NUMERICAL SIMULATION OF SOLITARY WAVE RUN-UP AND OVERTOPPING USING BOUSSINESQ-TYPE MODEL*

TSUNG Wen-Shuo

Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan 701, E-mail: slavik_1942@hotmail.com

HSIAO Shih-Chun

Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan 701 LIN Ting-Chieh

Tainan Hydraulics Laboratory, National Cheng Kung University, Tainan 701

(Received May 17, 2012, Revised August 30, 2012)

NUMERICAL SIMULATION OF SOLITARY WAVE RUN-UP AND OVERTOPPING USING BOUSSINESQ-TYPE MODEL*

TSUNG Wen-Shuo

Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan 701, E-mail: slavik_1942@hotmail.com

HSIAO Shih-Chun

Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan 701 LIN Ting-Chieh

Tainan Hydraulics Laboratory, National Cheng Kung University, Tainan 701

(Received May 17, 2012, Revised August 30, 2012)

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.

Boussinesq equations, solitary wave, run-up, shoreline, scale effect, overtopping

Introduction

The terrific catastrophe of Great Indian Ocean tsunami in 2004[1-3]and the 2011 Tohoku earthquake tsunami[4]raised significant concern all over the world. Based on field evidence, the water inundation and debris flow that accompany a tsunami wave affect water motion in coastal areas. The run-up/run-down heights are thus extensively studied in tsunami wave research.

Due to their simplicity, tsunami-like solitary waves and shallow water wave equations (SWEs) are frequently utilized to investigate the run-up and rundown heights of tsunami waves[1,5,6]. The informationobtained in these studies has led to significant progress in warning systems established for tsunami disaster mitigation. However, owing to the physical limitations of SWEs (i.e., hydrostatic pressure and uniform horizontal velocity profile assumptions), SWE-based models are unsuitable for problems in which both frequency dispersion and wave nonlinearity are of great concern[7]. In addition, large-scale experiments on solitary waves propagating over a mild slope have rarely been conducted due to the limits of laboratory facilities. Hence, the reliability of numerical model analyses is questionable[1,6].

Researchers have developed highly robust and accurate solvers for run-up and overtopping with the SWEs[8]. Nevertheless, few Boussinesq-based models have been used to study overtopping, one such model was proposed by Stansby and Feng[9]. One potential reason is that to simulate dispersive wind waves, the Boussinesq model usually requires a complicatednumerical scheme for accuracy which does not lend itself to capturing the complex flow patterns that are common with overtopping. Navier-Stokes-based approaches[10,11]have been shown to be very accurate in predicting overtopping rates on a small scale. However, solving the Navier-Stokes equations has led to an extremely high computational cost, and their using range is generally restricted to a low number of precise waves and structure conditions.

The objectives of this article are thus twofold: to investigate the shoreline properties caused by breaking and non-breaking solitary waves on plane beaches with various inclinations and to understand whether the Boussinesq-type equations can well simulate the wave overtopping the coastal structures. Numerical simulations based on a high-order Boussinesq model (COULWAVE, Cornell University long and intermediate wave modeling) by Kim et al.[12]and sets of laboratory experiments are conducted in the super flume of breaking solitary waves on 1:20 and 1:60 plane slopes are reported. We not only present new laboratory data of maximum run-up heights on two comparably gentle slopes but also measure the corresponding shoreline trajectory of swash motions for the numerical model calibration. The properties of scale effect on run-up heights of two comparable beaches between small flume[5]and large tank (1:20 and 1:60) are briefly discussed. In particular, present study intends to extend the results of Li and Raichlen[6]based on the nonlinear shallow water wave equations for an extensive concern of maximum run-up heights due to breaking and non-breaking solitary waves on plane beaches. By using the present numerical model with the available experiments and the given laboratory works, their results will be confirmed and extended.

The rest of this study is organized as follows. The Boussinesq-type model and numerical scheme are briefly described in Section 1. The present laboratory methods and scale effects are given in Section 2. In Section 3, the model simulation capability is examined with the available laboratory data and present experiments. Both breaking and non-breaking waves are considered. For the non-breaking cases, the energy dissipation algorithms built in this Boussinesq model are calibrated against the laboratory data by Synolakis[5], Li and Raichlen[6]and the given experiments, respectively. Section 4 then outlines a series of numerical experiments to widely discuss the shoreline kinematics of breaking and non-breaking solitary waves on sloping beaches. The properties of run-up, run-down and shoreline trajectory are also reported. The applicability of the empirical formula proposed by Hsiao et al.[1]for generally predicting the maximum run-up height for breaking solitary waves on plane slopes is also confirmed. Furthermore, solitary waves impacting and overtopping an impermeable seawall at various water depths are investigated. Comparisons are made with experimental data. Finally, Section 5 concludes this paper.

1. Numerical model

The fundamental Boussinesq modeling approach is introduced in this section. The governing equations with viscous effect, and the numerical scheme are presented.

1.1 Governing equations

The fully nonlinear, weakly dispersive, weakly rotational depth-integrated Boussinesq-type equations in conservation form are expressed in one horizontal dimension as given by Kim et al.[12]

where H=ζ+h is the total water depth,andrepresent the second-order terms in the depthintegrated x and y horizontal momentum equations respectively, and Hccontains the second-order terms in the continuity equation. The terms are given by

where

where (ψx,ψy)=ψThe second-order terms in the continuity equation are given by

The physical meanings of the high-order terms can be found in Ref.[12].

In the present study, a quadratic friction equation was used to approximate the bottom stress

For simulating subgrid-scale dissipation, the horizontal eddy viscosity is modeled using the Smagorinsky model, which is given by

The vertical eddy viscosity is given by

1.2 Numerical scheme

In this subsection, only a brief conceptual overview of the numerical solution is given. For more details, refer to Ref.[12].

Considering the extended Boussinesq-type equations, in which there are first- to third-order spatial derivatives, the time integration should be fourth-order accurate to prevent numerical truncation errors from the same order as included derivatives. A third-order Adams-Bashforth predictor and the fourth-order Adams-Moulton corrector scheme are used for the time integration.

The predictor step is

The corrector step is

where Ch=κ/6 is used, u*＇ is the friction velocity, κ is the Von Kármán constant, a value of 0.4 being adopted in this study.

After each predictor and corrector step, a matrix solve r is used to solve P and Q. All the computed physical values are cell-averagedvalues since a cellaveraged finite-volume method is implemented to represent the spatial numerical grid.

To calculate the leading-order (shallow water) terms in the governing equations, a fourth-order compact Monotone Upstream-centered Scheme for Conservation Laws-Total Variation Diminishing (MUSCL TVD) scheme is used to establish the interfacial physical values. Further, the Courant number determines the calculation time step used in the model, a value of 0.5 generally leads to stability and convergence, but for simulations with higher nonlinear waves, a value of as low as 0.1 may be required for stability. A value of 0.25 is utilized as default in this study.

1.3 Initial and boundary conditions

In the COULWAVE model, for a solitary wave, thecorresponding free surface elevations and velocities are described using conventional Boussinesq theory, in which the characteristic wave length is determined based on 95% solitary wave volume. In addition, the wave crest is located at a user-specified position. In the limited computational domain of a numerical model, the appropriate boundary conditions are needed. In this model, the sponge layer of the absorbing boundary used here absorbs both mass and wave energy, and has been shown to be an excellent absorber for all types of waves, with negligible reflection. Furthermore, in a numerical wave flume, the target structure must be placed far enough from the wavemaker to prevent the reflected wave from touching the left boundary (i.e., incident side). For this reason, the absorbing boundary condition is utilized for the left boundary to avoid re-reflection wave. In the numerical model, the users can switch between the Finite Difference Method (FDM) and the Finite Volume Method (FVM) solvers. The FDM is traditionally used, and is the scheme used by the original numerical code. TheFDM provides considerably high accuracy with relative fast computation. The downside of the FDM is that it is very sensitive to incisive fronts and shocks, making it prone to crashing. The FVM uses a highorder, shock-capturing, approximate Riemann solver for the leading-order flux terms, and is extremely stable and accurate. This scheme adds numerical dissipation for situations with barely resolved shocks, where the wave is not resolved enough for breaking to occur, the shock-capturing properties of the solver preserve the steep front and dissipate energy numerically. Note that the FVM approach requires 50% to 100% or more computational time compared with the FDM solution. Considering its stability, the FVM approach is used here in all overtopping cases.

2. Laboratory setup and procedure

2.1 Solitary waves over sloping beaches

The following variables and the given notations areutilized throughout this paper. x and z represent the abscissa and ordinate in the Cartesian coordinate system, respectively, H0is the initial wave height, η is the free surfaceelevation, h0is the still water depth, h is the local water depth,β is the slope of plane beach, Ruand Rdare the vertically maximum deviation of initial shoreline for the run-up and run-down motions, respectively. Also, we note that in the following investigationsg the gravitational acceleration, R the time history of water shoreline trajectory, ε=H/h0the wave nonlinearity, the superscript “*” the normalized physical quantity. Moreover, the dimensionless variables are defined as

Table 1(a) Experimental setup and wave condition[1]

The solitary wave run-up laboratory experiments were carried out in a super wave flume with dimensions of 300 m long, 5.0 m width and 5.2 m deep at the Tainan Hydraulics Laboratory (THL) in National Cheng Kung University. The target solitary wave was generated at one end of the wave flume by a programmable wavemaker. Two impermeable concrete slopes with inclinations of tan-1β=20 and tan-1β=60 were utilized. The correspondingwave conditions and experimental setup are summarized in Table 1(a). Totally, 15 and 80-92 wave gauges were deployed along the wave flume to record the local free surface elevation for the tan-1β=20 and tan-1β= 60, respectively. The reference time is defined as the time at which the incident wave crest passed the reference gauge. The trajectory of shoreline movements were accumulatively recorded by the six five-meter segment wires which were placed along the slope bottom. High coincidence of maximum run-up height is given compared the gauge data with the visual observations determined based on the instantaneous location of the wet-dry intersection between the water layer and the beach slope[1]. It should be noted that all waves employed in the present experiments broke during run-up and run-down courses based on laboratory observations. More measured data and analyses followed by the present numerical studies will be performed in the next section. Further laboratory information and discussions on data can be ascertained in the study by Hsiao et al.[1]

Fig.1Definition sketc[2h]of a solitary wave interacting with a sloping seawall

Table 1(b) Experime ntal con[2d]itions of a solitar y wave over a sloping seawall

2.2 Solitary waves impinging and overtopping an impermeable seawall

Figure 1 presents the layout of the experimental wave flume and the symbols for the corresponding physical variables used in analyses. This experiment was conducted by Hsiao and Lin[2]. Additionally,cR is the freeboard in Fig.1, which is defined as the vertical distance from the still water level to the seawall crown. The corresponding wave condition and experimental setup are summarized in Table 1(b). The overtopping experiments were conducted in a two-dimensional flume with a scale of 22 m (length), 0.5 m(width), 0.75 m (depth) at THL. The experimental topography was composed of two sections. One is a uniform and impermeable aluminum 1:20 slope starting at 10 m from the wavemaker (i.e., x=10m), the slope surface of which is smooth plexiglas, which significantly reduces friction. The other section is a trapezoidal object made of plexiglas with seaward and landward slopes of 1:4 and 1:1.8, respectively. The seawall model was rigidly mounted on the slope starting at a horizontal distance of 3.6 m from the beach toe (i.e., x=13.6m). Silicone was used to fill the gaps between the seawall, slope, and side glasswall to prevent infiltration. The free surface elevation along the flume during experiments was measured using 9 wave gauges. A reference gauge was placed at 1.1 m in front of the beach slope (i.e., x=8.9m).

Table 2 Typical comparison of scale effect on maximum run-up height

Fig.2 Comparison of run-up heights between small and large wave flumes. Present modeling (Δx=0.2m, Δt= 0.02 s, f=0.0025)

2.3 Scale effect on run-up height

The property of scale effect is of great interest for various coastal hydrodynamics. This section utilizes our data measured in a large scale tank, the available data obtained in a small flume and present simulated results to discuss the scale effect on run-up height. Table 2 compares run-up data correlated with similar normalized wave nonlinearities of breaking solitary waves climbing uppresent slopes(1:20and 1:60) and the well-known measured data bySynolakis[5]which concern breaking solitary waves travelling upon a 1:19.85 uniform slope in a small wave flume. Moreover, our numerical results for both of corresponding small- and large-scale problems are also incorporated for discussion. Note that the comparisons for gentle and relatively steep slopes are named A and B for the following descriptions, respectively.

Clearly, favorably good agreements in B are given (also see Fig.2 for the comparis ons with the formulaproposed by Synolakis[5]and our numerical results). The maximum deviation of run-up data is approximately 4.3%. The error may be partly due to the different bottom frictions in the two laboratory environments. According to the comparisons between the experimental data and present simulations, these results imply that the scale effects on the run-up problem could be possibly ignored at least for these two particular setups. However, it is also found that the data listed in Table 2 by Synolakis[5]are somewhat scattering for the issue of scale effect. In particular, the increase of water depth with the same wave nonlinearity the corresponding run-up height increase (i.e., =ε 0.188, ~3.8%) and furthermore, with the increase of wave nonlinearity the corresponding run-up height does not significantly increase (i.e., ε=0.188 and 0.193). Although the discrepancy is not prominent, these phenomena seem to respond to theexperimental observations of Jensen et al.[13]that they attributed the discrepancies of run-up heights due to the nearly the same wave nonlinearities with different water depths to the scale effects correlated with the viscosity and

Fig3 Normalized wave evolution (a1and b1) and shoreline trajectory (a2and b2) of a non-breaking solitary wave climbing up a 1: 2.75 plane slope

su rface tension (see Jensen et al.[13], p182).

Interestingly, for a comparison of Aand B we could not observe the same physical phenomena in A as was noted above (i.e., ε=0.07). It can be seen that the discrepancies of the run-up data correlated with similar normalized wave-nonlinearity in A are almost negligible. It reveals that for the same wave flume in our cases the scale effect could still be reasonably ignored. However, it must be also noted that since no available data performed in a small scale wave tank with a slope of 1:60, direct comparisons with ours are not feasible.

Overall, the analysesin this section suggest that the scale effect on run-up height in our large scale flume is reasonably negligible compared to the data from Synolakis[5]. In addition, the similar tendency is also observed in the present numerical simulations. However, for a small wave flume the scale effect on run-up height seems more prominent and causes the data somewhat scattered[5,13]. Whether or not, that this discrepancy is related to scale effect needs further investigation.

3. Numerical model validation

A series of[7,1n2u]merical tests using the COULWAVE model are presented to interpret the applicability of COULWAVE under one horizontaldime nsion conditions in this section. Both published laboratory works and the present experiments are used to examine the accuracy and stability of the model. Note that only one-layer model, which is identical to the extended Boussinesq equations, is applied in this paper. The numerical setup was the same as that employed in the laboratory experiments. Details of the experimental setups can be found in the original papers. The numerical parameters chosen in the model calculations are revealed in the corresponding figure captions. Besides, we emphasized that our numerical results are not grid dependent.

Fig.4 Normalized maximum run-up height of anon-breaking solitary wave on a 1: 2.08 sloping beach

Fig.5 Normalized wave profile snapshot of a breaking solitary wave on a sloping beach

3.1 Non-breaking solitary waves

Figure 3 shows the comparisonof present model results and the laboratory data of Zelt[14]. The numerical results of Zelt[14]based on the conventional Boussinesq equations are also introduced in the figures for a discussion. As was observed by Zelt[14]for the two chosen wave conditions, both of which did not break during the run-up process but one of which strongly broke during the run-down course (i.e., =ε 0.2). Apparently, excellent agreements are found between our numerical results and the laboratory data for the time histories of water surface elevations measured respectively in front of the slope (see Figs.3(a1) and 3(b1)) and for the corresponding shoreline trajectories (see Figs.3(a2) and 3(b2)). The discrepancies betw[1e4e]n COUL WAVE and the numerical results of Zelt are clearly observed especially for the case of relatively strong wave nonlinearity. The results indicate that both models can favorably simulate the wave evolution until maximum run-up heights are achieved while COULWAVE can much better describe the wave deformation during the run-down course when the wave breaks (i.e., oscillating tails). In fact, due to the imbalance of wave nonlinearity and frequency dispersivity on the sloping bottom, oscillating tails with dimensionless wavelength khO(2) at the reference gauge are generated. With such dimensionless wavelength, it is not surprising that the phase of oscillating tails based on the conventional Boussinesq equations does show large discrepancy with measureddata.

Furthermore, it is also worth noting that even our model satisfactorily captures the maximum run-up heights but somewhat underestimates the maximum run-do wn heights compared to the laboratory data for both the cases of small and strong nonlinearities (see Figs.3(a2) and 3(b2)). One reason for causing this is that for a non-breaking wave on a steep slope and the large curvature of the shoreline would lead to larger error[7]. Nevertheless, the overall results are quite favorable compared with the laboratory data.

Figure 4 further gives a comparison of maximum run-up heights between the present model results and the laboratory data from Li[15]. The waves employed in the experiments did not break for the complete evolution courses as was addressed by Li[15]. Three different bottom friction coefficients are introduced to discuss their influence on maximum run-up heights. Clearly, the maximum run-up heights are insensitive to the chosen bottom frictions for a steep slope and additionally the results indicate that COULWAVE can satisfactorily capture the maximum run-up heights.

In short, the model results have proved that the moving shoreline scheme and the energy dissipation algorithm employed in COULWAVE are validated to be satisfactory for a solitary wave climbing up a steep slope. Both wave nonlinearity and frequency dispersion can be well captured during the swash cycle by the present numerical model.

3.2 Breaking solitary waves

The shoreline behaviors of breaking and nonbreaking solitary waves are quite different. For a breaking wave event, the propagat ing bore after wave breaking will continue to climb up beach slope and the wave reflection phenomenon is relatively insignificant compared to a non-breaking wave[5]. Additionally, as waves propagate over a mild slope the effects of bottom friction play an important role on the swash motion owing to the thin layer of leading bore fronts dominates the run-up tongue and the long course of swash motion[7].

Figure 5 depicts the comparisons of wave profile snapshots between our numerical simulations and the laboratory data of breaking solitary waves climbing up two different sloping bottoms. The waves employed in Fig.5all broke during the run-up and run-down stages according to the laboratory observations[5,1]. The numerical model of Li and Raichlen[6]based on the nonlinear shallow water-wave equations (in this paper referred to WENO, see Li and Raichlen[6]for details) are also included for a discussion. Note that in the two experimental simulations a bottom friction coefficient f=0.0025 is chosen in the corresponding model calculations. Clearly, Fig.5 reveals that COULWAVE with the employed numerical parameters and the energy dissipation algorithms are validated to be favorable for both gentle slopes in comparison with the laboratory data. The wave life cycle on sloping beaches including propagation, breaking, bore formation, run-up and run-down can be reasonably recreated (see Hsiao et al.[1]for plentiful descriptions of wave evolution properties and physical deformation mechanisms). It can also be seen that for the case of tan-1β= 19.85 the COULWAVE shows a superior capability for the wave deformation modeling compared to the WENO model of Li and Raichlen[6]. The reason is similar to those given in Section 3.1, that is, the WENO model could not well capture the balance and kinematics of wave nonlinearity and phase dispersion effects of a strong nonlinear wave travelling upon a slope owing to the physical limitations of SWE[6].

Fig.6 Normalized maximum run-up height of a breakingsolitary wave on a sloping beach

Figure 6 represents the comparison of maximum run-up heights of the present model results with the laboratory data of breaking solitary waves moving on threesloping bottoms. Note th at the bottom friction coefficient for the three simulation cases is 0.0025. Obviously, the results indicate that COULWAVE can excellently simulate the maximum run-up heights for the slopes of 1:20 and 1:60 but slightly overestimate the maximum run-up heights for the slope of 1:15. The discrepancy is partially due to the same coefficients of the bottom friction effects used in two different laboratory environments. Nevertheless, fairly good agreements between the model results and measured data in the Fig.6 imply that the chosen bottom friction coefficient is reasonable for the available experimental environments.

Fig .7 Normalized shoreline trajectory time series of a breaking solitary wave on a plane slope

Moreover, Fig.7 compares the computed results with the laboratory data of the time series of shoreline motions for the cases of tan-1β=15, 20 and 60, res pectively. It is noted that the dataof shoreline trajectory(i.e., Fig.7(a), ε=0.3) were not measured by Synolakis[5]but were done by Li and Raichlen[6]with the same experimental conditionsusing a high-speed camera photography. Overall, favorably good agreements are given from the comparisons. We note that from Fig.7 the trajectories of the swash motions on the gentle and relatively gentle slopes are notably different. The maximum shoreline deviation during the run-down processes for the case of tan-1β=60 is nearly close to the original shoreline while for the cases of tan-1β=19.85 and 20 the hydraulic jumps are formed (see Figs.5(a6), 5(b6) and 7). It is because that as the leading bore front after wavebreaking continues to travel upon a gentle beach until the maximum run-up height and the saturation of wave kinetics and potential will be increasingly balanced owing to the incessant effects of gravitational force and the bottom frictions. Therefore as the retreated wave is dragged down the slope and the moving momentum is relatively insufficient compared to the run-up stage. Interestingly, we also observe that for the same slope the arrival time of the maximum run-up are almost the same for different incident wave heights (see Figs.7(b) and 7(c)). More discussions on shoreline movement will be given below.

4. Results and discussion

So far, the aforementioned section elucidates that COULWAVE can simulate propagating solitary wave kinematics on mild and steeper sloping beaches with high stability and accuracy. A seriesof numerical investigations was conducted to widespread study the shoreline properties for non-breaking and breaking solitary waves on various beach slopes.

A numerical slope was constructed with a range of 1≤tan-1β≤60 starting at x=20m in the computational domain. The offshore water depth is fixed at h=1.0m and three wave nonlinearities (ε=0.1, 0.2 and 0.3) were used to determine the characteristics of run-upand run- down heights. Areference gauge was dynamically placed at a distance from the beach toebased on the definition given by Synolakis[5](i.e., Eq.(21)) for mitigating wave reflection. The bottom friction effect on the shoreline movement of a solitary wave is less important for steeper bottoms where wave breaking does not occur. Furthermore, Hsiao et al.[1]experimentally confirmed that the breaking criterion by Grilli et al.[16]is appropriate for a fairly mild slope. Thus, the inclusion of bottom friction effects depends on whether wave breaking occurs during the swashing cycle. The formula given by Grilli et al.[16]is adopted in the numerical calculations

Fig.8 Normalized (a) run-up and (b) run-down of breaking and non-breaking solitary waves on sloping beaches

4.1 Run-up/run-down and shoreline properties

Figure 8 summarizes the calculated results of run-up and run-down of the present numerical experiments. The breaking limitation by Li[15], the run-up law by Synolakis[5], the modified run-up law byLi[15], the co rresponding laboratory data (see the caption of Fig.8) and the empirical formula proposed by Hsiao et al.[1]for generally predicting the maximum run-up heights of breaking solitary waves on sloping beaches (Eq.(22)) are also incorporated into the figures for a comparison.

Overall, the results in Fig.8 reveal that the maximum run-up/run-down height increases and decreases with the decreasing slope angle for non-breaking solitary waves and breaking waves, respectively. The same conclusion was drawn by Li and Raichlen[6], but only for the run-up. Good agreement can be found between our numerical modeling, the available experimental data, results from analytic approaches, and results calculated using Eq.(22) shown in Fig.8(a). Even though the results are quite promising, there are no available data to validate the run-down results for such a wide range of wave conditions and beach slopes. Nevertheless, it is instructive to observe how the run-down changes with the nonlinearity and slope. Clearly, the run-down height is insensitive to the initial wave conditions for beach slopes below 1:25. This is not surprising because the long course of propagation on a mild slope leads to energy dissipation. Therefore, near saturation on the shoreline is achieved. The run-down height is greatly influenced by wave nonlinearity for the slope of tan-1β＜20. In particular, the run-down height increases and decreases with decreasing beach slope until the maximum is reached for non-breaking and breaking events, respectively. This tendency is similar to that for run-up.

Fig.9 Normalized time series of shoreline motion

Figure 9 describes the wave nonlinearityeffects on the shoreline trajectories for breaking an d nonbreaking waves on sloping beaches. The difference of run-up/run-down heights between these two slopes increases with increasing wave nonlinearity. In particular, while the rundown for non-breaking wave is approximately proportional to wave nonlinearity, the wave nonlinearity appears to have no influences on run-down for the case of breaking wave. Furthermore, there is a clear time delay of wave arrival time with decreasing wave nonlinearity for the case of breaking wave. In contrast, the wave arrival time of run-up phenomenon is insensitive to wave nonlinearity which is identical to the findings in Fig.7.

4.2 Impingement and overtopping on a seawall due to solitary waves

In this subsection, numerical simulations were conducted to evaluate the ability of COULWAVE to simulate fluid motion under complex circumstances, such as a wave impacting and overtopping a seawall or breakwater. To our knowledge, only very few studies that used SWEs to simulate solitary wave overtopping are documented[17]. Therefore, more extensiveand comprehensive c[2]mparisons with experimental data of Hsiao and Lin were made in this subsection. Acceptable agreement with experimental data would validate the use of COULWAVE to predict overtopping volume in coastal areas.

Fig.10 Time series of free surface elevation due to wave overtopping the impermeable seawall

Three typical solitary wave cases with different breaking locations were used tosimulate approaching tsunami waves. In the first case, a violent bore rushes landward and subsequently contacts and overtops the seawall. In the second case, a wave directly collapses on the seawall with overtopping flow immediately generated. In the third case, a wave straightforwardly overtops the seawall crown and breaks behind the seawall. These cases were chosen because they exhibit the characteristics of tsunami waves illustrated in tsunami disaster reports.

T o acqu ire suit abl e compar iso ns, t he numerical flumesetupshouldbeidenticaltotheexperimentalsetup. For computational efficiency, an adequate numerical flume scale must be established. In wave condition of type 1, 150 grids were used to describe one wave length (i.e., Δx=0.015m ) and the slope starts at x=15.0m. 100 and 250 grids were used for simulating one wave length (i.e., Δx=0.027m , Δx=0.02m) in wave types 2 and 3, respectively. Therefore, the numerical reference gaugewas settled at x =13.9m. For describing the bottom friction effects in the laboratory, a value of roughness height Ks=10-4m was adopted throughout all simulations. Additionally, the total physical simulation time was 30 s. The corresponding time step was automatically adjusted during the calculations to satisfy the stability constrain by the advection process, in which the maximum time step was chosen to match the sampling rate of the experimental measurements. The aforementioned computational conditions were utilized in this portion.

Table 3(a) Total overtopping discharge for type 1 wave condition obtained from simulation and experiments[2]

Table 3(b)Total overtopping discharge for type 2 wave condition obtained from simulation and experiments

Table 3(c)Total overtopping discharge for type 3 wave condition obtained from simulation and experiments

Figure 10 shows the agreement betweenlaboratory data and numerical results of a time series in local free surface elevation along the flume. A sequence of oscillatory waves due to reflection by the seawall at about time t=11.5s can be clearly observed (see Figs.10(a2), 10(b2) and 10(c2)). The reflected wave amplitudes decrease as the freeboard decreases. A reasonable explanation is that in the bore formation and impinging case (i.e.,type 1), the wave breaks earlier compares to the other two cases, the wave breaks in front of the seawall and losses considerable wave energy. Most of the wave energy is thus blocked before the seawall and continuously interacts with the seawall front. In the meantime, a reflected tail wave is produced by the retreating fluid that does not overtop the crown completely. Furthermore, the time series data of free surface elevation shown in Figs.10(c4) and 10(c6) obviously respond the second wave breaking behind the seawall, in that distinct surface variations take place after the peak in the laboratory data. Even ifthis Boussinesq model spreads out an excellent ability to simulate wave breaking and overtopping at most conditions. There still were oscillating shocks in the free surface elevation of Fig.10(b6). This appearance of the oscillating shocks is due to the highly nonlinear processes caused by strong wave collapsing and backwashing. Nevertheless, the crucial and complex circumstances such as wave breaking and overtopping are always difficult and challenging problems for the numerical model with depth-integrated equations to resolve.

Fig.11 Comparison of calculated overtopping discharges with experimental data for three solitary wave conditions

In the experiment, the overtopping discharge is the water volume which gets across the seawall crest and reaches the leeward side of the seawall. Hence, after the wave overtops, the water body accumulates between the slope and the seawall. Thus, the experimen tal measurement of the water level is settling a wave gauge at the toe at the leeward side of the seawall. Once the water level is obtained, the overtopping discharge per unit width can be estimated using a basic geometrical relation. In the simulation, the setup must be identical to the experimental conditions for a valid comparison. The simulation results obtained using the Boussinesq equations and the experimental data are compared in Tables 3(a), 3(b), 3(c) and Fig.11. A fairly good match can be observed. The Boussinesq model is considerably accurate in predicting the propagation of a solitary wave overtopping and its computational cost is much lower than those of other numerical models. As a result, the current Boussinesq model can be adequately used to predict the process of solitary waves overtopping the seawall with highly efficiency and acceptable accuracy.

5. Concluding remarks

The shoreline motions of non-breaking and breaking solitary waves on plane beaches, and the interactions between tsunami-like solitary waves and an impermeable seawall on a 1:20 sloping beach have been investigated using a modified COULWAVE of Kim et al.[12]. Sets of laboratory data of run-up, shoreline motion and surface elevation of breaking solitary waves climbing up two plane slopes (tan-1β=20 and 60) are reported. In addition, the experimental data of solitary waves overtopping an impermeable seawall is also presented. The scale effect on run-up height of two comparably gentle slopes is discussed. Our analysis suggests that the scale effectcould be reasonably ignored based on limited comparison of experimental data and present numerical results, however, the data from Jensen et al.[13]show inconsistently with our analysis. No decisive conclusions can be drawn and further experimental investigations are needed to verify this issue. In the numerical part for both breaking and non-breaking waves, the model simulation capability is well-validated to be satisfactory by recreating the available experiments and the given laboratory works. The moving boundary scheme and energy dissipation algorithm built in the model are reasonably validated. Numerical simulations have been used to widely study the run-up and run-down heights of non-breaking and breaking solitary waves on various slopes. The maximum run-up/run-down height increases and decreases with increasing slope angle for non-breaking and breaking waves, respectively. The empirical formula proposed by Hsiao et al.[1]was confirmed to be suitable for applications to different nonlinearity tsunami waves on beach slopes with a wide range. The shoreline properties between non-breaking and breaking solitary waves are distinct owing to the wave saturation effects correlated to the energy dissipation processes. The effect of wave non-linearity and freeboard caused by interactions between waves and the seawall was identified. The overtopping discharge monotonically increased with the increasing wave nonlinearity for a given water depth. The results elucidate that the present depth-integrated model can be used as an efficient tool for predicting a wide spectrum of coastal problems.

Acknowledgements

This work was financially supported by the National Science Council (Grant NSC 101-2628-E-015-MY3). The authors acknowledge Dr. Hwang K. S. for his laboratory assistance and discussion. The authors gratefully thank the research team at the Tainan Hydraulics Laboratory of National Cheng Kung University for their support in providing experimental data.

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10.1016/S1001-6058(11)60318-1

* Biography: TSUNG Wen-Shuo (1987-), Male, Ph. D. Candidate

HSIAO Shih-Chun,

E-mail: schsiao@mail.ncku.edu.tw