Nonlinear Boussinesq Equation Research Articles

Boussinesq modeling of wave‐induced hydrodynamics in coastal wetlands

AbstractIn this paper, an improved formulation of the vegetation drag force, applicable for the fully nonlinear Boussinesq equations and based on the use of the depth‐varying, higher‐order expansion of the horizontal velocity, in the quadratic vegetation drag law has been presented. The model uses the same numerical schemes as FUNWAVE TVD but is based on the CACTUS framework. The model is validated for wave height and setup, against laboratory experiments with and without vegetation cover. The wave attenuation results using the improved formulation were compared with those using the first‐order reference velocity as well as with analytical solutions using linear wave theory. The analytical solution using the depth‐varying velocity, predicted by the linear wave theory, was shown to match the model results with the fully expanded velocity approach very well for all wave cases, except under near‐emergent and emergent conditions (when the ratio of stem height to water depth is greater than 0.75) and when the Ursell (Ur) number is less than 5. Simulations during peak storm waves, during Hurricane Isaac, showed that vegetation is very effective in reducing setup on platforms and in reducing the wave energy within the first few hundred meters.

Solitons and other exact solutions for variant nonlinear Boussinesq equations

In this article, some integration tools, namely the modified simple equation method, the general Expa-function method, the Jacobi elliptic equation expansion method, the Riccati equation expansion method and the G′/G-expansion method are used to construct exact solutions with parameters and the Jacobi elliptic function solutions for variant nonlinear Boussinesq equations. When these parameters are taken to be special values, the soliton solutions, the singular soliton solutions and trigonometric function solutions are derived from the exact solutions and the Jacobi elliptic function solutions. The obtained results confirm that the proposed methods are efficient techniques for analytic treatment of a wide variety of nonlinear partial differential equations in mathematical physics. We compare the results yielding from these integration tools. Also, a comparison between our results in this article and the well-known results are given.

Exact solutions of nonlinear conformable time-fractional Boussinesq equations using the $$\exp \left( { - \phi \left( \varepsilon \right)} \right)$$ exp - ϕ ε -expansion method

Nonlinear fractional Boussinesq equations are considered as an important class of fractional differential equations in mathematical physics. In this article, a newly developed method called the $$\exp \left( { - \phi \left( \varepsilon \right)} \right)$$ -expansion method is utilized to study the nonlinear Boussinesq equations with the conformable time-fractional derivative. Different forms of solutions, including the hyperbolic, trigonometric and rational function solutions are formally extracted. The method suggests a useful and efficient technique to look for the exact solutions of a wide range of nonlinear fractional differential equations.

Wave transformation and hydrodynamic characteristics of wave-breaking models coupled with Boussinesq equations

ABSTRACTThis paper investigates the different wave transformation and hydrodynamic characteristics of three different wave-breaking models – friction-type, eddy viscosity-type and surface roller-type models – coupled with the weakly nonlinear Boussinesq equations with improved linear dispersion characteristics based on comparisons between predicted and measured wave-breaking phenomena. The experimental cases consider not only different breaker types but also different bathymetry conditions, i.e. uniform slopes and a uniform slope with a submerged breakwater. Although all of the properly calibrated wave-breaking models yield equally good predictions of wave heights, mean water levels and trough levels of breaking and broken waves, the models show clear differences in their predictions of the undertow velocity, near-bottom wave orbital velocity profile, wave skewness, and wave asymmetry. The importance of such differences in the predictive capabilities of these widely used wave-breaking models should be highlighted, particularly when extending the model to the prediction of coastal morphodynamics around the surf zone.

New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method

In this paper, the nonlinear Boussinesq equations with the conformable time-fractional derivative are solved analytically using the well-established modified Kudryashov method. As a consequence, a number of new exact solutions for this type of equations are formally derived. It is believed that the method is one of the most effective techniques for extracting new exact solutions of nonlinear fractional differential equations.

Stability analysis of solutions for the sixth-order nonlinear Boussinesq water wave equations in two-dimensions and its applications

In the present study, we are concerned with the generalized Boussinesq equation including the singular sixth-order Boussinesq equation, which describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number less than but very close to 1/3. By using the extended auxiliary equation method, we obtained some new soliton like solutions for the two-dimensional sixth-order nonlinear Boussinesq equation with constant coefficients. These solutions include symmetrical, non-symmetrical kink solutions, solitary pattern solutions, Jacobi and Weierstrass elliptic function solutions and triangular function solutions. The stability analysis for these solutions is discussed.

Management of groundwater resources using surface pumps: Optimization using Genetic Algorithms and the Tabu Search method

The Visual MODFLOW software has been used to simulate the flow behavior, while both a Genetic Algorithm and a Tabu Search Algorithm have been used to maximize the extracted flowrates and investigate the optimal groundwater management strategy of an unconfined aquifer, for the case when surface pumps are used. The procedure above is applied in the south-west area of the Eidomeni-Evzoni phreatic aquifer, located near the borderline of Greece and the Former Yugoslav Republic of Macedonia (FYROM). The optimal total pumping flowrate has been estimated at 4560.9 m3/d. It is proven that the non-linear Boussinesq equation rather than its linearized counterpart should be used. It is also demonstrated that a coarse uniform grid might underestimate the head drawdown induced by the well; accordingly, a more sophisticated grid, with two refinement zones around each well, should be adopted. The optimization results indicate that the Tabu Search approach is computationally more efficient than the Genetic Algorithms, while both give the same results.

The 1979 Submarine Landslide-Generated Tsunami in Mururoa, French Polynesia

This paper aims at best describing the submarine landslide which induced partial submersion of the atolls of Mururoa and Fangataufa in 1979. More precisely, waves propagated along the south coast of Mururoa atoll and penetrated into its lagoon some minutes after the landslide triggering (t = 0 s), whereas a train of eight water waves reached the runway located on the north-east coast of Fangataufa (40 km south of Mururoa) between t = 7 min 30 s and t = 20 min. A numerical model based on shallow water equations is used to simulate the landslide as well as the associated tsunami. Saint-Venant equations are used to propagate the tsunami in coastal areas, whereas the offshore propagation is simulated by solving weakly nonlinear Boussinesq equations. Low- and high-resolution nested grids are used to simulate the tsunami propagation in deep sea and in shallow waters, respectively. Several scenarios have been tested to reproduce the observed water and run-up heights in the near and far fields. The best scenarios correspond to a landslide with a volume in the range (75–90 Mm3) (for a basal friction angle of 35°) and with a basal friction angle in the range (30°–40°) (for a volume of 80 Mm3). These results have been completed by a parametric study on the slide parameters.

Global well‐posedness of 2D nonlinear Boussinesq equations with mixed partial viscosity and thermal diffusivity

In this paper, we discuss with the global well‐posedness of 2D anisotropic nonlinear Boussinesq equations with any two positive viscosities and one positive thermal diffusivity. More precisely, for three kinds of viscous combinations, we obtain the global well‐posedness without any assumption on the solution. For other three difficult cases, under the minimal regularity assumption, we also derive the unique global solution. To the authors' knowledge, our result is new even for the simplified model, that is, F(θ) = θe2. Copyright © 2017 John Wiley & Sons, Ltd.

Exact Solutions of Atmospheric (2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations**Supported by the National Natural Science Foundation of China under Grant Nos. 11305031 and 11305106, and Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province under Grant No. Yq2013205

Exact solutions of the atmospheric (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space.

Approximation of surface–groundwater interaction mediated by vertical streambank in sloping terrains

New analytical solutions are derived to estimate the interaction of surface and groundwater in a stream–aquifer system. The analytical model consists of an unconfined sloping aquifer of semi-infinite extant, interacting with a stream of varying water level in the presence of a thin vertical sedimentary layer of lesser hydraulic conductivity. Flow of subsurface seepage is characterized by a nonlinear Boussinesq equation subjected to mixed boundary conditions, including a nonlinear Cauchy boundary condition to approximate the flow through the sedimentary layer. Closed form analytical expressions for water head, discharge rate and volumetric exchange are derived by solving the linearized Boussinesq equation using Laplace transform technique. Asymptotic cases such as zero slope, absence of vertical clogging layer and abrupt change in stream-stage can be derived from the main results by taming one or more parameters. Analytical solutions of the linearized Boussinesq equation are compared with numerical solution of corresponding nonlinear equation to assess the validity of the linearization. Advantages of using a nonlinear Robin boundary condition, and combined effects of aquifer parameters on the bank storage characteristic of the aquifer are illustrated with a numerical example.

Modeling Bed Evolution Using Weakly Coupled Phase-Resolving Wave Model and Wave-Averaged Sediment Transport Model

In this paper, we propose a model for the simulation of the bed evolution dynamics in coastal regions characterized by articulated morphologies. An integral form of the fully nonlinear Boussinesq equations in contravariant formulation, in which Christoffel symbols are absent, is proposed in order to simulate hydrodynamic fields from deep water up to just seaward of the surf zones. Breaking wave propagation in the surf zone is simulated by integrating the nonlinear shallow water equations with a high-order shock-capturing scheme. The near-bed instantaneous flow velocity and the intra-wave hydrodynamic quantities are calculated by the momentum equation integrated over the turbulent boundary layer. The bed evolution dynamics is calculated starting from the contravariant formulation of the advection-diffusion equation for the suspended sediment concentration in which the advective sediment transport terms are formulated according to a quasi-three-dimensional approach, and taking into account the contribution given by the spatial variation of the bed load transport. The model is validated against several tests by comparing numerical results with experimental data. The ability of the proposed model to represent the sediment transport phenomena in a morphologically articulated coastal region is verified by numerically simulating the long-term bed evolution in the coastal region opposite Pescara harbor (in Italy) and comparing numerical results with the field data.

Fully nonlinear Boussinesq equations for long wave propagation and run-up in sloping channels with parabolic cross sections

A general framework for derivation of long wave equations in narrow channels, and their transformation to Lagrangian coordinates is briefly established. Then, fully nonlinear Boussinesq equations are derived for channels of parabolic cross sections. The simplified version with normal nonlinearity is compared with corresponding models from the literature, and propagation properties are discussed. A Lagrangian run-up model is adapted to the fully nonlinear set. This model is tested by means of controlled residues and by a well-controlled comparison to exact analytic solutions from the literature. Then, run-up of solitary waves in simple geometries is simulated and compared to a semi-analytic solution that is derived for propagation and run-up in a composite channel. The dispersive model retains the higher run-up height in a parabolic channels, as reported in the recent literature for NLSW solutions, as compared to a rectangular channel.

Numerical simulation of wave transformation, breaking and runup by a contravariant fully non-linear Boussinesq equations model

In this paper we propose a new model based on a contravariant integral form of the fully non-linear Boussinesq equations (FNBE) in order to simulate wave transformation phenomena, wave breaking, runup and nearshore currents in computational domains representing the complex morphology of real coastal regions. The above-mentioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. The Boussinesq equation system is numerically solved by a hybrid finite volume-finite difference scheme. A high-order upwind weighted essentially non-oscillatory (WENO) finite volume scheme that involves an exact Riemann solver is implemented. The wave breaking is represented by discontinuities of the weak solution of the integral form of the non-linear shallow water equations (NSWE). On the basis of the shock-capturing high order WENO scheme a new procedure, for the computation of the structure of the solution of a Riemann problem associated with a wet/dry front, is proposed in order to simulate the run up hydrodynamics in swash zone. The capacity of the proposed model to correctly represent wave propagation, wave breaking, run up and wave induced currents is verified against test cases present in literature. The results obtained are compared with experimental measures, analytical solutions or alternative numerical solutions. The proposed model is applied to a real case regarding the simulation of wave fields and nearshore currents in the coastal region opposite San Mauro Cilento (Italy).

The approximate solutions of nonlinear Boussinesq equation

The homotopy analysis method (HAM) is introduced to solve the generalized Boussinesq equation. In this work, we establish the new analytical solution of the exponential function form. Applying the homotopy perturbation method to solve the variable coefficient Boussinesq equation. The results indicate that this method is efficient for the nonlinear models with variable coefficients.

Exact solution of certain time fractional nonlinear partial differential equations

Given a time fractional nonlinear partial differential equation, we show how to derive its exact solution using invariant subspace method. This has been illustrated through time fractional diffusion convection equation, time fractional nonlinear dispersive Boussinesq equation, time fractional reaction diffusion equation of second order, time fractional thin-film equation, and time fractional quadratic wave equation. Also, we explicitly shown that time fractional nonlinear partial differential equations admit more than one invariant subspaces which in turn helps to derive more than one exact solution.

Unsteady Groundwater Flow over Sloping Beds: Analytical Quantification of Stream–Aquifer Interaction in Presence of Thin Vertical Clogging Layer

AbstractIn this paper, new analytical solutions are presented to quantify interaction of water between a sloping aquifer and stream of varying heads in presence of a thin vertical sedimentary layer. The initial hydrogeological setting consist of an unconfined sloping aquifer of semi-infinite extent, a fully penetrating stream of varying water level, and a vertical layer of streambank deposits that acts as an interface between stream and aquifer. Unsteady groundwater flow is characterized by a nonlinear Boussinesq equation subject to Robin boundary condition (also referred to as third kind and Cauchy boundary condition). Unlike existing results, which focus only on step changes in stream stage, the current study accounts for gradual rise and decline in stream level. A closed-form analytical solution for water head distribution, discharge rate, and net volumetric exchange of water between stream and aquifer are developed from the solution of a linearized advection-diffusion equation. Performance of the anal...

Nonlinear Wave–Current Interactions in Shallow Water

We study here the propagation of long waves in the presence of vorticity. In the irrotational framework, the Green–Naghdi equations (also called Serre or fully nonlinear Boussinesq equations) are the standard model for the propagation of such waves. These equations couple the surface elevation to the vertically averaged horizontal velocity and are therefore independent of the vertical variable. In the presence of vorticity, the dependence on the vertical variable cannot be removed from the vorticity equation but it was however shown in that the motion of the waves could be described using an extended Green–Naghdi system. In this paper, we propose an analysis of these equations, and show that they can be used to get some new insight into wave–current interactions. We show in particular that solitary waves may have a drastically different behavior in the presence of vorticity and show the existence of solitary waves of maximal amplitude with a peak at their crest, whose angle depends on the vorticity. We also show some simple numerical validations. Finally, we give some examples of wave–current interactions with a nontrivial vorticity field and topography effects.

Feedback stabilization of a thermal fluid system with mixed boundary control

We consider the problem of local exponential stabilization of the nonlinear Boussinesq equations with control acting on portion of the boundary. In particular, given a steady state solution on an bounded and connected domain Ω⊂R2, we show that a finite number of controls acting on a part of the boundary through Neumann/Robin boundary conditions is sufficient to stabilize the full nonlinear equations in a neighborhood of this steady state solution. Dirichlet boundary conditions are imposed on the rest of the boundary. We prove that a stabilizing feedback control law can be obtained by solving a Linear Quadratic Regulator (LQR) problem for the linearized Boussinesq equations. Numerical result are provided for a 2D problem to illustrate the ideas.

Modeling wave processes over fringing reefs with an excavation pit

The excavation of reef-flat sand and aggregate for engineering projects is a common practice on atoll islands, with attendant coastal management and hazard mitigation issues. To assess the impact of reef-flat excavation pits on wave transformation, a numerical study is carried out based on one-dimensional weakly dispersive, highly nonlinear Boussinesq equations. Model simulations are compared to field observations made at Majuro Atoll, Republic of the Marshall Islands (Ford et al., 2013a), at a 75-m-wide fringing reef flat with a 17-m-wide, 4-m-deep excavation pit and at an adjacent unmodified reef flat. With a calibrated empirical breaking model and uniform friction coefficient, the numerical model is shown to satisfactorily reproduce the observed sea and swell (SS) and infragravity (IG) frequency band wave heights for a moderate amplitude wave event. By removing the lateral morphological differences of the field experiment, the model shows that an excavation pit reduces/increases shoreline IG/SS wave heights compared to numerical tests without the pit, in qualitative agreement with Ford et al. (2013a). An EOF analysis demonstrates that the reduction in IG wave heights is associated with modifications of the ¼ and ¾ wavelength IG standing modes across the reef caused by the pit. Model tests are used to evaluate the effects of cross-reef pit location and width on wave transformation. Shoreline wave heights are least affected by a pit located near the reef crest, and both SS and IG wave heights increase as the pit location moves shoreward. Shoreline wave heights are an increasing function of pit width as the increase in SS wave height exceeds the decrease in IG wave height produced by the pit. The effects of incident wave height and spectral bandwidth are also examined for the reef with and without pit. It is found that both shoreline SS and IG wave heights increase with the increasing incident wave height for both profiles. The numerical model results also demonstrate that the shoreline response is sensitive to the bandwidth of the incident wave spectrum, and larger IG shoreline wave heights are excited for narrower spectral bandwidths until wave reflection limits the forcing.