Solitary Wave Run-up Research Articles

Performance of submerged semi-circular breakwater under solitary wave in consideration of porous media

Submerged breakwater as the main structure of coastal engineering to protect coast is widely laid on the seabed because of its significant wave-dissipation performance. To reveal the interaction mechanism between waves and submerged breakwater, the hydrodynamic characteristics of a wave passing over different solid breakwaters were numerically and experimentally studied. Compared with the previous research, considering breakwater made up of porous media and the effects of extreme waves generated by hurricanes or tsunamis, this paper systematically investigates the effects of the semi-circular porous medium breakwater on the breaking solitary wave run-up based on a numerical model with the immersed boundary (IB) method. The computational capability of this model is verified firstly. Then, depending on the model, a series of cases are carried out to research the effects of different breakwaters on the run-up of solitary waves. The results show that the influence of porous breakwater on wave propagation is more significant than that of the solid breakwater. With the increase of the gravel particle median diameter, the maximum run-up height and the horizontal hydrodynamic force on the breakwater decrease, while the vertical hydrodynamic force on the breakwater increases. Increasing the size of the breakwater is beneficial to reducing the wave run-up height and the forces on the sloping beach. Compared with the single breakwater, the impact of the tandem breakwater on the wave propagation is more significant.

Predicting tsunami-like solitary wave run-up over fringing reefs using the multi-layer perceptron neural network

Modeling of tsunami wave interaction with coral reefs to date focuses mainly on the process-based numerical models. In this study, an alternative machine learning technique based on the multi-layer perceptron neural network (MLP-NN) is introduced to predict the tsunami-like solitary wave run-up over fringing reefs. Two hydrodynamic forcings (incident wave height, reef-flat water level) and four reef morphologic features (reef width, fore-reef slope, beach slope, reef roughness) are selected as the input variables and wave run-up on the back-reef beach is assigned as the output variable. A validated numerical model based on the Boussinesq equations is applied to provide a dataset consisting of 4096 runs for MLP-NN training and testing. Results analyses show that the MLP-NN consisting of one hidden layer with ten hidden neurons provides the best predictions for the wave run-up. Subsequently, model performances in view of individual input variables are accessed via an analysis of the percentage errors of the predictions. Finally, a mean impact value analysis is also conducted to evaluate the relative importance of the input variables to the output variable. In general, the adopted MLP-NN has high predictive capability for wave run-up over the reef-lined coasts, and it is an alternative but more efficient tool for potential use in tsunami early warning system or risk assessment projects.

Modeling solitary wave transformation and run-up over fringing reefs with large bottom roughness

Field observations over decades have found that the surface roughness of coral reefs is typically one or two order of magnitude larger than that of sandy beaches. To better reproduce the solitary wave transformation and run-up over fringing reefs with large bottom roughness, a numerical wave tank based on the CFD tool OpenFOAM® is developed in this study. The Reynolds-Averaged Navier-Stokes (RANS) equations are solved for two-phase incompressible flow with the k-ω SST model for the turbulence closure and VOF method for tracking the free surface. The reef surface with high friction is modeled by coupling a porous media model in the RANS equations. The model is first validated by our new performed laboratory experiments as well as a laboratory dataset in the literature. A fixed set of model parameters is found to be suitable for both experiments. Subsequently, the model is applied to evaluate the impacts of hydrodynamic, morphological and roughness element factors on the wave run-up on the back-reef beach, and an empirical formula is proposed to predict the wave run-up based on the simulations. Finally, the cross-reef variations of flow and vorticity fields under the breaking solitary waves are also examined via the numerical simulations.

A unified runup formula for solitary waves on a plane beach

Laboratory experiments to estimate solitary wave runup were performed on three different planar beaches with slopes of 1/4, 1/7 and 1/10, with varying wave heights and water depths. The non-dimensionalised vertical runup (runup over water depth) obtained from present laboratory data and several published sources were plotted against the surf similarity parameter. Best-fitting curves could be obtained for maximum runup on slopes ranging from 1/1 to 1/60 against the surf similarity parameter. Hence an approximate formula is presented for estimation of maximum solitary wave runup on planar beaches of different slopes regardless of whether the waves are breaking or non-breaking. The proposed empirical equation predicts the overall dataset, mostly within 20% scatter for all the laboratory data. Thus, the empirical formula can be used to obtain a quick estimation of maximum runup height for both breaking and non-breaking solitary waves in laboratory experiments, for disaster risk evaluation of wave runups and to verify numerical model runup results.

Staggered Conservative Scheme for 2-Dimensional Shallow Water Flows

Simulating discontinuous phenomena such as shock waves and wave breaking during wave propagation and run-up has been a challenging task for wave modeller. This requires a robust, accurate, and efficient numerical implementation. In this paper, we propose a two-dimensional numerical model for simulating wave propagation and run-up in shallow areas. We implemented numerically the 2-dimensional Shallow Water Equations (SWE) on a staggered grid by applying the momentum conserving approximation in the advection terms. The numerical model is named MCS-2d. For simulations of wet–dry phenomena and wave run-up, a method called thin layer is used, which is essentially a calculation of the momentum deactivated in dry areas, i.e., locations where the water thickness is less than the specified threshold value. Efficiency and robustness of the scheme are demonstrated by simulations of various benchmark shallow flow tests, including those with complex bathymetry and wave run-up. The accuracy of the scheme in the calculation of the moving shoreline was validated using the analytical solutions of Thacker 1981, N-wave by Carrier et al., 2003, and solitary wave in a sloping bay by Zelt 1986. Laboratory benchmarking was performed by simulation of a solitary wave run-up on a conical island, as well as a simulation of the Monai Valley case. Here, the embedded-influxing method is used to generate an appropriate wave influx for these simulations. Simulation results were compared favorably to the analytical and experimental data. Good agreement was reached with regard to wave signals and the calculation of moving shoreline. These observations suggest that the MCS method is appropriate for simulations of varying shallow water flow.

Numerical investigation of solitary wave run-up attenuation by patchy vegetation

Coastal vegetation is capable of decreasing wave run-up. However, because of regrowth, decay or man-made damage, coastal vegetation is always distributed in patches, and its internal distribution is often non-uniform. This study investigates the effects of patchy vegetation on solitary wave run-up by using a numerical simulation. A numerical model based on fully nonlinear Boussinesq equations is established to simulate the wave propagation on a slope with patchy vegetation. By using the model, the process of solitary wave run-up attenuation due to patchy vegetation is numerically analysed. The numerical results reveal that patchy vegetation can considerably attenuate the wave run-up in an effective manner. In addition, high-density patched vegetation can attenuate the solitary wave run-up more effectively than low-density patched vegetation can. For the same density, patchy vegetation with a uniform distribution has a better attenuation effect on wave run-up compared to that of patchy vegetation with a non-uniform distribution.

Wave Runup Loading Behind a Semipermeable Obstacle

This paper describes an investigation of the interaction of breaking solitary wave runup with a two-dimensional semipermeable (vegetated) obstacle and the resulting wave loading on a downstream cylinder. The OpenFOAM model, together with an IHFOAM module for mangrove–fluid interaction, are used for 276 numerical experiments, performed for a wide range of damping coefficients. In general, wave heights inside and just downstream of the obstacle were found to increase, compared with the bare earth case, with reflection from and inside the obstacle as the major factor. By contrast, wave loads decreased strongly when sheltered by the obstacle, with increasing damping coefficients leading to lower loads, but with considerable scatter. This paper provides several different methods to estimate wave runup loads behind semipermeable obstacles, with results presented in forms that may prove useful for design.

Well-balanced and shock-capturing solving of 3D shallow-water equations involving rapid wetting and drying with a local 2D transition approach

In this study, we develop a shock-capturing numerical model for solving the three-dimensional shallow-water equations with turbulence closure. Numerical discretization is performed using the Godunov-type finite-volume method in σ coordinates. An approximate Riemann solver is used for numerical flux evaluations. To efficiently and accurately resolve the wet–dry fronts, we propose a local 2D transition approach (i.e., switching from solving the three-dimensional shallow-water equations to solving the depth-averaged two-dimensional shallow-water equations) with using a rather small value of threshold water depth (1×10−6m) to distinguish dry cells from the wet ones. A fully-implicit discretization method of the bed-friction terms is developed, which was originally proposed to solve the depth-averaged two-dimensional shallow-water equations. A series of benchmark tests are used to assess the performance of the numerical models, showing that the proposed model is well-balanced and robust to resolve violent free-surface flows. The local 2D transition approach is found to significantly reduce the computational time by one to three orders in a dam-break wave propagation test, and by about 1∕2–2∕3 in a breaking solitary wave runup test, compared with a same simulation without using this approach.

Experimental Investigation on Solitary Wave Interaction With Vertical Porous Barriers

Abstract Tsunami waves pose a threat to the coastal zone, and numerous studies have been carried out in the past to understand them. Solitary waves have been extensively used in research because they approximate certain important characteristics of tsunami waves. The present study focusses on the interaction and run-up of solitary waves on coastal protection structures in the form of thin, rigid vertical porous barriers with special attention given to the degree of energy dissipation. To understand the physics of energy dissipation, solitary wave interaction with a porous barrier has been studied from the viewpoint of energy balance. Based on this, a relationship for the wave energy dissipation has been developed. The experimental data show that the plate porosity that gives the optimal energy dissipation lies within the 10–20% range. From the experiments, the phase shift that the solitary wave undergoes upon interaction with the porous barrier models has also been recorded. In addition, a formula is proposed for maximum wave run-up on the porous barrier, which should be useful in the planning, design, construction, and maintenance of coastal protection structures.

Numerical simulation of solitary waves overtopping on a sloping sea dike using a particle method

Sea dikes, as a commonly used type of coastal protection structures, are often attacked or damaged by violent waves overtopping under tsunamis and storm surges. In this study, the behavior of solitary waves traveling on a sloping sea dike is simulated, and solitary wave overtopping characteristics are analyzed using a complete Lagrangian numerical method, the moving particle semi-implicit (MPS) method. To better describe the complicated fluid motions during the wave overtopping process, the original MPS method is modified by introducing a new free surface detection method, i.e., the area filling rate identification method, and a modified gradient operator to provide higher precision. Meanwhile, the approximation method for sloping boundaries in particle methods is enhanced, and a smooth slope approximation method is proposed and recommended. To verify the improved MPS method, a solitary wave traveling over a steep sloping bed is studied. The entire solitary wave run-up and run-down processes and exquisite water movements are reproduced well by the present method, and are consistent with the corresponding experimental results. Subsequently, the improved MPS method is applied to investigate the overtopping process of a single solitary wave over a sloping sea dike. The results show that the hydraulic jump phenomenon is also possible to occur during the run-down motion of the solitary wave overtopping. Finally, the characteristics of the propagation and overtopping of two successive solitary waves on a sloping sea dike are discussed. The result manifests that the interaction between adjacent solitary waves affects wave overtopping patterns and overtopping velocities.

A Numerical Investigation of the Reduction of Solitary Wave Runup by A Row of Vertical Slotted Piles

To improve the current understanding of the reduction of tsunami-like solitary wave runup by the pile breakwater on a sloping beach, we developed a 3D numerical wave tank based on the CFD tool OpenFOAM® in this study. The Navier-Stokes equations were applied to solve the two-phase incompressible flow, combined with an LES model to solve the turbulence and a VOF method to capture the free surface. The adopted model was firstly validated with existing empirical formulas for solitary wave runup on the slope without the pile structure. It is then validated using our new laboratory observations of the free surface elevation, the velocity and the pressure around a row of vertical slotted piles subjected to solitary waves, as well as the wave runup on the slope behind the piles. Subsequently, a set of numerical simulations were implemented to analyze the wave reflection, the wave transmission, and the shoreline runup with various offshore wave heights, offshore water depths, adjacent pile spaces and beach slopes. Finally, an improved empirical equation accounting for the maximum wave runup on the slope was proposed by taking the presence of the pile breakwater into consideration.

Numerical investigation on tsunami wave mitigation on forest sloping beach

An explicit one-dimensional model based on the shallow water equations (SWEs) was established in this work to simulate tsunami wave propagation on a vegetated beach. This model adopted the finite-volume method (FVM) for maintaining the mass balance of these equations. The resistance force caused by vegetation was taken into account as a source term in the momentum equation. The Harten–Lax–van Leer (HLL) approximate Riemann solver was applied to evaluate the interface fluxes for tracing the wet/dry transition boundary. This proposed model was used to simulate solitary wave run-up and long-periodic wave propagation on a sloping beach. The calibration process suitably compared the calculated results with the measured data. The tsunami waves were also simulated to discuss the water depth, tsunami force, as well as the current speed in absence of and in presence of forest domain. The results indicated that forest growth at the beach reduced wave energy loss caused by tsunamis. A series of sensitivity analyses were conducted with respect to variable parameters (such as vegetation densities, wave heights, wave periods, bed resistance, and beach slopes) to identify important influences on mitigating tsunami damage on coastal forest beach.

An Efficient Two-Layer Non-Hydrostatic Model for Investigating Wave Run-Up Phenomena

In this paper, we study the maximum run-up of solitary waves on a sloping beach and over a reef through a non-hydrostatic model. We do a modification on the non-hydrostatic model derived by Stelling and Zijlema. The model is approximated by resolving the vertical fluid depth into two-layer system. In contrast to the two-layer model proposed by Stelling, here, we have a block of a tridiagonal matrix for the hydrodynamic pressure. The equations are then solved by applying a staggered finite volume method with predictor-corrector step. For validation, several test cases are presented. The first test is simulating the propagation of solitary waves over a flat bottom. Good results in amplitude and shape preservation are obtained. Furthermore, run-up simulations are conducted for solitary waves climbing up a sloping beach, following the experimental set-up by Synolakis. In this case, two simulations are performed with solitary waves of small and large amplitude. Again, good agreements are obtained, especially for the prediction of run-up height. Moreover, we validate our numerical scheme for wave run-up simulation over a reef, and the result confirms the experimental data.

Initial Stage of the Finite-Amplitude Cauchy–Poisson Problem

The nonlinear Cauchy–Poisson problem for an incompressible inviscid fluid to start flowing under gravity is investigated analytically. The general nonlinear initial/boundary-value problem is formulated, including both an initial surface deflection and an initial velocity generated by a pressure impulse on the surface. Two subproblems are: (1) a finite-amplitude surface deflection released from rest; (2) the fluid is forced into motion by a pressure impulse on the initially horizontal surface. Solutions for these two subproblems are given to the leading order. One exact solution is given for the fully nonlinear initial-value problem, where a surface pressure impulse is applied on a surface with finite initial deflection. The concept of the highest non-breaking wave is illustrated by dipole acceleration fields at a state of gravitational release from rest. This is done for two phenomena: run-up of a non-breaking solitary wave on a sloping beach, and free nonlinear sloshing in an open container.

Numerical investigation of breaking solitary wave runup on permeable sloped beach using a nonhydrostatic model

The wave profile of solitary wave has been widely used as a tsunami paradigm for decades to capture the main features of actual tsunamis. Since most of the fatalities were attributed to overland flow of tsunamis, the runup and rundown processes of solitary waves have been extensively studied experimentally and numerically. Comparing to previous research, this paper systematically investigates the hydrodynamic characteristics of breaking solitary wave runup on a permeable sloped beach using a nonhydrostatic model. Firstly, the computational capabilities of present numerical model in predicting solitary wave runup process and flow field in permeable media are well validated with experimental data. Then, a series of computational runs have been performed to study the breaking solitary wave on permeable sloped beach. Effects of three prominent factors on the maximum runup height of breaking solitary wave are considered in detail: the thickness (B), median diameter (d50), and porosity (n) of the permeable layers. The porosity of the beach has a substantial impact on the flow field for breaking waves running up a sloped beach. The maximum wave runup height can be substantially reduced if the permeability of the beach is considered. It is also found that the maximum wave runup height decreases with permeable layer thickness. However, the reduction rate gradually decreases. The influences of median diameter are very limited and can be ignored. In addition, based on our results, we also proposed a new empirical formula to calculate the maximum wave runup height of breaking solitary wave on permeable sloped beach. The findings of this paper are expected to be useful for developing an improved understanding on the runup processes of breaking solitary wave on sloped beach with some permeability.

Two-Layer Non-Hydrostatic Scheme for Simulations of Wave Runup

There are indisputable research supporting scientific argument that propagation of (tsunami) wave from intermediate depth towards shallower coastal area needs dispersive wave model. For tsunami wave simulation, efficiency of the numerical scheme is an important issue. In this paper, the two-layer non-hydrostatic model as developed previously in Pudjaprasetya et al. [2017] “A non-hydrostatic two-layer staggered scheme for transient waves due to anti-symmetric seabed thrust,” J. Earthquake Tsunami 11, 1–17, to study tsunami generation and propagation, is adopted. Restricting to 1+1 dimension, here, we focus on the performance of the scheme in simulating wave propagation in coastal areas, in particular predicting the run-up height. First, we conducted a simulation of harmonic wave over a sloping beach to conform the analytical shoreline motion by Carrier and Greenspan [1958] “Water waves of finite amplitude on a sloping beach,” J. Fluid Mech. 4, 97–109. The ability of the scheme in accommodating dispersion and non-linearity were shown via simulation of a solitary wave that propagates over a flat bottom. This solitary wave simulation provides an evaluation of the convergence aspect of the model. Further, several benchmark tests were conducted; a solitary wave over a sloping beach to mimic the experimental data by Synolakis [1986] “The run-up of solitary waves,” J. Fluid Mech. 185, 523–545, as well as solitary wave over a composite beach. Good agreement with laboratory data was obtained in terms of wave signal, whereas for relatively low amplitude, the solitary run-up height conforms the analytical formula. Moreover, the scheme is tested for simulating the Beji–Battjes experiment Beji, S. and Battjes, J. A. [1993] “Experimental investigation of wave propagation over a bar,” Coast. Eng. 19, 151–162. As well as wave focusing experiment by Kurnia et al. [2015] “Simulations for design and reconstruction of breaking waves in a wavetank,” Proc. ASME 2015 34th Int. Conf. Ocean, Offshore and Arctic Engineering, Newfoundland, Canada, 31 May–5 June 2015, pp. 2–7.

Pressures on a rubble-mound breakwater crown-wall for tsunami impact

Rubble mound-breakwaters are commonly constructed with a parapet or crown-wall at the crest. The design of these superstructures depends on the expected storm wave load history throughout the service life cycle. In tsunami-prone areas, this design must include tsunami actions since their loads can definitely exceed those of storm waves. However, tsunami loads are rarely accounted for. This study aimed at incorporating tsunami actions in the design of crown-walls of rubble-mound breakwaters for the first time. Within this scope, laboratory experiments on a scaled model of a typical Mediterranean rubble-mound breakwater typology under tsunami actions were conducted. This paper is the continuation of our previous paper, Aniel-Quiroga et al. (2018) [1], in which experiments were presented and a stability analysis of the armor units was conducted. This research paper presents the second part of the analysis focused on understanding the pressures that crown-walls of rubble-mound breakwaters must support due to tsunami-like actions. These pressures were measured and analyzed, providing the horizontal and uplift pressure time series and laws. The magnitude and timing of the maximum pressure peaks were identified. The maximum horizontal pressure is caused by the first impact of the tsunami. By contrast, the maximum uplift pressure is prompted by a pressure wave generated by the overtopped water falling into the leeside. This pressure wave penetrates the structure from the rear slope. As a result of this analysis, pressures were characterized, allowing the presentation of a new complete methodology that provides, for the tested structure, the design procedure of the crown-wall under tsunami actions. A new formulation to calculate the run-up of solitary waves on the tested rubble-mound breakwater slope is presented here.

Large eddy simulation modeling of tsunami-like solitary wave processes over fringing reefs

Abstract. Many low-lying tropical and subtropical reef-fringed coasts are vulnerable to inundation during tsunami events. Hence accurate prediction of tsunami wave transformation and run-up over such reefs is a primary concern in the coastal management of hazard mitigation. To overcome the deficiencies of using depth-integrated models in modeling tsunami-like solitary waves interacting with fringing reefs, a three-dimensional (3-D) numerical wave tank based on the computational fluid dynamics (CFD) tool OpenFOAM® is developed in this study. The Navier–Stokes equations for two-phase incompressible flow are solved, using the large eddy simulation (LES) method for turbulence closure and the volume-of-fluid (VOF) method for tracking the free surface. The adopted model is firstly validated by two existing laboratory experiments with various wave conditions and reef configurations. The model is then applied to examine the impacts of varying reef morphologies (fore-reef slope, back-reef slope, lagoon width, reef-crest width) on the solitary wave run-up. The current and vortex evolutions associated with the breaking solitary wave around both the reef crest and the lagoon are also addressed via the numerical simulations.

A Study of the Maximum Momentum Flux in the Solitary Wave Run-Up Zone over Back-Reef Slopes Based on a Boussinesq Model

This study utilized a shock-capturing Boussinesq model FUNWAVE-TVD to investigate the maximum momentum flux in the solitary wave run-up zone over back-reef slopes. Validation results of the present model were compared to the previous version of FUNWAVE using the eddy viscosity breaking model to demonstrate the advantages of the shock-capturing method in predicting the breaking solitary wave transformation and run-up over fringing reefs. A series of numerical experiments was designed comprehensively and performed then to obtain a new formulation for the envelope of the spatial distribution of the maximum momentum flux within the solitary wave run-up zone over back-reef beaches, which is different from the one used over uniformly-sloping beaches. Finally, the effects of the variation of reef parameters (i.e., the fore-reef slope angle, reef flat width, and water depth over the reef flat) on the maximum momentum flux at the initial shoreline were investigated to better understand the role of fringing reefs in the mitigation of tsunami hazard.

Momentum Conservative Scheme for Simulating Wave Runup and Underwater Landslide

This paper focuses on the numerical modelling and simulation of tsunami waves triggered by an underwater landslide. The equation of motion for water waves is represented by the Nonlinear Shallow Water Equations (NSWE). Meanwhile, the motion of underwater landslide is modeled by incorporating a term for bottom motion into the NSWE. The model is solved numerically by using a finite volume method with a momentum conservative staggered grid scheme that is proposed by Stelling & Duinmeijer 2003 [12]. Here, we modify the scheme for the implementation of bottom motion. The accuracy of the implementation for representing wave runup and rundown is shown by performing the runup of harmonic wave as proposed by Carrier & Greenspan 1958 [2], and also solitary wave runup of Synolakis, 1986 [14], for both breaking and non-breaking cases. For the underwater landslide, result of the simulation is compared with simulation using the Boundary Integral Equation Model (BIEM) that is performed by Lynett and Liu, 2002 [9].