Two classical bench mark cases have been selected to validate the model capability in solving the run up problem. They are one and two dimensional run up problems. The results are to be elaborated in detail.
The experimental study of the solitary wave run up on a plane beach has been investigated earlier by Synolakis (1986). He presented a variation of run up height for non-breaking and breaking solitary waves, which have then been used by several researchers such as Zelt, 1991, Lynett et al. (2002), Yamazaki et al. (2008) and Mahdavi and Talebbeydokhti (2009) in validating their models. Thus, the current lattice Boltzmann model will be validated using this accepted case. The purpose of this validation is to investigate the capability of the current model in simulating the one-dimensional run up problem
Figure 2 illustrates the solitary wave and sloped beach, where H is the solitary wave height, h0 is the still water level, β is the beach slope, L is the wavelength and R is the run up height measured above the still water level.
The profile of the solitary wave at t=0 is defined by Synolakis (1986). The initial wave crest is located at half wavelength from the toe of the beach, i.e,
For all numerical computation, ∆x=∆y=0.05 and ∆t=0.01s were used to calculate solitary wave with H/h0=0.0185 and H/h0=0.04 propagating up on a 1:19.85 slope beach. Based on Madsen and Sorensen (1992), the inclusion of the bottom friction becomes highly important when investigating the higher degree of breaking wave. In which, the smaller the total water depth, the more important bottom friction becomes. Therefore, the bed friction effect with the manning's coefficient value n=0.01 has been included in the model to improve the model capability to imitate the real fluid behaviours such in the experimental data. In term of the stability, the τ =1 has been picked for this case since it is the safest in term of fluid kinematic viscosity stability [18]. The variable t* in this case is based on
In the simulation, the extrapolation scheme has been applied nearest to the wet- dry boundary, which it depends on the minimum water depth that has been chosen for the program. In this study, the hmin=0.001 has been selected in terms of stability.
For the case H/h0=0.0185, the computed water surface profile at different times are shown and compared with the experimental data in Figure 3.
The lattice Boltzmann model predictions showed in common the surface profile pattern as recorded by Synolakis (1986).
Based on the figures, the wave is propagating on the sloping beach from t*=25 to t*=45 and accelerating down on the sloping beach from t*=50 to t*=70.
The evaluation of the surface profile for all comparison is greatly predicted.
The small disagreement numerical result can be found at t*=40, where the numerical result near the shoreline is slightly over predicted. It is believed to be happened as the selection of the hmin =0.001 and the grid size ∆x=∆y=0.05 are not the good combination for the particular case.
In general, the figures show that the model has a good capability in predicting the one-dimensional waves run up.
Figure 4 shows a comparison of breaking solitary waves run up with incident wave height H/h0=0.04.
From the figure, the motion of wave climbing onto the sloping beach can be seen from t*=20 to t*=38. After the fluid reaches its maximum level at t*=44, the run down process is taking place as the fluid accelerates down the beach
The all simulation results shown in the figure are well predicted. The small discrepancies are seen in Figure 4 at t*=44 and beyond, in which the breaking numerical results are slightly over predicted the experiments.
In terms of breaking phenomenon, a breaking equation has been derived by Synolakis (1986), in which the solitary wave H/h0=0.04 should break on 1:19.85 slope beaches. However, the numerical results shown in the Figure 4 are contradicted from the statement and it also supported by Synolakis (1986), in which the wave does not break in the laboratory. It also confirmed by Titov and Synolakis (1995).
Figure 3. Comparison between computed and experimented water levels at t*=25, 30, 35, 40, 45, 50, 55, 60, 65 and 70. The solid lines represent numerical results and symbols are experimental data by Synolakis (1987).
Figure 4. Comparison between computed and experimented water levels at t*=20, 26, 32, 38 44, 50, 56, 62. The solid lines represent numerical results and symbols are experimental data by Synolakis (1987).
6.2 Solitary wave run up on the conical island (two-dimensional)
A series of laboratory experiments to investigate the solitary waves run up around a conical island with water depth h=0.32m and h=0.42m has been done by Briggs et al. (1995). Three different solitary waves height per water depth H/h0=0.05, 0.1 and 0.2 were tested in the experiment.
According to the experiments, the length and width of the basin are 25m and 30m respectively. An island of conical frustum with 0.625m height and 1:4 side of slope was placed at the centre of the basin with the base and crest diameter are 7.2m and 2.2m correspondingly.
Figure 5 shows a schematic diagram of the conical island used in the experiment and the gauge locations which recorded the time series of surface elevation. The symbol - shows the locations of gauge 6, 9, 22 and 16.
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A
A
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1800
00
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2700
22
6
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Figure 5. (a) The plan views of the conical island and its gauges locations. (b) The side views of the island shape from A-A direction.
In the current numerical study, the experimental result with h=0.32m are used to validate the model. The incident wave heights used in the study are such as H/h0=0.045, 0.096 and 0.181, which slightly lower than the experimental data. The selected incident wave heights were believed closely agreed with the experimental results [5, 7, and 8].
The numerical computational setup consist of 250x300 square lattices, ∆x=∆y=0.1, ∆t=0.01s and τ =1. For the smooth concrete surface roughness, the manning's coefficient value, n=0.016 is used in the computation.
The technique of simple extrapolation by Lynett et al. (2002), which is generally known to be no doubt to use within any of numerical schemes, has been applied within the current model. Hence, hmin = 0.001 is used along with this wet and dry extrapolation scheme.
Figure 6 and 7 show the snapshots of the surface elevation transformation along the x-direction. The snapshots are purposely to show the run up and run down phenomenon in better way. The time selection for this figures are randomly picked.
Generally, the activities such as diffraction, refraction, breaking and shoaling occurred during the wave processes. Hence, all these activities can be seen in the Figures 6 and 7 respectively.
Diffracting and refracting waves are clearly shown in the front and the back of the island at the t=32.8s. Where the soliton breaks along the backside of the island as trapped waves intersect, Madsen and Sorensen (1992). Meanwhile, at t=31.4s shows the motion of the waves shoaling on the island.
In order to validate the model properly, the comparison of the computational results and the experimental data of surface elevation in time series results for the H/h0=0.045, 0.096 and 0.181 are shown in figure 8 (a), (b) and (c) respectively.
Overall, the present lattice Boltzmann model has predicted the waves run up and run down around the conical island quite good. However, for all comparison, the secondary depression waves are not predicted well, which is similar to Madsen and Sorensen (1992). The numerical results shown the current model is poorly predicting the depression following the main wave. It is clearly can be seen at Figure 8 for the initial waves case H/h0=0.096 and H/h0=0.181. In which the model was slightly under and over predicted the laboratory experiments at the tail of the island, Gauge 22 and Gauge 9 respectively.
Finally, a comparison result of maximum waves run up around the island for the H/h0=0.045, 0.096 and 0.181 has been plotted in the Figure 9. Generally, the computed results show a good agreement with the experimented, except for the case H/h0=0.181 where the result is clearly under predicted the laboratory experiments from angle 300 to 600.
From the results shown in the figures, it clearly indicated that the capability of the model in solving the two-dimensional problems is still need more investigation. An effort should be put in studying the application of the moving boundary towards the two-dimensional models.
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Figure 6. The waves processes around the island: (a) H/h0=0.045 and (b) H/h0=0.096.
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Figure 7. The waves processes around the island: (c) H/h0=0.181.
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Figure 8. Time series of solitary waves profile: (a) H/h0=0.045, (b) H/h0=0.096 and (c) H/h0=0.181. The solid line represents the numerical results.
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Figure 9. The maximum waves run up around the island: (a) H/h0=0.045, (b) H/h0=0.096 and (c) H/h0=0.181. The solid line represents the numerical results.
Summary
In this paper, a lattice Boltzmann model incorporating the nonlinear shallow water equation in predicting the 1D and 2D solitary waves run up is presented. The application of the moving shoreline towards the model in the two classical cases: 1) Solitary waves run up on plane beach and 2) the solitary waves run up around conical island has been validated properly. The results shown for the 1D and 2D cases are well agreed with the experimental data. Except for the lack of dispersion term in the governing equation which caused some discrepancies towards the wave breaking results. Moreover, due to the subcritical flow limitation in the model equation has brought the current model results poorly predicting the breaking phenomenon. In which, the breaking waves involved with the supercritical flows. Due to the average results shown in the 2D cases, an investigation should be made specifically on the ability of the model in simulating the subcritical flows. Hence, improve the current results in the future. Overall, the method which is simple and efficient presented in the current paper is capable of producing a good agreement in simulating the waves run up and run down phenomenon.
Abstract
A lattice Boltzmann model for the shallow water equations including the source terms equations such as bed slope and bed friction is used to predict the waves run up phenomenon. A simple linear extrapolation scheme has been applied within the model to treat the wet-dry shoreline boundaries. Due to the breaking waves, the standard subgrid-scale stress model is applying into the model as to predict the turbulence behaviour in the fluid flows. The results are compared with the available experimental data, which show that the model is capable of predicting the waves run up.
