Experiments from Dibajnia and Watanabe (1998), Dibajnia and Kioka(2000), and Dibajnia et al. (2001).
1. Introduction
Sand transport plays a very important role in many aspects of coastal and marine engineering. The balance of moving sands influences the construction of harbours, coastal defence, offshore wind turbine and oil rig, offshore platform and pipeline and many other engineering. Coastal sands may be carried by currents (such as tidal currents, winddrivencurrents, wavedrivencurrents, storm surge driven currents), or by waves (monsoon waves or typhoon waves), or influenced by bedform changes, or all of them acting together and interacting in general sea state.
We can easily consider a sediment budget for a coastal area where a control section or a control volume is selected. The changing rate of net accretion or erosion of the coastal area of sea bedlevel depends on the net transport rates at which sediments are entering or leaving the control section or the control volume. If the sum of the inflow sediment transport rates is larger than that of the outflows, the bedlevel will tend to accrete; if the sum of the inflow sediment transport rates is smaller than that of outflow rates, the bedform will erode.
Consequently, accurate prediction of sediment transport rates is an important element in coastal engineering, foundations of offshore structures and morphological studies for the coastal environment. The procedure of the coastal morphological modeling system is shown as Fig. 1. The prediction of net sediment transport rates is a subject of great importance to coastal engineers and morphological modellers concerned with mediumand longterm shoreline changes. The aim of this chapter is to provide models for calculating the hydrodynamics and dynamic quantities of sediment transports in coastal zone, especially for the applications in surf zone.
1.1. Threshold of sediment motion
In order to estimate the changes of sediment budget, a quantitative evaluation of the net sediment transport rates are required. The sediment transport rate is defined as the amount of sediment per unit time entering or leaving a control volume, which is the vertical plane of unit width perpendicular to the sea mean waterlevel. The mechanics of sediment motion is depended on the effects of sediment dynamics while the frictions exerted on the sea bed by the hydrodynamics forcing agents, such as waves and currents.
While the friction exerted on the sea bed by waves and currents, the sediment ‘entrainment’ is the sand grains are carried up from bed. The ‘bed load’ sediment transport is the entraining sand grains rolling, hopping and sliding along the bedform, and is dominant with the inertial force and drag force on grains less than gravity force for slow flow or large grains. Suspended load sediment transport is the portion of the entraining sediment that is carried by the larger flow (or the wave large enough) which settles slowly enough and moves with the stream.
To estimate the sediment transport rate is more difficult, it may be divided into two components by crossshore and longshore direction. The crossshore sediment transport is mainly carried by the skewness and asymmetry wave orbital motion and cyclemean water level around the surf zone. The longshore sediment transport is primarily dominated with wavedriven currents. In order to quantitate the sediment transport rate, included both bed load and suspended load, in each direction, an empirical relationship has been derived between ‘transport rate’ and the energeticbased components, such as bed shearstress, wavedriven currents, wave orbit velocity….. and so on. The agreement between measurements (or experiments in hydraulic laboratory) and calculations associated with the relationship has been widely applied and predicts accurately sediment transport along long, straightlike beaches.
The total load sediment transport models are widely applied with coastal engineering in last three decades. Although the models are relatively simple and easy to use, but there are some weaknesses when applied around the surf zone. We will discuss in next section.
1.2. Interaction between sediment motion and bed features for morphodynamics
Nearshore sandbars are the important and popular feature of natural beaches morphodynamics. The crossshore location of sandbars changes by the interactions between the sandbar and the sediment transport fluxes from waves and wavedrivencurrents. Hoefel and Elgar (2003) indicated the mechanics of waveinduced sediment transport and sandbar migration: large waves breaking on the sandbars caused offshore mean currents, which maximum near the sandbar crest, will lead sandbars moved offshore; small waves pitching forward on the sandbars made the onshore acceleration skewness of wave orbital velocities, which maximum near the sandbar crest, will lead the sandbar moved onshore.
The nearshore sandbars could protect shorelines from wave attack by dissipating wave energy offshore through sandbarcrestinduced wave breaking. In general coasts, the dynamic behavior of nearshore sandbars are similar to quasicycle for storm and seasons waves alternated. However, the formation and evolution of sandbars are very important to coastal planners and engineers. Prediction of the dynamic behaviour of nearshore sandbar systems could be of great importance, there are many studies about the evolutions and migrations of nearshore sandbars by numerical simulations in last decade (Hsu et al., 2006; Long et al., 2006; Ruessink et al., 2007; Drønen and Deigaard, 2007; Houser and Greenwood, 2007; Ruessink and Kuriyama, 2008; Ruessink et al., 2009; Pape et al., 2010; Almar et al., 2010).
According to Hoefel and Elgar (2003), two mechanisms are commonly used in the explanation of morphodynamics of sandbars migration. The first mechanism type is the migration of offshore sandbars. The offshore sandbars migrated seaward observed during storms were driven primarily by a maximum in the offshore mean current (Under highly energetic storm conditions, breaking waves cause near bottom seaward flows, also called “undertow”) near the sandbar crest. Offshore sandbar migration during storms results from feedback and interaction between breaking waves driven the “undertow” and bathymetric evolution (Elgar et al., 2001).
The second mechanism associates the migration of onshore sandbars. There are many studies have suggested mechanisms that could drive sandbars migration shoreward. Trowbridge and Young (1989) and Trowbridge and Madsen (1984) demonstrated that nonlinear wave boundary layer processes might play a role. Onshore sandbars migration might also be derived by the systematic changes in wave kinematics when passing over nearshore sandbars. As waves shoaling, their shapes are often described as “skewed” and “asymmetric” (Elgar, 1987), the mean water elevation depressed (the “Wave setdown”) leads mean currents been weak. The nonbreaking wave caused sediment transport over sandbars is driven predominately by wave asymmetric orbital velocities. Under the steep skewed and asymmetric waves, the water particle velocity is accelerated strongly as the asymmetric orbital velocity rapidly changes from maximum offshore to maximum onshore (e.g., Elgar et al., 1988).
In order to describe morphodynamics sandbars well, a key parameter for crossshore sediment transport under breaking and nearbreaking waves is well performed the nearbed skewed and asymmetric wave orbital velocity. Therefore, the three dimensionality of the hydrodynamic system should be considerable and must be taken into account. Most sediment transport models are based on phaseaveraged wave models, depthintegrated hydraulic models (currents and wave driven currents) and sediment transport formula. In the hydraulic models, nearshore currents have previously been predicted by using twodimensional models in the horizontal plane (2DH model). However, in the surf zone, the direction of current vectors near the water surface is different from that at the seabottom because of the effect of undertow velocities. Nearshore currents have spiral profiles in the vertical direction. Undertows also play an important role in the morphodynamical changes on a littoral beach such as the crossshore migration of longshore bars. In order to accurately predict the changes of sandbars migration, it is very important that the threedimensional distribution of nearshore currents is determined. Therefore, well predicted nonlinear wave dynamics, vertical current and sediment transport models can be a good tool for nearshore sandbar morphodynamics, especially for cases where crossshore transport mechanisms over sandbars are important. Drønen and Deigaard (2007) compared 2D horizontal depthintegrated approach and quasi3D numerical model with formations of alongshore bars on gradual slope beach by normal and oblique incident waves, and the quasi3D model produces a crescentic bar while the depth integrated model predicts almost straight sections of the bar interrupted by rip channels. Consequently, considering the accuracy and efficiency, a nonlinear waves model with quasi3D sediment transport model can be applied as morphodynamics sandbar models.
1.3. The importance of quasi3D sediment transport modeling
The numerical simulation of hydrodynamic and sediment transport processes form a powerful tool in the description and prediction of morphological changes and sediment budgets in the coastal zone. One of the key elements in a morphodynamics model is the correct quantification of local sand transport. Most sediment transport models are based on phaseaveraged wave models, depthintegrated hydraulic models (currents and wave driven currents) and sediment transport formula. In the hydraulic models, nearshore currents have previously been predicted by using twodimensional models in the horizontal plane (2DH model). However, in the surf zone, the direction of current vectors near the water surface is different from that at the seabottom because of the effect of undertow velocities. Nearshore currents have spiral profiles in the vertical direction. Undertows also play an important role in the morphological changes on a littoral beach such as the crossshore migration of longshore bars. In order to accurately predict the changes of beach profile, it is very important that the threedimensional distribution of nearshore currents is determined. Therefore, well predicted vertical current and sediment transport models can be a good tool for coastal area morphological modelling, especially for cases where crossshore transport mechanisms are important. Considering the accuracy and efficiency, a quasi3D model can be applied as a coastal profile model or a coastal area model.
Some models for determining the vertical distribution of nearshore currents have previously been proposed. de Vriend et al. (1987) presented a semianalytical model and suggested that a 3D model is required when the sediment transport in the crossshore direction becomes important; and then Svendsen and Lorenz (1989) proposed an analytical model composed of crossshore and longshore current velocities. In recent years, many quasi3D numerical models have been developed by extending 2DH model with onedimensional velocity profile model defined in the vertical direction (1DV model), have also been proposed (Sanchez et al., 1992; Briand and Kamphuis, 1993; Okayasu et al., 1994; Elfrink et al., 1996; Rakha, 1998; Kuroiwa et al,. 1998; Drønen and Deigaard, 2000; Davis and Thorne, 2002; Fernando and Pan, 2005; Drønen and Deigaard, 2007; Li et al., 2007). In these models, the mean flow is determined by the 2DH model, and the velocity profiles across water column in the vertical direction are resolved by using a 1DV model. While surface wave field and 3D flow field have been analyzed, sediment transport vectors in the horizontal plane can be calculated with the sediment transport profile across water column in the vertical direction.
The aim of this chapter is to develop an accurate model for estimation of local sediment transport rate of the nearshore both inside and outside of the surf zone. A twodimensional 2D fully nonlinear Boussinesq wave module is combined with a quasi3D hydrodynamic module (2DH and extended 1DV module). The 1DV hydrodynamic modules similar to those described by Elfrink et al. (1996) with surfaceroller concept and a oneequation turbulence model are developed. The calculation of sediment transport rates is based on the formula with wave asymmetric and ripplebed effects developed by Lin et al. (2009). The quasi3D hydrodynamic modules are validated and compared, for regular waves over fixed beds. The local sediment transport rates is also calculated and validated with experimental data.
2. 2DH waves and nearshore currents models
In this section, the twodimensional wave and nearshore current models are described as below:
2.1. Wave model
The wave model is based on the fully nonlinear Boussinesq equation developed by Wei et al. (1995); the equation is expressed by velocity with an arbitrary water depth. Bottom friction, wave breaking and subgrid lateral turbulent mixing as proposed by Kennedy et al. (2000), are also expressed by equations. The governing equations are shown as below:
In the above equations,
The eddy viscosity (
Kennedy et al. (2000) proposed the mixing length,
The onset and cessation of wave breaking using the parameter,
where
The parameter
where
2.2. 2DH nearshore current model
Based on computed characteristics of wave fields, the radiation stress terms can then be found and input into the depth integrated (2DH) nearshore current module, which solves the depthandwaveperiod averaged continuity and momentum equations at each local point on horizontal plane, for calculating wave driven current:
where U and V are depthintegrated nearshore current velocities in x and y direction respectively, S_{xx}, S_{xy} and S_{yy} are radiation stress tensor, g is acceleration due to gravity, ρ is water density, h is water depth, η is water surface elevation, τ_{xx}, τ_{xy} and τ_{yy} are Reynolds stress tensor, τ_{s} and τ_{b} are shear stress on surface and bottom. The friction factor for combined wavecurrent flow in bottom shear stresses and the mixing coefficient in Reynolds stresses are suggested by Chiang et al. (2010).
3. Quasi3D extended: 1DV velocity model
Fig. 4 depicts the coordination of the quasi3D hydraulic system. The x coordinate is defined in the crossshore direction towards shore. The y coordinate denotes the long shore direction. The Z coordinate is toward from sea bed to surface in depthdirection.
3.1. Numerical formulation
The distributions of the velocity profiles in the longshore and crossshore direction at each local point along the vertical water column are found through the following momentum equations:
where
The pressure gradient term may be divided into two components by inside/outside boundary layer. The pressure gradient term can be easily calculated from the variation of cyclemean water free surface. Within the boundary layer, we can calculate from the time differential of velocity at boundary layer:
where
The cyclemean waveinduced shear stress under waves is consist of wave motion component (τ_{w}), wave breaking surface roller component (τ_{r}), boundary layer streaming component (τ_{b}) and mean water surface changed (wave setup/setdown) component (τ_{su}), as shown in Fig. 5. The Boundary layer streaming components can be neglected, because it is small than the others. The total shear stress is defined as
The shear stress distribution due to wave motion (τ_{w}) is in accordance with the derivations of Deigaard and Fredsoe (1989):
where
The coefficient B is 1/12 while wave breaking and 1/8 in general.
The shear stress due to wave breaking (τ_{w}) can be calculated by the concept of surface roller. According to Svendsen (1984) and Deigaard et al. (1986), the shear stress of surface roller is assumed to be constant by experiment:
where
The shear stress due to variation of mean water level (τ_{su}) can be easily calculated by
The turbulence viscosity should be specified through certain turbulence models. In the present study, the oneequation kclosure is adopted as follows:
where
in which
where
The turbulence generated and energy dissipation at the water surface due to wave breaking is computed following Deigaard et al. (1991), as shown in Fig. 7 :
where constant
where
3.2. Boundary condition and key parameters
3.2.1. Eddy viscosity
According to Brøker et al. (1991), the eddy viscosity is calculated by assuming the total kinetic energy could be the sum of three contributions: the oscillatory near bed boundary layer (
The component of oscillatory near bed boundary layer (
where
The component of wave breaking (
where
The contribution of the timeaveraged currents (
where
3.2.2. Thickness of wave boundary layer
The thickness of wave boundary layer (
where
4. Sediment transport formula
The total sediment transport is consisted of bed load and suspended load suggested by Chiang et al. (2011):
where
4.1. Bed load sediment transport
Following Chiang et al. (2011), the instantaneous bed load transport rate due to wave asymmetric and ripplebed effects is given as:
where
where
where
In eq. (44),
According to Soulsby (1997), the total roughness (
In eq. (37), the grain related component is given as (Nielson, 1992 and Soulsby, 1997),
The form drag component associated with sandy ripples is defined (Davis and Villaret, 2003):
where the
In eq. (40),
4.2. Suspended load sediment transport
The suspended sediment transport can be calculated by integrated sediment concentration (C) of vertical water column from bottom to surface. The sediment concentration
where
where
After the suspended sediment concentration and vertical velocity profiles obtained, the instantaneous suspended transport rate can be evaluated as:
where
5. Model validation and discussion
The quasi3D sediment tranport model is validated against wave flume tests and existed numerical models.
5.1. Model validation with Cox and Kobayashi (1996)
The wave and quasi3D model system described above was firstly tested by regular wave with uniform sloped bed, and compared with experiment (Cox and Kobayashi, 1996). The test conditions are given: wave height 13.2cm, wave period 2.2 sec, the length of wave flume is 14.0m, width is 1.5m, depth is 30.0cm, slope1:35; sand medium diameter is 1.0mm, and bed roughness height
The comparisons of the vertical velocity profile at various position used by Cox and Kobayashi (1996) are also adopted here to assess the present model’s accuracy for wave propagating at a slope in wave flume. The velocity profiles of various position are shown as Fig. 9 ~ Fig. 14. The black triangles also indicate experimental data by Cox and Kobayashi (1996), the blue line is the results of present 1DV model, and the black line is the numerical results from Rakha (1998). Compared with experiments, they are shown good performance for the validation of distribution of the vertical velocity profile before and after wave breaking.
5.2. Model validation with Ting and Kirby (1994)
The Boussinesq wave model and quasi3D nearshore current model system described above was tested by regular wave with uniform sloped bed, and compared with experiment (Ting and Kirby, 1994, test 1). The test conditions are given: wave height 12.5cm, wave period 2.0 sec, the length of wave flume is 13.0m, width is 1.5m, depth is 30.0cm, slope1:35; sand medium diameter is 1.0mm, and bed roughness height ks=1mm. The numerical results of wave height and layouts of wave flume is shown as Fig. 15. In Fig. 15, the black triangles indicate experimental data by Ting and Kirby (1994), the red line is the results of present wave model, and the black line is the numerical results from Rakha (1998). The wave nonlinear effects and wave breaking and regenerating can be observed well, it’s also shown good agreement with experiments.
The comparisons of the vertical velocity profile at various position used by Ting and Kirby (1994) are also adopted here to assess the present model’s accuracy for wave propagating at a slope in wave flume. The velocity profiles of various position are shown as Fig. 16 ~ Fig. 21. The black triangles also indicate experimental data by Ting and Kirby (1994), the blue line is the results of present 1DV model, and the black line and dotted line are the numerical results from Rakha (1998) model A and model B. Compared with experiments, they are shown good performance for the validation of distribution of the vertical velocity profile before and after wave breaking. Because of fullnonlinear Bossinesq wave model and accuracy wave breaking dissipation terms in 1DV model, the present model performed well than Rakha (1998).
5.3. Validation for sediment transport calculations
To assess the present quasi3D sediment transport model’s ability of prediction for local sediment transport under combined waves and currents for a range conditions, a series of
experiments in the wave flume have been observed by Dibajnia and Watanabe (1998), Dibajnia and Kioka (2000), and Dibajnia et al. (2001). All of experimental data are shown as Table 1.
The comparison has been carried out between the periodaveraged net sediment transport and experimental results for different wavecurrent conditions and the results are reasonable accurate within a factor of 2, as shown in Fig. 22. There are 66% numerical results greater than experiments, and 8% numerical results out of compared accurate factor of 2. They are shown good performance to predict the local sediment transport rates in nearshore region.







0.02  163.8  3.9  0.67  0.29  0  0.133 
0.02  148.2  3.9  0.68  0.29  0  0.094 
0.02  136.1  3.9  0.68  0.29  0  0.079 
0.02  126.9  3.9  0.67  0.29  0  0.042 
0.02  103.9  3.7  0.59  0.41  0  0.014 
0.02  119.4  3.7  0.59  0.41  0  0.055 
0.02  128.5  3.7  0.59  0.41  0  0.048 
0.02  136.5  3.8  0.58  0.42  0  0.051 
0.02  140.5  3.5  0.68  0.32  0  0.08 
0.02  129  3.5  0.68  0.32  0  0.049 
0.02  121.2  3.6  0.68  0.31  0  0.061 
0.02  108.4  3.6  0.6  0.39  0  0.031 
0.02  144.8  3.8  0.58  0.42  0  0.072 
0.02  122.8  3.6  0.68  0.31  0  0.041 
0.02  117.1  3.5  0.59  0.41  0  0.032 
0.02  119.8  3.5  0.59  0.41  0  0.046 
0.02  114.6  3.6  0.60  0.4  0  0.044 
0.02  109.5  3.9  0.67  0.29  0  0.062 
0.02  126.3  3.5  0.59  0.41  0  0.032 
0.02  105.6  3.6  0.68  0.31  0  0.043 
0.02  137.3  3.5  0.67  0.33  0  0.086 
0.02  122.9  3.9  0.67  0.29  0  0.066 
0.02  112.7  3.7  0.59  0.41  0  0.046 
0.02  135.9  3.5  0.59  0.41  0  0.069 
0.02  119.2  3.9  0.68  0.3  16.3  0.117 
0.02  114.7  3.8  0.58  0.42  14.3  0.095 
0.02  116.1  3.7  0.59  0.41  5.6  0.063 
0.02  118.8  3.9  0.68  0.29  11  0.1 
0.02  189.5  4  0.65  0.31  0  0.21 
0.02  184.2  3.8  0.64  0.35  0  0.171 
0.02  167.5  3.8  0.65  0.33  0  0.134 
0.02  151.7  3.8  0.67  0.31  0  0.109 
0.02  183.2  3.6  0.65  0.34  0  0.142 
0.02  183.2  3.6  0.65  0.35  0  0.139 
0.02  179.3  3.6  0.65  0.34  0  0.123 
0.02  175.6  3.6  0.65  0.34  0  0.103 
0.02  165.5  3.6  0.65  0.33  0  0.125 
0.02  163.4  3.6  0.65  0.34  0  0.119 
0.02  155.6  3.6  0.66  0.33  0  0.082 
0.02  146.4  3.6  0.65  0.33  0  0.075 
0.055  275.3  4.2  0.57  0.43  0  1.184 
0.055  265  4.1  0.59  0.41  0  0.773 
0.055  239.1  4  0.62  0.37  0  0.683 
0.055  208.3  4  0.64  0.36  0  0.439 
0.055  264.8  3.6  0.59  0.41  0  0.634 
0.055  260.1  3.6  0.61  0.39  0  0.534 
0.055  250.1  3.6  0.62  0.38  0  0.56 
0.055  225.3  3.6  0.63  0.37  0  0.459 
0.08  280.5  4.2  0.57  0.43  0  1.373 
0.08  276.3  4.1  0.57  0.43  0  1.44 
0.08  270.2  3.6  0.57  0.43  0  1.137 
0.08  264.2  4  0.58  0.42  0  0.8 
6. Conclusion
In this chapter, a quasi3D numerical model has been developed to predict sand transport in the coastal region. The whole model system is consisted of a fully nonlinear Bossinesq wave module, 2HD depthintegral wavedriven current module, 1DV velocity profile module, and the sediment transport formula. Numerical results indicate that the wave breaking and regenerating are good agreement with experiments. The phenomenon of undertow for wave breaking or not is performed well, and the vertical velocity profiles are shown good accuracy with experiments. The numerical results of sediment transport have been compared with experiment and obtained are reasonably accurate within a factor of 2. The quasi3D sediment transport model system was then used to simulate several laboratory studies to test its ability to reproduce the important nearshore morphodynamic processes.
Acknowledgments
The work was partially financially supported by the National Science Council (Taiwan, R.O.C.) Project: NSC992218E320 002MY3.References
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