# Boussinesq Equation

Numerical Model of Breaking Regular Wave Runup Propagation by Using 1D Boussinesq Equation

 Yuliarko Sukardi B.E. Ocean Engineering Institute Technology of Bandung M. Eng. Coastal Engineering University of Tokyo

Numerical Model of Breaking Regular Wave Runup Propagation by Using 1D Boussinesq Equation
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.
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### Conclusions and Recommendations

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 rundown. Founded around the restrictions of the high-order numerical wave propagation model, the moving boundary scheme employs linear extrapolation of free surface and velocity through the wet–dry boundary, into the dry region. The linear extrapolation is simple to implement and can be straightforwardly incorporated into a numerical model.

It would seem that inclusion of an accurate dissipation term becomes increasingly important with increasing degree of wave breaking. In regard to breaking regular wave runup simulation, additional dissipation term acted as bottom friction was required for long term stability in surf zone area.

On the basis of comparisons with analytical solution, the Boussinesq model was proven to be accurate in simulating most of the wave transformations in the nearshore region like wave shoaling, breaking, and decay in the surf zone. The location of the wave breaking was accurately simulated by using the ratio of horizontal water particle velocity at arbitrary distance from still water level and wave celerity. Other secondary effects of the wave motion, such as wave setdown and wave setup as well as beach runup and rundown were found to be simulated by the model.

It is demonstrated that the present numerical model has very good numerical properties for mass and energy conservation. As a result, the model is more stable and can run long time without significantly spurious mass and energy fluctuation inside the domain that would eventually contaminate computational results. The numerical results compare excellently with the analytical solution and perfect mass conservation is accomplished for long-term computation.

The Boussinesq-type equations derived by Nwogu (1993) lead to a linear dispersion relation by matching a Pade approximant of that of the small amplitude wave theory and are able to simulate wave propagation from relatively deep to shallow water. Despite their improved dispersion relation, the extended Boussinesq equations are still restricted to situation with weakly nonlinear interactions, and thus extensions to the fully nonlinear Boussinesq equation are required in order to obtain a computational tool which is locally valid in the vicinity of a steep, almost breaking or breaking wave crest.

A high-order numerical scheme of this model requires the surface elevation and horizontal water particle velocity at the incident boundary to initiate the wave propagation into the computation domain. The wave absorption of waves approaching the incident boundary at may be improved by the use of sponge layer.

In regard to breaking regular wave runup simulation additonal dissipation term was required for long term stability replacing bottom friction term in surf zone area. The formula of additional dissipation term used in this simulation is adopted from sponge layer term but in more simplified form for convenience. However, one might recommend value of its coefficients in order to get optimum result and further can also be used universally in every wave condition.

The wave model analyses and investigations in this study were confined to one-dimensional propagation of regular waves. Applying the model to actual field problems is obviously the ultimate objective for all numerical models. Most field conditions are far from the idealizations and assumptions used in this study. However, within the limits of the wave model, it may be possible to apply this model in nearshore area with simple bed topography as those found in harbors and mildly sloping beach. It will be interesting if a computation are also simulated to model two-dimensional propagation of irregular waves including the sediment transport model and consider wave-structure interaction as well.

References can be found here.

BOUSSINESQ EQUATION's CONTENT

Joseph Valentin Boussinesq
He (born March 13, 1842 in Saint-André-de-Sangonis (Hérault département), died February 19, 1929 in Paris) was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat.

Master Research
This study was supported by the Asian Development Bank-Japan Scholarship Program.
Numerical Model of Breaking Regular Wave Runup Propagation by Using 1D Boussinesq Equation

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