Maria Ana Baptista

Miguel Miranda

P. Avilez-Valente

Vânia Lima

Portuguese

Repositório Aberto da Universidade do Porto

Lisboa, 21, 22 e 23 de junho de 20221737as Jornadas de Engenharia Hidrográfica / 2as Jornadas Luso-Espanholas de Hidrografia Fluid-structure interaction of  a leading depression N-wave shaped tsunami V. Lima (1, 2, 4), P. Avilez-Valente (3,4), M.A. Baptista (2,5), M. Miranda (2,6) (1) Instituto Hidrográfico, Rua das Trinas 49, 1249-093 Lisboa, Portugal. vania.lima@hidrografico.pt (2) Instituto Dom Luiz, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, Edifício C1 — Piso 1, 1749-016 Lisboa, Portugal. (3) Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal. (4) Centro Interdisciplinar de Investigação Marinha e Ambiental, Universidade do Porto, Terminal de Cruzeiros de Leixões, Av. General Norton de Matos s/n, 4450-208 Matosinhos, Portugal. (5) Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro 1, 1959-007 Lisboa, Portugal. (6) Instituto Português do Mar e da Atmosfera, Rua C do Aeroporto, 1749-077 Lisboa, Portugal.  Abstract: In this work, we present a 2DV study of the interaction of a tsunami wave with the Cascais marina breakwater. The numerical modelling is performed with the open-source computational fluid dynamics software OpenFOAM and the olaFlow toolbox. The tsunami wave, which resembles a leading depression N-wave, enters the domain from Southwest with a wave height of about 6 m. We show that the Cascais breakwater is overtopped by the tsunami wave and that vortices are generated on the sea-ward and leeward sides of the breakwater. We perform an analysis of the pressure distribution on the walls and crest of the breakwater and show that negative pressures arise on the leeward side of the breakwater. Key words: computational fluid dynamics, fluid-structure interaction, N-wave; tsunami.  1. THE N-WAVE CONCEPT A tsunami wave can be described as a long wave in shallow water. Such a wave has a wavelength 𝜆𝜆 which is much larger than the depth ℎ of the water it is prop-agating in, i.e. ℎ 𝜆𝜆⁄ ≪ 1. A common approach is to model tsunamis as solitary waves, both in experi-mental and numerical studies. Madsen et al. (2008) discussed the solitary wave paradigm in tsunami re-search and concluded that, in general, tsunamis do not generate solitary-like waves in the ocean or on the continental shelf. In fact, not all tsunami events are alike and reports of shoreline receding support the concept of N-shaped tsunami waves (cf. Figure 1). The N-wave concept was introduced by Tadepalli and Synolakis (1994), who suggested that an N-wave is the most adequate to describe the main tsunami wave. N-waves can be described as having a leading trough, the so-called leading depression N-wave (LDN-wave), or a leading crest, a leading elevation N-wave (LEN-wave). A conceptual N-wave was modelled by Tadepalli and Synolakis (1994, 1996) as 𝜂𝜂(𝑥𝑥, 𝑡𝑡) = 𝜀𝜀 (𝜃𝜃 − 𝜅𝜅 𝛿𝛿) 𝐻𝐻 sech2(𝜅𝜅 (𝑥𝑥 − 𝑐𝑐 𝑡𝑡 − 𝑥𝑥1)) , where 𝜀𝜀 is a scale factor necessary to define the wave height as 𝐻𝐻, 𝛿𝛿 = 𝑥𝑥2 − 𝑥𝑥1 is an eccentricity parameter, defined as the distance between 𝑥𝑥 =  𝑥𝑥2, the location of the inflexion point of the wave profile at 𝑡𝑡 = 0, i.e. 𝜂𝜂(𝑥𝑥2, 0) = 0, and 𝑥𝑥 = 𝑥𝑥1, the location at 𝑡𝑡 = 0 of the crest of a solitary wave of the same height 𝐻𝐻 and length 𝜆𝜆, 𝑐𝑐 is the wave velocity, and 𝜅𝜅 = 2𝜋𝜋 𝜆𝜆⁄  is a generalized wavenumber. The wave height 𝐻𝐻 of a N-wave is obtained as 𝐻𝐻 =  𝑎𝑎+ + 𝑎𝑎−, where 𝑎𝑎+ is the amplitude of the wave crest and 𝑎𝑎− is the amplitude of the wave trough.   Figure 1. N-wave profile and corresponding solitary wave of the same height 𝐻𝐻 and length 𝜆𝜆. 2. COMPUTATIONAL MODELLING To model nearshore processes in two- and three-di-mensional domains and to determine the properties and characteristics of tsunami interaction with coastal structures and surrounding areas, the open access nu-merical code OpenFOAM (OpenCFD Limited, 2019) and the olaFlow toolbox (Higuera, 2019) are used. OpenFOAM uses the finite volume method to solve computational fluid dynamics problems and includes several approaches to tackle turbulence. The olaFlow solver package (Higuera, 2019) consists of a set of solvers for the simulation of wave dynamics, enabling 7.as Jornadas de Engenharia Hidrográfica / 2.as Jornadas Luso-Espanholas de Hidrografia174Lisboa, 21 a 23 de junho de 2022 the generation and absorption of waves at the bound-aries and wave interaction with coastal structures. olaFlow solves the Volume Averaged Reynolds-Av-eraged Navier-Stokes (VARANS) equations for the two incompressible phases, applying the finite vol-ume (FV) discretisation method for numerical inte-gration, and the volume of fluid (VOF) method for phase definition. 3. SIMULATION AND RESULTS We modelled an earthquake event, located at the Horseshoe Abyssal Plain, triggering a tsunami with SW–NE incident direction (for more details see Lima, 2022). The modelled tsunami wave signal (see Figure 2) was retrieved at a point in the vicinity of the Cas-cais marina, at a depth 𝑑𝑑 = 16 m. At that point, the wave closely resembled the Tadepalli and Synolakis’ conceptual LDN-wave, with 𝑎𝑎+ = 4.44 m, 𝑎𝑎− =1.68 m, and an apparent period 𝑇𝑇 = 12 min. Assum-ing  𝑐𝑐 = √𝑔𝑔(𝑑𝑑 + 𝑎𝑎+), with the acceleration of grav-ity 𝑔𝑔 = 9.81 m/s2,  the apparent wavelength can be estimated as 𝜆𝜆 = 10.2 km. The free surface elevation time series (Figure 2) and the horizontal particle veloc-ity, considered as constant in the water column, com-puted as 𝑢𝑢(𝑡𝑡) = 𝑐𝑐 𝜂𝜂 ℎ⁄  , were used as boundary condition for the 2DV (two dimensional vertical) simulation.  Figure 2. Free surface elevation time series of the leading wave. The computational domain contemplates an initial 650 m-long slope, followed by the 50 m-wide break-water berm. The computational domain includes a cross-section through the marina southwest breakwa-ter, the marina basin, and the marina technical pier wall (cf. Figure 4). For the simulations, we consider a 3D canal, one cell wide (𝐿𝐿𝐿𝐿 = 10 m), with a total length 𝐿𝐿𝐿𝐿 = 830 m and height 𝐿𝐿𝐿𝐿 = 30 m. The cell resolution varies along the computational domain. In the 𝐿𝐿-direction, for instance, the cell size varies from ∆𝐿𝐿 = 1 m to ∆𝐿𝐿 = 5 m (cf. Table I). The still water depth was 𝑑𝑑 =16 m, the total run time was 715 s, and a time step Δ𝑡𝑡 = 1 s was used.  Figure. 3. Horizontal particle velocity time series.  Figure 4. 2DV computational domain for the Cascais marina (not to scale) and wave gauges locations. Table. I. Computational domain and mesh characteristics. Zone Slope Berm Breakwater Basin Technical pier 𝐿𝐿𝐿𝐿 (m) 650 50 10 100 20 𝐿𝐿𝐿𝐿 (m) 10 10 10 10 10 𝐿𝐿𝐿𝐿 (m) 10 13.5 21.5 11 20 ∆𝐿𝐿 (m) 5 1 1 1 5 ∆𝐿𝐿 (m) 10 10 10 10 10 ∆𝐿𝐿 (m) 0.5 0.5 0.5 0.5 0.5 The buoyancy-modified κ-ω SST turbulence model (Devolder et al, 2017) was used. This is a turbulence model tailored for simulation of wave propagation in VOF models. To mode correctly model the surface waves more correctly, those authors introduced an ad-ditional buoyancy term to eliminate the spurious tur-bulence generation, hence the wave height dissipation of surface waves.  They concluded that the inclusion in their model of a density gradient at the air-water interface better simulated the turbulent wave energy dissipation. Figure. 5 and Figure. 6 illustrate the propagation of the tsunami wave from the beginning of the simulation until after the wave overtop the breakwater at 𝑡𝑡 = 7.6 min. At 𝑡𝑡 = 3.7 min, the wave receding is perfectly perceptible. After 𝑡𝑡 = 10 min, the technical pier wall is flooded. The flow depth reaches 1 m both at the top of the breakwater and at the top of the technical pier (Figure. 7 and Figure. 8).  The streamlines are shown in Figure. 9 and Figure. 10.  A vortex is clearly identifiable over the seaward side berm of the breakwater.  A second vortex is observed on the leeward side of the breakwater. Higher veloci-ties are registered on the leeward side of the breakwa-ter and between that structure and the technical pier.on the leeward side of the breakwater. Higher ve-locities are registered on the leeward side of the 𝑢𝑢 (m/s) Lisboa, 21, 22 e 23 de junho de 20221757as Jornadas de Engenharia Hidrográfica / 2as Jornadas Luso-Espanholas de Hidrografia breakwater and between that structure and the tech-nical pier.  Figure. 5. Time evolution of the tsunami wave over the marina of Cascais. Top to bottom: 𝑡𝑡 = 5 s, 3.7 min, 6.7 min and 7.2 min.  Figure. 6. Time evolution of the tsunami wave over the marina of Cascais. Top to bottom: 𝑡𝑡 = 7.6 min, 8.0 min, 8.7 min and 9.4 min.  Figure. 7. Time series of the water level at breakwater crest (gauge wg6).  Figure. 8. Time series of the water level at the top of the technical pier (gauge wg11). Numerical pressure gauges were placed along the breakwater walls and crest (see Figure 4). Analysis of  the instantaneous pressure distribution along these surfaces showed that the pressures on the crest de-crease from the seaward to the leeward side, becom-ing negative, i.e. below atmospheric pressure, at the inner end of the crest and at the upper end of the lee-ward wall of the breakwater (cf. Figure 11, Figure. 12, Figure. 13).  Figure. 9. Streamlines on the seaward side of the breakwater for 𝑡𝑡 = 8.7 min.  Figure. 10. Streamlines within the marina basin, for 𝑡𝑡 = 8.7 min.  Figure 11. Instantaneous pressure distribution on the seaward side of the breakwater for 𝑡𝑡 = 8.2 min.  Figure. 12. Instantaneous pressure distribution on the breakwater crest for 𝑡𝑡 = 8.2 min.  Figure. 13 – Instantaneous pressure distribution on the leeward side of the breakwater for 𝑡𝑡 = 8.2 min. The vortices and recirculation effects observed on both sides of the breakwater point to the possible oc-currence of scouring, sand suspension and erosion, and consequent deballasting of the structure. This, to-gether with the negative pressure recorded on the lee-ward side of the breakwater, may lead to the collapse of the structure. The numerical simulations results presented herein have not been validated with any laboratory experi-ments nor with a real tsunami event. However, Open-FOAM has been previously used for this type of study (Morichon et al., 2021) and validated by Jiang et al. (2016), who performed small-scale laboratory exper-iments of tsunami-like solitary waves impacting a rectangular seawall. 7.as Jornadas de Engenharia Hidrográfica / 2.as Jornadas Luso-Espanholas de Hidrografia176Lisboa, 21 a 23 de junho de 2022 In a preliminary stage, an attempt was made to per-form 3D simulations, but these proved not to be sup-portable due to the limited cluster resources available, the large amount of time required for the simulations, the large size occupied by the output simulation re-sults and the limited post-processing capabilities. For this reason, it was chosen to perform the 2DV simulations. However, this will ignore some 3D ef-fects, such as currents generated along the breakwa-ters, and the 3D patterns of the vortices that may lead to asymmetric erosion along the breakwater walls. In addition, inundation of the marina basin will also oc-cur due to overtopping of the southeast breakwater. Finally, the effect on the northeast marina entrance is also overlooked. However, we believe that this 2DV analysis, although limited, can prove quite useful on a breakwater stability analysis of the breakwaters in the event of a tsunami impact. 4. CONCLUSIONS A 2DV study of the interaction of a tsunami wave with the defence structures of the Cascais marina was presented. It was found that the initial tsunami wave shows a profile very close to the conceptual N-wave model of Tadepalli and Synolakis, thus the N-wave profile is used in this study for the tsunami wave nu-merical modelling. The numerical modelling performed with Open-FOAM and the olaFlow toolbox showed that the Cas-cais breakwater is subject to overtopping by the tsu-nami wave and consequent inundation of the marina basin. We also show that vortices are generated on the seaward and leeward sides of the breakwater. Analysis of the pressure distribution on the walls and crest of the breakwater shows that the leeward side of the breakwater is subjected to negative pressures. To-gether with the recirculation effect generated by the vortices, this may contribute to scouring, sand sus-pension and erosion at those location, which may lead to the collapse of the breakwater. As future work, a full 3D simulation will be carried out to analyse how the vortices develop along the breakwater walls, to describe the currents generated within the marina, and to quantify the forces applied on the structures and their risk of collapse. The numerical simulation of the wave-shore interac-tion must be limited in terms of area, due to the con-straints in the computational resources available. Acknowledgements The first author was supported by FCT grant SFRH/BD/96725/2013. This research was partially supported by the Research Line ECOSERVICES, in-tegrated in the Structured Program of R&D&I INNO-VMAR: Innovation and Sustainability in the Manage-ment and Exploitation of Marine Resources (NORTE-01-0145-FEDER-000035), funded by the Northern Regional Operational Programme (NORTE 2020) through the European Regional Development Fund. REFERENCES Devolder, B., Rauwoens, P., and Troch, P. (2017). Application of a buoyancy-modified κ – ω SST turbulence model to simulate wave run-up around a monopile subjected to regular waves using OpenFOAM®. Coastal Engineering, 125, 81–94. Higuera P. (2019). olaflow. https://olaflow.github.io/, https://doi.org/10.5281/zenodo.1297012. Ac-cessed: 2022-05-08. Jiang, C., Liu, X., Yao, Y., Deng, B., and Chen, J. (2017). Numerical investigation of tsunami-like solitary wave interaction with a seawall. Journal of Earthquake and Tsunami, 11(01), 1740006. Lima, V.V. (2022). A Combined Model for Tsunami Wave Propagation, Dispersion, Breaking and Fluid-Structure Interaction. Doctoral disserta-tion. Faculty of Sciences of the University of Lis-bon, Portugal.  Madsen, P.A., Fuhrman, D.R., and Schäffer, H.A. (2008). On the solitary wave paradigm for tsuna-mis. Journal of Geophysical Research: Oceans, 113(C12). Morichon, D., Roeber, V., Martin-Medina, M., Bel-lafont, F., and Abadie, S. (2021). Tsunami impact on a detached breakwater: Insights from two nu-merical models. Journal of Waterway, Port, Coastal, and Ocean Engineering, 147(2), 05021001. OpenCFD Limited (2019). OpenFOAM® - The Open Source CFD Toolbox. https://www.open-foam.com/. Accessed: 2022-05-08. Tadepalli, S., and Synolakis, C.E. (1994). The run-up of N-waves on sloping beaches. Proceedings of the Royal Society: Mathematical and Physical Sciences, A445, 99–112. Tadepalli, S., and Synolakis, C.E. (1996). Model for the leading waves of tsunamis. Physical Review Letters, 77, 2141–2144. 

Fluid-structure interaction of a leading depression N-wave shaped tsunami

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Fluid-structure interaction of a leading depression N-wave shaped tsunami

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Repositório Aberto da Universidade do Porto