Boussinesq Equation

Abstract

2007-04-14T21:09:00.000-07:00

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 solving the Sommerfeld radiation and introducing sponge layer into the model.

Breaking regular wave runup propagation on a sloping beach is simulated. 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. The validity of the model was confirmed by comparing computations with analytical solutions and measured data.

Introduction

2007-04-13T21:00:00.000-07:00

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 beforehand the magnitude and direction of propagating waves to design the optimal layout of harbors.

As waves propagate toward shore, a combination of shoaling, refraction, reflection, diffraction, and breaking effects modify the waveform, and the wave characteristics will be very different from those in deep water. The ability to accurately predict wave transformation from deep to shallow water is vital to an understanding of coastal processes. By the refraction the directions of wave propagation as well as the wave height vary in accordance with the sea bottom topography. By the wave shoaling the wave height gradually change as a result of the change of the rate of energy flux due to the reduction in water depth, even if no refraction takes place.

When incoming waves are interrupted by a barrier such as a breakwater or an island, the waves curve around the barrier and penetrate into the sheltered area, this phenomenon is called diffraction. When waves reach a water of depth less than a few times the wave height, waves begin to break and the overall wave height decreases as the wave energy is dissipated.

The processes of wave breaking, runup, setdown and setup of the mean water level, turbulent energy production, generation of nearshore currents and generation of infragravity waves are important driving mechanisms for the transport of sediments and pollutants as well as erosions in coastal regions.

Most coastal erosion and accretion problems can be prevented or mitigated with some engineering knowledge of coastal sedimentation. Periodic dredging operation and channel navigation can be planned and managed more precisely and effectively. The understanding of effects caused by existing or newly constructed coastal structure is also some of the benefits derived from a detailed modeling of the wave and current phenomena in shallow water.

The effects of man-made or naturally caused changes around coastal area can be examined through physical modeling. This usually involves intricate and time consuming setting up of a scaled-down physical model and requires accurate measurements of wave data. With the availability of advanced computing facilities, numerical model has become an attractive alternative. The shift towards numerical modeling has resulted in rapid developments of numerical methods for solving water wave equations.

The greatest advance of numerical model over a physical model is its ability to determine the required information at a much shorter period with a comparatively larger degree of accuracy. The same numerical model may also be used repeatedly on different problems with only some minor modification in boundary conditions, bathymetry and incident wave condition.

It has long been a goal of coastal engineers to produce a computational wave model that is capable of accurately simulating wave motion from deep water through the surf zone. To do this, a model would have to include, among other things, nonlinear shoaling, refraction, diffraction, wave-wave interaction, breaking, and runup.

Early numerical model, owing perhaps to limitations in computing capability, were mostly based on the Airy (1845) linear wave theory wherein the pressure distribution in the vertical is assumed to be hydrostatic.

In the nearshore region, where water depth and shallow and amplitude may still be large, wave is highly nonlinear, characterized by asymmetric orbital motion of the water particle. The nonlinearity becomes increasingly dominant with decreasing water depth. Hence, it may be expected that numerical models based on the linear wave theory will not provide an accurate simulation of the nearshore wave phenomenon.

Linear theory also cannot simulate the secondary effects of the wave action such wave set-up and wave set-down. Nonlinear wave theories include the wave nonlinearities and simulate simultaneously the wave phenomenom and its secondary effects.

The mathematical description on nonlinear water wave involves the Laplace equation combined with nonlinear boundary conditions at the free-surface and at the sea bottom. This problem is complicated by the fact that the moving surface boundary is part of the solution. Direct numerical methods for solving the full equations exist but they are extremely time consuming and can only be applied to small scale problems. As it is currently impractical to perform a full solution over any significant domain, approximate models must be used. For this reason there is a strong interest in so-called Boussinesq-type formulations of the water wave problem.

The principle behind Boussinesq formulations is to incorporate the effects of non-hydrostatic pressure, while eliminating the vertical coordinate, thus significantly reducing the computational effort relative to a full three-dimensional solution. This principle was initially introduced by Boussinesq (1872), who derived new governing equations under the assumption that the magnitude of the vertical velocity increases polynomially from the bottom to the free surface which inevitably leads to some form of depth limitation in the accuracy of the embedded dispersive and nonlinear properties. Hence, Boussinesq-type equations are conventionally associated with relatively shallow water.

Due to increasing error in the modeled linear dispersion relation with increasing water depth, the standard Boussinesq equations are limited to relatively shallow water. Recently, efforts were made by a number of investigators to derive alternative forms of Boussinesq equations that can be applied in deeper water regions. In order to make the Boussinesq equations applicable in deeper water, many researchers have suggested the ways to extend the validity range of the equations. The extended Boussinesq equations normally have adjustable polynomial approximations for the exact dispersion relationship, a major improvement from the standard ones.

Recent developments in numerical schemes for solving Boussinesq-type equation have placed immense interest in nonlinear dispersive wave models based on the Boussinesq equations. One set of candidate models is the various Boussinesq-type equations (Peregrine 1967; Madsen and Sørensen 1992; Nwogu 1993; Wei et al. 1995). These can describe, to varying degrees of accuracy in representing nonlinearity and dispersion, most phenomena exhibited by non-breaking waves in finite depths.

However, as derived, they do not include dissipation due to wave breaking and thus become invalid in the surf zone. Therefore, it is necessary to implement the proper wave breaking model to the well developed Boussinesq wave model for simulating the surf zone phenomena in nearshore zone.

Although all of these efforts extend the usage of the Boussinesq-type equation model such that wave evolution from relatively deep water to the breaking point could be accurately captured, but the process of runup and rundown is not. There has been only limited success developing a general model which can theoretically predict the runup of arbitrary wave especially the wave break during approach to the shoreline. Even non-breaking waves are difficult to treat because the shoreline represents a moving boundary, which must be computed as part of the solution.

Objectives of the Study

The main objective of this study is to develop high-order numerical scheme that can be used to solve the improved nonlinear dispersive one-dimensional Boussinesq equations. This scheme has ability to accurately predict wave transformation from deep to shallow water and to simulate most of the phenomena of interest in the nearshore zone.

The numerical model based on these equations is also extended to simulate one-dimensional wave propagation in the shallow water region over a horizontal or mildly sloping bed to include wave transformation inside the surf zone.

A one-dimensional wave breaking criterion is used to determine the location of the breaking point. With the determination of the breaking point, by choosing a proper wave breaking model and implement it in the developed wave model, simulation of the surf zone wave transformation as well as secondary effects like wave set-down and wave set-up are also simulated.

Finally, a moving boundary technique utilizes linear extrapolation is developed to investigate wave runup and rundown. The moving boundary is tested for accuracy using one-dimensional analytical solutions.

References can be found here.

Boussinesq-type Equations

2007-04-12T21:09:00.000-07:00

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 Boussinesq-type equation models for studying wave propagation. Due to the wide popularity in the coastal engineering community, the equations derived by Peregrine (1967) are often referred to as the standard Boussinesq equations, which are able to describe the nonlinear transformation of irregular and multidirectional waves in shallow water. The vertical velocity is assumed to vary linearly over the depth.

The Boussinesq equations include the lowest-order effects of frequency dispersion and nonlinearity. They can thus account for the transfer of energy between different frequency components, changes in the shape of the individual waves and the evolution of wave groups, in the shoaling irregular wave train (e.g., Freilich and Guza 1984).

However, the standard Boussinesq equations (Peregrine, 1967) had two major limitations that they are only applicable to relatively shallow water depths:
(1) The depth-averaged model poorly described the frequency dispersion of wave propagation in intermediate depths.
(2) the weakly nonlinear assumption limited the largest wave height that could accurately be modeled.
To keep errors in the phase velocity less than 5%, the water depth has to be less than about one-fifth of the equivalent deep-water wavelength (McCowan 1987).

Recently, a number of attempts have been made to extend the range of applicability of the equations to deeper water by improving the dispersion characteristic of the equations. Witting (1984) used a different form of the exact, fully nonlinear, depth-integrated momentum equation for one horizontal dimension, expressed in terms of the velocity at free surface. A Taylor-series-type expansion was used to relate the different velocity variables in the governing equations, with the coefficients of the expansion determined to yield the best linear dispersion characteristics. By retaining terms up to the fourth order in dispersion, Witting obtained relatively accurate result for both deep and shallow water waves. However, the expansions presented by Witting are only valid in water of constant depth.

Murray (1989) and Madsen et al. (1991) examined the dispersion properties of various forms of the Boussinesq equations as well as Witting’s (1984) Pade approximation of the linear dispersion relation for Airy waves. Based on the excellent characteristic of Pade approximant, the writers have introduced an additional third-order term to the momentum equation to improve the dispersion properties of the Boussinesq equations. The third-order is derived from the long wave equations and reduces to zero in shallow water, resulting in the standard form of the equations for shallow water. The equations assume a constant water depth and, thus, are not applicable to shoaling waves.

On the other hand, by defining the dependent variable as the velocity at an arbitrary depth, Nwogu (1993) achieved a rational polynomial approximation to the exact linear dispersion relationship without the need to add higher order terms to the equations. Although the arbitrary location could be chosen to give a Pade approximation to the linear dispersion relationship, Nwogu (1993) chose an alternative value which minimized the error in the linear phase speed over certain depth range.

Both Madsen et al’s and Nwogu’s expressions give more accurate dispersion relation in intermediate water depths than the standard Boussinesq equations do. They have shown by examples that the extended equations are able to simulate wave propagation from much deeper water to shallow water. Of these two models, the Boussinesq equations derived by Nwogu (1993) are obtained through a consistent derivation from the continuity and Euler equations of motion.

Since Nwogu’s extended Boussinesq equations were proposed, it quickly became one of the most cited works in the field of water wave modeling by using Boussinesq equations. The popularity of Nwogu’s equations among coastal modelers is due to the relatively simple form of the equations as well as the balanced ability of representing wave nonlinearity and dispersion.

Liu (1994) and Wei et al. (1995) presented a set of highly nonlinear Boussinesq-type equations that not only can be applied to intermediate water depth but also are capable of simulating wave propagation with strong nonlinear interaction. Wei et al. (1995) have also developed a high-order numerical scheme to solve these equations.

Although the higher-order Boussinesq equations for the improvement of the nonlinear and dispersive properties in water waves have been attempted and successful in certain aspects, most of them involved numerous additional derivatives and hence made the accurate numerical solution increasingly difficult to obtain. For this reason, Nwogu’s extended Boussinesq equations remain to be a good choice for coastal wave modeling, especially for engineering applications.

References can be found here.

Wave Breaking Model

2007-04-11T21:09:00.000-07:00

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 natural step forward is an extension of the Boussinesq equations to cover the surf zone. The extension will require introduction of a wave breaking criterion and the introduction of energy dissipation.

One advantage of extending Boussinesq-type models to the surf zone is the ability to implicitly model interactions between hydrodynamics processes occurring at different time scale. Wave-induced currents and mean sea water level fluctuations are implicitly included in the wave propagation model and are derived from a time-average of the predicted velocities and surface wave elevation respectively, without having to explicitly calculate radiation stresses and separately solve a time-averaged hydrodynamic model.

Accordingly, there have been attempts to introduce wave breaking into Boussinesq models in order to model breaking phenomena such as wave height decay, wave-induced setup, and runup. The models essentially incorporate a dissipative term due to turbulence stresses or the presence of a surface roller into the momentum equation. The models differ on how they treat the onset of breaking and the rate of wave energy dissipation.

Simulation of wave breaking in Boussinesq models has been approached with a number of techniques, ranging from fairly ad-hoc additions of eddy viscosity formulations up to reasonably detailed calculations of the generation and transport of vorticity or turbulent kinetic energy under the breaking wave crest. Regardless of the formulation, each of the approaches can be thought of as a means for adding the breaking wave force term to the momentum equation.

At minimum, these terms must be scaled similarly in order to reproduce the correct amount of energy dissipation. They must also be localized in the region of the front face of the breaking wave, in order to provide the correct distribution of dissipation in the frequency domain and preserve the proper relative phasing of harmonic components in the wave. Beyond these requirements, the range of physical approaches taken has a great deal of variety.

Zelt (1991) used the Lagrangian finite-element Boussinesq wave model to study the breaking and runup of solitary waves. Wave dissipation due to turbulence generated by wave breaking is treated by an artificial viscosity term in the momentum equation. The horizontal velocity gradient is used as a breaking criterion to decide exactly when and where the dissipation becomes active.

Karambas and Koutitas (1992) extended the Boussinesq equations to cover the surf zone. The energy dissipation due to wave breaking was represented by including an eddy viscosity term in the depth-integrated momentum equation. The eddy viscosity coefficient is calculated from the numerical integration of the turbulent transport equation and the mixing length hypothesis. Crest elevation at breaking point with the wavelength in deep water was used as a breaking criterion.

Schaffer et al. (1994) used the concept of surface rollers to incorporate wave breaking in Boussinesq equations. The roller is considered as a volume of water being carried by the wave with wave celerity. The effect of the roller is induced in the vertical distribution of the horizontal velocity, which leads to an additional convective momentum term. The breaking criterion is related to the local slope of water surface and the thickness of the roller is determined from simple geometrical considerations.

Kabiling and Sato (1994) added a momentum mixing term using eddy viscosity to the momentum conservation equations to simulate the energy dissipation due to breaking. A breaking criterion using the ratio of the water particle velocity at the surface to the wave celerity was presented.

Nwogu (1996) extended the Boussinesq model to the surf zone by coupling the mass and momentum equations with one-equation model for the temporal and spatial evolution of the turbulent kinetic energy induced by wave breaking. The waves are assumed to start breaking when the horizontal component of the orbital velocity at the crest exceeds the phase velocity of the waves. The rate of wave energy dissipation, which is equal to the rate of production of turbulent kinetic energy, is assumed to be proportional to the vertical gradient of the horizontal water particle velocity at wave crest.

Svendsen (1996) describe a breaking wave model based on the classical assumptions for long, moderately nonlinear Boussinesq waves. The wave breaking is represented by including the vorticity generated by the breaking.

Kennedy (2000) used a momentum-conserving eddy viscosity technique to model breaking. This is somewhat like the eddy viscosity formulation Zelt (1991), but with extensions to provide a more realistic description of the initiation and cessation of wave breaking.

A time-dependent wave breaking model used by Sato and Kabiling (1994) is adopted and implemented as the wave breaking criterion and energy dissipation for numerical model. The models essentially incorporate a dissipative term due to turbulence stresses into the momentum equation. A time-dependent breaking model is also able to simulate the transition region after the onset of wave breaking, where there is a rapid decay in wave energy.

References can be found here.

Wave Runup Model

2007-04-10T21:09:00.000-07:00

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 date. Slots which are too wide relative to the model grid spacing admit too much fluid before filling during runup, and cause both a reduction in amplitude and a phase lag in modeled runup events. At the other extreme, slots which are too narrow tend to induce a great deal of numerical noise, leading to the need for intermittent or even fairly frequent filtering of swash zone solutions.

The runup of waves on beaches has been studied extensively by scientist and non scientist. Although research has improved our understanding of many aspects of this phenomenon, there has been only limited success developing a general model which can theoretically predict the runup of arbitrary wave especially the wave break during approach to the shoreline. Even non-breaking waves are difficult to treat because the shoreline represents a moving boundary, which must be computed as part of the solution.

In a wave runup problem, it is not easy to express moving boundary for a wave front with the fixed coordinate system in the Eulerian description which is used for most of wave models with finite difference method. Moving shoreline is difficult to treat because it represents a moving boundary which must be computed as part of the solution.

Alternately, several moving boundary techniques where the occurrence of a shoreline is traced on a fixed grid have been proposed. In one such method, Lynett et al (2002) determine the position of the shoreline on a beach face by linearly extrapolating the surface and velocities from the two wetted grid points closest to shore. Derivatives at wetted grid points are then computed using the regular finite difference stencil including extrapolated values, eliminating the need for conditional application of off-centered difference stencils near shore. The time stepping solution is then only performed for the wetted grid points.

The moving boundary algorithm is conceptually simple, easy to implement, and can be employed by different numerical schemes (i.e., finite difference and finite element) utilizing depth-integrated Boussinesq equations. The moving boundary technique utilizes linear extrapolation through the wet–dry boundary and into the dry region, thereby allowing the real boundary location to exist in-between nodal points.

Although this method does not impose any apparent constraints on mass conservation at the shoreline, results for runup of regular and solitary waves are seen to be quite accurate, indicating that the method holds promise for general application.

References can be found here.

Numerical Methodology

2007-04-09T21:00:00.000-07:00

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 by

These 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 staggered grid system.

First order derivatives of wave elevation employ the fourth-order accurate four-point central difference scheme. Advective term of wave velocity employs the second-order upwind scheme. Dispersive terms employ the second-order accurate three-point central difference scheme.

Discretization in time step of the governing equation consists of the third-order Adams-Bashforth predictor and fourth-order Adams-Moulton corrector schemes.

Boundary Condition

Appropriate boundary conditions are needed for the numerical model to run properly. The examples shown in the following involve two types of boundaries: incident wave boundary (offshore boundary) and absorbing boundary (onshore boundary).

Incident wave boundary (offshore boundary)

In offshore boundary the time series of incident wave elevation and the horizontal velocities are imposed and can be directly obtained from finite amplitude stokes or cnoidal waves theory. In Stokes wave theory, the wave steepness is assumed to be small and the relative water depth is assumed to be of order unity, which implies that this theory is valid for relatively deep water. In the cnoidal wave theory, both the square of the relative wave height and the relative water depth are assumed to be small, and the orders of these two quantities are assumed to the same. Therefore, cnoidal wave theory is applicable to relatively shallow water.

Isobe et al. (1982) made a detailed study of the ranges of validity of stokes wave and cnoidal wave theories. The applicable range of finite amplitude wave theory is based on Ursell parameter. Since the incident wave imposed in relatively deep water or offshore location, the Stokes wave is used here.

The wave profile drawn with a blue line and the horizontal water particle velocity drawn with a green line. The wave profile in the figure above exhibit a characteristic feature of finite amplitude waves, namely large, narrow crest and small, broad trough, as compared to linear wave theory. Since the numerical scheme is discretized in a staggered grid manner, the wave profile and horizontal water particle velocity is not plotted symmetrical to each other.

Absorbing boundary (onshore boundary)

1. Sommerfeld radiation or open boundary, which should absorb all energy arriving at the boundary from within the fluid domain. A perfect absorbing boundary should not allow wave reflection to occur.

2. Damping or sponge layer, which applied in order to reduce reflection in boundary as a short wave absorption.

References can be found here.

Wave Propagation Modeling

2007-04-08T21:09:00.000-07:00

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 Propagation on a Constant Depth

Simulation of Non-breaking Regular Wave Propagation on a Sloping Beach up to Breaking Location

Simulation of Breaking Regular Wave Propagation on a Sloping Beach up to Very Shallow Water Depth

Simulation of Non-breaking Regular Wave Runup Propagation on a Sloping Beach

Simulation of Breaking Regular Wave Runup Propagation on a Sloping Beach

References can be found here.

Simulation of Non-breaking Regular Wave Propagation on a Constant Depth

2007-04-07T21:09:00.001-07:00

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 shown

Simulation of Non-breaking Regular Wave Propagation on a Sloping Beach up to Breaking Location

Simulation of Breaking Regular Wave Propagation on a Sloping Beach up to Very Shallow Water Depth

Simulation of Non-breaking Regular Wave Runup Propagation on a Sloping Beach

Simulation of Breaking Regular Wave Runup Propagation on a Sloping Beach

Simulation of Non-breaking Regular Wave Propagation on a Sloping Beach up to Breaking Location

2007-04-06T21:09:00.001-07:00

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 symmetrical front and rare faces. With further decrease in water depth, the wave height increases rapidly to produce an asymmetrical wave profile; finally waves break.

In perturbation methods, the effect of the bottom slope is treated as a small perturbation within the framework of theories for a horizontal bottom. For a mildly sloping bottom, Kakutani (1971) extended the K-dV equation, which governs the transformation of first-order cnoidal waves traveling in uniform depth. Shuto (1974) treated wave height change based on Kakutani’s equation. The result in an approximate form is expressed as follows:

Comparison between analytical (Shuto, 1974) and numerical solution shows good agreement. The analytic solution is indicated by a red line while the numerical solution is indicated by a blue dotted-line. It may also conclude from figure below that Sommerfeld radiation and sponge layer are well effective in order to absorb long and short wave respectively.

Simulation of Non-breaking Regular Wave Propagation on a Constant Depth

Simulation of Breaking Regular Wave Propagation on a Sloping Beach up to Very Shallow Water Depth

Simulation of Non-breaking Regular Wave Runup Propagation on a Sloping Beach

Simulation of Breaking Regular Wave Runup Propagation on a Sloping Beach

Simulation of Breaking Regular Wave Propagation on a Sloping Beach up to Very Shallow Water Depth

2007-04-05T21:09:00.000-07:00

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-line.

Secondary effects like wave setdown seaward of the breaker and wave setup landward of the breaker are also clearly indicated. The minimum mean sea water level is about 5 % of wave height at the breaking location which is also inline with nonlinear properties derived from conservation equation.

Simulation of Non-breaking Regular Wave Propagation on a Constant Depth

Simulation of Non-breaking Regular Wave Propagation on a Sloping Beach up to Breaking Location

Simulation of Non-breaking Regular Wave Runup Propagation on a Sloping Beach

Simulation of Breaking Regular Wave Runup Propagation on a Sloping Beach

Simulation of Non-breaking Regular Wave Runup Propagation on a Sloping Beach

2007-04-04T21:09:00.001-07:00

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 below

Figure 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 Constant Depth

Simulation of Non-breaking Regular Wave Propagation on a Sloping Beach up to Breaking Location

Simulation of Breaking Regular Wave Propagation on a Sloping Beach up to Very Shallow Water Depth

Simulation of Breaking Regular Wave Runup Propagation on a Sloping Beach

Simulation of Breaking Regular Wave Runup Propagation on a Sloping Beach

2007-04-03T21:09:00.000-07:00

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 dissipative momentum flux, which can lead to an overflow in the iterative numerical scheme due to the fact that bottom friction term is inversely proportional to the total water depth. This statement is not restricted to simulations that contain a shoreline, any simulation with periodic wave breaking suffers from this difficulty.

The other reason is that as the wave propagates to shore the wave face becomes too steep and it creates instability during extrapolation of water surface in wave runup model. In regard to breaking regular wave runup simulation additional dissipation term was required for long term stability.

Simulation shows the propagation breaking regular wave runup on a sloping beach. 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.

Figure below shows the wave height distribution and mean sea water level obtained by the model and measurements. The overall agreement is seen to be fairly good but there some discrepancies: wave height is slightly underestimated around the breaking point and the calculated wave setup is also underestimated.

Simulation of Non-breaking Regular Wave Propagation on a Constant Depth

Simulation of Non-breaking Regular Wave Propagation on a Sloping Beach up to Breaking Location

Simulation of Breaking Regular Wave Propagation on a Sloping Beach up to Very Shallow Water Depth

Simulation of Non-breaking Regular Wave Runup Propagation on a Sloping Beach

Conclusions and Recommendations

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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.

Reference

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