Hybrid finite volume – finite difference scheme for 2DH improved Boussinesq equations

In this paper, a hybrid scheme based on a set of 2DH extended Boussinesq equations for slowly varying bathymetries is introduced. The numerical code combines the finite volume technique, applied to solve the advective part of the equations, with the finite difference method, used to discretize dispersive and source terms. Time integration is performed using the fourth-order Adams–Bashforth–Moulton predictor–corrector method; the Riemann problem is solved employing an approximate HLL solver, a fourth-order MUSCL-TVD technique is applied. Five test cases, for non-breaking and breaking waves, are reproduced to verify the model comparing its results to laboratory data or analytical solutions.     

Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations

A one-dimensional high-resolution finite volume model capable of simulating storm waves propagating in the coastal surf zone and overtopping a sea wall is presented. The model (AMAZON) is based on solving the non-linear shallow water (NLSW) equations. A modern upwind scheme of the Godunov-type using an HLL approximate Riemann solver is described which captures bore waves in both transcritical and supercritical flows. By employing a finite volume formulation, the method can be implemented on an irregular, structured, boundary-fitted computational mesh. The use of the NLSW equations to model wave overtopping is computationally efficient and practically flexible, though the detailed structure of wave breaking is of course ignored. It is shown that wave overtopping at a vertical wall may also be approximately modelled by representing the wall as a steep bed slope. The AMAZON model solutions have been compared with analytical solutions and laboratory data for wave overtopping at sloping and vertical seawalls and good agreement has been found. The model requires more verification tests for irregular waves before its application as a generic design tool.     

Wave characteristic and morphologic effects on the onshore hydrodynamic response of tsunamis

While the destruction caused by a tsunami can vary significantly owing to near- and onshore controls, we have only a limited quantitative understanding of how different local parameters influence the onshore response of tsunamis. Here, a numerical model based on the non-linear shallow water equations is first shown to agree well with analytical expressions developed for periodic long waves inundating over planar slopes. More than 13,000 simulations are then conducted to examine the effects variations in the wave characteristics, bed slopes, and bottom roughness have on maximum tsunami run-up and water velocity at the still water shoreline. While deviations from periodic waves and planar slopes affect the onshore dynamics, the details of these effects depend on a combination of factors. In general, the effects differ for breaking and non-breaking waves, and are related to the relative shift of the waves along the breaking–non-breaking wave continuum. Variations that shift waves toward increased breaking, such as steeper wave fronts, tend to increase the onshore impact of non-breaking waves, but decrease the impact of already breaking waves. The onshore impact of a tsunami composed of multiple waves can be different from that of a single wave tsunami, with the largest difference occurring on long, shallow onshore topographies. These results demonstrate that the onshore response of a tsunami is complex, and that using analytical expressions derived from simplified conditions may not always be appropriate.     

The application of a Godunov-type shock capturing scheme for the simulation of waves from deep water up to the swash zone

A two-dimensional vertical (2DV) non-hydrostatic boundary fitted model based on a Godunov-type shock-capturing scheme is introduced and applied to the simulation of waves from deep water up to the swash zone. The effects of shoaling, breaking, surf zone dissipation and swash motions are considered. The application of a Godunov-type shock-capturing algorithm together with an implicit solver on a standard staggered grid is proposed as a new approach in the 2DV simulation of large gradient problems such as wave breaking and hydraulic jumps. The complete form of conservative Reynolds averaged Navier–Stokes (RANS) equations are solved using an implicit finite volume method with a pressure correction technique. The horizontal advection of the horizontal velocity is solved by an explicit predictor–corrector method. Fluxes are predicted by an exact Riemann solver and corrected by a downwind scheme. A simple total variation diminishing (TVD) method with a monotonic upstream-centered scheme for conservation laws (MUSCL) limiter function is employed to eliminate undesirable oscillations across discontinuities. Validation of the model is carried out by comparing the results of the simulations with several experimental test cases of wave breaking and run-up and the analytical solution to linear short waves in deep water. Promising performance of the model has been observed.     

Shock-capturing Boussinesq-type model for nearshore wave processes

This paper describes the formulation and validation of a nearshore wave model for tropical coastal environment. The governing Boussinesq-type equations include the conservative form of the nonlinear shallow-water equations for shock capturing. A Riemann solver supplies the inter-cell flux and bathymetry source term, while a Godunov-type scheme integrates the evolution variables in time. The model handles wave breaking through momentum conservation with energy dissipation based on an eddy viscosity concept. The computed results show very good agreement with laboratory data for wave propagation over a submerged bar, wave breaking and runup on plane beaches as well as wave transformation over fringing reefs. The model accurately describes transition between supercritical and subcritical flows as well as development of dispersive waves in the processes.     

On the vertical distribution of

The aim of this paper is to present an analytical expression for the vertical distribution of the correlation between the horizontal ( ) and vertical ( ) wave velocity components. This quantity, , which appears explicitly in the time-averaged momentum balance equations, has been shown to play an important role in the vertical distribution of wave-induced currents.The proposed formulation for is based on an identity that relates the effective (wave) shear stress to the effective (wave) normal stresses ( ² and ²) and to the vorticity of the oscillatory flow gw. This general expression has been applied to simplified situations and has been shown to degenerate into other existing formulations with comparable simplifying assumptions, viz. irrotational waves in shallow water over an arbitrary bottom topography and breaking waves over a horizontal bottom.The model has also been applied to the case of waves interacting with a depth-varying current over a horizontal bottom, in which preliminary results have been obtained for a simplified situation invoking linear (small-amplitude) wave theory.     

An extended time-dependent numerical model of the mild-slope equation with weakly nonlinear amplitude dispersion

In the present paper, by introducing the effective wave elevation, we transform the extended ellip- tic mild-slope equation with bottom friction, wave breaking and steep or rapidly varying bottom topography to the simplest time-dependent hyperbolic equation. Based on this equation and the empirical nonlinear amplitude dispersion relation proposed by Li et al. (2003), the numerical scheme is established. Error analysis by Taylor expansion method shows that the numerical stability of the present model succeeds the merits in Song et al. (2007)’s model because of the introduced dissipation terms. For the purpose of verifying its performance on wave nonlinearity, rapidly vary- ing topography and wave breaking, the present model is applied to study: (1) wave refraction and diffraction over a submerged elliptic shoal on a slope (Berkhoff et al., 1982); (2) Bragg reflection of monochromatic waves from the sinusoidal ripples (Davies and Heathershaw, 1985); (3) wave transformation near a shore attached breakwater (Watanabe and Maruyama, 1986). Comparisons of the numerical solutions with the experimental or theoretical ones or with those of other models (REF/DIF model and FUNWAVE model) show good results, which indicate that the present model is capable of giving favorably predictions of wave refraction, diffraction, reflection, shoaling, bottom friction, breaking energy dissipation and weak nonlinearity in the near shore zone.     

Simplified method for the calculation of irregular waves in the coastal zone

A method applicable for the estimation of the wave parameters along a set bottom profile is suggested. It takes into account the principal processes having an influence on the waves in the coastal zone: the transformation, refraction, bottom friction, and breaking. The ability to use a constant mean value of the friction coefficient under conditions of sandy shores is implied. The wave breaking is interpreted from the viewpoint of the concept of the limiting wave height at a given depth. The mean and root-mean-square wave heights are determined by the height distribution function, which transforms under the effect of the breaking. The verification of the method on the basis of the natural data shows that the calculation results reproduce the observed variations of the wave heights in a wide range of conditions, including profiles with underwater bars. The deviations from the calculated values mostly do not exceed 25%, and the mean square error is 11%. The method does not require a preliminary setting and can be implemented in the form of a relatively simple calculator accessible even for an inexperienced user.     

The role of bottom friction in models of nonbreaking tsunami wave runup on the shore

The role of bottom friction in the runup of nonbreaking long waves on the shore is analyzed. The case of the normal incidence of monochromatic waves is considered. The relief of the model region consists of an even horizontal bottom area conjugated with a flat slope. The energy dissipation is estimated as the work of bottom friction forces over the wave field obtained using the known analytical solution based on the Carrier-Greenspan transforms. Estimates of energy losses for waves whose periods are typical for tsunami waves have been obtained. The energy dissipation is shown to be not concentrated in the shore line area as a rule. The question about the practicability of using partially reflecting boundary conditions on the coast to take into account the bottom friction in large-scale models of tsunami propagation is considered.     

Interaction of dispersive water waves with weakly sheared currents of arbitrary profile

A set of depth-integrated equations describing combined wave–current flows is derived and validated. To account for the effect of turbulence induced by interactions between waves and currents with arbitrary horizontal vorticity, new additional stress terms are introduced. These stresses are functions of a parameter b that relates the relative importance of wave radiation stress and bottom friction stress to the wave–current interaction. To solve the equations, a fourth-order MUSCL-TVD scheme with an approximate Riemann solver is adopted. As a first-order check of the model, the Doppler shift effect and wave dispersion over linearly sheared currents are analytically shown to be retained appropriately in the equation set. The model results are then validated through comparisons with three experimental data sets. First, based on the experiments of Kemp and Simons (1982, 1983), a reasonable functional form of b is estimated. Second, simulations examining the propagation of a weakly dispersive wave over a depth-uniform or linearly sheared current are performed. Finally, the model is applied to a more complex configuration where bichromatic waves interact with spatially varying currents. Simulated results indicate that the model is capable of predicting nearshore interactions of waves with currents of arbitrary vertical structure. One of the unique properties of the developed model is its ability to assimilate an external current field from any source, be it from a circulation model or an observation, and predict the interaction of a nonlinear and dispersive wave field with that current.     

Modeling wave�Ccurrent bottom boundary layers beneath shoaling and breaking waves

The boundary layer characteristics beneath waves transforming on a natural beach are affected by both waves and wave-induced currents, and their predictability is more difficult and challenging than for those observed over a seabed of uniform depth. In this research, a first-order boundary layer model is developed to investigate the characteristics of bottom boundary layers in a wave–current coexisting environment beneath shoaling and breaking waves. The main difference between the present modeling approach and previous methods is in the mathematical formulation for the mean horizontal pressure gradient term in the governing equations for the cross-shore wave-induced currents. This term is obtained from the wave-averaged momentum equation, and its magnitude depends on the balance between the wave excess momentum flux gradient and the hydrostatic pressure gradient due to spatial variations in the wave field of propagating waves and mean water level fluctuations. A turbulence closure scheme is used with a modified low Reynolds number k-ε model. The model was validated with two published experimental datasets for normally incident shoaling and breaking waves over a sloping seabed. For shoaling waves, model results agree well with data for the instantaneous velocity profiles, oscillatory wave amplitudes, and mean velocity profiles. For breaking waves, a good agreement is obtained between model and data for the vertical distribution of mean shear stress. In particular, the model reproduced the local onshore mean flow near the bottom beneath shoaling waves, and the vertically decreasing pattern of mean shear stress beneath breaking waves. These successful demonstrations for wave–current bottom boundary layers are attributed to a novel formulation of the mean pressure gradient incorporated in the present model. The proposed new formulation plays an important role in modeling the boundary layer characteristics beneath shoaling and breaking waves, and ensuring that the present model is applicable to nearshore sediment transport and morphology evolution.     

Internal generation of waves on an arc in a rectangular grid system

This paper presents a technique to generate waves at oblique angles in finite difference numerical models in a rectangular grid system by using internal generation technique [Lee, C., Suh, K.D., 1998. Internal generation of waves for time-dependent mild-slope equations. Coast. Eng. 34, 35–57.] along an arc-shaped line source. Tests were made for four different types of wave generation layouts. Quantitative experiments were conducted under the following conditions: the propagation of waves on a flat bottom, the refraction and shoaling of waves on a planar slope, and the diffraction of waves to a semi-infinite breakwater. Numerical experiments were conducted using the extended mild-slope equations of Suh et al. [Suh, K.D., Lee, C., Park, W.S., 1997. Time-dependent equations for wave propagation on rapidly varying topography. Coast. Eng. 32, 91–117.]. The fourth layout type consisting of two parallel lines connected to a semicircle showed the best solutions, especially for a small grid size. This technique is useful for the numerical simulation of irregular waves with broad-banded directional spectrum using conventional spectral wave models for the reasonable estimation of bottom friction and wave-breaking.     

A synchronously coupled tide–wave–surge model of the Yellow Sea

A coupled wave–tide–surge model has been established in this study in order to investigate the effect of tides, storm surges, and wind waves interactions during a winter monsoon on November 1983 in the Yellow Sea. The coupled model is based on the synchronous dynamic coupling of a third-generation wave model, WAM-Cycle 4, and the two-dimensional tide–surge model. The surface stress generated by interactions between wind and waves is calculated using the WAM-Cycle 4 directly based on an analytical approximation of the results obtained from the quasi-linear theory of wave generation. The changes of bottom friction factor generated by waves and current interactions are calculated by using simplified bottom boundary layer model. The model simulations showed that bottom velocity and effective bottom drag coefficient induced by combination of wave and current were increased in shallow waters of up to 50 m in the Yellow Sea during the wintertime strong storm conditions.     

Improvement on open boundaries on a time dependent numerical model of wave propagation in the nearshore region

An improvement on the simulation of outgoing waves on a time dependent numerical model for water wave propagation in the nearshore region is presented. The governing equations consist of a system of first order partial differential equations (PDEs), the equation of continuity and the equation of motion. A comparative study of first order radiation boundary conditions (BCs) and first order radiation BCs combined with sponge layers is presented for cases where outgoing waves leave the numerical domain of calculation through the open boundary. A reduction of spurious reflections from the numerical open boundaries can be obtained with an irrelevant increase in terms of computational cost.     

Static wave setup with emphasis on damping effects by vegetation and bottom friction

Wave setup can contribute significantly to elevated water levels during severe storms. In Florida we have found that wave setup can be 30% to 60% of the total 100-year storm surge. In areas with relatively narrow continental shelves, such as many locations along the Pacific Coast of the United States, wave setup can be an even larger proportionate contributor of anomalous water levels during major storms. Wave setup can be considered as comprising two components, with the first being the well-known static wave setup resulting from the transfer of breaking wave momentum to the water column. The second, oscillating component, is a result of nonlinear transfer of energy and momentum from the primary (linear) spectrum to waves with length and time scales on the order of the wave groups.Static wave setup is the focus of this paper with emphasis on effects due to internal or surface forces that act on the wave system and cause both dissipation of wave energy and transfer of momentum. In particular, the effects of wave damping by vegetation and bottom friction are considered. Linear wave theory is applied to illustrate these effects and, for shallow water waves, the setup is reduced by two-thirds the amount that would occur if the same amount of energy dissipation occurred in the absence of forces. Effects of nonlinear waves are then considered and it is found, for a shallow water wave of approximately one-half breaking height, that a wave setdown rather than setup occurs due to damping by vegetation and bottom friction.The problem of wave setup as waves propagate through vegetation was stimulated by studies to establish hazard zones associated with 100-year storm events along the shorelines of the United States. These storms can generate elevated water levels exceeding 4 to 6 m and can result in overland wave propagation. As these waves propagate through vegetation and damp, the question arose as to the contribution of this process to elevated mean water levels through additional wave setup.     

Surf zone dynamics simulated by a Boussinesq type model. Part II: surf beat and swash oscillations for wave groups and irregular waves

This is the second of three papers on the modelling of various types of surf zone phenomena. In the first paper the general model was described and it was applied to study cross-shore motion of regular waves in the surf zone. In this paper, part II, we consider the cross-shore motion of wave groups and irregular waves with emphasis on shoaling, breaking and runup as well as the generation of surf beats. These phenomena are investigated numerically by using a time-domain Boussinesq type model, which resolves the primary wave motion as well as the long waves. As compared with the classical Boussinesq equations, the equations adopted here allow for improved linear dispersion characteristics and wave breaking is modelled by using a roller concept for spilling breakers. The swash zone is included by incorporating a moving shoreline boundary condition and radiation of short and long period waves from the offshore boundary is allowed by the use of absorbing sponge layers. Mutual interaction between short waves and long waves is inherent in the model. This allows, for example, for a general exchange of energy between triads rather than a simple one-way forcing of bound waves and for a substantial modification of bore celerities in the swash zone due to the presence of long waves. The model study is based mainly on incident bichromatic wave groups considering a range of mean frequencies, group frequencies, modulation rates, sea bed slopes and surf similarity parameters. Additionally, two cases of incident irregular waves are studied. The model results presented include transformation of surface elevations during shoaling, breaking and runup and the resulting shoreline oscillations. The low frequency motion induced by the primary-wave groups is determined at the shoreline and outside the surf zone by low-pass filtering and subsequent division into incident bound and free components and reflected free components. The model results are compared with laboratory experiments from the literature and the agreement is generally found to be very good. Finally the paper includes special details from the breaker model: time and space trajectories of surface rollers revealing the breakpoint oscillation and the speed of bores; envelopes of low-pass filtered radiation stress and surface elevation; sensitivity of surf beat to group frequency, modulation rate and bottom slope is investigated. Part III of this work (Sørensen et al., 1998) presents nearshore circulations induced by the breaking of unidirectional and multi-directional waves.     

潜堤上破碎波浪传播变形的数值模型及其验证

为模拟潜堤上破碎波浪传播时产生能量的耗散这一特性,在改进的具有四阶色散的Boussinesq水波方程中中入二阶紊动粘性项,建立了考虑波浪破碎的水波数学模型.在非交错网格下建立了有限差分数值模型,并利用三阶Adams-Bash forth格式预报、四阶Adams-Mouton格式校正对数值模型进行求解.通过数值试验,模拟...     

An Efficient Approach for Simulation of Water Levels due to the Nonlinear Interaction of Tide and Surge Along the Coast of Bangladesh

The ultimate goal and highlight of this paper are to explore water levels along the coast of Bangladesh efficiently due to the nonlinear interaction of tide and surge by employing the method of lines(MOLs) with the aid of newly proposed RKAHeM(4, 4) technique. In this regard, the spatial derivatives of shallow water equations(SWEs) were discretized by means of a finite difference method to obtain a system of ordinary differential equations(ODEs) of initial valued with time as an independent variable. The obtained system of ODEs was solved by the RKAHeM(4, 4)technique. One-way nested grid technique was exercised to incorporate coastal complexities closely with minimum computational cost. A stable tidal oscillation was produced over the region of interest by applying the most influential tidal constituent M2 along the southern open boundary of the outer scheme. The newly developed model was applied to estimate water levels due to the non-linear interaction of tide and surge associated with the catastrophic cyclone April 1991 along the coast of Bangladesh. The approach employed in the study was found to perform well and ensure conformity with real-time observations.     

Theoretical and Experimental Study of Breaking Wave on Sloping Bottoms

This paper studies the continuous evolution of breaking wave for the surface water waves propagating on a sloping beach. A Lagrangian asymptotic solution is derived. According to the solution coupled with the wave breaking criteria and the equations of water particles motion, the wave deformation and the continuous wave breaking processes for the progressive water waves propagating on a sloping bottom can be derived. A series of experiments are also conducted to compare with the theoretical solution. The results show that the present solution can reasonably describe the plunging or spilling wave breaking phenomenon.