Boussinesq Equation

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Abstract
2007-04-15 06:09:00
This study is concerned with the development of a high-order numerical model to solve incompressible water wave motion based on improved nonlinear dispersive Boussinesq equations. A third-order Adams-Bashforth and a fourth-order Adams-Mouton predictor-corrector scheme was selected in an attempt to eliminate the truncation error terms that would be of the same form as the dispersive terms in the Boussinesq equations with second order schemes as in many other studies.Eddy viscosity type momentum correction term was added into the Boussinesq equation to simulate the energy loss due to wave breaking and to extend the model application to surf zone wave transformation. The location of the breaking point was determined through a wave breaking criterion using the ratio of horizontal water particle velocity to wave celerity.A moving boundary technique utilizing linear extrapolation is developed to investigate wave runup and rundown. Wave absorption at an open boundary was simulated by solvi...
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Introduction
2007-04-14 06:00:00
Coastal zone is one of the most valuable regions on earth from the viewpoint of the ecosystem and human welfare. The coastal zone has been recognized as natural resources for the activities of human beings and has been utilized for various purposes. Coastal zone has also an important ecological value due to plenty of diversity in habitats.These days the protection of coastal areas has become one of the more pressing environmental problems in many countries. Most of these problems are related to the wave and current phenomena in the nearshore zone. These phenomena have thus attracted immensely the interest of engineers in order to safeguard multi-faceted activities in the coastal zone.Understanding nearshore waves has become an indispensable tool in the estimation of forces for the proper design and construction of coastal infrastructures. Waves attack coastal structure and infiltrate harbor entrance, generating disturbance in the sheltered waters. Hence, it is also necessary to know...
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Boussinesq-type Equations
2007-04-13 06:09:00
Boussinesq-type equations are capable of providing accurate description of wave evolution in coastal regions. The onset of recent developments in the field of Boussinesq models was triggered by two events. The first was the increasing availability of the computer resources needed to run the models. The second was the development of variants of the theory which could be optimized to obtain better dispersion properties at larger values, thus allowing the model to treat a larger range of water depths.The earliest depth-averaged model that included both weakly dispersive and nonlinear effects was derived by Boussinesq in 1871. The equations were derived for horizontal bottoms only. Later, Mei and LeMehaute (1966) and Peregrine (1967) derived Boussinesq equations for variable depth. Mei and LeMehaute (1966) used the velocity at the bottom as the dependent variable, whereas Peregrine (1967) used the depth-averaged velocity.The past decade saw the advent and widespread applications of Bous...
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Wave Breaking Model
2007-04-12 06:09:00
As waves propagate toward shore, a combination of shoaling, refraction, reflection, and diffraction effects modify the waveform. With further decrease in water depth, the wave height increases rapidly and finally waves break.Wave breaking process is recognized as an irreversible transformation during which wave motion shifts from initially irrotational, simply-connected free surface dynamics to strong turbulence resulted from the intense vorticity generated by the folding of the free surface onto itself. Accompanying this special wave deformation, the dissipation of wave energy thus the decay of wave height is one of the principal characteristics of wave breaking.The Boussinesq-type equations, which include the weak non-linearity and frequency dispersion provide an accurate description of wave transformation processes outside the surf zone. However, the Boussinesq equations do not automatically lead to wave breaking in shallow water nor to predict the wave in the surf zone. Then a n...
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Wave Runup Model
2007-04-11 06:09:00
The waterline on a beach subjected to wave action is highly variable, and thus the physical domain in a nearshore Boussinesq model application changes in time. Although it is possible to utilize a time-dependent, shoreline-following grid system in order to resolve the fluid domain only up to the waterline, it is difficult to do so if the shoreline does not remain single valued or becomes multiply connected.For this reason, it is more standard to employ techniques whereby the entire region which is potentially wetted is treated as an active part of the computational grid. One of the earliest methods along this line is the “slot” method of Tao (1984), in which deep, narrow, flooded slots are added to each grid row, extending down at least to the lowest elevation that will be experienced during shoreface rundown.Kennedy et al (2000) employed the slot technique of Tao but modified it to better enforce mass conservation. Utilization of slot methods remains something of an art form to...
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Numerical Methodology
2007-04-10 06:00:00
A finite-difference numerical scheme that solves Nwogu’s one-dimensional Boussinesq equation on a staggered-grid system is presented. A staggered grid system is introduced with the first-order spatial derivatives being discretized by the fourth-order accurate finite-difference scheme. For the time derivatives, the fourth-order accurate Adams predictor–corrector method is used. As a result, both the numerical dissipation and dispersion are kept into higher-order.The new form of one-dimensional Boussinesq equations derived by Nwogu (1993) are given byThese equations are statements of conservation of mass and momentum, respectively. Compared to the Boussinesq equations based on depth-averaged velocity (referred to hereafter as standard Boussinesq equations) derived by Peregrine (1967), there is an additional dispersive term in continuity equation and the coefficients of dispersive terms in the momentum equations are different.The governing equations are finite-differenced on a stag...
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Wave Propagation Modeling
2007-04-09 06:09:00
Numerical experiments are developed and performed to evaluate the ability of the Boussinesq model to simulate the propagation of regular waves on a constant depth and on a sloping beach. The computations start from the still water condition.Given a beach bathymetry with a constant depth or a mildly sloping bottom, the elevation of a grid was defined as the vertical distance from the still water line. Grids located above the still water line have positive elevations while those located below have negative value.Numerical codes have been succesfully developed include:Simulation of Non-breaking Regular Wave Prop agation on a Constant DepthSimulation of Non-breaking Regular Wave Propagation on a Sloping Beach up to Breaking LocationSimulation of Breaking Regular Wave Propagation on a Sloping Beach up to Very Shallow Water DepthSimulation of Non-breaking Regular Wave Runup Propagation on a Sloping Beach Simulation of Breaking Regular Wave Runup Propagation on a Sloping Beach References ca...
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Simulation of Non-breaking Regular Wave Propagation on a Constant Depth
2007-04-08 06:09:00
Numerical experiment setup where initial wave properties and bathymetry condition are described. Wave absorbing boundary, Sommerfeld radiation and sponge layer, are applied at the right boundary. 5th order Stokes wave theory is selected and applied based on computed Ursell parameter as incident wave at the left boundary.The result shows that the Boussinesq equations are able to model the propagation of regular waves on a constant depth satisfactory. In addition to that, ideally an open absorbing boundary should allow wave components to pass through undisturbed. It may conclude from simulation below that Sommerfeld radiation and sponge layer are well effective in order to absorb long and short wave respectively.The maximum variation of wave height in the computation domain before any reflection from the absorbing boundary is less than 0.5% as shownSimulation of Non-breaking Regular Wave Propagation on a Sloping Beach up to Brea king LocationSimulation of Breaking Regular Wave Propagat...

Simulation of Non-breaking Regular Wave Propagation on a Sloping Beach up t
2007-04-07 06:09:00
Another case has been carried out in order to simulate propagation of non-breaking regular wave on a sloping beach. Wave shoaling is analyzed and the result of computation is compared with analytical solution. Given wave properties condition are exactly the same with the previous computation at the left boundary. At the right boundary, the shallow water depth is located at the breaking location and was calculated from Goda formula.In this simulation wave transformation increasing in their wave heights during propagate to shore are well simulated. With further decrease in water depth an asymmetrical wave profile are also clearly observed.If waves are incident normal to a beach with straight and parallel bottom contours, change in the wave profile is caused solely by change in water depth as so-called wave shoaling. Under this condition, wave shoaling over a sloping bottom can be observed to occur as follows: The waves first decrease and then gradually increase in height, maintaining ...
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Simulation of Breaking Regular Wave Propagation on a Sloping Beach up to Ve
2007-04-06 06:09:00
Simulation of breaking regular wave propagation on a sloping beach up to very shallow water is computed in order to test wave breaking model. Figure below shows numerical experiment setup where initial wave properties and bathymetry condition are described. At the left boundary, time history of wave elevation and horizontal water particle velocity based on 5th order Stokes wave are applied. In simulation below wave transformation increasing in their wave heights during propagate to shore are well simulated. With further decrease in water depth an asymmetrical wave profile are also clearly observed and once the wave exceed breaking location, their height is decreasing due to energy dissipation. Although there is a reflected wave due to introduction of discontinuity of dissipation breaking term, comparison between numerical and analytical solution show fairly good agreement. The analytic solution is indicated by a red line while the numerical solution is indicated by a blue dotted-lin...
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Simulation of Non-breaking Regular Wave Runup Propagation on a Sloping Beac
2007-04-05 06:09:00
As a first check of the moving boundary model, a monochromatic wave train is let to runup and rundown a plane beach. The model is compared with the analytic solution derived by Carrier and Greenspan (1958) for monochromatic long wave runup on a constant slope. Their derivation makes use of the non-linear shallow water equations, and thus for consistency the dispersive terms will be ignored in the numerical simulations for this comparison. Simulation of non-breaking regular wave runup propagation on a sloping beach is shown belowFigure below shows a comparison between analytical (Carrier and Greenspan, 1958) and numerical solution of vertical shoreline movement. The analytic solution is indicated by a red line while the numerical solution is indicated by a blue line. It is concluded that non-breaking regular wave runup is accurately predicted by the proposed model, yielding a validation of the moving boundary technique. Simulation of Non-breaking Regular Wave Propagation on a Constan...
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Simulation of Breaking Regular Wave Runup Propagation on a Sloping Beach
2007-04-04 06:09:00
Simulation of breaking regular wave runup propagation on a sloping beach is computed in order to test both wave breaking model and wave runup simultaneously. Figure below shows numerical experiment setup where initial wave properties and bathymetry condition are described. At the left boundary, time history of wave elevation and horizontal water particle velocity based on 5th order Stokes wave are applied. At the right boundary, moving shoreline boundary employ a linear extrapolation of free surface displacement and velocity components, are applied. The numerical model was initially evaluated using data obtained from experiments carried out by Nwogu (1993) in the three-dimensional wave basin of the Canadian Hydraulics Center. It would seem that inclusion of an accurate dissipation term becomes increasingly important with increasing degree of wave breaking. Introduction of a bottom friction alone as a dissipation term in a very small total water depth may create an equally large diss...
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Conclusions and Recommendations
2007-04-03 06:09:00
It is shown that a high-order numerical scheme developed in this study is basically stable and efficient. This scheme has the ability to predict wave transformation from deep to shallow water and guarantees that the leading order truncation error terms in the discretization form are not of the same form as the dispersive terms in the Boussinesq equations. Furthermore, using the space staggered-grid and second order upwind scheme for the convective terms improve the numerical stability.Eddy viscosity type momentum correction term was added into the Boussinesq equation to simulate the energy loss due to wave breaking and to extend the model application to surf zone wave transformation. The location of the breaking point was determined through a wave breaking criterion using the ratio of horizontal water particle velocity at arbitrary distance from still water level and wave celerity.A moving boundary technique utilizes linear extrapolation is developed to investigate wave runup and ru...
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Reference
2007-04-01 17:31:00
Carrier, G. F. and H. P. Greenspan (1958), Water waves of finite amplitude on a sloping beach, J. Fluid Mech. 4, pp. 97-109.Horikawa, K., Ed. (1988): Nearshore Dynamics and Coastal Process, University of Tokyo.Isobe, M. and N. C. Kraus (1983), Derivation of a third-order Stokes wave theory, Technical Report No. 83-1, Yokohama National University, Japan, 41 pp.Isobe, M. and N. C. Kraus (1983), Derivation of a second-order Cnoidal wave theory, Technical Report No. 83-2, Yokohama National University, Japan, 41 pp.Kennedy, A. B., Chen, Q., Kirby, J. T., Dalrymple, R. A. (2000), Boussinesq modeling of wave transformation, breaking, and runup. Part I: 1D. J. Waterw. Port Coast. Ocean Eng. 126 (1), 39– 47.Kirby, J. T., Wei, G., Chen, Q., Kennedy, A. B., Dalrymple, R. A. (1998), ‘‘FUNWAVE 1.0. Fully nonlinear Boussinesq wave model. Documentation and user’s manual.’’ Report CACR-98-06, Center for Applied Coastal Research, Department of Civil and Environment Engineering, Universit...
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