

_{x})

_{i, j}represent the numerical derivatives of the flow variable at cell (i, j). In order to preserve the stability of the model, a one-parameter family of generalized min mod limiters is used to calculate the numerical derivatives. For example,

_{x})

_{i, j}, ((hu)

_{x})

_{i, j}, and ((hv)

_{x})

_{i, j}can be obtained in a similar way. Similarly, the right-hand side values at the interface (i + 1/2, j) can be calculated by

^{−6}m in all our simulations).

_{b})

_{i}

_{+1/2, j}rather than the actual water surface elevation η in cell (i, j). However, fluxes at the cell interface (i − 1/2, j) are estimated using the actual water surface elevation η in cell (i, j), which produces a net flux in cell (i, j) and violates the well-balanced property of the scheme. The local bed modification is thus implemented to overcome this difficulty following Liang [21]. As show in Figure 2, the difference between the actual and fake water at interface (i + 1/2, j) is calculated by

#### 3.3. Central Upwind Scheme

#### 3.4. Treatments of the Source Terms

#### 3.5. Calculation of Sediment Transport

#### 3.6. Stability Criterion and Boundary Conditions

_{CFL}represents the Courant number, and a and b are given by:

## 4. Model Verification

#### 4.1. Case 1: Tsunami Run-Up onto a Complex Three-Dimensional (3D) Beach

**Figure 3.**Three-dimensional view of bed elevation and the locations of the three gauges. Bed elevation is presented in the colored contours and dots represent three recoded gauges. Red color represents high land and blue represents sea side.

**Figure 4.**The simulated inundation map at different times in a 3D view. Numerical results of water depth are presented in the colored contours, while grey color represents bed elevation. (

**a**–

**f**) represent water depth at the simulation times of 10 s, 12 s, 14, 16, 17 s, and 18 s, respectively.

**Figure 5.**Comparison between computed and measured water elevations at different gauges (η represents water elevation). Numerical results are plotted in red lines and experimental measurements are plotted in black circles. (

**a**–

**c**) represent the water elevation varying with time at gauges ch5, ch7, and ch9, respectively.

#### 4.2. Case 2: One-Dimensional Dam Break Flow over Moveable Bed

^{3}and a porosity of 0.28. The initial water depth in the upstream was 0.1 m and the downstream was dry. At the beginning of the experiment, the gate was instantly removed to release the water, which generated flood propagation and caused erosion downstream.

^{−1/3}. The adaptation length L

_{b}and adaptation coefficient α were 0.2 m and 2.0, respectively. The settling velocity of sand used in the model was 7.6 cm/s based on the experimental data. The time step was adapted by Equation (37).

**Figure 7.**Comparison between simulated and measured water elevation and bed profiles. (

**a**–

**c**) compare the results at different times of 0.3 s, 0.4 s, and 0.5 s, respectively.

## 5. Numerical Investigation of 2D Dam Breaching Processes with Overtopping Flow and Waves

#### 5.1. Breaching Processes Due to Overtopping Flow

^{3}/s was specified at the flume entrance, while a free overall flow condition was applied at the end of the flume.

**Figure 11.**Bed changes and water depth with no wave at different times. The simulated water depths are plotted in the colored contours in the 3D topography view. The simulated changes of bed elevation and breach width and depth can be seen from (

**a**) 139 s and (

**b**) 712 s.

#### 5.2. Breaching Processes Due to Wave Overtopping

#### 5.3. Discussion of Results

**Figure 12.**Bed changes and water depths with a wave at different times. The simulated water depths are plotted in the colored contours in the 3D topography view: (

**a**) bed elevation and water depth at 22.8s, the sandy dam was eroded along the entire length and sediment was deposited in the downstream; (

**b**) bed elevation and water depth at 28.2 s, the height of the dam was reduced quickly and erosion in the lower part of the dam downslope was increased.

**Figure 13.**Comparison of bed changes along the flow direction at different times with and without waves. Numerical results of bed elevation along the central flow direction are plotted in the colored lines: (

**a**) bed elevation changing with time in the case with no wave; (

**b**) bed elevation changing with time considering overtopping waves.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Donnelly, C.; Kraus, N.; Larson, M. State of knowledge on measurement and modeling of coastal overwash. J. Coast. Res.
**2006**, 22, 965–991. [Google Scholar] [CrossRef] - Kobayashi, N.; Buck, M.; Payo, A.; Johnson, B.D. Berm and dune erosion during a storm. J. Waterw. Port Coast. Ocean Eng.
**2009**, 135, 1–10. [Google Scholar] [CrossRef] - Figlus, J.; Kobayashi, N.; Gralher, C.; Iranzo, V. Wave overtopping and overwash of dunes. J. Waterw. Port Coast. Ocean Eng.
**2010**, 137, 26–33. [Google Scholar] [CrossRef] - Wu, W.M.; Altinakar, M.S.; Al-Riffai, M.; Bergman, N.; Bradford, S.F.; Cao, Z.X.; Chen, Q.J.; Constantinescu, S.G.; Duan, J.G.; Gee, D.M.; et al. Earthen embankment breaching. J. Hydraul. Eng.
**2011**, 137, 1549–1564. [Google Scholar] - Harten, A.; Lax, P.D.; Leer, B.V. On upstream differencing and godunov-type schemes for hyperbolic conservation laws. SIAM Rev.
**1983**, 25, 35–61. [Google Scholar] [CrossRef] - Toro, E.F. Shock-Capturing Methods for Free-Surface Shallow Flows; John Wiley: Chichester, Britain, 2001. [Google Scholar]
- Wu, W.; Marsooli, R.; He, Z. Depth-averaged two-dimensional model of unsteady flow and sediment transport due to noncohesive embankment break/breaching. J. Hydraul. Eng.
**2011**, 138, 503–516. [Google Scholar] [CrossRef] - Li, S.; Duffy, C.J. Fully coupled approach to modeling shallow water flow, sediment transport, and bed evolution in rivers. Water Resour. Res.
**2011**, 47. [Google Scholar] [CrossRef] - Xia, J.; Lin, B.; Falconer, R.A.; Wang, G. Modelling dam-break flows over mobile beds using a 2d coupled approach. Adv. Water Resour.
**2010**, 33, 171–183. [Google Scholar] [CrossRef] - Wu, W.; Wang, S.S. One-dimensional explicit finite-volume model for sediment transport. J. Hydraul. Eng.
**2008**, 46, 87–98. [Google Scholar] [CrossRef] - Cao, Z.; Pender, G.; Wallis, S.; Carling, P. Computational dam-break hydraulics over erodible sediment bed. J. Hydraul. Eng.
**2004**, 130, 689–703. [Google Scholar] [CrossRef] - Capart, H.; Young, D.L. Formation of a jump by the dam-break wave over a granular bed. J. Fluid Mech.
**1998**, 372, 165–187. [Google Scholar] [CrossRef] - Odd, N.; Roberts, W.; Maddocks, J. Simulation of lagoon breakout. In Proceedings of the 26th Congress-International Association for Hydraulic Research, London, UK, 11–15 September 1995; pp. 92–97.
- Kurganov, A.; Levy, D. Central-upwind schemes for the saint-venant system. ESAIM: Math. Modell. Numer. Anal.
**2002**, 36, 397–425. [Google Scholar] [CrossRef] - Wu, W. Computational River Dynamics; Taylor & Francis: London, UK, 2008. [Google Scholar]
- Pähtz, T.; Parteli, E.J.; Kok, J.F.; Herrmann, H.J. Analytical model for flux saturation in sediment transport. Phys. Rev. E
**2014**, 89, 052213. [Google Scholar] [CrossRef] - Pähtz, T.; Kok, J.F.; Parteli, E.J.; Herrmann, H.J. Flux saturation length of sediment transport. Phys. Rev. Lett.
**2013**, 111, 218002. [Google Scholar] [CrossRef] [PubMed] - Wu, W.; Wang, S.S.Y.; Jia, Y. Nonuniform sediment transport in alluvial rivers. J. Hydraul. Res.
**2000**, 38, 427–434. [Google Scholar] [CrossRef] - Wu, W.; Wang, S.S. A depth-averaged two-dimensional numerical model of flow and sediment transport in open channels with vegetation. Riparian Veg. Fluv. Geomorphol.
**2004**, 2004, 253–265. [Google Scholar] - Richardson, J.T. Sedimentation and fluidisation: Part i. Trans. Inst. Chem. Eng.
**1954**, 32, 35–53. [Google Scholar] [CrossRef] - Liang, Q. Flood simulation using a well-balanced shallow flow model. J. Hydraul. Eng.
**2010**, 136, 669–675. [Google Scholar] [CrossRef] - Kurganov, A.; Petrova, G. A second-order well-balanced positivity preserving central-upwind scheme for the saint-venant system. Commun. Math. Sci.
**2007**, 5, 133–160. [Google Scholar] [CrossRef] - Audusse, E.; Bouchut, F.; Bristeau, M.-O.; Klein, R.; Perthame, B. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput.
**2004**, 25, 2050–2065. [Google Scholar] [CrossRef] - Liang, Q.; Marche, F. Numerical resolution of well-balanced shallow water equations with complex source terms. Adv. Water Resour.
**2009**, 32, 873–884. [Google Scholar] [CrossRef] - Begnudelli, L.; Sanders, B.F. Unstructured grid finite-volume algorithm for shallow-water flow and scalar transport with wetting and drying. J. Hydraul. Eng.
**2006**, 132, 371–384. [Google Scholar] [CrossRef] - Wu, G.F.; He, Z.G.; Liu, G.H. Development of a cell-centered godunov-type finite volume model for shallow water flow based on unstructured mesh. Math. Problems Eng.
**2014**, 2014. [Google Scholar] [CrossRef] - Yoon, T.H.; Kang, S.-K. Finite volume model for two-dimensional shallow water flows on unstructured grids. J. Hydraul. Eng.
**2004**, 130, 678–688. [Google Scholar] [CrossRef] - Hu, K.; Mingham, C.G.; Causon, D.M. Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations. Coast. Eng.
**2000**, 41, 433–465. [Google Scholar] [CrossRef] - Song, L.; Zhou, J.; Guo, J.; Zou, Q.; Liu, Y. A robust well-balanced finite volume model for shallow water flows with wetting and drying over irregular terrain. Adv. Water Resour.
**2011**, 34, 915–932. [Google Scholar] [CrossRef] - Nikolos, I.K.; Delis, A.I. An unstructured node-centered finite volume scheme for shallow water flows with wet/dry fronts over complex topography. Comput. Methods Appl. Mech. Eng.
**2009**, 198, 3723–3750. [Google Scholar] [CrossRef] - Conclusions and Recommendations from the Impact Project WP2: Breach Formation. Available online: http://www.impact-project.net/cd4/cd/Presentations/THUR/17-1_WP2_10Summary_v2_0.pdf (accessed on 28 July 2015).
- Kuiry, S.N.; Ding, Y.; Wang, S.S. Modelling coastal barrier breaching flows with well-balanced shock-capturing technique. Comput. Fluids
**2010**, 39, 2051–2068. [Google Scholar] [CrossRef] - Reed, C.W.; Militello, A. Wave-adjusted boundary condition for longshore current in finite-volume circulation models. Ocean Eng.
**2005**, 32, 2121–2134. [Google Scholar] [CrossRef]

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

He, Z.; Hu, P.; Zhao, L.; Wu, G.; Pähtz, T.
Modeling of Breaching Due to Overtopping Flow and Waves Based on Coupled Flow and Sediment Transport. *Water* **2015**, *7*, 4283-4304.
https://doi.org/10.3390/w7084283

**AMA Style**

He Z, Hu P, Zhao L, Wu G, Pähtz T.
Modeling of Breaching Due to Overtopping Flow and Waves Based on Coupled Flow and Sediment Transport. *Water*. 2015; 7(8):4283-4304.
https://doi.org/10.3390/w7084283

**Chicago/Turabian Style**

He, Zhiguo, Peng Hu, Liang Zhao, Gangfeng Wu, and Thomas Pähtz.
2015. "Modeling of Breaching Due to Overtopping Flow and Waves Based on Coupled Flow and Sediment Transport" *Water* 7, no. 8: 4283-4304.
https://doi.org/10.3390/w7084283