Boussinesq equations Research Papers - Academia.edu (original) (raw)

In this work an improved Boussinesq model with a surface pressure term is discretized by a new approach. By specifying a single parameter the proposed discretization enables the user to run the program either in the long wave mode without dispersion terms or in the Boussinesq mode. Furthermore, the Boussinesq mode may be run either in the classical Boussinesq mode or in the improved Boussinesq mode by setting the dispersion parameter appropriately. In any one of these modes it is possible to specify a fixed or a moving surface pressure for simulating a moving object on the surface. The numerical model developed here is first tested by comparing the numerically simulated solitary waves with their analytical counterparts. The second test case concerns the comparison of the numerical solutions of moving surface pressures with the analytical solutions of the long wave equations for all possible modes (long wave, classical, and improved Boussinesq). a pressure disturbance moving at const...

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- Boussinesq equations

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 /ρgh 2 0 , 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.

In this document, we have proposed a second order solution to Boussinesq's problem (Boussinesq 1885), which allows us to account for the new experimental evidence (Ferretti & Bignozzi 2012, Ferretti 2012b) on the stress field induced by aircraft traffic in concrete pavements. In particular, the second order solution is able to describe the tensile state of stress acquired in the proximity of the contact area and not accounted for in the classical solution of Boussinesq's problem for a homogeneous linear-elastic and isotropic half-space. The second order solution also allows us to evaluate the effect of the elastic constants on the stress field, improving the solution of Boussinesq in this second case also.

- by Elena Ferretti
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- Boussinesq equations

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.

- by Theophanis V. Karambas and +1
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- Mediterranean, Tsunami, Tsunamis, Flooding
- by Anwar J M Jawad
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- Nonlinear Waves, Exact Solutions, Boussinesq equations, Tanh Method

This work presents results on the simulation of the generation and propagation of ship- borne waves, using an advanced nonlinear dispersive wave model based on the higher order Boussinesq-type equations. The model includes a single frequency dispersion term, expressed through convolution integrals; it is adapted to represent ship-borne waves by adopting the approach for wave generation by a moving local pressure disturbance, achieved through adding a respective pressure term in its governing equations. The model is tested against an analytical solution for the calculation of the angles of ship wakes and laboratory experiments of waves produced by a high-speed ship in a channel. Results com- pare well to data from both the aforementioned studies, confirming the model’s capabilities and highlighting its accuracy in the representation of ship-borne waves; the model’s applicability to practical coastal engineering applications is also investigated, through a set of numerical experiments of waves generated at sea and in a harbour.

- by Achilleas Samaras
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- Numerical Modelling, Ship Hydrodynamics, Water Waves, Ship Motions in Waves
- by Anwar J M Jawad
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- Mathematics, Exact Solutions, Boussinesq equations, Tanh Method

A class of model equations that describe the bi-directional propagation of small amplitude long waves on the surface of shallow water is derived from two-dimensional potential flow equations at various orders of approximation in two small parameters, namely the amplitude parameter a ¼ a=h 0 and wavelength parameter b ¼ ðh 0 =lÞ 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. These equations are also characterized by the surface tension parameter, namely the Bond number s ¼ C=qgh 2 0 , where C is the surface tension coefficient, q is the density of water, and g is the acceleration due to gravity. The traveling solitary wave solutions are explicitly constructed for a class of lower order Boussinesq system. From the Boussinesq equation of higher order, the appropriate equations to model solitary waves are derived under appropriate scaling in two specific cases: (i) b (ð1=3 À sÞ 6 1=3 and (ii) ð1=3 À sÞ ¼ OðbÞ. The case (i) leads to the classical Boussinesq equation whose fourth-order dispersive term vanishes for s ¼ 1=3. This emphasizes the significance of the case (ii) that leads to a sixth-order Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159] as a dispersive regularization of the ill-posed fourth-order Boussinesq equation.: S 0 0 2 0-7 2 2 5 (0 2) 0 0 1 8 0-5

Analytical study to an alpha-regularization of the Boussinesq system is performed using Fourier theory. Existence and uniqueness of strong solution are proved. Convergence results of the unique strong solution for the regularized Boussi-nesq system to the unique strong solution for the Boussinesq system are established as the regularizing parameter vanishes. The proofs are performed in the frequency space. We use energy methods, Friedrichs's approximation scheme, and Arselà-Ascoli compactness theorem.

- by Abdelkerim Chaabani and +1
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- Fluid Mechanics, PDE (Mathematics), nonlinear PDE, Boussinesq equations

For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy perturbation transform method(HPTM). The solution is compared with the exact solution. The comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height of water table. The results resemble well with the physical phenomena.

- by Trushit V Patel
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- Applied Mathematics, Numerical Analysis, Error Estimates, Boussinesq equations

The paper reports a study of the water surface profile of an entrapped air cavity while emptying water in an initially filled inclined duct. A one-dimensional (1D) model, which consists of the continuity and momentum equations applicable for open channel flow, pipe flow and air–water interface flow, is developed based on the finite volume method. A pressure drop model is proposed to reproduce a better profile around the cavity front, with a particular focus on air pressure changes inside the confined cavity to simulate a kind of transient flow with an entrapped air cavity. In contrast to the previous studies, the application of the model shows that when the pressure drop is not considered and the air pressure is not changed, the confined cavity soon vanishes. A comparison between the simulated and experimental results shows that the model is able to accurately reproduce the water surface profile of an entrapped air cavity while emptying inclined ducts.

- by Hamid Bashiri
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- Computational Fluid Dynamics, Hydraulics, Free surface flows, Pipeline

An updated version of a 2-DH post-Boussinesq wave model is introduced. The model is wavenumber free and as far as the linear dispersion relation is concerned, the approach is exact. It is implemented for the wave propagation and transformation due to shoaling, refraction, diffraction, bottom friction, wave breaking, wave-structure interaction, reflection, wave-current interaction, etc. in nearshore zones and specifically inside ports and in the vicinity of coastal structures. Thorough validation of the model is attempted by comparisons with output from classic laboratory-scale wave flume experiments as well as analytical solutions. Physical cases of both regular and irregular wave fields are numerically reproduced with acceptable accuracy. Results concerning a case study in a characteristic Greek port setup are also presented and seem encouraging for realistic scale simulations.

Two fundamental one-dimensional (1D) models are proposed and applied to simulate the transient flows with the propagation of an interface in a water-filled duct. The proposed models are developed to simulate the unsteady open channel flows based on finite-volume method (FVM). The models presented herein are based on the continuity and momentum equations of free surface and pressurized flows and the momentum equation of an interface between both flows. However, the highly simplified marker and cell (HSMAC) method with pressure iteration procedures is applied to the pressurized flow region. The numerical simulations are performed under the hydraulic conditions of previous experiments, and then simulated results were compared with the experimental data. It is pointed out that the solitary wave solution is able to reproduce the air cavity profile. In contrast to the hydrostatic model, results of the Boussinesq model compare reasonably well to the experimental observations.

- by Hamid Bashiri and +1
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- Finite Volume Methods, Hydraulics, Free surface flows, Pipeline
- by Claudio Estrada
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- Mathematical Sciences, Physical sciences, Boussinesq equations

Boussinesq equations with improved dispersion characteristics are used to simulate the generation and propagation of waves due to moving pressure fields. With surface pressure terms in the momentum equations the numerical scheme is first run for a moving 3-D hemispherical pressure field for a range of Froude numbers. The wedge angles obtained from simulations are compared with the values calculated from the analytical formulas of Havelock. Furthermore, two ship-like slender pressure fields, representing a moving catamaran, are employed to visualize the interaction of the waves generated.

- by Deniz Bayraktar and +1
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- Boussinesq equations, Moving pressure field, Wave pattern

The Euler's equations describing the dynamics of capillary-gravity water waves in two-dimensions are considered in the limits of small-amplitude and long-wavelength under appropriate boundary conditions. Using a double-series perturbation analysis, a general Boussi-nesq type of equation is derived involving the small-amplitude and long-wavelength parameters. A recently introduced sixth-order Boussinesq equation by Daripa and Hua [Appl. Math. Comput. 101 (1999), 159– 207] is recovered from this equation in the 1/3 Bond number limit (from below) when the above parameters bear a certain relationship as they approach zero.

Internal waves describe the (linear) response of an incompressible stably stratified luid to small perturbations. The inclination of their group
velocity with respect to the vertical is completely determined by their
frequency. Therefore the reflection on a sloping boundary cannot follow
Descartes' laws, and it is expected to be singular if the slope has the same
inclination as the group velocity. In this paper, we prove that in this
critical geometry the weakly viscous and weakly nonlinear wave equations
have actually a solution which is well approximated by the sum of the in-
cident wave packet, a reflected second harmonic and some boundary layer
terms. This result confirms the prediction by Dauxois and Young, and
provides precise estimates on the time of validity of this approximation.

- by Roberta Bianchini and +2
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- Nonlinear Internal Waves, Boussinesq equations, Internal waves
- by F Achallma
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- Boussinesq equations

In this paper, a numerical implementation of 1D Variational Boussinesq (VB) wave model in a staggered grid scheme is discussed. The staggered grid scheme that is used is based on the idea proposed by Stelling & Duinmeijer (2003) who implemented the scheme in a nondispersive Shallow Water Equations in a conservative form. Here, we extend the idea of the staggered scheme to be applied for VB wave model. To test the accuracy of the implementation, we test the numerical implementation of VB wave model for simulating propagation of solitary wave against analytical solution. Moreover, to test dispersiveness of the model, we simulate a standing wave against analytical solution. Results of simulations show a good agreement with analytical solutions.

- by Sul Kifli
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- Engineering, Applied Mathematics, Physics, Modeling and Simulation

This research aims to evaluate the influence of air on the water surface and pressure profiles in two-phase (air–water) flow. In previous studies, the authors used the continuity and momentum equations for water in a one-dimensional finite volume model. In the present study, the authors additionally incorporate both the mass conservation and momentum equations for incompressible fluids as basic equations for the air portion. Momentum equations were integrated for the free surface and pressurized flow regions after interface observations made at each time step. To correct velocity and pressure in the pressurized portion, the authors used the highly simplified mark-and-cell method. A pressure drop equation is used at the interface of pressurized and free surface flows to deduce the pressure. The Harten total variation diminishing scheme was used to avoid numerical oscillations. The results were compared with experimental data and numerical simulations without considering the air effect.

- by Hamid Bashiri
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- Computational Fluid Dynamics, Finite Volume Methods, Hydraulics, Free surface flows

Trial equation method is a powerful tool for obtaining exact solutions of nonlinear differential equations. In this paper, the improved Boussinesq is reduced to an ordinary differential equation under the travelling wave transformation. Trial equation method and the theory of complete discrimination system for polynomial are used to establish exact solutions of the improved Boussinesq equation.

- by Scientific Research Publishing
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- Mathematics, Boussinesq equations
- by gonzalo simarro
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- Coastal Engineering, Boundary Layers, Damping, Water Waves

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.

Karambas and Memos (2009) have presented a protocol version of a post-Boussinesq type wave model with a system of 2-DH equations for fully dispersive and weakly nonlinear irregular waves over any finite water depth. The model in its two-dimensional formulation, involves in total five terms in each momentum equation, including the classical shallow water terms and only one frequency dispersion term. The latter is expressed through convolution integrals, which are estimated using appropriate impulse functions. In this work, an updated version of the aforementioned model is introduced. It is implemented for wave propagation and transformation (due to shoaling, refraction, diffraction, bottom friction, wave breaking, runup, wave-structure interaction etc.) in nearshore zones and inside ports. One of the main goals is the model's thorough validation, thus it is tested against experimental data of wave transmission over and through breakwaters, uni-and multi-directional spectral wave ...

- by Vasilis Baltikas
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- Geology, Numerical Analysis, Ports and Harbours, Spectral Wave Modelling

Internal waves describe the (linear) response of an incompressible stably stratified fluid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection on a sloping boundary cannot follow Descartes' laws, and it is expected to be singular if the slope has the same inclination as the group velocity. In this paper, we prove that in this critical geometry the weakly viscous and weakly nonlinear wave equations have actually a solution which is well approximated by the sum of the incident wave packet, a reflected second harmonic and some boundary layer terms. This result confirms the prediction by Dauxois and Young, and provides precise estimates on the time of validity of this approximation.

- by Roberta Bianchini
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- Mathematics, Applied Mathematics, Physics, Pure Mathematics

In this paper, a numerical implementation of 1D Variational Boussinesq (VB) wave model in a staggered grid scheme is discussed. The staggered grid scheme that is used is based on the idea proposed by Stelling & Duinmeijer (2003) who implemented the scheme in a non-dispersive Shallow Water Equations in a conservative form. Here, we extend the idea of the staggered scheme to be applied for VB wave model. To test the accuracy of the implementation, we test the numerical implementation of VB wave model for simulating propagation of solitary wave against analytical solution. Moreover, to test dispersiveness of the model, we simulate a standing wave against analytical solution. Results of simulations show a good agreement with analytical solutions.

- by Dede Tarwidi and +1
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- Applied Mathematics, Modeling and Simulation, Numerical Methods, Boussinesq equations

The least-squares finite element method (LSFEM), based on
minimizing the l2-norm of the residual is now well established
as a proper approach to deal with the convection dominated
fluid dynamic equations. The least-squares finite element
method has a number of attractive characteristics such as the
lack of an inf-sup condition and the resulting symmetric
positive system of algebraic equations unlike Galerkin finite
element method (GFEM). However, the higher continuity
requirements for second-order terms in the governing
equations force the introduction of additional unknowns
through the use of an equivalent first-order system of
equations or the use of C1 continuous basis functions. These
additional unknowns lead to increased memory and
computational requirements that have limited the application
of LSFEM to large-scale practical problems.
A novel finite element method is proposed that employs a
least-squares method for first-order derivatives and a Galerkin
method for second order derivatives, thereby avoiding the
need for additional unknowns required by a pure LSFEM
approach. When the unsteady form of the governing equations
is used, a streamline upwinding term is introduced naturally
by the least-squares method. Resulting system matrix is
always symmetric and positive definite and can be solved by
iterative solvers like pre-conditioned conjugate gradient
method. The method is stable for convection-dominated flows
and allows for equal-order basis functions for both pressure
and velocity. The method has been successfully applied here
to solve complex buoyancy-driven flow with Boussinesq
approximation in a square cavity with differentially heated
vertical walls using low-order C0 continuous elements.

- by gonzalo simarro
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- Coastal Engineering, Water Waves, Boussinesq equations

We make use of the He’s semi-inverse method and symbolic computation to construct new exact traveling wave solutions for the (2 + 1)-dimensional Boussinesq and breaking soliton equations. Many new exact traveling wave solutions are successfully obtained, which contain soliton solutions. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.

- by sciepub.com SciEP
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- Boussinesq equations