A Unique Finite Element Modeling of the Periodic Wave Transformation over Sloping and Barred Beaches by Beji and Nadaoka's Extended Boussinesq Equations (original) (raw)
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This paper presents a numerical model based on one-dimensional Beji and Nadaoka's Extended Boussinesq equations for simulation of periodic wave shoaling and its decomposition over morphological beaches. A unique Galerkin finite element and Adams-Bashforth-Moulton predictor-corrector methods are employed for spatial and temporal discretization, respectively. For direct application of linear finite element method in spatial discretization, an auxiliary variable is hereby introduced, and a particular numerical scheme is offered to rewrite the equations in lower-order form. Stability of the suggested numerical method is also analyzed. Subsequently, in order to display the ability of the presented model, four different test cases are considered. In these test cases, dispersive and nonlinearity effects of the periodic waves over sloping beaches and barred beaches, which are the common coastal profiles, are investigated. Outputs are compared with other existing numerical and experimental data. Finally, it is concluded that the current model can be further developed to model any morphological development of coastal profiles.
CALCULATION OF SOLITARY WAVE SHOALING ON PLANE BEACHES BY EXTENDED BOUSSINESQ EQUATIONS
In this study, shoaling phenomenon is analyzed using Galerkin finite element approach. This numerical scheme is applied to the extended Boussinesq equations derived by for simulation of shoaling on plane beaches. For spatial discretization, quadratic elements with three-station Lagrange interpolation polynomials are used for horizontal velocity and the water surface elevation. However, for time discretization, two different numerical schemes are used. The first method is a combination of semi-implicit schemes with low-order backward finite difference for time integration and the second method is high-order Adam-Bashforth-Moulton predictor-corrector strategy. Based on this numerical approach, shoaling phenomenon caused by propagation of a solitary wave on sloped beaches is modeled and the results are compared with the available results from the fully nonlinear potential flow model. Considering the fact that the extended Boussinesq equations are affected by nonlinear effects, a non-dimensional parameter called "Asymmetric Parameter" is introduced. This parameter expresses the effects of the travelled distance of the solitary wave as well as the relative wave height on the resulting wave asymmetry. Finally, using this parameter, shoaling coefficient has been computed in an appropriate range.
Ocean Engineering, 2005
A non-linear wave propagation model, based on the higher order depth-integrated Boussinesqtype equations for breaking and non-breaking waves, was applied to predict irregular wave transformation in two horizontal dimensions. A new source function, adapted for the proposed equations, is introduced inside the computational domain, to generate the desired short-crested waves. The dissipation due to the roller is introduced in the momentum equation in order to simulate wave breaking. Bottom friction and sub-grid turbulent processes are also introduced in the model. At the open boundaries a damping layer is applied together with a radiation boundary condition. Model results are compared with experimental measurements, containing tests with normal or oblique to the shore long-and short-crested irregular waves. The comparisons show that the model is able to simulate successfully the non-linear evolution of a unidirectional or a multidirectional wave filed in the nearshore zone, under the effects of refraction, shoaling, and breaking.
In this study, free surface elevation is predicted by using a new finite element scheme. This numerical method solves a Nwogu Boussinesq equation system to simulate wave propagation in the complicated bathymetry of coastal regions. The numerical approach is based on a Galerkin finite element approach for spatial discretization and Adam-Bashforth- Moulton predictor-corrector strategy for time integration. Governing equations are rewritten in lower-order forms by introducing a novel form of auxiliary variable in order to make the application of the linear finite element method possible. Then, the stability of the suggested finite element schemes is studied using a theoretical analysis. For the validation of the present numerical method, five test cases are considered to show the capability of the numerical model for simulating the free surface elevation of wave propagation over different beach profiles where the nonlinear and dispersive effects are so important. The simulated results agree well with experimental observations.
A new shoreline boundary condition for a highly nonlinear 1DH Boussinesq model for breaking waves
2008
In order to model the wave motion and, in turn, the flow, within the nearshore region, in the last decades the derivation and the application of Boussinesq type of models have been extensively investigated. Nevertheless, in the framework of such depth integrated numerical models, the problems of modeling wave breaking and moving onshore boundary at the shoreline are not trivial and several approaches have been proposed to overcome these limits. In the present work an effort toward a more physical based model of the surf and the swash zone has been accomplished. In particular, starting from the work of Musumeci et al. (2005), a new model of the shoreline boundary condition has been implemented. The shoreline boundary condition is developed with a fixed grid method with a wet-dry interface and with a linear extrapolation (Lynett et al. 2002) near the wet-dry boundary has been used and coupled with the shoreline equations (Prasad and Svendsen, 2003). To validate the model a classical test which adopts a monochromatic wave train over a plane beach has been performed. In particular the analytical solution derived by Carrier and Greenspan (1958) has been used for comparison. The comparison between the analytical and numerical horizontal shoreline movements, gives a fairly good agreement. Other tests on breaking of solitary waves have been performed. The solitary wave shoreline oscillation is investigated by comparison with the experimental tests by Synolakis (1986). The results are in fairly good agreement with the experimental data.
Numerical Modeling of Wave Propagation, Breaking and Run-Up on a Beach
Lecture notes in computational science and engineering, 2009
A numerical method for free-surface flow is presented at the aim of studying water waves in coastal areas. The method builds on the nonlinear shallow water equations and utilizes a non-hydrostatic pressure term to describe short waves. A vertical boundary-fitted grid is used with the water depth divided into a number of layers. A compact finite difference scheme is employed that takes into account the effect of non-hydrostatic pressure with a few number of vertical layers. As a result, the proposed technique is capable of simulating relatively short wave propagation, where both frequency dispersion and nonlinear shoaling play an important role, in an accurate and efficient manner. Mass and momentum are strictly conserved at discrete level while the method only dissipates energy in the case of wave breaking. A simple wet-dry algorithm is applied for a proper calculation of wave run-up on the beach. The computed results show good agreement with analytical and laboratory data for wave propagation, transformation, breaking and run-up within the surf zone.
Modeling of wave run-up by using staggered grid scheme implementation in 1D Boussinesq model
A new numerical method is presented to study free surface waves in coastal areas. The method is based on the phase resolving variational Boussinesq model (VBM) which is solved on a computational staggered grid domain. In this model, the nonhydrostatic pressure term has been incorporated in order to correctly described short wave dynamics. In simulating run-up phenomena, a special treatment, so-called thin layer method, is needed for solving the elliptic equation of the Boussinesq model. As a result, the proposed scheme is capable of simulating various run-up phenomena with great accuracy. Several benchmark tests were conducted, i.e., run-up experiments by Synolakis (Int. J. Numer. Methods Fluids 43(12), 1329-1354 1987) for non-breaking and breaking case, a run-up case proposed by Carrier and Greenspan (J. Fluid Mech. 4(1), 97-109 1958) and a dam-break with shock wave Aureli et al. (J. Hydraul. Res. 38(3), 197-206 2000). Moreover, the ability of the numerical scheme in simulating dispersion and nonlinearity effects were shown via simulation of the broad band waves propagation, i.e., focusing wave and irregular wave. The propagation of regular wave above a submerged trapezoidal bar was shown to confirm Beji-Batjes experiment (Coast. , 151-162 1993). Moreover, the numerical model is tested for simulating regular wave breaking on a plane beach of Ting and Kirby (Coast. , 51-80 1995), and for simulating random wave over a barred beach of Boer (1996).
A shoreline boundary condition for a highly nonlinear Boussinesq model for breaking waves
Fuel and Energy Abstracts
A physically based strategy was used to model swash zone hydrodynamics forced by breaking waves within a Boussinesq type of model. The position and the velocity of the shoreline were determined continuously in space by solving the physically-based equations of the shoreline motion; moreover, a fixed grid method, with a wet–dry interface, was adopted for integrating the Boussinesq model. The numerical stability of the model was improved by means of an extrapolation method. To validate the proposed methodology, the classical analytical solution for the shoreline motion of a monochromatic wave train over a plane beach was considered. The comparison between the analytical and numerical horizontal shoreline movements provided a very good agreement. Several other tests on the run-up of non-breaking and breaking waves were performed as well. These tests showed that the proposed model was always in fairly good agreement with the experimental data, even in complex hydrodynamic situations like those forced by breaking solitary waves. In particular, in comparison with other state-of-the-art shoreline models, in all the analyzed cases the proposed model allowed much better predictions of the shoreline velocity to be obtained.► Physically based Lagrangian shoreline model for highly non-linear Boussinesq models. ► The number of calibration parameters are strongly reduced. ► Evolution of solitary breaking and non-breaking waves considering bottom friction. ► Better and more stable behavior than others method. ► The relative errors in maximum horizontal excursion of the shoreline are less than 1%.
Numerical Modeling and Experiments for Solitary Wave Shoaling and Breaking over a Sloping Beach
2004
This research deals with the validation of fluid dynamic models, used for simulating shoaling and breaking solitary waves on slopes, based on experiments performed at the Ecole Supérieure d'Ingénieurs de Marseille's (ESIM) laboratory. A separate paper, also presented at this conference, reports on experiments. In a first part of this work, a fully nonlinear potential flow model based on a Boundary Element Method (BEM) developed at the University of Rhode Island (URI), is used to generate and propagate solitary waves over a slope, up to overturning, in a set-up closely reproducing the laboratory tank geometry and wavemaker system. The BEM model uses a boundary integral equation method for the solution of governing potential flow equations and an explicit Lagrangian time stepping for time integration. In a second part, several Navier-Stokes (NS) models, developed respectively at MASTER-ENSCPB, IMFT, IRPHE, LSEET, and LaSAGeC, are initialized based on the BEM solution and used for modeling breaking solitary waves in a finely discretized region encompassing the top of the slope and the surfzone. The NS models are based on the Volume of Fluid Method (VOF). This paper mostly deals with the first part, which includes calibration and comparison of BEM results with experiments, for the generation of solitary waves in the physical wave tank. Thus, parameters of the physical wave tank were numerically matched, including tank geometry and motion of the wavemaker paddle corresponding to the generation of solitary waves. Use and coupling of the BEM and VOF models for the simulation of solitary wave breaking is discussed in the paper.