# Performance Evaluation of Seawalls in Mitigating a Real-World Tsunami Wave Using a Nonhydrostatic Numerical Wave Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations and Turbulence Model

#### Governing Equations

## 3. Wave Condition

^{2}(*)) to describe the tsunami wave profile recorded during the 2011 Tohoku tsunami. This method has also been applied in some recent studies [10,42,47,48]. The combination of three sech

^{2}(*) waves proposed by Chan and Liu [37] can be formulated as:

^{−1}, and ${t}_{i}$= [9.67 16.33 21.63] min by Chan and Liu [37], as shown in Figure 1. It seems that the observed tsunami wave profile can be accurately reproduced by the combined sech

^{2}(*) wave, whereas the solitary wave profile does not closely resemble the observations. In the following sections, the wave profile described by Equation (9) is referred to as a “tsunami-like wave”.

#### 3.1. Solitary Wave Runup Processes on a Plane Beach

#### 3.2. Solitary Wave Overtopping the Seawall

## 4. Results and Discussion

#### 4.1. Hydrodynamic Phenomena

#### 4.2. Effects of Wave Height and Water Depth

#### 4.3. Effects of the Side Slope of the Seawall

#### 4.4. Effects of Beach Slope

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Synolakis, C.E.; Bernard, E.N. Tsunami science before and beyond Boxing Day 2004. Philos. Trans. R. Soc. A.
**2006**, 364, 2231–2265. [Google Scholar] [CrossRef] [PubMed] - Pelinovsky, E.; Poplavsky, A. Simplified model of tsunami generation by submarine landslides. Phys. Chem. Earth.
**1996**, 21, 13–17. [Google Scholar] [CrossRef] - Tinti, S.; Piatanesi, A.; Bortolucci, E. The finite-element wave propagator approach and the tsunami inversion problem. Phys. Chem. Earth.
**1996**, 21, 27–32. [Google Scholar] [CrossRef] - Mori, N.; Takahashi, T. The 2011 Tohoku Earthquake Tsunami Joint Survey Group (2012) Nationwide post event survey and analysis of the 2011 Tohoku earthquake tsunami. Coast. Eng.
**2012**, 54, 1–27. [Google Scholar] - Yalciner, A.C.; Ozer, C.; Zaytsev, A.; Suppasri, A.; Mas, E.; Kalligeris, N.; Necmioglu, O.; Imamura, F.; Synolakis, C. Field survey on the coastal impacts of March 11, 2011 Great East Japan Tsunami. In Proceedings of the Seismic Protection of Cultural Heritage, Antalya, Turkey, 31 October–1 November 2011. [Google Scholar]
- Tomita, T.; Yeom, G.S.; Ayugai, M. Breakwater effects on tsunami inundation reduction in the 2011 off the Pacific coast of Tohoku Earthquake. J. Jpn. Soc. Civ. Eng. Ser. B 2 (Coast. Eng.)
**2012**, 68, I_156–I_160. [Google Scholar] [CrossRef] - Suppasri, A.; Muhari, A.; Ranasinghe, P. Damage and reconstruction after the 2004 Indian Ocean tsunami and the 2011 Great East Japan tsunami. J. Nat. Disaster Sci.
**2012**, 34, 19–39. [Google Scholar] [CrossRef] [Green Version] - Oshnack, M.E.; van de Lindt, J.; Gupta, R. Effectiveness of small onshore seawall in reducing forces induced by Tsunami bore: Large scale experimental study. J. Disaster Res.
**2009**, 4, 382–390. [Google Scholar] [CrossRef] - Prabu, P.; Bhallamudi, S.M.; Chaudhuri, A. Numerical investigations for mitigation of tsunami wave impact on onshore buildings using sea dikes. Ocean Eng.
**2019**, 187, 106159. [Google Scholar] [CrossRef] - Zhao, E.; Qu, K.; Mu, L. Numerical study of morphological response of the sandy bed after tsunami-like wave overtopping an impermeable seawall. Ocean Eng.
**2019**, 186, 106076. [Google Scholar] [CrossRef] - Didenkulova, I.; Pelinovsky, E. Tsunami wave run-up on a vertical wall in tidal environment. Pure Appl. Geophys.
**2018**, 175, 1387–1391. [Google Scholar] [CrossRef] - De Chowdhury, S.; Anand, K.V.; Sannasiraj, S.A. Nonlinear wave interaction with curved front seawalls. Ocean Eng.
**2017**, 140, 84–96. [Google Scholar] [CrossRef] - Zhang, M.; Ji, Y.; Wang, Y. Numerical investigation on tsunami wave mitigation on forest sloping beach. Acta Oceanol. Sin.
**2020**, 39, 130–140. [Google Scholar] [CrossRef] - Madsen, P.A.; Fuhrman, D.R.; Schäffer, H.A. On the solitary wave paradigm for tsunamis. J. Geophys. Res. Ocean.
**2008**, 113. [Google Scholar] [CrossRef] - Lu, Y.; Hua, L.I.U.; Wei, W. Numerical simulation of two-dimensional overtopping against seawalls armored with artificial units in regular waves. J. Hydrol. Ser. B.
**2007**, 19, 322–329. [Google Scholar] [CrossRef] - Goda, Y. Expected rate of irregular wave overtopping of seawalls. Coast. Eng. Jpn.
**1971**, 14, 43–51. [Google Scholar] [CrossRef] - Young, Y.L.; Xiao, H.; Maddux, T. Hydro-and morpho-dynamic modeling of breaking solitary waves over a fine sand beach. Part I: Experimental study. Mar. Geol.
**2010**, 269, 107–118. [Google Scholar] [CrossRef] - Park, H.; Cox, D.T.; Lynett, P.J. Tsunami inundation modeling in constructed environments: A physical and numerical comparison of free-surface elevation, velocity, and momentum flux. Coast. Eng.
**2013**, 79, 9–21. [Google Scholar] [CrossRef] - Synolakis, C.E. The runup of solitary waves. J. Fluid Mech.
**1987**, 185, 523–545. [Google Scholar] [CrossRef] - Li, Y.; Raichlen, F. Non-breaking and breaking solitary wave run-up. J. Fluid Mech.
**2002**, 456, 295–318. [Google Scholar] [CrossRef] [Green Version] - Lin, P.; Chang, K.A.; Liu, P.L.F. Runup and rundown of solitary waves on sloping beaches. J. Waterw. Port Coast. Ocean Eng.
**1999**, 125, 247–255. [Google Scholar] [CrossRef] - Irish, J.L.; Weiss, R.; Yang, Y. Laboratory experiments of tsunami run-up and withdrawal in patchy coastal forest on a steep beach. Nat. Hazards
**2014**, 74, 1933–1949. [Google Scholar] [CrossRef] - Yang, Y.; Irish, J.L.; Weiss, R. Impact of patchy vegetation on tsunami dynamics. J. Waterw. Port Coast. Ocean Eng.
**2017**, 143, 04017005. [Google Scholar] [CrossRef] - Watts, P.; Grilli, S.T.; Kirby, J.T. Landslide tsunami case studies using a Boussinesq model and a fully nonlinear tsunami generation model. Nat. Hazards Earth Syst. Sci.
**2003**, 3, 391–402. [Google Scholar] [CrossRef] [Green Version] - Aniel-Quiroga, I.; Vidal, C.; Lara, J.L. Pressures on a rubble-mound breakwater crown-wall for tsunami impact. Coast. Eng.
**2019**, 152, 103522. [Google Scholar] [CrossRef] - Gruwez, V.; Altomare, C.; Suzuki, T. Validation of RANS modelling for wave interactions with sea dikes on shallow foreshores using a large-scale experimental dataset. J. Mar. Sci. Eng.
**2020**, 8, 650. [Google Scholar] [CrossRef] - De Finis, S.; Romano, A.; Bellotti, G. Numerical and laboratory analysis of post-overtopping wave impacts on a storm wall for a dike-promenade structure. Coast. Eng.
**2020**, 155, 103598. [Google Scholar] [CrossRef] - Castellino, M.; Romano, A.; Lara, J.L. Confined-crest impact: Forces dimensional analysis and extension of the Goda’s formulae to recurved parapets. Coast. Eng.
**2021**, 163, 103814. [Google Scholar] [CrossRef] - Chen, H.; Yuan, J.; Cao, D. Wave overtopping flow striking a human body on the crest of an impermeable sloped seawall. Part II: Numerical modelling. Coast. Eng.
**2021**, 168, 103892. [Google Scholar] [CrossRef] - Hsiao, S.C.; Hsu, T.W.; Lin, T.C. On the evolution and run-up of breaking solitary waves on a mild sloping beach. Coast. Eng.
**2008**, 55, 975–988. [Google Scholar] [CrossRef] - Liang, D.; Jian, W.; Shao, S. Incompressible SPH simulation of solitary wave interaction with movable seawalls. J. Fluids Struct.
**2017**, 69, 72–88. [Google Scholar] [CrossRef] - Mokhtar, Z.A.; Mohammed, T.A.; Yusuf, B. Experimental investigation of tsunami bore impact pressure on a perforated seawall. Appl. Ocean Res.
**2019**, 84, 291–301. [Google Scholar] [CrossRef] - Xu, Z.; Melville, B.; Whittaker, C. Mitigation of tsunami bore impact on a vertical wall behind a barrier. Coast. Eng.
**2021**, 164, 103833. [Google Scholar] [CrossRef] - Kihara, N.; Niida, Y.; Takabatake, D. Large–scale experiments on tsunami–induced pressure on a vertical tide wall. Coast. Eng.
**2015**, 99, 46–63. [Google Scholar] [CrossRef] - Kocaman, S.; Ozmen–Cagatay, H. Investigation of dam-break induced shock waves impact on a vertical wall. J. Hydrol.
**2015**, 525, 1–12. [Google Scholar] [CrossRef] - Chen, X.; Hofland, B.; Altomare, C. Overtopping flow impact on a vertical wall on a dike crest. Coast. Eng. Proc.
**2014**, 34, 4. [Google Scholar] [CrossRef] [Green Version] - Chan, I.C.; Liu PL, F. On the runup of long waves on a plane beach. J. Geophys. Res. Ocean.
**2012**, 117, C08006. [Google Scholar] [CrossRef] - Madsen, P.A.; Schaeffer, H.A. Analytical solutions for tsunami runup on a plane beach: Single waves, N-waves and transient waves. J. Fluid Mech.
**2010**, 645, 27–57. [Google Scholar] [CrossRef] - Qu, K.; Ren, X.Y.; Kraatz, S. Numerical investigation of tsunami-like wave hydrodynamic characteristics and its comparison with solitary wave. Appl. Ocean Res.
**2017**, 63, 36–48. [Google Scholar] [CrossRef] - Qu, K.; Sun, W.Y.; Tang, H.S. Numerical study on hydrodynamic load of real-world tsunami wave at highway bridge deck using a coupled modeling system. Ocean Eng.
**2019**, 192, 106486. [Google Scholar] [CrossRef] - Qu, K.; Ren, X.Y.; Kraatz, S. Numerical analysis of tsunami-like wave impact on horizontal cylinders. Ocean Eng.
**2017**, 145, 316–333. [Google Scholar] [CrossRef] - Qu, K.; Lan, G.Y.; Sun, W.Y. Numerical study on wave attenuation of extreme waves by emergent rigid vegetation patch. Ocean Eng.
**2021**, 239, 109865. [Google Scholar] [CrossRef] - Rodi, W. Examples of calculation methods for flow and mixing in stratified fluids. J. Geophys. Res. Ocean.
**1987**, 92, 5305–5328. [Google Scholar] [CrossRef] - Ma, G.; Shi, F.; Kirby, J.T. Shock-capturing non-hydrostatic model for fully dispersive surface wave processes. Ocean Model.
**2012**, 43, 22–35. [Google Scholar] [CrossRef] - Ma, G.; Shi, F.; Hsiao, S.C. Non-hydrostatic modeling of wave interactions with porous structures. Coast. Eng.
**2014**, 91, 84–98. [Google Scholar] [CrossRef] - Boussinesq, J. Theorie des ondes et ramous qui se progagent le long dun canal rectangularie horizontal, en communiquant au liquid contenu dansce canal des vitesses sensiblement pareilles de la surface au. Math. Pures Appl.
**1872**, 17, 55–108. [Google Scholar] - Schimmels, S.; Sriram, V.; Didenkulova, I. Tsunami generation in a large scale experimental facility. Coast. Eng.
**2016**, 110, 32–41. [Google Scholar] [CrossRef] - Williams, I.A.; Fuhrman, D.R. Numerical simulation of tsunami-scale wave boundary layers. Coast. Eng.
**2016**, 110, 17–31. [Google Scholar] [CrossRef] [Green Version] - Hunt, A. Extreme Waves, Overtopping and Flooding at Sea Defences; University of Oxford: Oxford, UK, 2003. [Google Scholar]

**Figure 1.**Comparison of the time series of wave profiles from field observation, solitary wave, and tsunami-like wave.

**Figure 2.**Comparison of the time series of the wave profiles of the solitary wave and the tsunami-like wave with $h$= 1 m and $H$ = 0.2 m.

**Figure 4.**Comparisons of the time series of water surface elevation at different wave gauges for $H/h$ = 0.054; (

**a**) WG1; (

**b**) WG2; (

**c**) WG3; (

**d**) WG4; (

**e**) WG5; (

**f**) WG6; (

**g**) WG7; (

**h**) WG8; (

**i**) WG9; (

**j**) WG10; (

**k**) WG11; (

**l**) WG12.

**Figure 5.**Comparisons of the time series of water surface elevation at different wave gauges for $H/h$ = 0.208; (

**a**) WG1; (

**b**) WG2; (

**c**) WG3; (

**d**) WG4; (

**e**) WG5; (

**f**) WG6; (

**g**) WG7; (

**h**) WG8; (

**i**) WG9; (

**j**) WG10; (

**k**) WG11; (

**l**) WG12.

**Figure 6.**Comparisons of the time series of water surface elevation at different wave gauges for H/h = 0.338; (

**a**) WG1; (

**b**) WG2; (

**c**) WG3; (

**d**) WG4; (

**e**) WG5; (

**f**) WG6; (

**g**) WG7; (

**h**) WG8; (

**i**) WG9; (

**j**) WG10; (

**k**) WG11; (

**l**) WG12.

**Figure 9.**Comparisons of the time series of water surface elevation at different wave gauges; (

**a**) WG1; (

**b**) WG2; (

**c**) WG3; (

**d**) WG4; (

**e**) WG5.

**Figure 10.**Comparisons of the spatial distributions of water surface elevations at different time instances; (

**a**) t = 9 s; (

**b**) t = 10 s; (

**c**) t = 11 s; (

**d**) t = 12 s; (

**e**) t = 13 s.

**Figure 13.**Snapshots of the velocity contours of the water body at different time instances; (

**a**) t = 42.5 s; (

**b**) t = 52.1 s; (

**c**) t= 53.2 s; (

**d**) t = 55.8 s.

**Figure 15.**Comparisons of the temporal evolutions of wave energies of the whole water body in the computational domain; (

**a**) $KE$; (

**b**) $PE$; (

**c**) $TE$.

**Figure 25.**Snapshots of the velocity contours of the water body at the moment of the tsunami-like wave climbing over the seawall with different sidewall slopes; (

**a**) side slope = 1:1; (

**b**) side slope = 1:2; (

**c**) side slope = 1:3; (

**d**) side slope = 1:4; (

**e**) side slope = 1:5.

**Figure 30.**Snapshots of the velocity contours of the water body at the moment of the wave crest climbing over the seawall with different sidewall slopes; (

**a**) beach slope = 1:30; (

**b**) beach slope = 1:25; (

**c**) beach slope = 1:20; (

**d**) beach slope = 1:15; (

**e**) beach slope = 1:10; (

**f**) beach slope = 1:5.

**Figure 32.**Variations in maximum runup height of overtopping water surge bores with different beach slopes.

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**MDPI and ACS Style**

Huang, J.X.; Qu, K.; Li, X.H.; Lan, G.Y.
Performance Evaluation of Seawalls in Mitigating a Real-World Tsunami Wave Using a Nonhydrostatic Numerical Wave Model. *J. Mar. Sci. Eng.* **2022**, *10*, 796.
https://doi.org/10.3390/jmse10060796

**AMA Style**

Huang JX, Qu K, Li XH, Lan GY.
Performance Evaluation of Seawalls in Mitigating a Real-World Tsunami Wave Using a Nonhydrostatic Numerical Wave Model. *Journal of Marine Science and Engineering*. 2022; 10(6):796.
https://doi.org/10.3390/jmse10060796

**Chicago/Turabian Style**

Huang, J. X., K. Qu, X. H. Li, and G. Y. Lan.
2022. "Performance Evaluation of Seawalls in Mitigating a Real-World Tsunami Wave Using a Nonhydrostatic Numerical Wave Model" *Journal of Marine Science and Engineering* 10, no. 6: 796.
https://doi.org/10.3390/jmse10060796