Solitary Wave Run-up Research Articles

Wave Impact Simulations by an Improved ISPH Model

AbstractThis paper presents an improved incompressible smoothed particle hydrodynamics (ISPH) method for wave impact applications. In most conventional ISPH techniques the source term of the pressure Poisson equation (PPE) is usually treated by either a density invariant or a velocity divergence-free formulation. In this work, both the density invariant and velocity divergence free formulations are combined in a weighted average form to determine the source term. The model is then applied to two problems: (1) dam-breaking wave impact on a vertical wall and (2) solitary wave run-up and impact on a coastal structure. The computational results have indicated that the combined source term treatment can predict the wave impact pressure and force more accurately compared with using either formulation alone. It was further found that depending on the application case, the influence of the density invariant and divergence-free parts could be quite different. For the more violent wave impact case, the divergence-f...

Numerical modelling of wind effects on breaking solitary waves

Wind effects on breaking solitary waves are investigated in this study using a two-phase flow model. The model solves the Reynolds-averaged Navier–Stokes equations with the k−ϵ turbulence model simultaneously for the flows both in the air and water, with the air–water interface calculated by the volume of fluid method. First, the proposed model was validated with the computations of a breaking solitary wave run-up on a 1:19.85 sloping beach in the absence of wind, and fairly good agreement between the computational results and experimental measurements was obtained. Further, detailed information of the water surface profiles, velocity fields, vorticity, turbulent stress, maximum run-up, evolution of maximum wave height, energy dissipation, plunging jet and splash-up phenomena is presented and discussed for breaking solitary waves in the presence of wind. The inclusion of wind alters the air flow structure above water waves, increases the generation of vorticity and turbulent stress, and affects the solitary wave shoaling, breaking and run-up processes. Wind increases the water particle velocities and causes water waves to break earlier and seaward, which agrees with the previous experiment.

Numerical simulation of solitary and random wave propagation through vegetation based on VOF method

A vertical two-dimensional numerical model has been applied to solving the Reynolds Averaged Navier-Stokes (RANS) equations in the simulation of current and wave propagation through vegetated and non-vegetated waters. The k-ɛ model is used for turbulence closure of RANS equations. The effect of vegetation is simulated by adding the drag force of vegetation in the flow momentum equations and turbulence model. To solve the modified N-S equations, the finite difference method is used with the staggered grid system to solver equations. The Youngs’ fractional volume of fluid (VOF) is applied tracking the free surface with second-order accuracy. The model has been tested by simulating dam break wave, pure current with vegetation, solitary wave runup on vegetated and non-vegetated channel, regular and random waves over a vegetated field. The model reasonably well reproduces these experimental observations, the modeling approach presented herein should be useful in simulating nearshore processes in coastal domains with vegetation effects.

Boussinesq modelling of solitary wave and N-wave runup on coast

A total variation diminishing Lax–Wendroff scheme has been applied to numerically solve the Boussinesq-type equations. The runup processes on a vertical wall and on a uniform slope by various waves, including solitary waves, leading-depression N-waves and leading-elevation N-waves, have been investigated using the developed numerical model. The results agree well with the runup laws derived analytically by other researchers for non-breaking waves. The predictions with respect to breaking solitary waves generally follow the empirical runup relationship established from laboratory experiments, although some degree of over-prediction on the runup heights has been manifested. Such an over-prediction can be attributed to the exaggeration of the short waves in the front of the breaking waves. The study revealed that the leading-depression N-wave produced a higher runup than the solitary wave of the same amplitude, whereas the leading-elevation N-wave produced a slightly lower runup than the solitary wave of the same amplitude. For the runup on a vertical wall, this trend becomes prominent when the wave height-to-depth ratio exceeds 0.01. For the runup on a slope, this trend is prominent before the strong wave breaking occurs.

On the run-up and back-wash processes of single and double solitary waves — An experimental study

The run-up and back-wash processes of single and double solitary waves on a slope were studied experimentally. Experiments were conducted in three different wave flumes with four different slopes. For single solitary wave, new experimental data were acquired and, based on the theoretical breaking criterion, a new surf parameter specifically for breaking solitary waves was proposed. An equation to estimate maximum fractional run-up height on a given slope was also proposed. For double solitary waves, new experiments were performed by using two successive solitary waves with equal wave heights; these waves were separated by various durations. The run-up heights of the second wave were found to vary with respect to the separation time. Particle image velocimetry measurements revealed that the intensity of the back-wash flow generated by the first wave strongly affected the run-up height of the second wave. Showing trends similar to that of the second wave run-up heights, both the back-wash breaking process of the first wave and the reflected waves were strongly affected by the wave–wave interaction. Empirical run-up formula for the second solitary wave was also introduced.

Runup of solitary waves on a straight and a composite beach

Particle Image Velocimetry (PIV) and wave gauges have been used to investigate the runup of solitary waves at two different beaches. The first beach is straight with an inclination of 10°, whereas the second is a composite beach with a change in the 10° inclination to 4° at a vertex point above the equilibrium water level. Comparison with numerical simulations using a Navier–Stokes solver with zero viscosity has been performed for the composite beach. Four different amplitudes of incoming solitary waves are investigated.Measurements of the runup show that the composite beach gives a lower runup compared to the straight beach. Furthermore, the composite beach experiences a longer duration of the rundown compared to the straight beach. This is at least partially assumed to be a result of scaling effects, since the fluid above the vertex creates a relatively thinner runup tongue compared to the straight beach scenario.The appearance of a stagnation point at the beach boundary is clearly visible in both the PIV results and the numerical simulation. This stagnation point is originating at the lowermost part of the beach, and is moving upwards with time. It is found that the stagnation point moves faster upwards for the straight beach than for the composite beach. Further, the stagnation point is moving even faster in the numerical simulation, suggesting that the velocity with which the stagnation point moves is influenced by viscous scaling effects.Finally, the numerical simulation seems to capture the physics of the flow well, despite differences in the phase compared to the PIV results. This applies to both the flow field and the surface elevations.

Numerical study of vegetation damping effects on solitary wave run-up using the nonlinear shallow water equations

Vegetation damping effects on propagating water waves have been investigated by many researchers. This paper investigates the effects of damping due to vegetation on solitary water wave run-up via numerical simulation. The numerical model is based on an implementation of Morison's formulation for vegetation induced inertia and drag stresses in the nonlinear shallow water equations. The numerical model is solved via a finite volume method on a Cartesian cut cell mesh. The accuracy of the numerical scheme and the effects of the vegetation terms in the present model are validated by comparison with experiment results. The model is then applied to simulate a solitary wave propagating on a plane slope with vegetation. The sensitivity of solitary wave run-up to plant height, diameter and stem density is investigated by comparison of the numerical results for different patterns of vegetation. The numerical results show that vegetation can effectively reduce solitary wave propagation velocity and that solitary wave run-up is decreased with increase of plant height in water and also diameter and stem density.

Runup and boundary layers on sloping beaches

The present study is devoted to discrepancies between experimental and theoretical runup heights on an inclined plane, which have occasionally been reported in the literature. In a new study on solitary wave-runup on moderately steep slopes, in a wave tank with 20 cm water depth, detailed observations are made for the shoreline motion and velocity profiles during runup. The waves are not breaking during runup, but they do break during the subsequent draw-down. Both capillary effects and viscous boundary layers are detected. In the investigated cases the onshore flow is close to the transitional regime between laminar and turbulent boundary layers. The flow behaviour depends on the amplitude of the incident wave and the location on the beach. Stable laminar flow, fluctuations (Tollmien-Schlichting waves), and formation of vortices are all observed. Comparison with numerical simulations showed that the experimental runup heights were markedly smaller than predictions from inviscid theory. The observed and computed runup heights are discussed in the context of preexisting theory and experiments. Similar deviations are apparent there, but have often been overlooked or given improper physical explanations. Guided by the absence of turbulence and irregular flow features in parts of the experiments we apply laminar boundary layer theory to the inundation flow. Outer flows from potential flow models are inserted in a nonlinear, numerical boundary layer model. Even though the boundary layer model is invalid near the moving the shoreline, the computed velocity profiles are found to compare well with experiments elsewhere, until instabilities are observed in the measurements. Analytical, linear boundary layer solutions are also derived both for an idealized swash zone motion and a polynomial representation of the time dependence of the outer flow. Due to lacking experimental or theoretical descriptions of the contact point dynamics no two-way coupling of the boundary layer model and the inviscid runup models is attempted. Instead, the effect of the boundary layer on the maximum runup is estimated through integrated losses of onshore volume transport and found to be consistent with the differences between inviscid theory and experiments.

Boundary layer approach in the modeling of breaking solitary wave runup

The boundary layer is very important in the relation between wave motion and bed stress, such as sediment transport. It is a known fact that bed stress behavior is highly influenced by the boundary layer beneath the waves. Specifically, the boundary layer underneath wave runup is difficult to assess and thus, it has not yet been widely discussed, although its importance is significant. In this study, the shallow water equation (SWE) prediction of wave motion is improved by being coupled with the k–ω model, as opposed to the conventional empirical method, to approximate bed stress. Subsequently, the First Order Center Scheme and Monotonic Upstream Scheme of Conservation Laws (FORCE MUSCL), which is a finite volume shock-capturing scheme, is applied to extend the SWE range for breaking wave simulation. The proposed simultaneous coupling method (SCM) assumes the depth-averaged velocity from the SWE is equivalent to free stream velocity. In turn, free stream velocity is used to calculate a pressure gradient, which is then used by the k–ω model to approximate bed stress. Finally, this approximation is applied to the momentum equation in the SWE. Two experimental cases will be used to verify the SCM by comparing runup height, surface fluctuation, bed stress, and turbulent intensity values. The SCM shows good comparison to experimental data for all before-mentioned parameters. Further analysis shows that the wave Reynolds number increases as the wave propagates and that the turbulence behavior in the boundary layer gradually changes, such as the increase of turbulent intensity.

Numerical simulation of solitary wave run-up and overtopping using Boussinesq-type model

In this article, the use of a high-order Boussinesq-type model and sets of laboratory experiments in a large scale flume of breaking solitary waves climbing up slopes with two inclinations are presented to study the shoreline behavior of breaking and non-breaking solitary waves on plane slopes. The scale effect on run-up height is briefly discussed. The model simulation capability is well validated against the available laboratory data and present experiments. Then, serial numerical tests are conducted to study the shoreline motion correlated with the effects of beach slope and wave nonlinearity for breaking and non-breaking waves. The empirical formula proposed by Hsiao et al. for predicting the maximum run-up height of a breaking solitary wave on plane slopes with a wide range of slope inclinations is confirmed to be cautious. Furthermore, solitary waves impacting and overtopping an impermeable sloping seawall at various water depths are investigated. Laboratory data of run-up height, shoreline motion, free surface elevation and overtopping discharge are presented. Comparisons of run-up, run-down, shoreline trajectory and wave overtopping discharge are made. A fairly good agreement is seen between numerical results and experimental data. It elucidates that the present depth-integrated model can be used as an efficient tool for predicting a wide spectrum of coastal problems.

BED STRESS INVESTIGATION UNDER BREAKING SOLITARY WAVE RUNUP

The bed stress under breaking solitary wave runup was investigated in this study using the Simultaneous Coupling Method (SCM). The SCM couples the shallow water equation (SWE) with k-w model. The depth averaged velocity from SWE is applied as the upper boundary condition in k-w model for bed stress assessment from the boundary layer. It was found that the boundary layer approach provides more accurate bed stress estimation than the empirical method, which leads to a more accurate prediction of runup and wave profile. The accumulation of bed stress in during solitary wave runup was evaluated. The bed stress on the direction leaving the shoreline will have more impact in the overall process. However, during a short period of run up process, bed stress toward the shoreline may have significant effect as well.

Bed stress assessment under solitary wave run-up

Understanding and forecasting tsunami wave run-up is very important in mitigating tsunami hazards. The bed stress under wave motion governs viscous wave damping and sediment transport processes, which change coastal morphology. One of the most common methods used for simulation is the shallow water equation (SWE) model, often used with a Manning-style approach for modeling bottom friction. Boundary-layer approaches provide better information regarding bed stress, particularly since they are also valid for nonsteady flows. In this study, a simulation of wave run-up is carried out by simultaneous coupling of the SWE model with the k-ω model. The k-ω model is used near the flow boundary at the bottom, only for assessing the boundary layer shear stress. Free stream velocity and calculations of the free surface evolution are obtained from the SWE model. Both this method, and the conventional method, are applied to the canonical problem of a solitary wave propagating over a constant depth and then up a sloping beach (Synolakis, 1987). The new method is found to increase the computational accuracy and physical realism compared to the conventional Manning method. Comparison of bed shear stresses shows that the new method is able to accommodate the effect of deceleration, which leads to sign changes and a phase shift between the free stream velocity and the bed stress. Furthermore, it is found that during the run-up and run-down process, bed stress in the direction of leaving the shoreline is more dominant.

Improved efficiency of a non-hydrostatic, unstructured grid, finite volume model

An efficient depth-integrated non-hydrostatic unstructured grid finite volume model is presented and applied to several test cases, which involve the computation of free surface flows. The model solves the non-linear shallow water equations, with extra non-hydrostatic pressure terms to describe dispersion effects. The efficiency of the model is a major issue when it involves large spatial domains with high resolution meshes. Lumping of the pressure gradient of the horizontal velocities is employed, which reduces the number of non-zero elements in the sparse matrix of the Poisson equation by half. This greatly reduces both the memory requirements and the number of floating point operations required to solve the Poisson equations. In addition, for a model using a collocated grid, both the water surface and the non-hydrostatic pressure need to be defined at the incoming wave boundary. A non-hydrostatic pressure boundary, has been derived based on linear wave theory, that is accompanied with the regular incoming short wave for depth-integrated models. It can serve as a simple way to introduce short incoming waves in models with collocated grids. The model has been validated through several test problems including an oscillating basin, propagation of a solitary wave, wave propagation over a submerged bar, wave refraction and diffraction over an elliptical shoal, as well as solitary wave run-up on a conical island. The model gives good results for all test cases. We show that the lumping of the pressure gradient generates identical results to simulations without lumping, while the execution CPU time is reduced by around 30%, demonstrating a good computational efficiency of the model.

Interaction between Tsunami Waves and Isolated Conical Islands

Abstract Chen, J.M.; Liang, D., and Tang, H., 2012. Interaction between tsunami waves and isolated conical islands. A newly developed computer model, which solves the horizontal two-dimensional Boussinesq equations using a total variation diminishing Lax-Wendroff scheme, has been used to study the runup of solitary waves, with various heights, on idealized conical islands consisting of side slopes of different angles. This numerical model has first been validated against high-quality laboratory measurements of solitary wave runups on a uniform plane slope and on an isoliated conical island, with satisfactory agreement being achieved. An extensive parametric study concerning the effects of the wave height and island slope on the solitary wave runup has subsequently been carried out. Strong wave shoaling and diffraction effects have been observed for all the cases investigated. The relationship between the runup height and wave height has been obtained and compared with that for the case on uniform plane sl...

An experimental observation of a solitary wave impingement, run-up and overtopping on a seawall

A sequence of laboratory experiments using solitary waves was performed to model the effect of leading form of three types of tsunamis (a bore, an impinging wave and an overtopping wave) on a seawall on a sloping beach. The wave evolution process, impinging pressure along the seawall surface, total overtopping discharge behind the seawall and the maximum run-up height on the rear slope were measured and compared. Laboratory data were employed to re-examine relevant empirical formulae in the literature. The effect of the presence of the seawall in reducing maximum run-up height using the present setup was briefly discussed. The present data can be used for calibrating numerical and mathematical models.

浅水流方程式とｋ－ωモデルを組み合わせた孤立波遡上数値計算法の検証

浅水流方程式とｋ－ωモデルを組み合わせた孤立波遡上数値計算法の検証

Run-Up of Solitary Waves on Twin Conical Islands Using a Boussinesq Model

This paper investigates the interaction of solitary waves (representative of tsunamis) with idealized flat-topped conical islands. The investigation is based on simulations produced by a numerical model that solves the two-dimensional Boussinesq-type equations of Madsen and Sørensen using a total variation diminishing Lax–Wendroff scheme. After verification against published laboratory data on solitary wave run-up at a single island, the numerical model is applied to study the maximum run-up at a pair of identical conical islands located at different spacings apart for various angles of wave attack. The predicted results indicate that the maximum run-up can be attenuated or enhanced according to the position of the second island because of wave refraction, diffraction, and reflection. It is also observed that the local wave height and hence run-up can be amplified at certain gap spacing between the islands, owing to the interference between the incident waves and the reflected waves between islands.

Modeling of non-breaking and breaking solitary wave run-up using shock-capturing TVD-WAF scheme

A numerical model based on the Non-Linear Shallow Water (NLSW) equations is developed to study the propagation and run-up of non-breaking and breaking solitary waves. The Total Variation Diminishing version of Weighted Average Flux (TVD-WAF) explicit method in conjugation with the Harten-Lax-van Leer (HLL) approximate Riemann solver estimates the numerical flux in the governing equations. This approach permits an exact evaluation of wave speeds in the presence of dry bed situations. The bed topography and friction source terms are treated in a fully implicit way. The accuracy of the model is verified by recourse to the classical test problems with known analytical solutions as well as experimental data. The computed run-up heights show satisfactory agreement with existing run-up laws.

PROPAGATION AND RUNUP OF TSUNAMI WAVES WITH BOUSSINESQ MODEL

With certain profiles of bottom movements, orders of wave height of submarine earthquake-induced tsunami both in deep ocean and nearshore area have been studied using the Boussinesq equations. An earthquake of large magnitude generates a typical N-wave which can propagate long distance in open ocean without deformation. Since the magnitude and length of tsunami waves related to vertical and horizontal scale of geological movements, solitary wave and N-wave are extended to waves not tied to solitary property which represent tsunami waves better. In a horizontal one dimensional numerical wave flume, runup of solitary wave, N-wave, single crest and N-wave composed by a single crest and a single trough on a slope beach have been simulated. The results fit analytical solutions of nonlinear shallow water equations well. The Indian Ocean tsunami has been simulated with the horizontal two dimensional high order Boussinesq model. Comparison between numerical results and measured data from field survey validates the numerical model.

Oblique runup of non-breaking solitary waves on an inclined plane

When a wave of permanent form is obliquely incident on an inclined plane, the wave pattern becomes stationary in a frame of reference which moves along the shore. This enables a simplified mathematical description of the problem which is used herein as a basis for efficient and accurate numerical simulations. First, a nonlinear and weakly dispersive set of Boussinesq equations for the downstream evolution of such stationary patterns is derived. In the hydrostatic approximation, streamline-based Lagrangian versions of the evolution equations are developed for automatic tracing of the shoreline. Both equation sets are, in their present form, developed for non-breaking waves only. Finite difference models for both equation sets are designed. These methods are then coupled dynamically to obtain a single nonlinear model with dispersive wave propagation in finite depth and an accurate runup representation. The models are tested by runup of waves at normal incidence and comparison with a more general model for the refraction of a solitary wave on a slope. Finally, a set of runup computations for oblique solitary waves is performed and compared with estimates of oblique runup heights obtained from a combination of an analytic solution for normal incidence and optics. We find that the runup heights decrease in proportion to the square of the angle of incidence for angles up to 45°, for which the height is reduced by around 12% relative to that of normal incidence. In Appendix A, the validity of the downstream formulation is discussed in the light of solitary wave optics and wave jumps.