Bed stress assessment under solitary wave run-up
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2012
Received: 31 October 2010
Accepted: 21 February 2011
Published: 24 October 2012
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.
Key wordsWave run-up numerical simulation shallow water equation boundary layer bed stress
The Great Tsunami of 2004 demonstrated the impact of tsunami waves both near and far (Geist et al., 2006). While flooding is a primary concern, tsunami waves cause large amounts of sediment transport, and drastically change the coastal morphology. They are also known to cause erosion, as seen in the East Java tsunami (Tsuji et al., 1995). Sediment transport processes under a wave motion are closely related to the bed shear stress, which is influenced by the boundary layer beneath the wave itself (Vittori and Blondeaux, 2008).
Studies of the evolution of tsunami waves are most often conducted using the long-wave approximation of the governing equations of motion. This study also uses numerical solutions and laboratory experiments. Synolakis (1987) conducted a series of laboratory experiments measuring the run-up of solitary waves. That study had a significant impact in the field, because he also provided analytical results for what he referred to as the canonical problem, i.e., a solitary wave propagating over a constant depth and then running up a sloping beach. His laboratory results are often used as benchmarks for validating numerical models.
There are several types of numerical models for the study of tsunami wave run-up. The SWE model is commonly used in tsunami wave run-up simulation and is known to model tsunami evolution well (Liu et al., 1991) and it is known for its computational efficiency, relatively good accuracy, and ease of modifying. Titov and Synolakis (1998) have used the SWE model to model a range of real tsunami inundation, including the 1993 Okushiri tsunami. More recently, Adityawan (2007) applied the SWE model to various cases of wave run up in shallow water, including the inundation of the 2004 tsunami in Banda Aceh, Indonesia. Nevertheless, the SWE model provides few details of the flow near the boundary. Variants of the Boussinesq equation (Boussinesq, 1872; Peregrine, 1967; Madsen and Sϕrensen, 1992; Nwogu, 1993; Wei and Kirby, 1995) have also been applied to various cases of wave run-up. Such models provide a higher accuracy than the SWE for shorter waves, but require more computation time and are not as flexible as the SWE in the treatment of alternating wet/dry and boundaries. Surprisingly, they have not been demonstrated to provide more accurate run-up predictions for tsunami run-up even for landslide waves (Lynett et al., 2003).
Two-dimensional (one horizontal and one vertical dimension) models have been proposed solving Reynolds Averaged Navier Stokes (RANS) with turbulence closure using a two-equation model, such as given by NEWFLUME (Lin, et al., 1999) or CADMAS SURF (Isobe, et al., 1999). Liu et al. (2005) proposed a full 3-D model using RANS. All such models adopt the VOF method and are able to reproduce the flow behavior in detail. However, they require exceptionally long computation times and are temperamental in terms of stability, and, hence, are so far fairly limited in the application to modeling analytical results or laboratory experiments. Other than possibly the case of landslide generated waves very nearshore, the Boussinesq and RANS models do no better in terms of inundation prediction than the SWE model, which enables implementation in practical applications, but is limited in its calculation of bottom shear stresses.
The SWE model consists of two simultaneous equations: conservation of mass (also known as continuity) and conservation of momentum. The empirical method of Manning is commonly used to evaluate bed shear stress. Because it was originally developed for steady river flow, the near-bed flows under wave motion cannot always be explained using the Manning approximation method. In nature, the bed stress changes its sign in a deceleration phase to a direction opposite to the free stream velocity, as shown by Liu et al. (2007). Yet, the Manning method calculates bed stress in the same direction as the velocity. It is clear that, for a detailed analysis, the Manning method is not sufficient.
There have been several studies of bed stress under wave motion. Tanaka (1988) estimated the bottom shear stress under non-linear waves by a modified stream function theory and proposed a formula to predict bed load transport, except near the surf zone in which acceleration effects play an important role. Furthermore, Tanaka and Thu (1994) have shown the importance of friction and phase differences between velocity and bed stress under waves, exactly where the Manning method fails to explain the switch. In general, more complete bed stress formulations may incorporate both velocity- and acceleration-related terms, or may include phase lag (e.g., Kabiling and Sato, 1993; Nielsen, 2002).
Suntoyo and Tanaka have used a boundary-layer approach to assess bed stress under solitary waves. They have demonstrated good accuracy of their bed stress approximation in numerical computations. Two-equation models are often used to assess the boundary-layer properties with k-ϵ and k-ω being the most common. The k-ω model has the ability to accommodate the roughness effects as bed boundary conditions and is considered to be more accurate in assessing the boundary-layer properties (Suntoyo, 2006). Adityawan et al. (2009) used a 2D k-ω model to investigate boundary-layer properties under a periodic sinusoidal wave motion, with good results.
The objective of this study is to assess bed stress under solitary wave run-up using a simultaneous coupling of SWE with a k-ω model. Each model is benchmarked with reference to a test case. Both models are coupled by replacing the conventional Manning method with a direct assessment of bed stress from the boundary layer using a k-ω model. Thus, bed stress can be approximated directly from the boundary layer using a k-ω model. The method is also used to simulate the canonical problem of Synolakis (1987). Further analyses are carried out regarding bed stress assessment.
2. Model Development
2.1 Coupling method
The governing equations are SWE and a k-ω equation. The models are calculated separately at each time step; however, their results are exchanged between the models, enabling a simultaneous calculation. The basic premise for the calculation is to upgrade the SWE model by replacing the Manning terms often used in the SWE using the direct calculation of bed stress in the near bed region using a k-ω model.
2.2 SWE model
A wet-dry moving boundary condition is applied in the model to allow run-up simulation. A minimum threshold depth is selected. If the calculated water depth is lower than the threshold, then the water depth and velocity in the corresponding grid is given a zero value (dry cell).
2.3 The k-ω model
3. Numerical Simulation
The developed method consists of two different models. Each model is verified for an appropriate case. The k-ω model is used to simulate the bed stress fluctuation under random waves. The SWE model is used to simulate the run-up case.
3.1 k-ω model verification
3.2 SWE model verification
4. Bed Stress Assessment
An assessment of the bed shear stress under a solitary wave run-up was carried out in this study, using the conventional Manning method, and a new method based on the k-ω equations. Both methods were used to simulate the evolution of solitary waves for the canonical problem of Synolakis (1987) for a non-breaking wave condition, with satisfactory results. Comparisons of run-up heights and detailed free surface profiles particular at the moving shoreline, show that the new method exhibits smaller deviations, compared with laboratory measurements, than the Manning method. This is perhaps to be expected, since the Manning method was developed for steady unidirectional flows, and cannot reproduce well the bed stress under a wave motion, except with ad-hoc assumptions. The Manning method fails to explain the phase shift of the velocity peak to the bed stress. It also fails to predict the change of sign during the deceleration phase in the solitary wave profile. The method proposed in this study is able to assess bed stress behavior under wave motion. Bed stress comparisons between the Manning method and the new method shows that the Manning method tends to higher magnitudes in a shallow area and lower magnitudes in deeper areas. Further analysis of bed stress accumulation shows that the bed stress away from the shoreline will be more dominant during the entire runup process.
This research was supported by Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (No. 22360193), and partially supported by the Open Fund for Scientific Research from the State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, China. The first author is a scholarship holder under the auspices of the Indonesian Ministry of Education. The authors would also like to thank Professor Synolakis and an anonymous reviewer for their constructive review of the present paper.
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