Higher-order Boussinesq Equations Research Articles

Error analysis of the exponential wave integrator sine pseudo-spectral method for the higher-order Boussinesq equation

AbstractIn this paper, we derive a new exponential wave integrator sine pseudo-spectral (EWI-SP) method for the higher-order Boussinesq equation involving the higher-order effects of dispersion. The method is fully-explicit and it has fourth order accuracy in time and spectral accuracy in space. We rigorously carry out error analysis and establish error bounds in the Sobolev spaces. The performance of the EWI-SP method is illustrated by examining the long-time evolution of the single solitary wave, single wave splitting, and head-on collision of solitary waves. Numerical experiments confirm the theoretical results.

Solitary Wave Interaction with a Floating Pontoon Based on Boussinesq Model and CFD-Based Simulations

A mathematical model of solitary wave interaction with a pontoon-type rigid floating structure over a flat bottom is formulated based on Boussinesq-type equations under weakly nonlinear dispersive waves. Based on the higher-order Boussinesq equations, the solitary wave equation is derived, and a semi-analytical solution is obtained using the perturbation technique. On the other hand, brief descriptions of the application of wave2Foam and OceanWave3D on the aforementioned problem are presented. The analytical solitary wave profiles in the outer region are compared with Computational Fluid Dynamics (CFD) and OceanWave 3D model simulations in different cases. The comparison shows a good level of agreement between analytical, wave2Foam, and OceanWave3D. In addition, based on the wave2Foam and coupled OceanWave3D model, the horizontal, vertical wave forces, and the pressure distributions around the pontoon are analysed. Further, the effect of the Ursell number, pontoon length, and water depth on the solitary wave profiles are analysed based on the analytical solution. The paper validates each of the three models and performs intercomparison among them to assess their fidelity and computational burden.

Characteristics of the Soliton Molecule and Lump Solution in the 2 + 1 -Dimensional Higher-Order Boussinesq Equation

The soliton molecules, as bound states of solitons, have attracted considerable attention in several areas. In this paper, the 2 + 1 -dimensional higher-order Boussinesq equation is constructed by introducing two high-order Hirota operators in the usual 2 + 1 -dimensional Boussinesq equation. By the velocity resonance mechanism, the soliton molecule and the asymmetric soliton of the higher-order Boussinesq equation are constructed. The soliton molecule does not exist for the usual 2 + 1 -dimensional Boussinesq equation. As a special kind of rational solution, the lump wave is localized in all directions and decays algebraically. The lump solution of the higher-order Boussinesq equation is obtained by using a quadratic function. This lump wave is just the bright form by some detail analysis. The graphics in this study are carried out by selecting appropriate parameters. The results in this work may enrich the variety of the dynamics of the high-dimensional nonlinear wave field.

Group classification and exact solutions of a higher-order Boussinesq equation

We consider a family of sixth-order Boussinesq equations in one space dimension with an arbitrary nonlinearity. The equation was originally derived for a one-dimensional lattice model considering higher order effects, and later it was re-derived in the context of nonlinear nonlocal elasticity. In the sense of one-dimensional wave propagation in solids, the nonlinearity function of the equation is connected with the stress-strain relation of the physical model. Considering a general nonlinearity, we determine the classes of equations so that a certain type of Lie symmetry algebra is admitted in this family. We find that the maximal dimension of the symmetry algebra is four, which is realized when the nonlinearity assumes some special canonical form. After that we perform reductions to ordinary differential equations. In case of a quadratic nonlinearity, we provide several exact solutions, some of which are in terms of elliptic functions.

Painlevé Analysis, Soliton Molecule, and Lump Solution of the Higher-Order Boussinesq Equation

The Painlevé integrability of the higher-order Boussinesq equation is proved by using the standard Weiss-Tabor-Carnevale (WTC) method. The multisoliton solutions of the higher-order Boussinesq equation are obtained by introducing dependent variable transformation. The soliton molecule and asymmetric soliton of the higher-order Boussinesq equation can be constructed by the velocity resonance mechanism. Lump solution can be derived by solving the bilinear form of the higher-order Boussinesq equation. By some detailed calculations, the lump wave of the higher-order Boussinesq equation is just the bright form. These types of the localized excitations are exhibited by selecting suitable parameters.

Qualitative Description of Swashing Motion States on Mild Beach Slope

The swashing motion on mild beach slope is dominated by the motion of low frequency waves (LFWs). Companying such a motion, there are two types of swashing motion states, occurrence or no occurrence of LFW’s collision. The present study distinguishes the two states qualitatively by relating it to the number of generated LFWs for the case of two incident wave groups. A simplified swashing index is established theoretically for this purpose. A series of related experiments were performed to observe the generated out-going LFWs on different mild slope from 1:20 to 1:160 and to determine the critical value of the swashing index. Numerical simulations based on higher order Boussinesq equations are also performed to help the recognition of the LFWs generated in the experiment.

Analytical solution of groundwater waves in unconfined aquifers with sloping boundary

A new analytical solution is derived for tide-driven groundwater waves in coastal aquifers using higher-order Boussinesq equation. The homotopy perturbation solution is derived using a virtual perturbation approach without any pre-defined physical parameters. The secular term removal is performed using a combination of parameter expansion and auxiliary term. This approach is unique compared with existing perturbation solutions. The present first-order solution compares well with the previous analytical solutions and a 2D FEFLOW solution for a steep beach slope. This is due to the fact that the higher-order Boussinesq equation captures the streamlines better than ordinary Boussinesq equation based on Dupuit’s assumption. The slope of the beach emerges as an implicit physical parameter from the solution process.

Water wave solutions of Zufiria’s higher-order Boussinesq type equations and its stability

The weakly nonlinear Hamiltonian theory for two-dimensional irrotational shallow water waves is formulated. Zufiria’s higher-order Boussinesq model for waves on water of finite depth is obtained. The formulation Zufiria’s higher-order Boussinesq type equations are studied by transforming them into solvable ordinary differential equations. By implementing different types of extended direct algebraic method, we show that in fact this model has new exact solutions. An extended direct algebraic methods are applied to construct solitary wave solutions.

Simulation of tsunami generation, propagation and coastal inundation in the Eastern Mediterranean

Abstract. In the present work, an advanced tsunami generation, propagation and coastal inundation 2-DH model (i.e. 2-D Horizontal model) based on the higher-order Boussinesq equations – developed by the authors – is applied to simulate representative earthquake-induced tsunami scenarios in the Eastern Mediterranean. Two areas of interest were selected after evaluating tsunamigenic zones and possible sources in the region: one at the southwest of the island of Crete in Greece and one at the east of the island of Sicily in Italy. Model results are presented in the form of extreme water elevation maps, sequences of snapshots of water elevation during the propagation of the tsunamis, and inundation maps of the studied low-lying coastal areas. This work marks one of the first successful applications of a fully nonlinear model for the 2-DH simulation of tsunami-induced coastal inundation; acquired results are indicative of the model's capabilities, as well of how areas in the Eastern Mediterranean would be affected by eventual larger events.

Soft shore protection methods: The use of advanced numerical models in the evaluation of beach nourishment

Beach nourishment is one of the worldwide most common soft shore protection methods. However, the design of these projects is usually based on empirical equations and rules, leaving large margins of error regarding their expected efficiency. In the present work, an advanced wave and sediment transport numerical model is developed and tested in the evaluation of beach nourishment. Non-linear wave transformation in the surf and swash zone is computed by a non-linear breaking wave model based on the higher order Boussinesq equations, for breaking and non-breaking waves. The new Camenen and Larson (2007), transport rate formula for non-cohesive sediments (involving unsteady aspects of the sand transport phenomenon) is adopted for estimating the sheet flow sediment transport rates, as well as the bed load and suspended load over ripples. Suspended sediment transport rate is incorporated by solving the 2DH depth-integrated transport equation. Model results are compared with experimental data of both profile (cross-shore) and planform morphology evolution; the agreement between the two is considered to be quite satisfactory.

Modelling of climate change impacts on coastal flooding/erosion, ports and coastal defence structures

AbstractIn the present work, a numerical model for the redesign of the existing and future ports and coastal defence structures is presented. The model is mainly based on the higher order Boussinesq equations and describes nonlinear wave transformation in the surf and swash zone, wave structure interaction, breaking wave induced currents and morphological changes. The existing model is adapted to describe climate change impacts on coastal flooding/erosion, ports and coastal defence structures. Model applications result to the estimation of: wave overtopping over the breakwaters crests, wave entering the harbour basin through diffraction, coastal erosion and storm surge/wave flooding. The model provides coastal engineers with a useful tool for the redesign of the existing coastal structures.

Reproducing laboratory-scale rip currents on a barred beach by a boussinesq wave model

The pioneering work of Haller [8] on physically investigating bathymetry-controlled rip currents in the laboratory is a standard benchmark test for verifying numerical nearshore circulation models. In this paper, a numerical model based on higher-order Boussinesq equations was developed to reproduce the number of experiments involved in such an investigation, with emphasis on the effect of computational domain size on the numerical results. A set of Boussinesq equations with optimum linear properties and second-order full nonlinearity were solved using a higher-order finite difference scheme. Wave breaking, moving shoreline, bottom friction, and mixing were all treated empirically. The developed model was first run to simulate the rip current under full spatial and time-domain conditions. The computed mean quantities, including wave height, mean water level, and mean current, were compared with the experimental data and favorable agreements were found. The effects of computational domain size on the computation results were then investigated by conducting numerical experiments. The Willmott index was introduced to evaluate the agreements between the computed results and data. Inter-comparisons between the computation results and measurements demonstrated that the computational domain size significantly influenced the numerical results. Thus, running a Boussinesq wave model under full spatial and time-domain conditions is recommended to reproduce Haller's experiment.

Stability of Solitary Waves for the Generalized Higher-Order Boussinesq Equation

This work studies the stability of solitary waves of a class of sixth-order Boussinesq equations.

A 3-D time-domain coupled model for nonlinear waves acting on a box-shaped ship fixed in a harbor

A 3-D time-domain numerical coupled model is developed to obtain an efficient method for nonlinear waves acting on a box-shaped ship fixed in a harbor. The domain is divided into the inner domain and the outer domain. The inner domain is the area beneath the ship and the flow is described by the simplified Euler equations. The remaining area is the outer domain and the flow is defined by the higher-order Boussinesq equations in order to consider the nonlinearity of the wave motions. Along the interface boundaries between the inner domain and the outer domain, the volume flux is assumed to be continuous and the wave pressures are equal. Relevant physical experiment is conducted to validate the present model and it is shown that the numerical results agree with the experimental data. Compared the coupled model with the flow in the inner domain governed by the Laplace equation, the present coupled model is more efficient and its solution procedure is simpler, which is particularly useful for the study on the effect of the nonlinear waves acting on a fixed box-shaped ship in a large harbor.

Boussinesq-type equations for nonlinear evolution of wave trains

Nonlinear evolution of wave trains involves amplitude dispersion and four-wave resonant interaction and hence is difficult to describe using a simple wave equation such as the cubic Schrödinger equation or conventional Boussinesq equations. The present study develops a set of improved higher-order Boussinesq equations with a wide accuracy range of third-order nonlinear characteristics, including amplitude dispersion, and with superior performance for simulations of the nonlinear evolution of wave trains. The equations are obtained by enhancing the higher-order Boussinesq-type equations developed by Zou [Z.L. Zou, A new form of high-order Boussinesq equations, Ocean Eng. 27 (2000) 557–575] through introducing two nonlinear terms into the expression for the computation velocity. The new terms can improve the nonlinear property at higher order by adjusting their free parameters to match the theoretical solutions for amplitude dispersion and the third-order transfer function. Super- and sub-harmonics of bichromatic waves are also improved. The new equations are applied to simulate the nonlinear evolution of wave groups along a 1D wave tank with flat bottom, and nonlinear refraction and diffraction of regular wave trains over a cylindrical ramp, good effectiveness is found.

Further improvements to the higher-order Boussinesq equations: Bragg reflection

Enhancements for the Bragg reflection are introduced for three sets of 2D higher order Boussinesq equations to improve the prediction of the Bragg reflection. The extension of the approach to other sets of Boussinesq equations is discussed. The analytical solutions for the Bragg reflection over an infinite number of sinusoidal bars are derived for these Boussinesq models and compared to the exact theoretical solution in order to determine the optimized values of the parameters in the new enhancement terms. Numerical simulations are also carried out for the Bragg reflection over a finite number of sand bars and compared with corresponding measurements to validate the enhancements. Comparisons with other forms of Boussinesq models are made to discuss the applicability of different forms of Boussinesq models to rapidly varying topography with sand bars. The effects of the mild slope assumption on the prediction of Bragg reflection and of wave reflection on a plane self are also discussed.

A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity

In one space dimension, a non-local elastic model is based on a single integral law, giving the stress when the strain is known at all spatial points. In this study, we first derive a higher-order Boussinesq equation using locally non-linear theory of 1D non-local elasticity and then we are able to show that under certain conditions the Cauchy problem is globally well-posed.

A three-dimensional coupled numerical model of nonlinear waves in a harbor

A 3-D time-domain numerical coupled model for nonlinear waves acting on a ship in a harbor has been developed in the present study. The whole domain is divided into the inner domain and the outer domain. The inner domain is the area around the ship, where the flow is expressed by the Laplace equation and numerically solved by the finite element method. The other area is the outer domain, where the flow is described by the higher-order Boussinesq equations and numerically solved by the finite difference method. The matching conditions on the interfaces between the inner domain and the outer domain, the procedure of coupled solution, the length of common domain and the mesh generation in the inner domain are discussed in detail. The other coupled model with the flow in the inner domain governed by the simplified linear Euler equations and relevant physical experiment are adopted to validate the present coupled model, and it is shown that the numerical results of the present model agree with the experimental data, so the present model can be used for the study on the effect of nonlinear waves acting on a fixed ship in a large area and provide a reference for the time-domain simulation of nonlinear wave forces on an arbitrary object in a large harbor and the 3-D district computation in the future.

Alternative forms of the higher-order Boussinesq equations: Derivations and validations

An alternative form of the Boussinesq equations is developed, creating a model which is fully nonlinear up to O( μ 4) ( μ is the ratio of water depth to wavelength) and has dispersion accurate to the Padé [4,4] approximation. No limitation is imposed on the bottom slope; the variable distance between free surface and sea bottom is accounted for by a σ-transformation. Two reduced forms of the model are also presented, which simplify O( μ 4) terms using the assumption ε = O( μ 2/3) ( ε is the ratio of wave height to water depth). These can be seen as extensions of Serre's equations, with dispersions given by the Padé [2,2] and Padé [4,4] approximations. The third-order nonlinear characteristics of these three models are discussed using Fourier analysis, and compared to other high-order formulations of the Boussinesq equations. The models are validated against experimental measurements of wave propagation over a submerged breakwater. Finally, the nonlinear evolution of wave groups along a horizontal flume is simulated and compared to experimental data in order to investigate the effects of the amplitude dispersion and the four-wave resonant interaction.

Higher-order Boussinesq equations for two-way propagation of shallow water waves

Standard perturbation methods are applied to Euler's equations of motion governing the capillary-gravity shallow water waves to derive a general higher-order Boussinesq equation involving the small-amplitude parameter, α = a / h 0 , and long-wavelength parameter, β = ( h 0 / l ) 2 , where a and l are the actual amplitude and wavelength of the surface wave, and h 0 is the height of the undisturbed water surface from the flat bottom topography. This equation is also characterized by the surface tension parameter, namely the Bond number τ = Γ / ρ g h 0 2 , where Γ is the surface tension coefficient, ρ is the density of water, and g is the acceleration due to gravity. The general Boussinesq equation involving the above three parameters is used to recover the classical model equations of Boussinesq type under appropriate scaling in two specific cases: (1) | 1 3 − τ | ≫ β , and (2) | 1 3 − τ | = O ( β ) . Case 1 leads to the classical (ill-posed and well-posed) fourth-order Boussinesq equations whose dispersive terms vanish at τ = 1 3 . Case 2 leads to a sixth-order Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [P. Daripa, W. Hua, A numerical method for solving an illposed Boussinesq equation arising in water waves and nonlinear lattices, Appl. Math. Comput. 101 (1999) 159–207] as a dispersive regularization of the ill-posed fourth-order Boussinesq equation. The relationship between the sixth-order Boussinesq equation and fifth-order KdV equation is also established in the limiting cases of the two small parameters α and β.